on the accuracy of modal parameters identified from exponentially windowed, noise contaminated...
TRANSCRIPT
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
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