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An overview on vehicle dynamics

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DOI: 10.1007/s40435-013-0032-y

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Int. J. Dynam. Control (2013) 1:385–395
DOI 10.1007/s40435-013-0032-y

An overview on vehicle dynamics

Shaopu Yang · Yongjie Lu · Shaohua Li

Received: 29 September 2013 / Revised: 5 October 2013 / Accepted: 9 October 2013 / Published online: 29 October 2013
© Springer-Verlag Berlin Heidelberg 2013

Abstract As a basic theory of the vehicle industry, the vehi- technology and intelligent algorithm have been widely used
cle dynamics plays an important role in the development in the vehicle industry. The vehicle dynamics plays an
of the vehicle industry. In the past decades, great progress important role in the development of vehicle industry. The
was made in the theory and experiment of vehicle dynamics. early research of vehicle dynamics concentrated on various
This article summarizes recent advances in vehicle dynam- working conditions and service performances under exter-
ics. In vehicle dynamics, the vehicle body (sprung mass), nal excitation [1]. In the 1930s, researchers began to focus
the suspension component (spring and damper) and tire on the steering, suspension mechanics and driving stability.
(unsprung mass) are essential parts of the system. The mod- Lanchester Maurice and Segel studied the effects of the exter-
eling approaches and characteristics of the vehicle, tire and nal environment (such as road surface roughness, air flow, tire
driver model with the respect to handling and driving dynam- and driver) on the vehicle dynamics and the coupling inter-
ics are summarized in the paper. The important research action of these conditions [2]. In 1993 Segel [3] presented a
issues about the vehicle-pavement coupled dynamics are comprehensive summary on the achievements of the vehicle
discussed in detail. Several problems and directions for the dynamics before 1990 in the Proceedings of Institution of
further studying in vehicle dynamics are pointed out. Mechanical Engineers. In the following decades, the vehi-
cle ride comfort and handing stability research have been
Keywords Vehicle dynamics · System modeling · widely investigated. The handling dynamics deals with the
Vehicle-pavement coupled dynamics · Overview lateral dynamics or transverse dynamics of the vehicle, which
mainly refer to vehicle handling stability, vehicle sideslip
caused by tire lateral force, yawing and roll motion. The
1 Introduction handing stability of vehicle dynamics research went through
the development from experimental studies to the theoretical
The global vehicle industry and market structure experienced analysis, from the open-loop to the closed-loop. The repre-
an unprecedented scale of change in the 1990s. There has sentative monographs of vehicle handling dynamics include
been an increasing demand on vehicle safety, environmen- “Vehicle Handling Dynamics Theory and Application” by
tal protection and intelligent control. Thus, the advanced Abe [4], “Vehicle Handling Dynamics Theory” written by
technologies such as computer technology, virtual reality Guo [5]. The vehicle driving dynamics is divided into lon-
gitudinal dynamics and vertical dynamics, which includes
driving, braking and ride comfort. The problem of driving
S. Yang (B) · S. Li
slip and braking slip are solved by the study of vehicle longi-
School of Mechanical Engineering, Shijiazhuang Tiedao
University, Shijiazhuang 050043, Hebei, China tudinal tire force, which can also improve driving and braking
e-mail: yangsp@stdu.edu.cn efficiency. The ride comfort focuses on vehicle vibration and
pitch movement caused by vertical tire force. The representa-
Y. Lu (B)
tive monographs are “Vehicle Dynamics and Control” written
Key Laboratory of Traffic Safety and Control in Hebei,
Shijiazhuang 050043, China by Rajamani [6], “Vehicle Dynamics Theory and Applica-
e-mail: lu-yongjie@163.com tions” written by Zhang [7]. In addition, the field of vehicle

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386 S. Yang et al.

dynamics research also involves the longitudinal force of tire (a) o

δ –α
when vehicles are speeding up or braking and vehicle vibra-
tion caused by the working engine, etc.
In vehicle dynamics, the vehicle body (sprung mass),
the suspension component (spring and damper) and tire
α2
(unsprung mass) are essential parts of the system. The
ωz y
vehicle-road coupling is another important aspect of vehi-
α1
cle dynamics. In this paper, the vehicle system (full vehi-
cle, tire, driver) modeling methods are reviewed. The impor- V1 δ
V
α2 c
tant research issue about the vehicle-pavement coupled Fcx
V2 x
dynamics is discussed. Finally, several outstanding problems Fcy Fc FY 1
FY 2 b a
and the future development trend of vehicle dynamics are L
proposed.
(b)
2 Research progress of vehicle dynamics model

2.1 Vehicle dynamics model

The vehicle dynamics models went through the development

from the traditional lumped parameter model to the finite
element model (FEM), the dynamical substructure model
and the multi-body system dynamics model, from the linear
model to the non-linear model with the non-linear stiffness
and the non-linear damping.
Fig. 1 Lumped mass model of vehicle handling dynamics. a 2DOF
(lateral, yaw) model [8,9], b 3 DOF(longitudinal, lateral, yaw) model
2.1.1 Lumped parameter model [12]

For the lumped parameter modeling method, the finite degree

of freedom (DOF) model of a vehicle system is comprised research of handling stability went through the development
of mass, spring and damper elements. The examples include from the simulation of the steady state response characteris-
a quarter vehicle model with 2 DOF, half vehicle model with tics to the simulation of transient response characteristics and
four or five DOF, full vehicle model with 7 or 18 DOF, etc. cornering braking characteristics. Based on 250 references,
For the vehicle handling stability research, the number Vlk [14] summed up the truck handling stability during lon-
of DOF can be two, ten or more such as 2 DOF (lat- gitudinal traveling, lateral traveling and braking. Nagai et
eral, yaw), 3 DOF (longitudinal, lateral, yaw) and 4 DOF al. [15] put forward an active control model to control the
(longitudinal, lateral, yaw, roll). It is known that a 2 DOF front wheel steering angle, which can improve the vehicle
vehicle model is usually applied when the vehicle lateral dynamic performances under the condition of steering, brak-
acceleration is less than 0.3 g [8,9], as shown in Fig. 1. If ing, changing lane and wind disturbance. Zhao et al. [16]
the tire cornering properties is in the linear range, the vehi- reviewed the development history of vehicle dynamics and
cle model can be simplified as a 3 DOF model with lateral, the main research methods. Examples of vehicle handling
longitudinal and yaw movement. The model is also widely dynamics and inverse dynamics are given.
used to study vehicle handling dynamics while the vehi- For the vehicle driving dynamics, there is a quarter vehicle
cle encounters emergency collision avoidance [10,11]. Liu model of 2 DOF, which assumes the movement of the vehicle
proposed a four-wheel-steering vehicle nonlinear lateral front axle and rear axle to vibrate independently. The simplest
dynamics model of 3 DOF based on Pacejka model. The quarter vehicle model is one-dimensional model. It is widely
experimental study and simulation analysis for the vehicle applied in the research and design of the vehicle active and
single lane change, double lane change are also complete semi-active suspensions [17–19]. The lumped mass vehicle
[12]. Huh proposed a six-axis heavy vehicle model of 18 model of 7, 8, or more DOF is usually three-dimensional,
DOF with MATLAB/Simulink, in which the suspension and which can analyze vertical, pitch and roll vibrations of the
tires characteristics are nonlinear. It is found that the effects vehicle [20,21]. Cebon proposed a three-dimensional vehi-
of middle steering axle on the yawing angular velocity and cle model with 14 DOF considering the longitudinal tire
lateral acceleration cannot be ignored [13]. The simulation force, lateral tire force and the non-linear suspension [22].

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An overview on vehicle dynamics 387

(a) models lies in their simplicity and the possibility to obtain

analytic descriptions and to design active or semi-active con-
trols.

2.1.2 Virtual prototyping model

The finite element (FE) modeling demands much more cal-

culation time. The FE vehicle model is mainly used for
the dynamic design of complex parts such as vehicle body,
engine mount, etc. [25,26]. The dynamical substructure
approach includes the mechanical admittance method and
the modal synthesis method. The modal synthesis method
has been successfully applied to study the vehicle vibration
noise mechanism and vibration transfer characteristics of the
vehicle chassis and frame [27]. At present, the dynamical
(b) analysis modeling of the vehicle is usually combined with
the lumped mass method, FE method and modal synthesis
method.
With the development of computer technology and appli-
cation software, the functional virtual prototyping (FVP)
technology makes more and more complex models of the
vehicle [28]. The core contents of the FVP are the Multi-
Body System Kinematics and Dynamics modeling theory.
In recent years, some scholars begin to build the vehicle
model and analyze its dynamic behavior by using the FVP
software. The representative multi-body dynamics software
includes Simpack, ADAMS and TruckSim. Valášek et al.
[29] established a virtual prototype of triaxial heavy vehi-
cle model in Simpack software and designed the semi-active
(c) controller in MATLAB/Simulink. Hou et al. [30] and Ieluzzi
et al. [31] developed a vehicle model in ADAMS/CAR and
studied the semi-active suspension effectiveness for improv-
ing the vehicle vibration performance. Odhams developed
a complex multi-body model of heavy load hinged vehicle
using TruckSim software and studied the driving safety of
trailer vehicle under the operation of active steering system
[32]. Ren et al. [33], Yang et al. [34] and Lu et al. [35] inves-
tigated the dynamic behavior such as vehicle ride comfort,
handling stability and so on, using ADAMS software. The
most significant advantage of the multi-body vehicle model
is the parametric modeling. In order to make it clear which
vehicle parameter (vehicle geometric position, suspension
stiffness and damping characteristics) has the most impor-
Fig. 2 Lumped mass model for vehicle driving dynamics. a One- tant effect on the vehicle performance, several trial simula-
dimensional (quarter vehicle) model [19], b Two-dimensional (4 DOF) tions of the vehicle must be done. The traditional computa-
model [23,24], c Three-dimensional (7 DOF) full vehicle model [20,21] tional optimization approach becomes inadequate in dealing
with the complex multi-body model of the vehicle with a
Several typical vehicle models for the driving dynamics are lot non-linear parameters. A discrete optimization method
shown in Fig. 2. proposed by Fisher [36], known as the design of experiment
Although these lumped mass models are simplifications (DOE) for the optimization, has been adopted to study the
of the actual vehicle structure, they can capture the vehicle vehicle design. This method has been widely used in the
vibration characteristics and the effects of structural parame- detection, earthquake prediction and product design due to its
ters from the vehicle performance. The advantage of these effectiveness, robustness, and high-quality solutions [37,38].

123
388 S. Yang et al.

Lu presented an orthogonal optimization program for a multi- (a)

body vehicle model. After several virtual experiments and
range analysis, the most important influencing factor and its
range are screened out. Then a matching scheme of vehicle
parameters is proposed to achieve the ride comfort and the
road friendliness optimization [39].
Although the lumped mass models of a vehicle can be used
for the vehicle performance analysis and control study, they
are too simple to show the nonlinear behavior of the suspen-
sion system and describe the detail of the vehicle structure.
The virtual prototyping model (FE or multi-body) has advan-
tages such as user-oriented interactive environment and vir-
(b)
tual prototyping pretreatment function, which are important
to the development of vehicle dynamics.

2.2 Tire model

The tire is the link between a vehicle and the road surface. It
does not only support the vehicle load, but also attenuates the
shock from the uneven road surface. Thus, the tire dynamics
plays an important role in the vehicle performance.

2.2.1 Tire mechanical model

The research about the vehicle tire mechanics originates from

the aero tire research, which is dated back to the early 1930s. (c)
The tire mechanical model can be divided into three cat-
egories: pure physical model, empirical model and semi-
empirical model [40].
According to the tire physical structure and working
mechanism, the pure physical model is proposed in the form
of mathematical expression to describe a tire mechanical
characteristic. The pure physical models include Fiala, Gim
and Dugoff tire model, etc. In the Fiala model [41], the belt
and the buffer layers of a tire are simplified to an elastic
beam under the lateral concentrated force. The tire lateral
force and aligning torque is described by the dimension-
less expressions. Although the calculation error of the Fiala
model aligning torque is large, the cornering force has a better
(d)
accuracy within 5◦ slip angle, as shown in Fig. 3a. The Gim
model [42,43] is based on the Bergman 3D spring model,
in which the tire side slipping and longitudinal sliding is
governed by a kinetic equation of the infinitesimal element
between the tire and road surface. It is widely used in the
field of vehicle dynamics simulation and control study [44].
The Dugoff model expresses the friction factor between the
tire and the road as a function of the friction reduction factor
and the friction when both the tread sliding speed and vehicle
sliding velocity are zero. This gives a relationship between Fig. 3 Tire models. a Physical model (Fiala [41]), b Semi-empirical
the tire-road surface friction coefficient and the cornering or model (Swift) [52], c Semi-empirical model (UniTire) [5,47], d FE
longitudinal force in the Dugoff model [45]. The tire can have model [51]

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An overview on vehicle dynamics 389

large deformation under the loading, which is difficult to lished the road-surface roughness by using six sinusoidal
be described analytically. Therefore the pure physical model waves and calculated the tire force by using the point con-
can only serve as the basis for the semi-empirical model. tact model. The simulation result of the point contact tire
From experimental data of tire, an empirical model can model is rather similar to the measured tire force [54]. Collins
be directly built. The semi-empirical model has higher pre- [55] believed that the point contact model is adequate for the
cision combined with the theoretical model and the experi- use in the shimmy analysis of the tire yaw and structural-
mental data. The famous Magic Formula model is proposed torsion shimmy modes of vibration. Arunas and Jonas [56]
by Pacejka [40] based on extensive experiments. It is a semi- compared the results of the point-contact model, the flex-
empirical and semi-theoretical model, which is based on the ible ring model and the flexible band model based on the
tire physical prototype and the experimental data. It can be quarter vehicle model. Costanzi and Cebon [57] also used
used to describe the six-directional wheel force in the steady the point-contact tire model to calculate the tire dynamical
state. In order to extend the frequency range of the tire, force and the road deformation, which were validated by field
Pacekja introduced a dynamical rigid ring model based on data.
the Magic Formula model and proposed a Swift model, as However, the point contact model is only appropriate for
shown in Fig. 3b. The Magic Formula and Swift tire models the vertical loading condition, in which the range of road sur-
were developed at Delft University of Technology. They were face excitation frequency is between 0.1 and 0.2 Hz. When
implemented in the commercial software (ADAMS, SIM- the pavement wavelength is greater than 3 m, the model can-
PACK and MATLAB/Simulink), and named as MF-Tyre and not reflect the motions of shortwave (high frequency) excita-
MF-Swift, respectively. The mathematical expression of the tions [58]. The high-frequency filtering property of a tire is
tire lateral and longitudinal instantaneous deformation under called the enveloping characteristics, which is initially stud-
any movement is calculated by Guo and Lu [46] with a the- ied by Lippmann et al. [59]. The enveloping characteristics
oretical tire model in non-steady state. The calculation error means that a tire has a property to envelope the road profile
of a pure theoretical model caused by unavoidable simplifi- irregularities and attenuate the high spatial frequency compo-
cation can be compensated by the experiment and the data nents. Other tire models include line contact model or surface
fitting in a semi-empirical model. Guo et al. [47] also put for- contact model. The spring and damping elements along with
ward a semi-empirical model called UniTire Model that can the imprinting length are used to form a fixed imprinting tire
describe a tire mechanical characteristics in an E exponential model. The imprinting length is invariant and the confluence
form. It describes the six-component characteristics of a tire of the tire force is always in the direction of the wheel cen-
in a variety of working conditions, and has high precision ter, which also has an enveloping characteristic within the
simulation for the complex and extreme conditions. imprinting. With this model, the calculated value of vertical
The FE model of the tire has experienced from the sta- axle load under the high frequency excitation is lower than
tic and dynamic analysis to the thermal coupling, abrasion, the actual value [60]. Guo et al. [61–63] presented a flexi-
fatigue life prediction and vehicle-tire- pavement coupled ble roller contact tire model and carried out simulations of a
analysis. Narasimha used the ABAQUS software to inves- vehicle vibration system based on the rigid and the flexible
tigate the lateral and longitudinal forces of pneumatic tires roller contact tire model. Guan and Dong [64] put forward a
during steering and braking [48]. Gall and Tkacik [49] built vehicle system model with an enveloping tire model to study
a 3D FE model of tire with the tread, and found out that the the vehicle active suspension performances. The vehicle load
normal stress reaches the maximum value when the friction distribution on the road surface is assumed to be uniform and
coefficient is in the vicinity of 0.55. The distribution of nor- symmetrical with parabolic or trapezoidal distribution [65].
mal stress tends to be stable with the increasing of friction The true distribution of the loading on local contact area on
coefficient. It is suited for the analysis of transient rolling a moving vehicle is not symmetrical. A non-uniform distri-
contact, internal stress, modality, noise and so on [50,51]. bution model is introduced by Guo [66]. A modified elastic
However, there are still limitations for the FE method. For roller tire model is proposed by Yang et al. [67]. A two-
example, the quantitative relationship between variable para- dimensional vehicle-road-subgrade coupling system based
meters cannot be expressed explicitly. Several typical tire on the modified tire model is developed. The improved elas-
models are shown in Fig. 3. tic roller line contact model and point contact model of the
vehicle-road coupling system are compared.
2.2.2 The contact between tire and road There has been research work on experimental modeling
of a tire. The South Africa researchers of Council for Scien-
The relationship between the tire model and the road surface tific and Industrial Research (CSIR) developed a vehicle-road
are usually assumed as the point contact. It is the conve- surface pressure transducer array (VRSPTA), commonly
nient model for the dynamic analysis and widely used in the known as the “3-D Stress Sensor” [68–70]. Groenendik et al.
theoretical study of vehicle dynamics [53]. Kyongsu estab- [71] and Ronald [72] proposed the simplified mathematical

123
390 S. Yang et al.

formula of the tire contact stress distribution based on the to study the mechanism of accident using the optimal con-
tire test results of freely rolling tire. A measurement device trol theory. Guo et al. [87] proposed a closed-loop control
was developed and used to measure the tire ground pressure system of human-vehicle-road and established a driver pre-
distribution of light vehicles with different tread patterns in view follower model as well as a preview optimal curvature
different conditions of the tire inflation pressure and axle model. The research of driver behavior in China is in its early
loading [73,74]. A rig was developed for the tire impact test, stage.
allowing a drop mass with a round indenter to hit the pres-
surized tires with different impact energies [75]. Although
the cost of the test is relatively high, it provides important 3 Vehicle and road surface interaction
input to the mathematical tire model and analysis of physical
properties. In 1987, Frybal stated that “the dynamic issues of vehicle-
pavement interaction is a new branch of the science”. Since
2.3 Driver model this research field has emerged dealing with dynamics prob-
lems of vehicle, road and the interaction between them [88].
A number of models have been developed to describe the So far, the research field has not been fully developed yet
human driving behavior and evaluate the vehicle safety in except for a few publications. It is well recognized that the
braking and turning. An early driver model based on the cog- vehicle can cause the damage to the pavement. In 1958 the
nitive model was established in 1938 by Gibson and Crooks American association of highway and transportation officials
[76]. In the 1980s, the number of publications about the driver (AASHO) began to focus on the pavement damage caused
model had reached a peak. McRuer [77] is one of the schol- by the vehicle traveling. Based on many road tests under
ars who has a great influence on the driver model of control the different vehicle loadings, AASHO obtained the famous
theory. A long article written by MacAdam [78] expounded “fourth power law” [89]. The law states that there is a fourth
the driver’s limited driving ability in detail. Driver’s behavior power function relation between the fatigue damage of the
is not only limited to the control of extremities and the brain, pavement and the vehicle static axle load. The fourth power
but also related to the interaction with the outside world. It is law is important to the design of pavement structure. How-
a very complex process of perceiving location and speed as ever, the law doesn’t consider the dynamical loading effect
well as the action time, which involves the work of different of vehicle on the pavement. In the 1970s, West Germany
parts of the brain (identification of position, decision-making, highway department proposed the evaluation of “Road Stress
action to complete the task). Kinecke and Nielsen [79] sum- Factor” [90], which took into account the effect of dynam-
marized the driver’s behavior characteristics from the point ical axle load on the pavement. The evaluation pointes out
of view of dynamics and control. Driver’s behavior is nonlin- that the damage is largely caused by the vehicle axle load
ear, time varying, adaptive and random. Furthermore, driver’s (including the static axial load and dynamical axle load)
response has the property of foresight and hysteresis. The and the contact area between the tire and the road surface.
foresight is manifested as predicting the development of traf- Canadian roads and transport association organized a joint
fic situation and preventing or handling in advance. Hys- commission to research the vehicle quality and size. They
teresis shows up as delay in response to the events. Ranney studied the interaction between the truck and the pavement
puts forward several cognitive models of driving behavior. or bridge. The purpose is to study the effect of design, size
But, the model parameters are hard to be obtained [80]. Cac- and quality safety limit of the truck on the roads, bridges,
ciabue [81] summarized the various factors that contribute security and transport policy [91–93]. In 1987, the United
to modelling human behaviour in the specialized environ- States Congress adopted and launched the Strategic Highway
ment. Baron et al. [82] studied an optimal driver control. Research Program (SHRP), focusing on the research of four
Torsten and von Stryk [83] proposed a control system com- fields: asphalt road, pavement performance, concrete struc-
bining the driver and vehicle factors in the optimal control ture and road transportation [94]. In the early 1990s, TRL’s
theory. The driver’s transient choice of path and speed has transportation institute of the university of Cambridge, UK,
an influence on the vehicle model directly. But this factor studied on interactions between the vehicle and road surface
did not include driver’s motivation and personal characteris- to assess the driving safety, suspension structure design, fail-
tics. Cheng and Fujioka [84] used the hierarchical decision- ure mechanism of asphalt pavement, research and develop-
making system and the fuzzy system to design an appropriate ment of dynamical weighing equipment, and so on [95,96].
distance between vehicles. Michon [85] proposed a cogni- At the University of California, Beckeley, the damage caused
tive model of driver’s decision-making process, which can by the tire pressure and the axial load is studied theoreti-
achieve quantitative measurement and provide three types cally and experimentally. Kyongsu’s research showed that
of decisions: strategic, technical and operational. Yang [86] the dynamical tire force is much higher than the static tire
gave a longitudinal driver model and a lateral driver model force. Both of the theoretical analysis and the experimental

123
An overview on vehicle dynamics 391

results proved that the semi-active suspension using double (a)

linear disturbance decoupling observer can reduce the tire
dynamical force and the damage on the pavement [97]. The
European organization for the economic co-operation and
development (OECD) put forward the road traffic research
project entitled “Dynamic Interaction between Vehicles and
Infrastructure Experiment (DIVINE)” [98] and investigated
the interaction mechanism of vehicles, pavement and bridge.
A total of 19 countries had participated and conducted exper-
iments in the indoor accelerated pavement test facility (CAP-
TIF) in Canterbury of New Zealand, on the proving ground
in Virtta, Finland. (b)
With the modal superposition and integral transforma-
tion method, Seong-Min and McCullough [99] studied the
dynamical response of plates on the viscous Winkler foun-
dation under the moving loads with varying amplitudes.
Huang and Thambiratnam [100] also analyzed the dynam-
ical response of thin plates with the Winkler foundation
under the moving point harmonic load. Kim [101], Chen and
Huang [102] and many others studied the effects of vehi-
cle parameters on multilayer pavement’s dynamics with the
finite element method. Mamlouk and Brademeyer [103] pro-
posed a concept of vehicle-pavement interaction, which can (c)
be applied to weigh-in-motion, pavement design, and per-
formance and vehicle regulation. Cebon concluded that the
interaction of vehicle and road is a weak coupling system for
two reasons: (1) the displacement of the pavement is 0–1mm,
much smaller than the deflection of the tire and suspension
(10–20mm); (2) the speed of elastic waves in the road profile
is 100–600m/s, much greater than the vehicle running speed
(10–50 m/s) [104].
The research on the vehicle-road interaction in China is
mainly carried in the universities. As early as 1965, Fang Fig. 4 Contact relationship between tire and road surface, a Single
et al. recognized that the vibration of the vehicle is caused point-contact model [54–57], b Fixed foot print model [60], c Envelop-
by the pavement irregularity [105,106]. Sun and Deng [107] ing model (rigid, flexible) [63–65]
analyzed the influence of vehicle parameters on the dynamic
load in detail by using the random theory based on a simpli- layer rectangular thin plate on a nonlinear viscoelastic foun-
fied quarter vehicle model. Lv et al. [108] established a heavy dation, as shown in Fig. 4. It is found that the nonlinearity
vehicle model for optimization of improved vehicle ride com- of vehicle and the viscoelasticity of road material should
fort and reduced tire dynamic shock to the ground. Lv’ and be considered when the vehicle-road system response is
Dong [20] other achievements are summarized in the mono- studied [111].
graph entitled “Mechanical Analysis of Vehicle-Asphalt Despite of all these studies, the important issues about the
Pavement System”. Lu et al. [109] investigated a heavy interaction between the vehicle and road surface are still not
vehicle-tire-road coupled system and found that nonlinear- resolved. The contact mechanism between the tire and road
ities in the vehicle suspension contribute to the improved surface needs further investigations (Fig. 5).
ride comfort and reduced dynamic tire force. Yang et al.
[110] developed a 3D car-pavement-subgrade coupled model
and studied the effect of vehicle-pavement coupling on the 4 The prospects of vehicle dynamics research
vehicle body acceleration, suspension deformation, tire force
and the surface displacement by comparing the dynamical Through several decades of development, vehicle dynamics
response of this model and the traditional vehicle-road has achieved some success in handling stability, ride comfort
model. A nonlinear vehicle-road coupled system involves and road friendliness. Much further work is needed in the
a 7 DOF vehicle moving along a simply supported double- following areas.

123
392 S. Yang et al.

Fig. 5 The nonlinear z v

vehicle-road coupled system
Z1
Z2 m1 Z3

Zt2 Zt1 Zt4 Zt2

Ks2 Cs2 Ks11 Cs1
mt2 mt1 mt4 mt2
Kt2 Ct2 Kt1 Ct1
Zr

K C

O x

l2 l1

B
m1
mt4 mt3
dr

df
mt2 mt1

(1) Comprehensive nonlinear dynamic modeling of the (Grant Nos. 11102121, 51208319) and the Natural Science Foundation
vehicle system is needed. When the vehicle is running in of Hebei Province (Grant Nos. A2012210018, E2012210025).
transient maneuver, nonlinearities in the vehicle system
strongly influence the dynamics and should be consid-
ered. Furthermore, nonlinear components of the vehicle References
introduce complex dynamic phenomena including bifur-
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(3) The contact relationship between the tire and road sur- ence and Technology Press, Nanjing
face is usually modeled by the spring and damping ele- 6. Rajesh R (2005) Vehicle dynamics and control. Springer-Verlag,
New York
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