The Twenty-Sixth KKHTCNN Symposium on Civil Engineering
November 18-20, 2013, Singapore

Damage influence on vibration indices of a passing vehicle
Kyosuke YAMAMOTO1, Yoshinobu OSHIMA2, Kunitomo SUGIURA3
Department of Engineering Mechanics and Energy, University of Tsukuba,
1-1-1 Ten-No-Dai, Tsukuba 305-8573, Japan
yamamoto_k@kz.tsukuba.ac.jp
Department of Civil and Earth Resources Engineering, Kyoto University,
Kyoto Daigaku Katsura, Kyoto 606-8501, Japan2oshima.yoshinobu.3a@kyoto-u.ac.jp, 3sugiura.kunitomo.4n@kyoto-u.ac.jp
ABSTRACTThe new vibration index, spatial singular mode, is proposed in this paper. The numericalsimulation of VBI (Vehicle-Bridge Interaction) system is also carried out and shows the higher
sensitivity of the spatial singular mode to a local bridge damage than traditional vibration
indices.
INTRODUCTION
There still exist technical issues in vibration-based SHM (Structure Health Monitoring) inaccuracy and applicability to civil structures such as bridges. Although many traditional indicessuch as eigen-frequency are global, bridge damage often occurs in local. The spatial singularmode is proposed in this study to implement this settlement.
SPATIAL SINGULAR MODE
The spatial singular mode is a singular mode of corrected acceleration responses of the vehicle
travelling through the bridge. Only two accelerometers and GPS are installed on the vehicle.
Positions of vehicle axles are used to correct the vibration data.
Considering a half car model shown in Fig. 1, the equation of motion is described as𝐌𝒛̈ (t) + 𝐂𝒛̇ (𝑡) + 𝐊𝒛(𝑡) = 𝒇(𝑡)(1)where 𝒛(𝑡) and 𝒇(𝑡) are the (displacement responses of the vehicle) and input (the external
1Assistant Professor
Assositate Professor
3Professor
2

GFig. 1 VBI systemforce) which are given by
𝑧𝑠1 (𝑡)0𝑧𝑢1 (𝑡)𝑐𝑢 𝑢1̇ (𝑡) + 𝑘𝑢 𝑢1 (𝑡)}𝒛(𝑡) =, 𝒇 (𝑡 ) = {0𝑧𝑠2 (𝑡)𝑐𝑢 𝑢2̇ (𝑡) + 𝑘𝑢 𝑢2 (𝑡){𝑧𝑢2 (𝑡)}(2)where 𝑧𝑠𝑖 (𝑡), 𝑧𝑢𝑖 (𝑡) and 𝑢𝑖 (𝑡) are sprung-mass displacement, unsprung-mass displacement
and input displacement, respectively. The subscript 𝑖 denotes the front axle (𝑖 = 1) and therear axle (𝑖 = 2). 𝐌, 𝐂 and 𝐊 are mass, damping and stiffness matrices and given by𝑚𝑠 ⁄2𝐼 ⁄2𝑙
𝐌=[𝑃0000𝑚𝑢
0

𝑚𝑠 ⁄2− 𝐼𝑃 ⁄2𝑙00𝑐𝑠𝑙𝑐𝑠
𝐂 = [−𝑐𝑠000]0𝑚𝑢−𝑐𝑠−𝑙𝑐𝑠
𝑐𝑠 + 𝑐𝑢
0𝑐𝑠−𝑙𝑐𝑠0−𝑐𝑠−𝑐𝑠𝑙𝑐𝑠
0 ]𝑐𝑠 + 𝑐𝑢𝑘𝑠𝑙𝑘𝑠
𝐊=[−𝑘𝑠0−𝑘𝑠−𝑙𝑘𝑠
𝑘𝑠 + 𝑘𝑢0𝑘𝑠−𝑙𝑘𝑠
0−𝑘𝑠−𝑘𝑠𝑙𝑘𝑠]
0
𝑘𝑠 + 𝑘𝑢(3)

where 𝑚𝑠 , 𝐼𝑝 , 𝑘𝑠 , 𝑐𝑠 , 𝑙, 𝑚𝑢 , 𝑘𝑢 and 𝑐𝑢 are the sprung-mass, inertia, stiffness, damping,distance between the center of gravity and an axle, and the unsprung-mass, stiffness anddamping, respectively.
Assuming one-dimensional simple beam as the bridge on the 𝑥-coordinate, the input
displacement can be described as𝑢𝑖 = 𝑅(𝑥𝑖 (𝑡)) + 𝑦B(𝑥𝑖 (𝑡), 𝑡)(4)

where 𝑅(𝑥 ) and 𝑦B(𝑥, 𝑡) are the road unevenness and the bridge displacement. The vehicle
pathway is also on the 𝑥-coordinate, and 𝑥𝑖 (𝑡) denotes the axle position. The bridge response
can be decomposed into modal responses such as∞𝑦B(𝑥, 𝑡) = ∑ 𝜙𝑘 (𝑥 )𝑞𝑘 (𝑡)

(5)

𝑘=1

where 𝜙𝑘 (𝑥 ) and 𝑞𝑘 (𝑡) are 𝑘 -th mode shape and basis coordinate of the bridge.
Considering only first and second mode, the bridge component vector observed by thetravelling vehicle is given byTable 1

The standard parameters of vehicle

Mass𝑚𝑠Stiffness 𝑘𝑠
Damping 𝑐𝑠
Inertia
𝐼𝑃
Distance𝑙
Unsprung- Mass𝑚𝑢
Stiffness 𝑘𝑢
Damping 𝑐𝑢
Run speed𝑣
Sprung-Table 2

18,000[kg]1.0 × 106 [kg/s2]
1.0 × 104 [kg/s]
64958[kg m2]
1.875[m]1,100[kg]
3.5 × 106 [kg/s2]
3.0 × 104 [kg/s]
10.0[m/s]

𝜙1 (𝑥1 (𝑡))𝒚𝐁 (𝑡 ) = 𝚽 (𝑡 )𝒒 (𝑡 ) = [𝜙1 (𝑥2 (𝑡))The bridge parameters

Span𝐿
LengthFlexural
𝐸𝐼
Stiffness
Mass per𝜌𝐴
unit length
Reyleigh
𝛼coefficients 𝛽

30.0[m]1.56 × 1010[Nm]3,000[kg/m]
0.238
0.000

𝜙2 (𝑥1 (𝑡)) 𝑞1 (𝑡)]{}.
𝜙2 (𝑥2 (𝑡)) 𝑞2 (𝑡)(6)The unsprung-response vector 𝒛̈ 𝑢 (𝑡) is𝑧 (̈ 𝑡)𝒛̈ 𝑢 (𝑡) = { 𝑢1 } = 𝚽(𝑡)(𝑐𝑢 𝒒̇ (𝑡) + 𝑘𝑢 𝒒(𝑡)) + 𝝐(𝑡)𝑧𝑢2 (𝑡)(7)

where 𝝐(𝑡) is the error term including the sprung-mass response component and the roadprofile component. Shape function matrix is given by
12𝐿− (𝑥1 (𝑡) − )3𝐍 (𝑡 ) = [ 𝐿12𝐿− (𝑥2 (𝑡) − )𝐿
31𝐿(𝑥1 (𝑡) − )𝐿3 ].
1𝐿(𝑥2 (𝑡) − )𝐿3(8)where 𝐿 is the span length of the bridge. Substitution of Eq. (8) into Eq. (7), we obtain𝒛𝒖̈ (𝑡) = 𝐍(𝑡)𝐀(𝑐𝑢 𝒒̇ (𝑡) + 𝑘𝑢 𝒒(𝑡)) + 𝝐(𝑡).

(9)Assuming 𝐍 −1 (𝑡)𝝐(𝑡) = 𝐀𝝐̃(𝑡) and 𝝈(𝑡) = 𝑐𝑢 𝒒̇ (𝑡) + 𝑘𝑢 𝒒(𝑡) + 𝝐̃(𝑡), Eq. (9) becomes𝐍 −1 (𝑡)𝒛̈ 𝑢 (𝑡) = 𝐀𝝈(𝑡).

(10)𝐍 −1 (𝑡)𝒛̈ 𝑢 (𝑡) is the corrected vehicle response vector. Applying SVD (Singular ValueDecomposition) to 𝐍 −1 (𝑡)𝒛̈ 𝑢 (𝑡), 𝐀 can be estimated as the spatial singular mode.
NUMERICAL SIMULATION
To examine the efficiency of spatial singular mode, the numerical simulation of VBI system iscarried out. The parameters of the vehicle is shown in Table 1. The bridge is modeled by onedimensional finite beam elements. The parameters of the bridge are also shown in Table 2.
The first undamped natural frequency is 3.96[Hz]. The measurements are the acceleration

Intact
EI -20%
EI -40%EI -60%EI -80%

0.67

0.72
0.71

0.66

0.7

0.65

0.69
0.68

0.64

0.73

0.74

0.75

0.76

0.77

0.67
0.68

IntactEI -20%
EI -40%EI -60%EI -80%

0.69

0.7

0.71

0.72

0.73

(a) Road profile: Extra Good(b) Road profile: GoodFig. 4 The results of estimation for spatial singular mode (Damage Location: 3[m]=L/10)
responses of the unsprung-mass with 1% normal white noise. The bridge damage is introduced
by decreasing 𝐸𝐼 locally.

RESULT AND DISCUSSION
Two mode shapes at two locations can be obtained from two accelerometers, shown in Fig 3.
2The mode shape in this study is normalized to satisfy 𝐴1𝑘+ 𝐴22𝑘 = 1. Fig. 4 shows 𝐴11 and𝐴21 relationships in different road profiles for each bridge state. The damage location is L/10
and the width is 1[m]. The mode is calculated 10 times for each case by Monte CarloSimulation. The plot group for each bridge state are distinctively different. It means that thevariety of the spatial singular mode is larger than the errors of measurements.

CONCLUSION
This study proposes applying spatial singular mode to bridge damage detection and examines
the efficiency of the proposed method. According to the results, estimated spatial singular modechanges due to local damage larger than measurement errors. Thus, it is shown that highprobability of the proposed method for bridge damage detection.

ACKNOLEDGMENT
This study is done by "Grant-in-Aid for Young Scientists(B)” (No. 25820200)
REFERENCE
Oshima, Y., Yamamoto, K. and Sugiura, K.: Damage identification method of a bridge based on its estimateddisplacement by the responces of a passing vehicle, Journal of Structural Engineering. A, JSCE, Vol.57A, pp.646654, 2011.