POLITECNICO DI MILANO | JULY 2-6, 2023

Applicable schemes for the Vehicle-Bridge
Interaction System Identification method
YAMAMOTO Kyosuke, SHIN Ryota
University of Tsukuba(Japan)Introduction|The latent demand of our society• Bridge Screening for determining priorities and necessities of inspections• The vase number of bridges scattered over the wide area > the number of engineers• We need to allocate personnel and budget to damage-suspected bridges with a focus2

Solution|How to realize the bridge screening• Drive-by bridge monitoring can be an option for bridge screening• Sensors are installed only on traveling vehicles (Not in bridges)• Swift and cost-effective bridge diagnostics by passing sensor-equipped vehicles over bridges

Sensors are installedon traveling vehicles

Engineers inspect only
“damage-suspected” bridges

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Reviews|Development of Drive-by bridge monitoring• Measuring vehicle vibrations to extract bridge feature values• The first natural frequency of a bridge can be identified as a peak in Fourier’s power spectraof vehicle vibrations. (Yang et al, Sound and Vibration, 2004)• The mode shapes are also identified by using a multi-trailer system. (Yamamoto et al, JSCE
journal paper, 2012), (Yang et al, Mechanical Systems and Signal Processing, 2021)• The bridge damages can be detected / estimated by monitoring the variations of thesebridge feather values. However, you need to measure the intact values.

• Measuring vehicle vibrations to identify vehicle parameters and road profile• Drive-by monitoring for road pavement inspections• The vehicle parameters and road profiles can be simultaneously estimated. (Xue et al,
Mechanical Systems and Signal Processing, 2020), (Keenahan et al, Str. and Inf. Eng., 2020)• The parameters are optimized to decrease the road profile residual of front and rear wheels.
• This idea can be extemded to estimate vehicle and bridge parameters. (Yamamoto et al,Applied Sciences, 2023), (Shin et al, Sensors, 2023)
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Existing Studies|The VBI system identification method
Measure Vehicle Vibration（𝒘 ）Assuming the Mechanical Parameters（Vehicle：𝐌 , 𝐂 , 𝐊 、Bridge： 𝐌 , 𝐂 , 𝐊 ）Substitution
𝐌 𝒘

Repeat random-assuming

𝐂 𝒘

Substitution as the traffic loads
𝐊 𝒘

𝐌 𝒘

𝐂 𝒘

𝐊 𝒘

𝒇𝒃（Input Estimation Problem of Vehicle） （Dynamic Response Simulation of Bridge）

𝒇

Estimate Input Profile（𝒖） Estimate Bridge Deflection（𝒘 ）Estimate Road Profile（𝒓）Do road roughness estimated between
the front and rear wheels match?

NOYESEND

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Existing Studies|The VBI system identification method• The proposed method
1. Measure the vehicle vibration data
2. Assume the system paramters randomly
3. Equation of Motion of VBI system
4. Estimate the road profile
5. Evaluate the likelihood on road roughness
6. Repeat from 2• VBI (Vehicle-Bridge Interaction) system can be identified• Vehicle parameters: 𝑚 , 𝑐 , 𝑘 , 𝑚 , 𝑘 (𝑖: front/rear wheels)• Bridge parameters / responses: 𝜌𝐴, 𝐸𝐼 𝑥 , 𝛼, 𝛽, 𝑤 𝑥, 𝑡
• Road surface unevenness: 𝑅 𝑥 from 𝑟 𝑡
𝑅 𝑥 𝑡

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Technical Issue|Efficient Optimization Algorithm• To search the optimal solution (combination of vehicle-bridge paramters)that minimizes road unevenness residuals, we have several options:• MCMC (Monte Carlo Markov chain)• Randomly vary the candidate parameters incrementally• PSO (Particle Swarm Optimization)• Directionally vary the candidate parameters• Nelder-Mead method• Geometrically vary the candidate parameters

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Study Purpose|Optimization Algorithm• This study compares the MCMC, PSO and Nelder-Mead methods anddiscusses the applicability of these algorithms to the proposed scheme.
• The vehicle vibration data are numerically simulated

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MCMC method|Monte Carlo Markov chain• Randomly varying the parameters• wide range search• simplicity in implementation
However…
• high computational cost
• low efficiency𝑘

𝛥𝑅=100
𝛥𝑅=90

𝛥𝑅=80

𝛥𝑅=120
𝛥𝑅=70

𝛥𝑅=71
𝛥𝑅=69
Solution Space𝑚9

PSO method|Particle Swarm Optimization• Directionally varying the parameters• efficient search
However…
• high computational cost
• prone to local optima

𝑘𝛥𝑅=100

𝛥𝑅=90
𝛥𝑅=80

𝛥𝑅=80

𝛥𝑅=120𝛥𝑅=70

Global best

Solution Space𝑚10

PSO method|Particle Swarm Optimization• Directionally varying the parameters• efficient search
However…
• high computational cost
• prone to local optima𝑘

𝛥𝑅=100

𝛥𝑅=90
𝛥𝑅=80

𝛥𝑅=80

𝛥𝑅=120𝛥𝑅=60
𝛥𝑅=70
Solution Space

𝑚11

PSO method|Particle Swarm Optimization• Directionally varying the parameters• efficient search
However…
• high computational cost
• prone to local optima• dependent on the initial values

𝑘𝛥𝑅=90

𝛥𝑅=100𝛥𝑅=80
Global BestLocal Best

𝛥𝑅=80
𝛥𝑅=65

𝛥𝑅=120 Inertia
𝛥𝑅=90
𝛥𝑅=70

𝛥𝑅=60
𝛥𝑅=65

Solution Space𝑚12

Nelder-Mead method|Adaptive scheme• Geometrically varying the parameters• efficient search• low computational cost
• applicable even for small gradients

𝑘

𝛥𝑅=70

𝛥𝑅=100

Solution Space

𝛥𝑅=80

𝑚
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Nelder-Mead method|Adaptive scheme• Geometrically varying the parameters• efficient search• low computational cost
• applicable even for small gradients

𝑘

𝛥𝑅=90

𝛥𝑅=70

𝛥𝑅=100
𝛥𝑅=120

𝛥𝑅=50Solution Space

𝛥𝑅=80𝑚14

Nelder-Mead method|Adaptive scheme• Geometrically varying the parameters• efficient search• low computational cost
• applicable even for small gradients

𝑘𝛥𝑅=90

𝛥𝑅=70

𝛥𝑅=100
𝛥𝑅=120

𝛥𝑅=50
Solution Space

𝛥𝑅=80𝑚15

Numerical Simulation|to simulate vehicle vibrations• VBI system is modeled as Multibody-Continuum interaction system• Vehicle: Rigid-body and Suspension• Bridge: FE model using 1D finite beam elements
Body

30m spanbridge

𝑚𝑠8310 [kg]𝑑1

1.215 [m]Front TireMass𝑚𝑢 1469 [kg]Stiffness

𝑘𝑢 14,790,000 [kg/s2]

Mass

𝑚𝑢 2

751 [kg]

Stiffness𝑘𝑢 2

4,310,000 [kg/s2]

Rear from G 𝑑2

2.175 [m]Damping

𝑐𝑠1

24,200 [kg/s]

Suspension

Stiffness

𝑘𝑠1

456,000 [kg/s2]

Mass per unit length

𝜌𝐴

4400 [kg/m]

RearDamping

𝑐𝑠2

29,000 [kg/s]

Flexural Rigidity

𝐸𝐼𝑖

1.56×1011 [Nm2]

Suspension

Stiffness

𝑘𝑠2

431,000 [kg/s2]

𝛼0.7024

𝛽0.0052

Front

Disp.[𝑚]10 tonvehicle

Mass
Front from G

Rayleigh Damping

Bridge102

Rear Tire

0220

1001020304050

Vehicle position[𝑚]Road Unevenness
16Simulated Data|vehicle vibrations

50

Acc.[𝑚/𝑠 ]

0.5

250

Acc.[𝑚/𝑠 ]Acc.[𝑚/𝑠 ]Acc.[𝑚/𝑠 ]• Vehicle vibration data are simulated:

00.5

20

10010203040

202 201001020

Vehicle position[𝑚]

30

40

00.5

20

1001020

30

40

50

2 20

100102030

40

50017

Results|Appllying the proposed method with MCMC to the data
• Implementing the optimization process using MCMC method
1.6
1.4

𝑐1.2
1.0
1.00.8𝑚𝑚𝑐𝑘𝛼𝑘𝑚𝑘𝑚𝑘𝐸𝐼𝐸𝐼

0.6
0.40.2Body

Suspensions
Tires
Vehicle Parameters

𝐸𝐼𝜌𝐴

𝛽

Flexural Rigidity
Bridge Parameters

0.018

Results|Appllying the proposed method with PSO to the data
• Implementing the optimization process using PSO method
1.6
1.4
𝑐

1.2
1.0
1.00.8

𝑚𝑚𝑐𝑘

𝛼𝑘
𝑚𝑘𝑚𝑘

𝐸𝐼

𝐸𝐼𝐸𝐼

𝜌𝐴

𝛽0.6
0.4
0.2

Body

Suspensions
Tires
Vehicle Parameters

Flexural Rigidity
Bridge Parameters

0.019

Results|Appllying the proposed method with NM to the data
• Implementing the optimization process using Nelder-Mead method
1.6
1.4𝑐1.2
1.0
1.00.8𝑚𝑚𝑐𝑘𝛼𝑘𝑚𝑘𝑚𝑘

𝐸𝐼𝐸𝐼𝐸𝐼𝜌𝐴

𝛽0.6
0.4
0.2

Body

Suspensions
Tires
Vehicle Parameters

Flexural Rigidity
Bridge Parameters

0.020

Discussion|Comparison of three algorithms• Nelder-Mead is recommended for the optimization process• MCMC is costly and less accurate than both PSO and Nelder-Mead• PSO presents high accuracy but much more costly than Nelder-Mead
1.5
1.0
0.5

Lower accuracy
and large variability

0.0

1.5

1.5

1.0

1.00.5Better accuracy
and small variability

0.0

MCMC

0.5

Better accuracyand smallest variability

0.0

PSO

Nelder-Mead21

Conclusion|Applicability of Existing Optimization Schemes• The proposed method aims to simultaneously estimate vehicle and bridgeparameters and road unevenness only from vehicle vibration data.
• This method includes random search process for minimizing estimatedroad unevenness residual.
• significant computational cost due to the curse of dimensionality• Nelder-Mead method is recommended to use for the optimization process.
• Note that this validation is just based on numerical simulation• Necessary to validate this method through experiment22

Summary|Thank you for your attention
the proposed Drive-by Bridge Monitoring method:

Nelder-Mead method
is recommended

Vehicle vibrations are
simulated numerically
We can estimate
bridge parameters
and responses

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