Nonlinear dynamic analysis for coupled vehicle-bridge vibration system on nonlinear foundation

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Nonlinear dynamic analysis for coupled vehicle-bridge vibration system on nonlinear foundation ⁎

Shihua Zhou, Guiqiu Song , Rongpeng Wang, Zhaohui Ren, Bangchun Wen School of Mechanical Engineering & Automation, Northeastern University, 110819 China

A R T I C L E I N F O

ABSTRACT

Keywords: CVBVS Nonlinear dynamics Coupled vibration Bifurcation diagram Galerkin method

In this paper, the nonlinear dynamics of a parametrically excited coupled vehicle-bridge vibration system (CVBVS) is investigated, and the coupled system is subjected to a timedependent transverse load including a constant value together with a harmonic time-variant component. The dynamic equations of the CVBVS are established by using the generalized Lagrange's equation. With the Galerkin truncation method, a set of nonlinear ordinary differential equations are derived by discretizing the continuous governing equation. The influences of parametric excitation with nonlinear support stiffness, mass ratio, excitation amplitude and position relation on the dynamic behaviors are studied for the interaction between vehicle and the bridge. The analysis results indicate that the nonlinear dynamic characteristics are strongly attributed to the interaction of the coupled system. Nonlinear support stiffness of foundation and mass ratio can lead to complex dynamic behaviors such as jump discontinuous phenomenon, periodic, quasi-periodic and chaotic motions. Vibration amplitude increases depending on the position, where the maximum vibration displacement does not occur at the center of the bridge. The excitation amplitude has an obvious influence on the nonlinear dynamic behaviors and the increase of the excitation amplitude makes the vibration strengthen. The bifurcation diagram and 3-D frequency spectrum are used to analyze the complex nonlinear dynamic behaviors of the CVBVS. The presented results can provide an insight to the understanding of the vibration characteristics of the coupled vehicle-bridge vibration system in engineering.

1. Introduction Because heavy vehicles are widely used to communication and transportation with the development of the carrying capacity, the control technology and the configuration of vehicle suspension have been improved unceasingly. A detailed investigation on the vehicle suspension system to meet the industry development is required. Therefore, it is significant to establish and study the nonlinear dynamic behaviors of the vehicle suspension system by the nonlinear dynamic theory. Xiao [1] established a 4-DOF (degree of freedom) vehicle dynamic model considering the automobile suspension system with sinusoid excitation and hysteretic characteristic, and the chaotic movement was observed. Borowiec [2] carried out a dynamic analysis of the vehicle suspension system, numerical simulation and experiment were presented on three types of road surfaces. Borowiec [3] studied a quarter-car model by 2-DOF nonlinear oscillator and analyzed the transient vibration. Lindsey [4] described a new model for low-power active control of automotive suspension, and the steady state of the suspension system was investigated. ElMadany [5] developed a

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Corresponding author. E-mail addresses: [email protected] (S. Zhou), [email protected] (G. Song), [email protected] (R. Wang), [email protected] (Z. Ren), [email protected] (B. Wen). http://dx.doi.org/10.1016/j.ymssp.2016.10.025 Received 2 August 2016; Received in revised form 8 October 2016; Accepted 22 October 2016 Available online xxxx 0888-3270/ © 2016 Elsevier Ltd. All rights reserved.

Please cite this article as: Zhou, S., Mechanical Systems and Signal Processing (2016), http://dx.doi.org/10.1016/j.ymssp.2016.10.025

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kt1/ct1 kt2/ct2 ks1/cs1 ks2/cs2 F0 Fp ω0 B X0 Y1 Y2 Yb τ

Nomenclature m0 m1 m2 ρ I E L A kb1 kb2 cb

eccentric mass unsprung mass sprung mass density moment of inertia modulus of elasticity length of the beam cross section area of the beam linear stiffness of foundation nonlinear stiffness of foundation damping of foundation

linear stiffness/damping of tire nonlinear stiffness/damping of tire linear stiffness/damping of suspension nonlinear stiffness/damping of suspension external excitation mean external excitation amplitude rotational speed location of vehicle location of external excitation vibration displacement of unsprung vibration displacement of sprung vibration displacement of beam non-dimensional time

methodology for the design and evaluation of a slow-active vehicle suspension system. The vehicle dynamics, roadway excitations and performance measures were investigated. Lu [6] used numerical simulation and field test to investigate tire dynamic load with a nonlinear virtual prototype model of heavy duty vehicle. And the effects of vehicle speed, load, road surface roughness and tire stiffness on tire dynamic load and dynamic load coefficient were discussed. Dong [7] estimated the modal parameters and mass moments of inertia of an on-road vehicle by subspace identification method, and then the theoretical analysis was compared with Monte Carlo experiments. Múcka [8] used a linear planar model of automobile with 12-DOF to analyze the influence of tire–road contact model on the simulated vertical vibration response. The beam is one of the significant structures widely used in many technological devices and machines components. The dynamic behaviors of bridge or vibration platform, which are usually modeled as beam, have been investigated extensively with the influences on harmonic displacement excitation or stationary random excitation. However, due to. the different support materials or support position, the foundation often has the strong nonlinear characteristics. Therefore, the dynamic analysis of the beam resting on nonlinear foundation becomes an important research topic. Calım [9] analyzed the dynamic behaviors of beam on Pasternak-type viscoelastic foundation subjected to time-dependent loads. The dynamic responses of beam were investigated through various examples. Ansari [10] used the multiple scales method to study the vibration characteristics of a finite Euler-Bernoulli beam. The effects of damping and non-linear stiffness of the foundation as well as the magnitude of the moving load on the frequency responses were discussed. Huang [11] presented nonlinear vibration analysis of a curved beam subjected to uniform base harmonic excitation with both quadratic and cubic nonlinearities. Particular attention was paid to the anti-symmetric response with and without excitation by using incremental harmonic balance method (IHBM). A boundary element method formulation was developed for the dynamic analysis of Timoshenko beams by Carrer [12]. The four kinds of beams that were pinnedpinned, fixed-fixed, fixed-pinned and fixed-free, under uniformly distributed, concentrated, harmonic concentrated and impulsive loading, were analyzed. Akour [13] established a nonlinear beam model resting on linear elastic foundation and subjected to harmonic excitation with simply support at both ends. The damping coefficient, natural frequency and coefficient of the nonlinearity were investigated. Ghayesh [14] investigated the nonlinear forced vibration of a microbeam by employing the strain gradient elasticity theory, and frequency–response curves of the vibration system were demonstrated. Yang [15] analyzed the dynamic response of finite Timoshenko beam resting on a six parameter foundation subjected to a moving load. The effects of different truncation terms on the dynamic responses were discussed via the Runge–Kutta method. Zhao [16] focused on obtaining the direct expressions of steady-state two dimension temperature and displacement responses for the coupled thermoelastic vibrations of Timoshenko beam subjected to a heat flux and an external force, and the influences of some important physical parameters on the coupled multi-physics problem were discussed. Jorge [17] analyzed the vibration characteristics of beam on nonlinear elastic foundation, subjected to moving loads. The effects of the load's intensity and velocity and the foundation's stiffness were researched. Mbong [18] established a single-DOF nonlinear beam model, which is subjected to a combination of both low-frequency force and high-frequency force. The critical values of perturbation parameters for the onset of the chaotic motion were specified using Melnikov's method. Hasan [19] studied the nonlinear vibration of multi-mode flexible beam on an elastic foundation subjected to external harmonic excitation by using multi-level residue harmonic balance. The effects of various parameters such as vibration amplitude, foundation modulus coefficient, damping factor and excitation level etc., on the nonlinear behaviors were examined. Dynamic analysis of the CVBVS subjected to a load is significant in the design stage or during the assessment of coupled structures. And energy transfer from a bridge system to a nonlinear vehicle system has been a very important issue in recent studies. The dynamic behaviors of coupled system between vehicle and bridge have been studied by a number of researchers. Asnachinda [20] presented an identification of multiple vehicle dynamic axle loads on multi-span continuous bridge, and the accuracy of identified dynamic axle loads for all cases of study was within a relative percentage error of 13%. Zhang [21] analyzed the coupled vehicle-bridge system with vertical track irregularity by a new nonstationary, random vibration method. It showed that the proposed PEM-PIM method performed nonstationary random vibration analysis of coupled vehicle-bridge systems efficiently and accurately. Liu [22] studied the dynamic behavior of a suspension bridge due to moving vehicle loads with vertical support motion caused by earthquake. The interaction of both the moving loads and the seismic forces can substantially amplify the vibration response of longspan suspension bridge system. Xu [23] developed a new optimization method for the coupled vehicle-bridge system subjected to an uneven road surface excitation. The precise integration method was used to compute the vertical random vibrations for the coupled 2

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system. Yin [24] built an 18-DOF multi-body dynamics system and the bridge-subgrade transition zone, some dynamic responses of bridge-subgrade transition zone under moving vehicle with and without track irregularity were computed. Ding [25] focused on the coupled nonlinear vibration of vehicle-pavement system, where the pavement was modeled as a Timoshenko beam resting on a sixparameter foundation and the vehicle was simplified as a spring-mass-damper oscillator. Doménech [26] developed a coupled vehicle-bridge dynamic model in order to contribute to the problem of train-bridge interaction in high-speed railway lines. The results of a subsequent extensive sensitivity analysis were presented and discussed. Feng [27] proposed a new method to simultaneously identify bridge structural parameters and vehicle dynamic axle loads of a vehicle-bridge interaction system, and the effects of the vehicle speed, the number of sensors and initial estimates of structural parameters on the accuracy of the identification results were investigated. Fu [28] investigated the effects of these cracks on the vibration for a continuous beam bridge subjected to moving vehicles. The cracks were simulated by switching cracks, which can open and close fully instantaneously, and the beam behaviors were considered as a sequence of linear states. From the mentioned references above, although some research works on the coupled vehicle-bridge have been carried out based on the suspension system with linear or nonlinear spring and linear damper in recent years, effort in analyzing the coupled system with nonlinear foundation, nonlinear suspension system (nonlinear spring and nonlinear damping) and tire is rare. The motivation of this work is originated from the interests in the vibration flow-aiding device based on vibration exciter and researching the dynamic behaviors of the CVBVS. Therefore, the investigation on the nonlinear vibration of the CVBVS is a significant subject. In the present study, the complex dynamic behaviors of nonlinear transverse vibration for CVBVS are investigated. Based on the coupled model, the nonlinear dynamic equations is derived by using the generalized Lagrange's equation, which are discretized via the Galerkin method. The dynamic equations are solved by applying the Runge-Kutta integration method. The bifurcation diagram and 3-D frequency spectrum for vibration responses of the coupled system via the nonlinear support stiffness, mass ratio, excitation amplitude and position relation are respectively presented and discussed when other parameters are fixed. The structure of this paper is as follows. After this introduction in Section 1, a coupled model of vehicle-bridge vibration system is established in Section 2. And the dynamic equations of motion for the CVBVS are derived by using the Lagrange's equation. The Galerkin's method is used to discretize the equations. Section 3 presents the nonlinear dynamic behaviors of the CVBVS with the effects on key parameters by the bifurcation diagram, 3-D frequency spectrum, waveform, frequency, phase diagram and Poincaré map. Finally, some conclusions are drawn in Section 4.

2. Description of the coupled vehicle-bridge dynamic model In this section, a dynamic model of the vehicle system on the vibration offload platform subjected to an external harmonic excitation is developed. Furthermore, the vibration platform is modeled as a continuous Euler–Bernoulli beam resting on nonlinear foundation, and the vehicle is simplified to a 2 DOF model with cubic nonlinearity. The vertical vibration is produced by double eccentric shaft vibration exciter, which drives the vibration offload platform and accelerates the unloading of the remainder materials in the vehicle. In our work, due to the mass of the remainder materials relatively smaller to the mass of vehicle and beam, the mass of the remainder materials is neglected. Namely, the vibration mass of the CVBVS remains the same.

2.1. External excitation The external excitation for the CVBVS is assumed to change in vertical direction caused by double eccentric shaft vibration exciter. The eccentric mass and eccentric distance are the same, which are symmetry in y direction. When the two shafts are synchronous reverse rotation, the centrifugal forces in x direction offset each other and the component forces superpose each other in y direction, which forms a harmonic excitation. The Fig. 1 shows the external excitation model at any moment. Based on the above analysis, the external excitation in y direction can be defined as: 2

F=

2

2 ∑ (F0i + Fyi ) = ∑ (F0i + m 0rω i 0 sin ω0t ) i =1

= F0 + Fp sin ω0t

i =1

Fig. 1. The external excitation model.

3

(1)

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Fig. 2. Schematic of the CVBVS.

2.2. Equation of motion In order to analyze the CVBVS, a 2-DOF vehicle model coupled to a harmonically excited Euler–Bernoulli beam resting on nonlinear foundation is illustrated in Fig. 2. As seen in Fig. 2, the coupled dynamic equations of the CVBVS can be derived by Lagrange's equation. After many transformations and manipulations, the dynamic equations are given by: ∂Yb(X , t ) 3 ) ∂t 3 −cs1(Y2̇ − Y1̇ ) − cs2(Y2̇ − Y1̇ ) − 3 cs1(Y2̇ − Y1̇ ) + cs2(Y2̇ − Y1̇ ) + ks1(Y2

m1Y1̈ + ct1(Y1̇ − m 2Y2̈ +

∂Y 2(X , t ) ρA b 2 ∂t

+ cb

∂Yb(X , t ) ) ∂t

∂Yb(X , t ) ∂t

+ ct2(Y1̇ −

+ EI

∂Yb4(X , t ) 4 ∂X

+ k t1(Y1 − Yb(X , t )) + k t2(Y1 − Yb(X , t ))3 ks1(Y2 − Y1) − ks2(Y2 − Y1)3 = 0 − Y1) + ks2(Y2 − Y1)3 = 0

+ kb1Yb(X , t ) + kb2Yb3(X , t )

=Fin(X , t )δ (X − B ) + Fout (X , t )δ (X − X0 )

(2)

where δ(X-X0) is the Dirac delta function. Fin(X, t) is the interaction force between vehicle and beam. Fout(X, t) represents the external harmonic excitation. Therefore, the Dirac delta function and the excitations can be written: +∞

∫−∞

F (X )δ (X − X0 )dX = F (X0 )

Fin(X , t ) = − ct1(Y1̇ −

∂Yb(X , t ) ) ∂t

(3)

− ct2(Y1̇ −

∂Yb(X , t ) 3 ) ∂t

− k t1(Y1 − (Yb(X , t ))) − k t2(Y1 − (Yb(X , t )))3 − (m1 + m 2 )g

= − (m1 + m 2 )g + m1Y1̈ + m 2Y2̈ Fout (X , t ) = F0 + Fp sin(ω0t )

(4)

In order to improve the accuracy of the numerical calculation, the following dimensionless variables and parameters are defined as follows:

x=

X , L

y1 =

Y1 , L

y2 =

δ (x − b ) = Lδ (X − B ),

Y2 , L

yb =

Yb , L

ω=

ω0 , Ω

τ = Ωt ,

Ω=

δ (x − x 0 ) = Lδ (X − X0 )

1 L2

EI ρA

(5)

In this study, the Galerkin method is used to discretize the coupled equations of motion, and the vibration displacement yb(x, τ) of beam can be written in terms of generalized coordinate as follows: n

yb(x, τ ) =

∑ φr (x )ybr (τ )

(6)

r =1

where, φr(x) is the rth mode of the beam and φr(x)=sin(rπx), ybr(τ) is the vibration displacement of the rth mode, which is determined by solving an ordinary differential equation system. n represents the total number of models used for the beam consideration. The Galerkin scheme is applied by substituting Eq. (6) into Eq. (2) and multiplying the resultant equation by the corresponding eigenfunction as well as integrating over the interval of 0 and 1. The governing equations of dimensionless nonlinear vibration for the CVBVS are obtained: 4

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y1̈ +

ct1 (y ̇ m1Ω 1

n

− ∑i =1 φy ̇ )+ i bi

c

− ms1Ω (y2̇ − y1̇ ) − 1

y2̈ +

cs1 (y ̇ m2Ω 2

− y1̇ ) +

k b2L2 ⎛

n

cs2ΩL2 (y2̇ m1

cs2ΩL2 (y2̇ m2

⎞ n ⎛ 1 ∑i =1 ⎜∫ φφ dx y ̈ + i j ⎟ bi 0 ⎠ ⎝ +

ct2ΩL2 (y1̇ m1

cb ρAΩ

3

n

− ∑i =1 φy ̇ ) + i bi

− y1̇ )3 −

− y1̇ )3 +

ks1 m1Ω 2

ks1 m2Ω 2

(y2 − y1) −

(y2 − y1) +

⎞ n ⎛ 1 ∑i =1 ⎜∫ φφ dx y ̇ + i j ⎟ bi 0 ⎠ ⎝

1

⎞3

1

⎠

ρAΩ 2L2

y dx = ∫ φφ i j bi ⎟

⎜∑ ρAΩ 2 ⎝ i =1 0

⎡ 1 −⎢ 2 2 (m1 + m 2 )g + ⎣ ρAΩ L

1 (m1y1̈ ρAL

k t1 m1Ω 2

EI ρAΩ 2L 4

n

(y1 − ∑i =1 φy )+ i bi ks2L2 m1Ω 2

ks2L2 m2Ω 2

k t2L2 m1Ω 2

n

(y1 − ∑i =1 φy ) i bi

3

(y2 − y1)3 = 0

(y2 − y1)3 = 0

⎞ n ⎛ 1 ∑i =1 ⎜∫ φi′′′ ′φj dx ⎟ybi + 0 ⎠ ⎝

k b1 ρAΩ 2

⎞ n ⎛ 1 ∑i =1 ⎜∫ φφ dx y i j ⎟ bi 0 ⎠ ⎝

ω

[F0 + Fp sin( Ω0 τ )]φj (x 0 )

⎤ + m 2y2̈ )⎥φj (b ) ⎦

j = 1, 2, ⋯n

(7)

where, the parameters are defined as follows,

kbN =

kb2L2 ρAΩ

2

, μ=

Fp m2 , f = , Δx = b − x 0 ρAL p ρAL2Ω 2

(8)

Eq. (7) describe a nonlinear dynamic system, and the internal excitation Fin is an important effect factor between vehicle and bridge. The Runge-Kutta is used to solve approximate solutions of the CVBVS with parameterized nonlinear equations.

Fig. 3. Vibration responses of the y2 and yb at n=1 (- -), n=2 (o) and n=3 (—).

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Table 1 Parameters of the coupled vehicle-bridge vibration system. Item Beam Length Young's modulus Cross sectional area Moment of inertia Density Viscous damping Beam stiffness Vehicle Unsprung mass Sprung mass Suspension stiffness Suspension damping Tire stiffness Tire damping Load Amplitude Frequency

Notation

Value

L E A I ρ cb kb1/kb2

18 m 2.0×1011 N/m 7.7×10−3 m2 3.5×10−4 m4 7850 kg/m3 3.0×102 Ns/m2 4.8×107 N/m2, 3.0×106 N/m4

m1 m2 ks1/ks2 cs1/cs2 kt1/kt2 ct1/ct2

364 kg 1776 kg 2.8×106 N/m, 4.0×104 N/m, 3.7×107 N/m, 2.0×104 N/m,

F0 ω0

5×104 N 20 Hz

3.0×105 N/m3 5.0×104 N/m3 4.0×106 N/m3 1.0×104 N/m3

Fig. 4. Simulation condition schematic.

3. Nonlinear dynamic analysis of the CVBVS with different parameters 3.1. Convergence and verification The n in Eq. (7) represents the term of Galerkin truncation, which is an important parameter and affects the accuracy and stability of the coupled system. When n is smaller, the response of the vibration system will lose the stable value. When n is bigger, the computation time will be increased with the numerical iteration method. Therefore, the n is not the bigger the better. To study the convergence for the number of modes of the steady-state response, Ding [29] detailedly analyzed the convergence of the Galerkin method for the beam. Therefore, the effect of modes (n) considered in our work is discussed simply. The dynamic response with n=1, n=2 and n=3 are presented in Fig. 3. When only the first model (n=1) of the coupled system is considered, the vibration responses of y2 and yb are smaller relatively. For higher number of modes, namely, the first two models (n=2) and the first three modes (n=3), the dynamic responses of the coupled system display almost the same behavior. It can be seen from Fig. 3 that the first two modes (n=2) consideration can reasonably reflect the dynamic behaviors with adequate accuracy. Hence, in what follows, the analysis will be conducted with a first two modes of beam, namely, n=2. In the previous section, the dynamic model of CVBVS is described and the dynamic equations of motion are derived. Following subsections investigate the influences for several of the important factors affecting the dynamic responses of CVBVS, namely, nonlinear support stiffness kbN, mass ratio μ, excitation amplitude fp and position relation Δx. The bifurcation diagrams and 3-D 6

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Fig. 5. Bifurcation diagram and 3-D frequency spectrum of CVBVS at ω=1.95.

Fig. 6. Quasi-periodic motion at kbN=55.955×1010; (a) Waveform, (b) Frequency, (c) Phase diagram and (d) Poincaré map.

frequency spectrums are plotted and examined. The vibration responses mean generalized coordinate of sprung mass, y2, and the first generalized coordinate of the transverse motion, yb1, are demonstrated. The geometrical and physical parameters of the CVBVS are presented in Table 1, and a detailed simulation condition schematic is shown in Fig. 4.

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Fig. 7. Chaotic motion at kbN=103.075×1010; (a) Waveform, (b) Frequency, (c) Phase diagram and (d) Poincaré map.

3.2. The effect of nonlinear support stiffness kbN on dynamic response The nonlinear support stiffness (kbN) has an important influence on the energy transform from beam to vehicle system. In order to investigate the nonlinear dynamic properties of CVBVS with the nonlinear support stiffness, a constant value for all parameters expect the nonlinear support stiffness, and the corresponding bifurcation diagrams and 3-D frequency spectrums are displayed with kbN as control parameter in the interval [2.945, 206.15]×1010 at ω=1.58, ω=1.95 and ω=3.15, respectively. Fig. 5 displays the bifurcation diagrams and 3-D frequency spectrums for the vibration responses of the sprung mass y2 and the beam center yb1 at ω=1.95. It is worthwhile noting that the CVBVS shows complicated nonlinear phenomena with different nonlinear support stiffness. In the interval [2.945, 4.418]×1010, the coupled system presents periodic-5 motion and only five points are shown in bifurcation diagram. In corresponding 3-D frequency spectrum, the frequency multiplication components (f, 2f, 3f, 4f) appear. As the nonlinear support stiffness is increased further, namely 5.89×1010 < kbN < 17.67×1010, the system shows chaotic behavior. In Fig. 5(b2), the system presents a continuous frequency components, and the excitation frequency f is the main component. Increasing the kbN, periotic-2 motion occurs in the range of [17.67, 27.978]×1010, expect a few points where the motions become quasi-periodic and periodic-8 motions. By further increasing the nonlinear support stiffness, the CVBVS exists various types of motions, including periodic-1, periodic-3, periodic-5, periodic-6 and quasi-periodic motions are displayed in the interval [29.45, 57.428]×1010. The vibration responses of the CVBVS at kbN=55.955×1010 are displayed in Fig. 6. In can be seen from figures that the coupled system is quasi-periodic motion, and the f and 2 f are the main frequency components. The quasi-periodic can be verified by a closed circle in the Poincaré map as shown in Fig. 6(d1). With increasing kbN from 58.90×1010 to 117.8×1010, the system motion is from chaotic in the interval [58.9, 83.933]×1010, through periodic-3 motion in the range of [83.933, 89.833]×1010, to return to chaotic motion at

Fig. 8. Periodic-2 motion at kbN=17.67×1010; (a) Waveform, (b) Frequency, (c) Phase diagram and (d) Poincaré map.

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Fig. 9. Bifurcation diagrams and 3-D frequency spectrums of CVBVS at ω=1.58 and ω=3.15.

89.823×1010 < kbN < 117.8×1010. The waveform, frequency, phase diagram and Poincaré map at kbN=103.075×1010 are illustrated in Fig. 7, where it is clear that the system shows chaotic characteristic, and the continuous frequency components exists in Fig. 7(b1). The Poincaré map appears scattered points. Beyond that, a dominate periodic-1 motion is observed in the range of [117.8, 170.81]×1010, expect at kbN=117.8×1010 and kbN=119.273×1010, where the coupled system displays quasi-periodic and periodic-5 characteristics. Increasing the nonlinear support stiffness even further, the typical characteristic of periodic-2 motion is shown in the range of [172.283, 206.15]×1010, expect a few points where it presents periodic-4 motion. The related periodic-2 motion at kbN=176.7×1010 are shown in Fig. 8 by waveform, frequency, phase diagram and Poincaré map. The foregoing discussion concluded that the effect of the nonlinear support stiffness is a key parameter affecting the dynamic behaviors of the CVBVS. In order to investigate the significance effect on the nonlinear support stiffness under different rotational speed, the bifurcation diagrams and 3-D frequency spectrums of CVBVS at ω=1.58 and ω=3.15 are shown in Fig. 9. It is worthwhile

Table 2 Main frequency components comparison with different nonlinear support stiffness. Speed

Variation range×1010

Main frequency component

ω=1.58

2.945≤kbN≤20.615 20.615 < kbN≤50.065 50.065 < kbN≤67.735 67.735 < kbN≤197.315 197.315 < kbN≤206.15 2.945≤kbN≤14.725 14.725 < kbN≤26.505 26.505 < kbN≤41.23 41.23 < kbN≤117.8 117.8 < kbN≤173.755 173.755 < kbN≤206.15 2.945≤kbN≤206.15

f/3, 2f/3, f, 4f/3,5f/3, 2f, 7f/5, 8f/3, 3f, 10f/3, 4f, 5f, 6f f, 2f, 3f, 4f, 5f, 6f f, 2f, 3f, 4f, 5f, 6f and frequency demultiplication f, 2f, 3f, 4f, 5f, 6f f, 2f, 5f/2, 3f, 7f/2, 4f, 9f/2, 5f, 11f/2, 6f f, 2f, 3f, 4f f/2, f, 3f/2, 2f, 5f/2, 3f, 7f/2, 4f f, 2f, 3f, 4f f/3, f, 5f/3, 2f, 7f/5, 8f/3, 3f f, 2f, 3f, 4f f/2, f, 3f/2, 2f, 5f/3, 3f, 7f/2, 4f, 9f/2 f, 2f, 3f

ω=1.95

ω=3.15

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noting that the coupled system presents strong nonlinear characteristics with periodic motion, quasi-periodic motion, chaotic motion and the jump discontinuous phenomenon. In Fig. 9(a1) at ω=1.58, the coupled system exhibits chaotic motion in the range of [2.945, 11.78]×1010. As the nonlinear support stiffness is increased from kbN=11.78×1010 to kbN=22.088×1010, the system displays a simple periodic-3 motion thereafter with a periodic-1 motion in the interval [22.088, 48.593]×1010, where system exists obviously jump discontinuous phenomenon. As kbN is increased further, the quasi-periodic motion and chaotic motion are observed in the interval [50.065, 61.845]×1010. Then the coupled system presents different types of motions, such as periodic-1 motion, periodic-2 motion, periodic-3 motion and periodic-4 motion in the range of [63.318, 159.03]×1010 where the motion repeatedly jumps from one to the other. When the nonlinear support stiffness is increased from 160.503×1010 to 182.59×1010, the coupled system is from chaotic motion in the interval [160.503, 164.92]×1010, through periodic-1 motion in the range of [166.393, 175.228]×1010, to return to chaotic motion at 176.7×1010 < kbN < 182.59×1010. Whereafter, a direct transition to periodic-2 motion in the interval [184.063, 206.15]×1010. The bifurcation diagram and 3-D frequency spectrum of yb1 at ω=3.15 are displayed in Fig. 9(a2) and (b2). It is found that, compared to the previous conditions, the appearance of chaotic motion lags, and the chaotic region relative to the whole range of nonlinear support stiffness decreases, and the extent of the chaotic motion weaken. In the interval [2.945, 7.363]×1010, quasi-periodic motion is observed followed by a periodic-5 motion at kbN=7.363×1010. As the nonlinear support stiffness is increase further, the coupled system presents periodic-1 motion and chaotic motion in the range of [8.835, 48.593]×1010 where the motion exchanges alternately from one to the other. After a range of periodic-1 motion in the interval [50.065, 133.998]×1010, the system shows periodic-4 motion dominantly in the range of [136.943, 159.03]×1010, and the motion frequently changes from one to the other. When kbN > 159.03×1010, typical characteristic of quasi-periodic motion is presented. In addition, the main frequency components of the CVBVS are listed in Table 2. Comparing the dynamic responses of the CVBVS with varying rotational speed, under the low value of kbN, the higher order bifurcations increase, and the jump discontinuous phenomenon, periodic, quasi-periodic and chaotic motions frequently exchange. With the increasing of kbN, the vibration amplitude decreases, and the complex dynamic characteristics vanish gradually. In addition, the chaotic behavior lags, and the jump discontinuity phenomenon weakens with increasing rotational speed.

Fig. 10. Bifurcation diagram and 3-D frequency spectrum of CVBVS at ω=1.95.

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Fig. 11. Quasi-periodic at μ=3.45×10−2; (a) Waveform, (b) Frequency, (c) Phase diagram and (d) Poincaré map.

Fig. 12. Periodic-3 motion at μ=5.573×10−2; (a) Waveform, (b) Frequency, (c) Phase diagram and (d) Poincaré map.

3.3. The effect of mass ratio μ on dynamic response Due to the different carry passengers or freightage, the mass ratio between vehicle and bridge is an important parameter to evaluate performance of the coupled system. In this section, in order to give a detailed investigation of the CVBVS. A parametric study is performed for various mass ratio in the interval [1.6, 9.0]×10−2. And the other system parameters remain the same. The corresponding bifurcation diagrams and 3-D frequency spectrums of the CVBVS respect to mass ratio are shown in Fig. 10 at ω=1.95. With the increasing of the mass ratio from 1.6×10−2 to 1.858×10−2, the coupled system displays a large amplitude chaotic motion. As the mass ratio is increased, the motion of the coupled system becomes dominantly periodic-1 with an obvious jump discontinuity at μ=2.123×10−2 in the range of [1.858, 3.397]×10−2, expects a few points, namely, μ=1.911×10−2 and μ=1.964×10−2, where the motions become periodic-8 and periodic-5. For the mass ratio in the range of [3.45, 4.352]×10−2, the system shows quasiperiodic and chaotic motion in the interval [3.45, 3.822]×10−2, and it is replaced by periodic motion at 3.822×10−2 < μ < 3.981×10−2, finally, the chaotic motion in the range of [3.981, 4.352]×10−2 appears. The corresponding dynamic responses of the CVBVS are illustrated in Fig. 11 at μ=3.45×10−2 by waveform, frequency, phase diagram and Poincaré map, which indicates that the CVBVS displays a quasi-periodic motion. Increasing the mass ratio causes the motion of the coupled system to become simple periodic-1 in the range of [4.406, 4.883]×10−2 and quasi-periodic at 4.883×10−2 < μ < 5.149×10−2. Increasing the mass ratio further causes the motion to be dominate periodic-3 in the interval [5.149, 6.741]×10−2, expect for a few points where the system displays quasi-periodic motion at μ=5.945×10−2, μ=6.211×10−2 and μ=6.317×10−2. The related vibration responses of the CVBVS at μ=5.573×10−2 are shown in Fig. 12. It can be seen from figures that the system presents typical periodic-3 motion. As the mass ratio is increased further, the coupled system displays much richer dynamics. The motion is dominantly chaotic characteristics in the 11

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Fig. 13. Chaotic motion at μ=6.847×10−2; (a) Waveform, (b) Frequency, (c) Phase diagram and (d) Poincaré map.

Fig. 14. Bifurcation diagrams and 3-D frequency spectrums of CVBVS at ω=1.15 and ω=3.15.

range of [6.794, 7.059]×10−2, expect a periodic-1 motion at μ=7.006×10−2. And then the system shows different types of motion including periodic-4 at μ=7.113×10−2 and μ=7.166×10−2, periodic-1 in the range of [7.219, 9.0]×10−2 and periodic-9 at μ=8.758×10−2, where the periodic-1 motion is the predominant response. Fig. 13 shows the dynamic responses of the coupled system at μ=6.847×10−2. From figures, it indicates that the CVBVS is chaotic motion. Based on the previous investigation, it is worthwhile noting that the mass ratio μ is a significant parameter affecting the dynamic

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Table 3 Main frequency component comparison with different mass ratio. Speed

Variation range×10−2

Main frequency component

ω=1.15

1.60≤μ≤3.362 3.362 < μ≤4.6 4.6 < μ≤5.131 5.131 < μ≤6.193 6.193 < μ≤7.962 7.962 < μ≤8.67 1.60≤μ≤1.946 1.946 < μ≤3.539 3.539 < μ≤6.546 6.546 < μ≤8.67 1.60≤μ≤2.831 2.831 < μ≤3.362 3.362 < μ≤3.715 3.715 < μ≤4.07 4.07 < μ≤8.67

f, 2f, 3f, 4f, 5f, 6f, 7f, 8f and frequency demultiplication f, 2f, 3f, 4f, 5f, 6f, 7f, 8f f/2, f, 3f/2, 2f, 5f/2, 3f, 7f/2, 4f, 9f/2, 5f, 11f/2, 6f, f, 2f, 3f, 4f, 5f, 6f, 7f, 8f f/3, 2f/3, f, 4f/3, 5f/3, 2f, 7f/3, 8f/3, 3f, 10f/3, 11f/3, 4f f, 2f, 3f, 4f, 5f, 6f, 7f, 8f f/6, 4f/6, 5f/6, f, 7f/6, 10f/6, 11f/6, 2f, 13f/6,3f, etc. f, 2f, 3f, 4f f/3, 2f/3, f, 4f/3, 5f/3, 2f, 7f/3, 8f/3, 3f, etc. f, 2f f/4, 2f/4, 3f/4, f, 2f, 3f, etc. f/3, 2f/3, f, 4f/3, 5f/3, 2f, 7f/3, 8f/3, 3f f, 2f, 3f f, 2f, 3f and frequency demultiplication. f, 2f, 3f

ω=1.95

ω=3.15

Fig. 15. Bifurcation diagram and 3-D frequency spectrum of CVBVS at ω=1.95.

behaviors. In this section, the dynamic responses of the coupled system will be analyzed at ω=1.15 and ω=3.15. The corresponding bifurcation diagrams and 3-D frequency spectrums of the yb1 are shown in Fig. 14. The vibration responses of the coupled system exhibit rich nonlinear characteristics. In Fig. 14(a1), the system presents markedly chaotic characteristic in the intervals [2.07, 2.442]×10−2, [2.972, 3.503]×10−2 and [6.423, 7.431]×10−2 at ω=1.15. With the increasing of the rotational speed, the chaotic region narrows down and the periodic motion increases. Namely, the chaotic motion exists in the intervals [1.964, 2.123]×10−2, [3.822, 4.087]×10−2 and [6.688, 6.741]×10−2 at ω=3.15. In addition, it is observed that the exchange frequency of periodic, quasi-periodic and chaotic motions decreases. Under the condition of higher mass ration, the system appears periodic motion earlier with the

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Fig. 16. Period motion at fp=0.926×10−2; (a) Waveform, (b) Frequency, (c) Phase diagram and (d) Poincaré map.

Fig. 17. Quasi-periodic motion at fp=1.388×10−2; (a) Waveform, (b) Frequency, (c) Phase diagram and (d) Poincaré map.

increasing rotational speed. Comparing the 3-D frequency spectrums in Figs. 14 and 10, the frequency components have obvious changes, and the amplitude increases with the increasing rotational speed. In order to illustrate the frequency components of the system detailedly, the frequency features under different ω are listed in Table 3. Taking into consideration the vibration characteristics of the CVBVS with different rotational speed, the region of the chaotic motion relative to the whole mass ratio diminishes, but the exchange frequency of the motions strengthen with the increase of the rotational speed. At low value of μ, the rotational speed has an important effect on the dynamic response. By further increase in mass ratio no significant changes occur, and the simple periodic-1 motion of the beam. 3.4. The effect of excitation amplitude fp on dynamic response The external excitation is the main energy source, which has a significance effect on the dynamic behaviors of the CVBVS. The effects of excitation amplitude on the vibration characteristics of the CVBVS under different rotational speed are analyzed and other system parameters are fixed. The procedure is continued by varying the excitation amplitude regarded as the control parameter. Fig. 15 presents the bifurcation diagrams and 3-D frequency spectrums of CVBVS at ω=1.95 for y2 and yb1. In the interval [0.5, 0.926]×10−2, the coupled system displays periodic-1 motion, expect for a few points where the periodic-8 motion at fp=0.5×10−2 and quasi-periotic motion at fp=0.547×10−2 and fp=0.757×10−2 are presented. The related dynamic responses of the coupled system at fp=0.926×102 are illustrated in Fig. 16, which demonstrates that the coupled system is in a periodic-1 motion at this excitation amplitude. As the excitation amplitude is increased, the motion of the coupled system is the periodic-3 in the range of [0.968, 14

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Fig. 18. Chaotic motion at fp=2.609×10−2; (a) Waveform, (b) Frequency, (c) Phase diagram and (d) Poincaré map.

1.136]×10−2. Further increase of the excitation amplitude results in different types motion including periodic-6, quasi-periodic, and the responses repeatedly jump from one to the other at 1.136×10−2 < fp < 1.557×10−2. The waveform, frequency, phase diagram and Poincaré map of the CVBVS at fp=1.388×10−2 are displayed in Fig. 17, which demonstrates the coupled system undergoes quasiperiodic motion. Increasing the excitation amplitude even further, the motion of the CVBVS shows chaotic characteristic in the interval [1.599, 1.683]×10−2, after a narrow periodic motion in the range of [1.725, 1.809]×10−2, the motion of the CVBVS returns

Fig. 19. Bifurcation diagrams and 3-D frequency spectrums of CVBVS at ω=1.15 and ω=3.15.

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Table 4 Main frequency component comparison with different excitation amplitude. Speed

Variation range×10−2

Main frequency component

ω=1.15

0.5≤fp≤1.094 1.094 < fp≤2.525 2.525 < fp≤4.292 4.292 < fp≤4.965 4.965 < fp≤5.7 0.5≤fp≤0.673 0.673 < fp≤1.431 1.431 < fp≤1.851 1.851 < fp≤2.525 2.525 < fp≤3.45 3.45 < fp≤5.7 0.5≤fp≤3.0 3.0 < fp≤4.544 4.544 < fp≤5.7

f, 2f, 4f f, 2f, 3f, 4f, 5f, 6f f, 2f, 3f, 4f, 5f, 6f, 7f, 8f and continuous frequency f, 2f, 3f, 4f, 5f, 6f, 7f, 8f f/3, 2f/3, f, 4f/3, 5f/3, 2f, 7f/3, 8f/3, 3f, 10f/3, 11f/3, 4f etc. f/2, f, 3f/2, 2f, 5f/2, 3f f, 2f, 3f f/3, 2f/3, f, 4f/3, 5f/3, 2f, 7f/3, 8f/3, 3f f, 2f, 3f f/2, f, 3f/2, 2f, 5f/3, 3f, 7f/2, 4f, 9f/2 f, 2f, 3f f, 2f, 3f f, 2f, 3f and continuous frequency f, 2f, 3f

ω=1.95

ω=3.15

to chaotic response. In the interval [2.062, 2.482]×10−2, the periodic-1 motion is observed, expect a point where it is periodic-4 motion at fp=2.146×10−2. By further increasing the excitation amplitude, the response amplitude increases and the system displays chaotic characteristic in the range of [2.525, 3.366]×10−2, expect a point where the system becomes periodic motion at fp=2.945×10−2. Beyond that, the motion of the coupled system returns to periodic-1 and maintains that until fp=4.081×10−2. The related dynamic responses of the CVBVS at fp=2.609×10−2 are shown in Fig. 18. From the figures, it is found that the system presents chaotic motion. In the interval [4.123, 5.386]×10−2, the quasi-periodic motion is the predominate response. In addition, periodic-2 motion and periodic-3 motion are observed, and the motions frequency change from one to the other. Further increase of the excitation amplitude, the system returns to periodic-1 motion. In order to have a complete investigation of the CVBVS with the effect on excitation amplitude, the bifurcation diagrams and 3-D frequency spectrums of the coupled system with respect to excitation amplitude fp under different rotational speed conditions are displayed in Fig. 19. At the first glance, it is seen that the coupled system presents a simple dynamic characteristics at ω=1.15 and ω=3.15. In Fig. 19 (a1), when the rotational speed is low the second order resonance frequency (about ω=1.95), the coupled system shows periodic-1 motion in the interval [0.5, 2.483]×10−2. As the excitation amplitude is increased, it can be clearly found that the periodic motion, quasi-periodic motion and chaotic motion exist, and the motions frequently change from one to the other at 2.525×10−2 < fp < 2.945×10−2. Beyond that, the system turns into chaotic motion in the range of [2.945, 4.208]×10−2. Further increase of the excitation amplitude, after a narrow interval [4.25, 4.418]×10−2 where the coupled system displays predominantly quasi-periodic motion, the motion of the CVBVS becomes periodic-1 motion in the range of [4.418, 4.965]×10−2. When fp > 4.965×10−2, the system exhibits different types of motions, such as chaotic motion, quasi-periodic motion, periodic-1 motion and periodic-3 motion. In Fig. 19 (a2), in the range of [0.5, 1.893]×10−2, the coupled system shows a simple periodic-1 motion followed by a periodic-3 motion at fp=0.968×10−2 and quasi-periodic motion at fp=1.01×10−2. As the excitation amplitude is increased further, the coupled system follows the periodic-1 motion route to quasi-periodic motion in the internal [1.935, 3.029]×10−2, expect a point where the motion is periodic-5 at fp=4.754×10−2. Increasing the excitation amplitude causes the system to display predominantly chaotic behavior at 3.715×10−2 < fp < 4.46×10−2. The coupled system only exhibits periodic-1 motion, and the system experiences different jump discontinuous phenomena. In addition, the frequencies of the CVBVS are shown in Table 4. Analyzing the dynamic responses by varying rotational speed in this section, depending on the system parameters, the CVBVS displays different types of motions with fp as control parameter. The chaotic region relative to the whole range of excitation amplitude decreases, but the extent of the chaotic motion strength with the increasing rotational speed. Moreover, as the rotational speed closes to second order natural frequency, the dynamic behaviors of coupled system becomes more complex, and the periodic motion, quasi-periodic motion and chaotic motion more frequently exchange (Table 5).

Table 5 Main frequency component comparison with different position relation.

ω=1.07 ω=1.95

ω=2.98

Variation range

Main frequency component

−0.5 < Δx < −0.367 −0.367 < Δx < 0 −0.5 < Δx < −0.383 −0.383 < Δx < −0.317 −0.317 < Δx < 0 −0.5 < Δx < −0.35 −0.35 < Δx < −0.183 −0.183 < Δx < −0.133 −0.133 < Δx < 0

f, 2f, 3f, 4f, f, 2f, 3f, 4f, f, 2f, 3f, 4f f/6, 5f/6, f, f/3, 2f/3, f, f/2, f, 3f/2, f, 2f, 3f f/3, 2f/3, f, f, 2f, 3f

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5f 5f, 6f, 7f, 8f 7f/6, 11f/6, 2f, 13f/6, 3f, 19f/6, 4f, etc. 4f/3, 5f/3, 2f, 7f/3, 8f/3, 3f, 10f/3, 11f/3, 4f, etc. 2f, 5f/2, 3f 4f/3, 5f/3, 2f, 7f/3, 8f/3, 3f

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Fig. 20. Bifurcation diagram and 3-D frequency spectrum of CVBVS at ω=1.95.

3.5. The effect of position relation Δx on dynamic response In this section, the effects of position relation between vehicle and excitation on the dynamic behaviors of the CVBVS are investigated. The corresponding bifurcation diagrams and 3-D frequency spectrums of the coupled system respect to position relation under different rotational speed are shown in Fig. 20. It can be observed that the figures are symmetric base on the center of

Fig. 21. Quasi-periodic motion Δx=−0.367; (a) Waveform, (b) Frequency, (c) Phase diagram and (d) Poincaré map.

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Fig. 22. Period-3 motion at Δx=−0.117; (a) Waveform, (b) Frequency, (c) Phase diagram and (d) Poincaré map.

bridge. Therefore, the dynamic responses of the position relation in the range of [−0.5, 0] will be illustrated. For away from the center of beam, it can be seen that the coupled system presents periodic-1 motion in the interval [−0.5, −0.375], and only one point is shown in the bifurcation diagram for every position. As the distance is decreased between the vehicle and excitation, the quasiperiodic and chaotic responses are observed in the interval [−0.367, −0.258]. Fig. 21 presents the quasi-periodic motion at Δx=−0.367 via waveform, frequency, phase diagram and Poincaré map. Decreasing the distance sequentially, the coupled system shows different types dynamic responses, namely periodic-3 motion, periodic-7 motion and quasi-periodic motion, which frequently change from one to the other at −0.25 < Δx < −0.183. And then the motion of the system is replaced by chaotic with large amplitude in a narrow range of [−0.183, −0.133]. As the distance is decreased further, the motion becomes periodic-3 in the interval [−0.125, −0.1]. A simple periodic-3 motion is observed at Δx=−0.117 in Fig. 22. Beyond that, the coupled system follows the quasi-periodic motion route to chaotic motion at −0.092 < Δx < −0.033. Furthermore, the CVBVS exhibits a dominated periodic motion in the interval [−0.025, 0] followed by a quasi-periodic motion at Δx=0. In Fig. 23, the related dynamic responses of the CVBVS at Δx=−0.083 are demonstrated. It can be found that the coupled system presents typical chaotic motion. Based on the above analysis, it is can be found that the position relation between vehicle and harmonic excitation is an significant parameter influencing the dynamic behaviors of the CVBVS, and the vibration responses of the coupled system will be investigated under different rotational speed. The bifurcation diagrams and 3-D frequency spectrums of CVBVS with respect to position relation in different rotational are shown in Fig. 24. By comparing the previous analysis result, it is worthwhile noting that the system exhibits strong nonlinear characteristics with the effect on the position relation between vehicle and excitation. The amplitudes jump discontinuity and the alternation among periodic motion, quasi-periodic motion and chaotic motion are more frequent. And the appearance of the chaotic characteristic occurs earlier, the chaotic region relative to the whole range of Δx increases, and the

Fig. 23. Chaotic motion at Δx=−0.083; (a) Waveform, (b) Frequency, (c) Phase diagram and (d) Poincaré map.

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Fig. 24. Bifurcation diagrams and 3-D frequency spectrums of CVBVS at ω=1.07 and ω=2.98.

amplitude of chaotic motion strengthens with the increasing rotational speed. 4. Conclusions In this paper, the nonlinear dynamic characteristics of a parametrically excited coupled vehicle-bridge vibration system are investigated. The influences of nonlinear support stiffness (kbN), mass ratio (μ), excitation amplitude (fp) and position relation (Δx) on the dynamic behaviors are studied by Runge-Kutta method. The results of this paper are concluded as follows: (1) For the coupled system, the jump discontinuous phenomenon, periodic motion, quasi-periodic motion and chaotic behavior are exhibited, and the complex vibration and periodic vibration exchange alternately. Furthermore, it can be concluded that when the system is near the two times the critical speed, the bifurcation diagrams and 3-D frequency spectrums show more complex dynamic behaviors involving wide variety of motions. In addition, the path to periodic motion through quasi-periodic motion or chaotic motion is observed. (2) Under the low value of nonlinear support stiffness and mass ratio, the higher order bifurcations increase, the jump discontinuous phenomenon, periodic, quasi-periodic and chaotic motions frequently exchange. With the increasing of control parameters (kbN and μ), the vibration amplitudes decrease, and the complex dynamic characteristics vanish gradually. In addition, the chaotic behavior lags, and the jump discontinuity weakens with increasing rotational speed. The region of the chaotic motion relative to the whole control parameters diminishes, but the exchange frequency of the motions strengthens with the increasing of the rotational speed. (3) As mentioned above, with increasing excitation amplitude and decreasing distance, the CVBVS is prone to involve periodic motion, quasi-periodic response and chaotic behavior, and the extent of chaotic motion strengths with the increase rotational speed. In addition, the chaotic region relative to the whole range of excitation amplitude decreases, but the decreasing distance has a contrary effect of the excitation amplitude for the chaotic region.

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Acknowledgements The authors would like to thank the Collaborative Innovation Center of Major Machine Manufacturing in Liaoning, National Support Program, the key common technology research and demonstration of paving equipment for subgrade in alpine (No. 2015BAF07B07). The authors also gratefully acknowledge the support from the National Natural Science Foundation of China (No. 51475084). References [1] H.B. Xiao, M.X. Fang, Chaos in nonlinearity considered 4-degree automobile system, J. Dyn. Control. 6 (2008) 377–380. [2] M. Borowiec, A.K. Sen, G. Litak, J. Hunicz, G. Koszałka, A. Niewczas, Vibrations of a vehicle excited by real road profiles, Forsch. Ing. 74 (2010) 99–109. [3] M. Borowiec, G. Litak, Transition to chaos and escape phenomenon in two-degrees-of-freedom oscillator with a kinematic excitation, Nonlinear Dyn. 70 (2012) 1125–1133. [4] W.A. Lindsey, A.M. Spagnuolo, J.C. Chipman, M. Shillor, Numerical simulations of vehicle platform stabilization, Math. Comput. 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