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Ruebel-DissertationRuebel.pdf
```INAUGURAL-DISSERTATION
zur
Erlangung der Doktorwürde
der
Naturwissenschaftlich-Mathematischen Gesamtfakultät
der
Ruprecht-Karls-Universität
Heidelberg
vorgelegt von
Diplom-Mathematiker
Jan Rübel
aus Homburg
Tag der mündlichen Prüfung: 2. Juli 2009
Vibrations in Nonlinear
Rotordynamics
Modelling, Simulation, and Analysis
Gutachter: Prof. Dr. Dr. h.c. mult. Willi Jäger
Associate Professor Dr. Jens Starke
Abstract
Mechanical vibrations of rotor-bearing systems are an ubiquitous problem in mechanical engineering and the prediction of response frequencies and amplitudes with the
help of mathematical models is of major importance for the design of more efficient
and reliable machinery. In the present work a model for the dynamics of fast rotating,
elastic beams supported in hydrodynamic bearings is derived and its vibration behavior analyzed. Special focus is put on the influence of the nonlinear bearing reaction
forces on the dynamics. The continuous rotor is modeled using Euler-Bernoulli beam
theory under the inclusion of rotatory inertia and gyroscopic effects. For a general
class of support functions the existence of weak solutions to the equations of motion
is proved.
The pressure distribution in the oil-lubricated simple journal bearings is modeled by
the well known Reynolds’ equation. Its derivation from the Navier-Stokes equations
by an asymptotic expansion in the film thickness is reviewed and new correction
terms for fluid inertia effects are derived. Additional correction terms for the short
bearing approximation to Reynolds’ equation are also derived by making additional
assumptions on the bearings’ width-to-radius ratio. Furthermore, the pressure distribution and the bearing reaction forces are computed numerically in dependence of the
position and the velocity of the rotor inside the bearings.
The finite element method is applied to discretize the beam equation and the bearing
forces are included into the model as point forces in the bearing nodes. While the
classical lubrication theory leads to explicit equations of motion, the inertia corrections
lead to implicit equations of motion for the rotor-bearing system. The model is evaluated by comparing numerical simulations with experimental results obtained for a
passenger car turbocharger. For this example it is shown that the model equations describe the dynamics well, capturing most experimentally observed phenomena, such
as unbalance oscillation and self-excited oil whirl. Some differences between model
and experiment can be seen in the response frequency of the subharmonic oil whirl.
The inertia corrections yield a small improvement compared to the classical models.
A new phenomenological correction of the short bearing approximation based on the
adaptation of the average circumferential lubricant velocity is proposed and shown to
influence the whirl frequency ratio strongly.
ii
Continuation methods for periodic and quasiperiodic solutions are shown to be more
efficient and are hence more appropriate tools for the examination of the vibration
response behavior than direct numerical simulation. It is shown numerically, that the
static gravity load can be neglected for higher rotational frequencies. Combined with
a transformation to a co-rotating frame of coordinates this leads to a significant simplification, since the resulting ODE becomes autonomous, and the vibration response
can be computed by the continuation of periodic orbits instead of invariant tori. This
is applied successfully to study the parameter range where the inertia correction for
the short bearing is valid.
All in all, the presented model and its variations prove to be useful for future industrial
application in the design of more efficient turbomachinery. Parts of the presented research are already actively used for turbocharger design by the Toyota Central Research
and Development Laboratories.
Zusammenfassung
Mechanische Vibrationen von Rotor-Lager Systemen sind ein weit verbreitetes Problem und ihre Vorhersage mit Hilfe mathematischer Modelle ist von außerordentlicher Bedeutung für das Design effizienterer und zuverlässigerer Maschinen. In der
vorliegenden Arbeit wird ein Modell für die Dynamik rotierender, elastischer Balken
in hydrodynamischen Lagern hergeleitet und sein Vibrationsverhalten untersucht. Besonderes Augenmerk liegt dabei auf dem Einfluss der nichtlinearen Lagerkräfte. Der
kontinuierliche Balken wird mit Hilfe der Balkentheorie von Euler und Bernoulli unter
Berücksichtigung der Rotationsträgheit und der Kreiselkräfte modelliert. Die Existenz
schwacher Lösungen der Bewegungsgleichungen wird für eine allgemeine Klasse von
Lagerfunktionen gezeigt.
Die Druckverteilung in den ölgeschmierten Gleitlagern wird durch die ReynoldsGleichung beschrieben. Die Herleitung dieser klassischen Gleichung aus den NavierStokes-Gleichungen durch asymptotische Entwicklung nach der Filmdicke wird erneut durchgeführt. Dabei werden neue Korrekturterme für Trägheitseffekte in der
Schmierflüssigkeit durch Berücksichtigung höherer Ordnungen hergeleitet. Unter einer speziellen Annahme an des Verhältnis von Lagerbreite und -radius werden für
die Näherungslösung zur Reynolds-Gleichung, die ein kurzes Lager annimmt, ebenfalls Korrekturterme berechnet. Weiterhin werden die Druckverteilung und die daraus
resultierenden Lagerkräfte in Abhängigkeit von Rotorposition und Rotorgeschwindigkeit numerisch berechnet.
Durch Anwendung der Methode der finiten Elemente wird die Rotorgleichung diskretisiert. Dabei werden die Lagerkräfte als Punktkräfte in den Lagerknoten modelliert.
Während die klassischen Lubrikationsmodelle zu expliziten Systemen gewöhnlicher
Differentialgleichungen führen, führt die Berücksichtigung der Korrekturterme zu impliziten Bewegungsgleichungen für das Rotor-Lager-System. Die numerische Evaluation des Modells erfolgt durch Vergleich mit experimentellen Daten, die an einem
wird gezeigt, dass die Modellgleichungen die Schwingungen des Rotors gut beschreiben und die wesentlichen Effekte, insbesondere die unwuchterregte Schwingung und
die fluidinduzierte Instabilität (oil whirl), wiedergegeben werden. Die Verwendung
der Trägheitskorrekturen führt zu einer leichten Verbesserung der Qualität der Vor-
iv
hersage der Frequenz der Instabilität, bei der kleine Unterschiede zwischen Experiment und Simulation deutlich wurden. Desweiteren wird ein phänomenologisches
Modell basierend auf der Adaption der durchschnittlichen Umlaufgeschwindigkeit
der Schmierflüssigkeit eingeführt, durch welches die Frequenz der Instabilität stark
beeinflusst werden kann.
Kontinuationsmethoden für periodische und quasiperiodische Lösungen erweisen
sich als effizienter und daher geeigneter für die Untersuchung des Frequenzverhaltens als die direkte numerische Simulation. Durch Anwendung dieser Methoden wird
gezeigt, dass die statische Gewichtslast für große Rotationsfrequenzen vernachlässigt
werden kann. In Verbindung mit einer Transformation in ein mitrotierendes Koordinatensystem führt dies zu einer beträchtlichen Vereinfachung des Systems, welches
dadurch autonom wird. Durch diese Vereinfachung kann das Schwingungsverhalten
durch die Berechnung periodischer Orbits statt invarianter Tori ermittelt werden. Dies
wird bei der Untersuchung des Gültigkeitsbereiches der Trägheitskorrektur für das
kurze Lager erfolgreich angewendet.
Zusammenfassend lässt sich sagen, dass sich das beschriebene Modell und seine Varianten als geeignet und nützlich für die zukünftige industrielle Anwendung in der
Entwicklung effizienterer Turbomaschinen erweist. Einige Ergebnisse dieser Arbeit
werden bereits bei den Toyota Central Research and Development Laboratories erfolgreich in der Turboladerentwicklung eingesetzt.
Time is what happens when
nothing else happens.
Richard P. Feynman
Time is an illusion. Lunchtime
doubly so.
Acknowledgements
First of all I would like to express my gratitude to Professor Willi Jäger for giving me
the opportunity to work in his group and for granting me the academic freedom to
pursue my research. For his open mindedness, his constant support and also for his
patience I am very thankful.
Furthermore, I am also very grateful to Professor Jens Starke for initiating the collaboration with the Toyota Central Research and Development Laboratories (TCRDL) and
for organizing the research stays in Japan and at the Technical University of Denmark.
For his advice, critical remarks, and positive thinking I thank him a lot.
This PhD project has been financially supported by TCRDL for which I am very
thankful. My special thank goes to Dr Nobuyuki Mori, Dr Mizuho Inagaki, and
Dr Atsushi Kawamoto, as well as all the other members of the Structural Dynamics
Lab and the Design Engineering Lab for the good collaboration research. My two
stays as a visiting researcher in Nagakute gave me the opportunity for many good
discussions on rotordynamics and allowed me to learn a lot about the engineering
background and industrial research in general.
During the project I was given the opportunity to spend several weeks in inspiring
work exiles at the Department of Mathematics at the Technical University of Denmark. In particular, I have to thank Professor Wolfhard Kliem for the excellent organization, Professor Martin Bendsøe for fruitful discussions and financial support,
Michael Elmegård for proof reading and the helpful remarks during the final stage of
the thesis, and all the other faculty members for the warm welcome and the good atmosphere. One of the stays was supported by the European Research Training Network,
Homogenization and Multiple Scales (HMS 2000).
I am also indebted to Professor Hannes Uecker for the discussions about asymptotic
analysis and for his continuing interest in my thesis. Dr Frank Schilder not only kindly
provided the continuation software for quasiperiodic tori, the discussions with him
also improved my understanding of nonlinear dynamics.
My colleagues in the Applied Analysis group at the University of Heidelberg created
a good working atmosphere which I enjoyed a lot and for which I thank them all.
v
vi
In particular, I would like to thank Christian Reichert for being such a cooperative
office companion, Stefan Knauf for the proof reading and his support on deal.II, and
Eberhard Michel and Cristian Croitoru for their allways available computer advices.
I am also very grateful to Frank Strauß, my colleague on the joint research project
with TCRDL, for the good cooperation, for many mathematical and non-mathematical
discussions, and also for the proof reading. The research stays in Japan and Denmark
became even more enjoyable in his company.
Finally, I would like to thank my parents and Verena for their continuing support and
nearly endless patience throughout the last years.
Contents
1 Introduction
1
1.1
Rotordynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3
Main Results and Structure of the Thesis . . . . . . . . . . . . . . . . . . .
9
2 Continuous Rotor Model
21
2.1
Beam Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2
Potential Energy of the Rotating Beam . . . . . . . . . . . . . . . . . . . . 24
2.3
Kinetic Energy of the Rotating Beam . . . . . . . . . . . . . . . . . . . . . 26
2.4
Euler-Lagrange Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5
Modifications for Rigid Disks . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.6
Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.7
Complex Formulation of Equation of Motion . . . . . . . . . . . . . . . . 36
3 Existence of Solutions
37
3.1
Existence for Linear Support . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2
Existence for Nonlinear Support . . . . . . . . . . . . . . . . . . . . . . . 45
4 Finite Element Discretization
49
4.1
Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2
System Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3
Explicite Choice of Approximating Space . . . . . . . . . . . . . . . . . . 53
viii
CONTENTS
4.4
Rigid Disk Element Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.5
Assembling the Complete System . . . . . . . . . . . . . . . . . . . . . . . 56
4.6
Approximations for Unbalance Forcing . . . . . . . . . . . . . . . . . . . 59
5 Bearing Models
61
5.1
Derivation of Reynolds’ Equation and Inertia Correction . . . . . . . . . 63
5.2
Phenomenological Correction of Pressure Function . . . . . . . . . . . . 86
5.3
Alternative Bearing Models . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.4
Numerical Schemes for Reynolds’ Equation . . . . . . . . . . . . . . . . . 91
5.5
Calculation of Bearing Forces . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.6
Numerical Results and Comparison of Bearing Properties . . . . . . . . . 106
5.7
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6 Direct Numerical Simulation and Experimental Results
113
6.1
Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2
Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3
Analysis of Experimental Results . . . . . . . . . . . . . . . . . . . . . . . 119
6.4
Simulations for Large Rotor Model . . . . . . . . . . . . . . . . . . . . . . 123
6.5
Influence of Oilfilm Model on Dynamics . . . . . . . . . . . . . . . . . . . 127
6.6
Conclusions from the Simulations . . . . . . . . . . . . . . . . . . . . . . 134
7 Numerical Bifurcation Analysis
137
7.1
Short Overview of Continuation Methods and Bifurcation Theory . . . . 138
7.2
Reformulations of Equations of Motion . . . . . . . . . . . . . . . . . . . 140
7.3
Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.4
Numerical Bifurcation Analysis of Large Model . . . . . . . . . . . . . . 148
7.5
Numerical Bifurcation Analysis of Small System . . . . . . . . . . . . . . 150
7.6
Fixed Frame vs. Co-Rotating Frame . . . . . . . . . . . . . . . . . . . . . 154
7.7
Continuation of Quasiperiodic Oscillations . . . . . . . . . . . . . . . . . 157
CONTENTS
ix
7.8
Influence of inertia terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7.9
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
8 Conclusions and Outlook
173
A Euler Angles
177
B Tools from Functional Analysis
181
C Element Matrices
183
C.1 Shape Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
C.2 Element Matrices for Rigid Disks . . . . . . . . . . . . . . . . . . . . . . . 184
C.3 Rayleigh Beam Element Matrices . . . . . . . . . . . . . . . . . . . . . . . 185
D Specifications of Turbocharger Beam Models
187
D.1 13 Element Beam Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
D.2 3 Element Beam Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
E Detailed Formulas for the Inertia Correction p1
191
E.1 Short Bearing Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 191
E.2 Correction to Solution of Reynolds’ Equation . . . . . . . . . . . . . . . . 192
F Integration of Bearing Integrals
193
References
197
x
CONTENTS
Chapter 1
Introduction
The rotordynamic problems analyzed in this work have originated from a joint research project of the University of Heidelberg and the Toyota Central Research &
Development Laboratories. It was the purpose of the project to set up a model for a
passenger car turbocharger, to study its dynamics and parameter dependencies. To
look at things from a more general point of view, this work focuses on the modeling
aspect as well as on the nonlinear dynamics of turbomachinery in general. We will
derive the equations of motion from first principles and subsequently analyze them.
We prove existence for the equations of motion for a quite general set of bearing force
functions. It shows that especially the nonlinear effects of the bearing reaction forces
have a large influence on the dynamics. A large part of the work is therefore contributed to the derivation of laws for the bearing pressure distribution by a thin film
approximation to the Navier-Stokes equation and to the analysis of their dynamical
effects by the numerical solution of the corresponding equations and by numerical
continuation methods. The findings obtained from the numerical bifurcation analysis
of the proposed model provide a better understanding of the influence of the physical
parameters like damping, bearing clearance and width, or weight on the mechanical
vibration of the rotor, and hence on the emitted noise and the material fatigue. Thus
allowing the industrial engineers to improve future rotating machinery.
The purpose of this introduction is twofold. First it is to introduce the reader with
a background in mathematics into the engineering subject of rotordynamics and vice
versa. The second purpose is to give an outline of the structure of this thesis and to
summarize the main results. Therefore, additionally to the very short abstract at the
beginning of this thesis (page i), a more detailed summary of the main results is given
in Section 1.3 on page 9.
2
Chapter 1: Introduction
1.1 Rotordynamics
Rotordynamics is the discipline of mechanics that is concerned with the study of the
dynamics of systems containing parts that rotate with a significant angular momentum
(Crandall, 1995). Rotating mechanical systems are ubiquitous and examples range
from the dynamics of planets, satellites and spinning tops to machines such as turbines,
compressors, pumps, helicopters, gyroscopic wheels and computer hard drives. There
has been a keen interest in rotordynamics since the first steam engines and there is
an extensive literature, especially in engineering; for overviews we refer to (Childs,
1993; Gasch et al., 2002; Vance, 1988; Yamamoto & Ishida, 2001; Ehrich, 1999). Rotating
machinery is called turbomachinery, if the rotor is used to handle fluids or gases,
and energy is exchanged between the process fluid and the rotor. Typical examples
of turbomachinery are pumps and compressors, gas and steam turbines, as well as
turbochargers.
1.1.1 The Turbocharger
The turbocharger is a prime example of a nonlinear rotordynamical system. A turbocharger is a supercharging device used in many modern engines – especially diesel
engines – to increase engine power and to reduce fuel consumption. The turbocharger
Figure 1.1: The rotor of a turbocharger consists of the shaft, the compressor (left) and the
turbine wheel (right). The small brass rings are floating rings and mark the position of the
bearings.
for an internal combustion engine consists of a compressor which is powered by a
turbine. The turbine is driven by the exhaust gases of the previous combustions. The
central part in a typical passenger car turbocharger is the rotor which is depicted in Figure 1.1. It consists of a slender rotor shaft to which the compressor (or impeller) wheel
and the turbine wheel are attached. This rotor is supported by oil lubricated bearings
1.1 Rotordynamics
3
Figure 1.2: The rotor is contained in the casing which is attached to the engine block, here
on the upper left of the motor.
and contained in a casing which itself is attached to the engine block (Figure 1.2).
The supercharging of the engine, i.e. higher pressures of the air-fuel mixture inside the
cylinders, leads to a higher efficiency of the combustion and thus to reduced fuel consumption and/or more powerful motors. In recent years there has been a renaissance of
diesel engines in passenger cars and also in motor sports to which the increasing use of
turbochargers has contributed notably. The main working principle of a turbocharger
is that the exhaust gases from the cylinders drive the turbine wheel which transmits
the rotation to the compressor via the shaft. The pressure of the exhaust gases driving
the turbine is regulated by the wastegate. The rotational frequency of the shaft reaches
values as high as 120000 RPM, i.e. 2000 Hz.
Higher rotational velocities are desirable for an even better compression. However, it
is not possible to increase the frequencies ad infinitum, as the turbocharger and other
turbomachinery are subject to major mechanical vibration problems.
1.1.2 Vibration Problems in Rotordynamics
The two most common problems in rotordynamics of turbomachinery are the occurrence of large amplitude steady state synchronous vibrations and the occurrence of
subharmonic instabilities (San Andrés, 2006). The former have their source in the
unbalance of the rotor due to inevitable production inaccuracies. If e.g. the center
of gravity is not aligned with the rotation axis, the centrifugal forces will lead to a
periodic forcing of the system. Resonances occur for rotational speeds equal to one of
4
Chapter 1: Introduction
the rotor’s eigenfrequencies. These angular velocities are called critical speeds.
These undesirable oscillations cause noise, wear, material fatigue and failure due to
contact, which can have serious consequences since the rotational energy contained
in the system is very high. A common countermeasure to unbalance oscillation is the
balancing of the rotor to reduce the total unbalance or the introduction of damping.
Another possibility is to change the geometry of the rotor-bearing system, such that
the resonance frequencies are tuned out of the operation range of the device under
consideration. This can be done by shape optimization of the rotor and finding optimal
positions of the bearings as has been shown in a related parallel project (Strauß, 2005;
Strauß et al., 2007).
The subharmonic vibrations have their source in the nonlinear reaction forces of the
fluid bearings, seals and fly wheels interacting with the rotor. These vibrations can
also have very high amplitudes and therefore cause noise and wear. Their suppression
can be achieved again by changing the geometry of the system to tune the frequencies
or by introducing damping. The most desirable option is the elimination of the source
of instability. The instability caused by oil lubricated bearings is called oil whirl and
can be partially suppressed by using e.g. elliptical bearings instead of circular bearings
(San Andrés, 2006).
Several experiments were carried out at Toyota Central R&D Laboratories (TCRDL)
for a passenger car turbocharger. The turbocharger was driven by pressurized air and
operated at different rotational speeds from 7839 RPM to 101000 RPM (∼ 130−1700 Hz).
The x- and y-deflection of the shaft was measured by eddy current sensors at both ends
and in the middle of the rotor between the two simple journal bearings.
Usually, floating ring bearings are used for the high-speed turbochargers for diesel
engines. The use of the simpler plain circular journal bearings leads to more unstable
behavior and is not advised in practice. However, in the experiments which were
carried out for this project at TCRDL and in the modeling we will examine plain
circular journal bearings. The observed dynamics are rich, while the modeling stays
simple. The modeling techniques can be easily adapted to more complicated bearing
geometries.
Figure 1.3 shows a power spectrum of the observed vibrations and some orbits of the
impeller side tip of the rotor, measured at different driving speeds. The orbits show the
increasingly complex dynamics of the rotor. In the spectrum in Figure 1.3 we observe
two principal vibration modes: a harmonic part with a resonance peak at about 1000
Hz and a subharmonic part setting in at a threshold forcing frequency of about 400 Hz.
The frequency of this latter vibration is slightly less than half the forcing frequency.
For higher rotational speeds the frequency of the subharmonic increases further, but
the shift of the peaks away from the ω2 -line gets larger and the curve on which the
peaks are located seems to bend away from the ω2 -line.
5
1.2 Mathematical Modeling
Orbit at impeller side, ω=505Hz
Orbit at impeller side, ω=1503Hz
Frequency Diagram (Experiment)
0.1
0.1
Oil whirl
0
Unbalance
Oscillation
0.05
−0.1
y [mm]
y [mm]
0.1
−0.1
0
−0.1
0
0.1
x [mm]
Orbit at impeller side, ω=250Hz
0
2000
−0.1
0
0.1
x [mm]
Orbit at impeller side, ω=998Hz
1500
0.1
0.1
0
500
Driving
Frequency
[Hz] 0
−0.1
−0.1
0
x [mm]
y [mm]
y [mm]
1000
0
500
0.1
1000
Frequency [Hz]
1500
2000
0
−0.1
−0.1
0
x [mm]
0.1
Figure 1.3: Waterfall diagram of the response spectrum for varying driving frequencies and
orbits of impeller end of rotor measured in the experiments. Mainly two kinds of vibration
occur in the examined frequency range: the subharmonic oil whirl and the synchronous
unbalance oscillation of a bending mode.
Analyzing the two kinds of vibration we observe that the harmonic part is mainly due to
bending vibration with resonance of the first bending mode at approximately 1000 Hz.
In contrast to that the mode shape of the subharmonic vibration is a conical one. The
occurrence of the subharmonic vibration has been observed for a long time (Newkirk
& Taylor, 1925); see (Yamamoto & Ishida, 2001) for an overview. This self-excited
vibration with roughly half the driving frequency is called oil whirl and is caused by
the nonlinearity of the supporting oil film. In a series of papers (Muszynska, 1986;
Muszynska, 1987; Muszynska, 1988) the occurrence and the stability of this unwanted
phenomenon are studied. In the resonance region of the first bending mode of the
shaft the amplitude of the self excited oil whirl drops. This phenomenon is called
entrainment. It is known (Crandall, 1996) that the ratio of the subharmonic to the
driving frequency changes drastically when the angular velocity approaches twice
the critical speed and that the so-called oil whip occurs, a large amplitude response
with a frequency equal to two times the critical speed of the rotor. In the shown
experiment the transition happens slowly. The prediction of the frequency of the
subharmonic response and especially setting up a model that reproduces the shift of
the subharmonic frequency are two targets of this work.
1.2 Mathematical Modeling
Mathematical modeling has always played an important role in the study of rotordynamic problems. The prediction of the critical speeds of the more and more complicated rotors was the main interest in the early days of rotordynamics at the end of the
6
Chapter 1: Introduction
xg
k
e
xc
ω
Figure 1.4: The Laval rotor consists of a disk of mass m and a shaft of stiffness k which
is fixed at both ends. It rotates with angular velocity ω. The shaft pierces the disk in its
center xc . The center of mass xg has constant distance e to the center.
nineteenth and the beginning of the 20th century.
A simple and classical example for the usefulness of mathematical modeling in rotordynamics is Jeffcott’s model for the simple 2-D Laval rotor (Gasch & Pfützner, 1975).
The model has been introduced in 1919 and the rotor depicted in Figure 1.4 is therefore
also called Jeffcott rotor. It consists of an rotating elastic shaft with stiffness k which is
fixed at both ends. A disk of mass m is attached to it in the middle and rotates with the
shaft. The center of mass xg of the disk is supposed to be at distance e from the center
xc of the disk. The balance of the linear elastic force and the inertial force yields
mẍg = −kxc .
(1.1)
Due to the rotation of the disk the position of the center of gravity can be expressed by


 cos ωt 
 .
xg = xc + e 
(1.2)
sin ωt 
Hence the equation of motion for the shaft center is

 cos ωt
mẍc = −kxc + mω2 e 
sin ωt


 .

(1.3)
In this simple example the centrifugal force of the unbalance leads to a periodic forcing
term whose amplitude grows quadratically with the angular velocity. The general
analytic solution of this differential equation is


ω2 e  cos ωt 
(1.4)
x(t) = c1 cos ωc t + c2 sin ωc t + 2
 , c1 , c2 ∈ R2 .

ωc − ω2  sin ωt 
q
k
Spinning the rotor in the example with the angular velocity ωc =
unbounded solution, i.e. a resonance catastrophe. Therefore ωc is called critical speed.
1.2 Mathematical Modeling
7
In the presence of damping running the system near the critical speed results in a large
amplitude response. Despite being very simple, Jeffcott’s model exhibits already one
of the most important features of rotordynamics, that of unbalance excitation. Using
this model Jeffcott proved, that a rotor can be driven at super-critical speeds, i.e. at
rotational speeds higher than the critical speed which was doubted before.
The use of more complicated rotor geometries led to a multitude of different models
which take into account gyroscopic effects, asymmetries and varying cross sections of
the rotor and interactions with the bearings, both with ordinary and partial differential
equations. For a detailed overview of the history of rotordynamics we refer to the
introduction of (Yamamoto & Ishida, 2001) and to (Nelson, 2003).
In the last decades computer aided engineering (CAE) has become more and more
important in the industrial development process. In the automotive industry CAE
reduces the cost notably, since the number of prototypes produced for testing purposes can be decreased by using mathematical modeling and applying simulation and
optimization algorithms. In rotordynamics CAE is used to predict the vibrations of
the rotor. For this the calculation of the critical speeds as well as of the frequencies
of eventual self-excited oscillations is necessary. Only if these responses are known,
optimization of the vibration level is possible in a second step.
For the investigation of the critical speeds today the finite element method (FEM) is
most popular. This method has been developed parallely and rather independently by
engineers and mathematicians who have different viewpoints on the method. While
mathematicians see it as an approximation method for variational problems in special,
low-dimensional function spaces, engineers often see the finite elements as building
parts of a mesh that approximates the real object under consideration. Classical works
for FEM are (Strang & Fix, 1973) for the mathematical viewpoint, and (Clough, 1960) for
the engineering viewpoint. Finite elements for slender rotating beams are introduced
in (Nelson & McVaugh, 1976) and are now widely used for the computation of the
critical speeds.
Parallely with the increasing computing possibilities, not only the FEM has grown
in importance, but also a deeper investigation of the nonlinear dynamics of rotormachinery has been made possible. There is a multitude of nonlinear effects possible
in rotordynamics, from nonlinear beam theory for strongly bent rotors to the clearance effects in ball bearings, from hysteretic internal damping to the fluid-structure
interaction in the bearings, seals, and fly-wheels (Yamamoto & Ishida, 2001). Though
nonlinear effects like oil whirl and whip have been known for a long time (Newkirk
& Taylor, 1925), the use of modern simulation methods and numerical analysis has
contributed strongly to the theoretical understanding of these effects.
Especially the fluid-structure interaction has drawn major attention, as the development of high-speed turbomachinery beginning in the 1960s and 70s revealed several
8
Chapter 1: Introduction
rotor instability problems (Childs, 1993). Since the computation of the full 3 dimensional flow of the lubricant and of the pressure distribution in the bearing is a difficult
and computationally expensive task, many simplifications and approximations have
been proposed. Probably the most famous approximation is Reynolds’ equation which
reduces the problem to the solution of a scalar, elliptic equation in a two-dimensional
domain (Ockendon & Ockendon, 1995). Further simplifications are the so-called short
bearing or the long bearing approximation, which allow for an analytic formulation
of the pressure distribution (Sommerfeld, 1964; Yamamoto & Ishida, 2001). There is
a vast literature dedicated solely to the behavior of hydrodynamic journal bearings
bearings, e.g. considering cavitations (Elrod, 1981), turbulent flow (San Andrés et al.,
1995), higher order correction terms for Reynolds’ equation (Crandall & El-Shafei,
1993; El-Shafei, 1995). For an overview and further references we refer the reader to
(Lang & Steinhilper, 1978; Szeri, 1998; Childs, 1993; San Andrés, 2006).
The mathematical models for the bearing pressure distribution allow the computation
of the bearing reaction forces acting on the rotor. These models combined with the
finite element rotor models allow for the prediction of the instabilities, the frequency
response and the amplitudes of the vibrations of the rotor-bearing system. The method
of Lund for the computation of bearing coefficients from the bearing forces, which allow
to predict the onset of instability has to be mentioned in this context (Lund, 1987). Other
examples are (Muszynska, 1987; Muszynska, 1988), where a simple bearing model is
used to clarify the onset mechanism of oil whirl and whip, or (Hollis & Taylor, 1986),
where the limit cycles of fluid-induced oscillations are calculated by direct numerical
simulation. Modern finite element methods for the computation of flow and pressure
in the bearings allow an even better prediction of the rotor instabilities and response
frequencies (Holt et al., 2005).
The direct numerical simulation of the model equations is a popular approach for
the investigation of systems response. By solving multiple initial value problems it
allows the validation of the model equations for different parameter sets and also the
classification of solutions. However, the direct numerical simulation can be very time
consuming, not only because transient behavior has to be accounted for, but also due
to long data sets being necessary for the subsequent analysis of the solution with e.g.
Fourier analysis.
Numerical continuation and bifurcation methods are therefore an useful and efficient
alternative. These techniques provide efficient means of computing branches of solutions of algebraic or differential equations by taking prediction steps along the branch
and employing Newton type methods as corrector. Furthermore, they allow the detection and classification of bifurcations, i.e. qualitative changes of the solution. The
continuation of locus curves of bifurcations allows the partition of the parameter space
into domains of qualitatively equal behavior. Furthermore, continuation methods
allow the investigation of unstable solutions, and thus of hysteretic behavior, which
1.3 Main Results and Structure of the Thesis
9
is also not possible by direct numerical simulations, since unstable solutions can not
be reached by simulating the underlying system forward in time. They are therefore
better suited for extensive parameter studies. The theory of numerical continuation
and bifurcation methods is a broad field. We refer to the textbooks (Chow & Hale,
1982; Kuznetsov, 2004; Nayfeh, 2000; Nayfeh & Balachandran, 1995; Wiggins, 1990)
1.3 Main Results and Structure of the Thesis
The interest in an accurate model of the turbocharger to predict the vibration response
stood at the beginning of this work and the joint research project with TCRDL. The
prediction of response amplitude and frequency is of utter importance for the design
process. As the existing models did not describe the dynamics of the rotor sufficiently
exact and especially the frequency response was not captured correctly, it became
necessary to review the modeling process and to introduce some modifications and
extensions to the model. Special attention is given here to the modeling of the bearings
and to the model validation with techniques for the computation of periodic and
quasiperiodic responses.
The thesis is structured into three main parts. The first part consists of the Chapters
2, 3, and 4 where the modeling of rotating beams is reviewed, existence of solutions
of the derived partial differential equation is proved, and the finite element method is
applied to discretize the equation of motion of the rotor. The second part is identical
with Chapter 5 where the lubrication theory for the hydrodynamical journal bearings
is derived. Higher order terms from an asymptotic expansion of the Navier-Stokes
equations are used to compute correction terms for Reynolds’ equation and the short
bearing approximation. The third part consists of Chapters 6 and 7. There, we validate
the model by direct numerical simulations and use continuation methods for periodic
orbits and quasiperiodic tori to compute the frequency response and its parameter
dependence. In the conclusions in Chapter 8 we summarize the results again with a
special view to future developments and applications.
1.3.1 First Part: The Model for the Rotor
The partial differential equations for the transverse motion of a continuous, isotrope,
rotating beam with varying cross-section are the foundation of our model. These
equations are derived in Chapter 2 by the Lagrangian formalism using Euler-Bernoulli
beam theory and taking into account rotatory inertia and gyroscopic effects following
(Yamamoto & Ishida, 2001) and (Nelson & McVaugh, 1976).
The equations obtained for the lateral deflections u and v of the beam as shown in
10
Chapter 1: Introduction
y
x
v
u
z
Figure 1.5: The displacement of the rotating beam in x and y direction at the axial position
z is described by u(z) and v(z).
Figure 1.5 are
(EIa u′′ )′′ + µü − (Ia ü′ )′ − ω(Ip v̇′ )′ + cu̇ = µω2 (rG,1 cos ωt − rG,2 sin ωt),
′′ ′′
′ ′
′ ′
2
(EIa v ) + µv̈ − (Ia v̈ ) + ω(Ip u̇ ) + cv̇ = µω (rG,1 sin ωt + rG,2 cos ωt) − µg,
(1.5)
(1.6)
where ′ denotes the derivative with respect to the axial position z, and ˙ the derivative
with respect to time. Ia and Ip are the beams cross-sectional diametral and polar
moments of inertia, E is Young’s modulus, µ is the mass per length, c the external
viscous damping factor, and g the gravitational acceleration. rG denotes the position
of the center of gravity of a cross-section relative to the rotation axis. One sees that
analogously to the Jeffcott rotor, the misalignment of center of mass and the rotation
axis leads to the appearance of a harmonic forcing term.
The shaft is supported by bearings. The reaction forces of these bearings are modeled
as boundary or transmission conditions to the equations of motion. The bending
moment and the shear force have jumps of the size of the reaction forces at the location
of the bearings ([ f ]z = f (z+ ) − f (z− ) denotes the jump of f at z)
h
i
= fbi ,1 ,
(1.7)
−Ia ü′ − ωIp v̇′ + (EIa u′′ )′
zb
h
i i
−Ia v̈′ + ωIp u̇′ + (EIa v′′ )′
= fbi ,2 ,
(1.8)
zbi
′′
[−EIa u ]zb = tbi ,1 ,
(1.9)
[−EIa v′′ ]zb = tbi ,2 .
(1.10)
i
i
Together with appropriate initial conditions these 6 equations state the initial/boundary
value problem for the motion of the rotor.
In Chapter 3 we prove existence and uniqueness of weak solutions of these equations
for a quite general class of bearings. To facilitate notation in that chapter, we use a
11
1.3 Main Results and Structure of the Thesis
complex formulation for the displacement of the form w = u + iv, where u and v are
the lateral displacements from (1.5) to (1.10). We define general bearing function of
the following form
f : C2 ⊃ Bcr (0) × C → C,
(1.11)
(x + iy, v + iw) 7→ eiγ ( f n (r, ṙ, γ̇) + i f t (r, ṙ, γ̇)),
where cr is the bearing clearance and
q
r=
x2 + y2 , γ = arg(x + iy), ṙ = v cos γ + w sin γ, γ̇ = (w cos γ − v sin γ)/r.
(1.12)
Note that f is defined only on Bcr (0) × C to model the confinement of the shaft inside
the bearings. With these bearing functions we prove the following existence theorem.
Theorem. Let f be a locally Lipschitz, nonlinear support function as in (1.11). Let a, b ∈
Ω = [0, L] and let the coefficients µ, E, Ip , Ia ∈ L∞ (Ω) be strictly positive. Furthermore let
g ∈ H1 (0, T; L2 (Ω)) and let w0 ∈ H3 (Ω) with |w0 (a)|, |w0 (b)| < cr , and w1 ∈ H2 (Ω).
There exists a short time weak solution to the initial/boundary value problem (1.5)-(1.10) with
support function f , i.e. there is a T > 0 and
w ∈ C0 (0, T; H2 (Ω)),
1
with w′ ∈ C0 (0, T; Hm
(Ω)),
(1.13)
such that for all ψ ∈ H2 (Ω) and for almost all t ∈ [0, T]
ZL h
i
µwtt ψ̄ + Ia wttx ψ̄x − iωIp wtx ψ̄x + EIa wxx ψ̄xx dx−
0
X
Z
f (w(xi ), wt (xi ))ψ̄(xi ) =
xi ∈{a,b}
gψ̄dx.
(1.14)
Furthermore w satisfies the initial conditions:
w(0) = w0 ∈ H3 (Ω) and wt (0) = w1 ∈ H2 (Ω).
(1.15)
The proof is split in two parts. In the first part we use Galerkin’s method to prove
existence and uniqueness for the case of linear support functions. In a second step we
then use a fixed point argument to prove existence in the nonlinear case.
In Chapter 4 the finite element method is applied to Equations (1.5)-(1.10). We use
standard 3rd order Hermite polynomials to compute the well known system stiffness
matrix K, gyroscopic matrix G, damping matrix C and mass matrix M for beam elements
of constant diameter. A rotor with varying cross-sections can be composed from several
such simple beam elements of constant diameter like in Figure 1.6. The individual
beam element matrices are then assembled to yield a system of ordinary equations
describing the motion of the nodal coordinates of whole beam.
12
Chapter 1: Introduction
Beam model
50
40
30
width [mm]
20
10
0
−10
−20
−30
−40
−50
0
20
40
60
80
100
120
length [mm]
Figure 1.6: Detailed beam model of turbocharger: the rotor shaft is modelled with 13 finite
elastic beam elements shown in blue, the turbine and impeller wheels are modelled as rigid
disks and are shown with dashed red lines, and the positions of the bearings are indicated
by the black triangles.
1.3.2 Second Part: The bearing model
As already mentioned in the previous sections the influence of the bearings is crucial for
entire dynamics of the rotor. We assume plain circular journal bearings throughout this
work, since bearings of this kind were also used in the experimental setup at TCRDL.
The shaft rotates inside this cylindrical bearing and the thin clearance between shaft
and bearing is filled with a lubricant fluid as shown in Figure 1.7. The rotation of the
shaft creates a circular flow pattern by dragging along the fluid. This flow pattern
causes the impedance of the bearing to loads on the shaft by causing higher pressures
in narrowing regions of the bearing. This creates reaction forces that oppose the
movement. The reaction forces can be calculated from the pressure distribution inside
the bearing by integration over the bearing surface. For a thin fluid film of thickness
h bounded by two moving surfaces with relative velocities Vϕ and Vr the pressure
distribution p is given approximately by the well known Reynolds’ equation
1
δ2 ∂ϕ (h3 ∂ϕ p) + ∂z (h3 ∂z p) = −12(Vr + Vϕ ∂ϕ h)
2
for
(ϕ, z) ∈ [0, 2π] × [0, 1].
(1.16)
In Chapter 5 we review the derivation of this elliptic, second order equation from the
Navier-Stokes equations by asymptotic analysis with the ratio ε of bearing clearance
and bearing radius as small parameter (Ockendon & Ockendon, 1995).
The short bearing approximation to Reynolds’ equation is usually derived (Yamamoto
& Ishida, 2001) from Reynolds’ equation by an additional asymptotic expansion under
13
1.3 Main Results and Structure of the Thesis
r=0
ur
uϕ
ϕ
r = −h(ϕ)
γ
VZ
cr
Rj
R
e
bearing casing
lubricant (oil)
Z
O
rotating shaft
ω
rotating shaft
lubricant film (oil)
bearing casing
lubricant (oil)
bearing casing
W
Figure 1.7: Sketch of simple journal bearing; view in axial direction (left) and lateral
direction (right); the radial bearing clearance cr is exaggerated for illustration
the assumption that the ratio δ of the axial length to the radius of the bearing is small.
We derive both, Reynolds equation and the short bearing approximation in Chapter 5
in one step from the Navier-Stokes equations by assuming a special relation between
the two scaling parameters. With this special scaling we can identify inertia correction
terms from higher order terms of the asymptotic expansion. All in all we consider four
cases in this work which differ by higher order terms and by the ratio of the bearing
width W to the bearing radius R:
1. The classical Reynolds’ equation: The ratio ε = cRr is small, while the ratio δ =
is of order 1; no higher order terms are considered.
W
R
2. The short bearing approximation: the ratio δ = W
R is also small in addition to ε.
To derive the approximate equations in one step we set ε = Kδ2 . Then the
equations simplify even more and an analytical solution for the bearing forces
can be obtained.
3. Reynolds’ equation with inertia corrections: In the derivation of Reynolds’ equation only terms of zeroth order in ε are considered. The inertia correction takes
into account also terms of order ε1 .
4. Short bearing with inertia corrections: As above the additional assumption of
small δ and ε = Kδ2 allows for further simplification and analytical solution for
the pressure distribution also for the higher order terms.
While the results for points 1 and 2 are well known, the corrected solutions from point
3 and 4 obtained in Chapter 5 are new. We summarize them in two statements.
14
Chapter 1: Introduction
Statement 1.1. The pressure solution for the zeroth order short bearing approximation corresponding to point 2 in the list is
p0 = −6z(z − 1)
(γ′ − 21 )κ sin ϕ + κ′ cos ϕ
(1 − κ cos ϕ)3
.
(1.17)
The inertia correction for the short bearing pressure distribution has the following structure
p1 = F0 + RF1 + γ′ F2 + Rγ′ F3 + R(γ′ )2 F4 + Rγ′′ F5
+Rκ′′ F6 + κ′ F7 + Rκ′ F8 + Rκ′ γ′ F9 + R(κ′ )2 F10 ,
(1.18)
where the F j are rational functions of h, κ cos ϕ and κ sin ϕ. The detailed formula is given in
Equation (5.61). An alternative formulation is given in Appendix E.1. We call p = p0 + εp1
the first order short bearing approximation. This approximation corresponds to point 4 in the
list above.
While Statement 1.1 holds for bearings where the ratios ε = cRr and δ = W
R are both
2
small and approximately fulfill the additional relation ε = Kδ , the following statement
holds for bearings where only ε = cRr is small and which can have arbitrary width W.
Statement 1.2. The pressure distribution in a circular hydrodynamic bearing with lubrication
film thickness h0 = 1 − κ cos ϕ is determined by three equations with the same differential
operator
3∂ϕ h0
L(·) = δ2 ∂2ϕ (·) + δ2
∂ϕ (·) + ∂2z (·)
(1.19)
h0
and varying right hand sides
ε0 :
L(p0 ) = f0 (κ, κ̇, γ̇),
ε1 :
(1.20)
L(p1 ) = Ψ(κ, κ̇, γ̇, κ̈, γ̈, p0 , ∇p0 , ∇2 p0 , ∇3 p0 , ∇2 (∂t p0 )),


∂
h
∂
h
3∂
∂
h


t
0
ϕ
0
ϕ
t
0

 .
+
L(∂t p0 ) = ∂t f0 − δ2 ∂ϕ p0 

2
h
h0
where
(1.21)
(1.22)
Here the pi are scalar functions defined on Ω = (0, 1) × (0, 2π). The boundary conditions are
pi (ϕ, 0) = pi (ϕ, 1) = 0
i
i
p (0, z) = p (2π, z)
for ϕ ∈ (0, 2π),
for z ∈ (0, 1).
(1.23)
(1.24)
The parameters κ, γ, κ̇, γ̇, etc. are given by the current shaft position and velocity in polar
coordinates. The function f0 is given in Eq. (5.79), while a detailed expression for Ψ can be
found in the Appendix E.2. The pressure distribution p0 is called the zeroth order solution and
corresponds to point 1 in the list. The pressure distribution p1 is called the inertia correction
and we call p = p0 + εp1 the first order solution to Reynolds’ equation, corresponding to point
3 in the list.
1.3 Main Results and Structure of the Thesis
15
These higher order effects lead to a nonlinear dependence of the pressure distribution, and hence the bearing reaction forces, on the rotational frequency of the shaft.
Reynolds’ equation as well as the zeroth order short bearing approximation only yield
a linear dependence of the pressure on the driving frequency. In Chapter 6 simulations
show that this nonlinearity has an influence on the frequency of the subharmonic vibration. In some parameter regions it decreases that frequency, thus reproducing the
effect observed in the experiments to a certain degree.
In addition to these analytically justified bearing models, we introduce a phenomenological correction for the zeroth order short bearing approximation
W 2 ρν (γ̇ − ω + s(ω))κ sin ϕ + κ̇′ cos ϕ
z̄
z̄
2
p̄0 = −6
.
(1.25)
−1
W W
(1 − κ cos ϕ)3
c2r
The correction term s(ω) changes the average circumferential lubricant velocity which
is equal to ω2 in the usual short bearing approximation. This average velocity has been
identified as an important parameter for the onset and the frequency of the oil whirl
(Muszynska, 1986). By introducing this correction, we provide a simple and effective
model which can be tuned to show a certain frequency response, as shown by the
simulations in Chapter 6.
Finally, in Chapter 5 we also introduce some numerical schemes for the solution of
Reynolds equation. In particular, we give variational formulations of (1.20)-(1.22) and
use the finite element toolbox deal.II (Bangerth et al., 2008) to compute solutions on
an adaptive grid. These are used in the numerical simulation for the calculation of the
bearing forces. For a more detailed summary of the modeling of the bearings we refer
the reader to the conclusions section 5.7 of Chapter 5.
1.3.3 Third Part: Numerical Analysis
Application of the finite element method with standard C1 -elements to the equation
of motion yields a system of coupled ordinary differential equations for the nodal
coordinates q
Mq̈ + (C + ωG)q̇ + Kq = Funb (t) + F g + Fbear
(1.26)
with system mass, damping, gyroscopic and stiffness matrices M, C, G, and K, 2π/ωperiodic unbalance forcing term Funb (t), static load F g and bearing reaction forces Fbear .
For simpler bearing models like the zeroth order short bearing approximation, the
forces depend on the deflection qb and velocity q̇b of the node inside the bearing
Fbear = Fbear (qb , q̇b ),
(1.27)
while for bearing models that include inertial terms, they also depend on the acceleration q̈b
Fbear = Fbear (qb , q̇b , q̈b ),
(1.28)
16
Chapter 1: Introduction
thus yielding an implicit differential equation.
This model reproduces the experimental results quite well as is shown in Chapter 6
by direct numerical simulation with standard implicit integration methods for stiff
problems like backward differentiation formulas (BDF).
Orbit Impeller Side 350 Hz (Simulation)
Orbit Impeller Side 1500 Hz (Simulation)
Frequency diagram (simulation)
0.1
0.1
0
0
0.05
−0.1
−0.1
0
−0.1
0
0.1
2000
0.1
−0.1
Orbit Impeller Side 50 Hz (Simulation)
0
0.1
Orbit Impeller Side 1000 Hz (Simulation)
1500
0.1
0.1
1000
0
0
500
Driving
Frequency
[Hz] 0
−0.1
−0.1
0
0.1
0
500
1500
1000
Frequency [Hz]
2000 −0.1
−0.1
0
0.1
Figure 1.8: Simulated orbits and waterfall diagram showing the response spectrum for 13
element beam model. The parameters are chosen similar to the experimental configuration
used for the results in Fig. 1.3. The main experimentally observed vibration effects of
subharmonic oil whirl and synchronous unbalance vibration are captured in the model.
Figure 1.8 depicts the simulation results corresponding to the experiment shown in
Figure 1.3. It can be seen that the simulated and the experimental results agree to a
great deal and that the main dynamical features of the experiment are reproduced.
The complexity of the orbits increases with the rotational speed and also the computed
amplitudes are only slightly larger than in the experiment. Entrainment can be observed around a rotational speed of 1000 Hz where the first resonance of the harmonic
response occurs. There are however two differences between the simulations and the
experiments. In the simulations there appears a second peak in the harmonic response,
which is caused by the resonance of a second bending mode. This can not be observed
in the experiments. Furthermore, the ratio of the oil whirl frequency and the driving
frequency remains constant 21 and the shift to lower frequency ratios does not occur.
Further simulations reveal that the inertia corrections of the bearing functions have a
small effect on the frequency shift. The right panel in Figure 1.9 shows the response
spectra from two simulations of a smaller system. It can be clearly seen, that the use
of the inertia correction in the bearing function reduces the subharmonic frequency.
The effect of phenomenological correction (1.25) is more pronounced, as can be seen
from the left panel in Figure 1.9. By choosing an appropriate correction term, it
allows to reproduce a measured frequency behavior of a certain bearing type without
detailed modeling of the bearing, and without the computational effort of solving
17
1.3 Main Results and Structure of the Thesis
1900Hz
−3
sigma=0.1
x 10
8
short b.
inertia corr.
2000
rot. speed [Hz]
7
6
1500
5
4
1000
3
2
500
Amp=0.01mm
0
0
500
1000
1500
frequency [Hz]
2000
1
0
920
940
960
frequency [Hz]
980
Figure 1.9: Left: The phenomenological correction of the short bearing approximation can
be used to influence the frequency shift of the subharmonic response frequency. Right:
The first order correction of the short bearing approximation has a small effect on the
subharmonic response. The blue peak shows the response frequency of the system with an
uncorrected, the green peak with inertia corrected bearing function. The system is driven
with 1900Hz.
partial differential equations in each time or optimization step. The results of Chapter
6 are summarized in more detail in Section 6.6
Direct simulation is useful to validate the model, for the investigation of the parameter
dependence of the solutions, however, continuation methods yield better results, because they allow to investigate also unstable solutions as well as bifurcations. As we
have seen before, the rotor-bearing systems examined in this work exhibit responses
with more than one frequency. Additionally to the harmonic response to the forcing
with frequency ω, there appears also a subharmonic, self-excited oscillation with frequency ω2 ≤ ω2 . In Chapter 7 we analyze the onset of this self-excited oscillation and
show numerically that the instability appears through a torus bifurcation of a stable
2π
ω -periodic orbit. We compute locus curves of the bifurcations, i.e. stability boundaries, in several parameters such as driving frequency, external damping and bearing
clearance. It shows that the self-excited oscillation can be suppressed by increasing
external damping, or by decreasing the bearing clearance.
A transformation of the equations of motion to a frame of coordinates co-rotating with
angular velocity ω
Mp̈+(2ωMH+G+C)ṗ+(K−ω2 M+ωGH+ωCH)p = F̃bear (p, ṗ)+Fgr cos(ωt)+ω2 Funb (1.29)
together with the neglection of the static (e.g. gravity) load Fgr leads to a significant
simplification of the equations of motion, since the system becomes autonomous.
This has the effect that quasi-periodic solutions with one of the basic frequencies
equal to ω in the fixed frame of coordinates are transformed to periodic solutions
in the co-rotating frame. This simplification has been applied for the computation
of the periodic orbits shown in Figure 1.10 with the software package for numerical
18
Chapter 1: Introduction
1000
y 0
−1000
1000
x
0
−1000
0
200
1000
1200
ω=826.0Hz
200
0
0
0
y
200
−200
−200
−200
0
200
x
ω=949.9Hz
−200
0
200
x
ω=1000.0Hz
0
200
x
ω=1392.7Hz
0
0
y
0
y
200
−200
0
x
200
1800
−200
200
−200
1600
1400
−200
200
−200
ω
ω=898.7Hz
200
y
y
ω=405.0Hz
y
800
600
400
−200
−200
x
0
200
−200
0
x
200
Figure 1.10: Continuation of periodic solutions of Eq. (7.22) in absence of constant load
w.r.t. driving frequency; the two lower lines of graphs show the detailed orbits drawn in
red in the top figure.
continuation AUTO (Doedel et al., 2000).
A justification for the simplification of neglecting the static load is also given in Chapter 7 through the computation and continuation of the corresponding quasiperiodic
tori of the non-simplified system. In Figure 1.11 we show the locus curves of quasiperiodic tori with constant rotation number in the driving frequency- bearing clearance
domain. The rotation number given by the color of the respective branch. A comparison with the frequency of the corresponding periodic orbits like those depicted
in Figure 1.10 shows significant differences only for small rotational frequencies outside the usual range of operation of the turbocharger. The continuation method for
quasiperiodic tori has been applied to a ’real world’ problem of this size for the first
time in this work and has been published in (Schilder et al., 2007). It is based on a
Fourier method proposed in (Schilder & Peckham, 2007).
19
1.3 Main Results and Structure of the Thesis
1150
0.491
1100
Driving Frequency [Hz]
1050
0.485
1000
0.479
950
0.474
900
850
0.468
800
0.462
750
700
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Bearing Clearance [mm]
Figure 1.11: Locus curves of quasiperiodic tori with constant rotation number indicated
by the color. The dashed line shows the locus of a torus or Neimark-Sacker bifurcation.
The continuation methods combined with the simplification of neglecting gravity and
transforming into a co-rotating frame is also used to study the influence of the inertia.
The implicity of the equation of motion in that case makes necessary an adaptation of
the numerical method since AUTO only can deal with explicit equations. We employ
an internal Newton method to solve the implicit equation (1.28) for q̈ in every iteration
step of the corrector method of the continuation method. Even though this increases
the total number of Newton steps, this is still more efficient than direct numerical
simulations. It shows that only very few internal Newton steps are necessary. The
results show that the effect of the short bearing inertia correction can be observed over
a large parameter range in bearing width (W = 0.25 mm−2 mm) and driving frequency
(ω = 0 Hz − 2000 Hz). However, the validity of the first order correction for the short
bearing approximation is not given for bearings of the length used in the experimental
set-up (W = 5.4). Instead, the full Reynolds’ equation with inertia correction from
1.21 should be used. This is numerically very expensive, which makes a bifurcation
analysis of the same detailedness impossible. However, a proof of principle for the
method proposed is given in the last section of Chapter 7. Therefore it could be used
in the development and validation of future models for rotordynamical systems. The
reader is referred to Section 7.9 at the end of Chapter 7 for a more detailed summary
of the results of the numerical bifurcation analysis.
The conclusions in Chapter 8 close this thesis with a discussion of the results in the
light of further developments and applications.
20
Chapter 1: Introduction
Chapter 2
Continuous Rotor Model
In this chapter we derive partial differential equations that model the transverse motion
a slender rotating beam. The model is based on the theory of linear elasticity and
includes gyroscopic effects. In Section 2.1 the geometry of the beam is defined and
special attention is paid to the imperfections in the geometry which can be caused
during production and which lead to static unbalance excitation. Afterwards we derive
the equations of motion for a rotating Rayleigh beam by means of the Lagrangian
formalism in Sections 2.2 to 2.4. We obtain two partial differential equations for
the transversal displacements which include rotatory inertia and gyroscopic terms.
Further modifications for the inclusion of fly wheels into the model are made in Section
2.5. Angular misalignment of the rigid disks modeling the wheels leads to dynamic
unbalance excitation. Finally we show how bearings can be added to the model (Sec.
2.6) and introduce a complex valued notation which reduces the number of equations
(Sec. 2.7).
2.1 Beam Geometry
A beam is a three-dimensional solid body with a long-stretched geometry, meaning
that it has two small dimensions compared to the third. In the following the beam is
supposed to be homogeneous and isotropic, i.e. all the material properties like density
or Young’s modulus do not depend on position. It is supposed to rotate about its
center line with constant angular velocity ω. In the undeformed state the rotational
axis is identical with the z-axis. In the center of each cross-section we fix a coordinate
system spanned by the cross-section’s principal axis of inertia e1 (z) and e2 (z), and the
normal vector e3 (z)). This coordinate system is rotating with the same angular velocity.
In the following we will consider a circular beam of total length L with varying cross-
22
Chapter 2: Continuous Rotor Model
y
y
x
A1
A0
R0
rG(0)
R1
rG(L)
x
z
rG(z)
Figure 2.1: Left: Location of the center of mass along the axis of the beam; the eccentricity
at axial position z is given by rG (z). Right: Shape of a cross-section with circular inner part
A0 and perturbed exterior A1 . The size of the perturbation is exaggerated for illustration.
sectional shape. The reference configuration is given by
q
D = {(x, y, z) ∈ R3 |z ∈ [0, L], x2 + y2 ≤ R(z, ϕ) ≪ L},
(2.1)
where ϕ = arg(x + iy) and the shape function R : [0, L] × [0, 2π] → R is a piecewise
continuous in z and in ϕ, bounded, strictly positive function. Where it is more convenient, e.g. in sums, we will also use the notation r = (x, y, z) = (x1 , x2 , x3 ) for points in
D.
A major source of vibration in real life rotating machine elements is unbalance which is
caused by small imperfections in the geometry due to production and by unbalanced
loads on the beam. The small imperfections of the rotor can be modelled either by
assuming non perfect geometry or by allowing small density variations. We will take
the first point of view and assume homogeneous density ρ and the shape function R
to be of the form
R(z, ϕ) = R0 (z) + R1 (z, ϕ) > 0.
(2.2)
We take R0 > 0 and R1 ≥ 0, and further more we consider the non-circular perturbation
R1 to be small, i.e. for all (z, ϕ) ∈ [0, L] × [0, 2π] we have
R1 (z) R − R0
=
≪ 1.
R
R
(2.3)
Therefore its contribution to the area A(z) of each cross-section is also small,
R 2π R R0 +R1
r dr dϕ
A1 (z)
R
0
= R 2π R R0 +R
≪ 1.
0
1
A(x)
r dr dϕ
0
(2.4)
0
The mass per length µ is given by
µ(z) = ρA(z).
(2.5)
23
2.1 Beam Geometry
We define the center of mass (rG,1 , rG,2 )(z) of a cross-section A(z) orthogonal to the z-axis
in the body-fixed coordinate frame of the principal axis of inertia by
rG,α (z) =
1
µ(z)
=
1
µ(z)
=
1
µ(z)
Z
ρxα dx dy
(2.6)
A(z)
Z
Z
1
ρxα dx dy +
µ(z)
A0 (z)
ρxα dx dy
(2.7)
A1 (z)
Z
ρxα dx dy
for α = 1, 2.
(2.8)
A1 (z)
The first integral in (2.7) vanishes due to the circular shape of the unperturbed crosssection with radius R0 . Only the perturbation part of radius R1 gives a nonzero
contribution. However, the smallness of R1 compared to the entire radius obviously
also yields the smallness of rG,α compared to the shaft radius. As shown in Fig. 2.1 rG
describes the eccentricity of the centers of mass of the cross-sections. The coordinate
system of the principal axis of inertia is rotating with an angular velocity ω with respect
to the coordinate system which is fixed in space. Hence the coordinates rG of the center
of mass in the rotating system and the position of the center of mass rg in the fixed
system are related
rG = A(ωt)rg ,
(2.9)
where the matrix A is the rotation matrix

 cos ωt sin ωt
A(ωt) = 
− sin ωt cos ωt


 .

(2.10)
The perturbation of the shape also leads to a perturbation of the principle moments
of inertia of the cross-sections. We have defined the slender beam to be small in two
dimensions and elongated in the third. Like before, we now look at infinitesimal
cross-sections orthogonal to the z-axis. The tensor of inertia J of such an infinitesimal
cross-section in a fixed coordinate system whose origin is located at its center is defined
by
Z
ρ(δij xk xk − xi x j ) dx1 dx2 dx3 .
Jij =
(2.11)
A(x3 )
By using the fact that the center is located at the origin (x3 = 0) and by the decomposi-
24
Chapter 2: Continuous Rotor Model
tion of the area A(x3 ) into the circular part A0 and the perturbed part A1 we obtain
J31 = J32 = 0,
Z
Z
J33 =
ρxα xα dx1 dx2 dx3 =: ρIp (x3 ) dx3 +
A(x3 )
Z
A1 (x3 )
Z
J12 = −
ρx1 x2 dx1 dx2 = −
A(x3 )
ρx1 x2 dx1 dx2 dx3 := −ρ∆I12 dx3 ,
A1 (x3 )
Z
Z
ρx22 dx1 dx2 dx3 =: ρIa (x3 ) dx3 +
J11 =
A(x3 )
ρx22 dx1 dx2 dx3 ,
A1 (x3 )
Z
Z
ρ(x)x21 dx1
J22 =
ρ(x21 + x22 ) dx1 dx2 dx3 ,
dx2 dx3 =: ρIa (x3 ) dx3 +
A(x3 )
ρx21 dx1 dx2 dx3 .
A(x3 )
We see that J11 , J22 , and J33 can be expressed in terms of a the circular cross-sections area
moments of inertia, the diametral moment Ia and the polar moment Ip , respectively,
and small perturbations caused by the shape perturbation R1 . Hence the inertia tensor
of the cross-section has the following form

−∆I12
0
 Ia + ∆I11


J = ρ  −∆I12
Ia + ∆I22
0

0
0
Ip + ∆I33



 dz.


(2.12)
The coordinate systems spanned by the cross-section’s principal axis of inertia (e1 (z),
e2 (z), e3 (z)) in which I is diagonal depend on z. However, from J31 = J32 = 0 we
can deduce that the third principal axis e3 is always collinear with the z-axis. After
diagonalization of the tensor, let the principal moments of inertia be J1 , J2 and J3 = J33
and let principal area moments be I1 , I2 and I3 . We then have Ji = ρIi dz. Due to
the small variations in shape, the diametral moments I1 and I2 differ only by small
perturbations ∆Ii from the diametral area moment Ia = πR4 /4 of a perfect disk. We
will assume that
I1 = Ia + ∆I,
and I2 = Ia − ∆I.
(2.13)
This assumption can be fulfilled by adapting the radius R in such a way that a perfect
disk of radius R has an area moment equal to the mean of I1 and I2 . The polar area
moment for a circular cross-section with uniform density distribution is Ip = πR4 /2.
2.2 Potential Energy of the Rotating Beam
We will now derive an expression for the elasic energy stored in a deformed rotating
beam. In order to describe deformations of the reference configuration D we define
the displacement field u : D → R3 . This vector field u describes the translation of a
25
2.2 Potential Energy of the Rotating Beam
point x of the reference configuration to its new position x̃, i.e. for all x ∈ D we have
x̃ = x + u(x).
(2.14)
In the following we will assume that the displacement of the beam is small. This
assumption leads to the classical Euler-Bernoulli beam theory which can be found in
many textbooks. In (Landau & Lifschitz, 1983) it is derived by physical reasoning,
in (Trabucho & Viaño, 1995) the displacement field is derived by strict asymptotic
analysis based on a variational formulation and using only the two assumptions that
the overall displacement is small and that two dimensions of the beam are small.
Shear effects can be neglected for slender beams. The shaft of the turbocharger we
want to model has a slenderness ratio s = L/R ≈ 70 for which it is sufficient to use
Euler-Bernoulli-theory. The inclusion of such shear effects would lead to Timoshenko
beam theory (Han et al., 1999) which is not studied in further detail in this work. A
continuous rotating Timoshenko beam model can be found in (Eshleman & Eubanks,
1969) and in (Nelson, 1980) where also a finite element discretization for such beams
is presented.
First we consider the displacement field (u1 , u2 , u3 ) : D → R3 obtained from EulerBernoulli theory in the rotating frame of the bodies principal axis of inertia. In the
absence of axial forces it can be described (Trabucho & Viaño, 1995) by
u1 (x, y, z) = d1 (z),
u2 (x, y, z) = d2 (z),
u3 (x, y, z) = x∂z d1 + y∂z d2 .
(2.15)
Cross-sections orthogonal to the center line stay planar and orthogonal to the center line
after deformation. The displacement of the center line is hence sufficient to describe
the whole displacement of the beam. If the displacement is small, the elastic energy of
a bent Euler-Bernoulli beam is given (Landau & Lifschitz, 1983) by
1
Ue =
2
ZL
EI1 (∂zz d1 )2 + EI2 (∂zz d2 )2 dz.
(2.16)
0
Here E is Young’s modulus. I1 and I2 are the cross-sectional diametral moments of
inertia.
If we assume that the beam rotates about its center line with constant angular velocity
ω we can transform the above expression to a fixed coordinate system in which the
deformation of the centerline is described by the functions u for the deformation in the
x-direction and by v for the deformation in the y-direction. We have


 
 d1 
 

 = A(ωt)  u  .
 d 
 v 
2
(2.17)
26
Chapter 2: Continuous Rotor Model
If we denote by ′ the derivative with respect to z, we obtain from (2.16)
1
Ue =
2
ZL
EI1 (cos ωtu′′ + sin ωtv′′ )2 + EI2 (− sin ωtu′′ + cos ωtv′′ )2 dz
0
ZL 1
E (I1 cos2 ωt + I2 sin2 ωt)u′′ + (I2 cos2 ωt + I1 sin2 ωt)v′′
=
2
0
+ 2(I1 − I2 )u′′ v′′ cos ωt sin ωt dz
1
=
2
ZL
EIa (u′′2 + v′′2 ) + E∆I (u′′2 − v′′2 ) cos 2ωt + 2u′′ v′′ sin 2ωt dz.
(2.18)
0
If the rotor is also subject to gravity (or other static loads which are given by a potential)
we have to add the potential to the elastic energy (2.18)
ZL
µgv dz.
U = Ue + Upot = Ue +
(2.19)
0
where the second equation is for gravity acting in the −y-direction and g denotes the
gravitational acceleration.
2.3 Kinetic Energy of the Rotating Beam
Euler-Bernoulli theory states that planar cross-sections of the undistorted beam which
are orthogonal to the center line stay planar and orthogonal to the center line also in
the bent state. Our derivation of the kinetic energy of the rotating beam is based upon
this observation.
In the unbent reference state each point in the beam is described by its position along
the axis of the rotor (z-coordinate) and its position on the cross-section orthogonal to
the center line at the point’s z-coordinate. The position of the center relative to the
z-axis is denoted by r0 (z, t) and the center line has the coordinates (r0 , z). In the unbent
state therefore r0 = 0. At each point on the z-axis we choose the principal axes of inertia
of the cross-section e1 (z, t), e2 (z, t) and the normal vector e3 (z, t) as moving orthonormal
basis for that cross-section. As stated above this normal vector is always collinear
with the tangent vector of the center line. Using the sum convention the position of
the material point relative to the center line is given by r = ξα eα (z, t). Repeated Greek
indices are summed over 1 and 2, and repeated Roman indices are summed over 1, 2,
and 3.
The new position X(x, t) of a material point x at time t is given by the displacement
r0 (z, t) of the center point transverse to the z-axes and the position relative to the center
27
2.3 Kinetic Energy of the Rotating Beam
point
X(x, t) = r0 (z, t) + ξα eα (z, t).
(2.20)
The cross-section in which the point x lies in the undistorted beam remains rigid and
orthogonal to the center line. The position of the point on the disk is given by ξα eα (z, t)
in terms of the rotating coordinate system. The disk rotates with the angular velocity
Ω(z, t) about the momentary axis of rotation spanned by Ω = Ωi ei .
The kinetic energy of the beam is given by the integral over the velocities in each point
Z
1
T=
ρ(x)(∂t X(x, t))2 dx3 .
(2.21)
2
D
Using the above assumptions on the deformation we obtain by integration over the
cross-sections
Z
1
ρ(x)(ṙ0 (z, t) + Ω(z, t) × ξi ei (z, t))2 dx3
(2.22)
T =
2
D
Z
1
=
ρ(x)(ṙ0 (z, t) · ṙ0 (z, t) + 2ṙ0 (z, t) · (Ω(z, t) × ξi ei (z, t))
2
D
=
+(Ω(z, t) × ξi ei (z, t)) · (Ω(z, t) × ξi ei (z, t))) dx3
Z
1
ρ(x)(kṙ0 (z, t)k2 + 2(ṙ0 (z, t) × Ω(z, t)) · ξi ei (z, t)
2
D
+(Ω1 ξ2 − Ω2 ξ1 )2 + (ξ21 + ξ22 )Ω3 ) dx3
=
1
2
ZL
µ(u̇2 + v̇2 ) + 2µṙ0 · (Ω × rg (z, t)) + I1 Ω21 + I2 Ω22 + I3 Ω23 dz
(2.23)
0
ZL
Tt + Tunb + Trot dz,
=:
(2.24)
0
where we used the property of the ei being the principal axis of inertia of the disk. The
energy splits into three parts. The first term in (2.23) gives the translational kinetic
energy Tt . The term containing the eccentricity rg (z, t) of the cross-section appears
when integrating the second term in the third line. It gives the rotational energy stored
in the rotation of the center of mass of the cross-section about the center line and is
denoted by Tunb . The third term gives the rotational energy Trot of the inclined disk.
Starting with the last term we will now derive expressions for the rotational energy
in terms of the displacement and the inclination of the center line. To do so, we have
to express the momentary angular velocity Ω in such terms. This can be achieved by
introducing Euler angles describing the postion of the disk.
The position of the cross-section relative to the origin is given by the Euler angles
(γ, β, φ) (Nelson & McVaugh, 1976). The three angles describe three successive rotations
28
Chapter 2: Continuous Rotor Model
∂ zv
∂ zu
z
t
β
γ
e3
γ
ex’
e2
y
ϕ
e1
x
Figure 2.2: Euler angles γ, β and φ describe the orientation of the rotating disk.
of the disk whose principal axis of inertia (e1 , e2 , e3 ) are assumed to be initially collinear
with the space coordinate system (ex , ey , ez ). The first rotation leaves the y-axis fixed
image ex′ of the vector ex fixed and rotates the disk by β. The third rotation which
corresponds to the spin of the disk rotates the coordinate system about the image of ez
under the first two equations by the angle φ.
From Figure 2.2 we see that the momentary angular velocity is given by
Ω = γ̇ey + β̇ex′ + φ̇e3 .
(2.25)
In this equation ey is the unit vector in y-direction, e3 is the normal vector to the plane
(and also spans a principal axis of inertia), and ex′ is the unit vector along the image
of the x-axis after the first rotation. From (2.25) we can obtain the expression for the
current angular velocity in terms of the Euler angles. The detailed calculations of what
follows are given in the Appendix A where we obtain
Ω1 = γ̇ sin ϕ cos β + β̇ cos ϕ,
(2.26)
Ω2 = γ̇ cos ϕ cos β − β̇ sin ϕ,
(2.27)
Ω3 = −γ̇ sin β + ϕ̇,
(2.28)
and for the third term in the integral from Equation (2.23) which gives the rotational
29
2.4 Euler-Lagrange Equations
energy of the infinitesimal cross-section
Trot =
Ia 1
I1 Ω21 + I2 Ω22 + I3 Ω23 = γ̇2 cos2 β + β̇2
2
2
Ip 2
ϕ̇ − 2 sin βγ̇ϕ̇ + sin2 βγ̇
(2.29)
+
2
∆I 2
+
(β̇ − γ̇2 cos2 β) cos 2ϕ + 2β̇γ̇ cos β sin 2ϕ .
2
We now use the assumption that the total deformation and hence also the inclination
of the beam is small. Using γ ≈ u′ , β ≈ −v′ and φ = ωt as derived in Appendix A, by
dropping second order terms we get
2Trot ≈Ia (u̇′2 + v̇′2 ) + Ip ω2 + 2ωu̇′ v′
(2.30)
+ ∆I (v̇′2 − u̇′2 ) cos 2ωt − 2u̇′ v̇′ sin 2ωt .
The second term from the integral in (2.23) evaluates to
Tunb = 2µṙ0 · (Ω × rg (z, t)) =2µ(−u̇Ω3 rG,2 + v̇Ω3 rG,1 )
=2µ(−u̇(ϕ̇ − γ̇ sin β)rg,2 + v̇(ϕ̇ − γ̇ sin β)rg,1 )
≈2µω(v̇rg,1 − u̇rg,2 ).
(2.31)
Here again the terms v′ rg,2 and u′ rg,2 are of second order and are therefore dropped.
Combining (2.23) with (2.30) and (2.31) we finally obtain the expression for the kinetic
energy we will use in the following.
1
T=
2
ZL
µ(u̇2 + v̇2 ) + 2µω(v̇rg,1 − u̇rg,2 )
0
+ Ia (u̇′2 + v̇′2 ) + Ip ω2 + 2ωu̇′ v′
+ ∆I (v̇′2 − u̇′2 ) cos 2ωt − 2u̇′ v̇′ sin 2ωt dz.
(2.32)
′2
Common Euler-Bernoulli beam theory
does not
include the rotatory inertia term Ia(u̇ +
v̇′2 ) and the gyroscopic term Ip ω2 + 2ωu̇′ v′ . The rotatory inertia term would also
appear in the non-rotating case. The beam theory which includes this rotatory inertia
term is called Rayleigh beam (Han et al., 1999) and is considered in this work.
2.4 Euler-Lagrange Equations
The derivation of the equation of motion is based on Hamilton’s principle (José &
Saletan, 1998). The equations of motion are the Euler-Lagrange equations of the action
functional
Zt1
S=
L(q, q̇, ∇q, ∇2 q, t, x).
(2.33)
t0
30
Chapter 2: Continuous Rotor Model
The Lagrangian L depends on the functions q and their temporal and spatial derivatives. It is worth noting that higher order derivatives are allowed which leads to a
more general formulation of the Euler-Lagrange equations. The Lagrangian is given
by
L = T − U,
where T is the kinetic energy and U the potential energy of the system under consideration. In continuum mechanics the Lagrangian is calculated from a Lagrangian
density
Z
Z
2
L=
L(q, q̇, ∇q, ∇ q, t, x) =
T −U
Ω
Ω
which is again calculated from the densities of the kinetic and the potential energy
respectively.
The Euler-Lagrange equations in the continuum case do not lead to ordinary differential equations as it is the case for Lagrangians describing systems of point masses,
but to partial differential equations. The generalized Euler-Lagrange equations for
functionals involving higher order derivatives are
n+1
n+1
X
∂L X d
∂L
d2
∂2 L
−
+
= 0,
∂qi
dy j ∂(∂ y j qi )
dy j dyk ∂(∂ y j ∂ yk qi )
j=1
(2.34)
j,k=1
where y = (t, x). Depending on the problem one also obtains natural boundary conditions.
The dissipation of energy by viscous damping can be accounted for by adding the
Rayleigh dissipation function FR to the Lagrangian density L
1
FR (q̇) = ckq̇k2
2
with the distributed damping parameter c. The Euler-Lagrange equations are then
modified as follows (see e.g. (José & Saletan, 1998; Yamamoto & Ishida, 2001))
n+1
n+1
X
∂L X d
∂L
d2
∂2 L
∂FR
−
+
= 0.
−
∂qi
dy j ∂(∂ y j qi )
dy j dyk ∂(∂ y j ∂ yk qi ) ∂t qi
j=1
(2.35)
j,k=1
In the previous sections we have derived formulas for the kinetic and the potential
energy (cf. Eqs. (2.32) and (2.19)) of a rotating shaft with 2 distinct area moments of
inertia I1 and I2 differing by 2∆I and eccentricity rG of the center of gravity. In the
following we will assume that the two diametral moments of inertia are equal, i.e. that
∆I = 0.
31
2.4 Euler-Lagrange Equations
The Lagrangian is thus given by
1
L=
2
ZL
µ(u̇2 + v̇2 ) + 2µω(v̇rg,1 − u̇rg,2 ) + Ia (u̇′2 + v̇′2 ) + Ip ω2 + 2ωu̇′ v′
0
− EIa (u′′2 + v′′2 ) − 2µgv + c(u̇2 + v̇2 ) dz.
(2.36)
The derivatives with respect to the generalized coordinates are:
∂L
∂u
d ∂L
dt ∂u̇
d ∂L
dz ∂u′
d2 ∂L
dzdt ∂u̇′
d2 ∂L
dz2 ∂u′′
∂FR
∂u̇
∂L
∂v
d ∂L
dt ∂v̇
d ∂L
dz ∂v′
d2 ∂L
dzdt ∂v̇′
d2 ∂L
dz2 ∂v′′
∂FR
∂v̇
= 0,
= µü − µωṙg,2 ,
= 0,
= (Ia ü′ )′ + ω(Ip v̇′ )′ ,
= −(EIa u′′ )′′ ,
= cu̇,
= −µg,
(2.37)
= µv̈ + µωṙg,1 ,
(2.38)
= ω(Ip u̇′ )′ ,
(2.39)
= (Ia v̈′ )′ ,
(2.40)
= −(EIa v′′ )′′ ,
(2.41)
= cv̇.
(2.42)
Using the generalized Euler-Lagrange equations 2.35 we obtain
(EIa u′′ )′′ + µü − (Ia ü′ )′ − ω(Ip v̇′ )′ + cu̇ = µωṙg,2 ,
′′ ′′
′ ′
′ ′
(EIa v ) + µv̈ − (Ia v̈ ) + ω(Ip u̇ ) + cv̇ = −µωṙg,1 − µg,
(2.43)
(2.44)
and by using the expression (2.9) for the unbalance we can replace the time derivatives
on the right hand side to obtain the equation of motion for the shaft:
(EIa u′′ )′′ + µü − (Ia ü′ )′ − ω(Ip v̇′ )′ + cu̇ = µω2 (rG,1 cos ωt − rG,2 sin ωt),
′′ ′′
′ ′
′ ′
2
(EIa v ) + µv̈ − (Ia v̈ ) + ω(Ip u̇ ) + cv̇ = µω (rG,1 sin ωt + rG,2 cos ωt) − µg.
(2.45)
(2.46)
Additionally, from the variation of the Lagrangian we obtain the natural boundary
conditions on {0, L} × [0, T]
Ia ü′ + ωIp v̇′ − (EIa u′′ )′ = 0,
(2.47)
EIa u′′ = 0,
(2.49)
′′
(2.50)
′
′
′′ ′
Ia v̈ − ωIp u̇ − (EIa v ) = 0,
EIa v = 0.
(2.48)
This system of equations together with the initial conditions
u(z, 0) = u0 (z),
u̇(z, 0) = u1 (z),
(2.51)
v(z, 0) = v0 (z),
v̇(z, 0) = v1 (z),
(2.52)
describes the motion of a free shaft without bearings. The two equations (2.45) and
(2.46) are coupled and one sees that the coupling vanishes for ω = 0, i.e. when the
32
Chapter 2: Continuous Rotor Model
beam is not rotating the vibrations in x and y directions are mutually independent.
The antisymmetric coupling reflects the gyroscopic moments in the rotating beam
which are caused by changes of the angular momentum vector Ip ω due to the whirling
of the rotor (Yamamoto & Ishida, 2001).
In Section 2.6 we will show how bearings can be included into the model generally and
in Chapter 5 we will derive a detailed model for the forces exerted by oil lubricated
journal bearings. The natural boundary conditions (2.47) and (2.48) show the absence
of shearing forces at the ends of the beam, while the boundary conditions (2.49) and
(2.50) stand for the absence of bending moments. We see that a harmonic forcing term
appears on the right hand side of Equations (2.45) and (2.46) which depends linearly on
the eccentricity of the center of mass and grows quadratically with the angular velocity.
This forcing is called static unbalance excitation (Gasch & Pfützner, 1975; Yamamoto
& Ishida, 2001). It is called static because the eccentricity of the center of mass can
be detected also when the rotor is at rest, in contrast to dynamic unbalance which is
caused by angular misalignment of flywheels as described in the following section.
The quadratic dependence of the unbalance on the angular velocity makes balancing
very important for fast rotating bodies, such as the rotors used in turbomachinery. The
technique of balancing is described in (Yamamoto & Ishida, 2001) for continuous and
rigid rotors with different kinds of unbalance.
2.5 Modifications for Rigid Disks
The rotor of a turbocharger consists not only of the shaft but also of the compressor
wheel and of the turbine wheel. The vibration of the blades and the gasdynamic interaction of either of them shall be neglected in this work. Furthermore the vibrations
of the wheel structure are negligible as the eigenfrequencies are much higher than
the driving frequency and are not excited. It is therefore common in studies of rotordynamics to model the wheels as rigid disks which are attached to the elastic shaft
(Yamamoto & Ishida, 2001; Vance, 1988; Childs, 1993).
As we have seen before, a cross-section of a Rayleigh beam remains planar and can
be considered rigid. Its dynamics can be described by four variables: x- and ydisplacement u and v and the two inclination angles γ ≈ u′ and β ≈ −v′ . In contrast to
the above, the rigid disks which model the fly wheels are not necessarily perpendicular
to the shaft axis and so-called dynamic unbalance excitation can occur (Yamamoto &
Ishida, 2001). In Figure 2.3 such a situation is depicted. The Euler angles β, γ, and ϕ
describing the position of the body system relatively to the space system are defined
as above. The disk is attached at the axial coordinate zd and is initially inclined to
the shaft center line by the angle τ with phase η. When the center line inclines by the
33
2.5 Modifications for Rigid Disks
τ
e3
z
t
e2
τ
e3
η
θ2
β
γ
z
θ1
y
e2
η
e1
y
ϕ
e1
ex’
x
x
Figure 2.3: Dynamic unbalance: The disk is initially inclined to the shaft center line by the
angle τ with phase η (left). During movement (right) the position of the tangential vector
t relative to the body coordinate system e1 , e2 , e3 remains constant.
angles θ1 and θ2 away from the z-axis the following relations hold to first order
u′ (zd ) ≈ θ1 = γ + τ cos(φ + η),
′
v (zd ) ≈ −θ2 = −β + τ sin(φ + η).
(2.53)
(2.54)
We plug these relations into the Equation (2.29) for the rotational energy of a rotating
disk in terms of the Euler angles and we obtain after dropping second order terms in τ
2Trot =Ia u̇′2 + v̇′2 + 2ωτ(u̇′ sin(ωt + η) − v̇′ cos(ωt + η))
+ I3 ω2 + 2ωu̇′ v′ − 2ωτ(u̇′ sin(ωt + η) − ωv′ sin(ωt + η))
+ ∆I (v̇′2 − u̇′2 ) cos 2ωt − 2u̇′ v̇′ sin 2ωt
+ 2τω∆I u̇′ sin(ωt − η) − v̇′ cos(ωt − η) .
(2.55)
Furthermore the disks also contribute to the kinetic energy stored in transversal motion.
This contribution is not influenced by the misalignment τ, but only by the eccentricity
of the center of mass rg as in Equation 2.32 where we have to replace mass density µ
by mass md
2Tt =md (u̇(zd )2 + v̇(zd )2 ) + 2md ω(v̇(zd )rdg,1 − u̇(zd )rdg,2 ).
(2.56)
Adding the two contributions (2.55) and (2.56) yields the total kinetic energy of the
disk
Td = Tt + Trot .
(2.57)
Since the disk is considered not elastic, only its weight contributes to the potential
energy
Ud = md gv(zd ).
(2.58)
Hence the Lagrangian of the disk results as
L d = T d − Ud
(2.59)
34
Chapter 2: Continuous Rotor Model
which has to be added to the Lagrangian of the shaft (2.36) to obtain the Lagrangian
for the composite system. Note that the Lagrangian of the disk only depends on the
displacement and the inclination of the centerline at the position zd and is not given
by an integral.
However it can be included into the analytic formulation of Section 2.4 by splitting
up the integration interval at the points zdi . We obtain additional interface conditions
between the different parts of the shaft. Furthermore we have to demand continuity
of the functions u and v and their first spatial derivatives u′ and v′ in the points zdi
Consider the combined Lagrangian
ZL
L+
L=
0
Nd
X
Ldi ,
(2.60)
i=1
where L is the Lagrangian density of the continuous shaft (2.36). Ldi is the Lagrangian
of the ith rigid disk fixed in zdi , which is only depending on deflection and inclination
in that point. Variation of the action functional (2.33) yields the same partial differential
equations (2.45) and (2.46) on each interval Ii = [zi , zi+1 ] as for the case without disks.
The additional interface conditions at zdi are
h
i
i
,
(2.61)
= mdi ü(zdi ) − mdi ωṙdg,2
Ia ü′ + ωIp v̇′ − (EIa u′′ )′
zdi
h
i
i
(2.62)
− mdi g,
= mdi v̈(zdi ) + mdi ωṙdg,1
Ia v̈′ − ωIp u̇′ − (EIa v′′ )′
zdi
′′
[EIa u ]zd −
i
ωIpdi v̇′ (zdi )
= (Iadi − Ipdi )ω2 τdi cos(ωt + ηdi ),
[EIa v′′ ]zd − Iadi v̈′ (zdi ) + ωIpdi u̇′ (zdi ) = (Iadi − Ipdi )ω2 τdi sin(ωt + ηdi ).
i
(2.63)
(2.64)
The notation [u]z stands for the jump of u at z, i.e. [u]z = lim u(x) − lim u(x).
xցz
xրz
We see that the inertial force of the disk and the static unbalance forcing term add to the
by the disk due to gyroscopic effects and dynamic unbalance add to the bending
moments. The additional forcing term in the moment balance (2.63) and (2.64) is
the already mentioned dynamic unbalance excitation which depends on the angle of
misalignment τ and also grows quadratically with the angular velocity. In contrast
to the static unbalance, misalignment can only be detected when the rotor is rotating
because the gyroscopic effects appear only during movement.
2.6 Bearings
As mentioned before we will derive a detailed model for oil lubricated journal bearings
in Chapter 5. Here we describe for reasons of completeness how to include bearings
into the continuous model of the rotating shaft in a more abstract way. The system
35
2.6 Bearings
(2.45)-(2.48) as it is written in Section 2.4 is not statically determined, i.e. it remains
unchanged under rigid body transformations since linear transformations are in the
kernel of the corresponding differential operator. The introduction of bearings solves
this problem and leads to the existence of a unique solution of our system as we shall
see in Chapter 3.
We shall consider two bearings which support the rotating shaft. They are located at
two disctinct axial positions a, b ∈ [0, L] along the rotor. Each of the bearings reacts
on the movement of the shaft by reaction forces fb and reaction moments tb which
we consider to be localized to the position of the bearing, i.e. we neglect the axial
dimension of the bearing. Hence, the support forces and moments in the two distinct
points {a, b} ⊂ [0, L] are concentrated point forces.
Here, as in Section 2.5, the bearing forces are added as transmission conditions for the
forces at their respective postions. For this we define I1 = [0, zb1 ], I2 = [zb1 , zb2 ], and
I3 = [zb2 , L]. Hence on each Ii we have
(EIa u′′ )′′ + µü − (Ia ü′ )′ − ω(Ip v̇′ )′ + cu̇ = µω2 (rG,1 cos ωt − rG,2 sin ωt),
(EIa v′′ )′′ + µv̈ − (Ia v̈′ )′ + ω(Ip u̇′ )′ + cv̇ = µω2 (rG,1 sin ωt + rG,2 cos ωt) − µg.
(2.65)
(2.66)
In addition to the boundary conditions (2.47) - (2.50) at z = 0 and z = L, we get the
force and moment transmission conditions at zb1 and zb2 :
h
i
= fbi ,1 ,
−Ia ü′ − ωIp v̇′ + (EIa u′′ )′
zb
h
i i
−Ia v̈′ + ωIp u̇′ + (EIa v′′ )′
= fbi ,2 ,
zbi
(2.67)
(2.68)
[−EIa u′′ ]zb = tbi ,1 ,
(2.69)
[−EIa v′′ ]zb = tbi ,2 .
(2.70)
i
i
As an example, the bearing response force function for linear spring and damper
support with stiffness k and damping coefficient c is

 fb,1 (u, v, u̇, v̇)

 f (u, v, u̇, v̇)
b,2
 
  −ku − cu̇
 = 
  −kv − cv̇


 .

(2.71)
For other kinds of support the response function fb may be chosen differently, however
throughout this work they are considered as concentrated point forces. Furthermore,
we will not consider bearings such as spiral springs which exert moments on the shaft,
therfore tbi = 0 for the rest of this work. As we will see later (cf. Sec. 5), the nonlinear
bearing forces of the oil lubricated bearing forces also respond only to the displacement
and not to the inclination of the shaft.
36
Chapter 2: Continuous Rotor Model
2.7 Complex Formulation of Equation of Motion
The number of equations of motion can be reduced by introducing a complex-valued
notation using C ∼ R2 . For this we set
w = u + iv,
Fb = fb,1 + i fb,2 ,
funb = rG,1 + irG,2 ,
fgrav = −iµg,
(2.72)
and obtain by adding Equations (2.65) and (2.65)
µẅ − (Ia ẅ′ )′ + iω(Ip ẇ′ )′ + cẇ + (EIa w′′ )′′ = fgrav + µω2 funb eiωt .
(2.73)
Analogously we obtain
h
−Ia ẅ′ + iωIp ẇ′ + (EIa w′′ )′
i
zbi
= Fb ,
[EIa w′′ ]zb = 0,
i
(2.74)
(2.75)
from the boundary conditions (2.47)-(2.50). This formulation reflects nicely the symmetries and antisymmetries of the original equation. Furthermore it reduces notably
the notational effort and will therefore be used in Chapter 3 where we prove existence
of solutions to Equation (2.73).
Chapter 3
Existence of Solutions
In this chapter we want to examine the existence of solutions of the equation of motion
for a rotating beam. The equation has been derived in Section 2.6 of Chapter 2. The
equation for the deflection u(t, x) ∈ C of the beam at point x ∈ Ω = [0, L] at time t ∈ [0, T]
is
µutt − (Ia uttx )x + iω(Ip utx )x + (EIa uxx )xx = g in (0, T) × Ω,
EIa uxx = 0 on (0, T) × ∂Ω,
Ia uttx − iωIp utx − (EIa uxx )x = 0 on (0, T) × ∂Ω,
u(0, x) = u0 (x)
and
ut (0, x) = u1 (x)
for x ∈ Ω,
(3.1)
(3.2)
(3.3)
(3.4)
where g : [0, T] × Ω → C is a periodic driving force. Supporting the shaft at the two
distinct interior points {a, b} ⊂ Ω leads transmission conditions for the moments and
the forces at the location of the support:
[EIa uxx ]x j = 0
[ Ia uttx − iωIp utx − (EIa uxx )x ]x j = f (u(t, x j ), ut (t, x j ))
for (t, x j ) ∈ (0, T) × {a, b},
(3.5)
for (t, x j ) ∈ (0, T) × {a, b},
(3.6)
where f : C2 → C gives the reaction forces of the bearings depending on deflection and
velocity of the beam and [.]z denotes the jump of the term at that point. The coefficients
µ, Ip , Ia , and E are considered to be in L∞ (Ω) and do not depend on time. Additionally
they are strictly positive, i.e. there are constants such that
0 < µ ≤ µ(x) ≤ µ < ∞,
0 < Ip ≤ Ip (x) ≤ Ip < ∞,
0 < Ia ≤ Ia (x) ≤ Ia < ∞,
0 < E ≤ E(x) ≤ E < ∞,
for almost every x ∈ Ω. Furthermore ω > 0 is constant.
38
Chapter 3: Existence of Solutions
We will first prove the existence of solutions for the case of linear support functions
corresponding to spring support using Galerkin’s method (Evans, 1998; Zeidler, 1990).
In a second step we will consider nonlinear support as it is the case for the fluid film
bearings. Throughout the chapter, C will denote a generic positive real constant.
3.1 Existence for Linear Support
In the case of linear support the reaction forces in the support are as follows
f (u(x, t), ut (x, t)) = −ku(x, t),
(3.7)
where k > 0 is a constant.
Multiplication of Equation (3.1) by ϕ̄ ∈ H2 (Ω) (the overbar denoting the complex
conjugate) and partial integration leads to the following weak formulation:
ZL h
i
µutt ϕ̄ + Ia uttx ϕ̄x − iωIp utx ϕ̄x + EIa uxx ϕ̄xx dx+k(u(a)ϕ̄(a)+u(b)ϕ̄(b)) =
Z
gϕ̄dx (3.8)
0
for all ϕ ∈ H2 (Ω). By the Sobolev embedding theorem H2 (Ω) ֒→ C1 (Ω) continuously
for one-dimensional Ω and every element in H2 is equal to a continuously differentiable
function after changing it on a subset of measure 0. Therefore the evaluation of u at
the points a and b is possible.
We can simplify the weak formulation (3.8) by introducing three sesquilinear forms.
These are
a :H2 (Ω) × H2 (Ω) → C,
(3.9)
ZL
EIa (x)uxx v̄xx dx + k(u(a)v̄(a) + u(b)v̄(b)),
a(u, v) =
(3.10)
0
which takes into account all terms related to stiffness, and
m :H1 (Ω) × H1 (Ω) → C,
(3.11)
ZL
µ(x)uv̄ + Ia (x)ux v̄x dx,
m(u, v) =
(3.12)
0
which takes all the inertia terms, and
b :H1 (Ω) × H1 (Ω) → C,
(3.13)
ZL
b(u, v) = −iω
Ip (x)ux v̄x dx,
0
(3.14)
39
3.1 Existence for Linear Support
which is the gyroscopic term. Equation (3.8) then reads as
m(utt , v) + b(ut , v) + a(u, v) = g, v L2 (Ω) .
(3.15)
It follows directly from the boundedness and the strict positivity of the coefficients µ
and Ia that the form m(., .) defines a scalar product on H1 (Ω) which is equivalent to the
standard scalar product in the sense that the induced norms are equivalent.
Definition 3.1. Let m(., .) be the sesquilinear form defined in (3.11). We equip the space
H1 (Ω) with the equivalent scalar product (u, v)m = m(u, v) for u, v ∈ H1 (Ω) to obtain
the space
1
Hm
(Ω) := (H1 (Ω), (., .)m = m(., .)).
(3.16)
From the bounds on µ(x) and Ia (x) we can immediately deduce:
1 (Ω) is a continuous bijection with continuous
Proposition 3.2. The identity id : H1 (Ω) → Hm
inverse.
In the following we do not consider the usual Gelfand triple with L2 (Ω) as pivot space
but the triple
′
1
1
H2 (Ω) ⊂ Hm
(Ω) = Hm
(Ω) ⊂ H2 (Ω)′ ,
(3.17)
1 (Ω) with its dual and embed it into H 2 (Ω)′ by
where we identify Hm
hu, viH2 (Ω)′ ×H2 (Ω) = (u, v)m = m(u, v) for
1
u ∈ Hm
(Ω), v ∈ H2 (Ω).
(3.18)
The sesquilinear form a(., .) is examined in the following lemma.
Lemma 3.3. Let u, v ∈ H2 (Ω), Ω = [0, L] ⊂ R1 bounded, a, b ∈ Ω with 0 ≤ a < b ≤ L,
0 < ρ < ρ(x) < ρ < ∞ for almost every x ∈ Ω and k > 0. The sesquilinear form
ZL
ρ(x)uxx v̄xx dx + k(u(a)v̄(a) + u(b)v̄(b))
a(u, v) =
(3.19)
0
is continuous and coercive.
Proof. We have
 21
 L

Z


|a(u, v)| ≤ ρ  |uxx |2 dx


0
 21
 L

Z


 |vxx |2 dx + 2kkuk∞ kvk∞ ≤ CkukH2 kvkH2


(3.20)
0
by Hölder’s inequality and the boundedness of ρ for the first term and from the continuos embedding of H2 (Ω) into C1 (Ω) for the second term. Thus a(u, v) is continuous.
40
Chapter 3: Existence of Solutions
For u ∈ H2 (Ω) we obtain by applying the triangular and Hölder’s inequality
Zx
|u(x)|2 = |u(a) +
2
x
Z
ux (t)dt|2 ≤ 2|u(a)|2 + 2 ux (t)dt
a
a
 x
 x
2

Z
Z






≤ 2|u(a)|2 + 2  |ux (t)|dt ≤ 2|u(a)|2 + 2|x − a|  |ux (t)|2 dt




a
a
2
≤ 2|u(a)| +
2Lkux k2L2 (Ω)
and hence by integration over Ω
kukL2 (Ω) ≤ 2L|u(a)|2 + 2L2 kux k2L2 (Ω) .
(3.21)
From the generalized Poincaré equation (B.1) from the appendixRwe get the following
1
estimate for the first derivative by setting B = [a, b] and ūxB = |B|
u (s)ds:
Ω x
kux k2L2 (Ω) ≤ 2kux − ūx,B k2L2 (Ω) + 2kūx,B k2L2 (Ω)
2


Z
2 

2

≤ 2kux − ūx,B kL2 (Ω) + 2  ux (y)dy


|B|
(3.22)
(3.23)
B
C
4|Ω|
≤ 2 kuxx k2L2 (Ω) +
(|u(a)|2 + |u(b)|2 ).
|B|
|B|2
(3.24)
From these two estimates for kukL2 (Ω) and kux kL2 (Ω) we obtain the coercitivity of the
sesquilinear form:
kuk2H2 (Ω) = kuk2L2 (Ω) + kux k2L2 (Ω) + kuxx k2L2 (Ω)
≤ kuxx k2L2 (Ω) + 2L|u(a)|2 + (2L2 + 1)kux k2L2 (Ω)
≤ Ckuxx k2L2 (Ω) + C′ (|u(a)|2 + |u(b)|2 )
≤ C(ρkuxx k2L2 (Ω) + k(|u(a)|2 + |u(b)|2 ))
≤ Ca(u, u).
(3.25)
(3.26)
(3.27)
(3.28)
(3.29)
The properties of the forms can be used to give some a priori estimates on the weak
solution.
Lemma 3.4. The solution of the linear Equation (3.8) fulfills the following a priori estimates:
kukL∞ (0,T;H2 (Ω)) + kut kL∞ (0,T;Hm1 (Ω)) + kutt kL2 (0,T;H2 (Ω)′ )
≤C g 2
+ ku0 kH2 (Ω) + ku1 kH1 (Ω) .
2
L (0,T;L (Ω))
(3.30)
41
3.1 Existence for Linear Support
Proof. Using ut as test function in Equation (3.8), we obtain
Z
Z
(µutt ūt +Ia uttx ūtx −iωIp utx ūtx +EIa uxx ūtxx )dx+k(u(a)ūt (a)+u(b)ūt (b)) =
gūt dx. (3.31)
Ω
With sesquilinear forms used in Eq. (3.15) this can be written in a simplified manner
m(utt , ut ) + b(ut , ut ) + a(u, ut ) = g, ut L2 (Ω) .
(3.32)
Adding the complex conjugate of the equation and observing that
Z
d
d
µut ūt + Ia utx ūtx dx
m(ut , ut ) =
dt
dt
Z Ω
=
µ (utt ūt + ut ūtt ) + Ia (uttx ūtx + utx ūttx ) dx
Ω
= m(utt , ut ) + m(utt , ut ),
and analogously
d
a(u, u) = a(u, ut ) + a(u, ut ),
dt
and that
b(ut , ut ) + b(ut , ut ) = 0,
we get
d
(m(ut , ut ) + a(u, u)) = g, ut L2 (Ω) + g, ut L2 (Ω) .
(3.33)
dt
Using the Hölder inequality and kut kL2 (Ω) ≤ kut kH1 (Ω) < Cm(ut , ut ) and 0 ≤ a(u, u) we
get
d
(m(ut , ut ) + a(u, u)) ≤ C( gL2 (Ω) + m(ut , ut ) + a(u, u)).
(3.34)
dt
We apply
inequality (B.3) with η(t) = m(ut , ut ) + a(u, u), φ(t) = C and
Gronwall’s
ψ(t) = C g(t)L2 (Ω) and obtain the a priori estimate

Zt

Ct 
m(ut , ut ) + a(u, u) ≤ e m(u1 , u1 ) + a(u0 , u0 ) + C

0



g(t)L2 (Ω)  .

(3.35)
From the equivalence of a(., .) to the standard scalar product on H2 (Ω) (shown in lemma
3.3) we obtain the estimate for the norms for almost all t ∈ [0, T]
kukH2 (Ω) + kut kHm1 (Ω) ≤ C kgkL2 (0,T;L2(Ω)) + ku0 kH2 (Ω) + ku1 kHm1 (Ω) .
(3.36)
To get an estimate on kutt kL2 (0,T;H2 (Ω)′ ) we use the continuity of m(., .), b(., .) and a(., .).
We observe that for v ∈ H2 (Ω)
| hutt , vi | = | (utt , v)Hm1 (Ω) | = |m(utt , v)|
= | g, v L2 (Ω) − a(u, v) − b(ut , v)|
≤ gL2 (Ω) + C kukH2 (Ω) + C′ kutx kL2 (Ω) kvkH2 (Ω) .
(3.37)
(3.38)
(3.39)
42
Chapter 3: Existence of Solutions
Here we used the equivalence of m(., .) to the standard scalar product in the first line,
the Equation (3.15) in the second, and the continuity of a(., .) and b(., .) in the third line.
From this we deduce that
(3.40)
kutt kH2 (Ω)′ ≤ C gL2 (Ω) + kukH2 (Ω) + kut kHm1 (Ω)
and after integration from 0 to T we obtain
ZT
kutt k2L2 (0,T;H2 (Ω)′ )
kutt k2H2 (Ω)′
=
(3.41)
0
2
2
2
≤ C g L2 (0,T;L2 (Ω)) + ku0 kH2 (Ω) + ku1 kH1 (Ω)
m
(3.42)
using the above estimates (3.36) for kukH2 (Ω) . This completes the proof.
Now we have all the ingredients to prove the existence of a weak solution of the linear
problem.
Theorem 3.5. Let a, m, b be the bilinear forms defined in (3.10),(3.11) and (3.14) and let
g ∈ L2 (0, T; L2 (Ω)).
There exists a weak solution of initial/boundary-value problem (3.1)-(3.4) with linear support
function f as in (3.7), i.e. we can find a function
u ∈ L∞ (0, T; H2 (Ω)),
with
1
u′ ∈ L∞ (0, T; Hm
(Ω))
and u′′ ∈ L2 (0, T; H2 (Ω)′ ) (3.43)
that fulfills
hutt , vi + b(ut , v) + a(u, v) = (g, v)L2 (Ω)
(3.44)
for all v ∈ H2 (Ω) and for almost all t ∈ [0, T]. Furthermore u satisfies the initial conditions:
u(0) = u0 ∈ H2 (Ω) and
1
ut (0) = u1 ∈ Hm
(Ω).
(3.45)
Proof. We use Galerkin’s method to construct a weak solution. For this we choose a
basis of {wk }k ⊂ H2 (Ω) to construct approximate solutions.
1 (Ω) and therefore also span({wk } ), we can find coefficients
Since H2 (Ω) is dense in Hm
k
αkn and βkn such that
un0 =
un1 =
n
X
k=1
n
X
k=1
αkn w j → u0
in
H2 (Ω),
(3.46)
βkn w j → u1
in
1
Hm
(Ω).
(3.47)
We want to construct for each n ∈ N approximate solutions of the form
un (t) =
n
X
j=0
j
dn (t)w j
(3.48)
43
3.1 Existence for Linear Support
to the problems projected onto the subspace spanned by n of the testfunctions wk
for all k = 1, . . . , n,
(3.49)
m(untt , wk ) + b(unt , wk ) + a(un , wk ) = g, wk 2
L (Ω)
where the coefficients fulfill the initial conditions
j
dn (0) = αnk
j′
dn (0) = βnk .
and
(3.50)
If the solution has the form (3.48) we can put it into the equation and obtain a system
j
of n linear differential equations for the coefficients dn : [0, T] → R
n h
X
i j′′
j′
j
dn (t)m(w j , wk ) + dn (t)b(w j , wk ) + dn a(w j , wk ) = g, wk
L2 (Ω)
j=0
(3.51)
for all k = 1, . . . , n. It can be written in matrix form
′
Md′′
n + Bdn + Adn = gn ,
(3.52)
where the matrices
are given by Mk, j = m(w j , wk ), Bk, j = b(w j , wk ), Ak, j = a(w j , wk ), and
gn = ( g, w1 , . . . , g, wn )T . This system can be solved uniquely because the matrix M
is nonsingular if the {wk }k are linearly independent. This follows from the fact that
1 (Ω) (cf. Lemma B.3 in the appendix). We therefore have
m(., .) is a scalar product on Hm
a unique solution vector dn ∈ C2 (0, T; Rn ) fulfilling the initial conditions (3.50).
The approximate solutions clearly also fulfill for all n the a priori estimate (3.30) we
have proved in Lemma 3.4. Therefore the sequence of approximate solutions {un }n
is bounded in L2 (0, T; H2 (Ω)), {unt } is bounded in L2 (0, T; H1 (Ω)) and {untt } is bounded
in L2 (0, T; H2 (Ω)′ ). By the theorem of Banach-Alaoglu-Bourbaki (Brezis, 1999) we
therefore find a weakly convergent subsequence {unk }k and u ∈ L2 (0, T; H2 (Ω)) with
ut ∈ L2 (0, T; H1 (Ω)) and utt ∈ L2 (0, T; H2 (Ω)′ ), such that
u nk ⇀ u
n
ut k ⇀ ut
weakly in
L2 (0, T; H2 (Ω)),
weakly in
L2 (0, T; H1 (Ω)),
(3.53)
unttk ⇀ utt
weakly in L2 (0, T; H2 (Ω)′ ).
P
Now we consider a test function v = lk=1 ξk (t)wk ∈ C1 (0, T; H2 (Ω)) with smooth ξk (t).
The weak convergence (3.53) implies for nk > l
ZT
hutt , vi + b(ut , v) + a(u, v) − (g, v)L2 (Ω) dt
0
ZT
E
D
nk
nk
utt − utt , v + b(ut − ut , v) + a(u − unk , v)dt
≤
0
|
{z
→0
}
by (3.53)
ZT n
n
m(uttk , v) + b(ut k , v) + a(unk , v) − (g, v)L2 (Ω) dt.
+
|
{z
}
0
=0
by (3.49)
(3.54)
44
Chapter 3: Existence of Solutions
As {w j } j has been chosen dense in H2 (Ω), functions of the form of the test function
v are dense in L2 (0, T; H2 (Ω)) and hence Equation (3.54) remains valid for all v ∈
L2 (0, T; H2 (Ω)). Furthermore it follows that for almost all t ∈ [0, T] (3.44) is fulfilled for
all v ∈ H2 (Ω).
It remains to show that the initial conditions are fulfilled. For this we note first
1 (Ω)) and u′′ ∈ L2 (0, T; H 2 (Ω)′ ) we can
that from u ∈ L∞ (0, T; H2 (Ω)), u′ ∈ L∞ (0, T; Hm
1 (Ω)) and u ∈ C(0, T; H 1 (Ω)′ ) (Evans, 1998). We choose a
conclude that u ∈ C(0, T; Hm
t
m
2
′
function ψ ∈ C (0, T; R) with ψ(T) = ψ (T) = 0 and set v = ψwk ∈ C2 (0, T; H2 (Ω)). We
can then deduce from Equation (3.54) by partial integration over t that
ZT
ZT
k
′
k
hu, vtt i + b(ut , v) + a(u, v) dt − ψ(0)m(ut (0), w ) + ψ (0)m(u(0), w ) =
(g, v)L2 (Ω) dt
0
0
and from (3.49) that
ZT
hunk , vtt i + b(unt k , v) + a(unk , v)dt
0
ZT
−
ψ(0)m(unt k (0), wk )
+ ψ′ (0)m(unk (0), wk ) =
(g, v)L2 (Ω) dt.
0
From these two equations we can deduce by (3.53) that for all wk from our basis:
n
lim ψ(0)m(ut k (0), wk ) − ψ′ (0)m(unk (0), wk ) = ψ(0)m(ut (0), wk ) − ψ′ (0)m(u(0), vt (0)).
nk →∞
1 (Ω)
But above we have chosen the projected initial conditions (3.50) such that in Hm
unt k (0) =
nk
X
j=1
βkn w j → u1
and therefore by continuity of m(., .) and the arbitrariness of ψ(0)
m(u1 , wk ) = lim m(unt k (0), wk ) = m(ut (0), wk ) for all
nk →∞
Analogously we have chosen unk =
P nk
j=1
k.
(3.55)
βkn w j → u0 in H2 (Ω) and hence
m(u0 , wk ) = lim m(unk (0), wk ) = m(u(0), wk )
nk →∞
k.
(3.56)
x ∈ Ω.
(3.57)
for all
1 (Ω) we get the equalities
From the density of span({wk }k ) in Hm
u0 = u(0) and
This finishes our proof.
u1 = ut (0)
for almost all
3.2 Existence for Nonlinear Support
45
Remark 3.6. Uniqueness of the solution can be shown using the technique in (Evans,
1998), Chapter 7.2, Theorem 4. There, the test function
 Rs

 t u(τ) dτ 0 ≤ t ≤ s
v(t) = 
 0
s≤t≤T
(3.58)
is used to show that the only weak solution of the homogeneous hyperbolic equation
with zero initial conditions is indeed u ≡ 0.
Remark 3.7. The regularity of the solution can be improved by taking g ∈ H1 (0, T; L2 (Ω)),
u0 ∈ H3 (Ω) and u1 ∈ H2 (Ω). Differentiation of the equation of motion (3.1) and the
boundary/transmission conditions with respect to time and testing with utt yields
1
u ∈ L∞ (0, T; H3 (Ω)), u′ ∈ L∞ (0, T; H2 (Ω)), u′′ ∈ L∞ (0, T; Hm
(Ω)), u′′′ ∈ L∞ (0, T; H2 (Ω)′ ).
(3.59)
The proof is analogous to (Evans, 1998), Chapter 7.2, Theorem 5. In particular this
1 (Ω)), which is needed in the
regularity yields u ∈ C0 (0, T; H2 (Ω)) and u′ ∈ C0 (0, T; Hm
next section.
3.2 Existence for Nonlinear Support
In this section we want to extend our existence result from the previous section to
nonlinear support functions, more precisely to support functions that model hydrodynamic bearings. We prove short time existence of solutions to the nonlinear equations
using local Lipschitz continuity and a fixed point argument.
The support functions are derived in Chapter 5. Their detailed form is not important
here and we will write them in the following convenient way.
f : C2 ⊃ Bcr (0) × C → C,
(3.60)
(x + iy, v + iw) 7→ eiγ ( f n (r, ṙ, γ̇) + i f t (r, ṙ, γ̇)),
where
q
r=
x2 + y2 , γ = arg(x + iy), ṙ = v cos γ + w sin γ, γ̇ = (w cos γ − v sin γ)/r.
(3.61)
The functions f n and f t give the normal and tangential forces acting on the rotating
beam in the bearing. Their special form is given in Equations (5.159) and (5.160) in
Section 5.5.1. For r < cr the functions f n and f t are locally Lipschitz-continuous as
can be seen from the analytical expressions in Section 5.5.1. For r → cr however
e−iγ f n → −∞, so that we do not have a global Lipschitz-constant for f . The local
Lipschitz condition is the essential ingredient in the proof of the following existence
theorem for the equation of motion with nonlinear support funcitons.
46
Chapter 3: Existence of Solutions
Theorem 3.8. Let f be a locally Lipschitz, nonlinear support function as in (3.60). Let
a, b ∈ Ω = [0, L] and let the coefficients µ, Ia , Ip , EIa ∈ L∞ (Ω) be strictly positive. Furthermore
let g ∈ H1 (0, T; L2 (Ω)) and let u0 ∈ H3 (Ω) with |u0 (a)|, |u0 (b)| < cr , and u1 ∈ H2 (Ω).
There exists a short time weak solution to the initial/boundary value problem (3.1)-(3.4) with
support function f , i.e. there is a T > 0 and
u ∈ C0 (0, T; H2 (Ω)),
with
1
u′ ∈ C0 (0, T; Hm
(Ω)),
(3.62)
such that for all v ∈ H2 (Ω) and for almost all t ∈ [0, T]
ZL h
i
µutt v̄ + Ia uttx v̄x − iωIp utx v̄x + EIa uxx v̄xx dx −
0
X
Z
f (u(xi ), ut (xi ))v̄(xi ) =
xi ∈{a,b}
gv̄dx.
(3.63)
Furthermore u satisfies the initial conditions:
u(0) = u0 ∈ H3 (Ω)
and ut (0) = u1 ∈ H2 (Ω).
(3.64)
Proof. In the proof we will consider the nonlinear equation as a perturbation of the
linear case. In order to apply Banach’s theorem we will show that the solution operator
of a regularized linear equation with fixed nonlinearity on the right hand side is indeed
a contraction. The unique fixed point will then be a solution to the nonlinear problem.
Since f fulfills a local Lipschitz condition we find δ > 0, such that for xi ∈ {a, b}
| f (ξ, ζ) − f (ξ′ , ζ′ )| ≤ L(|ξ − ξ′ | + |ζ − ζ′ |) and
ξ, ξ′ ∈ Bcr (0)
(3.65)
for all ξ, ξ′ ∈ Bδ (u0 (xi )) and ζ, ζ′ ∈ Bδ (u1 (xi )).
Let k > 0 be a constant. We define the bilinear forms a, m, b as in (3.10), (3.11), and
(3.14) using k as coefficient for the form a:
Z
X
a(u, v) =
EIa uxx v̄xx dx + k
u(xi )v̄(xi ).
(3.66)
xi ∈{a,b}
Ω
We can then rewrite the equation of motion (3.63).
X
f (u(t, xi ), ut (t, xi ))v̄(xi ) + ku(t, xi )v̄(xi ) (3.67)
hutt , vi + b(ut , v) + a(u, v) = (g, v)L2 −
xi ∈{a,b}
such that it resembles the linear equation in Theorem 3.5. Fixing the function u on the
right hand side and solving the resulting linear equation yields the operator on which
we will apply the fixed point theorem.
Consider the Banach space
1
X = {u ∈ C0 (0, T; H2 (Ω)) | u′ ∈ C0 (0, T; Hm
(Ω))}
(3.68)
47
3.2 Existence for Nonlinear Support
with the norm
kukX = sup kukH2 (Ω) + sup ku′ kHm1 (Ω) .
0≤t≤T
(3.69)
0≤t≤T
We will use Banach’s theorem on the following closed subset M ⊂ X
M = {u ∈ X| |u(t, xi ) − u0 (xi )| ≤ δ, |u′ (t, xi ) − u1 (xi )| ≤ δ for 0 ≤ t ≤ T, xi ∈ {a, b}}. (3.70)
The operator A : M → M is then defined as follows: For a given u ∈ M we define a
linear equation by
X
f (u(t, xi ), ut (t, xi ))v̄(xi ) + ku(xi )v̄(xi ) (3.71)
hwtt , vi + b(wt , v) + a(w, v) = (g, v)L2 −
xi ∈{a,b}
which has a unique weak solution w ∈ X. From the proof of Theorem 3.5 and from
1 (Ω)) and that the
Remark 3.7 we know that w ∈ C0 (0, T; H2 (Ω)) and w′ ∈ C0 (0, T; Hm
initial conditions are fulfilled. By eventually reducing T > 0 we can therefore assure
that |w(xi ) − u0 (xi )| ≤ δ and |w′ (xi ) − u1 (xi )| ≤ δ. Hence w ∈ M and we set
Au = w.
(3.72)
Now we show that A is a contraction. Choose u, ũ ∈ M and set w = Au, w̃ = Aũ
and ∆w = w − w̃. For notational convenience we set ui = u(t, xi ), wi = w(t, xi ), etc.
Analogously to the proof of the a-priori estimate in Lemma 3.4 we test (3.71) with w′
and w̃′ and the complex conjugated equation with w′ and w̃′ . Substraction then yields
d
m(∆w′ , ∆w′ ) + a(∆w′ , ∆w′ )
dt
X
≤
ℜ f (ui , u′i ) − f (ũi , ũ′i ) + k(ui − ũi ) ∆w′i
xi ∈{a,b}
X 1
| f (ui , u′i ) − f (ũi , ũ′i )|2 + k2 |ui − ũi |2 + |∆w′i |2
2
xi ∈{a,b}
X 1 2
L2 |ui − ũi | + |u′i − ũ′i | + k2 |ui − ũi |2 + |∆w′i |2
≤
2
xi ∈{a,b}
X
(L2 + k2 )|ui − ũi |2 + L2 |u′i − ũ′i |2 + |∆w′i |2
≤
≤
xi ∈{a,b}
′
′
′
′ 2
2
≤ C ku − ũkH2 (Ω) + ku − ũ kH1 (Ω) + m(∆w , ∆w ) + a(∆w, ∆w) .
m
1 (Ω) into C1 (Ω)
The last estimate is due to the Sobolev embeddings of H2 (Ω) and Hm
and C0 (Ω), respectively. As in the proof of Lemma 3.4 we can now apply Gronwall’s
inequality to conclude
sup (k∆w′ k2H1 (Ω) + k∆wk2H2 (Ω) ) ≤ eCT T sup (ku − ũk2H2 (Ω) + ku′ − ũ′ k2H1 (Ω) ).
0≤t≤T
m
m
0≤t≤T
(3.73)
This is equivalent to
1
kAu − AũkX ≤ CT 2 ku − ũkX ,
(3.74)
48
Chapter 3: Existence of Solutions
and by choosing T > 0 sufficiently small we can make A a contraction. This yields
the existence of a fixed point in M and hence a short time solution of the nonlinear
equation.
For simpler nonlinear support functions like e.g. f (u) = −eiγ |u|3 one could also prove
long time existence, because global a-priori estimates can be derived easily, since
d
4
′
′
¯
dt |u| = f (u)ū + f (u)u . The complicated nature of the support function for hydrodynamic bearings does not permit such a simple approach. However, since the forces are
always restoring a proof for global existence should still be possible, but is not in the
scope of this work.
Chapter 4
Finite Element Discretization
In Chapter 2 we have derived the equations of motion (2.45) and (2.46) of a homogenous, isotropic, rotating beam by the Lagrangian formalism. We are now going to use
the variational formulation of the equations of motion (2.36) to derive approximative
equations by the finite element method. The derivation is based on (Yamamoto &
Ishida, 2001) which is itself based on (Nelson & McVaugh, 1976). Good introductions
to the use of the finite element method in engineering can also be found in the books
(Kikuchi, 1986) and (Meirovitch, 1986). For the mathematical background see e.g.
(Rannacher, 2000; Rannacher, 2001; Braess, 2007; Strang & Fix, 1973).
4.1 Variational Formulation
We derive the finite element formulation for a rotating Rayleigh beam to which several
rigid disks are attached and which rotates in bearings. For this, consider the equations
of motion (2.45) and (2.46) for u, v ∈ H2 (Ω) together with boundary conditions at zp1 = 0
and zpN+1 = L and transmission conditions at the points zpi ∈ Ω̊ = (0, L), i = 2, . . . N to
model the point forces fp and moments mp occuring at the positions of rigid disks and
and bearings
(EIa u′′ )′′ + µü − (Ia ü′ )′ − ω(Ip v̇′ )′ + cu̇ = funb,1 ,
′′ ′′
′ ′
′ ′
(EIa v ) + µv̈ − (Ia v̈ ) + ω(Ip u̇ ) + cv̇ = funb,2 − µg,
h
i
= fpi ,1 ,
Ia ü′ + ωIp v̇′ − (EIa u′′ )′
zpi
h
i
= fpi ,2 ,
Ia v̈′ − ωIp u̇′ − (EIa v′′ )′
zpi
(4.1)
(4.2)
(4.3)
(4.4)
[EIa u′′ ]zp = tpi ,1 ,
(4.5)
[EIa v′′ ]zp = tpi ,2 .
(4.6)
i
i
50
Chapter 4: Finite Element Discretization
The notation [g]z denotes the jump of g at point z. To obtain a variational formulation
we multiply 4.1 with the test function η ∈ H2 (Ω) and (4.2) with the test function
ξ ∈ H2 (Ω). Partial integration over Ω = [0, L] then yields
Z
EIa u′′ η′′ + µüη + Ia ü′ η′ + ωIp v̇′ η′ + cu̇η dz
Ω
+
N X
′′
′
′
′
pi
i=1
Z
zpi+1 Z
=
funb,1 η dz,
(EIa u ) η − (EIa u )η − Ia ü − ωIp v̇ z
′′ ′
(4.7)
Ω
EIa v′′ ξ′′ + µv̈ξ + Ia v̈′ ξ′ − ωIp u̇′ ξ′ + cv̇ξ dz
Ω
+
N X
i=1
zpi+1 Z
= ( funb,2 − µg)ξ dz.
(EIa v ) η − (EIa v )η − Ia v̈ + ωIp u̇ z
′′ ′
′′
′
′
′
pi
(4.8)
Ω
The boundary terms that appear here, also appear in (4.3) - (4.6) and can therefore be
replaced with the corresponding forces and moments
Z
EIa u′′ η′′ + µüη + Ia ü′ η′ + ωIp v̇′ η′ + cu̇η dz
Ω
+
N+1
X
Z
fpi ,1 (zpi )η(zpi ) + tpi ,1 η (zpi ) =
funb,1 η dz,
′
i=1
Z
(4.9)
Ω
EIa v′′ ξ′′ + µv̈ξ + Ia v̈′ ξ′ − ωIp u̇′ ξ′ + cv̇ξ dz
Ω
+
N+1
X
i=1
Z
fpi ,2 (zpi )η(zpi ) + tpi ,2 η (zpi ) = ( funb,2 − µg)ξ dz.
′
(4.10)
Ω
Adding the two equations we obtain the variational formulation of our equation
Z  T 
 η   µ
  
 ξ  
µ
Ω

 η′′
+  ′′
ξ

N+1
X

+

i=1
  
  ü   η′
   + 
  v̈   ξ′
T 

  EIa
  u′′
 
 
 
EIa   v′′
T 
 
η(zpi )   fpi ,1  
 
+
ξ(zpi )   fpi ,2  
T 
  Ia
 
 
Ia
 

  ü′   η
 
 
  v̈′  +  ξ
T 
  c
 
 
c
 
T 

  η′  
 
ωI
p
 + 
 
 
  ξ′   −ωI

p
T 
 Z 
 η
η′ (zpi )   tpi ,1 

 
 =
 ξ
′
ξ (zpi )
tpi ,2
Ω
 
  u̇ 
  
  v̇ 

u̇′ 
 dz
v̇′ 
T 
 
funb,1
 
  f
− µg
unb,2
(4.11)


 dz.

4.2 System Matrices
We observe that the integral part of the functional on the left hand side of this equation is
the sum of five functionals. We will now discuss the transformation of these functionals
51
4.2 System Matrices
in the finite element approximation. The five functionals are the translatory inertia
functional
 
Z  T 
  ü 
 η   µ

   dz,


mt (ü, v̈, η, ξ) =
(4.12)
  
ξ
µ   v̈ 
Ω
the rotatory inertia functional
Z  ′ T 
 η   Ia
 

mr (ü, v̈, η, ξ) =
 ξ′  
Ia
Ω


  ü′ 

 
  v̈′  dz,
(4.13)
the damping functional
Z  T 
 η   c
  
c(u̇, v̇, η, ξ) =
 ξ  
c
 
  u̇ 
   dz,
  v̇ 
(4.14)
Ω
the stiffness functional
Z  ′′
 η

k(u, v, η, ξ) =
 ξ′′
Ω
T 
  EIa
 
EIa

  u′′
  ′′
v


 dz,

(4.15)
and finally, the gyroscopic functional which introduces the coupling

Z  ′ T 
 η  
ωIp   u̇′



g(u̇, v̇, η, ξ) =
 ′  
  ′
ξ
−ωIp
v̇


 dz.

(4.16)
Ω
To derive approximative equations we chose a finite dimensional subspace Vn ⊂ H2 (Ω)
and a basis {w̃i }i=1,...,n of Vn . The vectors
 



 w̃i   0 

{wi }i=1,...,2n = 
(4.17)
 0  ,  w̃ 

i
then form a basis for Vn × Vn . We are seeking solutions with separated variables of the
form


2n
 u(z, t)  X
qi (t)wi (z)
(4.18)
 =

v(z, t)
i=1
which we will write in matrix notation as
 
 u 
  = Wq,
 v 
(4.19)
where q = (q1 , . . . , q2n ) is the vector of the time dependent coefficients and W =
(w1 , . . . , w2n ) is the matrix which has the basis vectors as columns. By putting (4.19) into
(4.11) and using the wi as test functions we obtain a system of 2n ordinary differential
52
Chapter 4: Finite Element Discretization
equations for q. This is the explicite formulation of the Galerkin approximation in
equation (3.52) in chapter 3.






Z






I
c
µ
a





 W q̇
 W q̈ + W ′T 
 W ′ q̈ + W T 
W T 




Ia
c 
µ
Ω




 EIa

 ′′
ωIp  ′
′′T 
′T 



+W 
(4.20)
 W q + W 
 W q̇ dz
EIa 
−ωIp





 Z
N
X
 fpi ,1 
 tpi ,1 


f
unb,1
T
′
T
T
 + W (zp ) 
 dz.
 =
+
W(zpi ) 
W 
i




f
t
(
f
−
µg)
p
,2
p
,2
unb,2
i
i
i=1
Ω
The integrals can be evaluated and yield the system matrices. Each of the functionals
(4.12)-(4.16) defines a matrix which acts on the vector of coefficients q or its time
derivatives. We have


Z


µ

 W q̈ dz = Mt q̈,
(4.21)
mt (W q̈, W) =
W T 
µ 
Ω
where
Z
Mti, j
µwTi w j dz,
=
i, j = 1, . . . , 2n.
(4.22)
i, j = 1, . . . , 2n,
(4.23)
cwTi w j dz,
i, j = 1, . . . , 2n,
(4.24)
′′
EIa w′′T
i w j dz,
i, j = 1, . . . , 2n,
(4.25)
Ω
Analogously we obtain the matrices
Z
r
′
Mi, j =
Ia w′T
i w j dz,
Ω
Z
Ci, j =
Ω
Z
Ki, j =
Ω
(4.26)
which are all symmetric. The matrix defined by the gyroscopic functional is slightly
more complicated, due to the coupling. By Wu we denote the first row of W, and by
Wv we denote its second row. We have


Z  ′ T 
 Wu  
ωIp   Wu′ 



g(W q̇, W) =
(4.27)
 ′  
  ′  dzq̇
−ωIp
Wv
Wv
Ω
Z
=
Ω
Z

T 
 −Wv′   Wu′
 
ωIp 
Wu′   Wv′


 dzq̇

ωIp (Wu′T Wv′ − Wv′T Wu′ ) dzq̇ = Gq̇,
=
Ω
(4.28)
(4.29)
53
4.3 Explicite Choice of Approximating Space
where
Z
′
′T ′
ωIp (w′T
i,u w̃ j,v − wi,v w j,u ) dz,
Gi, j =
i, j = 1, . . . , 2n.
(4.30)
Ω
In the approximate Equation (4.20) the sum
N+1
X
i=1

 fp ,1
W(zpi )T  i
fpi ,2




 + W ′ (zp )T  tpi ,1
i
 t

pi ,2




(4.31)
containing the transmission terms transforms into a vector that gives the projected
forces and moments which might also depend on displacments and inclinations in the
zpi . It can be further simplified if we chose the basis elements in an appropriate way
(cf. Sec. 4.3). Also the integral of the unbalance forcing
Z
funb =
Ω

 funb,1
W T 
funb,2


 dz

(4.32)
yields a vector valued forcing term which in principle can be calculated explicitly if
we know the value of the eccentricity along the shaft. This, unfortunately, is rarely the
case and the unblance has to be estimated in practice. Analogously, only if the mass
density µ is known we can calculate the gravity load explicity
Z
fgrav =
Ω

 0
W T 
−µg


 dz.

(4.33)
4.3 Explicite Choice of Approximating Space
Until now, we have left open the choice of the approximating subspace Vn and of the
basis elements wi . We will now give a derivation of the explicit system matrices based
on (Yamamoto & Ishida, 2001) and (Nelson, 1980). For this the beam is decomposed
into N sections of constant diameter, cross-section and material properties like density
and Young’s modulus. We will first derive the approximate equations for such a
uniform beam element and then accumulate them into a system for the whole beam.
Now, let us consider the kth beam element bounded by zk and zk+1 of lenght lk = zk+1 −zk .
We want to describe the movement of the beam element by the nodal displacements
(uk , vk ) and (uk+1 , vk+1 ) and the nodal inclination angles (βk , γk ) and (βk+1 , γk+1 ) in zk
and zk+1 , respectively, as shown in Figure 4.1. In the Appendix A it is shown that
β ≈ −v′ and γ ≈ u′ (see also Section 2.3). The displacement in x-direction u(z) is
approximated by a cubic polynomial which is uniquely determined by the four values
(uk , γk , uk+1 , γk+1 ), and so is the displacement in y-direction v(z) by (vk , βk , vk+1 , βk+1 ).
We can solve this interpolation problem with the use of the Hermite polynomials Ψki
54
Chapter 4: Finite Element Discretization
y
γ k+1
x
γk
βk+1
vk+1
vk
βk
uk+1
uk
zk
zk+1
z
Figure 4.1: A single beam element is described by the nodal displacements (uk , vk ) and
uk+1 , vk+1 and the nodal inclination angles (βk , γk ) and (βk+1 , γk+1 ).
(Stoer & Bulirsch, 2002; Meirovitch, 1986)
!2
!3
z − zk
z − zk
+2
,
Ψ1 (z) = 1 − 3
lk
lk

!
!2 


z
−
z
z
−
z
k
k
 ,
+
Ψ2 (z) = (z − zk ) 1 − 2

lk
lk
!2
!3
z − zk
z − zk
Ψ3 (z) = 3
−2
,
lk
lk

!
!3 
 z − zk
z − zk 

+
Ψ4 (z) = lk −
 ,
lk
lk
(4.34)
(4.35)
(4.36)
(4.37)
and obtain the following representation for the displacements in [zk , zk+1 ]
u(z) = Ψ1 (z)uk + Ψ2 (z)γk + Ψ3 (z)uk+1 + Ψ4 (z)γk+1 ,
(4.38)
v(z) = Ψ1 (z)vk − Ψ2 (z)βk + Ψ3 (z)vk+1 − Ψ4 (z)βk+1 .
(4.39)
Note that the Hermite polynomials are also the solutions of the static problem with
unit displacement of one of the variables, e.g. Ψ1 solves EIa u′′′′ = 0 with boundary
condition uk = 1 and γk = γk+1 = uk+1 = 0. In the engineering literature these
interpolating functions are called shape functions.
We set
q̃k = (uk , vk , βk , γk , uk+1 , vk+1 , βk+1 , γk+1 )T
and rewrite (4.38) and (4.39) in matrix form
  

 u   Ψ1 0
0
Ψ2 Ψ3 0
0
Ψ4 
  = 
 q̃k = Wk q̃k .
 v   0 Ψ −Ψ
0
0 Ψ3 −Ψ4 0 
2
1
(4.40)
(4.41)
55
4.4 Rigid Disk Element Matrices
If we put this into equation (4.20), we obtain the system translatory mass matrix by
calculating all the integrals in (4.21):








µlk 
t

Mk =
420 






156
0
156
Sym.
2
0 −22lk
4lk
22lk
0
0
4l2k
54
0
0 13lk
156
0
54 −13lk
0
0 156
2
0
13lk −3lk
0
0 22lk 4l2k
0
0 4l2k
−13lk
0
0 −3l2k −22lk










 .








(4.42)
The other matrices are obtained by evaluating equations (4.23) - (4.30) and are given
in the Appendix C. We obtain the equation of motion for one beam element
Mq̃¨k + (G + C)q̃˙ k + Kq̃k = F,
(4.43)
where the vector F = funb + fgrav is the sum of unbalance forcing and gravity load. For
a uniform element we can calculate the gravity load vector
fgrav = −
iT
µg h
0, 6lk , −l2k , 0, 0, 6lk , l2k , 0 .
12
(4.44)
4.4 Rigid Disk Element Matrices
We now focus our attention on the discrete terms in the variational formulation (4.20)
which give the virtual work of the forces and moments exerted by rigid disks. To
facilitate notation we first consider only the Lagrangian of the rotating rigid disk at
node k which we have derived in (2.59). Here again, we assume that the two crosssectional moments of inertia are equal, i.e. ∆I = 0
L=
Ia 2
γ̇ + β̇2 + 2ωτ(γ̇ sin(ωt + η) + β̇ cos(ωt + eta))
2
I3 2
+
ω − 2ωγ̇β − 2ωτ(γ̇ sin(ωt + η) + ωβ sin(ωt + η))
2
md 2
+
(u̇ + v̇2k ) + md ω(v̇k rdg,1 − u̇k rdg,2 ) − md gvk .
2 k
(4.45)
Variation of the action functional with Rayleigh dissipation function FR = c(u̇2 + v̇2 )
leads to 4 ordinary differential equations for the nodal coordinates qk = (uk , vk , βk , γk )
md ük + cu̇k = md ω2 (rG,1 cos ωt − rG,2 sin ωt),
2
md v̈k + cv̇k = md ω (rG,1 sin ωt + rG,2 cos ωt) − md g,
Ia β̈k + ωI3 γ̇k = ω2 τ(Ia − I3 ) sin(ωt + η),
Ia γ̈k − ωI3 β̇k = ω2 τ(Ia − I3 ) cos(ωt + η).
(4.46)
(4.47)
(4.48)
(4.49)
56
Chapter 4: Finite Element Discretization
We can rewrite this in matrix notation
d
d
+ fgrav
,
Md q̈k + (Cd + Gd )q̇k = funb
where the mass matrix Md is given by

 m 0 0 0

 0 m 0 0
d
M = 
 0 0 Ia 0

0 0 0 Ia
(4.50)




 ,



(4.51)
the gyroscopic matrix Gd by


 0 0 0 0 


 0 0 0 0 
d

 ,
G = ω 

 0 0 0 I3 


0 0 −I3 0
and the damping matrix by

 c 0

 0 c
d
C = 
 0 0

0 0
0
0
0
0
0
0
0
0
(4.52)




 .



(4.53)
d and gravity f d
The vectors of unbalance forces funb
grav are
d
funb

 md ω2 (rG,1 cos ωt − rG,2 sin ωt)

 md ω2 (rG,1 sin ωt + rG,2 cos ωt)
= 

ω2 τ(Ia − I3 ) sin(ωt + η)

ω2 τ(Ia − I3 ) cos(ωt + η)








and
d
fgrav

 0

 −md g
= 
 0

0




 .



(4.54)
4.5 Assembling the Complete System
So far we have considered either one beam section with two nodes or a node to which a
rigid disk is attached. We will now show how the elements can be combined to derive
the approximative equation for the complete beam with N elements. Corresponding
to the choice we have made in Section 4.3, we take Vn to be the space of continuously
differentiable functions which are piecewise cubic polynomials
Vn = {u ∈ C1 ([0, L])|u|[zk ,zk+1 ] ∈ P3 ([zk , zk+1 ]),
(4.55)
which is dense in H2 (Ω) and has the dimension 2(N + 1). We set
q = (q1 , . . . , qN+1 ) = (u1 , v1 , β1 , γ1 , . . . , uN+1 , vN+1 , βN+1 , γN+1 ),
(4.56)
where qk is the vector of nodal coordinates for the kth node. We choose the position
of the nodes such that the set of nodes includes the set of points where interface
4.5 Assembling the Complete System
57
conditions are given in equations (4.3)-(4.6). The individual element displacement
vectors q̃k = (qk , qk+1 ) can be obtained from this via q̃k = Ak q where


 0 0 . . . 0 1 0 . . . 0 0 . . . 0 


 0 0 . . . 0 0 1 . . . 0 0 . . . 0 


8×4(N+1)
.
(4.57)
Ak =  .
..  ∈ R
.
.
.
 .

.
.




0 0 ... 0 0 0 ... 1 0 ... 0
Here, N + 1 is the total number of nodes and the block with the identity matrix starts
in the 4(k − 1) + 1th column. We define the continuously differentiable matrix function
W on [0, L] by
W(z) = Wk (z)Ak for z ∈ Ik ,
(4.58)
where each Wk is defined as in equation (4.41) on the respective interval [zk , zk+1 ]. From
the definition of the Wk it is clear that W is in Vn × Vn . Hence the overall displacement
is given by
 
 u 
  = Wq.
(4.59)
 v 
In terms of Section 4.1, the columns of W are the basis elements wi ∈ Vn × Vn , i =
1, . . . , 4(N + 1). They are chosen in such a way that in node k they have have either unit
displacement (w4(k−1)+1 and w4(k−1)+2 ) or unit inclination (w4(k−1)+3 and w4(k−1)+4 ).
By putting the definition (4.58) of W into the variational formulation and the formulas
for the system matrices (4.21) - (4.30), we obtain expressions for the complete system
matrices






Z
n−1
X

 Ia

 µ
 c
 ′
′T
T
T
T
Wk 
Ak
 Wk q̈ + Wk 
 Wk q̇
 Wk q̈ + Wk 
Ia
µ
c 
k=1
Ik

 EIa


EIa

Z
n−1
X

T
T
Wk 
Ak
=
+Wk′′T
k=1
Ik



!
 ′′

ωIp  ′
 W q + W ′T 
 Wk q̇ dz Ak + Fp
 k
k  −ωI
p


funb,1
 dz.
funb,2 − µg 
(4.60)
The vector Fp gives the nodal forces and moments exerted by rigid disks or bearings.
From the definition of the Wk we see that Wk (zpi ) , 0 only for zpi = zk or zpi = zk+i


 1 0 0 0 0 0 0 0 
(4.61)
Wk (zk ) = 
 ,
0 1 0 0 0 0 0 0 


 0 0 0 0 1 0 0 0 
 ,

(4.62)
Wk (zk+1 ) = 
0 0 0 0 0 1 0 0 


 0 0 0 1 0 0 0 0 
′

 ,
Wk (zk ) = 
(4.63)
0 0 −1 0 0 0 0 0 




0
0
0
0
0
0
0
1

 ,
Wk′ (zk+1 ) = 
(4.64)
0 0 0 0 0 0 −1 0 
58
Chapter 4: Finite Element Discretization
and hence Fp can be written as
Fp = [ fp1 ,1 , fp1 ,2 , tp1 ,1 , tp1 ,2 , . . . , fpn ,1 , fpn ,2 , tpn ,1 , tpn ,2 ]T .
(4.65)
The bearing forces will be under consideration in Chapter 5. The disk forces have been
considered in Section 4.4 and can be written as


 fdk ,1 


 fd ,2 
 k  = Md q̈ + (Gd + Cd )q̇ − f d
(4.66)


k k
k
k k
unb,k − fgrav,kd .
 tdk ,1 


tdk ,2
The complete mass matrix can hence be written as
M=
n−1
X
ATk Msk Ak
n
X
+
k=1
BTk Mdk Bk ,
(4.67)
k=1
where Msk is the mass matrix of the kth element, analogously for the gyroscopic,
damping and stiffness matrices and Mdk is the mass matrix of the disk attached at the
kth node. The matrix


 0 . . . 0 1 0 0 0 0 . . . 0 


 0 . . . 0 0 1 0 0 0 . . . 0 
 ∈ R4×4N

(4.68)
Bk = 

 0 . . . 0 0 0 1 0 0 . . . 0 


0 ... 0 0 0 0 1 0 ... 0
places the nodal matrices at the right position in the system matrix like the Ak do for
the element matrices. Hence from (4.60) we get the equation of motion for the entire
system
Mq̈ + (G + C)q̇ + Kq = Funb + Fgr + Fbear ,
(4.69)
where the matrices are defined as above and the force vectors are combined from the
nodal and the element forces
Funb = Fdunb + Fsunb
d1
dn T
= [ funb
, . . . , funb
] +
n−1
X
Z
ATk
k=1
Ik
Fgr = Fdgrav + Fsgrav
n−1
X
Z
ATk
k=1
h
d1
dn
, . . . , fgrav
fgrav


 dz,

(4.70)
(4.71)
(4.72)
d1
dn T
, . . . , fgrav
] +
= [ fgrav
=

 funb,1
T
Wk 
funb,2
iT
−
n−1
X
k=1
Ik
ATk


0
T
Wk 
−µ(z)g


 dz

iT
µg h
0, 6lk , −l2k , 0, 0, 6lk , l2k , 0 .
12
(4.73)
(4.74)
The last equation only holds for uniform mass distribution in each element. The vector
of bearing forces has the form
iT
h
(4.75)
Fbear = 0, . . . , 0, fb1 ,1 , fb1 ,2 , tb1 ,1 , tb1 ,2 , 0, . . . , 0, fb2 ,1 , fb2 ,2 , tb2 ,1 , tb2 ,2 , 0, . . . , 0 .
59
4.6 Approximations for Unbalance Forcing
The entries in the vector are at the postions of the bearing nodes. The special form of
the bearing response functions depends on the mechanical properties of the bearing
as described in Section 2.6 and in Chapter 5.
4.5.1 An Example
Beam model
50
40
30
width [mm]
20
10
0
−10
−20
−30
−40
−50
0
20
40
60
80
100
120
length [mm]
Figure 4.2: Detailed beam model of turbocharger: the rotor shaft is modelled with 13 finite
elastic beam elements shown in blue, the turbine and impeller wheels are modelled as rigid
disks and are shown with dashed red lines, and the positions of the bearings are indicated
by the black triangles.
In Figure 4.2 a typical beam modell for a turbocharger is depicted. It consists of 13
elements and hence has 14 nodes. There are 2 rigid disks attached in node 4 and
12 respectivley (indicated by dashed lines), and the bearings (indicated by triangles)
are located at node 7 and node 9. This model will be used later on for numerical
simulations. A detailed description and all the paramters are given in the Appendix D.
4.6 Approximations for Unbalance Forcing
The unbalance force Funb in equation (4.69) includes terms coming from the rigid disks
and integral terms coming from an inhomogenous mass distribution along the shaft.
In practice it is very difficult to estimate the inhomogenity which is usually also very
small compared to the unbalance that is caused by the rigid disks. In the following the
integral terms will therefore be neglected and only the static and dynamic unbalance
caused by the misalignment of the rigid disks will be considered.
60
Chapter 4: Finite Element Discretization
Furthermore the moments cause by the dynamic unbalance are replaced by pairs of
forces located in neighbouring nodes. This formulation is equivalent (Yamamoto &
Ishida, 2001) and facilitates notation. The unbalance force vector in node k is then
characterized by its amplitude ak and its phase ψk in the x − y-plane perpendicular to
the shaft
funb,1 = ω2 ak cos(ωt + ψk ),
(4.76)
funb,2 = ω2 ak sin(ωt + ψk ).
(4.77)
The amplitude ak is calculated from the static unbalance and the force pairs replacing
the dynamic unbalance. For static unbalance only we have
ak = md krG k,
(4.78)
and for purely dynamic unbalance at node k we have
ak−1 = ak+1 =
τ(Ip − I3 )
zk+1 − zk−1
.
(4.79)
For the example from Section 4.5.1 the following constellation is considered where the
dynamic unbalance dominates the static unbalance
a3 = 1.35 × 10−7 kgm,
ψ3 = π,
(4.80)
ψ5 = 0,
(4.81)
a11 = 2.01 × 10−7 kgm,
ψ11 = 0,
(4.82)
ψ13 = π.
(4.83)
−7
a5 = 1.50 × 10 kgm,
−7
a13 = 2.07 × 10 kgm,
Chapter 5
Bearing Models
In the construction of turbomachinery, the bearings play an important role. Not only
do they contain the moving parts in their designated position, but they also provide
the necessary damping to prevent resonance catastrophes. In this chapter we derive
models for oil lubricated journal bearings and the reaction forces they exert on the
rotor.
In this work we consider plain journal bearings as they were used in the experiments
conducted at the Toyota Central R&D Laboratories, whereas in commercial high-speed
turbochargers usually floating ring bearings are used (San Andrés, 2006). The plain
journal bearings (c.f. Figs. 5.2 and 5.3) have the advantage that they are relatively
easy to model and that a closed form analytical solution of the pressure distribution
is possible in some special cases. For more complicated bearing geometries this is no
longer possible. The analytical formulation of the pressure distribution and hence also
the bearing reaction forces allows for a faster numerical integration of the equation of
motion of the rotor.
The derivation of bearing characteristics is an important branch of rotordynamics in
particular since the discovery of lubricant induced instabilities, i.e. oil whirl (Newkirk
& Taylor, 1925). The basic theory can be found in most textbooks on turbomachinery,
e.g. in (Childs, 1993; Vance, 1988; Yamamoto & Ishida, 2001). It is based on Reynolds’
equation for thin films of lubricant as we show in Section 5.1. The special case of long
bearings was already studied by Sommerfeld (Sommerfeld, 1964). The special case of
short bearings was introduced by Ocvirk (Childs, 1993). These two special cases allow
for an analytical formulation of the bearing reaction forces and are therefore of great
theoretical value. There is an extensive literature covering the calculation of bearing
coefficients, i.e. the coefficients of the linearization of the bearing functions (Lund,
1987; Szeri, 1998). These coefficients can be used for the prediction of the onset of the
lubricant induced instabilities (Muszynska, 1986; Muszynska, 1988; Crandall, 1995).
The evaluation of the model which was set up during the collaboration with TCRDL
62
Chapter 5: Bearing Models
Frequency Diagram (Experiment)
Unbalance
Oscillation
0
2000
0.2
0.5
Amp [mm]
Ampl [mm]
Oil whirl
0.05
Spectrum modified bearing model, sigma=0.1
Amplitudes Compressor side, simulation
0.1
0
2000
1500
0
1500
2000
1500
1000
1000
0.1
1000
500
Driving
Frequency
[Hz] 0
0
500
1000
Frequency [Hz]
1500
2000
500
rot. speed
0 0
[Hz]
500
500
1000 1500 2000
frequencies [Hz]
rot. speed [Hz]
0 0
500
1000 1500
frequency [Hz]
2000
Figure 5.1: Comparison of experimental results (left) to simulations with Ocvirk short
bearing approximation (center) and modified short bearing (right); a perturbation term as
described in Sec. 5.2) yields a frequency shift of the oil whirl similar to the experimental
results.
showed that already the common simple journal bearing model explains the main
experimentally observed vibration phenomena quite well. The oil whirl, as well as
the resonance of the bending mode occur at nearly the same frequencies as in the
experiment as shown in the left and center diagram of Figure 5.1. The same holds for
the computed amplitudes and mode shapes. The detailed results of the simulations
are presented in Chapter 6, where also the comparison with the experimental data can
be found.
However, in the course of the evaluation of the model it showed that the simple journal
bearing model does not reproduce entirely correct the frequency of the subharmonic
self-excited oscillation, i.e. the oil whirl. The simulation results and the experimental
results differed, as can also be seen in Chapter 6. In the simulations the frequency of
the subharmonic oscillation was close to half the driving frequency until the driving
frequency reached twice the natural frequency of the beam. Then the frequency of the
self-excited oscillation locks to the natural frequency of the beam and reaches very high
amplitudes. The large amplitude oscillation is called oil whip. This transition behavior
is common and also observed in (Muszynska, 1986; Muszynska, 1988; San Andrés,
2006).
In the experiments the ratio of whirl frequency to driving frequency shifts from 21 to
lower values in the driving frequency range above the first critical frequency of the
beam. In order to describe this shift properly several other bearing models where considered and compared in simulations. Starting point is a modification of the pressure
distribution function (cf. Section 5.2) which shows that the frequency of the self-excited
oscillation can be directly influenced. Introduction of a perturbation which is quadratic
in the angular velocity ω leads to a shift of the whirl ratio that is very similar to the
one that is observed in the experiment (cf. Fig. 5.1).
In order to derive this perturbation from first principles the influence of second order
corrections to Reynolds’ equation is studied. As shown in 5.1.5 these inertia corrections
lead to a quadratic dependence of the pressure distribution on the angular velocity. The
5.1 Derivation of Reynolds’ Equation and Inertia Correction
63
influence of inertia corrections on linearized bearing coefficients for journal bearings
has been studied in (Nataraj et al., 1994; Szeri, 1998; El-Shafei, 1995). A derivation of
the second order correction terms for squeeze film dampers is shown in (Crandall & ElShafei, 1993). The mathematical convergence theory for Reynolds theory is presented
in (Nazarov, 1990; Bayada & Chambat, 1986; Duvnjak & Marušić-Paloka, 2000) for
plain journal bearings with Newtonian lubricant and for thin films of non Newtonian
liquids in (Bourgeat et al., 1993; Duvnjak & Marušić-Paloka, 2000). The inertial effects
are studied in (Assemien et al., 1994)
The chapter is structured as follows. First we give the derivation of Reynolds’ equation
from the Navier-Stokes equation by an asymptotic expansion. The short bearing
approximation is derived by giving a relation of the film thickness to the bearing
length. This assumption of a relation between the two parameters is then also used to
derive the second order correction terms. In Section 5.2 we give a motivation for the
phenomenological perturbation term mentioned above. For completeness we list in
Section 5.3 some other bearing models which were also considered during the project,
such as finite length approximation, or Reynolds’ equation with Reynolds’ boundary
conditions.
5.1 Derivation of Reynolds’ Equation and Inertia Correction
We derive an equation for the pressure distribution inside a circular journal bearing.
This is a hydrodynamic bearing where a circular shaft rotates inside a circular bearing
casing and is supported by a thin layer of lubricant, usually oil in the case of the
turbocharger. It is schematically shown in the Figures 5.2 and 5.3. The bearing casing
is at rest. The rotation of the shaft creates a circular flow pattern by dragging along
the fluid. This flow pattern causes the impedance of the bearing to loads on the shaft
by causing higher pressures in narrowing regions of the bearing. This creates reaction
forces that oppose the movement. The reaction forces can be calculated from the
pressure distribution inside the bearing by integration over the bearing surface.
The radius R of the bearing and the radius R j of the rotating shaft differ by a small
distance cr , the radial bearing clearance (cf. Figs. 5.2 and 5.3). The axial length of the
bearing is called bearing width W. For small inclinations of the shaft the film thickness
h(ϕ) varies only with the circumferential angle ϕ and depends on the momentary
position of the shaft’s center Z which is considered to move with the velocity VZ . The
position of Z is given in polar coordinates by the eccentricity e and precession angle γ.
The angular velocity of the shaft is denoted by ω. Note that in the figures the bearing
clearance is exaggerated for illustration purposes. In the examined real turbocharger
the clearance is 0.02 mm, while the radius of the shaft is 3 mm.
The derivation is based on an asymptotic expansion of the Navier-Stokes equations
64
Chapter 5: Bearing Models
r=0
uϕ
ϕ
ur
r = −h(ϕ)
γ
VZ
cr
Rj
R
e
Z
O
rotating
shaft
ω
lubricant
film (oil)
bearing
casing
Figure 5.2: Sketch of simple journal bearing; view in axial direction; the radial bearing
clearance cr is exaggerated for illustration
with respect to the small parameter ε = cRr . This asymptotic expansion leads to approximations for the flow and the pressure in a thin film. The equation governing the
pressure is called Reynolds’ equation. We consider four cases in this work which differ
by higher order terms and by the ratio of the bearing width W to the bearing radius R:
1. The classical Reynolds’ equation: The ratio ε = cRr is small, while the ratio δ =
is of order 1; no higher order terms are considered.
W
R
2. The short bearing approximation: the ratio δ = W
R is also small in addition to ε.
To derive the approximate equations in one step we set ε = Kδ2 . Then the
equations simplify even more and an analytical solution for the bearing forces
can be obtained.
3. Reynolds’ equation with inertia corrections: In the derivation of Reynolds’ equation only terms of zeroth order in ε are considered. The inertia correction takes
into account also terms of order ε1 .
5.1 Derivation of Reynolds’ Equation and Inertia Correction
65
bearing casing
lubricant (oil)
rotating shaft
lubricant (oil)
bearing casing
W
Figure 5.3: Sketch of simple journal bearing; view in lateral direction
4. Short bearing with inertia corrections: As above the additional assumption of
small δ and ε = Kδ2 allows for further simplification and analytical solution for
the pressure distribution also for the higher order terms.
We derive in detail the equations for the pressure distributions in these four cases in
the following sections.
5.1.1 Scaling of the Navier Stokes Equations
The Navier-Stokes equations for an incompressible fluid (Ockendon & Ockendon,
1995) are given by
∂t ū + (ū · ∇)ū = −∇p̄ + ν∆ū,
div ū = 0.
(5.1)
(5.2)
The bearings have cylindrical geometry. Therefore we write (5.1) and (5.2) in cylindrical coordinates and non-dimensionalize them using the following variables. The
dimensional variables carry an overbar, the non-dimensionalized ones carry a tilde:
r̄
z̄
, z̃ = , ϕ̃ = ϕ̄,
R
R
p̄
R2 ω
ū
,
R̃
=
ũ =
, p̃ =
.
Rω
ν
ρR2 ω2
τ = ωt, r̃ =
(5.3)
(5.4)
The non-dimensionalized NSE for an incompressible fluid in cylindrical coordinates
66
Chapter 5: Bearing Models
r-component:
∂τ ũr + (ũr ∂r̃ ũr +
ϕ-component:
ũϕ
r̃
∂ϕ̃ ũr + ũz ∂z̃ ũr −
ũ2ϕ
r̃
)
∂2ϕ̃ ũr ũ
2∂ϕ̃ ũr
∂r̃ ũr
1 2
r
2
+ ∂z̃ ũr + 2 − 2 −
= −∂r̃ p̃ + (∂r̃ ũr +
),
r̃
r̃
r̃
r̃2
R̃
ũϕ
ũr
∂ϕ̃ ũϕ + ũz ∂z̃ ũϕ )
ũϕ + ũr ∂r̃ ũϕ +
r̃
r̃
∂2ϕ̃ ũϕ ũϕ 2∂ϕ̃ ũr
∂r̃ ũϕ
∂ϕ̃ p̃ 1 2
2
= −
),
+ (∂r̃ ũϕ +
+ ∂z̃ ũϕ + 2 − 2 +
r̃
r̃
r̃
r̃
r̃2
R̃
(5.5)
∂τ ũϕ + (
z-component:
∂τ ũz + (ũr ∂r̃ ũz +
= −∂z̃ p̃ +
Continuity equation:
∂r̃ ũr +
1
R̃
ũϕ
r̃
(5.6)
∂ϕ̃ ũz + ũz ∂z̃ ũz )
(∂2r̃ ũz
2
∂r̃ ũz ∂ϕ̃ ũz
+ 2 + ∂2z̃ ũz ),
+
r̃
r̃
ũr ∂ϕ̃ ũϕ
+
+ ∂z̃ ũz = 0.
r̃
r̃
(5.7)
(5.8)
Two of the three dimensions of the problem are considered small compared to the
radius of the bearing: the radial clearance cr of the bearing and the axial length W
of the film. We therefore introduce the following scalings of the nondimensionalized
variables in Eqs. (5.5)-(5.8):
r̃ − 1
z̃
W
cr
,δ= ,r=
, z = , ϕ = ϕ̃,
R
R
ε
δ
ũr
ε
ũz
ur = , uz = , uϕ = ũϕ , R = εR̃, p = R 2 p̃.
ε
δ
δ
The scaling for the axial coordinate z is chosen different from the scaling for the radial
coordinate in order to obtain a limit for different ratios of these two dimensions. The
scaling for the radial component includes a transformation such that r = 0 at the
bearing casing and r = −h(ϕ) on the journal surface. The differential operators in the
new variables fulfill
1
1
∂r̃ = ∂r , ∂z̃ = ∂z .
ε
δ
Writing NSE in the new coordinates then yields


u2ϕ 

1
1
1

∂τ εur + εur ( ∂r εur ) +
uϕ ∂ϕ εur + δuz ∂z εur −
ε
1 + εr
δ
1 + εr 
1
1
1
1
1 1 2
= − ∂r p̃ +
∂r εur +
( ∂r εur ) + 2 ∂2z εur
2
ε
1
+
εr
ε
δ
R̃ ε
!
2∂
εu
εu
ϕ
r
1
r
,
∂2 εur −
−
+
(1 + εr)2 ϕ
(1 + εr)2 (1 + εr)2
ε=
67
5.1 Derivation of Reynolds’ Equation and Inertia Correction
uϕ
1
1
εur
uϕ + εur ( ∂r uϕ ) +
∂ϕ uϕ + δuz ∂z uϕ
1 + εr
ε
1 + εr
δ
1
1
1
1
1
1 1 2
∂ u +
∂2 u + 2 ∂2z uϕ
= −
∂ϕ p̃ +
( ∂r u ϕ ) +
2 r ϕ
2 ϕ ϕ
1 + εr
1
+
εr
ε
ε
(1
+
εr)
δ
R̃
!
uϕ
2∂ϕ εur
−
,
+
(1 + εr)2 (1 + εr)2
∂τ u ϕ +
∂τ δuz
1
1
1
+ εur ( ∂r δuz ) +
uϕ ∂ϕ δuz + δuz ∂z δuz
ε
1 + εr
δ
!
1
1
1
1
1 2
1 1 2
2
= − ∂z p̃ +
∂r δuz +
∂ δuz .
( ∂r δuz ) + 2 ∂z δuz +
δ
1 + εr ε
δ
(1 + εr)2 ϕ
R̃ ε2
With the scaling for the Reynolds number R = εR̃ we obtain:


u2ϕ 

1

u ϕ ∂ϕ u r + u z ∂z u r −
R ∂τ ur + ur ∂r ur +
1 + εr
(1 + εr)ε 
R
ε
1
1
ε
∂r p̃ + ∂2r ur +
∂2 u r
∂r ur + 2 ∂2z ur +
ε
1 + εr
ε2
δ
(1 + εr)2 ϕ
!
2∂ϕ εur
ε
ur −
,
−
(1 + εr)2
(1 + εr)2
=−
R ∂τ u ϕ +
uϕ
ε
u r u ϕ + u r ∂r u ϕ +
∂ϕ u ϕ + u z ∂z u ϕ
1 + εr
1 + εr
1
ε
1
R
∂ϕ p̃ + ∂2r uϕ +
∂r u ϕ +
=−
∂2 u ϕ
1 + εr
ε
1 + εr
(1 + εr)2 ϕ

2∂ϕ ε2 ur
εuϕ 
ε 2
−
+ 2 ∂z u ϕ +
 ,
δ
(1 + εr)2 (1 + εr)2 
εR ∂τ uz + ur ∂r uz +
1
u ϕ ∂ϕ u z + u z ∂z u z
1 + εr
!
ε2 2
ε2
ε
Rε
2
∂r u z + 2 ∂z u z +
∂ϕ u z .
= − 2 ∂z p̃ + ∂r uz +
1 + εr
δ
δ
(1 + εr)2
Finally by introducing the scaling for the pressure p = R δε2 p̃ and ordering by orders of
ε and δ we get:
∂r u r +
1
ε
ur +
∂ϕ uϕ + ∂z uz = 0,
1 + εr
1 + εr
(5.9)
68
Chapter 5: Bearing Models
ε2
−∂r p = − 2
δ
ε3
+ 2
δ
ε4
− 4
δ
ε4
− 2
δ
∂2r uϕ =


u2ϕ 
 2
∂r ur + R


1 + εr 
R ∂τ u r + u r ∂r u r +
(5.10)
1
1
u ϕ ∂ϕ u ϕ + u z ∂z u z −
∂r u r
1 + εr
1 + εr
∂2z ur
 2

 ∂ϕ ur − ur − 2∂ϕ ur 

 ,


(1 + εr)2
δ2
∂ϕ p
1 + εr
+ε R ∂τ uϕ + ur ∂r uϕ +
(5.11)
uϕ
1 + εr
ε2 2
− 2 ∂z u ϕ
δ
uϕ ur
+ε2 R
1 + εr
∂2ϕ uϕ + 2ε∂ϕ ur − uϕ
,
−ε2
(1 + εr)2
∂2r uz
− ∂z p = ε R ∂τ u z + u r ∂r u z +
∂ϕ u ϕ + u z ∂z u ϕ −
uϕ
1
∂r u ϕ
1 + εr
1
∂ϕ u z + u z ∂z u z −
∂r u z
1 + εr
1 + εr
ε2 2
∂ uz
δ2 z
∂2ϕ uz
2
−ε
.
(1 + εr)2
−
(5.12)
5.1.2 Boundary Conditions
We impose the following no slip boundary conditions on our system. On the outer
surface:
ũr = 0, ũϕ = 0, ũz = 0 on Γ1 = {r̃ = 0, ϕ̃ ∈ [0, 2π], z̃ ∈ [0, W]},
(5.13)
and on the journal surface:
ũr = Ṽr (ϕ, t), ũϕ = Ṽϕ (ϕ, t), ũz = 0 on Γ2 = {r̃ = −h̃(ϕ, t), ϕ̃ ∈ [0, 2π], z̃ ∈ [0, W]},
(5.14)
and at the openings at both ends:
p̃ = 0
on Γ3 ∪ Γ4 = {r̃ ∈ [−h̃(ϕ), 0], ϕ̃ ∈ [0, 2π], z̃ ∈ {0, W}}.
(5.15)
The velocities Ṽr (ϕ, t) and Ṽϕ (ϕ, t) of the journal surface and the oil film thickness
h̃(ϕ, t) can be expressed in terms of the position and velocity of the journal center (cf.
69
5.1 Derivation of Reynolds’ Equation and Inertia Correction
ϕ
Vr
Vϕ
h(ϕ)
A
a
l
eγ̇
Rj
cr
R
γ
ė
e
Z
O
ω
Figure 5.4: Sketch of simple journal bearing; view in axial direction
figure 5.4). Let Ee be the unit vector in the direction of the eccentricity ẽ and Eγ be the
corresponding orthonormal unit vector in the tangential direction. The velocity ṼZ of
the journal center can be written as
ṼZ = ẽ˙Ee + γ̇ẽEγ .
Let Er and Eϕ be the local unit vectors at the point A. Then we have
Ee = cos ϕEr − sin ϕEϕ ,
Eγ = sin ϕEr + cos ϕEϕ ,
and from this
ṼZ = (ẽ˙ cos ϕ + γ̇ẽ sin ϕ)Er + (γ̇ẽ cos ϕ − ẽ˙ sin ϕ)Eϕ .
Taking into concern the rotation of the shaft with with angular velocity ω we can write
the velocity ṼA at point A as ṼA = ωEz × a + ṼZ where a = −ẽEe + lEr . We can determine
70
Chapter 5: Bearing Models
l from the relation
kak2 = R2j = l2 + ẽ2 − 2lẽ cos ϕ,
q
l± = ẽ cos ϕ ± R2j − ẽ2 sin2 ϕ.
Taking the positive solution l = l+ we get
q
R2j − ẽ2 sin2 ϕ)Er
q
= −ẽ cos ϕEr + ẽ sin ϕEϕ + (ẽ cos ϕ + R2j − ẽ2 sin2 ϕ)Er
q
= ẽ sin ϕEϕ + R2j − ẽ2 sin2 ϕEr .
a = −ẽEe + (ẽ cos ϕ +
Hence
q
ṼA = ωẽ sin ϕ(Ez × Eϕ ) + ω
R2j − ẽ2 sin2 ϕ(Ez × Er )
+(ẽ˙ cos ϕ + γ̇ẽ sin ϕ)Er + (γ̇ẽ cos ϕ − ẽ˙ sin ϕ)Eϕ
= Er (−ωẽ sin ϕ + ẽ˙ cos ϕ + ẽ˙γ sin ϕ)
q
+Eϕ (ω R2j − ẽ2 sin2 ϕ − ẽ˙ sin ϕ + γ̇ẽ cos ϕ)
= Ṽr Er + Ṽϕ Eϕ .
(5.16)
(5.17)
For the film thickness h̃ we have
h̃ = R − l
= R − ẽ cos ϕ −
= R − ẽ cos ϕ −
q
q
R2j − e2 sin2 ϕ
(R − cr )2 − e2 sin2 ϕ.
(5.18)
We now nondimensionalize and scale the boundary conditions in the same way as we
did for the NSE:
Vr =
Ṽϕ
1 Ṽr
ẽ
1 ẽ
, Vϕ =
,κ=
= , τ = ωt.
ε Rω
Rω
ε R cr
The derivative with respect to τ will be denoted by ′ . From this and Eqs. (5.16) and
(5.18) we deduce the following equations for Vr
cr ωVr = −ωcr κ sin ϕ + ωcr κ′ cos ϕ + ωcr κγ′ sin ϕ
Vr = −κ sin ϕ + κ′ cos ϕ + κγ′ sin ϕ
= Vr0 ,
(5.19)
5.1 Derivation of Reynolds’ Equation and Inertia Correction
71
for Vϕ
q
RωVϕ = ω R2j − ẽ2 sin2 ϕ − cr ωκ′ sin ϕ + ωcr κγ′ sin ϕ
r
c2r 2
(R − cr )2
Vϕ =
−
κ sin2 ϕ − εκ′ sin ϕ + εκγ′ cos ϕ
2
2
R
R
q
=
1 − 2ε + ε2 − ε2 κ2 sin2 ϕ − εκ′ sin ϕ + εκγ′ cos ϕ
1
= (1 − ε − ε2 κ2 sin2 ϕ + O(ε3 )) − εκ′ sin ϕ + εκγ′ cos ϕ
2
1
= 1 + ε(−1 − κ′ sin ϕ + κγ′ cos ϕ) − ε2 κ2 sin2 ϕ + O(ε3 )
2
= Vϕ0 + εVϕ1 + O(ε2 ),
(5.20)
and for h(ϕ, t)
h(ϕ, t) =
=
=
=
=
h̃(ϕ) 1 h̃(ϕ)
=
R ε
cr
r
1
1
− κ cos ϕ − ( − 1)2 − κ2 sin2 ϕ
ε
ε
1
1
1
− κ cos ϕ − ( − 1 − ε κ2 sin2 ϕ + O(ε2 ))
ε
ε
2
1 2
1 − κ cos ϕ + ε κ sin2 ϕ + O(ε2 )
2
h0 + εh1 + O(ε2 ).
(5.21)
5.1.3 Reynolds’ Equation and Short Bearing Approximation
We now want to solve the scaled NSE (5.9)-(5.12). We still have two free parameters
in the equations which we will assume to be small in the following. ε = cRr is the ratio
of radial clearance cr and bearing radius R and is a measure for the film thickness. It
is considered small in all bearing theories. δ = W
R is the ratio of the bearing width W
and the radius R. This parameter is considered small in the so called short bearing
approximation. In the present model we have ε ≪ δ ≪ 1. We therefore assume in the
following relation between the two parameters
ε = Kδ2 .
(5.22)
72
Chapter 5: Bearing Models
This leads to the following equations


u2ϕ 
 2


−∂r p = −Kε ∂r ur + R
1 + εr 
1
1
+Kε2 R ∂τ ur + ur ∂r ur +
u ϕ ∂ϕ u ϕ + u z ∂z u z −
∂r ur − K∂2z ur
1 + εr
1 + εr
 2

∂
u
−
u
−
2∂
u
 ϕ r
r
ϕ r

−ε3 K 
 ,
2
(1 + εr)
!
1
1
2
2
∂ϕ p −
∂r uϕ − K∂z uϕ
∂r u ϕ = ε
K(1 + εr)
1 + εr
uϕ
∂ϕ u ϕ + u z ∂z u ϕ
+εR ∂τ uϕ + ur ∂r uϕ +
1 + εr
∂2ϕ uϕ + 2ε∂ϕ ur − uϕ
uϕ ur
− ε2
+ε2 R
,
1 + εr
(1 + εr)2
uϕ
1
2
2
∂r u z − ∂z p = ε R ∂τ u z + u r ∂r u z +
∂ϕ u z + u z ∂z u z −
∂r uz − K∂z uz
1 + εr
1 + εr
∂2ϕ uz
2
,
−ε
(1 + εr)2
and
∂r u r +
ε
1
ur +
∂ϕ uϕ + ∂z uz = 0.
1 + εr
1 + εr
We expand u and p into a series in ε
uε = u0 + εu1 + O(ε2 ) and
pε = p0 + εp1 + O(ε2 )
and insert the series into the scaled equations above, keeping in mind that
∞
X
1
=
(−εr)n .
1 + εr
n=0
Collecting terms of same order in ε we get
ε0 :
∂r p0 = 0,
(5.23)
∂2r u0ϕ = 0,
(5.24)
∂2r u0z − ∂z p0 = 0,
(5.25)
∂r u0r + ∂ϕ u0ϕ + ∂z u0z = 0,
(5.26)
5.1 Derivation of Reynolds’ Equation and Inertia Correction
ε1 :
−∂r p1 = −K(∂2r u0r + R(u0ϕ )2 ),
∂2r u1ϕ =
∂ϕ p0
K
73
(5.27)
− ∂r u0ϕ − K∂2z u0ϕ
+ R ∂τ u0ϕ + u0r ∂r u0ϕ + u0ϕ ∂ϕ u0ϕ + u0z ∂z u0ϕ ,
(5.28)
∂2r u1z − ∂z p1 = −∂r u0z − K∂2z u0z
+ R ∂τ u0z + u0r ∂r u0z + u0ϕ ∂ϕ u0z + u0z ∂z u0z ,
∂r u1r + u0r − r∂ϕ u0ϕ + ∂ϕ u1ϕ + ∂z u1z = 0.
(5.29)
(5.30)
The boundary conditions for the functions appearing in the expansions for the pressure
and the velocity can be calculated from the boundary conditions for the NSE (5.13)
- (5.15) and their expansions (5.19) - (5.21). From uεr (0, ϕ, z) = 0 on the outer surface
we deduce u0r (0, ϕ, z) = u1r (0, ϕ, z) = 0. Analogously u0ϕ (0, ϕ, z) = u1ϕ (0, ϕ, z) = 0 and
u0z (0, ϕ, z) = u1z (0, ϕ, z) = 0. The boundary conditions at the inner surface r = −hε are
more interesting. For notational convenience the dependence of the velocities from z
and ϕ are not explicitly written down in the next equations. We get
Vr = uεr (−hε )
= u0r (−h0 − εh1 + O(ε2 )) + εu1r (−h0 − εh1 + O(ε2 )) + O(ε2 )
= u0r (−h0 ) − εh1 ∂r u0r (−h0 ) + εu1r (−h0 ) + O(ε2 ).
From (5.19) we see that Vr = Vr0 and therefore
u0r (−h0 ) = Vr0 ,
(5.31)
u1r (−h0 )
(5.32)
=
h1 ∂r u0r (−h0 ).
For u0ϕ , u1ϕ , u0z and u1z we derive in the same way that
u0ϕ (−h0 ) = Vϕ0 ,
u1ϕ (−h0 )
u0z (−h0 )
u1z (−h0 )
=
Vϕ1
+
(5.33)
h1 ∂r u0ϕ (−h0 ),
(5.34)
= 0,
(5.35)
= h1 ∂r u0z (−h0 ).
(5.36)
In combination with these boundary conditions the equations (5.23) - (5.30) can be
solved analytically which shall be done in the following sections.
74
Chapter 5: Bearing Models
5.1.4 Approximate Solution of Order ε0
We start with the order ε0 . The first equation (5.23) tells us that the pressure does not
depend on the radial variable r. Equations (5.24) and (5.25) can therefore be integrated
over r using the boundary conditions at r = 0 and r = −h0 . We obtain the velocity
profiles:
r
,
h0
r(r + h0 )
∂z p0 .
2
u0ϕ = −Vϕ0
(5.37)
u0z =
(5.38)
From the continuity equation (5.26) we can calculate ∂r u0r :
r(r + h0 ) 2 0
r
+
∂z p .
−∂r u0r = ∂ϕ u0ϕ + ∂z u0z = ∂ϕ −Vϕ0
h0
2
(5.39)
We use again the fact that p0 does not depend on r and integrate both sides with respect
to r to eliminate the radial coordinate
−h
Z0 (ϕ)
−Vr0
"
#−h
h0 2 0
0 r
2 0 1 3
∂ϕ −Vϕ
dr + ∂z p
r + r
h0
6
4
0
=
0
(5.40)
−h
Z0 (ϕ)
= ∂ϕ
0
=
!
h3
0 h0
0 r
dr + Vϕ
∂ϕ h0 + 0 ∂2z p0
−Vϕ
h0
h0
12
h0
∂ϕ −Vϕ0
!
2
+
Vϕ0 ∂ϕ h0
+
h30
12
∂2z p0 .
(5.41)
(5.42)
R g(α)
R g(α)
Leibniz’ rule ∂α 0
f (x, α) dx = 0 ∂α f (x, α) dx + f (g(α), α)∂α g was used for interchanging derivation and integration. From this we get for the pressure
!
1 0
12 h0
0
0
0
2 0
(5.43)
∂ϕ Vϕ + Vϕ ∂ϕ h0 − Vϕ ∂ϕ h0 − Vr
∂z p =
2
h30 2
!
1 0
12 h0
0
0
∂ϕ Vϕ − Vϕ ∂ϕ h0 − Vr .
(5.44)
=
2
h30 2
Integration over z and using p = 0 for z ∈ {0, 1} then yields
!
6z(z − 1) h0
1 0
0
0
p =
∂ϕ Vϕ − Vϕ ∂ϕ h0 − Vr .
2
2
h30
0
(5.45)
To obtain the pressure profile in the lowest order in ε we finally use the ε0-approximation
Vr0 , Vϕ0 and h0 from (5.19) and (5.20) and (5.21).
h0 = 1 − κ cos ϕ,
and finally get
Vϕ0 = 1,
Vr0 = −κ sin ϕ + κ′ cos ϕ + κγ′ sin ϕ
5.1 Derivation of Reynolds’ Equation and Inertia Correction
75
Statement 5.1. The pressure solution for the zeroth order short bearing approximation
corresponding to point 2 in the list in Section 5.1 is
− 12 κ sin ϕ + κ sin ϕ − κ′ cos ϕ − κγ′ sin ϕ
(5.46)
p0 = 6z(z − 1)
(1 − κ cos ϕ)3
(γ′ − 21 )κ sin ϕ + κ′ cos ϕ
= −6z(z − 1)
.
(5.47)
(1 − κ cos ϕ)3
Usually the short bearing approximation is obtained by first deriving Reynolds’ equation for thin films and then in a second step assuming also smallness of the bearing
width. Here, due to the special scaling it can be derived in one step. In Figure 5.5 the
distributions for different values of κ, κ̇ are depicted to give an idea of the geometric
nature of the solutions.
Figure 5.5: Zeroth order short bearing pressure distributions p0 for varying values of κ and
κ̇; left: κ = 0.1, κ̇ = 0, middle: κ = 0.9, κ̇ = 0, right: κ = 0.5, κ̇ = 0.4.
For zero radial velocity the pressure distribution shows lower pressures in the region
behind (in mathematically positive direction) the point of smalles film-thicknes and
higher pressures in front of this bottle neck. For large eccentricities the region of large
pressure variation is concentrated around this point of smallest film-thickness. These
pressure differences sustain the circular whirling motion of the shaft. For nonzero
radial velocity there appears a large pressure opposing the outward movement and
avoiding collision.
From the solution for the pressure we can calculate the velocity profiles u0r , u0ϕ and u0z
by (5.37), (5.38), and (5.39)
u0ϕ = −
r
,
h0
(5.48)
76
Chapter 5: Bearing Models
r(r + h0 )
∂z p0
2
−6r(r + h0 )
1
1
′
′
=
z
−
γ
−
κ
sin
ϕ
+
κ
cos
ϕ
,
2
2
h30
Zr
(−∂ϕ u0ϕ − ∂z u0z ) dρ
=
u0z =
u0r
(5.49)
0
Zr
=
=
ρ
ρ(ρ + h0 ) 2 0
∂z p dρ
∂
h
−
ϕ
0
2
h20
0


 2r3 + 3r2 h0  ′ 1 r2

 γ − κ sin ϕ + κ′ cos ϕ .
κ
sin
ϕ
−



2
2h20
h30
−
(5.50)
5.1.5 Approximate Solution of Order ε1
The equations (5.27) - (5.30) for the order ε1 have almost the same structure as the
zeroth order approximation. The difference lies only in the right hand side terms
which depend on the lower order solutions p0 and u0 . To simplify the notation we
define
∂ϕ p0
Tϕ0 (r, ϕ, z, τ) :=
− ∂r u0ϕ − K∂2z u0ϕ
K
+R ∂τ u0ϕ + u0r ∂r u0ϕ + u0ϕ ∂ϕ u0ϕ + u0z ∂z u0ϕ ,
Tz0 (r, ϕ, z, τ)
−∂r u0z
:=
− K∂2z u0z
+R ∂τ u0z + u0r ∂r u0z
+ u0ϕ ∂ϕ u0z + u0z ∂z u0z .
(5.51)
(5.52)
Equations (5.28) and (5.29) can then be rewritten as
∂2r u1ϕ = Tϕ0 ,
(5.53)
∂2r u1z − ∂z p1 = Tz0 .
(5.54)
From the equation for the pressure (5.27) and the solutions of lower order from the
previous section we see that
∂r ∂z p1 = ∂z ∂r p1 = K(∂z ∂2r u0r + R∂z (u0ϕ )2 ) = 0.
Therefore ∂z p1 does not depend on r and we can again use an analogous procedure
like in the previous section to derive an equation for the pressure.
We integrate equation (5.54) twice over r and use the boundary conditions (5.36) to
obtain


Zr Zρ
Z−h0Zρ




r(r + h0 )
r 
0
0
1
0
1
Tz (s) ds dρ +
∂z p +
Tz (s) ds dρ . (5.55)
uz =
−h1 ∂r uz (−h0 ) +

2
h0 
0
0
0
0
77
5.1 Derivation of Reynolds’ Equation and Inertia Correction
For u1ϕ we have the boundary conditions u1ϕ (0) = 0 and u1ϕ (−h0 ) = Vϕ1 + h1 ∂r u0ϕ (−h0 )
(5.34). Hence the circumferential velocity v1ϕ is given by


Zr Zρ
Z−h0Zρ




r  1
Tϕ0 (s) ds dρ +
u1ϕ =
Tϕ0 (s) ds dρ
(5.56)
−Vϕ − h1 ∂r u0ϕ (−h0 ) +

h0 
0
0
0
0
and does only depend on the boundary condition and known zeroth order terms.
For u1r we have the boundary conditions u1r (0) = 0 and u1r (−h0 ) = h1 ∂r u0r (−h0 ) (5.32). By
integrating the continuity equation (5.30) over r we then obtain
Z−h0
−h1 ∂r u0r (−h0 ) =
−∂r u1r dr
(5.57)
0
Z−h0
Z−h0
Z−h0
0
0
1
=
(ur − r∂ϕ uϕ ) dr +
∂ϕ uϕ dr +
∂z u1z dr
0
0
(5.58)
0
Z−h0
Z−h0
Z−h0
r(r + h0 )
0
0
1
2 1
dr
=
(ur − r∂ϕ uϕ ) dr +
∂ϕ uϕ dr + ∂z p
2
0
0
0
Z−h0Zr Zρ
∂z Tz0 (s) ds dρ dr
+
0
Z−h0
+
0
0
0


Z−h0Zρ



r 
0
0
∂z Tz (s) ds dρ dr.
−h1 ∂z ∂r uz (−h0 ) +

h0 
0
(5.59)
0
Here we used again that h0 and h1 do not depend on the axial coordinate z. Thus we
have eliminated the radial coordinate again and we have

Z−h0

12 
2 1
0
∂z p =
−h1 ∂r ur (−h0 ) − (u0r − r∂ϕ u0ϕ ) dr
h3 
0
0
Z−h0
Z−h0Zr Zρ
−
∂z Tz0 (s) ds dρ dr
∂ϕ u1ϕ dr −
0
0
0
0


Z−h0Zρ



h0 
0
0
∂z Tz (s) ds dρ
− −h1 ∂z ∂r uz (−h0 ) −

2 
0
0
=: R(z, ϕ).
Using the zero boundary conditions for the pressure we can integrate the right hand
side term and we get the following solution for the pressure in the order ε1 :
Z1 Zζ
Zz Zζ
p1 (z, ϕ, τ) =
R(s, ϕ) ds dζ − z
0
0
R(s, ϕ) ds dζ.
0
0
(5.60)
78
Chapter 5: Bearing Models
All the integrals on the right hand side exist and can be solved analytically. Hence
the solution for the pressure p1 can be computed explicitly and can be written in the
following form


 51

2 

p = R(κ ) −
z(z
−
1)
cos
ϕ

35h20


 3

′ ′

+ Rκ γ −
z(z
−
1)
sin
ϕ(47κ
cos
ϕ
+
14)

70h20



 3
′

z(z − 1) sin ϕ(10κ cos ϕ + 7)
+ Rκ 
2
35h0

 1
+ κ′  5 z(z − 1) 6h0 κ sin2 ϕ(2κ cos ϕ + 1)
2h0
1
′ 2



(z2 − z − 1)
(−κ cos2 ϕ + 2κ2 cos3 ϕ + 3κ sin2 ϕ − cos ϕ) 
+
K
3
′′
+ Rκ −
z(z − 1) cos ϕ
5h0
3
′′
z(z − 1)κ sin ϕ
+ Rγ −
5h0




81

2
2 

+ Rγ′2 −
z(z
−
1)κ
sin
ϕ

70h20



 1
2
2
′
z(z − 1)κ 253κ sin ϕ − 7 cos ϕ + 7κ cos ϕ 
+ Rγ 
140h20

 1
′
+ γ  5 z(z − 1) 6κ2 sin ϕh0 (κ cos2 ϕ − cos ϕ + 3κ sin2 ϕ)
2h0


(z2 − z − 1)
2
2
2
2
κ sin ϕ(3κ sin ϕ − 1 − 4κ cos ϕ + 5κ cos ϕ) 
+
K




1
2
2

+ R −
z(z − 1)κ −14 cos ϕ + 14κ cos ϕ + 65κ sin ϕ 
140h20

 1
+ − 5 z(z − 1)κ sin ϕ 2h0 (5 − 10κ cos ϕ + 5κ2 cos2 ϕ + 9κ2 sin2 ϕ)
4h0


(z2 − z − 1)
2
2
2
2
(3κ sin ϕ − 1 − 4κ cos ϕ + 5κ cos ϕ)  .
−
K
(5.61)
This rather inconveniently long expression can also be written in the form


4
X

z(z − 1) 
z(z − 1)G +
Hi  ,
p1 =
5

h0
i=0
(5.62)
where the terms G and Hi are expression in terms of hi , Vi , and p0 . They are given in
Appendix E.1.
79
5.1 Derivation of Reynolds’ Equation and Inertia Correction
Statement 5.2. The inertia correction for the short bearing pressure distribution has the
following structure
p1 = F0 + RF1 + γ′ F2 + Rγ′ F3 + R(γ′ )2 F4 + Rγ′′ F5
+Rκ′′ F6 + κ′ F7 + Rκ′ F8 + Rκ′ γ′ F9 + R(κ′ )2 F10 ,
(5.63)
where the F j are rational functions of h, κ cos ϕ and κ sin ϕ. The detailed formula is given in
Equation (5.61). An alternative formulation is given in Appendix E.1. We call p = p0 + εp1
the first order short bearing approximation. This approximation corresponds to point 4 in the
list of bearings models in Section 5.1.
An analysis of the performed scalings shows that the term RF1 leads to an ω2 dependence of the pressure
p=
p̄
ε
ε
Rp̃ = 2 R 2 2 ,
2
δ
δ ρR ω
R=ε
R2 ω
.
ν
The rescaled formula for the pressure is
ρR2 ω2 δ2 ν
p
ε2 R 2 ω
ρνωδ2
p
=
ε2
ρνω 0
≈
(p + εp1 )
Kε
ρνγ̇
ρνω 0 1 ρνωF0 + ω2 ρRcr F1 + ρνγ̇F2 +
p +
F3 + ωρRcr γ̇F4
=
Kε
K
K
+γ̇2 Rcr F5 + Rcr γ̈F6 + ρRcr κ̈F7 + ρνκ̇F8
ρνκ̇
+
F9 + ωρRcr κ̇F10 + ρRcr κ̇γ̇F11 + ρRcr κ̇2 F12 .
K
p̄ =
In Figure 5.6 the inertia corrected pressure distributions for different parameter values
are depicted. It can be seen in the lower line that for large values of W, i.e. small
values of K, the first order solution differs significantly from the zeroth order solution.
In these cases the validity of the short bearing approximation is in question and a
solution without this approximation is needed.
5.1.6 Solution Without Short Bearing Approximation
The short bearing approximation is only valid for W
R -ratios up to 1 (Childs, 1993). For
W
bearings with very high R -ratios the Sommerfeld or long bearing approximation is
valid, but as most modern bearings do not have such high ratios this approximation is
rarely used (San Andrés, 2006). In the case where neither approximation can be used,
the classical Reynolds equation has to be solved numerically. We will now derive the
80
Chapter 5: Bearing Models
Figure 5.6: First order pressure distributions p0 + εp1 for varying values of κ, κ̇ and W;
upper line: W = 0.25 (K = 1), lower line: W = 5.4 (K = 0.002); left: κ = 0.1, κ̇ = 0, middle:
κ = 0.9, κ̇ = 0, right: κ = 0.9, κ̇ = 0.
equations for zeroth and first order in ε corresponding to the points 1 and 3 in the
list in Section 5.1. In Section 5.4.1 we give a simple finite difference scheme for their
solution.
In equations (5.9) - (5.12) we expand p and u again into series in ε and collect terms of
the same order. Thus we get
ε0 :
∂r p0 = 0,
(5.64)
∂2r u0ϕ − δ2 ∂ϕ p0 = 0,
(5.65)
∂2r u0z − ∂z p0 = 0,
(5.66)
∂r u0r + ∂ϕ u0ϕ + ∂z u0z = 0,
(5.67)
5.1 Derivation of Reynolds’ Equation and Inertia Correction
ε1 :
81
∂r p1 = 0,
(5.68)
∂2r u1ϕ − δ2 (∂ϕ p1 − r∂ϕ p0 ) = −∂r u0ϕ
+ R ∂τ u0ϕ + u0r ∂r u0ϕ + u0ϕ ∂ϕ u0ϕ + u0z ∂z u0ϕ ,
(5.69)
∂2r u1z − ∂z p1 = −∂r u0z +
+ R ∂τ u0z + u0r ∂r u0z + u0ϕ ∂ϕ u0z + u0z ∂z u0z ,
(5.70)
∂r u1r + u0r − r∂ϕ u0ϕ + ∂ϕ u1ϕ + ∂z u1z = 0.
(5.71)
To simplify the notation in the equations for the first order corrections we define the
following terms which collect all lower order terms:
Tϕ0 = −δ2 r∂ϕ p0 − ∂r u0ϕ + R ∂τ u0ϕ + u0r ∂r u0ϕ + u0ϕ ∂ϕ u0ϕ + u0z ∂z u0ϕ ,
(5.72)
Tz0 = −∂r u0z + R ∂τ u0z + u0r ∂r u0z + u0ϕ ∂ϕ u0z + u0z ∂z u0z .
(5.73)
Analogously to Section 5.1.4 we use the fact that the pressure does not depend on the
radial coordinate to integrate velocity equations over r
r(r + h0 )
r
∂ϕ p0 − Vϕ0 ,
2
h0
r(r + h0 )
∂z p0 .
2
uϕ = δ2
(5.74)
uz =
(5.75)
The boundary conditions remain the same as before. From the continuity equation we
get
−∂r u0r = ∂ϕ u0ϕ + ∂z u0z ,
(5.76)
and integration then yields
−Vr0
Z−h0
=
∂ϕ u0ϕ + ∂z u0z dr
(5.77)
0
=
h0
1
1
1 2
δ ∂ϕ (h30 ∂ϕ p0 ) + ∂z (h30 ∂z p0 ) + Vϕ0 ∂ϕ h0 − ∂ϕ Vφ0 .
12
12
2
2
(5.78)
The last term vanishes since ∂ϕ Vφ0 = 0. We obtain the classical Reynolds equation, an
elliptic equation for the zeroth order pressure p0 :
1
δ2 ∂ϕ (h30 ∂ϕ p0 ) + ∂z (h30 ∂z p0 ) = −12(Vr0 + Vϕ0 ∂ϕ h0 ).
2
(5.79)
By means of the numerical methods described in Section 5.4.1 we can compute approximate solutions for this second order elliptical equation. For different parameter
82
Chapter 5: Bearing Models
Figure 5.7: Solution p0 of zeroth order Reynolds equation for varying values of κ, κ̇, and
W; upper line: W = 0.25 (K = 1), lower line: W=5.4 (K = 0.002); left: κ = 0.1, κ̇ = 0, middle:
κ = 0.9, κ̇ = 0, right: κ = 0.9, κ̇ = 0.
values of κ, κ̇, and bearing width W these are depicted exemplarily in Figure 5.7. The
figures show similar behavior for different values of the bearing width in contrast to
the behavior of the short bearing approximation in the previous section 5.1.5. Here
we see zones of elevated pressure in front of the point of smallest film-thickness and
reduced pressure behind this point. Note that the different scales are due the nondimensionalization. In the scaled case with units the pressure is higher for the longer
bearing, the scaling factor for the pressure being
ρνω
Kε
=
ηωW 2
.
c2r
For the first order equations we proceed analogously to obtain another elliptic equation
for p1 . The right hand side of this equation however is more complicated due to the
terms Tϕ0 and Tz0 . By integration of the equations for the velocities u1ϕ and u1z we get
u1ϕ
Zr Zs
r(r
+
h
)
0
Tϕ0 (t, ϕ, z, s) ds dρ
= δ2
∂ϕ p1 +
2
0 0


Z−h0Zs




r  1
0
0
Tϕ (t, ϕ, z, s) ds dρ ,
+ −Vϕ − h1 ∂r uϕ (−h0 ) +

h0 
0
0
(5.80)
(5.81)
83
5.1 Derivation of Reynolds’ Equation and Inertia Correction
Zr Zs
r(r + h0 )
1
Tz0 (t, ϕ, z, s) ds dρ
∂z p +
=
2
0 0


Z−h0Zs



r 
0
0
Tz (t, ϕ, z, s) ds dρ .
+ −h1 ∂r uz (−h0 ) +

h0 
u1z
0
(5.82)
(5.83)
0
Using the continuity equation and integrating over r we obtain
−Vr1
Z−h0
=
∂ϕ u1ϕ + ∂z u1z + u0r − r∂ϕ u0ϕ dr
(5.84)
0
=
1 2
1
δ ∂ϕ (h30 ∂ϕ p1 ) + ∂z (h30 ∂z p1 )
(5.85)
12
12


Zr Zs
Z−h0 Zr Zs



0
0
∂
 dr
T
(t,
ϕ,
z,
s)
ds
dρ
T
(t,
ϕ,
z,
s)
ds
dρ
+
∂
+
z
 ϕ
z
ϕ


0 0
0 0
0
 

Z−h0  
Z−h0Zs
 
 r 
 
+
∂ϕ  −Vϕ1 − h1 ∂r u0ϕ (−h0 ) +
Tϕ0 (t, ϕ, z, s) ds dρ  dr
 h0 
 
0

Z−h0 
 r
+
∂z 
 h0
0
0
0


Z−h0Zs




0
−h1 ∂r u0 (−h0 ) +
Tz (t, ϕ, z, s) ds dρ dr
z



0
0
Z−h0
u0r − r∂ϕ u0ϕ dr.
+
0
This is again an elliptic equation for p1 with a rather complicated right hand side
that depends only on the zeroth order solutions and the boundary conditions. The
velocities u0r , u0ϕ and u0z can also be computed in terms of p0 . After some computations
we obtain a right hand side that only depends on p0 , film thickness, and the boundary
conditions
δ2 ∂2ϕ p1 + δ2
3∂ϕ h0
h0
∂ϕ p1 + ∂2z p1 = Ψ(p0 , h0 , h1 , Vϕ0 , Vϕ1 ).
(5.86)
The the complete formula for Ψ is given in Appendix E.2. In the right hand side of
this equation there appear time derivatives of h0 and p0 . The time derivative of h0
can easily be computed from (5.21) while the time derivative of the pressure needs
the solution of an auxiliary partial differential equation. Differentiating the classical
Reynolds equation (5.79) with respect to time we obtain
 0 1 0

 (Vr + 2 Vϕ ∂ϕ h0 ) 
3∂
h
ϕ
0

∂ϕ ∂t p0 + ∂2z ∂t p0 = − 12∂t 
δ2 ∂2ϕ ∂t p0 + δ2
(5.87)

h0
h30
− δ2 ∂ϕ p0 (
3∂ϕ ∂t h0
h
+
∂t h0 ∂ϕ h0
h20
).
84
Chapter 5: Bearing Models
This equation involves the same differential operator as Equations 5.79 and 5.86 and a
modified right hand side with known lower order terms. To summarize the previous
considerations we formulate the following statement.
Statement 5.3. The pressure distribution in a circular hydrodynamic bearing with lubrication
film thickness h0 = 1 − κ cos ϕ is determined by three equations with the same differential
operator
3∂ϕ h0
∂ϕ (·) + ∂2z (·),
(5.88)
L(·) = δ2 ∂2ϕ (·) + δ2
h0
and varying right hand sides
ε0 :
L(p0 ) = f0 (κ, κ̇, γ̇),
(5.89)
ε1 :
L(p1 ) = Ψ(κ, κ̇, γ̇, κ̈, γ̈, p0 , ∇p0 , ∇2 p0 , ∇3 p0 , ∇2 (∂t p0 )),
(5.90)
where
L(∂t p0 ) = ∂t f0 − δ2 ∂ϕ p0 (
3∂ϕ ∂t h0
h
+
∂t h0 ∂ϕ h0
h20
).
(5.91)
Here the pi are scalar functions defined on Ω = (0, 1) × (0, 2π). The boundary conditions are
pi (ϕ, 0) = pi (ϕ, 1) = 0
pi (0, z) = pi (2π, z)
for ϕ ∈ (0, 2π),
for z ∈ (0, 1).
(5.92)
(5.93)
The parameters κ, γ, κ̇, γ̇, etc. are given by the current shaft position and velocity in polar
coordinates. The function f0 is given in Eq. (5.79), while a detailed expression for Ψ can be
found in the Appendix E.2. The pressure distribution p0 is called the zeroth order solution
and correspondes to point 1 in the list in Section 5.1. The pressure distribution p1 is called
the inertia correction and we call p = p0 + εp1 the first order solution to Reynolds’ equation,
corresponding to point 3 in the list.
The fact that we need the same differential operator in equations (5.89)-(5.91) simplifies
the numerical solution because we only need one discretization. Furthermore we have
to compute L−1 only once and can apply it subsequently to the three different right
hand sides.
In Figure 5.8 we exemplarily depict different inertia corrected solutions p0 + εp1 . Compared to the corrections in the short bearing case (cf. Fig. 5.6) the corrections in this case
are small. For both cases of W the geometry is very similar to the zeroth order case. In
Section 5.6 we thoroughly compare the different pressure distributions and also give
some numerical data that shows the convergence of the short bearing approximation
to the solution of the full Reynolds equation in zeroth and first order.
5.1 Derivation of Reynolds’ Equation and Inertia Correction
85
Figure 5.8: Inertia corrected solutions p0 + εp1 of Reynolds equation for varying values of
κ, κ̇, and W; upper line: W = 0.25, lower line: W = 5.4; left: κ = 0.1, κ̇ = 0, middle: κ = 0.9,
κ̇ = 0, right: κ = 0.9, κ̇ = 0.
5.1.7 Remark on Existence and Regularity
The existence of a weak solution of Reynolds’ equation (5.79) is guaranteed by applying
the Lax-Milgram Lemma to the continuous bilinear form
a : H × H → R,
Z
a(u, v) =
h30 (δ2 ∂ϕ u∂ϕv + ∂z u∂z v),
(5.94)
(5.95)
Ω
where u and v belong to the Hilbert space
1,2
H = Hper
∩ { f | TΓ2 f = 0}
(5.96)
with TΓ2 being the trace operator onto the Dirichlet boundary
Γ2 = {(ϕ, z) ∈ ∂Ω | z ∈ {0, 1}, ϕ ∈ [0, 2π]}.
(5.97)
The film thickness h0 is in C∞ (Ω̄) and we have 0 < h0 < 2 on the whole of Ω̄, assuring
continuity. Since on one part of the boundary the functions of H have prescribed value,
the Poincaré inequality
(5.98)
kukL2 ≤ Ck∂z ukL2 ≤ Ck∇ukL2
holds and the coercitivity βkuk2L2 ≤ a(u, u) of the bilinear form is assured.
The right hand sides of the Reynolds’ equations for the time derivative ∂t p0 and for the
first order correction p1 involve higher order derivatives of the zeroth order solution
86
Chapter 5: Bearing Models
p0 . Therefore some regularity considerations are required. The coefficient h0 is in
C∞ (Ω) which gives us all the interior regularity we need (Evans, 1998). Because we
can continuate it periodically, the Dirichlet boundary at z = 0 and z = 1 is in C∞ , so we
also have regularity up to the boundary.
5.2 Phenomenological Correction of Pressure Function
In Section 5.1 we derived four bearing theories from first principles by asymptotic
expansions of the Navier-Stokes equations. In Chapter 6 we see that simulations
which only use the zeroth order approximations for the pressure do not reproduce the
frequency of the oil whirl as observed in the experiments. The inertia corrections yield
a small effect and lead to a reduced whirl frequency. However, the observed effect is
still small compared to the experimental result. In the following we describe a simple
fifth bearing model that is used to examine the influence of the pressure function on
the whirl frequency. It is obtained by the following phenomenological considerations.
The pressure distribution of the short bearing solution is
W 2 ρν (γ̇ − ω )κ sin ϕ + κ̇′ cos ϕ
z̄
z̄
2
p̄0 = −6
.
−1
W W
(1 − κ cos ϕ)3
c2r
(5.99)
for z ∈ [0, W]. Using this pressure function in the calculation of the bearing forces
during numerical integration of the equation of motion of the rotor, we observe a
subharmonic instability with a frequency of approximately ω2 .
To test the influence of the term G = (γ̇ − ω2 ) in the expression for p0 on the frequency
of the self excited oscillation we modify the term G and use the term
ω
G̃ = γ̇ − + s(ω)
(5.100)
2
instead. The shift function s has a strong influence on the subharmonic frequency. A
linear term s = σω with σ ∈ [0, 12 ] leads to a linear frequency shift, while the correction
function s(ω) = σω2 introduces a quadratic dependence on ω into the pressure function.
By tuning the coefficient σ it is possible to control the shift. Using the modified pressure
distribution
W 2 ρν (γ̇ − ω + s(ω))κ sin ϕ + κ̇′ cos ϕ
z̄ z̄
2
−1
,
(5.101)
p̄0 = −6
2
W W
(1 − κ cos ϕ)3
cr
to calculate the bearing reaction forces, the experimentally observed nonlinear frequency shift can be observed also in the numerical simulations shown in Chapter 6.
Figure 5.9 depicts three pressure distributions for varying values of κ and κ̇ for illustration. In Section 5.6 we compare this pressure distribution and the resulting bearing
forces to the usual short-bearing approximation.
87
5.2 Phenomenological Correction of Pressure Function
Figure 5.9: Phenomenologically corrected short bearing pressure distributions p0 for varying values of κ and κ̇; left: κ = 0.1, κ̇ = 0, middle: κ = 0.9, κ̇ = 0, right: κ = 0.5,
κ̇ = 0.4.
The structure of the term G is explained in (Crandall, 1995). Recall the derivation
of Reynolds’ equation (cf. Sec. 5.1.4) where we obtained as principal flow pattern a
Couette flow with a linear velocity profile growing from 0 on the bearing casing to Rω
on the shaft. Integration of the continuity equation (5.26) over the radial coordinate
and using the expression (5.38) for u0z yields
−h
Z0 (ϕ)
−h
Z0 (ϕ)
−Vr0
∂ϕ u0ϕ (r, ϕ) dr
=
∂z u0z (r, ϕ, z) dr
+
0
0
−h
Z0 (ϕ)
u0ϕ (r, ϕ) dr + u(−h0 (ϕ), ϕ)∂ϕ h0 (ϕ) +
= ∂ϕ
h30
12
∂2z p0
0
=
−∂ϕ (h0 (ϕ)u0ϕ (ϕ))
+
Vϕ0 ∂ϕ h0 (ϕ)
+
h30
12
∂2z p0
= (Vϕ0 − u0ϕ (ϕ))∂ϕ h0 (ϕ) − h0 (ϕ)∂ϕ u0ϕ (ϕ) +
h30
12
∂2z p0 ,
where uϕ (ϕ) is the radially averaged circumferential lubricant velocity.
In Section 5.1.1 we have rescaled time such that the angular velocity is of order O(1).
In order to show the appearance of the ω/2-term, in the following paragraphs we go
back to dimensionalized time. Hence Vϕ0 = ω and Vr0 = −ωκ sin ϕ + κ̇ cos ϕ + κγ̇ sin ϕ.
Integration over z yields
0
p
=
=
=
6z(z − 1)
0
0
0
0
−Vr + (uϕ (ϕ) − Vϕ )∂ϕ h(ϕ) + h(ϕ)∂ϕ uϕ (ϕ)
h3
6z(z − 1)
ωκ sin ϕ − κ̇ cos ϕ − κγ̇ sin ϕ + (u0ϕ (ϕ) − ω)κ sin ϕ
h3
+(1 − κ cos ϕ)∂ϕ u0ϕ (ϕ)
6z(z − 1)
0 (ϕ) − γ̇)κ sin ϕ − κ̇ cos ϕ + (1 − κ cos ϕ)∂ u0 (ϕ) . (5.102)
(u
ϕ ϕ
ϕ
h3
88
Chapter 5: Bearing Models
In the short bearing approximation the circumferential velocity has a linear Couette
profile (cf. Eq. (5.37)) which rises from zero velocity on the casing to Vϕ0 = ω on the
shaft
r
uϕ =
ω,
(5.103)
h(φ)
and hence the radially averaged circumferential velocity is
1
u0ϕ (ϕ) = ω
2
(5.104)
and does not depend on ϕ. Therefore the last term in (5.102) vanishes and we get
−6z(z − 1)
ω
(γ̇ − )κ sin ϕ + κ̇ cos ϕ .
p =
2
h3
0
(5.105)
In (Muszynska, 1986; Muszynska, 1987; Muszynska, 1988) the fluid average circumferential velocity is identified as an important parameter for the onset and the frequency
of the oil whirl. Experimental results are given which show a decrease of the average
circumferential velocity for large eccentricities of the shaft. Hence
G̃ = (γ̇ −
ω
+ s(ω))
2
(5.106)
can be interpreted as a direct change of the lubricants average circumferential velocity.
In Chapter 6 we see that this manipulation of the term G = (γ′ − 21 ) leads to a change
in the subharmonic response frequency. This observation is also the motivation for
the derivation of the inertia correction. As shown in Section 5.1.5 the inclusion of first
order corrections leads to an ω2 -dependence of the pressure and hence the bearing
forces.
5.3 Alternative Bearing Models
Several other modifications of the bearing model were also tested for their ability to
reproduce the experimental results, especially the frequency shift of the oil whirl. As
the simulations in Chapter 6 show, it was not possible to reproduce the shift by any of
them. We describe them here for completeness.
5.3.1 The Finite Bearing
In the short bearing theory the axial pressure distribution is approximated by a
parabolic profile of order 2. Another possibility is to assume a separation of the
variables and approximate the axial pressure distribution by a parabolic function of
order m (Inagaki, 2002):
p(ϕ, ζ) = (1 − (2ζ − 1)m )v(ϕ).
(5.107)
89
5.3 Alternative Bearing Models
If we plug this into equation (5.79) and integrate both sides with respect to ζ from 0 to
1, we obtain
m 2
δ2
(5.108)
3h hϕ vϕ + h3 vϕϕ − 4mh3 v = −12 G sin ϕ + E cos ϕ ,
m+1
where we set E = κ′ and G = κ(γ′ − 12 ). It is possible to simplify this equation by setting
w = h2 v. Inserting this into equation (5.108) one gets:
hwϕϕ − hϕ wϕ − (2hϕϕ + ah)w = −12b G sin ϕ + E cos ϕ ,
(m+1)
(5.109)
(m+1)
where a = 4 δ2 and b = mδ2 . Expanding w into a Fourier series and neglecting terms
of order 3 and higher one gets an approximation of the solution:
w=
2
X
Kn cos(nϕ) +
n=0
2
X
Cn sin(nϕ),
(5.110)
n=1
where
K0 =
K1 =
K2 =
C1 =
C2 =
−24κbE
,
− 4a − 4 − 6κ2
2
−48bE
= K0 ,
2
2
κ
3aκ − 4a − 4 − 6κ
(a − 2)
−24κbE(a − 2)
= K0
,
2
2
(a + 4)
(3aκ − 4a − 4 − 6κ )(a + 4)
−48bG
,
2
aκ − 4a − 2κ2 − 4
−24κbG(a − 2)
κ (a − 2)
= C1
.
2
2
(aκ − 4a − 2κ − 4)(a + 4) 2 (a + 4)
3aκ2
Having this, the approximate pressure function is
p = (1 − (2ζ − 1)m )
w
.
h2
(5.111)
Figure 5.10 show the pressure distributions for varying values of κ, κ̇, and W. The
geometry is most notably different for small eccentricities (κ = 0.1). In Section 5.6
we compare the resulting forces. The finite bearing approximation proposed here
underestimates the pressure in most cases, leading to too small bearing forces. This
is due to the large error made by truncating the Fourier series. This approximation is
therefore not considered in the numerical simulations in Chapter 6.
5.3.2 Reynolds’ Boundary Conditions
In the calculation of the bearing forces in Section 5.5 the so-called Gümbel boundary
conditions are used. They provide a simple cavitation model by setting the pressure
to zero where negative pressure occurs. In our search for an oilfilm model that reproduces the frequency shift the use of these simplified boundary conditions appeared
90
Chapter 5: Bearing Models
Figure 5.10: Finite bearing pressure distributions p0 for varying values of κ, κ̇, and W;
upper line: W = 0.25, lower line: W = 5.4; left: κ = 0.1, κ̇ = 0, middle: κ = 0.9, κ̇ = 0, right:
κ = 0.5, κ̇ = 0.4.
to be another possible explanation. A more realistic cavitation model is given by
Reynolds’ boundary conditions (Yamamoto & Ishida, 2001). So we reformulate the
nondimensional Reynolds’ equation (5.79)
1
δ2 ∂ϕ h3 ∂ϕ p̄ + ∂ζ h3 ∂ζ p̄ = −12 (γ′ − )κ sin ϕ + κ′ cos ϕ
2
(5.112)
using Reynolds’ boundary conditions
p(0, z) = p(2π, ζ)
∀ζ ∈ [0, 1],
p(ϕ, 0) = p(ϕ, 1) = 0
p ≥ 0,
∀ϕ ∈ [0, 2π],
(5.113)
(5.114)
(5.115)
which leads to a free boundary problem for the pressure. The third boundary condition
is responsible for the free boundary problem structure because it does not fix the point
of oilfilm rupture.
This kind of problem can be solved numerically by a projection method like projective
successive overrelaxation (PSOR) (Deuflhard & Hohmann, 1993). Simply spoken, in
this method one computes the solution for the unconstrained problem and then projects
the solution on the subspace containing the feasible functions, that is the subspace of
positive functions in our case. In Section 5.4.1.3 we describe the method in detail
and give the appropriate choice of iteration matrices which assure convergence. In
Figure 5.11 the pressure distribution is again shown for varying parameter values.
91
5.4 Numerical Schemes for Reynolds’ Equation
Figure 5.11: Pressure distributions p0 computed with Reynolds’ boundary conditions for
varying values of κ, κ̇, and W; upper line: W = 0.25, lower line: W = 5.4; left: κ = 0.1,
κ̇ = 0, middle: κ = 0.9, κ̇ = 0, right: κ = 0.5, κ̇ = 0.4.
5.4 Numerical Schemes for Reynolds’ Equation
In the previous section a variety of different bearing models were given. For some of
them it is possible to give analytical solutions, e.g. for the short bearing approximation
(5.47). Others involve the solution of partial differential equations, i.e. Reynolds’ equation with different right hand sides for the zeroth and first order approximation (5.79)
and (5.86) and the iterative solution of this equation with other boundary conditions
by the PSOR method in section 5.3.2. In this work three different methods have been
used to solve these PDEs which we will describe in the following. The differential
operator in all three cases is given by
(5.116)
L̃(·) = δ2 ∂ϕ h30 (ϕ)∂ϕ (·) + h30 (ϕ)∂2z (·).
The operator L̃ above is given in divergence form and therefore useful in the weak
formulation of the problem which is used later for the finite element discretization.
The software package deal.II (Bangerth et al., 2007; Bangerth et al., 2008) is used for
the discretization and the setup of the linear equations.
The film thickness h0 depends only on the circumferential angle ϕ. Therefore we can
also divide L̃ by h30 and by δ2 to obtain the operator L which allows for an easier
discretization by finite differences
L(·) = ∂2ϕ (·) +
3∂ϕ h0
1 2
∂ϕ (·).
∂
(·)
+
z
h0
δ2
(5.117)
92
Chapter 5: Bearing Models
5.4.1 Finite Differences for Reynolds’ Equation
5.4.1.1
5-point Laplacian and Second Order Approximation
A very simple discretization of L can be calculated by applying the five-point discretization of the Laplace operator and central differences for the first order derivative.
We obtain
3κ sin ϕi 1 p(ϕi+1 , ζ j ) − p(ϕi−1 , ζ j )
1 − κ cos ϕi 2∆ϕ
1 + δ2 2 p(ϕi+1 , ζ j ) + p(ϕi−1 , ζ j ) − 2p(ϕi , ζ j )
∆ϕ
1 +
p(ϕ
,
ζ
)
+
p(ϕ
,
ζ
)
−
2p(ϕ
,
ζ
)
i
j+1
i
j−1
i
j
∆2ζ
−12 (γ′ − 12 )κ sin ϕi + κ′ cos ϕi
,
=
(1 − κ cos ϕi )3
δ2
(5.118)
where ϕi = i∆ϕ , i = 1, . . . , N and ϕζ = j∆ζ , j = 1, . . . , M.
We set
pi, j = p(ϕi , ζ j ),
gi =
3κ sin ϕi
,
1 − κ cos ϕi
′ − 1 )κ sin ϕ + κ′ cos ϕ
−12
(γ
i
i
1
2
fi = 2
,
δ
(1 − κ cos ϕi )3
and
d=
∆2ϕ
∆2ζ δ2
.
From this we get
1
1
(pi+1, j + pi−1, j + d(pi, j+1 + pi, j−1 ) − 2(1 + d)pi, j ) +
gi (pi+1, j − pi−1, j ) = fi .
2
2∆ϕ
∆ϕ
The periodic boundary condition p(0, z) = p(2π, z) leads to the following special relations:
∂ϕ p1, j ≈
∂ϕ pN, j ≈
1
(p2, j − pN, j ),
2∆ϕ
1
(p1, j − pN−1, j ).
2∆ϕ
So the discretized equation can be written in matrix form as
Cp = Ap + ∆ϕ Bp = F
(5.119)
93
5.4 Numerical Schemes for Reynolds’ Equation
with

 L1


 dI
A = 




 L2


L2
1 
B = 
2 





 −2 − 2d 1




..
..


.
.
1
 , where L1 = 


..
..

.
. dI 



1
dI L1
dI
L1
..
.
..
.
L2


 g1







 , where L2 = 






g2
..
.
gN−1
1
..
.
..
.
1
1
−2 − 2d





 ,





 
1
−1 
  0

 

..
 

.
  −1 0
 ,
 · 

.
.
 
.
.
.
. 1 
 

 
1
−1 0
and ( f1 , . . . , fN ) M-times repeated to form the right hand side
F = ∆2ϕ f1 , . . . , fN , . . . , f 1, . . . , fN
T
.
Since the 5-point-Laplacian and the central differences both are second order approximations this discretization also is of second order.
5.4.1.2
9-point-Laplacian and 4th Order Approximation
The 9-point-Laplacian on a uniform grid with step-size h
(9)
∆h uh =
1 4u
+
4u
+
u
−
20u
i±1,
j
i,
j±1
i±1,
j±1
i,
j
6h2
(5.120)
yields a fourth order approximation for the Poisson equation after a modification of
the right hand side (Rannacher, 2000) using the well known 5-point Laplacian
1
(5)
f˜ = f + h2 ∆h f.
12
(5.121)
To apply this to a discretization of the differential operator L of Reynolds’ equation
(5.117) with different step-sizes h1 and h2 in each direction we modify the discretization.
We set
L1 = θpi, j + αpi±1, j + βpi, j±1 + γpi±1, j±1
(5.122)
with
γ =
α =
h21 + δ2 h22
,
(5.123)
,
(5.124)
12δ2 h21 h22
1 − 2γh21
h21
1 − 2γh22 δ2
,
h22 δ2
θ = −2(α + β + 2γ).
β =
(5.125)
(5.126)
94
Chapter 5: Bearing Models
Furthermore we set
L2 =
where gi =
3∂ϕ h0 (ϕi )
h0 (ϕi ) ,
gi pi−2, j − 8pi−1, j + 8pi+1, j − pi+2, j ,
12h1
(5.127)
1 −4pi, j + pi±1, j + pi, j±1
12
(5.128)
and
L3 =
to come to the following result.
Proposition 5.4. Let p be the solution of
L(p) = F
where L is given by (5.117) and F =
f
.
δ2 h30
(5.129)
We then have
L1 p + (I − L3 )L2 p − (I − L3 )F = O(h4 ),
(5.130)
i.e. the consistency order is τ = 4.
Proof. Taylor expansion and substitution of Equation (5.129) show that
1
1 2
∂ p + (h21 ∂2ϕ + h22 ∂2z )(F − g∂ϕ p) + O(4),
12
δ2 z
L2 p = g∂ϕ p + O(4).
L1 p = ∂2ϕ p +
(5.131)
(5.132)
Furthermore, for any π ∈ C4 we have
L3 π =
1 2 2
(h ∂ + h22 ∂2z )π + O(4).
12 1 ϕ
Hence the result follows.
(5.133)
The computational effort can be reduced by using the symmetry in the z-direction. We
take Dirichlet conditions at z = 0 and symmetry conditions at z = 21 . In the ϕ-direction
we take periodic boundary conditions. The vector of unknowns is
p = (p0,1 , p1,1 , . . . , pN−1,1 , p0,2 , . . . , pN−1,M )T
(5.134)
where the first index i of pi, j stands corresponds to the grid in ϕ-direction and the second
index to the grid in z-direction. Using the above scheme we obtain the following system
matrices



 A B


.

 B A
..


L1 = 
(5.135)
 ,
.. ..

.
. B 



2B A
95
5.4 Numerical Schemes for Reynolds’ Equation
where

 θ α


 α θ
A = 
..

.


α

α 


..

.
 ,

..
. α 

α θ
and

 β γ


 γ β
B = 
..

.


γ

γ 


..

.
 ,

..
. γ 

γ β
and
L2 =
1
IdM ⊗ C,
12h1
(5.136)
where

 g1



C = 



g2
..
.
gN−1
 

8 −1
1 −8 
  0
 

  −8 0
8 −1
1 
 
 ,
 · 

.. .. .. ..
 

.
.
.
.
 

 

8 −1
1
8
0
and

 D IdN


1  IdN D
(I − L3 ) =

..
12 
.


..
.
..
.
2IdN


IdN 
 8 1






 , where D =  1 8

..


.
IdN 



D
1

1 


..

.
 .

..
. 1 

1 8
As already mentioned in the derivation (cf. Section 5.1.6), the solution of Reynolds’
equation and its inertia correction requires the solution of three PDEs with identical
differential operator and varying right hand side. Therefore it is advantageous to use a
direct solution method like LU-decomposition for the corresponding linear equation,
since the decomposition of the system matrix can be stored and used three times
consecutively. In our case we use the LU-decomposition for sparse matrices which
is implemented in the software package UMFPACK (Davis, 2004; Davis, 2007) and
which is included in MATLAB. Comparison with iterative methods like bicgstab
or gmres showed a slight advantage of the sparse LU-decomposition in the required
computational time.
In the solution process the matrices for the discretized operator L1 , L2 , and L3 are set
up and the LU-decomposition is computed. Then the solution p0 of the zeroth order
of Reynolds’ equation is computed with the right hand side
−12 (γ′ − 21 )κ sin ϕi + κ′ cos ϕ
.
(5.137)
f0 =
δ2 h30
The right hand side of the equation for ∂t p0 involves spatial derivatives of p0 (cf.
(5.87)) and the one for p1 involves higher order derivatives of p0 and of ∂t p0 (cf. (E.9)).
96
Chapter 5: Bearing Models
Due to the periodicity of the solution we can compute higher order approximations
of the ϕ-derivatives also at the boundary by prolongation. In order to compute the
z-derivatives, we use the differential equation to calculate second order derivatives,
while the first and third derivative are computed by finite differences.
The solution of the linear equations give an approximation to the pressure distribution
p = p0 + εp1 .
(5.138)
The resulting forces on the rotating shaft are computed by numerical integration of the
pressure distribution.
5.4.1.3
Projection Method for Constrained Problem
In Section 5.3.2 the positivity contstraints to Reynolds’ equation known as Reynolds’
boundary conditions are introduced as an alternative bearing model. The projective
successive overrelaxation method (PSOR) can be used to solve this free boundary
problem (Deuflhard & Hohmann, 1993). The PSOR algorithm for the constrained
problem works as follows. The matrix C = A + ∆ϕ B from Eq. (5.119) is decomposed
into a diagonal, a lower and an upper triangular matrix C = D+L+R. We set Q = D+ρL,
S = (1 − ρ)D − ρR and F̃ = ρF. In every iteration step k we solve:
Qp̃k = Spk−1 + F̃,
pk = max(p̃k , 0),
(5.139)
(5.140)
where the first step is the usual successive overrelaxation step and the second step
is the projection on the subspace of positive functions. For a choice of ρ ∈ [1, 2] this
procedure converges to the solution of the free boundary problem because the SOR
converges (Deuflhard & Hohmann, 1993) and the projection does not increase the
spectral radius of the iteration mapping. Due to the fact that we have to solve the free
boundary problem in every time step the numerical integration becomes very slow.
The simulation results in Chapter 6 show no significant effect on the frequency of the
self-excited oscillation.
5.4.2 Finite Element Approximation
The pressure distribution inside the bearings shows large variations in the circumferential ϕ-direction in a small interval around ϕ = 0 and almost no variation elsewhere.
In order to resolve this with finite differences a very small stepsize has to be chosen,
which increases the computational cost. To overcome this problem a finite element
approach with local refinement is used alternatively to improve the speed of the calculations. The theory of finite elements for elliptic equations is well developed and can
be found in textbooks like (Braess, 2007; Ciarlet, 1978; Brenner & Scott, 1994; Strang &
97
5.4 Numerical Schemes for Reynolds’ Equation
Fix, 1973) or in (Rannacher, 2000). We use the finite element software library deal.II
(Bangerth et al., 2007; Bangerth et al., 2008).
The weak formulations of the three Reynolds equations for zeroth order pressure p0
(derived from Eq. (5.79)), its time derivative ∂t p0 (from (5.87)), and first order pressure
p1 (cf. (5.85)) are
"
"
1 3˜ 0˜
−
=−
f ψ dϕ dz,
(5.141)
h ∇p ∇ψ dϕ dz
12 0
Ω
Ω
"
"
"
1 3˜ 0˜
˜ dϕ dz,
˜ 0 ∇ψ
−
=−
h ∇∂t p ∇ψ dϕ dz
∂t f ψ dϕ dz +
(5.142)
∂t h30 ∇p
12 0
Ω
Ω
Ω
"
"
1 3˜ 1˜
h0 ∇p ∇ψ dϕ dz
=
−W0 ψ + Wϕ ∂ϕ ψ + Wz ∂z ψ dϕ dz,
(5.143)
−
12
Ω
Ω
where
W0 = h1 ∂r u0r (−h0 ) + ∂ϕ h0 Vϕ1 + h1 ∂r uϕ (−h0 ) − u0r + r∂ϕ uv ph0 ,


Z−h0Zr
Z−h0Zr Zs




1  1
0
0
Tϕ (ρ, ϕ, z) dρ dr ,
Tϕ (ρ, ϕ, z) dρ ds dr + h0 −Vϕ − h1 ∂r uϕ (−h0 ) +
Wϕ =

2 
0
0
0
Z−h0Zr Zs
Wz =
0
0
0
0
0


Z−h0Zr



1 
Tz0 (ρ, ϕ, z) dρ ds dr + h0 −h1 ∂r uz (−h0 ) +
Tz0 (ρ, ϕ, z) dρ dr .

2 
0
0
˜ denotes the operator
Here we use the notations from Section 5.1.6 and additionally ∇
1
T
˜ = ( ∂ϕ , ∂z ) . ψ is a test function.
∇
δ
In deal.II we use second order quadratic elements and Gaussian quadrature with 3
quadrature points in each direction. The local refinement is done following a-posteriori
error estimation with the built-in Kelly-estimator which measures the local error of the
solution of the Poisson problem by integrating over the jump of the gradients along
the faces of each cell (Bangerth et al., 2008). The treatment of the periodic boundary
conditions is not trivial since it must be kept track of the hanging nodes at different
levels of refinement along periodic boundary. This is done following the suggestions
of (Bangerth, 2002) by recursively checking the refinement of adjacent elements along
the periodic boundary and interpolating hanging nodes in case of different refinement
levels.
We want to use the same discretization of the differential operator for the three different
equations. Therefore, we first compute the solution of the zeroth order Equation (5.141)
on a relatively coarse grid that has been 3 times globally refined and which has 64
nodes. Then we subsequently refine 9 times locally by applying the a-posteriori error
estimation and refining the 30% of the cells with highest local error, while coarsening
the lowest 3%. The obtained system matrix is then also used to solve Equations
98
Chapter 5: Bearing Models
cycles
DOF
3
4
5
6
7
8
9
10
11
12
1126
2253
4262
8021
15674
29621
55805
105926
200516
378415
I1
κ = 0.7
-0.181363
-0.183821
-0.184569
-0.184602
-0.184642
-0.184649
-0.184641
-0.184637
-0.184635
-0.184634
I2
DOF
0.0735314
0.0737043
0.0736962
0.0737123
0.0737650
0.0737705
0.0737767
0.0737791
0.0737798
0.0737802
1093
2225
4639
8831
16042
31122
58818
115497
216140
407322
I1
κ = 0.98
-2.05621
-2.19322
-2.21716
-2.22607
-2.22823
-2.22937
-2.22937
-2.22968
-2.22975
-2.22978
I2
0.348900
0.348822
0.353029
0.354908
0.355328
0.355737
0.355813
0.355882
0.355925
0.355933
Table 5.1: Values of the bearing integrals converge with increasing number of local
refinement cycles for κ = 0.7 and slower for κ = 0.98. The slow convergence is due to
cut-off of the negative part of the solution.
(5.142) and (5.143). In Chapter 6 the numerical solution of the equation of motion of
a rotating, elastic shaft supported by hydrodynamic bearings requires the solution of
the three equations (5.141)-(5.143) in every time-step. The system matrix depends on
the thickness h0 of the lubricant film and has to be updated in every timestep. If we
solve the three systems in every time step, the direct LU-decomposition for sparse
matrices from the UMFPACK library is faster than the usual iterative solvers, because
it has to be done only once in the first linear equation and can be reused to solve the
second and the third. The process can still be accelerated by using the fact that during
several time-steps of the ode-solver the system matrix does not change drastically.
If we use the LU-decomposition as a preconditioner for e.g. the cg-solver for some
time steps the convergence of the iterative solver is very good, since we use a matrix
close to the inverse as preconditioner. We recompute the LU-decomposition as soon
as the iterative solver makes more than a prescribed number of iteration steps (in the
numerical examples shown later this is 10). The number of necessary refinement cylces
is determined by examining the convergence of the bearing integrals
Z
Z
p cos ϕ dr dϕ,
I1 =
Ω
p sin ϕ dr dϕ,
I2 =
(5.144)
Ω
for different values of κ as shown in Table 5.1. There it can be seen, that 10 refinements
lead to a reasonably small error (≈ 1e − 4) in the intgrals. The resulting mesh and a
series of solutions p0 , p1 , and ∂t p0 is shown in Figure 5.12. The package VisIt is used
here for the visualization of mesh and surfaces.
The process of local refinement takes up a lot of computation time, therefore we restrain
it to those time-steps when a new Jacobian matrix for the implicit solver is computed.
5.4 Numerical Schemes for Reynolds’ Equation
99
Figure 5.12: Solutions and mesh computed with deal.II after 11 refinement cylces; upper
left: p0 , upper right: p1 , lower left: ∂t p0 , lower right: mesh. The domain shown is
[−π, π] × [0, 21 ].
The zone of large variation in the pressure distribution is located most of the time
around the line ϕ = 0 (cf. Figure 5.12), so there is always a fine grid around this line,
even if the pressure distribution changes slightly. Since a new Jacobian is computed
each time the error gets large and convergence is slow, this is a good heuristic for the
timing of the recomputation of the grid.
100
Chapter 5: Bearing Models
5.5 Calculation of Bearing Forces
Having calculated the pressure distribution in the journal bearing we can now compute the force acting on the journal center. Negative pressures lead to cavitation in
the oil film. A very simple but popular cavitation model are the so called Gümbel
boundary conditions which are often used in the rotordynamics literature (Childs,
1993; Yamamoto & Ishida, 2001; San Andrés, 2006). When calculating the force by integration of the pressure over the journal surface only the domain where the pressure
is positive is considered. The above mentioned Reynolds’ boundary conditions are a
more complicated and computationally more expensive model.
For all the lubrication models proposed in this chapter the resulting bearing forces are
computed by integrating the positive part of the pressure over the bearing surface.
Fbear

 FN
= 
FT
 Z
Z2π ZW 
 cos ϕ

 =
p̄+ 
p̄+ ν dσ = −

sin ϕ
(5.145)
0
0
Ω


 R dϕ dz.

The analytical formulation of the pressure distribuation in the short bearing approximation also allows for the analytical computation of these integrals.
5.5.1 Analytical Solution for the Lowest Order in Short Bearing Approximation
The surface integral for the lowest order approximation can be solved analytically using
Sommerfeld’s variable transformation (Lang & Steinhilper, 1978). The (dimensional)
normal and tangential forces are computed from the (dimensional) pressure p̄ by
ZW Z2π
FN = −
0
(5.146)
p̄(ϕ, z̄/W) + sin ϕR dϕ dz̄.
(5.147)
0
ZW Z2π
FT = −
0
p̄(ϕ, z̄/W) + cos ϕR dϕ dz̄,
0
A variable transformation z = z̄/W leads to an integral over the nondimensional
pressure.
FN
WRρνω
= −
Kε
Z1 Z2π
(p0 )+ cos ϕ dϕ dz,
0
FT = −
WRρνω
Kε
(5.148)
0
Z1 Z2π
(p0 )+ sin ϕ dϕ dz.
0
0
(5.149)
101
5.5 Calculation of Bearing Forces
The pressure distribution obtained from the short bearing approximation to Reynolds’
equation (cf. Eq. (5.47) in Sec. 5.1.4) can be written in a complex notation, which shows
the positive pressure region in a more distinctive way
p0 = −6z(z − 1)
=
=
(γ′ − 12 )κ sin ϕ + κ′ cos ϕ
(1 − κ cos ϕ)3
−6z(z − 1)
1
′
′
ℜ κ − i γ − κ eiϕ
2
(1 − κ cos ϕ)3
s
2 −6z(z − 1)
1
i(ϕ+ψ)
′2 + γ′ −
κ
κ
ℜ
e
.
2
(1 − κ cos ϕ)3
(5.150)
(5.151)
(5.152)
The phase angle ψ is defined by
1
ψ = arg κ̇ − i γ̇ − κ .
2
(5.153)
Hence p0 (ϕ, z) > 0 if the real part of the complex number ei(ϕ+ψ) is positive, i.e.
Re ei(ϕ+ψ) > 0
⇔
π π
.
ϕ+ψ∈ − ,
2 2
(5.154)
So the integration in Eqs. (5.148) and (5.149) runs from
ϕ1 = −
π
π
− ψ to ϕ2 = − ψ,
2
2
(5.155)
as the first two terms in (5.152) are all positive since 0 < κ < 1.
One can now integrate the equations (5.148) and (5.149) using Sommerfeld’s variable
transformation
1 − κ2
.
(5.156)
1 − κ cos ϕ =
1 − κ cos θ
However, the calculations are lengthy and can be found in Appendix F. The integration
boundaries transform as follows:
√
θ1 = arctan(κ − cos ϕ1 , − 1 − κ2 sin ϕ1 ),
√
θ2 = arctan(κ − cos ϕ2 , − 1 − κ2 sin ϕ2 ).
(5.157)
(5.158)
With these boundaries the integrals result in
FN
FT
W 3 Rρνω 1
′
′
= −
A1 κ − A2 (γ − )κ ,
2
c2r
3
W Rρνω
1
′
′
A
(γ
−
,
= −
)κ
−
A
κ
3
2
2
c2r
(5.159)
(5.160)
(5.161)
102
Chapter 5: Bearing Models
where
A1 =
A2 =
A3 =
(κ2 + 12 )(θ2 − θ1 ) − 2κ(sin θ2 − sin θ1 ) + 14 (sin 2θ2 − sin 2θ1 )
5
(1 − κ2 ) 2
−κ(cos θ2 − cos θ1 ) + 41 (cos 2θ2 − cos 2θ1 )
(1 − κ2 )2
(2(θ2 − θ1 ) − sin 2θ2 + sin 2θ1 )
3
4(1 − κ2 ) 2
,
,
.
(5.162)
(5.163)
(5.164)
This nice analytical formulation allows for a much faster numerical solution of the
equation of motion, since the numerical evaluation of the surface integrals is computationally much more expensive.
5.5.2 Calculation of Bearing Forces for Inertia Corrected Pressure
The (dimensional) normal and tangential forces are computed from the (dimensional)
pressure p̄ by
ZW Z2π
FN = −
0 0
ZW Z2π
FT = −
0
p̄(ϕ, z̄/W) + cos ϕR dϕ dz̄,
(5.165)
p̄(ϕ, z̄/W) + sin ϕR dϕ dz̄.
(5.166)
0
The same variable transformation for z̄ like in Section 5.5.1 leads to an integral over
the nondimensional pressure
ρνωδ2
FN = − 2
ε
Z1 Z2π
p(ϕ, z) + cos ϕR dϕ W dz
(5.167)
0
0
WRρνωδ2
=−
ε2
Z1 Z2π 

−6z(z − 1) Ψ(ϕ)

h30
0
0

+
4
X

1  2
Hi (ϕ) cos ϕ dϕ dz
+ε 5 z (z − 1)2 G(ϕ) + z(z − 1)
h0
i=0
+

Z2π 
4
X
 Ψ(ϕ)

 1
WRρνω
1
1


=−
Hi (ϕ) cos ϕ dϕ,
 3 + ε 5  G(ϕ) −
Kε
6
h
h 30
0
0
FT = −
ρνωδ2
ε2
Z1 Z2π
0
0
0
p(ϕ, z) + sin ϕR dϕ W dz
Z2π 
WRρνω
1
 Ψ(ϕ)

+ε 5
=−

3
Kε
h0
h0
0
(5.168)
i=0
(5.169)
+

4
X

 1
 G(ϕ) − 1
Hi (ϕ) sin ϕ dϕ.

30
6
i=0
(5.170)
103
5.5 Calculation of Bearing Forces
The integration along the z-axis can easily be computed while the solution of the
ϕ-integral is rather complicated and therefore done numerically.
5.5.3 Order Improvement for Numerical Integration
The computation of the bearing reaction forces (5.148) and (5.149) can be done analytically in the simple case of the short bearing approximation (cf. Sec. 5.5.1). Already in
the case of short bearing approximation with inertia correction the analytical solution
of the involved integrals becomes too complicated. When the full Reynolds equation
(with or without inertia correction) is solved numerically with the methods in Sections
5.4.1.2 and 5.4.2, the computation of the bearing forces also requires the numerical
solution of the integrals
Z1 Z2π
(p)+ cos ϕ dϕ dz,
FN
WRρνω
= −
Kε
FT
Z1 Z2π
WRρνω
(p)+ sin ϕ dϕ dz,
= −
Kε
0
0
(5.171)
0
(5.172)
0
which are not linear in the pressure distribution because of the truncation of the
negative part. Furthermore we want to use the given grid points for the quadrature
having computed the values there already.
As can be seen in Sec. 5.1 the solution of Reynolds equation show periodic behavior
with sign-change in the circumferential ϕ-direction and parabolic shape in the axial
z-direction without sign-changes for most values of z on the grid. We first compute
all the integrals with respect to z and then use the resulting sequence on the ϕ-grid to
approximate the integral in the ϕ-direction. In the case of a uniform grid which we have
for a finite differences discretization (cf. Sec. 5.4.1) we can compute the integrals using
Simpson’s rule and the error is of order 4 in the grid sizes h1 or h2 (Stoer & Bulirsch,
2002) in those cases where no sign-changes occur. The truncation of the negative part
leads to a problem in the computation of the integrals essentially because of the loss
of differentiability. In the following we describe a work-around for this problem.
For sufficiently differentiable functions f the accumulated trapezoidal rule
X
1
h( f (a) + f (b)) + h
f (a + ih)
2
N−1
Th =
(5.173)
i=1
can be corrected to yield an accuracy of order
h4
where h = (b − a)/N) and ξ ∈ (a, b)
f ′ (b) − f ′ (a)
b − a (4)
+ h4
f (ξ).
(5.174)
12
720
For periodic functions the derivatives at the end-points are equal and the trapezoidal
rule is always of order 4 (Stoer & Bulirsch, 2002).
I( f ) = Th − h2
104
Chapter 5: Bearing Models
Since the zeros of the pressure distribution are not necessarily located at grid-points,
the application of the trapezoidal rule to the truncated pressure (p)+ has no longer
order 4. Figure 5.13 shows that the truncation entails the inclusion of the triangular
1
0
0
1
fi , f’i
xi
x+
0
1
1
0
0
01
1
x*
1
0
0
1
xi+1
1
0
0
1
fi+1 , f’i+1
Figure 5.13: Correction of the quadrature by integration of an interpolant
area below the dashed line into the trapezoidal sum, while the exact integral only
includes the area below the black curve up to its zero at x∗ . This can be overcome by
approximating the positive part of the integral on the cell in which the sign change
occurs with the help of an interpolant, which is shown in red. This leads again to an
error of order h41 . In detail we do the following to correct the approximation of the
integral. We therefore determine the intervals where sign changes occur and do the
following steps for all of them:
• compute derivatives fi′ and fi′ + 1 with central differences,
• compute the coefficients of the interpolating cubic Hermite polynomial H3 which
are given by
a0 = fi ,
a1 = fi′ ,
a2 =
′
−3 fi + 3 fi+1 2 fi′ + fi+1
−
,
h
h2
a3 =
′
fi′ + fi+1
h2
+
2 fi − 2 fi+1
,
h3
(5.175)
• find the zero x+ of polynomial H3 that lies in [xi , xi+1 ],
• compute the integrals of the interpolant from the boundaries of the interval to
105
5.5 Calculation of Bearing Forces
the zero
Z
x+
a3 + a2 + a1 +
x + )x + )x + a0 )x+ ,
4
3
2
xi
Z xi+1
′
′
f − fi+1
fi + fi+1
H3 (x) dx = h(
I2 =
+h i
) − I1 ,
2
12
x+
I1 =
H3 (x) dx = (((
• correct the trapezoidal sum Th over the whole interval [0, 2π] by

h2 f ′
h fi


if
fi > 0,
 − 2 − 12i + I1
Ih = Th + 
2 ′

 − h fi+1 + h fi+1 + I if f > 0.
2
2
12
(5.176)
i+1
Proposition 5.5. The resulting approximation to the integral is of order 4, i.e. |I − Ih | = O(h4 ).
Proof. Outside the intervals where sign changes occur the approximation order of the
corrected trapezoidal sum is 4 (cf. Eq. (5.174)). We need to give an estimate for the
error of the above procedure inside these intervals.
′ be the function values and derivatives at the boundary of the
Let fi , fi′ , fi+1 and fi+1
interval (xi , xi+1 ) of length h. The interpolation error of the Hermite interpolation is
given by (Stoer & Bulirsch, 2002)
1 (iv) k f k∞ (x − xi )2 (x − xi+1 )2 .
(5.177)
4!
With this the error of the integrals of the truncated functions can be estimated.
| f (x) − H3 (x)| ≤
Zxi+1
Zxi+1
Zxi+1
+
+
+
+
|( f − H3 )| dx ≤ Ck f (iv) k∞ h5 . (5.178)
|( f − H3 )| dx ≤
|I − Ih | = | ( f − H3 ) dx| ≤
xi
xi
xi
Hence the integral of the interpolant up to its zero is a sufficient approximation to the
truncated function f inside the intervals where sign changes occur.
When calculating the pressure distribution on a uniform grid by finite difference we use
this method to correct the integration. Numerical results show that the computation
time is not drastically increased while convergence is faster. In Figure 5.14 we compare
the convergence of the above method to the convergence of the simple trapezoidal rule
for the integration of the functions
!+
sin(ϕ − α)
cos(ϕ),
f =
(1 − c cos(ϕ))3
and
sin(ϕ − α)
g=
(1 − c cos(ϕ))3
!+
sin(ϕ),
which occur in the computation of the bearing forces. The relative errors of the
corrected method converge much faster to zero than those of the trapezoidal method.
106
Chapter 5: Bearing Models
0
10
rel. Error
−5
10
−10
10
corr. trapez f
trapez f
corr. trapez g
trapez g
−15
10
−6
10
−4
−2
10
10
0
10
h
Figure 5.14: Error of the numerical quadrature: Corrected trapezoidal method shows faster
convergence than trapezoidal method .
5.6 Numerical Results and Comparison of Bearing Properties
In the previous sections different models for the pressure distribution in a plain journal
bearing and numerical methods to compute them were introduced. In this section
we compare the results and take a look at the bearing reaction forces caused by the
movement of the shaft in the lubricant.
5.6.1 Convergence Considerations for Short Bearing Approximation
In Section 5.1 Reynolds’ equation is derived as a thin film approximation to the NavierStokes equations. The short bearing approximation additionally considers the axial
dimension of the bearing being small. We introduce the additional relation ε = Kδ2
where ε = cRr is the ratio of radial bearing clearance cr to bearing radius R and δ = W
R
the ratio of bearing width W to radius. By letting them simultaneously become small
the convergence of the solution of Reynolds’ equation (in two variables) to the short
bearing approximation can be observed. Figure 5.15 shows the relative difference
between the solutions of Reynolds’ equation pir and the short bearing approximation
pis in the L2 -Norm
kpr − ps kL2
(5.179)
error =
kpr kL2
for the zeroth and the first order. The error is shown in dependence of ε for a fixed ratio
of K = cWr R2 = 1 which corresponds to a very short bearing. As expected from Section 5.1
the convergence rate is roughly quadratic for the first order approximation and linear
107
5.6 Numerical Results and Comparison of Bearing Properties
0
10
0
ε ,κ=0.1
1
ε ,κ=0.1
0
ε ,κ=0.5
−2
10
1
ε ,κ=0.5
0
ε ,κ=0.9
1
ε ,κ=0.9
2
rel. L error
−4
10
−6
10
−8
10
−10
10
−4
10
−3
−2
10
ε
10
−1
10
kpr −ps k 2
Figure 5.15: The relative difference kpr k 2L between the solutions of Reynolds’ equation
L
and the short bearing approximation for zeroth and first order and for different eccentricity
values; the ratio of clearance to the square of width is constant K = δε2 = 1.
in ε for the zeroth order approximation. The validity for different eccentricity ratios
is also demonstrated in the figure. However, for large eccentricities the convergence
becomes poorer.
cr R
Since the bearings used in practice usually do not fulfill this special ratio K = W
2 = 1,
it is also necessary to examine the convergence behavior for different ratios in order to
know the limits of usability of the approximation. In Figure 5.16 the relative error in
dependence of δ is depicted for several values of ε. In the zeroth order approximation
the ratio K does not appear in the expression for the pressure (5.47), so there is no
difference in the convergence of the zeroth order solutions for different ε and only the
one for ε = cRr = 0.0067 is depicted. We have convergence of the solutions of Reynolds’
equation to the short bearing approximation for decreasing δ = W
R , the convergence
being better for small ε and also better for the first order equation than for the zeroth
order. However, it can also be seen, that for large δ, i.e. longer bearings of a width to
radius ratio of 1 or larger, the short bearing solution differs largely from the solution
of Reynolds equation no matter how small ε. Here the difference is bigger for the first
order equation than for the zeroth order. This corresponds with (Childs, 1993) and
(San Andrés, 2006) where is stated that the short bearing approximation is valid only
for W
R -ratios smaller than 1.
108
Chapter 5: Bearing Models
6
10
ε=3.3e−04
ε=1.7e−03
ε=3.3e−03
ε=6.7e−03
ε=1.7e−02
ε=3.3e−02
ε=6.7e−02
order 0
4
10
2
10
0
rel. L2 error
10
−2
10
−4
10
−6
10
−8
10
−10
10
−3
10
−2
−1
10
Figure 5.16: The relative difference
10
δ
kpr −ps kL2
kpr kL2
0
10
1
10
between the solutions of Reynolds’ equation
and the short bearing approximation (order ε0 and order ε1 ) for varying width and different
values of ε; the relative eccentricity is κ = 0.9.
5.6.2 Comparison of Pressure Distributions and Resulting Forces
The surface plots in the Figures 5.5 - 5.11 depict the pressure distributions arising from
the solution of Reynolds’ equation, the short bearing approximation and alternative
bearing models, as derived in the Sections 5.1 - 5.3. The comparison of the various
pressure distributions is facilitated by looking at the z-averaged pressure distributions
as shown in Figure 5.17. It can be seen clearly that for the very short bearing (first line:
W = 0.24 mm, K = 1) and for the short bearing (second line: W = 1 mm, K = 0.06) the
z-averaged pressure distributions are almost identical with slightly larger differences
for larger eccentricities. The only exception is the finite bearing approximation (5.111),
which underestimates the pressure in all cases and is therefore not considered in
the simulations in Chapter 6. For the medium sized bearing as it is used in the
turbocharger of TCRDL the first order short bearing approximations shows drastically
different behavior than the first order solution of Reynolds’ equation, as seen in the
convergence considerations in the last section. While the zeroth order approximation
overestimates the pressure, the correction term (5.61) in this case becomes larger than
the zeroth order solution and therefore dominates the pressure distribution, leading
to a physically wrong solution where the pressure is negative in front of the point of
smallest film thickness.
109
5.6 Numerical Results and Comparison of Bearing Properties
ε = 6.67e−03, δ = 8.16e−02, Rey = 151.3,
κ = 0.1, κ′ = 0, κ′′ = 0, γ′ = 0, γ′′ = 0
ε = 6.67e−03, δ = 8.16e−02, Rey = 151.3,
κ = 0.5, κ′ = 0.4, κ′′ = 0.10125, γ′ = 0.45, γ′′ = 0
0.06
2.5
0
−0.02
2
rey0
rey1
shb0
shb1
fin
reybc
phen
40
z−av. pressure
0.02
60
rey0
rey1
shb0
shb1
fin
reybc
phen
3
z−av. pressure
z−av. pressure
0.04
ε = 6.67e−03, δ = 8.16e−02, Rey = 151.3,
κ = 0.9, κ′ = 0, κ′′ = 0, γ′ = 0, γ′′ = 0
3.5
rey0
rey1
shb0
shb1
fin
reybc
phen
1.5
1
20
0
−20
0.5
−0.04
−40
0
−3
−2
−1
0
φ
1
2
3
−0.5
−4
4
ε = 6.67e−03, δ = 3.33e−01, Rey = 151.3,
κ = 0.1, κ′ = 0, κ′′ = 0, γ′ = 0, γ′′ = 0
−2
−1
0
φ
1
2
3
−60
−4
4
−3
rey0
rey1
shb0
shb1
fin
reybc
phen
3
2.5
z−av. pressure
0.02
0
−1
0
φ
1
2
3
4
60
3.5
rey0
rey1
shb0
shb1
fin
reybc
phen
0.04
−2
ε = 6.67e−03, δ = 3.33e−01, Rey = 151.3,
κ = 0.9, κ′ = 0, κ′′ = 0, γ′ = 0, γ′′ = 0
ε = 6.67e−03, δ = 3.33e−01, Rey = 151.3,
κ = 0.5, κ′ = 0.4, κ′′ = 0.10125, γ′ = 0.45, γ′′ = 0
0.06
z−av. pressure
−3
2
rey0
rey1
shb0
shb1
fin
reybc
phen
40
z−av. pressure
−0.06
−4
1.5
1
20
0
−20
−0.02
0.5
−40
−0.04
0
−3
−2
−1
0
φ
1
2
3
−0.5
−4
4
−2
−1
0
φ
1
2
3
−60
−4
4
−3
ε = 6.67e−03, δ = 1.80e+00, Rey = 151.3,
κ = 0.5, κ′ = 0.4, κ′′ = 0.10125, γ′ = 0.45, γ′′ = 0
ε = 6.67e−03, δ = 1.80e+00, Rey = 151.3,
κ = 0.1, κ′ = 0, κ′′ = 0, γ′ = 0, γ′′ = 0
rey0
rey1
shb0
shb1
fin
reybc
phen
0.02
2.5
z−av. pressure
0.04
0
2
0
φ
1
2
3
4
rey0
rey1
shb0
shb1
fin
reybc
phen
600
400
1.5
1
0.5
−0.02
−1
800
rey0
rey1
shb0
shb1
fin
reybc
phen
3
−2
ε = 6.67e−03, δ = 1.80e+00, Rey = 151.3,
κ = 0.9, κ′ = 0, κ′′ = 0, γ′ = 0, γ′′ = 0
3.5
0.06
z−av. pressure
−3
z−av. pressure
−0.06
−4
200
0
−200
−400
0
−0.04
−600
−0.5
−0.06
−4
−3
−2
−1
0
φ
1
2
3
4
−1
−4
−3
−2
−1
0
φ
1
2
3
4
−800
−4
−3
−2
−1
0
φ
1
2
3
4
Figure 5.17: z-averaged pressure distributions from solutions to Reynolds’ equation (rey0,
rey1), short bearing approximation(shb0, shb1), finite bearing (fin), Reynolds’ equation with
free boundary conditions (reybc), and the phenomenological correction (phen ): First line:
very short bearing (W = 0.24, K = 1); second line: short bearing (W = 1, K = 0.06); third
line: medium sized bearing (W = 5.4, K = 0.002); in each column different values of κ, κ̇,
and γ̇ are used.
This affects also the bearing reaction forces as shown in Figure 5.18. We depict the
normal force along the line from the center of the bearing to center of the shaft and the
orthogonal tangential force in dependence of the eccentricity κ for a rotating shaft with
rotational frequency ν = 1400 Hz and whirling with an angular velocity of γ̇ = 0.4ω.
In the three columns of the figure different bearing lengths are considered, all other
parameters being equal. Again it can be seen that for a long bearing (W = 5.4 mm) the
first order solution shows a drastically different behavior due to the large difference of
the pressure distributions shown in Figure 5.17. This also leads to a totally different
dynamic behavior of the shaft as can be seen in the simulations in Chapter 6.
The forces arising from the solutions of Reynolds’ equation only show little differences.
In Figure 5.19 surface plots of the bearing forces arising from the zeroth order short
bearing approximation are shown exemplarily in dependence of κ and γ̇ for κ̇ = 0, and
W = 1. The forces are zero for γ̇ = 12 . This is the case for the zeroth order Reynolds’
110
Chapter 5: Bearing Models
Normal Force
0
−10
2
F
−10
3
−10
4
−10
5
−10
0
−10
3
−10
rey0
rey1
shb0
shb1
fin
reybc
phen
F
2
0
rey0
rey1
shb0
shb1
fin
reybc
phen
−1
−2
−3
F
1
−10
−10
−7
5
−8
−9
6
0.4
κ
0.6
0.8
1
rey0
rey1
shb0
shb1
fin
reybc
phen
−6
−10
0.2
−4
−5
4
−10
0
0.2
0.4
κ
0.6
Normal Force
6
x 10
Normal Force
1
−10
0.8
0
1
0.2
0.4
F
10
rey0
rey1
shb0
shb1
fin
reybc
phen
5
10
4
rey0
rey1
shb0
shb1
fin
reybc
phen
1
0.5
0
10
−0.5
10
1
0
0.8
1
2
10
1
F
3
0.8
10
F
4
10
x 10
Tangential Force
6
0.6
Tangential Force
7
Tangential Force
5
10
κ
−1
3
0.2
0.4
κ
0.6
0.8
1
10
0
0.2
0.4
κ
0.6
0.8
0
1
rey0
rey1
shb0
shb1
fin
reybc
phen
0.2
0.4
κ
0.6
Figure 5.18: Normal and tangential forces acting on a rotating shaft (rot. frequency ν =
1400 Hz) orbiting the bearing center on a circular orbit of radius κ; the angular velocity is
γ̇ = 0.4ω; the other parameters are R = 3 mm, cr = 0.02 mm, Reynolds number R = 152;
the bearing width varies from the left to the right: W = 0.245 mm, W = 1 mm, and
W = 5.4 mm. Depicted are the forces resulting from the solutions to Reynolds’ equation
(rey0, rey1), short bearing approximation (shb0, shb1), finite bearing (fin), Reynolds’ equation
with free boundary conditions (reybc), and the phenomenological correction (pheno). The
right column is not in logarithmic scale due to the sign changes of the first order short
bearing approximation shb1.
Tangential Force
Normal Force
6
6
x 10
x 10
0
2
−1
1
−2
0
−3
−1
−4
−5
0
1
0.5
0.5
1
κ
0
γ′
−2
0
1
0.5
0.5
1
κ
0
γ′
Figure 5.19: Surface plots of normal and tangential force computed for zeroth order short
bearing approximation for varying κ and γ̇ with fixed κ̇ = 0, W = 1.
equation as well.
For the short bearings the first order approximations differ only a little from the zeroth
order approximations. They are however not identical and especially the tangential
force of the first order solution is smaller than that of the zeroth order solution. In view
of the question for the rotor self excited whirl frequency the behavior of the tangential
111
5.6 Numerical Results and Comparison of Bearing Properties
force is of special interest, since it accelerates or decelerates the rotor along the circular
whirling orbit. For the first order solutions the zero of the tangential force moves to
lower values of γ̇ as shown in the Figure 5.20 which backs the hypothesis that the
inertia correction leads to a reduction of the whirl frequency.
Shift of Zeros
0.5005
r1, W=0.25
r1, W=1
r1, W=5.4
s1, W=0.25
s1, W=1
s1, W=5.4
0.5
0.4995
0.499
0.4985
0.498
0.4975
0.497
0.4965
0
1000
2000
3000
4000
5000
Figure 5.20: Approximate value of γ̇ for which the tangential force vanishes in dependence
of ω for κ = 0.9, κ̇ = 0.
The phenomenological correction of the zeroth order short bearing approximation
(5.101) also shows similar behavior. In Figure 5.21 we depict the normal and the
tangential force for W = 1, γ̇ = 0.4, and κ̇ = 0, in dependence of κ and the correction
parameter σ for a linear correction s(ω) = σω.
Tangential Force
Normal Force
5
x 10
5
x 10
4
0
2
−2
F
F
−4
0
−6
−2
−8
−10
0
0.2
0.1
0.5
1 0
κ
σ
−4
0
0.2
0.5
0.1
1 0
κ
σ
Figure 5.21: Normal and tangential force from phenomenological pressure correction in
dependence of κ and σ.
From Equation (5.101) it is easy to see that the parameter σ shifts the zero of the
tangential force. In the figure the tangential force is zero for σ = 0.1. Hence we can use
112
Chapter 5: Bearing Models
the shift parameter to control for which value of γ̇ the tangential force vanishes, e.g. in
the linear case
1
γ̇ + σ =
⇒ ft = 0.
(5.180)
2
5.7 Conclusions
In view of the convergence results and the comparison of the resulting forces we can
draw the following conclusions.
• For very short and short bearings the solution of Reynolds equation converges
to the derived short bearing approximation and yields a model for the pressure
distribution that is easier and faster computable.
• The first order short bearing approximation can be used for very short bearings
(K ≈ 1), while for medium sized bearings the first order Reynolds equation
should be the model of choice for the pressure distribution in the plain journal
bearing.
• The numerical evidence points to an influence of the inertia correction on the
whirling frequency via a change in the zero of the tangential force. This influence
on the tangential force is also present for the phenomenological correction to the
short bearing approximation
• The finite bearing approximation underestimates the pressure and the forces and
will therefore not be considered for the direct numerical simulations.
• The free boundary problem for zeroth order Reynolds equation with Reynolds
boundary conditions yields solutions that are similar to those with Gümbel
conditions, however the computation time is considerably larger. In Chapter 6
we see that there is no influence on the whirling frequency.
The derivation of the bearing model is an important part of the modelling of the
rotor-bearing system as the influence on the dynamics is considerable. It is therefore
naturally an intensively studied field with a large variety of models around. We do
not seek completeness here, which would be beyond the scope of this work, but we
want to show in the following that already the relatively simple models presented here
lead to complex dynamic behavior. The methods we will use in Chapters 6 and 7 can
then also be applied to more complicated bearing that e.g. involve turbulence, surface
roughness, or variable viscosity.
Chapter 6
Direct Numerical Simulation and
Experimental Results
In the previous chapters we have derived the various parts of a model for rotating
beams constrained by simple journal bearings. In Chapter 2 a PDE model was derived
for the lateral motion of a continuous Rayleigh beam. In Chapter 4 we have used
the finite element method to discretize the equations and we have arrived at an ODE
system for the nodal displacement of the beam. In Chapter 5 the model for the
reaction forces of the oil lubricated bearings has been derived from the Navier-Stokes
equations by asymptotic analysis. In the present chapter we combine these results to
obtain finally the equations of motion for the rotating beam, and we solve these with
numerical integration methods. The results of the simulations are compared to the
experiments performed at the Toyota Central Research and Development Laboratories
(TCRDL). As we shall see, the equations of motion are either a stiff system of ordinary
differential equations if no inertia correction is applied, or an implicit system if the
inertia correction is used in the bearing model. Therefore appropriate implicit methods
such as backward differentiation formulas (BDF) (Hairer & Wanner, 1996) or numerical
differentiation formulas (NDF) (Shampine & Reichelt, 1997; Shampine, 2002) have to
be used.
6.1 Equations of Motion
The equations of motion for a rotating beam are a system of ordinary differential
equations for the nodal displacement of the beam elements used for the discretization
of the rotor as in Equation (4.69).
The state space is R4(N+1) , where N is the number of finite beam elements. The state
114
Chapter 6: Direct Numerical Simulation and Experimental Results
vector is
x = (x1 , . . . , x4(N+1) )
= (q1 , . . . , qN+1 ) = (u1 , v1 , β1 , α1 , . . . , uN+1 , vN+1 , βN+1 , αN+1 ),
(6.1)
(6.2)
where uk and vk are the lateral displacements in node k, and βk and αk the inclinations
as shown in Section 4.3 of Chapter 4. For small displacements we have approximately
βk ≈ −v′k and αk ≈ u′k (cf. Appendix A). Furthermore, we need the following notations
which facilitate the formulation of the equations of motion. The polar coordinates of
the nodal displacement are
q
rk = u2k + v2k , γk = arg(uk + ivk ).
(6.3)
The nodal lateral displacements at the bearing nodes are denoted by xbi = (ubi , vbi )T .
The vector xb = (ub1 , vb1 , ub2 , vb2 )T contains the displacements at all the bearing nodes.
As seen in the previous chapters the rotating shaft is subject to several external forces.
First, there are constant loads, such as gravity, which we will denote in the following
by Fgr . Second, there are time-periodic forces Funb (t) due to the unbalance of the rotor,
as given in (4.76) and (4.77) with nodal unbalance amplitude ak and phase ψk . They
have the following structure



a1 cos(ωt + ψ1 )





a1 sin(ωt + ψ1 )




0




..




.
Funb (t) = ω2 
(6.4)
 .
 a

 N+1 cos(ωt + ψN+1 ) 


 aN+1 sin(ωt + ψN+1 ) 




0




0
Third, there are the bearing reaction forces. Depending on the bearing model the
system is either explicit or implicit.
6.1.1 Explicit Case
When no inertia correction is applied, i.e. the zeroth order short bearing approximation
(Eqs. (5.159), (5.160)) or the zeroth order solution to Reynolds equation (Eq. (5.145)) are
used as models for the pressure distribution in the bearings, the equation of motion
(4.69) is explicit. It is also explicit for the alternative bearing models proposed in
Section 5.3, like the phenomenological model, the finite bearing approximation, and
the solution with Reynolds boundary conditions. In these cases the bearing reaction
force Fbear has the following form
T
Fbear (xb ) = 0, . . . , fb1 ,1 (xbi , ẋbi ), fb1 ,2 (xb1 , ẋb1 ), 0, . . . , fb2 ,1 (xb1 , ẋbi ), fb2 ,2 (xb2 , ẋb2 ), 0, . . . , 0 .
(6.5)
115
6.1 Equations of Motion
It is computed from the normal and the tangential forces at the bearing nodes bi , i ∈
{1, 2}, by




 fn (rbi , ṙbi , γ̇bi ) 
 fbi ,1 
 ,
(6.6)

 (xbi , ẋbi ) = T(γbi ) 
f (r , ṙ , γ̇ ) 
f
bi ,2
t
bi
where T(γ) is the two-dimensional rotation matrix

 cos γ − sin γ
T(γ) = 
sin γ cos γ
bi
bi


 .

(6.7)
Combining the above, we finally obtain the equation of motion which can be written
in second order form as
Mẍ + (ωG + C)ẋ + Kx = Fgr + Funb (t) + Fbear (xb , ẋb ),
(6.8)
or in first order form with y = (ẋ, x)T

 M


I








 ẏ =  −(ωG + C) −K  y +  Fgr + Funb (t) + Fbear (yb )




I
0
0


 .

(6.9)
6.1.2 Implicit Case
When the inertia corrected lubrication models (Eq. (5.63) or (5.90)) are used, the bearing
reaction forces additionally depend on the acceleration of the shaft at the bearing node.
We have
Fbear = 0, . . . , 0, fb1 ,1 (xb1 , ẋb1 , ẍb1 ), fb1 ,2 (xb1 , ẋb1 , ẍb1 ) , 0, . . .
T
(6.10)
. . . , 0, fb2 ,1 (xb2 , ẋb2 , ẍb2 ), fb2 ,2 (xb2 , ẋb2 , ẍb2 ), 0, . . . , 0
with

 fbi ,1

 f
bi ,2




 (xb , ẋb , ẍb ) = T(γb )  fn (rbi , ṙbi , γ̇bi , r̈bi , γ̈bi )
 i i i
i 
ft (rbi , ṙbi , γ̇bi , r̈bi , γ̈bi )


 .

(6.11)
This leads to an implicit equation of motion
Mẍ + (ωG + C)ẋ + Kx = Fgr + Funb (t) + Fbear (xb , ẋb , ẍb ),
(6.12)
which we can be rewritten in first order form

 M


I





 −(ωG + C) −K 

 ẏ = 
 y +  Fgr + Funb (t) + Fbear (yb , ẏb )




I
0
0


 .

(6.13)
116
Chapter 6: Direct Numerical Simulation and Experimental Results
By introducing the projection Pb onto the components of the state space describing the
lateral displacement of the bearing nodes and the dummy variable a
Pb ẍ = ẍb = a,
(6.14)
we can rewrite this implicit equation as a differential-algebraic equation of degree 1
with singular mass matrix.


 M 0 0   ẍ


 0 I 0   ẋ

 

Pb 0 0
ȧ
 
  −(ωG + C) −K 0
 
 = 
I
0
0
 
 
0
0 −I
  
  ẋ   Fgr + Funb (t) + Fbear (xb , ẋb , a)
   
0
  x  + 
   
a
0



 .


(6.15)
6.1.3 Simple Model
Additionally, as in (Hollis & Taylor, 1986), a very simple model can be used for examination of the influence of the bearing function on the dynamics. We consider a
rotating cylinder of mass M of the same length as a journal bearing. The forces acting
on the cylinder are the bearing forces, unbalance forcing, gravity and eventually some
viscous damping with damping coefficient D ≥ 0. With x ∈ R2 being the vector of
lateral displacement, the equation of motion for the lateral displacement is


Fbear,1
Mẍ + Dẋ = 
Fbear,2 − Mg




 + funb ω2  cos ωt
 sin ωt



 .

(6.16)
One can interpret this as a model describing just the motion of the bearing node subject
to the bearing forces and an additional periodic forcing, but without interaction with
the neighboring nodes. Depending on the lubrication model, Equation (6.16) can be
either explicit or implicit.
6.2 Numerical Methods
In the previous section we stated equations of motion that arise from our models. All
these equations show the characteristics of stiff equations (Hairer & Wanner, 1996).
Especially the spectrum of the Jacobian is very stretched out over the negative half
plane. In Figure 6.1, we exemplarily depict the eigenvalues of the Jacobian of the
explicit system (6.8) using a 13 element beam as in Section 4.5.1 and the short bearing
lubrication model (5.159), (5.160). We see that it has eigenvalues of large absolute value
on the negative real axis as well as close to the imaginary axis. Therefore the use of
A-stable implicit methods is appropriate, where the whole negative half plane is in
the region of stability (Hairer & Wanner, 1996). Semiexplicite equations of the form
(6.15) can be solved by Matlab’s ode15s solver or Hairer and Wanner’s RADAU5 (Hairer
117
6.2 Numerical Methods
7
3
x 10
Spectrum of Jacobian (ω=2010 Hz)
2
Imag(λ)
1
0
−1
−2
−3
−3
−2.5
−2
−1.5
−1
Real(λ)
−0.5
0
0.5
6
x 10
Figure 6.1: Spectrum of the Jacobian of the explicit system (6.8) for a 13 element beam
model with short bearing lubrication model at ω = 2010 Hz. Most of the eigenvalues are
located along the imaginary axis with a small negative real part and a few eigenvalues
have large negative real parts, which leads to the stiffness of the equations.
& Wanner, 1996), among others. The implicit equation can be solved with Matlab’s
ode15i. The code DASPK can be used to solve both variants.
In this work, we mainly apply Matlab’s ode15s code and the code DASPK, which is
a successor of the popular code DASSL. DASPK uses backward differentiation formulas
(BDF) of variable order for the integration of stiff differential(-algebraic) systems. These
implicit multistep methods are described in (Brenan et al., 1989) and also in (Deuflhard
& Bornemann, 1994). DASPK can also compute consistent initial conditions. The DASPK
routine is included in the free software Octave (Eaton, 2002) which is mostly compatible
with Matlab.
The code ode15s used in Matlab is described in (Shampine & Reichelt, 1997). It uses
numerical differentiation formulas (NDF) which are a modification of the BDF having
an additional free parameter that can be used to either enlarge the region of stability
while maintaining the truncation error, or to reduce the error while keeping the size
of the region of stability close to the one of the BDF. In ode15s the second target is
chosen leading to smaller regions of stability. For order 2, however, both methods are
A-stable. When using ode15s it is necessary to set the maximal order of the method to
2, since otherwise numerical instabilities occur. ode15s can compute consistent initial
conditions.
In both codes the user can supply a routine for the Jacobian matrix of the system, which
118
Chapter 6: Direct Numerical Simulation and Experimental Results
is otherwise computed by finite differences. The computation of the bearing reaction
forces can be computationally expensive, especially when the Reynolds equation has
to be solved to compute them. This makes the computation of the Jacobian by finite
differences expensive although the non-linearity only appears in a few (4) components
and depends only on the coordinates of the bearing nodes. We therefore supply a
routine that makes use of the special structure of the system to compute the Jacobian
more efficiently. For this, we compute the matrices



Kbi = 




Cbi = 




Mbi = 

∂ fbi ,1
∂xbi ,1
∂ fbi ,2
∂xbi ,1
∂ fbi ,1
∂xbi ,2
∂ fbi ,2
∂xbi ,2
∂ fbi ,1
∂ẋbi ,1
∂ fbi ,2
∂ẋbi ,1
∂ fbi ,1
∂ẋbi ,2
∂ fbi ,2
∂ẋbi ,2
∂ fbi ,1
∂ẍbi ,1
∂ fbi ,2
∂ẍbi ,1
∂ fbi ,1
∂ẍbi ,2
∂ fbi ,2
∂ẍbi ,2



 ,

(6.17)



 ,

(6.18)



 ,

(6.19)
using central differences. These are subsequently added to the corresponding components of the matrix of the constant linear part of Equations (6.9) or (6.13), respectively,
and to the mass matrix of (6.13)
 


 −(ωG + C) −K   Cb Kb 
 + 
 ,
J = 
I
0   0 0 
 


  −Mb
 M

 − 
 ,
B = 


I
0 
(6.20)
(6.21)
where

 . . .



K b1


Kb = 





..
.
K b2












. . 
.
(6.22)
is zero except for the blocks Kb1 and Kb2 which are located at the indices corresponding
to the variables describing the lateral displacement of the bearing nodes. Cb and Mb are
formed analogously. While still using finite differences to approximate the derivative
of the non-linearity, we can hence reduce the number of necessary computations of
the bearing forces to 24 regardless of the dimension of the system. Both, the Jacobian
J and the mass matrix B are sparse. The use of the sparsity pattern and other sparse
matrix routines in Matlab also leads to a considerable speed-up of the computations.
6.3 Analysis of Experimental Results
119
6.3 Analysis of Experimental Results
Simulations have been done for several different parameter sets. These sets reflect
the different set-ups used in the experiments at TCRDL using the turbocharger rotor
similar to the one depicted in Figure 6.2, but without floating ring bearings. The
experiments have been carried out in 2002 at TCRDL by Mizuho Inagaki. The standard
Figure 6.2: The rotor of a turbocharger used in the experiments at TCRDL. It consists of
the shaft, the compressor (left) and the turbine wheel (right). The bearing positions are
indicated here by the two floating rings (golden color). The experiments were done for
plain journal bearings without floating rings.
Figure 6.3: Positions of eddy current sensors measuring lateral x-y-displacement at three
points along the shaft (impeller, middle, turbine) and axial displacement at the right end.
120
Chapter 6: Direct Numerical Simulation and Experimental Results
Orbit at impeller side, ω=505Hz
Orbit at impeller side, ω=1503Hz
Frequency Diagram (Experiment)
0.1
0.1
Oil whirl
0
Unbalance
Oscillation
0.05
−0.1
y [mm]
y [mm]
0.1
−0.1
0
−0.1
0
0.1
x [mm]
Orbit at impeller side, ω=250Hz
0
2000
−0.1
0
0.1
x [mm]
Orbit at impeller side, ω=998Hz
1500
0.1
0.1
0
500
Driving
Frequency
[Hz] 0
−0.1
−0.1
0
x [mm]
0.1
y [mm]
y [mm]
1000
0
500
1000
Frequency [Hz]
1500
2000
0
−0.1
−0.1
0
x [mm]
0.1
Figure 6.4: Experimental results: Waterfall diagram of the response spectrum of rotor for
varying driving frequencies (center); orbits of impeller end of rotor for 4 different driving
frequencies (left and right). Mainly two kinds of vibration occur in the examined frequency
range: the subharmonic oil whirl and the synchronous unbalance oscillation of a bending
mode.
experimental case used two bearings of 5.4 mm width, 0.02 mm radial clearance,
80◦ C oil pan temperature and an oil supply pressure of 0.001 Pa. The dimensions
of the rotor and the unbalance condition are given in the Appendix D. During the
experiments the rotor is driven by pressurized air at a given rotational velocity. The
lateral displacement in x- and y-direction is measured by eddy current sensors at three
positions along the rotor. Additionally, the axial displacement is measured at one end.
The sensor positions are shown in Figure 6.3.
The results of the measurements with the above experimental set-up are shown in
Figure 6.4. In the central waterfall diagram for each rotational speed ω (y-axis) the
power spectrum of the motion of the impeller is plotted with the observed frequencies
ν along the x-axis. The two diagonals indicate the ν = ω and 2ν = ω lines. To the
left and to the right of the waterfall diagram orbits of the impeller end of the rotor are
plotted for four different rotational frequencies. One observes increasingly complex
behavior. In Chapter 7 an analysis of the Poincaré sections of the orbits also reveals
quasiperiodic behavior.
In the waterfall diagram it can be seen that above a threshold of about 400 Hz there are
mainly two frequencies present in the power spectrum, while there is only one present
below 400 Hz. The one frequency always present is equal to the driving frequency,
which is due to a harmonic response to unbalance forcing. Around 1000 Hz there is
a peak with large amplitudes, which is caused by the resonance of the first bending
mode of the rotor. The other frequency is a subharmonic, large amplitude vibration
which sets in above 500 Hz. Around the resonance of the bending mode the amplitude
of this mode drops, only to increase again above the critical speed of 1000 Hz. This
121
6.3 Analysis of Experimental Results
Bending mode with 999Hz (ω=998Hz)
0.01
0.01
0.005
0.005
displacement [mm]
displacement [mm]
Conical mode with 464Hz (ω=998Hz)
0
−0.005
−0.01
0
−0.005
−0.01
0
20
40
60
length [mm]
80
100
120
0
20
40
60
length [mm]
80
100
120
Figure 6.5: The running modes measured in experiment (set-up 1) are an allmost rigid
conical mode (left) and a bending mode (right). Red markers indicate the measuring
positions, points in between are unknown.
phenomenon is known as entrainment. The subharmonic vibration is called oil whirl,
as it is a rotor instability which is caused by the oil-lubricated bearings. This instability
has been known for a long time (Newkirk & Taylor, 1925) and has been investigated
thoroughly in (Muszynska, 1986; Muszynska, 1988; Crandall, 1995; Hollis & Taylor,
1986) among others. In Figure 6.4, but also in Figure 6.6 below, one can see that the
whirl frequency is about half the driving frequency for rotational speeds below the
critical speed of 1000 Hz. For higher rotational speeds the frequency of the oil whirl
still increases, but no longer linearly. The ratio of whirl frequency to rotational speed
drops from 21 at 500 Hz to approximately 52 at 1500 Hz and one observes a shift away
from the ω2 -line.
The running mode shapes can also be calculated from the experimental data. They
are shown in Figure 6.5. There are two main running modes at a rotational speed of
998 Hz: a conical mode of the subharmonic oil whirl with a frequency of less than half
the rotor speed and a bending mode synchronous with the rotation.
As mentioned above several experiments with varying parameters were carried out.
The experimental parameters are given in Table 6.1. In Figure 6.6 the Campbell diagrams for 4 different situations are depicted. In these diagrams, the x-coordinate of
the centers of the circles gives the response frequency, the y-coordinate the driving
frequency, and the radii of the represent the amplitudes of the underlying Fourier
modes.
The comparison of the cases 1 and 2 shows that a variation of the unbalance parameter
has little influence on the occurrence and the frequency of the oil whirl. In case 2, the
amplitudes of the oil whirl are smaller, while the entrainment is more pronounced.
The measurements in case 3 and 4 show that the oil whirl is suppressed by a reduction
the bearing clearance or the bearing width. However, these modifications lead to
122
Campbell diagramm, experimental case 1
1800
Campbell diagramm, experimental case 2
1800
1600
1600
1400
1400
rot. speed [Hz]
rot. speed [Hz]
Chapter 6: Direct Numerical Simulation and Experimental Results
1200
1000
800
600
400
200
1000
800
600
400
Amp=0.10mm
200
0
0
Amp=0.10mm
0
0
500
1000
1500
frequency [Hz]
Campbell diagramm, experimental case 3
1800
500
1000
1500
frequency [Hz]
Campbell diagramm, experimental case 4
1800
1600
1600
1400
1400
rot. speed [Hz]
rot. speed [Hz]
1200
1200
1000
800
600
400
200
0
0
1200
1000
800
600
400
Amp=0.10mm
500
1000
frequency [Hz]
1500
200
0
0
Amp=0.10mm
500
1000
frequency [Hz]
1500
Figure 6.6: Campbell diagram showing amplitudes (radius of circles) and frequency (position of center) of the oscillations at impeller side for 4 different experimental set-ups.
For the experimental parameters see Table 6.1. Circles on the diagonal are due to harmonic response, circles to the left of the line indicating half-frequency response are due to
self-excited vibration.
larger unbalance oscillations in the resonance region of the bending mode. These large
oscillations caused the limitation of these two experiments to the speed region below
60000 RPM (1000 Hz), since bearing failures and extreme noise occurred.
Case
#
1
2
3
4
[mm]
0.02
0.02
0.01
0.02
bearing width
[mm]
5.4
5.4
5.4
3.8
a3 , ψ3 , a5 , ψ5 a11 , ψ11 a13 , ψ13
1.35, π 1.50, 0 2.01, 0 2.07, π
0.64, 0 0.75, 0 0.98, π 1.09, π
1.35, π 1.50, 0 2.01, 0 2.07, π
1.35, π 1.50, 0 2.01, 0 2.07, π
Table 6.1: Experimental parameters for the results shown in Figure 6.6.
123
6.4 Simulations for Large Rotor Model
Beam model
Beam model
50
40
40
30
30
20
width [mm]
width [mm]
20
10
0
−10
−20
10
1
3
2
4
0
−10
−20
−30
−30
−40
−40
−50
0
20
40
60
80
100
120
0
length [mm]
20
40
60
80
100
length [mm]
Figure 6.7: A detailed 13 element beam model (left) and a less detailed 3 element beam
model (right) used in numerical simulations. The 3 element model is also used in the
investigation of oil-film influence and numerical bifurcation analysis. The rigid discs
modelling impeller and turbine are shown in red dashed lines and the positions of the
bearings are indicated by the black triangles.
6.4 Simulations for Large Rotor Model
In the joint research project with TCRDL direct numerical simulations were performed
for several rotor models of varying element number. These were a larger model with
13 beam elements (cf. Section 4.5.1) that was also considered for shape optimization
in (Strauß, 2005), as well as a smaller model with 4 beam elements that is also used
in the investigation of the influence of the lubrication model in Section 6.5 and in the
bifurcation analysis in Chapter 7.
The analytic formulation of the bearing reaction forces in the short bearing approximation allows for their fast computation. This reduces the computation time for direct
numerical simulation considerabely. The 13 element rotor model is depicted again in
Figure 6.7 with the positions of the short bearings indicated by triangles. The bearing
and unbalance parameters are chosen as in the experimental case 1 (cf. Table 6.1) with
a dynamic viscosity of η = 0.049 Pas. The integration of Equation (6.9) with bearing
force functions given by the zeroth order short bearing approximation (5.159) and
(5.160) is done by applying ode15s with maximal order set to 2 and a relative tolerance
of 10−2 . The only remaining free parameter that is not known from the experiment is
the viscous damping coefficient that is used in the damping matrix C in Equation (6.9).
It is set to 9 Ns/m in this simulation.
The results are shown in Figure 6.8 and one can see from the waterfall diagram that
the main dynamical features of the experiment are reproduced. The complexity of the
orbits increases with the rotational speed and also the amplitudes are only slightly
larger than in experiment. Entrainment can be observed around a rotational speed
124
Chapter 6: Direct Numerical Simulation and Experimental Results
Orbit Impeller Side 350 Hz (Simulation)
Orbit Impeller Side 1500 Hz (Simulation)
Frequency diagram (simulation)
0.1
0.1
0.1
0
0
0.05
−0.1
−0.1
0
−0.1
0
2000
0.1
−0.1
Orbit Impeller Side 50 Hz (Simulation)
0
0.1
Orbit Impeller Side 1000 Hz (Simulation)
1500
0.1
0.1
1000
0
0
500
Driving
Frequency
[Hz] 0
−0.1
−0.1
0
1500
1000
Frequency [Hz]
500
0
2000 −0.1
0.1
−0.1
0
0.1
Figure 6.8: Simulated orbits and waterfall diagram for 13 element beam model and parameters as in experimental set-up 1 with external damping coefficient 9 Ns/m. The main
experimentally observed vibration effects of subharmonic oil whirl and synchronous unbalance vibration are captured in the model.
Conical mode with 487Hz (ω=1000Hz)
0.01
0.01
0.005
0.005
displacement [mm]
displacement [mm]
Bending mode with 1023Hz (ω=1000Hz)
0
−0.005
−0.01
0
−0.005
−0.01
0
20
40
60
length [mm]
80
100
120
0
20
40
60
length [mm]
80
100
120
Figure 6.9: The running modes from simulation with ext. damping coefficient 9 Ns/m are
similar to those from the experiment. Associated to the subharmonic oil whirl is an allmost
rigid conical mode, while the harmonic part has the curved shape of a first bending mode.
of 1000 Hz where the first resonance of the harmonic response occurs. The running
modes can also be calculated and are shown in Figure 6.9. In good agreement with the
experiments the subharmonic mode is a conical mode, while the harmonic response
consists of a bending mode.
There are however two differences between the simulations and the experiments. In the
simulations there appears a second peak in the harmonic response, which is caused by
the resonance of a second bending mode. This can not be observed in the experiments.
Furthermore, the ratio of the oil whirl frequency and the driving frequency remains
constant 12 and the shift to lower frequency ratios does not occur. This phenomenon
will be addressed later in Section 6.5.
125
6.4 Simulations for Large Rotor Model
Simulation: case 1, ext. damping: 1E2 g/s
0.05
0
0
Rotational speed [Hz]
Rotational speed [Hz]
Simulation: case 1, ext. damping: 1E1 g/s
0.05
1500
1000
500
0
0
500
1000
1500
1000
500
1500
0
Frequency [Hz]
500
0
1000
1500
Frequency [Hz]
Simulation: case 1, ext. damping: 1E4 g/s
Simulation: case 1, ext. damping: 1E3 g/s
0.05
0.04
0.02
0
Rotational speed [Hz]
Rotational speed [Hz]
0
1500
1000
500
0
0
500
1000
1500
1000
500
1500
0
Frequency [Hz]
0
Simulation: case 1, ext. damping: 1E5 g/s
500
1000
1500
Frequency [Hz]
Simulation: case 1, ext. damping: 1E6 g/s
0.05
0.01
0.005
0
Rotational speed [Hz]
Rotational speed [Hz]
0
1500
1000
500
0
0
500
1000
Frequency [Hz]
1500
1500
1000
500
0
0
500
1000
1500
Frequency [Hz]
Figure 6.10: Waterfall diagrams for simulations using the same set-up as in the experimental case 1; external damping coefficient is varying from 10−2 Ns/m to 103 Ns/m. For
small external damping the entrainment effect for driving frequencies around 1000 Hz is
strongly developed, while for higher damping the self-excited oscillation can be completely
suppressed. For D = 103 Ns/m (lower left) the subharmonic part of the response is shifted
slightly to lower frequencies, i.e. to the left of the 12 ω-line.
In order to further evaluate the model and to find an appropriate external damping
parameter more simulations with this model have been carried out. The external
damping coefficient is varied from 10−2 Ns/m to 104 Ns/m to study its influence. The
126
Chapter 6: Direct Numerical Simulation and Experimental Results
results are shown in waterfall diagrams in Fig. 6.10. Note that significant features of
the experiment such as the half-frequency oil whirl, harmonic unbalance resonance
and entrainment are reproduced. The variation of the damping coefficient shows
that the oil whirl instability can be completely suppressed by increasing the viscous
damping. Furthermore the frequency ratio of oil whirl to driving frequency can also
be influenced. In the central panel of the lower row the ratio of the two frequencies is
still constant, but slightly less than 12 . It can be said that the external damping plays
a big role. Its influence is examined in more detail in Chapter 7 where continuation
methods are used to determine the exact values for which the oil whirl is suppressed.
The frequency shift seems to depend not only on the external damping but also on the
lubrication model used. This will be investigated partly in the next section.
Simulation: case 3, ext. damping: 5x9E4 g/s
0.01
0.005
0.005
0
0
Rotational speed [Hz]
Rotational speed [Hz]
Simulation: case 3, ext. damping: 4x9E3 g/s
0.01
1500
1000
500
0
0
500
1000
1500
1000
500
1500
0
Frequency [Hz]
0
500
1000
1500
Frequency [Hz]
Figure 6.11: Simulation results with higher viscosity η = 0.2188 Pas; external damping
36 Ns/m (left) and 450 Ns/m (right): the higher viscosity leads to a stronger damping and
to the suppression of the self-excited part of the vibration response; stronger damping also
suppresses the harmonic part.
Simulation: case 5, ext. damping: 5x9E4 g/s
0.01
0.005
0.005
0
0
Rotational speed [Hz]
Rotational speed [Hz]
Simulation: case 5, ext. damping: 4x9E3 g/s
0.01
1500
1000
500
0
0
500
1000
Frequency [Hz]
1500
1500
1000
500
0
0
500
1000
1500
Frequency [Hz]
Figure 6.12: Simulation results with reduced radial bearing clearance cr = 0.01 mm: the
subharmonic part of the response is suppressed for frequencies above 1000 Hz for external
damping of 36 Ns/m (left); for external damping of 450 Ns/m (right) it vanishes completely.
The figures 6.11, 6.12 and 6.13 show results from simulations done with different parameter values, such as higher viscosity, smaller radial clearance or smaller bearing
127
6.5 Influence of Oilfilm Model on Dynamics
Simulation: case 6, ext. damping: 1E3 g/s
0.05
0
0
Rotational speed [Hz]
Rotational speed [Hz]
Simulation: case 6, ext. damping: 1E2 g/s
0.05
1500
1000
500
0
0
500
1000
Frequency [Hz]
1500
1500
1000
500
0
0
1000
500
1500
Frequency [Hz]
Figure 6.13: Simulation results with reduced bearing length B = 3.8 mm; external damping
0.1 Ns/m (left) and 1 Ns/m (right). For both damping values the subharmonic response is
quite large in contrast to the experiment.
length. In all of these simulations we can observe self-excited oscillations and unbalance oscillation. Entrainment is also present for smaller external damping. Especially
the effect of the reduction of the bearing clearance is reproduced very well. For both
values of the damping parameter the oil whirl is almost entirely suppressed. In contrast to that, in the simulations the reduction of the bearing width leads to oil whirl with
an even larger amplitude, while the corresponding experiment shows no self-excited
oscillations. This result could be due to a stronger influence of the oil-inlet which is
neglected in the model, due to inclination of the shaft, or also due to a rotor damaged
by previous experiments.
6.5 Influence of Oilfilm Model on Dynamics
In the last section the simulations of the larger 13 beam element model with short bearing lubrication model already showed quite good agreement with the experimental
data. The main difference between the calculated and the measured power spectrum
is the shift of the frequency of the oil whirl away from the line ν = ω2 . The results in Figure 6.10 show that increased damping leads to such a shift but also to the suppression
of the oil whirl.
6.5.1 Simulations with Phenomenological Bearing Model
From looking at the equations for the bearing reaction forces of the short bearing (5.159)
and (5.160), we see that the forces vanish for value of γ̇ = ω2 and κ̇ = 0. This leads to the
reasonable conjecture, that the whirl frequency can be influenced by the lubrication
model. The phenomenological model proposed in Section 5.2 is a simple modification
of the short bearing lubrication model. In Section 5.2 the factor ω2 is identified as the
128
Chapter 6: Direct Numerical Simulation and Experimental Results
modif. sh. bearing
2000
1500
1500
rot. speed [Hz]
rot. speed [Hz]
sh. bear
2000
1000
500
1000
500
Amp=0.01mm
0
0
500
1000
1500
frequency [Hz]
Amp=0.01mm
2000
0
0
500
1000
1500
frequency [Hz]
2000
Figure 6.14: Modified bearing function leads to frequency shift; left: short bearing solution
(s ≡ 0), right: ω2 -term changed to 0.46ω (s = 0.04ω) in Eq. (6.23); blue: lines ν = ω and ν = ω2 ,
red: (ν = 0.46ω)-line coincides with actual frequency of self-excited oscillation (D = 0.001,
M = 100, funb = 2.3 · 10−7 ).
lubrication fluid’s average circumferential velocity and a correction term is introduced
that changes this velocity. The resulting pressure function then looks as follows
W 2 ρν (γ̇ − ω + s(ω))κ sin ϕ + κ̇′ cos ϕ
z̄
z̄
2
−1
p̄0 = −6
.
(6.23)
W W
(1 − κ cos ϕ)3
c2r
As a first test for this phenomenological model we integrate the simple model (6.16)
using first short bearing approximation and second the corrected pressure from (6.23)
with s(ω) = 0.04ω. The comparison of the results depicted in Figure 6.14 shows that
the linearly shifted peaks of the self-excited oscillation of the corrected model lie on
ω
the ν = 2.3ω
5 line, while the peaks for the uncorrected model lie on the ν = 2 line. This
clearly indicates a direct influence of the correction on the whirl frequency.
To further investigate the influence of the modification term s(ω) we simulate a smaller
rotor model with 3 beam elements as depicted in Figure 6.7 using the phenomenologically corrected short bearing pressure function (6.23) with a quadratic correction term
2
s = σω
ω0 . The sole reason for introducing ω0 = 1000 Hz into the term is to normalize it,
so that not too small values of σ have to be used.
In Figure 6.15 the Campbell diagrams a depicted for varying values of σ ∈ [0, 0.2]. For
small values of σ the spectrum is similar to the uncorrected one in the top left corner.
All diagrams show the presence of a harmonic response with a resonance around 900
Hz. The line ν = ω on which the circles of the harmonic part lie and the line ν = ω2 are
indicated in blue. The circles indicating the self-excited oscillation are always located
2
on the curves ν = ω2 − σω
ω0 which are depicted in red. As σ grows, the red lines, and
hence the peaks of the self-excited oscillation, bend away more and more from the
blue line ν = ω2 , a behavior similar to the experiment. Additionally it can be observed
that the large amplitude oscillation which is present at 1800 Hz for σ = 0, is pushed
129
6.5 Influence of Oilfilm Model on Dynamics
sigma=0.02
2000
2000
1500
1500
rot. speed [Hz]
rot. speed [Hz]
sigma=0
1000
500
1000
500
Amp=0.01mm
500
1000
1500
frequency [Hz]
sigma=0.05
0
0
2000
2000
2000
1500
1500
rot. speed [Hz]
rot. speed [Hz]
0
0
Amp=0.01mm
1000
500
500
1000
1500
frequency [Hz]
sigma=0.1
1000
500
Amp=0.01mm
500
1000
1500
frequency [Hz]
sigma=0.15
Amp=0.01mm
0
0
2000
2000
2000
1500
1500
rot. speed [Hz]
rot. speed [Hz]
0
0
1000
500
500
1000
1500
frequency [Hz]
sigma=0.2
2000
1000
500
Amp=0.01mm
0
0
2000
500
1000
1500
frequency [Hz]
2000
Amp=0.01mm
0
0
500
1000
1500
frequency [Hz]
2000
Figure 6.15: Campbell diagrams for simulation of small beam model with phenomenological correction of short bearing. σ is the tuning parameter. The subharmonic part of the
2
response allways lies on the curve ν(ω) = ω2 − s(ω) = ω2 − σω
with ω0 = 2π · 1000 rad
.
ω0
s
towards higher frequencies and finally out of the range of simulated rotational speeds.
This illustrates again the important influence of the lubrication model on the whole
dynamics. In (Crandall, 1995) it is shown by considering force equilibria that the
onset frequency of the oil-whip, the large amplitude oscillation which starts in our
example at around 1800 Hz, is double the eigenfrequency of the first bending mode.
In (Muszynska, 1986) the fluid’s average circumferential velocity is identified as additional critical speed. The coalescence of these two critical speeds leads to this harmful
130
Chapter 6: Direct Numerical Simulation and Experimental Results
phenomenon that is the oil-whip. In this case the self-excitated vibration forces the
bending oscillation causing the large amplitudes.
6.5.2 Varying the Lubrication Model
The last section has shown the importance of the lubrication model and especially
of the average circumferential fluid velocity for the dynamical behavior of the rotorbearing system. In Chapter 5 we have derived several bearing models which are more
realistic than the short bearing approximation. In order to reproduce the experimentally observed shift of the ratio of the whirl frequency to the driving frequency these
models have tested with the small 3 beam element rotor model which has 32 degrees
of freedom (DOF), as well as with the simple model with 4 degrees of freedom (6.16).
According to (Childs, 1993) and (Yamamoto & Ishida, 2001) the zeroth order short
bearing approximation is valid for ratios of bearing length to bearing radius W
R < 1.
The bearings of the turbocharger of which the vibration behavior was examined in the
5.4
experiments has a ratio W
R = 3 and so the short bearing approxmation tends to give
erroneous results. As a first step it is therefore interesting to investigate the behavior of
the rotor when the bearing reaction forces are computed from the solution of Reynolds
equation itself without the simplification of assuming a short bearing. Furthermore
the choice of the boundary conditions which serve as a simple cavitation model could
influence the vibration behavior. Two sets of boundary conditions described in more
detail in Sections 5.3.2 and 5.5 are under consideration in the following simulations:
• the Gümbel boundary conditions prescribe the periodicity of the pressure in the
circumferential direction and environment pressure (p = 0) at the bearing ends.
After the solution the pressure is set to zero in the regions where it is negative.
• the Reynolds’ boundary conditions additionally demand that p ≥ 0 inside the
domain which leads to a free boundary problem whose solution is quite time
consuming and which has to solved every time the bearing forces are evaluated.
6.5.2.1
Simulations with Reynolds’ Boundary Conditions
Since the solution of the free boundary boundary problem with the PSOR algorithm
(cf. Section 5.4.1.3) is very time consuming, the simulations for this lubrication model
are done with the 4 DOF model (6.16). Matlab’s ode15s is used as integrator and the
5-point Laplacian (cf. Sec. 5.4.1) is used for the discretization of Reynolds’ equation.
This computational setup leads to very long computation times which definitely could
be improved a lot by applying more sophisticated numerical methods. However, the
results of this simple numerical experiment imply that the influence of the Reynolds’
boundary conditions on the vibrations is neglectable. The frequency response is shown
131
6.5 Influence of Oilfilm Model on Dynamics
Campbell diagram, Reynolds BC
rot. speed [Hz]
2000
1500
1000
500
Amp=0.001mm
0
0
500
1000
1500
frequency [Hz]
2000
Figure 6.16: Spectrum from simulation of simple model (6.16) with Reynolds’ boundary
condition the lubrication equation shows no shift of subharmonic response.
in Figure 6.16 for a few driving frequencies. The parameters used in this simulation
−7
are D = 0.5 Ns
m , u = 2.1 · 10 kg m and M = 0.1 kg. The results show no significant shift
of the subharmonic response.
6.5.2.2
Reynolds’ Equation With/Without Inertia Correction
Although they provide a cruder cavitation model, the Gümbel conditions allow for
a faster evaluation of the bearing forces, which accelerates the computation. In the
following simulations Reynolds’ equation with Gümbel boundary condition is used in
the two variants derived in Section 5.1.6. We compare the solutions of the explicit model
(6.9), that uses the zeroth order Reynolds’ equation (5.89), with those of the implicit
model (6.13), where the first order inertia corrected version of Reynolds’ equation
(5.89) is used. For the rotor itself the smaller 3 element model is used in both cases
with the parameters as given in Appendix D.2. For the explicit model Matlab’s ode15s
solver for stiff problems is used. For the implicit model the solver routine DASPK is
called from an octave script. The partial differential equations for the evaluation of the
pressure distribution in each step are solved with the deal.II package as described in
Section 5.4.2. The computations are again quite time consuming. The most critical part
is the computation of the pressure distribution from Reynolds’ equation. As we have
seen in Sections 5.4.2, many degrees of freedom are needed to compute the bearing
forces with sufficient accurateness. Less accuracy in the finite element solver often
leads to failure in the ode solvers, which in that case run into discontinuities leading
to nonconvergence of the underlying Newton method.
132
Chapter 6: Direct Numerical Simulation and Experimental Results
1000Hz
1200Hz
0.03
0.03
Reynolds
inertia corr.
0.025
0.02
0.02
0.015
0.015
0.01
0.01
0.005
0.005
0
0
500
1000
1500
frequency [Hz]
1400Hz
2000
0.025
Reynolds
inertia corr.
0.025
0
0
500
1000
1500
frequency [Hz]
1600Hz
2000
0.012
Reynolds
inertia corr.
0.02
Reynolds
inertia corr.
0.01
0.008
0.015
0.006
0.01
0.004
0.005
0.002
0
0
500
1000
1500
frequency [Hz]
2000
0
0
500
1000
1500
frequency [Hz]
2000
Figure 6.17: 3 element beam mode: comparison of the spectra of simulation results with
zeroth order (blue) and first order (green) Reynolds’ equation. No significant frequency
shift of the subharmonic can be observed.
The results are shown in Figure 6.17. The power spectrum of the rotor vibration
taken from the orbit of the turbine is shown for four different rotational speeds from
the region in which the frequency shift is observed in the experiment. The blue
peaks show the frequency response of the explicit system with zeroth order Reynolds’
equation. The green peaks show the response of the implicit system with the inertia
corrected version of Reynolds’ equation. A frequency shift which is as significant as in
the experiment cannot be observed. Computations over a longer time interval which
allow for a better frequency resolution in the spectra are not done, since a significant
frequency comparable to the one observed in experiment would be visible at this scale
and the calculation time (in the order of several days for one rotational velocity) is
prohibitively long.
6.5.2.3
Simulations Using Inertia Corrected Short Bearing Approximation
Faster computations are possible when we use the short bearing approximation which
allows for an analytic formula for the pressure distribution in the case of inertia cor-
133
6.5 Influence of Oilfilm Model on Dynamics
100Hz
−3
x 10
short b.
inertia corr.
7
900Hz
−3
x 10
short b.
inertia corr.
8
6
6
5
4
4
3
2
2
1
49
49.5
−3
x 10
50
50.5
frequency [Hz]
1400Hz
51
435
440
−3
x 10
short b.
inertia corr.
8
0
445 450 455
frequency [Hz]
1900Hz
460
8
465
short b.
inertia corr.
7
6
6
5
4
4
3
2
2
1
0
680
690
700
710
frequency [Hz]
720
0
920
940
960
frequency [Hz]
980
Figure 6.18: Comparison of subharmonic responses of simple model (6.16) for classical
short bearing (blue) and inertia corrected short bearing (green). A frequency shift toward lower frequencies of about 1% compared to response of uncorrected system can be
observed.
rected pressure, as well as in the case of zeroth order approximation. In Section 5.6 it
is demonstrated that bearing parameters similar to those form the experiment (R = 3
mm, W = 5.4 mm, cr = 0.02 mm) lead to erroneous results in the pressure distribution
especially for the inertia corrected pressure. To evaluate the influence of the inertia
correction on the frequency of the oil whirl, the simple model (6.16) is simulated (M = 1
g, D = 1e − 3g/s, u = 3e − 7gmm) with both lubrication models (5.47) and (5.63) and
a bearing configuration for which the short bearing approximation and the inertia
correction are valid (R = 3 mm, W = 0.25 mm, cr = 0.02 mm). DASPK is used for
time integration in both cases, the explicit case using the zeroth order short bearing
approximation and the implicit case using inertia corrected first order short bearing
approximation.
The results of these simulations are depicted in Figure 6.18. They show that the
subharmonic response of the inertia corrected model (green) displays a frequency shift
to the response of the model using the classical short bearing approximation (blue).
The shift, however, is not of the same magnitude as observed in the experiments,
but it is relatively small. Compared to the subharmonic response of the uncorrected
134
Chapter 6: Direct Numerical Simulation and Experimental Results
model with its ω2 -response, the response of the inertia corrected system is between
0.5% and 1% lower. This clearly shows that the inertia correction has an influence on
the self-excited oscillation, but it also indicates that the large frequency shift observed
in experiment can not be fully explained by this correction.
6.6 Conclusions from the Simulations
To conclude this chapter we summarize here the results and comment on the quality
of the proposed model.
• The simulations show that the presented beam model together with the short
bearing approximation for the hydrodynamic bearings reproduces the dynamical behavior of the turbocharger quite well. In good agreement with the experimental results it exhibits the harmonic unbalance response, the self-excited
subharmonic oscillation known as oil whirl, and the same vibration modes.
• In the model with short bearing the dependence on parameters such as bearing
width, radial clearance, and oil viscosity is similar to the experimental response
to parameter variation.
• The external damping constant is a parameter unknown in experiment. Its
influence can be used to suppress the self-excited oscillation and to influence its
frequency.
• The frequency of the oil whirl can be influenced by varying the lubricant’s average
circumferential velocity. By introducing a phenomenological correction term into
the short bearing approximation it is possible to prescribe the frequency of this
self-excited oscillation.
• The ratio of oil whirl frequency to forcing frequency differs between experiment
and simulation. The variation of the lubrication model does not have a large
effect on this ratio. The use of zeroth order Reynolds’ equation neither with
Gümbel boundary conditions, nor with Reynolds’ boundary conditions instead
of the short bearing approximation leads to a reduction of the whirl-forcing ratio.
• The inertia correction of Reynolds equation also leads to no significant shift of
the whirl-forcing ratio, while the computation time is considerably longer due
to the required solution of 3 partial differential equations in each time step.
• The inertia correction of the short bearing approximation has a small influence on
the frequency of the oil whirl. The ratio of oil whirl frequency to forcing frequency
is close to 12 for the uncorrected classical short bearing approximation. For the
inertia corrected short bearing approximation the ratios drops by approximately
1%.
6.6 Conclusions from the Simulations
135
These results suggest that the short bearing approximation is a good compromise for
simulations where the computation time is critical, such as e.g. in an shape optimization
framework. The inclusion of the phenomenological correction for the circumferential
lubricant velocity allows to reproduce a measured frequency behavior of a certain
bearing type without detailed modeling of the bearing, and without the computational
effort of solving partial differential equations in each time or optimization step.
However, the determination of the overall frequency response of the system by direct
numerical simulation is somewhat inefficient, since it is first necessary to compute also
the transient behavior at the beginning and to determine the end of the transient region.
Secondly the accuracy of the Fourier spectra depends on the length of the simulated
interval. In order to compute the frequency response more efficiently, continuation
methods seem to be more appropriate which follow the periodic orbit in parameter
space and which do not require the computaition of transients. Such methods are
introduced and applied to the above models in the following Chapter 7.
136
Chapter 6: Direct Numerical Simulation and Experimental Results
Chapter 7
Numerical Bifurcation Analysis
The direct numerical simulation of the model equations is a popular approach for the
investigation of systems response. By solving multiple initial value problems it allows
for validation of the model equations over a broader parameter range and also for the
classification of solutions. However, as we have seen in Chapter 6, the direct numerical
simulation can be very time consuming, not only because transient behavior has to
be accounted for, but also due to long data sets being necessary for the subsequent
analysis of the solution with e.g. Fourier analysis.
Numerical continuation and bifurcation methods are an useful and efficient alternative. These techniques not only provide efficient means of computing certain types of
solutions, but also allow the detection and classification of bifurcations, i.e. qualitative
changes of the solution. They are therefore better suited for extensive parameter studies. Bifurcation theory is a very broad and flourishing field. We refer to the textbooks
(Chow & Hale, 1982; Kuznetsov, 2004; Nayfeh, 2000; Nayfeh & Balachandran, 1995;
Wiggins, 1990) and the references there for more information on bifurcation theory.
There are some well-established software-packages for the computation of equilibria
and periodic solutions such as AUTO (Doedel et al., 2000), CONTENT (Kuznetsov &
Levitin, 1997), or MATCONT (Dhooge et al., 2004). These packages provide methods
for detecting pitchfork, transcritical, period-doubling, and Neimark-Sacker bifurcations. Furthermore, loci of such bifurcations can be computed in a two-parameter
plane.
In this chapter the software-package AUTO 2000 (Doedel et al., 2000) is used to study
the parameter dependency and the bifurcation behavior of equilibria and also of periodic solutions of the equations that model the dynamics of a fast rotating body in
hydrodynamic bearings. Additionally, a method proposed in (Schilder & Peckham,
2007) and implemented in the package TORCONT (Schilder, 2004) is used to continue
the quasi-periodic solutions which also have been observed in the direct numerical
simulations and in the experiments.
138
Chapter 7: Numerical Bifurcation Analysis
7.1 Short Overview of Continuation Methods and Bifurcation
Theory
Continuation methods are based on the Implicit Function theorem (Chow & Hale,
1982). For a given solution (x0 , λ0 ) ∈ X × Λ of an equation
F(x, λ) = 0
(7.1)
with a Fréchet differentiable function F : X × Λ → Z, where X and Z are Banach
spaces, and the parameter set Λ is an open set in a Banach space, the boundedness
of the derivative Dx F asserts the existence of a a function x∗ (λ) that parametrizes
the solutions in a neighborhood of (x0 , λ0 ), i.e. for (x, λ) in that neighborhood we
have F(x, λ) = 0 iff x = x∗ (λ). In points where the differentiability condition fails the
qualitative behavior of the solutions can change. These points are called bifurcation
points.
In this chapter we will mainly come across two types of bifurcations: the Hopf bifurcation and the Neimark-Sacker or torus bifurcation. The two bifurcations are closely
related, since both are occurring at the onset of self-excited vibrations. The Hopf bifurcation describes the bifurcation of a periodic orbit from a branch of equilibrium
solutions, while the torus bifurcation describes the bifurcation of an invariant torus
from a branch of periodic orbits leading to the appearance of a second frequency in
the solution.
The Hopf bifurcation theorem (Guckenheimer & Holmes, 1983; Kuznetsov, 2004) states
that generically, if an ordinary differential equation ẋ = f (x, µ), x ∈ Rn , µ ∈ R has
an equilibrium at (x0 , µ0 ) at which a pair of complex conjugate eigenvalues of the
Jacobian Dx f (x, µ0 ) crosses the imaginary axes transversally, there exists a family of
periodic orbits close to the equilibrium towards one side of the bifurcation value of the
parameter, i.e. for µ > µ0 or for µ < µ0 . The stability of the equilibrium changes in µ0
and the periodic orbit is always of a different stability type than the equilibrium on its
side of the bifurcation value.
The Neimark-Sacker bifurcation is the equivalent bifurcation for fixed points of maps.
It occurs when the one-parameter family of maps fµ : x 7→ fµ (x) has a fixed point (x0 , µ0 )
at which the Jacobian Dx fµ0 has a pair of complex conjugate eigenvalues λ, λ̄ which
cross the unit circle transversally at µ0 . Furthermore, the additional non-resonance
conditions λn , 1 for n = 1, 2, 3, 4 are required. If these conditions are fulfilled there
exists an invariant circle of the map fµ on one side of the bifurcation value. This
is of particular importance for the bifurcation of limit cycles of ordinary differential
equations. The stability of a limit cycle is given by the Floquet multipliers, which are
the eigenvalues of the stroboscopic or Poincaré map of the cycle. The limit cycle corresponds to a fixed point of the Poincaré map and if a pair of multipliers crosses the unit
circle, this fixed point undergoes a Neimark-Sacker bifurcation and an invariant circle
7.1 Short Overview of Continuation Methods and Bifurcation Theory
139
of the Poincaré map appears. This invariant circle corresponds to a two-dimensional
invariant torus that bifurcates from the limit cycle. Under the additional assumption of
normal hyperbolicity the torus persists for small parameter changes (Fenichel, 1971).
The dynamics on this invariant torus can either be periodic or quasi-periodic. The
periodic case is structurally stable, while the quasi-periodic flow can be destroyed by
arbitrarily small parameter variations. However the measure of the set of parameter
values where quasiperiodic behavior on the torus can occur is non-zero and there is
hence a non-zero probability to observe it (Wiggins, 1990).
The Implicit Function theorem is the foundation for the numerical continuation methods that are used in the following. The solutions that are continued can have different
natures. In the case of equilibrium solutions of an ordinary differential equation
ẋ = f (x, λ) with f : Rn × R → Rn , starting from a known equilibrium (x0 , λ0 ) with
f (x0 , λ0 ) = 0 and nonsingular Jacobian, a Newton-type method can be used to compute a nearby equilibrium, the existence of which is guaranteed by the above theorem.
The case of the continuation of periodic solutions of a differential equation can be
regarded as an infinite dimensional analogon of the above. Consider the function F
going from the space of continuously differentiable functions to R2n+1 , and mapping a
function ϕ on the residual of the boundary value problem
F : C1 ([0, 1]) × R × Λ → R2n+1 ,


 ϕ̇ − T f (ϕ, λ) 


F(ϕ, T, λ) =  ϕ(1) − ϕ(0)  .


Ψ(ϕ)
(7.2)
(7.3)
Then a zero of F is a solution to the boundary value problem, i.e. a periodic solution of
the original equation. The first component of F is the differential equation, the second
is the boundary condition which asserts periodicity, and the third is a phase condition
which chooses on particular solution out of the possible shifted ones. Differentiability
conditions on analogous to the equilibrium case then allow a continuation in parameter
space.
A detailed overview of the numerical methods used in AUTO can be found in (Beyn
et al., 2002; Doedel et al., 1991a; Doedel et al., 1991b; Kuznetsov, 2004). AUTO
uses pseudo-arclength continuation (Keller, 1977) which allows the continuation also
around folds. The system F(x, λ) = 0, λ ∈ R with a zero in (x0 , λ0 ) is augmented with
an additional equation which fixes the stepsize along the solution branch
F(x1 , λ1 ) = 0,
′
′
(x1 − x0 )x + (λ1 − λ0 )λ − ∆s = 0,
(7.4)
(7.5)
where (x0 , λ0 ) is the current position on the branch, (x1 , λ1 ) the unknown next point,
and (x′ , α′ the normalized tangent vector of the branch in (x0 , λ0 ). The advantage of
this formulation is, that the Jacobian of the left hand side is always nonsingular as long
140
Chapter 7: Numerical Bifurcation Analysis
as the solution is regular, i.e. the Jacobian DF = (DFx , DFλ ) of the original equation has
rank n and only one solution branch passes through the solution (Beyn et al., 2002).
Bifurcations are detected by test functions which have zeros at the bifurcation points,
e.g. the determinant of the Jacobian of the augmented system (7.4) and (7.5) is used as
test function for a branch point. In AUTO the real part of the complex eigenvalue with
smallest absolute value of the real part is used as test function for Hopf bifurcation
points.
The computation of invariant tori is currently an active area of research. An overview
of recent contributions can be found in the introduction of (Schilder et al., 2005). The
recently developed package TORCONT (Schilder, 2004) is used for the continuation
in invariant tori with quasiperiodic solutions in Section 7.7.3. The technique is based
on the computation of a Fourier approximation of the invariant circle of the Poincaré
map. More details and references are given in Section 7.7.2.
7.2 Reformulations of Equations of Motion
The equations of motion for rotordynamical systems are derived in Chapters 2 to 5.
They are an explicit system of ordinary differential equations
Mẍ + (ωG + C)ẋ + Kx = Fgr + Funb (t) + Fbear (xb , ẋb ),
(7.6)
if the bearing force function does not depend on the nodal acceleration of the bearing
nodes. In first order form it reads







 M
 −(ωG + C) −K 
 Fgr + Funb (t) + Fbear (yb ) 
 ẏ = 

 y + 
 .
(7.7)




I 
I
0 
0
If the bearing force function does depend on the nodal acceleration, the system becomes
implicit
Mẍ + (ωG + C)ẋ + Kx = Fgr + Funb (t) + Fbear (xb , ẋb , ẍb ).
The first order form is






 M
 −(ωG + C) −K 

 y +  Fgr + Funb (t) + Fbear (yb , ẏb )
 ẏ = 



I
I
0
0
(7.8)


 .

(7.9)
From Chapter 4 we recall that the mass matrix M is symmetric and positive definite, the
damping matrix C and the stiffness matrix K are symmetric and positive semidefinite,
and the gyroscopic matrix G is skew-symmetric. Fgr is a static load (e.g. gravity),
Fbear is the nonlinear bearing reaction force, and Funb = uω2 cos(ωt + ψ) is the periodic
unbalance forcing.
141
7.2 Reformulations of Equations of Motion
7.2.1 Transformations for AUTO and Internal Newton Method
For the software AUTO, the user has to supply a file which returns the right hand side
of an explicit, autonomous, ordinary differential equation for given state vector and
parameters. As described in the AUTO user manual (Doedel et al., 2000) the transformation of a non-autonomous periodically forced system like (7.7) to an autonomous
system can be done by coupling a nonlinear oscillator unidirectionally to the original
system, like e.g. the Hopf normal form
ẋ = x + ωy − x(x2 + y2 ),
(7.10)
ẏ = −ωx + y − y(x2 + y2 ),
(7.11)
which has the asymptotically stable solution x = sin(ωt), y = cos(ωt). The periodic
forcing term funb = u cos ωt in (7.7) can then be replaced by funb = uy to obtain an
autonomous system.
For a given state vector y the user supplied right hand side function has to return ẏ.
It is however not possible to transform (7.9) analytically into an explicit expression for
ẏb , the variables describing deflection and velocity at the bearing nodes. In order to
apply AUTO to (7.9), we have to solve the equation for ẏ numerically. This is possible
by applying an internal Newton method. Since the nonlinearity is only in the bearing
nodes this can be done relatively efficiently by a decomposition into variables which
affect and are affected directly by the nonlinearity and variables that are only affected
by the linear term. For this we separate y and ẏ into those variables y1 and ẏ1 that
describe displacements and velocities at the bearing and those variables y2 and ẏ2 that
describe the rest. By rearranging the system matrices we can put (7.9) in the following
form


 

 
 

 M11 M12   ẏ1   A11 A12   y1   Fbear (y1 , ẏ1 )   G1 
(7.12)

 
 = 
 
 + 
 + 

M21 M22   ẏ2   A21 A22   y2  
0
G2 
 


 Fbear (y1 , ẏ1 )   R1 



 ,
= 
(7.13)
 + 
0
R2 
where A and G are the respective permutations of



 −(ωG + C) −K 
 Fgr + Funb (t)


Ã = 
 , G̃ = 
I
0
0


 ,

(7.14)
and
R1 = A11 y1 + A12 y2 + G1 ,
(7.15)
R2 = A21 y1 + A22 y2 + G2 .
(7.16)
We can eliminate ẏ2 from the first line of Equation (7.12) by using
ẏ2 = (M22 )−1 R2 − (M22 )−1 M21 ẏ1
(7.17)
142
Chapter 7: Numerical Bifurcation Analysis
and obtain the following low-dimensional equation for ẏ1


 Fbear (y1 , ẏ1 ) 
−1
 + R1 − M12 (M22 )−1 R2 .
(M11 − M12 (M22 ) M21 ) ẏ1 = 

0
(7.18)
This equation has to be solved numerically with Newton’s method. The iteration
matrix is
J = M11 − M12 (M22 )−1 M21 − Mb ,
(7.19)
where the Jacobian Mb of Fbear with respect to ẏ1 is approximated by central finite
differences. The result of the last successful step is a good starting value for the
internal Newton method, in practice leading to convergence after a few iteration steps.
This internal Newton method is less numerically efficient than solving the implicit
equation directly, because one ends up in doing two Newton methods: one every time
the right hand side is evaluated, and one in AUTO’s continuation procedure. But since
AUTO does not allow implicit equations, this is a reasonable work-around, especially
since generally only very few internal iterations are needed.
7.2.2 Formulation in Co-Rotating Frame
So far we considered the equation of motion (7.6) in a fixed frame coordinate system.
The transformation to a co-rotating frame however yields some further insight into
the dynamics of the system. Let qi = (ui , vi , βi , αi ) be the nodal coordinates in the fixed
frame as above and pi the coordinates in a frame that is rotating about the z-axis with
rotational speed ω. Then





 T(ωt)
 cos ωt − sin ωt 
 pi , where T(ωt) = 

qi = 
(7.20)
 sin ωt cos ωt  .
T(ωt) 
Setting

 T(ωt)


..
P = 
.


T(ωt)







and


 0 −1



 1 0





.

 ,
.
H = 
.




0 −1 



1 0
(7.21)
and substituting q = Pp in (7.6) we obtain the following equation for y:
Mp̈ + (2ωMH + G + C)ṗ + (K − ω2 M + ωGH + ωCH)p = F̃bear (p, ṗ) + Fgr cos(ωt) + ω2 Funb .
(7.22)
The bearing function Fbear in (7.6) only depends on the eccentricity r, the radial velocity
ṙ and the angular velocity γ̇ of the journal center (cf. (5.159), (5.160))




 Fb,x 
 Fn (r, ṙ, γ̇) 


 .

Fb = 
(7.23)
 = T(γ) 
Fb,y 
Ft (r, ṙ, γ̇) 
143
7.2 Reformulations of Equations of Motion
If we denote the polar coordinates in the co-rotating frame by e and ψ we have the
following relations: e = r, ė = ṙ, γ = ψ + ωt and γ̇ = ψ̇ + ω. Therefore the transformed
bearing function F̃bear in (7.22) has the following form

 Fn (r, ṙ, γ̇)
= T−1 (ωt)Fbear (q, q̇) = T−1 (ωt)T(γ) 
Ft (r, ṙ, γ̇)


 Fn (e, ė, ψ̇ + ω) 
−1

= T (ωt)T(ωt)T(ψ) 
Ft (e, ė, ψ̇ + ω) 


 Fn (e, ė, ψ̇ + ω) 
= T(ψ) 
 .
Ft (e, ė, ψ̇ + ω) 
F̃bear




(7.24)
(7.25)
(7.26)
In the implicit case the transformed bearing function is of the form

 Fn (e, ė, ψ̇ + ω, ψ̈)
= 
Ft (e, ė, ψ̇ + ω, ψ̈)
F̃bear


 .

(7.27)
The formulation in the co-rotating can sometimes be more convenient than the fixed
frame version as the harmonic unbalance forcing transforms to a constant term and the
formerly static gravity load term becomes an harmonic forcing term. In the absence
of gravity load (e.g. in a vertical rotor) or any other constant load the analysis of the
dynamics is therefore simpler because the system is autonomous. Periodic orbits of the
Orbits in fixed frame
0.2
y [mm]
0.1
0
−0.1
−0.2
0.2
0
−0.2
400
200
0
x [mm]
600
800
1000
1200
1400
1600
1800
ω [Hz]
Orbits in rotating frame
0.2
y [mm]
0.1
0
−0.1
−0.2
0.2
0
−0.2
x [mm]
0
200
400
600
800
1000
1200
1400
1600
1800
ω [Hz]
Figure 7.1: Comparison of simulated orbits seen in fixed frame of coordinates (top) and
co-rotating frame (bottom); quasi-periodic orbits in the fixed frame become periodic orbits
in the co-rotating frame of reference.
system with unbalance forcing in the static coordinate system transform to equilibria
in the co-rotating frame and invariant tori transform to periodic orbits. This can be
144
Chapter 7: Numerical Bifurcation Analysis
seen from the following considerations. Let
x = aeiω0 t + beiω1 t
(7.28)
be a quasi-periodic signal with the two basic frequencies ω0 and ω1 . The transformed
signal in a rotating frame with ω0 then has the form
z = e−iω0 t x = a + bei(ω1 −ω0 )t .
(7.29)
Hence a periodic solution with frequency ω0 (i.e. b = 0) will transform to a constant
solution, while the term with frequency ω1 is transformed into another periodic term
with the frequency ω1 − ω0 . In Section 7.8 the period length Trot of a solution to the
equation of motion in the co-rotating frame of coordinates is computed with AUTO.
To calculate the basic frequency ω1 in the fixed frame of coordinates for ω1 < ω0 , we
use
2π
2π
Trot =
⇔ ω1 = ω0 −
.
(7.30)
ω0 − ω1
Trot
4π
Hence if ω1 = ω20 the period is Trot = 4π
ω0 , and if Trot < ω0 , the subharmonic frequency is
ω0
also reduced ω1 < 2 , as it is observed experimentally for the oil whirl.
The simplification achieved through the transformation is illustrated in Figure 7.1,
where we see a comparison of simulation results of a system without gravity load in
the fixed frame and in the co-rotating frame for varying rotational speed. The orbits in
the lower graph, seen in the co-rotating frame, are much simpler. Neglecting gravity
is also justified in our example for large rotational speeds, because Fgr becomes small
compared to unbalance above frequencies of approximately 500 Hz as can be seen later
in this chapter.
7.3 Linear Stability Analysis
As a first step in the bifurcation analysis we perform a linear stability analysis of the
rotor-bearing system. From the equations of motion (7.7) and (7.9) we see that in the
absence of the constant load, i.e. fgrav = 0, and of the unbalance forcing ( funb = 0),
there exists an equilibrium in the origin. For non-vanishing constant load fgrav , 0 the
equilibrium is no longer in the origin. If the bearing function is complicated one has to
calculate it numerically. The real parts of the eigenvalues of the Jacobian of the right
hand side of the equation of motion evaluated at the equilibrium point give its linear
stability. The computation of the zeros of the right hand side can be complicated if the
system is large, i.e. if many beam elements are considered. However, since nonlinearity
is brought into the system only by the bearing function which itself only depends on
the nodal coordinates of at the bearing positions, a decomposition of the equation of
motion can be used to facilitate the numerical computation of the equilibrium.
145
7.3 Linear Stability Analysis
From (7.6) we see that the condition for an equilibrium in x∗ is
0 = Kx∗ + Fgr + Fbear (x∗b , 0).
(7.31)
K is the stiffness matrix of the beam, its null space is spanned by the rigid body motions
of the beam (cf. Eq. 4.25), and therefore Kxr = 0 for a rigid translation xr of the beam.
Hence we can decompose the equilibrium position x∗ into a rigid translation xr and a
elastic bending deflection xe with fixed zero deflection at the bearing nodes (xeb = 0):
x∗ = xr + xe .
(7.32)
Substituting this into (7.31), the forces in the bearings are then given by
Fbear (xrb ) = −Kxe + Fgr .
(7.33)
Ordering K and xe in such a way that the zero components of xe are in the first rows
we can write the last equation in more details:

 K1



− 
 K3


K2
K4

 
  0   Fgr,1
 
 
 
 
 
 
  x̃e  +  F
 
  gr,2
 
 
 
  Fbear (xr )
 
 
 = 
 
 
0
 




 .



(7.34)
From this we can calculate the bending deflection
x̃e = K4−1 Fgr,2
(7.35)
and the resulting bearing forces by elimination of xe from the equation
Fbear (xrb ) = −K2 x̃e + Fgr,1
=
−K2 K4−1 Fgr,2
= F̃.
+ Fgr,1
(7.36)
(7.37)
(7.38)
Then the rigid translation in each of the bearings can be calculated by solving numerically the nonlinear equation
Fbear (xrb ) = F̃,
(7.39)
e.g. with a Newton type method. The entire rigid translation xr is calculated from
the translations at the bearings xrb by linear interpolation. The lateral shaft deflection
in equilibrium is then given by adding again the rigid translation and the bending
deflection.
Since (7.39) decomposes even further into two two-dimensional problems, one for each
bearing, this reduced problem is much easier to solve than the direct problem (7.31),
also for large systems with many finite elements.
146
Chapter 7: Numerical Bifurcation Analysis
Knowing the equilibrium position, the bearing stiffness and damping coefficients, as
well as the derivative with respect to the nodal acceleration ẍb in the case of the implicit
system (7.9)
∂Fbear,i
)i, j ,
∂xb, j
∂Fbear,i
)i, j ,
= (
∂ẋb, j
∂Fbear,i
= (
)i, j ,
∂ẍb, j
Koil = (
(7.40)
Coil
(7.41)
Moil
(7.42)
can be determined by calculating the central difference quotients of the oil film forces
around the equilibrium as in Equation (6.17) in Section 6.2.
With these coefficients one can do a linear stability analysis of the equilibrium position
by analyzing the linearized right hand side of the equation of motion without periodic
forcing


 −(M − Moil )−1 (ωG + C + Coil ) −(M − Moil )−1 (K + Koil ) 
 y

ẏ = 

I
0
= By.
(7.43)
Note that in the case of the explicit equation of motion (7.7) Moil = 0. The eigenvalues
of the matrix B in Eq. (7.43) determine the stability of the equilibrium. By computing
the equilibria and the corresponding eigenvalues for varying values of the rotational
velocity, dependence of the equilibria’s stability on this parameter can be studied.
Using the old equilibrium as a starting value for the Newton iteration of the next one is
a good guess and leads to fast convergence. If all eigenvalues have negative real part,
the system is linearly stable and for small forcing the expected response is harmonic.
If one or more eigenvalues have positive real part, the corresponding equilibrium is
unstable and self-excited oscillations may occur. A Hopf bifurcation occurs at a given
parameter value, if a pair of two eigenvalues crosses the imaginary axes transversally
at this parameter (Wiggins, 1990).
7.3.1 Hopf Bifurcation at Onset of Oil Whirl
Like in Chapter 6 we mainly study two models of a turbocharger, a larger one with
thirteen beam elements resulting in an equation of motion with 112 dimensions and
a smaller one with only three beam elements whose equation of motion has only 32
dimensions (cf. Fig. 6.7).
For rotational speeds between 10 Hz and 2010 Hz we calculate the equilibria of the
large model with 13 finite beam elements and the corresponding eigenvalues of the
Jacobian. In Figure 7.2 the real and the imaginary parts of the eigenvalues are plotted
147
7.3 Linear Stability Analysis
Realparts of eigenvalues of all modes
Re(λ/2π)
20
0
Frequency [Hz]
−20
0
500
1000
1500
Rotational speed [Hz]
Frequencies of modes
2000
500
1000
1500
Rotational speed [Hz]
2000
2000
0
−2000
0
Figure 7.2: Plot of the eigenvalues of the linearized dynamical system for the 13 beam
element model. Top: Real parts. Bottom: Frequencies of the eigenmodes given by the
imaginary parts divided by 2π. Two eigenmodes subsequently lose stability at driving
frequencies of around 100 and around 350 Hz, respectively.
in dependency of the rotational velocity ω. In the upper plot we see that two curves
cross the zero line, one around 100 Hz and one at about 350 Hz, which indicates
some vibration modes becoming unstable. In the lower plot the red line indicates
the forcing frequency. Crossings of the red line with blue eigenvalue curves indicate
driving frequencies where harmonic resonances occur. This is in agreement with the
simulation results shown in Figure 6.8 in Section 6.4, where a harmonic resonance of
the first bending mode appears at 1000Hz and a second harmonic resonance peak is
observed around 1600 Hz. In Figure 7.3 the orbits of the eigenvalues in the complex
plane are shown. One can see the pair of eigenvalues crossing the imaginary axis into
the positive half plane. This shows numerically the presence of a Hopf bifurcation at
the onset of the oil whirl.
This linear stability analysis is common in rotordynamics and is used e.g. in (San Andrés,
2006) or (Childs, 1993) to predict the stability of rotordynamic systems. However in
the presence of self-excited oscillations, the linear stability analysis can lead to wrong
conclusions, since we perform it along the then unstable equilibrium which is not
observed in the physical reality. Nevertheless, the correct perdiction of the resonance
peaks shows that the solutions with self-excited oscillation are still reasonably close to
the equilibrium, such that a linear stability analysis makes sense.
For the smaller system we observe similar behavior as can be seen in Figure 7.4. Again
two pairs of complex eigenvalues cross the imaginary axis. One pair at around 50 Hz
and the other one around 300 Hz. In the lower figure on the left, one can see that again
there is a harmonic resonance near 1000 Hz.
148
Chapter 7: Numerical Bifurcation Analysis
Orbits of eigenvalues
1000
Im(λ)
500
0
−500
−1000
−10
−5
0
5
Re(λ)
10
15
20
Figure 7.3: Close up of the origin of the complex plane with spectrum of the Jacobian of
the 13 element model: two pairs of complex conjugate eigenvalues cross the imaginary
axis indicating two subsequent Hopf bifurcations. Eigenvalues with negative real parts
are plotted in red, eigenvalues with positve real parts in blue, for small ω both pairs start
close to the origin in the negative half plane.
Orbits of eigenvalues
Realparts of eigenvalues of all modes
1000
0
Frequency [Hz]
−20
0
500
500
1000
1500
Rotational speed [Hz]
Frequencies of modes
2000
Im(λ)
Re(λ/2π)
20
0
2000
−500
0
−2000
0
500
1000
1500
Rotational speed [Hz]
2000
−1000
0
50
100
150
Re(λ)
200
250
Figure 7.4: Results of linear stability analysis of 3 element model; left: plots of real and
imaginary parts of the eigenvalues of the Jacobian; right: close-up of origin of the complex
plane showing the orbits of 2 pairs of complex conjugate eigenvalues crossing the imaginary
axis with varying ω. Eigenvalues with negative real part are plotted in red, those with
positive real part in blue; for small ω both pairs start close to the origin in the negative half
plane.
7.4 Numerical Bifurcation Analysis of Large Model
The software-package AUTO (Doedel et al., 2000) has been developed to perform
parameter continuation for a multitude of problems. In this section it is used to
investigate closer the onset of instability of the larger, 112-dimensional model of the
turbocharger (cf. 6.7) with 13 beam elements. Unfortunately the dimension of our
problem makes it difficult to use AUTO for the calculation of the bifurcations the
periodic orbits of our model (7.7) with unbalance forcing. This is mainly due to the
149
7.4 Numerical Bifurcation Analysis of Large Model
0.5
x 10
−3
Damping D=10Ns/m
0
Displ y [mm]
−0.5
1
−1
−1.5
−2
−2.5
−3
x 10
0
H
H
Displ y1 [mm]
−3
0.5
Damping D=50Ns/m
H
−0.5
−1
−1.5
−2
−2.5
0
100
200
300
ω [Hz]
400
500
−3
0
500
1000
ω [Hz]
1500
2000
Figure 7.5: y-deflection at impeller side of the equilibrium state of the large model (N = 112)
for varying rotational speed (x-axis) with lower (36 Ns/m;left) and higher (180Ns/m;right)
damping factor and no unbalance forcing; to the right of the first Hopf bifurcation (H) the
fixed point is unstable.
size and the stiffness of the problem and the resulting convergence problems for the
Newton methods used in the algorithm. In the absence of unbalance it is possible to
calculate the equilibrium and its stability for different damping factors and to locate
the Hopf bifurcations which mark the onset of the oil whirl.
Loci of Hopf Bifurcations
5
3
x 10
damping [g/s]
2.5
2
1.5
1
0.5
0
0
200
400
600
ω [1/s]
800
1000
Figure 7.6: Curves of Hopf Bifurcations in the ω-D plane; above and left of the red curve
the equilibrium is asymptotically stable; between the curves 2 unstable, below blue curve
4 unstable directions
The starting point for all continuations in the following is always the trivial equilibrium
which exists in the absence of unbalance and static load (Fgr = 0 and Funb = 0). For
150
Chapter 7: Numerical Bifurcation Analysis
increasing the Fgr this equilibrium becomes non trivial. The equilibrium branch is
followed until Fgr reaches the same value it has in the simulations. Starting from
this point, the equilibrium is now continued in the driving frequency from 10 Hz to
2000 Hz.
The results of the continuation are shown in the Figure 7.5 for two different external
damping factors. Both graphs in the figure show the y-deflection of the rotor at the
impeller side in the equilibrium state for a range of rotational speeds. The distance of
the equilibrium from the origin decreases with increasing driving frequency. Several
Hopf bifurcations are detected along the branches and marked with red H. As soon as
the rotational speed passes the frequency of the first Hopf bifurcation the equilibrium
solution becomes unstable and oil whirl occurs. For the lower value of the damping
factor a second Hopf bifurcation is detected at a higher rotational speed. Note that
for the higher damping (180Ns/m) the second Hopf bifurcation vanishes and that the
onset of self-excited oscillation is pushed to higher frequencies. The parameters used
to obtain the results in Figure 7.5 were the same as those used in the simulation shown
in Figure 6.11, except for the unbalance excitation, which is neglected here.
The prediction of the onset of instability is of crucial importance for the design of the
turbocharger and also of other rotors. A continuation of the first Hopf bifurcation in
parameter space would yield a good tool for this purpose. However, the size of the
system poses some problems to the numerical methods of AUTO and we encountered
convergence problems when we tried this continuation. As an alternative it is possible
to detect the loci of the Hopf bifurcations by several runs of AUTO which cover
the parameter region of interest. For this we compute an equilibrium branch by
continuating the starting solution in the damping parameter. From well chosen points
on this branch we repeat the continuation in ω, and we can hence detect the location
of the Hopf bifurcations on branches with different damping. For the results shown
in Figure 7.6 such a search strategy was applied. Connecting the points of the first
Hopf bifurcations (red) and those of the second Hopf bifurcations (blue) linearly we
obtain a partition of the parameter space. In the left upper corner, above the red curve
the equilibrium is asymptotically stable. Between the two curves there are 110 stable
directions and 2 unstable ones. Below the blue curve there are 4 unstable directions.
These results agree with our simulations in Section 6.4 (cf. Fig. 6.10) where the selfexcited subharmonic vibration disappears for a large external damping factor D.
7.5 Numerical Bifurcation Analysis of Small System
As mentioned in Section 7.4 there are some convergence problems in AUTO for the
large 112 dimensional system. To analyze the qualitative behavior in more detail we
use again the smaller 3-element beam model for some calculations (cf. 6.7) which has
be shown to have similar dynamical properties in Section 6.5. It consists only of 3 beam
151
7.5 Numerical Bifurcation Analysis of Small System
elements with 2 disks attached at each end of the rotor and two journal bearings. The
system therefore has 32 degrees of freedom (cf. Fig. 6.7). As explained in Section 7.2.1 2
more degrees of freedom have to be coupled to the system to make up for the periodic
forcing, leading to a 34 dimensional system.
7.5.1 Unforced System in Fixed Frame of Coordinates
First we want to examine the bifurcation behavior of the equilibrium in the absence
of unbalance forcing. For this we proceed exactly as above. Starting from the trivial
equilibrium with small ω, the static load is increased until it reaches the value from the
simulations. Then the driving frequency is increased and the equilibrium is continued
until ω = 2000 Hz.
Damping D=10Ns/m
Loci of Hopf Bifurcaction
100
ext damping D [Ns/m]
1
max displ y [mm]
0.2
0.15
0.1
0.05
0 H H
H
−0.05
0
500
1000
ω [Hz]
1500
2000
80
60
40
20
0
0
500
ω [Hz]
1000
1500
Figure 7.7: Bifurcations of equilibrium of small system. Left: 3 subsequent Hopf bifurcations (marked with H) are detected along equilibrium branch (black) for varying driving
frequency; colored branches show the maximal amplitudes of the periodic orbits emerging
at the Hopf points; only the leftmost one (red) is stable. Right: Locus curve of the Hopf
bifurcations in the frequency-damping factor domain; the first Hopf bifurcation from the
left picture lies on the black curve, the second and third Hopf bifurcations on the red curve
at the intersection with D = 10 Ns/m.
The results for the comparable smaller system are similar to the ones obtained for the
larger system, but more detailed as shown in Figure 7.7. For the unforced system
we observe three Hopf bifurcations from the equilibrium for rotational frequencies
between 0 and 2000 Hz. It is possible to follow the periodic orbits emerging from
the Hopf points. Unfortunately, there are again convergence problems for larger
amplitudes of the periodic orbits, and they can not be continued through the whole
frequency domain. Their maximal amplitude in the y-direction is shown in the colored
branches in the left diagram. At the first point (lowest frequency) a supercritical Hopf
bifurcation occurs and a stable periodic orbit branches off from the fixed point which is
unstable from then on for all higher frequencies. The frequency of the stable periodic
orbit is half the driving frequency as expected from the simulations. The two Hopf
152
Chapter 7: Numerical Bifurcation Analysis
Orbits of eigenvalues
600
400
Im(λ)
200
0
−200
−400
−600
−20
0
20
40
Re(λ)
60
80
100
Figure 7.8: Results of linear stability analysis: trajectories of eigenvalues in the complex
plane. One pair of eigenvalues crosses the imaginary axis twice with increasing driving
frequency; first from negative (red) to positive (blue), and then back. For low driving
frequencies both pairs are close to the origin.
points that are detected for higher rotational speeds seem to be linked by an unstable
periodic orbit that emerges at the Hopf bifurcation in the middle and merges again
with the unstable fixed point at the third Hopf point (higher frequency value). Since the
calculations stop due to convergence problems it is not possible to link them in AUTO.
However dimensional considerations indicate this, because the stable eigenspace of
the fixed point has dimension 32 until the first Hopf bifurcation, dimension 30 between
the first and the second one, the dimension decreases again to 28 between the second
and third Hopf point and increases to 30 again for frequencies higher than at the third
Hopf point. This periodic orbit is unstable, as already the periodic orbit emerging
from the first Hopf point is stable and stays so. This conclusion is supported by a
linear stability analysis. As can be seen from Fig. 7.8 one pair of eigenvalues crosses
the imaginary axis twice, once in positive for a lower frequency and once in negative
direction as the frequency increases. Another pair crosses the imaginary axis and stays
in the right half-plane. The latter corresponds to the stable periodic orbit emerging at
the first Hopf point, while the former corresponds to the unstable periodic orbit that
exists between the second and third Hopf point.
This interpretation is also backed by the continuation of the loci of the Hopf bifurcation
points. For this smaller system it is possible to track the locus curve in the two
parameters damping factor and driving frequency. This is shown in the right diagram
of Figure 7.7. The diagram shows the frequency-damping pairs at which the onset of
self-excited oscillation occurs, i.e. the locus curve of first Hopf bifurcation (black), and
those values of damping and frequency where the secondary Hopf bifurcation occurs.
One can see that for a damping factor larger than approximately 90 Ns/m no Hopf
bifurcation occurs anymore. This corresponds to the simulations in Section 6.4 and to
the results of the bifurcation analysis for the large system (cf. Fig. 7.6), where for higher
153
7.5 Numerical Bifurcation Analysis of Small System
damping factors no self-excited oscillation is observed. The second Hopf bifurcation is
suppressed already for smaller values of the external damping factor. The frequencies
where the red branch of Hopf bifurcations crosses the value D = 10 Ns/m in the right
panel coincide with the frequencies of the second and third Hopf bifurcation in the left
panel.
7.5.2 Forced System in Fixed Frame of Coordinates
In the presence of unbalance excitation the dynamics is slightly different. We start
again with the trivial equilibrium for Fgr = 0 and Funb = 0. As above, we trace the
equilibrium when we increase Fgr to the value specified by the design of the rotor. As
explained in Section 7.2.1 the non-autonomous forcing is replace by a two-dimensional
oscillator with a stable limit cycle, that is unidirectionally coupled to the system. By
increasing Funb to the value specified in the design, we couple in the forcing. This has
the effect that the equilibrium no longer exists. For rotational frequencies below the
frequency of the first Hopf bifurcation, a stable periodic orbit takes its place. It has
the same frequency as the forcing. AUTO allows for a continuation of the periodic
orbit with the rotational frequency as continuation parameter. With rising rotational
frequency the amplitude of the periodic orbit increases as expected from Equation (7.7)
where the amplitude of Funb grows quadratically in ω.
Damping D=10Ns/m
−3
0.05
10
0.04
8
x 10
Damping D=50Ns/m
max displ y1 [mm]
0.03
1
max displ y [mm]
T
0.02
T
T
0.01
0 T
−0.01
0
6
4
2
0
500
ω [Hz]
1000
1500
−2
0
T
500
ω [Hz]
1000
1500
Figure 7.9: Frequency-response diagrams for the smaller system with unbalance excitation, at the leftmost torus bifurcation (T) this periodic orbit gets unstable and self excited
oscillations appear which leads to a quasiperiodictours branching off; left: low damping,
right: higher damping.
In Figure 7.9 the amplitude of this periodic orbit is depicted against the rotational
frequency for two different values of the external damping factor D. For low damping
we observe three torus bifurcations marked with a red T, while for the higher value
only two of them are observed in the frequency range covered. An examination of the
34 Floquet multipliers that are computed by AUTO shows that all but one multiplier
lie inside the unit circle and one is equal to 1. This shows that the periodic orbit
154
Chapter 7: Numerical Bifurcation Analysis
is asymptotically stable for driving frequencies lower than the one of the first torus
bifurcation on the branch. At the first torus bifurcation a pair of Floquet multipliers
with non-zero imaginary part leaves the unit circle, the periodic orbits gets unstable
and a stable invariant torus bifurcates from it (Kuznetsov, 2004). At the second torus
bifurcation another pair of Floquet multipliers leaves the unit circle. At the third torus
bifurcation one pair reenters the unit circle. For higher external damping, the first
torus point which is observed on the branch also marks a pair of Floquet multipliers
leaving the unit circle. Again this has the consequence of a stable invariant torus
bifurcating from the periodic orbit which is hence unstable. However, at the second
torus bifurcation point this pair of multipliers returns into the unit circle and the
periodic orbit regains its stability. There is no crossing of a second pair of multipliers
detected. This behavior can be explained by looking at the Hopf bifurcation diagram
on the right of Figure 7.7 which shows that the region of instability of the fixed point
is like a tongue in the frequency-damping factor domain, i.e. for large damping as
well as for very small and for very large frequencies the equilibrium is asymptotically
stable. While this is not a proof for the situation in the forced case, the results of the
averaging theorem (K., 1969; Guckenheimer & Holmes, 1983) suggest that the loci of
the torus bifurcations show a similar behavior. This means that that the second torus
bifurcation which is observed in the low damping case is suppressed in the higher
damping case, and that the second torus bifurcation is the reversion of the first one.
Unfortunately it is not possible to follow the locus of the torus bifurcations themselves
in the frequency-damping factor domain. Again, the relatively high dimension of the
system leads to convergence problems in the Newton methods AUTO employs.
7.6 Fixed Frame vs. Co-Rotating Frame
As already shown in Section 7.2.2 a formulation in a coordinate frame co-rotating with
angular velocity ω can be a simplification of the system (cf. Eq. (7.22)), if there is no
constant load applied to the rotor. In this case the transformed system becomes autonomous, periodic orbits with period T = 2π
ω become fixed points, and quasiperiodic
solutions with one of the basic frequencies equal to ω become limit cycles. In the
absence of a constant load, e.g. for a vertical rotor, this makes numerical continuation
of these quasiperiodic solutions possible. However, since the trivial equilibrium is unstable in the absence of gravity, some preliminary continuations are necessary to reach
a starting solution for continuation in the driving frequency. For this, starting from the
trivial equilibrium, we increase the amplitude Funb and follow the equilibrium branch.
A Hopf bifurcation is detected along the branch for high values of the unbalance. The
stable periodic orbit emerging from it is continued backwards in the forcing amplitude
until the smaller value of Funb used in the simulation is reached. This stable periodic
orbit is the starting solution for a continuation in the driving frequency. Alternatively,
155
7.6 Fixed Frame vs. Co-Rotating Frame
one could also start from a solution computed by direct numerical simulation. However, the proposed method of reaching the starting solution by 2 continuation runs is
faster, and we can also use the other solutions computed during the process for further
continuation runs.
1000
y 0
−1000
1000
x
0
−1000
0
200
400
1200
200
0
0
0
y
200
−200
−200
0
200
x
ω=949.9Hz
−200
0
200
x
ω=1000.0Hz
0
200
x
ω=1392.7Hz
0
0
y
0
y
200
−200
0
x
200
1800
−200
200
−200
1600
1400
−200
200
−200
ω
ω=898.7Hz
200
−200
y
1000
ω=826.0Hz
y
y
ω=405.0Hz
800
600
−200
−200
x
0
200
−200
0
x
200
Figure 7.10: Continuation of periodic solutions of Eq. (7.22) in absence of constant load
w.r.t. driving frequency; the bottom line shows the detailed orbits drawn in red in the
top figure. The reduced amplitudes around driving frequencies of 1000 Hz are due to the
entrainment phenomenon also observed in the simulations.
Figure 7.10 shows the results of this continuation of the stable periodic orbits of the
system (7.22) in the co-rotating frame. These are equivalent to the quasiperiodic
solutions of the system (7.7) in the fixed frame for which both, the subharmonic and
the harmonic response are present. The orbits depicted in red in the upper diagram are
shown again in the lower row for better visibility. The entrainment of the subharmonic
in the region of resonance of the first bending mode can be observed nicely in this frame
of reference. Between 700 Hz and 1000 Hz and again at about 1500 Hz, the amplitude
of the subharmonic responce decreases significantly. This entrainment effect can also
156
Chapter 7: Numerical Bifurcation Analysis
be observed in the simulations and in the experiments in the vicinity of the resonance
of the first bending mode.
For some parameter regions the entrainment becomes so strong that the subharmonic
is completely suppressed. This is shown in the left diagram of Figure 7.11. It shows
the locus of the Hopf bifurcation which is at the onset of the oil whirl in the frequencybearing clearance ω − cr domain. In the region above the blue line the periodic orbit
Locus of Hopf bifurcation
Loci of Torus and PD bifurcations
0.01
0.005
500
1000
ω [Hz]
1500
2000
0.1
bearing clearance [mm]
0.015
0
0
Loci of Torus and PD bifurcations
0.02
bearing clearance [mm]
bearing clearance [mm]
0.02
0.015
0.01
0.005
0
0
500
1000
ω [Hz]
1500
2000
0.08
0.06
0.04
0.02
0
0
500
ω [Hz]
1000
1500
Figure 7.11: Left: Locus of the Hopf bifurcation that marks the onset of oil whirl in the
ω − cr domain for the system (7.22) in a co-rotating frame of coordinates; center and right:
Corresponding Loci of torus and period doubling bifurcations of the system in fixed frame
of coordinates. The right diagram shows a larger part of the parameter range and the locus
of the second torus and period doubling bifurcation.
exists and is stable. Below it, there exits a stable fixed point which corresponds to
a stable periodic orbit with rotational frequency omega in the fixed frame, i.e. an
harmonic response. For small values of cr the subharmonic response disappears in a
region around 900 Hz.
In order to compare this result from the slightly simplified case of zero load in the corotating frame to the full problem in the fixed frame, we have computed the location
of the onset and suppression of the oil whirl in the presence of an harmonic unbalance
forcing and a static load in the fixed frame of coordinates (7.7). The center and the
right diagram in Figure 7.11 show the results. For this, we proceed as in Section 7.5.2
to reach the small stable 2π
ω -periodic orbit present for small values of ω. Again the
technique of sweeping the parameter domain with branches started from previously
computed solutions allows to draw a locus curve of the secondary bifurcations.
The curve of Hopf bifurcations is replaced by the locus curve of torus bifurcations
whose shape is allmost identical to the former. In some parameter regions the torus
curve splits into two curves of period doubling bifurcations where a Floquet multiplier passes through −1 on each branch. The point of intersection of the torus curve
two multipliers pass through −1. This situation is known as 1 : 2-resonance. The
complicated bifurcation behavior in the vicinity of such a resonance is described in
(Kuznetsov, 2004) and in the references given in that book, especially (Arnol’d, 1987;
Gambuado, 1985). The diagram in the center shows the same part of the ω − cr domain
as the left diagram, while the right diagram shows a larger parameter region which
157
7.7 Continuation of Quasiperiodic Oscillations
0.1
Oil Whirl
Unbalance
Oscillation
0
2000
1500
1000
D
Fre
riving
quen
cy [H
z]
0.05
500
0
0
1500
1000
500
Response Frequency [Hz]
2000
(a)
(b)
Figure 7.12: Power spectrum of vibrations measured in an experiment for ramping up the
driving frequency of the rotor from 130 Hz to 1700 Hz; panel (a) shows a waterfall diagram
and panel (b) a logarithmic intensity plot.
shows the curve of secondary torus bifurcations from the unstable periodic orbit, like
they where observed in Figure 7.9 of Section 7.5.2.
This result shows that neglecting gravity and transformation to a co-rotating frame
yields a significant simplification of the equation of motion, while the prediction of the
onset of the oil whirl and entrainment remains unaffected.
7.7 Continuation of Quasiperiodic Oscillations
In this section the continuation method for invariant tori presented in (Schilder &
Peckham, 2007) is applied to the 3 beam element model of the turbocharger with
constant gravity load and unbalance forcing in a co-rotating frame of coordinates.
Loci of invariant tori with fixed rotation number are computed and compared to
periodic solutions of the system without gravity. Parts of the content of this section
are joint work with Frank Schilder, Jens Starke, Mizuho Inagaki, Hinke Osinga, and
Bernd Krauskopf, which has been published in (Schilder et al., 2007).
7.7.1 Poincaré Section for Experimental Data
The experimental results presented in Section 6.3 of Chapter 6 showed the presence
of two principal vibration modes which are due to unbalance and oil whirl. These
vibration modes have been described in detail there. For convenience we show again
a typical power spectrum from the experimental results in two different ways in
Figure 7.12. One can clearly observe the two principal vibration modes as peaks in the
waterfall diagram in panel (a) and as darker lines in the intensity plot in panel (b). The
158
Chapter 7: Numerical Bifurcation Analysis
0.01
Displ y [mm]
0.1
ω=595 Hz
(b)
0.05
0.05
0
0
0
−0.005
−0.05
−0.05
−0.005
0.1
0
0.005
0.01
−0.05
0.1
ω=998 Hz
(d)
−0.1
−0.1
(e)
0
0.05
0.1
−0.1
−0.1
(f)
0.05
0.05
0
0
0
−0.05
−0.05
−0.05
−0.05
0
0.05
Displ x [mm]
0.1
−0.1
−0.1
−0.05
0
0.05
Displ x [mm]
−0.05
0.1
ω=1092 Hz
0.05
−0.1
−0.1
ω=918 Hz
(c)
0.005
−0.01
−0.01
Displ y [mm]
0.1
ω=131 Hz
(a)
0.1
−0.1
−0.1
0
0.05
0.1
ω=1683 Hz
−0.05
0
0.05
0.1
Displ x [mm]
Figure 7.13: Orbits of turbine dynamics (gray) overlaid with their Poincaré sections (black)
measured for different rotational speeds (note the different scale for ω = 131 Hz). Especially
for driving frequencies of 595 Hz and 918 Hz the invariant circles on the tori are clearly
visible, while for higher driving frequencies the invariant circles are broken up and show
phase locking.
ratio of the frequency of the subharmonic to the driving frequency is approximately
1
2 for low driving frequencies. For driving frequencies higher than 1000 Hz the ratio
drops significantly below 12 .
Figure 7.13 shows the increasingly complex behavior of the orbits measured at the
turbine end of the shaft. For small rotational speeds (250 Hz) the orbit is periodic with
a small amplitude. As the driving frequency rises above 400 Hz a second frequency
appears, which results in quasiperiodic dynamics on a torus. Such behavior is best
analyzed by stroboscopic or Poincaré maps: We mark the position of the turbine every
time the impeller crosses the x-axis from positive to negative values. The periodic orbit
of the turbine end of the shaft that we observe for low rotational speeds corresponds
to a fixed point of the stroboscopic map; see Figure 7.13 (a). For increasing rotational
speeds the Poincaré map shows invariant circles indicating the existence of invariant
tori; see Figures 7.13 (b) and (c). For even higher speeds the invariant circles show
phase locking, i.e. the quasiperiodic solution is replaced by stable periodic orbits of
possibly longer periods on the invariant torus; see Figures 7.13 (d)–(f).
159
7.7 Continuation of Quasiperiodic Oscillations
7.7.2 Computation of Quasiperiodic Oscillations
A vibration with two or more (but finitely many) incommensurate frequencies is a
quasiperiodic solution of an ODE. Such quasiperiodic solutions appear e.g. in coupled
or forced oscillators. A quasiperiodic solution never repeats and densely covers an
invariant torus in phase space. The experimental data shown in Figures 7.13 (b) and (c)
provide an example for such a behavior. In (Schilder & Peckham, 2007) a method is
presented for the computation of quasiperiodic solutions with two incommensurate
frequencies; for further references see also (Ge & Leung, 1998; Schilder et al., 2006;
Schilder et al., 2005). The basic idea of this method is to compute an invariant circle
of the period-2π/ω1 stroboscopic map, which is the intersection of the torus with the
plane t = 0. Here, in the case of a forced oscillator, ω1 = ω is the forcing frequency and
time is interpreted as an angular variable modulo the forcing period. By construction,
the invariant circle has rotation number ̺ = ω2 /ω1 , where ω2 is the additional response
frequency of the occurring vibration. The invariant circle with rotation number ̺ is a
solution of the so-called invariance equation
u(θ + 2π̺) = g(u(θ)),
(7.44)
where u is a 2π-periodic function and g is the period-2π/ω1 stroboscopic map of (7.22).
We approximate the invariant circle u with a Fourier polynomial of the form
uN (θ) = c1 +
N
X
c2k sin kθ + c2k+1 cos kθ
(7.45)
k=1
and compute the real coefficient vectors c1 , . . . , c2N+1 by collocation at the uniformly
distributed points θk , k = −N, . . . , N on the circle S1 . The stroboscopic map g is
computed with the second-order fully implicit midpoint rule as the solution of a twopoint boundary value problem. For this, we demand for k = −N, . . . , N, that uN (θk )
and uN (θk + 2πρ) are connected by the flow of the equation of motion, i.e.
ẋk = T1 f (xk ),
(7.46)
xk (0) = uN (θk ),
(7.47)
xk (1) = uN (θk + 2πρ),
(7.48)
P1 (x) = 0,
(7.49)
P2 (UN ) = 0.
(7.50)
P1 and P2 are scalar phase conditions which fix an initial point of one solution from
{x−N , . . . , xN } and {u−N , . . . , uN }, since tori, like periodic solutions, are only unique up
to phase shifts. For more details we refer to (Schilder & Peckham, 2007) where the
method is introduced, and to Figure 7.14, where the functions xk and uN are depicted
on the torus.
To start the continuation of the torus in the two continuation parameters driving
frequency ω1 and bearing clearance cr , it is necessary to construct starting solutions.
160
Chapter 7: Numerical Bifurcation Analysis
Solution Segments
x(t0+T1)
x(t0)
x(t )
0
u (θ)
N
(a)
(b)
Figure 7.14: Illustration of the invariance equation (7.44). The solution curve starting at the
point x(t0 ) crosses the invariant circle again in the point x(t0 + T1 ) after one period (a). In
angular coordinates on the invariant circle we have x(t0 + T1 ) = u(θ0 + 2π̺). If we identify
the circles at both ends of the tube, we obtain a torus (b).
These seed solutions for our subsequent continuations of tori are computed with the
method of homotopy. To this end, we introduce an artificial parameter λ ∈ [0, 1] as an
amplitude of the gravitational forcing:
M ÿ + (ωG + C) ẏ + Ky = e
Fb (y, ẏ) + λA(ωt)T Fg + ω2 Funb .
(7.51)
In the following the case λ = 0 is referred to as the zero-gravity system and the case λ = 1
as the Earth-gravity system. The principle of homotopy is to compute a torus for λ = 0,
where (7.51) is autonomous and a torus is easily constructed, and then to follow this
torus as λ is slowly increased up to λ = 1. In the autonomous case, we can construct
an invariant torus directly from a N-th order Fourier approximation of the form (7.45)
of a T2 = 2π/ω2 -periodic solution of the zero-gravity system with frequency ω2
x(t) = uN (ω2 t) = uN (θ),
(7.52)
where θ = ω2 t. For the period-2π/ω1 stroboscopic map of this solution we have by
definition
x(t + T1 ) = g(x(t)) = g(uN (θ)),
(7.53)
and we also have
x (t + T1 ) = x
θ
ω2
1
2π
θ + 2π
=x
= uN θ + 2π̺ .
+
ω2 ω1
ω2
ω1
(7.54)
So the Fourier approximation already fulfills the invariance equation. Finally, the
solution segment of the seed solution which connects a starting point x(t0 ) with the
endpoint x(t0 +T1 ) is given by x(t) = uN (θ), where (θ−t0 /ω2 ) ∈ [0, 2π̺] (cf. Figure 7.14).
For the system under consideration it turns out that the zero-gravity tori are such
accurate approximations to the Earth-gravity tori that the latter can be computed in
just one homotopy step. In panels (b) and (c) of Figure 7.15 a series of computed tori in
the zero-gravity system and in the Earth-gravity system is depicted for varying radial
bearing clearance cr and driving frequency ω. The zero-gravity tori are almost identical
7.7 Continuation of Quasiperiodic Oscillations
161
to the Earth-gravity tori. This is a first indication that neglecting gravity is a valid and
powerful simplification of the model equation, since the computation of invariant tori
(for λ = 1) is a much harder problem than the analysis of periodic solutions (for λ = 0).
Hence, a reduction of this numerical complexity is desirable.
Furthermore, the behavior of periodic solutions with respect to parameter variations
can be studied by changing the parameters independently and perform one-parameter
continuation. For quasiperiodic tori this is not true, since quasiperiodic solutions
are not structurally stable, i.e. a quasiperiodic vibration with two incommensurate
frequencies can be changed into a phase-locked state by arbitrarily small changes
in any parameter (Wiggins, 1990; Kuznetsov, 2004; Strogatz, 2000). Therefore, the
continuation of quasiperiodic tori requires two free parameters to follow solutions with
fixed frequency ratio (rotation number), and their loci are curves in a two-parameter
plane. For example, note the slight shift in some of the positions of Earth-gravity tori
in Figure 7.15 (a) with respect to the seed solutions. The union of the locus-curves of
quasiperiodic tori covers a set of large measure (Glazier & Libchaber, 1988; Kuznetsov,
2004; Strogatz, 2000) in parameter space. In other words, there is a non-zero probability
to observe quasiperiodic behavior in physical systems.
According to the above construction, the computation of periodic solutions for λ = 0 is
equivalent to a computation of invariant tori for λ = 0. It shows that these tori are good
approximations of the tori for λ = 1 for certain parameters. The computation of the
invariant tori allows one to identify such parameter regions where neglecting gravity
is a sound assumption, i.e. where the tori of the Earth-gravity system and the periodic
solutions of the zero-gravity system do not differ significantly. In these regions one
can obtain the response behavior of the turbocharger model much easier by studying
the periodic solutions of the autonomous ODE (7.51) with λ = 0.
7.7.3 Computational Results
To test the validity of the zero-gravity assumption we sweep the two-parameter plane
of radial bearing clearance cr and forcing frequency ω with a large number of curves
of Earth-gravity tori with fixed rotation number to obtain a picture as complete as
possible. We then compare these results with the respective computations of periodic
solutions for the zero-gravity approximation. All the computations were performed
on Equation (7.51) in co-rotating coordinates. Note that, due to the shift in frequencies,
the rotation numbers ̺ f in the fixed frame and ̺r in the co-rotating frame systems are
related via ̺ f = |̺r − 1|. In the results below we find ̺r ≤ 1, thus ̺ f = 1 − ̺r . This
corresponds to the results in (7.30) where a similar transformation is obtained for the
period length T2 of the periodic solutions in the zero-gravity system.
The starting point for the whole computational process is a Fourier approximation of
the periodic solution of the zero-gravity system for ω = 1000 Hz and cr = 0.02 mm. It is
162
Chapter 7: Numerical Bifurcation Analysis
1150
0.4
0.4
0.3
0.3
1000
950
1
2
3
4
900
850
Displ y [mm]
1050
Displ y [mm]
Driving Frequency [Hz]
1100
0.2
0.1
2
4
3
0
1
0.2
0.1
4
2
3
0
1
800
−0.1
−0.1
750
700
0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Bearing Clearance [mm]
(a)
−0.2
−0.3
−0.2
−0.1
0
0.1
Displ x [mm]
(b)
0.2
0.3
−0.2
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
Displ x [mm]
(c)
Figure 7.15: Positions of the seed solutions (label ×) and corresponding earth-gravity
tori (label ◦) in the (cr , ω) plane (a). The invariant circles for Earth gravity (b) and the
corresponding periodic solutions for zero gravity (c) for the starting positions along the
row near ω ≈ 894 Hz. The tori at the labeled positions are shown in Figure 7.16.
obtained by simulation and a subsequent Fourier transformation with N = 15 Fourier
modes in (7.45). Then, by continuation of this periodic solution with respect to the
forcing frequency or the bearing clearance, we cover the parameter space with a set of
starting points for subsequent torus continuation. These starting solutions are shown
in Figure 7.15 (a) as the columns and rows of crosses. The covered parameter range is
cr ∈ [0.01 mm, 0.08 mm], which is the design margin of the used journal bearing, and
ω ∈ [700 Hz, 1200 Hz], which is a principal range of operation for the turbocharger.
Furthermore, the entrainment occurs in this frequency range in the experiment.
As explained in the previous section, initial approximations of tori in the Earth-gravity
system can be constructed from these periodic solutions. These tori are computed with
N = 15 Fourier modes and M = 100 Gauß collocation points and the mesh size is kept
fixed for all subsequent computations. In the homotopy step we keep the radial bearing
clearance cr fixed and take the forcing frequency ω as a secondary free parameter. The
obtained starting positions of tori are marked with circles in Figure 7.15 (a). Note
that most of the starting positions coincide with the seed positions. The differences in
the forcing frequencies mean that tori with a certain rotation number are observed for
slightly different rotational speeds in the two systems. In other words, the response
frequencies differ somewhat.
It shows that the distribution of starting solutions is dense enough to cover the (cr , ω)plane with loci of tori with fixed rotation numbers so that meaningful conclusions
can be drawn. If the locus curves were to scarce in the parameter plane, more seed
solutions with different rotation numbers would have to be computed.
In panels (b) and (c) of Figure 7.15 we compare the two types of solutions. Both graphs
illustrate the change of the invariant circle in the stroboscopic map as the bearing
clearance is increased and the forcing frequency is kept (approximately) constant
ω ≈ 900 Hz. The two sets of circles are clearly very similar. The full tori for the
starting positions labeled 1 to 4 are shown in Figure 7.16 together with a plot of the
163
7.7 Continuation of Quasiperiodic Oscillations
0.3
(a)
0.2
0
Displ y1 [mm]
Displ x2 [mm]
0.05
−0.05
lx
Disp 1
0.2
[mm
−0.4
]
0.3
0.2
0.1
0
−0.1
−0.4
−0.2
−0.2
Displ y 1 [mm]
0
0.2
0.4
0
0.2
0.4
0
0.2
0.4
0
0.2
0.4
Displ x1 [mm]
0.3
(b)
0.2
0
Displ y1 [mm]
Displ x2 [mm]
−0.1
−0.2
0
−0.05
lx
Disp 1
0.2
0.1
0
−0.1
−0.2
0
[mm
−0.2
]
−0.4
0.3
0.2
0.1
0
−0.1
−0.4
−0.2
−0.2
Displ y 1 [mm]
0.05
Displ x1 [mm]
0.3
(c)
0.2
0
Displ y1 [mm]
Displ x2 [mm]
0
−0.2
0.05
−0.05
Disp
0.2
0.1
0
−0.1
l x1
−0.2
0
[mm
−0.2
]
−0.4
0.3
0.2
0.1
0
−0.1
−0.4
−0.2
−0.2
Displ y 1 [mm]
0.05
Displ x1 [mm]
0.3
(d)
0.2
0
Displ y1 [mm]
Displ x2 [mm]
0.1
−0.05
l x1
Disp
0.2
0.1
0
−0.1
−0.2
0
]
[mm
−0.2
−0.4
0.3
0.2
0.1
0
−0.1
Displ y 1 [mm]
−0.2
−0.4
−0.2
Displ x1 [mm]
Figure 7.16: The left-hand column of (a)–(d) shows starting tori with labels 1, 2, 3 and 4,
respectively, along the row ω = 894 Hz in Figure 7.15 (a). The corresponding x- and ydisplacements at the first FEM-node are shown in the right-hand column. The dark closed
curve is the invariant circle of the period-2π/ω1 stroboscopic map.
164
Chapter 7: Numerical Bifurcation Analysis
x-y-displacements at node 1. A comparison of the results in Figure 7.16 (b) with the
experimentally observed orbits and Poincaré sections depicted in Figure 7.13 (c) shows
that, even though our finite beam-element model with oil-film forces is quite coarse,
the numerical results are in good qualitative agreement with the experiment. Note
that the Poincaré sections are defined differently in these figures.
A comparison of the two sets of resulting two-parameter curves is shown in Figure 7.17
with loci of tori with fixed rotation number in panel (a) and loci of periodic solutions
with fixed frequency ratio in panel (b). The color bar indicates the rotation numbers
that are associated with these curves. For easier interpretation they are shifted back
to fixed-frame frequencies. The second response frequency ω2 is the product of this
shifted rotation number with the driving frequency shown on the vertical axis
ω2 = ρ f ω.
(7.55)
The curves match very well: Only in a band around 900 Hz there are some visible
differences, which are small. At a first glance we observe that in the region covered the
frequency ω2 of the self-excited oscillation is approximately half the driving frequency,
in accordance with the experimental data (cf. Figure 7.12). The line with constant
cr = 0.02 mm and varying driving frequency is of particular interest, since this is the
value of the bearing clearance used in the current design of the turbocharger. Along
this cross-section the rotation number decreases initially, stays almost constant for
ω ∈ [830 Hz, 970 Hz] and then starts to increase again. This behavior occurs in the
same region as the shift of the oil whirl response frequency away from the straight line
ω2 = 0.5ω in Figure 7.12. However, in contrast to the experiment, it returns to ρ f = 12
for even higher frequencies.
Figure 7.17 also shows two dashed bifurcation curves, namely, a locus of NeimarkSacker bifurcations (a) and the corresponding locus of Hopf bifurcations (b). These
curves are parts of the curves shown in Figure 7.11 and match very well each other in
this frequency range. For small bearing clearance, to the left of the Neimark-Sacker
curve, the response is periodic and has the same frequency as the forcing. In the zerogravity system this corresponds to an equilibrium solution. If this curve is crossed
from left to right the quasiperiodic response is born and its amplitude grows rapidly
as the bearing clearance is further increased. This can be seen from the invariant
circle in Figure 7.15 (b) and the increasingly larger tori in Figure 7.16. Again the
zero-gravity system exhibits very similar behavior, as is illustrated with panels (b)
and (c) of Figure 7.15, where periodic solutions are compared with invariant circles
of tori along the line ω = 894 Hz. These results indicate that for the range of forcing
frequencies considered here a reduction of the bearing clearance could dramatically
reduce the amplitude of the quasiperiodic vibration or even suppress the second
frequency completely.
Figure 7.17 (a) shows a total of 51 curves of tori and along each curve we computed
200 tori, which is the reason why some curves end in the middle of the figure. The
165
7.7 Continuation of Quasiperiodic Oscillations
1150
(a)
0.491
1100
0.485
1000
0.479
950
0.474
900
850
Rotation number
Driving Frequency [Hz]
1050
0.468
800
0.462
750
700
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
1150
(b)
0.491
Driving Frequency [Hz]
1050
0.485
1000
0.479
950
0.474
900
850
0.468
800
Forcing−response frequency ratio
1100
0.462
750
700
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Bearing Clearance [mm]
Figure 7.17: Curves of quasiperiodic tori with fixed rotation number of the Earth-gravity
system (a) and curves of periodic solutions with fixed frequency ratio of the zero-gravity
system (b). The diagram gives an overview of the second response frequency as a function
of the bearing clearance and the forcing frequency. The color bar indicates the rotation
number or frequency ratio associated with each curve, which was shifted back to fixedframe frequencies for easier interpretation. The dashed curve in panel (a) is the locus of
the Neimark-Sacker bifurcations, and the dashed curve in panel (b) is the locus of Hopf
bifurcations (cf. Fig. 7.11). The second frequency is suppressed to the left of these curves,
that is, there are no tori for bearing clearances smaller than ≈ 0.01 mm.
166
Chapter 7: Numerical Bifurcation Analysis
computation of the tori took approximately four weeks on an Intel Xeon CPU 2.66GHz,
that is, the average time to compute one torus is about four minutes. The computation
of the corresponding curves of periodic solutions in panel (b) with the same number
of solutions along each curve was completed within 24 hours, that is, the computation
of one periodic solution takes about nine seconds. Therefore, neglecting the gravity
forcing in (7.51) yields a drastic gain in computation time, while the accuracy of the
results is only slightly affected. As our computations show, the qualitative behavior
of the two systems virtually coincides for the investigated parameters, and one might
ask whether the introduced approximation error is at all significant. The results clearly
suggest that one could perform an analysis of periodic solutions of the zero-gravity
system and look at the Earth-gravity system only for reference and verification.
7.8 Influence of inertia terms
In the previous sections we have seen that continuation methods can be efficiently
used to compute the frequency response of forced oscillating systems. In Section 7.7.3
it was demonstrated the transformation of the equation of motion into a co-rotating
frame of reference and the subsequent neglecting of the gravity forcing provides a
very useful and yet still accurate simplification in the parameter range of interest.
We will now use this knowledge to examine the influence of the inertia correction
to the short bearing solution of the pressure distribution in the journal bearing. The
exact formula for this correction is given in Statement 5.2 in Chapter 5 and we will
not repeat it here. The important difference to the previous sections is the implicit
nature of the equation of motion, because the corrected pressure distribution depends
on the nodal acceleration. In Section 7.2.1 it is shown how this can be overcome by
an internal Newton method, so that we still can apply AUTO for the continuation
of the periodic orbits. The correction parameter σ ∈ [0, 1] is introduced to perform
homotopies from the uncorrected pressure distribution p = p0 to the inertia corrected
pressure distribution p = p0 + εp1 by p(σ) = p0 + σεp1 . Hence, the bearing forces depend
on σ and are equal to (5.159) and (5.160) for σ = 0 and to (5.168) and (5.170) for σ = 1.
7.8.1 Simple 4-D System
In order to illustrate the influence of the pressure correction and also its validity
over a larger range of parameters, we examine the simple 4-D bearing model (6.16).
Transformed to a co-rotating frame of coordinates the equations of motion are
1 −D(ẋ − ωy) + F̃bear + ω2 Funb ,
M
1 −D( ẏ + ωx) + F̃bear .
ÿ = ω2 y − 2ωẋ +
M
ẍ = ω2 x + 2ω ẏ +
167
7.8 Influence of inertia terms
Note that here Funb is transformed to a constant load. The gravity forcing has already
been dropped.
To investigate the ratio of forcing frequency and response frequency directly, we additionally scale the time such that in the fixed frame the forcing has period 1
2πτ = ωt,
(7.56)
which results in
′′
x
y′′
πF̃bear,1 (σ, W)
1
D
= 4π πx + y +
+ πFunb
− (x′ − πy) +
M ω
ω2
!!
πF̃bear,2 (σ, W)
D ′
1
′
− (y + πx) +
= 4π πy − x +
,
M ω
ω2
′
!!
,
(7.57)
(7.58)
where ′ denotes the derivative with respect to τ. We identify 4 parameters in the
system which we will use for continuation in the following: the driving frequency
ω, the amplitude Funb of the unbalance forcing, the correction factor σ which couples
in the inertia correction, and the bearing width W, which we use to examine the
range of validity of the inertia correction. All other parameters, especially the bearing
radius and clearance, remain fixed at the usual values (R = 3 mm and cr = 0.02 mm,
D = 0.001 g/s, M = 1 g).
The starting values for the parameters are ω = 16π, σ = 0 and W = 0.25 mm. For
Funb = 0 the system has a trivial fixed point for any value of ω, which is unstable for
most values of ω. Continuation of the equilibrium with increasing Funb as continuation
parameters yields a non-trivial branch of equilibria. On this branch a Hopf bifurcation
occurs after which the equilibria are stable. This happens for relatively large values
of Funb . The stable equilibria of the system in the co-rotating frame correspond to
periodic orbits of the system in the fixed frame, which are phase-locked to the driving
frequency in the case of strong forcing. In the next step we continue the stable periodic
orbit that emerges at the Hopf bifurcation point backwards in Funb and pick a solution
with Funb = 10−3 mm g. Now, we split the continuation process. First, a family A
of periodic orbits is computed with respect to the driving frequency. Then, another
family B of periodic orbits is computed after the inertia correction is switched on,
i.e. the starting solution is continued until σ = 1, and then the driving frequency is
increased. From each family we pick 20 solutions in steps of 100 Hz on the branches.
All these are continued in the bearing width from W = 0.25 mm, a value for which the
additional assumption (5.22) on the relation of width and radius is fulfilled, to W = 2.
In Figure 7.18 we show the results of these continuations. We plot the continuation
branches in the driving frequency – bearing width plane (ω − W). The period length
of the corresponding periodic orbit is color encoded with shades of red indicating a
period length T close to 2 and shades of blue indicating lower values of T. The graph
on the left hand side shows the families of periodic solutions without inertia correction
initiating at W = 0.25 from the previously computed branch A. The graph on the right
168
Chapter 7: Numerical Bifurcation Analysis
inertia corrected short bearing
2
1.5
1.5
1
0.5
0
0
2
1.99
bearing width [mm]
bearing width [mm]
short bearing w/o inertia correction
2
500
1000
ω [Hz]
1500
2000
1.98
1
1.97
0.5
0
0
1.96
1.95
500
1000
ω [Hz]
1500
2000
Figure 7.18: Branches of periodic solutions in the ω − W plane with the corresponding
period length color encoded. The thicker points show the actually computed solutions.
Left graph: Along the branches without inertia correction the period length varies very
little and stays close to 2. Right graph: The solutions to the inertia corrected problem in
the co-rotating frame coordinates show reduced period lengths, especially around W = 1.5
mm and ω = 1000 Hz, which corresponds to increased period lengths the fixed frame, i.e.
reduced response frequencies.
hand side shows the families of solutions initiating from the branch B computed before
with the inertia correction switched on.
The period length of the uncorrected model show almost no dependence on ω and W
along the branches and the period length remains 2 over the whole parameter range.
This corresponds to a self-excited oscillation with frequency ω2 = ω2 which we have
also observed in most of the simulations.
It can clearly be seen, that already for W = 0.25 the period length of the inertia corrected
solutions is reduced. In the fixed frame of coordinates this corresponds to a reduction of
the frequency of the self-excited oscillation, as we have shown in (7.30). For increasing
W the frequency shift gets stronger until approximately W = 1.5. From this value on,
the period length increases again on the branches with higher driving frequency ω
and decreases further on those with lower ω. For small W the frequency shift between
the branches on the left and those on the right is constant. This corresponds to a
linear shift of the subharmonic response. For higher W however, the difference in
period length between the branches first increases with omega until around ω = 1000
Hz. This corresponds to the ’bending’ away of the oil whirl peaks from the ω2 -line
in Figure 7.12. For ω > 1000 Hz the difference in period length decreases again, a
behavior not observed in experiment. Note that the value W = 5.4 mm used in the
previous simulations cannot be reached along the read branches as the computations
break down on the way, due to strongly increasing periods. This can be explained by
the fact that especially the inertia correction for the short bearing (5.63) is no longer
valid for larger values of W and gives erroneous results, while the uncorrected short
bearing solution still yields quite reasonable results. In Section 5.6 the analysis of the
asymptotic behavior of the bearing integrals showed similar results.
7.8 Influence of inertia terms
169
However, for small values of the bearing width for which both orders of the short
bearing approximation are valid, the inertia correction leads to a significant reduction
of the subharmonic response frequency over the whole frequency range. Allthough,
there can be other explanations for the shift of the frequency ratio, the results in this
section show that the influence of the inertia correction has to be considered as a
possible explanation of the this experimentally observed behavior.
7.8.2 Inertia Correction for 3-Element Model
In the previous section we have seen, that the inertia correction has an effect on the
frequency of the self-excited oscillation. We have noticed there, that in the case of the
zeroth order approximation of the bearing pressure distribution the frequency ratio
of self-excited oscillation and forcing do not depend on the bearing width, but they
do in the case of first order approximation. For larger values of W the results tend
to be erroneous as we have also seen in Section 5.6. However, values for W small
enough for inertia correction to be valid do not suffice to contain a rotor with the given
specifications.
However, to show the applicability of the method also to the 32-dimensional model
of the turbocharger, we computed continuations of periodic orbits in the correction
parameter σ for 2 different driving frequencies (ω = 20 Hz, 1200 Hz) using a bearing
width of W = 5.4 mm as it was also used in the experiments. The equation of motion
is given by (7.22) and like in the previous section we introduce the inertia correction
parameter σ with which we do a homotopy from bearing function given by the zeroth
order short bearing approximation to the first order approximation.
The starting solutions are chosen from the family of periodic orbits which has been
computed in Section 7.6 with the zeroth order short bearing approximation as bearing
model (5.159),(5.160), and which is depicted in Figure 7.10. Along this family of
periodic orbits, the ratio of the frequency of the self excited oscillation and the forcing
is not constant. This figure corresponds to the cut along the line cr = 0.02 in panel (b)
of Figure 7.17 which is described in Section 7.7.3. We observe that the ratio drops
significantly in the frequency range around 900 Hz where we have a resonance of
the first bending mode (cf. the linear stability analysis in Fig. 7.4). Note, that the
frequencies of the solution seen in the co-rotating frame and in the fixed frame of
coordinate transform into each other according to Equation (7.30). To facilitate the
comparison with the experimental results, we show the frequency ratio calculated in
the fixed frame.
From the starting solution we initiate the continuation by increasing the parameter
σ. The continuation process is quite slow because of the internal Newton method.
It is not possible to reach the value σ = 1, because the period length of the periodic
orbit becomes very large and the computations stop due to convergence problems. In
170
Chapter 7: Numerical Bifurcation Analysis
0.5
ω2/ω
0.49
0.48
0.47
0.46
0.45
0
500
1000
ω
1500
Figure 7.19: Ratio of the frequency of the self-excited oscillation and the forcing frequency
for family of periodic orbits depicted in Figure 7.10. In contrast to the experimental result,
the ratio increases to 0.5 again after the entrainment region around 1000Hz.
ω=1200 Hz
500
0
0
y
y
ω=20 Hz
50
−50
50 0
−50
x
0.04
0.02
σ
0
0.06
−500
500 0
−500
x
ω2/ω
0.55
0.5
0.45
0
2
7
6
−3
x 10
5
4
3
σ
0.55
2
ω /ω
0.6
1
0.01
0.02
0.03
σ
0.04
0.05
0.06
0.5
0.45
1
2
3
4
σ
5
6
7
−3
x 10
Figure 7.20: 2 families of periodic orbits with different driving frequency computed by
continuation in the inertia correction factor σ; the lower graphs show the dependence of
the frequency of the self-excited oscillation from σ. The shift of the frequencies goes to the
opposite direction as the one observed in the experiments.
Figure 7.20 we depict the two families of periodic solutions in dependence of σ. While
the orbit structure does not change very much in both cases, the ratio of the frequencies
increases with σ. For the low forcing frequency the change of the ratio is smaller than
for the large forcing frequency.
These results are in contrast to what we observe in the experiment. There, no shift away
from the frequency ratio 21 is observable for low driving frequencies, while for higher
driving frequencies the ratio drops below 12 . So, the results show that for larger values
of the bearing width the inertia correction of the short bearing pressure distribution
7.9 Conclusions
171
does not explain the frequency shift. However, this is not a surprising result, since it
was already shown in Section 5.6 that the validity of the short bearing approximation,
and especially of the first order correction p1 5.63 is not given for W = 5.4 mm.
Nevertheless, these computations show that the continuation of periodic orbits is also
possible if we apply the internal Newton method to solve the implicitly given equation
of motion. The steps taken by the predictor of AUTO are small enough, such that the
last value computed for ẋ in the user supplied function is also good starting point for
the internal Newton method, and convergence is achieved in very few iteration steps.
It can hence be used in the evaluation of other bearing models even if these involve
implicit terms, and can therefore assist in the future development of a lubrication
model which describes the frequency shift of the subharmonic response better than
the lubrication models used here.
7.9 Conclusions
In this chapter we analyzed the bifurcation behavior of several models for fast rotating
bodies in journal bearings. Two models of a turbocharger were used as examples. A
four dimensional prototypical model was also analyzed. We summarize our findings:
7.9.1 Torus Bifurcation at the onset of Oil Whirl
The continuation of the periodic orbits with respect to the driving frequency show that
a torus bifurcation occurs at the onset of the oil whirl of the rotor of the turbocharger.
The periodic orbit of the harmonic response becomes unstable at the critical frequency
and a stable invariant torus bifurcates from it. This behavior is observed in both, the
large 112-dimensional and the smaller 32-dimensional turbocharger model. In the case
without unbalance forcing a Hopf bifurcation occurs at a frequency close to the critical
frequency of the forced case.
7.9.2 Computation of Locus Curves and Suppression of Whirl
The locus curve of the Hopf bifurcation at the onset of instability can be computed directly with AUTO in the damping-driving frequency domain as well as in the bearing
clearance-driving frequency domain. The locus curve of the torus bifurcation has to be
detected by sweeping the parameter space with branches of periodic solutions. Comparison of the two curves shows no significant differences. The self-excited oscillation
can be suppressed by large external damping or for small values of the bearing clearance. From a physical point of view, a large bearing clearance leads to the undamping
of the conical mode via oil-film forces.
172
Chapter 7: Numerical Bifurcation Analysis
7.9.3 Computation of Quasiperiodic Tori Justifies Neglecting Static Loads
The continuation of quasiperiodic tori in the two parameters bearing clearance and
the driving frequency verified that gravitational forces can be neglected for higher
rotational speeds. This leads to a dramatic reduction of computational effort if the
model is formulated in co-rotating coordinates, because invariant tori of the Earthgravity system are well approximated by tori constructed from periodic solutions of
the zero-gravity system. These periodic solutions can be computed much simpler
and the available efficient methods like AUTO can be applied to substantially more
detailed models of a turbocharger, or other machinery with fast rotating parts. For
systems where the static loads cannot be considered to be small perturbations, the
computation of invariant tori is necessary and the method proposed in (Schilder &
Peckham, 2007) and applied above works well for moderately large systems. On the
one hand it is more memory consuming than direct simulation. On the other hand it
also unstable tori and hence the analysis of hysteresis effects.
7.9.4 Validity of Inertia Correction of Short Bearing Approximation
The effect and the validity of the inertia correction of the short bearing approximation
of pressure distribution in the journal bearings can be studied efficiently by performing
a homotopy between zeroth order and first order short bearing approximation. The
internal Newton method is applied to solve the implicit equation of motion inside
the user supplied function for AUTO and performs well. For the 4-D toy problem
with small bearing width fulfilling (5.22), the inertia correction leads to a significant
reduction of the ratio of the frequency of the self-excited oscillation and the driving
frequency as observed in the experiments. This reduction persists under a change
of the bearing width up to values of W = 2 mm. While the validity of the inertia
correction of the short bearing approximation is not given for the 32-dimensional
turbocharger model with bearing width W = 5.4 mm, the proposed procedure of using
a co-rotating frame of reference, neglecting gravity, and applying the internal Newton
method proves to be applicable for the continuation of such systems. It could be used
in the future to study the behavior of other models for systems with fast rotating parts
which include lubrication models leading to implicit equations of motion.
Chapter 8
Conclusions and Outlook
The following closing remarks summarize the most important results of the present
thesis and show up some possible directions for future research.
The starting point for the thesis was the need for a more detailed and exact model for
the prediction of the vibration of a passenger car turbocharger. Vibrations in rotating
machinery are a common problem and the prediction of the frequencies and the amplitudes of the occurring oscillations and their dependence on physical parameters such
as bearing geometry, lubricant viscosity or temperature, and rotational frequency is of
utter importance for the development of more efficient and reliable designs. Therefore, a general model for rotating beams in oil-lubricated bearings has been developed
in this work and has been successfully applied to predict the lateral vibrations of a
turbocharger in simple journal bearings.
By applying Euler-Bernoulli beam theory and the Lagrange formalism we have derived
a model for a continuous, isotropic rotor of varying diameter to which rigid disks
modelling fly-wheels are attached. Special attention has been given to the effects of
rotatory inertia and gyroscopic effects which become more and more influential on
the rotors eigenfrequencies for rising rotational frequencies. The misalignment of the
center of gravity of the cross-sections of the beam with axis of rotation leads to a periodic
unbalance forcing. The bearing reaction forces are modeled as nonlinear point forces
at two points along the axis of the rotor. For the numerical simulation of the model
equations, the finite element discretization of the equations for the continuous beam
has been realized by the approximation with piecewise cubic polynomials, leading
to Rayleigh beam elements. The resulting ordinary differential equation describes the
lateral motion in the finite element nodes. The modeling approach in this work is valid
for small displacements and neglects shear deformation which is justified for slender
beams. For non-slender beams with larger diameter-to-length ratio the inclusion of
shear effects by using e.g. Timoshenko beam theory is necessary.
One of the main results of the work is the proof of existence of solutions for the equa-
174
Chapter 8: Conclusions and Outlook
tions of motion. It is given for the quite general class of locally Lipschitz-continuous
support-functions by first applying Galerkin’s method to the problem with linear
spring and damper support, and then using a Banach fixed point argument to obtain
existence also in the nonlinear case. A special difficulty is the confinement of the rotor
into the bearings and the absence of a growth condition for the bearing functions for
large eccentricities. Hence, the result for the nonlinear support only yields short time
existence and no statement is made about possible collisions with the bearing casing.
For simpler polynomial bearing functions it should be possible to prove the existence
of solutions global in time which are bounded at the positions of the bearings such
that no collisions occur, because global a priori estimates can be easily obtained. The
more complicated bearing functions for hydrodynamic bearings do not show such a
simple growth behavior for large eccentricities. They are however also restoring forces
and therefore a proof of long time existence should also be possible with some more
technical considerations.
Another main result of the thesis is the derivation of inertia corrections to the thin
film equations used in the modeling of the bearings and the examination of their influence on the rotor self-excited oscillation. In previous models, the reaction forces
of the oil-lubricated journal bearings were modeled by using Reynolds’ equation for
the pressure distribution in the thin lubricant film and subsequent integration over
the journal surface. In the present work the short bearing approximation was applied in first simulations of the system. This simplification allows for an analytical
expression of the forces and hence fast numerical evaluation. These simulations yield
already qualitatively good results. The main oscillation phenomena observed in the
experiments are captured in the model. The harmonic unbalance oscillation is present
and a resonance of a bending mode occurs at the same driving frequency as in the
experiment. The fluid-induced self-excited oscillation, the well known rotor instability called oil whirl, also appears in the model, and the onset frequency is close to the
experimentally measured one.
The frequency of the self-excited subharmonic oscillation, however, is predicted too
high. In the experiment the ratio of subharmonic frequency driving frequency is 12
between the onset of the instability and the resonance of the bending mode, and it
drops to approximately 25 with increasing rotational frequencies above the resonance
of the bending mode. In the simulations this ratio remained constant equal to 12 for all
rotational frequencies. This small difference between model and experiment resulted
in a deeper investigation of the derivation of the lubrication model.
of a phenomenological correction term into the short bearing pressure solution, which
allowed for a tuning of the subharmonic response frequency. There is a one-to-one
correspondence between that velocity and the subharmonic response frequency. The
success of the phenomenological model showed that the frequency of the self-excited
175
oscillation depends strongly on the lubrication model. In order to give also a modification of the old lubrication model based on first principles, we have derived new inertia
correction terms for Reynolds’ equation and also for the short bearing approximation
based on an additional relation between bearing width and radius. The introduction of
these correction terms into the equation of motion complicates its numerical solution,
because it is changed from an explicit to an implicit ordinary differential equation,
but it can still be solved by applying the appropriate implicit methods like e.g. DASPK
(Brenan et al., 1989). For bearing dimensions as in the experimental setup, the inertia
corrections of Reynolds’ equation showed no detectable effect on the frequency of
the oil whirl. For very short bearings however an small decreasing effect could be
observed in the simulations.
Numerical continuation methods for periodic orbits and quasiperiodic orbits have
proved to be more efficient for performing extensive parameter studies of oscillating
systems. Using the package AUTO (Doedel et al., 2000) locus curves of the Hopf bifurcations and the torus bifurcations at the onset of the oil whirl were computed in the
unforced and forced case, respectively, thereby allowing the direct determination of
the regions of stability of the rotor. A transformation of the equations of motion to
a co-rotating frame of coordinates together with neglecting the static load makes the
system autonomous, a significant simplification. For the example of the turbocharger,
neglecting the static load is justified for larger rotational velocities. This is shown by
computing the quasiperiodic solutions of the non-autonomous system by applying a
recently introduced Fourier method for the continuation of tori and comparing them
with the periodic ones of the simplified autonomous system. By applying this simplification it was also possible to examine the validity of the inertia correction in the short
bearing approximation over a larger range of parameters. All in all, the application
of continuation methods together with the simplification of the system to co-rotating
coordinates and the neglect of the static load is shown to be a powerful tool in the
investigation of rotordynamic models and should be considered as an alternative to
the time integration methods popular in current CAE methods.
The phenomenological correction of the bearing reaction forces seems to be a good
alternative starting point for further research. Its influence on the frequency response
is much stronger than the inertia corrections and changes in the response frequency
similar to the experiment could be achieved. There are some possible further developments. First, an even more detailed analysis of the lubrication model taking
into account thermic effects on the bearing geometry, the effect of the oil inlet, or
non-Newtonian behavior of the lubricant (San Andrés & Kerth, 2004) could lead to
a change in the average circumferential velocity. Also cavitation effects, turbulence
or secondary flows, i.e. liquid flowing in the direction opposite the rotation of the
shaft, have been neglected in this work, but could reduce that velocity and hence the
frequency of the oil whirl.
176
Chapter 8: Conclusions and Outlook
Another interesting application for the phenomenological correction is its application
in situations where fast evaluation of the bearing response is important and has to
be done very often, like in an optimization setting. In (Strauß, 2005) the shape of the
rotor has been optimized with respect to the unbalance response for a similar model
but with linear spring and damper bearings. The nonlinear effects of the bearings,
especially the fluid induced instability, were not considered there. To include them,
the computation of periodic obits, e.g. by a boundary value method, is necessary which
is computationally expensive. If the bearing geometry is not part of the optimization,
the frequency response behavior for the oil whirl which depends mainly on that
geometry could be prescribed by applying the phenomenologically corrected model
during the optimization. After the optimization or at some intermediate stage a more
detailed and computationally more expensive lubrication model could than be used
for verification only.
The research presented in this thesis has contributed successfully to the developement
of a model for rotating machinery and the examination of the parameter dependencies
of the occuring oscillations. Parts of the results are also currently applied at the Toyota
Central Research and Development Laboratories (TCRDL) to compute the response
vibration of a turbocharger in floating ring bearings and are included in a pending
patent application (Rübel et al., 2006) in Japan.
Appendix A
Euler Angles
∂ zv
∂ zu
z
t
β
γ
e3
γ
ex’
e2
y
ϕ
e1
x
Figure A.1: Euler angles of a rotating disc
In Section 2.3 we use Euler angles to describe the position of a disk in space. The
Euler angles used here and in (Nelson & McVaugh, 1976) differ slightly from those
commonly used in textbooks like (Nolting, 1989; José & Saletan, 1998) and in parts of
the engineering literature (Yamamoto & Ishida, 2001). We shall therefore give here the
explicit derivation for the expression of the momentary angular velocity Ω in terms of
the Euler angles.
178
Appendix A: Euler Angles
The position of the cross-section relative to the origin is given by the Euler angles
(γ, β, φ) shown in Figure A.1. The three angles describe three successive rotations of
the disk whose principal axis of inertia (e1 , e2 , e3 ) are assumed to be initially collinear
with the space coordinate system (ex , ey , ez ). The first rotation leaves the y-axis fixed
and rotates the disk by the angle γ about this axis. The second rotation leaves the image
ex′ of the vector ex fix and rotates the disk by β. The third rotation which corresponds
to the spin of the disk rotates the coordinate system about the image of ez under the
first two equations by the angle φ. The rotation matrices for the three rotations are
R1
R2
R3


 cos γ 0 − sin γ 


 ,
=  0
1
0



sin γ 0 cos γ


0
0 
 1


=  0 cos β sin β  ,


0 − sin β cos β


 cos ϕ sin ϕ



 .
=  − sin ϕ cos ϕ



0
0
1
(A.1)
(A.2)
(A.3)
Successive application of the rotations yields the transformation matrix which allows
the calculation of the angular velocity in the body system (e1 , e2 , e3 )
R = R3 R2 R1
(A.4)


 cos ϕ cos γ + sin ϕ sin γ sin β sin ϕ cos β − cos ϕ sin γ + sin ϕ sin β cos γ 


=  − sin ϕ cos γ + cos ϕ sin γ sin β cos ϕ cos β sin ϕ sin γ + cos ϕ sin β cos γ  .


cos β sin γ
− sin β
cos β cos γ
From Figure A.1 we see that the momentary angular velocity is given by
Ω = γ̇ey + β̇ex′ + φ̇e3 .
(A.5)
In this equation ey is the unit vector in y-direction, e3 is the normal vector to the plane
(and also spans a principal axis of inertia), and ex′ is the unit vector along the image of
the x-axis after the first rotation. The coordinates of ey in the rotating body coordinate
system are R · ey , those of ex′ are given by RR−1
· (1, 0, 0)T . Hence we can write
1



 sin ϕ cos β cos ϕ 0   γ̇ 

 

Ω =  cos ϕ cos β − sin ϕ 0   β̇  .

 

ϕ̇
− sin β
0
1
(A.6)
In Section 2.3 the Euler angles and the kinetic energy of the rotating disk are expressed
in terms of the shaft’s prescribed spin velocity ω which is assumed to be constant and
the tangential vector t = (∂z u, ∂z v, 1) = (u′ , v′ , 1) of the center curve r0 (z, t). We give
the detailed derivations here for completeness. To simplify the notation we denote the
179
derivative of a given quantity f with respect to z by f ′ , and the derivative with respect
to t by f˙.
From Euler-Bernoulli theory we know that the tangential vector to the center-line is
orthogonal to the disk and hence collinear to e3 . From this and Fig. A.1 we can derive
that
tan γ = u′ ,
tan β =
√
(A.7)
−v′
1 + u′2
.
(A.8)
The angle φ of the rotation about the axis spanned by e3 equals the spin of the disk if
there is no torsion. We shall assume this and we set
φ = ωt.
(A.9)
If the inclination of the shaft is small, so are γ and β and we have
γ = u′ + O(u′3 ),
(A.10)
β = −v′ + O(v′3 , v′ u′2 ).
(A.11)
Furthermore from (A.7) and (A.8) we get the following for the angular velocities:
u̇′
= u̇′ + O(u′2 ),
1 + u′2
v̇′ + v̇′ u′2 − u′ v′ u̇′
= −v̇′ + O(u′2 , u′ v′ , v′2 ),
β̇ = − √
′2
′2
′2
1 + u (1 + u + v )
ϕ̇ = ω.
γ̇ =
(A.12)
(A.13)
(A.14)
The rotational energy of a rigid body rotating about an axis through its center of gravity
is given by
1
Trot =
I1 Ω21 + I2 Ω22 + I3 Ω23 ,
(A.15)
2
where Ω is the angular velocity in the coordinate system spanned by the principal axis
of inertia. We want to express this energy in terms of u and v. First we plug in the
expressions for Ω from Eq. (A.6).
2Trot = I1 (γ̇ sin ϕ cos β + β̇ cos ϕ)2
+I2 (γ̇ cos ϕ cos β − β̇ sin ϕ)2
+I3 ϕ̇ − γ̇ sin β 2
1
(I1 + I2 ) γ̇2 cos2 β + β̇2
=
2 +I3 ϕ̇2 − 2γ̇ϕ̇ sin β + γ̇2 sin2 β
1
+ (I1 − I2 ) (β̇2 − γ̇2 cos2 β) cos 2ϕ + 2β̇γ̇ cos β sin 2ϕ .
2
(A.16)
180
Appendix A: Euler Angles
Then we insert the expressions for γ and β from above and neglect all terms of order
u′2 , v′2 , u′ v′ , and higher,
2Trot ≈
1
(I1 + I2 ) u̇′2 + v̇′2
2 +I3 ω2 + 2ωu̇′ v′
1
+ (I1 − I2 ) (v̇′2 − u̇′2 ) cos 2ωt − 2u̇′ v̇′ sin 2ωt .
2
(A.17)
Appendix B
Tools from Functional Analysis
In (Jost, 1998) we find the following variant of Poincaré’s inequality.
Lemma B.1 (Generalized Poincaré Inequality). Let Ω ⊂ Rd be convex, u ∈ H1,p (Ω). Then
for every measurable B ⊂ Ω with |B| > 0 we have
 1p


 1p
Z

Z

C 



 |u − ūB |p  ≤
 |Du|p  ,




|B| 
Ω
where ūB =
R
1
B B u(x)dx
(B.1)
Ω
is the average of u on B.
In (Evans, 1998) Gronwall’s inequality is used for proving a priori estimates of solutions
of PDEs. We cite the differential version here.
Lemma B.2 (Gronwall’s Inequality). Let η : [0, T] → R+ be absolutely continuous and
satisfy for almost all t the differential inequality
η′ ≤ φ(t)η(t) + ψ(t)
(B.2)
with nonnegative, summable functions φ : [0, T] → R+ and ψ : [0, T] → R+ . Then η satisfies


Rt
( 0 φ(s)ds) 

η(t) ≤ e
η(0) +

Zt
0



ψ(s)ds .

(B.3)
Lemma B.3. The mass matrix appearing in the ODE of the Galerkin approximation (3.52) is
nonsingular.
Proof. M is defined by applying the scalar product m(., .) to the elements of the orthogonal basis {wk }k ⊂ H2 (Ω) which are chosen as approximation function,
M jk = m(wk , w j ) for
1 ≤ j, k ≤ n.
(B.4)
182
Appendix B: Tools from Functional Analysis
Suppose M is singular. Then the columns of M are linear dependent and there are
λ1 , . . . , λn , with at least one λi , 0, such that the columns can be combined to 0,
 n

n
X
X

k
j
k
j

0=
λk m(w , w ) = m 
λk w , w  for 1 ≤ j ≤ n.
(B.5)
k=1
k=1
Now we form the scalar product of this vector with the vector λ̄ = (λ̄1 , . . . , λ̄n )T and
obtain
2

  n

n
n
n
n
X
X
X
 X
X


k
j
k
j
j




0=
λ̄ j m 
λk w , w  = m 
λk w ,
λ j w  = λ j w ,

 j=1
k=1
k=1
j=1
j=1
1 (Ω)
Hm
since m(., .) is a scalar product. Hence
n
X
λ j w j = 0,
j=1
which is a contradiction, because the {wk }k=1,...,n are elements of an orthogonal basis
and hence linearly independent. Therefore, M is nonsingular.
Appendix C
Element Matrices
In Chapter 4 we derive the element matrices for a finite rotating Rayleigh beam element
and for rigid disks by the use of Hermite polynomials as shape functions. The exact
form of the functions and of the matrices is given in this appendix.
C.1
Shape Functions
The Hermite polynomials which are used for the interpolation of the displacement of
a beam element with prescribed nodal displacements uk , vk , uk+1 , vk+1 and inclinations
βk , γk , βk+1 , and γk+1 are given by
!2
!3
z − zk
z − zk
Ψ1 (z) = 1 − 3
+2
,
lk
lk

!
!2 


z
−
z
z
−
z
k
k
 ,
+
Ψ2 (z) = (z − zk ) 1 − 2

lk
lk
!2
!3
z − zk
z − zk
Ψ3 (z) = 3
−2
,
lk
lk

!
!3 
 z − zk 2
z − zk 

Ψ4 (z) = lk −
+
 ,
lk
lk
and the displacement is given by
  
 u   Ψ1 0
0
Ψ2 Ψ3 0
0
Ψ4 (z)
  = 
 v   0 Ψ −Ψ
0
0 Ψ3 −Ψ4 (z)
0
2
1


 qk = Wk qk ,

(C.1)
(C.2)
(C.3)
(C.4)
(C.5)
where qT = (uk , vk , βk , γk , uk+1 , vk+1 , βk+1 , γk+1 ). The derivative in axial direction is given
by


 
 u′   Ψ′1 0
0
Ψ′4 (z) 
0
Ψ′2 Ψ′3 0

 = 
(C.6)
 qk = Wk′ qk
 v′   0 Ψ′ −Ψ′ 0
′ −Ψ′ (z)
0
0
Ψ
3
2
4
1
184
Appendix C: Element Matrices
with

!2 

 z − zk
z
−
z
k
 ,
−
+


lk
lk
!2
z − zk
z − zk
′
Ψ2 (z) = 1 − 4
,
+3
lk
lk

!2 
6  z − zk
z − zk 
′
Ψ3 (z) = 
−
 ,
lk
lk
lk
!2
z − zk
z − zk
′
Ψ4 (z) = −2
.
+3
lk
lk
6
Ψ′1 (z) =
lk
For the second derivatives we have

 
 u′′   Ψ′′
0
0
Ψ′′
(z)
Ψ′′
0
0
Ψ′′
3
2
4

 =  1
 v′′   0 Ψ′′ −Ψ′′ 0
′′
′′
0
0 Ψ3 −Ψ4 (z)
2
1
(C.7)
(C.8)
(C.9)
(C.10)


 qk = W ′′ qk .

k
(C.11)
with
!
6
z − zk
= 2 −1 + 2
,
lk
lk
z − zk
4
,
Ψ′′
2 (z) = l + 6
l2k
k
!
z − zk
6
′′
,
Ψ3 (z) = 2 1 − 2
lk
lk
2
z − zk
Ψ′′
.
4 (z) = − l + 6
l2k
k
Ψ′′
1 (z)
(C.12)
(C.13)
(C.14)
(C.15)
C.2 Element Matrices for Rigid Disks
The mass matrix Md for the rigid disk elements is given by

 m 0 0

 0 m 0
d
M = 
 0 0 I

0 0 0
0
0
0
I




 ,



(C.16)
the gyroscopic matrix Gd by


 0 0 0 0 


 0 0 0 0 
d
 ,

G = ω 

 0 0 0 Ip 


0 0 −Ip 0
(C.17)
185
C.3 Rayleigh Beam Element Matrices
and the damping matrix by

 c 0

 0 c
d
C = 
 0 0

0 0
0
0
0
0
0
0
0
0




 ,



(C.18)
where c is the damping coefficient for the external damping.
C.3
Rayleigh Beam Element Matrices
The mass matrix is split into two parts, the matrix Mt for the translational inertia

 156


0 156
Sym.



0 −22l
4l2

µl  22l
0
0
4l2

Mt =
420  54
0
0
13l 156


0
54 −13l
0
0 156


2

0
13l −3l
0
0 22l 4l2

−13l
0
0 −3l2 −22l
0
0 4l2










 ,








(C.19)
and the matrix Mr for the rotatory inertia

 36


0
36
Sym.


2

0 −3l 4l

2


µr  3l
0
0 4l2

Mr =

120l  −36
0
0 −3l 36


0
−36
3l
0
0 36


2

0 −3l −l
0
0 3l 4l2

3l
0
0 −l2 −3l 0
0 4l2










 .







(C.20)
For our choice of damping, the damping matrix is a multiple of the translatory mass
matrix



 156




0 156
Sym.




2


0 −22l
4l




0
0
4l2
cl  22l

(C.21)
C=
 .


420  54
0
0 13l 156





0
54 −13l
0
0 156




2
2


0
13l −3l
0
0 22l 4l


2
2
−13l
0
0 −3l −22l
0
0 4l
186
Appendix C: Element Matrices
The skew symmetric gyroscopic matrix is

0


 −36
0
Skew–sym.


 3l
0
0

ωIp 
0
3l −4l2
0

G=

30l 
0 −36
3l
0
0

 36
0
0
3l −36
0


2
 3l
0
0 −l
−3l
0
0

2
0
3l
l
0
0
−3l 4l2 0
and the stiffness matrix is

 12


0
12
Sym.


2

0 −6l 4l



0
0 4l2
EIa  6l
K = 3 
l  −12
0
0 −6l 12


0 −12 6l
0
0 12


2

0 −6l 2l
0
0 6l 4l2

6l
0
0 2l2 −6l 0
0 4l2










 .


















 ,








(C.22)
(C.23)
Appendix D
Specifications of Turbocharger
Beam Models
D.1 13 Element Beam Model
In Chapters 4 and 6 we consider a beam model consisting of 13 Euler beam elements.
Turbine and impeller are modeled with rigid disks attached in different nodes. A
further refinement is the use of a two layer shaft model. Considering two layers of
different material along the shaft allows for a more detailed description of the shaft. We
consider now the following shaft with 13 elements that have the following parameters:
Beam model
50
40
30
width [mm]
20
10
0
−10
−20
−30
−40
−50
0
20
40
60
80
100
length [mm]
Figure D.1: Detailed beam model of turbocharger.
120
188
Appendix D: Specifications of Turbocharger Beam Models
Inner Shaft
Outer Shaft
#
Length
[mm]
∅
[mm]
Young’s
modulus
[N/m2 ]
Density
[kg/m3 ]
1
2
3
4
5
6
7
8
9
10
11
12
13
10.0
4.5
15.2
6.0
7.1
9.5
12.65
12.5
6.65
11.2
6.6
9.6
5.0
4.1
4.1
4.1
4.1
8.0
6.2
6.0
6.0
6.0
9.9
14.2
11.0
7.0
7800
7800
7800
7800
7800
7800
7800
7800
7800
7800
0
0
7800
2.058e11
2.058e11
2.058e11
2.058e11
2.058e11
2.058e11
2.058e11
2.058e11
2.058e11
2.058e11
2.058e11
2.058e11
2.058e11
∅
[mm]
Young’s
modulus
[N/m2 ]
Density
[kg/m3 ]
6.0
10.0
25.0
0
0
0
7.35e10
7.35e10
7.35e10
The parameters of the rigid disks are the following:
Impeller
Turbine
Mass [kg]
Diametral inertia
moment Id [kgm2 ]
Polar inertia moment Ip
[kgm2 ]
1.3328e-2
4.3414e-2
1.2740e-6
3.1360e-6
2.1560e-6
5.8800e-6
A picture of this model is shown in Figure D.1, showing the beam elements from left
to right and also the positions of the bearings. The thinner lines represent the outer
shaft elements and the red dashed lines symbolize the two rigid disks.
From this geometry data one can calculate the effective masses and moments of inertia
of each element, which are needed for the setup of the system’s matrices. They are
given by the following formulas:
µ = πl r2i ρi + (r2o − r2i )ρo ,
π 4
Ip =
ρi ri + ρo (r4o − r4i ) ,
2
π 4
ρi ri + ρo (r4o − r4i ) ,
Ia =
4
π
EY Id = =
EY,iρi r4i + EY,oρo (r4o − r4i ) .
4
(D.1)
(D.2)
(D.3)
(D.4)
These effective parameters are used in the calculation of the system matrices and the
equation of motion.
189
D.2 3 Element Beam Model
The unbalance parameters for the above beam model are
a3 = 1.35 × 10−7 kgm,
a5 = 1.50 × 10−7 kgm,
ψ3 = π,
(D.5)
ψ5 = 0,
(D.6)
−7
ψ11 = 0,
(D.7)
−7
ψ13 = π
(D.8)
a11 = 2.01 × 10 kgm,
a13 = 2.07 × 10 kgm,
where ai is the unbalance amplitude, and ψi is the phase of the unbalance vector in
node i.
D.2 3 Element Beam Model
In Chapters 6 and 7 we consider a simplified smaller 3 element beam model. This
model (c.f. Figure D.2) has only one layer and it has the following specifications.
Shaft
#
Length
[mm]
∅
[mm]
Young’s
modulus
[N/m2 ]
Density
[kg/m3 ]
1
2
3
35
25
35
6
6
10
7800
7800
7800
2.058e11
2.058e11
2.058e11
Beam model
40
30
width [mm]
20
10
1
3
2
4
0
−10
−20
−30
−40
0
20
40
60
80
length [mm]
Figure D.2: 4 element beam model of turbocharger.
The parameters of the two rigid disks attached at both ends are
100
190
Appendix D: Specifications of Turbocharger Beam Models
Impeller
Turbine
Mass [kg]
Diametral inertia
moment Id [kgm2 ]
Polar inertia moment Ip
[kgm2 ]
2.0e-2
3.0e-2
0.5e-6
0.7e-6
2e-6
3e-6
The unbalance parameters for the above beam model are
a1 = 1.35 × 10−7 kgm,
a4 = 2.07 × 10−7 kgm,
ψ3 = π,
(D.9)
ψ13 = π.
(D.10)
Appendix E
Detailed Formulas for the Inertia
Correction p1
E.1 Short Bearing Approximation
The detailed solution for the pressure p1 can be computed by solving the integrals in
(5.60). Additionally to the expression on Equation (5.61) the solution can be written in
a more convenient form in terms of p0 , hi , and Vi (i = 0, 1)


4
X

z(z − 1) 
 .
z(z − 1)G +
p1 =
H

i
h50 
i=0
(E.1)
Here the following abbreviations are used
1
Ψ(t, ϕ) := (γ′ − )κ sin ϕ + κ′ cos ϕ,
2
1 3
3
3
1
G(t, ϕ) :=
( (∂ϕ h0 )2 Ψ − h0 ∂ϕ Ψ∂ϕ h0 + h20 ∂2ϕ Ψ − h0 Ψ∂2ϕ h0 ),
K 2
2
2
2
1 H4 (t, ϕ) := − R −3∂2ϕ h0 + 24∂τ Ψ − 10∂τ ∂ϕ h0 + 2∂ϕ Ψ h40 ,
40
3
1
3
81 2 13
2
H3 (t, ϕ) := R
Ψ∂τ h0 + ∂τ h0 ∂ϕ h0 + (∂ϕ h0 ) − Ψ + Ψ∂ϕ h0
10
2
20
70
20
3
1
+ 3Ψ − 4∂ϕ h0 + 3∂ϕ Vϕ h0 ,
h2
1
−3KVϕ1 ∂ϕ h0 − ∂2ϕ Ψ − 3K∂ϕ h1 0 ,
2
K
3
h0
3
,
H1 (t, ϕ) := 18Kh1 Ψ + Ψ∂2ϕ h0 + ∂ϕ h0 ∂ϕ Ψ
2
2
K
Ψ
3
H0 (t, ϕ) := − (∂ϕ h0 )2 .
2
K
H2 (t, ϕ) :=
(E.2)
(E.3)
(E.4)
(E.5)
(E.6)
(E.7)
(E.8)
192
Appendix E: Detailed Formulas for the Inertia Correction p1
E.2 Correction to Solution of Reynolds’ Equation
In Section 5.1.6 the equation for the correction term p1 to the solution of Reynolds’
equation is derived. The right hand side of this elliptical equation is given here.
Ψ(p0 , h0 , h1 , Vϕ0 , Vϕ1 ) =
9 2 02 4
9
∂ϕ p0 ∂3ϕ p0 −
(∂ p ) )δ
Rh40 ((−
560
560 ϕ
3
9
3 2 0 2 0
9
+ (−
(∂ϕ ∂z p0 )2 −
∂z p0 ∂2ϕ ∂z p0 −
∂ϕ p ∂z p −
∂ϕ p0 ∂2z ∂ϕ p0 )δ2
140
560
280
560
9
9 2 02
−
∂z p0 ∂3z p0 −
(∂ p ) )
560
560 z
3
3
(∂ϕ p0 )2 ∂2ϕ h − ∂ϕ p0 ∂2ϕ p0 ∂ϕ h0 )δ4
80
16
3
9
+ (− ∂ϕ p0 ∂ϕ h0 ∂2z p0 − ∂ϕ h0 ∂z p0 ∂ϕ ∂z p0 )δ2 )
40
80
+Rh0 3 ((−
9 4
1
1 3 0 0 2
2
δ ∂ϕ p0 (∂ϕ h0 )2 + ( ∂t ∂2z p0 +
∂ p Vϕ )δ
40
10
120 ϕ
1
1 2
∂z ∂ϕ p0 Vϕ0 + ∂t ∂2z p0 )
+
120
10
+Rh20 (−
1 2 0 0
1
1
1
∂ϕ p Vϕ ∂ϕ h0 + ∂ϕ p0 ∂t ∂ϕ h0 + ∂2ϕ p0 ∂t h0 + ∂t ∂ϕ p0 ∂ϕ h0 )δ2
24
4
4
2
1
1 2 0 0
2 0
+ ∂t h0 ∂z p − ∂z p Vϕ ∂ϕ h0 )R
4
12
3 2 2 0 1 2 0
− δ ∂ϕ p − ∂z p )
2
2
+h0 (((
+ (δ2 ∂ϕ p0 ∂ϕ h0 ∂t h0 R − 5δ2 ∂ϕ p0 ∂ϕ h0 )
3 0 2 2
1
(Vϕ ) ∂ϕ h0 + Vϕ0 ∂t ∂ϕ h0 )R
20
2
2 0
+ (−3h1 ∂ϕ p − 3∂ϕ h1 ∂ϕ p0 )δ2 − 3h1 ∂2z p0 )
+h−1
0 ((
3 0 2
(V ) (∂ϕ h0 )2 )R
10 ϕ
+ 6∂ϕ Vϕ1 − 6h1 δ2 ∂ϕ p0 ∂ϕ h0 − 8Vϕ0 ∂ϕ h0 )
0
+h−2
0 ((Vϕ ∂t h0 ∂ϕ h0 +
0
1
+h−3
0 (−6∂ϕ h1 Vϕ − 6∂ϕ h0 Vϕ ).
(E.9)
Appendix F
Integration of Bearing Integrals
with Sommerfeld Variable
Transformation
We want to solve the following integrals that appear in the calculation of the bearing
forces
FN = −
FT
W 3 Rρνω
c2r
Z1 Z2π
(p0 )+ dϕ dz,
0
(F.1)
0
Z1 Z2π
W 3 Rρνω
(p0 )+ dϕ dz,
= −
c2r
0
where
p0 = −6z(z − 1)
(F.2)
0
(γ′ − 12 )κ sin ϕ + κ′ cos ϕ
(1 − κ cos ϕ)3
.
(F.3)
We set
a = κ′ ,
(F.4)
1
b = κ(γ′ − ),
2
W 3 Rρνω
,
C =
c2r
ψ = arg(a − ib).
(F.5)
(F.6)
(F.7)
The pressure is positive for
ϕ ∈ [ϕ1 , ϕ2 ] = [−ψ −
π
π
, −ψ + ].
2
2
(F.8)
194
Appendix F: Integration of Bearing Integrals
Hence, the integrals become
Z1 Zϕ2
FN = −C
0 ϕ1
−6z(z − 1)
b sin ϕ cos ϕ + a cos2 ϕ
dϕ dz,
(1 − κ cos ϕ)3
(F.9)
−6z(z − 1)
b sin2 ϕ + a cos ϕ sin ϕ
dϕ dz.
(1 − κ cos ϕ)3
(F.10)
Z1 Zϕ2
FT = −C
0 ϕ1
The integration with respect to z is readily done and yields
Zϕ2
FN = −C
ϕ1
b sin ϕ cos ϕ + a cos2 ϕ
dϕ,
(1 − κ cos ϕ)3
Z1 Zϕ2
FT = −C
0 ϕ1
b sin2 ϕ + a cos ϕ sin ϕ
dϕ.
(1 − κ cos ϕ)3
(F.11)
(F.12)
To solve the integrals with respect to ϕ we introduce the variable transformation which
is due to Sommerfeld
1 − κ2
.
(F.13)
1 − κ cos ϕ =
1 − κ cos θ
This yields
κ − cos θ
,
1−
√ κ cos θ
− 1 − κ2 sin θ
,
sin ϕ =
1 − κ cos θ
√
− 1 − κ2 sin θ
),
ϕ = arctan(
κ − cos θ
√
√
dϕ
(κ − cos θ)(− 1 − κ2 cos θ) + 1 − κ2 sin2 θ
=
dθ
(1−κ2 ) sin2 θ
1 + (κ−cos θ)2 (κ − cos θ)2
√
( 1 − κ2 (sin2 θ + cos2 θ − κ cos θ))
=
(κ − cos θ)2 + (1 − κ2 ) sin2 θ
√
− 1 − κ2 κ cos θ
=
κ2√− 2κ cos θ + cos2 θ + sin2 θ − κ2 sin2 θ
1 − κ2
=
.
1 − κ cos θ
cos ϕ =
(F.14)
(F.15)
(F.16)
(F.17)
(F.18)
(F.19)
(F.20)
Using the transformation formula we obtain the new integral boundaries
√
θ1 = arctan(κ − cos ϕ1 , − 1 − κ2 sin ϕ1 ),
√
θ2 = arctan(κ − cos ϕ2 , − 1 − κ2 sin ϕ2 ).
(F.21)
(F.22)
195
The integrals transform to
Zϕ2
FN = −C
ϕ1
b sin ϕ cos ϕ + a cos2 ϕ
dϕ
(1 − κ cos ϕ)3
 √
√
Zθ2   κ − cos θ 2
1 − κ2 sin θ(κ − cos θ)  1 − κ2 (1 − κ cos θ)3

dθ
−b
= −C a

1 − κ cos θ
(1 − κ cos θ)2
(1 − κ2 )3 (1 − κ cos θ)
θ1

Zθ2 
 (κ − cos θ)2
sin θ(κ − cos θ) 

= −C a
−b
 dθ,
5
(1 − κ2 )2
(1 − κ2 ) 2
(F.23)
θ1
and analogously
Zϕ2
FT = −C
ϕ1
b sin2 ϕϕ + a cos ϕ sin ϕ
dϕ
(1 − κ cos ϕ)3

Zθ2 

 sin2 θ
sin
θ(κ
−
cos
δ)
 dθ.
−a
= −C b

3
(1 − κ2 )2
(1 − κ2 ) 2
(F.24)
θ1
The three resulting integrals are
Zθ2
A1 =
5
θ1
=
(κ − cos θ)2
(1 − κ2 ) 2
dθ
(F.25)
(κ2 + 21 )(θ2 − θ1 ) − 2κ(sin θ2 − sin θ1 ) + 41 (sin 2θ2 − sin 2θ1 )
5
Zθ2
A2 =
θ1
Zθ2
A3 =
θ1
(1 − κ2 ) 2
,
−κ(cos θ2 − cos θ1 ) + 14 (cos 2θ2 − cos 2θ1 )
sin θ(κ − cos θ)
dθ
=
,
(1 − κ2 )2
(1 − κ2 )2
sin2 θ
(1 −
3
κ2 ) 2
dθ =
(2(θ2 − θ1 ) − sin 2θ2 + sin 2θ1 )
3
4(1 − κ2 ) 2
,
(F.26)
(F.27)
(F.28)
which finally gives the following formulation for the forces
FN = −C(aA1 − bA2 ),
FT = −C(bA3 − aA2 ).
(F.29)
(F.30)
196
Appendix F: Integration of Bearing Integrals
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