Research Article Modeling and Simulation of the Vibration ...downloads.hindawi.com/journals/sv/2016/1982390.pdf · is study can provide a method for the modeling and simulation of

Research ArticleModeling and Simulation of the Vibration Characteristics of theIn-Wheel Motor Driving Vehicle Based on Bond Graph

Di Tan and Qiang Wang

School of Transportation and Vehicle Engineering, Shandong University of Technology, Zibo 255049, China

Correspondence should be addressed to Di Tan; [email protected]

Received 12 July 2015; Revised 18 September 2015; Accepted 11 October 2015

Academic Editor: Matteo Aureli

Copyright © 2016 D. Tan and Q. Wang.This 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.

Bond graph theory is applied to the modeling and analysis of the vibration characteristics of the in-wheel motor driving vehicle.First, an 11-degree-of-freedom vibration model of the in-wheel motor driving vehicle is established based on bond graph, and thenthe correctness of the model is verified. Second, under the driving condition of class B road excitations and a speed of 50Km/h, thevibration characteristics of the in-wheelmotor driving vehicle are simulated and analyzed, and the activity of each part in the systemis then calculated. Third, these parts that have less of an effect on the vibration characteristics of an in-wheel motor driving vehicleare identified according to the magnitude of the activity, and then the model is simplified by removing these parts. Finally, thereliability of the simplified model is verified by comparing the vibration characteristics of the model before and after simplification.This study can provide a method for the modeling and simulation of the vibration characteristics of the in-wheel motor drivingvehicle.

1. Introduction

Vehicle vibration has always been a popular research topicin the field of vehicles, because the analysis of the vehiclevibration characteristics can effectively evaluate the vehicleride performance. At present, most studies adopt Newton’ssecond law method to model and analyze the vibrationproblems of vehicles.

The bond graph (BG)methodwas proposed by AmericanH. M. Paynter, a professor in the late 1950s. The BG methodcan be used to solve the modeling problem of the systeminvolved in the multienergy field from the power perspective[1]. Compared with the traditional modeling method, BGtheory has the following advantages: (1) the BG theory canbe applied in a unified way to address the multienergy formsof the system; (2) the modular modeling method simplifiesthe modeling of the complex system; and (3) based on themodeling idea on the energy transfer, it is easy to simplify themodel [2]. Currently, only a few researchers have performedresearch using the BGmethod in the field of vehicle vibration.In one study [3], a 5-degree-of-freedom vibration model

for a traditional vehicle was established based on the BGmethod, and the simulation results were compared with realvehicle test data. In addition, the BG method was used toestablish the power train model of an electric vehicle with atraditional chassis structure and deduce the state equation ofsystem [4]. The BG theory was also applied for the analysisof the ride comfort of the terrain crane, and a 2-degree-of-freedom vibration model was established using the BGmethod [5]. In another study [6], the active hydropneumaticsuspension system of a multiaxle vehicle with a traditionalchassis was analyzed based on the BG method. In summary,BG theory has been applied to vehicle dynamics analysisto some extent, but most of the research has focused ontraditional vehicles. For the in-wheel motor driving vehicle(IWMDV), the chassis structure is different compared tothat of the traditional vehicles; in addition, the motor, speedreducer, and brakes are all integrated into the wheel. The BGmethod is a physical object-oriented method. The differencein the chassis structure represents the differences in thephysical and the BG model.

Hindawi Publishing CorporationShock and VibrationVolume 2016, Article ID 1982390, 14 pageshttp://dx.doi.org/10.1155/2016/1982390

2 Shock and Vibration

Suspension arm

Control cable

Import and exportof the cooling fluid

Hub bearing

Drum brake

Tire

Rim

Electronic controldevice

Hub bearing

Hub unit

Stator of motor

Rotor of motor

Figure 1: Basic structure of the electric wheel.

Based on the above advantages, the BG theory is appliedto the vibration analysis of the IWMDV in this paper. An11-degree-of-freedom IWMDV physical model is developedfirst using the BG method for modular modeling and assem-bly. Next, the mathematical model is deduced, for whichthe validity is verified using Newton’s second law. Basedon the developed model, the vibration characteristics ofthe IWMDV are analyzed, and the activity of each part iscalculated in the system. Taking advantage of the BGmethod,model reduction and model simplification are performed,and several of the parts that have little influence on the vehiclevibration are removed from the original physical model.Furthermore, the reliability of the model after simplification(MAS) is verified by the comparison analysis with the modelbefore simplification (MBS).

2. Vehicle Vibration Model

2.1. Electric Wheel Structure. As shown in Figure 1, the basicstructure of the electric wheel is mainly composed of thefollowing components: a motor stator, a motor rotor, brakes,wheel hub bearings, a rim, and a tire. In this structure, themotor housing and the rim are directly combined in onebody; therefore, by controlling the motor rotor rotation, thevehicle can be directly driven [7].

2.2. Physical Model of the Vehicle. To facilitate analysis, thefollowing hypotheses are applied to the vehicle physicalmodel:

(1) the vehicle body is regarded as a rigid body;(2) the operating conditions are set for uniform linear

driving;(3) the wheel hub bearing is equivalent to a spring-

damper;(4) the spring force is simplified as a linear function of

displacement, and the damping force is simplified asa linear function of velocity;

(5) the vehicle body has 3 degrees of freedom (vertical,pitch, and roll), and the unsprung mass has 8 degreesof freedom in the vertical direction; that is, the vehiclehas a total of 11 degrees of freedom.

Using the above hypotheses, the 11-degree-of-freedomphysical model is established, as shown in Figure 2.

In Figure 2, 𝑚𝑏 is the body mass; 𝑙𝑓 is the distance fromthe center of the body mass to the front axle; 𝑙𝑟 is the distancefrom the center of the body mass to the rear axle; 𝑑 is thetrack width; 𝜃 is the vehicle pitching angle; 𝜑 is the vehicleroll angle; 𝑘𝑖1 and 𝑘𝑖4 are the tire stiffness and the suspensionstiffness, respectively; 𝑐𝑖1 and 𝑐𝑖4 are the tire damping and thesuspension damping, respectively;𝑚𝑖1 is the total mass of thetire, rim, and motor rotor; 𝑚𝑖2 is the total mass of the motorstator and the brake, and so forth; 𝑘𝑖2 and 𝑘𝑖3 are the bearingstiffness; 𝑐𝑖2 and 𝑐𝑖3 are the bearing damping; 𝑧𝑖1 and 𝑧𝑖2 arethe vertical displacements of the corresponding mass; 𝑧𝑖 isthe vertical displacement of the connection point between thesuspension and vehicle body; 𝑖 = 𝑎, 𝑏, 𝑐, and 𝑑 represents theright front wheel, left front wheel, right rear wheel, and leftrear wheel, respectively; and 𝑧𝑗 is the displacement input ofthe road surface roughness, where 𝑗 = 1, 2, 3, and 4.

2.3. The BG Model of the Vehicle

2.3.1. Basic BG Modeling Method. For a mechanical system,the BG model includes five basic elements: R elementrepresents damping, C element represents spring, I elementrepresents mass or inertia, Se element represents the forcesource, and Sf element represents the velocity source; Relement is the energy dissipation element and C and Ielements are the energy storage elements. The five elementsare linked by constant velocity 1-junctions, constant force0-junctions, and transformer TF-junctions. The basic BGmodeling method is as follows:

(1) Create 1-junctions for each absolute velocity andrelative velocity in the model.

(2) Create 0-junctions and transformer TF-junctionsbetween correlative 1-junctions to establish the rela-tionships between the correlative absolute velocitiesand between the correlative relative velocity andabsolute velocity using the relationship between thevelocities with power flow direction.

(3) Connect the five basic elements to the corresponding1-junctions with suitable causality.

2.3.2. Modular Modeling. Modular modeling not only makesthe establishment of the model clearer and easier but alsomakes modification of the model more convenient. Thevehicle physical model shown in Figure 2 is modeled usingthe BG method.

In this modeling approach, a model can be divided intoseveral subsystems; each subsystem is modeled, and thenall subsystem models are combined to obtain the wholeBG model of the vehicle [8]. The physical model shown inFigure 2 can be divided into five subsystems: the body systemand four quarter car suspension systems. First, the BGmodels

Shock and Vibration 3

zc

zdz

y

𝜃

𝜑d/2 d/2

d/2 d/2

lr

mb

lf

za

x

kc4 cc4kd4

cd4

zd2za2

ka4ca4

kb4

cb4zb2

zc2

mc2

md2ma2

kb2

mb2

cb3kb3

kc2 kc3

kd2 kd3cd3

ka2 ka3 cb2

zb1

zc1cc2 cc3

mc1

md1ma1 kb1 cb1 z2

mb1ca3ca2

za1

zd1cd2

z3 kc1 cc1

kc1 cc1 z4z1

ka1 ca1

zb

Figure 2: 11-degree-of-freedom physical model.

of the left front (LF), right front (RF), left rear (LR), and rightrear (RR) quarter car suspension systems are set up, as shownin Figure 3.

Next, the body system BGmodel is established, as shownin Figure 4.

2.3.3. BGModel Combination. The reserved ports LF, RF, LR,andRR in Figure 3 are first connectedwith the correspondingports LF, RF, LR, and RR in Figure 4. Then, the whole BGmodel of the vehicle can be obtained and further simplifiedas shown in Figure 5.

2.4. Mathematical Model of the Vehicle

2.4.1. Definition of the BG Symbols. Using the generalizedvariables defined in BG theory [1], formechanical translation,the generalized variables 𝑞 and 𝑝 represent displacement andmomentum, respectively, whereas, for mechanical rotation, 𝑞and 𝑝 represent angle and angular momentum, respectively.In addition, the generalized variables 𝑞 and �� are the deriva-tives of 𝑞 and 𝑝 with respect to time, respectively. Likewise,for machine translation, 𝑞 and �� represent velocity and force,respectively, and for mechanical rotation, 𝑞 and �� representangular velocity and torque, respectively.

Each element in the BG model has a subscript, and eachsubscript corresponds to a different element. The elements 𝑞,𝑝, 𝑞, and ��with different subscripts refer to the correspondingsymbol meaning of the corresponding element.

2.4.2. State Equation of the BG Model. According to the BGmodel of the vehicle, the causality and power flow directionare analyzed by the BG method, and then the vehicle stateequation of the BG model is expressed as follows [9].

The vertical velocity equation of𝑚𝑎1 is

𝑞4 = 𝑞𝑎 −𝑝6

𝑚𝑎1. (1)

The vertical velocity equation of𝑚𝑎2 is

𝑞14 = 𝑞12 =𝑝6

𝑚𝑎1−𝑝17

𝑚𝑎2. (2)

The vertical velocity equation of𝑚𝑏1 is

𝑞65 = 𝑞𝑏 −𝑝60

𝑚𝑏1. (3)

The vertical velocity equation of𝑚𝑏2 is

𝑞56 = 𝑞53 =𝑝60

𝑚𝑏1−𝑝49

𝑚𝑏2. (4)

The vertical velocity equation of𝑚𝑐1 is

𝑞106 = 𝑞𝑐 −𝑝102

𝑚𝑐1. (5)

The vertical velocity equation of𝑚𝑐2 is

𝑞99 = 𝑞96 =𝑝102

𝑚𝑐1−𝑝91

𝑚𝑐2. (6)

The vertical velocity equation of𝑚𝑑1 is

𝑞85 = 𝑞𝑑 −𝑝81

𝑚𝑑1. (7)

The vertical velocity equation of𝑚𝑑2 is

𝑞75 = 𝑞77 =𝑝81

𝑚𝑑1−𝑝70

𝑚𝑑2. (8)


LF

cb4 R

46

C47

145

0

1/kb4

R cb2

48

149

mb2 R cb3

54

53 52

1/kb2

1/kb1

cb1mb1

Sf qb

I

5750

C561550

58

I60

6163

0621C

1

59

51

01C

R

6465

1/kb3

(a) LF

RF

ca4 R

20

21C1/ka4

1 190

18

171

ma2R ca3I

15 13

100 1

14 C1/ka3

R ca2

11

12C1/ka2

19

16

0

7ca1R

31/ka1

C4

112

0

1

5

6

8

I ma1

Sf qa

(b) RF

LR

cd4 R

66

67C1/kd4

168 0

69

70R cd2

78

77C1/kd2

cd1R

176

71

0

79

86

851/kd1

C 184

083

82

81

Sf qd

I md1

80

073

74

175 C

1/kd3

1

1

72

Imd2 R cd3

(c) LR

RR

cc4 R

88

89C1/kc4

R cc2

95

96C

1/kc2

194

0

101

93

cc1 R

105

1061/kc1

C 1104

0

103

107

102

Sf qc

I1

100

mc1

097

98

991 C

1/kc3

I91

90

1

92

087

1

R cc3mc2

(d) RR

Figure 3: BG model of the four quarter car suspension systems.

The vertical motion equation of the four endpoints of thevehicle body can be obtained:

𝑞21 =𝑝17

𝑚𝑎2− 𝑙𝑓

𝑝26

𝐼𝑦−𝑑

2

𝑝44

𝐼𝑥−𝑝35

𝑚𝑏, (9)

𝑞47 =𝑝49

𝑚𝑏2− 𝑙𝑓

𝑝26

𝐼𝑦+𝑑

2

𝑝44

𝐼𝑥−𝑝35

𝑚𝑏, (10)

𝑞89 =𝑝91

𝑚𝑐2+ 𝑙𝑟

𝑝26

𝐼𝑦−𝑑

2

𝑝44

𝐼𝑥−𝑝35

𝑚𝑏, (11)

𝑞67 =𝑝70

𝑚𝑑2+ 𝑙𝑟

𝑝26

𝐼𝑦+𝑑

2

𝑝44

𝐼𝑥−𝑝35

𝑚𝑏. (12)

The vertical force equation of𝑚𝑎1 is

��6 = 𝑐𝑎1 𝑞4 + 𝑘𝑎1𝑞4 − 𝑐𝑎2 𝑞12 − 𝑘𝑎2𝑞12 − 𝑐𝑎3 𝑞14 − 𝑘𝑎3𝑞14. (13)

The vertical force equation of𝑚𝑎2 is

��17 = 𝑐𝑎2 𝑞12 + 𝑘𝑎2𝑞12 + 𝑐𝑎3 𝑞14 + 𝑘𝑎3𝑞14 − 𝑐𝑎4 𝑞21

− 𝑘𝑎4𝑞21.(14)


LR LF

RR RF

043

36

31

0

30TF

TF

TF

−2/d

−2/d

TF

29

26

I Iy

1

28

3332

37 39

25

142

44

I Ix

41

24

TF−2/d

TF

22

2327

0

135

34

I mb

0

40

38

TF

TF−2/d

−1/lr1/lf

−1/lr 1/lf

Figure 4: BG model of the vehicle body system.

The vertical force equation of𝑚𝑏1 is

��60 = 𝑐𝑏1 𝑞65 + 𝑘𝑏2𝑞65 − 𝑐𝑏3 𝑞56 − 𝑘𝑏3𝑞56 − 𝑐𝑏2 𝑞53

− 𝑘𝑏2𝑞53.(15)

The vertical force equation of𝑚𝑏2 is

��49 = 𝑐𝑏3 𝑞56 + 𝑘𝑏3𝑞56 + 𝑐𝑏2 𝑞53 + 𝑘𝑏2𝑞53 − 𝑐𝑏4 𝑞47

− 𝑘𝑏4𝑞47.(16)

The vertical force equation of𝑚𝑐1 is

��102 = 𝑐𝑐1 𝑞106 + 𝑘𝑐1𝑞106 − 𝑐𝑐2 𝑞96 − 𝑘𝑐2𝑞96 − 𝑐𝑐3 𝑞99

− 𝑘𝑐3𝑞99.(17)

The vertical force equation of𝑚𝑐2 is

��91 = 𝑐𝑐2 𝑞96 + 𝑘𝑐2𝑞96 + 𝑐𝑐3 𝑞99 + 𝑘𝑐3𝑞99 − 𝑐𝑐4 𝑞89

− 𝑘𝑐4𝑞89.(18)

The vertical force equation of𝑚𝑑1 is

��81 = 𝑐𝑑1 𝑞85 + 𝑘𝑑1𝑞85 − 𝑐𝑑2 𝑞77 − 𝑘𝑑2𝑞77 − 𝑐𝑑3 𝑞75

− 𝑘𝑑3𝑞75.(19)

The vertical force equation of𝑚𝑑2 is

��70 = 𝑐𝑑2 𝑞77 + 𝑘𝑑2𝑞77 + 𝑐𝑑3 𝑞75 + 𝑘𝑑3𝑞75 − 𝑐𝑑4 𝑞67

− 𝑘𝑑4𝑞67.(20)

The vertical force equation of the vehicle body is asfollows:

��35 = 𝑐𝑎4 𝑞21 + 𝑘𝑎4𝑞21 + 𝑐𝑐4 𝑞89 + 𝑘𝑐4𝑞89 + 𝑐𝑑4 𝑞67

+ 𝑘𝑑4𝑞67 + 𝑐𝑏4 𝑞47 + 𝑘𝑏4𝑞47.(21)

The pitching moment equation of the vehicle body is

��26 = 𝑙𝑓 (𝑐𝑎4 𝑞21 + 𝑘𝑎4𝑞21 + 𝑐𝑏4 𝑞47 + 𝑘𝑏4𝑞47)

− 𝑙𝑟 (𝑐𝑐4 𝑞89 + 𝑘𝑐4𝑞89 + 𝑐𝑑4 𝑞67 + 𝑘𝑑4𝑞67) .

(22)

The roll torque equation of the vehicle body is

��44 =𝑑

2(𝑐𝑎4 𝑞21 + 𝑘𝑎4𝑞21 + 𝑐𝑐4 𝑞89 + 𝑘𝑐4𝑞89 − 𝑐𝑑4 𝑞67

− 𝑘𝑑4𝑞67 − 𝑐𝑏4 𝑞47 − 𝑘𝑏4𝑞47) .

(23)

3. Validation of the BG Model

3.1. The Mathematical Model Developed Using the NewtonMethod. Based on the physical model in Figure 2, the math-ematical model is derived using Newton’s second law; adescription is given as follows.

The vertical motion equations of the unsprungmasses are

𝑚𝑎1��𝑎1 = 𝑐𝑎1 (��1 − ��𝑎1) + 𝑘𝑎1 (𝑧1 − 𝑧𝑎1)

− (𝑐𝑎2 + 𝑐𝑎3) (��𝑎1 − ��𝑎2)

− (𝑘𝑎2 + 𝑘𝑎3) (𝑧𝑎1 − 𝑧𝑎2) ,

𝑚𝑎2��𝑎2 = (𝑐𝑎2 + 𝑐𝑎3) (��𝑎1 − ��𝑎2)

+ (𝑘𝑎2 + 𝑘𝑎3) (𝑧𝑎1 − 𝑧𝑎2)

− 𝑐𝑎4 (��𝑎2 − ��𝑎) − 𝑘𝑎4 (𝑧𝑎2 − 𝑧𝑎) ,


𝑚𝑏1��𝑏1 = 𝑐𝑏1 (��2 − ��𝑏1) + 𝑘𝑏1 (𝑧2 − 𝑧𝑏1)

− (𝑐𝑏3 + 𝑐𝑏2) (��𝑏1 − ��𝑏2)

− (𝑘𝑏3 + 𝑘𝑏2) (𝑧𝑏1 − 𝑧𝑏2) ,

𝑚𝑏2��𝑏2 = (𝑐𝑏2 + 𝑐𝑏3) (��𝑏1 − ��𝑏2)

+ (𝑘𝑏2 + 𝑘𝑏3) (𝑧𝑏1 − 𝑧𝑏2) − 𝑐𝑏4 (��𝑏2 − ��𝑏)

− 𝑘𝑏4 (𝑧𝑏2 − 𝑧𝑏) ,

𝑚𝑐1��𝑐1 = 𝑐𝑐1 (��3 − ��𝑐1) + 𝑘𝑐1 (𝑧3 − 𝑧𝑐1)

− (𝑐𝑐3 + 𝑐𝑐2) (��𝑐1 − ��𝑐2)

− (𝑘𝑐3 + 𝑘𝑐2) (𝑧𝑐1 − 𝑧𝑐2) ,

𝑚𝑐2��𝑐2 = (𝑐𝑐2 + 𝑐𝑐3) (��𝑐1 − ��𝑐2)

+ (𝑘𝑐2 + 𝑘𝑐3) (𝑧𝑐1 − 𝑧𝑐2) − 𝑐𝑐4 (��𝑐2 − ��𝑐)

− 𝑘𝑐4 (𝑧𝑐2 − 𝑧𝑐) ,

𝑚𝑑1��𝑑1 = 𝑐𝑑1 (��4 − ��𝑑1) + 𝑘𝑑1 (𝑧4 − 𝑧𝑑1)

− (𝑐𝑑3 + 𝑐𝑑2) (��𝑑1 − ��𝑑2)

− (𝑘𝑑3 + 𝑘𝑑2) (𝑧𝑑1 − 𝑧𝑑2) ,

𝑚𝑑2��𝑑2 = (𝑐𝑑2 + 𝑐𝑑3) (��𝑑1 − ��𝑑2)

+ (𝑘𝑑2 + 𝑘𝑑3) (𝑧𝑑1 − 𝑧𝑑2)

− 𝑐𝑑4 (��𝑑2 − ��𝑑) − 𝑘𝑑4 (𝑧𝑑2 − 𝑧𝑑) .

(24)

The vertical, pitch, and roll motion equations of thevehicle body are

𝑚𝑏�� = 𝑐𝑎4 (��𝑎2 − ��𝑎) + 𝑘𝑎4 (𝑧𝑎2 − 𝑧𝑎) + 𝑐𝑏4 (��𝑏2 − ��𝑏)

+ 𝑘𝑏4 (𝑧𝑏2 − 𝑧𝑏) + 𝑐𝑐4 (��𝑐2 − ��𝑐) + 𝑘𝑐4 (𝑧𝑐2 − 𝑧𝑐)

+ 𝑐𝑑4 (��𝑑2 − ��𝑑) + 𝑘𝑑4 (𝑧𝑑2 − 𝑧𝑑) ,

𝐼𝑦𝜃 = 𝑙𝑓 [𝑐𝑎4 (��𝑎2 − ��𝑎) + 𝑘𝑎4 (𝑧𝑎2 − 𝑧𝑎)

+ 𝑐𝑏4 (��𝑏2 − ��𝑏) − 𝑘𝑏4 (𝑧𝑏2 − 𝑧𝑏)] − 𝑙𝑟 [𝑐𝑐4 (��𝑐2 − ��𝑐)

+ 𝑘𝑐4 (𝑧𝑐2 − 𝑧𝑐) + 𝑐𝑑4 (��𝑑2 − ��𝑑) − 𝑘𝑑4 (𝑧𝑑2 − 𝑧𝑑)] ,

𝐼𝑥𝜙 =

𝑑

2[𝑐𝑎4 (��𝑎2 − ��𝑎) + 𝑘𝑎4 (𝑧𝑎2 − 𝑧𝑎)

+ 𝑐𝑐4 (��𝑐2 − ��𝑐) + 𝑘𝑐4 (𝑧𝑐2 − 𝑧𝑐) − 𝑐𝑏4 (��𝑏2 − ��𝑏)

− 𝑘𝑏4 (𝑧𝑏2 − 𝑧𝑏) − 𝑐𝑑4 (��𝑑2 − ��𝑑) − 𝑘𝑑4 (𝑧𝑑2 − 𝑧𝑑)] .

(25)

LFLR

RR RF

Body

Figure 5: The whole BG model of the vehicle.

The verticalmotion equations of the four endpoints of thevehicle body are

��𝑎 = �� + 𝑙𝑓𝜃 +

1

2𝑑��,

��𝑏 = �� + 𝑙𝑓𝜃 −

1

2𝑑��,

��𝑐 = �� − 𝑙𝑟𝜃 +

1

2𝑑��,

��𝑑 = �� − 𝑙𝑟𝜃 −

1

2𝑑��.

(26)

3.2. Model Validation. To verify the correctness of the BGmodel in Figure 5, the state equations deduced by the BGmodel are translated into the form of motion equationsdeduced byNewton’s second law.The transformationmethodis based on the relationship between the symbol in thestate equations and the symbol in the Newtonian equations;accordingly, the state equations are converted into the New-tonian equations.

From (1), (2), and (13), the following expression isobtained:

𝑚𝑎1��𝑎1 = 𝑐𝑎1 (��1 − ��𝑎1) + 𝑘𝑎1 (𝑧1 − 𝑧𝑎1)

− (𝑐𝑎2 + 𝑐𝑎3) (��𝑎1 − ��𝑎2)

− (𝑘𝑎2 + 𝑘𝑎3) (𝑧𝑎1 − 𝑧𝑎2) .

(27)


𝑚𝑎2��𝑎2 = −𝑐𝑎4 (��𝑎2 − ��𝑎) − 𝑘𝑎4 (𝑧𝑎2 − 𝑧𝑎)

+ (𝑐𝑎2 + 𝑐𝑎3) (��𝑎1 − ��𝑎2)

+ (𝑘𝑎2 + 𝑘𝑎3) (𝑧𝑎1 − 𝑧𝑎2) .

(28)


𝑚𝑏1��𝑏1 = 𝑐𝑏1 (��2 − ��𝑏1) + 𝑘𝑏1 (𝑧2 − 𝑧𝑏1)

− (𝑐𝑏3 + 𝑐𝑏2) (��𝑏1 − ��𝑏2)

− (𝑘𝑏3 + 𝑘𝑏2) (𝑧𝑏1 − 𝑧𝑏2) .

(29)



𝑚𝑏2��𝑏2 = −𝑐𝑏4 (��𝑏2 − ��𝑏) − 𝑘𝑏4 (𝑧𝑏2 − 𝑧𝑏)

+ (𝑐𝑏2 + 𝑐𝑏3) (��𝑏1 − ��𝑏2)

+ (𝑘𝑏2 + 𝑘𝑏3) (𝑧𝑏1 − 𝑧𝑏2) .

(30)


𝑚𝑐1��𝑐1 = 𝑐𝑐1 (��3 − ��𝑐1) + 𝑘𝑐1 (𝑧3 − 𝑧𝑐1)

− (𝑐𝑐3 + 𝑐𝑐2) (��𝑐1 − ��𝑐2)

− (𝑘𝑐3 + 𝑘𝑐2) (𝑧𝑐1 − 𝑧𝑐2) .

(31)


𝑚𝑐2��𝑐2 = −𝑐𝑐4 (��𝑐2 − ��𝑐) − 𝑘𝑐4 (𝑧𝑐2 − 𝑧𝑐)

+ (𝑐𝑐2 + 𝑐𝑐3) (��𝑐1 − ��𝑐2)

+ (𝑘𝑐2 + 𝑘𝑐3) (𝑧𝑐1 − 𝑧𝑐2) .

(32)


𝑚𝑑1��𝑑1 = 𝑐𝑑1 (��4 − ��𝑑1) + 𝑘𝑑1 (𝑧4 − 𝑧𝑑1)

− (𝑐𝑑3 + 𝑐𝑑2) (��𝑑1 − ��𝑑2)

− (𝑘𝑑3 + 𝑘𝑑2) (𝑧𝑑1 − 𝑧𝑑2) .

(33)


𝑚𝑑2��𝑑2 = −𝑐𝑑4 (��𝑑2 − ��𝑑) − 𝑘𝑑4 (𝑧𝑑2 − 𝑧𝑑)

+ (𝑐𝑑2 + 𝑐𝑑3) (��𝑑1 − ��𝑑2)

+ (𝑘𝑑2 + 𝑘𝑑3) (𝑧𝑑1 − 𝑧𝑑2) .

(34)

From (9), (10), (11), (12), and (21), the vertical motionequation of the vehicle body is obtained:

𝑚𝑏�� = 𝑐𝑎4 (��𝑎2 − ��𝑎) + 𝑘𝑎4 (𝑧𝑎2 − 𝑧𝑎) + 𝑐𝑏4 (��𝑏2 − ��𝑏)

+ 𝑘𝑏4 (𝑧𝑏2 − 𝑧𝑏) + 𝑐𝑐4 (��𝑐2 − ��𝑐)

+ 𝑘𝑐4 (𝑧𝑐2 − 𝑧𝑐) + 𝑐𝑑4 (��𝑑2 − ��𝑑)

+ 𝑘𝑑4 (𝑧𝑑2 − 𝑧𝑑) .

(35)

From (9), (10), (11), (12), and (22), the pitch motionequation of the vehicle body is obtained:

𝐼𝑦𝜃 = 𝑙𝑓 [𝑐𝑎4 (��𝑎2 − ��𝑎) + 𝑘𝑎4 (𝑧𝑎2 − 𝑧𝑎)

+ 𝑐𝑏4 (��𝑏2 − ��𝑏) − 𝑘𝑏4 (𝑧𝑏2 − 𝑧𝑏)] − 𝑙𝑟 [𝑐𝑐4 (��𝑐2 − ��𝑐)

+ 𝑘𝑐4 (𝑧𝑐2 − 𝑧𝑐) + 𝑐𝑑4 (��𝑑2 − ��𝑑) − 𝑘𝑑4 (𝑧𝑑2 − 𝑧𝑑)] .

(36)

From (9), (10), (11), (12), and (23), the roll motionequation of the vehicle body is obtained:

𝐼𝑥𝜙 =

𝑑

2[𝑐𝑎4 (��𝑎2 − ��𝑎) + 𝑘𝑎4 (𝑧𝑎2 − 𝑧𝑎)

+ 𝑐𝑐4 (��𝑐2 − ��𝑐) + 𝑘𝑐4 (𝑧𝑐2 − 𝑧𝑐) − 𝑐𝑏4 (��𝑏2 − ��𝑏)

− 𝑘𝑏4 (𝑧𝑏2 − 𝑧𝑏) − 𝑐𝑑4 (��𝑑2 − ��𝑑) − 𝑘𝑑4 (𝑧𝑑2 − 𝑧𝑑)] .

(37)

From (9), (10), (11), and (12), the verticalmotion equationsof the four endpoints of the vehicle body are obtained:

��𝑎 = �� + 𝑙𝑓𝜃 +

1

2𝑑��,

��𝑏 = �� + 𝑙𝑓𝜃 −

1

2𝑑��,

��𝑐 = �� − 𝑙𝑟𝜃 +

1

2𝑑��,

��𝑑 = �� − 𝑙𝑟𝜃 −

1

2𝑑��.

(38)

In comparing this mathematical model with the mathe-matical model deduced using Newton’s second law, both arethe same; as a result, the correctness of the BG model isverified.

4. Model Simplifications andComparative Analysis

Figure 2 shows that there are many parts in the physicalmodel; this larger number of parts will affect the subsequentvibration characteristic analysis. Because some parts haveless of an effect on vehicle vibration characteristics, theseparts can be identified through simulation and analysis andsubsequently removed, thereby simplifying the model.

4.1. Model Simplification. The simplified model approach isas follows: according to the change in power of each part inthe system over time, the sum of the energy absorption andrelease of each part at a given period of time is calculated.The relevant calculation formula of the power for each part iscalculated by

𝑊(𝑡) = ∫𝑃 (𝑡) 𝑑𝑡, (39)

where𝑊(𝑡) represents the sum of the energy absorption andrelease of each part in the system at a given period of time and𝑃(𝑡) represents the power changes over time of each part.

The above approach is used to judge the activity of eachpart in the system. This activity is represented by the sum ofenergy absorption and release by each part at a given periodof time as a percentage of the sum of energy absorption andrelease by the whole system (all parts) at this period of time.The low activity of some parts illustrates that the impact ofthe parts on the vehicle vibration characteristics is small; thus,the activity of these less relevant parts is easily identified, andsuch parts can be removed to simplify the model.


The left front wheelThe left rear wheel

The right front wheelThe right rear wheel

−0.04

−0.02

0

0.02

0.04

Disp

lace

men

t (m

)

2 4 6 8 100Time (s)

−0.04

−0.02

0

0.02

0.04

Disp

lace

men

t (m

)

2 4 6 8 100Time (s)

Figure 6: Road random input of four wheels.

4.1.1.The Vehicle Parameters. Theparameters of the IWMDVmodel are given as shown in Table 1.

4.1.2. Time Domain Model of the Road Surface RoughnessExcitation. In the vehicle vibration characteristics analysis,the road correlation of the four wheels should be considered.In this paper, a four-wheel correlation time domain model ofroad roughness excitations is built up [10].The state equationof the four-wheel related road excitation is given as follows:

�� (𝑡) = 𝐴𝑍 (𝑡) + 𝐵0𝑆 (𝑡) , (40)

where

𝑍 (𝑡) = [𝑧1 (𝑡) 𝑧2 (𝑡) 𝑧3 (𝑡) 𝑧4 (𝑡) 𝑥1 (𝑡) 𝑥2 (𝑡)]𝑇;

�� (𝑡) = [��1 (𝑡) ��2 (𝑡) ��3 (𝑡) ��4 (𝑡) ��1 (𝑡) ��2 (𝑡)]𝑇;

𝐴

=

[[[[[[[[[[[[[[[[[

[

𝑎1 0 0 0 0 0

𝑒−2𝜋𝑛00𝑑𝑢

𝑑−𝑢

𝑑0 0 0 0

−12𝑢

𝑙+ 𝑎1 0 0 0 0 0

𝑒−2𝜋𝑛00𝑑𝑢

𝑑−(

12𝑢

𝑙+𝑢

𝑑) 0 0 0 1

−12𝑢

𝑙0 0 0 0 1

72𝑢2

𝑙20 0 0 −

12𝑢2

𝑙2−6𝑢

𝑙

]]]]]]]]]]]]]]]]]

]

;

𝐵0 = [𝑏1 0 𝑏1 0 0 0]𝑇;

𝑏1 = 2𝜋𝑛0√𝑆𝑞 (𝑛0) 𝑢;

𝑎1 = −2𝜋𝑛00𝑢;

(41)

Table 1: Parameters of the IWMDV.

Variable Value Unit𝑚𝑏

1580 kg𝑘𝑎1/𝑘𝑏1/𝑘𝑐1/𝑘𝑑1 400/400/400/400 kN/m𝑘𝑎2/𝑘𝑏2/𝑘𝑐2/𝑘𝑑2 5120/5120/5120/5120 kN/m𝑘𝑎3/𝑘𝑏3/𝑘𝑐3/𝑘𝑑3 5000/5000/5000/5000 kN/m𝑘𝑎4/𝑘𝑏4/𝑘𝑐4/𝑘𝑑4 80/80/80/80 kN/m𝑐𝑎1/𝑐𝑏1/𝑐𝑐1/𝑐𝑑1

100/100/100/100 N⋅s/m𝑐𝑎2/𝑐𝑏2/𝑐𝑐2/𝑐𝑑2 0/0/0/0 N⋅s/m𝑐𝑎3/𝑐𝑏3/𝑐𝑐3/𝑐𝑑3 0/0/0/0 N⋅s/m𝑐𝑎4/𝑐𝑏4/𝑐𝑐4/𝑐𝑑4 5000/5000/5000/5000 N⋅s/m𝑚𝑎1/𝑚𝑏1/𝑚𝑐1/𝑚𝑑1 80/80/80/80 kg𝑚𝑎2/𝑚𝑏2/𝑚𝑐2/𝑚𝑑2

50/50/50/50 kg𝐼𝑥/𝐼𝑦 1100/3000 kg⋅m2

𝑑 1.3 m𝑙𝑓/𝑙𝑟 1.19/1.2 m

𝑆(𝑡) is the Gaussian white noise; 𝑆𝑞(𝑛0) is the road roughnesscoefficient; 𝑛0 is the reference frequency; 𝑛00 is the road cutoffspatial frequency; 𝑢 is the vehicle speed; 𝑙 is the length of thevehicle; and 𝑙 = 𝑙𝑓 + 𝑙𝑟.

4.1.3. Simulation Analysis. In this paper, the speed 𝑢 is50 km/h, the road excitation simulated is a class B road, theroad roughness coefficient is 𝑆𝑞(𝑛0) = 64 × 10−6m3, the roadspace cutoff frequency is 𝑛00 = 0.01m−1, and the four-wheelcorrelation input of the road surface roughness excitation isobtained using Matlab/Simulink simulation [11], as shown inFigure 6.


z

y

𝜃

𝜑d/2

d/2 mb

d/2

d/2 x

kc4 cc4kd4

cd4 ka4 ca4

kb4

cb4

mc

mdma

mb

kc1kd1

z3

z4 z1 ka1

kb1z2

lr lf

Figure 7: The simplified physical model.

Table 2: Sum of the energy absorption and release of each part inthe system within 10 s.

PartSum of the energy

absorption and releasewithin 10 s

Percentage (%)

𝑘𝑎1/𝑘𝑏1/𝑘𝑐1/𝑘𝑑1 1178.9/1351.1/1198.2/1218.6 6.96/7.97/7.07/7.19𝑘𝑎2/𝑘𝑏2/𝑘𝑐2/𝑘𝑑2 9.1/10.9/9.2/8.1 0.05/0.06/0.05/0.05𝑘𝑎3/𝑘𝑏3/𝑘𝑐3/𝑘𝑑3 8.9/10.6/9/7.9 0.05/0.06/0.05/0.05𝑘𝑎4/𝑘𝑏4/𝑘𝑐4/𝑘𝑑4 180.8/334.6/185/134.8 1.07/1.98/1.09/0.8𝑐𝑎1/𝑐𝑏1/𝑐𝑐1/𝑐𝑑1 46/44.4/47.1/44.7 0.27/0.26/0.28/0.26𝑐𝑎2/𝑐𝑏2/𝑐𝑐2/𝑐𝑑2 0/0/0/0 0/0/0/0𝑐𝑎3/𝑐𝑏3/𝑐𝑐3/𝑐𝑑3 0/0/0/0 0/0/0/0𝑐𝑎4/𝑐𝑏4/𝑐𝑐4/𝑐𝑑4 578.7/611.9/543.5/469 3.42/3.61/3.21/2.71𝑚𝑎1/𝑚𝑏1/𝑚𝑐1/𝑚𝑑1 613/684.8/652.7/695.2 3.62/4.04/3.85/4.1𝑚𝑎2/𝑚𝑏2/𝑚𝑐2/𝑚𝑑2 395/436.3/420.2/444.2 2.33/2.58/2.48/2.62𝑚𝑏 4360.1 25.73

Substituting the above road surface roughness excitationinto the BG model of the vehicle, the sum of energy absorp-tion and release of each part in the system within 10 s canbe calculated by (39). Then, the power for each part can becalculated as a percentage of the sum of energy absorptionand release of the whole system during this period. Thestatistical data are listed in Table 2.

As observed from the data in Table 2, the percentage ofthe parts 𝑘𝑖2, 𝑘𝑖3, 𝑐𝑖1, 𝑐𝑖2, and 𝑐𝑖3 is less than 0.5%, whichindicates that the activity of these parts is low. As a result,the impact of these parts on the vehicle vibration character-istics can be ignored, and the model can be simplified andultimately obtained (as shown in Figure 7). In this simplifiedmodel, the mass of 𝑚𝑎2, 𝑚𝑏2, 𝑚𝑐2, and 𝑚𝑑2 is combined withthe mass of𝑚𝑎1,𝑚𝑏1,𝑚𝑐1, and𝑚𝑑1, respectively, yielding the

combination masses𝑚𝑎,𝑚𝑏,𝑚𝑐, and𝑚𝑑, respectively, for thewhole in-wheel motor as a rigid body.

4.2. Simplified Mathematical Model. The BG model of thesimplified physical model is established in this section.Because this model is similar to the BG model shown inFigure 5, we can eliminate the parts with low activity fromFigure 5 directly. The simplified BG model can be obtained,which is shown in Figure 8. In this figure, in addition tothe symbol 𝑚𝑎, 𝑚𝑏, 𝑚𝑐, and 𝑚𝑑 represent the summation of𝑚𝑎2 & 𝑚𝑎1,𝑚𝑏2 & 𝑚𝑏1,𝑚𝑐2 & 𝑚𝑐1, and𝑚𝑑2 & 𝑚𝑑1, respec-tively; the other symbols are the same as described in Figure 5.

Similarly, according to Figure 8, the state equations ofMAS are obtained as follows.

The vertical velocity equation of𝑚𝑎 is

𝑞2 = 𝑞𝑎 −𝑝4

𝑚𝑎. (42)

The vertical velocity equation of𝑚𝑏 is

𝑞10 = 𝑞𝑏 −𝑝12

𝑚𝑏. (43)

The vertical velocity equation of𝑚𝑐 is

𝑞18 = 𝑞𝑐 −𝑝20

𝑚𝑐. (44)

The vertical velocity equation of𝑚𝑑 is

𝑞50 = 𝑞𝑑 −𝑝52

𝑚𝑑. (45)

The vertical force equation of𝑚𝑎 is

��4 = 𝑘𝑎1𝑞2 − 𝑘𝑎4𝑞8 − 𝑐𝑎4 𝑞8. (46)


qd Sf

49

50C1/kd1 0

51

52Imd

53

5455cd4 R 1 0

1

C 561/kd4

1/kc4 C46

47cc4 R 145

0

21

20Imc

1

19

018

17

C

qc Sf

I Ix

44

−2/d

TF43 42

141

−2/d

TF40

38

36

TF 32

33

25

39

2837

TF

TF31 −2/d

TF

30 29

26

I Iy

124

TF22

23

TF

−2/d

Sf qb

9

010

11

C 1/kb1

1 12I mb

13

014

1 16

1534

C

R

1/kb4

cb4

135

27

I m

8

06 1

7R

C 1/ka4

ca4

5

14

3

I ma

02

1

Sf qa

C 1/ka11/kc1

−1/lr

−1/lr1/lf

1/lf

Figure 8: The simplified BG model.

The vertical force equation of𝑚𝑏 is

��12 = 𝑘𝑏1𝑞10 − 𝑘𝑏4𝑞16 − 𝑐𝑏4 𝑞16. (47)

The vertical force equation of𝑚𝑐 is

��20 = 𝑘𝑐1𝑞18 − 𝑘𝑐4𝑞46 − 𝑐𝑐4 𝑞46. (48)

The vertical force equation of𝑚𝑑 is

��52 = 𝑘𝑑1𝑞50 − 𝑘𝑑4𝑞55 − 𝑐𝑑4 𝑞55. (49)

The vertical force equation of the vehicle body is

��26 = 𝑐𝑎4 𝑞8 + 𝑘𝑎4𝑞8 + 𝑐𝑏4 𝑞16 + 𝑘𝑏4𝑞16 + 𝑐𝑐4 𝑞46

+ 𝑘𝑐4𝑞46 + 𝑐𝑑4 𝑞55 + 𝑘𝑑4𝑞55.(50)

The pitching moment equation of the vehicle body is

��44 =𝑑

2(𝑐𝑎4 𝑞8 + 𝑘𝑎4𝑞8 + 𝑐𝑐4 𝑞46 + 𝑘𝑐4𝑞46 − 𝑐𝑏4 𝑞16

− 𝑘𝑏4𝑞16 − 𝑐𝑑4 𝑞55 − 𝑘𝑑4𝑞55) .

(51)

The roll torque equation of the vehicle body is

��26 = 𝑙𝑓 (𝑐𝑎4 𝑞8 + 𝑘𝑎4𝑞8 + 𝑐𝑏4 𝑞16 + 𝑘𝑏4𝑞16)

− 𝑙𝑟 (𝑐𝑐4 𝑞46 + 𝑘𝑐4𝑞46 + 𝑐𝑑4 𝑞55 + 𝑘𝑑4𝑞55) .(52)

The vertical velocity equations of the four endpoints ofthe vehicle body are the same as (9), (10), (11), and (12).

4.3. Comparison and Analysis. Consistent with the previousmodel parameters, the contrast results of the four-wheeldynamic tire loads, the four-wheel suspension dynamictravel, the body vertical acceleration (BVA), the body pitchingangle acceleration (BPAA), and the body roll angle accelera-tion (BRAA) between the MBS and the MAS are obtained byusing Matlab/Simulink simulation analysis. The simulationresults are depicted in Figures 9–13. Figure 9 includes fourfigures: the contrast results of the left front wheel tire dynamicload (LFWTDL), the right front wheel tire dynamic load(RFWTDL), the left rearwheel tire dynamic load (LRWTDL),and the right rear wheel tire dynamic load (RRWTDL).


MBSMAS

−4000

−3000

−2000

−1000

0

1000

2000

3000

4000

LFW

TDL

(N)

2 4 6 8 100Time (s)

MBSMAS

−4000

−3000

−2000

−1000

0

1000

2000

3000

4000

RFW

TDL

(N)

2 4 6 8 100Time (s)

MBSMAS

−4000

−3000

−2000

−1000

0

1000

2000

3000

4000

LRW

TDL

(N)

2 4 6 8 100Time (s)

MBSMAS

−4000

−3000

−2000

−1000

0

1000

2000

3000

4000RR

WTD

L (N

)

2 4 6 8 100Time (s)

Figure 9: Contrast results of the tire dynamic loads.

Figure 10 includes four figures: the contrast results of the leftfront wheel suspension dynamic course (LFWSDC), the rightfront wheel suspension dynamic course (RFWSDC), the leftrear wheel suspension dynamic course (LRWSDC), and theright rear wheel suspension dynamic course (RRWSDC).

From the above simulation results, the root mean squarevalues (RMSVs) of the dynamic tire loads, suspensiondynamic course, BVA, BPAA, and BRAA between MBS andMAS are compared and analyzed; the contrasting data of theMBS and MAS are listed in Table 3.

The statistical data in Table 3 indicates that the changein vibration response variables of the MBS and MAS aswell as the RMSV of the LFWTDL, RFWTDL, LRWTDL,and RRWTDL is larger relatively, whereas changes in theother vibration response variables are smaller. The maximal

Table 3: Contrasting data of the MBS and MAS.

RMSV MBS MAS Difference (%)RFWTDL (N) 885.6 934.0 5.47LFWTDL (N) 1025.6 1052.6 2.63RRWTDL (N) 914.2 957.6 4.75LRWTDL (N) 911.9 950.5 4.23RFWSDC (m) 0.0032 0.0033 3.13LFWSDC (m) 0.0058 0.0058 0RRWSDC (m) 0.0036 0.0037 2.78LRWSDC (m) 0.0028 0.0028 0BVA (m/s2) 0.3167 0.3214 1.48BPAA (rad/s2) 0.5380 0.533 0.93BRAA (rad/s2) 0.5937 0.6099 2.73


MBSMAS

−0.02

−0.01

0

0.01

0.02LF

WSD

C (m

)

2 4 6 8 100Time (s)

MBSMAS

−0.02

−0.01

0

0.01

0.02

RFW

SDC

(m)

2 4 6 8 100Time (s)

MBSMAS

−0.02

−0.01

0

0.01

0.02

LRW

SDC

(m)

2 4 6 8 100Time (s)

MBSMAS

−0.02

−0.01

0

0.01

0.02RR

WSD

C (m

)

2 4 6 8 100Time (s)

Figure 10: Contrast results of the suspension dynamic course.

−1.5

−1.2

−0.9

−0.6

−0.3

0

0.3

0.6

0.9

1.2

1.5

BVA

(m/s2)

2 4 6 8 100Time (s)

MASMBS

Figure 11: Contrast results of BVA.


MBSMAS

−2

−1.6

−1.2

−0.8

−0.4

0

0.4

0.8

1.2

1.6

2

BPA

A (r

ad/s2)

2 4 6 8 100Time (s)

Figure 12: Contrast result of BPAA.

MBSMAS

BRA

A (r

ad/s2)

−3

−2.5

−2

−1.5

−1

−0.5

00.5

11.5

22.5

3

2 4 6 8 100Time (s)

Figure 13: Contrast results of BRAA.

difference is only 5.47%. These observations illustrate thereliability of the MAS based on activity.

5. Conclusions

Based on themodeling and analysis of the IWMDV vibrationcharacteristics, the following conclusions can be drawn:

(1) The BG theory is found to be clearly superior inperforming dynamics analysis. The BG method is aphysical object-oriented method, and the BG modu-lar modeling approach simplifies the modeling of thecomplex system; thus, the BG model is advantageousfor performing model modification. Using the BGmodel, the state equations of the model are deduced

directly, which is beneficial for subsequent analysis ofthe IWMDV vibration characteristics.

(2) By calculating the activity of each part in the system,the parts that are less influential to the IWMDVvibra-tion characteristics can be effectively identified andthen removed to simplify the model.The reliability ofthe MAS is verified based on the contrastive analysis.

(3) Without considering the in-wheel motor excitation,the in-wheel motor can be analyzed as a whole rigidbody, and the tires can be simplified as a spring in thevehicle vibration analysis.

In summary, this study can provide a method for themodeling and simulation of the vibration characteristics ofthe IWMDV.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This research is sponsored by National Natural ScienceFoundation of China (Grant no. 51405273) and supported byShandongProvinceHigher Educational Science andTechnol-ogy Program (Grant no. J14LB08) and Doctoral Program forShandongUniversity of Technology (Grant no. 4041-413039).

References

[1] D. C. Karnopp, D. L. Margolis, and R. C. Rosenberg, SystemDynamics: Modeling and Simulation of Mechatronic Systems,John Wiley & Sons, New York, NY, USA, 2006.

[2] L. Louca, J. Stein, G. Hulbert et al., “Proper model generation:an energy-based methodology,” in Proceedings of the 3rd Inter-national Conference on Bond Graph Modeling and Simulation(ICBGM ’97), vol. 29, pp. 44–49, Phoenix, Ariz, USA, January1997.

[3] J. R. Chen, Y. Lin, M. K. Liu, and Y. H. Peng, “Application ofbond graph theory in vibration analysis of vehicles,”AutomotiveEngineering, vol. 15, no. 1, pp. 26–33, 1993.

[4] W. Tao, W. Qing, and L. Yon, “Modeling and simulation forpowertrain of electric vehicle based on bond graph,” Transac-tions of the Chinese Society of Agricultural Engineering, vol. 27,no. 12, pp. 64–68, 2011.

[5] Q. Zhang, Analysis of ride comfort of all terrain crane based onbond theory [Ph.D. thesis], Taiyuan University of Science andTechnology, Taiyuan, China, 2014.

[6] B. Ma, D. Wu, H. Wen et al., “Fuzzy logic control research ofhydro-pneumatic suspension system of multi-axle vehicle bypower bond graph theory,” Journal of ShiheziUniversity (NaturalScience), vol. 33, no. 2, pp. 258–264, 2015.

[7] M. Zeraoulia, M. E. H. Benbouzid, and D. Diallo, “Electricmotor drive selection issues for HEV propulsion systems: acomparative study,” IEEE Transactions on Vehicular Technology,vol. 55, no. 6, pp. 1756–1764, 2006.

[8] R. Loureiro, R. Merzouki, and B. O. Bouamama, “Bond graphmodel based on structural diagnosability and recoverabilityanalysis: application to intelligent autonomous vehicles,” IEEETransactions onVehicular Technology, vol. 61, no. 3, pp. 986–997,2012.


[9] Z. Wang, Y. Gao, and Y. Wang, “The transformation of systemstate-space equations based on bond graph theory,”MechanicalScience and Technology, vol. 18, no. 1, pp. 54–56, 1999.

[10] Z. Lijun and Z. Tianxia, “General nonstationary random inputmodel of road surface with four wheels correlated,” Journal ofVibration and Shock, vol. 18, no. 1, pp. 54–56, 1999.

[11] S. Chen, J. Li, and Z. Fang, “Modeling and simulation of Auto-motive suspension system based on bond graph,” ComputerSimulation, vol. 24, no. 9, pp. 245–249, 2007.

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