influenceoffluidviscousdamperonthedynamicresponseof suspensionbridgeunderrandomtrafficload · 2020....

Research ArticleInfluence of Fluid Viscous Damper on the Dynamic Response ofSuspension Bridge under Random Traffic Load

Yue Zhao ,1 Pingming Huang,1 Guanxu Long,1 Yangguang Yuan,2 and Yamin Sun3,4

1School of Highway, Chang’an University, Xi’an 710064, China2School of Civil Engineering, Xi’an University of Architecture and Technology, Xi’an 710055, China3School of Architecture and Civil Engineering, Xi’an University of Science and Technology, Xi’an 710054, China4Postdoctoral Research Station on Civil Engineering, Xi’an University of Science and Technology, Xi’an 710054, China

Correspondence should be addressed to Yue Zhao; [email protected]

Received 30 October 2019; Accepted 24 April 2020; Published 26 May 2020

Academic Editor: Marco Corradi

Copyright © 2020 Yue Zhao et al. )is is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Fluid viscous dampers (FVDs) are widely used in long-span suspension bridges for earthquake resistance. To analyze efficientlythe influences of FVDs on the dynamic response of a suspension bridge under high-intensity traffic flow, a bridge-vehicle couplingmethod optimized by isoparametric mapping and improved binary search in this work was first developed and validated.Afterwards, the traffic flow was simulated on the basis of monitored weigh-in-motion data. )e dynamic responses of bridge wereanalyzed by the proposed method under different FVD parameters. Results showed that FVDs could positively affect bridgedynamic response under traffic flow. )e maximum accumulative longitudinal girder displacement, longitudinal girder dis-placement, and longitudinal pylon acceleration decreased substantially, whereas the midspan girder bending moment, pylonbending moment, longitudinal pylon displacement, and suspender force were less affected. )e control efficiency of maximumlongitudinal girder displacement and accumulative girder displacement reached 33.67% and 57.71%, longitudinal pylon ac-celeration and girder bending moment reached 31.51% and 7.14%, and the pylon longitudinal displacement, pylon bendingmoment, and suspender force were less than 3%. )e increased damping coefficient and decreased velocity exponent can reducethe bridge dynamic response. However, when the velocity exponent was 0.1, an excessive damping coefficient brought littleimprovement and may lead to high-intensity work under traffic flow, which will adversely affect component durability. )ebenefits of low velocity exponent also reduced when the damping coefficient was high enough, so if the velocity exponent has to beincreased, the damping coefficient can be enlarged to fit with the velocity exponent. )e installation of FVDs influences dynamicresponses of bridge structures in daily operations and this issue warrants investigation. )us, traffic load should be considered inFVD design because structural responses are perceptibly influenced by FVD parameters.

1. Introduction

FVDs were first used in the machinery and military in-dustries. On account of their excellent energy dissipatingcapacity, they have been applied in structural vibrationcontrol under earthquakes.With structural vibration controlas a key problem in safety operation of long-span bridges,FVDs are installed on many bridges to mitigate vibrationand to serve as an effective seismic control device. In 1989,FVDs were first used on the Golden Bridge located in USAfor increasing its seismic resistance. )e identification testorganized by the Highway Innovative Technology

Evaluation Center had been a valid reference in subsequentapplication and specification preparation of FVDs [1]. )etechnology was first brought to China on the ChongqingEgongyan Yangtze River Bridge in 2000 and is now widelyused in Yangtze River bridges [2]. Since FVDs are integratedwith the bridge, they have various complex effects on bridgeresponses, such as girder displacement, pylon bendingmoment, and expansion joint displacement.)e designationof optimum FVD parameters focuses on the response underearthquake excitation and has been investigated compre-hensively. )e research methods generally included nu-merical analyses, shake table tests, and data monitoring

HindawiAdvances in Civil EngineeringVolume 2020, Article ID 1857378, 19 pageshttps://doi.org/10.1155/2020/1857378

mailto:[email protected]://orcid.org/0000-0002-1455-7606https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2020/1857378

[3–5]. Different techniques have been proposed by scholarsto ascertain the optimal parameters that can achieve idealstructural response [6–8].

Although FVDs’ effects under earthquake excitationhave been widely studied, the dynamic responses of bridges,which are also vulnerable to vibrations excited by wind,temperature, and vehicles, have not been thoroughly ex-amined. )e most common dynamic response in the bridgelife cycle is the frequent low-speed reciprocating motion,which is mainly caused by wind and traffic flow [9]. )ecorresponding service life of bridge accessories, such asexpansion joints, is also influenced by this frequent recip-rocating motion. FVDs can also absorb and dissipate energyimported from wind and traffic flow because of their dis-sipation capacity. )e working conditions of supports, ex-pansion joints, and suspenders can be improved byconsidering the daily longitudinal girder displacement indetermining FVD parameters; the frequent reciprocatingmotion of main girder also can be effectively improved byusing FVDs [10]. For a long-span bridge, the wind load maybe more remarkable than vehicle load in transverse responseof the bridge. Research shows that the dynamic responses oflong-span suspension bridge under high winds and runningtrains are mainly influenced by strong winds when the windis strong, but when the wind comes to low intensity, trainload plays a decisive role in vertical displacement [11]. Forthe bridge dynamic response under wind and traffic flow, thewind load plays a major role in bridge lateral displacement.)e vertical displacement is mainly controlled by the vehicleload [12]. Furthermore, scholars have made some studies onwind-induced vibration control and found that FVDs havedifferent influences on various response indicators, whichhas an optimum interval for damping parameters [13, 14]. Inaddition to the wind-induced vibration, the vehicle-inducedvibration of long-span bridges is also remarkable and theinfluences of FVDs are still indistinct. )e daily movementof bridge structures under social vehicles is an importantfactor related to bridge accessories’ durability [5]. Variedstructural response indexes present different variationtrends with changing parameters of FVDs, such as accu-mulative girder displacement, girder midspan bendingmoment, pylon longitudinal acceleration, and pylon bend-ingmoment.)e specific influences of FVDs on the dynamicresponse of long-span suspension bridges under traffic load,which can be further referenced in the optimization of FVDparameters, should hence be analyzed.

Because the traffic load of the highway bridge is acomplicated random process with strong randomness ofvehicle type, weight, and quantity, the calculation matrix ofrandom traffic flow-bridge coupling analysis is huge and willtake a lot of computing resources. )us, a high efficiencyrandom traffic flow-bridge analysis system needs to beestablished. )e traditional vehicle-bridge coupling analysissystem, which is based on a numerical method, mainlyincludes the modal superposition method and full couplingtheory [15].)emodal superposition method is more simpleand practical than the latter, but the high-order modes ofstructures are hard to obtain, thereby negatively affecting thecalculation precision [16]. )e full coupling analytical

method has the advantages of clear physical meaning andhigh precision. However, this method requires huge amountof time and lots of computing resources when applied tolong-span bridges under large and highly random trafficflow. )is paper developed a vehicle-bridge interactionsystem with high efficiency to analyze the dynamic responseof a bridge with FVDs under a large and highly randomtraffic flow. )is system was established based on theinterhistory iteration method and optimized by iso-parametric mapping and improved binary search. )en, thetraffic flow was simulated in accordance with the monitoreddata. Finally, the responses of the large-span bridge undertraffic flow with different parameters of FVDs were studied.

2. Establishment of Vehicle-BridgeInteraction System

2.1. Finite Element Model of Bridge. )e bridge model isestablished by the space truss model in ANSYS 15.0 [17].Spine-beam, double-beam, and triple-beam models arewidely used in the integral analysis of long-span bridges, andthe grillage model and multiscale model are also usedaccording to different requirements [18, 19]. Main cablesand suspenders are usually simulated by the Link10 element,which can simulate the elasticity and plasticity of thestructure. )e pylon and girder are simplified into a two-node beam element and modeled by Beam4 element. )erailings and deck paving are simulated by the Mass21 ele-ment. )e entire structure is distributed into a number ofelements and connected through the node on the boundaryof adjacent elements. Given the demand for refined simu-lation of the vehicle-bridge interaction, a grillage model isutilized in this article to construct the prototype bridge toobtain detailed component response. )e detailed infor-mation and feasibility of the model are proposed and val-idated in succeeding chapters.

2.2. Dynamic Vehicle Models. A vehicle is mainly composedof a car body, wheels, a shock mitigation system, and adamped system connecting different components. A vehicledynamic model needs to be established before the vehicle-bridge coupling analysis is conducted. With these demands,some hypothesis is unavoidable in constructing the dynamicmodels. )e mass of damper and spring components areignored compared with the quality of the vehicle body.Vehicle model is generally divided into different rigid bodiesthat are connected by axle mass blocks and by elastic anddamped components. According to previous research, thecommon Chinese highway vehicles can be divided into 17types of 5 categories in accordance with vehicle axle distance,axle number, axle load, and vehicle load based on weigh-in-motion data and Chinese automobile model manual [20, 21].)e vehicle classification is shown in Table 1.

)e dynamic models are also provided based on differentvehicle type. Taking three-axle vehicle as an example, thevehicle dynamic model of three-axle vehicle (double rearaxle) is shown in Figure 1 [20]. )e rigid body is usuallydeemed to have 6 degrees of freedom. As the vibration of the

2 Advances in Civil Engineering

vehicle in advancing direction is tiny compared to thedistance travelled, the vibration of the vehicle body in ad-vancing direction has few effects on the bridge. )e bridge ismainly influenced by the integral movement of the vehicle inadvancing direction. In order to reduce the complexity ofvehicle model and the calculation, the degree of freedom inadvancing is generally neglected in vehicle-bridge interac-tion analysis [22]. )us, 5 degrees of freedom (vertical,horizontal, head nodding, side rolling, and head shaking) areconsidered for the integral vehicle. Each wheel has 2 degreesof freedom (vertical and horizontal). For the trailer, hori-zontal and head nodding degrees of freedom are neglected toderive the equation. Each vehicle body has 3 degrees offreedom, and each wheel has 1 degree of freedom.

2.3. Load Distribution Based on Quadrilateral IsoparametricMapping. For the dynamic analysis under large randomtraffic flow, the positioning and loading of the contact pointbetween the vehicle and the bridge should be a crucial issue.Bridge decks are typically simulated by grillage, solid, or shellelements in most models. )e wheel load is simplified to aconcentrated force acting on the bridge deck and thendistributed to four adjacent nodes by four-node

isoparametric mapping [23], which is the same as that ofdisplacement in a two-dimensional plane, as shown inFigures 2 and 3.

)e coordinate mapping equations are as follows:

x � 4

i�1Ni(ξ, η)xi,

y � 4

i�1Ni(ξ, η)yi,

Ni(ξ, η) �14

1 + ξξi( 1 + ηηi( ,

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

where x and y denote the coordinates of the force point of thewheel load P; xi and yi denote the ith quadrilateral elementpoint’s horizontal and vertical positions; ξ and η denote thecorresponding values that x and y map from the quadri-lateral element to the parent element; ξi and ηi denote knownquantities, namely, the ith node’s horizontal and verticalpositions of the parent element; Ni(ξ, η) denotes a functionof isoparametric point (ξ, η); function value Ni denotes theith quadrilateral element point’s load distribution coefficient.

Table 1: Vehicle classification [20, 21].

Type Model number Number of axles DescriptionT1 V1, V2 2 Sedan car and off-road vehicleT2 V3, V4 2 Microbus and midsize truckT3 V5, V6 2 Two-axle bus or truck

T4

V7 3 )ree-axle busV8 3 )ree-axle truck (double front axle)V9 3 )ree-axle truck (double rear axle)V10 4 Four-axle truck

T5

V11 3 )ree-axle trailerV12 4 Four-axle trailerV13 5 Five-axle trailer (double front axle)

V14, V15 5 Five-axle trailer (triple front axle with different axle distance)V16, V17 6 Six-axle trailer (different axle distance)

θvrZvr

K2vuL C2vuL

K2vlL C2vlL

Z2vaL

X

Z C3vlLK

3vlL

C3vuLK3vuL

L2 L1

Z3vaL Z1vaL

C1vlLK1vlL

C1vuLK1vuL

Driver seat

L3

(a)

b1b1

Zvr

Y

Z1vaR

C1vlRK1vlR

C1vuRK1vuRK1vuL C

1vuL

K1vlL C1vlL

Z1vaL

ZY1vaL

K1yuLC1yuLK

1ylL

C1ylL

K1yuRY1vaRC

1yuR K

1ylRC1ylR

h 1

h v

b2

Yvr

Φvr

(b)

Figure 1: Dynamic analysis vehicle model of three-axle vehicle (double rear axle) [20].

Advances in Civil Engineering 3

)e unknown quantities ξ and η of bilinear equation (1)can be solved by Newton iteration to obtain the distributioncoefficient Ni. Afterwards, the automatic loading of thewheel load can be realized.)e iteration process is as follows:

ξk+1

ηk+1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ �

ξk

ηk

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ − J ξ

k, ηk

− 1

4

i�1Ni ξ

k, ηk xi − x

4

i�1Ni ξ

k, ηk yi − y

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

J �

z 4i�1 Ni(ξ, η)xi

zξz

4i�1 Ni(ξ, η)xi

zη

z 4i�1 Ni(ξ, η)yi

zξz

4i�1 Ni(ξ, η)yi

zη

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)

where J denotes a Jacobian matrix; ξk and ηk denote thecalculated value of the kth iteration. )e iteration is stopped

when ξk+1

− ξk

ηk+1 − ηk

��

��< ε with the iteration error ε set to 10−8.

)e position information of four adjacent nodes is in-dispensable for realizing the automatic loading of the wheelload P. Binary search can effectively halve the scope infinding specific elements in ordered arrays. )us, it can beextended for searching the coordinate range of the wheelloading. For example, the process of searching the longi-tudinal wheel loading point x is shown in Figure 4, and thepositioning for the lateral loading point y is the same.

)e detailed steps are as follows.

① )e node coordinates of the lane on which the ve-hicles drove are imported and sorted in ascendingorder to form a one-dimensional matrix.

② Initialization: the value of “front” is set to 1; the valueof “last” is set to the number of columns of the matrixformed in step 1.

③ Calculation of “mid”: “mid”� “round”((“front”+“last”)/2); “round” means rounding off.

④ )e search ends when “mid”� “last.” )e upper limitof the wheel x-coordinate in the one-dimensionalmatrix is “mid,” and the lower limit is “mid”− 1.Otherwise, whether the wheel x-coordinate is be-tween “front” and “mid” is determined. If it happens,then “last”� “mid;” otherwise, “front”� “mid.”

⑤ Step 3 is executed until “mid”� “last.”

Equation (1) can be solved by Newton iteration afterfinding four adjacent nodes of the wheel load P. )en, theinterpolated coefficient at each node can be obtained tocalculate the distribution load. In order to reduce thecomputation of bridge-vehicle coupling, only the results ofdisplacement and velocity of load acting point are output.Afterwards, based on the known interpolated coefficient, thevelocity and displacement of load acting point are calculatedto solve the bridge-vehicle interaction force.

2.4. Road Surface Roughness. Road surface roughness is animportant factor that could cause a vehicle’s vertical vi-bration, which is directly related to the safety and comfort ofthe vehicle. Surface roughness can be described as animplementation of random processes and expressed by apower spectral density function. )e recommended powerspectral density function is adopted in this study [24]. )efunction Gd (n) is expressed as

Gd(n) � Gd n0( n

n0

−ω

, (3)

FF1 F2

F3F4

Figure 2: Contact point load distribution.

P (x, y)

1 (–1, –1) 2 (1, –1)

3 (1, 1)4 (–1, 1)η

ξ x

y

1 (x1, y1) 2 (x2, y2)

3 (x3, y3)4 (x4, y4)

Figure 3: Isoparametric mapping.

Sort one–dimensional matrix

Element nodecoordinates

Start

mid = round ((front + last)/2)

mid = lastx ≥ t (front)

&x ≤ t (mid)

ANSYSmodel

Longitudinalcoordinate x of

wheelfront = 1last = length (t)

last = mid

front = mid

Yes

No

Upper limit is “mid”,lower limit is “mid – 1”

Finish

Yes

No

Figure 4: Flowchart of improved binary search.


where Gd (n) denotes the power spectral density function(m3/cycle); n denotes spatial frequency per unit length(m−1); n0 denotes reference spatial frequency, n0 � 0.1m−1;Gd (n0) denotes power spectral density under referencespatial frequency; and ω denotes the frequency exponent,and ω� 2.

Road surface roughness is assumed to be a zero-meanstationary Gaussian random process. )us, it can be gen-erated through Fourier inversion as follows:

r(x) � N

k�1

��

2Gd nk( Δn

cos 2πnkx + θk( , (4)

where θk denotes the random phase angle uniformly dis-tributed in [0, 2π]; nk denotes the kth reference spatialfrequency.

)e wheel-bridge relationship is assumed to be the pointcontact in constructing the vehicle-bridge interaction sys-tem. When the wheel comes into contact with the roadsurface, the vertical displacement of vehicle and bridge is thesame and the bridge’s vertical displacement is deemed ad-ditional surface roughness to the vehicle. )us, the addi-tional surface roughness caused by bridge displacement androad surface roughness is superposed to constitute theequivalent roughness which is inputted as the vertical ex-citation source. For the ith wheel, the formula consideringbridge displacement is expressed as

Zie � Z

ib + ri(x), (5)

where Zie denotes the vertical equivalent roughness con-sidering bridge displacement, ri (x) denotes the surfaceroughness at the wheel-bridge contact point, and Zib denotesthe displacement at wheel-bridge contact point.

2.5. Motion Equations of Vehicle-Bridge Interaction. )einterhistory iteration method is adopted to construct thehighway vehicle-bridge coupling analysis system given thetechnique’s remarkable efficiency under high-intensitytraffic flow [25]. Unlike time step iteration, each step of theinterhistory method involves full-time calculation. )ebridge dynamic response under external load can be com-puted by any commercial analysis software with severaliteration steps, thereby addressing the limitation of theprogramming calculation in the traditional method.

)e bridge and vehicle dynamic equations can beexpressed as follows:

Mv€Zv + Cv

_Zv + KvXv � Fv,

Mb€Zb + Cb

_Xb + KbXb � Fb,

⎧⎨

⎩ (6)

where Mv, Cv, and Kv denote the global mass, damping, andstiffness matrices of the train subsystem, respectively;Mb,Cb,and Kb denote the global mass, damping, and stiffnessmatrices of the bridge subsystem, respectively; Xv and Xbdenote the displacement vectors of the train subsystem andthe bridge subsystem, respectively; _Zv and €Zv denote thevelocity vector and acceleration vector of vehicle, respec-tively; _Zb and €Zb denote the velocity vector and acceleration

vector of bridge, respectively; and Fv and Fb denote the forcevectors of vehicle and bridge, respectively.

Wheels are assumed to always be in contact with thebridge deck while the vehicle is moving, so the shockmitigation system and suspension system are deformed withbridge vertical deformation and surface roughness. )edynamic equations of the vehicle-bridge coupling systemcan be expressed as (7). )e interaction force between thevehicle and the bridge is a function of vehicle motion, bridgemotion, and road surface roughness stated by Zv, Zb, and i.

Fv � Fvi Zv,_Zv,

€Zv, Zb,_Zb,

€Zb, i ,

Fb � Fbi Zv,_Zv,

€Zv, Zb,_Zb,

€Zb, i .

⎧⎪⎨

⎪⎩(7)

)e coupling relationship between the vehicle and bridgesubsystems is established by simultaneously solving (6) and(7) and then solved by the interhistory iteration method.)eiteration process is realized by MATLAB R2014a [26], whilethe bridge model constructed in ANSYS 15.0 is used tooutput the bridge response. Each step of interhistory iter-ation for the bridge and the vehicle is separate and involvesfull-time calculation. )us, the method will occupy reducedcomputer storage, generate accurate computations, and havethe advantages of a clear aim and ease of operation. )econvergence is checked by the interaction force between thevehicle and the bridge and defined as ‖Fiv − F

i−1v ‖< 0.1. )e

detailed process is plotted in Figure 5.

3. Validation of the Established AnalysisSystem by Field Load Test

3.1. Prototype Bridge and the FE Model. To validate the ve-hicle-bridge interaction analysis system and to analyze theinfluence of FVDs under random traffic flow, a three-di-mensional finite element model of a long-span suspensionbridge is constructed in ANSYS 15.0 as the research object(Figure 6). )e bridge has a span of 896m, a steel truss maingirder with the deck slab width of 26m, and a rigid centralbuckle. )e detailed information of the bridge is as follows.

(1) )e sag-span ratio of main cable is 1 :10. )e lon-gitudinal distance in each suspender is 12.8m. )esteel girder adopts warren truss stiffening beam withthe height of 6.5m and width of 26m, the minimumsegment length is 6.4m, and the standard segmentlength is 12.8m. )e pylon adopts a concrete portalframe structure. One pylon is 117.6m high andanother pylon is 122.2m high.

(2) )e main cable consists of 127 galvanized high-strength prefabricated parallel steel wire strands;each steel wire strand consists of 127 steel wires witha diameter of 5.1mm. Suspender is pin jointed withmain cable and girder, and each consists of 91 gal-vanized high-strength prefabricated parallel steelwires with a diameter of 5.2mm.

(3) )e material parameters have been modifiedaccording to the measured features of the bridge:elastic modulus of main cable is 1.94×1011 Pa,


elastic modulus of girder is 2.13 ×1011 Pa, elasticmodulus of pylon is 3.48×1010 Pa, the density ofpylon is 2663 kg/m3, the density of the truss elementis 8646 kg/m3, and the density of main cable is8020 kg/m3.

)e pylon, central buckle, and main girder are modeledusing Beam4 element. In order to simulate the lane load,four longitudinal girders were built in the model. Each laneload was exerted on the corresponding longitudinal girder.)e main cables and suspenders are simulated using Link10element, and the railings and deck paving are simulatedusingMass21 element.)e boundary conditions of the pylonfoundation are simulated as fixed connection. )e linedisplacement of the main cable at the anchorage point isconstrained in three directions. )e simulation of the vertexand tangential point of the cable saddle are realized throughthe node coupling. )e vertical and horizontal displacementof the end of the girder is limited and the rotational degree offreedom is released.

)e damping effect is an output resistive force thatdepends on the relative velocity of the damping device.Equation (8) is recommended to calculate the produceddamping force [27]. FVDs are simulated using Combin37element. As shown in Figure 7, the FVDs are longitudinallysettled on both sides of the midspan as a connection of thegirder and the pylon, with two dampers on each side.

F � K|v|αsign(v), (8)

where F denotes the damping force, K denotes the dampingcoefficient, ] denotes the relative velocity between the endsof the damping device itself, sign denotes the sign function,and α denotes the velocity exponent.

)e data of the bridge were collected when the prototypebridge just finished the construction. Existing researchshows that the conditions of road surface roughness of newbridges in China are most A grade in accordance with ISO8608 [24, 28]. So the road surface roughness is set to A gradeand generated based on specified parameters in ISO 8608, asshown in Figure 8.

3.2. Accuracy Validation by Field Load Test. )e numericalanalysis result is compared with the result of dynamic ex-periment to verify the accuracy of the proposed method.)ebasis of the validation is the uniformity of theoretical modeland actual bridge. )us, the FE model in this article wasconstructed according to measured material properties andcable forces. Referring to existing researches, the accuracy ofFE model can be considered as acceptable when the errorsbetween the calculated and measured frequencies are lessthan 5% [29, 30]. )e measured frequencies of the bridge arecompared with the numerical results in Table 2. It can beseen that the numerical results are comparable to the

114m 896m

Main cable

Steel truss girder Midspan

Suspender

Approachbridge

GravityanchorCentralbuckle

2.14%

Tunnelanchorage

208m

Figure 6: Information of bridge model (unit: (m)).

No

Start

Vehicle motion history

Vehicle-bridgeinteraction force history

Set bridge initialdisplacement as 0

Road surfaceroughness

Bridge motion history

Vehiclemotion state

Bridgemotion state

Yes

Finish

Bridgesubsystem

Vehiclesubsystem

Convergencecheck

Figure 5: Flowchart of interhistory iteration.


measured results. )e maximum frequency error is 4.37%and all errors are less than 5%, which proves the correctnessof the FE model.

)en the dynamic strain and vertical acceleration of mid-span are measured to validate the proposed method. )emeasuring point of the dynamic strain is set on the top chord ofthe steel trusses in the middle of midspan. )e measuring pointof vertical acceleration is set on the bridge deck at the same place.

)e driving position of trucks and measuring points areshown in Figure 9; the detailed vehicle dynamic parametersare shown in Table 3 [31]. )ree test conditions are settledfor the validation: (1) two three-axle trucks passing throughthe bridge at a constant speed of 20 km/h, (2) two three-axletrucks passing through the bridge at a constant speed of30 km/h, and (3) two three-axle trucks passing through thebridge at a constant speed of 50 km/h.

Table 2: Comparison of the bridge frequencies.

Vibrating direction Order Measured result Numerical result Error (%)

Horizontal 1 0.071 0.07 1.412 0.183 0.175 4.37

Vertical

1 0.131 0.134 2.292 0.185 0.182 1.623 0.292 0.282 3.424 0.394 0.38 3.55

Torsional 1 0.348 0.342 1.722 0.434 0.425 2.07

Combin 37 element

Link 10

Beam 4

Beam 4Fluid viscous damper

Node JNode I

Spring

Axialforce

Figure 7: Information of bridge FE model.

Dec

k ro

ughn

ess (

mm

)

10

5

0

–5

–100 100 200 300 400 500

Position on bridge (m)600 700 800 900 1000

Figure 8: Road surface roughness of A grade.


)e measured results are extracted and compared withthose obtained by the analytical solution on the basis of theproposed method. )e comparison of dynamic strain resultsunder different vehicle velocities is shown in Figure 10. It can beseen that the changing trends of two curves are consistent inFigure 10. By comparison, the peak value of the dynamic strainof 50km/h is higher than 20km/h and 30km/h.)e peak valueof measured dynamic strain of 50 km/h is 30.03 while that of20km/h and 30km/h is 26.6 and 27.9. It indicates that themaximum dynamic strain increases with the vehicle velocity. InFigure 10(a), the error of peak values of two curves is 1.35%. InFigure 10(b), the error of peak values of two curves is 1.12%. InFigure 10(c), the error of peak values of two curves is 0.3%.)ecomparison results indicate that the validity and accuracy of theproposed method are acceptable.

)e midspan acceleration of measured result and theproposed method is shown in Figure 11. As the noise ofmeasured acceleration is more obvious than that of strain andcannot be eliminated, the comparison of acceleration is notgood as that of strain. But the variation trends of measuredresult and the proposed method are also consistent. )econsistence of changing trends of the vertical acceleration alsoproves the accuracy of the proposed method.

4. Traffic Flow Simulation Based onMonitored Data

)e response of long-span suspension bridges under trafficflow is mainly influenced by traffic flow density and totalvehicle weight. )e investigation focuses on the FVDs’ in-fluence on the bridge dynamic responses under traffic flow;thus the key variables are FVD parameters while the trafficflow needs to remain unchanged. Monitored traffic data

gathered by weigh-in-motion system are processed to formtypical traffic load to obtain the response of the studiedstructure [32]. )e data collection site lies in Zhangjiawanbridge at the G104 highway in Zhejiang province of China,which is a coastal developed area. )e vehicles include cars,vans, trucks, and trailers. )e WIM data were measured intwo lanes and collected more than 10,000 vehicles throughone-month monitoring. In order to eliminate the sick data,the vehicles have low weight (less than 0.5 t) and high speed(more than 120 km/h) filtered in preprocessing the datacoming from weigh-in-motion system.

First, the monitored data are statically initialized toseparate vehicles in carriageway and passing laneaccording to the statistical property of lane distribution. Asthe statistical analysis in hours can reflect the randomnessand the peak hours of traffic flow, hourly traffic flow isoften used in traffic load analysis [32, 33]. )e WIM datawere divided into 24 hours each day and the total vehicleweight of each period was calculated. For the bridge, thedensity of dynamic response is directly influenced by theexternal load. In order to obtain the most obvious re-sponse, the vehicles with maximum total vehicle weight inan hour are screened out as the basic data of the traffic flow.Figure 12 shows the maximum total vehicle weight of eachhour in a month. )e left road of two lanes of the ad-vancing direction in China is passing lane and the right iscarriageway. )e maximum vehicle weights in an houralong the passing lane and the carriageway are 1115.31 tand 759.78 t, respectively. )e corresponding vehicle dataare chosen as the basic data to simulate traffic flow. )edetailed information of vehicle type is shown in Figure 13;it can be seen that the proportion of small and mediumcars in traffic is larger than trucks.

Table 3: Technical parameters of the vehicle [31].

Vehicle model Axle number Axleweight (kg)Upper stiffness

coefficient (kN/m)Upper damping

coefficient (kN·s/m)Lower stiffness

coefficient (kN/m)Lower damping

coefficient (kN·s/m)

)ree-axle truckFront axle 8000 1577 26.6 3146 22.4Middle axle 13500 2362 20 2362 33.4Rear axle 13500 2362 20 2362 33.4

114 1122 128 1122 114

714

656 1288 6562600

Suspender

Main cable

2.0%Straingauge

Straingauge

Accelerometer

192

Figure 9: Information of bridge model and dynamic test (unit: cm).


Axi

al st

rain

(με)

40

0 20 40 60 80 100Time (s)

120 140 160

20

0

–20

–40

Measured resultProposed method

(a)

–20Axi

al st

rain

(με)

40

20

0

–400 20 40 60 80 100

Time (s)


(b)

–20Axi

al st

rain

(με)

40

20

0

–400 10 20 30

Time (s)40 50


60

(c)

Figure 10: )e comparison of dynamic strain results. (a) 20 km/h; (b) 30 km/h; (c) 50 km/h.

0.0050

0.0025

0.0000

Acce

lera

tion

(m/s

2 )

–0.0025

–0.00500 20 40 60 80 100

Time (s)120


140 160

(a)

Figure 11: Continued.


)e detailed simulation process is shown in Figure 14. Asthe prototype bridge is bidirectional with four lanes and themonitored data only have two lanes, the chosen vehicle data

0.02

0.01

0.00

Acce

lera

tion

(m/s

2 )–0.01

–0.020 20 40 60 80

Time (s)


100

(b)

0.06

0.03

0.00

Acce

lera

tion

(m/s

2 )

–0.03

–0.060 10 20 30 5040

Time (s)


60

(c)

Figure 11: )e comparison of midspan acceleration results. (a) Under 20 km/h; (b) under 30 km/h; (c) under 50 km/h.

Tota

l veh

icle

wei

ght (t)

1200

Passing laneCarriageway

1115.31

759.68

1000

800

600

400

0 2 4 6 8 10 12 14 16Time (h)

18 20 22 24

Figure 12: Extreme value of vehicle weight in hours.

Prop

ortio

n (%

)

10089.20

Passing laneCarriageway

23.68

2.35

31.58

3.76

13.16

4.46

18.42

0.24

13.16

80

60

40

20

0

Vehicle typeT1 T2 T3 T4 T5

Figure 13: )e vehicle type of selected data.


are used to simulate sparse flow and dense flow separatelythat can be loaded on both sides of the bridge simulta-neously. )e traffic condition is divided into sparse flow anddense flow that the headway of the former exceeds 3 s andthat of the latter is less than 3 s [34]. Statistical analysis on theWIM data indicates that the vehicle headway of the denseand sparse flow obeys gamma distribution. )e key pa-rameters are shown in Table 4. )e corresponding vehicleheadways are generated according to statistical properties.Figure 15 shows the vehicle headways of dense flow inpassing lane. )en, combined with vehicle data, a bidirec-tional traffic flow with a dense upstream flow and a sparsedownstream flow is formed to load on the bridge.

5. FVD Effect under Traffic Flow

According to existing research, the service life of bridgeaccessories, such as expansion joints and FVDs, is closelyrelated to the deformation and fatigue properties of thecomponents under external loads [35]. An accumulativefriction slide distance that exceeds the design value is animportant precipitating factor, and also it will lead tocomponent damage in which traffic flow accounts for mostof the accumulative displacement. Maximum girder dis-placement is a critical factor that determines whether theworking condition of a bridge accessory is in the designrange. )us, the longitudinal accumulative girder dis-placement and maximum longitudinal girder displacement

are selected as the typical indexes of the vibration controleffect of FVDs under traffic flow. In JT/T 926-2014, thedamping velocity exponent is specified into seven classes(0.1, 0.2, 0.3, 0.4, 0.5, 0.6, and 1), and no limitation exists forthe damping coefficient, whose value commonly ranges from1000 kN/(m/s)α to 20000 kN/(m/s)α [13, 27]. )e dampingcoefficient is set to 1000 kN/(m/s)α, 2500 kN/(m/s)α,5000 kN/(m/s)α, 7000 kN/(m/s)α, 10000 kN/(m/s)α,15000 kN/(m/s)α, and 20000 kN/(m/s)α to analyze theFVDs’ effects with representative combinations ofparameters.

250

200

Gamma distribution:α = 7.7043, β = 3.7311

150

Generated data

Cou

nt

100

50

00 1 2 3

Headway (s)4 5 6 7

Figure 15: Vehicle headways of dense flow in passing lane.

Selected trafficdata

Vehicleweight

Dense flow Sparse flow

Bidirectionaltraffic flow

Vehiclelength

Axleload

Axledistance

Physicalcharacteristicsof traffic flow

Vehicletype

Vehiclespeed

Timeheadway

Lanedistribution

Figure 14: Flowchart of traffic flow generating.

Table 4: Parameters of vehicle headway.

Operation state Distribution type LaneDistribution parametersα β

Sparse flow Gamma distribution Carriageway 0.2954 0.0038Passing lane 0.4005 0.0149

Dense flow Gamma distribution Carriageway 7.4072 3.5829Passing lane 7.7043 3.7311


Nonlinear time history analysis under traffic flow isconducted using the bridge-vehicle interaction systemestablished in Section 2. )e main girder of the suspensionbridge is directly influenced by FVDs, so the girder responsevaries as shown in Figures 16–18. )e installation of FVDsprovides the following benefits: longitudinal displacementand accumulative displacement decrease noticeably with anenlargement in the damping coefficient and a decrease in thevelocity exponent.)e girder midspan bendingmoment alsodecreases with increased damping coefficient and decreasedvelocity exponent. When the velocity exponent exceeds 0.4,

the influence on girder midspan bending moment is notremarkable. But when the velocity exponent is below 0.4 andthe damping coefficient exceeds 7000 kN/(m/s)α, the girdermidspan bending moment noticeably declines with in-creased damping coefficient and decreased velocityexponent.

Figures 19 and 20 show that the variation trend oflongitudinal pylon displacement and acceleration increasesaccordingly with the increase in the velocity exponent or thedecrease in the damping coefficient. Figure 21 shows that thevariation trend of the pylon bending moment is relatively

5

4Ac

cum

ulat

ive l

ongi

tudi

nal

gird

er d

ispla

cem

ent (

m)

3

2

1200040006000

800010000Damping coefficient (kN/(m/s) α)

1200014000

1600018000

20000

None1

0.60.5

0.40.3

Velocity

expone

nt

0.20.1

4.400

3.900

3.400

2.900

2.400

1.900

Figure 16: Accumulative longitudinal girder displacement Wgc.

Max

imum

long

itudi

nal

gird

er d

ispla

cem

ent (

m)

0.65

0.60

0.55

0.50

0.45

0.40

0.352000

4000 60008000

Damping coefficient (kN/(m/s) α)

1000012000

1400016000

1800020000

None1

0.60.5

0.4

Velocity

expone

nt

0.30.2

0.1

0.580

0.540

0.490

0.440

0.390

Figure 17: Maximum longitudinal girder displacement Wgm.


complex; it increases with the damping coefficient when thevelocity exponent α< 0.3 but decreases and then increaseswith increased damping coefficient when 0.3≤ α< 1.

Figure 22 shows that the suspender force is also influ-enced by FVDs, which is caused by the reciprocating motionof the main girder. )e maximum suspender forces wereinduced by vehicles only; it may cause a relative displace-ment between the girder and the main cable. )e changinglaw is similar to that of maximum longitudinal pylondisplacement.

)e extreme values of each index are shown in Table 5.)e FVDs evidently influence the longitudinal girder

displacement and pylon acceleration. )e maximum controlefficiency of the girder displacement reaches 33.67%. )eaccumulated longitudinal girder displacement can be re-duced by 57.71%. )ese results mean that FVDs can re-markably decrease the daily reciprocating motion of relatedcomponents. Although the maximum reduced rate of thepylon longitudinal displacement is only 2.02%, the maxi-mum pylon acceleration can be reduced by 31.51%. )epylon bending moment and suspender force are less affectedin that the maximum control efficiencies are less than 2%.

Once the damping coefficient and velocity exponent aredetermined, the damping force under normal working

Max

imum

grid

er m

id-s

pan

bend

ing

mom

ent (

kN·m

)

1.29×103

1.26

1.23

1.20

1.17

1.142000

400060008000

1000012000

1400016000

1800020000

None

1.250

1.230

1.210

1.190

1.170

Velocity

exponent

10.6

0.50.4

0.30.2

0.1

Damping coefficient (kN(m/s) α)

Figure 18: Maximum midspan girder bending moment Mg.

Max

imum

long

itudi

nal p

ylon

disp

lace

men

t (m

)

0.160

0.159

0.158

0.157

0.156

0.155

0.1542000

40006000

800010000

1200014000

16000

None

0.1588

0.1581

0.1572

0.1563

0.15541

0.6

Velocity

expone

nt0.5

0.40.3

0.20.1

1800020000


Figure 19: Maximum longitudinal pylon displacement Wtm.


conditions can be calculated by (8) on the basis of thecorresponding damping velocity shown in Figure 23. Asshown in Figure 24, the damping force of a partial dampingcoefficient and velocity exponent is around or exceeds thespecified value for a single FVD in Chinese standard [27].)e damping coefficient has an apparent influence on thedamping force of the FVDs only when the velocity exponentis relatively small. Preceding results reveal that the longi-tudinal girder displacement is the most significant responseof the structure, but its variation tendency is nonlinear. Alarge damping coefficient improves the FVD effects, but theeffect of improvement is not proportional to the damping

coefficient. An excessive damping coefficient brings in-creased damping force, so the design force of FVD must behigh enough to burden the external load. Otherwise, theFVD would be working in limit state under the dense trafficflow, which is adverse to its durability.

By contrast, low velocity exponent results in an evidentimprovement, especially for FVD performance and girderdisplacement, thereby potentially making FVD a valid vi-bration mitigation device in daily operation. Figures 25 and26 show the time history curves of the maximum girderdisplacement and accumulative girder displacement whenthe velocity exponent is 0.1. )e accumulative girder

Max

imum

long

itudi

nal

pylo

n ac

cele

ratio

n (m

/s2)

0.016

0.014

0.012

0.010

0.0082000

None

0.015

0.014

0.012

0.011

0.0101

0.60.5

Velocity

expone

nt0.40.3

0.20.1

4000 60008000

1000012000

1400016000

1800020000


Figure 20: Maximum longitudinal pylon acceleration at.

Max

imum

pyl

onbe

ndin

g m

omen

t (kN

·m)

×1051.716

1.712

1.708

1.704

1.7002000

40006000

800010000

1200014000

16000

Damping coefficient (kN/(m/s) α) 18000 20000

None

1.717

1.714

1.711

1.708

1.7051

0.60.5

0.4

Velocity

expone

nt0.3

0.20.1

Figure 21: Maximum pylon bending moment Mt.


displacement decreases noticeably when the damping co-efficient C< 5000 kN/(m/s)α, but when the damping coef-ficient C> 5000 kN/(m/s)α, the accumulative girderdisplacement decreases slightly with increased damping

coefficient. )e benefits of an excessive damping coefficientare not proportional to the costs of it.

But for the low velocity exponent, the manufacturingprocess of this type of FVD is complicated and its durability

Susp

ende

r for

ce (k

N)

3.98×102

3.96

3.94

3.92

3.902000 None

3.973

3.961

3.946

3.931

3.916

Velocity

expone

nt

10.6

0.50.4

0.30.2

0.1

40006000 8000

1000012000Damping coefficient (kN/(m/s) α)

1400016000

1800020000

Figure 22: Maximum suspender force Fs.

Table 5: Peak values of bridge dynamic response.

Wgm (m) Wgc (m) Mg (kN·m) Wtm (m) at (m/s2) Mt (kN·m) Fs (kN)

Minimum value 0.3931 1.8845 1157 0.1554 0.010 170548 391.65Maximum value 0.5741 4.2352 1246 0.1586 0.0143 171652 397.15Without damper 0.5926 4.4561 1246 0.1586 0.0146 171481 397.25Maximum control efficiency 33.67 57.71 7.14 2.02 31.51 0.54 1.41

Dam

ping

velo

city

(m/s

)

0.048

0.045

0.042

0.039

0.033

0.036

0 2500 5000 7500 10000 12500 15000 17500 20000Damping coefficient (kN/(m/s)α)

α = 0.1α = 0.2α = 0.3

α = 0.5α = 0.6α = 1

α = 0.4

Figure 23: Maximum value of damping velocity Vc.


16000

14000

12000

10000

8000

Dam

ping

forc

e (kN

)

6000

4000

2000

0 2500 5000 7500Damping coefficient (kN/(m/s)α)

10000 12500 15000 17500 200000

α = 0.1α = 0.2α = 0.3

α = 0.5α = 0.6α = 1

α = 0.4

Figure 24: Maximum value of damping force F.

Max

imum

gird

er d

ispla

cem

ent (

m)

0.45

0.30

0.15

0.00

–0.15

–0.30

–0.45

–0.600 50

–0.46

C = 1000C = 2500C = 5000

C = 10000C = 15000

C = 20000C = 7000None

Unit: kN/(m/s)α

–0.48

–0.50

–0.06

–0.09

–0.12

–0.1530 40 215 220 225 230

100Time (s)

150 200 250 300

Figure 25: Time history curve of maximum girder displacement (α � 0.1).

Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

C = 1000C = 2500C = 5000

C = 7000C = 10000

C = 20000None

C = 15000

0 50 100 150 200Time (s)

250 300

Unit: kN/(m/s)α

4

3

2

1

0

Figure 26: Time history curve of accumulative girder displacement (α � 0.1).


may be limited. So the influence of the velocity exponent also isinvestigated.When the damping coefficient is 5000 kN/(m/s)α,the time history curves of the maximum girder displacementand accumulative girder displacement are shown in Figures 27and 28. It can be seen that the decreasing of velocity exponenthas little influence on the maximum girder displacement whilethe common vibration is reduced. According to Figure 28, theaccumulative girder displacement can be reduced obviouslywith the decreased velocity exponent. )at is because trafficload is different from the seismic excitation which has thecharacteristics of short time but strong intensity. Traffic load isgenerally a low intensity but long termprocess.)e low velocity

exponent brings evident benefits for the bridge response undertraffic flow.

According to the results, the damping coefficientC> 5000 kN/(m/s)α and the velocity exponent α� 0.1 are thebest parameters for FVDs under traffic flow. )e dampingcoefficient does not need to be too high. However, the FVDsapplication should consider a variety of factors, such as windand earthquake. So if the velocity exponent has to be in-creased, combined with Figure 16, the benefit of low velocityexponent reduced when the damping coefficient is highenough and the damping coefficient can be enlarged to fitwith the velocity exponent for the best effect of reducing theresponse under traffic flow.

6. Conclusions

In this study, bridge dynamic response with FVDs undertraffic flow was analyzed on the basis of monitored data. Forrapid analysis under high-intensity traffic flow, a bridge-vehicle coupling method optimized by isoparametricmapping and improved binary search was developed andvalidated.)en, traffic data with extreme total vehicle weightin an hour were screened out from monitoring data tosimulate traffic flow. Lastly, the simulation analysis of thebridge with FVDs under traffic flow based on monitoreddata was accomplished. Various indicators of the bridge areaffected to different degrees by variations in the FVD pa-rameters. )e results show that vehicle-bridge couplinganalysis method optimized by isoparametric mapping andimproved binary search based on the interhistory methodwas validated. )e method’s findings were consistent withthe experimental results; therefore, the proposed methodcan be used in vehicle-bridge coupling analysis under high-intensity traffic flow.

For the prototype bridge, the results show that FVDs’settlement positively impacts bridge dynamic responseunder traffic flow. )e traffic load should be considered inFVD design, as the dynamic response is remarkably influ-enced by FVD parameters. A reasonable FVD design caneffectively reduce the accumulative longitudinal girderdisplacement, maximum longitudinal girder displacement,and longitudinal pylon acceleration whose maximum con-trol efficiency, respectively, reaches 57.71%, 33.67%, and31.51%, the girder bending moment also can decrease by7.14%, and the maximum control efficiencies of other in-dexes were less than 3%.

)is study also shows an increase in the damping co-efficient and a decrease in the velocity exponent that canreduce the response of key indexes. However, an excessivedamping coefficient will bring negligible improvement andmay lead to high-intensity work under traffic flow, therebyadversely affecting component durability. A low velocityexponent will result in evident improvement, especially forFVD performance and girder displacement, and plays a rolein maintaining low-velocity movement. )e benefits of lowvelocity exponent reduced when the damping coefficient ishigh enough. When the velocity exponent has to be in-creased, the damping coefficient can be enlarged to fit withthe velocity exponent.

Max

imum

grid

er d

ispla

cem

ent (

m)

0.4

0.2

0.0

–0.2

–0.6

–0.4

–0.47

α = 0.1α = 0.2α = 0.3

α = 0.4α = 0.5

α = 1None

α = 0.6

–0.06

–0.09

–0.12

–0.15220 230

–0.48

–0.49

–0.50 30 32 34 36

0 50 100 150Time (s)

200 250 300

Figure 27: Time history curve of maximum girder displacement(C� 5000 kN/(m/s)α).

Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

4

3

2

1

00 50 100 150

Time (s)200 250 300

α = 0.1α = 0.2α = 0.3

α = 0.4α = 0.5

α = 1None

α = 0.6

Figure 28: Time history curve of accumulative girder displacement(C� 5000 kN/(m/s)α).


Note that the research and conclusions in this paperfocus on the FVDs’ influence on the bridge response undertraffic flow. In a subsequent research program, FVDs effectson bridge dynamic response under the combined action ofwind and traffic loads and the comprehensive optimizationof FVD parameters will be investigated.

Appendix

KivuL andKivuR denote the vertical spring stiffness of theith axle in upper suspension (N/m):

KivuL � K

ivuR. (A.1)

KiyuL andKiyuR denote the lateral spring stiffness of theith axle in upper suspension (N/m):

KiyuL � K

iyuR. (A.2)

KivlL andKivlR denote the vertical spring stiffness of the

ith axle in lower suspension (N/m):

KivlL � K

ivlR. (A.3)

KiylL andKiylR denote the lateral spring stiffness of the

ith axle in lower suspension (N/m):

KiylL � K

iylR. (A.4)

CivuL andCivuR denote the vertical damping of the ithaxle in upper suspension (N·s/m):

CivuL � C

ivuR. (A.5)

CiyuL andCiyuR denote the lateral damping of the ith axlein upper suspension (N·s/m):

CiyuL � C

iyuR. (A.6)

CivlL andCivlR denote the vertical damping of the ith axle

in lower suspension (N·s/m):

CivlL � C

ivlR. (A.7)

CiylL andCiylR denote the lateral damping of the ith axle

in lower suspension (N·s/m):

CiylL � C

iylR. (A.8)

Li denotes the distance from the center of mass of carbody to the ith axle.b1 denotes half of the lateral wheel spacing.h1 denotes the vertical distance from the center of massof car body to the upper plane of the spring.hv denotes the vertical distance from the center of massof car body to road surface.

Data Availability

)e raw/processed data required to reproduce these findingscannot be shared at this time as the data also form part of anongoing study.

Conflicts of Interest

)e authors declare that they have no conflicts of interest.

Acknowledgments

)e authors appreciate the financial support of the NationalNatural Science Foundation of China (Grant no. 51878058)and the Fundamental Research Funds for the CentralUniversities (Grant no. 300102219402).

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