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Kushner2004-ResonanceRotatingComponents.pdf
ROTATING COMPONENT MODAL ANALYSIS
AND RESONANCE AVOIDANCE RECOMMENDATIONS
by
Frank Kushner
Consulting Engineer
Delmont, Pennsylvania
Frank Kushner is an independent
Consulting Engineer specializing in
dynamics and acoustics testing and
analysis. He has had 35 years of experience
with industrial turbomachinery, mostly all
while with The Elliott Company, and four
years previous experience with the jet
engine combustion section development
group at Pratt & Whitney Aircraft. He is a
previous author for the Ninth, 25th, 29th,
31st, and 32nd Turbomachinery Symposia,
as well as for the ASME.
After obtaining a B.S. degree (Mechanical Engineering, 1965)
from Indiana Institute of Technology, Mr. Kushner received his
M.S. degree (Mechanical Engineering, 1968) from Rensselaer
Polytechnic Institute. He is a registered Professional Engineer in
the State of Pennsylvania and a member of the ASME and the
Vibration Institute. Mr. Kushner holds patents for a blade damping
mechanism and a method to prevent one-cell rotating stall in centrifugal compressors.
Figure 1. Location of Rotor Blade Failures—Data for 58 Electric
Utilities. (Courtesy EPRI, Palo Alto, CA)
ABSTRACT
Rotating blade and disk fatigue failures cause a fairly small percentage of machine shutdowns, but can require extensive repairs
and downtime. This tutorial focuses on steam turbines, centrifugal
compressors, and axial compressors primarily for refineries and
process plants. For steam turbines, axial compressors, and openimpeller centrifugal compressors, the main concern is resonance
giving blade rather than disk vibration. For covered impellers in
centrifugal compressors disk modes are the main response for
problems, albeit failures are much more rare.
Reviews of methods are to give insight to design evaluation, calculation, and test procedures. Cases will be given where designs
were suspect with high potential for cause of fatigue, with other
cases due to minor resonance combined with either manufacturing
defects, operation, and/or environment such as liquid ingestion and
corrodants. For some cases, test data will be correlated with
theories provided, while others will use references, some of which
will be shown to perhaps require consideration of alternative
probable causes. The material will utilize both proven concepts and
cases for lessons learned.
McCloskey (2002) gives a comprehensive summary of utility
turbine blade and disk troubleshooting including many references.
There are similar issues with jet engine designs, their land-based
derivatives, and other gas turbines including microturbines (refer to
Srinivasan, 1997), but these also have many other issues such as
low cycle fatigue analysis and interaction of components with
much different operational modes and life cycles (Layton and
Marra, 2000). Reports give values between 40 and 50 percent of
gas turbine outages due to blade fatigue; besides other sources,
combustion pulsation can also give rise to alternating pressure
forces. Gas expanders such as those for fluid cat cracker (FCC)
processes are a special case with the combustion occurring in the
process, but there have been limited blade fatigue problems not
associated with corrosion, such as described by Dowson, et al.
(1995), and catalyst dumps into the turbine inlet casing.
Turboexpanders, fans, and some liquid pumps should have similar
issues to those that are to be covered, but extensive reviews will
only include machinery based on direct experience.
Some of the materials to be presented primarily for fluidhandling turbomachinery for the oil, petrochemical, and related
industries are:
INTRODUCTION
High cycle fatigue (HCF) and its prevention is one of the most
serious issues for fluid-handling turbomachinery for the oil, petrochemical, and related industries. Relative to this type of industrial
machinery, electric utility turbine/generator sets can have consequences that are often greater; in some cases, blade and/or disk
fracture can be catastrophic and a safety issue. In some reports,
failures of turbine blades are identified as the leading causes of
unplanned outages for these steam turbines. Accidents of lowpressure end turbine blades suffer a high percentage of failures;
Bhaduri, et al. (2003), provide some Electric Power Research
Institute data as reproduced in Figure 1.
• Basic coverage of resonance response.
• Methods to analyze cantilevered and shrouded blade designs.
• Disk modes and disk critical speeds.
• Test procedures and correlation with finite element analysis
(FEA).
• Excitation sources with designs for minimizing loads.
• Campbell diagrams—do’s and don’ts for both blades and disks.
143
144
• Interaction
PROCEEDINGS OF THE THIRTY-THIRD TURBOMACHINERY SYMPOSIUM • 2004
resonance with proper interference diagrams for
disks.
• Effects of mistuning of blade natural frequencies on response
causing fictitious interference diagrams.
A historical review for the all important interaction resonance
effect between blades and vanes for disks will be offered, starting
with Kushner (1979). Another objective is to assist with explanations as to why there must be exceptions taken to the extreme
specifications for resonance avoidance such as those in API 612
(2003) specifications. There are numerous resonance criteria to
consider, including many that give negligible response and thus
alternating stress. For many potential resonant speeds, environmental effects that reduce endurance strength can sometimes lead
to questions of design adequacy. Other resonant points must
always be avoided for all designs, and some have limits that can
only be exceeded if there are improper operation and/or environmental factors. Besides forced response for turbine and axial
compressor blades instability with flow interaction reducing
damping is an issue in component design, whereas instability does
occur but is extremely rare for radial flow impellers.
Cases will be given where designs were suspect with high
potential for cause of fatigue, with others due to minor resonance
combined with either manufacturing defects, operation, and/or
environment such as liquid ingestion and corrodants. These will
include several cases for steam turbines and multistage centrifugal
compressors along with two axial compressors, and a plant air
compressor. For some, available test data will be used; some other
results will be supplemented with reviews in the literature. In some
cases, alternative possible causes will be given for the cited references.
Reliability of turbomachinery components requires that aspects
of fitness-for-purpose be properly addressed at various stages of its
design, fabrication, testing, and operation. Besides economic considerations, safety considerations influence the inherent quality
and reliability of the machine. Safety basically requires proper
functioning without presenting hazards to personnel and property.
It is possible to build a highly reliable compressor or turbine that
ends up being economically competitive, i.e., using the author’s
personal motto—”there is an optimum to everything.” Design
specifications are needed to incorporate optimum fitness-forservice attributes. Those specifications must include review of any
potential resonance that could cause fatigue failures. This tutorial
will hopefully present some important views on how components
should be designed so that they meet the service requirements for
the required life (20 years per some specifications). Of course components also must be fabricated with specified materials, inspected
to show they conform to design concepts, and are operated and
maintained properly. Whenever there are failures even from misoperation, it is always wise to use the knowledge gained to achieve
optimum designs for “fitness-for-service.”
Besides the rotor/bearing system lateral modes of all turbomachinery, and potential severe torsional system resonance for some
with motors and/or gears, an equally critical aspect is resonance
avoidance for structures on the rotor. If the frequency at which
variable loads act on a bladed disk coincides with one of the
important natural frequencies of the structure, a very dangerous
situation can occur, leading to rapid failure due to high cycle
fatigue. High cycle fatigue (HCF) is defined as that which results
in cracking or fracture from a large number of stress cycles well
below the yield strength of the material, and is associated with
fatigue lives greater than about 10,000 cycles. Low cycle fatigue
(LCF) such as from thermal effects during startup is normally not
an issue for blades and disks in machinery to be covered herein.
However, severe forces from improper operation such as continuous surging and/or excessive liquid ingestion can cause failure in
much less than a million vibration cycles. For many observers,
even 10 million cycles (corresponding to threshold endurance
strength for many defect-free, ductile materials) appear to be an
unattainable high number of cycles for setting a lower limit.
However, consider that structures discussed in this tutorial respond
typically in the range or 200 to 10,000 cycles per second.
Depending on the severity of excitation and resonant amplification
for the structure, a failure can occur in a few minutes to several
hours if operating at high loads very close to, let alone exactly at,
a resonant operating speed. There have been cases of failures in the
first day of operation after startup, as shown by one of the case
studies. Resonant response is avoided or minimized by controlling
the stimulus or by changing the blade frequency. The stimulus can
be controlled by changing the frequency of excitation, changing
the strength of the stimulus or its relative phase with respect to the
blade or bladed disk. The most common method of modifying
blade frequency is to tailor the thickness of the blade or disk along
with other variables affecting frequency: aspect ratio, taper,
number and thickness of blades, and even radius ratio. More drastic
modifications are to use one of many damping methods, e.g.,
change to a friction damping mechanism at part span or by using a
tip shroud incorporating damping and/or phase cancellation.
In some cases, natural frequency variations even for constant
speed applications of load may be greater than an acceptable band;
thus the blades must be designed with sufficient stiffness, higher
strength materials, and/or higher damping to limit stresses
whenever blades operate at resonance. HCF failures can otherwise
occur when these conditions are not met, or if a forcing function
becomes excessive, such as from upstream or downstream blade or
vane damage, operation at off-design causing rotating stall, surge,
flutter, or other damage, e.g., from blade tip rubs. In some cases
there can be simultaneous sources of excitation affecting the same
or different modes. Manufacturing tolerances and fabrication
variation can influence natural frequencies for a blade assembly;
corrosion, wear, and erosion can also reduce resonant frequency
margins and/or endurance strength over time. Either or a combination of these conditions can result in high-cycle fatigue initiation
and propagation. There also can be a cause of crack initiation,
followed by secondary resonant points that can assist in propagation. The cracked structure has lower stress intensity and required
number of cycles for crack growth compared to initiation. If
inspections do not find cracks to abate further propagation, a
structure then could easily eventually reach its fast fracture limit.
Many machines have been shut down due to changes in monitored
vibration data before a catastrophic wreck. Thankfully ductile
materials mitigate this concern. In some cases cracks could lower
the resonant frequency such that the structure moves away from the
resonant speed. This change could give some relief, especially for
constant-speed machines such as turbine/generator sets. In other
cases, sometimes after a long period, resonance and/or the same or
another speed change or off-design upset giving high loads can
reoccur, causing the “beech-mark” patterns on a fractured surface.
The tutorial will include an initial precursory description of
resonance, using examples for blades and disks. There will then be
a more general description of modal analysis of rotating structures
that should apply to other machinery, outlining which vibratory
modes are especially important. The risks of resonance for
different applications vary; each major type of rotating machine,
centrifugal and axial handling gaseous fluids will then be more
fully covered in the case studies.
REVIEW OF BLADE AND DISK RESONANCE
Machinery discussed in this tutorial typically has potential for
excitation of either blade or disk modes; however some designs
could have coupled disk/blade modes as do many jet engine
designs and gas turbine derivatives. As in all aspects of vibration,
it is very important to always consider phase, not just amplitude
and frequency. As the author likes to repeat, you do not push a
child on a swing as he is coming toward you, unless you want him
to stop and do his homework.
ROTATING COMPONENT MODAL ANALYSIS
AND RESONANCE AVOIDANCE RECOMMENDATIONS
145
Natural frequencies with finite element programs are utilized for
design, with much less testing as compared to 20 years ago. Some
correlation tests as described in APPENDIX A are always beneficial, especially for critical resonant modes such as for blade leading
edge mode for impellers to be reviewed later. The equation of
motion for a single degree of freedom model is given in Figure 2.
Figure 2. Equations of Motion for Single Degree of Freedom.
For modal analysis using finite element analysis, the same
equations of motion apply; for multiple degrees of freedom they take
on a matrix form. For complex structures, the analysis requires
careful modeling (Figure 3) and proper techniques including
boundary conditions. In order to obtain relative stresses for higher
order modes, the entire impeller may require modeling. Modal
analysis of a cyclic symmetric structure to determine nodal diameter
frequencies can be computed by modeling only one sector per the
finite element computer code (Kohnke, 1999). Nodal diameter mode
shapes contain lines of zero out-of-plane displacements crossing the
entire disk. These lines of zero displacements are commonly called
nodal diameters. The number of nodal diameters, n, is an integer that
determines the variation in the value of a single degree of freedom
at points spaced at a circumferential angle—equal to the basic sector
angle (Theta). For a number of nodal diameters equal to n, this
variation is described by the function, cos (n [Theta]). This definition allows a varying number of waves to exist around the
circumference for a given nodal diameter.
Constraint relationships can be defined to relate the displacements of one edge of the cut boundary to the other edge. This
allows for calculation of natural frequencies related to a given
number of nodal diameters. The basic sector is used twice to satisfy
the required constraint relationships and to obtain nodal displacements. The analysis results will show pairs of frequencies for each
nodal diameter solution. Using this technique, it is possible to
obtain solutions for up to a number of nodal diameters, n, up to n
= B/2. For an even number of blades, B, this would be B/2; for an
odd number of blades this would be (B-1)/2. Methods are given to
evaluate these frequencies for both static and dynamic (stress
stiffened) conditions. More complex modes require a model of the
entire impeller as shown in Case B-6 below.
Cantilevered Blade Design and Resonance Avoidance
Rotating blades in axial machinery have the most difficult task
of HCF avoidance, as they are most compliant, not only in axial
compressors but also in nonshrouded stages of both steam turbines
and centrifugal compressors. Freestanding, cantilevered blades
come in many forms and sizes. Shorter blades usually do not have
much twisting along the profile length and have three fairly independent modes: fundamental tangential, fundamental axial, and
Figure 3. FEA Model of Covered Impeller with Full Inducer and
Curved, 3-D Blades.
fundamental torsional (refer to APPENDIX A). Higher modes have
nodal lines and have phase changes along the length and width and
thus have inherent phase cancellation for most cases. The fundamental
tangential mode is not only the most compliant, but typically experiences the highest excitation forces as the direction of force (a
percentage of steady forces) aligns with the predominant steady gas
force that “drives” the machine. Many blades are canted so that tangential forces can assist in excitation of the fundamental axial mode,
but axial forces are lower and axial modes are less compliant giving
lower dynamic stresses. Torsional modes also are of lower concern as
there is a node line along the blade, so that the leading edge exciting
force is nearly out of phase with that on the trailing side. For short
blades, as compared to the tangential bending modes, axial and
torsional modes are at least on the order of one tenth of the concern for
short blades; there were perhaps one or two field problems in 30 years
in the author’s former company. There could of course be exceptions,
such as from axial forces on a highly loaded reaction-bladed stage.
Medium length blades such as the middle stages of steam turbines
typically have much fewer problems (refer to Figure 1). The
Campbell diagram in Figure 4 shows that nozzle passing frequency is
avoided; resonance of the first modes would only be equal to higher
harmonics of speed where severe nonuniformity of nozzle throats is
the main issue along with potential excessive corrosion in transition
stages (one of the cases below). Higher order modes, not shown in
Figure 4, could also present problems with excessive corrosion.
Longer cantilevered blades such as those in the rear of steam
turbines (refer to Figure 5) and the front-end of axial compressors
can have resonance potential for all three directions. In modern
machines, longer blades have significant twist and have larger chord
widths; thus the modes for the three directions are coupled. In fact
there can be higher axial motion as compared to tangential at the
blade tip for the lowest frequency mode; thus higher axial forces for
a reaction-type stage of a steam turbine could give relatively more
response at resonance. In addition, the fundamental modes also have
potential for flow-incidence related flutter. The longer freestanding
blades in steam turbine-generator exhaust ends usually must have
the first few modes carefully tuned. Gas expanders should have less
risk, especially since inlet connections are axial reducing harmonic
excitation, other than from support struts; the high number of inlet
vanes typically would minimally excite higher order modes.
146
PROCEEDINGS OF THE THIRTY-THIRD TURBOMACHINERY SYMPOSIUM • 2004
Figure 4. Campbell Diagram for Turbine Blade, 128 Inlet Nozzles.
amplitude and phase changes, can be done for higher modes. An
example is shown in APPENDIX B where the response factor can
be forced to give zero response. With the many varieties of designs,
it is impossible to obtain complete cancellation for every stage, but
typically designs can be utilized with resonant factors down to
0.20, i.e., dynamic stress reduced by a factor of five, as compared
to freestanding blades. However shrouding blades into packets also
gives some more coupled modes to be concerned with as shown by
Weaver and Prohl (1956) and many others. For example, the
turbines on the Queen Elizabeth 2 ocean liner had failures due to
higher modes for nonoptimized packet configurations. An out-ofphase mode in resonance with upstream vane passing frequency
could again have the number of blades per packet optimized giving
a different value than for the fundamental mode.
Some manufacturers seem to standardize with a number such as
five blades per packet, but it is sometimes very important to
optimize a given stage. For example, a high pressure, high temperature Curtis stage with 64 rotating blades and 71 inlet vanes should
have eight blades per packet, not four per packet—if a fundamental
mode is resonant with 71 times speed. If it is not resonant, it could
be acceptable to use four blades per packet. Besides shrouding
shorter blades to reduce response at high frequencies from inlet
vane passing frequency, “long-arc” blading can be used to minimize
response of fundamental modes (refer to Ortolano, et al., 1981; and
McCloskey, 2002). In a low-pressure turbine stage, instead of using
24 packets with four blades in each, a six-per-revolution resonance
could have six packets with 16 blades per packet around the circumference. APPENDIX B can be used to check that the response
factor would be zero when using N equal to 6. If the shroud could
not physically fit over the tangs during assembly but could fit with
eight packets, the response would still have fairly low response.
Friction Damping Methods
Figure 5. Looking from Tip of Long Blades with Zigzag Damping
Pins.
Freestanding, cantilevered blades used for open-impellers also
have much more risk for HCF than fixed-fixed blades in covered
impellers. Inlet guide vane selection must consider the fundamental bending and sometimes the higher modes with nodal lines.
Number of inlet guide vanes thus becomes a design variable for
integrally geared compressors and radial-inflow turbines. As
impeller sizes increase the leading edge flapping mode can become
a design issue for covered impellers that can be missed. The fixedfixed mode is not very compliant, i.e., as compared to a
cantilevered blade. Many designs likely could run with having first
order resonance as there are thousands running with resonance of
the same mode, but excited by two times upstream vanes. Cases
such as described by Singh, et al. (2003), and Phillips, et al. (2003),
likely were due to other mitigating factors. However, there will
always be the issues:
• Was operation within the specified operating map, and
• How much liquid was really being ingested to potentially
aggravate the wakes from upstream vanes.
Shrouded Turbine Blades
In order to mitigate resonance, phase cancellation has been used
for many years in steam turbines and other machines. Weaver and
Prohl (1956) give one of the best reviews on this subject. By tying
blades into packets, phase cancellation can be optimized for a
given mode, so that some blades in a packet are out of phase with
the exciting forces. This design feature is likely the most important
reason over the years for reducing the number of HCF incidents in
steam turbines. Response from partial-admission causing nonuniform loading is also greatly reduced by shrouding. APPENDIX B
gives a simplified method to check for optimum blade number per
packet for the fundamental mode where all blades vibrate in phase
with the same amplitude. Similar procedures, considering
Besides shrouding, part-span snubbers and lashing wire designs
have greatly reduced failures in steam turbines. The patent on
zigzag damping pins for long twisted blades has expired and has
been used by many others. An excellent design, especially for
shorter blades, and used for decades—is the integral-shroud with
rolled-in lashing wire; a more recent design is shown in Figure 6
with improved tip sealing availability. In effect the design has
many advantages with the integral shrouds acting as snubbers.
They are assembled with zero or negligible gaps at the tips; the
small gaps that occur at speed and temperature are usually small
enough to limit motion especially in the flexible tangential
direction. In addition, rolled-in lashing wire provides a large
damping force where there is high relative motion between blade
tips. Highly loaded blades can have two rolled-in lashing wires as
shown in the figure, with the ends of each wire meeting at blades
that are 180 degrees opposite.
Another trend today is to use more z-lock shrouds (Figure 7) for
tapered, twisted blades, with friction damping at the tip contact
surfaces, eliminating the need for rolled-in wire. The trend likely
began with designs in the 1980s such as the power turbines for gas
turbine derivatives. With a continuous shroud, the modes to
consider are coupled modes where the blades vibrate with mode
shapes defined by nodal diameters. Thus instead of individual
blade modes, the most responsive disk-type critical speeds must be
avoided as described later for disks. For example, the threediameter mode of a design may have high stress if the frequency
was equal to three times operating speed; for either of both sets of
modes defined for motion in the predominant tangential and then
axial direction. In Figure 8 is shown an interference diagram for a
design where the first mode could have resonance for the five- or
six-diameter mode for the first family of modes in the speed range.
The second family of modes, mostly tangential, could have the
nine- through 11-diameter mode in resonance. Thus it would not be
wise to have high excitation sources at 5, 6, 9, 10, or 11 times
speed. For some designs, the blades must be preloaded to ensure
that under centrifugal growths and blade untwisting, there is
ROTATING COMPONENT MODAL ANALYSIS
AND RESONANCE AVOIDANCE RECOMMENDATIONS
147
fatigue failures. The fundamental, two-diameter mode shape is
shown in Figure 9. As this disk rotates, at a certain speed each part
of the disk will arrive at an area in phase with a stationary exciting
force. As shown in the Campbell diagram (Figure 10) the resonant
operating speed is equal to the natural frequency divided by the
number of diametrical nodal lines. In Figure 10, the “backward
wave” line, |fr – 2x|, intersects the x-axis when the disk frequency
is equal to two per revolution. Note that fr, frequency at speed,
includes effect of temperature, fluid mass loading, and centrifugal
loading on stationary frequency, fs. The diagram only shows one of
the two frequencies for each “n” diameter mode due to dynamical
imperfection (Tobias and Arnold, 1957), with nodal lines shifted
by (90/n) degrees. Figure 11 has some data for a more recent
impeller with a zoomed fast Fourier transform (FFT) spectrum
analysis of a rap test. At speed, the two frequencies will likely get
further apart as the cover rotates circumferentially with respect to
the hub (refer to plots in Kushner, 1979).
Figure 6. Blade Integral Tip Shroud with Two Rolled-In Lashing
Wires for Damping.
sufficient contact for the surfaces, which in turn are also properly
coated. Other advantages of z-lock shrouds are: permit ease of
using tip seals, eliminate inherent aerodynamic losses of lashing
wires and snubbers in the steam path, eliminate areas in open
shrouds and lashing wire designs where deposits and corrosive
medium can accumulate.
Figure 9. Mode Shape of Two-Diameter Disk Mode for Covered
Impeller.
Figure 7. Schematic of Turbine Blade Z-Lock Shrouds with Friction
Damping.
Figure 10. Campbell Diagram for Disk Critical Speed—
Fundamental Two-Diameter Mode.
Figure 8. Interference Diagram for Z-Lock Integral-Shroud Blade
Design.
DISK MODE RESONANCE
Disk Critical Speeds
The most destructive resonance is that for disk critical speeds as
was found in the 1920s by Wilfred Campbell (1924). In fact some
of those disks had axial rubs from the high amplitudes involved.
Disks had basically become too thin in steam turbines, leading to
Nodal lines remain fixed on the rotating disk because there are
two frequencies for each mode, with normal phase changes
depending on damping in traversing each resonant peak. With a
large enough difference between the two frequencies, resonance at
each one of the peaks only has a small component of the other
similar mode. A measurement of a rotating, vibrating disk from a
particular stationary point would show two frequencies for both of
the modes: the frequency measured by an accelerometer or strain
gauge attached to the disk, plus and minus the number of nodal
148
PROCEEDINGS OF THE THIRTY-THIRD TURBOMACHINERY SYMPOSIUM • 2004
Figure 11. Zoom Frequency Spectrum from Impact Test Showing
Two Frequencies for Same Mode.
diameters times rotating speed. At a disk critical speed, the lower
frequency reaches zero; i.e., the average amplitude of the disk
around the circumference in stationary coordinates would form a
standing wave in stationary space. Thus even a local high-pressure
nonuniformity is able to excite the disk; it is in phase with each
point (and thus every blade) on the disk as the disk rotates and
vibrates. At all other speeds, there would be phase cancellation
over each revolution of the disk. There have not been many disk
critical speed problems since the time-honored publication of
Campbell (1924); however there must be continuous scrutiny.
Rotating Blade/Stationary Vane Interaction Resonance
For other resonant points away from disk critical speeds on
Campbell diagrams, to determine whether forces would cancel for
the entire disk at this speed, one must review the number of
rotating blades interacting with the number of stationary elements.
For almost all combinations, there is phase cancellation of forces.
For a mode such as a five-diameter mode for a 15-bladed impeller
shown in Figure 3, other than for stationary vane numbers of 10 or
20, exciting forces would cancel as the impeller spins and vibrates.
Forces do not cancel if the natural frequency is equal to 10 times
speed with 10 stationary vanes, or with either 10 or 20 vanes if the
frequency is equal to 20 times speed. This is because either 10 or
20 vanes give a difference of five with 15, the number of rotating
blades. The parametric equations from Kushner (1979) giving
cases where there is not phase cancellation are reiterated as
follows:
VIBRATION EXCITATION AND RESPONSE
• Not at disk critical speeds:
Effects of Mistuning on Blade Modes
(a ) y ⋅ S ± z ⋅ B = n
(b) y ⋅ S = h
( c) f r = y ⋅ S ⋅ ω
(1)
• At disk critical speeds:
(a ) For B > 1
(b) y ⋅ S = h = n
( c) f r = n ⋅ ω
where:
B
=
S
=
fr
=
h
=
n
=
y&z =
ω
=
extreme excitation case of disk critical speeds, as then it does not
matter how many blades are on the disk; each blade is in phase
with the force. The extreme case for Equation (2) shows the added
danger of having both a disk critical speed and the number of stationary elements equal to the number of nodal diameters. By
contrast, interaction resonance has to have a certain combination of
numbers besides having operation at a specific resonant speed. The
highly beneficial recommended standard of not using equal
numbers for blades and vanes is based on Equation (1). That is,
when n = 0, circle and torsional modes are not excited unless y S = z B. Typically, only values of one and two are used in
Equation (1) for integers, “y” and/or “z”; higher factors would give
much lower excitation and/or response (refer to calculations in
Kushner, 1979). In reviewing these equations, it must be considered that the loading is absorbed by the blade surfaces that transfer
the steady-state loads. One of the least understood causes of disk
vibration is acoustic interaction with acoustic pressure waves that
directly excite the disk. The interaction of pressure waves between
disk surfaces and the adjacent stationary wall may also be
important, as shown by Eckert (1999) and Ni (1999); the instability of the two-diameter mode from acoustic interaction described
could thus also give forced excitation with matching phase angles
with the disk modes.
Equation (1b) is actually implied by (1a) and (1c); it is inserted
to add emphasis that resonance is at a harmonic related to the
number of stationary elements, not the number of rotating blades.
This point is sometimes misunderstood, e.g., response for very
flexible fan disks can show amplitude peaks at blade passing
frequency on bearing housings due to resonance of one-circle and
one-diameter modes (Baade, 1998). For impellers with splitter
blades, both full or splitter blade numbers and their sums should be
evaluated for excitation, as did NASA (1990). To date, the plus
sign in Equation (1a) is not important for radial compressors; it
results in high values of “n” diameters, difficult to excite, let alone
find in testing.
Because of the use of higher values of numbers used and the
high differences between numbers of blades and vanes, axial flow
turbines typically have very low risk of blade/vane interaction
resonance for disk modes. It should also be noted that any design
that has a disk critical speed or interaction resonance below
operating speed would have a very limited number of cycles while
traversing the low speeds. Steady-state stress will be lower and
reduced power giving much lower exciting loads will further limit
risk.
(2)
Number of rotating blades
Number of stationary elements
Natural frequency at speed, Hz
Harmonic of speed
Number of diameter nodal lines
Integers > 0
Rotating speed, Hz
Note that the natural frequency at speed, fr is affected by centrifugal loads and in some cases by thermal, pressure, or fluid mass
loading changes from ambient. Equation (2) is also given for the
As was concluded by Slater, et al. (1999), “Detailed finite
element analysis of tuned bladed assemblies can be prone to large
errors.” Mistuning, variations of blade natural frequencies in a
rotating row, is the reason why there are often “rouge” blade
failures (refer to cases below). For forced excitation Bladh, et al.
(2002), show that mistuned responses can exceed tuned response
levels by nearly 200 percent, if appropriate levels of mistuning and
interblade coupling are present. Whitehead (1996) has also
provided much research on the subject. There have been cases in
the literature where coupled blade/disk modes have been discussed
for short blades, using interaction resonance affects similar to those
described above. However, for packets of relatively short blades, it
is not recommended to use such interference diagrams as prescribed by Dello (1987) and Singh, et al. (1988 and 1994).
Mistuning of blades as described in many publications surely
prevents the occurrence of such nodal diameters for many but not
all cases, in that some blades can be largely decoupled from the
disk, especially for the predominant tangential direction. By not
considering mistuning, this type of graphing procedure for
shrouded packets of rotating blades can sometimes assert “safe”
operation at resonant speeds of packeted blades. For short blades in
ROTATING COMPONENT MODAL ANALYSIS
AND RESONANCE AVOIDANCE RECOMMENDATIONS
packets, the natural frequencies for a particular mode can easily be
a few percent different from packet to packet. With resonance at
nozzle passing frequency, a nodal diameter pattern may actually
not be important at all. There are inherent variations involved in
manufacturing tolerances and assembly differences. Differences
can also increase with time of operation due to corrosion, erosion
or deposits, and in rare cases from foreign object damage
(McCloskey, 2002). Thus each packet can be excited separately;
e.g., resonant with nozzle passing frequency; geometry patterns
that define response are numbers of blades per packet and ratio of
nozzle to rotating blade numbers for phase cancellation (refer to
APPENDIX B). In fact, just as for “rouge” cantilevered blades
without shrouds there will be increased response of some packets,
in effect acting as vibration absorbers for others (refer to analysis
for shrouded blades in Bladh, 1999).
The interference/nodal diameter plotting method reviews
optimum number of blades per packet primarily based on coupled
blade/disk modes. However, the inference is that all blade packets
will have the same frequency for the nodal diameter patterns, and
does not consider individual packet modes. Reality is that there can
be different frequencies for the same mode, including the fundamental, albeit close together; thus in some stages such as those
with locking pieces instead of locking buckets, it is necessary to
have different numbers of blades per packet that gives added differences. If there ever was a troublesome mode of an entire
structure, i.e., blades all coupled through the disk that they are
attached to, then the method of nodal diameters may have merit as
has been well known. Plotting of frequency versus nodal diameter
is a design technique used for decades to avoid disk critical speeds
and interaction resonance, so the two techniques are similar for
those cases. The method uses the same principles as explained
above for Equation (1), except that it uses a graphing method that
is applicable to completely bladed disks—in effect having one
packet. Both methods are discussed by Wang, et al. (1999). The
graphing method is also similar to that offered by Wildheim (1979)
for modes dominated by the disk.
With long tapered, twisted blades with relatively flexible
disks, the graphing method may be useful for disk-dominating
modes. As explained by Wagner and Griffin, (1996a, 1996b), it
is only the modes that have out-of-plane motion that are effectively coupled. For designs with very rigid disks or drum
construction, individual packets of short, nontwisted blades,
phase cancellation analysis is the method to use, especially for
the tangential modes. The prime reason of having shrouds for
reducing response is phase cancellation for individual packets,
again tried and proven for decades, even prior to Weaver and
Prohl (1956). Case No. 2 in Dello (1987) will be used for phase
cancellation comparisons to minimize resonant response. The
case assumes a stage in the turbine having 108 blades, 18
packets, and six blades per packet, with frequency of the fundamental tangential mode equal to 46 upstream nozzles times
operating speed. In Figure 12 is the interference diagram for this
case. The diagram implies that since the shape of the force with
46 nodal diameters does not match the shape for 18 packets,
resonance near 15,000 rpm shown really does not occur at the
point encircled. However, per equations in APPENDIX B, the
optimum number would be either seven or 12 blades per packet.
In Table 1 is a summary of resonant response factors. Since
seven blades are not divisible into 108 blades, nine blades could
be selected. If this choice gives assembly problems, then a combination of five and seven per packet to total 108 blades would
be better than using 18, six-bladed packets. Note that with six
blades per packet, response would be about three times higher
than for nine per packet. The interference graphing method
based on nodal diameters says a true resonance condition does
not exist for 18 packets except for up to nine nodal diameters.
The blades in a packet may not fail as the resonant response
factor is still below 0.20, but resonance will exist and could
149
indeed fail if loads were high enough, or if endurance strength
becomes lower with time in operation. The graphing method
does not give the result that it is best to use either nine blades per
packet or a combination of five and seven blades per packet. A
slight change in numbers could result in using an optimum
common number of blades per packet for the stage using
APPENDIX B. Thus this case shows that response is reduced
based on the interference diagram graphing method. But it
cannot be said that there will be “safe” operation at resonant
speeds of the first in-phase modes with six blades per packet.
Figure 12. Interference Diagram for First Mode with Limit of Nine
Nodal Diameters.
Table 1. Fundamental Mode Resonant Response Factors, 108
Blades, 46 Nozzles.
Number of Blades Per Packet
Resonant Response Factor
3
0.26
4
0.21
5
0.08
6
0.17
7
0.01
8
0.12
9
0.06
A similar example, Case 1 in Singh, et al. (1988), has the first
three modes in resonance with nozzle passing frequency. Resonant
response factor for six blades is very low, but it is not true that “…a
true resonance will not occur.” Reviewing that case, one could
potentially select three blades per packet and the resonant response
factor would be excessive, near 0.33, too high for resonance with
nozzle passing frequency (refer to Table 2). For the number used in
the example, six blades per packet, the “rocking mode” would also
have a fairly high response factor.
Table 2. Fundamental Mode Resonant Response Factors, 150
Blades, 78 Nozzles.
Number of Blades Per Packet
Resonant Response Factor
3
0.33
4
0.06
5
0.19
6
0.06
7
0.13
8
0.06
9
0.09
Another case described in Singh, et al. (1994), was for higherorder axial modes of six-bladed packets, affected by deposits and
corrosion from sodium and/or chlorine. Resonance was with the
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PROCEEDINGS OF THE THIRTY-THIRD TURBOMACHINERY SYMPOSIUM • 2004
second harmonic of nozzle passing frequency (exciting force
should only be about one-half as strong as those at nozzle passing
frequency). The solution, changing the design and reducing
governor speed range to remove most (but not all) resonant conditions, was likely only needed because of excessive corrosion,
similar to one of the cases to be given below. Corrosion pits such
as at the fatigue initiation site are not supposed to occur if contaminants are controlled within specified limits.
The first case in Dello (1987) has the same stage configuration
as for the second discussed above, i.e., 108 blades and 46 nozzles,
18 packets, six blades per packet but the fundamental tangential
mode is instead resonant with a low harmonic of speed, five per
revolution. The interference graphing method based on nodal
diameter patterns results in only recommending a change as long
as there are less than nine groups of blades. An example is shown
in Figure 13 for eight packets, the point shown at four nodal
diameters (8 packets/2) is below resonance for five nodal
diameters. The resonance still exists however for eight packets at
the point encircled. Equations in APPENDIX B can be used with
setting value for excitation source number, S = 5. By using combinations of 13 to 16 blades per packet to give either seven or eight
groups, response factor would then be between 0.31 and 0.50,
much better than freestanding blades value of 1.0. However by
using the “long arc” method previously described, if the Goodman
factor is very high, it would be optimum to have five groups of
blades tied together, two with 21 and three with 22 blades per
packet giving a resonant response factor near zero.
for an unshrouded cantilevered blade, R would be close to 1.0,
while an optimized packet of blades can usually be reduced to at
least 0.2 for the fundamental modes (refer to APPENDIX B).
Another way to effectively use this method for vibratory stress
values that are acceptable is to use empirical Goodman factors
based on experience.
For blade flutter avoidance: Σ aerodynamic and mechanical
damping must be > zero. Empirical factors are utilized for reduced
frequency factors, in effect limiting how flexible the blades can be,
i.e., limiting the fundamental modes to a minimum value for
different machinery types. In addition operation must have guidelines, e.g., for surge and choke for axial compressors, and exhaust
pressure limits for steam turbine exhaust ends. Precise analytical
methods incorporating computational fluid dynamics (CFD)
methods are still in development; in fact, especially for transonic
stages, there are still issues relative to inclusion of complete
viscous and turbulence models even for steady-state conditions.
Flutter aeroelastic calculations depend on the solution of unsteady
nonlinear fluid mechanics, a Navier-Stokes solver, a solver for
eigenvalues with structural equations, as well as a transient structural dynamic solver. Thus both dynamic mesh algorithms for the
fluid mechanics and structural interaction are required. Nowinski
and Panovsky (2000) and Panovsky and Kielb (2000) provide
analysis and experimental data for low-pressure turbine stages for
jet engines that once again verify that mistuning of blades greatly
assists in preventing flutter. However, mistuning used to assist in
guarding against flutter such as in steam turbine fixed-speed units
will reduce the separation margins for any tuning for resonance
avoidance. The authors also show how movement of the torsional
axis from forward to aft position of the profile can greatly increase
the propensity of flutter for the torsional mode.
Sources of blade damping are:
• Aerodynamic
(viscous, normally positive but can become
negative at poor incidence angles).
• Friction
(lashing wire, midvane snubbers, platform dampers,
root/disk land interface).
• Hysteretic (material)—significant for many of the 12 percent
Figure 13. Interference Diagram for Stage with 108 Blades: Two
15-Bladed Packets, Six 13-Bladed Packets.
Other machines besides those with open impellers now use
blisks, blades machined integrally with the disk. These typically
have less damping, especially for higher modes. Mistuning of
blades is also much lower in blisks than for fabricated assemblies
that have larger differences in geometry; thus flutter could be a
larger concern, but “rouge” blade resonance amplitude increase
due to mistuning would be of less concern.
For any design, the importance of damping cannot be under
emphasized as: forced response vibration amplitude = Σ excitation
forces/Σ aerodynamic and mechanical damping. As the resulting
amplitude can also be correlated with dynamic stress, the resulting
equation that can be used is:
σ v = σ s ⋅ S ⋅ R ⋅ (π / δ )
where:
σv
=
σs
=
S
=
R
=
(π/δ) =
δ
=
(3)
Vibratory stress
Steady-state gas bending stress
Stimulus factor for the harmonic at resonance
Response factor
Amplification factor, where
Damping log decrement
In effect, in Equation (3), the response is a function of a resonant
response factor times a stimulus factor for the gas bending forces
in turn multiplied by the amplification factor that is determined by
available damping. The response factor, R is both affected by the
mode shape and response to the force based on phase angle. Thus
chromium, martensitic steel alloys, small for most other steel
alloys, and negligible for titanium.
Mean stress can have a significant effect to lower material
damping, but still is often used for added conservatism, especially
for mechanical drives where highly responsive modes cannot be
tuned as can be done for units only loaded at one rotating speed.
Total system damping is one of the most difficult aspects of
Equation (3). Values in Table 3 from Kielb (2001) are for materials
with very low hysteretic damping and show a wide disparity.
Table 3. Amplification Factors from Jet Engine Strain Gauge Data.
Mode
Fundamental Bending
Second and Third Bending
First and Second Torsional
Stripe Modes (Plate-type modes)
Total Q
30-130
80-100
20-190
60-150
All cases in discussions below involved rotating component
failures. Stationary vanes only have gas loads and potential thermal
effects to give steady-state stresses. In centrifugal compressors
inlet and return channel vanes have low excitation forces; in axial
machines that use 12 percent chromium steel, material damping is
also higher as compared to other machinery with other steel alloys.
Without centrifugal loads, there typically is much lower mean
stress, thus higher damping values as compared to rotating blades.
Many vanes, especially in steam turbines and the high-pressure
sections of industrial axial compressors are also shrouded, giving
reduced compliance with forced excitation as well as having
inherent phase cancellation. Resonance of stationary vanes is much
less likely but can occur and must sometimes be scrutinized, such
as:
ROTATING COMPONENT MODAL ANALYSIS
AND RESONANCE AVOIDANCE RECOMMENDATIONS
151
• Avoidance of resonance with rotor blade passing frequency of
• Ensure that first stage blades avoid very low harmonic excita-
• Rotating stall with multiple stall cells in compressors.
• Avoid
cantilevered diffuser vanes immediately downstream of rotating
blades in a centrifugal design, and
CFD analysis can be utilized to predict magnitudes of excitation
forces. As to whether the resulting excitation forces are important
for a given design, a Fourier analysis can be performed such as
given in APPENDIX C. Excitation can come from non-uniform
flow and pressure due to construction geometry besides the inherent
wake effects from adjacent vanes. An example would be harmonic
excitation at multiples of speed from poorly designed inlets or
exhausts. The force distribution can be separated into pulse trains
for the analysis. As speeds increase, shock waves become a consideration for normal operation, besides having potential for transient
conditions or those inadvertently outside operating performance
maps. Differences in manufacturing such as variations in upstream
throat areas between vanes can give significant excitation, as can
erosion and corrosion differences circumferentially. For compressors, aerodynamic sources also can include rotating stall and surge
at one end of the operation map and choke flow on the other.
Turbines are inherently more stable, but also more prone to have
changes to both excitation forces and endurance strength due to
erosion, corrosion, and deposits. In addition, partial admission
stages give a rich signature for many harmonics of speed. In effect,
as a blade enters the arc it can be in phase, vibrate a number of
cycles, and then enter the arc in phase for the next revolution, i.e.,
in exact harmonic resonance. The higher the number of cycles of
vibration per revolution, the lower the response as there are more
cycles having a decay of amplitude via damping mechanisms.
Partial admission can also aggravate the excitation from the worst
offender, wake-passing frequency. The predominant source of excitation for short blades is passing frequency, number of upstream
wakes per revolution. Downstream wakes are usually of little
concern, along with most cases of difference frequencies between
upstream and downstream vanes. Passing frequency is based on the
spacing between wakes; Fourier analysis can be utilized for nonuniform spacing of nozzles and vanes.
CASE STUDIES
It is important to acknowledge that there is in an optimum to
everything and risk taking is necessary to reach that goal. As part
of the risk taking required to compete, for bladed disk resonance,
the ramifications of a fracture for a customer can be tremendous.
Thus the author at his former company recommended fairly conservative approaches that will be discussed for the various types of
machinery to be reviewed. For example, others have taken more
risk employing many more freestanding blades. In reviewing field
problems and field failures, for original design and/or machinery
operation, the optimum may not always be achieved. A large part
of the improvement process is to learn from a limited number of
mistakes. These case studies and accompanying references
hopefully will assist the reader. A reason for achieving success was
given in a January 2004 quotation from Lt. Col. Keith “Sully”
Sullivan. Per the director of USA Central Command Joint Search
and Rescue Center/Saudi Arabia: “…but sometimes unless you do
something wrong, it’s hard to make changes, and we did a lot of
things right.”
Axial Compressors
Problems with blade resonance for industrial axial compressors
should be less than with counterpart machines, axial turbines. That
is, risk is only less if design and operation minimizes rotating stall,
and of course a proper surge control system is used to eliminate
heavy repetitive surging. Continuous operation at extremely high
flow producing stage choking is also taboo (refer to typical performance map in Gresh, 2001). There are several prime design
issues:
tion—dictated not only by loading (Goodman factor), but by how
well the inlet design minimizes circumferential flow distortion;
resonance of fundamental modes with upstream vane
passing frequency;
• Aerodynamic
design must be free of rotating stall at all
operating points including those with variable guide vane settings.
Others in the past have described failures with a poor inlet
design without proper straightening vanes, followed by adjustable
prewhirl vanes. These vanes assist in minimizing nonuniform
flow—the prewhirl vanes also provide flow equalization besides
giving adjustable flow angles required by the first rotating row.
Proper operation requires that variable vanes be closed at startup to
eliminate rotating stall. By retaining subsonic designs, conservatism is ensured; transonic designs are much more demanding as
relative tip speed increases. Modern designs have more need to
have design procedure to avoid blade flutter of first stage blades,
high-pressure blades in choke, and acoustic resonance interaction.
Instead of old National Advisory Committee for Aeronautics
(NACA) circular-arc blades, using controlled diffusion airfoils
(CDA) blades that are actually designed for transonic conditions
greatly assists in not exceeding limits. Improvements by using
CDA blades are described by Kilgore (1987). For high flow
operation, a design with dewhirl stationary vanes in the discharge
casing can be used to guard against cases where process system
resistance is such that choking can be within the compressor. For
industrial designs, acoustic resonance should be easily avoided
with checks as in Parker and Stoneman (1985).
Mazzawy (1980) describes some surge data on gas turbine compressors, still and likely always taboo for jet engines. Issues
including measurement of rotating stall are included in Kushner
(1996) where there is pressure pulsation data for both surge and
stall on a fairly recent axial compressor. In fact the use of dynamic
pressure probes:
• Eliminated
the need for surging at all in setting the surge
controls, and
• Verified a proper design without any rotating stall at much lower
flows than the surge-control set points over the complete range of
vane settings at full speed.
Case A-1
Turbocompressors with axial compressors are used to pressurize
the boilers of ships, including seven US Navy Wasp-class carriers.
The first of many highly successful units had a fatigue crack found
on the first stage rotor blades following initial full load testing to
confirm performance of the longer blading. Analysis of wake excitation as a percentage of steady-state loads verified that it was
severe enough to cause the failure even though the struts supporting the inlet casing were over one chord length removed from the
rotor blades. A simple fix was to increase the number of struts in
the casing that had previously been used with a rotor with smaller
blades. The lesson learned was that using an existing part from an
older model must be reviewed for interaction with components in
the new design. The change was quickly implemented for the first
and 27 subsequent blowers running with no field problems.
Case A-2
In the 1980s, several rotating blade and variable guide vane
failures occurred on a 12-stage axial compressor in FCC service.
There was no corrosion issue from contaminated open-inlet atmosphere as has occurred in other plants, requiring coating of some of
the front-end stages. Extensive tests were done, including a final
test using casing-mounted dynamic pressure probes and strain
gauges on several rows of blades. The cause was proven to be
excessive surging although the rotating blade roots were not
initially optimized and were eventually redesigned, giving an even
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PROCEEDINGS OF THE THIRTY-THIRD TURBOMACHINERY SYMPOSIUM • 2004
higher safety factor. The inherent process problem was that the
blowoff valve was so undersized that a second blowoff valve had
to be added. Initial, continuous, and excessive surging also
damaged the linkage for the variable guide vanes, and actually
yielded the casing due to transient thermal stress. It was argued that
by having skewed roots, adding to steady-state stress, the
endurance strength was compromised.
The user and their consultant were insisting on using high mean
stress due to the skewed roots as a limiting factor using a modified
Goodman or Soderberg diagram. It was argued then and is often
exemplified such as by Wang, et al. (2000), that ductile alloys do
not always require extreme conservatism. This statement applies to
defect-free specimens, unlike the problems with small initial
cracks or defects. In addition, there was actual test data for the AISI
403 material showing that actual fatigue limits were higher than the
modified Goodman values. The stress concentration was partly due
to geometry and partly due to loading effects, with only the latter
required to be applied to the mean stress. It was concluded that the
blades would have still failed with the redesigned blades if
excessive surging had continued. In addition, the blades would not
have failed without excessive surging as shown by similar compressors with higher relative loads and identical resonant points,
but were not changed and have exceeded many years of operation.
The field operations group refused to actually surge the compressor during final tests with strain gauges; but data verified that there
was not rotating stall or damaging acoustic pressure pulsation frequencies up to the surge control point. In addition it was verified
that excitation at exact resonance points for low speed harmonics
gave acceptable stresses—even for the original skewed roots.
Speed was changed extremely slowly to obtain maximum peak
amplitudes. After surging was eliminated, occasional failures
occurred whenever some of the variable linkage failed (also later
improved), giving nonuniform flow impulses. The fundamental
mode of the first rotating row was close to resonance with four per
revolution. As the train was a mechanical drive with a fairly wide
speed range, it was impossible to tune all the blades from low
harmonic excitation. If the train had been a constant speed unit,
proper separation margin could have been feasible for the first four
stages but would have required each blade to be accurately tuned.
As the harmonic number increases, it becomes exceedingly more
difficult to tune even the fundamental mode where there is some
speed change with load. Tests were done on all the blades for the
first four rows using a fixture. The fixture had mating rotor drum
lands such that a contact force at the blade root of approximately
50,000 lb could be exerted using the hydraulic cylinder to equal
centrifugal force at speed. The data were used to verify that the first
row indeed avoided resonance of the fundamental mode with four
per rev, while the next three rows had higher harmonic resonance.
Note that variation in fundamental mode frequencies from tolerances was near ±1 percent.
In addition to the concern for surge and low harmonic
resonance, dynamic pressure probe data showed there was
resonance of acoustic modes of the air stream within the compressor as described by Parker and Stoneman (1985). Description of
similar acoustic natural frequencies for analysis of piping noise
and interaction with structural modes are explained by Kushner, et
al. (2002). In this case the acoustic waves are altered by the
presence of the vanes and blades downstream of the inlet. However
it was shown that relative frequencies based on the number of
nodal diameters of the modes would be well above the blades’ fundamental modes as they traversed through the standing acoustic
wave. Exciting frequency relative to the rotating blades is the
standing wave frequency plus and minus the number of nodal
diameters times speed for each of the acoustic modes. Strain gauge
tests also confirmed that there were no responsive acoustic
resonance peaks in the data. Using one-fifth scale-model testing of
the inlet casing, it was found that the inlet struts excited the
acoustic modes. Benefits to all the testing were that it verified the
excellent steady-state flow characteristics of the inlet. A method
was found and then verified to alleviate the high frequency noise
by treating the trailing edges of the struts to prevent the vortices
from each strut to communicate with each other. The case proved
once again a proper surge control system is mandatory for axial
compressors; in addition, rotor blade redesign and optimization of
root/rim interface resulted in minimal increased cost for the
increase in reliability. The strut treatment method greatly reduced
ambient noise in subsequent compressors. Many lessons were
learned by the vendor and user; in addition, the contractor had a
lesson in design of critical surge controls for axial compressors.
Centrifugal Compressors
For the axial compressor in the previous case study, the rotor
was of drum construction, so disk modes were a moot point. In
many other axial compressors that have disks, disk critical speeds
can be an issue; some lightweight engines have thin disks and must
run up and down through disk resonance points. By contrast,
industrial centrifugal compressors almost always avoid disk critical
speeds.
Case B-1
A unit that did not avoid disk critical conditions was a plant air
compressor as detailed by Kushner (2000). In Figure 14 is shown
a comparison of the initial and final design. FEA analysis and
modal testing showed proper thickening and tapering of the disk
hub solved the problem. It requires fairly thin impellers to have
disk critical speeds, even for open impellers. Both modal testing
and FEA analysis for an unshrouded, plant-air compressor impeller
was required to eliminate the four-diameter mode critical speed.
Figure 14. Campbell Diagram for Open Impeller, Removing Disk
Critical Speed.
Use of four inlet vanes or struts would be especially disconcerting, as would be the second harmonic of two stationary elements
equally spaced. It does not matter how many blades would be on
the disk; they are the elements that transfer unsteady forces to the
disk, just as they absorb steady loads. For this case, four vanes
were not used; the number was 15. At this disk critical speed, each
blade is in phase with the effective sinusoidal pattern at four per
revolution. If the four-diameter mode frequency were resonant
with five per revolution, the effective forces would cancel out in
each revolution. For the actual impeller, the only stationary vanes
were in the diffuser, 15 in number with no critical interaction
resonance above 12,000 rpm. It was concluded that nonuniform
flow from the downstream volute had sufficient Fourier component
for fourth engine order to cause cracking at the periphery of the
disk. The solution was to modify disk thickness and to increase the
taper, accommodating “flow cuts” both for the disk and blade
profiles. The revised impeller operates safely below disk critical
speeds. Open impellers could have more risk for disk critical
resonance as shown in Figure 15. Other research with blade
ROTATING COMPONENT MODAL ANALYSIS
AND RESONANCE AVOIDANCE RECOMMENDATIONS
153
failures for open impellers such as during the series of laboratory
tests by Haupt, et al. (1985), to Jin, et al. (1995), may primarily be
due to excessively thin sections for both blades and disks, and use
of aluminum as compared to commercial designs.
Figure 17. Pressure Fluctuations Just Before Compressor Surge for
Open Impeller.
Figure 15. Comparison of Diametral Mode Frequencies.
Case B-2
Besides more concern for disk critical speed, open impellers are
more prone to damage from surging and rotating stall. The case of
a failure just after startup given by Kushner (1996) was a definite
case of continuous surging, albeit milder than in a multistage compressor, where a measurement is shown in Figure 16. However the
air compressor was highly loaded and there was extremely high
nonuniform flow excitation from the volute while approaching
surge. (Figure 17). Thus the lower harmonics of speed gave
resonance at four per rev, adding to the impulse loading from surge
itself. Analysis and data showed that the volute was sized properly.
A poor discharge volute can give high harmonic excitation from
nonuniform circumferential flow. It was fortuitous that rotor
vibration data were being tape-recorded and thus verified
numerous surge cycles, perhaps for hours. Continuous service has
been achieved over many years with the special zigzag damping
pins between blades described in the reference for even greater
reliability in case of excessive surging. An improved aerodynamic
impeller with 3-D curved blades (Aungier, 2000) offset the
inherent pressure losses due to use of pins in the air stream.
second harmonic of vane passing (32 times speed), not the first
harmonic of vane passing (16 times speed), interacting with 27
impeller blades, exciting the five-diameter mode. This resonance is
shown in Figure 18; similar interaction resonance is a common
occurrence with numerous covered impellers with dry gas throughout the world. Even with liquid ingestion, the more responsive
three-diameter mode with resonance point shown in Figure 19 did
not respond, conclusively verifying the interaction resonance
equations given above. Strain gauge data at speed and pressure
showed little effect of mass loading on impeller natural frequencies; effects such as the extremely large reductions due to fluid
mass for high pressure compressors (Gill, et al., 1999) are likely a
function of design variations.
Figure 18. Campbell Diagram for Mode with Interaction
Resonance, Impeller with Excessive Liquid Ingestion.
Figure 16. Multiple Surge Pressure Pulsations in Discharge
Volute; 30,000 HP, Four-Stage Centrifugal Compressor.
Case B-3
The damage to many impellers in four identical refrigeration
compressors was due to interaction resonance accompanied with
liquid ingestion as described by Kushner (2000) and Kushner, et al.
(2000). Similar liquid ingestion is likely the reason for many gas
compressor failures throughout history. Based on experience with
many similar geometries, relative speeds, and loads, it is nearly
impossible that the impeller would fail unless liquids aggravated
the wakes. First of all, excitation was from upstream vanes, not the
much more severe force from diffuser vanes at the impeller tip (the
tip of the impeller is also much more compliant to exciting forces
than at the eye—especially for higher modes). Secondly, it was the
For a similar design without liquid ingestion, a two-diameter
mode could be questioned for dry gas, but two-diameter mode
interaction can be avoided by ensuring an even number for the inlet
vanes and an odd number for the rotating blades. Analysis for this
case gave the impetus for what is believed the industry’s first publication on the subject, Kushner (1979), later backed up by Jay, et
al. (1983), followed by many others such as NASA (1990). The
same year, Wildheim (1979) published a similar equation to
Equation (1). Previous pioneering research into disk modes such as
by Ewins (1973) may have included the same discovery in the jet
engine industry that had much more proprietary methods prior to
engine companies forming and working within consortia. For axial
flow jet engines, however, due to much higher blade numbers and
differences between numbers, Equation (1) may not be required as
is for radial designs. The parametric Equation (1) could have been
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PROCEEDINGS OF THE THIRTY-THIRD TURBOMACHINERY SYMPOSIUM • 2004
Figure 19. Campbell Diagram for Mode Without Interaction
Resonance, Impeller with Excessive Liquid Ingestion.
useful for many previous impeller failures such as those unexplained in Bultzo (1975). The equations are sometimes still not
applied properly as described by Kushner (2000).
Case B-4
Not explained by interaction resonance are some covered
impeller failures that occur near the hub and/or at or near the tip.
Many cracks initiate at the toe of the blade welds. It is possible for
an impeller to actually have an unstable response for the fundamental two-diameter mode as shown by Kushner (2000). Note that
in this case operation of the same compressor as Case B-3 had an
inherent overload operating condition with an erroneous opening
of the recycle valve at high flow. Impeller fatigue cracks initiated
at the eye where testing showed maximum values for the twodiameter mode. (FEA analysis was in its infancy in the 1970s.)
Liquids that even reached the last stage could also have been associated with heavy gases in the mixture causing a
super-condensation effect, such as described for ethylene plant
reactors (Parkinson, et al., 1999). Centrifugal compressors are
easier to design from the stability standpoint as they do not have
steep head rise to surge characteristics as do axial compressors
affecting aerodynamic damping. One would look at the structure of
an impeller compared to a single long axial compressor blade and
never surmise that the entire disk would flutter.
The responding two-diameter mode was shown to respond only
with heavy liquid ingestion; transient response reaching the
endurance limit was proven by strain gauges. The two-diameter
mode is the fundamental mode where vibratory moments are
balanced within the disk. The mode had no interaction resonance
and the transient response was not at any harmonic of speed. In
addition for gas processes, system resistance normally prevents
severe impeller overload at high flow. Compressors in parallel and
other operations with reduced system resistance, including startup,
where an impeller can actually choke should be avoided for significant time periods accumulating high number of cycles. This is
especially true if there is likelihood of liquid ingestion, eventually
eliminated to solve Cases B-3 and B-4.
by Borer, et al. (1997), for overloading a compressor when a
parallel unit tripped offline, driving the running unit into deep
choke. The immediate solution that was suggested for the blast
furnace blower was to improve controls for the turbine. It was then
easier to immediately slow down the string when the “wind” was
not needed for the furnaces, i.e., not just open the blowoff valves.
A mitigating factor was that a larger blowoff valve also had been
installed to protect the larger furnace under emergency shutoff of
its air supply. Thus the compressor would have been driven deeper
into choke than before the rerate; the stage that choked first may
also have moved due to the rerate to a stage that was less resistant.
Fatigue cracks likely initiated due to overload condition, then
easily propagated at the resonant speed where twice inlet passing
frequency was equal to the blade leading-edge frequency. Besides
the turbine control change, it was suggested to increase discharge
piping system resistance with a longer-range goal of increasing
strength and natural frequencies of impellers and blades. After
reporting on probable cause using dynamic pressure probes near
the impeller, the operators refused to intentionally test off-design
curve at full surge or at the overload condition.
Case B-6
An impeller that failed in its first day of operation, described by
Kushner, et al. (2000), was also due to liquids and operating in
overload due to erroneous instrumentation. There was no prime
interaction resonance so thus the cracks near the impeller tip were
deemed to be due from resonance with inlet vanes of a complex
plate mode. A small section of the impeller hub actually tore off,
giving high rotor vibration initiating a trip. After scrutiny of the
mode, there was a component of a three-diameter pattern shown in
Figure 20, and the difference between 18 inlet vanes and 15 rotating
blades was equal to three, satisfying Equation (1). This along with
likely shock waves and liquids aggravating the vane wakes helps to
explain why the mode responded to give high alternating stress.
Case B-5
Following a rerate, a large blast furnace air blower suffered
several fatigue failures after many years of service. Instead of the
reason given by Phillips, et al. (2003), other data showed that the
most likely cause was operation in deep choke when the blowoff
valve was opened at high flow. It was much more important for the
operators to protect the newly installed furnace than the spared
compressor. Granted the impeller had resonance of a leading edge
mode with two times inlet vane passing frequency, but this
resonance is a common fact in the history of compressors. As this
case does not involve liquids, it is perhaps similar to that described
Figure 20. Mode Shape for Impeller Higher-Order Plate Mode,
Having a Three-Diameter Pattern.
ROTATING COMPONENT MODAL ANALYSIS
AND RESONANCE AVOIDANCE RECOMMENDATIONS
Case B-7
Impeller plate modes sometimes are involved in other failures that
do not have the same intense analysis as Case B-5; it sometimes is
decided it is best to make a change with the best engineering
judgement. Experiencing recurring failures with many years between
failures is also sometimes a reason to have a quick fix. Some
customers just opt to repair the rotor for spared units, as has one
customer with two units in parallel. One method to make a change in
the past is to scallop out sections of the impeller and cover between
blades (aerodynamic effects are small). The method did not work for
Case B-3; scalloped impellers still had interaction resonance and
liquid wakes were still present. For this case there was no interaction
resonance, so the modification shown in Figure 21 was made.
155
as there is some benefit of hysteretic damping as shown in Figure 22,
although steady-state (mean) stress reduces values as shown in
Figure 23. Some designs are highly scrutinized so that materials
such as titanium and other variations of corrosion-resistant steels can
be utilized in a specific design or modification for a problem such as
those in the transition zone experiencing corrosion fatigue. The risk
can be higher for some variables however so that care must be used,
such as avoiding flutter conditions where material damping adds to
the positive side of the equation. Recently there have been some
examples where blade lives have “only” been around 20 years for
some long back-end blades employing zigzag damping pins shown
in Figure 5. In some cases the most economical solutions would be
to add more conservatism by employing z-lock shrouds or integral
shrouds with rolled-in lashing wire.
Figure 22. Typical Hysteretic Damping, AISI 403, Cantilever Beam
in Bending.
Figure 21. Covered Impeller Tip Modification, Scalloped Cover
and Hub.
Case B-8
High flow compressors employing covered impellers have the
most risk of resonance with blade leading edge modes. For the
highest flow impellers, potential resonance was found for a new unit
with the inlet already designed and manufactured. It was determined
that resonance would be avoided as it occurred well below operating
speed, so that there would only be occasional ramps up through the
resonant point. A check was made that the lower speed would not be
used during performance testing, often done with a different gas and
running speed, but at equivalent aerodynamic tip speed.
Turbines
Radial-inflow turbines with unshrouded impellers have similar
concerns to compressors, but inlet vane excitation can be worse as to
excitation of disk modes due to their location - near the compliant
impeller tip. For steam turbines, as related in a lecture by the expert
Den Hartog (1954), a golden rule for long blades is never to have to
have the fundamental mode equal to two per rev, especially for
turbine generator sets with severe electrical impulses. There have
been catastrophic failures due to transient torsional excitation at
twice line frequency; in effect the torsional vibration of the entire
rotor is exciting the base of the blades as there were on a shaker
table. Otherwise the other engineering standards for long blades
(and others) are made by a myriad of individual vendors with many
designs, shrouding, and friction damping methods (refer for example
to Sohre, 1975). Most designs still use 12 percent chromium steels,
Figure 23. Damping Capacity of High-Strength 12 Percent
Chromium Steel Versus Alternating Stress.
Case C-1
A turbine that had reduced endurance strength in the transition
zone suffered a failure with an axial mode having first order
resonance with special inlet nozzle vanes used to add rigidity to the
diaphragm. Failed blades were at the ends of the packets, a sign
that an out-of-phase mode was most likely involved. This is not
always true as the ends of the packets have somewhat higher
steady-state stresses, so that another mode could have the end
blades above the endurance limit. Many years earlier there had
been one case for an even shorter Curtis stage blade that was
suspected to be an axial mode resonance; it never was proven but
the packeted design was successfully changed to the more
forgiving design of mating integral shrouds with rolled-in lashing
wire. There was not a corrosion issue for the Curtis stage as there
was for this case. Both “stress corrosion” and “corrosion fatigue”
156
PROCEEDINGS OF THE THIRTY-THIRD TURBOMACHINERY SYMPOSIUM • 2004
can become a contentious issue for steam turbines whenever there
is a field failure for a stage in the transition zone. Some of these
incidents are likely the reason the API 612 (2003) committee used
such an impossible hard line for the Fifth Edition with respect to
resonance. The higher modes with phase changes along the blade
are safe even with resonance as response is typically negligible. It
is also impossible to avoid resonance with all harmonics up to 15
times speed (no relief is given in the current edition by showing
acceptable stresses as was in the Fourth Edition).
It is known that corrosion fatigue is a special case of stress
corrosion caused by the combined effects of cyclic stress and
corrosion (refer to McCloskey, 2002). Turbine metals are not
immune from some reduction of its resistance to cyclic stressing if
the metal is in a corrosive environment. By far the highest propensity is where the water droplets first form, at the transition stage.
Water, metal, and the presence of a corrosive medium such as
chlorides are the three required elements. The first two are a given,
the third—the corrosive medium—is almost always the argument,
e.g., makeup water treatment of high quality as specified, or
another source of contamination. In this case, the pitting was just
starting; however, corrosion fatigue is often encountered not as a
visible degradation of the metal but as a premature failure of a
component under cyclic conditions.
The excitation and offending vibratory mode still should not have
been a problem for the small amount of corrosion; it was finally
determined that there was also transient water ingestion into the
stage, aggravating the wakes. Damage from corrosion fatigue is
greater than the sum of the damage from both cyclic stresses and
corrosion. Fatigue corrosion failure occurs in two stages. First the
combined action of corrosion and cyclic stresses damages the metal
by pitting and possibly small crack formation; then endurance
strength and stress risers exist to such a degree that fracture by
cyclic stressing ultimately occurs. The second stage is essentially
the fatigue stage in which failure proceeds by propagation of the
crack and is controlled primarily by stress concentration effects and
the physical properties of the metal. Even if the corrosive medium
is completely removed, it can be too late to avoid failure with the
same excitation. The transient water ingestion likely caused the
final fracture. This mechanical drive unit, as for all other likely
causes was going in and out of resonance, randomly accumulating
many cycles until some blades failed. Even though the blades were
shrouded, individual packets could have had much higher response
than other packets, from mistuning described earlier. Fracture due
to corrosion fatigue occurs at a stress far below the fatigue limit in
laboratory air, even though the amount of corrosion is extremely
small. The mode for this case was an out-of-phase mode shown to
be resonant in Figure 24. Using FEA, the crack initiation site correlated with peak stresses for this out-of-phase mode. As there were
operating issues, it was best to provide a more rugged design
employing integral shrouds with rolled-in lashing wire. A more
recent type of coating was also considered, but in this case a rerate
due to plant changes had already been necessary so that improvements in design and operation fit into the plans.
Figure 24. Campbell Diagram for Medium-Length Steam Turbine
Blade.
Case C-2
A variable-speed steam turbine last-stage blade using 12 percent
chromium steel, with zigzag lashing pins experienced two “rouge”
failures; only one blade cracked on two rotors with long lives
between failures, i.e., years apart. Failure was at a small stress riser
as the surface had limited liquid impingement damage near the
stellite overlay. FEA analysis confirmed that the failure site
showed a high dynamic stress for a higher mode with a nodal line
near the zigzag pins. The excitation was from nozzle passing
frequency, normally not causing problems for high frequency
modes of long blades due to phase cancellation. However, every
mode needs some damping; in this case records were found that the
exhaust pressure was often well below limits, which can greatly
reduce aerodynamic damping of last stage blades, in fact likely
causing it to go negative. The zigzag pins and some inherent
material damping would prevent flutter for the fundamental modes;
however as the pins were located near a nodal line for the mode
shape, forced excitation and response were compromised by the
high aerodynamic incidence angles during low exhaust pressure
conditions. The options were to take some risk and just have operations monitor that maintaining that exhaust limits are adhered to,
or also get a more robust design. The user opted for an improved
design. In this case, the loss of damping occurred at low exhaust
pressure; it can also be greatly reduced with abnormally high
pressure while at or near design speed.
Case C-3
A rerated and repaired steam turbine had the next-to-last stage
assembled where due to large number of blades, tolerances gave a
condition that all packets would not have the same number of
blades. However it was only the nonresonant fundamental mode
that would have a low response factor for the different packet based
on APPENDIX B. The one packet with a different number of
blades would definitely have somewhat different frequencies from
mistuning; however, higher modes are much less likely to cause
problems. Similar calculations for the first out-of-phase modes
showed that response factor would actually be less for the one
different packet, compensating for the potential increase in
amplitude from mistuning effects. Many designs have slightly
different numbers per packet in a stage and thus should be
compared for critical cases, adding to other reasons for mistuning.
This could explain why in some cases only one or perhaps a few
packets experience fatigue cracks.
Case C-4
A radial turbine stage had two different upstream sources of
excitation for a fundamental mode of a cantilevered blade.
Calculations were made to show that the resultant excitation was as
shown in Figure 25. This rare cause of resonant excitation was
identified as being due to use of 15 stationary inlet vanes interacting with 12 other equally spaced elements upstream of the vanes.
Besides giving harmonic excitation at 15 times and 12 times speed,
the sources will cause interaction depending on spacing between
them and the strength of each source. Interaction affects the local
gas loads on the turbine blades downstream, accentuating various
harmonics of speed. As the difference between 15 and 12 was
three, three times speed could be a prime source of excitation; and
as verified by Fourier analysis, multiples of three could also
produce significant excitation. Using the variation shown in Figure
25, the harmonic peak-to-peak excitations from the predominant
sources are 5.5 percent for the 15 per rev component, and 2.2
percent for the 12 per rev component. The interaction between the
two sources of nonuniformity results in a value equal to 1.9 percent
excitation for the three times speed, and 1.2 percent excitation for
the six per rev component. Natural frequencies of the blades
avoided harmonics of 3, 12, and 15, but the second mode was
resonant with six per revolution.
ROTATING COMPONENT MODAL ANALYSIS
AND RESONANCE AVOIDANCE RECOMMENDATIONS
157
overall damping. Better online liquid monitoring and prevention
procedures should be developed for both compressors and turbines.
• For steam turbines, API 612, Fifth Edition (2003), specification
is overly restrictive as to handling limits for blade resonance. It
should be revised to add consideration of what modes really need
scrutiny, with verification that stresses will be within Goodman
factor limits.
• Additional
research and implementation of new techniques
replacing glass bead shot peening such as laser shock-peening, and
low plasticity burnishing as well as new coatings will aid in precluding fatigue failures.
• Case studies such as those given will hopefully lead to more preFigure 25. Excitation Force Around Circumference for Radial
Inflow Turbine.
vention of inherent design limitations, as well as ensuring proper
operation of all fluid-handling turbomachinery for the oil, petrochemical, and related industries.
• Steam turbine transition stages require more attention in both
design and achievement of minimal corrosion.
Finite element analysis was used to show that maximum
dynamic stress at the crack initiation point (identified through
fracture mechanics analysis) correlated for the second mode. A
small modification that also alleviated a stress concentration factor
was utilized to both avoid the resonance problem and the need for
major design revisions to the structural components. Rather than a
complete redesign, the solution was successful for the constantspeed application.
CONCLUSIONS
• The methods described using parametric Equations (1) and (2)
should be utilized whenever possible, not only for failure analysis
but also in the design phase such as in selecting number of vanes
for vaned diffusers and for all new bladed-disk designs.
• Equal numbers of vanes and blades should be always avoided;
often while also reviewing aerodynamic effects, a more optimum
design can utilize a difference of one between stationary vane and
rotating blade numbers. Even and odd combinations are usually a
better choice.
• Typically, an odd blade number is best used for the impeller for
better mode splitting of diameter modes; also a prime number
appears to be a better choice for mistuning effects for open
impellers.
• Although
covered centrifugal compressor impellers are definitely much more rugged than open impellers, first order resonance
of the first blade leading edge mode requires as much scrutiny as
the first few modes for freestanding cantilevered blades in other
machinery.
• There is always a need to scrutinize operating and environmen-
tal parameters and the design, not only in design specifications, but
also in problem resolution.
• Extensive
finite element analysis can be used to correlate
vibratory modes and location of peak stresses with fracture
analysis documenting crack initiation sites for a failed specimen.
• It is sometimes necessary to opt to use a much more rugged design
when investigation of a field failure is inconclusive. Past unresolved
problems should be addressed as more knowledge is gained.
• Operation of machinery at too low or too high flow cause many
failures of compressors and for the longer steam turbine blade
designs. These conditions also could present added interaction
resonance concerns, especially for high gas density and loads in
radial impellers. Running in deep stonewall should be avoided; if
the high flow condition occurs, limits should be used for transients
with limited number of cycles, just as is recommended for surge.
• Liquid ingestion must also be avoided since it intensifies engine
order resonance, and has caused at least one verified case of loss of
• Because mistuning of blades is unavoidable, it is recommended
to use care in replacing phase analyses as those in APPENDIX B
with the graphing method suggested by others in conjunction with
nodal diameters, especially for modes of short blades.
• Interference
diagrams using nodal diameters should be
employed for cases of blades all coupled together with a continuous shroud or locked-up with z-shrouds, or if conditions are such
that there is a disk-dominating mode of concern.
• There definitely is an optimum to everything; increased research
and development will get turbomachinery manufacturers closer to
that achievement for high cycle fatigue (HCF) analysis.
APPENDIX A
Modal Analysis Test Procedures
Avitabile (2001) provides a good review of experimental modal
analysis. For machinery components, use of a two-channel fast
Fourier analyzer is recommended. Modal frequency tests are to be
described. A small accelerometer is attached to the test object.
Frequency response functions, FRFs, are obtained while striking
the object at various points, using a small hammer that has a force
gauge in the head. The FRFs provide ratios of response to applied
dynamic force; at the peaks of response/force ratios in the
frequency spectrum, mode shapes are then obtained from extracted
data for impacts at all required points. The size of the hammer as
well as the hardness of the tip determines whether proper
amplitude and frequency window of the impulse force is generated.
To measure tangential, torsional, and axial natural frequencies of
a single blade mounted in a vise, at the top land of the root, a small
accelerometer is attached with glue or thin wax to the blade (near
the root to minimize effect of accelerometer mass). Both the
hammer and accelerometer must have proper frequency response
capabilities; data are verified using coherence checks between the
input and response for each test. In Figure A-1 is a model used
showing the points that were utilized for striking the blade to
obtain the tangential and torsional modes; hammer blows are
approximately perpendicular to the blade neutral axis at each
position along the blade height. Typically, only the points along the
leading edge would be used for obtaining axial mode shapes,
striking the blade parallel to the neutral axis at each position along
the blade height.
“Rap tests” can also be utilized with microphone response
without the accelerometer attached; data verify negligible effect of
accelerometer mass. “Rap tests” can also be utilized for expeditious checks to correlate with values obtained with FEA analysis.
Modal frequencies are listed in Table A-1; refer to modal plots in
Figures A-2, A-3, and A-4 (fundamental axial mode shape would
have entire length of blade in phase).
158
PROCEEDINGS OF THE THIRTY-THIRD TURBOMACHINERY SYMPOSIUM • 2004
Figure A-4. Modal Test of Single Blade—Second Bending Mode.
Figure A-1. Modal Test Point Definition for Tangential and
Torsional Modes.
Table A-1. Modal Frequencies Obtained for Example MediumLength, Nontwisted Blade.
MODE
Fundamental Tangential
Fundamental Torsional
Fundamental Axial
Second Tangential
FREQUENCY - Hz
1580
3210
4950
5770
Frequencies are then obtained from extracted data from amplitude
Vs frequency spectrums. The amplitudes are those for ratio of
amplitude to force (FRFs), so the peaks are the natural frequencies.
The frequency values are used to verify predicted values. For the
pure diametral modes without nodal circles, only points near the
tip are usually needed for impacts. The extracted data would show
the mode shapes for the frequency peaks.
APPENDIX B
Blade Packet Optimization for the Fundamental Mode
The equation to calculate response factor, R, for a packet of
blades where all blades are in phase with each other and have the
same amplitude is:
(
R = X 2 + Y2
)
1/ 2
/p
(B-1)
where:
X = Σ (from i = 1, to i = p) cos [(i1) • α]
Y = Σ (from i = 1, to i = p) sin [(i1) • α]
α = 360 • (S/B)
α = Relative phase angle, degrees
S = Number of stationary vanes or nozzles
B = Number of rotating blades per row
p = Number of blades per packet
Figure A-2. Modal Test of Single Blade—Fundamental Tangential
Mode.
Figure A-3. Modal Test of Single Blade—Torsional Mode.
For an impeller, diametral modes, two-diameter through B/2
diameters are usually obtained, where B is the number of impeller
blades, but many other modes can also be found. An accelerometer
would be placed in the axial direction near the tip of the impeller.
The impeller would be impacted with the instrumented hammer at
a number of points, both in line with blades and in between blades.
These equations can be consolidated for a single equation to more
readily find the optimum number of blades per packet, p:
j = k⋅ S− B /B
(B-2)
Set k = 1, k = 2, k = 3, etc., and calculate values of j. Then optimum
value of blades per packet, p, is the value used for k whenever j is
equal to, or is closest to, a whole integer. Multiples of the optimum
value of p can also be used, as long as the shroud can be assembled
with higher number of blades per packet. An example is given in
Table B-1. In this case, using k = 4 or 8 results in an integer (not
always the case—the closer to an integer the better). The resonant
response factor can be verified using Equation (B-1), the response
factor for p = 4 or p = 8 would be zero, i.e., perfect phase cancellation. The row of 64 blades could thus be assembled with either 16
packets of four blades, or eight packets of eight blades. Assume that
this was a fairly long blade at the transition zone of a steam turbine,
and there was no chance of resonance with an in-phase mode. If an
out-of-phase mode was a cause of an unusual field problem, such as
with excessive corrosion, then it could be much better to choose two
packets with five blades and nine packets having six blades.
APPENDIX C
Harmonic Excitation—Fourier Analysis
For a pulse train made up of a number of rectangular pulses, m,
over 360 degrees (one revolution), to calculate Sh, peak-to-peak
exciting force at harmonic, h:
ROTATING COMPONENT MODAL ANALYSIS
AND RESONANCE AVOIDANCE RECOMMENDATIONS
Table B-1. Values of “J” for Selected Values of Integer “K” for S
= 80, B = 64.
Integer "k"
"j"
1
0.25
2
0.50
3
0.75
4
1.00
5
1.25
6
1.50
7
1.75
8
2.00
(
Sh = C 2 + D2
)
1/ 2
Baade, P. K., 1998, “Vibration Control of Propeller Fans,” Sound
and Vibration, July, pp. 16-26.
(C-1))
(C-2)
where:
am = Fraction of maximum force of any pulse
βm = Rectangular width of pulse, degrees
For a sine wave shown in Figure C-1, at resonance, the exciting
force is in-phase with the response over the entire vibration cycle.
As the force is out-of-phase over part of the cycle, a square wave
for example would only have 0.637 times the exciting force of a
sine wave at the first harmonic. For the square wave, a rectangular
pulse with β = 180 degrees, Equation (C-1) simplifies to:
[
]
Sh = 2 / (h ⋅ π ) ⋅ sin(h ⋅ 180 / 2)
Bhaduri, A. K., Albert, S. K., Ray, S. K., and Rodriguez, P., 2003,
“Recent Trends in Repair and Refurbishing of Steam Turbine
Components,” Sadhana (India), 28, Parts 3 & 4, June/August,
pp. 395–408.
Bladh, R., Castanier, M. P., and Pierre, C., 1999, “Reduced Order
Modeling and Vibration Analysis of Mistuned Bladed Disk
Assemblies with Shrouds,” ASME Journal of Engineering for
Gas Turbines and Power, 121, (3), pp. 515-522.
Bladh, R., Pierre, C., Castanier, M. P., and Kruse, M. J., 2002,
“Dynamic Response Predictions for a Mistuned Industrial
Turbomachinery Rotor Using Reduced-Order Modeling,”
Journal of Engineering for Gas Turbines and Power, 124, pp.
311-324.
and:
]
API 612, 2003, “Special Purpose Steam Turbines for Refinery
Services,” Fifth Edition, American Petroleum Institute,
Washington, D.C.
Avitabile, P., 2001, “Experimental Modal Analysis, A Simple NonMathematical Presentation,” Sound and Vibration, January.
where:
h = Harmonic of speed
Am = Peak-to-peak force of each pulse
θm = Angle from zero degrees to center of each pulse
[
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D = Σ from 1 to m of Am [sin(h • θm)]
Am = am 2 / (h ⋅ π ) ⋅ sin( h ⋅ βm / 2)
159
(C-3)
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ACKNOWLEDGEMENT
The author expresses much gratitude to all those at The Elliott
Company who provided their support and assistance for many of
the case studies, with special thanks to Bob Strickland for his
advice on methods used for finite element analysis.
162
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