U.S. patent application number 11/445572 was filed with the patent office on 2007-04-12 for fundamental mistuning model for determining system properties and predicting vibratory response of bladed disks.
Invention is credited to Drew M. Feiner, Jerry Howard Griffin.
Application Number | 20070083338 11/445572 |
Document ID | / |
Family ID | 33457543 |
Filed Date | 2007-04-12 |
United States Patent
Application |
20070083338 |
Kind Code |
A1 |
Griffin; Jerry Howard ; et
al. |
April 12, 2007 |
Fundamental mistuning model for determining system properties and
predicting vibratory response of bladed disks
Abstract
A reduced order model called the Fundamental Mistuning Model
(FMM) accurately predicts vibratory response of a bladed disk
system. The FMM software may describe the normal modes and natural
frequencies of a mistuned bladed disk using only its tuned system
frequencies and the frequency mistuning of each blade/disk sector
(i.e., the sector frequencies). The FMM system identification
methods--basic and advanced FMM ID methods--use the normal (i.e.,
mistuned) modes and natural frequencies of the mistuned bladed disk
to determine sector frequencies as well as tuned system
frequencies. FMM may predict how much the bladed disk will vibrate
under the operating (rotating) conditions. Field calibration and
testing of the blades may be performed using traveling wave
analysis and FMM ID methods. The FMM model can be generated
completely from experimental data. Because of FMM's simplicity, no
special interfaces are required for FMM to be compatible with a
finite element model. Because of the rules governing abstracts,
this abstract should not be used to construe the claims.
Inventors: |
Griffin; Jerry Howard;
(Pittsburgh, PA) ; Feiner; Drew M.; (Pittsburgh,
PA) |
Correspondence
Address: |
JONES DAY
500 GRANT STREET
SUITE 3100
PITTSBURGH
PA
15219-2502
US
|
Family ID: |
33457543 |
Appl. No.: |
11/445572 |
Filed: |
June 2, 2006 |
Related U.S. Patent Documents
|
|
|
|
|
|
Application
Number |
Filing Date |
Patent Number |
|
|
10836422 |
Apr 30, 2004 |
7082371 |
|
|
11445572 |
Jun 2, 2006 |
|
|
|
60474083 |
May 29, 2003 |
|
|
|
Current U.S.
Class: |
702/56 ;
702/81 |
Current CPC
Class: |
G01H 1/006 20130101 |
Class at
Publication: |
702/056 ;
702/081 |
International
Class: |
G01L 7/00 20060101
G01L007/00 |
Goverment Interests
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH
[0002] The invention in the present application was made under a
grant from the United States Air Force Research Laboratory,
Contract No. F33615-01-C-2186. The United States federal government
may have certain rights in the invention.
Claims
1. A method, comprising: obtaining frequency response data of each
blade in a bladed disk system to a traveling wave excitation;
transforming data related to spatial distribution of said traveling
wave excitation and said frequency response data; and determining a
set of mistuned modes and natural frequencies of said bladed disk
system using data obtained from said transformation.
2. The method of claim 1, wherein said traveling wave excitation
has a spatially-invariant frequency profile.
3. The method of claim 1, wherein said obtaining includes measuring
amplitude and phase of displacement of each said blade as a
function of said traveling wave excitation.
4. The method of claim 1, wherein said determining includes
converting said data obtained from said transformation into a set
of physical coordinates.
5. The method of claim 1, further comprising: calculating mistuning
of a blade in said bladed disk system and nominal frequency of said
bladed disk system when tuned by using said set of mistuned modes
and natural frequencies.
6. The method of claim 5, further comprising validating a finite
element model of said bladed disk system using said nominal
frequencies of the tuned bladed disk system.
7. The method of claim 5, wherein calculating said nominal
frequencies includes calculating said nominal frequencies using a
finite element model of said bladed disk system treating each blade
in said bladed disk system as identical and also using said set of
mistuned modes and natural frequencies.
8. The method of claim 5, wherein said calculating includes
calculating said nominal frequencies for an isolated family of
modes of said bladed disk system when tuned.
9. The method of claim 5, wherein said calculating is performed
iteratively.
10. The method of claim 5, wherein said blade includes a
corresponding blade-disk sector in said bladed disk system.
11. The method of claim 10, wherein a mean value of mistuning of at
least one blade-disk sector is zero.
12. The method of claim 11, wherein said calculating includes
solving: where {tilde over (B)} is a stacked matrix composed from
the elements of {right arrow over (.beta.)}.sub.j, which is a
vector containing weighting factors that describe the j.sup.th
mistuned mode as a sum of tuned modes; is a stacked matrix of
.OMEGA..degree..GAMMA., where .OMEGA..degree. is a diagonal matrix
of the nominal frequencies of said bladed disk system when tuned
and .GAMMA..sub.j is a matrix composed from the elements in the
vector {right arrow over (.gamma.)}.sub.j where .gamma. .fwdarw. j
= .OMEGA..degree. .times. .beta. .fwdarw. j ##EQU38## [ B .about. 2
.times. ( .OMEGA..degree..GAMMA. .about. ) 0 c .fwdarw. ]
.function. [ .lamda..degree. .fwdarw. .omega. _ .fwdarw. ] = [ r _
.fwdarw. 0 ] ##EQU39## c is a row vector whose first element is 1
and whose remaining elements are zero; {right arrow over (.omega.)}
is a vector of mistuning parameters; .lamda..degree. is a vector of
the tuned frequencies squared; and {right arrow over (r)} is the
vector given by the following .DELTA. .times. .times. .omega. .psi.
( s ) = p = 0 N - 1 .times. e - I .times. .times. s .times. .times.
p .times. .times. 2 .times. .times. .pi. N .times. .omega. .fwdarw.
p ##EQU40## [ ( .omega. 1 2 .times. I - .OMEGA..degree. 2 ) .times.
.beta. .fwdarw. 1 ( .omega. 1 2 .times. I - .OMEGA..degree. 2 )
.times. .beta. .fwdarw. 2 ( .omega. 1 2 .times. I - .OMEGA..degree.
2 ) .times. .beta. .fwdarw. m ] ##EQU41## where I is the identity
matrix, and .omega..sub.1 is the natural frequency of the 1.sup.st
mistuned mode.
13. The method of claim 12, wherein the vector {right arrow over
(.omega.)} is related to a physical sector mistuning by the
equation P= where .DELTA..omega..sub..omega..sup.(s) is the sector
frequency deviation of the s.sup.th blade-disk sector; and
.omega..sub.p is a p.sup.th mistuning parameter in the vector
{right arrow over (.omega.)}.
14. The method of claim 1, further comprising: obtaining nominal
frequencies of said bladed disk system when tuned; and calculating
mistuning of a blade in said bladed disk system from said nominal
frequencies and said mistuned modes and natural frequencies.
15. The method of claim 14, wherein said obtaining includes
obtaining said nominal frequencies for an isolated family of modes
of said bladed disk system when tuned.
16. The method of claim 14, wherein said obtaining includes
obtaining said nominal frequencies using a finite element
analysis.
17. The method of claim 16, wherein said finite element analysis
includes finite element analysis of a tuned, cyclic symmetric model
of a single blade-disk sector in said bladed disk system.
18. The method of claim 14, wherein said blade includes a
corresponding blade-disk sector in said bladed disk system.
19. The method of claim 1, wherein said obtaining includes:
rotating said bladed disk system; and exciting said rotating bladed
disk system with pressure fluctuations.
20. The method of claim 1, wherein said blade includes a
corresponding blade-disk sector in said bladed disk system.
21. The method of claim 1, wherein said transforming is performed
from a physical coordinates domain to a modal analysis domain.
22. The method of claim 21, wherein said transforming is performed
according to the following equations: f _ .fwdarw. = DFT .times. {
f .fwdarw. } h _ .fwdarw. = DFT .times. { h .fwdarw. } ##EQU42##
where {right arrow over (f)} is a vector that describes the spatial
distribution of said traveling wave excitation; {right arrow over
(h)} is a vector that describes the frequency response of each
measurement point to said traveling wave excitation; {right arrow
over (f)} is a vector that is a discrete Fourier Transform of the
force vector {right arrow over (f)}; and {right arrow over (h)} is
a vector that is a discrete Fourier Transform of the response
vector {right arrow over (h)}.
23. A computer-readable data storage medium containing a program
code, which, when executed by a processor, causes said processor to
perform the following: receive frequency response data of each
blade in a bladed disk system to a traveling wave excitation;
transform data related to spatial distribution of said traveling
wave excitation and said frequency response data; and determine a
set of mistuned modes and natural frequencies of said bladed disk
system using data obtained from said transformation.
Description
REFERENCE TO RELATED APPLICATION
[0001] This application is a divisional of copending U.S.
application Ser. No. 10/836,422 filed Apr. 30, 2004 and entitled
"Fundamental Mistuning Model for Determining System Properties and
Predicting Vibratory Response of Bladed Disks", which claims
priority benefits of the earlier filed U.S. provisional patent
application Ser. No. 60/474,083, titled "Fundamental Mistuning
Model for Determining System Properties and Predicting Vibratory
Response of Bladed Disks," filed on May 29, 2003, the entirety of
which is hereby incorporated by reference.
BACKGROUND
[0003] The present disclosure generally relates to identification
of mistuning in rotating, bladed structures, and, more
particularly, to the development and use of reduced order models as
an aid to the identification of mistuning.
[0004] It is noted at the outset that the term "bladed disk" is
commonly used to refer to any (blade-containing) rotating or
non-rotating part of an engine or rotating apparatus without
necessarily restricting the term to refer to just a disk-shaped
rotating part. Thus, a bladed disk can have externally-attached or
integrally-formed blades or any other suitable rotating
protrusions. Also, this rotating mechanism may have any suitable
shape, whether in a disk form or not. Further, the term "bladed
disk" may include stators or vanes, which are non-rotating bladed
disks used in gas turbines. Various types of devices such as fans,
pumps, turbochargers, compressors, engines, turbines, and the like,
may be commonly referred to as "rotating apparatus."
[0005] FIG. 1 illustrates a bladed disk 10 which is representative
of those used in gas turbine engines. One such exemplary gas
turbine 12 is illustrated in FIG. 1. Bladed disks used in turbine
engines are nominally designed to be cyclically symmetric. If this
were the case, then all blades would respond with the same
amplitude when excited by a traveling wave. However, in practice,
the resonant amplitudes of the blades are very sensitive to small
changes in their properties. The small variations that result from
the manufacturing process and wear cause some blades to have a
significantly higher response and may cause them to fail from high
cycle fatigue (HCF). This phenomenon is referred to as the
mistuning problem, and has been studied extensively.
[0006] FIG. 2 represents an exemplary selection of nodal diameter
modes 13-15 in a bladed disk. In the zero nodal diameter mode 13
(part (a) in FIG. 2), all the blades move in phase with one
another, while in the higher nodal diameter modes 14-15, the blades
move out of phase. FIG. 2(b) illustrates a mode 14 with five (5)
nodal diameters, whereas FIG. 2(c) illustrates a mode 15 with ten
(10) nodal diameters. The displacement of the blades as a function
of angular position is given in these modes by functions
sin(n.theta.) and cos(n.theta.), where "n" defines the number of
nodal diameters. For a given value of "n", the corresponding sine
and cosine modes both have the same natural frequency. The only
nodal diameter modes which do not have repeated frequencies are the
cases of n=0 and n=N/2, where N is the number of blades on the
disk.
[0007] FIG. 3 is an exemplary nodal diameter map 16 of a bladed
disk's natural frequencies. Thus, the natural frequencies of a
bladed disk are plotted as a function of the number of nodal
diameters in their corresponding mode. When plotted in this
fashion, the frequencies cluster into families of modes. Each
family consists of a set of N nodal diameter modes. Each mode 17-19
at the right in FIG. 3 represents the blade deformation in the
corresponding family. Although the relative amplitudes of the
blades varies from one nodal diameter to the next, the deformation
within a given blade remains uniform throughout all modes within a
given family, at least for families which are isolated in frequency
from their neighbors. The blade deformation in the first few
families generally resembles the simple bending and torsion modes
of a cantilevered plate. Many families of modes have most of their
strain energy stored in the blades. These families appear
relatively flat in FIG. 3, because the added strain energy
introduced with additional nodal diameters has a minimal effect on
their natural frequency. In contrast, mode families with large
strain energy in the disk increase their frequency rapidly from one
nodal diameter to the next.
[0008] Mistuning can significantly affect the vibratory response of
a bladed disk. This sensitivity stems from the nature of the
eigenvalue problem that describes a disk's modes and natural
frequencies. An eigenvalue is equal to the square of the natural
frequency of a mode. The eigenvalues of a bladed disk are
inherently closely spaced due to the system's rotationally periodic
design, as can be seen from the plot in FIG. 3. Therefore, the
eigenvectors (mode shapes) of a bladed disk are very sensitive to
the small perturbations caused by mistuning. In the case of very
small mistuning, the blade displacements in the modes are given by
distorted sine and cosine waves, while large mistuning can alter
the modes to such an extent that the majority of the motion will be
localized to just one or two blades. FIG. 4 illustrates exemplary
forced response tracking plots 20-21 of a tuned bladed disk system
(plot 20) and the mistuned disk system (plot 21). The plot 21
illustrates blade amplitude magnification caused by mistuning.
[0009] To address the mistuning problem, researchers have developed
reduced order models (ROMs) of the bladed disk. These ROMs have the
structural fidelity of a finite element model of the full rotor,
while incurring computational costs that are comparable to that of
a mass-spring model. In numerical simulations, most published ROMs
have correlated extremely well with numerical benchmarks. However,
some models have at times had difficulty correlating with
experimental data. These results suggest that the source of the
error may lie in the inability to determine the correct input
parameters to the ROMs.
[0010] The standard method of measuring mistuning in rotors with
attachable blades is to mount each blade in a broach block and
measure its natural frequency. The difference of each blade's
natural frequency from the mean value is then taken as a measure of
the mistuning. However, the mistuning measured through this method
may be significantly different from the mistuning present once the
blades are mounted on the disk. This variation in mistuning can
arise because each blade's frequency is dependent on the contact
conditions at the attachment. Not only may the blade-broach contact
differ from the blade-disk contact, but the contact conditions can
also vary from slot-to-slot around the wheel or disk. Therefore, to
accurately measure mistuning, it is desirable to develop methods
that can infer the mistuning from the vibratory response of the
blade-disk assembly as a whole. In addition, many blade-disk
structural systems are now manufactured as a single piece in which
the blades cannot be physically separated from the disk. In the gas
turbine industry they are referred to as blisks (for bladed disks)
or IBRs (for integrally bladed rotors). Thus, in the case of IBRs,
the conventional testing methods of separately measuring individual
blade frequencies cannot be applied and, therefore, it is desirable
to develop methods that can infer the properties of the individual
blades from the behavior of the blade-disk assembly as a whole.
[0011] Therefore, to accurately measure mistuning, it is desirable
to develop methods or reduced order models that apply to individual
blades or the blade-disk assembly as a whole. It is further
desirable to use the mistuning values obtained from the
newly-devised reduced order models to verify finite element models
of the system and also to monitor the frequencies of individual
blades to determine if they have changed because of cracking,
erosion or other structural changes. It is also desirable that the
obtained mistuning values can be analytically adjusted and used
with the reduced order model to predict the vibratory response of
the structure (or bladed disk) when it is in use, e.g., when it is
rotating in a gas turbine engine, an industrial turbine, a fan, or
any other rotating apparatus.
SUMMARY
[0012] In one embodiment, the present disclosure contemplates a
method that comprises obtaining a set of vibration measurements
that provides frequency deviation of each blade of a bladed disk
system from the tuned frequency value of the blade and nominal
frequencies of the bladed disk system when tuned; and calculating
the mistuned modes and natural frequencies of the bladed disk
system from the blade frequency deviations and the nominal
frequencies.
[0013] In another embodiment, the present disclosure contemplates a
method that comprises obtaining nominal frequencies of a bladed
disk system when tuned; measuring at least one mistuned mode and
natural frequency of the bladed disk system; and calculating
mistuning of at least one blade (or blade-disk sector) in the
bladed disk system from only the nominal frequencies and the at
least one mistuned mode and natural frequency.
[0014] In a further embodiment, the present disclosure contemplates
a method that comprises measuring a set of mistuned modes and
natural frequencies of a bladed disk system; and calculating
mistuning of at least one blade in the bladed disk system and
nominal frequencies of the bladed disk system when tuned by using
only the set of mistuned modes and natural frequencies.
[0015] In a still further embodiment, the present disclosure
contemplates a method that comprises obtaining frequency response
data of each blade in a bladed disk system to a traveling wave
excitation; transforming data related to spatial distribution of
the traveling wave excitation and the frequency response data; and
determining a set of mistuned modes and natural frequencies of the
bladed disk system using data obtained from the transformation.
[0016] According to the methodology of the present disclosure, a
reduced order model called the Fundamental Mistuning Model (FMM) is
developed to accurately predict vibratory response of a bladed disk
system. FMM may describe the normal modes and natural frequencies
of a mistuned bladed disk using only its tuned system frequencies
and the frequency mistuning of each blade/disk sector (i.e., the
sector frequencies). If the modal damping and the order of the
engine excitation are known, then FMM can be used to calculate how
much the vibratory response of the bladed disk will increase
because of mistuning when it is in use. The tuned system
frequencies are the frequencies that each blade-disk and blade
would have were they manufactured exactly the same as the nominal
design specified in the engineering drawings. The sector
frequencies distinguish blade-disks with high vibratory response
from those with a low response. The FMM identification
methods--basic and advanced FMM ID methods-use the normal (i.e.,
mistuned) modes and natural frequencies of the mistuned bladed disk
measured in the laboratory to determine sector frequencies as well
as tuned system frequencies. Thus, one use of the FMM methodology
is to: identify the mistuning when the bladed disk is at rest, to
extrapolate the mistuning to engine operating conditions, and to
predict how much the bladed disk will vibrate under the operating
(rotating) conditions.
[0017] In one embodiment, the normal modes and natural frequencies
of the mistuned bladed disk are directly determined from the disk's
vibratory response to a traveling wave excitation in the engine.
These modes and natural frequency may then be input to the FMM ID
methodology to monitor the sector frequencies when the bladed disk
is actually rotating in the engine. The frequency of a disk sector
may change if the blade's geometry changes because of cracking,
erosion, or impact with a foreign object (e.g., a bird). Thus,
field calibration and testing of the blades (e.g., to assess damage
from vibrations in the engine) may be performed using traveling
wave analysis and FMM ID methods together.
[0018] The FMM software (containing FMM ID methods) may receive the
requisite input data and, in turn, predict bladed disk system's
mistuning and vibratory response. Because the FMM model can be
generated completely from experimental data (e.g., using the
advanced FMM ID method), the tuned system frequencies from advanced
FMM ID may be used to validate the tuned system finite element
model used by industry. Further, FMM and FMM ID methods are simple,
i.e., no finite element mass or stiffness matrices are required.
Consequently, no special interfaces are required for FMM to be
compatible with a finite element model.
BRIEF DESCRIPTION OF THE DRAWINGS
[0019] For the present disclosure to be easily understood and
readily practiced, the present disclosure will now be described for
purposes of illustration and not limitation, in connection with the
following figures, wherein:
[0020] FIG. 1 illustrates a bladed disk which is representative of
those used in gas turbine engines;
[0021] FIG. 2 represents an exemplary selection of nodal diameter
modes in a bladed disk;
[0022] FIG. 3 is an exemplary nodal diameter map of a bladed disk's
natural frequencies;
[0023] FIG. 4 illustrates exemplary forced response tracking plots
of a tuned bladed disk system and the mistuned disk system;
[0024] FIG. 5 illustrates near equivalence of sector modes from
various nodal diameters;
[0025] FIG. 6 illustrates an exemplary three dimensional (3D)
finite element model of a high pressure turbine (HPT) blade-disk
sector;
[0026] FIG. 7 shows tuned system frequencies of the first families
of modes of the blade-disk sector modeled in FIG. 6;
[0027] FIG. 8 illustrates the tuned frequencies of the fundamental
family of modes of the system modeled in FIG. 6, along with the
frequencies as determined by ANSYS.RTM. software and the best-fit
mass-spring model;
[0028] FIGS. 9 (a) and (b) depict representative results of using
FMM with a realistic mistuned bladed disk modeled in FIG. 6;
[0029] FIGS. 10(a) and (b) show a representative case of the blade
amplitudes as a function of excitation frequency for a 7.sup.th
engine order excitation predicted by the mass-spring model,
ANSYS.RTM. software, and FMM;
[0030] FIG. 11 illustrates the leading edge blade tip displacements
for the third family of modes shown in FIG. 7;
[0031] FIGS. 12(a)-(c) illustrate FMM and ANSYS.RTM. software
predictions of blade amplitude as a function of excitation
frequency for a 2.sup.nd engine order excitation of 2.sup.nd,
3.sup.rd, and 4.sup.th families respectively;
[0032] FIGS. 13(a)-(c) illustrate FMM and ANSYS.RTM. software
predictions of blade amplitude as a function of excitation
frequency for a 7.sup.th engine order excitation of 2.sup.nd,
3.sup.rd, and 4.sup.th families respectively;
[0033] FIG. 14 represents an exemplary finite element model of a
twenty blade compressor;
[0034] FIG. 15 illustrates the natural frequencies of the tuned
compressor modeled in FIG. 14;
[0035] FIG. 16 shows the comparison between the sector mistuning
calculated directly by finite element simulations of each mistuned
blade/sector and the mistuning identified by basic FMM ID;
[0036] FIG. 17 schematically illustrates a rotor 29 with
exaggerated stagger angle variations as viewed from above;
[0037] FIG. 18 shows a representative mistuned mode caused by
stagger angle mistuning of the rotor in FIG. 14;
[0038] FIG. 19 illustrates a comparison of mistuning determination
from basic FMM ID and the variations in the stagger angles;
[0039] FIG. 20 depicts a comparison of mistuning predicted using
advanced FMM ID with that obtained using the finite element
analysis (FEA);
[0040] FIG. 21 shows a comparison of the tuned frequencies
identified by advanced FMM ID and those computed directly with the
finite element model;
[0041] FIG. 22 illustrates an exemplary setup to measure transfer
functions of test rotors and also to verify various FMM ID
methods;
[0042] FIG. 23 illustrates natural frequencies of a test compressor
with no mistuning;
[0043] FIG. 24 illustrates a typical transfer function from a test
compressor obtained using the test setup shown in FIG. 22;
[0044] FIG. 25 illustrates a comparison of mistuning from each FMM
ID method with benchmark results for a test rotor SN-1;
[0045] FIG. 26 shows a comparison of tuned system frequencies for
the test rotor SN-1 from advanced FMM ID and the finite element
model (FMM) using ANSYS.RTM. software;
[0046] FIGS. 27 and 28 are similar to FIGS. 25 and 26,
respectively, but illustrate the identified mistuning and tuned
system frequencies for a different test rotor SN-3;
[0047] FIGS. 29 (a) and (b) show a comparison, for rotors SN-1 and
SN-3 respectively, of the mistuning identified by FMM ID with the
values from benchmark results from geometric measurements;
[0048] FIG. 30 illustrates a comparison of tuned system frequencies
from advanced FMM ID and ANSYS.RTM. software for torsion modes of
rotors SN-1 and SN-3;
[0049] FIG. 31(a) depicts FMM-based forced response data, whereas
FIG. 31(b) depicts the experimental forced response data;
[0050] FIGS. 32(a) and (b) respectively show relative blade
amplitudes at forced response resonance for the resonant peaks
labeled {circle around (1)} and {circle around (2)} in FIG.
31(a);
[0051] FIG. 33 depicts cumulative probability function plots of
peak blade amplitude for a nominally tuned and nominally mistuned
compressor;
[0052] FIG. 34 shows mean and standard deviations of each sector's
mistuning for a nominally mistuned compressor;
[0053] FIG. 35 illustrates a lumped parameter model of a rotating
blade;
[0054] FIG. 36 shows a comparison of mistuning values analytically
extrapolated to speed with an FEA (finite element analysis)
benchmark;
[0055] FIG. 37 illustrates the effect of centrifugal stiffening on
tuned system frequencies;
[0056] FIG. 38 illustrates the effect of centrifugal stiffening on
mistuning;
[0057] FIG. 39 depicts frequency response of blades to a six engine
order excitation at 40,000 RPM rotational speed;
[0058] FIGS. 40(a), (b) and (c) show a comparison of the
representative mode shape extracted from the traveling wave
response data with benchmark results for a stationary
benchmark;
[0059] FIG. 41 depicts comparison of the natural frequencies
extracted from the traveling wave response data with the benchmark
results for the stationary benchmark of FIG. 40;
[0060] FIG. 42 shows a calibration curve relating the effect of a
unit mass on a sector's frequency deviation in a stationary
benchmark;
[0061] FIG. 43 shows the comparison between the mass mistuning
identified through traveling wave FMM ID with the values of the
actual masses placed on each blade tip;
[0062] FIGS. 44(a) and (b) show tracking plots of blade amplitudes
as a function of excitation frequency for two different
acceleration rates;
[0063] FIGS. 45(a) and (b) illustrate the comparison of the
mistuning determined through the traveling wave system
identification method with benchmark values for two different
acceleration rates; and
[0064] FIG. 46 illustrates an exemplary process flow depicting
various blade sector mistuning tools discussed herein.
DETAILED DESCRIPTION
[0065] Reference will now be made in detail to some embodiments of
the present disclosure, examples of which are illustrated in the
accompanying drawings. It is to be understood that the figures and
descriptions of the present disclosure included herein illustrate
and describe elements that are of particular relevance to the
present disclosure, while eliminating, for the sake of clarity,
other elements found in typical bladed disk systems, engines, or
rotating devices. It is noted here that the although the discussion
given below is primarily with reference to a blade-disk sector, the
principles given below can be equally applied to just the blade
portion of the blade-disk sector as can be appreciated by one
skilled in the art. Therefore, the terms "blade" and "blade-disk
sector" have been used interchangeably in the discussion below, and
no additional discussion of the blade-only application is presented
herein.
[1] DERIVING THE FUNDAMENTAL MISTUNING MODEL (FMM)
[0066] The more general form of the modal equation for the
Fundamental Mistuning Model (FMM), derived below, is applicable to
rotating, bladed apparatus. The generalized FMM formulation differs
in two ways from the original FMM derivation described in the paper
by D. M. Feiner and J. H. Griffin titled "A Fundamental Model of
Mistuning for a Single Family of Modes," appearing in the
Proceedings of IGTI, ASME Turbo Expo 2002 (Jun. 3-6, 2002),
Amsterdam, The Netherlands. This paper is incorporated herein by
reference in its entirety. First, the following derivation no
longer approximates the tuned system frequencies by their average
value. This allows for a much larger variation among the tuned
frequencies. Second, rather than using the blade-alone mode as an
approximation of the various nodal diameter sector modes, a
representative mode of a single blade-disk sector is used below.
Consequently, the approach now includes the disk portion of the
mode shape, and thus allows for more strain energy in the disk.
Although mistuning may be measured as a percent deviation in the
blade-alone frequency (as in the above mentioned paper), in the
following discussion mistuning is measured as a percent deviation
in the frequency of each blade-disk sector. The sector frequency
deviations not only capture mistuning in the blade, but can also
capture mistuning in the disk as well as variations in the ways the
blades are attached to the disk.
[0067] In the discussion below, section 1.1 describes how the
subset of nominal modes (SNM) approach is used to reduce the order
of the mistuned free-response equations and formulates the problem
in terms of reduced order sector matrices, section 1.2 relates the
sector matrices to mistuned sector frequencies, and section 1.3
simplifies the resulting mathematical expressions.
[0068] 1.1 Reduction of Order
[0069] Consider a mistuned, bladed disk in the absence of an
excitation. The order of its equation of motion may be reduced
through a subset of nominal modes (SNM) approach. The resulting
reduced order equation can be written as (see, for example, the
discussion in the above mentioned Feiner-Griffin paper):
[(.OMEGA..degree..sup.2+.DELTA.{circumflex over
(K)})-.omega..sub.j.sup.2(I+.DELTA.{circumflex over (M)})]{right
arrow over (.beta.)}.sub.j=0 (1) .OMEGA..degree..sup.2 is a
diagonal matrix of the tuned system eigenvalues (an eigenvalue is
equal to the square of the natural frequency of a mode), and I is
the identity matrix. .DELTA.{circumflex over (K)} and
.DELTA.{circumflex over (M)} are the variations in the modal
stiffness and modal mass matrices caused by stiffness and mass
mistuning. The vector {right arrow over (.beta.)}.sub.j, contains
weighting factors that describe the j.sup.th mistuned mode as a
limited sum of tuned modes, i.e., {right arrow over
(.phi.)}.sub.j=.PHI..degree.{right arrow over (.beta.)}.sub.j (2)
where .PHI..degree. is a matrix whose columns are the tuned system
modes.
[0070] Note that to first order, (I+.DELTA.{circumflex over
(M)}).sup.-1.apprxeq.(I-.DELTA.{circumflex over (M)}). Thus by
pre-multiplying (1) by (I+.DELTA.{circumflex over (M)}).sup.-1 and
keeping only first order terms, the expression becomes
(.LAMBDA..degree.+A) {right arrow over
(.beta.)}.sub.j=.omega..sub.j.sup.2{right arrow over
(.beta.)}.sub.j (3) where A=.DELTA.{circumflex over
(K)}-.DELTA.{circumflex over (M)}.OMEGA..degree..sup.2 (4)
[0071] The next section relates the matrix A to the frequency
deviations of the mistuned sectors.
[0072] 1.2 Relating Mistuning to Sector Frequency Deviations
[0073] Relating A to frequency deviations of the sectors is a
three-step process. First, the mistuning matrix is express in terms
of the system mode shapes of an individual sector. Then, the system
sector modes are related to the corresponding mode of a single,
isolated sector. Finally, the resulting sector-mode terms in A are
expressed in terms of the frequency deviations of the sectors.
[0074] 1.2.1 Relating Mistuning to System Sector Modes
[0075] Consider the mistuning matrix, A, in equation (4). This
matrix can be expressed as a sum of the contributions from each
mistuned sector. A ^ = s = 0 N - 1 .times. A ^ ( s ) ( 5 ) ##EQU1##
where the superscript denotes that the mistuning corresponds to the
s.sup.th sector. The expression for a single element of A.sup.(s)
is A.sub.mn.sup.(s)={right arrow over
(.phi.)}.degree..sub.m.sup.(s)H(.DELTA.K.sup.(s)-.omega..degree..sub.n.su-
p.2.DELTA.M.sup.(s)) {right arrow over
(.phi.)}.degree..sub.n.sup.(s) (6) where .DELTA.K.sup.(s) and
.DELTA.M.sup.(s) are the physical stiffness and mass perturbations
of the s.sup.th sector. The modes {right arrow over
(.phi.)}.degree..sub.m.sup.(s) and {right arrow over
(.phi.)}.degree..sub.n.sup.(s); are the portions of the m.sup.th
and n.sup.th columns of .PHI..degree. which describe the s.sup.th
sector's motion. The term .omega..degree..sub.n.sup.2 is the nth
diagonal element of .OMEGA..degree..sup.2. Equation (6) relates the
mistuning to the system sector modes. In the next section, these
modes are related to the mode of a single isolated blade-disk
sector.
[0076] 1.2.2 Relating System Sector Modes to an Average Sector
Mode
[0077] The tuned modes in equation (6) are expressed in a complex
traveling wave form. Thus, the motion of the s.sup.th sector can be
related to the motion of the 0.sup.th sector by a phase shift.
Thus, equation (6) can be restated as A ^ mn ( s ) .times. e I
.times. .times. s .function. ( n - m ) .times. 2 .times. .pi. N
.times. .PHI. -> m .cndot. .function. ( 0 ) .times. H .function.
( .DELTA. .times. .times. K ( s ) - .omega. n .smallcircle. 2
.times. .DELTA. .times. .times. M ( s ) ) .times. .PHI. -> n
.cndot. .function. ( 0 ) ( 7 ) ##EQU2## Because the tuned modes
used in the SNM formulation are an isolated family of modes, the
sector modes of all nodal diameters look nearly identical as can be
seen from FIG. 5, which illustrates near equivalence of sector
modes from various nodal diameters. Therefore, one can approximate
the various sector modes by an average sector mode. Applying the
average sector mode approximation for the system sector modes in
equation (7), A.sub.mn.sup.(s) can be written as A ^ mn ( s ) = (
.omega. m * .times. .omega. n * .omega. V .cndot.2 ) .times. e I
.times. .times. s .function. ( n - m ) .times. 2 .times. .pi. N
.function. [ .psi. -> .cndot. .function. ( 0 ) H .function. (
.DELTA. .times. .times. K ( s ) - .omega. n .cndot.2 .times.
.DELTA. .times. .times. M ( s ) ) .times. .psi. -> .cndot.
.function. ( 0 ) ] ( 8 ) ##EQU3## where {right arrow over
(.psi.)}.degree..sup.(0) is the average tuned system sector mode,
and .omega..degree..sub..psi. is its natural frequency. In
practice, {right arrow over (.psi.)}.degree..sup.(0) can be taken
to be the median modal diameter mode. The factor
(.omega..degree..sub.m.omega..degree..sub.n)/(.omega..degree..sub..psi..s-
up.2) scales the average sector mode terms so that they have
approximately the same strain energy as the sector modes they
replace.
[0078] 1.2.3 Introduction of Sector Frequency Deviation
[0079] The deviation in a sector frequency quantity may be used to
measure mistuning. To understand this concept, consider an
imaginary "test" rotor. In the test rotor every sector is mistuned
in the same fashion, so as to match the mistuning in the sector of
interest. Since the test rotor's mistuning is cyclically symmetric,
its mode shapes are virtually identical to those of the tuned
system. However, there will be a shift in the tuned system
frequencies. For small levels of mistuning, the frequency shift is
nearly the same in all of the tuned system modes and can be
approximated by the fractional change in the frequency of the
median nodal diameter mode. This may typically be the case for an
isolated family of modes in which the strain energy is primarily in
the blades. If there is a significant amount of strain energy in
the disk then the frequency of the modes may change significantly
as a function of nodal diameter and the modes may not be isolated
(i.e., the modes may cover such a broad frequency range that they
may interact with other families of modes). However, in the
following, the fractional shift in the median nodal diameter's
frequency is taken as a measure of mistuning and is defined as the
sector frequency deviation.
[0080] The bracketed terms of in equation (8) are related to these
frequency deviations in the following manner. Consider a bladed
disk that is mistuned in a cyclic symmetric fashion, i.e., each
sector undergoes the same mistuning. Its free-response equation of
motion is given by the expression
[(K.degree.+.DELTA.K)-.omega..sub.n.sup.2(M.degree.+.DELTA.M)]{right
arrow over (.phi.)}.sub.n=0 (9) Take the mode {right arrow over
(.phi.)}.sub.n to be the mistuned version of the tuned median nodal
diameter mode, {right arrow over (.psi.)}.degree.. Here, {right
arrow over (.psi.)}.degree. is the full system mode counterpart of
the average sector mode {right arrow over (.psi.)}.degree..sup.(0).
Because mistuning is symmetric, the tuned and mistuned versions of
the mode are nearly identical. Substituting {right arrow over
(.psi.)}.degree. for {right arrow over (.phi.)}.sub.n and
pre-multiplying by {right arrow over (.psi.)}.degree..sup.H yields,
(.omega..degree..sub..psi..sup.2+{right arrow over
(.psi.)}.degree..sup.H .DELTA.K{right arrow over
(.psi.)}.degree.)-.omega..sub.n.sup.2(1+{right arrow over
(.psi.)}.degree..sup.H .DELTA.M{right arrow over
(.psi.)}.degree.)=0 (10) These terms may be rearranged to isolate
the frequency terms, {right arrow over
(.psi.)}.degree..sup.H(.DELTA.K-.omega..sub.n.sup.2.DELTA.M){right
arrow over
(.psi.)}.degree.=.omega..sub.j.sup.2-.omega..degree..sub..psi..sup.2
(11) Because the mistuning is symmetric, each sector contributes
equally to equation (11). Thus, the contribution from the 0.sup.th
sector is, .psi. -> .cndot. .function. ( 0 ) .times. H
.function. ( .DELTA. .times. .times. K - .omega. n 2 .times.
.DELTA. .times. .times. M ) .times. .psi. -> .cndot. .function.
( 0 ) = 1 N .times. ( .omega. j 2 - .omega. .psi. .cndot.2 ) ( 12 )
##EQU4## By factoring the frequency terms on the right-hand side of
equation (12), it can be shown that .psi. -> .cndot. .function.
( 0 ) .times. H .function. ( .DELTA. .times. .times. K - .omega. n
2 .times. .DELTA. .times. .times. M ) .times. .psi. -> .cndot.
.function. ( 0 ) .apprxeq. 2 .times. .omega. .psi. .smallcircle. 2
.times. .DELTA..omega. .psi. N ( 13 ) ##EQU5## where
.DELTA..omega..sub..psi. is the fractional change in {right arrow
over (.psi.)}'s natural frequency due to mistuning, given by
.DELTA..omega..sub..psi.=(.omega..sub..psi.-.omega..degree..sub..psi.)/.o-
mega..degree..sub..psi.. Note that, by definition, as is a sector
frequency deviation. Equation (13) can be substituted for the
bracketed terms of equation (8), resulting in an expression that
relates the elements of the sector "s" mistuning matrix to that
sector's frequency deviation, A ^ mn ( s ) = 2 .times. .omega. m *
.times. .omega. n * N .times. e I .times. .times. s .function. ( n
- m ) .times. 2 .times. .pi. N .times. .DELTA. .times. .times.
.omega. .psi. ( s ) ( 14 ) ##EQU6## where the superscript on
.DELTA..omega..sub..psi. Any is introduced to indicate that the
frequency deviation corresponds to the s.sup.th sector. These
sector contributions may be summed to obtain the elements of the
mistuning matrix, A ^ mn = 2 .times. .omega. m 0 .times. .omega. n
0 .function. [ 1 N .times. s = 0 N - 1 .times. e I .times. .times.
sp .times. 2 .times. .pi. N .times. .DELTA..omega. .psi. ( s ) ] (
15 ) ##EQU7##
[0081] 1.3 The Simplified Form of the Fundamental Mistuning Model
Modal Equation
[0082] The bracketed term in equation (15) is the discrete Fourier
transform (DFT) of the sector frequency deviations. If one uses the
dummy variable p to replace the quantity (n-m) in equation (15),
then the p.sup.th DFT of the sector frequency deviations is given
by .omega. _ p = [ 1 N .times. s = 0 N - 1 .times. e I .times.
.times. sp .times. 2 .times. .pi. N .times. .DELTA. .times. .times.
.omega. .psi. ( s ) ] ( 16 ) ##EQU8## where .omega..sub.p denotes
the p.sup.th DFT. By substituting equation (16) into equation (15),
A may be expressed in the simplified matrix form A=2.OMEGA..degree.
.OMEGA..OMEGA..degree. (17) where .OMEGA. -> = [ .omega. _ 0
.omega. _ 1 .omega. _ N - 1 .omega. _ N - 1 .omega. _ 0 .omega. _ N
- 2 .omega. _ 1 .omega. _ 2 .omega. _ 0 ] ( 18 ) ##EQU9## .OMEGA.
is a matrix which contains the discrete Fourier transforms of the
sector frequency deviations. Note that .OMEGA. has a circulant
form, and thus contains only N distinct elements. .OMEGA..degree.
is a diagonal matrix of the tuned system frequencies.
[0083] Substituting equation (17) into equation (3) produces the
most basic form of the eigenvalue problem that may be solved to
determine the modes and natural frequencies of the mistuned system.
(.OMEGA..degree..sup.2+2.OMEGA..degree.
.OMEGA..OMEGA..degree.){right arrow over
(.beta.)}.sub.j.sup.2{right arrow over (.beta.)}.sub.j (19)
[0084] Equations (18) and (19) represent the functional form of the
Fundamental Mistuning Model. Here, .OMEGA..degree..sup.2 is a
diagonal matrix of the nominal system eigenvalues, ordered in
accordance with the following equation. { .PHI. -> 0
.smallcircle. ( s ) .PHI. -> 1 .smallcircle. ( s ) .PHI. -> 2
.smallcircle. ( s ) .PHI. -> N 2 .smallcircle. ( s ) .PHI. ->
N 2 + 1 .smallcircle. ( s ) .PHI. -> N - 1 .smallcircle. ( s ) (
0 ) ( 1 .times. B ) ( 2 .times. B ) ( N 2 ) ( ( N 2 - 1 ) .times. F
) ( 1 .times. F ) } ( 20 ) ##EQU10## where the second row in
equation (20) indicates the nodal diameter and direction of the
corresponding mode that lies above it. "B" denotes a backward
traveling wave, defined as a mode with a positive phase shift from
one sector to the next, and "F" denotes a forward traveling wave,
defined as a mode with a negative phase shift from one sector to
the next. Note that the modes are numbered from 0 to N-1.
[0085] As mentioned before, the eigenvalues are equal to the
squares of the natural frequencies of the tuned system. This
.OMEGA..degree..sup.2 matrix contains all the nominal system
information required to calculate the mistuned modes. Note that the
geometry of the system does not directly enter into this
expression. The term representing mistuning in equation (1),
2.OMEGA..degree..OMEGA. .OMEGA..degree., is a simple circulant
matrix that contains the discrete Fourier transforms of the blade
frequency deviations, pre- and post-multiplied by the tuned system
frequencies.
[0086] The eigenvalues of equation (19) are the squares of the
mistuned frequencies, and the eigenvectors define the mistuned mode
shapes through equation (2). Because the tuned modes have been
limited to a single family and appear in .PHI..degree. in a certain
order, one can approximately calculate the distortion in the
mistuned mode shapes without knowing anything specific about
.PHI..degree.. The reason for this is the assumption that all of
the tuned system modes on the zero.sup.th sector look nearly the
same, i.e. {right arrow over
(.phi.)}.degree..sub.m.sup.(0).apprxeq.{right arrow over
(.phi.)}.degree..sub.n.sup.(0). Further, when the tuned system
modes are written in complex, traveling wave form, the amplitudes
of every blade in a mode are the same, but each blade has a
different phase. Therefore, the part of the mode corresponding to
the s.sup.th sector can be written in terms of the same mode on the
0.sup.th sector, multiplied by an appropriate phase shift, i.e.,
.PHI. -> n .smallcircle. ( s ) = e I .times. .times. sn .times.
2 .times. .pi. N .times. .PHI. -> ( n ) .smallcircle. ( 0 ) ( 21
) ##EQU11## where i= {square root over (-1)}. Equation (21) implies
that if the j.sup.th mistune mode is given by {right arrow over
(.beta.)}.sub.j=[.beta..sub.j0, .beta..sub.j1 . . .
.beta..sub.j,N-1].sup.T then the physical displacements of the
n.sup.th blade in this mode are proportional to x n = m = 0 N - 1
.times. .beta. jm .times. e I .times. .times. mn .times. 2 .times.
.pi. N ( 22 ) ##EQU12##
[0087] 1.4 Numerical Results
[0088] A computer program was written to implement the theory
presented in sections (1.1) through (1.3). The program also
incorporated a simple modal summation algorithm to calculate the
bladed disk's forced response. The modal summation assumed constant
modal damping. The basic modal summation algorithm was chosen to
benchmark the forced response because a similar algorithm may be
used as an option in the commercial finite element analysis
ANSYS.RTM. software, which was used as a benchmark. It is observed,
however, that FMM may be used with more sophisticated methods for
calculating the forced response, such as the state-space approach
used in a subset of nominal modes (SNM) analysis discussed in Yang
M.-T. and Griffin, J. H., 2001, "A Reduced Order Model of Mistuning
Using a Subset of Nominal Modes," Journal of Engineering for Gas
Turbines and Power, 123(4), pp. 893-900.
[0089] It is noted that when a beam-like blade model is used (to
minimize the computational cost), FMM could accurately calculate a
bladed disk's mistuned response based on only sector frequency
deviations, without regard for the physical cause of the mistuning.
However, in the discussion below, a more realistic geometry is
analyzed using FMM.
[0090] FIG. 6 illustrates an exemplary three dimensional (3D)
finite element model 22 of a high pressure turbine (HPT) blade-disk
sector. There were 24 sectors in the full system. This model was
developed by approximating the features of an actual turbine blade
and provided a reasonable test of FMM's ability to represent a
realistic blade geometry. FIG. 7 shows tuned system frequencies of
the first families of modes of the blade-disk system modeled in
FIG. 6. As can be seen from FIG. 7, the system of FIG. 6 had an
isolated first bending family of modes with closely spaced
frequencies. As a benchmark, a finite element analysis was
performed of the full mistuned rotor using the ANSYS.RTM. software.
The bladed disk was mistuned by randomly varying the elastic moduli
of the blades with a standard deviation that was equal to 1.5% of
the tuned system's elastic modulus.
[0091] Then, an equivalent mass-spring model was constructed with
one degree-of-freedom per sector as described in Rivas-Guerra, A.
J., and Mignolet, M. P., 2001, "Local/Global Effects of Mistuning
on the Forced Response of Bladed Disks," ASME Paper 2001-GT-0289,
International Gas Turbine Institute Turbo Expo, New Orleans, La.
Each mass was set to unity and the stiffness parameters were
obtained through a least squares fit of the tuned natural
frequencies. FIG. 8 illustrates the tuned frequencies of the
fundamental family of modes of the system modeled in FIG. 6, along
with the frequencies of the ANSYS.RTM. software and the best-fit
mass-spring model. It is noted that while the mass-spring model was
able to capture the higher nodal diameter frequencies fairly well,
it had great difficulty with the low nodal diameter frequencies.
This discrepancy arises because the natural frequencies of the
single degree-of-freedom mass-spring system have the form
.omega..sub.n= {square root over ([k+4k.sub.c sin.sup.2
(n.pi./N)]/m)} (23) where m is the blade mass, k and k.sub.c are
the base stiffness and coupling stiffness, n is the nodal diameter
of the mode, and N is the number of blades. However, the actual
frequencies of the finite element model have a significantly
different shape when plotted as a function of nodal diameters. In
contrast, FMM takes the actual finite element frequencies as input
parameters, and therefore it matches the tuned system's frequencies
exactly.
[0092] The mass-spring model was then mistuned by adjusting the
base stiffness of the blades to correspond to the modulus changes
used in the finite element model. The mistuned modes and forced
response were then calculated with both FMM and the mass-spring
model, and compared with the finite element results using the
ANSYS.RTM. software. FIGS. 9 (a) and (b) depict representative
results of using FMM with a realistic mistuned bladed disk modeled
in FIG. 6. As can be seen from FIG. 9(a), the mistuned frequencies
predicted by FMM and ANSYS.RTM. software were quite similar.
However the mass-spring model had some significant discrepancies,
especially in the low frequency modes. FMM and ANSYS.RTM. software
also predicted essentially the same mistuned mode shapes as can be
seen from FIG. 9(b). In contrast, the mass-spring model performed
poorly when matching the finite element mode shapes, even on modes
whose frequencies were accurately predicted. For example, the mode
plotted in FIG. 9(b) corresponded to the 18.sup.th frequency. From
FIG. 9(a), it is seen that the mass-spring model accurately
predicted this frequency. However, it is clear from FIG. 9(b) that
the mass-spring model still did a poor job of matching the finite
element mode shapes.
[0093] The predicted modes were then summed to obtain the system's
forced response to various engine order excitations. As noted
before, gas turbine engines are composed of a series of bladed
disks (see, for example, FIG. 1). When a bladed disk is operating
in an engine, it is subjected to pressure loading from the flow
field which excites the blades. As the flow progresses through the
engine, it passes over support struts, inlet guide vanes, and other
stationary structures which cause the pressure to vary
circumferentially. Therefore, the excitation forces are periodic in
space when considered from a stationary reference frame. As a
periodic excitation, the pressure variations can be spatially
decomposed into a Fourier series. Each harmonic component drives
the system with a traveling wave at a frequency given by the
product of its harmonic number and the rotation speed. The harmonic
number of the excitation is typically referred to as the Engine
Order, and corresponds physically to the number of excitation
periods per revolution. Each of the engine order excitations may be
generally considered separately, because they drive the system at
different frequencies.
[0094] FIGS. 10(a) and (b) show a representative case of the blade
amplitudes as a function of excitation frequency for a 7.sup.th
engine order excitation predicted by the mass-spring model,
ANSYS.RTM. software, and FMM. For clarity, only the high
responding, median responding, and low responding blades are
plotted. It is again seen that the mass-spring model provided a
poor prediction of the system's forced response. However, the
results from FMM agreed well with those computed by ANSYS.RTM.
software, as shown in FIG. 10(b). The prediction by FMM of the
highest blade response differed from that predicted by ANSYS.RTM.
software by only 1.6%. Thus, FMM may be used to provide accurate
predictions of the mode shapes and forced response of a turbine
blade with a realistic geometry.
[0095] 1.5 Other Considerations
[0096] From the foregoing discussion, it is seen that the
Fundamental Mistuning Model was derived from the Subset of Nominal
Modes theory by applying three simplifying assumptions: only a
single, isolated family of modes is excited; the strain energy of
that family's modes is primarily in the blades; and the family's
frequencies are closely spaced. In addition, one corollary of these
assumptions is that the blade's motion looks very similar among all
modes in the family. As demonstrated in the previous section, FMM
works quite well when these assumptions are met. However, these
ideal conditions are usually found only in the fundamental modes of
a rotor. The higher frequency families are often clustered close
together, have a significant amount of strain energy in the disk,
and span a large frequency range. Furthermore, veerings are quite
common, causing a family's modes to change significantly from one
nodal diameter to the next. Therefore, there may be situations
where FMM may not work effectively in high frequency regions.
[0097] The realistic HPT model of FIG. 6 may be used to further
study FMM's performance, without the need carefully assign modes to
families. Therefore, some crossings shown in FIG. 7 may in fact be
veerings. However, because such errors are easily made in practice,
it is useful to include them in the study. For reference, four mode
families are numbered along the right side of FIG. 7. It is noted
that except for the fundamental modes, the families (in FIG. 7)
undergo a significant frequency increase between 0 and 6 nodal
diameters. The steep slopes in this region suggest that the modes
have a large amount of strain energy in the disk. Furthermore, the
high modal density in this area makes it likely that some modes
were assigned to the wrong family. Therefore, the modes of a single
family may likely change significantly from one nodal diameter to
the next. To show this behavior, FIG. 11 illustrates the leading
edge blade tip displacements for the third family of modes shown in
FIG. 7. FIG. 11 shows how the circumferential (O) and out-of-plane
(z) motion of the blade tip's leading edge changes from the 0 nodal
diameter mode to the 12 nodal diameter mode. Observe that the 0 and
z components of the mode shape change significantly between 0 and 6
nodal diameters. In such case, the assumptions of FMM are violated
throughout these low nodal diameter regions. Thus, FMM may not
accurately predict the mistuned frequencies or shapes of these
modes. As a result, FMM may not provide accurate forced response
predictions when these modes are heavily excited.
[0098] To illustrate FMM performance in such situations, FMM was
used to predict the forced response of families 2, 3, and 4 to a
2.sup.nd engine-order excitation because that engine order would
primarily excite the low nodal diameter modes of each family, and
those modes violate the assumptions of FMM. The FMM predictions
were compared with finite element results calculated in ANSYS.RTM.
software. FIGS. 12(a)-(c) illustrate FMM and ANSYS.RTM. software
predictions of blade amplitude as a function of excitation
frequency for a 2.sup.nd engine order excitation of 2.sup.nd,
3.sup.rd, and 4.sup.th families respectively. For clarity, each
plot in FIGS. 12(a)-(c) shows only the low responding blade, the
median responding blade, and the high responding blade. As
expected, the FMM results differed significantly from the
ANSYS.RTM. software response in both peak amplitudes and overall
shape of the response. Thus, when a mode lies in a region where
there is uncertainty as to what family a mode belongs, veering, or
high slopes on the frequency vs. nodal diameter plot, FMM may not
always accurately predict its mistuned frequency or mode shape.
That is, FMM may not work effectively for engine orders that excite
modes in these regions.
[0099] However, there are regions in high frequency modes where FMM
may perform quite well. It is seen from the Frequency vs. Nodal
Diameter plot in FIG. 7 that the slopes over the high nodal
diameter regions are very small, indicating that the modes have
most of their strain energy in the blades. Furthermore, the flat
regions are well isolated from other families of modes. Therefore,
the blade's motion is very similar from one nodal diameter to the
next. This can be seen in FIG. 11, which indicates that the 0 and z
components of the blade tip motion remain fairly constant over the
higher nodal diameter regions. In that case, the FMM assumptions
are satisfied for these high nodal diameter modes, and FMM may
capture the physical behavior of these modes better than it did for
low engine orders.
[0100] To illustrate FMM performance in the situation described in
the preceding paragraph, FMM was used to predict the forced
response of families 2, 3, and 4 to a 7.sup.th engine order
excitation. The FMM results were compared against a finite element
benchmark performed in ANSYS.RTM. software. FIGS. 13(a)-(c)
illustrate FMM and ANSYS.RTM. software predictions of blade
amplitude as a function of excitation frequency for a 7.sup.th
engine order excitation of 2.sup.nd, 3.sup.rd, and 4.sup.thfamilies
respectively. For clarity, each plot in FIGS. 13(a)-(c) shows only
the low responding blade, the median responding blade, and the high
responding blade. In all three cases in FIG. 13, the FMM
predictions captured the overall shape of the response curves as
well as the peak amplitudes to within 6% of the ANSYS.RTM. software
performance. Therefore, for this test case, it is observed that
when a mode lies in a flat region at the upper end of a Frequency
vs. Nodal Diameter plot, its response can be reasonably well
predicted by FMM.
[2] SYSTEM IDENTIFICATION METHODS
[0101] It is seen from the discussion hereinbefore that the
Fundamental Mistuning Model provides a simple, but accurate method
for assessing the effect of mistuning on forced response, generally
in case of an isolated family of modes. However, FMM can be used to
derive more complex reduced order models to analyze mistuned
response in regions of frequency veering, high modal density and
cases of disk dominated modes. These complex models may not
necessarily be limited to an isolated family of modes.
[0102] The following description of system identification is based
the Fundamental Mistuning Model. As a result, the FMM based
identification methods (FMM ID) (discussed below) may be easy to
use and may require very little analytical information about the
system, e.g., no finite element mass or stiffness matrices may be
necessary. There are two forms of FMM ID methods discussed below: a
basic version of FMM ID that requires some information about the
system properties, and a somewhat more advanced version that is
completely experimentally based. The basic FMM ID requires the
nominal frequencies of the tuned system as input. The nominal
frequencies of the tuned system (i.e., natural frequencies of a
tuned system with each sector being identical) may be calculated
using a finite element analysis of a single blade-disk sector with
cyclic symmetric boundary conditions applied to the disk. Then,
given (experimental) measurements of a limited number of mistuned
modes and frequencies, basic FMM ID equations solve for the
mistuned frequency of each sector. It is noted that the modes
required in basic FMM ID are the circumferential modes that
correspond to the tip displacement of each blade around the wheel
or disk.
[0103] The advanced form of FMM ID uses (experimental) measurements
of some mistuned modes and frequencies to determine all of the
parameters in FMM, i.e. the frequencies that the system would have
if it were tuned as well as the mistuned frequency of each sector.
Thus, the tuned system frequencies determined from the second
method (i.e., advanced FMM ID) can also be used to validate finite
element models of the nominal system.
[0104] 2.1 Basic FMM ID Method
[0105] As noted before, the basic method uses tuned system
frequencies along with measurements of the mistuned rotor's system
modes and frequencies to infer mistuning.
[0106] 2.1.1 Inversion of FMM Equation
[0107] The FMM eigenvalue problem is given by equation (19), which
is reproduced below. (.OMEGA..degree..sup.2+2.OMEGA..degree.
.OMEGA..OMEGA..degree.) .beta..sub.j=.omega..sub.j.sup.2
.beta..sub.j (24) The eigenvector of this equation, {right arrow
over (.beta.)}.sub.j, contains weighting factors that describe the
j.sup.th mistuned mode as a sum of tuned modes. The corresponding
eigenvalue, .omega..sub.j.sup.2, is the j.sup.th mode's natural
frequency squared. The matrix of the eigenvalue problem contains
two terms, .OMEGA..degree. and .OMEGA.. .OMEGA..degree. is a
diagonal matrix of the tuned system frequencies, ordered by
ascending inter-blade phase angle of their corresponding mode. The
notation a .OMEGA..degree..sup.2 is shorthand for
.OMEGA..degree..sup.T.OMEGA..degree., which results in a diagonal
matrix of the tuned system frequencies squared. The matrix .OMEGA.
contains the discrete Fourier transforms (DFT) of the sector
frequency deviations.
[0108] As discussed earlier, FMM treats the rotor's mistuning as a
known quantity that it uses to determine the system's mistuned
modes and frequencies. However, if the mistuned modes and
frequencies are treated as known parameters, the inverse problem
could be solved to determine the rotor's mistuning. This is the
basis of FMM ID methods.
[0109] The following describes manipulation of the FMM equation of
motion to solve for the mistuning in the rotor. Thus, in equation
(24), all quantities are treated as known except .OMEGA., which
describes the system's mistuning. Subtracting the
.OMEGA..degree..sup.2 term from both sides of equation (24) and
regrouping terms yields 2.OMEGA..degree.
.OMEGA.[.OMEGA..degree.{right arrow over
(.beta.)}.sub.j]=(.omega..sub.j.sup.2I-.OMEGA..degree..sup.2){right
arrow over (.beta.)}.sub.j (25) The bracketed quantity on the
left-hand side of equation (25) contains a known vector, which may
be denoted as {right arrow over (.gamma.)}.sub.j, {right arrow over
(.gamma.)}.sub.j=.OMEGA..degree.{right arrow over (.beta.)}.sub.j
(26) Thus, {right arrow over (.gamma.)}.sub.j contains the modal
weighting factors, {right arrow over (.beta.)}.sub.j scaled on an
element-by-element basis by their corresponding natural
frequencies. Substituting {right arrow over (.gamma.)}.sub.j into
equation (25) yields 2.OMEGA..degree.[ .OMEGA.{right arrow over
(.gamma.)}.sub.j]=(.omega..sub.j.sup.2I-.OMEGA..degree..sup.2){right
arrow over (.beta.)}.sub.j (27) After some algebra, it can be shown
that the product in the bracket in equation (27) may be rewritten
in the form .OMEGA.{right arrow over (.gamma.)}.sub.j=.GAMMA..sub.j
{right arrow over (.omega.)} (28) where the vector {right arrow
over (.omega.)} equals [ .omega..sub.0, .omega..sub.1 . . .
.omega..sub.N-1].sup.T. The matrix .GAMMA..sub.j is composed from
the elements in {right arrow over (.gamma.)}.sub.j and has the form
.GAMMA. j = [ .gamma. _ j .times. .times. 0 .gamma. _ j .times.
.times. 1 .gamma. _ j .function. ( N - 1 ) .gamma. _ j .times.
.times. 1 .gamma. _ j .times. .times. 2 .gamma. _ j .times. .times.
0 .gamma. _ j .function. ( N - 1 ) .gamma. _ j .times. .times. 0
.gamma. _ j .function. ( N - 2 ) ] ( 29 ) ##EQU13## where
.gamma..sub.jn denotes the n.sup.th element of the vector {right
arrow over (.gamma.)}.sub.j; the {right arrow over (.gamma.)}.sub.j
elements are numbered from 0 to N-1. Note that each column of
.GAMMA..sub.j is the negative permutation of the previous
column.
[0110] Substituting equation (28) into (27) produces an expression
in which the matrix of mistuning parameters, .OMEGA., has been
replaced by a vector of mistuning parameters, {right arrow over
(.omega.)}. 2.OMEGA..degree..GAMMA..sub.j {right arrow over
(.omega.)}=(.omega..sub.j.sup.2I-.OMEGA..degree..sup.2){right arrow
over (.beta.)}.sub.j (30) Pre-multiplying equation (30) by
(2.OMEGA..degree..GAMMA..sub.j).sup.-1 would solve this expression
for the DFT (Discrete Fourier Transform) of the rotor's mistuning.
Furthermore, the vector {right arrow over (.omega.)} can then be
related to the physical sector mistuning through an inverse
discrete Fourier transform. However, equation (30) only contains
data from one measured mode and frequency. Therefore, error in the
mode's measurement may result in significant error in the predicted
mistuning. To minimize the effects of measurement error, multiple
mode measurements may be incorporated into the solution for the
mistuning. Equation (30) may be constructed for each of the M
measured modes, and the modes may be combined into the single
matrix expression, [ 2 .times. .OMEGA. .smallcircle. .times.
.GAMMA. 1 2 .times. .OMEGA. .smallcircle. .times. .GAMMA. 2 2
.times. .OMEGA. .smallcircle. .times. .GAMMA. m ] .times. .omega. _
-> = [ ( .omega. 1 2 .times. I - .OMEGA. .smallcircle. 2 )
.times. .beta. -> 1 ( .omega. 2 2 .times. I - .OMEGA.
.smallcircle. 2 ) .times. .beta. -> 2 ( .omega. m 2 .times. I -
.OMEGA. .smallcircle. 2 ) .times. .beta. -> m ] ( 31 ) ##EQU14##
For brevity, equation (31) may be rewritten as {tilde over (L)}
{right arrow over (.omega.)}= {right arrow over (r)} (32) where
{tilde over (L)} is the matrix on the left-hand side of the
expression, and {right arrow over (r)} is the vector on the
right-hand side. The ".about." is used to indicate that these
quantities are composed by vertically stacking a set of
sub-matrices or vectors.
[0111] It is noted that the expression in equation (32) is an
overdetermined set of equations. Therefore, it may not be possible
to solve for {right arrow over (.omega.)} by direct inverse.
However, one can obtain a least squares fit to the mistuning, i.e.
.omega. _ -> = Lsp .times. { L ~ , r -> ~ } ( 33 ) ##EQU15##
Equation (33) produces the vector {right arrow over (.omega.)}
which best-fits all the measured data. Therefore, the error in each
measurement is compensated for by the balance of the data. The
vector {right arrow over (.omega.)} can then be related to the
physical sector mistuning through the inverse transform,
.DELTA..omega. .psi. ( s ) = p = 0 N - 1 .times. e - I .times.
.times. sp .times. 2 .times. .pi. N .times. .omega. _ p ( 34 )
##EQU16## where .DELTA..omega..sub..psi..sup.(s) is the sector
frequency deviation of the s.sup.th sector. The following section
describes how equations (33) and (34) can be applied to determine a
rotor's mistuning.
[0112] 2.1.2 Experimental Application of Basic FMM ID
[0113] To solve equation (33) and (34) for the sector mistuning,
one must first construct {tilde over (L)} and {right arrow over
(r)} from the tuned system frequencies and the mistuned modes and
frequencies. The tuned system frequencies can be calculated through
finite element analysis of a tuned, cyclic symmetric, single
blade/disk sector model. However, the mistuned modes and
frequencies must be obtained experimentally.
[0114] The modes used by basic FMM ID are circumferential modes,
corresponding to the tip displacement of each blade on the rotor.
In case of isolated families of modes, it may be sufficient to
measure the displacement of only one point per blade. In practice,
modes and frequencies may be obtained by first measuring a complete
set of frequency response functions (FRFs). Then, the modes and
frequencies may be extracted from the FRFs using modal curve
fitting software.
[0115] The mistuned frequencies obtained from the measurements
appear explicitly in the basic FMM ID equations as .omega..sub.j.
However, the mistuned modes enter into the equations indirectly
through the modal weighting factors {right arrow over
(.beta.)}.sub.j. Each vector {right arrow over (.beta.)}.sub.j is
obtained by taking the inverse discrete Fourier transform of the
corresponding single point-per-blade mode, i.e., .beta. jn = m = 0
N - 1 .times. .PHI. jm .times. e - I .times. .times. mn .times. 2
.times. .pi. N ( 35 ) ##EQU17## The quantities obtained from
equation (35) may then be used with the tuned system frequencies to
construct {tilde over (L)} and {right arrow over (r)} as outlined
hereinbefore. Finally, equations (33) and (34) may be solved for
the sector mistuning. This process is demonstrated through the two
examples in the following section.
[0116] 2.1.3 Numerical Examples for Basic FMM ID
[0117] The first example considers an integrally bladed compressor
whose blades are geometrically mistuned. The sector frequency
deviations identified by basic FMM ID are verified by comparing
them with values directly determined by finite element analyses
(FEA). The second example highlights basic FMM ID's ability to
detect mistuning caused by variations at the blade-disk interface.
This example considers a compressor in which all the blades are
identical, however they are mounted on the disk at slightly
different stagger angles. The mistuning caused by the stagger angle
variations is determined by FMM ID and compared with the input
values.
[0118] 2.1.3.1 Geometric Blade Mistuning
[0119] FIG. 14 represents an exemplary finite element model 26 of a
twenty blade compressor. Although the airfoils on this model are
simply flat plates, the rotor design reflects the key dynamic
behaviors of a modern, integrally bladed compressor. The rotor was
mistuned through a combination of geometric and material property
changes. Approximately one-third of the blades were mistuned
through length variations, one-third through thickness variations,
and one-third through elastic modulus variations. The magnitudes of
the variations were chosen so that each form of mistuning would
contribute equally to a 1.5% standard deviation in the sector
frequencies.
[0120] A finite element analysis of the tuned rotor was first
performed to generate its nodal diameter map. FIG. 15 illustrates
the natural frequencies of the tuned compressor modeled in FIG. 14,
i.e., the tuned rotor's nodal diameter map. It is observed from
FIG. 15 that the lowest frequency family of first bending modes was
isolated (as denoted by the rectangle portion 27) for the basic FMM
ID analysis. The sector mistuning of this rotor was then determined
through two different methods: finite element analyses (FEA) of the
mistuned sectors using the commercially available ANSYS finite
element code, and basic FMM ID.
[0121] The finite element calculations serve as a benchmark to
assess the accuracy of the basic FMM ID method. In the benchmark, a
finite element model was made for each mistuned blade. In the model
the blade was attached to a single disk sector. The frequency
change in the mistuned blade/disk sector was then calculated with
various cyclic symmetric boundary conditions applied to the disk.
It was found that the phase angle of the cyclic symmetric
constraint had little effect on the frequency change caused by
blade mistuning. The values described herein were for a disk phase
constraint of 90 degrees, i.e., for the five nodal diameter
mode.
[0122] A finite element model of the full, mistuned bladed disk was
also constructed and used to compute its mistuned modes and natural
frequencies. The modes and frequencies were used as input data for
basic FMM ID. In another embodiment, the mistuned modes and
frequencies may be obtained through a modal fit of the rotor's
frequency response functions. Typically, the measurements may not
detect modes that have a node point at the excitation source. To
reflect this phenomenon, all mistuned modes that had a small
response at blade one were eliminated. This left 16 modes and
natural frequencies to apply to basic FMM ID.
[0123] The mistuned modes and frequencies were combined with the
tuned system frequencies of the fundamental mode family to
construct the basic FMM ID equations (31). These equations were
solved using a least squares fit. The solution was then converted
to the physical sector frequency deviations through the inverse
transform given in equation (34).
[0124] FIG. 16 shows the comparison between the sector mistuning
calculated directly by finite element simulations of each mistuned
blade/sector and the mistuning identified by basic FMM ID. As is
seen from FIG. 16, the two results are in good agreement.
[0125] 2.1.3.2 Stagger Angle Mistuning
[0126] One of the differences between basic FMM ID and other
mistuning identification methods is its measure of mistuning. Basic
FMM ID uses a frequency quantity that characterized the mistuning
of an entire blade-disk sector, whereas other methods in the
literature consider mistuning to be confined to the blades as can
be seen, for example, from the discussion in Judge, J. A., Pierre,
C., and Ceccio, S. L., 2002, "Mistuning Identification in Bladed
Disks," Proceedings of the International Conference on Structural
Dynamics Modeling, Madeira Island, Portugal. The sector frequency
approach used by FMM not only identifies the mistuning in the
blades, but it also captures the mistuning in the disk and the
blade-disk interface. To highlight this capability, the following
example considers a rotor in which the blades are identical except
they are mounted on the disk with slightly different stagger
angles. FIG. 17 schematically illustrates a rotor 29 with
exaggerated stagger angle variations as viewed from above.
[0127] In case of the compressor 26 in FIG. 14, its rotor was
mistuned by randomly altering the stagger angle of each blade with
a maximum variation of .+-.4.degree.. Otherwise the blades were
identical. The modes of the system were then calculated using the
ANSYS.RTM. finite element code. FIG. 18 shows a representative
mistuned mode caused by stagger angle mistuning of the rotor in
FIG. 14. It is seen in FIG. 18 that the mode was localized (in the
higher blade number region), indicating that varying the stagger
angles does indeed mistune the system.
[0128] The mistuned modes and frequencies calculated by ANSYS.RTM.
software were then used to perform a basic FMM ID analysis of the
mistuning. The resulting sector frequency deviations are plotted as
the solid line in FIG. 19, which illustrates a comparison of
mistuning determination from basic FMM ID and the variations in the
stagger angles. The circles in FIG. 19 correspond to the stagger
angle variations applied to each blade. The vertical axes in FIG.
19 have been scaled so that the maximum frequency and angle
variation data points (blade 14) are coincident. This was done to
highlight the fact that the stagger angle variations are
proportional to the sector frequency deviations detected by basic
FMM ID. Thus, not only can basic FMM ID substantially accurately
detect mistuning in the blades, as illustrated in the previous
example, but it can also substantially accurately detect other
forms of mistuning such as variation in the blade stagger
angle.
[0129] 2.2 Advanced FMM ID Method
[0130] As discussed before, the basic FMM ID method provides an
effective means of determining the mistuning in an IBR. The basic
FMM ID technique requires a set of simple vibration measurements
and the natural frequencies of the tuned system. However, at times
neither the tuned system frequencies nor a finite element model
from which to obtain them are available to determine an IBR's
mistuning. Furthermore, even if a finite element model is
available, there is often concern as to how accurately the model
represents the actual rotor. Therefore, the following describes an
alternative FMM ID method (advanced FMM ID) that does not require
any analytical data. Advanced FMM ID requires only a limited number
of mistuned modes and frequency measurements to determine a bladed
disk's mistuning. Furthermore, the advanced FMM ID method also
identifies the bladed disk's tuned system frequencies. Thus,
advanced FMM ID not only serves as a method of identifying
mistuning of the system, but can also provide a method of
corroborating the finite element model of the tuned system
[0131] 2.2.1 Advanced FMM ID Theory
[0132] Advanced FMM ID may be derived from the basic FMM ID
equations. Recall that a step in the development of the basic FMM
ID theory was to transform the mistuning matrix .OMEGA. into a
vector form. Once the mistuning was expressed as a vector, it could
then be calculated using standard methods from linear algebra. A
similar approach is used below to solve for the tuned system
frequencies. However, the resulting equations are nonlinear, and
require a more sophisticated solution approach.
[0133] 2.2.1.1 Development of Nonlinear Equations
[0134] Consider the basic FMM ID equation (30). Moving the
.OMEGA..degree..sup.2 term to the left-hand side of the equation,
the expression becomes .OMEGA..degree..sup.2{right arrow over
(.beta.)}.sub.j2.OMEGA..degree..GAMMA..sub.j {right arrow over
(.omega.)}=.omega..sub.j.sup.2{right arrow over (.beta.)}.sub.j
(36) It is assumed that from measurement of the mistuned modes and
frequencies, {right arrow over (.beta.)}.sub.j and .omega..sub.j in
equation (36) are known. All other quantities are unknown. It is
noted that although .GAMMA..sub.j is not known, the matrix contains
elements from {right arrow over (.beta.)}.sub.j. Therefore, some
knowledge of the matrix is available.
[0135] After some algebra, one can show that the term
.OMEGA..degree..sup.2{right arrow over (.beta.)}.sub.j in equation
(36) may be re-expressed as .OMEGA..degree..sup.2{right arrow over
(.beta.)}.sub.j=B.sub.j{right arrow over (.lamda.)}.degree. (37)
where {right arrow over (.lamda.)}.degree. is a vector of the tuned
frequencies squared, and B.sub.j is a matrix composed from the
elements of {right arrow over (.mu.)}.sub.j. If .eta. is defined to
be the maximum number of nodal diameters on the rotor, i.e.
.eta.=N/2 if N is even or (N-1)/2 if N is odd, then {right arrow
over (.lamda.)}.degree. is given by .lamda. -> .times. .degree.
= [ .omega. 0 .times. .times. ND .smallcircle. 2 .omega. 1 .times.
.times. ND .smallcircle. 2 .omega. .eta. .times. .times. ND
.smallcircle. 2 ] ( 38 ) ##EQU18## For N even, the matrix B.sub.j
has the form B j = [ .beta. j .times. .times. 0 .beta. j .times.
.times. 1 .beta. j .times. .times. 2 .beta. jm .times. .beta. j
.times. .times. 2 .beta. j .times. .times. 1 ] ( 39 ) ##EQU19## A
similar expression can be derived for N odd.
[0136] Substituting equation (37) into (36) and regrouping the
left-hand side results in a matrix equation for the tuned
frequencies squared and the sector mistuning, [ B j 2 .times.
.OMEGA. .smallcircle. .times. .GAMMA. j ] .function. [ .lamda.
-> .omega. _ -> ] = .omega. j 2 .times. .beta. -> j ( 40 )
##EQU20## Equation (40) contains information from only one of the M
measured modes and frequencies. However, equation (40) can be
constructed for each measured mode, and combined into the single
matrix expression [ B 1 2 .times. .OMEGA. .smallcircle. .times.
.GAMMA. 1 B 2 2 .times. .OMEGA. .smallcircle. .times. .GAMMA. 2 B M
2 .times. .OMEGA. .smallcircle. .times. .GAMMA. M ] .function. [
.lamda. -> .omega. _ -> ] = [ .omega. 1 2 .times. .beta.
-> 1 .omega. 2 2 .times. .beta. -> 2 .omega. M 2 .times.
.beta. -> M ] ( 41 ) ##EQU21## Thus, equation (41) represents a
single expression that incorporates all of the measured data. For
brevity, equation (41) is rewritten as [ B ~ 2 .times. ( .OMEGA.
.smallcircle. .times. .GAMMA. ~ ) ] .function. [ .lamda. ->
.omega. _ -> ] = r -> ' ~ ( 42 ) ##EQU22## where {tilde over
(B)} is the stacked matrix of B.sub.j, the term () is the stacked
matrix of .OMEGA..degree..GAMMA..sub.j, and {right arrow over (r)}
is the right-hand side of equation (41).
[0137] An additional constraint equation may be required because
the equations (42) are underdetermined. To understand the cause of
this indeterminacy, consider a rotor in which each sector is
mistuned the same amount. Due to the symmetry of the mistuning, the
rotor's mode shapes will still look tuned, but its frequencies will
be shifted. If one has no prior knowledge of the tuned system
frequencies, there may not be any way to determine that the rotor
has in fact been mistuned. The same difficulty arises in solving
equation (42) because there may not be any way to distinguish
between a mean shift in the mistuning and a corresponding shift in
the tuned system frequencies. To eliminate this ambiguity,
mistuning may be defined so that it has a mean value of zero.
[0138] Mathematically, a zero mean in the mistuning translates to
prescribing the first element of {right arrow over (.omega.)} to be
zero. With the addition of this constraint, equation (42) takes the
form [ B ~ 2 .times. ( .OMEGA. .smallcircle. .times. .GAMMA. ~ ) 0
c -> ] .function. [ .lamda. -> * .omega. _ -> ] = [ r
-> ~ 0 ] ( 43 ) ##EQU23## where {right arrow over (c)} is a row
vector whose first element is 1 and whose remaining elements are
zero.
[0139] 2.2.1.2 Iterative Solution Method
[0140] If the term () in equation (43) were known, then a least
squares solution could be obtained for the tuned eigenvalues {right
arrow over (.lamda.)}.degree. and the DFT of the sector mistuning
{right arrow over (.omega.)}. However, because () is based in part
on the unknown quantities {right arrow over (.lamda.)}.degree., the
equations in (43) are nonlinear. Therefore, an alternative solution
method may be devised. In a solution described below, an iterative
approach is used to solve the equations in (43).
[0141] In iterative form, the least squares solution to equation
(43) can be written as [ .lamda. -> * .omega. _ -> ] ( k ) =
Lsq .times. { [ B ~ 2 .times. ( .OMEGA. .smallcircle. .times.
.GAMMA. ~ ) ( k - 1 ) 0 c -> ] , [ r -> ~ 0 ] } ( 44 )
##EQU24## where the subscripts indicate the iteration number. For
each iteration, a new matrix () may be constructed based on the
previous iteration's solution for {right arrow over
(.lamda.)}.degree.. This process may be repeated until a converged
solution is obtained. With a good initial guess, this method may
typically converge within a few iterations.
[0142] To identify a good initial guess, in case of analyzing an
isolated family of modes, it is observed that generally the
frequencies of isolated mode families tend to span a fairly small
range. Therefore, one good initial guess is to take all of the
tuned frequencies to be equal to one another, and assigned the
value of the mean tuned frequency, i.e. {right arrow over
(.lamda.)}.degree..sub.(0)=.omega..degree..sub.avg.sup.2 (45)
However, the value of .omega..degree..sub.avg is not known and
therefore cannot be directly applied to equation (44).
Consequently, equation (43) may be slightly modified to incorporate
the initial guess defined by equation (45). In equation (43), if
the tuned frequencies are taken to be equal to
.omega..degree..sub.avg, then the term () may be expressed as (
.OMEGA. .smallcircle. .times. .GAMMA. ~ ) = .omega. avg
.smallcircle. .times. .GAMMA. ~ ( 46 ) ##EQU25## where {tilde over
(.GAMMA.)} is the matrix formed by vertically stacking the M
.GAMMA..sub.j matrices.
[0143] The matrix .GAMMA..sub.j is also related to the tuned
frequencies. As a result, the elements of each matrix .GAMMA..sub.j
simplify to the form .omega..degree..sub.avg.beta..sub.jn. This
allows one to rewrite .GAMMA..sub.j as
.GAMMA..sub.j=.omega..degree..sub.avgZ.sub.j (47) where Z.sub.j is
composed of the elements .beta..sub.jn arranged in the same pattern
as the .gamma..sub.jn elements shown in equation (29). Thus,
consolidating all .omega..degree..sub.avg terms, equation (46) can
be written as ( .OMEGA. .smallcircle. .times. .GAMMA. ~ ) = .omega.
avg .smallcircle. 2 .times. Z ~ ( 48 ) ##EQU26## where {tilde over
(Z)} is the stacked form of the Z.sub.j matrices.
[0144] Substituting equation (48) into equation (43) and regrouping
terms results in the expression [ B ~ 2 .times. Z ~ 0 c -> ]
.function. [ .lamda. -> * .omega. avg .smallcircle. 2 .times.
.omega. _ -> ] = [ r -> ~ 0 ] ( 49 ) ##EQU27## Note that the
.omega..degree..sub.avg.sup.2 term was grouped with the vector
{right arrow over (.omega.)}. Thus, all the unknown expressions are
consolidated into the single vector on the left-hand side of
equation (49). These quantities can be solved through a least
squared fit of the equations. This represents the 0.sup.th
iteration of the solution process. The {right arrow over
(.lamda.)}.degree. terms of the solution may then be used as an
initial guess for the first iteration of equation (44).
[0145] In practice, the mistuned modes and frequencies may be
measured using the technique described for basic FMM ID in Section
2.1.3.2. The next section presents a numerical example that
demonstrates the ability of the advanced FMM ID method to identify
the frequencies of the tuned system as well as mistuned sector
frequencies.
[0146] 2.2.2 Numerical Test Case for Advanced FMM ID
[0147] This section presents a numerical example of the advanced
FMM ID method that identifies the tuned system frequencies as well
as the mistuning. This example uses the same geometrically mistuned
compressor model 26 (FIG. 14) as that used for the basic FMM ID
method. The tuned system frequencies and sector mistuning
identified by advanced FMM ID are then compared with finite element
results.
[0148] The modes and natural frequencies of the mistuned bladed
disk were calculated using a finite element model of the mistuned
system. The physical modes were then converted to vectors of modal
weighting factors, {right arrow over (.beta.)}, through equation
(35). The weighting factors were used to form the elements of
equation (49) which was solved to obtain an initial estimate of the
tuned system frequencies. This initial estimate was used as an
initial guess to iteratively solve equation (44). The solution
vector contained two parts: a vector of the tuned system
frequencies squared, and a vector of the DFT of the sector
frequency deviations. The sector mistuning was converted to the
physical domain using the inverse transform in equation (34).
[0149] The resulting sector frequency deviations were compared with
the benchmark finite element analysis (FEA) values. FIG. 20 depicts
a comparison of mistuning predicted using advanced FMM ID with that
obtained using the finite element analysis (FEA). The results in
FIG. 20 were obtained using the same procedure as that discussed in
section 2.1.3 above. FIG. 21 shows a comparison of the tuned
frequencies identified by advanced FMM ID and those computed
directly with the finite element model (i.e., FEA). In each of
FIGS. 20 and 21, the results obtained using advanced FMM ID were in
good agreement with those from FEA.
[3] SYSTEM IDENTIFICATION: APPLICATION
[0150] FIG. 22 illustrates an exemplary setup 32 to measure
transfer functions of test rotors and also to verify various FMM ID
methods discussed hereinbefore. As discussed earlier, the advanced
FMM ID method uses the measurements of the mistuned rotor's system
modes and natural frequencies. The term "system mode" is used
herein to refer to the tip displacement of each blade as a function
of blade's angular position. The system modes may be obtained using
a standard modal analysis approach: measure the bladed disk's
transfer functions, and then curve-fit the transfer functions to
obtain the mistuned modes and natural frequencies. The setup 32 in
FIG. 22 may be used to perform standard transfer function
measurements. As illustrated in FIG. 22, the rotor to be tested
(rotor 34) may be placed on a foam pad 36 to approximate a free
boundary condition. Then, one of the rotor blades may be excited
over the frequency range of interest using an excitation source 38
(for example, a function generator coupled to an audio amplifier
and loudspeaker) and the response of each blade may be measured
with a laser vibrometer 40 coupled to a spectrum analyzer 42, which
can be used to analyze the output of the laser vibrometer 40 to
determine the transfer function. The devices 38, 40, and 42 may be
obtained from any commercially available sources as is known in the
art. For example, the companies that make the function generator
and spectrum analyzer include Hewlett-Packard, Agilent, and
Tektronix. The laser vibrometer may be a Polytec or Ometron
vibrometer.
[0151] All of the devices 38, 40, 42 used in the test setup 32 are
shown coupled (directly or indirectly through another device) to a
computer 44, which may be used to operate the devices as well as to
analyze various data received from the devices. The computer 44 may
also store the FMM software 46, which can include software to
implement any or all of the FMM ID methods. It is understood by one
skilled in the art that the FMM software module 46 may be stored on
an external magnetic, electro-magnetic or optical data storage
medium (not shown) such as, for example, a compact disc, an optical
disk, a floppy diskette, etc. The data storage medium may then be
supplied to the appropriate reader unit in the computer 44 or
attached to the computer 44 to read the content of the data storage
medium and supply the FMM software to the computer 44 for
execution. Alternatively, the FMM software module 46 may reside in
the computer's internal memory such as, for example, a hard disk
drive (HDD) from which it can be executed by the computer's
operating system. It is apparent to one skilled in the art that the
computer 44 may be any computing unit including, for example, a
stand-alone or networked IBM-PC compatible computer, a computing
work station, etc.
[0152] It is noted here that for the sake of convenience and
brevity the following discussion uses the term "FMM ID" to refer to
any of the basic as well as the advanced FMM ID methods without
specifically identifying each one. However, based on the context of
the discussion and the discussion presented hereinbefore, it would
not be difficult for one skilled in the art to comprehend which one
of the two FMM ID methods is being referred to in the
discussion.
[0153] To investigate applicability of FMM ID methods to real
experimental data from actual hardware, the methods were applied to
a pair of transonic compressors whose corresponding test rotors
were designated as SN-1 and SN-3. A single blade/disk sector finite
element model of the tuned compressor was provided by Pratt and
Whitney. By solving this model with free boundary conditions at the
hub and various cyclic symmetric boundary conditions on the radial
boundaries of the disk, a nodal diameter map of the tuned rotor was
generated as illustrated in FIG. 23. The free boundary conditions
at the hub represented the boundary conditions in the experiment:
an IBR supported by a soft foam pad and is otherwise unconstrained.
In FIG. 23, each of the first two families of modes (designated by
reference numerals 50 and 52) have isolated frequencies. These
correspond to first bending and first torsion modes, respectively.
Because FMM ID is equally applicable for isolated families of
modes, both the first bending and first torsion modes were suitable
candidates for system identification analysis.
[0154] FIG. 24 illustrates a typical transfer function from
compressor SN-1 obtained using the test setup 32 shown in FIG. 22.
Note that due to the high modal density, it was necessary to
measure the compressor frequency response with a very high
frequency resolution. This process was repeated for both
compressors over two frequency bands to capture the response of
both the first bending and first torsion modes. The commercially
available MODENT modal analysis package was then used to curve-fit
the transfer functions. This resulted in measurements of the
mistuned first bending and torsion modes of each rotor, along with
their natural frequencies. Because the blade that was excited was
at a low response point in some modes, two or three of the modes in
each family were not measurable. In any event, the measured
mistuned modes and natural frequencies were used to demonstrate the
applicability of FMM ID to actual hardware.
[0155] 3.1 FMM ID Results
[0156] The measured modes and frequencies were used to test both
forms of the FMM ID method. The basic and advanced FMM ID methods
were applied to each rotor, for both the first bending and torsion
families of modes. The tuned frequencies required by basic FMM ID
were the same as those depicted in FIG. 23. To assess the accuracy
of FMM ID, the results were compared to benchmark data.
[0157] 3.1.1 Benchmark Measure of Mistuning
[0158] To assess the accuracy of the FMM ID method, the results
must be compared to benchmark data. However, because the test
rotors were integrally bladed, their mistuning could not be
measured directly. Therefore, an indirect approach was used to
obtain the benchmark mistuning. Pratt and Whitney personnel
carefully measured the geometry of each blade on the two rotors and
calculated the frequencies that it would have if it were clamped at
its root. Because each blade had a slightly different geometry, it
also had slightly different frequencies. Thus, the variations in
the blade frequencies caused by geometric variations were
determined. This data was put in a form that could be compared with
the values identified by FMM ID. First, the frequency variations as
a fraction of the mean were calculated so that the deviation in the
blade frequencies could be determined. These in turn were related
to the sector frequency deviations determined by FMM ID. For modes
with most of their strain energy in the blade, sector frequency
deviations can be obtained from blade frequency mistuning by simple
scaling, i.e.
.DELTA..omega..sub..psi.=.alpha.(.DELTA..omega..sub.b) (50) where
.alpha. is the fraction of strain energy in the blade for the
average nodal diameter mode.
[0159] 3.1.2 FMM ID Results for Bending Modes
SN-1 Results
[0160] The measured mistuned modes and natural frequencies for the
compressor SN-1 were used as input to both versions of FMM ID. In
the case of basic FMM ID, the tuned system frequencies of the first
bending family from FIG. 23 were also used as input. FIG. 25
illustrates a comparison of mistuning from each FMM ID method with
benchmark results for a test rotor SN-1. FIG. 25 thus shows the
sector frequency deviations identified by each FMM ID method along
with the benchmark results. Both FMM ID methods were in good
agreement with the benchmark. This may imply that the mistuning was
predominantly caused by geometric variations and that the
variations were accurately captured by Pratt and Whitney.
[0161] To make the comparisons easier, all mistuning in FIG. 25 was
plotted as the variation from a zero mean. However, it is noted
that rotor SN-1 had a mean frequency 1.3% higher than that of the
tuned finite element model. This DC shift was detected by basic FMM
ID as a constant amount of mistuning added to each blade's
frequency. However, because the advanced FMM ID formulation does
not incorporate the tuned finite element frequencies, it had no way
to distinguish between a mean shift in the mistuning and a
corresponding shift in the tuned system frequencies. Therefore, in
advanced FMM ID, mistuning was defined to have a zero mean, and
then a corresponding set of tuned frequencies was inferred.
[0162] FIG. 26 shows a comparison of tuned system frequencies for
the test rotor SN-1 from advanced FMM ID (i.e., identified by
advanced FMM ID) and the finite element model (FO) using ANSYS.RTM.
software. It is seen from FIG. 26 that the FMM ID frequencies were
approximately 17 Hz higher than the finite element values. This
corresponds to a 1.3% shift in the mean of the tuned system
frequencies that compensated for fact that the blade mistuning now
had a zero mean. To facilitate the comparison of the finite element
and FMM ID results, the mean shift was subtracted and then the
results were then plotted as circles on FIG. 26. After this
adjustment, it is seen that the distribution of the tuned
frequencies determined by FMM ID agreed quite well with the values
calculated from the finite element model. Advanced FMM ID
additionally identified the fact that SN-1 had slightly higher
average frequencies than the FEM model--a fact that could be
important in establishing frequency margins for the stage.
[0163] It is observed from the sector frequency deviations of SN-1
shown in FIG. 25 that the mistuning varied from blade-to-blade in a
regular pattern. The decreasing pattern of mistuning and the jump
in the pattern may suggest that the mistuning might have been
caused by tool wear during the machining process and that an
adjustment in the process was made during blade manufacturing.
SN-3 Results
[0164] The basic and advanced FMM ID methods were then applied in a
similar manner to rotor SN-3's family of first bending modes. The
identified mistuning and tuned system frequencies are shown in
FIGS. 27 and 28, respectively. For comparison purposes, the
mistuning was again plotted with a zero mean, and a corresponding
mean shift was subtracted from the predicted tuned system
frequencies. The predictions for rotor SN-3 from both FMM ID
methods were also in good agreement with the benchmark results. It
is noted that in FIG. 27, the blades were numbered so that blade-1
corresponded to the high frequency sector. A similar numbering
scheme (not illustrated here) was also implemented for SN-1 for
comparison.
[0165] 3.1.3 FMM ID Results for Torsion Modes
[0166] In this section, FMM ID's ability to identify mistuning in
the first torsion modes is examined. For brevity, only the results
for advanced FMM ID are presented. Advanced FMM ID was applied to
each test rotor's family of torsion modes. FIGS. 29 (a) and (b)
show a comparison, for rotors SN-1 and SN-3 respectively, of the
mistuning identified by FMM ID with the values from benchmark
results obtained by Pratt & Whitney from geometric
measurements. The agreement between FMM ID and benchmark results is
good. In FIG. 29, the blades were numbered in the same order as in
FIG. 27, which represents the numbering for SN-3 but, although not
shown, a similar numbering for SN-1 was also employed. Thus, the
mistuning patterns in the torsion modes looked very similar to
those observed for the bending modes, e.g., the blades with the
highest and lowest frequencies were the same for both sets of
modes. This suggests that the mistuning in SN-1 and SN-3 systems
might have been caused by relatively uniform thickness variations
in the blades because such mistuning would affect the frequencies
of both types of modes in a very similar manner.
[0167] In addition to identifying the mistuning in these rotors,
advanced FMM ID also simultaneously inferred the tuned system
frequencies of the system's torsion modes, as shown in FIG. 30,
which illustrates a comparison of tuned system frequencies from
advanced FMM ID and ANSYS.RTM. software for torsion modes of rotors
SN-1 and SN-3. Thus, FMM ID worked well on both the torsion and
bending modes of the test compressors.
[0168] 3.2 Forced Response Prediction
[0169] The mistuning identified in section 3.1 was used to predict
the forced response of the test compressors (SN-1, SN-3) to a
traveling wave excitation. The results were compared with benchmark
measurements done by Pratt & Whitney.
[0170] Pratt and Whitney has developed an experimental capability
for simulating traveling wave excitation in stationary rotors.
Their technique was applied to SN-1 to measure its first bending
family's response to a 3E excitation (third engine order
excitation). The response of SN-1 was then predicted using FMM ID
methods. To make the prediction, the mistuning and tuned system
frequencies identified by advanced FMM ID (as discussed in section
3.1) were input to the FMM reduced order model discussed
hereinabove under part [1]. FMM calculated the system's mistuned
modes and natural frequencies. Then, modal summation was used to
calculate the response to a 3E excitation. The modal damping used
in the summation was calculated from the half-power bandwidth of
the transfer function peaks.
[0171] FIG. 31(a) depicts FMM-based forced response data, whereas
FIG. 31(1b) depicts the experimental forced response data. Thus,
the plots in FIG. 31 show the comparison of the benchmark forced
response results with that predicted by FMM. For clarity, only the
envelope of the blade response is shown in FIGS. 31 (a), (b). Also,
the plots in FIG. 31 have been normalized so that the maximum
response is equal to one. In general, the two curves in FIG. 31
agree reasonably well. To observe how well the response of
individual blades was predicted, the relative responses of the
blades at two resonant peaks were compared. The peaks are labeled
{circle around (1)} and {circle around (2)} in FIG. 31(a). FIGS.
32(a) and (b) respectively show relative blade amplitudes at forced
response resonance for the resonant peaks labeled {circle around
(1)} and {circle around (2)} in FIG. 31 (a). The relative amplitude
of each blade as determined by FMM and experimental methods is
plotted for both resonant peaks in FIG. 32. The agreement between
FMM and experimental predictions was reasonably good. Thus, the FMM
based method may be used to not only capture the overall shape of
the response, but also to determine the relative amplitudes of the
blades at the various resonances.
[0172] 3.3 Cause and Implications of Repeated Mistuning Pattern
[0173] The mistuning in bladed disks is generally considered to be
a random phenomenon. However, it is seen from the discussion in
section 3.1 that both test rotors SN-1 and SN-2 had very similar
mistuning patterns that were far from random. If such repeated
mistuning matters are found to be common among IBRS, it may have
broad implications on the predictability of these systems.
[0174] 3.3.1 Cause of Repeated Mistuning
[0175] The similarity between the mistuning patterns identified in
SN-1 and SN-3 is highly suggestive that the mistuning was caused by
a consistent manufacturing effect. In addition, it was observed
that the mistuning in the torsion modes followed the same trends as
in the bending modes. Thus, the dominant form of mistuning may most
likely be caused by relatively uniform blade-to-blade thickness
variations. Blade thickness variations may be analyzed using
geometry measurements of a rotor to extract the thickness of each
blade at different points across the airfoil. Then, a calculation
may be performed to determine how much each point's thickness
deviated from the average values of all corresponding points. The
results can be expressed as a percentage variation from the mean
blade thickness. It was found that a 2% change in blade thickness,
produced about a 1% change in corresponding sector frequency, which
is consistent with beam theory for a beam of curved
cross-section.
[0176] It is observed that tool wear may cause blade thickness
variations. For example, if the blades were machined in descending
order from blade 18 to blade 1 (e.g., the 18 blades in rotor SN-1),
then, due to tool wear, each subsequent blade would be slightly
larger than the previous one. This effect would cause the sector
frequencies to monotonically increase around the wheel. Any
frequency jump or discontinuity observed (e.g., the jump at blade
15 in FIG. 25) may be the result of a tool adjustment made during
the machining process.
[0177] 3.3.2 Implications of Repeated Mistuning
[0178] The repeating mistuning patterns caused by machining effects
may allow prediction of the response of a fleet (e.g., of
compressors) through probabilistic methods. For example, consider
an entire fleet of the transonic compressors, two of which--SN-1
and SN-2--were discussed hereinbefore. If it is incorrectly assumed
that the mistuning in these rotors was completely random, then one
would estimate that the sector frequency deviation of each sector
has a mean of zero and a standard deviation of about 2%. Assuming
these variations, FMM was used to perform 10,000 Monte Carlo
simulations to represent how a fleet of engines would respond to a
3E excitation. The data from Monte Carlo simulations was used to
compute the cumulative probability function (CPF) of the maximum
blade amplitude on each compressor in the fleet. FIG. 33 depicts
cumulative probability function plots of peak blade amplitude for a
nominally tuned and nominally mistuned compressor. The CPF of a
fleet of engines with random mistuning had a standard deviation of
2% as shown by the dashed line in FIG. 33. It is observed from FIG.
33 that the maximum amplitude varied widely across the fleet,
ranging in magnification from 1.1 to 2.5.
[0179] However, the test rotors were in fact nominally mistuned
with a small random variation about the nominal pattern. Because
the random variation was much smaller than that considered above,
the fleet's response was more predictable. To illustrate this
point, the nominal mistuning pattern (of the fleet of rotors) was
approximated as the mean of the patterns measured for the two test
rotors SN-1 and SN-2. Based on this approximated pattern, it was
found that the sector frequency deviations differed from the
nominal values with a standard deviation of only 0.2%, as shown in
FIG. 34, which shows mean and standard deviations of each sector's
mistuning for a nominally mistuned compressor. Making use of the
fact that the rotors were nominally mistuned, the Monte Carlo
simulations were repeated. The CPF of the maximum amplitude on each
rotor was then calculated. The calculated results were plotted as
the solid line on FIG. 33. It is observed that by accounting for
nominal mistuning, the range of maximum amplitudes is significantly
reduced. Thus, by measuring and making use of nominal mistuning
when it occurs, a test engineer may predictably determine the
fleet's vibratory response behavior from the vibratory response of
a specific IBR that is tested in a spin pit, rig test or
engine.
[4] MISTUNING EXTRAPOLATION FOR ROTATION
[0180] The FMM ID methods presented earlier in part [3] determine
the mistuning in a bladed disk while it is stationary. However,
once the rotor is spinning, centrifugal forces can alter its
effective mistuning. However, an analytical method, discussed
below, may be used for approximating the effect of rotation speed
on mistuning.
[0181] 4.1 Mistuning Extrapolation Theory
[0182] Centrifugal effects cause the sector frequency deviations
present under rotating conditions to differ from their values when
the bladed disk is not rotating. To approximate the effect of
rotational speed on mistuning, a lumped parameter model 54 of a
rotating blade, as shown in FIG. 35, may be considered. The
pendulum 56 mounted on a torsion spring 58 represents the blade,
while the circular region 60 of the system represents a rigid disk.
Thus, the blade is modeled as a pendulum 56 of mass "m" and length
"1" which is mounted to a rigid disk 60 through a torsional spring
"k" 58. The disk 60 has radius "L" and rotates at speed "S".
[0183] It can be shown that the blade's natural frequency in this
system is given by the expression .omega. .function. ( S ) 2 = k ml
2 + L l .times. S 2 ( 51 ) ##EQU28## where S is the rotation speed
in radians/sec, and the notation .omega.(S) indicates the natural
frequency at speed S. Notice that the quantity k/ml.sup.2 is the
natural frequency of the system at rest. Therefore, equation (51)
can be rewritten in the more general form
.omega.(S).sup.2=.omega.(0).sup.2+rS.sup.2 (52) where r is a
constant.
[0184] Take .omega. to be a mistuned frequency in the form
.omega.(S)=.omega..degree.(S)[1+.DELTA..omega.(S)]. Substituting
this expression into equation (52) and keeping only the first order
terms implies
.omega.(S).sup.2.apprxeq..omega..degree.(S).sup.2+2(.DELTA..omeg-
a.(0)).omega..degree.(0).sup.2 (53) where .omega..degree.(S) is the
tuned frequency at speed.
[0185] Taking the square root of expression (53) and again keeping
only the first order terms one obtains an expression for the
mistuned frequency at speed, .omega. .function. ( S ) .apprxeq.
.omega. .smallcircle. .function. ( S ) .times. { 1 + .DELTA.
.times. .times. .omega. .function. ( 0 ) .function. [ .omega.
.smallcircle. .function. ( 0 ) 2 .omega. .smallcircle. .function. (
S ) 2 ] } ( 54 ) ##EQU29## Subtracting and dividing both sides of
the expression (54) by .omega..degree.(S) yields an approximation
for the mistuned frequency ratio at speed, i.e. .DELTA. .times.
.times. .omega. .function. ( S ) .apprxeq. .DELTA. .times. .times.
.omega. .function. ( 0 ) .function. [ .omega..degree. .function. (
0 ) 2 .omega..degree. .function. ( S ) 2 ] ( 55 ) ##EQU30## In the
case of system modes in which the strain energy is primarily in the
blades, the tuned system frequencies tend to increase with speed by
the same percentage as the blade alone frequencies. Therefore,
expression (55) can also be approximated by noting how a frequency
of the tuned system changes with speed, e.g., .DELTA..omega.
.function. ( S ) .apprxeq. .DELTA. .times. .times. .omega.
.function. ( 0 ) .function. [ .omega. .psi. .smallcircle.
.function. ( 0 ) 2 .omega. .psi. .smallcircle. .function. ( S ) 2 ]
( 56 ) ##EQU31## where .omega..degree..sub..psi. is the average
tuned system frequency. Expression (56) may then be used to adjust
the sector frequency deviations measured at rest for use under
rotating conditions.
[0186] 4.2 Numerical Test Cases
[0187] This section presents two numerical tests of the mistuning
extrapolation theory. The first example uses finite element
analysis of the compressor SN-1 discussed hereinbefore (see, for
example, FIG. 22) to assess the accuracy of expression (56). Then,
the second example demonstrates that this result may be combined
with FMM ID and the FMM forced response software to predict the
response of a rotor at speed.
[0188] 4.2.1 Compressor SN-1
[0189] As mentioned earlier, Pratt & Whitney personnel made
careful measurements of each blade's geometry and used this data to
construct accurate finite element models of all 18 airfoils in
SN-1. Thus, these finite element models captured the small
geometric variations from one blade to the next.
[0190] Two of the airfoil models were randomly selected for use in
this test case. For the purpose of this study, the first airfoil
represented the tuned blade geometry, and the second represented a
mistuned blade. Then, both blades were clamped at their root, and
their natural frequencies were calculated using finite element
analysis (FEA) software ANSYS.RTM.. The values were obtained for
the first three modes corresponding to first bending, first
torsion, and second bending respectively. The calculations were
then repeated with the addition of rotational velocity loads to
simulate centrifugal effects. Through this approach, the natural
frequencies of both blades were obtained at five rotation speeds
ranging from 0 to 20,000 RPM.
[0191] Next, the frequency deviation of the mistuned blade was
calculated by subtracting the tuned frequencies from the mistuned
values, and then dividing each result by its corresponding tuned
frequency. FIG. 36 shows a comparison of mistuning values
analytically extrapolated to speed with an FEA (finite element
analysis) benchmark. The results plotted as lines in FIG. 36
represent benchmark values on which to assess the accuracy of the
analytical mistuning extrapolation method. Using expression (55),
the frequency deviations calculated for the stationary rotor were
extrapolated to the same rotational conditions considered in the
benchmark calculation. The extrapolated results are shown as
circles on FIG. 36. The agreement between extrapolated results and
the results using the FEA benchmark were good for all three modes.
Thus, expression (55) may be used to analytically extrapolate blade
frequency deviations to rotating conditions. It is noted that
expression (55) was used here rather than expression (56) because
the calculated natural frequencies represented an isolated blade
and not a blade/disk sector. However, for the cases where FMM is
applicable, a blade-alone frequency differs from an average sector
frequency by a multiplicative constant. Thus, expression (56) may
also be suitable for mistuning extrapolation.
[0192] 4.2.2 Response Prediction at Speed
[0193] This section uses a numerical test case that shows how FMM
ID, expression (56), and the FMM forced response software can be
combined to predict the response of a bladed disk under rotating
conditions.
[0194] The geometrically mistuned rotor illustrated in FIG. 14 had
a 6.sup.th engine order crossing with the first bending modes at a
rotational speed of 20,000 RPM. However, to create a more severe
test case, it is assumed that the crossing occurred at 40,000
RPM.
[0195] To use FMM to predict the rotor's forced response at this
speed (40,000 RPM), the FMM prediction software must be provided
with the bladed disk's tuned system frequencies and the sector
frequency deviations that are present at 40,000 RPM. As part of the
discussion in section (2.1.3.1) above, these two sets of parameters
were determined for at-rest condition using ANSYS and basic FMM ID
respectively. However, because both of these properties change with
rotation speed, they must first be adjusted to reflect their values
at 40,000 RPM.
[0196] To adjust the tuned system frequencies for higher rotational
speed, tuned system frequencies were recalculated in ANSYS.RTM.
software using the centrifugal load option to simulate rotational
effects. FIG. 37 illustrates the effect of centrifugal stiffening
on tuned system frequencies. As shown in FIG. 37, the centrifugal
stiffening caused the tuned system frequencies to increase by about
30%. Then, the change in the five nodal diameter, tuned system
frequency and expression (56) were used to analytically extrapolate
the sector frequency deviations to 40,000 RPM. The adjusted
mistuning, along with the original mistuning values identified at
rest, are plotted in FIG. 38, which illustrates the effect of
centrifugal stiffening on mistuning. It is seen from FIG. 38 that
the centrifugal loading reduces the mistuning ratios by 40%.
[0197] The adjusted parameters were then used with the FMM forced
response software to calculate the rotor's response to a 6E
excitation using the method described hereinabove in parts [1] and
[2]. As a benchmark, the forced response was also calculated
directly in ANSYS.RTM. software using a full 360.degree. mistuned
finite element model. Tracking plots of the FMM and ANSYS.RTM.
software results are shown in FIG. 39, which depicts frequency
response of blades to a six engine order excitation at 40,000 RPM
rotational speed. For clarity, the response of only three blades is
shown in FIG. 39: the high responding blade, the median responding
blade, and the low responding blade. It is observed from FIG. 39
that each blade's peak amplitude and the shape of its overall
response as predicted by FMM agree well with the benchmark results.
Thus, FMM ID, the mistuning extrapolation equation, and FMM may be
combined to identify the mistuning of a rotor at rest, and use the
mistuning to predict the system's forced response under rotating
conditions.
[5] SYSTEM IDENTIFICATION FROM TRAVELING WAVE RESPONSE
MEASUREMENTS
[0198] Traditionally, mistuning in rotors with attachable blades is
measured by mounting each blade in a broach block and measuring its
natural frequency. The difference of each blade's frequency from
the mean value is then taken as a measure of its mistuning.
However, this method cannot be applied to integrally bladed rotors
(IBRs) whose blades cannot be removed for individual testing. In
contrast, FMM ID system identification techniques rely on
measurements of the bladed disk system as a whole, and are thus
well suited to IBRs.
[0199] FMM ID may also be used for determining the mistuning in
conventional bladed disks. Even when applied to bladed disks with
conventionally attached blades, the traditional broach block method
of mistuning identification is limited. In particular, it does not
take into account the fact that the mistuning measured in the
broach block may be significantly different from the mistuning that
occurs when the blades are mounted on the disk. This variation can
arise because each blade's frequency is dependent on the contact
conditions at the attachment. In the engine, the attachment is
loaded by centrifugal force from the blade which provides a
different contact condition than the clamping action used in broach
block tests. This difference is accentuated in multi-tooth
attachments because different teeth may come in contact depending
on how the attachment load is applied. In addition, the contact in
multi-tooth attachments may be sensitive to manufacturing
variations and, consequently, vary from one location to the next on
the disk. The discussion given below addresses these issues by
devising a method of system identification that can be used to
directly determine mistuning while the stage is rotating, and can
also identify mistuning from the response of the entire system
because the blades are inherently coupled under rotating
conditions. The method discussed below provides an approach for
extracting the mistuned modes and natural frequencies of the bladed
disk under rotating conditions from its response to naturally
occurring, engine order excitations. The method is a coordinate
transformation that makes traveling wave response data compatible
with the existing, proven modal analysis algorithms. Once the
mistuned modes and natural frequencies are known, they can be used
as input to FMM ID methods.
[0200] 5.1 Theory
[0201] Both of the FMM ID mistuning identification methods require
the mistuned modes and natural frequencies of the bladed disk as
input. Under stationary conditions, they can be determined by
measuring the transfer functions of the system and using standard
modal analysis procedures. One way of measuring the transfer
functions is to excite a single point (e.g., on a blade) with a
known excitation and measure the frequency response of all of the
other points that define the system. However, when the bladed disk
is subjected to an engine order excitation all of the blades are
simultaneously excited and it may not be clear how the resulting
vibratory response can be related to the transfer functions
typically used for modal identification. As discussed below, if the
blade frequency response data is transformed in a particular manner
then the traveling wave excitation constitutes a point excitation
in the transform space and that standard modal analysis techniques
can then be used to extract the transformed modes. Once the
transformed modes are determined, the physical modes of the system
can be calculated from an inverse transformation.
[0202] 5.1.1 Traditional Modal Analysis
[0203] Standard modal analysis techniques are based on measurements
of a structure's frequency response functions (FRFs). These
frequency response functions are then assembled as a frequency
dependant matrix, H(.omega.), in which the element
H.sub.i,j(.omega.) corresponds to the response of point i to the
excitation of point j as discussed, for example, in Ewins, D. J.,
2000, Modal Testing: Theory, Practice, and Application, Research
Studies Press Ltd., Badlock, UK, Chapter 1. Traditional modal
analysis methods require that one row or column of this frequency
response matrix be measured. In the test cases discussed
hereinbelow the mistuned modes correspond to a single isolated
family of modes. For example, the lower frequency modes such as
first bending and first torsion families often have frequencies
that are relatively isolated. When this is the case the "modes" of
interest may be defined in terms of how the blade displacements
vary from one blade to the next around the wheel and can be
characterized by the response of one point per blade. Thus, the
standard modal analysis experiment may be performed in one of two
ways when measuring the mistuned modes of a bladed disk. First, the
structure's frequency response may be measured at one point on each
blade, while it is excited at only one blade. This would result in
the measurement of a single column of H(.omega.). Alternatively, a
row of H(.omega.) may be obtained by measuring the structure's
response at only one blade and exciting the system at each blade in
turn. In either of these acceptable test configurations, the
structure is excited at only one point at a time. However, in a
traveling wave excitation, all blades are excited simultaneously.
Thus, the response of systems subjected to such multi-point
excitations cannot be directly analyzed by standard SISO (single
input, single output) modal analysis methods.
[0204] 5.1.2 General Multi-Point Excitation Analysis
[0205] As discussed above, a traveling wave excitation is not
directly compatible with standard SISO modal analysis methods.
Further, a traveling wave excites each measurement point with the
same frequency at any given time. The method discussed below may be
applicable to any multi-input system, in which the frequency
profile is consistent from one excitation point to the next;
however, the amplitude and phase of the excitation sources may
freely vary spatially. It is noted that suitable excitation forms
include traveling waves, acoustic pressure fields, and even shakers
when appropriately driven.
[0206] In typical applications, the ij element of the frequency
response matrix H(.omega.) corresponds to the response of point i
to the excitation of point j. However, to analyze frequency
response data from a multi-point excitation, H.sub.i,j(.omega.) may
be viewed in a more general fashion. Thus, in a more general sense,
the i,j element describes the response of the i.sup.th coordinate
to an excitation at j.sup.th coordinate. Although these coordinates
are typically taken to be the displacement at an individual
measurement point, this need not be the case.
[0207] The structure's excitation and response can instead be
transformed into a different coordinate system. For example, an N
degree-of-freedom coordinate system can be defined by a set of N
orthogonal basis vectors which span the space. In this
representation, each basis vector is a coordinate. Thus, to perform
modal analysis on multi-point excitation data, it may be desirable
to select a coordinate system in which the excitation is described
by just one basis vector. Within this newly defined modal analysis
coordinate system, the structure is subjected to only a single
coordinate excitation. Therefore, when the response measurements
are expressed in this same domain, they represent a single column
of the FRF matrix, and can be analyzed by standard SISO modal
analysis techniques. The following section describes how this
approach may be applied to traveling wave excitations.
[0208] 5.1.3 Traveling Wave Modal Analysis
[0209] Consider an N-bladed disk subjected to a traveling wave
excitation. It is assumed that the amplitude and phase of each
blade's response is measured as a function of excitation frequency.
In practice, these measurements may be made under rotating
conditions with a Non-intrusive Stress Measurement System (NSMS),
whereas a laser vibrometer may be used in a stationary bench test.
For simplicity, only consider one measurement point per blade is
considered.
[0210] It is assumed that the blades are excited harmonically by
the force {right arrow over (f)}(.omega.)e.sup.i.omega.t, where the
vector {right arrow over (f)} describes the spatial distribution of
the excitation force. Similarly, the response of each measurement
point is given by h(.omega.)e.sup.i.omega.t. The components of
{right arrow over (f)} and {right arrow over (h)} are complex
because they contain phase as well as magnitude information. It is
this excitation and response data from which modes shapes and
natural frequencies may be extracted. However, for this data to be
compatible with standard SISO modal analysis methods, it must
preferably first be transformed to an appropriate modal analysis
coordinate system.
[0211] As indicated in the immediately preceding section, an
appropriate coordinate system that would allow this to occur is one
in which the spatial distribution of the force, {right arrow over
(f)}, is itself a basis vector. For simplicity, only the phase
difference that occurs from one blade to the next is included in
the equation (57) below. In the case of higher frequency
applications, it may be necessary to also include the spatial
variation of the force over the airfoil if more than one family of
modes interact. The spatial distribution of a traveling wave
excitation has the form: f E = F .smallcircle. .function. [ e 0 e -
I .function. ( 2 .times. .times. g N ) .times. E e - I .function. (
N - 1 ) .times. ( 2 .times. .times. g N ) .times. E ] ( 57 )
##EQU32## where E is the engine order of the excitation. Therefore,
a coordinate system whose basis vectors are the N possible values
of {right arrow over (f)}, corresponding to all N distinct engine
order excitations, 0 through N-1, may be used as a basis. The basis
vectors are complete and orthogonal.
[0212] The vectors {right arrow over (f)} and {right arrow over
(h)} are transformed into this modal analysis coordinate system by
expressing them as a sum of the basis vectors. Denoting the basis
vectors as the set {{right arrow over (b)}.sub.0, {right arrow over
(b)}.sub.1, . . . , {right arrow over (b)}.sub.N-1}, this summation
takes the form, f .fwdarw. = m = 0 N - 1 .times. f _ m .times. b
.fwdarw. m ( 58 .times. .times. a ) h .fwdarw. .function. ( .omega.
) = m = 0 N - 1 .times. h _ .function. ( .omega. ) m .times. b
.fwdarw. m ( 58 .times. .times. b ) ##EQU33## where the
coefficients f.sub.m and h(.omega.).sub.m describe the value of the
m.sup.th coordinate in the modal analysis domain. To identify the
values of these coefficients, orthogonality may be used. This is a
general approach that may be applicable for any orthogonal
coordinate system. However, for the case of traveling wave
excitations, the coordinate transformation may be simplified.
[0213] Consider the n.sup.th element of the vectors in equations
(58). For convenience, let all vector indices run from 0 to N-1.
Thus, these elements may be expressed as, f n = m = 0 N - 1 .times.
f _ m .times. e - I .function. ( 2 .times. .times. g N ) .times. m
.times. .times. n ( 59 .times. .times. a ) ##EQU34## h .function. (
.omega. ) n = m = 0 N - 1 .times. h _ .function. ( .omega. ) m
.times. e - I .function. ( 2 .times. .times. g N ) .times. m
.times. .times. n ( 59 .times. .times. b ) ##EQU35## where the
exponential term is the n.sup.th component of the basis vector
{right arrow over (b)}.sub.m. Equation (59) is the inverse discrete
Fourier Transform (DFT.sup.-1) of f. This relation allows to state
the transformation between physical coordinates and the modal
analysis domain in the simpler form, f .fwdarw. = DFT - 1 .times. {
f _ .fwdarw. } ( 60 .times. .times. a ) h .fwdarw. = DFT - 1
.times. { h _ .fwdarw. } ( 60 .times. .times. b ) ##EQU36## and
conversely, f _ .fwdarw. = DFT .times. { f .fwdarw. } ( 61 .times.
.times. a ) h _ .fwdarw. = DFT .times. { h .fwdarw. } ( 61 .times.
.times. b ) ##EQU37## where DFT is the discrete Fourier Transform
of the vector.
[0214] By applying equation (61), the force and response vectors
may be transformed to the modal analysis coordinate system. Due to
the present selection of basis vectors, the resulting vector {right
arrow over (f)} will contain only one nonzero term that corresponds
to the engine order of the excitation, i.e., a 5E excitation (fifth
engine order excitation) will produce a nonzero term in element 5
of {right arrow over (f)}. This indicates that within the modal
analysis domain, only the E.sup.th coordinate has been excited.
Therefore, {right arrow over (h)}(.omega.) represents column E of
the FRF matrix.
[0215] The transformed response data, {right arrow over
(h)}(.omega.), may now be analyzed using standard SISO modal
analysis algorithms. The resulting modes will also be in the modal
analysis coordinate system, and must be converted back to physical
coordinates though an inverse discrete Fourier transform given, for
example, in equation (60). These identified (mistuned) modes and
natural frequencies may in turn be used as inputs to FMM ID methods
to determine the mistuning of a bladed disk from its response to an
engine order excitation.
[0216] There are two further details of this method. First, for the
purpose of convenience of notation, the indices of all matrices and
vectors are numbered from 0 to N-1. However, most modal analysis
software packages use a numbering convention that starts at 1.
Therefore, an E.sup.th coordinate excitation in the present
notation corresponds to an (E+1).sup.th coordinate excitation in
the standard convention. This must be taken into account when
specifying the "excitation point" in the modal analysis software.
Second, the coordinate transformation described herein is based on
a set of complex basis vectors. Because the modes are extracted in
the modal analysis domain they will be highly complex, even for
lightly damped systems. Thus it may be necessary to use a modal
analysis software package that can properly handle highly complex
mode shapes. In one embodiment, the MODENT Suite by ICATS was used.
Information about MODENT may be obtained from Imregun, M., et al.,
2002, MODENT 2002, ICATS, London, UK, http://www.icats.co.uk.
[0217] 5.2 Experimental Test Cases
[0218] This section presents two experimental test cases of the
traveling wave system identification technique. In the first
example, an integrally bladed fan (IBR) was excited with a
traveling wave while it was in a stationary configuration. Because
the IBR was stationary, it was easier to make very accurate
response measurements using a laser vibrometer. Thus, this example
may serve as a benchmark test of the traveling wave identification
theory. Then, in the second example, the method's effectiveness on
a rotor that is excited in a spin pit under rotating conditions is
explored. The amplitude and phase of the response were measured
using an NSMS system; NSMS is a-non-contacting measurement method
which is commonly used in the gas turbine industry for rotating
tests. The NSMS technology may be used with the traveling wave
system identification technique to determine the IBR's mistuning
from its engine order response.
[0219] 5.2.1 Stationary Benchmark
[0220] An integrally bladed fan was tested using the traveling wave
excitation system at Wright Patterson Air Force Base's Turbine
Engine Fatigue Facility as discussed in Jones K. W., and Cross, C.
J., 2003, "Traveling Wave Excitation System for Bladed Disks,"
Journal of Propulsion and Power, 19(1), pp. 135-141. Because the
facility's test system used an array of phased electromagnets to
generate a traveling wave excitation, the bladed disk remained
stationary during the test. The experiment was performed with the
fan placed on a rubber mat to approximate a free boundary
condition. First, the IBR was intentionally mistuned by fixing a
different mass to the leading edge tip of each blade with wax. The
masses ranged between 0 and 7 g, and were selected randomly. Then,
to obtain a benchmark measure of the mistuned fan's mode shapes, a
standard SISO modal analysis test was performed. Specifically, a
single electromagnet was used to excite one blade over the
frequency range of the first bending modes while the response was
measured at all sixteen (16) blades with a Scanning Laser Doppler
Vibrometer (SLDV). The modes were then extracted from the measured
FRFs using the commercially available MODENT modal analysis
package.
[0221] Next, to validate the traveling wave modal analysis method,
the fan was excited using a 5.sup.th engine order traveling wave
excitation. Again, the response of each blade was measured using
the SLDV. The blade responses to the traveling wave excitation were
transformed using equation (61) and then analyzed with MODENT to
extract the transformed modes. Because MODENT numbers its
coordinates starting at 1 (0E), a 5E excitation corresponds to the
excitation of coordinate 6. Therefore, in the mode extraction
process, it was specified that the excitation was applied at the
6.sup.th coordinate. Lastly, equation (60) was used to transform
the resulting modes back to physical coordinates.
[0222] The modes measured through the traveling wave test were then
compared with those from the benchmark analysis. FIGS. 40(a), (b)
and (c) show a comparison of the representative mode shape
extracted from the traveling wave response data with benchmark
results. FIG. 40 thus shows several representative sets of mode
shape comparisons that range from nearly tuned-looking modes (e.g.,
FIG. 40(a)) to modes that are very localized (e.g., FIG. 40(c)). In
all cases in FIG. 40, the modes from the traveling wave and SISO
benchmark methods agreed quite well. In addition, the natural
frequencies were also accurately identified as can be seen from
FIG. 41, which depicts comparison of the natural frequencies
extracted from the traveling wave response data with the benchmark
results. Thus, the traveling wave modal analysis method may be used
to determine the modes and natural frequencies of a bladed disk
based on its response to a traveling wave excitation.
[0223] It is discussed below that the resulting modes and natural
frequencies can be used with FMM ID methods to identify the
mistuning in the bladed disk. Because most of the mistuning in the
stationary benchmark fan was caused by the attached masses, to a
large extent the mistuning was known. Therefore, these mass values
may be used as a benchmark with which to assess the accuracy of the
FMM ID results.
[0224] Because the mass values are to be used as a benchmark, the
mistuning caused by the masses must be isolated from the inherent
mistuning in the fan. Therefore, a standard SISO modal analysis was
first performed on the rotor fan with the masses removed, and the
resulting modal data was used as input to FMM ID (e.g., advanced
FMM ID). This resulted in an assessment of the IBR's inherent
mistuning, expressed as a percent change in each sector's
frequency.
[0225] Next, an FMM ID analysis was performed of the modes and
frequencies extracted from the traveling wave response of the rotor
with mass-mistuning. The resulting mistuning represented the total
effect of the masses and the IBR's inherent mistuning. To isolate
the mass effect, the rotor's nominal mistuning was subtracted.
Again, the resulting mistuning was expressed as a percent change in
each sector's frequency.
[0226] To compare these mistuning values with the actual masses
placed on the blade tips, each sector frequency change may be first
translated into its corresponding mass. A calibration curve to
relate these two quantities was generated through two independent
methods. First, the calibration was determined through a series of
finite element analyses in which known mass elements were placed on
the tip of a blade, and the finite element model was used to
directly calculate the effect of the mass elements on the
corresponding sector's frequency. It is noted that in this method a
single blade disk sector of the tuned bladed disk with cyclic
symmetric boundary conditions applied to the disk was used.
Further, changing the phase in the cyclic symmetric boundary
condition had only a slight effect on the results (the results
shown in FIG. 42 corresponded to a phase constraint of 90 degrees).
While this method was sufficient in this case, there are often
times when a finite element model is not available. For such cases,
a similar calibration curve can be generated experimentally by
varying the mass on a single blade, and repeating the FMM ID
analysis. This experimental method was performed as an independent
check of the calibration. Both approaches gave very similar
results, as can be seen from the plot in FIG. 42, which shows a
calibration curve relating the effect of a unit mass on a sector's
frequency deviation in a stationary benchmark. For the range of
masses used in this experiment, it was found that mass and sector
frequency change were linearly related as shown in FIG. 42. The
calibration curve of FIG. 42 was then used to translate the
identified sector frequency changes into their corresponding
masses.
[0227] FIG. 43 shows the comparison between the mass mistuning
identified through traveling wave FMM ID with the values of the
actual masses placed on each blade tip (i.e., the input mistuning
values). As can be seen from FIG. 43, the agreement between the
mistuning obtained using the traveling wave system identification
method and the benchmark values is quite good. Thus, by combining
the traveling wave modal analysis method with FMM ID, the mistuning
in a bladed disk from its traveling wave response can be
determined.
[0228] 5.2.2 Rotating Test Case
[0229] In the example in section 5.2.1, the traveling wave modal
analysis method was verified using a stationary benchmark rotor.
However, if the method is to be applicable to conventional bladed
disks, it may be desirable to make response measurements under
rotating conditions. This second test case assesses if the
measurement techniques commonly used in rotating tests are
sufficiently accurate to be used with FMM ID to determine the
mistuning in a bladed disk.
[0230] For this second case, another rotor fan was considered. To
obtain a benchmark measure of the rotor's mistuning in its first
bending modes, an impact hammer and a laser vibrometer were used to
perform a SISO modal analysis test. The resulting modes and natural
frequencies were then used as input to FMM ID to determine the
fan's mistuning.
[0231] Next, the fan was tested in the spin pit facility at NASA
Glenn Research Center. The NASA facility used an array of permanent
magnets to generate an eddy current excitation that drove the
blades. The blade response was then measured with an NSMS system.
For this test, the fan was driven with a 7E excitation, over a
rotational speed range of 1550 to 1850 RPM. The test was performed
twice, at two different acceleration rates. The NSMS signals were
then processed to obtain the amplitude and phase of each blade as a
function of its excitation frequency. FIGS. 44(a) and (b) show
tracking plots of blade amplitudes as a function of excitation
frequency for two different acceleration rates. The NSMS system
measured the amplitude and phase of each blade once per revolution.
Thus, the data taken at the slower acceleration rate (FIG. 44(b))
had a higher frequency resolution than that obtained from the
faster acceleration rate (FIG. 44(a)). However, in both cases, the
data was significantly noisier than the measurements obtained in
the previous example (in section 5.2.1) using an SLDV.
[0232] Next, the traveling wave system identification method was
applied to extract the mode shapes from the response data. First,
the measurements were transformed to the modal analysis domain by
using equation (61), and the mode shapes and natural frequencies
were extracted with MODENT. The extracted modes were then
transformed back to the physical domain through equation (60).
Finally, the resulting modes and frequencies were used as input to
FMM ID to identify the fan's mistuning.
[0233] The mistuning identified from the two spin pit tests was
then compared with the benchmark values. FIGS. 45(a) and (b)
illustrate the comparison of the mistuning determined through the
traveling wave system identification method with benchmark values
for two different acceleration rates. In the case of the faster
acceleration rate (FIG. 45(a)), the trends of the mistuning pattern
were identifiable, but the mistuning values for all blades were not
accurately determinable. The reduced accuracy may be attributed to
difficulty in extracting accurate mode shapes from data with such
coarse frequency resolution. However, the frequency resolution of
the data measured at a slower acceleration rate (FIG. 45(b)) was
three times higher than the case for faster acceleration. Thus,
when FMM ID was applied to this higher resolution data set, the
agreement between the traveling wave based ID and the benchmark
values was significantly improved, as can be seen from FIG. 45(b).
Thus, with adequate frequency resolution, NSMS measurements can be
used to determine the mistuning of a bladed disk under rotating
conditions.
[0234] Thus, NSMS measurements (from traveling wave excitation) may
be used to elicit system mode shapes (blade number vs.
displacement) and natural frequencies. The modes and natural
frequency data may then be input to, for example, advanced FMM ID
to infer frequency mistuning of each blade in a bladed disk and,
thus, to predict the disk's forced response.
[0235] There are a number of advantages to performing system
identification based on a bladed disk's response to a traveling
wave excitation. First, it allows the use of data taken in a spin
pit or stage test to determine a rotor's mistuning. In this way,
the identified mistuning may include all effects present during the
test conditions, i.e., centrifugal stiffening, gas loading,
mounting conditions, as well as temperature effects. The effect of
centrifugal loading on conventional bladed disks may also be
analyzed using FMM ID.
[0236] Although FMM ID theoretically only needs measurements of one
or two modes, the method's robustness and accuracy may be greatly
improved when more modes are included. For certain bladed disks, a
single traveling wave excitation can be used to measure more modes
than would be possible from a single point excitation test. For
example, in a highly mistuned rotor that has a large number of
localized modes, it may be hard to excite all of these modes with
only one single point excitation test, because the excitation
source will likely be at a node of many of the modes. Therefore, to
detect all of the mode shapes, the test must be repeated at various
excitation points. However, if the system is driven with a
traveling wave excitation, all localized modes can generally be
excited with just a single engine order excitation. The more
localized a mode becomes in physical coordinates, the more extended
it will be in the modal analysis coordinate system. Thus in highly
mistuned systems, one engine order excitation can often provide
more modal information than several single point excitations.
[0237] The traveling wave system identification method may form the
basis of an engine health monitoring system. If a blade develops a
crack, its frequency will decrease. Thus, by analyzing blade
vibration in the engine, the traveling wave system identification
method could detect a cracked blade. A health monitoring system of
this form may use sensors, such as NSMS, to measure the blade
vibration. The measurements may be filtered to isolate an engine
order response, and then analyzed using the traveling wave system
identification method to measure the rotor's mistuning, which can
be compared with previous measurements to identify if any blade's
frequency has changed significantly, thus identifying potential
cracks. It may be possible to develop a mode extraction method that
does not require user interaction--i.e., an automated modal
analysis method which is tailored to a specific piece of
hardware.
[0238] The traveling wave system identification method discussed
hereinabove may be extended to any structure subjected to a
multi-point excitation in which the driving frequencies are
consistent from one excitation point to the next. This allows
structures to be tested in a manner that more accurately simulates
their actual operating conditions.
[6] CONCLUSION
[0239] FIG. 46 illustrates an exemplary process flow depicting
various blade sector mistuning tools discussed herein. The flow
chart in FIG. 46 summarizes how the FMM and FMM ID methods
discussed hereinbefore may be used to predict bladed disk system
mistuning in stationary as well as rotating disks. For simplicity,
the FMM discussion presented hereinbefore addressed mistuning in
mode families that are fairly isolated in frequency (i.e., first
two or three families of modes). Modeling mistuning in these modes
may be relevant to the problem of flutter as discussed in
Srinivasan, A. V., 1997, "Flutter and Resonant Vibration
Characteristics of Engine Blades," Journal of Engineering for Gas
Turbines and Power, 119, 4, pp. 742-775. However, as mentioned
before, the applicability of various FMM methodologies discussed
hereinbefore may not be necessarily limited to an isolated family
of modes.
[0240] Referring to FIG. 46, measurements (block 68) of the mode
shapes and natural frequencies of a mistuned bladed disk (block 70)
may be input to various FMM ID methods (block 72) to infer the
mistuning in each blade/disk sector. The advanced FMM ID method can
also calculate the natural frequencies that the system would have
if it were tuned, i.e., was perfectly periodic. The detailed
discussion of blocks 68, 70, and 72 has been provided under parts 1
through 3 hereinabove. Because mode shapes measurements are
generally made on stationary systems, the resulting mistuning
often-corresponds to a non-rotating bladed disk. However,
centrifugal forces that are present while the disk rotates can
alter the mistuning. Thus, mistuning extrapolation (block 74) may
be performed to correct the mistuning from a stationary rotor for
the effects of centrifugal stiffening. Mistuning extrapolation has
been discussed under part-4 hereinabove. The FMM methodology
(including FMM ID methods) may be coupled with a modal summation
algorithm to calculate the forced response (block 76) of a bladed
disk based on the mistuning identified in the previous steps. The
discussion of forced response analysis (block 76) has been provided
hereinabove at various locations under parts 1 through 3. Further,
as discussed under part-5 above, the mode shapes and natural
frequencies of a bladed disk may be extracted from its response to
a traveling wave excitation (blocks 78, 80). Thus, by combining the
traveling wave modal analysis technique (block 80) (which may use
NSMS measurements identified at block 78) with the FMM ID system
identification methods, mistuning in a bladed disk can be
determined while the disk is under actual operating conditions.
[0241] The vibratory response of a turbine engine bladed disk is
very sensitive to mistuning. As a result, mistuning increases its
resonant stress and contributes to high cycle fatigue. The
vibratory response of a mistuned bladed disk system may be
predicted by the Fundamental Mistuning Model (Fe) because of its
identification of parameters--the tuned system frequencies and the
sector frequency deviations--that control the mode shapes and
natural frequencies of a mistuned bladed disk. Neither the geometry
of the system nor the physical cause of the mistuning may be
needed. Thus, FMM requires little or no interaction with finite
element analysis and is, thus, extremely simple to apply. The
simplicity of FMM provides an approach for making bladed disks less
sensitive to mistuning--at least in isolated families of modes. Of
the two parameters that control the mistuned modes of the system,
one is the mistuning itself which has a standard deviation that is
typically known from past experience. The only other parameters
that affect the mistuned modes are the natural frequencies of the
tuned system. Consequently, the sensitivity of the system to
mistuning can be changed only to the extent that physical changes
in the bladed disk geometry affect these frequencies. For example,
if the disk were designed to be more flexible, then the frequencies
of the tuned system would be spread over a broader range, and this
may reduce the system's sensitivity to mistuning.
[0242] The FMM ID methods use measurements of the mistuned system
as a whole to infer its mistuning. The measurements of the system
mode shapes and natural frequencies can be obtained in laboratory
test through standard modal analysis procedures. The high
sensitivity of system modes to small variations in mistuning causes
measurements of those modes themselves to be an accurate basis for
mistuning identification. Because FMM ID does not require
individual blade measurements, it is particularly suited to
integrally bladed rotors. The basic FMM ID method requires the
natural frequencies of the tuned system as an input. The method is
useful for comparing the change in a components mistuning over
time, because each calculation will be based about a consistent set
of tuned frequencies. The advanced FMM ID method, on the other
hand, does not require any analytical data. The approach is
completely experimental and determines both the mistuning and the
tuned system frequencies of the rotor.
[0243] Effects of centrifugal stiffening on mistuning may be
identified on a stationary IBR using FMM ID and FMM, and
extrapolated to engine operating conditions to predict the system's
forced response at speed. Further, in conventional bladed disks,
centrifugal forces may cause changes in the contact conditions at
the blade/disk attachment to substantially alter the system's
mistuning. This behavior may not be accounted for in the mistuning
extrapolation method. In that case, the mode shapes and natural
frequencies of a rotating bladed disk may be extracted from
measurements of its forced response (e.g., traveling wave
excitation) and the results may then be combined with FMM ID to
determine the mistuning present at operating conditions.
[0244] It is observed that, besides centrifugal loading, other
factors may also be present in the engine that can affect its
mistuned response. These may include: temperature effects, gas
bending stresses, how the disk is constrained in the engine, and
how the teeth in the attachment change their contact if the blades
are conventionally attached to the disk. Except for the constraints
on the disk, these additional effects may be relatively unimportant
in integrally bladed compressor stages. The disk constraints can be
taken into account by performing the system identification (using,
for example, an FMM ID method) on the IBR after the full rotor is
assembled. Thus, the FMM ID methodology presented herein may be
used to predict the vibratory response of actual compressor stages
so as to determine which blades may be instrumented, interpret test
data, and relate the vibratory response measured in the CRF to the
vibration that will occur in the fleet as a whole.
[0245] The traveling wave modal analysis method discussed
hereinbefore may detect the presence of a crack in an engine blade
by analyzing blade vibrations because a crack will decrease a
blade's frequency of vibration. This method, thus, may be used with
on-board sensors to measure blade response during engine
accelerations. The measurements may be filtered to isolate an
engine order response, and then analyzed using the traveling wave
system identification method. The identified mistuning may then be
compared with previous results to determine if any blade's
frequency has changed significantly, thus identifying potential
cracks. The FMM and FMM ID methods may be applied to regions of
higher modal density using Subset of Nominal Modes (SNM)
method.
[0246] The foregoing describes development of a reduced order model
called the Fundamental Mistuning Model (FMM) to accurately predict
vibratory response of a bladed disk system. FMM may describe the
normal modes and natural frequencies of a mistuned bladed disk
using only its tuned system frequencies and the frequency mistuning
of each blade/disk sector (i.e., the sector frequencies). If the
modal damping and the order of the engine excitation are known,
then FMM can be used to calculate how much the vibratory response
of the bladed disk will increase because of mistuning when it is in
use. The tuned system frequencies are the frequencies that each
blade-disk and blade would have were they manufactured exactly the
same as the nominal design specified in the engineering drawings.
The sector frequencies distinguish blade-disks with high vibratory
response from those with a low response. The FMM identification
methods-basic and advanced FMM ID methods--use the normal (i.e.,
mistuned) modes and natural frequencies of the mistuned bladed disk
measured in the laboratory to determine sector frequencies as well
as tuned system frequencies. Thus, one use of the FMM methodology
is to: identify the mistuning when the bladed disk is at rest, to
extrapolate the mistuning to engine operating conditions, and to
predict how much the bladed disk will vibrate under the operating
(rotating) conditions.
[0247] In one embodiment, the normal modes and natural frequencies
of the mistuned bladed disk are directly determined from the disk's
vibratory response to a traveling wave excitation in the engine.
These modes and natural frequency may then be input to the FMM ID
methodology to monitor the sector frequencies when the bladed disk
is actually rotating in the engine. The frequency of a disk sector
may change if the blade's geometry changes because of cracking,
erosion, or impact with a foreign object (e.g., a bird). Thus,
field calibration and testing of the blades (e.g., to assess damage
from vibrations in the engine) may be performed using traveling
wave analysis and FMM ID methods together.
[0248] It is noted that because the FMM model can be generated
completely from experimental data (e.g., using the advanced FMM ID
method), the tuned system frequencies from advanced FMM ID may be
used to validate the tuned system finite element model used by
industry. Further, FMM and FMM ID methods are simple, i.e., no
finite element mass or stiffness matrices are required.
Consequently, no special interfaces are required for FMM to be
compatible with a finite element model.
[0249] While the disclosure has been described in detail and with
reference to specific embodiments thereof, it will be apparent to
one skilled in the art that various changes and modifications can
be made therein without departing from the spirit and scope of the
embodiments. Thus, it is intended that the present disclosure cover
the modifications and variations of this disclosure provided they
come within the scope of the appended claims and their
equivalents.
* * * * *
References