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Castanier2006-Mistuning--AIAA-16345-832.pdf
JOURNAL OF PROPULSION AND POWER
Vol. 22, No. 2, March–April 2006
Modeling and Analysis of Mistuned Bladed Disk Vibration:
Status and Emerging Directions
Matthew P. Castanier
University of Michigan, Ann Arbor, Michigan 48109-2125
and
Christophe Pierre
McGill University, Montreal, Québec H3A 2K6, Canada
The literature on reduced-order modeling, simulation, and analysis of the vibration of bladed disks found in
gas-turbine engines is reviewed. Applications to system identification and design are also considered. In selectively
surveying the literature, an emphasis is placed on key developments in the last decade that have enabled better
prediction and understanding of the forced response of mistuned bladed disks, especially with respect to assessing
and mitigating the harmful impact of mistuning on blade vibration, stress increases, and attendant high cycle
fatigue. Important developments and emerging directions in this research area are highlighted.
I.
Introduction
was broad in scope and rich in insight, and it provides a more global
view of the research issues for bladed disks. The 1999 survey by
Slater et al.11 focused on forced response of bladed disks and covered
many of the latest advances at that time. However, many important
research developments have occurred since then, particularly with
respect to finite element-based reduced-order modeling.
In this paper, the literature on bladed disk vibration is selectively
surveyed, with an emphasis on key developments in the last decade
that have enabled better prediction and understanding of the forced
response of mistuned bladed disks. In addition, various modeling
and analysis topics of recent and emerging interest are highlighted.
In Sec. II, background material on the vibration of tuned and mistuned bladed disks is covered. In Sec. III, some fundamental issues
for bladed disk vibration are reviewed briefly. In Sec. IV, finiteelement-based reduced-order modeling techniques are surveyed,
and emerging research topics are noted. In Sec. V, key issues with
respect to analysis and assessment of mistuning sensitivity are discussed. In Sec. VI, uncertainties inherent in bladed disk vibration
modeling are noted, and it is suggested that a larger framework be
pursued in the future for uncertainty and reliability analysis. Conclusions are summarized in Sec. VII.
T
URBINE engine rotors, or bladed disks, are rich dynamical
systems that are known to suffer from severe vibration problems. Although a bladed disk is typically designed to have identical
blades, there are always random deviations among the blades caused
by manufacturing tolerances, wear, and other causes. This is called
mistuning. Even though mistuning is typically small (e.g., blade
natural frequency differences on the order of a few percent of the
nominal values), mistuned bladed disks can have drastically larger
forced response levels than the ideal, tuned design. The attendant
increase in stresses can lead to premature high cycle fatigue (HCF)
of the blades. HCF is a major cost, safety, and reliability issue for
gas-turbine engines. For example, in 1998 it was estimated by the
U.S. Air Force that about 55% of fighter jet engine safety Class A
mishaps (over $1 million in damage or loss of aircraft) and 30% of
all jet engine maintenance costs were due to HCF.1 It is clearly of
great interest to be able to predict—and, ultimately, to reduce—the
maximum blade response as a result of mistuning. The comprehensive modeling, analysis, and understanding of bladed disk vibration
is thus critical to reducing the occurrence of HCF and improving
the performance and reliability of turbine engines.
Bladed disk vibration first received significant attention from the
research community in the late 1960s and the 1970s. Notable early
work was done by Whitehead,2 Wagner,3 Dye and Henry,4 and
Ewins.5−8 The bladed disk vibration literature has been surveyed by
Srinivasan9,10 and Slater et al.11 The 1997 survey by Srinivasan10
II.
A.
Background
Vibration of Individual Blades
To describe the vibration of bladed disks, it is helpful to first consider the vibration of individual blades. For a system with inserted
Matthew P. Castanier is an Associate Research Scientist in the Department of Mechanical Engineering at the University of Michigan. He received his
Ph.D. in Mechanical Engineering from the University of Michigan in 1995. His research interests are in the area of structural dynamics and vibration,
including reduced-order modeling, low- to mid-frequency vibration and power flow in complex structures, localization and related phenomena in
periodic or cyclic structures, and vibration of mistuned bladed disks in turbine engines.
Christophe Pierre is Dean of the Faculty of Engineering at McGill University in Montréal, where he is also Professor of Mechanical Engineering
and holds the Canada Research Chair in Structural Dynamics and Vibration. He received the Diplôme d’Ingénieur from the Ecole Centrale de Paris,
France in 1982, the M.S. from Princeton University in 1984, and the Ph.D. from Duke University in 1985. His research interests include vibrations,
structural dynamics, and nonlinear dynamics. He has done pioneering work on mode localization in disordered periodic structures, for which he has
received the 2005 Myklestad Award from the American Society of Mechanical Engineers. He currently works on reduced-order modeling of complex
structures, mid-frequency dynamics, component mode synthesis, nonlinear modal analysis, and dry friction damped systems, with application to
turbomachinery bladed disks and automotive body structures. He has published more than 100 journal articles and has given numerous invited
lectures internationally.
c 2005 by Matthew P. Castanier.
Received 28 February 2005; revision received 15 August 2005; accepted for publication 25 August 2005. Copyright Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal or internal use,
on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code
0748-4658/06 $10.00 in correspondence with the CCC.
384
CASTANIER AND PIERRE
385
blades, a blade-alone finite element model could correspond to an
actual blade component. However, it could also be defined by partitioning the finite element model of a bladed disk into disk and blade
components. This is the typical approach for defining the blade and
disk components of a one-piece bladed disk, which is called a blisk
or integrally bladed rotor.
An important characteristic of bladed disk vibration is the set of
blade-alone modes defined by holding the blade fixed at its interface
with the disk. These are also referred to as cantilevered blade modes.
There are several types of cantilevered blade modes, which are similar to the modes of a rectangular plate cantilevered at a narrow edge.
These mode types include: flexural (F) bending, torsion (T), stripe
(S) or chordwise bending, and edgewise (E) bending. The letters
given in parentheses are examples of the shorthand often used for
referring to these modes. For instance, the first (lowest frequency)
flexural mode is denoted 1F, the second flexural mode is 2F, and so
forth. These cantilevered blade modes are important because they
closely resemble the blade motion in a vibrating bladed disk for
blade-dominated system modes or for forced-response cases.
B.
Vibration of Tuned Bladed Disks
Fig. 2 Free-vibration natural frequencies vs nodal diameters for the
industrial bladed disk.
A finite element model of a bladed disk used in an industrial gas
turbine is shown in Fig. 1. In the idealized, tuned case, a bladed
disk has identical blades. Defining a sector as one blade plus the
corresponding segment of the disk, a tuned N -bladed disk is composed of N identical sectors. Such a structure is said to have cyclic
symmetry. By applying the appropriate phase conditions at the interfaces with adjacent sectors, then a model of just one sector is
sufficient to predict the vibration of the entire cyclic structure.12
This makes it relatively inexpensive to run finite element analyses
of tuned systems.
The free linear vibrations of cyclic structures have a notable characteristic: each system mode shape consists of identical motion in
each sector except for a fixed sector-to-sector phase difference,
which is called an interblade phase angle for bladed disks. Thus,
looking at a point in the same relative location in each sector, the
mode shapes are sinusoidal in the circumferential direction. For a
bladed disk, this leads to nodal lines across the disk called nodal
diameters, and the system modes are referred to as nodal diameter
modes.
The natural frequencies calculated by finite element analysis for
the industrial bladed disk are plotted as a function of the number of
Fig. 3
Fig. 1
Finite element mesh for an industrial bladed disk.
Three nodal diameter-mode shape of a tuned bladed disk.
nodal diameters in Fig. 2. Lines are drawn to help visualize families
of blade-dominated and disk-dominated modes. Modes dominated
by blade motion tend to appear as horizontal lines in the plot, and
the associated blade-alone mode is marked on the right-hand side.
In contrast, the modal stiffness of a disk increases rapidly as the
number of nodal diameters is increased, and so disk-dominated system modes appear as slanted lines in the plot. Note that there are
several veerings between disk- and blade-dominated modes. These
veering regions will be discussed in Sec. V.A.
A mode shape for the tuned system is shown in Fig. 3. This mode
features three nodal diameters.
In terms of forced response, bladed disks are subject to engine order excitation. Engine order excitation is the effective traveling wave
excitation that a bladed disk experiences as it passes through disturbances in the flowfield for each revolution. For example, engine
order seven excitation corresponds to each blade passing through
seven evenly spaced forcing peaks per revolution. This type of forcing only excites modes with a harmonic index that matches the
engine order. For example, the nodal diameter three mode shown in
Fig. 3 would be excited by engine order three excitation. Because
there are 29 blades in the system, this mode would also be excited by
engine order 26 (=29 − 3), engine order 32 (=29 + 3), and so forth.
386
C.
CASTANIER AND PIERRE
Vibration of Mistuned Bladed Disks
As just mentioned, the tuned case is an idealization. In reality,
there are always small, random deviations in the blade properties
because of factors such as manufacturing and material tolerances
and in-operation wear. These blade-to-blade discrepancies are called
mistuning. Mistuning destroys the cyclic symmetry of the system.
Therefore, a single sector model can no longer be used to predict
the vibration of the full system.
Blade mistuning can have a drastic effect on the vibration of a
bladed disk. To illustrate this, a mode shape for the industrial bladed
disk with mistuning is shown in Fig. 4. The mistuned mode shape is
not a pure nodal diameter mode, but instead has multiple harmonic
content so that it can be excited by all engine orders of excitation.
Furthermore, the mode shape shows localization of the vibration
about a few blades.
Because of the spatial confinement of vibration energy, certain
blades in a mistuned system can suffer a significant increase in
forced-response vibration amplitudes compared to the ideal (tuned)
system.2,4,5,13 For certain operating conditions and levels of mistuning, the confinement of vibration energy to a few blades can lead to
a large magnification of the maximum blade forced-response amplitude and stress. Note that the maximum blade amplitude refers to
the largest resonant response for any blade in the frequency range
of interest. As an example, consider the statistics of the resonant
response of the industrial bladed disk of Fig. 1 in a frequency band
corresponding to a single family of blade-dominated modes. Simulation results for the 99.9th-percentile amplitude magnification—the
ratio of mistuned to tuned maximum blade amplitude that is expected
to be exceeded by only 0.1% of the population of mistuned bladed
disks—are shown in Fig. 5 for various engine orders of excitation
and levels of mistuning. For some combinations of mistuning and
engine order, the amplitude magnification exceeds a value of 2.5 for
this system in the frequency range of interest, which corresponds to
a 150% increase compared to the tuned vibration level.
III.
Fundamental Issues
To capture the basic vibration characteristics, bladed disks have
often been modeled as cyclic chains of spring-mass oscillators. The
simplest such model of an N -sector bladed disk is a chain of N
single-degree-of-freedom oscillators coupled by linear springs. Additional oscillators can be added at each sector to have both blade
and disk degrees of freedom (DOF). The mistuning is typically modeled as small, random perturbations to the stiffnesses of the blade
DOF. Although simple, these lumped parameter models can evidence much of the rich dynamics of bladed disk systems, and they
can be used for statistical investigations of mistuned forced-response
characteristics (for example, see the excellent and pioneering work
by Griffin and Hoosac14 ).
In this section, some fundamental issues, which were mostly investigated and explained with simple lumped parameter models,
are covered. This will help illuminate the discussions in subsequent
sections.
A.
Fig. 4
Localized mode shape of a mistuned bladed disk.
Fig. 5
Coupling, Disorder, and Mode Localization
An ideal bladed disk is a cyclic structure, which is a subset of
a more general class of structures called periodic structures. A periodic structure consists of a set of identical substructures that are
dynamically coupled in some manner. The mode shapes of a periodic
structure are characterized by sinusoidal shapes that are extended
throughout the structure. However, even very slight, random perturbations to the substructures, called disorder, can have a drastic effect
on the vibration of the structure. In particular, the mode shapes can
become confined in a small region of the structure. This phenomenon
The 99.9th-percentile amplitude magnification.
387
CASTANIER AND PIERRE
is known as localization, and it was discovered by Anderson15 in
the field of solid-state physics.
Although vibration localization phenomena16 had been observed
in the engineering literature (for example, see the important early
contributions of Ewins5−7 in bladed disk vibration), it was not recognized as Anderson localization until the work of Hodges17 in
1982. Hodges showed that mode localization increases monotonically with increasing strength of disorder or decreasing strength of
mechanical coupling between substructures. That is, the degree of
mode localization depends only on the disorder-to-coupling ratio.
B.
Interblade Coupling, Mistuning, and Forced Response Increases
When harmonic excitation is applied to a single substructure in
a disordered periodic structure, the forced-vibration response will
tend to be localized around the excitation source and decay away
from the source. The spatial decay rate is asymptotically exponential, and so it looks like a damped response. However, it is different
from damping in that the decay is due to confinement of the energy
in a small region of the structure, not dissipation. Furthermore, the
localization of the vibration energy will increase with increasing
disorder-to-coupling ratio. From a wave perspective, this confinement is caused by partial reflections at each connection between
substructures. From a modal perspective, it is because modes localized about the source will be more strongly excited by the excitation
as the mode localization increases, and modes localized away from
the source will not be excited.
For bladed disks, disorder is called mistuning, and the coupling is
related to the structural coupling (through the disk or shrouds) or the
aerodynamic coupling between blades. Collectively, this is called
interblade coupling. However, mistuned bladed disks are a notably
special case of periodic structures for several reasons. First, they are
cyclic structures. Second, they receive forcing from the fluid at each
blade, which means that every substructure in the system is excited.
Third, the excitation is engine order excitation. For an ideal, cyclic
bladed disk, engine order excitation will excite only those modes
with a number of nodal diameters that match the harmonic index of
the excitation. For a mistuned bladed disk, the modes have multiple
harmonic content, so that many modes will be excited by engine
order excitation. The modes that retain significant harmonic content
matching the engine order of excitation will be strongly excited.
Although mode localization increases monotonically with respect
to the mistuning-to-coupling ratio, this is not true of maximum
forced-response levels. It has been demonstrated that vibration amplitude magnification tends to exhibit a peak value with respect
to mistuning strength or coupling strength.5,18−22 For example, see
Fig. 6, which shows the 99th, 50th, and 1st percentile amplitude
magnification for the industrial bladed disk subject to engine order excitation in the frequency range of the 2F modes. The 99thpercentile forced-response level increases with increasing mistuning up to a certain level, but a further increase in mistuning actually
results in lower forced-response amplitudes. Using a single-DOFper-sector bladed disk model, Óttarsson and Pierre22 determined
that this is because moderately weak interblade coupling is required
for significant increases in forced-response amplitudes. Consider
changes in the coupling as the mistuning level is held the same. If
the interblade coupling is weak, then each blade acts as an individual mistuned oscillator, and the mistuned response does not deviate
significantly from the tuned response. As coupling increases, an
avenue is created for the blades to communicate vibration energy,
which raises the possibility of vibration energy being transferred to
and confined around certain blades. That is, vibration energy can
be transferred to the maximum responding blade from other blades
in the assembly, potentially resulting in great increases in forced
response for some blades. Further increases in coupling allow energy to be readily exchanged among blades and eliminate the spatial
confinement of vibration energy in the system, yielding tuned-like
response levels for large coupling values.
In a lumped parameter model, it is straightforward to define the
coupling strength and thus the mistuning-to-coupling ratio. However, it is not as simple to quantify the coupling strength for an actual bladed disk, for which interblade structural coupling depends
on the relative participation of disk-dominated and blade-dominated
modes in the response. This issue will be addressed in Sec. V.A.
C.
Accelerated Monte Carlo Simulation
To estimate the statistics of the forced response for a population
of randomly mistuned bladed disks with the same nominal design,
a Monte Carlo simulation can be performed. First, given a value
for the standard deviation of random mistuning, the mistuned blade
stiffnesses for one realization of a mistuned bladed disk are assigned
by a pseudo-random-number generator. Second, a frequency sweep
is performed to find the largest peak response amplitude of any
blade on the bladed disk. Third, this process is repeated for many
realizations of mistuned rotors.
Depending on the mistuning strength and the sensitivity of the
system, it can require many thousands or even tens of thousands
of realizations to estimate the probability density function of the
worst-case blade response, especially for capturing the tails of the
distribution (e.g., the 99th percentile). Furthermore, each realization
requires a frequency sweep and a search for the maximum resonant
response, and it is often of interest to also sweep through many values
of mistuning strength. Even using a lumped-parameter model, such
a statistical analysis is computationally expensive. Therefore, Sinha
and Chen23 and Mignolet and coworkers24−26 have proposed various methods for predicting the distribution of the mistuned forced
response without using Monte Carlo simulations.
However, the variable of interest is the largest forced-response
amplitude found for any blade in a frequency sweep. Therefore, the
theory of extreme value statistics27,28 can be applied to the problem.
A remarkable result from this area of probability theory is that the
distribution of the maximum of a set of independently and identically
distributed random trials approaches one of three extreme value
distributions as the number of trials becomes large. It was shown by
Castanier and Pierre29 that the distribution of the largest-responding
blade amplitudes will asymptotically approach the third extreme
value distribution, which is the Weibull distribution. This conclusion
has also been confirmed by Mignolet et al.25
Therefore, the statistics of the mistuned forced response can be
estimated by fitting the Monte Carlo results from relatively few realizations (e.g., 50 realizations) to the Weibull distribution, which
reduces computational costs by orders of magnitude. This accelerated Monte Carlo simulation procedure was presented by Castanier
and Pierre29 and by Bladh et al.30
The statistical results shown in Figs. 5 and 6 for the industrial
bladed disk were estimated using this accelerated Monte Carlo procedure. In addition, finite element-based reduced-order modeling
was used to calculate the mistuned forced response of the industrial rotor for each realization, at relatively low cost. The topic of
reduced-order modeling is discussed next.
IV.
Fig. 6
Amplitude magnification as a function of mistuning strength.
Reduced-Order Modeling
The lumped-parameter models discussed in the preceding section can be thought of as fundamental or qualitative models of
388
CASTANIER AND PIERRE
mistuned bladed disks. For predicting the vibration response of an
actual bladed disk used in a turbine engine, it is much better to
take advantage of finite element models. A finite element model is
typically generated for only one sector of a bladed disk. Assuming
that all of the sectors are identical, cyclic symmetry routines can be
used to calculate the free and forced response much more efficiently
than modeling the entire system. However, not only does mistuning cause a possibly drastic change in the bladed disk dynamics,
but it destroys the cyclic symmetry as well. Therefore, modeling
just one sector is not sufficient; a full bladed disk model is needed.
Modern industrial finite element models of a full bladed disk can
be on the order of millions of degrees of freedom. Even with accelerated Monte Carlo simulation, using finite element analysis to
predict the statistics of the mistuned forced response is not feasible.
Therefore, reduction techniques are used to generate reduced-order
models from a parent finite element model for a frequency range of
interest.
A.
Component-Mode-Based Methods
The first generation of finite element-based reduced-order models
(ROMs) were based on component mode synthesis31−33 (CMS) or
similar component-mode-based techniques. In a CMS approach, the
modes of each component are calculated separately. CMS methods
are referred to as fixed-interface, free-interface, or hybrid methods,
depending on the boundary conditions used to calculate the component modes. Fixed-interface methods also include a set of static
shapes, or constraint modes, to capture the interface motion. The
CMS system model is then synthesized from the component models by enforcing compatibility conditions at the interfaces between
the components. This divide-and-conquer scheme provides significant computational savings. Furthermore, for bladed disks the disk
and each of the N blades can be treated as separate components. This
yields a system model with generalized coordinates corresponding
to individual blade modes, which provides a convenient framework
for modeling blade mistuning.
In 1983, the application of CMS to reduced-order modeling of
bladed disk vibration was investigated by Irretier.34 Irretier employed the free-interface CMS method of Craig and Chang.35,36
He considered a 24-bladed disk and two finite element models, one
with 576 DOF and the other with 1584 DOF. By selecting 15 modes
for the disk and four modes for each blade, he generated a 135-DOF
ROM. The free-response results from the ROM showed good agreement with those from the finite element models. In 1985, Zheng and
Wang37 used free-interface CMS to model groups of blades coupled
through shrouds. They found that CMS provided good accuracy
with a significant reduction in computational time relative to finite
element analysis. Despite the promising findings of these initial
investigations, it would be almost a decade before important new
contributions were made in this research area.
In 1994, Óttarsson38 made the observation that the vibration of
a blade in a bladed disk could be represented with a basis of cantilevered blade modes plus a set of shapes corresponding to the
blade deformation induced by the disk vibration. He suggested calculating the latter set by using finite element analysis to find the
modes of the disk with massless blades attached (a bladed disk with
mass density set to zero for all blade elements), which automatically defines the disk-induced static shapes for the blades. These
disk-induced shapes eliminated the need to include constraint modes
at the disk-blade interface, even though the blade component modes
were solved with the disk-blade interface held fixed, which kept the
final ROM size low. This approach also made it trivial to synthesize the full bladed disk model from the component modes because
the disk “component” modes included blade motion that satisfied
displacement compatibility at the disk-blade interface. Therefore,
this was, in effect, a hybrid CMS technique that was tailored to
modeling mistuned bladed disks. This method was presented, extended, and applied in several studies by Pierre and coworkers,30,39,40
and it was found that a ROM on the order of 10N could provide
good accuracy relative to the parent finite element model of an N bladed disk. In this same time frame, Yang and Griffin41 introduced
a component-based reduced-order modeling technique that synthe-
sized the disk and blade motion through an assumption of rigid-blade
base motion. That is, the surface defined by the intersection of disk
and blade DOF was treated as a rigid body, which also simplified
the synthesis process and obviated the need to calculate constraint
modes at the disk-blade interface. These methods represented a leap
in predictive capabilities because they made it possible to generate
reduced-order models systematically from finite element models,
while generally retaining good accuracy and capturing the effects
of mistuning. However, both approaches suffered from the coarse
way in which the coupling at the disk-blade interface was captured.
The disk-blade coupling is critically important because it relates directly to the interblade coupling, which has been shown to be a key
factor for a bladed disk’s sensitivity to mistuning.22
To better capture the motion at the disk-blade interface—up to the
discretization of the finite element mesh—one can use the Craig–
Bampton method32 of CMS because there is a one-to-one correspondence between Craig–Bampton constraint modes and physical finite
element DOF in each interface between components. This aspect,
combined with the fact that the blade modes are calculated with a
fixed interface and are thus compatible with the traditional bench test
measurements of blade-alone natural frequencies, makes the Craig–
Bampton method appealing for modeling bladed disks. However,
retaining all of these interface DOF for each blade renders the CMS
model size large relative to a free-interface approach, especially
for a fine mesh and/or a large number of blades. Nevertheless, this
problem can be overcome by performing a (secondary) modal analysis on the CMS model to condense out the interface DOF.36,42−46
This approach was adopted by Bladh et al.,47,48 who cast the Craig–
Bampton method into a form suitable for a bladed disk with a cyclic
disk component and retrieved extremely compact ROMs (on the
order of N ) through the modal analysis on the intermediate CMS
model. Mistuning was implemented by direct manipulation of the
cantilevered blade modal properties in the CMS coordinates, and
the eigenvectors for the CMS matrices were used to transform the
mistuning to the final ROM coordinates. The same type of approach
was also explored for modeling multistage systems.49
Another approach to CMS modeling of a bladed disk (or any
cyclic structure) is to treat each sector as a separate component.
This strategy has been investigated by Tran50 and Moyroud et al..51
Tran used CMS techniques with component interface reduction to
keep the system model size low. Moyroud et al. did not consider
interface reduction, but they explored issues such as modeling mistuned and shrouded bladed disks. For shrouded bladed disks, they
found that a sector-wise application of the classical Craig–Bampton
method was an excellent approach for handling a full range of friction constraints at shroud interfaces. However, in the case of fully
stuck shrouds, which is analogous to the unshrouded case in terms
of modeling, they found that a modal reduction approach that had
been introduced recently by Yang and Griffin52 was clearly more
efficient. This approach is discussed next.
B.
System-Mode-Based Methods
In 2001, Yang and Griffin52 presented an important new reducedorder modeling technique that they called the subset of nominal
modes (SNM) method. Their modeling approach was based on their
earlier observation that a selected set of tuned system modes provides an excellent basis for representing the vibration of mistuned
bladed disks if the mistuning is slight.53 That is, the reduced-order
model can be constructed with classical modal analysis by selecting a frequency range that includes a family of blade-dominated
system modes, typically including at least one mode or mode pair
per nodal diameter to span the space of the mistuned modes. This
is essentially a simple way to provide a de facto Fourier basis for
representing the vibration of the bladed disk. The mistuning is then
included in the model by assuming a form of the mistuned blade
structural matrices and using an appropriate coordinate transformation to express the mistuning in system modal coordinates. In light
of the component-mode-based approaches covered in the preceding
section, a key insight from the work of Yang and Griffin was that it
was not necessary to use component modes to model a bladed disk
to introduce mistuning.
CASTANIER AND PIERRE
The fact that mistuned bladed disk modes are essentially linear
combinations of tuned system modes had been recognized previously, and in some studies this property has been used to investigate
the interpretation and implications of the harmonic content of mistuned modes.54,55 However, the SNM method was the first reducedorder modeling technique to fully exploit this idea to simply and
directly generate an ROM as small as N DOF for an N -bladed disk,
which is a minimal size for representing a mistuned system. Furthermore, the ROMs generated with this method retain high accuracy
relative to the parent finite element model, but at a fraction of the
cost. Thus, this technique began what could be called a second generation of reduced-order modeling methods, which are generally
system-mode-based methods that yield ROMs on the order of N
DOF.
Feiner and Griffin56 subsequently derived a simplified form of
SNM for the case of an isolated family of blade-dominated modes.
In their method, the only required input is a set of tuned system natural frequencies and the blade-alone frequency deviations caused by
mistuning. Because of its simplicity, they called this a fundamental model of mistuning (FMM). For an example case, they found
good agreement with finite element results for operating conditions
that excited isolated blade-dominated modes, although the accuracy
broke down near veerings of disk- and blade-dominated modes and
in regions with higher modal density. Nevertheless, the minimal input requirements make it a good candidate for applications in which
high fidelity is less important than the ability to obtain simple and
quick estimates of mistuning effects.
Petrov et al.57 proposed an alternative approach for efficient vibration modeling of mistuned bladed disks. In their formulation,
the mistuned system forced-response vector was expressed in terms
of the tuned system forced-response vector and a modification matrix. The modification matrix was constructed using the frequencyresponse-function (FRF) matrix of the tuned system and a mistuning matrix defined as the perturbed part of the system dynamic
stiffness matrix. They noted that it was an exact expression and
that it could be solved by considering only a subset of the system
DOF, referred to as active coordinates, which are DOF where mistuning is applied or where forced-response levels are of interest.
This would seem to imply that all physical blade DOF might need
to be retained in the model. However, they enabled the possibility
of only including a few active DOF per blade by introducing socalled mistuning elements: lumped masses, dampers, and springs
that are attached to selected blade DOF to represent the effects
of mistuning. In addition, they calculated the tuned FRF matrix
efficiently by representing the tuned model with a set of system
modes. They noted that the reduced-order model in active coordinates was only as accurate as this modal representation of the
tuned system. In this sense, this is a system-mode-based method,
although the order reduction is achieved through the formulation
based on the relationship between the tuned and mistuned forced
response.
Lim et al.58 introduced a method that makes use of both tuned
system modes and blade component modes to generate reducedorder models of mistuned bladed disks. As in the SNM method,52
a selected set of tuned system modes is used to form a basis for the
ROM. However, for the purposes of modeling mistuning, the bladealone motion is represented by a set of cantilevered blade modes and,
optionally, the Craig–Bampton constraint modes32 for the DOF that
are held fixed in the cantilevered blade model. For this reason, it
is called the component mode mistuning (CMM) method. Modal
participation factors are calculated to relate the blade component
modes to the blade motion in the tuned system modes, and this relation is used to project the individual blade mistuning onto the ROM.
This mistuning projection approach is an extension of the method
developed by Bladh et al.40 for reduced-order modeling of mistuned
bladed disks with shrouds. A notable feature of the CMM method
is that it can handle various types of blade mistuning in a systematic
manner, including nonuniform variations of individual blades that
lead to different frequency mistuning patterns for different types of
blade-alone modes.
C.
389
New Methods and Emerging Topics
Even for the latest state-of-the-art reduced-order models, there
are still several simplifying assumptions that limit their capabilities
for predicting the response of actual bladed disks in turbine engines.
To illustrate this, consider Fig. 1, which shows the bladed disk as
it is typically modeled for structural dynamic analyses: in vacuo,
and isolated from other stages. In reality, a bladed disk is usually
one stage of a multistage rotor, and it is subject to the effects of the
fluid flow, which provides not only excitation but also damping and
interblade coupling. Nonlinear phenomena associated with various
physical and design aspects of turbine engine rotors provide additional complexities. Therefore, there are still many future research
challenges and opportunities. In the following subsections, some
emerging topics for vibration modeling of mistuned bladed disks
are highlighted.
1.
Combined Structural and Aerodynamic Coupling
Because the interblade coupling is so crucial to the mistuning
sensitivity of a bladed disk, for some systems the inclusion of the
coupling through the fluid might be critical to generating a meaningful model of the mistuned response. The vibration of mistuned
bladed disks with both structural and aerodynamic coupling has
been examined in several studies to date.59−64 However, recently
developed reduced-order modeling methods provide a new opportunity to investigate this issue more thoroughly. Furthermore, the
SNM,52 FMM,56 and CMM58 methods all generate ROMs in tuned
system modal coordinates, which allows easy incorporation of aerodynamic coupling. In fact, the CMM and SNM formulations were
presented with unsteady aerodynamic terms included in the system
equations of motion. New research in this area has been recently
initiated by Kielb et al.65,66 using the FMM method, and He et al.67
using the CMM method.
2.
Modeling of Nonlinearities and Blade Damage
Although the reduced-order modeling methods discussed earlier
in this section were formulated for linear vibration analysis, there
are many examples of important nonlinear phenomena in bladed
disks. There are nonlinearities caused by contact at shroud interfaces, at dovetail attachments for inserted blades,68,69 and caused by
rubbing between blade tips and the engine casing.70 There are also
nonlinearities introduced by vibration reduction elements such as
impact dampers71,72 and dry friction dampers.
Recently, several investigators have been developing comprehensive computational tools for the study of structural systems with
strong, nonanalytic nonlinearities. For example, efficient hybrid
frequency time73 and other methods have been developed for the
prediction of multiharmonic, steady-state response of bladed disks
with nonlinearities.74−76 The significant advances made by various
investigators for reduced-order modeling of mistuned linear systems and for nonlinear analysis enable the exploration of the physical interactive mechanisms between mistuning and nonlinearities
for a bladed disk. Such investigations would yield a better understanding of the vibration of realistic (i.e., mistuned or damaged)
structures as they operate in environments modeled so as to better describe realistic operating conditions (e.g., frictional boundary
interfaces).
Other nonlinearities and complexities include the effects of
rotation,77,78 cracked blades, and geometric blade damage (e.g.,
caused by a bird strike or a missing material in the blade tip). The
last item is interesting in that such geometric changes to the blade
result in blade-mode-shape mistuning. One consequence of this is
that the number of tuned-system normal modes required to describe
the mistuned-system normal modes increases dramatically, which
makes it impractical to use system-mode-based methods such as
SNM52 or CMM.58 To address this, a reduced-order modeling technique for bladed disks with geometric damage was recently developed by Lim et al.79 This technique employs a mode-acceleration
formulation with static mode compensation to account for the effects of the geometric changes on the mode shapes. This and other
specialized techniques could be combined with mistuning models
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CASTANIER AND PIERRE
to assess the reliability and safety of bladed disks that suffer inoperation blade damage.
3.
Multistage Modeling
Bladed disks are typically modeled as isolated systems, but in
general they are actually connected to adjacent stages in a multistage
rotor. A recent study by Bladh et al.80 has shown that connecting
a second stage to a single-stage finite element model of a bladed
disk can lead to significant changes for predictions of the maximum
blade response. For some operating conditions, dramatic changes
in the first stage’s sensitivity to mistuning were observed. This was
explained by the presence of the adjacent stage, which alters bladeto-blade coupling through the disk and thus mistuning sensitivity.
Furthermore, it was found that applying constraints to the boundary
degrees of freedom of a single stage could not faithfully capture the
boundary conditions of the actual stage-to-stage connection.
Therefore, it appears that multistage modeling can be essential
to improving the predictive capabilities of mistuned bladed disk
simulations. There are several fundamental research issues to be
addressed by future research. For example, the coupling of multiple
stages destroys the cyclic symmetry of the system, because of the
different number of blades for each stage. This means that even tuned
finite element models of the multistage assembly possess no special
symmetry, which mandates the use of effective reduced-order modeling methods. In addition, the convenient nodal diameter modes of
a tuned bladed disk can no longer be readily identified from singlesector analyses, because multistage systems do not possess cyclic
symmetry. It is also likely that some modes will have significant
strain energy in multiple stages. Thus, the characterization, identification, and classification of multistage system modes will need
to be explored. Finally, the influence of the stage-to-stage connections on the mistuned forced response could potentially be used to
improve the robustness of rotor designs with respect to their sensitivity to blade mistuning. This opens a new set of design possibilities
that can only be exploited through the development of multistage
modeling capabilities.
4.
Fundamental and Alternative Models
The discussions in the preceding subsections have largely focused on handling complexities by incorporating new capabilities
in bladed disk modeling. However, another way to handle complexity is to avoid it or to attack it from a different angle. By developing
simpler frameworks or alternative formulations, new benefits and insights can be gained. An obvious example of this is the Fundamental
Model of Mistuning developed by Feiner and Griffin,56 which has
already been found to be readily applicable to the problem of mistuning identification (see Sec. V.C). Another example is the use of
vibration power flow analysis81 to examine the exchange of vibration energy among blades.22,82,83 Such an alternative formulation
can also be used as a preprocessor (e.g., to identify operating conditions that can cause high blade forced response) or as a complement
to finite element-based reduced-order models.
V.
Analysis and Assessment with Respect
to Mistuning Sensitivity
The preceding section discussed the development of reducedorder vibration modeling methods. The next logical step is to be
able to apply these tools and develop additional techniques to assess
and improve the bladed disk design with respect to its sensitivity
to mistuning. In this section, some research issues in this area are
discussed.
A.
Frequency Veerings and Structural Coupling
As mentioned earlier, the increase in maximum forced response
for mistuned bladed disks is caused by the transfer and significant
confinement of vibration energy to a few blades. Thus, the interblade
coupling plays a crucial role in the dynamics of the system because
it governs the communication of vibration energy among blades.
Physically, blades are coupled through the disk, through shrouds,
and through the fluid. For the unshrouded case, the structural cou-
pling through the disk is typically the dominant mechanism for energy transfer. Therefore, blade-blade coupling is largely dependent
on disk-blade coupling, which is related to the interaction between
the disk and blade dynamics. This can be deduced, at least to a certain extent, from an examination of a plot of natural frequencies vs
number of nodal diameters.
Consider the 29-blade industrial rotor shown in Fig. 1. The plot
of free-vibration natural frequencies vs number of nodal diameters
is shown in Fig. 2. The lines are drawn to help visualize the various
families of modes. The groups of modes that appear as horizontal
lines are blade-dominated modes, and the blade mode family (1F,
1T, etc.) for each is labeled on the plot. Blade-dominated modes tend
to feature little disk motion, and thus they have weak interblade coupling. The sloped lines correspond to disk-dominated modes, which
have strong interblade coupling. Note that there are several regions
where the disk and blade modes appear to veer away from each other.
These modes in the veering regions tend to feature mixed disk-blade
motion. Therefore, the blades have significant vibration response if
these modes are excited, and there is also a mechanism for transferring energy between blades through the disk.22 This combination of
conditions can lead to the blade vibration energy being localized in
a few blades when mistuning is present, which results in high values of mistuned forced response. Therefore, the frequency veering
regions are of great interest and can provide important information
with respect to the system’s sensitivity to mistuning.
The “curve veering” phenomenon has been studied in the field of
vibration, with many applications to models of blades and turbine
engine rotors, for the last three decades.19,20,30,38,84−106 This work
was sparked by a 1974 article by Leissa,84 in which he commented
on an apparent curve veering aberration in the natural frequency loci
of rectangular cantilever plates for varying aspect ratio shown in an
earlier study by Claassen and Thorne.107 Leissa rather poetically described the changes in mode shapes in these veering regions as follows: “. . . in the ‘transition zones’ of veering away the mode shapes
and nodal patterns must undergo violent change—figuratively
speaking, a dragonfly one instant, a butterfly the next, and something
indescribable in between.” Leissa questioned whether the curve
veerings were real or an artifact of the discretization used to generate
numerical solutions. In 1981, Kuttler and Sigilitto85 used an approximate solution method for rectangular plates that indicated that such
curve veerings are indeed real. Perkins and Mote confirmed the existence of curve veerings for continuous models and established
criteria for distinguishing between crossings and veerings.86
Gottlieb87 considered the case of two pendulums coupled by a
spring and examined frequency veerings as the ratio of pendulum
lengths was varied. He noted that a crossing occurs only for the
decoupled case, and that for coupling greater than zero there exists
a curve veering. In 1988, Pierre88 considered eigenvalue loci veering for the two-pendulum system, a chain of coupled oscillators, and
other models of periodic structures. He used perturbation methods to
examine connections among eigenvalue loci veering, coupling, and
localization. He showed that weak coupling leads to both close veerings with sharp curvature and strongly localized modes in the free response of disordered structures. That same year, Wei and Pierre19,20
investigated veerings in the free response of mistuned cyclic chains
of oscillators, as a lumped-parameter representation of the vibration
of bladed disks. The eigenvalue loci veering investigated for these
coupled-oscillator systems occurred with respect to varying disorder or mistuning strength. Furthermore, there exists no veering with
respect to varying nodal diameters until the model includes at least
two DOF per sector (e.g., one disk DOF and one blade DOF).
In 1992, Afolabi and Alabi93 discussed the veering of eigenvalue
loci vs number of nodal diameters for bladed disks in terms of catastrophe theory,108,109 relating the veerings to bifurcation diagrams.
They noted that there is a crossing of blade and disk modes for
the uncoupled case; a close, sharp veering for weak coupling between blades; and a wide, shallow veering for strong coupling. They
pointed out that, in the veering region, the “forced vibration amplitudes of the individual blades are likely to change greatly under the
smallest change in the system’s parameters,” and recommended that
researchers devote more attention to these veering regions.
CASTANIER AND PIERRE
391
sis techniques that were developed for simple models of disordered
periodic and cyclic structures during the last few decades.
Another aspect of the importance of blade-to-blade coupling
through the disk is that it is possible to influence the blade response,
including the sensitivity to mistuning, through variation of the disk
design parameters. Slater and Blair110 investigated this strategy by
examining the effect of small disk design modifications on mode
localization in a simple finite element model of a bladed disk. Bladh
et al.80 also used finite element analysis for a simple finite element
model to consider the influence of both multistage coupling and
changes in nominal disk stiffness on natural frequency veerings and
mistuning sensitivity. In general, though, this topic has received
surprisingly little attention to date in bladed disk vibration research.
B.
Fig. 7 Natural frequencies vs nodal diameters from the continuous
interblade phase angle calculation.
To further investigate frequency veerings with respect to number
of nodal diameters, a method was developed by the Bladh et al.30,100
to obtain continuous natural frequency curves. The approach taken
is to consider the phase relationship between adjacent blades or the
interblade phase angle. The interblade phase angle φh takes on a
discrete set of values for the modes of an N -blade rotor:
φh = h(2π /N )
h = 0, 1, . . . , int[N /2]
(1)
where h is the number of nodal diameters or the harmonic number.
From the preceding equation, it is seen that the interblade phase
angle is a function of the ratio h/N . Therefore, a continuous nodal
diameter mode description can be obtained either by letting h assume noninteger values or, equivalently, by specifying integer values
of h and N that yield the same ratio. Choosing the latter option, one
can use cyclic symmetry solvers in commercial finite element software to compute these intermediate interblade phase angle modes,
as long as the finite element code does not check for geometric consistency. For example, the frequency plot in Fig. 7 was obtained
using MSC.NASTRAN by setting the number of sectors to 580,
yielding 20 data points per nodal diameter for the 29-blade rotor.
Comparing this plot with Fig. 2, note that several veerings can now
be seen in detail in Fig. 7.
Although the continuous curves might seem nonphysical, it is
important to recognize that the geometry of the fundamental sector
remains unchanged, and thus any interblade phase angle for which
the first and N th sector have identical motion at their shared boundary will correspond to an actual nodal diameter mode of the original
system. From this perspective, the integer nodal diameter modes are
sampled values on the frequency curves of the system with an infinite number of sectors. Furthermore, veerings that are not apparent
for the case of integer nodal diameters are seen in the continuous
veering plot, and it is then possible to better determine whether the
actual nodal diameter modes are disk dominated (sloped part of a
curve), blade dominated (flat part of a curve), or mixed (near center
of veering). Thus, quantifying the veering characteristics and determining the proximity of the nodal diameters to these veerings
can yield key information for quantifying blade-to-blade coupling
strength and predicting mistuning sensitivity.30,103
Certainly the prospect of using tuned free-response information
to make rapid predictions regarding mistuned forced response and
mistuning sensitivity is a tantalizing prospect. Not surprisingly,
this has received significant attention in the literature in recent
years.30,102,103,105,106 Geometric characteristics of veering regions
can also be used to identify representative lumped-parameter models with two or more DOF per sector.102,105 These models typically
capture vibration characteristics in the frequency range around a
family of blade-dominated modes, and the systematic identification
of their parameters opens the door for applying fundamental analy-
Mistuning Patterns and Intentional Mistuning
When considering the forced-response statistics for a population
of randomly mistuned bladed disks, there is often a large range
of forced-response levels. This is true even when focusing on the
largest responding blade for any blade in the system, or the extreme
value statistics. Consider the results shown in Fig. 6, which show
selected percentiles of the vibration amplitude magnification caused
by mistuning. There is a significant gap between the 99th and first
percentiles. Clearly, the particular pattern of the mistuning has a
large effect on the forced-response amplification, even for small
mistuning.
It is natural to wonder what sort of mistuning patterns lead to high
or low responses. In fact, there have been several studies on using
optimization methods to find the worst or best overall mistuning
patterns in terms of aeroelastic stability111−114 or forced -response
amplification.115−118 There have also been studies on finding the best
blade arrangement when a set of mistuned blades are given,119,120
although interest in this strategy is likely to decrease as the turbine
engine industry moves away from manufacturing bladed disks with
inserted blades and toward single-piece, integrally bladed designs.
Finding the worst-case mistuning pattern is useful because it involves finding the maximum blade forced response for the entire
population of mistuned bladed disks. This provides a measure of
the mistuning sensitivity and the reliability of the nominal bladed
disk design. Furthermore, investigating the types of mistuning patterns that lead to high response, and their effect on system vibration
characteristics such as mode shapes, can provide important insight.
The best-case mistuning pattern is of interest, of course, because
it minimizes the forced-response level. However, the best-case mistuning pattern does not necessarily represent an optimal or nearoptimal design because the actual response can be greatly affected
by the additional, random mistuning that will inevitably occur in the
manufacturing process or during engine operation. In fact, Crawley and Hall111 showed that the performance of the optimal pattern
could be very sensitive to small changes in the mistuning values.
The heart of the design challenge, then, is to determine a mistuning pattern that can be implemented in the nominal design that
will make the system less sensitive to random mistuning. This is
usually referred to as intentional mistuning. Intentional mistuning
is a design strategy that is appealing from many viewpoints. First,
because structures with cyclic symmetry can be so sensitive to small
mistuning, it makes sense to move away from a design with identical
sectors. Second, the amplitude magnification often exhibits a peak
with respect to mistuning strength at a small value of mistuning, as
seen in Fig. 6. This suggests that it might be best to start with sufficient mistuning to put the nominal design point on the other side of
the peak. Third, mistuning tends to increase the stability of a rotor
with respect to flutter,16,59,111 which means that intentional mistuning can have multiple benefits for increasing the safety of a design.
Intentional mistuning as a design strategy seems to have been first
considered, either directly or indirectly, in a few studies performed
20–30 years ago. In 1975, El-Bayoumy and Srinivasan121 investigated the effect on the system mode shapes and forced response of
having two distinct blade types, as defined by two different nominal
natural frequencies, arranged in various patterns around the disk.
However, they did not consider additional random mistuning. In
1980, Ewins122 discussed the possible advantages of bladed disk
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CASTANIER AND PIERRE
designs in which blades are grouped into “packets” of shrouded
blades. This type of design introduces a special form of mistuning, and Ewins explored the beneficial effects of “detuning” the
response of certain modes. The vibration modes of packeted bladed
disks were investigated further by Ewins and Imregun,123 using both
computational and experimental methods.
In 1984, Griffin and Hoosac14 conducted what appears to be the
first research investigation into the effectiveness of using intentional
mistuning in the design of a randomly mistuned bladed disk. They
investigated the case of two blade types placed in alternating slots
around the disk. They implemented this mistuning by setting the
mean weights of the two blade types to be 20% apart (±10% of
the original weight). From Monte Carlo simulations with and without intentional mistuning, they found that this “alternate mistuning” pattern reduced the worst-case blade amplitude significantly.
In 1985, Crawley and Hall111 also considered intentional mistuning,
as well as random mistuning, and they even found optimal patterns
of intentional mistuning. However, the focus of their work was the
aerodynamic stability (flutter) of the rotor.
More recently, Rzadkowski124 investigated the transient nozzle
excitation of mistuned bladed disks. By examining several configurations of a set of nominally different blades, Rza̧dkowski found
that a random configuration led to the greatest increase in stresses,
whereas a periodic (harmonic) blade arrangement was the best distribution in terms of minimizing the largest stresses. Yiu and Ewins119
simulated many randomly mistuned realizations of a simple model
of a 36-bladed disk, and they used discrete Fourier transforms to find
the harmonic components of the best and worst mistuning patterns.
However, they did not include intentional mistuning in the design.
In 1997, new research into the use of intentional mistuning was
initiated by the work of Castanier and Pierre.29 In this and subsequent
studies,54,55,125 they investigated the effect of introducing intentional
mistuning into the nominal design of a rotor. To examine a range
of designs, harmonic patterns of intentional mistuning were considered. In addition, square-wave patterns of intentional mistuning,
which only require two different blade types, were proposed as a
more practical design alternative. For all of the designs, the effect
of random mistuning on the statistics of the maximum blade forcedresponse amplitudes were investigated. It was found that certain
patterns of intentional mistuning were very effective in reducing
the maximum vibration levels. Furthermore, for effective patterns
of intentional mistuning the vibration level was relatively flat with
respect to the random mistuning level, indicating a decrease in sensitivity to random mistuning. In general, it was found that intentional
mistuning made the rotor design much more robust with respect to
random mistuning.
The degree of mode localization and the harmonic content
(Fourier decomposition) of the mode shapes were also considered for both the original design and designs with intentional
mistuning.54,55 For the original design, it was found that the worstcase response occurs at a resonant frequency for which the corresponding mode: 1) exhibits some moderate degree of localization
and 2) has a strong nodal diameter component that matches the
engine order of the forcing. (This same phenomenon was later observed by Kenyon et al.,126 who referred to the critical mode shape
changes as mode distortion.) By adding intentional mistuning to the
design, the modes became more localized. However, the dominant
nodal diameter components of the modes were eliminated, making
the system less susceptible to being strongly excited by engine order
excitation.
In the last few years, there have been several studies on determining effective patterns of intentional mistuning and/or mistuning
patterns that lead to high or low response levels.82,117,118,120,127−134
Kenyon and Griffin128 analyzed the effect of harmonic mistuning
on a rotor’s sensitivity to small perturbations in the mistuning.
Jones and O’Hara130 found that a linear pattern (one period of a
sawtooth-wave pattern) of intentional mistuning was very effective.
Choi et al.131 and Hou and Cross134 considered the optimization of
intentional mistuning patterns in order to reduce forced response
amplitudes. Lim et al.82 proposed some rules for reducing the design space for intentional mistuning, so that a small set of promising
intentional mistuning patterns can be identified without requiring a
full optimization process. The effectiveness of intentional mistuning has also been validated experimentally in recent studies for the
case of a square-wave pattern135 and a linear pattern.130
C.
Mistuning Identification
An important practical consideration for bladed disk vibration
research is how to identify the mistuning that is actually present in
a manufactured bladed disk. For rotor stages with inserted blades,
the blade-alone natural frequencies can be measured directly to determine mistuning values to be used in simulations as well as to
estimate mistuned blade structural dynamic properties.136−138 However, for an integrally bladed disk—a one-piece bladed disk—the
blades cannot be removed from the assembly. Therefore, mistuning identification techniques based on experimental measurements
of system response have been developed recently to determine the
individual blade mistuning pattern for each blade-dominated mode
family of interest.
The first such mistuning identification technique was developed
by Judge et al.139−141 To identify individual blade mistuning from
the vibration response of an entire bladed disk two sources of information were used: a finite element model of the bladed disk and a set
a measurements of the response of the actual bladed disk. To formulate the identification algorithm, an adapted form of the CMS-based
reduced-order modeling technique developed by Bladh et al.47 was
employed. In the mistuning identification process, secondary modal
analysis was used to condense out only the disk and disk-blade interface portions of the CMS model. This yielded an extremely reduced
model that retained blade modal stiffnesses explicitly. Then a small
set of experimental measurements of the system response was used
to determine the mistuning for the isolated blade modal stiffnesses.
Either mode shape measurements or forced-response measurements
can be used.141 Similar techniques have recently been developed by
Feiner and Griffin142,143 based on the FMM method and Lim and
coworkers144,145 based on the CMM method. The technique of Lim
et al. also incorporated a reduced-order model updating procedure
to improve the accuracy of the identification procedure.
Mistuning identification can also be used to assess the quality
of the manufacturing process, to flag possible tooling problems,143
and to perform structural health monitoring of turbine engine rotors. An example of the last item is to identify a cracked blade.
Cracks or other types of blade damage can have very small effects on
the blade-alone frequencies yet still cause strong changes in some
of the system mode shapes.79 Therefore, these system-responsebased identification methods are promising for damage detection
applications.
VI.
Uncertainty and Reliability Analysis
Because blade mistuning is random, most forced-response analyses of bladed disks are probabilistic. However, mistuning is not the
only form of uncertainty in the system. Certainly, the research topics
discussed in Sec. IV.C are examples of phenomena that could potentially be of great importance to the mistuned blade response but are
often neglected in vibration models. Thus, modeling assumptions
and limitations introduce significant uncertainty for forced-response
predictions. Moreover, even when some aspect of the dynamics
is well modeled, certain parameter values might not be precisely
known. This is yet another form of uncertainty. An example is the
estimation of mistuning values used for the blades of a particular
rotor stage.141,146
In general, the modeling and computational tools developed to
date would be enhanced by incorporating fundamental analysis of
the propagation of uncertainty into free- and forced-response predictions. That is, researchers developing vibration analysis methods
should attempt to account for the sensitivity of the mistuned response
predictions to various types of parameter uncertainties and modeling assumptions, such as 1) the nominal system parameters used in
the model (tuned blade-alone natural frequencies, assumed damping
coefficients, etc.); 2) boundary conditions imposed at the stage-tostage connections of a bladed disk; 3) the relative strength of aerodynamic and structural coupling, or the implications of neglecting
CASTANIER AND PIERRE
aerodynamic coupling; and 4) other types of mistuning besides traditional blade frequency or modal stiffness mistuning, such as forcing
mistuning (i.e., perturbations about pure engine order excitations),
geometric blade mistuning, damping mistuning,147 blade stagger
angle mistuning,142,148 and disk mistuning. One key contribution of
this uncertainty analysis would be a capability for assigning confidence limits to numerical simulation results for mistuned systems.
Another contribution would be a framework for determining the relative importance of neglected phenomena and thereby choosing a
level of modeling fidelity that is commensurate with the goals of the
simulation. In some cases, it might be found that a simpler model
can provide sufficiently useful predictions for, say, a design sensitivity study, and that a more complex model would not be worth
the additional computational expense given unavoidable modeling
limitations and uncertainties.
Finally, it would be interesting to investigate the use of reliability
methods developed in other disciplines to applications in vibration
analysis of mistuned bladed disks. This poses great challenges because bladed disks are highly sensitive to small parameter changes
and the blade response varies nonlinearly with respect to the parameter space. It might turn out that accelerated Monte Carlo simulation
is the best option for characterizing the statistics and bounds of the
mistuned forced response. On the other hand, the variety of vibration
modeling and analysis tools that have been developed to date provide
a wealth of opportunity for exploring new methods for assessing and
improving the reliability and robustness of turbine engine rotors.
VII.
Conclusions
Recently, significant research progress has been made in structural modeling, analysis, and understanding of mistuned bladed disk
vibration. Much insight to date has been gained from the use of simple lumped parameter models. However, such models are best suited
to exploring fundamental issues. In the last decade, a variety of finite element-based reduced-order modeling approaches have been
developed that allow bladed disk vibration to be captured with extremely compact yet accurate models relative to the parent finite
element model. This makes it feasible to predict mistuned forced
response for a bladed disk in the design stage and to assess the
sensitivity of the system to mistuning. These reduced-order models have also been used to examine the effect of different types of
mistuning patterns on forced-response amplification, to explore the
use of intentional mistuning in the nominal design, and to identify
blade mistuning from experimental measurements. There are still
many potentially important phenomena that are typically neglected
in vibration models of mistuned bladed disks, such as aerodynamic
coupling and nonlinearities caused by friction and contact. Emerging research topics include combining efficient models developed
in aeroelasticity and nonlinear vibration with reduced-order mistuning models, modeling the effects of blade damage, and modeling
multistage systems.
Acknowledgment
The authors would like to thank Sanghum Baik and Sang Heon
Song for generating the images of the tuned and mistuned mode
shapes.
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