The Application of the SVD-FDD Hybrid Method toBridge Mode Shape Estimation
Sakai Masaki
Graduate Student (M1)
Department of Engineering Mechanics and Energy,
Graduate School of Systems and Information Engineering,

University of Tsukuba

Yamamoto Kyosuke
Assistant Professor
Research Group of Engineering Mechanics and Energy,
Faculty of Engineering, Information and Systems

University of Tsukuba

1Our Theme

Easy and Reliable Bridge Inspections(Low-cost & Labor-saving)

Anyone can do it(Objective: ex) vibration-based SHM)

Same result no matter who does it
Visual Check, or Hammering test
are accurate but labor-consuming

Target Bridges and Methodology

Vibration-based SHM for Short-span Bridges
Sensors are installed on the bridges

More than 700,000 in Japan

Mechanical Characteristics

Traffic-induced Vibration is predominant
Predominant Frequencies and Mode Shapes do not match the Natural ones

Merit on using Traffic-induced Vibration

Damage Information may be easily foundLocal damage usually affects the higher order modal parameters
But, how to extract
the damage informationfrom measured data?

Evaluation Indicator

Mode Shape is a good Damage Indicator
Defects locally reduce the flexural rigidity 𝐸𝐼 and mass 𝜌𝐴

Evaluation Indicator

Mode Shape is a good Damage Indicator
Defects locally reduce the flexural rigidity 𝐸𝐼 and mass 𝜌𝐴and Mode Shape is sensitive to local damages

Local Damage

What is Mode Shape?

Mode Shape Matrix is Orthogonal
Free vibration

Equation of Motion−𝜔 𝐌𝜶 e𝐌𝐊 𝜶

+ 𝐊𝜶 e=𝜔= 𝟎 Mode Shape𝜶Natural Frequency

Mode Shape Matrix

Eigen-value
Decomposition
Natural Frequency

How to estimate?

Operational Modal Analysis is needed
To estimate the modal parameters from transient responses

the SVD method

the FDD method

Singular Value Decomposition
SVD is like Eigen-Value Decomposition
often used for Data-Compression

Frequency Domain Decomposition
FDD is like Cross-Power method
for mode shape and Frequency

Applicable to estimate
Bridge Mode Shapebut No Consensus

Very Popular method
for BMS estimation
but complicatedthe SVD method

SVD assumes thatis uncorrelated
Diagonalization

Assuming that 𝒒 𝑡 is uncorrelated,
𝐃 becomes to be diagonal

the FDD method

FDD assumes thatis independent

If 𝜔 = 𝜔 , the shape of 𝒚 𝜔 matches the 𝑘-th mode shape
Diagonalization

cross-power spectrum

𝐒If 𝑞 𝜔 is not zero,𝑞 𝜔 (𝑙 = 1~𝑛, except 𝑘) are zero

𝜔 is called Singular Spectrum, of which peaks are estimated natural frequency

The 𝑘-th order estimated mode shape 𝐀 𝜔

can be found by detecting 𝑘-th peak 𝜔Different Assumptions

SVD and FDD uses different assumptions
If the bridge is in the state of free vibration or excited by white noise loads,
the results from SVD and FDD will be the same
Traffic-induced vibration does not satisfy this condition

the Size of Difference can be used to evaluate damage?
(Hybrid Method)

The Purpose of this StudyTo Compare the Mode Shapes estimated by SVD and FDD
Similarity is evaluated by MAC

Bridge Vibration Data is simulated by numerical simulation
Vehiclem s Ix Iy
ycs4
mu4

Ly
x2

cs2 ks2
cs1
ks1
ks4 mu2
ks3 mu1
ku2 cs3 mu3 ku1
ku4x1
Lxku3
E 𝜈BridgeRoad RoughnessxB2

1.00
-1.0
-2.0
2.0

4.0

5.0

6.0

7.0

8.0

9.0

10

Acceleration [m/s2 ]

2.0

2.0

2.0

B3

1.0
00

-1.0

-1.0

-2.0

4.0

5.0

7.0

8.0

9.010Time [s]

B1

1.0

6.0

B1
4.0

5.0

6.0

7.0

8.0

9.0

B2B3

B4B5

2.0

:sensors
B60B7B8B9B10

-1.0
2.0

5.0

6.0

7.0

8.0

9.0

10

B5

4.0

5.0

6.0

7.0

8.0

9.0

10

B10

1.00
-1.0

4.0

5.0

6.0

7.0

8.0

9.0

10

B7

1.0

-2.0
2.0

2.0

B8

1.0
00

-1.0

-1.0

-1.0

4.0

5.0

6.0

7.0

8.0

9.0

10

-2.0

4.0

5.0

6.0

7.0

8.0

9.0

4.0

5.0

6.0

7.0

8.0

10

-2.0

9.0

10B9

1.00

-2.0

4.0

-1.0-2.0

10

B6

1.0

-2.0

2.00

-1.0
2.0-2.0
1.0

0-2.0

B4

1.0

4.0

5.0

6.0

7.0

8.0

9.0

10

Estimated Bridge Mode Shapes (Low Order)

The SVD and FDD’s estimates are similar in Low Order

0-1.0
0

1
SVDestimated1.0
Amplitude

Amplitude

1.0

FDDestimated20

FDDestimated

10 20 30 0
10 20 30
(Opposite-side)(Pathway-Side)
Location [m]-1.00SVDestimated
10 20 30 0
10 20 30
(Opposite-side)(Pathway-Side)
Location [m]Estimated Bridge Mode Shapes
FDD
1100.5-101

100-1-11100-1-11100-1-1

10-1FDDCorrect
SVD

10-1SVD

Correct

However, those inHigher Order
are much different
Comparing them with Correct values
the accuracy of SVD is better

Singular Spectrum from FDD
The hidden (not-excited) mode

Correct NaturalFrequencies

Singular value

106105
104
103

S1102

S2101
0102030405060

70

80

90

100

Frequency [Hz]SS of FDD cannot also find the natural frequencies
that SVD can estimate

Conclusion
Traffic-induced Vibration is numerically simulated to compare
the applicability of SVD and FDD to Operational Modal Analysis
SVD is not popular for Operational Modal Analysis
but shows good accuracy to estimate Bridge Mode Shapes
from Traffic-induced Vibrations
FDD is a popular method for Operational Modal Analysis
but it cannot estimate the hidden (not-excited) modes
SVD can estimate the hidden modes from the orthogonality of Mode Matrix