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


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


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



OMAC 2008: 15

Noise also contributes to the Response

Measurement Noise

Computational Noise

Force Spectrum Structural SystemCombined Spectrum



OMAC 2008: 16


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


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









Keynote



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


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


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


)()}({)}({)(}{}{)(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


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


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


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


Structural System


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


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:


]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









Keynote



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

omac 2008: 1 operational modal analysis conference 2008 modal testing of structures under...

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