# Study of Suspension Parameters Matching to Enhance Vehicle Ride Comfort on Bump Road

Authors: Jin Gao; Xiaoping Qi
Journal: Shock and Vibration
 (2021)

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## Abstract

In this study, the parameters of the MacPherson front suspension and the E-type multilink rear suspension are matched to enhance the vehicle ride comfort on bump road. Vehicle vibration and suspension stiffness are analyzed theoretically. In the simulation study, the influence of the front and rear wheels on the vehicle vibration is considered, so the time-domain curves of the front and rear seat rail accelerations are processed by adding windows with two different window functions. The resulting ΔRmsLocal and ΔRmsGlobal are used as evaluation indexes of the vehicle ride comfort. The sensitivity analysis yields the magnitude of the influence of the suspension parameters on the evaluation indexes. In addition, the trends of ΔRmsLocal and ΔRmsGlobal with bushing stiffness at different vehicle speeds are discussed. The results show that longitudinal ΔRmsLocal and ΔRmsGlobal of the seat rails are influenced by the bushings mostly, while the vertical ΔRmsLocal and ΔRmsGlobal of the seat rails are influenced by the spring and shock absorber mostly. The trends of ΔRmsLocal and ΔRmsGlobal with bushing stiffness are influenced by the speed of the vehicle. Finally, the vehicle ride comfort is enhanced after optimization and matching of the suspension parameters by NSGA-II optimization algorithm.


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## Body

## 1. Introduction

Vehicle ride comfort refers to keeping the impact of vibrations and shocks within certain ranges when the vehicle is excited by the road. The study of vehicle ride comfort has always been a subject of interest for scholars.Several scholars have carried out extensive theoretical studies on vehicle ride comfort [1–5]. Wang et al. [1] developed a frequency-based modelling approach for vehicles equipped with a hydraulic interconnected suspension (HIS) system. Guntur et al. [2] focused on the effect of the asymmetric characteristics of the damping forces on the ride comfort of a typical passenger car under different road excitations. Jiao et al. [4] proposed a 4-DOF dynamic model for the two-axle heavy truck to investigate the low-frequency vibration characteristics of the suspension system. Zhao et al. [6] established a vibration simulation model of a three-axle goods vehicle passing through a speed bump. The vibration characteristics of the vehicle at different speeds were studied through the simulation, the vibration decay time at different speeds was obtained, and the minimum distance between the speed bump and the vehicle scale was determined.Several scholars have conducted a lot of studies to improve vehicle ride comfort [7–20]. Qi et al. [8] proposed a new type of suspension including a hydraulically interlinked suspension and electronically controlled air springs, enabling a high level of performance in terms of both handling stability and ride comfort. Tan et al. [9] proposed a pitch-roll-interconnected hydropneumatic suspension system to improve the stability and attenuate the vibration for the lying patients. Ebrahimi-Nejad et al. [10] derived the vibration control equations for the vehicle suspension system and compared the effects of different damping coefficients and spring stiffnesses on the vehicle ride comfort. Finally, multiobjective optimization was carried out with the TOPSIS method to improve the ride comfort of the vehicle. Li et al. [11] developed a 10-DOF model considering the lying patient and the driver to investigate the ride comfort of the ambulance. After multiobjective optimization, vehicle ride comfort bump and random road have been improved. Kaldas et al. [12] developed a novel optimization technique for optimizing the damper top mount characteristics to improve vehicle ride comfort. Based on the Bolza-Meyer criterion, Phu [17] proposed a new optimal control law with sliding mode control. The proposed controller is applied to a vehicle seat suspension system with magneto-rheological damper to evaluate vibration control performances. Through simulation studies, the proposed optimal controller is shown to have the advantages of less power and faster convergence. He et al. [18] developed and simulated a multibody dynamics model based on nonlinear damping and equivalent damping and optimized with an optimization method based on nonlinear damping and intelligent algorithms to improve the vehicle ride comfort. Phu et al. [19] designed a new controller based on a modified Riccati-like equation. Simulation experiments are carried out on both bump and random road. The results show that the controller can control the acceleration and displacement at the driver position effectively. Phu et al. [20] proposed a novel adaptive control method to deal with the dead zone and time delay issues in actuators of vibration control systems. The unwanted vibrations due to external excitations are well controlled despite of the presence of dead zone and time delay in actuators.Many studies by scholars have focused on the vibration of vehicles on random road. Global and local vibrations are not considered separately in the study of vibration characteristics. Most of the studies improve vehicle ride comfort only by matching spring stiffness of suspension and damping coefficient of shock absorber. The optimization of vehicle ride comfort based on specific suspension structures has not yet been well explored.This study takes into account the following aspects: (a) taking a full vehicle model with a MacPherson front suspension and an E-type multilink rear suspension as an example, the influence of the bushing is analyzed from a force perspective, and the contribution of bushing stiffness to suspension stiffness is discussed. (b) The vehicle is simulated on bump road. Local and global vibrations are considered, so the time-domain curves of the front and rear seat rails are windowing with different window functions and the resulting ΔRmsLocal and ΔRmsGlobal are used as evaluation indexes. (c) Sensitivity analysis resulted in the most influential suspension parameters for ΔRmsLocal and ΔRmsGlobal. The conclusions resulting from the theoretical analysis are verified. The trend of ΔRmsLocal and ΔRmsGlobal with bushing stiffness is further discussed. (d) Considering the coupling of the full vehicle vibration, the suspension parameters are matched reasonably at different vehicle speeds using the NSGA-II algorithm.The paper is organized as follows: Section2 develops a vehicle vibration model and discusses the ride comfort characteristics of the vehicle. Section 3 analyzes the influence of suspension’s parameters on suspension’s stiffness. Section 4 verifies the conclusions from part two and part three through simulation. In Section 5, the front and rear suspension parameters are matched by an optimization algorithm to improve the ride comfort of the vehicle. Section 6 presents the conclusions of this study.


## 2. Vehicle Vibration Characteristics Analysis

### 2.1. Vertical Longitudinal Vibration Model

Road excitation may cause vertical, pitch, and longitudinal vibrations of the vehicle. The vertical longitudinal coupling model [21] is shown in Figure 1. The model consists of sprung mass represented by the body mb, unsprung mass represented by the front axle and wheel m1, rear axle, and wheel m2. There is relative vertical and longitudinal movement between the body and the wheels. The body can pitch around the y axes, Iy representing the pitch inertia of the body around the y axe. Keqfz and Keqrz are the vertical equivalent stiffnesses of the front and rear suspensions, respectively; Keqfx and Keqrx are the longitudinal equivalent stiffnesses of the front and rear suspensions, respectively; Beqfz and Beqrz are the vertical equivalent damping coefficients of the front and rear suspensions, respectively; Beqfx and Beqr are the longitudinal equivalent damping coefficients of the front and rear suspensions, respectively; Q is the road excitation; Ztf and Ztr are the vertical displacements of the front and rear wheels, respectively; Xtf and Xtr are the longitudinal displacements of front and rear wheels, respectively; Zf and Zr are the vertical displacements of the front and rear body, respectively; Xf and Xr are the longitudinal displacements of the front and rear body, respectively; Zb is the vertical displacement at the centre of mass of the body; Xb is the longitudinal displacement at the centre of mass of the body.Figure 1
Coupling longitudinal vertical vehicle model.Neglecting the damping of the tyres, the forces between the wheels and the body consist of elastic and damping forces. According to the Lagrange equation, which is expressed as shown in equation (1), the equation of vehicle coupling vibration can be derived(1)ddt∂L∂q˙i−∂L∂qi=−∂R∂q˙i,i=1,2,3,…n.L is a Lagrangian function(2)L=T−U,where T,U, and R are the kinetic energy, potential energy, and dissipative energy of the system, respectively.Generalized coordinates are defined as(3)XtfXtrZtfZtrXbZbφ.Generalized velocity is defined as(4)X˙tfX˙trZ˙tfZ˙trX˙bZ˙bφ˙.Based on the above analysis of the vehicle model, the potential energy of vehicle vibration can be expressed by the stiffness characteristics of the system(5)U=12KtzfZtf−qzf2+KtzrZtr−qzr2+KeqfZf−Ztf2+KeqrZr−Ztr2+KtxfXtf−qxf2+KtxrXtr−qxr2+KeqfXf−Xtf2+KeqrXr−Xtr2.The kinetic energy of the vehicle vibration can be found as(6)T=12mbZ˙b2+mbX˙b2+Iyφ˙2+m1Z˙tf2+m2Z˙tr2+m1X˙tf2+m2X˙tr2.The dissipation energy of vehicle vibrations can be expressed by the damping characteristics of the system(7)R=12BeqfZ˙f−Z˙tf2+BeqrZ˙r−Z˙tr2+BeqfX˙f−X˙tf2+BeqrX˙r−X˙tr2.The geometric constraints of the model satisfy the following equations:(8)Zf=Zb−Xb−Xtf+aφ,Z˙f=Z˙b−Xb−Xtf+aφ˙,Zr=Zb+Xb−Xtr+bφ,Z˙r=Z˙b+Xb−Xtr+bφ˙,Xf=Xb−Zb−Ztf+hbwφ,Z˙f=X˙b−Zb−Ztf+hbwφ˙,Xr=Xb−Zb−Ztr+hbwφ,Z˙f=X˙b−Zb−Ztr+hbwφ˙,where hbw is the wheel’s centre and vehicle body’s centre of gravity when the wheel is not in deformation.Substituting equations (2) and (5)–(8) into (1) yields the differential equation of vehicle vibration.(9)m1X¨tf+BeqfxX˙tf−BeqfxX˙b−Beqfxc1φ˙+KeqfxXtf−KeqfxXb−Keqfxc1φ+KtxfXtf−qxf=0,m2X¨tr+BeqrxX˙tr−BeqrxX˙b−Beqrxc2φ˙+KeqrxXtr−KeqrxXb−Keqrxc2φ+KtxrXtr−qxr=0,m1Z¨tf+BeqfzZ˙tf−BeqfzZ˙b+Beqfzb1φ˙+KeqfzZtf−KeqfzZb+Keqfzb1φ+KtzfZtf−qzf=0,m2Z¨tr+BeqrzZ˙tr−BeqrzZ˙b−Beqrzb2φ˙+KeqrzZtr−KeqrzZb−Keqrzb2φ+KtzrZtr−qzr=0,mbX¨b+Beqfx+BeqfxX˙b−Beqfxc1+Beqrxc2φ˙−BeqfxX˙tf−BeqrxX˙tr+Keqfx+KeqrxXb−Keqfxc1+Keqrxc2φ−KeqfxXtf−KeqrxXtr=0,mbZ¨b+Beqfz+BeqfzZ˙b+Beqrzb2−Beqfzb1φ˙−BeqfzZ˙tf−BeqrzZ˙tr+Keqfz+KeqrzZb+Keqrzb2−Keqfzb2φ−KeqfzZtf−KeqrzZtr=0,Iyφ¨+Beqfzb12+Beqrzb22+Beqfxc12+Beqrxc22φ˙+BeqrzZ˙b−Z˙trb2−BeqfzZ˙b−Z˙tfb1+BeqfxX˙tf−X˙bc1+BeqrxX˙tr−X˙bc2+Keqfzb12+Keqrzb22+Keqfxc12+Keqrxc22φ+KeqrzZb−Ztrb2−KeqfzZb−Ztfb1+KeqfxXtf−Xbc1+KeqrxXtr−Xbc2=0,where(10)b1=Xb−Xtf+a,b2=Xb−Xtr+b,c1=Zb−Ztf+hbw,c2=Zb−Ztr+hbw.Writing (9) in matrix form yields(11)MP¨+CP˙+KP=KtQ,where M is the mass matrix; K is the stiffness matrix; C is the damping matrix; and Kt is the stiffness matrix of the tyre.(12)M=diagm1m2m1m2mbmbIy,K=Keqfx000−Keqfx0−Keqfxc10Keqrx00−Keqrx0−Keqrxc200Keqfz00−KeqfzKeqfzb1000Keqrz0−Keqrz−Keqrzb2−Keqfx−Keqrx00Keqfx+Keqfx0−Keqfxc1+Keqrxc200000Keqfz+KeqfzKeqrzb2−Keqfzb1Keqfxc1Keqrxc2Keqfzb1−Keqrzb2−Keqfxc1−Keqrxc2Keqrzb2−Keqfzb1Keqfzb12+Keqrzb22+Keqfxc12+Keqrxc22,C=Beqfx000−Beqfx0−Beqfxc10Beqrx00−Beqrx0−Beqrxc200Beqfz00−BeqfzBeqfzb1000Beqrz0−Beqrz−Beqrzb2−Beqfx−Beqrx00Beqfx+Beqfx0−Beqfxc1+Beqrxc200000Beqfz+BeqfzBeqrzb2−Beqfzb1Beqfxc1Beqrxc2Beqfzb1−Beqrzb2−Beqfxc1−Beqrxc2Beqrzb2−Beqfzb1Beqfzb12+Beqrzb22+Beqfxc12+Beqrxc22,Kt=KtxfKtxrKtzfKtzr000T.A Laplace variation of equation (11) yields the transfer function matrix of the system vibration(13)Gs=Ms2+Cs+K−1Kt.Substitutings=jωj=−1 into equation (13) yields the frequency response function matrix of the vibration response(14)Hjω=K−Mω2+jωC−1Kt.


### 2.2. Vehicle Longitudinal and Vertical Vibrations

The frequency response characteristics of the coupled model can be found through equation (14). If the coupling between the front and rear masses of the vehicle is small, the study of vehicle vibration characteristics can be carried out separately for the front and rear suspensions.The response characteristics of the vehicle vibration can be analyzed using the single wheel model shown in Figure2.Figure 2
Two-degree-of-freedom two-mass vibration model: (a) vertical; (b) longitudinal.
(a)(b)Figure2(a) shows the 2-DOF vertical vibration model of the vehicle. m1 and m2 are the sprung and unsprung masses, respectively; Beqz and Keqz are the equivalent damping coefficient and equivalent stiffness of the suspension, respectively; Ktz is the vertical stiffness of the tyre.The vibrations of the system can be represented by the displacements of these two masses, so giving two coupled differential equations:(15)m1Z¨1+BeqzZ˙1−Z˙2+KeqzZ1−Z2=0,m2Z¨2−BeqzZ˙1−Z˙2−KeqzZ1−Z2+KtzZ2−Z3=0.The frequency response functions of the tyre dynamic loadFd, the suspension dynamic deflection σd, and the acceleration Z¨1 of the suspension vertical vibration response to the road excitation velocity Z¨3 can be calculated, respectively.(16)fZ¨1∼Z˙3=jωrk+2rkξλj2ω1−λ21+rk−λ2/rm−1+rk−1+rm/rmλ22ξλj,fσd∼Z˙3=rk−λ2rk+2ξλrkjjω1−λ21+rk−λ2/rm−1+2jωξλrk+1+rmλ2/rmω,fFd∼Z˙3=Ktzrk−λ2rk+2ξλrkjjω1−λ21+rk−λ2/rm−1−rk−1+rm/rmλ22ξλω,where(17)rk=KtzKeqz,rm=m1m2,ω0=Keqzm1,λ=ωω0,ξ=Beqz2Keqzm1,where rk is the vertical stiffness ratio; ξ is the damping ratio in the vertical direction; rm is the mass ratio; ω0 is the vertical natural frequency of the suspension; and λ is the frequency ratio.Figure2(b) shows the 2-DOF longitudinal vibration model of the vehicle. Ktx is the longitudinal stiffness of the tyre; m1′ is considered to be only the front unsprung mass. The rear unsprung mass can be considered to be fixed to the sprung mass and thus equal to m2′ [5]; Keqx is the equivalent longitudinal stiffness of the suspension. Similar to the vertical vibration, the differential equation of vehicle longitudinal vibration can be expressed as(18)m2′X¨3−BeqxX˙2−X˙3−KeqxX2−X3=0,m1′X¨2+BeqxX˙2−X˙3+KeqxX2−X3=KtxX1−X2.The frequency response functions of the accelerationX¨3 to the road excitation velocity X˙1 can be calculated(19)fX¨3∼X˙1=jωrk′−2rrk′ξ′λ′ω1−λ′21+rk′−λ′2/rm′−1+rk′−1+rm′/rm′λ′22ξ′λ′j,where(20)rk′=KtxKeqx,rm′=m2′m1′,ω0′=Keqxm2′,λ′=ω′ω0′,ξ′=Beqx2Keqxm2′,where rk′ is the longitudinal stiffness ratio; ξ′ is the longitudinal damping ratio; rm′ is the mass ratio; ω0′ is the longitudinal natural frequency of the suspension; and λ′ is the frequency ratio.From equation (16), the frequency response function of vertical vibration shows that the vertical natural frequency and damping ratio of suspension make an impact on response of the vertical vibration acceleration, the suspension dynamic deflection, and the tyre dynamic load. From equation (19), it can be seen that the longitudinal natural frequency and damping ratio of suspension make an impact on the response of longitudinal vibration acceleration.From equation (17), it can be seen that the vertical natural frequency and vertical damping ratio of the suspension are influenced by the vertical stiffness of the suspension. From equation (20), it can be seen that the longitudinal natural frequency and longitudinal damping ratio of the suspension are influenced by the longitudinal stiffness of the suspension. From equation (9) it can be seen that the front and rear suspension vibration characteristics make an impact on the pitching behaviour of the vehicle.Consequently, well-matched suspension stiffness and damping can enhance the vehicle ride comfort. In this study, a full vehicle model with the MacPherson front suspension and the E-type multilink rear suspension is used as an example to analyze the suspension stiffness and to optimize the vibration characteristics.


## 2.1. Vertical Longitudinal Vibration Model

Road excitation may cause vertical, pitch, and longitudinal vibrations of the vehicle. The vertical longitudinal coupling model [21] is shown in Figure 1. The model consists of sprung mass represented by the body mb, unsprung mass represented by the front axle and wheel m1, rear axle, and wheel m2. There is relative vertical and longitudinal movement between the body and the wheels. The body can pitch around the y axes, Iy representing the pitch inertia of the body around the y axe. Keqfz and Keqrz are the vertical equivalent stiffnesses of the front and rear suspensions, respectively; Keqfx and Keqrx are the longitudinal equivalent stiffnesses of the front and rear suspensions, respectively; Beqfz and Beqrz are the vertical equivalent damping coefficients of the front and rear suspensions, respectively; Beqfx and Beqr are the longitudinal equivalent damping coefficients of the front and rear suspensions, respectively; Q is the road excitation; Ztf and Ztr are the vertical displacements of the front and rear wheels, respectively; Xtf and Xtr are the longitudinal displacements of front and rear wheels, respectively; Zf and Zr are the vertical displacements of the front and rear body, respectively; Xf and Xr are the longitudinal displacements of the front and rear body, respectively; Zb is the vertical displacement at the centre of mass of the body; Xb is the longitudinal displacement at the centre of mass of the body.Figure 1
Coupling longitudinal vertical vehicle model.Neglecting the damping of the tyres, the forces between the wheels and the body consist of elastic and damping forces. According to the Lagrange equation, which is expressed as shown in equation (1), the equation of vehicle coupling vibration can be derived(1)ddt∂L∂q˙i−∂L∂qi=−∂R∂q˙i,i=1,2,3,…n.L is a Lagrangian function(2)L=T−U,where T,U, and R are the kinetic energy, potential energy, and dissipative energy of the system, respectively.Generalized coordinates are defined as(3)XtfXtrZtfZtrXbZbφ.Generalized velocity is defined as(4)X˙tfX˙trZ˙tfZ˙trX˙bZ˙bφ˙.Based on the above analysis of the vehicle model, the potential energy of vehicle vibration can be expressed by the stiffness characteristics of the system(5)U=12KtzfZtf−qzf2+KtzrZtr−qzr2+KeqfZf−Ztf2+KeqrZr−Ztr2+KtxfXtf−qxf2+KtxrXtr−qxr2+KeqfXf−Xtf2+KeqrXr−Xtr2.The kinetic energy of the vehicle vibration can be found as(6)T=12mbZ˙b2+mbX˙b2+Iyφ˙2+m1Z˙tf2+m2Z˙tr2+m1X˙tf2+m2X˙tr2.The dissipation energy of vehicle vibrations can be expressed by the damping characteristics of the system(7)R=12BeqfZ˙f−Z˙tf2+BeqrZ˙r−Z˙tr2+BeqfX˙f−X˙tf2+BeqrX˙r−X˙tr2.The geometric constraints of the model satisfy the following equations:(8)Zf=Zb−Xb−Xtf+aφ,Z˙f=Z˙b−Xb−Xtf+aφ˙,Zr=Zb+Xb−Xtr+bφ,Z˙r=Z˙b+Xb−Xtr+bφ˙,Xf=Xb−Zb−Ztf+hbwφ,Z˙f=X˙b−Zb−Ztf+hbwφ˙,Xr=Xb−Zb−Ztr+hbwφ,Z˙f=X˙b−Zb−Ztr+hbwφ˙,where hbw is the wheel’s centre and vehicle body’s centre of gravity when the wheel is not in deformation.Substituting equations (2) and (5)–(8) into (1) yields the differential equation of vehicle vibration.(9)m1X¨tf+BeqfxX˙tf−BeqfxX˙b−Beqfxc1φ˙+KeqfxXtf−KeqfxXb−Keqfxc1φ+KtxfXtf−qxf=0,m2X¨tr+BeqrxX˙tr−BeqrxX˙b−Beqrxc2φ˙+KeqrxXtr−KeqrxXb−Keqrxc2φ+KtxrXtr−qxr=0,m1Z¨tf+BeqfzZ˙tf−BeqfzZ˙b+Beqfzb1φ˙+KeqfzZtf−KeqfzZb+Keqfzb1φ+KtzfZtf−qzf=0,m2Z¨tr+BeqrzZ˙tr−BeqrzZ˙b−Beqrzb2φ˙+KeqrzZtr−KeqrzZb−Keqrzb2φ+KtzrZtr−qzr=0,mbX¨b+Beqfx+BeqfxX˙b−Beqfxc1+Beqrxc2φ˙−BeqfxX˙tf−BeqrxX˙tr+Keqfx+KeqrxXb−Keqfxc1+Keqrxc2φ−KeqfxXtf−KeqrxXtr=0,mbZ¨b+Beqfz+BeqfzZ˙b+Beqrzb2−Beqfzb1φ˙−BeqfzZ˙tf−BeqrzZ˙tr+Keqfz+KeqrzZb+Keqrzb2−Keqfzb2φ−KeqfzZtf−KeqrzZtr=0,Iyφ¨+Beqfzb12+Beqrzb22+Beqfxc12+Beqrxc22φ˙+BeqrzZ˙b−Z˙trb2−BeqfzZ˙b−Z˙tfb1+BeqfxX˙tf−X˙bc1+BeqrxX˙tr−X˙bc2+Keqfzb12+Keqrzb22+Keqfxc12+Keqrxc22φ+KeqrzZb−Ztrb2−KeqfzZb−Ztfb1+KeqfxXtf−Xbc1+KeqrxXtr−Xbc2=0,where(10)b1=Xb−Xtf+a,b2=Xb−Xtr+b,c1=Zb−Ztf+hbw,c2=Zb−Ztr+hbw.Writing (9) in matrix form yields(11)MP¨+CP˙+KP=KtQ,where M is the mass matrix; K is the stiffness matrix; C is the damping matrix; and Kt is the stiffness matrix of the tyre.(12)M=diagm1m2m1m2mbmbIy,K=Keqfx000−Keqfx0−Keqfxc10Keqrx00−Keqrx0−Keqrxc200Keqfz00−KeqfzKeqfzb1000Keqrz0−Keqrz−Keqrzb2−Keqfx−Keqrx00Keqfx+Keqfx0−Keqfxc1+Keqrxc200000Keqfz+KeqfzKeqrzb2−Keqfzb1Keqfxc1Keqrxc2Keqfzb1−Keqrzb2−Keqfxc1−Keqrxc2Keqrzb2−Keqfzb1Keqfzb12+Keqrzb22+Keqfxc12+Keqrxc22,C=Beqfx000−Beqfx0−Beqfxc10Beqrx00−Beqrx0−Beqrxc200Beqfz00−BeqfzBeqfzb1000Beqrz0−Beqrz−Beqrzb2−Beqfx−Beqrx00Beqfx+Beqfx0−Beqfxc1+Beqrxc200000Beqfz+BeqfzBeqrzb2−Beqfzb1Beqfxc1Beqrxc2Beqfzb1−Beqrzb2−Beqfxc1−Beqrxc2Beqrzb2−Beqfzb1Beqfzb12+Beqrzb22+Beqfxc12+Beqrxc22,Kt=KtxfKtxrKtzfKtzr000T.A Laplace variation of equation (11) yields the transfer function matrix of the system vibration(13)Gs=Ms2+Cs+K−1Kt.Substitutings=jωj=−1 into equation (13) yields the frequency response function matrix of the vibration response(14)Hjω=K−Mω2+jωC−1Kt.


## 2.2. Vehicle Longitudinal and Vertical Vibrations

The frequency response characteristics of the coupled model can be found through equation (14). If the coupling between the front and rear masses of the vehicle is small, the study of vehicle vibration characteristics can be carried out separately for the front and rear suspensions.The response characteristics of the vehicle vibration can be analyzed using the single wheel model shown in Figure2.Figure 2
Two-degree-of-freedom two-mass vibration model: (a) vertical; (b) longitudinal.
(a)(b)Figure2(a) shows the 2-DOF vertical vibration model of the vehicle. m1 and m2 are the sprung and unsprung masses, respectively; Beqz and Keqz are the equivalent damping coefficient and equivalent stiffness of the suspension, respectively; Ktz is the vertical stiffness of the tyre.The vibrations of the system can be represented by the displacements of these two masses, so giving two coupled differential equations:(15)m1Z¨1+BeqzZ˙1−Z˙2+KeqzZ1−Z2=0,m2Z¨2−BeqzZ˙1−Z˙2−KeqzZ1−Z2+KtzZ2−Z3=0.The frequency response functions of the tyre dynamic loadFd, the suspension dynamic deflection σd, and the acceleration Z¨1 of the suspension vertical vibration response to the road excitation velocity Z¨3 can be calculated, respectively.(16)fZ¨1∼Z˙3=jωrk+2rkξλj2ω1−λ21+rk−λ2/rm−1+rk−1+rm/rmλ22ξλj,fσd∼Z˙3=rk−λ2rk+2ξλrkjjω1−λ21+rk−λ2/rm−1+2jωξλrk+1+rmλ2/rmω,fFd∼Z˙3=Ktzrk−λ2rk+2ξλrkjjω1−λ21+rk−λ2/rm−1−rk−1+rm/rmλ22ξλω,where(17)rk=KtzKeqz,rm=m1m2,ω0=Keqzm1,λ=ωω0,ξ=Beqz2Keqzm1,where rk is the vertical stiffness ratio; ξ is the damping ratio in the vertical direction; rm is the mass ratio; ω0 is the vertical natural frequency of the suspension; and λ is the frequency ratio.Figure2(b) shows the 2-DOF longitudinal vibration model of the vehicle. Ktx is the longitudinal stiffness of the tyre; m1′ is considered to be only the front unsprung mass. The rear unsprung mass can be considered to be fixed to the sprung mass and thus equal to m2′ [5]; Keqx is the equivalent longitudinal stiffness of the suspension. Similar to the vertical vibration, the differential equation of vehicle longitudinal vibration can be expressed as(18)m2′X¨3−BeqxX˙2−X˙3−KeqxX2−X3=0,m1′X¨2+BeqxX˙2−X˙3+KeqxX2−X3=KtxX1−X2.The frequency response functions of the accelerationX¨3 to the road excitation velocity X˙1 can be calculated(19)fX¨3∼X˙1=jωrk′−2rrk′ξ′λ′ω1−λ′21+rk′−λ′2/rm′−1+rk′−1+rm′/rm′λ′22ξ′λ′j,where(20)rk′=KtxKeqx,rm′=m2′m1′,ω0′=Keqxm2′,λ′=ω′ω0′,ξ′=Beqx2Keqxm2′,where rk′ is the longitudinal stiffness ratio; ξ′ is the longitudinal damping ratio; rm′ is the mass ratio; ω0′ is the longitudinal natural frequency of the suspension; and λ′ is the frequency ratio.From equation (16), the frequency response function of vertical vibration shows that the vertical natural frequency and damping ratio of suspension make an impact on response of the vertical vibration acceleration, the suspension dynamic deflection, and the tyre dynamic load. From equation (19), it can be seen that the longitudinal natural frequency and damping ratio of suspension make an impact on the response of longitudinal vibration acceleration.From equation (17), it can be seen that the vertical natural frequency and vertical damping ratio of the suspension are influenced by the vertical stiffness of the suspension. From equation (20), it can be seen that the longitudinal natural frequency and longitudinal damping ratio of the suspension are influenced by the longitudinal stiffness of the suspension. From equation (9) it can be seen that the front and rear suspension vibration characteristics make an impact on the pitching behaviour of the vehicle.Consequently, well-matched suspension stiffness and damping can enhance the vehicle ride comfort. In this study, a full vehicle model with the MacPherson front suspension and the E-type multilink rear suspension is used as an example to analyze the suspension stiffness and to optimize the vibration characteristics.


## 3. Stiffness Characteristics Analysis of Suspensions

### 3.1. Stiffness of the Suspension

Neglecting the influence of the bushings and antiroll bar on the suspension stiffness, the suspension stiffness satisfies the following relationship:(21)Keq=Ksi2,where Keq is the suspension stiffness; Ks is the spring stiffness; and i is the installation ratio.Taking McPherson suspension as an example, equation (21) is described in detail.Neglecting the mass of the suspension strut and the control arm, considering the vertical displacement of the vehicle only, the quarter-vehicle model of the MacPherson suspension can be simplified to the model shown in Figure3 [22]. A is the connection point between the MacPherson suspension strut and the control arm; B is the connection point between the control arm and the body. The global coordinate system is established with the point B as the reference point, Z-axis and Y-axis along with the vertical and horizontal directions, respectively.Figure 3
Suspension forces schema for determining equivalent spring force.In a MacPherson suspension, the position and orientation of the control arm determine the relative movement between the wheel assembly and the chassis. The movement of the system can be seen as a rotational movement around a transient centreO. F1 is the tangential force acting on the control arm at point A; F2 is the spring force acting on the control arm at point A; β is the angle between F1 and F2; F4 is the tangential force acting at point C; F3 is the equivalent spring force acting at the point C; α is the angle between F3 and F4; lOA and lOC are the lengths of point A and point C to point R, respectively; Z1 and Z2 are the vertical displacements of the wheel and the suspension, respectively.The following relationships can be obtained:(22)F1=F2cosβ,F4=F3cosα,F1lOA=F4lOC,lOAlOC=Z2cosβZ1cosα,F3=Z2Z1cosβcosα2F2.In the linear range, the deformation is proportional to the force applied(23)F3=KeqZ1,F2=KsZ2.The linear equivalent stiffnessKeq and installation ratio i can be obtained from equations (22) and (23)(24)Keq=Z2Z1cosβcosα2Ks,i=Z2Z1cosβcosα.From equation (24), it can be seen that the vertical stiffness Keq of the suspension is influenced by the spring stiffness Ks and the installation ratio i.For a multilink rear suspension, the vertical stiffness also satisfies equation (21).The above analysis does not consider the stiffness of the bushings, whereas, for suspension systems, elastic elements such as bushings are installed to absorb impacts. The vertical stiffness of the suspension is influenced by the spring stiffness and bushing stiffness. The deformation of the bushing makes an impact on the installation ratioi. As for the longitudinal stiffness of the suspension, in addition to being influenced by the longitudinal component of the suspension springs and shock absorber, it is mainly influenced by the stiffness of the bushings. Therefore, the suspension will be analyzed specifically with considering the stiffness of the bushings. In the actual driving of a vehicle, the excitation signals transmitted to the bushings are mainly of low-frequency excitation. Therefore, the influence of bushing stiffness on vehicle ride comfort can be studied by considering the static stiffness of the bushing only.This paper presents the further analysis of the bushings by establishing the static equilibrium equation of the suspension.


### 3.2. Static Equilibrium Equations for MacPherson Front Suspension

Figure4 shows the schematic diagram of the MacPherson suspension. The control arm and steering tie rod are connected to the wheel knuckle by ball joint. Control arm is connected to the body by bushings. When an external load is applied to the tyre contact point, the bushings and spring of suspension will deform, and the tie rod will move to a new position, thus achieving a new static equilibrium.Figure 4
Schematic of MacPherson suspension.This section will calculate the forces and moments of the bushings with the methods in reference [23].The relative displacement of the bushing’s inner tube to the bushing’s outer tube is required. For the bushings of the MacPherson suspension control arm, the displacement of the outer tube can be considered as zero as the outer tube of the bushing is fixed to the body. Therefore, the displacement of the inner tube is only required. Firstly, taking pointC as a reference point to derive the new coordinates of the inner tube of A and B,(25)A∗=XαYβZγA0−C0+C∗,B∗=XαYβZγB0−C0+C∗,where A∗,B∗, and C∗ are the new coordinates of the inner tube of bushings A,B, and C in the global coordinate system, respectively; A0,B0, and C0 are the initial coordinates of the inner tube of bushings A,B, and C in the global coordinate system, respectively; Xα,Yβ, and Zγ are the 3×3 rotation matrixes when the control arm is rotated by an angle of α,β, and γ about the x-, y-, and z-axes of its rigid coordinate system, respectively.The new coordinates of the ball jointsD and G, which are connected to the wheel knuckle, and the new coordinates of point P1 can also be estimated using point C as a reference point.(26)D∗=XαkYβkZγkD0−C0+C∗,G∗=XαkYβkZγkG0−C0+C∗,P1∗=XαkYβkZγkP10−C0+C∗.where Xαk,Yβk. and Zγk are the 3×3 rotation matrixes when the wheel knuckle is rotated by an angle of αk,βk, and γk about the x-, y-, and z-axes of its rigid coordinate system, respectively.Thus, the translational deformation of bushingA and bushing B in the global coordinate system rA′ and rB′ is expressed as(27)rA′rB′=A∗−A0B∗−B0+rA′0rB′0,where rA′0 and rA′0 are the initial translational displacements of bushings A and B in the global coordinate system.The forces acting on the bushings due to the displacements of bushingsA and B in the global coordinate system can be expressed as(28)FAFB=KA,t00KB,trA′rB′+Δ¯B,tΔ¯C,t,where Δ¯B,t and Δ¯C,t are the force-displacement correction vectors of bushing A and bushing B in the global coordinate system, respectively; KA,t and KB,t are the linear stiffness matrixes for bushing A and bushing B in the global coordinate system, respectively.KA,t can be calculated from the linear stiffness matrix K¯A,t of the bushing A in the local coordinate system. KB,t can be calculated from the linear stiffness matrix K¯B,t of the bushing B in the local coordinate system.(29)KA,t=RSTK¯A,tRS,KB,t=RSTK¯B,tRS.RS is the transformation matrix from the local to the global coordinate system of the bushings A and B.The rotational deformationsθA′ and θB′ of the bushings A and B in the global coordinate system can be expressed as(30)θA′θB′=θAθB+θA0θB0,where θA and θB are rotation angles at load input and θA0θB0 are rotation initial displacements.The moments of the bushingA and bushing B in the global coordinate system can be calculated(31)MAMB=KA,r00KB,rθA′θB′+Δ¯A,rΔ¯B,r,where Δ¯A,r and Δ¯B,r are the moment-angle correction vectors of bushing A and bushing B in the global coordinate system, respectively; KA,r and KB,r are the linear translational stiffness matrixes of bushing A and bushing B in the global coordinate system, respectively.KA,r can be calculated from the linear stiffness matrix K¯A,r of the bushing A in the local coordinate system. KB,r can be calculated from the linear stiffness matrix K¯B,r of the bushing B in the local coordinate system.(32)KA,r=RSTK¯A,rRS,KB,r=RSTK¯B,rRS.According to equations (28) and (31), the reaction forces and moments acting on the control arms of bushing A and bushing B can be derived as follows:(33)RFiRMi=I0R˜I−Fi−Mi.Subscripti represents bushings A and B; R˜ is the (3 × 3) skew-symmetric matrix of the position vector for bushings A and B(34)R˜=0zyz0−x−yx0,where x,y, and z represent the displacements of the bushing in the global coordinate system in the directions of the x-, y-, and z-axes, respectively.The control arm is in equilibrium under the reaction forces, the reaction moments of the bushings, and the ball joints of the wheel knuckle, and therefore the following relationships can be obtained:(35)RFA+RFB+FC=0,RMA+RMB=0.The wheel knuckle is in equilibrium under the reaction forces and moments from the control arms, the tie rod, and the external load at the tyre contact patch and therefore satisfies the following relationships:(36)FP1+RFC+RFD+RFG=0,MP1+RMD+RMG=0,where MP1 is the external force input to the tyre from the road; RFG and RMG are the spring reaction force and reaction moment, respectively.Since the length of the steering tie rod is constant during the suspension kinematics, a constraint can be given(37)LED′−LED=0.The spring deformation satisfies the following constraints:(38)LGH′+ΔLGH−LGH=0,where LGH′ is the new length of the spring; LGH is the initial length of the spring; and ΔLGH is the displacement of the spring. ΔLGH can be calculated by(39)ΔLGH=fsLED′−fs0LGHKs,where fs0 is the initial scale factor of spring force; fs is the new scale factor of spring force; and Ks is the stiffness of spring.According to (35)–(38), the static equilibrium equation for the front suspension can be expressed as(40)RFA+RFB+FCRMA+RMBFP1+RFC+RFD+RFGMP1+RMD+RMGLED′−LEDLGH′+ΔLGH−LGH=0.There are 14 equations in equation (40) and the number of unknowns is 14. Therefore, equation (40) can be solved


### 3.3. Static Equilibrium Equations for E-Type Multilink Rear Suspensions

Figure5 shows a diagram of the E-type multilink rear suspension. S1 is the upper control arm; S2 is the rear lower control arm; S3 is the front lower control arm; S4 is the trailing arm; Bii=1∼3 are bushings for the connecting rod to the body; Aii=1∼3 are the bushings for connecting the wheel knuckle to the rod; A4 is the bushing for the trailing arm to the body.Figure 5
Schematic of MacPherson suspension.This section will calculate the forces and moments of the bushings with the methods in reference [23, 24].The new position after the bushing movement was calculated first. Since the outer tube of bushingAi is connected to the wheel knuckle and the inner tube connected to the rod, both the inner and outer tubes of bushing Ai will move. The displacement of the centre point of both the inner and outer tubes needs to be taken into account. As the outer tubes of bushing Bi and bushing A4 are fixed to the body, the displacement at the centre of the outer tube will be considered as zero and the displacement at the centre of the inner tube of the bushing will be taken into account.The centre points of the inner and outer tubes of the bushing coincide at the initial position.Ai0i=1∼3 is the coordinate of the centre point of the inner and outer tubes of bushing Aii=1∼3 in the global coordinate system at the initial position. Bi0 is the coordinate of the centre point of the inner and outer tubes of the bushing Bii=1∼3 in the global coordinate system at the initial position.ChooseBi0 as the reference point and calculate the new coordinates of the centre point of the inner tube Ali∗ of the bushing Ai in the global coordinate system.(41)Ali∗=XαlYβlZγlAi0−Bi0+Bli∗,where Bli∗ is the new coordinate of the centre point of the inner tube of bushing Bi in the global coordinate system; Xαl,Yβl and Zγl are the 3×3 rotation matrixes when the rod is rotated by an angle of αl,βl, and γl about the x-, y-, and z-axes of its rigid coordinate system, respectively.ChooseP1 as the reference point and calculate the new coordinates of the centre point of the inner tube Al4∗ of bushing A4 in the global coordinate system.(42)Aki∗=XαkYβkZγkAi0−P1+P1∗,where P1∗ is the new coordinate of the centre point P1 in the global coordinate system.By choosingP1 as the reference point, the new coordinates of the inner tube Al4∗ of the bushing A4 in the global coordinate system can be calculated(43)Al4∗=XαkYβkZγkAl40−P1+P1∗,where Xαk,Yβk, and Zγk are the 3×3 rotation matrixes when the wheel knuckle is rotated by an angle of αk,βk, and γk about the x-, y-, and z-axes of its rigid coordinate system, respectively.The new coordinates of the centre points of the inner and outer tubes of the bushing can be calculated by equations (41)–(43).Based on the preceding analysis, the translational and rotational deformations of bushingBi in the global coordinate system can be expressed as(44)rBi′θB′=Bli∗−Bi0θBli+rBi′0θBi′0,where θBli is the rotational angle of bushing Bi in the global coordinate system under the force of the rod; rBi′0 and θBi′0 are the initial translational displacements and the initial rotational angles of bushing Bi.The forces and moments applied to bushingBi in the global coordinate system can be expressed as(45)FBliMBli=KBi,t00KBi,rrBi′θBi′+Δ¯Bi,tΔ¯Bi,r,where Δ¯Bi,t and Δ¯Bi,r are the moment-angle and the force-displacement correction vectors for bushing Bi in the global coordinate system, respectively; KBi,t and KBi,r are the linear translational stiffness matrixes for bushing Bi in the global coordinate system, respectively.KBi,t and KBi,r can be calculated from the linear stiffness matrices K¯Bi,t,K¯Bi,r of bushing Bi in the local coordinate system.(46)KBi,t=RS′TK¯Bi,tRS′,KBi,r=RS′TK¯Bi,rRS′.RS′ is the (3 × 3) transformation matrix from the global coordinate system to the local coordinate system of bushing Bi.The reaction forcesRFBli and moments RMBli applied to the rod by bushing Bi in the global coordinate system are therefore(47)RFBliRMBli=−FBliMBli=−KBi,t00−KBi,rrBi′θBi′−Δ¯Bi,tΔ¯Bi,r.The deformation of bushingAii=1∼3 satisfies the following relationships:(48)rAi′θAi′=Ali∗−Ai0θlAi−Aki∗−Ai0θkAi+rAi′0θAi′0,where θlAi and θkAi are the rotational angles in relative to the rod and in relative to the wheel knuckle of the bushing Ai in the global coordinate system when the load is applied, respectively; rAi′0 and θAi′0 are the initial translational displacements and the initial rotational angles of bushing Ai.Thus, the forces and moments applied to the inner tube of bushingAi by the rod in the global coordinate system can be calculated(49)FAliMAli=KAi,t00KAi,rrAi′θAi′+Δ¯Ai,tΔ¯Ai,r,where Δ¯Ai,t and Δ¯Ai,r are the moment-angle and the force-displacement correction vectors for bushing Ai in the global coordinate system, respectively; KAi,t and KAi,r are the linear translational stiffness matrix and rotational stiffness matrix of bushing Ai in the global coordinate system, respectively.KAi,t and KAi,r can be calculated from the linear stiffness matrices K¯Ai,t,K¯Ai,r of the bushing Ai in the local coordinate system.(50)KAi,t=Rs″TK¯Ai,tRs″,KAi,r=Rs″TK¯Ai,rRs″,where Rs″ is the (3 × 3) transformation matrix from the global coordinate system to the local coordinate system of bushing Ai.Combined with equation (49), the reaction force and reaction moment applied to the rod by the inner tube of bushing Ai can be expressed as(51)RFAliRMAli=I0R˜AiI−FAli−MAli,where I is the (3 × 3) unity matrix; R˜Ai is the (3 × 3) skew-symmetric matrix of the position vector in the global coordinate system for inner tube of bushing Ai changing with the rods.Substituting (49) into (51) gives(52)RFAliRMAli=−KAi,t0−R˜AiKAi,t−KAi,rrAi′θAi′−Δ¯Ai,tR˜AiΔ¯Ai,t+Δ¯Ai,r,R˜Ai=0z′y′z′0−x′−y′x′0,where x′, y′, and z′ are the values of the x, y, and z directions of the position vector for the inner tube of bushing Ai with the rod, respectively.Since the inner tube of bushingAi is loaded against the outer tube, the forces and moments applying to the outer tube of the bushing Ai from the wheel knuckle are(53)FAkiMAki=−FAliMAli.Taking the moments atP1 and combining with equations (49) and (53), the reaction force and moment applied by the bushing Ai to the wheel knuckle can be derived.(54)RFAkiRMAki=I0R˜′AiI−FAki−MAki,where I is the (3 × 3) unity matrix; R˜Ai′ is the (3 × 3) skew-symmetric matrix of the position vector in the global coordinate system for the inner tube of bushing Ai changing with the wheel knuckle.(55)R˜Ai′=0z1′y1′z1′0−x1′−y1′x1′0.Combining equations (49) and (53), equation (54) can be written as(56)RFAkiRMAki=KAi,t0R˜Ai′KAi,tKBi,rrAi′θAi′+Δ¯Ai,tR˜Ai′Δ¯Ai,t+Δ¯Ai,r.Similarly, the translational and rotational deformations of bushingA4 in the global coordinate system can be expressed as(57)rA4′θA4′=Al4∗−A40θAl4+rA4′0θA4′0,where θAl4 is the rotational angle of the bushing A4 in relative to the rod in the global coordinate system when the load is applied; rA4′0 and θA4′0 are the initial translational displacements and the initial rotational angles of bushing Ai in the global coordinate.Similarly, the forces and moments applied by the rod on the inner tube of bushingA4 can be derived. Taking the moments at P1, the reaction force and moment applied by the bushing A4 to the wheel knuckle can be derived.(58)RFAl4RMAl4=I0R˜A4′I−FAl4−MAl4,where R˜A4′ is the (3 × 3) skew-symmetric matrix of the position vector in the global coordinate system for the inner tube of bushing A4 changing with the wheel knuckle.Consequently, the sum of each reaction forceRF, reaction moment RM, and external force applied to the wheel knuckle have a value of zero in the global coordinate system. The static equilibrium equation for the wheel knuckle can be expressed as(59)FP1+∑i=13RFAki+RFAl4=0,∑i=13RMAki+RMAl4=0,where FP1 is the external force applied to the point P1 on the wheel.RodSii=1,3 is in equilibrium under the reaction forces and moments from the bushings Ai and Bii=1,3. Thus, the static equilibrium equation for the rod Sii=1,3 can be derived as(60)RFAli+RFBli=0,RMAli+RMBli=0.The lower rear control armS2 is connected to the body by the spring. The ends of S2 are connected to the wheel knuckle and the body by bushings A2 and B2. The equations for the static equilibrium of the lower rear control arm S2 can be derived as(61)RFBl2+RFAl2+FA0=0,RMAl2+RMBl2+MA0=0.FA0 and MA0 are the forces and moments applied by the spring with B2 as the reference point. The initial length, the length after deformation, and the displacement of the spring satisfy the following equation:(62)LA0B0′+ΔLA0B0−LA0B0=0,where LA0B0′ is the length of the spring after deformation; LA0B0 is the initial length of the spring; and ΔLA0B0 is the displacement of the spring. ΔLA0B0 can be calculated from(63)ΔLA0B0=fsLA0B0∗−fs0LA0B0Ks,where fs0 is the initial scale factor of spring force; Ks is the stiffness of spring.According to (58)–(61), the static equilibrium equation of the E-type multilink rear suspension can be derived(64)FP1+∑i=13RFAki+RFAl4∑i=13RMAki+RMAl4RFAli+RFBliRMAli+RMBliRFBl2+RFAl2+FA0RMAl2+RMBl2+MA0LA0B0′+ΔLA0B0−LA0B0i=1,3=0.


### 3.4. Bushing Stiffness and Suspension Stiffness

Longitudinal and vertical forces with intervals of [0, 1500] are applied to the front and rear wheel centre, respectively.The forces of bushingB and bushing A4 are shown in Figure 6. Taking the example of bushing A, FAX represents the force of bushing A in the X direction.Figure 6
The forces of bushing: applying longitudinal force (a); applying vertical force (b).
(a)(b)Due to the different stiffness of the bushings, the bushings generate different displacements. Figure7 shows the displacements of bushing B and bushing A4. DAX represents the displacement of bushing A in the X direction.Figure 7
The displacements of bushing: applying longitudinal force (a); applying vertical force (b).
(a)(b)As can be seen in Figure7, the bushings have initial forces and initial displacements in the initial position. Therefore, the values of the change in forces and displacements are used to analyze the influence on bushings. The relevant results are obtained, as shown in Figures 8 and 9.Figure 8
Variation of bushing forces: applying longitudinal force (a); applying vertical force (b).
(a)(b)Figure 9
Variation of bushing displacements: applying longitudinal force (a); applying vertical force (b).
(a)(b)BushingsA and B are the bushings of front suspension. Bushings Aii=1∼4 and Bii=1∼4 are the bushings of rear suspension.As can be seen from Figure8(a), when longitudinal forces are applied, FAY,FA2X,FA2X,FB2X,FA3X,FB2X are more variable. As can be seen from Figure 8(b), when vertical forces are applied, the forces on the front suspension bushings have little variation, while the forces on the rear suspension bushings FA1X,FB1X,FA2X,FB2X,FA3X,FB3X have larger variations.As can be seen in Figure9(a), when applying longitudinal forces, DBY and DA4X are more variable. As can be seen in Figure 9(b), displacements of bushings on the front suspension are minor when the vertical force is applied. For the rear suspension, DB3Z and DA4Z are more variable.The displacements of the bushings will influence the displacement of the wheel centre. Based on the preceding analysis, varying the stiffness of bushingB in the Y direction, the stiffness of bushing A4 in the X direction and Z direction, and the stiffness of bushing B3 in the Z direction, the longitudinal displacement vs. longitudinal force curve of wheel centre and the vertical displacement vs. vertical force curve of wheel centre can be obtained, as shown in Figure 10. As can be seen in Figure 10, the force vs. displacement curve changes significantly when changing the stiffness of the mentioned bushings.Figure 10
Force vs. displacement curves for suspensions with different stiffnesses of bushing: longitudinal (a); vertical (b).
(a)(b)The vertical stiffness of suspension can be expressed by the ratio of the vertical load to the vertical displacement of the wheel centre. Longitudinal stiffness is usually defined as the ratio of the longitudinal force to the longitudinal displacement of the wheel centre [5]. The slope of the curve in the linear range, i.e., the linear stiffness of the suspension, can be thus obtained, as shown in Tables 1 and 2.Table 1
Stiffness of front suspension.
CaseBush stiffnessLongitudinal stiffnessBaseline (N/mm)TBY152TBY−50% (N/mm)16685Table 2
Stiffness of rear suspension.
CaseStiffnessLongitudinal stiffnessVertical stiffnessBaseline (N/mm)TA4X=330,TB3Z=100,TA4Z=200443.7924.67TA4X−50% (N/mm)165261.32—TB3Z−50% (N/mm)50—24TA4Z−50% (N/mm)100—23.29The stiffness of bushingB in the Y direction is denoted as TBY. The stiffness of bushing A4 in the X direction is denoted as TA4X. The stiffness of bushing A4 in the Z direction is denoted as TA4Z. The stiffness of bushing B3 in the Z direction is denoted as TB3Z.Table1 shows the longitudinal stiffness of the front suspension for different stiffness TBY. As seen in Table 1, reducing the stiffness TBY by 50% reduces the longitudinal stiffness of the front suspension by 44.1%, thus showing that the stiffness TBY contributes more than 50% to the longitudinal stiffness of the front suspension. Table 2 shows the longitudinal and vertical stiffness of the rear suspension for different stiffnesses of bushing. As seen in Table 2, reducing the stiffness TA4X by 50% reduces the longitudinal stiffness of the rear suspension by 41.1%, so the contribution of the stiffness TA4X to the longitudinal stiffness of the rear suspension is greater than 50%. Reducing the stiffness TB3Z and TA4Z by 50% reduces the rear suspension vertical stiffness by 2.8% and 5.6%, respectively, so it can be seen that the stiffness of these two bushings contributes little to the vertical stiffness of the rear suspension.From the analysis in Sections2 and 3, it is clear that the stiffness of the bushings will influence the vibration characteristics of the vehicle.This paper will analyze the vehicle ride comfort on bump road through simulation study.


## 3.1. Stiffness of the Suspension

Neglecting the influence of the bushings and antiroll bar on the suspension stiffness, the suspension stiffness satisfies the following relationship:(21)Keq=Ksi2,where Keq is the suspension stiffness; Ks is the spring stiffness; and i is the installation ratio.Taking McPherson suspension as an example, equation (21) is described in detail.Neglecting the mass of the suspension strut and the control arm, considering the vertical displacement of the vehicle only, the quarter-vehicle model of the MacPherson suspension can be simplified to the model shown in Figure3 [22]. A is the connection point between the MacPherson suspension strut and the control arm; B is the connection point between the control arm and the body. The global coordinate system is established with the point B as the reference point, Z-axis and Y-axis along with the vertical and horizontal directions, respectively.Figure 3
Suspension forces schema for determining equivalent spring force.In a MacPherson suspension, the position and orientation of the control arm determine the relative movement between the wheel assembly and the chassis. The movement of the system can be seen as a rotational movement around a transient centreO. F1 is the tangential force acting on the control arm at point A; F2 is the spring force acting on the control arm at point A; β is the angle between F1 and F2; F4 is the tangential force acting at point C; F3 is the equivalent spring force acting at the point C; α is the angle between F3 and F4; lOA and lOC are the lengths of point A and point C to point R, respectively; Z1 and Z2 are the vertical displacements of the wheel and the suspension, respectively.The following relationships can be obtained:(22)F1=F2cosβ,F4=F3cosα,F1lOA=F4lOC,lOAlOC=Z2cosβZ1cosα,F3=Z2Z1cosβcosα2F2.In the linear range, the deformation is proportional to the force applied(23)F3=KeqZ1,F2=KsZ2.The linear equivalent stiffnessKeq and installation ratio i can be obtained from equations (22) and (23)(24)Keq=Z2Z1cosβcosα2Ks,i=Z2Z1cosβcosα.From equation (24), it can be seen that the vertical stiffness Keq of the suspension is influenced by the spring stiffness Ks and the installation ratio i.For a multilink rear suspension, the vertical stiffness also satisfies equation (21).The above analysis does not consider the stiffness of the bushings, whereas, for suspension systems, elastic elements such as bushings are installed to absorb impacts. The vertical stiffness of the suspension is influenced by the spring stiffness and bushing stiffness. The deformation of the bushing makes an impact on the installation ratioi. As for the longitudinal stiffness of the suspension, in addition to being influenced by the longitudinal component of the suspension springs and shock absorber, it is mainly influenced by the stiffness of the bushings. Therefore, the suspension will be analyzed specifically with considering the stiffness of the bushings. In the actual driving of a vehicle, the excitation signals transmitted to the bushings are mainly of low-frequency excitation. Therefore, the influence of bushing stiffness on vehicle ride comfort can be studied by considering the static stiffness of the bushing only.This paper presents the further analysis of the bushings by establishing the static equilibrium equation of the suspension.


## 3.2. Static Equilibrium Equations for MacPherson Front Suspension

Figure4 shows the schematic diagram of the MacPherson suspension. The control arm and steering tie rod are connected to the wheel knuckle by ball joint. Control arm is connected to the body by bushings. When an external load is applied to the tyre contact point, the bushings and spring of suspension will deform, and the tie rod will move to a new position, thus achieving a new static equilibrium.Figure 4
Schematic of MacPherson suspension.This section will calculate the forces and moments of the bushings with the methods in reference [23].The relative displacement of the bushing’s inner tube to the bushing’s outer tube is required. For the bushings of the MacPherson suspension control arm, the displacement of the outer tube can be considered as zero as the outer tube of the bushing is fixed to the body. Therefore, the displacement of the inner tube is only required. Firstly, taking pointC as a reference point to derive the new coordinates of the inner tube of A and B,(25)A∗=XαYβZγA0−C0+C∗,B∗=XαYβZγB0−C0+C∗,where A∗,B∗, and C∗ are the new coordinates of the inner tube of bushings A,B, and C in the global coordinate system, respectively; A0,B0, and C0 are the initial coordinates of the inner tube of bushings A,B, and C in the global coordinate system, respectively; Xα,Yβ, and Zγ are the 3×3 rotation matrixes when the control arm is rotated by an angle of α,β, and γ about the x-, y-, and z-axes of its rigid coordinate system, respectively.The new coordinates of the ball jointsD and G, which are connected to the wheel knuckle, and the new coordinates of point P1 can also be estimated using point C as a reference point.(26)D∗=XαkYβkZγkD0−C0+C∗,G∗=XαkYβkZγkG0−C0+C∗,P1∗=XαkYβkZγkP10−C0+C∗.where Xαk,Yβk. and Zγk are the 3×3 rotation matrixes when the wheel knuckle is rotated by an angle of αk,βk, and γk about the x-, y-, and z-axes of its rigid coordinate system, respectively.Thus, the translational deformation of bushingA and bushing B in the global coordinate system rA′ and rB′ is expressed as(27)rA′rB′=A∗−A0B∗−B0+rA′0rB′0,where rA′0 and rA′0 are the initial translational displacements of bushings A and B in the global coordinate system.The forces acting on the bushings due to the displacements of bushingsA and B in the global coordinate system can be expressed as(28)FAFB=KA,t00KB,trA′rB′+Δ¯B,tΔ¯C,t,where Δ¯B,t and Δ¯C,t are the force-displacement correction vectors of bushing A and bushing B in the global coordinate system, respectively; KA,t and KB,t are the linear stiffness matrixes for bushing A and bushing B in the global coordinate system, respectively.KA,t can be calculated from the linear stiffness matrix K¯A,t of the bushing A in the local coordinate system. KB,t can be calculated from the linear stiffness matrix K¯B,t of the bushing B in the local coordinate system.(29)KA,t=RSTK¯A,tRS,KB,t=RSTK¯B,tRS.RS is the transformation matrix from the local to the global coordinate system of the bushings A and B.The rotational deformationsθA′ and θB′ of the bushings A and B in the global coordinate system can be expressed as(30)θA′θB′=θAθB+θA0θB0,where θA and θB are rotation angles at load input and θA0θB0 are rotation initial displacements.The moments of the bushingA and bushing B in the global coordinate system can be calculated(31)MAMB=KA,r00KB,rθA′θB′+Δ¯A,rΔ¯B,r,where Δ¯A,r and Δ¯B,r are the moment-angle correction vectors of bushing A and bushing B in the global coordinate system, respectively; KA,r and KB,r are the linear translational stiffness matrixes of bushing A and bushing B in the global coordinate system, respectively.KA,r can be calculated from the linear stiffness matrix K¯A,r of the bushing A in the local coordinate system. KB,r can be calculated from the linear stiffness matrix K¯B,r of the bushing B in the local coordinate system.(32)KA,r=RSTK¯A,rRS,KB,r=RSTK¯B,rRS.According to equations (28) and (31), the reaction forces and moments acting on the control arms of bushing A and bushing B can be derived as follows:(33)RFiRMi=I0R˜I−Fi−Mi.Subscripti represents bushings A and B; R˜ is the (3 × 3) skew-symmetric matrix of the position vector for bushings A and B(34)R˜=0zyz0−x−yx0,where x,y, and z represent the displacements of the bushing in the global coordinate system in the directions of the x-, y-, and z-axes, respectively.The control arm is in equilibrium under the reaction forces, the reaction moments of the bushings, and the ball joints of the wheel knuckle, and therefore the following relationships can be obtained:(35)RFA+RFB+FC=0,RMA+RMB=0.The wheel knuckle is in equilibrium under the reaction forces and moments from the control arms, the tie rod, and the external load at the tyre contact patch and therefore satisfies the following relationships:(36)FP1+RFC+RFD+RFG=0,MP1+RMD+RMG=0,where MP1 is the external force input to the tyre from the road; RFG and RMG are the spring reaction force and reaction moment, respectively.Since the length of the steering tie rod is constant during the suspension kinematics, a constraint can be given(37)LED′−LED=0.The spring deformation satisfies the following constraints:(38)LGH′+ΔLGH−LGH=0,where LGH′ is the new length of the spring; LGH is the initial length of the spring; and ΔLGH is the displacement of the spring. ΔLGH can be calculated by(39)ΔLGH=fsLED′−fs0LGHKs,where fs0 is the initial scale factor of spring force; fs is the new scale factor of spring force; and Ks is the stiffness of spring.According to (35)–(38), the static equilibrium equation for the front suspension can be expressed as(40)RFA+RFB+FCRMA+RMBFP1+RFC+RFD+RFGMP1+RMD+RMGLED′−LEDLGH′+ΔLGH−LGH=0.There are 14 equations in equation (40) and the number of unknowns is 14. Therefore, equation (40) can be solved


## 3.3. Static Equilibrium Equations for E-Type Multilink Rear Suspensions

Figure5 shows a diagram of the E-type multilink rear suspension. S1 is the upper control arm; S2 is the rear lower control arm; S3 is the front lower control arm; S4 is the trailing arm; Bii=1∼3 are bushings for the connecting rod to the body; Aii=1∼3 are the bushings for connecting the wheel knuckle to the rod; A4 is the bushing for the trailing arm to the body.Figure 5
Schematic of MacPherson suspension.This section will calculate the forces and moments of the bushings with the methods in reference [23, 24].The new position after the bushing movement was calculated first. Since the outer tube of bushingAi is connected to the wheel knuckle and the inner tube connected to the rod, both the inner and outer tubes of bushing Ai will move. The displacement of the centre point of both the inner and outer tubes needs to be taken into account. As the outer tubes of bushing Bi and bushing A4 are fixed to the body, the displacement at the centre of the outer tube will be considered as zero and the displacement at the centre of the inner tube of the bushing will be taken into account.The centre points of the inner and outer tubes of the bushing coincide at the initial position.Ai0i=1∼3 is the coordinate of the centre point of the inner and outer tubes of bushing Aii=1∼3 in the global coordinate system at the initial position. Bi0 is the coordinate of the centre point of the inner and outer tubes of the bushing Bii=1∼3 in the global coordinate system at the initial position.ChooseBi0 as the reference point and calculate the new coordinates of the centre point of the inner tube Ali∗ of the bushing Ai in the global coordinate system.(41)Ali∗=XαlYβlZγlAi0−Bi0+Bli∗,where Bli∗ is the new coordinate of the centre point of the inner tube of bushing Bi in the global coordinate system; Xαl,Yβl and Zγl are the 3×3 rotation matrixes when the rod is rotated by an angle of αl,βl, and γl about the x-, y-, and z-axes of its rigid coordinate system, respectively.ChooseP1 as the reference point and calculate the new coordinates of the centre point of the inner tube Al4∗ of bushing A4 in the global coordinate system.(42)Aki∗=XαkYβkZγkAi0−P1+P1∗,where P1∗ is the new coordinate of the centre point P1 in the global coordinate system.By choosingP1 as the reference point, the new coordinates of the inner tube Al4∗ of the bushing A4 in the global coordinate system can be calculated(43)Al4∗=XαkYβkZγkAl40−P1+P1∗,where Xαk,Yβk, and Zγk are the 3×3 rotation matrixes when the wheel knuckle is rotated by an angle of αk,βk, and γk about the x-, y-, and z-axes of its rigid coordinate system, respectively.The new coordinates of the centre points of the inner and outer tubes of the bushing can be calculated by equations (41)–(43).Based on the preceding analysis, the translational and rotational deformations of bushingBi in the global coordinate system can be expressed as(44)rBi′θB′=Bli∗−Bi0θBli+rBi′0θBi′0,where θBli is the rotational angle of bushing Bi in the global coordinate system under the force of the rod; rBi′0 and θBi′0 are the initial translational displacements and the initial rotational angles of bushing Bi.The forces and moments applied to bushingBi in the global coordinate system can be expressed as(45)FBliMBli=KBi,t00KBi,rrBi′θBi′+Δ¯Bi,tΔ¯Bi,r,where Δ¯Bi,t and Δ¯Bi,r are the moment-angle and the force-displacement correction vectors for bushing Bi in the global coordinate system, respectively; KBi,t and KBi,r are the linear translational stiffness matrixes for bushing Bi in the global coordinate system, respectively.KBi,t and KBi,r can be calculated from the linear stiffness matrices K¯Bi,t,K¯Bi,r of bushing Bi in the local coordinate system.(46)KBi,t=RS′TK¯Bi,tRS′,KBi,r=RS′TK¯Bi,rRS′.RS′ is the (3 × 3) transformation matrix from the global coordinate system to the local coordinate system of bushing Bi.The reaction forcesRFBli and moments RMBli applied to the rod by bushing Bi in the global coordinate system are therefore(47)RFBliRMBli=−FBliMBli=−KBi,t00−KBi,rrBi′θBi′−Δ¯Bi,tΔ¯Bi,r.The deformation of bushingAii=1∼3 satisfies the following relationships:(48)rAi′θAi′=Ali∗−Ai0θlAi−Aki∗−Ai0θkAi+rAi′0θAi′0,where θlAi and θkAi are the rotational angles in relative to the rod and in relative to the wheel knuckle of the bushing Ai in the global coordinate system when the load is applied, respectively; rAi′0 and θAi′0 are the initial translational displacements and the initial rotational angles of bushing Ai.Thus, the forces and moments applied to the inner tube of bushingAi by the rod in the global coordinate system can be calculated(49)FAliMAli=KAi,t00KAi,rrAi′θAi′+Δ¯Ai,tΔ¯Ai,r,where Δ¯Ai,t and Δ¯Ai,r are the moment-angle and the force-displacement correction vectors for bushing Ai in the global coordinate system, respectively; KAi,t and KAi,r are the linear translational stiffness matrix and rotational stiffness matrix of bushing Ai in the global coordinate system, respectively.KAi,t and KAi,r can be calculated from the linear stiffness matrices K¯Ai,t,K¯Ai,r of the bushing Ai in the local coordinate system.(50)KAi,t=Rs″TK¯Ai,tRs″,KAi,r=Rs″TK¯Ai,rRs″,where Rs″ is the (3 × 3) transformation matrix from the global coordinate system to the local coordinate system of bushing Ai.Combined with equation (49), the reaction force and reaction moment applied to the rod by the inner tube of bushing Ai can be expressed as(51)RFAliRMAli=I0R˜AiI−FAli−MAli,where I is the (3 × 3) unity matrix; R˜Ai is the (3 × 3) skew-symmetric matrix of the position vector in the global coordinate system for inner tube of bushing Ai changing with the rods.Substituting (49) into (51) gives(52)RFAliRMAli=−KAi,t0−R˜AiKAi,t−KAi,rrAi′θAi′−Δ¯Ai,tR˜AiΔ¯Ai,t+Δ¯Ai,r,R˜Ai=0z′y′z′0−x′−y′x′0,where x′, y′, and z′ are the values of the x, y, and z directions of the position vector for the inner tube of bushing Ai with the rod, respectively.Since the inner tube of bushingAi is loaded against the outer tube, the forces and moments applying to the outer tube of the bushing Ai from the wheel knuckle are(53)FAkiMAki=−FAliMAli.Taking the moments atP1 and combining with equations (49) and (53), the reaction force and moment applied by the bushing Ai to the wheel knuckle can be derived.(54)RFAkiRMAki=I0R˜′AiI−FAki−MAki,where I is the (3 × 3) unity matrix; R˜Ai′ is the (3 × 3) skew-symmetric matrix of the position vector in the global coordinate system for the inner tube of bushing Ai changing with the wheel knuckle.(55)R˜Ai′=0z1′y1′z1′0−x1′−y1′x1′0.Combining equations (49) and (53), equation (54) can be written as(56)RFAkiRMAki=KAi,t0R˜Ai′KAi,tKBi,rrAi′θAi′+Δ¯Ai,tR˜Ai′Δ¯Ai,t+Δ¯Ai,r.Similarly, the translational and rotational deformations of bushingA4 in the global coordinate system can be expressed as(57)rA4′θA4′=Al4∗−A40θAl4+rA4′0θA4′0,where θAl4 is the rotational angle of the bushing A4 in relative to the rod in the global coordinate system when the load is applied; rA4′0 and θA4′0 are the initial translational displacements and the initial rotational angles of bushing Ai in the global coordinate.Similarly, the forces and moments applied by the rod on the inner tube of bushingA4 can be derived. Taking the moments at P1, the reaction force and moment applied by the bushing A4 to the wheel knuckle can be derived.(58)RFAl4RMAl4=I0R˜A4′I−FAl4−MAl4,where R˜A4′ is the (3 × 3) skew-symmetric matrix of the position vector in the global coordinate system for the inner tube of bushing A4 changing with the wheel knuckle.Consequently, the sum of each reaction forceRF, reaction moment RM, and external force applied to the wheel knuckle have a value of zero in the global coordinate system. The static equilibrium equation for the wheel knuckle can be expressed as(59)FP1+∑i=13RFAki+RFAl4=0,∑i=13RMAki+RMAl4=0,where FP1 is the external force applied to the point P1 on the wheel.RodSii=1,3 is in equilibrium under the reaction forces and moments from the bushings Ai and Bii=1,3. Thus, the static equilibrium equation for the rod Sii=1,3 can be derived as(60)RFAli+RFBli=0,RMAli+RMBli=0.The lower rear control armS2 is connected to the body by the spring. The ends of S2 are connected to the wheel knuckle and the body by bushings A2 and B2. The equations for the static equilibrium of the lower rear control arm S2 can be derived as(61)RFBl2+RFAl2+FA0=0,RMAl2+RMBl2+MA0=0.FA0 and MA0 are the forces and moments applied by the spring with B2 as the reference point. The initial length, the length after deformation, and the displacement of the spring satisfy the following equation:(62)LA0B0′+ΔLA0B0−LA0B0=0,where LA0B0′ is the length of the spring after deformation; LA0B0 is the initial length of the spring; and ΔLA0B0 is the displacement of the spring. ΔLA0B0 can be calculated from(63)ΔLA0B0=fsLA0B0∗−fs0LA0B0Ks,where fs0 is the initial scale factor of spring force; Ks is the stiffness of spring.According to (58)–(61), the static equilibrium equation of the E-type multilink rear suspension can be derived(64)FP1+∑i=13RFAki+RFAl4∑i=13RMAki+RMAl4RFAli+RFBliRMAli+RMBliRFBl2+RFAl2+FA0RMAl2+RMBl2+MA0LA0B0′+ΔLA0B0−LA0B0i=1,3=0.


## 3.4. Bushing Stiffness and Suspension Stiffness

Longitudinal and vertical forces with intervals of [0, 1500] are applied to the front and rear wheel centre, respectively.The forces of bushingB and bushing A4 are shown in Figure 6. Taking the example of bushing A, FAX represents the force of bushing A in the X direction.Figure 6
The forces of bushing: applying longitudinal force (a); applying vertical force (b).
(a)(b)Due to the different stiffness of the bushings, the bushings generate different displacements. Figure7 shows the displacements of bushing B and bushing A4. DAX represents the displacement of bushing A in the X direction.Figure 7
The displacements of bushing: applying longitudinal force (a); applying vertical force (b).
(a)(b)As can be seen in Figure7, the bushings have initial forces and initial displacements in the initial position. Therefore, the values of the change in forces and displacements are used to analyze the influence on bushings. The relevant results are obtained, as shown in Figures 8 and 9.Figure 8
Variation of bushing forces: applying longitudinal force (a); applying vertical force (b).
(a)(b)Figure 9
Variation of bushing displacements: applying longitudinal force (a); applying vertical force (b).
(a)(b)BushingsA and B are the bushings of front suspension. Bushings Aii=1∼4 and Bii=1∼4 are the bushings of rear suspension.As can be seen from Figure8(a), when longitudinal forces are applied, FAY,FA2X,FA2X,FB2X,FA3X,FB2X are more variable. As can be seen from Figure 8(b), when vertical forces are applied, the forces on the front suspension bushings have little variation, while the forces on the rear suspension bushings FA1X,FB1X,FA2X,FB2X,FA3X,FB3X have larger variations.As can be seen in Figure9(a), when applying longitudinal forces, DBY and DA4X are more variable. As can be seen in Figure 9(b), displacements of bushings on the front suspension are minor when the vertical force is applied. For the rear suspension, DB3Z and DA4Z are more variable.The displacements of the bushings will influence the displacement of the wheel centre. Based on the preceding analysis, varying the stiffness of bushingB in the Y direction, the stiffness of bushing A4 in the X direction and Z direction, and the stiffness of bushing B3 in the Z direction, the longitudinal displacement vs. longitudinal force curve of wheel centre and the vertical displacement vs. vertical force curve of wheel centre can be obtained, as shown in Figure 10. As can be seen in Figure 10, the force vs. displacement curve changes significantly when changing the stiffness of the mentioned bushings.Figure 10
Force vs. displacement curves for suspensions with different stiffnesses of bushing: longitudinal (a); vertical (b).
(a)(b)The vertical stiffness of suspension can be expressed by the ratio of the vertical load to the vertical displacement of the wheel centre. Longitudinal stiffness is usually defined as the ratio of the longitudinal force to the longitudinal displacement of the wheel centre [5]. The slope of the curve in the linear range, i.e., the linear stiffness of the suspension, can be thus obtained, as shown in Tables 1 and 2.Table 1
Stiffness of front suspension.
CaseBush stiffnessLongitudinal stiffnessBaseline (N/mm)TBY152TBY−50% (N/mm)16685Table 2
Stiffness of rear suspension.
CaseStiffnessLongitudinal stiffnessVertical stiffnessBaseline (N/mm)TA4X=330,TB3Z=100,TA4Z=200443.7924.67TA4X−50% (N/mm)165261.32—TB3Z−50% (N/mm)50—24TA4Z−50% (N/mm)100—23.29The stiffness of bushingB in the Y direction is denoted as TBY. The stiffness of bushing A4 in the X direction is denoted as TA4X. The stiffness of bushing A4 in the Z direction is denoted as TA4Z. The stiffness of bushing B3 in the Z direction is denoted as TB3Z.Table1 shows the longitudinal stiffness of the front suspension for different stiffness TBY. As seen in Table 1, reducing the stiffness TBY by 50% reduces the longitudinal stiffness of the front suspension by 44.1%, thus showing that the stiffness TBY contributes more than 50% to the longitudinal stiffness of the front suspension. Table 2 shows the longitudinal and vertical stiffness of the rear suspension for different stiffnesses of bushing. As seen in Table 2, reducing the stiffness TA4X by 50% reduces the longitudinal stiffness of the rear suspension by 41.1%, so the contribution of the stiffness TA4X to the longitudinal stiffness of the rear suspension is greater than 50%. Reducing the stiffness TB3Z and TA4Z by 50% reduces the rear suspension vertical stiffness by 2.8% and 5.6%, respectively, so it can be seen that the stiffness of these two bushings contributes little to the vertical stiffness of the rear suspension.From the analysis in Sections2 and 3, it is clear that the stiffness of the bushings will influence the vibration characteristics of the vehicle.This paper will analyze the vehicle ride comfort on bump road through simulation study.


## 4. Analysis of Vehicle Ride Comfort on Bump Road

Simulation study on bump road: the parameters of the simulated vehicle are shown in Table3, with a bump height of 25mm and a width of 100mm. The vehicle will generate vertical and longitudinal vibrations when passing the bump road. As can be seen in Figure 11, the reaction force of the bump road on the tyre can be simplified to the force through the wheel centre FT and a moment around the wheel centre MT. MT will be balanced by the reaction moment generated from the changing of the kingpin caster angle. FT can be divided into the component force F1 perpendicular to the line and the component force F2 acting along the line between the centre of longitudinal inclination O and the wheel centre. F1 makes the tyres to move upwards, resulting in vertical vibrations. F2 makes the tyres to move backwards, resulting in longitudinal vibrations.Table 3
Vehicle’s parameters.
ParameterValueParameterValuem (kg)1815.25a (mm)1352.6l (mm)2620.0b (mm)1247.4TBY (N/mm)332TA4X (N/mm)330Ksf (N/mm)21Ksr (N/mm)26Bsf (N·s/mm)1.44Bsr (N·s/mm)0.976Figure 11
Bump event.Using the continuous RMS method, the transient vibrations are taken into account by integrating over short periods of time.This paper calculates longitudinal and vertical acceleration RMS curves with two different calculation windows.(65)Rms Local=1Mloc∫t0t0+Mlocaw2tdt1/2,Rms Global=1MGlo∫t0t0+MGloaw2tdt1/2,Mloc=0.52×3.6lv=1.872lv,MGlo=1.05×3.6lv=3.78lv,where l is the wheelbase of the vehicle.Rms Local is the RMS curve of longitudinal and vertical accelerations calculated using a narrow calculation window, which is used to separate the influence of the impact of the front and the rear wheels. Rms Global is the RMS curve of longitudinal and vertical accelerations calculated using a large calculation window, which is used to consider the influence of the impact due to all wheels.ΔRms Local is the maximum difference between Rms Local during bump passing and before the bump passing. ΔRms Global is the maximum difference between Rms Global during bump passing and before the bump passing. ΔRms Global and ΔRms Global are used to study the transient signals.Figure12 shows the curves of the longitudinal and vertical acceleration time-domain signal, local variation Rms Local, and global variation Rms Global of the front and rear seat rails when the vehicle passes over the bump road at the speed of 30 km/h.Figure 12
Signal and indexes: front seat rail in the longitudinal direction (a); front seat rail in the vertical direction (b); rear seat rail in the longitudinal direction (c); rear seat rail in the vertical direction (d).
(a)(b)(c)(d)As can be seen in Figure12, the curve of local variation Rms Local has two significant peaks due to the influence of the front and rear wheels. The value of ΔRms Local is greater than ΔRms Global.LongitudinalΔRms Local and longitudinal ΔRms Global of the front seat rail are greater than the rear suspension. The vertical ΔRmsLocal and vertical ΔRmsGlobal of the rear seat rail are greater than the front suspension.The influence of the front and rear suspension parameters on theΔRmsLocal and ΔRmsGlobal of the front and rear seat rails was analyzed by sensitivity. The top 10 parameters with high impact are shown in Figure 13.Figure 13
The effect of suspension parameters on the indexes: longitudinalΔRmsLocal (a); vertical ΔRmsLocal (b); longitudinal ΔRmsGlobal (c); vertical ΔRmsGlobal (d).
(a)(b)(c)(d)As can be seen in Figure13, the stiffness TBY of the bushing B has a significant influence on the longitudinal ΔRmsLocal and ΔRmsGlobal of the front seat rail. The stiffness of the spring and the damping of the shock absorber of the front suspension have a greater influence on the vertical ΔRmsLocal and ΔRmsGlobal of the front seat rail. The stiffness TA4X of bushing A4 has a significant influence on the longitudinal ΔRmsLocal and ΔRmsGlobal of the rear seat rail. The stiffness of the spring and the damping of the shock absorber of the rear suspension have a greater influence on the vertical ΔRmsLocal and ΔRmsGlobal of the rear seat rail.From the analysis of the suspension stiffness characteristics in Section3, it is clear that the longitudinal stiffness of the front suspension is influenced by the stiffness TBY of bushing B mostly and the longitudinal stiffness of the rear suspension is influenced by the stiffness TA4X of bushing A4 mostly, while the bushing stiffness has little influence on the vertical stiffness of suspension. Combined with (28), it can be seen that the vertical stiffness of suspension is influenced by the spring stiffness mostly.From the analysis of the vehicle vibration characteristics in Section2, it can be seen that the natural frequency and damping ratio of the suspension influence the acceleration relative to the road excitation response. The natural frequency of the suspension is influenced by the stiffness of suspension. The damping ratio of the suspension is influenced by the stiffness of spring and the damping of the shock absorber. Therefore, the results obtained in Figure 13 are consistent with the previous derivation.The trends of longitudinalΔRmsLocal and ΔRmsGlobal of the front and rear seat rails with bushing stiffness are shown in Figure 14. Figures 14(a) and 14(b) show the trends of longitudinal ΔRmsLocal and ΔRmsGlobal of the front seat rail with bushing stiffness TBY at different vehicle speeds. As can be seen from both graphs, the variation of ΔRmsLocal and ΔRmsGlobal with the bushing stiffness TBY is not a linear relationship. The values of longitudinal ΔRmsLocal and ΔRmsGlobal are different for the same vehicle speed with different bushing stiffness TBY. On the other hand, for the same bushing stiffness TBY,the values of longitudinal ΔRmsLocal and ΔRmsGlobal are different for different vehicle speeds. Figures 14(c) and 14(d) show the trends of longitudinal ΔRmsLocal and ΔRmsGlobal of the rear seat rail with the bushing stiffness TA4X at different vehicle speeds. As can be seen from the two graphs, both ΔRmsGlobal and ΔRmsLocal of the rear seat rail do not have a linear relationship with the bushing stiffness TA4X. The values of ΔRmsLocal and ΔRmsGlobal of the rear seat rail are also influenced by the bushing stiffness TA4X and the speed of the vehicle.Figure 14
Trends of the indexes with bushing stiffness at different vehicle speeds:TBY vs. front longitudinal ΔRmsLocal (a); TBY vs. front longitudinal ΔRmsGlobal (b); TA4X vs. rear longitudinal ΔRmsLocal (c); TA4X vs. rear longitudinal ΔRmsGlobal (d).
(a)(b)(c)(d)


## 5. Optimal Design of Vehicle Ride Comfort

### 5.1. Multiobjective Optimization Mathematical Model

It is clear from Section2 that the safety of the vehicle needs to be guaranteed while improving the vehicle ride comfort. As a result, the dynamic deflection of suspensions σdf and σdr needs to be used as a constraint. The spring stiffnesses Ksf and Ksr of the front and rear suspension, the damping coefficients Bsf and Bsr of the front and rear shock absorbers, the stiffness of the bushing TBY for the front suspension, and the stiffness of the bushing TA4X of the rear suspension are therefore used as optimization variables. From the analysis of Figure 14, it can be seen that the stiffness of the bushings that make ΔRmsLocal and ΔRmsGlobal lower are approximately the same, and the value of ΔRmsLocal is greater than the value of ΔRmsGlobal. To reduce the amount of calculation, this paper takes the ΔRmsLocal of the front and rear seat rails in the longitudinal and vertical directions as optimization objective 1, the RMS of the dynamic loads σFdf,σFdr of the front and rear suspensions as optimization objective 2, and the RMS of pitch angle acceleration σp for the body as optimization objective 3. The equation of multiobjective optimization mathematical model can be expressed as(66)minimize:ΔRmsL1,ΔRmsL2,ΔRmsL3,ΔRmsL4,minimize:σFdf,σFdr,minimize:σp,subject to:σdf≤13fdf,σdr≤13fdr,subject to:18.9≤Ksf≤23.1,1.296≤Bsf≤1.584,subject to:23.4≤Ksr≤28.6,0.878≤Bsr≤1.074,subject to:166≤TBY≤498,165≤TA4X≤495,where ΔRmsL1 is the longitudinal ΔRmsLocal of the front seat rail; ΔRmsL2 is the vertical ΔRmsLocal of the front seat rail; ΔRmsL3 is the longitudinal ΔRmsLocal of the rear seat rail; ΔRmsL4 is the vertical ΔRmsLocal of the rear seat rail; and fdf and fdr are the allowable values of dynamic deflection of the front and rear suspensions, respectively. The values of fdf and fdr in this paper are 50mm and 30mm, respectively.


### 5.2. Multiobjective Optimization Algorithm

The genetic algorithm is a group search algorithm that continuously finds and compares optimal solutions and eliminates poor solutions by iteratively simulating the process of biological evolution. NSGA-II is a more classical modern genetic algorithm that uses an elite retention strategy to better retain elite individuals, addressing the problem of standard genetic algorithms, where the last generation of individuals is discarded in the iteration [25]. In addition, it uses a fast nondominated ranking method to reduce computational complexity. The algorithm then reengages in the selection, crossover mutation, and elite retention until the maximum number of evolutionary generations is reached, at which point the Pareto frontier is obtained. Through the above analysis, this paper uses the NSGA-II algorithm to optimize the above multiobjective optimization problem. The main parameters of the NSGA-II algorithm optimization are shown in Table 4.Table 4
Main parameters of NSGA-II algorithm.
ParameterValuePopulation size12Number of generations20Crossover probability0.9Mutation distribution index20Max failed runs100Failed run penalty1000


### 5.3. Analysis of Optimization Results

Figure15 shows the partial set of Pareto solutions obtained by the optimization algorithm at a vehicle speed of 30 km/h. Figure 15(a) shows the Pareto set of solutions for the RMS of the dynamic load on the front wheels with respect to the vertical ΔRmsLocal of the front seat rail. Figure 15(b) shows the Pareto set of solutions for the RMS of the dynamic load on the rear wheels with respect to the vertical ΔRmsLocal of the rear seat rail. Table 5 shows the values of the optimization objectives and optimization variables before and after the optimization.Figure 15
The projection of Pareto sets: onto dynamic loads Rms and verticalΔRmsL1 (a); onto dynamic loads Rms and vertical ΔRmsL4 (b).
(a)(b)Table 5
Comparison of optimization results.
ParametersBefore optimizationAfter optimizationPercentage change (%)ΔRmsL1 (g)0.3540025430.249441268−29.54ΔRmsL2 (g)0.3031069210.298594014−1.49ΔRmsL3 (g)0.3530697980.248816026−29.53ΔRmsL4 (g)0.4577499960.441953026−3.45Fdf (kN)0.3673863720.3692608410.51Fdr (kN)0.3725443520.3737400120.32σp (g)0.0369605140.035220008−4.71Ksf (N/mm)2120.28365283−3.41Bsf (N·s/mm)2.161.997156851−7.54Ksr (N/mm)2624.0494012−7.50Bsr (N·s/mm)0.960.9660299560.63TBY (N/mm)332278.4620487−16.13TA4X (N/mm)330494.46268149.84After optimization, the longitudinalΔRmsLocal of the front seat rail decreases by 25.94% and the vertical ΔRmsLocal decreases by 1.49%. The longitudinal ΔRmsLocal of the rear seat rail decreases by 29.53% and the vertical ΔRmsLocal decreases by 3.45%. The RMS of the vehicle pitch angle acceleration decreases by 4.71%. The increase in the RMS value of the dynamic load of the front tyres was 0.51% and the increase in the RMS value of the dynamic load of the rear tyres was 0.32%. After optimization, the RMS of the dynamic loads on the front and rear tyres increased less and the safety of the vehicle could be guaranteed. In summary, the optimization has improved the vibration characteristics of the vehicle on bump road and the safety has not worsened.The optimization is carried out at different vehicle speeds. TheΔRmsLocal and ΔRmsGlobal of both the longitudinal and vertical directions of the seat rail are obtained. The results of before and after optimization are shown in Figure 16. As can be seen from the graph, the values of ΔRmsLocal and ΔRmsGlobal of both the longitudinal and vertical directions of the front and rear seat rails are reduced after optimization. Table 6 shows the optimization results of the optimization variables for different vehicle speeds. As can be seen from the table, the values of the optimization variables that make the vehicle ride comfort better at each speed are different. The suspension parameters need to be matched reasonably to ensure the vehicle ride comfort.Figure 16
Comparison of results before and after optimization at different vehicle speeds: longitudinal (a); vertical (b).
(a)(b)Table 6
Results for optimized variables after optimization.
Parameters30 km/h40 km/h50 km/h60 km/hKsf (N/mm)20.2836528322.7802604219.2716403820.67443909Bsf (N·s/mm)1.9971568511.9476298571.944216681.947121335Ksr (N/mm)24.049401223.7641242323.6654970126.44512566Bsr (N·s/mm)0.9660299560.9138980480.9289115810.895457426KTYB (N/mm)278.4620487486.843411301.8880437376.3872639KTXA4 (N/mm)494.462681494.9839004486.6719285493.9985812


## 5.1. Multiobjective Optimization Mathematical Model

It is clear from Section2 that the safety of the vehicle needs to be guaranteed while improving the vehicle ride comfort. As a result, the dynamic deflection of suspensions σdf and σdr needs to be used as a constraint. The spring stiffnesses Ksf and Ksr of the front and rear suspension, the damping coefficients Bsf and Bsr of the front and rear shock absorbers, the stiffness of the bushing TBY for the front suspension, and the stiffness of the bushing TA4X of the rear suspension are therefore used as optimization variables. From the analysis of Figure 14, it can be seen that the stiffness of the bushings that make ΔRmsLocal and ΔRmsGlobal lower are approximately the same, and the value of ΔRmsLocal is greater than the value of ΔRmsGlobal. To reduce the amount of calculation, this paper takes the ΔRmsLocal of the front and rear seat rails in the longitudinal and vertical directions as optimization objective 1, the RMS of the dynamic loads σFdf,σFdr of the front and rear suspensions as optimization objective 2, and the RMS of pitch angle acceleration σp for the body as optimization objective 3. The equation of multiobjective optimization mathematical model can be expressed as(66)minimize:ΔRmsL1,ΔRmsL2,ΔRmsL3,ΔRmsL4,minimize:σFdf,σFdr,minimize:σp,subject to:σdf≤13fdf,σdr≤13fdr,subject to:18.9≤Ksf≤23.1,1.296≤Bsf≤1.584,subject to:23.4≤Ksr≤28.6,0.878≤Bsr≤1.074,subject to:166≤TBY≤498,165≤TA4X≤495,where ΔRmsL1 is the longitudinal ΔRmsLocal of the front seat rail; ΔRmsL2 is the vertical ΔRmsLocal of the front seat rail; ΔRmsL3 is the longitudinal ΔRmsLocal of the rear seat rail; ΔRmsL4 is the vertical ΔRmsLocal of the rear seat rail; and fdf and fdr are the allowable values of dynamic deflection of the front and rear suspensions, respectively. The values of fdf and fdr in this paper are 50mm and 30mm, respectively.


## 5.2. Multiobjective Optimization Algorithm

The genetic algorithm is a group search algorithm that continuously finds and compares optimal solutions and eliminates poor solutions by iteratively simulating the process of biological evolution. NSGA-II is a more classical modern genetic algorithm that uses an elite retention strategy to better retain elite individuals, addressing the problem of standard genetic algorithms, where the last generation of individuals is discarded in the iteration [25]. In addition, it uses a fast nondominated ranking method to reduce computational complexity. The algorithm then reengages in the selection, crossover mutation, and elite retention until the maximum number of evolutionary generations is reached, at which point the Pareto frontier is obtained. Through the above analysis, this paper uses the NSGA-II algorithm to optimize the above multiobjective optimization problem. The main parameters of the NSGA-II algorithm optimization are shown in Table 4.Table 4
Main parameters of NSGA-II algorithm.
ParameterValuePopulation size12Number of generations20Crossover probability0.9Mutation distribution index20Max failed runs100Failed run penalty1000


## 5.3. Analysis of Optimization Results

Figure15 shows the partial set of Pareto solutions obtained by the optimization algorithm at a vehicle speed of 30 km/h. Figure 15(a) shows the Pareto set of solutions for the RMS of the dynamic load on the front wheels with respect to the vertical ΔRmsLocal of the front seat rail. Figure 15(b) shows the Pareto set of solutions for the RMS of the dynamic load on the rear wheels with respect to the vertical ΔRmsLocal of the rear seat rail. Table 5 shows the values of the optimization objectives and optimization variables before and after the optimization.Figure 15
The projection of Pareto sets: onto dynamic loads Rms and verticalΔRmsL1 (a); onto dynamic loads Rms and vertical ΔRmsL4 (b).
(a)(b)Table 5
Comparison of optimization results.
ParametersBefore optimizationAfter optimizationPercentage change (%)ΔRmsL1 (g)0.3540025430.249441268−29.54ΔRmsL2 (g)0.3031069210.298594014−1.49ΔRmsL3 (g)0.3530697980.248816026−29.53ΔRmsL4 (g)0.4577499960.441953026−3.45Fdf (kN)0.3673863720.3692608410.51Fdr (kN)0.3725443520.3737400120.32σp (g)0.0369605140.035220008−4.71Ksf (N/mm)2120.28365283−3.41Bsf (N·s/mm)2.161.997156851−7.54Ksr (N/mm)2624.0494012−7.50Bsr (N·s/mm)0.960.9660299560.63TBY (N/mm)332278.4620487−16.13TA4X (N/mm)330494.46268149.84After optimization, the longitudinalΔRmsLocal of the front seat rail decreases by 25.94% and the vertical ΔRmsLocal decreases by 1.49%. The longitudinal ΔRmsLocal of the rear seat rail decreases by 29.53% and the vertical ΔRmsLocal decreases by 3.45%. The RMS of the vehicle pitch angle acceleration decreases by 4.71%. The increase in the RMS value of the dynamic load of the front tyres was 0.51% and the increase in the RMS value of the dynamic load of the rear tyres was 0.32%. After optimization, the RMS of the dynamic loads on the front and rear tyres increased less and the safety of the vehicle could be guaranteed. In summary, the optimization has improved the vibration characteristics of the vehicle on bump road and the safety has not worsened.The optimization is carried out at different vehicle speeds. TheΔRmsLocal and ΔRmsGlobal of both the longitudinal and vertical directions of the seat rail are obtained. The results of before and after optimization are shown in Figure 16. As can be seen from the graph, the values of ΔRmsLocal and ΔRmsGlobal of both the longitudinal and vertical directions of the front and rear seat rails are reduced after optimization. Table 6 shows the optimization results of the optimization variables for different vehicle speeds. As can be seen from the table, the values of the optimization variables that make the vehicle ride comfort better at each speed are different. The suspension parameters need to be matched reasonably to ensure the vehicle ride comfort.Figure 16
Comparison of results before and after optimization at different vehicle speeds: longitudinal (a); vertical (b).
(a)(b)Table 6
Results for optimized variables after optimization.
Parameters30 km/h40 km/h50 km/h60 km/hKsf (N/mm)20.2836528322.7802604219.2716403820.67443909Bsf (N·s/mm)1.9971568511.9476298571.944216681.947121335Ksr (N/mm)24.049401223.7641242323.6654970126.44512566Bsr (N·s/mm)0.9660299560.9138980480.9289115810.895457426KTYB (N/mm)278.4620487486.843411301.8880437376.3872639KTXA4 (N/mm)494.462681494.9839004486.6719285493.9985812


## 6. Conclusions

In this study, the parameters of the MacPherson front suspension and the E-type multilink rear suspension are matched reasonably to enhance the vehicle ride comfort on bump road. The following conclusions are drawn:(1)
The longitudinal stiffness of the front suspension is influenced by the stiffnessTBY of the bushing B mostly. The longitudinal stiffness of the rear suspension is influenced by the stiffness TA4X of the bushing A4 mostly.(2)
ΔRmsLocal and ΔRmsGlobal of the front and rear seat rails are used as the vibration evaluation indexes. The vertical ΔRmsLocal and ΔRmsGlobal of the seat rails are influenced by the stiffness of spring and the damping of the shock absorbers mostly. The longitudinal ΔRmsLocal and ΔRmsGlobal of the seat rails are influenced by the stiffness TBY of the bushing B and the stiffness TA4X of the bushing A4 mostly.(3)
The trends ofΔRmsLocal and ΔRmsGlobal with bushing stiffness are influenced by the speed of the vehicle.(4)
Optimal matching of suspension parameters by NSGA-II algorithm at different speeds: at the speed of 30 km/h, the longitudinalΔRmsLocal and the vertical ΔRmsLocal of the front seat rail decrease by 25.94% and 1.49%, respectively. The longitudinal ΔRmsLocal and vertical ΔRmsLocal of the rear seat rail decrease by 29.53% and 3.45%, respectively. The other indexes are within reasonable range. The matching combination of suspension parameters that makes the vehicle ride comfort better is different at different vehicle speeds.



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*Source: 5806444-2021-12-16.xml*