On the Accuracy of Modal Parameters Identified from Exponentially

Windowed, Noise Contaminated Impulse Responses for a System with a

Large Range of Decay Constants

Matthew S. Allen

Jerry H. GinsbergGeorgia Institute of Technology

George W. Woodruff School of

Mechanical EngineeringNovember, 2004

2

OutlineOutline

Background: Introduction to Experimental Modal Analysis Measuring Frequency Response Functions

Persistent vs. Impulsive Excitations Difficulties in testing a system with a range of

decay constants in the presence of noise. Exponential Windowing

Experiment: Noise contaminated data Effect of exponential window on accuracy

Conclusions

3

F

…

Experimental Modal AnalysisExperimental Modal Analysis

A Linear-Time-Invariant (LTI) system’s response is a sum of modal contributions. r r r

Natural Frequency Damping Ratio Mode Vector (shape) In EMA we seek to identify

these modal parameters from response data.

4

EMA ApplicationsEMA Applications

Applications of EMA Validate a Finite Element

(FE) model Characterize damping Diagnose vibration

problems Simulate vibration

response Detect damage Find dynamic material

properties Control design …

010

2030

4050

60

-202

-6-4-20

Mode Shape for Mode # 1 , at 3.8433 Hz

X

Z

Y

010

2030

4050

60

-8-6-4-202

-6-4-202

Mode Shape for Mode # 2 , at 4.8223 Hz

X

Z

Y

5

EMA Theory – Measuring FRFsEMA Theory – Measuring FRFs

Two common ways of measuring the Frequency Response Periodic or Random

Excitation Impulse Excitation.

Impulse method is often preferred: Doesn’t modify the

structure Cost High force amplitude Noisy Data

H()U Y

H()

H()

FFT

6

+

Range of Decay Constants: (Range of Decay Constants: (rrrr))

+NoiseResponse

7

Range of Decay Constants: (Range of Decay Constants: (rrrr))

Noise dominates the response of the quickly decaying modes at late times.

+ +Slow Fast Noise

Early Response

Late Response

80 500 1000 1500 2000 2500 3000 3500 4000

10-9

10-8

10-7

10-6

Frequency (rad/s)

|H(

)|

Magnitude FRF

Range of Decay Constants: (Range of Decay Constants: (rrrr))

+ +Slow Fast Noise

9

Exponential WindowingExponential Windowing

Exponential Windows (EW) are often applied to reduce leakage in the FFT.

Effect on modal parameters: Adds damping – (can be precisely

accounted for) Other windows (Hanning, Hamming,

etc…) have an adverse effect.

An EW also causes the noise to decay, reducing the effect of noise at late times.

Could this result in more accurate identification of the quickly decaying modes?

time (s)

w(t

)

time (s)

w(t)

10

Range of Decay ConstantsRange of Decay Constants

Prototype System: Modes 7-11 have large decay constants. The FRFs in the vicinity of these modes are noisy.

Mode Decay Nat.Constant Freq.

1 -0.66 68.02 -0.16 265.93 -0.48 386.34 -0.27 829.55 -0.79 1033.86 -0.61 1697.07 -3.12 1937.98 -18.80 2479.79 -18.13 2511.0

10 -5.35 2995.711 -5.21 3380.0

c1

22

c3

c4

L

2

1

L

w

w

c

x = 01

x = 0

Frame Structure

11

FFT

Windowing ExperimentWindowing Experiment

Apply windows with various decay constants to noise contaminated analytical data.

Estimate the modal parameters using the Algorithm of Mode Isolation (JASA, Aug-04, p. 900-915)

Evaluate the effect of the window on the accuracy of the modal parameters.

Repeat for various noise profiles to obtain statistically meaningful results.

c1

22

c3

c4

L

2

1

L

w

w

c

x = 01

x = 0

AMI

Modal Parameters

time (s)

w(t)

Noisy Data Window

12

Sample Results: Damping RatioSample Results: Damping Ratio

Two distinct phenomena were observed. Increase in scatter – (Lightly damped modes.) Decrease in bias – (Heavily damped modes.)

These are captured by the standard deviation and mean of the errors respectively.

1 10-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Results for Mode 2 (30 Trials)

% E

rro

r in

Da

mp

ing

Ra

tio

Exponential Factor / Decay Constant0.01 0.1 1

-4

-2

0

2

4

6

8

10Results for Mode 11 (30 Trials)

% E

rro

r in

Da

mp

ing

Ra

tio

Exponential Factor / Decay Constant

Standard Deviation Mean

13

10-3

10-2

10-1

100

101

-2

-1

0

1

2

3

4

5

6

7

8

Exponential Factor / Decay Constant

% B

ias

in N

atu

ral F

req

ue

nc

y

Damping Ratio Errors: Mean (.)

1 2 3 4 5 6 7 8 91011

10-3

10-2

10-1

100

101

0

0.3

0.6

0.9

1.2

1.5

Exponential Factor / Decay Constant

% S

tan

da

rd D

ev

iati

on

of

Na

tura

l Fre

qu

en

cy

Damping Ratio Errors: Standard Deviation (o)

1 2 3 4 5 6 7 8 91011

% S

catt

er in

Dam

ping

Rat

io

Results: Damping RatioResults: Damping Ratio%

Bia

s in

Dam

ping

Rat

io

Largest errors were the bias errors in modes 8-11.

These decreased sharply when an exponential window was applied.

14

Results: Natural FrequencyResults: Natural Frequency

10-2

10-1

100

101

-0.015

-0.01

-0.005

0

0.005

0.01

Exponential Factor / Decay Constant

% B

ias

Err

or

in N

atu

ral F

req

ue

nc

y

Natural Frequency Errors: dots - mean

1 2 3 4 5 6 7 8 91011

10-3

10-2

10-1

100

101

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Exponential Factor / Decay Constant

% S

ca

tte

r E

rro

r in

Na

tura

l Fre

qu

en

cy

Natural Frequency Errors: circles - std

1 2 3 4 5 6 7 8 91011

15

10-3

10-2

10-1

100

101

102

10-3

10-2

10-1

100

Exponential Window Factor / Modal Decay Constant

No

ise

St.

De

v. /

Ma

x M

od

al A

mp

litu

de

Effect of Exponential Window on NSR

1 2 3 4 5 6 7 8 91011

Noise Level vs. Exponential FactorNoise Level vs. Exponential Factor

Bias errors are related to the Signal to Noise Ratio. Bias is small

when the signal is 20 times larger than the noise.

SNR attains a maximum when the window factor equals the modal decay constant.

16

ConclusionsConclusions

Exponential windowing improves the SNR of the FRFs in the vicinity of each mode, so long as the window factor is not much larger than the modal decay constant.

Damping Ratio: Bias Errors in the damping estimates are small so

long as the SNR is above 20 (see definition.) Natural Frequency:

EW has a small effect so long as the exponential factor is smaller than the modal decay constant.

Similar Results for Mode Shapes & Modal Scaling.

17

Questions?Questions?

010

2030

4050

60

-202

-6-4-20

Mode Shape for Mode # 1 , at 3.8433 Hz

X

Z

Y

0 500 1000 1500 2000 2500 3000 3500 400010

-9

10-8

10-7

10-6

Frequency (rad/s)

|H(

)|

Magnitude FRF

18

10-3

10-2

10-1

100

101

-2

-1

0

1

2

3

4

5

6

7

8

Exponential Factor / Decay Constant

% B

ias

in N

atu

ral F

req

ue

nc

y

Damping Ratio Errors: Mean (.)

1 2 3 4 5 6 7 8 91011

10-3

10-2

10-1

100

101

0

0.3

0.6

0.9

1.2

1.5

Exponential Factor / Decay Constant

% S

tan

da

rd D

ev

iati

on

of

Na

tura

l Fre

qu

en

cy

Damping Ratio Errors: Standard Deviation (o)

1 2 3 4 5 6 7 8 91011

Results: Damping RatioResults: Damping Ratio%

Bia

s in

Dam

ping

Rat

io

% S

catt

er in

Dam

ping

Rat

io

Observations: Exponential windowing did

not decrease the scatter significantly for modes 8-11.

The scatter for modes 1-7 increased sharply for large exponential factors.

Exponential factors as large as the modal decay constant could be safely used.

19

EMA TheoryEMA Theory

Two common ways of measuring the Frequency Response Apply a broadband

excitation and measure the response.

Apply an impulsive excitation and record the response until it decays.

Equation of Motion

Frequency Domain

Frequency Response

Modal Parameters

20

Effect of Exponential Window on SNREffect of Exponential Window on SNR

Damping added by the exponential window decreases the amplitude of the response in the frequency domain.

The amplitude of the noise also decreases.

The net effect can be increased or decreased noise.

261 262 263 264 265 266 267 268 269 270

10-7

10-6

Frequency (rad/s)

|H(

)|

Magnitude FRF for Various Damping Factors

0-0.049-0.098 -0.16 -0.21 -0.26 -0.32 -0.37 -0.42 -0.49 -0.54 -0.75 -1.3 -2 -4

IncreasingDamping

0 500 1000 1500 2000 2500 3000 3500-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5x 10

-8 Magnitude of Noise for Various Exponential Factors

Frequency (rad/s)

Mag

nit

ud

e

0-0.048814-0.097628 -0.16216 -0.21097 -0.25978

210 500 1000 1500 2000 2500 3000 3500 4000

10-9

10-8

10-7

10-6

Frequency (rad/s)

|H(

)|

Magnitude FRF

Range of Decay Constants: (Range of Decay Constants: (rrrr))

Noise dominates the response of the quickly decaying modes at late times.

A shorter time window reduces the noise in these modes, though it also results in leakage for the slowly decaying modes.

+ +Slow Fast Noise

Early Response

Late Response