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Modal parameter estimation of low-rise building
using sine sweep vibration tests
Le Thai HoaWind Engineering Research Center
Tokyo Polytechnic University
Contents
1. Sine Sweep Force (Measured & Simulated)
2. Frequency Response Functions (FRF)3. Smoothing Techniques for FRF4. Modal Parameter Estimation [DX1 only]
Objectives
o Estimating modal parameters (natural frequencies and damping ratios) using sine sweep vibration data
o Sine sweep input force has been measured and simulated theoretically
o Identifying Frequency Response Functions (FRFs) from measured/theoretical input and measured response
Exciter
Exciter
o Linear sine sweep forceo Constant sweep force (Constant amplitude)o Variable frequency rangeo Starting frequency fo= 2Hzo Ending frequency fe 6Hzo Sweep rate =0.01 Hz/s
Reference (Exciter)S-ACCTABLE-ACC: Input acceleration TABLE-DISP: Input displacement
Sensors
Exciter
1F
2F
2F
PU4-X(CH4)
PU
4-Y
(CH
5)
X
Y
PU6-X(CH7)
PU
6-Y
(CH
8)
PU5-X(CH6)
PU1-X (CH1)
X
Y
PU3-X(CH3)
PU2-X(CH2)
Sampling rate: 100Hz
Accelerometer
Sine sweep excitation
Measured sweep forceSimulated sweep force
Measurement of sine sweep force
0 100 200 300 400 500 600 700 800-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Time (s)
Acc
e. (
m/s
2 )
Table - Disp
Table - Disp
0 100 200 300 400 500 600 700 800-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
Time (s)
Acc
e. (
m/s
2 )
Table - ACC
Table - ACC
0 2 4 6 8 10 12 14 16 18 200
0.002
0.004
0.006
0.008
0.01
0.012
Frequency (Hz)
PS
D
Table - Disp
Table - Disp
2.207Hz
2.573Hz3.013Hz
3.428Hz
3.843Hz4.283Hz
4.697Hz
5.137Hz
5.552Hz5.943Hz
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
2
2.5
3
3.5x 10
-3
Frequency (Hz)
PS
D
S - ACC
S - ACC3.697Hz
10.02Hz
DX1-Small Amplitude
Input displacement
Input acceleration
PSD
PSD
Input / Output (PSD)
0 1 2 3 4 5 6 7 8 9 1010
-10
10-8
10-6
10-4
10-2
100
Frequency (Hz)
PS
D
Segment 0-360seconds
Pu4xTable-dips2.207Hz
2.622Hz 3.452Hz
3.037Hz 3.843Hz
4.258Hz 5.088Hz
4.673Hz
3.697Hz
0 1 2 3 4 5 6 7 8 9 1010
-10
10-8
10-6
10-4
10-2
100
Frequency (Hz)
PS
D
Segment 360-720seconds
Pu4xTable-dips
3.062Hz
3.526Hz
3.672Hz3.965Hz
4.405Hz
4.844Hz
5.283Hz
5.772Hz
InputOutput
0 100 200 300 400 500 600 700 800-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Time (s)
Acce
. (m
/s2 )
PU4X - 2F
PU4X
0-360s 360-720s
InputResponse
DX1-Small Amplitude
Spectral leakage due toperiodic excitation
Input / Output (Wavelet)Input
Response
0 100 200 300 400 500 600 700 800-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Time (s)
Acc
e.
(m/s
2 )
Table - Disp
Table - Disp
0 100 200 300 400 500 600 700 800-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Time (s)
Acc
e.
(m/s
2 )
PU4X - 2F
PU4X
100÷
300s
300÷
500s
500÷
700s
DX1-Small Amplitude
Input / Output (Wavelet)Input
Response120 140 160 180 200 220 240 260 280
-0.2
-0.1
0
0.1
0.2
0.3
Time (s)
Dis
p.
(m
)
Table - Disp
Input displacement
320 340 360 380 400 420 440 460 480
-0.2
-0.1
0
0.1
0.2
0.3
Time (s)
Dis
p. (m
)
Table - Disp
Input displacement
520 540 560 580 600 620 640 660 680
-0.2
-0.1
0
0.1
0.2
0.3
Time (s)
Dis
p. (m
)
Table - Disp
Input displacement
120 140 160 180 200 220 240 260 280-0.1
-0.05
0
0.05
0.1
Time (s)
Acce
. (m
/s2 )
PU4X - 2F
Acceleration
320 340 360 380 400 420 440 460 480-0.1
-0.05
0
0.05
0.1
Time (s)
Acce
. (m
/s2 )
PU4X - 2F
Acceleration
520 540 560 580 600 620 640 660 680-0.1
-0.05
0
0.05
0.1
Time (s)
Acce
. (m
/s2 )
PU4X - 2F
Acceleration
o Sine sweep excitation:
Simulation of sine sweep force
o Linear sweep:
: Argument (rad): Amplitude (m)
: Instantaneous frequency
: Starting frequency (Hz)
: Ending frequency (Hz)
: Sweep rate (Hz/s)
o Linear argument:
o Input sweep:
Amplitude Sweep Initial frequency& phase
Simulation of sine sweep force
o Setting initial parameters
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
X: 0Y: -0.259
Time (s)
Dis
p.(m
)
Input: Measured
Measured displacement
0 50 100 150 200 250 300 350 400-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Time (s)
Dis
p.(m
)
Theoretical Input
Theoretical displacement
To=0.5s0.18s
0 1 2 3 4 5 6 7 8 9 1010
-20
10-15
10-10
10-5
100
Frequency (Hz)
Mag
nitu
de
PSD
Theoretical input
2.20Hz
2.62Hz
3.03Hz
3.45Hz
3.86Hz
4.28Hz
4.69Hz
5.11Hz
5.50Hz
Simulated sine sweep input
Initial condition (phase)
Comparison bt. simulation & measure
0 50 100 150 200 250 300 350 400-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Time (s)
Dis
p.(m
)
Theoretical Input
Theoretical displacement
0 1 2 3 4 5 6 7 8 9 1010
-20
10-15
10-10
10-5
100
Frequency (Hz)
Mag
nitu
de
PSD
Theoretical input
2.20Hz
2.62Hz
3.03Hz
3.45Hz
3.86Hz
4.28Hz
4.69Hz
5.11Hz
5.50Hz
0 1 2 3 4 5 6 7 8 9 10-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Time (s)
Dis
p.(m
)
Input displacement
MeasuredTheoretical
Simulated and measured input
Simulated input
0 1 2 3 4 5 6 7 8 9 1010
-20
10-15
10-10
10-5
100
Frequency (Hz)
Mag
nitu
de
PSD
Measured inputTheoretical inputMeasured output
Phase difference happens
Frequency Response Function (FRF)
Measured FRFTheoretical FRFSmoothing techniques for FRF
FRFo FRFs: Relationship between input forces x(t) and output responses y(t) in the frequency domain
x(t)InputsExciter
Second order FRFs:
Phase:
Coherence:
Type1: Cross spectrum
Type2: Auto spectrum
Measured FRFDX1-Small Amplitude
0 1 2 3 4 5 6 7 8 9 1010
-5
10-4
10-3
10-2
10-1
100
Frequency (Hz)
Mag
nitu
de
FRF
PU1X/Input
0 1 2 3 4 5 6 7 8 9 1010
-6
10-5
10-4
10-3
10-2
10-1
100
Frequency (Hz)
Mag
nitu
de
FRF
PU2X/Input
0 1 2 3 4 5 6 7 8 9 1010
-3
10-2
10-1
100
101
Frequency (Hz)
Mag
nitu
de
FRF
PU4X/Input
0 1 2 3 4 5 6 7 8 9 1010
-4
10-3
10-2
10-1
100
101
Frequency (Hz)
Mag
nitu
de
FRF
PU5X/Input
Floor 1
Floor 2
3.67Hz
3.67Hz 3.67Hz
3.67Hz
Measured FRF
0 1 2 3 4 5 6 7 8 9 1010
-3
10-2
10-1
100
101
Frequency (Hz)
Ma
gn
itu
de
FRF
PU4X/Input
0 1 2 3 4 5 6 7 8 9 10-10
-5
0
5
10
15
20
25
Frequency (Hz)
De
gre
e
Phase
PU4X/Input
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1Coherence
PU4X/Input
DX1-Small Amplitude
FRF
Phase
Coherence
PU4X and Measured Input
3.67Hz
Theoretical FRFDX1-Small Amplitude
0 1 2 3 4 5 6 7 8 9 1010
-6
10-4
10-2
100
102
Frequency (Hz)
Mag
nitu
de
FRF
PU1X/Measured InputPU1X/Theoretical Input
Floor 2
0 1 2 3 4 5 6 7 8 9 1010
-6
10-4
10-2
100
102
Frequency (Hz)
Mag
nitu
de
FRF
PU2X/Measured InputPU2X/Theoretical Input
0 1 2 3 4 5 6 7 8 9 1010
-6
10-4
10-2
100
102
104
Frequency (Hz)
Mag
nitu
de
FRF
PU4X/Measured InputPU4X/Theoretical Input
0 1 2 3 4 5 6 7 8 9 1010
-6
10-4
10-2
100
102
104
Frequency (Hz)
Mag
nitu
de
FRF
PU5X/Measured InputPU5X/Theoretical Input
3.67Hz 3.67Hz
3.67Hz 3.67Hz
Floor 1
Theoretical FRF
0 1 2 3 4 5 6 7 8 9 1010
-6
10-4
10-2
100
102
104
Frequency (Hz)
Magn
itude
FRF
PU4X/Measured InputPU4X/Theoretical Input
DX1-Small Amplitude
FRF
Phase
Coherence
0 1 2 3 4 5 6 7 8 9 10-30
-20
-10
0
10
20
30
40
50
Frequency (Hz)
De
gre
e
Phase
PU4X/Measured InputPU4X/Theoretical Input
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
Frequency (Hz)
Coherence
PU4X/Measured InputPU4X/Theoretical Input
PU4X and Measured Input
3.67Hz
Smoothing techniques for FRFo Single Block Technique (SBT): One blocko Block Overlapping Technique (BOT): Many blockso Frequency Averaging Technique (FAT): Many blocks
0 50 100 150 200 250 300 350 400-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Time (s)
Acc
e. (
m/s
2 )
PU4X - 2F
PU4X
BOT
FAT
SBT
2N data blocks
1 data blocks
Block=4096 samplesTotal 10 blocksFrequency resolution
Blo
ck 1
Blo
ck 2
nfft nfft samples
No overlappingnfft nfft samples
50% overlapping
Smoothing FRF
0 1 2 3 4 5 6 7 8 9 1010
-4
10-3
10-2
10-1
100
101
Frequency (Hz)
Mag
nitu
de
FRF: PU4X/Input
Single blockNo overlapping50% overlapping
0 1 2 3 4 5 6 7 8 9 1010
-4
10-3
10-2
10-1
100
Frequency (Hz)
Mag
nitu
de
FRF: PU4X/Input
50% block overlapping
DX1-Small Amplitude DX1-Medium Amplitude
0 1 2 3 4 5 6 7 8 9 1010
-4
10-3
10-2
10-1
100
101
Frequency (Hz)
Mag
nitu
de
FRF: PU4X/Input
Single blockNo overlapping50% overlapping
0 1 2 3 4 5 6 7 8 9 1010
-3
10-2
10-1
100
Frequency (Hz)
Mag
nitu
de
FRF: PU4X/Input
50% block overlapping
Effects of smoothing techniques on frequencies and damping
(DX1 - Small amplitude)
Smoothing techniques
Natural frequency
[Hz]
Damping ratio[%]
Single block 3.67 0.27Block overlapping (No overlapping)
3.67 0.28
Block overlapping (50%overlapping)
3.67 0.57
Natural frequencies & Damping ratio estimation
Half-power bandwidth method (HPB) Least-squares complex frequency domain method (LSCF)
Half power bandwidth (SDOF system)
2 2.5 3 3.5 4 4.5 5 5.5 60
0.2
0.4
0.6
0.8
1
1.2
1.4FRF: Input/Output
Frequency (Hz)
Mag
nitu
de
3.67Hz
1.32Hz
0.93Hz
2 2.5 3 3.5 4 4.5 5 5.5 60
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6FRF: Input/Output
Frequency (Hz)
Mag
nitu
de
1.54Hz
1.09Hz
3.67Hz
DX1-Small Amplitude
DX1-Medium amplitude
Amplitude f [Hz] [%]Small 3.67 0.27Medium 3.67 0.26Large no data no data
Lease squares complex frequency domain (MDOF system)
o Relationship bt. input force and output response: Complex value FRF matrix
m: Number of measured pointsn: Number of excited pointsf: Frequency variable
o Least squares solution
o FRF matrix identified by measured inputs/ outputs
Min
Further work
• Natural frequencies can be estimated using identified FRFs. Smoothing should be applied to reduce noise
• Both theoretical or measured inputs can be used to identified FRFs
• Damping estimation of the first mode can be obtained by the half power method, however, comprehensive approach via LSCF should be used
• Next work will be based on the LSCF method for estimating damping ratios
Thank you for your attention
