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Transcript of OMAC 2008: 1 Operational Modal Analysis Conference 2008 Modal testing of structures under...
OMAC 2008: 1
Operational Modal Analysis Conference 2008
Modal testing of structures under operational conditions.
Real Boundary Conditions
Real Operational Loads
OMAC 2008: 2
OMAC 2008 Schedule
09:00 – 10:40 Technical Session 1 – Introduction to OMA
What is Modal Analysis
Traditional Modal Analysis versus Operational Modal Analysis
Applications of OMA – Some Case Studies
11:10 - 13:00 Technical Session 2 – Analysis of a Real Structure
Preliminary (Automatic) Modal Analysis
The Frequency Domain Decomposition (FDD) Technique
The Stochastic Subspace Identification Technique.
Keynote
14:15 – 15:15 Technical Session 3 – Testing in Practice Generating Test Geometry and Plan the Test Testing using Reference Sensors and Moving Sensors
15:35 – 16:30 Technical Session 4 – Rounding Off Last Questions and Answers
OMAC 2008: 3
What is Modal Analysis – Modal Information
One of the few things in life that will never go out of fashion:
The Newton’s 2nd Law of Motion
f = M a
Force Mass Acceleration
Modal Information provides a systematic and decoupled way of describing how a structure responds when forces are applied to it.
Sir Isaac Newton (1642-1727)
OMAC 2008: 4
What is Modal Analysis – Modal Information
What does modal information mean ?
ExcitationResponse
ForceMotion
InputOutput
H() = = =
The Frequency Response Function
2
122
12,)( jjjj
n
j j
Hjj ifH
Modal Decomposed Frequency Response Function
OMAC 2008: 5
What is Modal Analysis – What are Modes?
11st Mode Shape
22nd Mode Shape
OMAC 2008: 6
What is Modal Analysis – What are Modes?
Damping
Frequency f
Frequency DomainTime Domain
OMAC 2008: 7
Traditional Modal Technology
Input
Output Time DomainFrequency Domain
Frequency Response H1(Response,Excitation) - Input (Magnitude)Working : Input : Input : FFT Analyzer
0 200 400 600 800 1k 1,2k 1,4k 1,6k
100m
10
[Hz]
[(m/s²)/N]Frequency Response H1(Response,Excitation) - Input (Magnitude)Working : Input : Input : FFT Analyzer
0 200 400 600 800 1k 1,2k 1,4k 1,6k
100m
10
[Hz]
[(m/s²)/N]
Autospectrum(Excitation) - InputWorking : Input : Input : FFT Analyzer
0 200 400 600 800 1k 1,2k 1,4k 1,6k
100u
1m
10m
100m
1
[Hz]
[N] Autospectrum(Excitation) - InputWorking : Input : Input : FFT Analyzer
0 200 400 600 800 1k 1,2k 1,4k 1,6k
100u
1m
10m
100m
1
[Hz]
[N]
Autospectrum(Response) - InputWorking : Input : Input : FFT Analyzer
0 200 400 600 800 1k 1,2k 1,4k 1,6k
1m
10m
100m
1
10
[Hz]
[m/s²] Autospectrum(Response) - InputWorking : Input : Input : FFT Analyzer
0 200 400 600 800 1k 1,2k 1,4k 1,6k
1m
10m
100m
1
10
[Hz]
[m/s²]
Time(Excitation) - InputWorking : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m
-200
-100
0
100
200
[s]
[N] Time(Excitation) - InputWorking : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m
-200
-100
0
100
200
[s]
[N]
Time(Response) - InputWorking : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m
-80
-40
0
40
80
[s]
[m/s²] Time(Response) - InputWorking : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m
-80
-40
0
40
80
[s]
[m/s²]
FFT
FFT
Impulse Response h1(Response,Excitation) - Input (Real Part)Working : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m
-2k
-1k
0
1k
2k
[s]
[(m/s²)/N/s]Impulse Response h1(Response,Excitation) - Input (Real Part)Working : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m
-2k
-1k
0
1k
2k
[s]
[(m/s²)/N/s]
InverseFFT
ExcitationResponse
ForceMotion
InputOutput
H() = = =
FrequencyResponseFunction
ImpulseResponseFunction
OMAC 2008: 8
Shaker excitation
Small homogenous structures Quick Polyreference technique Fast method - no fixtures required
......................
......................
......................
...H......HH
H
n11211
.............H
...................
.............H
.............H
H
1n
21
11
Out
In Out
In
Multichannel response orresponse points may be moved
Large or complex structures Various excitation signals possible Time consuming - installation work
to be done
Hammer excitation
Excitation moved
Traditional Modal Technology
OMAC 2008: 9
Traditional Modal Technology - Limitations
No external excitation during testing – Test Rig Required!
Improper boundaries and excitation levels have to be accepted sometimes.
Hammers and shakers limits applications: Modes of symmetric structures are difficult to find due to the single input.
Large structures impossible to excite artificially.
OMAC 2008: 10
Determination of Modal Model by response testing only
– No measurement of input forces required
– Measurement procedure similar to Operational Deflection Shapes (ODS)
Determination of Modal Model in-situ under operational conditions
– True boundary conditions.
– Correct excitation level giving correct Modal Model in case of amplitude dependent non-linearities.
Used in Civil Engineering applications
– Bridges and buildings
– Off-shore platforms etc.
Used in Mechanical Engineering applications
– On-road and in-flight testing etc.
– Rotating Machinery
Operational Modal Analysis – (OMA)
OMAC 2008: 11
Operational Modal Analysis - Procedure
Determination of Modal parameters based on natural excitation
Measurement of responses in a number of DOF’s
– simultaneously
– by roving accelerometers with one or more fixed accelerometers as references
Fixed ReferenceAccelerometers
Accelerometers are moved for each data set
OMAC 2008: 12
Measured
Responses
Stationary
Zero Mean
Gaussian
White Noise
Model of the combined system is estimated from measured responses
Excitation Filter
(linear, time-invariant)
Structural System
(linear, time-invariant)
Unknown excitation forces
Combined System
Modal Model of Structural System extracted from estimated model of Combined System
Operational Modal Analysis – Combined Model
OMAC 2008: 13
If the system is excited by white noise
the output spectrum contains full information of the structure
as all modes are excited equally
But this is in general not the case!
Structural SystemForce SpectrumOutput Spectrum
Operational Modal Analysis
Operational Modal Analysis – Combined Model
OMAC 2008: 14
In general the excitation has a spectral distribution
Modes are weighted by the spectral distribution of the input force
Both the “modes” originating from the excitation signal and
the structural modes are observed as “modes” in the Response
Structural SystemForce SpectrumCombined Spectrum
Operational Modal Analysis
Operational Modal Analysis – Combined Model
OMAC 2008: 15
Noise also contributes to the Response
Measurement Noise
Computational Noise
Force Spectrum Structural SystemCombined Spectrum
Operational Modal Analysis
Operational Modal Analysis – Combined Model
OMAC 2008: 16
Operational Modal Analysis – Combined Model
Rotating parts creates Harmonic vibrations
Measurement Noise
Computational Noise
Force Spectrum
The “Modes” in the combined spectrum contains information of The system under test (Physical Modes) Input Force (Non-physical “Modes”) Noise (Non-physical “Modes”) Harmonics (Non-physical “Modes”)
Rotating Parts
Structural SystemCombined Spectrum
Operational Modal Analysis
OMAC 2008: 17
Operational Modal Analysis – Advantages
OMA is MIMO whereas traditional modal testing in general is SISO or SIMO.
In case of a symmetric structures with closely spaced modes, MIMO technology is the only choice.
Test procedures are in general easier. No hammers, No shakers.
Modal parameters can be obtained in the serviceability state:
Needed for slightly non-linear structures
Needed if e.g. operating (rotating) machinery is present.
In case of larger structures, traditional modal testing is simply impossible.
No hammers or shakers can excite the structure due to the mass and the low frequency modes.
Possible applications:
FEM updating / validation
Damage detection
Structural Health Monitoring
Vibration level estimation
Fatigue estimation.
OMAC 2008: 18
Case Study – Launch Vehicle – FEM Validation
Launch Vehicle Control System Verification
Control system is based on FE model
During launch vibration data is acqiured and transmitted to the ground.
Based on OMA analysis the FE model is verified due the different states of the launch.
Customer: Boeing Integrated Defense Systems, Delta IV, CA, USA
OMAC 2008: 19
Case Study – On Road Testing
Customer: Mazda, Hiroshima, Japan
OMAC 2008: 20
Case Study – Qutub Minar, New Delhi, India
Customer: Mazda, Hiroshima, Japan
OMAC 2008: 21
OMAC 2008 Schedule
09:00 – 10:40 Technical Session 1 – Introduction to OMA
What is Modal Analysis
Traditional Modal Analysis versus Operational Modal Analysis
Applications of OMA – Some Case Studies
11:10 - 13:00 Technical Session 2 – Analysis of a Real Structure
Preliminary (Automatic) Modal Analysis
The Frequency Domain Decomposition (FDD) Technique
The Stochastic Subspace Identification Technique.
Keynote
14:15 – 15:15 Technical Session 3 – Testing in Practice Generating Test Geometry and Plan the Test Testing using Reference Sensors and Moving Sensors
15:35 – 16:30 Technical Session 4 – Rounding Off Last Questions and Answers
OMAC 2008: 22
Frequency Domain Decomposition – Step by Step
Frequency Domain Decomposition (FDD) Theory Practise Automation
Enhanced Frequency Domain Decomposition (EFDD) Theory Practise
Curve-fit Frequency Domain Decomposition (CFDD) Theory Practise
OMAC 2008: 23
Frequency Domain Decomposition - Theory
Showing that y the dynamic deflection is a linear combination of the Mode Shapes, the coefficients being the Modal Coordinates.
y(t) = 1 q1(t) + 2 q2(t) + 3 q3(t) + + n qn(t)
m
rrr tqtqty
1
)()()(
= ++ + +
OMAC 2008: 24
Frequency Domain Decomposition - Theory
T
yytytyEC )()()(
Hqq
HH
yyCtqtqEC )()()()(
m
rrr tqtqty
1
)()()(
Hqqyy
fGfG )()(
Linear system response:
Covariance function of system response:
Spectral density function obtained by Fourier Transformation:
Spectrum is decouple and described by superposition of Single-Degree-Of-Freedom
models
G(f)
f
OMAC 2008: 25
Frequency Domain Decomposition - Theory
Theory says...
Perform an SVD
Extract Mode Shape
H
n
yyfU
fs
fs
fs
fUfG )(
)(.00
....
0.)(0
0.0)(
)()( 2
1
n
fufufufU )}({,,)}({,)}({)(21
Hqqyy
fGfG )()(
OMAC 2008: 26
Frequency Domain Decomposition - Theory
)()}({)}({)(}{}{)(111111 p
H
pppq
H
pyyfsfufufGfG
Hn
pq
pq
pq
npyy
fG
fG
fG
fG
n
}{.}{}{
0)(.00
....
0.0)(0
0.0)(
}{.}{}{)(2121
2
1
G(f)
f
In the vicinity of the resonance peak of a well-seperated mode:
Hnppp
pn
p
p
npppfufufu
fs
fs
fs
fufufu )}({.)}({)}({
0)(.00
....
0.0)(0
0.0)(
)}({.)}({)}({21
2
1
21
... the first singular vector is a good approximation to the mode shape!
OMAC 2008: 27
Frequency Domain Decomposition - Practice
Modes are found by picking the peaks of the modes at 1st singular value. Mode shapes estimate is given by the associate singular vector.
OMAC 2008: 28
Frequency Domain Decomposition - Practice
2 is a good place for estimating shape 1 from singular vector v1
3 is a good place for estimating shape 2 from singular vector v1
s1
s2
3
v1s1 v2s2
a11 a22
v1s1
v2s2
a11
a22
v2s2
v1s1
a22
a11
2
1
1
2 3
1
1
1
2
2
2
In case of non-orthogonal mode shapes:
OMAC 2008: 29
Frequency Domain Decomposition - Automation
Modal Coherence
Discriminator function:
Low modal coherence: Noise High modal coherence: Modal dominance
10101)}({)}({)( fufufd H
10.8
0
OMAC 2008: 30
Frequency Domain Decomposition - Automation
The measured responses of a structure are approximate Gaussian distributed if just a few independent broad-banded random inputs excites the structure.
The measured response will only be approximately Gaussian in case of a large number of different harmonic exciation sources.
Detection Procedure:
1. Bandpass filter.
2. Normalize to zero-mean andunit variance.
3. Calculate Kurtosis:
4. If γ is significantly different from 3 then most likely not Gaussian.
Hamonic Indicators
Gaussian Probability
Density Function
Sinusoidal Probability Density
Function
4
4
,
xE
x
OMAC 2008: 31
Frequency Domain Decomposition - Automation
Modal Domain
Mode property – defined for all modes Defines the frequency region dominated by the mode
Definition:The frequency range around
a peak where: Modal coherence is
higher than a certain
threshold, say 0.8, No harmonics present. Damping resonably low.
OMAC 2008: 32
Frequency Domain Decomposition - Automation
Procedure:
Define a search set as all 1st. Singular values from DC to Nyquist. Identify the highest peak in the current search set. Check if peak could be physical (High modal coherence, no
harmonic, low damping). If so, establish a modal domain. If not, establish a noise domain. Exclude modal/noise domain from search set. Continue until:
1. Search set is empty.
2. Peak is below defined noise floor.
3. A specified number of modes are
found.
OMAC 2008: 33
Enhanced Frequency Domain Decomposition - Theory
IFFT performed to calculate Correlation Function of SVD function Frequency and Damping estimated from Correlation Function Mode shape is obtained from weighted sum of singular vectors
s1
s2
••
• • • ••
•••
•
0•
i
MAC =
8,0}{}{}{}{
}{}{
00
2
0
i
H
i
H
i
H
Improved shape estimation from weighted sum:
i
iiweights}{}{
Select MAC rejection level(default 0,8):
OMAC 2008: 34
Enhanced Frequency Domain Decomposition - Practice
OMAC 2008: 35
Curve-fit Frequency Domain Decomposition - Theory
s1
s2
••
• • • ••
•••
•
0•
i
)(
)(
1)(
4
2
2
1
4
2
2
10
fA
fB
eAeA
eBeBBfH
fTfT
fTfT
Algorithm:1. Estimate SDOF spectrum G(f).
2. Calculate half-power spectrum P(f) of G(f)
3. Construct the following matrices:
4. Solve the following regression problem:
5. Estimates are then given by:
6. Frequency and damping are obtain from the roots of A(f)
)(
.
.
)(
)(
,
1)()(
.....
.....
1)()(
1)()(
1
0
4242
424
1
2
1
424
0
2
0
1111
0000
fP
fP
fP
B
eeefPefP
eeefPefP
eeefPefP
Ac
TfTfTfTf
TfTfTfTf
TfTfTfTf
c
)Im(
)Re(
)Im(
)Re(ˆ1
c
c
c
c
B
B
A
A
TBBBAA21021
Perform SDOF curve-fitting on estimated SDOF model in frequency domain to estimate Frequency and Damping.
OMAC 2008: 36
Curve-fit Frequency Domain Decomposition - Practice
Perform SDOF curve-fitting on estimated SDOF model in frequency domain to estimate Frequency and Damping.
Mode shape is obtained from weighted sum of singular vectors
s1
s2
••
• • • ••
•••
•
0•
i
OMAC 2008: 37
Frequency Domain Decomposition - Conclusions
Frequency Domain Decomposition:
Simple peak picking technique that quickly estimates modes even in case of hundreds of measurement channels.
Mode shapes are estimated by removing influence of other modes by utilization of the Singular Value Decomposition.
Can easially be automated.
Enhanced Frequency Domain Decomposition:
Frequency and damping determined on the basis of identification of the SDOF model of the mode.
The SDOF model is estimated in frequency domain by utilizing the Modal Assurance Criterion.
Frequency and damping is estimated from the time domian equivalent SDOF model.
Curve-fit Frequency Domain Decomposition:
Curve-fit frequency and damping directly in frequency domain.
OMAC 2008: 38
SSI procedure Generate compressed input format
– Select total number of modes (structural, harmonics, noise) based on apriori knowledge
– Select Identification Class» Unweighted Principal Components (UPC); Principal Components (PC);
Canonical Variate Analysis (CVA)
Estimate Parameters from Stabilization diagram– Select interval of model order candidates (use SVD diagram)
– Estimate models (adjust tolerance criteria)
– Select the optimal model (use validation) Select and link modes across data sets
Stochastic Subspace Identification (SSI) Classes of Identification Data Driven: Use of raw time data Covariance Driven: Use of Correlation functions
OMAC 2008: 39
Combined System Model used in SSI
Measured
Responses
Stationary
zero mean
Gaussian
White Noise Excitation Filter
(linear, time-invariant)
Structural System
(linear, time-invariant)
Unknown excitation forces
Combined System A
ttt
ttt
vCxy
wAxx
1State Equation
Observation (Output) equation
Model of the dynamics of the system
Model of the output of the system
wt: Process noise - vt: Measurement noise - Model order: Dimension of A
Discrete-time Stochastic State Space Model
t
tt
v
yxC
tw
OMAC 2008: 40
Stochastic Subspace Identification (SSI)
ttt
ttt
vCxy
wAxx
1 wt: Process noise
vt: Measurement noise
ttt
ttt
exCy
KexAx
ˆ
ˆˆ 1
ttt
ttit
ezy
ezz
1
Modal decomposition
i Eigenvalues
Modal frequency
and damping
Left hand mode shapes
Physical Modes
Right hand mode shapes
Non-physical Modes
Modal distribution of eInitial modal amplitudes
Discrete-time Stochastic State Space Model
Innovation form et: Innovation (white noise)
K: Kalman gain (noise model)
Modal parameter extraction from SSI
1i VVA
t1
t xVz
OMAC 2008: 41
Conclusion: If can be determined then A and C can be
optimally predicted using least squares estimation
1ˆ,ˆ tt xx
Least squares estimation of A and C
Stochastic Subspace Identification (SSI)
ttt
ttt
exCy
KexAx
ˆ
ˆˆ 1
x x
x
xx
x
tx̂
1ˆ txtyIItttt
Itttt
xCyxC
xAxxA
ˆ:ˆ
ˆˆ:ˆ 1
Assuming properties: Zero mean Gaussian stochastic process Modeled by a state space formulation
Then a least squares estimation gives
Gaussian white noise residuals
Result :
t
tIItt
It eKe &
ttt yxx ,ˆ,ˆ 1
x x tx̂xx
x x
Error
OMAC 2008: 42
Estimation of state vectors:
Stochastic Subspace Identification (SSI)
]ˆˆˆ[ˆ11 jiiii xxxX
S1
O: Compressed input format matrixW1 , W2: Weighting matrices
S1 : Subspace matrix
Selected subspace
Singular valueSta
te s
pac
e d
imen
sio
n
s1
s2
s3
s4
s5
s6
21112/1
12/1
1111ˆ WXWVSSUVSU ii
TT iX̂ is calculated from
T
T
V
V
2
1 2121 UUOWW
0......0
.0.
.0.
.0..
...0
...00
...00
0...000
2
1
s
s
s
TVSU 111SVD:
OMAC 2008: 43
Stochastic Subspace Identification (SSI) Parametrical Modal estimation requiring apriory knowledge of Model Order Physical Modes as well as Non-physical Modes are estimated
How can we separate Physical Modes from Non-physical Modes?
Physical modes are repeated for multiple Model orders!
Stabilization Diagram
Frequency
Number of modes in the model
4567 +
+++
XX X
X
Stable Modes
X
XX
X Remaining modes are considered as unstable
X Estimated parameters not fulfilling apriori knowledge of damping
+ Stable modes are repeated in two consecutive models fulfilling user defined criteria
X
X
XXX
Stable Modes not fulfilling Damping apriori knowledge
XX
X X +X
OMAC 2008: 44
Selection of State Space Dimension Error diagram
Model vs. measurements
Stabilization diagram
OMAC 2008: 45
Selecting proper model order for SSI
Final Prediction Error
– Fitting error decreases with increasing model order
– Parameter uncertainty increases with increasing model order
Final P
rediction Error
Model Order
Fitting Error
Parameter Uncertainty
Optimum Choice
OMAC 2008: 46
Effects of time varying systems, 2
Run up/down tests gives good modal parameters,but might have bad SSI validation
Loading System Structural SystemInput
Stochastic Deterministic
Output
Time Invariant Time Varying
Tim
e In
vari
ant ~ Spectral Density Correct
~ Clear Peaks~ Good Results in FDD EFDD and SSI~ Good SSI validation Covariance equivalance
~ Spectral Density Incorrect~ Incorrect Modal parameters of all methods~ Bad validation in SSI no Covariance equivalence AVOID!
Tim
e V
aryi
ng
~ Spectral Density peaks correct, valleys not~ Good modal parameters results in all methods~ Bad validation in SSI no Covariance equivalence
AVOID!
Structural System
Lo
ad
ing
Sy
ste
m
OMAC 2008: 47
OMAC 2008 Schedule
09:00 – 10:40 Technical Session 1 – Introduction to OMA
What is Modal Analysis
Traditional Modal Analysis versus Operational Modal Analysis
Applications of OMA – Some Case Studies
11:10 - 13:00 Technical Session 2 – Analysis of a Real Structure
Preliminary (Automatic) Modal Analysis
The Frequency Domain Decomposition (FDD) Technique
The Stochastic Subspace Identification Technique.
Keynote
14:15 – 15:15 Technical Session 3 – Testing in Practice Generating Test Geometry and Plan the Test Testing using Reference Sensors and Moving Sensors
15:35 – 16:30 Technical Session 4 – Rounding Off Last Questions and Answers
OMAC 2008: 48
The ARTeMIS Software Solution
TEAC LX
ARTeMISTestor
ARTeMISExtractor
VibrationData
File TransferOr OLE
ARTeMIS
Solution
OMAC 2008: 49
Data Acquisition – Total Sample Time and Sampling Frequency
)( fS
)(ty minf
f
max
1
f
min
1
f
t
Tfs
1
Tf
2
1
maxf
T Sampling Interval, s
ttotal Total measurement time, s
fs Sampling Frequency, Hz
fν Nyquist Frequency, Hz
fmin Frequency of lowest mode of interest
fmax Frequency of highest mode of interest
maxmax 5.28.0 ffff s
1000,1
min
xf
xttotal
OMAC 2008: 50
Acquire High Quality Data
Check / Optimize effective dynamic range:
– Can be increased by oversampling / decimation.
– Check valleys of spectra and SVD.
[dB | (1 V)² / Hz]
Frequency [Hz]
0 200 400 600 800 1000-150
-120
-90
-60
-30
Singular Values of Spectral Density Matrixof Data Set Measurement 1
[dB | (1 V)² / Hz]
Frequency [Hz]
0 200 400 600 800 1000-120
-100
-80
-60
-40
Magnitude of Spectral Density betw eenTransducer #6 and Transducer #6 of Data Set Measurement 1
[dB]
Frequency [Hz]
0 20 40 60 80-80
-60
-40
-20
0
20
Singular Values of Spectral Density Matrixof Data Set Setup 1