Research Journal of Applied Sciences, Engineering and Technology 6(21): 3946-3952, 2013
ISSN: 2040-7459; e-ISSN: 2040-7467
© Maxwell Scientific Organization, 2013
Submitted: December 20, 2012
Accepted: January 25, 2013
Published: November 20, 2013
Nonlinear Responses of a Two Dimensional Vehicle-pavement System
Shaohua Li and Haoyu Li
Mechanical Engineering School, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
Abstract: The vehicle and pavement are usually investigated separately in vehicle dynamics and pavement
dynamics. In this work, a new research scheme is proposed to link the vehicle and pavement model by tire loads and
compute the nonlinear dynamic responses by analytical methods. A two-DOF nonlinear vehicle and a BernoulliEuler beam on a nonlinear elastic foundation with two simply supported ends compose the nonlinear vehiclepavement system. The nonlinear tire loads are analytically gained using the averaging method. Then the nonlinear
vibration equation of the pavement is obtained using Galerkin method and solved using the multiple scales method.
The theoretical solutions are verified by numerical results and the effects of system parameters on pavement
vibration are also studied. It is found that the pavement responses excited by tire loads attenuate quickly and small
pavement mass, large foundation damping or foundation stiffness may decrease the pavement vibration.
Keywords: Beam on nonlinear elastic foundation, method of multiple scales, nonlinear dynamic, tire loads, vehiclepavement system
INTRODUCTION
In vehicle dynamics, the responses of vehicles with
nonlinear suspensions were widely studied. Stensson et
al. (1994) analyzed nonlinear phenomena in a vehicle
suspension, including multi-solutions, sub-resonance
and the system’s sensitivity in numerical integral. Li
et al. (2004) investigated chaos in a hysteretic nonlinear
vehicle suspension both by Melnikov method and many
numerical methods. Georgios et al. (2008) studied the
semi-active control based on a nonlinear vehicle with
MRF damper. Raghavendra and Pravin (2009)
researched the chaotic motion of a one degree of
freedom quarter-car model with nonsymmetric
potential. However, these researches didn’t consider
responses of pavement induced by the tire load of a
nonlinear vehicle.
In pavement dynamics, responses of a beam or a
plate on a nonlinear elastic foundation under moving
load was attracted much attention. The moving load is
often a constant load or a harmonic load with a fixed
frequency. Kargarnovin et al. (2005) obtained response
of infinite beams supported by nonlinear visco-elastic
foundations subjected to harmonic moving loads using
a perturbation method and investigated influences of
the load speed and frequency on the beam responses.
Santee investigated the stability of a beam on nonlinear
elastic foundation and obtained the critical boundary of
system instability with the method of Melnikov (Santee
and Gonalves, 2006). Yang et al. (2006) studied
nonlinear vibration and singularities of a rectangular
thin plate on nonlinear elastic foundation. Xiao et al.
(2008) investigated bifurcation and chaos of rectangular
moderately thick cracked plates on an elastic
foundation subjected to periodic load. However the tire
load acting on pavement is consist of several
frequencies and depends on vehicle suspension
vibration. The tire load obtained from vehicle dynamic
responses are seldom considered in present studies on
pavement dynamics.
In this study, a nonlinear vehicle-pavement system
is modeled by a two-DOF nonlinear vehicle and a
Bernoulli-Euler beam on the nonlinear elastic
foundation. The analytical solutions of wheel and
vehicle body displacement are obtained by the
averaging method. Then the tire loads of the nonlinear
vehicle are gained. Subjected to this tire load, a
nonlinear ordinary differential equation of the pavement
is obtained using Galerkin method. Then the analytical
pavement displacement is gained using the multiple
scales method. Effects of system parameters on
pavement vibration are also studied, including
pavement elastic module, foundation damping, linear
foundation stiffness and nonlinear foundation stiffness.
MATHEMATIC MODEL
A nonlinear vehicle-pavement system is modeled
by a two-DOF nonlinear vehicle and a Bernoulli-Euler
beam on a nonlinear elastic foundation with two simply
supported ends, as shown in Fig. 1. It is assumed that
the vehicle runs at a constant speed from the midpoint
of the beam to right direction.
Corresponding Author: Shaohua Li, Mechanical Engineering School, Shijiazhuang Tiedao University, Shijiazhuang 050043,
China, Tel/Fax: 0086-311-87935554
3946
Res. J. Appl. Sci. Eng. Technol., 6(21): 3946-3952, 2013
v
y2
Fc
y1
m2
K2
m1
K1
C
yr
1
C
Fig. 1: The nonlinear vehicle-pavement model
Tire load of the nonlinear vehicle: Letting
dimensionless displacement x1 = y1/B0 , x2 = y2/B0 and
dimensionless time τ = Ωt, one gets the dimensionless
vehicle system:
The vehicle’s equations of motion are expressed by:
m1 y1 " K2 ( y1  y2 )  Fc  K1 ( y1  y0 )  C1 ( y1 ' y0 ')  0 (1)

m2 y2 " K2 ( y2  y1 )  Fc  0
where, m1, m2 are the masses of vehicle body and
wheel respectively; y2, y1 are the body and wheel’s
vertical displacements respectively; K1, K2 are stiffness
of tire and suspension, respectively; C1 is tire’s
damping coefficient.
The harmonic road roughness y0 is expressed by:
y0  B0 sin t
(2)
where, B0 is the amplitude of road surface roughness, Ω
= 2 πv/ L0, v and L0 are the vehicle running speed and
the wavelength of road roughness relatively.
Fc is the nonlinear skyhook damping control force
of vehicle suspension. A revised Bingham model is
applied here (Yang and Li, 2005):
Fc  C2 y2 ' Fy sgn( y2 ' V0 )
EI
 yr
y
 y
 K3 yr  K 4 yr3  C r  m 2r  F ( x  L / 2  vt )
x 4
t
t
2
(4b)
dx ,
d
K2 ,
K
B12  2 1 , B13  C1 ,
 2 m1
 m1
m1
K
C
F
F
C
B15  2 , B16  y , B21  2 2 , B22  2 , B23  y ,
 m2
m1
m2
m2
m1
V
x0  0 .
B0
where,
x 
B11 
Since the coefficients B23 are much smaller than
other coefficients, the system (4) is weak nonlinear and
may be investigated by the averaging method. Letting:
 xi  ai cos  i

 x i   ai sin  i
(3)
Here C2 is the viscous damping coefficient, Fy is
controlling force and V0 is the velocity when MRF
damping force is zero.
The vertical vibration equation of the pavement
under the moving vehicle loads can be obtained as
follows:
4
x1  B11 ( x1  x2 )  B12 x1  B13 x1  B15 x2  B16 sgn( x2  x20 )  B12 sin  B13 cos


x2  B21 ( x2  x1 )  B22 x2  B23 sgn( x2  x20 )  0

where, φi = τ – θi, i = 1, 2.
The averaging equations can be derived as follows:
a1  P1
 
a11  R1

a 2  P2
a   R
2
 2 2
(4a)
where E, I, x, yr, K3, K4, C, L are the modulus of
elasticity, cross-sectional moment of inertia, vehicle’s
position in running direction, beam’s vertical
displacement, linear foundation stiffness, nonlinear
foundation stiffness, foundation damping and the
beam’s length respectively.
(5)
(6)
where,
1 2
f1 sin 1d1
2 0
1
1
1
1
1
 B11a2 sin(1  2 )  B13 sin 1  B12 cos 1  B13a1  B15 a2 cos(1  2 )
2
2
2
2
2
P1 
3947 Res. J. Appl. Sci. Eng. Technol., 6(21): 3946-3952, 2013
f 2  ( B21  1)a2 cos 2  B21a1 cos 1  B22 a2 sin 2  B23 sgn(a2 sin 2  x20 )
S1
In order to obtain the second term of P2 and R2, the
following
techniques
are
applied.
Letting
S 1   a 2 sin  2  x 20 , S 2   a 2 sin  2  x 20 ,
2
0
2π-α
π+α
S 3  B 23 sgn(  a 2 sin  2  x 20 ) . Functions S1, S2
and S3 are shown in Fig. 2.
According to – α2 sin φ2 + ‫ݔ‬ሶ 20 = 0, α is gained:
  arcsin(x 20 / a 2 )
(a) S1
The value of φ2 is 0, α, π – α, π + α, and 2π - α at
point 1~5 respectively. The four angles divide a period
of 0~2π into five parts. Thus the second term of P2 and
R2 can be derived:
S2
2
0
α
(7)
2
1
2
π-α
 B 23 sgn(  a 2 sin  2  x 20 ) sin  2 d 2
0

 
 
B
 23 ( sin  2 d 2  (  sin  2 ) d  2 
(  sin  2 ) d  2

 
2 0

(b) S2

2
  sin  2 d 2  2  sin  2 d 2 )



2 
2 B 23

cos   
2 B 23
2
a 22  x 20
a 2
(8a)
S3
1

2
1
B23
2
x 2




0
1 2
f 2 sin  2 d 2
2 0
1
1
1
  B22 a 2  B21a1 sin(1   2 ) 
2
2
2
  cos  2 d 2  2  cos  2 d 2 )
2 B 23

sin   
2 B 23 x 20
a 2
(8b)
(9a) B11a2 cos(1 2 )  B13 cos1  B12 sin1  (1 B11  B12 )a1  B15a2 sin(1 2 )  0 (9b)

2
0 B23 sgn(a2 sin  2  x 20 ) sin  2 d 2
1 2
f 2 cos  2 d  2
2 0
1
1
1 2
 (1  B 21 ) a 2  B 21 a1 cos(  1   2 ) 
B 23 sgn(  a 2 sin  2  x 20 ) cos  2 d  2
2
2
2 0
R2  

f1 cos 1d1
1
1
1
1
1
B11a2 cos(1   2 )  B13 cos 1  B12 sin 1  (1  B11  B12 )a1  B15 a2 sin(1   2 )
2
2
2
2
2
P2 

2
B11a2 sin(1 2 )  B13 sin 1  B12 cos1  B13a1  B15a2 cos(1 2 )  0

Fig. 2: Function S1, S2 and S3
2

2 
Letting ߙሶ 1 = ߙሶ 2 = ߠሶ1 = ߠሶ2 = 0 , the stable solution
equation of system (6) is obtained:
(c) S3
1
2
0
4
3
R1  
 B 23 sgn(  a 2 sin  2  x 20 ) cos  2 d 2

 
 
B
  23 ( cos  2 d  2  (  cos  2 ) d  2 
(  cos  2 ) d  2

 
2 0
5
x 20
2


f1   B11a2 cos 2  ( B11  B12 )a1 cos 1  B13a1 sin 1
 B12 sin   B13 cos   B15 a2 sin 2  B16 sgn(a2 sin 2  x20 )

4B23 a22  x202

0
B22a2  B21a1 sin(1 2 ) 
 a2


4B23 x20
0
(1 B21 )a2  B21a1 cos(1 2 ) 
 a2

(9c, 9d)
Eliminating θ1 – θ2 from Eq. (9c) and (9d) yields:
( B22 a 2  4 B23
2
a 22  x 20
a 22
2 2 2 2
) 2 a 22  (a 22  B21a 22  4 x 20 B23 ) 2  B21
 a1 a 2
(10)
From Eq. (9c) and Eq. (9d) one has:
3948 Res. J. Appl. Sci. Eng. Technol., 6(21): 3946-3952, 2013
sin(   )  ( B  a 2  4 B a 2  x 2 ) / ( B  a a ) (11)
1
2
22
2
23
2
20
21
1 2

2
cos(1   2 )  [( B21  1) a2  4 B23 x20 ] / ( B21 a1a2 )
Moving the terms including sin θ1 or cosθ1 from
the left hand of Eq. (9a) and (9b) to the right hand,
eliminating θ1-θ2
from the two equations and
considering Eq. (11), one get:
(B112  B152 )a22  (1 B112  B122  B132  2B11B12  2B11  2B12 )a12 
(2B11  2B112  2B11B12  2B13 B15 )
B21
Due to the weak nonlinearity of system (16), the
responses can be gained by the multiple scale method.
Since the excitation frequency is much smaller than the
inherence frequency of system (16), only the nonresonance condition is considered.
One may express the solution in terms of different
time scales as:
xr (t )  xr 0 (T0 , T1 )   xr1 (T0 , T1 )
(17)
( a22  B21 a22  4x20 B23 )
(12)
where, T0 = τ, T1 = ετ.
Substituting Eq. (17) into Eq. (16) and equating
coefficients of like powers of ε leads to:
From Eq. (10) and Eq. (12) one may get a1 and a2.
Substituting a1 and a2 into Eq. (10) leads to θ1 and θ2.
Thus the analytical displacements of wheel and vehicle
body are ascertained:
 D0 2 xr 0  r2 xr 0   cos( 1T0  1 )   cos( 2T0  1 )   cos( T0 )
 2
2
3
 D0 xr1  r xr1  2D0 D1 xr 0  D0 xr 0  3 xr 0

(2B11B13  2B15  2B11B15  2B12 B15 )
(B22 a22  4B23 a22  x202 )  B122  B132
B21
x1  a1 cos(  1 ),
x 2  a 2 cos(   2 )
(13)
The analytical tire load acting on pavement can be
expressed as:
F  K1( y1  y0 )  C1( y1  y0 )  (m2  m1)g  F1 sin(t  1   )  F2 sin(t   )  (m1  m2 )g
(14)
(18)
The solution for the first line of Eq.(18) is given
the form:
xr 0 (T0 , T1 )  A(T1 )e jrT0  B1e jr1T0  B2e jr2T0  B3e jrT0  cc
(19)
F1  ( K1B0 a1 ) 2  (C1B0 a1) 2 , F2  ( K1B0 ) 2  (C1B0) 2 ,  tan 1 (C1 / K1 )
where, cc denotes the complex conjugate of the
preceding terms and A  ae j  / 2 ， B1   e j / (r 2  12 ),
B2   e j / ( r 2   22 ), B3   / ( r 2   2 ) .
Responses of the nonlinear pavement: Letting yr =
U(t) sin πx / L, a nonlinear ordinary equation can be
gained as follows:
Substituting Eq. (19) into the second line of
Eq. (18) leads to the condition for eliminating the
secular term:
EI 4  K3 L4
 ( L / 2  vt )
C
3K
U(t )  U (t ) 
U (t )  4 U (t )3  F sin
4M
M
ML4
L
3 3 A 2 A  2 j r D1 A  j r A  6 3 AB1 B1  6 3 AB2 B2  6 3 AB3 B3  0
where,
1
(15)
2
(20)
Substituting the tire load Eq. (14) into Eq. (3),
letting dimensionless displacement xr = U/ B0 and
dimensionless time τ = Ωt, one get the dimensionless
system:
Separating the real and imaginary parts of Eq. (20),
one obtains:
1

 D1a   2 a

 D1  3 3 (a 2  2 B12  2 B22  2 B32 )
8r


xr   x   xr   x   cos( 1  1 )   cos( 2  1 )  6 cos 
2
r r
3
3 r
(16)
where,
r  ( EI 4  KL4 ) / (2 ML4 )
  C / ( M )
 3  3K 2 B02 / (4 M  2 )
From Eq. (21), one can get the first order
approximate solution of the dimensionless system (16):
xr (t)= α cos (ωr τ + β) + B1 cos (γ1τ + θ1) + B2 cos
(γ2 τ + θ1) + B3 cos (γτ)
(22)
     / 2,
2
4
(21)
2
5
1  tan 1 ( 5 /  4   / 2),
where,
 6  (m1  m2 ) g / ( B02 M 2 )
 1  1  L0 / (2L),  2  1  L0 / (2L)  L0 / (2L)
a  c1e T1 /2 ,   3 3 (c12 e T1  2 B12T1  2 B22T1  2 B32T1 ) / 8r  c2
Here
Thus the analytical pavement displacement under
the moving vehicle loads is gained:
4  (F1 cos(   )  F2 cos ) / (B0 M 2 )
5  (F1 sin( )  F2 sin)/ (B0M2 )
yr  B0 xr (t )sin  x / L
3949 (23)
Res. J. Appl. Sci. Eng. Technol., 6(21): 3946-3952, 2013
5
x 10
-0.995
-1
F/N
-1.005
-1.01
-1.015
-1.02
-1.025
10
15
20
t/s
25
30
Fig. 3: Numerical verification of the analytical tire load
yr/m
t/s
Fig. 4: The theoretical solution and numerical solution of
pavement displacement

With the rise of pavement length or pavement
mass, the amplitude of pavement displacement
increases. Large elastic modulus, foundation
damping or foundation stiffness will reduce
pavement vibration.
a r/m
Numerical simulation: Parameters of the nonlinear
vehicle-pavement system (Wang et al., 2005; Deng,
2000) are m2 = 10109 kg, m1 = 190 kg, K1 = 2060000
N/m, C1 = 900 Ns2/m, K2 = 75000 N/m, C2 = 9000
Ns2/m, C3 = 1000 Ns2/m, v0 = 0.1, Fy = 1000 N, B0 = 0.5
m, Ω = 2Hz, L0 = 20 m, E = 1.6×109 N/m2, I = 5×104 4
m , K3 = 48×106 N/m2, K4 = -4.8×106 N/m2, C =
0.3×105 Ns/m2, M = 1500 Kg, L = 120 m.
To verify the analytical tire load from averaging
method, the numerical tire load is obtained by the
Rugga-Kutta method according to Eq. (1), as shown in
the dashed line in Fig. 3. The solid line is the analytical
tire load calculated by Eq. (14). From Fig. 3 it can be
seen that the analytical tire load is very near to the
numerical one. Thus the analytical tire load can be used
to investigate the pavement response.
The solid line in Fig. 4 shows the analytical
solution of the pavement displacement obtained by
Eq. (23). The dashed line From Fig. 4 in Fig. 4 is the
numerical pavement displacement obtained by
integrating Eq. (16) numerically. It can be found that
the pavement response excited by vehicle loads
attenuates quickly and the analytical pavement
displacement is very near to the numerical one. Thus
the verification of the analytical pavement response
Eq. (23) is tested.
Varying the road excitation frequency Ω from 0.5
rad/s to 22 rad/s, the amplitude frequency response
curve of pavement displacement is gained from
Eq. (23), as shown in Fig. 5. It can be seen from Fig. 5
that with the rise of the road excitation frequency, two
peaks occur in the pavement response which is
associate with the inherence frequency of the vehicle
body and the tire. Letting L0 = 10 m, these two peak
will occur at the running speed 14km/h and 100 km/h.
Thus it can be concluded that as the running speed
increases, the increase in road vibration is not
monotonous. In other words, the heavy vehicle running
at low speed may lead to more severe road vibration
than that running at high speed.
Figure 6 shows the effects of six system parameters
on pavement vibration, including pavement length,
pavement mass, elastic modulus, foundation damping,
linear foundation stiffness and nonlinear foundation
stiffness. It can be seen that
Ω/Hz
Fig. 5: The amplitude frequency response curve of pavement displacement
3950 Res. J. Appl. Sci. Eng. Technol., 6(21): 3946-3952, 2013
-3.6
-3.8876
-3.65
-3.8876
-3.7
-3.8876
-3.75
y r /mm
y r/mm
-3.8876
-3.8876
-3.8
-3.8876
-3.85
-3.8876
-3.9
-3.8876
-3.95
-3.8876
50
100
150
-4
500
200
L/m
1000
1500
2000
M/kg/m
(a) The effect of pavement length
2500
3000
(b) The effect of pavement mass
-2.4
-3.7142
-2.6
-3.7142
-2.8
-3.7142
-3
y r /mm
y /mm
-3.7142
r
-3.7142
-3.7142
-3.2
-3.4
-3.6
-3.7142
-3.8
-3.7142
-4
-3.7142
0
2
4
6
8
E/N/m
10
12
14
2
-4.2
16
1
1.5
2
2.5
C/Ns/m2
3
5
x 10
(d) The effect of foundation damping,
-3.7142
-2
-3.7142
-2.5
-3.7142
-3
-3.7142
-3.5
-3.7142
y r/mm
-1.5
r
y /mm
0.5
x 10
(c) The effect of elastic modulus
-3.7142
-4
-4.5
-3.7142
-5
-3.7142
-5.5
0
10
-3.7142
3
4
5
6
7
K3/N/m2
8
9
10
7
x 10 (e) The effect of foundation stiffness
0
1
2
3
K4/N/m4
4
5
6
7
(f) The effect of nonlinear foundation stiffness
Fig. 6: The effect of system parameters on pavement vibration
3951 7
x 10
Res. J. Appl. Sci. Eng. Technol., 6(21): 3946-3952, 2013

The effect of pavement mass, foundation damping
and foundation stiffness is greater than that of the
other three parameters.
CONCLUSION
Nonlinear dynamics of a vehicle-pavement system
is investigated with the method of averaging, Galerkin
and multiple scales. Some numerical results are
presented as well. It can be concluded from this work
that:



The analytical solutions of nonlinear vehicle using
averaging method can be used to compute the tire
load acting on pavement with enough accuracy.
The pavement response excited by vehicle loads
attenuate quickly and the heavy vehicle running at
low speed may lead to more severe vibration than
that running at high speed.
Small pavement mass, large foundation damping or
foundation stiffness may decrease the amplitude of
pavement vibration.
ACKNOWLEDGMENT
The National Natural Science Foundation of China
under Grant No. 11072159, 11102121 and the Natural
Science Foundation of Hebei province under Grant No.
E2012210025，E2011210055 support this study.
REFERENCES
Deng, X.J., 2000. Subgrade and Pavement Engineering.
China Communications Press, Beijing.
Georgios, T., W.S. Charles and G. Emanuele, 2008.
Hybrid balance control of a magnetorheological
truck suspension. J. Sound Vib., 317(3-5): 514-536.
Kargarnovin, M.H., D. Younesian and D.J. Thompson,
2005. Response of beams on nonlinear viscoelastic
foundations to harmonic moving loads. Comput.
Struct., 83(2005): 1865-1877.
Li, S.H., S.P. Yang and W.W. Guo, 2004. Investigation
on chaotic motion in hysteretic nonlinear
suspension system with multi-frequency excitation.
Mech. Res. Commun., 31: 229-236.
Raghavendra D.N. and M.S. Pravin, 2009. Establishing
the limiting conditions of operation of magnetorheological fluid dampers in vehicle suspension
systems. Mech. Res. Commun., 36: 957-962.
Santee, D.M. and P.B. Gonalves, 2006. Oscillations of a
beam on a non-linear elastic foundation under
periodic loads. Shock Vib., 13(4-5): 273-284.
Stensson, A., C. Asplund and L. Karlsson, 1994.
Nonlinear behaviour of a MacPherson strut wheel
suspension. Vehicle Syst. Dyn., 23(2): 85-106.
Wang, J.J., W. Zhang and W.X. Wu, 2005. Analysis of
dynamic response of simply supported girder
bridge under heavy moving vehicles. Central South
Highway Eng., 30(2): 55-62.
Xiao Y.G., Y.M. Fu and X.D. Zhao, 2008. Bifurcation
and chaos of rectangular moderately thick cracked
plates on an elastic foundation subjected to
periodic load. Chaos. Soliton. Fract., 35(3):
460-465.
Yang, S.P. and S.H. Li, 2005. A hysteresis model for
MR damper. Int. J. Nonlin. Sci. Num., 6(2):
139-144.
Yang, Z.A., X.J., Zhao and X.Y. Xiao, 2006. Nonlinear
vibration and singularities analysis of a thin
rectangular plate on nonlinear elastic foundation. J.
Vib. Shock, 25(5): 69-73.
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