Effects of Parameters on Dynamics of a Nonlinear ...

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JOURNAL OF COMPUTERS, VOL. 6, NO. 12, DECEMBER 2011

Effects of Parameters on Dynamics of a Nonlinear Vehicle-Road Coupled System Shaohua Li Shijiazhuang Tiedao University, Shijiazhuang, Hebei, 050043, China Email: [email protected]

Yongjie Lu and Haoyu Li Shijiazhuang Tiedao University, Shijiazhuang, Hebei, 050043, China Email: [email protected], [email protected]

Abstract—The nonlinear vehicle-road coupled system is modeled as a seven DOF vehicle moving along a simply supported double-layer rectangular thin plate on a nonlinear viscoelastic foundation. The vehicle suspension stiffness, suspension damping and tire stiffness are described by the nonlinear model. The material of the upper pavement surface is modeled as the nonlinear viscoelastic Leaderman constitutive relation. The dynamical response of the vehicle-road coupled system is obtained numerically by the quick direct integral method and four steps Runge-Kutta method. The effects of system parameters on vehicle body vertical acceleration and pavement displacements are also obtained. It is found that the nonlinearity of vehicle and the viscoelasticity of road material should be considered when study the vehicle-road system responses. Index Terms—vehicle-road system, dynamics, nonlinearity, viscoelastic material, numerical method

I. INTRODUCTION In the past tens years, vehicle and pavement are investigated in vehicle dynamics and road dynamics separately. In vehicle dynamics, road surface roughness is generally regarded as excitations to vehicle. Some scholars investigated the road damage due to dynamical tire forces based on two or four DOF vehicle model [1-2]. Other scholars modeled the nonlinear characteristic of the suspension and tire, and studied the nonlinear dynamics of vehicle system based on quarter car model [3-5]. However, the present investigations on vehicle dynamics didn’t consider the vibration of pavement. The study on road damage due to tire forces based on the nonlinear vehicle system is seldom found. In road dynamics, the vehicle is generally regarded as moving loads acting on the pavement, and the pavement is modeled as a beam, plate and multi-layer system on elastic or viscoelastic foundation. The viscoelasticity and nonlinearity of pavement material have drawn more and more attention [6-8]. However, these researches on road dynamics seldom consider the characteristic of vehicle suspension. In this work, a nonlinear vehicle-road coupled system is modeled as a seven DOF vehicle moving along a simply supported double-layer rectangular thin plate on

© 2011 ACADEMY PUBLISHER doi:10.4304/jcp.6.12.2656-2661

a nonlinear viscoelastic foundation. Using Galerkin’s method, the quick direct integral method and four steps Runge-Kutta method, the nonlinear dynamics of the vehicle-road coupled system is analyzed and the effects of system parameters on vehicle body vertical acceleration and pavement displacements are studied. II. SYSTEM MODELING A seven DOF vehicle and a double-layer rectangular thin plate on viscoelastic foundation with four simply supported boundaries are employed to model vehicle and pavement, as shown in Fig.1. The nonlinear tire spring force is expressed by Ftk = K t Z t + β 1 K t Z t2 (1) where, K t is the linear tire stiffness, β 1 is the nonlinear tire stiffness coefficient, and Z t is the relative vertical displacement between wheel and road surface. The nonlinear spring force of vehicle suspension is modeled as Fsk = K sl Z s + β 2 K sl Z s2 + β3 K sl Z s3 (2) where, K sl is the linear suspension stiffness, β 2 and

β 3 are the square and cubic nonlinear coefficients of suspension stiffness, and Z s is the relative vertical displacement between wheel and suspension. The hydraulic damper of vehicle suspension is modeled by 1.25 (3) F = C (1 + β sig ( Z )) Z sc

sl

4

s

s

where Csl is the linear suspension damping coefficient,

β 4 is the asymmetry coefficient, and Z s is the relative vertical velocity between wheel and suspension. The Leaderman constitutive relation is applied to model nonlinearity and viscoelasticity of the material of asphalt topping, σ = E 0 (ε ( x, z , t ) + β 5 ε ( x, z , t ) 2 + β 6 ε ( x, z , t ) 3 ) (4) t + ∫ E (t − τ )(ε ( x, z , τ ) + β 4 ε ( x, z , τ ) 2 + β 5 ε ( x, z , τ ) 3 )dτ 0

where, E0 is the initial elastic modulus， β 5 is the square

JOURNAL OF COMPUTERS, VOL. 6, NO. 12, DECEMBER 2011

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nonlinear coefficient, β 6 is the cubic nonlinear coefficient. E(t) is the relax function derived from Burgers model for asphalt mix, which is expressed by E (t ) = Ae −αt + Be − βt (5) The road subgrade is modeled by nonlinear Kelvin foundation, and the reaction force of the subgrade is P = KZ r + β 7 KZ r3 + CZ r (6)

and m3 are moment of inertia of vehicle body in pitching and rolling directions respectively. mt1 , mt 2 , mt 3 , mt 4 are wheel’s masses. df is half of front wheel track, dr is half of rear wheel track, and l1 +l2 is wheel space. Ft1 , Ft 2 , Ft 3 , Ft 4 are four tire forces which are expressed by Fts = k ts [ x ts − rts − w( x ts , y ts , t )] + c ts [ x ts − rts −

where, K is the foundation response modulus, β 7 is the cubic nonlinear coefficient, and C is the foundation damping coefficient. The seven degree of freedom (7DOF) vehicle equations can be obtained by Dalembert’s principle, M v Zv + C v Z v + K v Z v = Rv (7)

where, w(xts,yts,t) is pavement displacement of the point under the tire. rts is the road surface roughness satisfying the following functions 2πvt 2π rt1 = rt 3 = A sin( ), rt 2 = rt 4 = A sin[ (vt + l1 + l 2 )] L0 L0 Here, A is the amplitude of road surface roughness, and L0 is the wavelength of road roughness. ksi , Csi are nonlinear coefficients of suspension

v

z

Zt2

Ks2

Cs2

Z3 Zt1

Ks11

mt1 Kt1

Ct2

Cs1

Zt4

Cs1

mt2 Kt2

Z1

m1

Z2

mt4 Ct1

∂w( x ts , y ts , t ) ] ∂t

Zt2

mt2

Zr

stiffness and suspension damping respectively, which are expressed by k si = k sli + β 2 k sli ( Z b1 − Z ti ) + β 3 k sli ( Z b1 − Z ti ) 2 c si = c sli (1 + β 4 sig ( Z b1 − Z ti ) Z b1 − Z ti

C

K

x

O

l2

where i = 1 ~ 4 . The upper surface’s stress is nonliear and can be derived, t E (t − τ ) E1 ⎧ [ g 1 ( x, y , z , t ) + g 1 ( x , y , z , τ ) dτ ] ⎪σ x = 2 0 E1 1 − µ1 ⎪ ⎪⎪ t E (t − τ ) E1 [ g 2 ( x, y , z , t ) + g 2 ( x, y, z , τ )dτ ] ⎨σ y = 2 0 E1 1 µ − ⎪ 1 ⎪ t E (t − τ ) ⎪τ xy = G1 ( g 3 ( x, y, z , t ) + g 3 ( x, y, z , τ )dτ ) 0 ⎪⎩ G1

∫

B

l1

m1

∫

mt3 df

dr

mt4

0.25

mt2

mt1

(8)

∫

L

y

Fig.1 The nonlinear vehicle-road coupled system

where, M v = diag[m1 m2 m3 mt1 mt 2 mt 3 mt 4 ] , ⎤ ⎡4 −ks1 −ks2 −ks3 −ks4 ⎥ ⎢∑ksi −ks1l1 +ks2l2 −ks3l1 +ks4l2 −ks1d f −ks2dr +ks3d f +ks4dr ⎥ ⎢i=1 2 2 2 2 ks1l1 +ks2l2 +ks3l1 +ks4l2 ks1l1d f −ks2l2dr −ks3l1d f +ks4l2dr ks1l1 −ks2l2 ks3l1 −ks4l2 ⎥ ⎢ 2 2 2 2 ⎢ ks1d f +ks2dr +ks3d f +ks4dr ks1d f ks2dr −ks3d f −ks4dr ⎥ ⎥ Kv = ⎢ 0 0 0 ⎥ ks1 ⎢ ⎢ 0 0 ⎥ ks2 ⎥ ⎢ ⎢ 0 ⎥ ks3 ⎥ ⎢ ks4 ⎦ ⎣

Symmetry

⎤ ⎡4 −cs1 −cs2 −cs3 −cs4 ⎥ ⎢∑csi −cs1l1 +cs2l2 −cs3l1 +cs4l2 −cs1d f −cs2dr +cs3d f +cs4dr ⎥ ⎢i=1 2 2 2 2 cs1l1 +cs2l2 +cs3l1 +cs4l2 cs1l1d f −cs2l2dr −cs3l1d f +cs4l2dr cs1l1 −cs2l2 cs3l1 −cs4l2 ⎥ ⎢ 2 2 2 2 ⎢ cs1d f +cs2dr +cs3d f +cs4dr cs1d f cs2dr −cs3d f −cs4dr ⎥ ⎥ Cv = ⎢ cs1 0 0 0 ⎥ ⎢ ⎢ cs2 0 0 ⎥ ⎥ ⎢ ⎢ cs3 0 ⎥ ⎥ ⎢ cs4 ⎦ ⎣ Symmetry

Rv = [0 0 0 Ft1 Z v = [ z b1

zb2

zb3

Ft 2

Ft 3

Ft 4 ]T ,

z t1

zt 2

zt 3

zt 4 ]T .

zb1 , zb 2 , zb3 are vehicle body’s vertical, pitching and

rolling displacements. zt1 , z t 2 , z t 3 , zt 4 are wheel’s vertical displacements. m1 is mass of vehicle body, m2

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where, E1，G1，μ1 are elastic modulus, shear modulus, Poisson ratio of the upper surface respectively. The expressions of g1, g2 and g3 are ∂2w ∂2w ∂2w + β 5 z 2 ( 2 ) 2 − β 6 z 3 ( 2 )3 2 ∂x ∂x ∂x 2 2 ∂2w ∂ ∂ w w + µ1 (− z 2 + β5 z 2 ( 2 ) 2 − β 6 z 3 ( 2 )3 ∂y ∂y ∂y g1 ( x, y, z , t ) = − z

g 2 ( x, y , z , t ) = − z

∂ 2w ∂2w ∂2w + β 5 z 2 ( 2 ) 2 − β 6 z 3 ( 2 )3 2 ∂y ∂y ∂y

∂2w ∂2w ∂2w + β 5 z 2 ( 2 ) 2 − β 6 z 3 ( 2 )3 2 ∂x ∂x ∂x ∂2w ∂2w 2 ∂2w 3 g 3 ( x, y, z , t ) = −2 z ) − 8β 6 z 3 ( ) + 4β 5 z 2 ( ∂x∂y ∂x∂y ∂x∂y

+ µ1 (− z

Using elastic dynamic theory, the partial differential equation of the nonlinear double-layer thin plate on nonlinear viscoelastic foundation subjected by moving vehicle loads can be derived,

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D x1 (

JOURNAL OF COMPUTERS, VOL. 6, NO. 12, DECEMBER 2011

∂4w ∂4w ∂ 4w + ) + 2 ( D y1 + 2 D xy1 ) 2 2 4 4 ∂x ∂y ∂x ∂ y

t ⎧ −α (t −τ ) G (τ )dτ ⎪⎪ x1ij = 0 − Aαe ⎨ t ⎪ x = − Bβe − β (t −τ ) G (τ )dτ ⎪⎩ 2ij 0 The first derivation of Eq.(13) is, ⎧⎪ x1ij = −αx1ij − Aα ( A1ij U ij + A2ij U ij2 + A3ij U ij3 ) ⎨ 2 3 ⎪⎩ x 2ij = − βx 2ij − Bβ ( A1ij U ij + A2ij U ij + A3ij U ij ) Substituting Eq.(12) in Eq.(11) leads to M ijUij + CijU ij + K1ijU ij + K 2ijU ij2 + K 3ijU ij3

∫ ∫

∂ 3w ∂ 2w ∂ 4w ∂ 3w ∂ 2w ∂ 4w + 2 D x 2 [( 3 ) 2 + + ( 3 )2 + ] ∂x ∂ x 2 ∂x 4 ∂y ∂ y 2 ∂y 4 + 2 D y 2 [(

∂ 3w 2 ∂ 2 w ∂ 4w ∂ 3w ∂ 2w ∂ 4w + ( 2 )2 + ) + ] 2 2 2 2 ∂x ∂y ∂y ∂x ∂y ∂x ∂y ∂ x 2 ∂x 2 ∂y 2

+ 4 D xy 2 [

∂ 3w ∂ 3w ∂ 2w ∂ 4w + ] 2 2 ∂x ∂y ∂ x∂y ∂ x∂ y ∂ x 2 ∂y 2

∂ 2w ∂ 3w ∂ 2w ∂ 3w ∂ 2w ∂ 4w ∂ 2w ∂ 4w + 3Dx3[2 2 ( 3 ) 2 + 2 2 ( 3 ) 2 + ( 2 ) 2 ] + ( 2 )2 ∂x ∂ x ∂y ∂ y ∂x ∂x 4 ∂y ∂y 4 + 3D y3[2

∂ 2 w ∂ 3w 2 ∂ 2w ∂ 3w ( ) + 2 2 ( 2 )2 2 2 ∂ y ∂x ∂y ∂x ∂ x ∂ y

+ x1ij + x2ij = Rij

∂2w ∂4w ∂2w ∂4w + ( 2 )2 2 2 + ( 2 )2 2 2 ] ∂y ∂x ∂y ∂x ∂ x ∂y + 6 D xy 3 [ 2 +

∂ 2w ∂ 3w ∂ 3w ∂ 2w 2 ∂ 4w +( ) ] ∂x∂ y ∂x∂ y 2 ∂ x 2 ∂y ∂x∂ y ∂ x 2 ∂ y 2

∫ E (t − τ )( t

0

+ Kw + C

∂ 2 D Exy

(9)

∂ D E1 ∂ DE 2 +2 + ) dτ ∂x 2 ∂ x∂y ∂y 2 2

2

4 ∂w ∂2w ∂ 2w + ρ h 2 = L ( w ) + ρ h 2 = ∑ Ftiδ ( x − xti )δ ( y − y ti ) ∂t ∂t ∂t i =1

where, the expressions of D xi , D yi , D xyi , D E1 , D E 2 , DExy , ρh can be found in [10] and the value of these seven coefficients depends on the pavement parameters. Displacement of double-layer thin plate with four simply supported boundaries can be expressed as NM NN nπy mπx (10) w( x, y, t ) =

∑∑ U

mn (t ) sin

m =1 n =1

L

sin

B

where, L and B are the pavement’s length and width. By Galerkin’s method, equation (18) can be discretized into a set of ordinary differential equations with integral item, M ijUij + CijU ij + K1ijU ij + K 2ijU ij2 + K 3ijU ij3 (11) t + ∫ E (t − τ )G (τ )dτ = Rij 0

where i = 1 ~ NM , j = 1 ~ NN ， LB LB ρh, Cij = C, 4 4

M ij =

LB LB K , K 2ij = D 2ij , K 3ij = D3ij + β7 K, 4 4

K 3ij = D1ij + G (τ ) =

L

∫∫ 0

B

(

∂ 2 D E1

0

∂x 2

+2

∂ 2 D Exy ∂x∂y

+

∂ 2 DE 2 ∂y 2

) sin(iπx / L) sin( jπy / B )dxdy

Rij =

L

B 4

0

0

∫ ∫ ∑ F δ (x − x ts

ts )δ ( y −

y ts ) sin(iπx / L) sin( jπy / B)dxdy

s =1

The expressions of D1ij , D 2ij , D3ij , A1ij , A2ij , A3ij are omitted and can be found in [10]. Due to the integral item in Eq.(11)，the following transform [11-12] is applied, x1ij + x 2ij =

t

∫ E (t − τ )G(τ )dτ 0

Substituting Eq.(5) in Eq.(12), one get

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(12)

(14)

(15)

Thus Eq.(11) turns into one second-order ordinary differential equation Eq.(15) and two first-order ordinary differential equations Eq.(14). Rewrite Eq.(14) and Eq.(15) as matrix equations, ⎧⎪ X 1 = −αX 1 − Aα ( A1U + A2U 2 + A3U 3 ) = f (t , X 1 , U ) ⎨ 2 3 ⎪⎩ X 2 = − β X 2 − Bβ ( A1U + A2U + A3U ) = f (t , X 2 , U )

M r U + C r U + K r U = R r − X 1 − X 2

(16) (17)

where K r = [ K1 + K 2U + K 3U ] . 2

Eq.(11) and Eq.(7) compose the governing equations of the nonlear vehicle-road coupled system. Rewrite Eq.(1) and Eq.(17) in the following form [ M ]{Z} + [C ]{Z } + [ K ]{Z } = {R} (18) where Rv 0 ⎤ ⎡M ⎧ ⎫ ⎧Z v ⎫ , {R} = ⎨ [M ] = ⎢ v ⎬， {Z } = ⎨ ⎬ . ⎥ ⎩U ⎭ ⎣ 0 Mr ⎦ ⎩ Rr − X 1 − X 2 ⎭

III. THE NUMERICAL METHOD BY COMPUTER Considering the time-varing of the coupled system’s stiffness and damping matix, the quick direct integral method [13-14] and four steps Runge-Kutta method are used to solved Eq.(18) and obtain the system’s responses. The computer processes are as follows, (1) Giving initial conditions Let initial displacement and initial velocity of Eq.(18) be ⎪⎧{Z }0 = {Z (0)} ⎨  ⎪⎩{Z }0 = {Z (0)} . Initial acceleration can be obtained from Eq.(18), {Z}0 = [ M ]−1 ({R}0 − [ K ]0{Z }0 − [C ]0{Z }0 )

= A1ij U ij + A2ij U ij2 + A3ij U ij3

(13)

.

Let initial displacement of Eq.(16) be ⎧{ X 1 }0 = { X 1 (0)} ⎨ ⎩{ X 2 }0 = { X 2 (0)}

. (2) Computing displacement, velocity, and acceleration of the vehicle-road coupled system when t = (n + 1)∆t . Let ϕ = ψ = 0 when n=0, and ϕ = ψ = 1 / 2 when n ≥ 1 . Giving Integration time step ∆t , one can build the following relations, ⎧⎪{Z }n+1 = {Z }n + {Z }n ∆t + (1 / 2 + ψ ){Z}n ∆t 2 −ψ {Z}n−1 ∆t 2 ⎨  ⎪⎩{Z }n+1 = {Z }n + (1 + ϕ ){Z}n ∆t − ϕ{Z}n−1 ∆t

(19)

JOURNAL OF COMPUTERS, VOL. 6, NO. 12, DECEMBER 2011

∆t ∆t , X 1n + f 11 , U n ), 2 2

∆t ∆t , X 1n + f 12 , U n ), f 14 = f 1 (t n + ∆t , X 1n + ∆tf 13 , U n ) 2 2 ∆t ∆t f 21 , U n ), = f 2 (t n , X 1n , U n ), f 22 = f 2 (t n + , X 1n + 2 2 ∆t ∆t f 22 , U n ), f 24 = f 2 (t n + ∆t , X 1n + ∆tf 23 , U n ) = f 2 (t n + , X 1n + 2 2

f 13 = f 1 (t n + f 21 f 23

System equations (18) when t = (n + 1)∆t are [ M ]{Z}n+1 + [C ]n+1{Z }n+1 + [ K ]n+1{Z }n+1 = {R}n+1 (21) Substituting Eq.(19) and Eq.(20) into Eq.(21) one can obtain {Z}n +1 = [ M ]−1 ({R}n +1 − [ K ]n +1{Z }n − ([C ]n +1 + [ K ]n +1 ∆t ){Z }n − {(1 + ϕ )[C ]n +1 + (1 / 2 + ψ )[ K ]n +1 ∆t}{Z}n ∆t + (ϕ[C ]n +1 + ψ [ K ]n +1 ∆t ){ A}n −1 ∆t Thus the displacement, velocity and acceleration of the system when t = (n + 1)∆t can be gained.

(3) Repeating the process (2) ， we can get the displacement, velocity and acceleration of the system step by step.

0.52

0.0187

0.51

0.0187 MZr /mm

f 11 = f 1 (t n , X 1n , U n ), f 12 = f 1 (t n +

0.5

0.0186

0.49 0.48

0.0185 0

50 β1

0.0185

100

0

50 β1

100

Fig.2 The effect of square nonlinear tire stiffnessβ1 0.0186

0.5015

0.0186

0.501

MZr/mm

where

hardly influence the pavement displacement. (2) Effects of pavement nonlinear parametersβ5、β6 and β7 on responses of vehicle and pavement is very small. So these three nonlinear parameters may be omitted in order to simplify the calculation course. (3) In four viscoelastic parameters of pavement asphalt topping, the effect of E1 on system response is greater than that of E3, and the effect ofη2 on system response is greater than that of η3. Small E1, big E3, bigη2, or big η3 may not only improve vehicle running comfort but also extend road service life.

MA1 /m/s 2

(20)

MA1 /m/s 2

∆t ⎧ { X } = { X 1}n + ( f11 + 2 f12 + 2 f13 + f14 ) ⎪⎪ 1 n +1 6 ⎨ ⎪{ X } = { X } + ∆t ( f + 2 f + 2 f + f ) 2 n 21 22 23 24 ⎪⎩ 2 n +1 6

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0.5005 0.5

0.0186 0.0186

0

1000 β2

0.0186

2000

0

1000 β2

2000

Fig.3 The effect of square nonlinear suspension stiffnessβ2

VI. EFFECTS OF SYSTEM PARAMETERS

the greatest, effects of square nonlinear tire stiffnessβ1 and square nonlinear suspension stiffness β 2 are the second, and the effect of cubic nonlinear suspension stiffness β3 is the least. Bigβ1 or small β 4 is benefit to both vehicle running comfort and road service life. Smallβ2 may improve the vehicle running comfort but

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MZr/mm

MA1 /m/s 2

0.0186

0.5004 0.5004

0.0186 0.0186

1000 β3

0.0186

2000

0

1000 β3

2000

Fig.4 The effect of cubic nonlinear suspension stiffnessβ3 0.019

0.7 MZr/mm

MA1 /m/s 2

0.8

0.6 0.5 0.4

0

0.5 β4

0.0188 0.0186 0.0184

1

0

0.5 β4

1

Fig.5 The effect of suspension damper asymmetry coefficientβ4 0.5003

0.0161

0.5003

0.0161

MZr/mm

Effects of seven nonlinear parameters and four viscoelastic parameters on vehicle body vertical acceleration and pavement displacement are studied, as shown in Fig.2~Fig.11. Main conclusions are listed here, (1) In four nonlinear parameters of vehicle system, the effect of suspension damper asymmetry coefficient β 4 is

0.0186

0

MA1 /m/s 2

The system parameters are chosen according to [10]. m1=15280kg, m2=3×105kg﹒m2, m3=0.6×105kg﹒m2, mt1= mt3=190kg, mt2= mt4=380kg, Ksl1=Ksl3=370×103 N/m, Ksl2=Ksl4=920×103 N/m, Csl1= Csl3=12000 N﹒s2/m, Csl2=Csl4=30000 N﹒s2/m, Ktl1= Ktl3=0.73×106N/m, Ktl2= Ktl4=1.46×106N/m, Ct1= Ct3=600N﹒s2/m, Ct2= Ct4=900N﹒s2/m，l1=3.29m, l2=1.48m, lf=1.90m, lr=1.80m,β1=0.01, β2=0.1, β3=0.6, β4=1/3, v=10m/s, L=600m, B=24m, L0=2.3m, h1=0.09m, E1=2400MPa, E3=2400Mpa, η2=3159.32 Mpa﹒s,η3=509.61 Mpa﹒s,γ1=0.35, ρ1=2.613×103 kg/m3, h2=0.2m，E2=1100MPa, γ2=0.35,ρ2=2.083×103 kg/m3, K=48×106 N/m2, C=0.3×104 N﹒s/m2, β5=0.1, β6=0.1, β7=0.01.

]

0.5004

0.5003 0.5003

0

5 β5

10 4

x 10

0.0161 0.0161

0

5 β5

Fig.6 Effect of square nonlinear pavement topping elasticβ5

10 x 10

4

JOURNAL OF COMPUTERS, VOL. 6, NO. 12, DECEMBER 2011

0.5003

0.0161

0.5003

0.0161 MZr/mm

MA1 /m/s 2

2660

0.5003

0.0161

0.5003 0.5003

0.0161 0

5 β6

0.0161

10

0

5 β6

4

x 10

10 4

x 10

Fig.7 Effect of cubic nonlinear pavement topping elasticβ5 0.0186 0.0186

0.5004

MZr/mm

MA1 /m/s 2

0.5004

0.5004 0.5004

0.0186 0.0186

0

1000

2000

0

1000

β7

2000

β7

responses of the coupled system is studied. It is found that (1) Big tire nonlinearity β 1 or small asymmetry coefficient of suspension damper β 4 is benefit to both vehicle running comfort and road service life. Small suspension square nonlinearity β 2 may improve the vehicle running comfort but hardly influence the pavement displacement. Effects of suspension cubic nonlinearity β 3 and pavement nonlinear parametersβ5、β6 andβ7 on responses of vehicle and pavement is very small. (2) The effect of pavement viscoelasticity on pavement responses should not be neglected. Small E1, big E3, big η 2, or big η 3 may not only improve vehicle running ride comfort but also extend road service life.

0.03

0.5003

0.025

MZr/mm

MA1 /m/s 2

Fig.8 Effect of nonlinear foundation stiffnessβ7 0.5003

0.5002 0.5002

0

1000

2000 3000 E1 /MPa

This work was supported by the National Natural Science Foundation of China under Grant No. 11072159 and the Natural Science Foundation of Hebei province under Grant No. E2010001095.

0.02

0.015

4000

ACKNOWLEDGMENT

0

1000

2000 3000 E1 /MPa

REFERENCES

4000

[1]

Fig.8 Effect of pavement topping elastic modulus E1 0.021

0.5003 0.5003

[2]

0.0205

0.5003

MZr/mm

MA1 /m/s 2

0.5003

0.02

[3]

0.0195 0

2000 E3 /MPa

0.019

4000

0

2000 E3/MPa

4000

[4]

Fig.9 Effect of pavement topping elastic modulus E3 0.022 0.021

0.5003

MZr/mm

MA1 /m/s 2

0.5004

0.5003

0.5003

[5]

0.02 0.019

0

5000 η2 /MPa.s

0.018

10000

[6] 0

5000 η2 /MPa.s

10000

Fig.10 Effect of pavement topping dampingη2 [7] 0.0195 0.0194

0.5003

MZr/mm

MA1/m/s 2

0.5003

0.5002

0.5002

[8]

0.0194 0.0193

0

5000

10000

0.0193

[9] 0

η3/MPa.s

5000

10000

η3/MPa.s

Fig.11 Effect of pavement topping dampingη3

[10]

V. CONCLUSIONS Not only considering the nonlinearity of vehicle and pavement, but also considering the viscoelasticity of asphalt topping, a nonlinear vehicle-road coupled system is modeled and the effects of parameters on dynamic

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[11]

[12]

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Shaohua Li Hebei Province, China. Birthdate: July, 1973. is Vehicle Engineering Doctor of Technology, graduated from School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University. And research interests on dynamical interaction between vehicle and road. She is a vice-professor of School of Mechanical Engineering, Shijiazhuang Tiedao University. Yongjie Lu Hebei Province, China. Birthdate: September, 1981. is Vehicle Engineering Doctor of Technology, graduated from School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University. And research interests on vehicle dynamics. She is a instructor of School of Mechanical Engineering, Shijiazhuang Tiedao University. Haoyu Li Hebei Province, China. Birthdate: November, 1971. is Vehicle Engineering Doctor of Technology, graduated from School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University. And research interests on dynamical interaction between vehicle and road. She is a professor of School of Mechanical Engineering, Shijiazhuang Tiedao University.

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