Numerical Study of the Effect of Measurement Noise on the Accuracy of Bridge Parameter Estimation in VBI System Identification - IAENG

Proceedings of the World Congress on Engineering 2021
WCE 2021, July 7-9, 2021, London, U.K.

 Numerical Study of the Effect of Measurement
 Noise on the Accuracy of Bridge Parameter
 Estimation in VBI System Identification
 Shin Ryota, Inoue Jun, Okada Yukihiko and Yamamoto Kyosuke, Member, IAENG

 minimized.
 Abstract— In this study, we propose a method to Murakami[3] proposes a method for simultaneously
 simultaneously estimate a vehicle's parameters and a bridge by estimating the vehicle, bridge, and road surface from the
 estimating the road surface roughness of the front and rear vibration data of multiple vehicles. The road profile input to
 wheels from the vibration data of a running vehicle and
 minimizing the difference between them. It is possible to
 each of the multiple vehicles is estimated from the vehicle
 estimate the vehicle, bridge, and road surface unevenness vibration alone. The combination of vehicle and bridge
 simultaneously from vehicle vibration. In this study, we focused parameters that minimizes the error in the obtained road
 on the shape of the objective function of the bridge parameters. profile is searched. Since the shape of the objective function
 We found that the objective function of the bending stiffness is is unknown, the vehicle and bridge parameters and the road
 always downward convex. In contrast, the unit weight of the surface roughness are estimated based on particle swarm
 bridge is multimodal.
 optimization(PSO). This method's features are: 1) only
 Index Terms— structure health monitoring , vehicle response vehicle vibration is used without bridge vibration, and 2) not
 analysis, vehicle-bridge Interaction System only vehicle parameters, and road surface profile but also
 bridge parameters are estimated simultaneously. However,
 the actual vehicle vibration is greatly influenced by the engine
 I. INTRODUCTION vibration. Besides, since the time that multiple vehicles travel
 on a bridge simultaneously is short, it is more practical to use
 D UE to the aging of infrastructure structures related to
 logistics networks, the development of maintenance
 techniques for road bridges has become an urgent issue.
 a single vehicle as in Nagayama et al.[2]
 In this study, we improve the Murakami model's
 Therefore, bridge monitoring methods using signals have estimation method by constructing a model that considers the
 been proposed. They can be classified into bridge response effects of external disturbances such as engine vibration. Also,
 analysis and vehicle response analysis. Bridge response we clarify the shape of the objective function of the bridge
 analysis uses sensors mounted directly on the bridge, parameters estimated by this method. The effect of
 enabling accurate investigation of the bridge's health, but is measurement noise on each parameter's estimated values is
 expensive. In vehicle response analysis[1], sensors are determined, and its influence on the objective function is
 mounted only on the vehicle. The acceleration response clarified. The experiments were conducted using numerical
 obtained when the vehicle runs over an infrastructure experiments.
 structure can be used for indirect damage detection. Therefore,
 it can be implemented at a relatively low cost. II. BASIS THEORY OF THE PROPOSED METHOD
 Nagayama et al.[2] proposed a method for estimating the A. Vehicle-Bridge Interaction (VBI)
 road profiles of the front and rear wheels by measuring the
 rigid behavior of the vehicle using a smartphone and VBI is a phenomenon that occurs when a vehicle runs over
 combining the Kalman filter, RTS (Rauch-Tung-Striebel) a bridge. It is caused by the vehicle and bridge systems'
 smoothing, and the Robbins-Monro algorithm. Assuming that equations of motion, the contact force between the vehicle
 the same road profile is input to the front and rear wheels, the and the bridge, and the unevenness of the road surface. When
 genetic algorithm is used to simultaneously identify the a vehicle enters a bridge, the unevenness of the road surface
 vehicle's parameters so that the difference between the first shakes the vehicle. Then, the vehicle shakes the bridge.
 estimated road profiles of the front and rear wheels is In addition, the shaken bridge and the uneven road surface
 shake the vehicle in succession. The conceptual diagram of
 Manuscript received March 23, 2021; revised April 24, 2021. This VBI is shown in Fig. 1.
 work was partly supported by JSPS KAKENHI Grant Number 19H02220
 (Grant-in-Aid for Scientific Research(B)).
 Shin Ryota is a doctoral student of University of Tsukuba, Ibaraki
 prefecture, Japan (e-mail: s2130108@s.tsukuba.ac.jp).
 Inoue Jun is a master student of University of Tsukuba, Ibaraki
 prefecture, Japan (e-mail: s1920887@s.tsukuba.ac.jp).
 Okada Yukihiko is an associate professor of University of Tsukuba,
 Ibaraki, Japan (e-mail: okayu@sk.tsukuba.ac.jp).
 Yamamoto Kyosuke is an assistant professor of University of Tsukuba,
 Ibaraki, Japan (e-mail: yamamoto_k@kz.tsukuba.ac.jp).

ISBN: 978-988-14049-2-3 WCE 2021
ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

Proceedings of the World Congress on Engineering 2021
WCE 2021, July 7-9, 2021, London, U.K.

 INP

 forced dis lacement P ̈ ( ) = − 1 ( ̇ 1 ( ) − ̇ 1 ( ))
 Vehicle System ehicle i rations
 System Parameters and − 2 ( ̇ 2 ( ) − ̇ 2 ( ))
 (1)
 − 1 ( 1 ( ) − 1 ( ))
 Sensors installed
 oad Profile only on a tra eling ehicle − 2 ( 2 ( ) − 2 ( ))

 ̈ ( ) = 3 − 1 × 1 ( ̇ 1 ( ) − ̇ 1 ( ))
 + 2 × 2 ( ̇ 2 ( ) − ̇ 2 ( ))
 P INP (2)
 ridge i rations contact forces − 1 × 1 ( 1 ( ) − 1 ( ))
 =
 Bridge System + 2 × 2 ( 2 ( ) − 2 ( ))
 System Parameters and

 Fig.1 Conceptual image of Vehicle-Bridge Interaction ̈ ( ) = ( ̇ ( ) − ̇ ( ))
 + ( ( ) − ( )) (3)
 − ( ( ) − ( ))

 Here, the displacement and rotation at the center of gravity
 of the rigid body are expressed using the displacements at the
 front and rear wheel positions.

 2 1 ( ) + 1 2 ( )
 ( ) = (4)
 1 + 2

 1 ( ) − 2 ( )
 ( ) = ( )
 Fig. 2. Dynamical model diagram assumed in this study 1 + 2
 (5)
 1 ( ) − 2 ( )
 B. Vehicle System ≅
 1 + 2
 In this study, the vehicle model is the half-car model shown
 in Fig. 2. A rigid body of mass is used as the vehicle body,
 Equation (1) through Equation (5) can be summarized as
 connected to the ground by a suspension modeled as a spring
 follows.
 and dampers (spring constant: , damping: ) and tires v ̈ ( ) + v ̇ ( ) + v ( ) = v ( ) (6)
 modeled as a single spring (spring constant: ).Here, 
 represents the axle, and the front wheels are 1, and the rear 2 1 
 wheels are 2. The excitation force due to engine vibration is
 1 + 2 1 + 2
 represented by . Between the suspension and the tires, there
 
 is a mass , which is called the unsprung mass, while the v = (7)
 car's body is called the sprung mass. Since the sprung mass is 1 + 2 1 + 2
 a rigid body, it can be expressed by two equations of motion, 1
 one for translation and one for rotation. The vertical [ 2 ]
 displacement vibration at the center of gravity on the spring
 1 2 − 1 − 2
 is ( ), the rotation is ( ), the displacement vibration at
 1 1 − 2 2 − 1 1 2 2
 the front and rear wheel positions is ( ). The displacement v = [ − ] (8)
 vibration under the spring is ( ) .The first-order time 1 1
 − 2 2
 derivative is denoted by ( ̇ ) and the second-order time
 derivative by ( ̈ ).The distance from the center of gravity to v
 the front and rear wheels is , and the distance to the engine 1 2 − 1 − 2
 is 3 .The equations of motion for the vehicle model are then 1 1 − 2 2 − 1 1 2 2 (9)
 as follows. =
 − 1 1 + 1
 [ − 2 2 + 2 ]
 TABLE I
 THE VEHICLE SYSTEM PARAMETERS ( ) = [ 1 ( ), 2 ( ), 1 ( ), 2 ( )] (10)
 Mass:ms 9.00×103 [kg]
 Unsprung-Mass:mu1,mu2 5.00×102 [kg*m2] v ( ) = [ ( ), 3 ( ), 1 1 ( ), 2 2 ( )] (11)
 Damping (Sprung-mass):cs1,cs2 2.00×103 [kg/s]
 Stiffness(Sprung-mass):ks1,ks2 4.50×103 [kg/s2] v , v and v are the mass, damping, and composite
 Stiffness(Upsprung-mass):ku1,ku2 6.00×104 [kg/s2] matrices of the vehicle. v is the external force term, which
 Distance between axles: d1+d2 3.00 [m] consists of the excitation force f due to engine vibration and
 the input displacement to the tire. The parameters of the
 vehicle are shown in TABLE I.

ISBN: 978-988-14049-2-3 WCE 2021
ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

Proceedings of the World Congress on Engineering 2021
WCE 2021, July 7-9, 2021, London, U.K.

 C. Bridge System
 The bridge is assumed to be a simple one-dimensional
 beam. Suppose the vibration of the bridge is ( , ). In that (19)
 case, the bending stiffness is , the mass per unit length is
 , and the external force is , the equation of motion of the
 bridge is expressed by equation (12).
 Both components are assumed to be zero outside the
 element. The component of the deformation vector at the
 2 2
 ̈ ( , ) + ( ( , )) = (12) node is given by equation (20) using the deflection
 2 2
 and the deflection angle ( ).
 2
 =∑ ( − ( )) ( ) (13) 
 =1 (20)
 
 Here, the external force is the concentrated external force
 in the vertical direction at ( ) for each axle position of the In this case, the approximate solution of the solution y(x,t)
 front and rear wheels, and ( ) is the concentrated external is expressed by equation (21).
 force in the vertical direction. However, is Dirac's delta
 function and satisfies the following conditions. ∙ (21)

 The weights can be similarly transformed into equation
 (14) (22) by substituting ( ) = ( ) ∙ into the weighted
 residue equation.
 (15)
 T ( B ̈ ( ) + B ( ) − ( )) = 0 (22)
 The weighted residue equation in Eq. (12) is given by Eq.
 (16) below. B and B refer to the mass and composite matrix of the
 bridge, respectively. ( ) is a vector of deformations with
 
 2 4 components of deflection and deflection angle at each node.
 ∫ ( 2
 + 4 − ) d = 0 (16) ( ) is a vector of external forces with concentrated external
 0 
 force components and moment load of force at each node.
 The equality condition of equation (15) is obtained for any ,
 Let denote the weights. We change equation (16) to the
 and the finite element equation shown below is obtained by
 weak formulation. introducing a damping term.
 
 2 2 2 B ̈ + B ̇ + B = ( ) (23)
 ∫ ( + − ) d = 0 (17)
 0 2 2 2
 The parameters of the bridge are shown in TABLE II.
 A one-dimensional finite element beam model with
 Hermite interpolation function is applied to numerically TABLE Ⅱ
 calculate the bridge vibration. The Hermite interpolation THE BRIDGE SYSTEM PARAMETERS
 function can be defined for an element coordinate system Length 30 [m]
 (normalized to − ≤ ≤ ). Number of Elements 6
 Mass per unit length
 3000 [kg/m]
 values of all elements:
 Flexual Rigidities of all
 1.56×1011 [N×m2]
 elements: 
 (18)
 D. Vehicle Bridge Interaction System
 In general, the response of a vehicle and a bridge is
 modeled by their outputs with each other as inputs. First, the
 vehicle vibration is determined using only the road profile.
 When the whole coordinate system is inside a beam Then, the bridge vibration is calculated by determining the
 element consisting of nodes and +1 , where ∆ = input load to the bridge from the vehicle vibration and the
 +1 − , the components of the basis function vector vehicle model. The bridge vibration is then added to the road
 of the interpolation are expressed as follows. profile to obtain the input displacement to the vehicle. By
 repeating this process, the input displacements to the vehicle
 and bridge are obtained. The input profile and ground force,
 which are the vehicle and bridge inputs, are described below.

ISBN: 978-988-14049-2-3 WCE 2021
ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

Proceedings of the World Congress on Engineering 2021
WCE 2021, July 7-9, 2021, London, U.K.

 D1. Input profile mechanical parameters can be estimated from vehicle
 The input profile u(t) in this study is given by the sum of vibration alone by the proposed method. The VBI system is
 the road surface profile r(t) and the bridge profile y ̃(t) and is modeled using a rigid-Bodies-Spring Model (RBSM) vehicle
 expressed by equation (24). and a one-dimensional Finite Element Method bridge. In the
 numerical experiments, the vehicle model and the bridge
 ̃( )
 ( ) = ( ) + (24) model are separated. Their respective inputs are obtained
 using the Newmark-β method and the iterati e method.
 Here, the road profile is the unevenness of the road surface
 at the axle position. When the road surface roughness is ( ), E1. Newmark-β method
 and the axle position is ( ), equation (25) is obtained. The vehicle vibration ( ) and bridge ( ) vibration are
 obtained by applying the Newmark-β method to the
 respective equations of motion (Equations (6) and (23)). The
 ( ) = ( ( )) (25) equations of motion are expressed in the MCK system as
 follows.
 ( ) is a component of ( ).On the other hand, the bridge
 profile is the bridge vibration ̃ ( ) = ( ( ), ) at the axle ̈ ( ) + ̇ ( ) + ( ) = ( ) (31)
 position ( ) . The bridge vibration ( ) is the vector of
 deformations at each node. Therefore, the basis functions The time function ( ) is discretized to , where is the
 used in the finite element method are used to obtain the bridge displacement response of the bridge or vehicle, and ∆ is the
 displacement ̃ ( ) at the axle position ( ) . Using the time increment.
 transformation matrix ( ), the bridge vibration at the axle
 position is represented by Equation (26), and ( ) is shown = ( Δ ) (32)
 in Equation (27).
 In this study, the update equations of the Newmark-β
 ̃( ) = 
 T ( ) ( )
 (26) method are assumed to be equations (33) and (34).

 ̇ = ̇ −1 + ( − )∆ ̈ −1 + ∆ ̈ (33)
 ( ( )) ( = − )
 ( ( )) ( = )
 ( ) = (27) = −1 + ∆ ̇ −1 + ( − ) ∆ 2 ̈ −1
 (34)
 ( ( )) ( = + ) + (∆ )2 ̈ 
 { ( ( )) ( = + ) The vehicle and the bridge's vibrations are calculated from
 their respective equations of motion using the following
 D2. Contact force equations.
 The ground force, which is the bridge's input, corresponds
 to the elastic force acting on the spring with tire stiffness .
 ̈ = −1 (35)
 However, in the equation of motion of the vehicle (Equation
 (6)), the gravity term disappears because the equation is based
 ∆ (∆ )2
 on a balanced position. However, in calculating the elastic = [ + + ] (36)
 force, the effect of gravity must be taken into account because 4
 it is set to zero when the vehicle is at its natural length. Noting
 that the center of the vehicle's rotation is the center of gravity, = + 1 + 2 (37)
 the front and rear wheels' ground forces can be written as
 equation (28). = ( Δ ) (38)

 2 ̈ −1
 1 ( ) = ( − ̈ 1 ) + 1 ( − ̈ 1 ) 1 = − ̇ −1 − ∆ (39)
 1 + 2
 (28)
 1 
 2 ( ) = ( − ̈ 2 ) + 2 ( − ̈ 2 ) ̈ −1
 1 + 2 2 = − −1 − ̇ −1 ∆ − (∆ )2 (40)
 4
 The external force vector acting on the bridge is Equation
 (30), where ( ) represents the fulcrum reaction force. The vehicle's speed is constant at 10 [m/s], and it runs on
 the road surface for 8 seconds. If the starting point of the
 ( ) = ( )[ 1 ( ) 2 ( )] + ( ) (30) vehicle's frontal area position is -20[m], the bridge is between
 0[m] and 30[m]. Fig. 3 and Fig. 4 show the behavior of the
 vehicle and the bridge obtained from the numerical
 E. Implementation of numerical simulation simulation.
 In this study, vehicle vibration data during driving is
 numerically reproduced to investigate whether the

ISBN: 978-988-14049-2-3 WCE 2021
ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

Proceedings of the World Congress on Engineering 2021
WCE 2021, July 7-9, 2021, London, U.K.

 surface roughness matches, we can expect that the parameters
 will eventually approach the correct values. In this study, we
 refer to such a parameter identification method as "VBI
 system identification". However, the shape of the objective
 function is not known. Besides, it is not always the case that
 ra eling osition m there is only one combination of system parameters for the
 vehicle and the bridge when the road surface roughness
 Fig.3 Unsprung, sprung displacement and input profile
 × 0
 estimated for the front and rear wheels coincides. Therefore,
 0
 the shape of the objective function is checked by selecting
 0 one parameter and varying its value. This method is
 equivalent to estimating the optimization parameters in a
 0
 0 4 brute force fashion in system identification. Although this
 ra eling ime s method is computationally expensive, it is a reliable way to
 Fig.4 Displacement of the bridge (center section) search for the correct combination of values. Next, the VBI
 system identification is also performed using the gradient
 descent method, which is the simplest and least
 F. VBI System Identification computationally expensive method. Finally, the results are
 Assume that the vehicle vibration ̈ ( ) has been obtained compared and discussed.
 as measured data. By substituting the vehicle vibration data
 and randomly assumed initial values of the vehicle and bridge F2. Gradient descent method
 parameters into the VBI system, the road profile ( ) can be In this study, the gradient descent method is used to search
 obtained. First, the measured vehicle vibration data, ̈ ( ), is for the optimal solution of the parameters. The gradient
 substituted into the Newmark-β method to o tain ̇ ( ) and descent method is one of the gradient method algorithms for
 ( ) . Assuming that the vehicle's system parameters are the continuous optimization problem to find the minimum
 random, v , v , and v can also be obtained. At this point, value of a function. In the gradient descent method, the
 all the left-hand sides of Equation (1), the equation of motion parameters are brought closer to the solution using an
 of the vehicle, have been obtained, and v ( ) can be obtained. iterative method; when the solution is at ℎ in the th
 In equation (11), which expresses v ( ), 1 , 2 , 1 , and 2 iteration, the position of the parameters in the + th
 in the equation have already been assumed, so , 3 , 1 , and iteration is expressed by equation (31).
 2 can be obtained. In other words, the excitation force and
 the input profile ( ) due to engine vibration are estimated. ℎ − ℎ −1
 ℎ +1 = ℎ − −1 (31)
 Next, the vehicle vibration data ̈ ( ) and the assumed − −1
 vehicle system parameters are substituted into Equation (28)
 to obtain the ground forces 1 ( ) and 2 ( ). If the system III. RESULTS AND DISCUSSION
 parameters of the bridge are also assumed to be random, the
 bridge vibration ( ) can be obtained using Equation (23), Numerical simulations are performed based on TABLE I
 which is the equation of motion of the bridge, and the and TABLE II parameters to obtain the vehicle vibration.
 Newmark-β method. he o tained ridge i ration ( ) can Vary from 0.5 to 1.5 times the correct value, and vary 
 be substituted into equation (26) to obtain the bridge profile from 1 to 2 times the correct value, to find the value of the
 ̃( ).
 objective function and examine the shape of each
 Now, subtracting ̃( ) from ( ) obtained earlier, ( ) parameter's error function. The shape of the error function for
 can be estimated. Furthermore, the road surface unevenness each parameter is examined. The model in this study is a
 ( ) is position-synchronized with the axle position ( ), multivariate function. However, for the sake of clarity, the
 results are presented as a single variable.
 and the road surface unevenness ( )（ ( ( )) = ( )）.
 When the vehicle goes straight, the front and rear wheels
 follow the same path, so the road surface irregularities 1 ( ) A. Shape of the error function without noise
 and 2 ( ) are equal. However, in this estimation process, the The results of the error functions obtained for different
 vehicle and bridge parameters are assumed randomly. The bridge parameters ( , ) are shown in Fig. 5 to Fig. 6.
 values of 1 ( ) and 2 ( ) are expected to be different. The red dotted line is the correct value for each parameter.
 The red circle is the minimum value of the error function,
 F1. The error function i.e., the optimal solution when calculated by the brute force
 Consider the optimization problem of minimizing the method. has six parameters, but there is no significant
 squared error of 1 ( ) and 2 ( ). The objective function is difference in each of them, so the objective function of
 shown in Equation (30). only 1 is visualized as a representative value. It can be
 seen that the stiffness of the bridge is convex downward
 and there is no multimodality. Furthermore, the optimal
 = ∑| 1 ( ) − 2 ( )|2 (30) solution (the minimum value of the error function) was
 consistent with the correct value. However, the mass per
 If all the randomly assumed parameters are correct, the two unit length of the bridge, , was convex near the correct
 calculated road surface roughness will match. Therefore, if solution and resembled a parabola but was multimodal in a
 we can update the parameters so that the calculated road broader range. Therefore, the optimal solution for the

ISBN: 978-988-14049-2-3 WCE 2021
ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

Proceedings of the World Congress on Engineering 2021
WCE 2021, July 7-9, 2021, London, U.K.

 stiffness of the bridge can be found by a simple method 0
 such as the gradient descent method. In order to find , .
 other optimization methods may need to be combined. .
 0
 . .
 .
 .
 .
 . .
 . .
 .
 .

 .
 . . . . . . . . . . Fig.8 Objective function shape of (noise is added)
 0
 On the other hand, the estimated value of was the
 Fig.5 Objective function shape of 1 correct value even when noise was added.
 0
 .
 IV. SUMMARY AND FUTURE ISSUES
 .
 In this study, we developed a model that took into account
 . the disturbance and clarified the effect of measurement noise
 on the bridge parameter estimates in VBI system
 .
 identification from the objective function's shape. The
 . parameters were obtained in a brute force fashion. The results
 showed that the shape of the EI was convex downward. Also,
 .
 the vertices match the correct values. Therefore, it is
 . suggested that can be estimated by a simple method such
 as finding the minimum value of the error function.
 In addition, was found to be multimodal. Therefore,
 Fig.6 Objective function shape of parameter optimization using only the gradient descent
 method cannot reach all parameters' correct values. To solve
 B. Effect of added noise on the error function this problem, it is necessary to improve parameter estimation
 Next, we examine the case where noise is added to the methods' accuracy and efficiency by combining other
 vehicle vibration. Since this study's noise is supposed to be optimization methods such as PSO and MCMC (Markov
 the engine vibration and observation noise, it is assumed that chain Monte Carlo) methods, as Murakami has done.
 all sensors are subjected to the same magnitude of noise.
 Therefore, we used a brute force equation to find the optimal REFERENCES
 solution when 0.01 percent of noise (uniform random [1] Yang, Y.B., Lin, C.W., and Yau, J.D., “Extracting ridge
 number) is added to the vehicle vibration's maximum frequency from the dynamic response of a passing
 amplitude under the spring. For each parameter ( , ), the ehicle”, Journal of Sound and Vibration, Vol.272,
 VBI parameter identification results for each noise are shown pp.471-493, 2004.
 in Fig. 7 to Fig. 8. It was found that the optimal solution [2] omonori Nagayama, Bo Yu Zhao and Kai Xue. “Half
 obtained by adding noise deviated from the correct value. car model Identification and road profile estimation
 using vibration res onses of a running ehice”, Journal
 However, even with the addition of noise, the shape of EI's
 of Japan Society of Civil Engineers, E1 (Pavement
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 Engineering), Vol. 75, No. 1, pp. 1–16, 2019 (in
 0 Japanese).
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 [3] Murakami Syo. “Parameter identification of vehicles,
 . bridges, and roads using multi-vehicle vibration data
 . ased on article swarm o timization”, ni ersity of
 Tsukuba, Bachelor's Thesis, 2018.
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 0

 Fig.7 Objective function shape of 1 (noise is added)

ISBN: 978-988-14049-2-3 WCE 2021
ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)