# Collaborative Optimization for Torsion Beam Opening Direction and Rubber Bushing Installation Angle of Torsion Beam Suspension to Improve Frequency Response Characteristics of the Vehicle

Authors: Jin Gao
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The structure of torsion beam suspension is analyzed, and based on the real vehicle, the torsion beam suspension coupled with flexible torsion beam and the multibody dynamics model of vehicle is established. The influence mechanism of torsion beam opening angle and rubber bushing installation angle on roll center and roll stiffness of suspension is systematically analyzed. Under different opening angles of torsion beam and different rubber bushing installation angles, a comparative analysis of the changes of the toe angle and camber angle is carried out when the torsion beam suspension is moved and subjected to force. Based on simulation of the sine swept steering input of vehicle, the frequency response characteristic index of vehicle is taken as the optimization objectives, the opening angle of the torsion beam and rubber bushing installation angle of the vehicle are taken as the optimization variables, and performance of the whole vehicle is optimized by using multiobjective optimization algorithm. The optimized vehicle and original vehicle are compared with the frequency response characteristics and steady-state characteristics. The results show the frequency response and steady-state characteristics of the optimized vehicle have been improved.


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## Body

## 1. Introduction

Torsion beam suspension is a more common form of vehicle rear suspension, and its characteristics have an important influence on the handling stability of the vehicle. When the suspension is in motion and under force, the suspension characteristics are mainly manifested as the kinematics and compliance characteristics of the suspension. The kinematics characteristic mainly refers to the change of the suspension performance when suspension is in motion. It mainly analyzes the wheel travel in the parallel and opposite direction. The change characteristics of the wheel alignment angle caused by the deformation of the rubber bushing in the suspension are called the compliance characteristic. This paper will conduct in-depth and detailed research on the relationship between the torsion beam suspension structure and the kinematics and compliance characteristics, as well as the relationship with the frequency response characteristics of vehicle. There is research about the frequency response of vehicle and suspension. Zhang et al. proposed a robust fault-tolerant output feedback control strategy with finite-frequency constraint to synchronously control the active suspension and dynamic vibration absorber (DVA) for IWM-driven Evs [1]. Liu et al. address the challenges of ride comfort improvement and in-wheel-motor vibration suppression in in-wheel-motor-driven electric vehicle [2].At present, many experts have conducted research on torsion beam suspension. Fichera et al. used a linear method in view of component modal synthesis to represent a flexible torsion beam with a multibody model and compared and analyzed the method with the nonlinear finite element method. The final conclusion is that the accuracy of the multibody model is poor, but its calculation speed is faster [3]. Wang et al. optimized the torsion beam with multiobjective in view of lightweight problem of torsion beam. Based on the establishment of the torsion beam topology optimization model, the mass and maximum deformation are used as the optimization objectives, and strength, torsion stiffness, and main vibration mode frequency are used as constraints to optimize. Finally, under the condition of reducing the mass of torsion beam, its strength and maximum deformation and other parameters still meet the requirements of use [4]. Vasconcelos et al. used the finite element method to numerically simulate the torsion beam suspension to evaluate the influence of torsion beam on the motion characteristics of torsion beam suspension. Through analysis, the results show when the orientation of lateral beam faces the rear end of the vehicle, the change in the camber angle is small [5]. Hyun et al. studied a torsion beam multibody dynamics model based on coupling the hard points and characteristics of the torsion beam and used this model to predict the change of dynamic response. Finally, through the test, the results show that even if the shape of the torsion beam is uncertain, the torsion beam model based on the beam element can easily predict the system according to the changes in the hard points and the performance of the torsion beam [6]. Zhan et al. used three-dimensional digital image correlation (DIC) technology to test the deformation of the weld zone of torsion beam suspension system. The test results show under the action of cyclic loading, there is obvious strain concentration at the weld. The fatigue damage of the torsion beam rear suspension system is simulated by the finite element analysis method. The simulation results show that the location of fatigue crack is in the strain concentration area of the test work. The reinforcement test was carried out in the proving ground. The position of fatigue crack of the torsion beam suspension system of the test vehicle is consistent with the results of DIC and finite element analysis [7]. Chen et al. modeled and analyzed the torsion beam suspension system. After establishing the corresponding multibody dynamics model, the finite element method was used for verification. On this basis, the influence of geometric parameters on the torsion beam suspension was studied. Finally, the torsion beam suspension is optimized to improve the vehicle handling stability [8]. Based on the torsion beam suspension, Sinokrot et al. have studied two methods to solve the nonlinear torsional deformation problem of torsion beams. They are the subconstruction method and the cosimulation method. Finally, the two methods are compared and analyzed, the results show that the simulation time required by the subconstruction method is shorter than that of the cosimulation method, and the communication interval of the cosimulation method has a great influence on the simulation accuracy and simulation time [9]. Choi et al. proposed a multidisciplinary method of automatic optimization of torsion beams and generated optimal design parameters such as thickness and shape. The application of multidisciplinary analysis models such as modal analysis, dynamic analysis of rolling modal and fatigue analysis shortens the analysis time while enhancing the durability and K&C characteristics of the torsion beam suspension [10]. Carneiro et al. used load to generate DOE and fatigue analysis to evaluate the impact of load changes of the components on the durability of the rear suspension [11]. Several scholars have studied the torsion beam suspension characteristics and the effect on the whole vehicle roll characteristics [12–15]. Some scholars have also studied the optimization of torsion beam suspension and the optimization algorithms [16–21].The above researches on torsion beam suspension mainly includes three aspects. One is to study the influence of torsion beam parameters on suspension performance, and the other is to carry out finite element analysis on torsion beam to study its durability and lightness quantification and other issues, the third is to research and analyze the modeling of torsion beam suspension. For the research of these experts, there are still some issues about torsion beam suspension that have not been studied in depth:(1)
The structure of the torsion beam suspension, especially the opening angle of torsion beam and the rubber bushing installation angle of the torsion beam suspension, has not been systematically analyzed in detail.(2)
The above papers did not execute a systematic and detailed research on the influence mechanism of the opening angle of torsion beam and the rubber bushing installation angle of torsion beam suspension on the kinematics characteristic and compliance characteristic.(3)
The influence of the opening angle of torsion beam and the installation angle of the rubber bushing of the torsion beam suspension on the frequency response characteristics of vehicle has not been studied in depth.(4)
The opening angle of torsion beam and the installation angle of the rubber bushing of the torsion beam suspension beam have complex effects on the frequency response characteristics. The above researches did not consider the influence of rubber bushing installation angle and the opening angle of torsion beam on the frequency response characteristics at the same time.For the problems that the above researches have not considered, this paper analyzes the structure of the torsion beam suspension in detail and systematically and analyzes the influence mechanism of the opening angle of torsion beam and the rubber bushing installation angle of torsion beam suspension on the suspension kinematics and compliance characteristics. Then analyzed the influence mechanism of the opening angle of torsion beam and the rubber bushing installation angle on the frequency response characteristics of the vehicle. And on this basis, the opening angle of torsion beam and the rubber bushing installation angle of torsion beam suspension are taken as optimization variables, and the frequency response characteristic index is taken as the optimization objective. The multiobjective optimization algorithm is used to coordinately optimize the opening angle of torsion beam and the rubber bushing installation angle, so as to improve the frequency response characteristics of the vehicle.


## 2. Structure and Motion Analysis of Torsion Beam Suspension

For the torsion beam suspension, changes in the opening angle of torsion beam and the installation angle of the rubber bushing will affect the shear center, roll center, and roll stiffness of suspension, and then affect the frequency response characteristics. In order to analyze its influence mechanism, it is necessary to analyze the structure of the torsion beam suspension.


### 2.1. Structural Analysis of Torsion Beam Suspension

The structure of the torsion beam suspension is shown in Figure1(a). A beam is connected between the two longitudinal arms of the suspension, similar to an H shape. When the wheel is impacted, the beam will twist and part of the impact will be absorbed, thereby reducing the impact transmission to the other wheel. In Figure 1(b), point P1 is the point where the torsion beam suspension is connected to the vehicle body, point P2 is the shear center of the torsion beam suspension, point P3 is the wheel center, and point P4 is the point where the wheel is in contact with the ground. Coordinates defining each direction of torsion beam suspension are shown in Figure 1(b) [12].Figure 1
Schematic diagram of torsion beam suspension structure.
(a)(b)In order to facilitate the analysis of torsion beam suspension, the torsion beam suspension is equivalent to a semitrailing arm suspension. The pointP1 is set as the origin of the coordinates of the subcoordinate system, the horizontal axis and vertical axis of the subcoordinate system are parallel to the coordinates of the global coordinate system. And n is recorded as the unit vector from origin of the global coordinate system to the point P1. Suppose half of the torsion beam rotates angle θ about the unit vector n, the unit vector n is going in the direction from P1 to P2. The coordinate transformation matrix A can be expressed by Euler parameter ρ: (1)n=P2−P1P2−P1=nx,ny,nzT,ρ=λ0,λ1,λ2,λT=cosθ2,nxsinθ2,nysinθ2,nzsinθ2,A=A11A12A13A21A22A23A31A32A33,=λ02+λ12−λ22−λ322λ1λ2−λ3λ02λ1λ3+λ2λ02λ1λ2+λ3λ0−λ12+λ22−λ32+λ022λ2λ3−λ1λ02λ1λ3−λ2λ02λ2λ3+λ1λ0λ02−λ12−λ22+λ32.For the opening angle of torsion beam, there are typically 4 opening angles, respectively 0°, 90°, 180°, and 270°. In order to facilitate analysis, these four typical opening angles of torsion beam are defined as the opening directions shown in Figure2.Figure 2
Diagram of opening angle of torsion beam.The installation direction of the rubber bushing of the torsion beam suspension is expressed by the installation angle. The rubber bushing installation angle of torsion beam suspension is shown in Figure3.Figure 3
Diagram of rubber bushing installation angle.


### 2.2. Determine the Shear Center Position of the Torsion Beam Suspension

The shear center of the torsion beam is a point where the object is subjected to bending moment but not torque when the line of action of the transverse force passes through the point. When the line of action of the force does not apply at this point, the beam is subjected to both bending moment and torque.For the cross section of beam, the shear force is uniformly distributed along the thickness of the thin wall direction. A transverse forceFsz acts on the beam, and its line of action passes by the shear center of the cross section. There are bending stress and shear stress on the cross section. Figure 4 shows the cross section of the torsion beam suspension, where the side part of the torsion beam suspension is an annular area composed of a rectangular part and an arc part. Yan et al. analyzed the shear stress of the beam of torsion beam [13] and the shear stress τ of the section can be obtained as follows:(2)τ=FsyMzIzl1.Figure 4
Schematic diagram of cross section of beam.In (2), Fsy is the shear force of the cross section, Mz is the static moment of side relative to the z axis, Iz is the moment of inertia of entire section relative to the z axis, l1 is the thickness of the section. According to (2), to obtain the size of the shear stress at each point of the section, it needs to obtain the Mz and the Iz of each point.In Figure4, η is the half angle of the circular arc, r3 is the radius of the arc segment of the center line of section, l2 is the length of the rectangle, l1 is the thickness of the section. According to the principle of plane coordinate transformation, if the coordinate system yoz is rotated counter-clockwise by an angle π/2−η, the coordinate of the y′o′z′ coordinate system is Py′,Pz′, then:(3)Py=Pz′sinη+Pz′cosη.In the rectangular part, taking any element of lengthξ, ξ<l2, as shown in Figure 4, the static moment of this element relative to z-axis is as follows:(4)Mz0=∫S1ydS=∫S1y′sinη+z′cosηdS.In (4), S1 is the area of the shaded part of the rectangle. It can be obtained after calculation and sorting out:(5)Mz0=r3ξl1sinη+l1cosξl2−12ξ2.Taking another small segment ofη1 in the circular arc, 0<η1<η, the static moment of the area from the element to the edge of the section relative to the z axis is as follows:(6)Mz1=∫S2ydS+∫S3ydS.In (6), S2 is the area of the whole rectangular part, S3 is the area of a small circular arc of the shaded part. After calculation and sorting out, there is(7)Mz1=r3l1l2sinη+12l1l21cosη+r32l1cosη−η1−cosη.After calculating the static moment of the section of torsion beam, the moment of inertia of the torsion beam suspension is calculated. The moment of inertia of the rectangular part relative to thez-axis is calculated as(8)Iz0=∫S2y2dS=∫S2y′sinη+z′cosη2dS.After calculation and sorting out:(9)Iz0=13l1l23cos2η+r3l1l22cosηsinη+13l1l23r32+14l12sin2η.The moment of inertia of the circular arc is as follows:(10)Iz1=∫S4y2dS=l1r33η−cosηsinη.The moment of inertia of the whole section relative to thez axis is as follows:(11)Iz=2Iz0+Iz1.After the static moment and the moment of inertia of torsion beam suspension are calculated, the position of shear center is determined by combining with (11). The shear stress τ0=FszMz0/Izl1 at any point of the rectangular part and the shear stress τ1=FszMz1/Izl1 at the circular arc part can be obtained through the static moment and the moment of inertia. As the cross section is symmetric, and the force passes through the shear center, the shear force of the whole cross section takes a torque on the center o of the arc, the cross section does not torsion, and the torque balance is as follows:(12)FMzaz=∫Sr3τdS.In (12), FMz is the total shear force on the cross section, its value is F; τ=τ0+τ1; S is the area of the entire section. Combining equations (11-14), substituting the dimensional data of the section, az can be calculated, so that the position of the shear center can be determined.


### 2.3. Analysis of Roll Center and Equivalent Roll Stiffness of Torsion Beam Suspension

In order to calculate the roll stiffness of torsion beam, it is necessary to derive the torsion stiffness of torsion beam. Assuming that the torsion beam is a tubular structure, its length is recorded asl4, the closed cross section is constant, and both ends are twisted. On the basis of the linear theory of beam, under the assumption of constant shear force, the torsion stiffness of the torsion beam is expressed as:(13)Kttorqueθ=4S52Gl1l3l4.In (13), S5 is the shadow area in the middle line of the closed section of the torsion beam, l3 is the total length of boundary. G and l1 are the shear modulus and thickness of the torsion beam, respectively. For torsion beam suspension, the cross-sectional area of the center line will change along with its distance from the center. Therefore, the torsion stiffness of the torsion beam can be calculated. In this paper, it is assumed that the beam is a series connection with a finite length Δi in the same section. The equivalent torsion stiffness of the torsion beam is approximately:(14)1Ktors=∑i1Kti=∑iliΔi4Si2Gl1,Ktors=∑i1Kti−1=∑iliΔi4S52Gl1−1.Assuming that the torsion beam with rubber bushing is subjected to a vertical external force at both ends, and the force is in the opposite direction, the work done by the force on the torsion beam is equal to the sum of the torsional potential energy of the torsion beam and the torsion deformation potential energy of the rubber bushing.Under most circumstances, the rubber bushing is installed on the vehicle body at an angle to increase the characteristics of understeer tendency, as shown in Figure3, when considering the influence of rubber bushing on the roll stiffness of the torsion beam suspension, it is assumed that the axial torsion angle θbtz of rubber bushing is half of the torsion angle of torsion beam, and the radial torsion angle θbtc is as large as the wheel camber angle. So there are(15)θbtz=θtwist2,θbtc=θcamber.Assuming that both external forces are at the center of the wheel, the potential energy of suspension when it rolls is denoted byUp, which includes the torsional potential energy of the beam, the torsion potential energy of the rubber bushing, and the work done by the external force Fp. There are(16)Up=12Ktorsθtwist2+52Kbtzθbtz2+52Kbtcθbtc2−2Fpςz,where, Kbtz and Kbtc, respectively, represent the axial torsion stiffness and radial torsion stiffness of the bushing; ςz represents the displacement produced by Fp.In the stationary state, the potential energy should satisfy(17)dUpdςz=0=Ktorsθtwistdθtwistdςz+2Kbtzθbtzdθbtzdςz+2Kbtcθbtcdθbtcdςz−2Fpdςz.In the linear region, the relationship between the torsion angle of the torsion beam, the torsion angle of the rubber bushing, and vertical displacement of the wheel center is expressed as(18)θtwist=dθtwistdςzςz,The radial and axial rotation angles of the rubber bushing can be respectively expressed as(19)θbtz=dθbtzdςzςz=12dθtwistdςzςz,θbtc=dθcamberdςzςz.From equations (21–23), we can get(20)Ktorsdθtwistdςz2+2Kbtzdθbtzdςz2+2Kbtcdθbtcdςz2ςz−2FP=0.According to (20), the equivalent spring stiffness at wheel center is(21)Keq=FPςz=12Ktorsdθtwistdςz2+2Kbtzdθbtzdςz2+2Kbtcdθbtcdςz2.Therefore, roll stiffnessKϕr of the torsion beam suspension can be expressed by wheelbase dr and the equivalent spring stiffness Keq, and after considering the spring stiffness Ksr in the suspension, it can be obtained(22)Kϕr=12Keq+Ksrdr2.The height of static roll center can be calculated based on instantaneous velocity center method and geometric drawing method, as shown in Figure5. The plane P passes through the center of the wheel and is perpendicular to the ground, the connecting lines l5 and l6 of Ol, Or and the shear center S intersect plane P at Ql and Qr, respectively, the connecting lines l7 and l8 between Qr, Ql, and the ground point M,N have an intersection HH; this point is the roll center of suspension.Figure 5
Kinematic analysis of torsion beam suspension.In Figure5, it can be seen that the position of shear center of the torsion beam suspension will affect the position of roll center of the torsion beam suspension. The roll center height of the suspension directly affects the roll stiffness. Therefore, the opening angle of torsion beam and the installation angle of the rubber bushing affect the roll stiffness of suspension by affecting the shear center of suspension.In order to complete the simulation test analysis of torsion beam suspension, the corresponding multirigid body dynamics model of the torsion beam suspension is established according to a certain actual vehicle model. Because the actual torsion beam suspension will produce torsional deformation under the force, the torsion beam of torsion beam suspension is flexible. Therefore, the multirigid body dynamics model of suspension coupled with flexible torsion beam is finally established, as shown in Figure6.Figure 6
Multibody dynamics model of torsion beam suspension.On the basis of the above theoretical analysis, the analysis of opposite wheel travel simulation is carried out on the established torsion beam suspension. As shown in Figure7 is roll center height of torsion beam suspension under different opening angles of torsion beam. As shown in Figure 8 is roll center height of torsion beam suspension under different rubber bushing installation angles.Figure 7
Roll center height of torsion beam suspension under different opening angles of torsion beam.Figure 8
Roll center height of torsion beam suspension under different rubber bushing installation angles.As shown in Figures9 and 10, the roll stiffness of torsion beam suspension under different opening angles of the torsion beam and different rubber bushing installation angles are obtained.Figure 9
Roll stiffness of torsion beam suspension under different opening angles of torsion beam.Figure 10
Roll stiffness of torsion beam suspension under different rubber bushing installation angles.In Figure9, from the previous analysis, it can be seen that when the opening angle of torsion beam of the torsion beam suspension changes, roll stiffness of vehicle will change, and the shear center of the suspension will change as well, which will affect the height of the roll center. Therefore, the roll stiffness will change with the opening angle of torsion beam.In Figure10, when the installation angle of the rubber bushing is changed, 2Kbtzdθbtz/dςz2 and 2Kbtcdθbtc/dςz2 in formula (21) will change accordingly, resulting in a change of Keq, and finally, the roll stiffness Kϕr of the suspension will change.


## 2.1. Structural Analysis of Torsion Beam Suspension

The structure of the torsion beam suspension is shown in Figure1(a). A beam is connected between the two longitudinal arms of the suspension, similar to an H shape. When the wheel is impacted, the beam will twist and part of the impact will be absorbed, thereby reducing the impact transmission to the other wheel. In Figure 1(b), point P1 is the point where the torsion beam suspension is connected to the vehicle body, point P2 is the shear center of the torsion beam suspension, point P3 is the wheel center, and point P4 is the point where the wheel is in contact with the ground. Coordinates defining each direction of torsion beam suspension are shown in Figure 1(b) [12].Figure 1
Schematic diagram of torsion beam suspension structure.
(a)(b)In order to facilitate the analysis of torsion beam suspension, the torsion beam suspension is equivalent to a semitrailing arm suspension. The pointP1 is set as the origin of the coordinates of the subcoordinate system, the horizontal axis and vertical axis of the subcoordinate system are parallel to the coordinates of the global coordinate system. And n is recorded as the unit vector from origin of the global coordinate system to the point P1. Suppose half of the torsion beam rotates angle θ about the unit vector n, the unit vector n is going in the direction from P1 to P2. The coordinate transformation matrix A can be expressed by Euler parameter ρ: (1)n=P2−P1P2−P1=nx,ny,nzT,ρ=λ0,λ1,λ2,λT=cosθ2,nxsinθ2,nysinθ2,nzsinθ2,A=A11A12A13A21A22A23A31A32A33,=λ02+λ12−λ22−λ322λ1λ2−λ3λ02λ1λ3+λ2λ02λ1λ2+λ3λ0−λ12+λ22−λ32+λ022λ2λ3−λ1λ02λ1λ3−λ2λ02λ2λ3+λ1λ0λ02−λ12−λ22+λ32.For the opening angle of torsion beam, there are typically 4 opening angles, respectively 0°, 90°, 180°, and 270°. In order to facilitate analysis, these four typical opening angles of torsion beam are defined as the opening directions shown in Figure2.Figure 2
Diagram of opening angle of torsion beam.The installation direction of the rubber bushing of the torsion beam suspension is expressed by the installation angle. The rubber bushing installation angle of torsion beam suspension is shown in Figure3.Figure 3
Diagram of rubber bushing installation angle.


## 2.2. Determine the Shear Center Position of the Torsion Beam Suspension

The shear center of the torsion beam is a point where the object is subjected to bending moment but not torque when the line of action of the transverse force passes through the point. When the line of action of the force does not apply at this point, the beam is subjected to both bending moment and torque.For the cross section of beam, the shear force is uniformly distributed along the thickness of the thin wall direction. A transverse forceFsz acts on the beam, and its line of action passes by the shear center of the cross section. There are bending stress and shear stress on the cross section. Figure 4 shows the cross section of the torsion beam suspension, where the side part of the torsion beam suspension is an annular area composed of a rectangular part and an arc part. Yan et al. analyzed the shear stress of the beam of torsion beam [13] and the shear stress τ of the section can be obtained as follows:(2)τ=FsyMzIzl1.Figure 4
Schematic diagram of cross section of beam.In (2), Fsy is the shear force of the cross section, Mz is the static moment of side relative to the z axis, Iz is the moment of inertia of entire section relative to the z axis, l1 is the thickness of the section. According to (2), to obtain the size of the shear stress at each point of the section, it needs to obtain the Mz and the Iz of each point.In Figure4, η is the half angle of the circular arc, r3 is the radius of the arc segment of the center line of section, l2 is the length of the rectangle, l1 is the thickness of the section. According to the principle of plane coordinate transformation, if the coordinate system yoz is rotated counter-clockwise by an angle π/2−η, the coordinate of the y′o′z′ coordinate system is Py′,Pz′, then:(3)Py=Pz′sinη+Pz′cosη.In the rectangular part, taking any element of lengthξ, ξ<l2, as shown in Figure 4, the static moment of this element relative to z-axis is as follows:(4)Mz0=∫S1ydS=∫S1y′sinη+z′cosηdS.In (4), S1 is the area of the shaded part of the rectangle. It can be obtained after calculation and sorting out:(5)Mz0=r3ξl1sinη+l1cosξl2−12ξ2.Taking another small segment ofη1 in the circular arc, 0<η1<η, the static moment of the area from the element to the edge of the section relative to the z axis is as follows:(6)Mz1=∫S2ydS+∫S3ydS.In (6), S2 is the area of the whole rectangular part, S3 is the area of a small circular arc of the shaded part. After calculation and sorting out, there is(7)Mz1=r3l1l2sinη+12l1l21cosη+r32l1cosη−η1−cosη.After calculating the static moment of the section of torsion beam, the moment of inertia of the torsion beam suspension is calculated. The moment of inertia of the rectangular part relative to thez-axis is calculated as(8)Iz0=∫S2y2dS=∫S2y′sinη+z′cosη2dS.After calculation and sorting out:(9)Iz0=13l1l23cos2η+r3l1l22cosηsinη+13l1l23r32+14l12sin2η.The moment of inertia of the circular arc is as follows:(10)Iz1=∫S4y2dS=l1r33η−cosηsinη.The moment of inertia of the whole section relative to thez axis is as follows:(11)Iz=2Iz0+Iz1.After the static moment and the moment of inertia of torsion beam suspension are calculated, the position of shear center is determined by combining with (11). The shear stress τ0=FszMz0/Izl1 at any point of the rectangular part and the shear stress τ1=FszMz1/Izl1 at the circular arc part can be obtained through the static moment and the moment of inertia. As the cross section is symmetric, and the force passes through the shear center, the shear force of the whole cross section takes a torque on the center o of the arc, the cross section does not torsion, and the torque balance is as follows:(12)FMzaz=∫Sr3τdS.In (12), FMz is the total shear force on the cross section, its value is F; τ=τ0+τ1; S is the area of the entire section. Combining equations (11-14), substituting the dimensional data of the section, az can be calculated, so that the position of the shear center can be determined.


## 2.3. Analysis of Roll Center and Equivalent Roll Stiffness of Torsion Beam Suspension

In order to calculate the roll stiffness of torsion beam, it is necessary to derive the torsion stiffness of torsion beam. Assuming that the torsion beam is a tubular structure, its length is recorded asl4, the closed cross section is constant, and both ends are twisted. On the basis of the linear theory of beam, under the assumption of constant shear force, the torsion stiffness of the torsion beam is expressed as:(13)Kttorqueθ=4S52Gl1l3l4.In (13), S5 is the shadow area in the middle line of the closed section of the torsion beam, l3 is the total length of boundary. G and l1 are the shear modulus and thickness of the torsion beam, respectively. For torsion beam suspension, the cross-sectional area of the center line will change along with its distance from the center. Therefore, the torsion stiffness of the torsion beam can be calculated. In this paper, it is assumed that the beam is a series connection with a finite length Δi in the same section. The equivalent torsion stiffness of the torsion beam is approximately:(14)1Ktors=∑i1Kti=∑iliΔi4Si2Gl1,Ktors=∑i1Kti−1=∑iliΔi4S52Gl1−1.Assuming that the torsion beam with rubber bushing is subjected to a vertical external force at both ends, and the force is in the opposite direction, the work done by the force on the torsion beam is equal to the sum of the torsional potential energy of the torsion beam and the torsion deformation potential energy of the rubber bushing.Under most circumstances, the rubber bushing is installed on the vehicle body at an angle to increase the characteristics of understeer tendency, as shown in Figure3, when considering the influence of rubber bushing on the roll stiffness of the torsion beam suspension, it is assumed that the axial torsion angle θbtz of rubber bushing is half of the torsion angle of torsion beam, and the radial torsion angle θbtc is as large as the wheel camber angle. So there are(15)θbtz=θtwist2,θbtc=θcamber.Assuming that both external forces are at the center of the wheel, the potential energy of suspension when it rolls is denoted byUp, which includes the torsional potential energy of the beam, the torsion potential energy of the rubber bushing, and the work done by the external force Fp. There are(16)Up=12Ktorsθtwist2+52Kbtzθbtz2+52Kbtcθbtc2−2Fpςz,where, Kbtz and Kbtc, respectively, represent the axial torsion stiffness and radial torsion stiffness of the bushing; ςz represents the displacement produced by Fp.In the stationary state, the potential energy should satisfy(17)dUpdςz=0=Ktorsθtwistdθtwistdςz+2Kbtzθbtzdθbtzdςz+2Kbtcθbtcdθbtcdςz−2Fpdςz.In the linear region, the relationship between the torsion angle of the torsion beam, the torsion angle of the rubber bushing, and vertical displacement of the wheel center is expressed as(18)θtwist=dθtwistdςzςz,The radial and axial rotation angles of the rubber bushing can be respectively expressed as(19)θbtz=dθbtzdςzςz=12dθtwistdςzςz,θbtc=dθcamberdςzςz.From equations (21–23), we can get(20)Ktorsdθtwistdςz2+2Kbtzdθbtzdςz2+2Kbtcdθbtcdςz2ςz−2FP=0.According to (20), the equivalent spring stiffness at wheel center is(21)Keq=FPςz=12Ktorsdθtwistdςz2+2Kbtzdθbtzdςz2+2Kbtcdθbtcdςz2.Therefore, roll stiffnessKϕr of the torsion beam suspension can be expressed by wheelbase dr and the equivalent spring stiffness Keq, and after considering the spring stiffness Ksr in the suspension, it can be obtained(22)Kϕr=12Keq+Ksrdr2.The height of static roll center can be calculated based on instantaneous velocity center method and geometric drawing method, as shown in Figure5. The plane P passes through the center of the wheel and is perpendicular to the ground, the connecting lines l5 and l6 of Ol, Or and the shear center S intersect plane P at Ql and Qr, respectively, the connecting lines l7 and l8 between Qr, Ql, and the ground point M,N have an intersection HH; this point is the roll center of suspension.Figure 5
Kinematic analysis of torsion beam suspension.In Figure5, it can be seen that the position of shear center of the torsion beam suspension will affect the position of roll center of the torsion beam suspension. The roll center height of the suspension directly affects the roll stiffness. Therefore, the opening angle of torsion beam and the installation angle of the rubber bushing affect the roll stiffness of suspension by affecting the shear center of suspension.In order to complete the simulation test analysis of torsion beam suspension, the corresponding multirigid body dynamics model of the torsion beam suspension is established according to a certain actual vehicle model. Because the actual torsion beam suspension will produce torsional deformation under the force, the torsion beam of torsion beam suspension is flexible. Therefore, the multirigid body dynamics model of suspension coupled with flexible torsion beam is finally established, as shown in Figure6.Figure 6
Multibody dynamics model of torsion beam suspension.On the basis of the above theoretical analysis, the analysis of opposite wheel travel simulation is carried out on the established torsion beam suspension. As shown in Figure7 is roll center height of torsion beam suspension under different opening angles of torsion beam. As shown in Figure 8 is roll center height of torsion beam suspension under different rubber bushing installation angles.Figure 7
Roll center height of torsion beam suspension under different opening angles of torsion beam.Figure 8
Roll center height of torsion beam suspension under different rubber bushing installation angles.As shown in Figures9 and 10, the roll stiffness of torsion beam suspension under different opening angles of the torsion beam and different rubber bushing installation angles are obtained.Figure 9
Roll stiffness of torsion beam suspension under different opening angles of torsion beam.Figure 10
Roll stiffness of torsion beam suspension under different rubber bushing installation angles.In Figure9, from the previous analysis, it can be seen that when the opening angle of torsion beam of the torsion beam suspension changes, roll stiffness of vehicle will change, and the shear center of the suspension will change as well, which will affect the height of the roll center. Therefore, the roll stiffness will change with the opening angle of torsion beam.In Figure10, when the installation angle of the rubber bushing is changed, 2Kbtzdθbtz/dςz2 and 2Kbtcdθbtc/dςz2 in formula (21) will change accordingly, resulting in a change of Keq, and finally, the roll stiffness Kϕr of the suspension will change.


## 3. Analysis of Suspension Motion and Force

Based on the establishment of the model, the kinematics characteristic and compliance characteristic suspension can be analyzed and studied through the combination of theory and simulation.


### 3.1. Analysis of Wheel Center Movement

On the basis of the analysis in Section2.1, defining a vector whose coordinates are x′, y′, and z′, and then the motion at the wheel center can be deduced.(23)w3′=P3−P1=w3x′w3y′w3z′=P3xP3yP3z−P1xP1yP1z,ς=ςxςyςzT=P1+Aw3′.Therefore, the vertical displacement at the wheel center is(24)ςz=A31w3x′+A32w3y′+A33w3y′+P1z=2λ1λ3−λ2λw3x′+2λ2λ−3λ1λ0w3y′+−λ12−λ22+λ32+λ02wz′+P1z.Defining the unit vector at the wheel axis aswp′, and wp′=010T, as shown in Figure 11. According to the theory of coordinate transformation, the coordinates of the unit vector of the wheel center can be obtained as(25)wP=AwP′=wPxwPywPz=2λ1λ2−λ3λ0−λ12+λ22−λ32+λ022λ2λ3+λ1λ.Figure 11
Schematic diagram of the toe angle and camber angle.Figure11 shows the schematic diagram of the change of the toe angle and the camber angle after the torsion beam suspension is twisted at the θ angle [14]. According to Figure 11, the calculation equations of the toe angle and camber angle can be obtained:(26)αe=−tan−1wpxwpy,θcamber=−tan−1wpzwpx2+wpy2.Setting 4 groups of typical opening angles of torsion beam, and use the established torsion beam suspension to perform opposite wheel travel simulation under different opening angles of torsion beam, the toe angle and camber angle as shown in Figures12 and 13 are obtained. Set 3 groups of different rubber bushing installation angle values and then perform opposite wheel travel simulation of the torsion beam suspension, the change of toe angle of the wheel as shown in Figure 14 is obtained.Figure 12
The change of the toe angle with wheel travel under different opening angles of torsion beam.Figure 13
The change of the camber angle with wheel travel under different opening angles of torsion beam.Figure 14
The change of the toe angle with the wheel travel under different rubber bushing installation angles.For the opening angle of torsion beam, when it changes, the shear center of the torsion beam suspension will change accordingly, which in turn causes the suspension to change and ultimately affects the kinematics characteristic of suspension. For the installation angle of the rubber bushing, when it changes,2Kbtzdθbtz/dςz2 and 2Kbtcdθbtc/dςz2 in (21) will change accordingly, resulting in a change in Keq, as a result, the roll stiffness Kϕr of the suspension is changed, and the kinematics characteristic of the suspension is also changed. Therefore, changes in the opening angle of torsion beam and the installation angle of the rubber bushing will cause toe angle and camber angle to change differently when the wheel bounces. In Figures 12 and 13, the change diagram of the toe angle and camber angle has different changes under different opening angles of torsion beam. In Figure 14, when the rubber bushing installation angle is different, the change of toe angle is different.For the opposite wheel travel simulation of the suspension, the wheel that jumps down is the inner wheel when the vehicle turns, and the wheel that jumps up is the outer wheel when the vehicle turns. For the rear suspension, when the vehicle turns, the toe angle of inner wheel increases negatively and outside wheel increases positively, the vehicle will increase the understeer tendency. In Figure12, when the opening angle of torsion beam is 90°, the understeer tendency of the vehicle is greater than that of the other three groups of opening angles.


### 3.2. Force Analysis of Torsion Beam Suspension

When the wheels of the vehicle are subjected to longitudinal force and lateral force, the torsion beam suspension of the vehicle will produce corresponding deformation, which will affect the performance of the suspension and the performance of the whole vehicle. Since this article mainly conducts the sine swept steering input simulation, the longitudinal speed remains constant, so the torsion beam suspension is mainly affected by the lateral force at this time. Figure15 shows the deformation of torsion beam suspension when it is subjected to lateral force (in all equations assume a small angle) [15].Figure 15
Deformation of torsion beam suspension when subjected to lateral force.When the rubber bushing installation angle is 0, for the lateral axis steering compliance characteristic and the compliance characteristic of the wheel center, (27) and (28) can be derived:(27)ΔδaxleFyr=2Ltdt2Kbl,(28)ΔyFyr=12Ksy+2Lt2dt2Kbl,where, Ksy indicates that the torsion beam suspension itself has a certain lateral stiffness; Kbl represents the radial stiffness of the bush along the x direction; Fyr,Δy, respectively, represent the force and displacement of the tire along the y direction.Assuming that the shear center of torsion beam suspension coincides with the center of mass, the side view of the left rear wheel vibration system when the vehicle is vibrating is shown in Figure16.Figure 16
Motion and force diagram of torsion beam suspension.In Figure16. s,Ls,Lt are the distance from the beam center, the shear center, the installation point of the trailing arm and the wheel to the installation point of the trailing arm and the body, respectively; Ksr,Csr are the stiffness of the spring and damping of the shock absorber respectively; Ktr is stiffness of tire; φp is the pitch angle of the body in a certain position; ϕo is the roll angle of the body in a certain position; ZtfL,ZtfR,ZrL,ZrR are the vertical displacement of the connecting point of left and right trailing arms and the vehicle body, and the vertical displacement of the left and right wheel center, respectively; Lb is the distance between the installation points of the left and right trailing arms on the body; L4 is the distance between the installation points of the left and right trailing arms on the wheels; L1,L2,L3 are the distance between the lower fulcrum of the spring damper, the longitudinal swing arm at the wheel installation point, and the wheel center to the axis of rotation, respectively; mwr is the quality of the wheel; FwL is the vertical force acting on the wheel center by the ground; FrL is the vertical force on the lower fulcrum of the spring damper.When moving vertically, the pitch angleφt of the center of mass of torsion beam suspension in the longitudinal section plane is(29)φt=zrL−zrR−ztfL−ztfR2Lt.The roll angleϕr of torsion beam suspension is(30)ϕr=LsLtzrL−zrRL4−ϕo,The torsion angleφb of torsion beam suspension is approximately:(31)φb=zrL−zrR−ztfL−ztfRLt.The vertical forceFtrL, FtrR on the mounting point of the left and right trailing arms on the wheel, and the vertical force FtfL, FtfR on the mounting point of the left and right trailing arms on the vehicle body are(32)FtrL=KtorsφbLt,FtrR=−KtorsφbLt,FtfL=L4FtrLLb,FtfR=L4FtrRLb.It can be seen that the larger the span of the body installation point, the more conducive to reducing the force of the rubber bushing and improving the fatigue life. When the left and right wheels are excited in different directions and displacements, the torque absorbed by the torsion beam suspension is(33)Mtors=KtorsΔzL−ΔzRLt=FtrLLt.From the kinematics analysis of the rear suspension system, it can be seen when rotating along the line from the front hinge point to the shear center, the static torque balance equation can be established along the axis of rotation as(34)FwLL3=FtrLL2+FrLL1.It can be seen that the shear center of the torsion beam suspension and the position of hinge point have an important influence on the force relationship.The change of the opening angles of torsion beam causes the change of roll center of suspension, which leads to the change ofφb in (32). As a result, the force of the suspension changes, and finally, the C characteristic of the suspension will change. So in Figures 17 and 18, when the opening angles of torsion beam is different, the toe angle and camber angle will change differently when the force is applied.Figure 17
The toe angle changes with the wheel lateral force under different opening angles of torsion beam.Figure 18
The change of the camber angle with wheel lateral force under different opening angles of torsion beam.In Figure19, when the installation angle of the rubber bushing is changed, the axial torsion angle θbtz, the radial torsion angle θbtc, and the force of the rubber bushing will change, and eventually, the C characteristic of the suspension will change as well.Figure 19
The change of the toe angle with the wheel lateral force under different rubber bushing installation angles.


## 3.1. Analysis of Wheel Center Movement

On the basis of the analysis in Section2.1, defining a vector whose coordinates are x′, y′, and z′, and then the motion at the wheel center can be deduced.(23)w3′=P3−P1=w3x′w3y′w3z′=P3xP3yP3z−P1xP1yP1z,ς=ςxςyςzT=P1+Aw3′.Therefore, the vertical displacement at the wheel center is(24)ςz=A31w3x′+A32w3y′+A33w3y′+P1z=2λ1λ3−λ2λw3x′+2λ2λ−3λ1λ0w3y′+−λ12−λ22+λ32+λ02wz′+P1z.Defining the unit vector at the wheel axis aswp′, and wp′=010T, as shown in Figure 11. According to the theory of coordinate transformation, the coordinates of the unit vector of the wheel center can be obtained as(25)wP=AwP′=wPxwPywPz=2λ1λ2−λ3λ0−λ12+λ22−λ32+λ022λ2λ3+λ1λ.Figure 11
Schematic diagram of the toe angle and camber angle.Figure11 shows the schematic diagram of the change of the toe angle and the camber angle after the torsion beam suspension is twisted at the θ angle [14]. According to Figure 11, the calculation equations of the toe angle and camber angle can be obtained:(26)αe=−tan−1wpxwpy,θcamber=−tan−1wpzwpx2+wpy2.Setting 4 groups of typical opening angles of torsion beam, and use the established torsion beam suspension to perform opposite wheel travel simulation under different opening angles of torsion beam, the toe angle and camber angle as shown in Figures12 and 13 are obtained. Set 3 groups of different rubber bushing installation angle values and then perform opposite wheel travel simulation of the torsion beam suspension, the change of toe angle of the wheel as shown in Figure 14 is obtained.Figure 12
The change of the toe angle with wheel travel under different opening angles of torsion beam.Figure 13
The change of the camber angle with wheel travel under different opening angles of torsion beam.Figure 14
The change of the toe angle with the wheel travel under different rubber bushing installation angles.For the opening angle of torsion beam, when it changes, the shear center of the torsion beam suspension will change accordingly, which in turn causes the suspension to change and ultimately affects the kinematics characteristic of suspension. For the installation angle of the rubber bushing, when it changes,2Kbtzdθbtz/dςz2 and 2Kbtcdθbtc/dςz2 in (21) will change accordingly, resulting in a change in Keq, as a result, the roll stiffness Kϕr of the suspension is changed, and the kinematics characteristic of the suspension is also changed. Therefore, changes in the opening angle of torsion beam and the installation angle of the rubber bushing will cause toe angle and camber angle to change differently when the wheel bounces. In Figures 12 and 13, the change diagram of the toe angle and camber angle has different changes under different opening angles of torsion beam. In Figure 14, when the rubber bushing installation angle is different, the change of toe angle is different.For the opposite wheel travel simulation of the suspension, the wheel that jumps down is the inner wheel when the vehicle turns, and the wheel that jumps up is the outer wheel when the vehicle turns. For the rear suspension, when the vehicle turns, the toe angle of inner wheel increases negatively and outside wheel increases positively, the vehicle will increase the understeer tendency. In Figure12, when the opening angle of torsion beam is 90°, the understeer tendency of the vehicle is greater than that of the other three groups of opening angles.


## 3.2. Force Analysis of Torsion Beam Suspension

When the wheels of the vehicle are subjected to longitudinal force and lateral force, the torsion beam suspension of the vehicle will produce corresponding deformation, which will affect the performance of the suspension and the performance of the whole vehicle. Since this article mainly conducts the sine swept steering input simulation, the longitudinal speed remains constant, so the torsion beam suspension is mainly affected by the lateral force at this time. Figure15 shows the deformation of torsion beam suspension when it is subjected to lateral force (in all equations assume a small angle) [15].Figure 15
Deformation of torsion beam suspension when subjected to lateral force.When the rubber bushing installation angle is 0, for the lateral axis steering compliance characteristic and the compliance characteristic of the wheel center, (27) and (28) can be derived:(27)ΔδaxleFyr=2Ltdt2Kbl,(28)ΔyFyr=12Ksy+2Lt2dt2Kbl,where, Ksy indicates that the torsion beam suspension itself has a certain lateral stiffness; Kbl represents the radial stiffness of the bush along the x direction; Fyr,Δy, respectively, represent the force and displacement of the tire along the y direction.Assuming that the shear center of torsion beam suspension coincides with the center of mass, the side view of the left rear wheel vibration system when the vehicle is vibrating is shown in Figure16.Figure 16
Motion and force diagram of torsion beam suspension.In Figure16. s,Ls,Lt are the distance from the beam center, the shear center, the installation point of the trailing arm and the wheel to the installation point of the trailing arm and the body, respectively; Ksr,Csr are the stiffness of the spring and damping of the shock absorber respectively; Ktr is stiffness of tire; φp is the pitch angle of the body in a certain position; ϕo is the roll angle of the body in a certain position; ZtfL,ZtfR,ZrL,ZrR are the vertical displacement of the connecting point of left and right trailing arms and the vehicle body, and the vertical displacement of the left and right wheel center, respectively; Lb is the distance between the installation points of the left and right trailing arms on the body; L4 is the distance between the installation points of the left and right trailing arms on the wheels; L1,L2,L3 are the distance between the lower fulcrum of the spring damper, the longitudinal swing arm at the wheel installation point, and the wheel center to the axis of rotation, respectively; mwr is the quality of the wheel; FwL is the vertical force acting on the wheel center by the ground; FrL is the vertical force on the lower fulcrum of the spring damper.When moving vertically, the pitch angleφt of the center of mass of torsion beam suspension in the longitudinal section plane is(29)φt=zrL−zrR−ztfL−ztfR2Lt.The roll angleϕr of torsion beam suspension is(30)ϕr=LsLtzrL−zrRL4−ϕo,The torsion angleφb of torsion beam suspension is approximately:(31)φb=zrL−zrR−ztfL−ztfRLt.The vertical forceFtrL, FtrR on the mounting point of the left and right trailing arms on the wheel, and the vertical force FtfL, FtfR on the mounting point of the left and right trailing arms on the vehicle body are(32)FtrL=KtorsφbLt,FtrR=−KtorsφbLt,FtfL=L4FtrLLb,FtfR=L4FtrRLb.It can be seen that the larger the span of the body installation point, the more conducive to reducing the force of the rubber bushing and improving the fatigue life. When the left and right wheels are excited in different directions and displacements, the torque absorbed by the torsion beam suspension is(33)Mtors=KtorsΔzL−ΔzRLt=FtrLLt.From the kinematics analysis of the rear suspension system, it can be seen when rotating along the line from the front hinge point to the shear center, the static torque balance equation can be established along the axis of rotation as(34)FwLL3=FtrLL2+FrLL1.It can be seen that the shear center of the torsion beam suspension and the position of hinge point have an important influence on the force relationship.The change of the opening angles of torsion beam causes the change of roll center of suspension, which leads to the change ofφb in (32). As a result, the force of the suspension changes, and finally, the C characteristic of the suspension will change. So in Figures 17 and 18, when the opening angles of torsion beam is different, the toe angle and camber angle will change differently when the force is applied.Figure 17
The toe angle changes with the wheel lateral force under different opening angles of torsion beam.Figure 18
The change of the camber angle with wheel lateral force under different opening angles of torsion beam.In Figure19, when the installation angle of the rubber bushing is changed, the axial torsion angle θbtz, the radial torsion angle θbtc, and the force of the rubber bushing will change, and eventually, the C characteristic of the suspension will change as well.Figure 19
The change of the toe angle with the wheel lateral force under different rubber bushing installation angles.


## 4. The Analysis of Frequency Response Characteristics of Vehicle

The frequency response characteristics of vehicle is an important part of the transient response characteristics of vehicle. This paper will analyze the frequency response characteristics of vehicle. The transient characteristics index of vehicle includes the yaw rate gain relative to the steering wheel angle, the roll angle gain relative to the lateral acceleration, the resonance frequency, the delay time of the lateral acceleration relative to the steering wheel angle, and the delay time of the yaw rate relative to the lateral acceleration. The resonance frequency is the frequency corresponding to the maximum yaw rate gain relative to the steering wheel angle.


### 4.1. Derivation of Frequency Response Index of Vehicle

Since this paper is based on the sine swept steering input simulation, the longitudinal speed of the vehicle remains constant, so only the three degrees of freedom of the lateral, roll, and yaw of vehicle are considered. Figure20 shows the change of side slip angle βf1,βf2,βr1,βr2 of tires during movement of vehicle. In Figure 20, the longitudinal speed V and yaw rate ψ˙ of the vehicle will produce corresponding components in the longitudinal and lateral directions of the vehicle. For the front and rear wheels of vehicle, the lateral component of the speed of vehicle V is Vsinβ, since the side slip angle β is small, the lateral velocity component is approximately equal to Vβ. df,dr, respectively, represent the wheelbase of the front and rear wheels, lf,lr, respectively, represent the distance from the center of mass to the front and rear axis, lq represents the wheelbase.Figure 20
Schematic diagram of the movement of the vehicle.For the yaw rate of vehicle, its component in the longitudinal direction of the front wheels of vehicle is−dfψ˙/2, the lateral component of the front wheel is lfψ˙, its component in the longitudinal direction of the rear wheel of the vehicle is −drψ˙/2, and its component in the lateral direction of the rear wheel is −lrψ˙.When the vehicle rolls, both the front and the rear wheel will have corresponding side slip angles due to the roll of vehicle, which are denoted as∂αf/∂ϕϕ and ∂αr/∂ϕϕ, respectively, among them, ∂αf/∂ϕ and ∂αr/∂ϕ indicate that when the suspension produces a unit roll angle, the rear wheel produce the corresponding side slip angle. ϕ represents the roll angle of the body. And the front wheel itself has a turning angle δ input by the steering wheel, so the final side slip angles of the front and rear wheels are as follows(35)Front wheel:δ+∂αf∂ϕϕ−β−lfVψ˙,Rear wheel:∂αr∂ϕϕ−β+lrVψ˙.The schematic diagram of the external force received by the vehicle wheel is shown in Figure21. In the figure, Mx,Mz represent the moments around the x-axis and y-axis, respectively. The lateral force received by tire of vehicle consists of two parts in total. The first is the lateral force generated by the side slip angle of the wheel, and the other part is the camber caused by the roll of vehicle, which generates a corresponding lateral force at the wheel. The lateral force generated Ff,Fr by the side slip angle of wheel is calculated by (36) and (37).(36)Ff=Kfδ+∂αf∂ϕϕ−β−lfVψ˙,(37)Fr=Kr∂αr∂ϕϕ−β−lrVψ˙,Kf and Kr in (36) and (37) represent the side slip stiffness of the front and rear tires.Figure 21
Schematic diagram of the external force received by tire of vehicle.And the lateral camber thrust forceFcf,Fcr can be calculated by (38) and (39).(38)Fcf=Kcf∂θcamberf∂ϕϕ,(39)Fcr=Kcr∂θcamberr∂ϕϕ,In the (38) and (39), Kcf,Kcr, respectively, represent the camber stiffness of the front and rear tires, ∂θcamberf/∂ϕ, ∂θcamberr/∂ϕ represent the lateral camber thrust coefficients of the front and rear suspension, respectively.According to the balance conditions of force and moment, the motion equation of the vehicle with 3-DOF (Degree of Freedom) can be obtained:(40)mVdβdt+ψ˙−mshsd2ϕdt2=2Kfδ+∂αf∂ϕϕ−β−lfVψ˙+2Kr∂αr∂ϕϕ−β−lrVψ˙−2Kcf∂θcamberf∂ϕϕ+Kcr∂θcamberr∂ϕϕ,Idψ˙dt−Ixzd2ϕdt2=2Kfδ+∂αf∂ϕϕ−β−lfVψ˙lf−2Kr∂αr∂ϕϕ−β+lrVψ˙lr−2lfKcf∂θcamberf∂ϕϕ−lrKcr∂θcamberr∂ϕϕ,Iϕd2ϕdt2−Ixzdψ˙dt−mshsVdβdt+ψ˙=−Kϕ+msghsϕ−Cϕdϕdt.In (40), I represents the moment of inertia of the vehicle around the z axis, Iϕ represents the moment of roll inertia of the vehicle body around the roll axis, Ixz represents the product of inertia of the vehicle on the x and y axis, hs is the distance from the sprung mass of the vehicle to the roll center, lf,lr respectively represent the distance from the center of mass to the front and rear axis, m is the mass of the whole vehicle, ms is the sprung mass, Kϕ is the roll stiffness of the vehicle, ϕ is the roll angle of the vehicle, and Cϕ is the equivalent damping coefficient of roll motion.After the steering wheel angle, center of mass slip angle, roll angle, and yaw rate are transformed by Laplace, there are(41)δs=∫0∞e−stδtdt,βs=∫0∞e−stβtdt,ϕs=∫0∞e−stϕtdt,ψ˙s=∫0∞e−stψ˙tdt.By bringing (41) into (40), and then performing Laplace transforms on (40), the equation group after Laplace transforms can be obtained.By solving the equation group by Laplace transforms, the corresponding transform function of the vehicle can be obtained, for example, the transform function of yaw rate gain relative to the steering wheel angleψ˙/δs, the transform function of roll angle gain relative to the lateral acceleration ϕ/aysϕ/ays. In the obtained transfer function, make s=jω where j=−1，the frequency response characteristic function of the vehicle is obtained. Such as: ψ˙/δjω, β/δjω, ϕ/ayjω. Then resonance frequency (the frequency value corresponding to the maximum value of the yaw rate gain), the phase lag of the lateral acceleration relative to the steering wheel angle and the phase lag of the yaw rate relative to the lateral acceleration. The phase lag angle can be converted into the corresponding delay time.


### 4.2. The Influence Mechanism of the Opening Angle of Torsion Beam and the Installation Angle of the Rubber Bushing on the Frequency Characteristics of Vehicle

To sum up, the influence mechanism of the opening angle of torsion beam of the torsion beam suspension and the installation angle of the rubber bushing on the vehicle frequency characteristics mainly includes the following aspects:(1)
The opening angle of torsion beam affects the roll center position of the suspension by affecting the shear center of the torsion beam suspension. As a result, the parameterhs in equation (40) changes, and the frequency characteristic parameters of the vehicle also change accordingly.(2)
The opening angle of torsion beam of the torsion beam suspension and the rubber bushing installation angle will affect the roll stiffness of the suspension, thereby affecting the roll stiffnessKϕ of entire vehicle, and ultimately affect the frequency characteristic parameters of the entire vehicle.(3)
Due to the change in the roll stiffness of the torsion beam suspension, the amount of load transfer between the left and right wheels of the suspension will also change accordingly, which will affect the side slip stiffnessKr of the tire.When the vehicle rolls, load transfer occurs between the left and right wheels, and the load transfer is closely related to the roll stiffness of the suspension. When there is load transfer between the left and right wheels of the suspension, the side slip stiffness of the wheels will be affected, which will affect the force of the wheels, and ultimately affect the steering characteristics of the vehicle. When the vehicle is subjected to the lateral force, for the rear axle of a vehicle, the greater the load transfer between left and right wheels, the weaker the understeer characteristics of the vehicle [16].The opposite wheel travel simulation test is performed on the torsion beam suspension, and the load transfer amount between the left and right wheels under different opening angles of torsion beam as shown in Figure22 is obtained.(4)
During the movement of the suspension, the roll angle of the suspension will change the side slip angle of wheel, that is, the∂αf/∂ϕϕ and ∂αf/∂ϕϕ in Eq. (40) will change.(5)
From the previous analysis, it can be seen that different opening angle of torsion beam and rubber bushing installation angles cause different changes in the wheel alignment angles when the suspension wheels are stressed, thereby affecting the frequency response characteristics of the vehicle.Figure 22
Load transfer amount between left and right wheels under different opening angles of torsion beam.


### 4.3. The Influence Trend of the Opening Angle of Torsion Beam and the Rubber Bushing Installation Angle on the Frequency Response Characteristics of Vehicle

According to the previous analysis and the actual parameters of a certain vehicle, the corresponding multibody dynamics model of the whole vehicle is established. The main parameters of the established multibody dynamics model of the original whole vehicle are shown in Table1.Table 1
Vehicle parameters.
ParameterValueSprung mass (kg)1097.00Unsprung mass (kg)551.98Center of mass to front wheelbase (m)0.7099Center of mass to rear wheelbase (m)1.8501Opening angle of torsion beam (°)270.00Spring stiffness of front suspension ( )2.1 × 104Height from center of mass to ground (m)0.378Distance between front wheels (m)1.594Distance between rear wheels (m)1.52Wheelbase (m)2.56Rubber bushing installation angle (°)−25.50Spring stiffness of rear suspension ( )2.27 × 104Since the frequency when the driver turns the steering wheel is about 0.5 Hz, the gain value and delay time at 0.5 Hz are mainly analyzed. As shown in Figures23 to 27, the frequency response characteristic index corresponding to 0.5 Hz changes with the change of the opening angle of torsion beam.Figure 23
The trend of resonant frequency changing with the opening angle of torsion beam.Figure 24
The trend of the roll angle gain relative to the lateral acceleration at 0.5 Hz as the opening angle of torsion beam changes.Figure 25
Trend of the delay time of lateral acceleration relative to the steering wheel angle at 0.5 Hz as the rubber bushing installation angle changes.Figure 26
The trend of the delay time of yaw rate relative to the lateral acceleration at 0.5 Hz as the rubber bushing installation angle changes.Figure 27
The trend of yaw rate gain at 0.5 Hz as the rubber bushing installation angle changes.Figure23 shows the change trend of the resonance frequency with the opening angle of torsion beam. In Figure 23, as the opening angle of torsion beam increases from 70° to 110°, the resonance frequency of the vehicle roughly first increases and then decreases with some fluctuations in the middle. In Figure 24, as the opening angle of torsion beam increases from 70° to 110°, the roll angle gain at 0.5 Hz relative to the lateral acceleration roughly first decreases and then increases.Figure25 shows the trend of the delay time t1 of lateral acceleration relative to the steering wheel angle at 0.5 Hz as the rubber bushing installation angle increases. Figure 26 shows the trend of the delay time t2 of the yaw rate relative to the lateral acceleration at 0.5 Hz as the rubber bushing installation angle increases.In Figure25, as the rubber bushing installation angle increases, the absolute value of the delay time of the lateral acceleration at 0.5 Hz first showed a decreasing trend and then an increasing trend. When the rubber bushing installation angle reaches about 65°, the absolute value of the delay time of the lateral acceleration at 0.5 Hz is slightly reduced.In Figure26, as the rubber bushing installation angle increases, the absolute value of the delay time of the yaw rate at 0.5 Hz relative to the lateral acceleration first decreases, and then increases again. When the rubber bushing installation angle reaches about 65°, the absolute value of the delay time of the yaw rate at 0.5 Hz relative to the lateral acceleration is slightly reduced.Figure27 shows the trend of the yaw rate gain at 0.5 Hz relative to the steering wheel angle as the installation angle of the rubber bushing increases. In Figure 27, as the installation angle of rubber bushing increases, the yaw rate gain relative to the steering wheel angle at 0.5 Hz shows a decreasing trend first, and then an increasing trend. When the rubber bushing installation angle reaches about 45°, the yaw rate gain relative to the steering wheel angle at 0.5 Hz is slightly reduced.


## 4.1. Derivation of Frequency Response Index of Vehicle

Since this paper is based on the sine swept steering input simulation, the longitudinal speed of the vehicle remains constant, so only the three degrees of freedom of the lateral, roll, and yaw of vehicle are considered. Figure20 shows the change of side slip angle βf1,βf2,βr1,βr2 of tires during movement of vehicle. In Figure 20, the longitudinal speed V and yaw rate ψ˙ of the vehicle will produce corresponding components in the longitudinal and lateral directions of the vehicle. For the front and rear wheels of vehicle, the lateral component of the speed of vehicle V is Vsinβ, since the side slip angle β is small, the lateral velocity component is approximately equal to Vβ. df,dr, respectively, represent the wheelbase of the front and rear wheels, lf,lr, respectively, represent the distance from the center of mass to the front and rear axis, lq represents the wheelbase.Figure 20
Schematic diagram of the movement of the vehicle.For the yaw rate of vehicle, its component in the longitudinal direction of the front wheels of vehicle is−dfψ˙/2, the lateral component of the front wheel is lfψ˙, its component in the longitudinal direction of the rear wheel of the vehicle is −drψ˙/2, and its component in the lateral direction of the rear wheel is −lrψ˙.When the vehicle rolls, both the front and the rear wheel will have corresponding side slip angles due to the roll of vehicle, which are denoted as∂αf/∂ϕϕ and ∂αr/∂ϕϕ, respectively, among them, ∂αf/∂ϕ and ∂αr/∂ϕ indicate that when the suspension produces a unit roll angle, the rear wheel produce the corresponding side slip angle. ϕ represents the roll angle of the body. And the front wheel itself has a turning angle δ input by the steering wheel, so the final side slip angles of the front and rear wheels are as follows(35)Front wheel:δ+∂αf∂ϕϕ−β−lfVψ˙,Rear wheel:∂αr∂ϕϕ−β+lrVψ˙.The schematic diagram of the external force received by the vehicle wheel is shown in Figure21. In the figure, Mx,Mz represent the moments around the x-axis and y-axis, respectively. The lateral force received by tire of vehicle consists of two parts in total. The first is the lateral force generated by the side slip angle of the wheel, and the other part is the camber caused by the roll of vehicle, which generates a corresponding lateral force at the wheel. The lateral force generated Ff,Fr by the side slip angle of wheel is calculated by (36) and (37).(36)Ff=Kfδ+∂αf∂ϕϕ−β−lfVψ˙,(37)Fr=Kr∂αr∂ϕϕ−β−lrVψ˙,Kf and Kr in (36) and (37) represent the side slip stiffness of the front and rear tires.Figure 21
Schematic diagram of the external force received by tire of vehicle.And the lateral camber thrust forceFcf,Fcr can be calculated by (38) and (39).(38)Fcf=Kcf∂θcamberf∂ϕϕ,(39)Fcr=Kcr∂θcamberr∂ϕϕ,In the (38) and (39), Kcf,Kcr, respectively, represent the camber stiffness of the front and rear tires, ∂θcamberf/∂ϕ, ∂θcamberr/∂ϕ represent the lateral camber thrust coefficients of the front and rear suspension, respectively.According to the balance conditions of force and moment, the motion equation of the vehicle with 3-DOF (Degree of Freedom) can be obtained:(40)mVdβdt+ψ˙−mshsd2ϕdt2=2Kfδ+∂αf∂ϕϕ−β−lfVψ˙+2Kr∂αr∂ϕϕ−β−lrVψ˙−2Kcf∂θcamberf∂ϕϕ+Kcr∂θcamberr∂ϕϕ,Idψ˙dt−Ixzd2ϕdt2=2Kfδ+∂αf∂ϕϕ−β−lfVψ˙lf−2Kr∂αr∂ϕϕ−β+lrVψ˙lr−2lfKcf∂θcamberf∂ϕϕ−lrKcr∂θcamberr∂ϕϕ,Iϕd2ϕdt2−Ixzdψ˙dt−mshsVdβdt+ψ˙=−Kϕ+msghsϕ−Cϕdϕdt.In (40), I represents the moment of inertia of the vehicle around the z axis, Iϕ represents the moment of roll inertia of the vehicle body around the roll axis, Ixz represents the product of inertia of the vehicle on the x and y axis, hs is the distance from the sprung mass of the vehicle to the roll center, lf,lr respectively represent the distance from the center of mass to the front and rear axis, m is the mass of the whole vehicle, ms is the sprung mass, Kϕ is the roll stiffness of the vehicle, ϕ is the roll angle of the vehicle, and Cϕ is the equivalent damping coefficient of roll motion.After the steering wheel angle, center of mass slip angle, roll angle, and yaw rate are transformed by Laplace, there are(41)δs=∫0∞e−stδtdt,βs=∫0∞e−stβtdt,ϕs=∫0∞e−stϕtdt,ψ˙s=∫0∞e−stψ˙tdt.By bringing (41) into (40), and then performing Laplace transforms on (40), the equation group after Laplace transforms can be obtained.By solving the equation group by Laplace transforms, the corresponding transform function of the vehicle can be obtained, for example, the transform function of yaw rate gain relative to the steering wheel angleψ˙/δs, the transform function of roll angle gain relative to the lateral acceleration ϕ/aysϕ/ays. In the obtained transfer function, make s=jω where j=−1，the frequency response characteristic function of the vehicle is obtained. Such as: ψ˙/δjω, β/δjω, ϕ/ayjω. Then resonance frequency (the frequency value corresponding to the maximum value of the yaw rate gain), the phase lag of the lateral acceleration relative to the steering wheel angle and the phase lag of the yaw rate relative to the lateral acceleration. The phase lag angle can be converted into the corresponding delay time.


## 4.2. The Influence Mechanism of the Opening Angle of Torsion Beam and the Installation Angle of the Rubber Bushing on the Frequency Characteristics of Vehicle

To sum up, the influence mechanism of the opening angle of torsion beam of the torsion beam suspension and the installation angle of the rubber bushing on the vehicle frequency characteristics mainly includes the following aspects:(1)
The opening angle of torsion beam affects the roll center position of the suspension by affecting the shear center of the torsion beam suspension. As a result, the parameterhs in equation (40) changes, and the frequency characteristic parameters of the vehicle also change accordingly.(2)
The opening angle of torsion beam of the torsion beam suspension and the rubber bushing installation angle will affect the roll stiffness of the suspension, thereby affecting the roll stiffnessKϕ of entire vehicle, and ultimately affect the frequency characteristic parameters of the entire vehicle.(3)
Due to the change in the roll stiffness of the torsion beam suspension, the amount of load transfer between the left and right wheels of the suspension will also change accordingly, which will affect the side slip stiffnessKr of the tire.When the vehicle rolls, load transfer occurs between the left and right wheels, and the load transfer is closely related to the roll stiffness of the suspension. When there is load transfer between the left and right wheels of the suspension, the side slip stiffness of the wheels will be affected, which will affect the force of the wheels, and ultimately affect the steering characteristics of the vehicle. When the vehicle is subjected to the lateral force, for the rear axle of a vehicle, the greater the load transfer between left and right wheels, the weaker the understeer characteristics of the vehicle [16].The opposite wheel travel simulation test is performed on the torsion beam suspension, and the load transfer amount between the left and right wheels under different opening angles of torsion beam as shown in Figure22 is obtained.(4)
During the movement of the suspension, the roll angle of the suspension will change the side slip angle of wheel, that is, the∂αf/∂ϕϕ and ∂αf/∂ϕϕ in Eq. (40) will change.(5)
From the previous analysis, it can be seen that different opening angle of torsion beam and rubber bushing installation angles cause different changes in the wheel alignment angles when the suspension wheels are stressed, thereby affecting the frequency response characteristics of the vehicle.Figure 22
Load transfer amount between left and right wheels under different opening angles of torsion beam.


## 4.3. The Influence Trend of the Opening Angle of Torsion Beam and the Rubber Bushing Installation Angle on the Frequency Response Characteristics of Vehicle

According to the previous analysis and the actual parameters of a certain vehicle, the corresponding multibody dynamics model of the whole vehicle is established. The main parameters of the established multibody dynamics model of the original whole vehicle are shown in Table1.Table 1
Vehicle parameters.
ParameterValueSprung mass (kg)1097.00Unsprung mass (kg)551.98Center of mass to front wheelbase (m)0.7099Center of mass to rear wheelbase (m)1.8501Opening angle of torsion beam (°)270.00Spring stiffness of front suspension ( )2.1 × 104Height from center of mass to ground (m)0.378Distance between front wheels (m)1.594Distance between rear wheels (m)1.52Wheelbase (m)2.56Rubber bushing installation angle (°)−25.50Spring stiffness of rear suspension ( )2.27 × 104Since the frequency when the driver turns the steering wheel is about 0.5 Hz, the gain value and delay time at 0.5 Hz are mainly analyzed. As shown in Figures23 to 27, the frequency response characteristic index corresponding to 0.5 Hz changes with the change of the opening angle of torsion beam.Figure 23
The trend of resonant frequency changing with the opening angle of torsion beam.Figure 24
The trend of the roll angle gain relative to the lateral acceleration at 0.5 Hz as the opening angle of torsion beam changes.Figure 25
Trend of the delay time of lateral acceleration relative to the steering wheel angle at 0.5 Hz as the rubber bushing installation angle changes.Figure 26
The trend of the delay time of yaw rate relative to the lateral acceleration at 0.5 Hz as the rubber bushing installation angle changes.Figure 27
The trend of yaw rate gain at 0.5 Hz as the rubber bushing installation angle changes.Figure23 shows the change trend of the resonance frequency with the opening angle of torsion beam. In Figure 23, as the opening angle of torsion beam increases from 70° to 110°, the resonance frequency of the vehicle roughly first increases and then decreases with some fluctuations in the middle. In Figure 24, as the opening angle of torsion beam increases from 70° to 110°, the roll angle gain at 0.5 Hz relative to the lateral acceleration roughly first decreases and then increases.Figure25 shows the trend of the delay time t1 of lateral acceleration relative to the steering wheel angle at 0.5 Hz as the rubber bushing installation angle increases. Figure 26 shows the trend of the delay time t2 of the yaw rate relative to the lateral acceleration at 0.5 Hz as the rubber bushing installation angle increases.In Figure25, as the rubber bushing installation angle increases, the absolute value of the delay time of the lateral acceleration at 0.5 Hz first showed a decreasing trend and then an increasing trend. When the rubber bushing installation angle reaches about 65°, the absolute value of the delay time of the lateral acceleration at 0.5 Hz is slightly reduced.In Figure26, as the rubber bushing installation angle increases, the absolute value of the delay time of the yaw rate at 0.5 Hz relative to the lateral acceleration first decreases, and then increases again. When the rubber bushing installation angle reaches about 65°, the absolute value of the delay time of the yaw rate at 0.5 Hz relative to the lateral acceleration is slightly reduced.Figure27 shows the trend of the yaw rate gain at 0.5 Hz relative to the steering wheel angle as the installation angle of the rubber bushing increases. In Figure 27, as the installation angle of rubber bushing increases, the yaw rate gain relative to the steering wheel angle at 0.5 Hz shows a decreasing trend first, and then an increasing trend. When the rubber bushing installation angle reaches about 45°, the yaw rate gain relative to the steering wheel angle at 0.5 Hz is slightly reduced.


## 5. Optimization Analysis of Opening Angle of Torsion Beam and Rubber Bushing Installation Angle of Torsion Beam Suspension

On the basis of the previous analysis, under the sine swept steering input of the vehicle, the frequency response characteristic index of the vehicle is taken as the optimization objective, the opening angle of torsion beam and the rubber bushing installation angle of the torsion beam suspension are used as the optimization variables, the multiobjective optimization algorithm is used to optimize the handling stability of the vehicle.


### 5.1. Multiobjective Optimization

Through the previous analysis, multiobjective optimization is carried out for the opening angle of torsion beam and the installation angle of the rubber bushing. The opening angle of torsion beam and the rubber bushing installation angle are the optimization variables, and the frequency response characteristic index of the vehicle is the optimization objective, NSGA-II multiobjective optimization algorithm is taken as optimization method [17–19]. Since the frequency when the driver turns the steering wheel is about 0.5 Hz, the gain and delay time at 0.5 Hz are used as optimization objectives [20]. The specific optimization objectives are(42)f1=minψ˙δjω0.5Hz,f2=minϕayjω0.5Hz,f3=maxfψ˙,f4=mint10.5Hz,f5=mint20.5Hz.Subject to:(43)0≤αo≤360,−90≤αi≤90.where, α0 and αi are the torsion beam opening angle and the bushing installation angle, respectively.For the NSGA-II algorithm, the current population is first used as the parent population in the evolutionary process, the next generation of offspring population is obtained through crossover and a mutation process, and then the two generations are merged together. In determining the selection of individuals, the NSGA-II algorithm uses an elite strategy that divides the individuals in the population into a number of sequentially controlled Pareto front. For the NSGA-II algorithm, the search capability is greatly improved because individuals at the Pareto front are selected preferentially.After optimization, the iterative diagrams of each optimization objective are shown in Figures28 to 33. The distribution of the common and Pareto solutions of the multiobjective optimization problem can be observed in the iteration diagram.Figure 28
Iterative graph of yaw rate gain relative to the steering wheel angle at 0.5 Hz.Figure 29
Iterative graph of roll angle gain relative to the lateral acceleration at 0.5 Hz.Figure 30
Iterative diagram of the maximum yaw rate gain relative to the steering wheel angle.Figure 31
Iterative diagram of the delay time of the lateral acceleration relative to the steering wheel angle at 0.5 Hz.Figure 32
Iterative diagram of the delay time of yaw rate relative to the lateral acceleration at 0.5 Hz.Figure 33
Iterative diagram of resonance frequency.Figure28 shows the iterative graph of the yaw rate gain relative to the steering wheel angle at 0.5 Hz. In Figure 28, as the optimization progresses, the yaw rate gain at 0.5 Hz is gradually decreasing, and finally reaches a state of convergence. When the yaw rate gain value becomes smaller, the frequency response characteristics of the vehicle will be improved, so the change trend of the yaw rate gain meets the optimization requirements.Figure29 shows the iterative graph of the roll angle gain relative to the lateral acceleration at 0.5 Hz. In Figure 29, as the optimization progresses, the roll angle gain relative to the lateral acceleration is decreasing. The smaller roll angle gain is conducive to improve the frequency response characteristics of the vehicle, so the change trend of the roll angle gain meets the optimization requirements.Figure30 shows the iterative graph of the maximum yaw rate gain relative to the steering wheel angle. In Figure 30, as the optimization progresses, the maximum yaw rate gain gradually decreases and converges within a range. Similarly, the decrease of the maximum yaw rate gain is conducive to improve the frequency characteristics of the vehicle, so the change trend of the maximum yaw rate gain also meets the optimization requirements.Figure31 shows the iterative graph of the delay time of lateral acceleration relative to the steering wheel angle at 0.5 Hz. In Figure 31, as the optimization progresses, the absolute value of the delay time of the lateral acceleration relative to the steering wheel angle at 0.5 Hz gradually decreases and finally converges. The reduction of the delay time of lateral acceleration is conducive to improve the reaction speed of the vehicle, so the change trend of the delay time of lateral acceleration in the optimization process meets the optimization requirements.Figure32 is the iterative diagram of the delay time of yaw rate relative to the lateral acceleration at 0.5 Hz. In Figure 32, as the optimization progresses, the absolute value of the delay time of yaw rate relative to the lateral acceleration gradually decreases and converges to a range. The delay time of the yaw rate relative to the lateral acceleration becomes smaller, which is conducive to improve the reaction speed of the vehicle. Therefore, the change trend of the delay time of the yaw rate meets the optimization requirements.In Figure33, the resonant frequency value of the vehicle changed within a range and finally did not reach the state of convergence.For multiobjective optimization problems, the objectives may conflict and the trade-offs need to be made between them. For the optimization results in this study, not all the objectives are consistent with the optimization direction, but all are within acceptable limits.


### 5.2. Comparative Analysis of the Optimized Vehicle and the Original Vehicle

After optimizing the vehicle, a solution in the Pareto solution set is chosen to compare with the original vehicle to test whether the optimization is reasonable and effective. The comparison includes comparing the transient characteristics and steady-state characteristics of the vehicle. The comparison of frequency response characteristics of vehicle is mainly based on the comparison of the frequency response characteristics of vehicle. Among them, the comparative analysis of the frequency response characteristics of the vehicle is mainly based on the sine swept steering input simulation, and the comparison of the steady-state characteristics is mainly based on the constant radius simulation. The vehicle model is adjusted according to the selected Pareto solution parameters, and then the sine swept steering input simulation test is performed. The comparison result of the frequency response characteristic of the vehicle is shown in Table2.Table 2
Comparison results of vehicle frequency response characteristics before and after optimization.
ParameterVariable1 (°)Variable2 (°)f1 (s−1)f2 (s−1)f3 (Hz)f4 (s)f5 (s)Original vehicle270−25.50.34816.21710.58590.18840.0935Optimized vehicle97.5−28.80.25675.74320.92770.16120.1206In Table2, variable 1 is the opening angle of the torsion beam, variable 2 is the rubber bushing installation angle, among the five optimization objectives, the yaw rate gain relative to the steering wheel angle at 0.5 Hz (f1), the roll angle gain relative to the lateral acceleration at 0.5 Hz (f2), the resonance frequency (f3), and the delay time of the lateral acceleration relative to the steering wheel angle at 0.5 Hz (f4) are improved compared to the original vehicle, and the delay time of the yaw rate relative to the lateral acceleration (f5) is worse than the original vehicle, but the value is still within an acceptable range.The comparison diagrams of the frequency response characteristics of the optimized vehicle and the original vehicle are shown in Figures34 to 37.Figure 34
Yaw rate gain relative to the steering wheel angle.Figure 35
Roll angle gain relative to the lateral steering wheel acceleration.Figure 36
Delay time of lateral acceleration relative to the steering wheel angle.Figure 37
Delay time of yaw rate relative to the lateral acceleration.Figure34 shows the curve of yaw rate gain relative to the steering wheel angle of the vehicle before and after the optimization. Figure 35 shows the curve of roll angle gain relative to the lateral acceleration of the vehicle before and after optimization. In Figure 34, when the frequency is in the range of 1 Hz, the yaw rate gain value of the optimized vehicle is smaller than that of the original vehicle. A smaller yaw rate gain value is conducive to improve the handling stability of the vehicle. In terms of yaw rate gain, it becomes better after optimization. In Figure 35, when the frequency is around 0.5 Hz, the roll angle gain of the optimized vehicle is reduced compared to the original vehicle, indicating that the roll angle gain value of the vehicle has also been optimized.Figure36 shows the delay time of the lateral acceleration relative to the steering wheel angle, and Figure 37 shows the delay time of the yaw rate relative to the lateral acceleration. In Figure 36, around 0.5 Hz, the absolute value of the delay time of the lateral acceleration of the optimized vehicle relative to the steering wheel angle is smaller than that of the original vehicle, indicating that the response of the optimized vehicle becomes faster. In Figure 37, for the optimized vehicle, the absolute value of the delay time of the yaw rate at 0.5 Hz is slightly larger than that of the original vehicle, indicating that the response of the vehicle becomes slightly slower after optimization.In addition to the comparative analysis of the frequency response characteristics of the vehicle, the steady-state characteristics of the vehicle studied in this paper are compared and analyzed. After the constant radius steering simulation test of the vehicle, the curve of the steering wheel angle relative to the lateral acceleration as shown in Figure38 and the curve of roll angle relative to the lateral acceleration as shown in Figure 39 are obtained.Figure 38
Curve of steering wheel angle of vehicle relative to the lateral acceleration.Figure 39
Curve of roll angle of vehicle relative to the lateral acceleration.In Figure38, under the same lateral acceleration, the steering wheel angle of the optimized vehicle is larger than that of the original vehicle, indicating that the optimized vehicle needs to input a larger steering wheel angle to maintain the turning radius of the vehicle, indicating that the understeer characteristics of vehicle tend to increase [21]. In Figure 39, under the same lateral acceleration, the roll angle of the optimized vehicle is reduced compared to the original vehicle, indicating that the roll stability of the optimized vehicle has been improved.This paper conducts a comparative analysis of preoptimization and optimization of the researched vehicle, including a comparative analysis of frequency response characteristics and steady-state response characteristics. The comparison results show that for the vehicle studied in this paper, while improving its frequency response characteristics, the steady-state characteristics are still keeping in a good state, indicating that the optimization of the researched vehicle in this paper is feasible for improving the handling stability.


## 5.1. Multiobjective Optimization

Through the previous analysis, multiobjective optimization is carried out for the opening angle of torsion beam and the installation angle of the rubber bushing. The opening angle of torsion beam and the rubber bushing installation angle are the optimization variables, and the frequency response characteristic index of the vehicle is the optimization objective, NSGA-II multiobjective optimization algorithm is taken as optimization method [17–19]. Since the frequency when the driver turns the steering wheel is about 0.5 Hz, the gain and delay time at 0.5 Hz are used as optimization objectives [20]. The specific optimization objectives are(42)f1=minψ˙δjω0.5Hz,f2=minϕayjω0.5Hz,f3=maxfψ˙,f4=mint10.5Hz,f5=mint20.5Hz.Subject to:(43)0≤αo≤360,−90≤αi≤90.where, α0 and αi are the torsion beam opening angle and the bushing installation angle, respectively.For the NSGA-II algorithm, the current population is first used as the parent population in the evolutionary process, the next generation of offspring population is obtained through crossover and a mutation process, and then the two generations are merged together. In determining the selection of individuals, the NSGA-II algorithm uses an elite strategy that divides the individuals in the population into a number of sequentially controlled Pareto front. For the NSGA-II algorithm, the search capability is greatly improved because individuals at the Pareto front are selected preferentially.After optimization, the iterative diagrams of each optimization objective are shown in Figures28 to 33. The distribution of the common and Pareto solutions of the multiobjective optimization problem can be observed in the iteration diagram.Figure 28
Iterative graph of yaw rate gain relative to the steering wheel angle at 0.5 Hz.Figure 29
Iterative graph of roll angle gain relative to the lateral acceleration at 0.5 Hz.Figure 30
Iterative diagram of the maximum yaw rate gain relative to the steering wheel angle.Figure 31
Iterative diagram of the delay time of the lateral acceleration relative to the steering wheel angle at 0.5 Hz.Figure 32
Iterative diagram of the delay time of yaw rate relative to the lateral acceleration at 0.5 Hz.Figure 33
Iterative diagram of resonance frequency.Figure28 shows the iterative graph of the yaw rate gain relative to the steering wheel angle at 0.5 Hz. In Figure 28, as the optimization progresses, the yaw rate gain at 0.5 Hz is gradually decreasing, and finally reaches a state of convergence. When the yaw rate gain value becomes smaller, the frequency response characteristics of the vehicle will be improved, so the change trend of the yaw rate gain meets the optimization requirements.Figure29 shows the iterative graph of the roll angle gain relative to the lateral acceleration at 0.5 Hz. In Figure 29, as the optimization progresses, the roll angle gain relative to the lateral acceleration is decreasing. The smaller roll angle gain is conducive to improve the frequency response characteristics of the vehicle, so the change trend of the roll angle gain meets the optimization requirements.Figure30 shows the iterative graph of the maximum yaw rate gain relative to the steering wheel angle. In Figure 30, as the optimization progresses, the maximum yaw rate gain gradually decreases and converges within a range. Similarly, the decrease of the maximum yaw rate gain is conducive to improve the frequency characteristics of the vehicle, so the change trend of the maximum yaw rate gain also meets the optimization requirements.Figure31 shows the iterative graph of the delay time of lateral acceleration relative to the steering wheel angle at 0.5 Hz. In Figure 31, as the optimization progresses, the absolute value of the delay time of the lateral acceleration relative to the steering wheel angle at 0.5 Hz gradually decreases and finally converges. The reduction of the delay time of lateral acceleration is conducive to improve the reaction speed of the vehicle, so the change trend of the delay time of lateral acceleration in the optimization process meets the optimization requirements.Figure32 is the iterative diagram of the delay time of yaw rate relative to the lateral acceleration at 0.5 Hz. In Figure 32, as the optimization progresses, the absolute value of the delay time of yaw rate relative to the lateral acceleration gradually decreases and converges to a range. The delay time of the yaw rate relative to the lateral acceleration becomes smaller, which is conducive to improve the reaction speed of the vehicle. Therefore, the change trend of the delay time of the yaw rate meets the optimization requirements.In Figure33, the resonant frequency value of the vehicle changed within a range and finally did not reach the state of convergence.For multiobjective optimization problems, the objectives may conflict and the trade-offs need to be made between them. For the optimization results in this study, not all the objectives are consistent with the optimization direction, but all are within acceptable limits.


## 5.2. Comparative Analysis of the Optimized Vehicle and the Original Vehicle

After optimizing the vehicle, a solution in the Pareto solution set is chosen to compare with the original vehicle to test whether the optimization is reasonable and effective. The comparison includes comparing the transient characteristics and steady-state characteristics of the vehicle. The comparison of frequency response characteristics of vehicle is mainly based on the comparison of the frequency response characteristics of vehicle. Among them, the comparative analysis of the frequency response characteristics of the vehicle is mainly based on the sine swept steering input simulation, and the comparison of the steady-state characteristics is mainly based on the constant radius simulation. The vehicle model is adjusted according to the selected Pareto solution parameters, and then the sine swept steering input simulation test is performed. The comparison result of the frequency response characteristic of the vehicle is shown in Table2.Table 2
Comparison results of vehicle frequency response characteristics before and after optimization.
ParameterVariable1 (°)Variable2 (°)f1 (s−1)f2 (s−1)f3 (Hz)f4 (s)f5 (s)Original vehicle270−25.50.34816.21710.58590.18840.0935Optimized vehicle97.5−28.80.25675.74320.92770.16120.1206In Table2, variable 1 is the opening angle of the torsion beam, variable 2 is the rubber bushing installation angle, among the five optimization objectives, the yaw rate gain relative to the steering wheel angle at 0.5 Hz (f1), the roll angle gain relative to the lateral acceleration at 0.5 Hz (f2), the resonance frequency (f3), and the delay time of the lateral acceleration relative to the steering wheel angle at 0.5 Hz (f4) are improved compared to the original vehicle, and the delay time of the yaw rate relative to the lateral acceleration (f5) is worse than the original vehicle, but the value is still within an acceptable range.The comparison diagrams of the frequency response characteristics of the optimized vehicle and the original vehicle are shown in Figures34 to 37.Figure 34
Yaw rate gain relative to the steering wheel angle.Figure 35
Roll angle gain relative to the lateral steering wheel acceleration.Figure 36
Delay time of lateral acceleration relative to the steering wheel angle.Figure 37
Delay time of yaw rate relative to the lateral acceleration.Figure34 shows the curve of yaw rate gain relative to the steering wheel angle of the vehicle before and after the optimization. Figure 35 shows the curve of roll angle gain relative to the lateral acceleration of the vehicle before and after optimization. In Figure 34, when the frequency is in the range of 1 Hz, the yaw rate gain value of the optimized vehicle is smaller than that of the original vehicle. A smaller yaw rate gain value is conducive to improve the handling stability of the vehicle. In terms of yaw rate gain, it becomes better after optimization. In Figure 35, when the frequency is around 0.5 Hz, the roll angle gain of the optimized vehicle is reduced compared to the original vehicle, indicating that the roll angle gain value of the vehicle has also been optimized.Figure36 shows the delay time of the lateral acceleration relative to the steering wheel angle, and Figure 37 shows the delay time of the yaw rate relative to the lateral acceleration. In Figure 36, around 0.5 Hz, the absolute value of the delay time of the lateral acceleration of the optimized vehicle relative to the steering wheel angle is smaller than that of the original vehicle, indicating that the response of the optimized vehicle becomes faster. In Figure 37, for the optimized vehicle, the absolute value of the delay time of the yaw rate at 0.5 Hz is slightly larger than that of the original vehicle, indicating that the response of the vehicle becomes slightly slower after optimization.In addition to the comparative analysis of the frequency response characteristics of the vehicle, the steady-state characteristics of the vehicle studied in this paper are compared and analyzed. After the constant radius steering simulation test of the vehicle, the curve of the steering wheel angle relative to the lateral acceleration as shown in Figure38 and the curve of roll angle relative to the lateral acceleration as shown in Figure 39 are obtained.Figure 38
Curve of steering wheel angle of vehicle relative to the lateral acceleration.Figure 39
Curve of roll angle of vehicle relative to the lateral acceleration.In Figure38, under the same lateral acceleration, the steering wheel angle of the optimized vehicle is larger than that of the original vehicle, indicating that the optimized vehicle needs to input a larger steering wheel angle to maintain the turning radius of the vehicle, indicating that the understeer characteristics of vehicle tend to increase [21]. In Figure 39, under the same lateral acceleration, the roll angle of the optimized vehicle is reduced compared to the original vehicle, indicating that the roll stability of the optimized vehicle has been improved.This paper conducts a comparative analysis of preoptimization and optimization of the researched vehicle, including a comparative analysis of frequency response characteristics and steady-state response characteristics. The comparison results show that for the vehicle studied in this paper, while improving its frequency response characteristics, the steady-state characteristics are still keeping in a good state, indicating that the optimization of the researched vehicle in this paper is feasible for improving the handling stability.


## 6. Conclusions

This paper analyzes the influence mechanism of the opening angle of torsion beam and the rubber bushing installation angle on the suspension and vehicle performance, the variation trend of the frequency response characteristic index with the opening angle of torsion beam and the rubber bushing installation angle under the sine swept steering input is obtained. Finally, multiobjective optimization of the opening angle of torsion beam and the installation angle of the rubber bushing is carried out. The optimized vehicle and the original vehicle are compared and analyzed for the frequency response characteristics and the steady-state characteristics. The results show that the handling stability of the vehicle is improved. The specific conclusions are as follows:(1)
The opening angle of torsion beam of the torsion beam suspension will affect the shear center of the suspension, which in turn affects the roll center of the suspension. The installation angle of the rubber bushing will affect the roll stiffness and force of the suspension. Therefore, the opening angle of torsion beam and the installation angle of the rubber bushing will affect the kinematics and compliance characteristics.(2)
The opening angle of torsion beam and the rubber bushing installation angle of the torsion beam suspension will affect the position of roll center and the roll stiffness of the vehicle, thereby affecting the side slip stiffness and force of the wheel.(3)
The three degree of freedom dynamic model of the vehicle is deduced, and the frequency response characteristic index is obtained. On this basis, the influence mechanism of the opening angle of torsion beam and the installation angle of the rubber bushing of the torsion beam suspension on the frequency response index of the vehicle is revealed. After using the established vehicle dynamics model to analysis, the change rule of the frequency response characteristics index with the opening angle of torsion beam and the rubber bushing installation angle is obtained.(4)
Multiobjective optimization is performed on the frequency response characteristic indicators, and the corresponding optimization target iteration diagram and optimization results are obtained. A result of the optimization solution is chosen and compared with the original vehicle. It is found that when the opening angle of torsion beam is 97.5°, the rubber bushing installation angle is −28.8°, the frequency characteristics and steady-state characteristics of the vehicle have been improved.



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*Source: 9250064-2022-09-17.xml*