# Analysis and Optimization of Suspension Spring Stiffness for Improving Frequency Response Characteristics of the Vehicle

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

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## Abstract

The work puts forward a way to research the kinematics of front double wishbone suspension by virtue of analytical method. A way of discussing the kinematics of rear five-link suspension is put forward by means of space coordinate transformation. The influence of suspension spring stiffness on the suspension characteristics is analyzed. The vehicle frequency response index is deduced in view of theoretical model of equivalent handling stability. The relationship between the vehicle frequency characteristic index and the frequency of the steering wheel angle under the variation of spring stiffness is gained, and the influence mechanism of spring stiffness on the vehicle frequency characteristics is explained. The suspension spring stiffness is the optimization variable, the vehicle frequency characteristic index is the optimization objective, and genetic algorithm is used to optimize the vehicle frequency characteristic.


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## Body

## 1. Introduction

The handling stability determines the degree of convenience of the vehicle handling and safe driving at high speed. It is critical to make the car have better the handling stability during the development of the car [1, 2]. As one of the important parts of the car, the suspension of the vehicle can effectively absorb the vibration and effect caused by the uneven road surface during the driving of the vehicle, so as to ensure that the vehicle can run smoothly. As an important parameter in suspension design, spring stiffness has a great influence on the handling stability. It is very meaningful to study how it affects the handling stability of the car. Helical spring has the advantages of compact structure, small mass, high energy density, flexible structure, etc. and is widely used in the field of passenger cars [3, 4].Currently, many scholars have done various research with respect to suspension springs. Tey and Rahizar put forward an optimization method for the dynamic handling performance of racing cars based on a constrained multi-objective evolutionary algorithm method. The optimization method involves a software-in-the-loop optimization (SiLO) algorithm between a high-fidelity model modeled in VI-Grade and the optimization algorithms. The optimization process included adjusting the suspension design variables and introducing a custom constraint function to improve the convergence speed of the optimization process. The optimized results show that the handling performance has been made better [5]. Atig et al. put forward three analysis models to describe the bending stress of leaf spring and evaluated the stress distribution generated by the vertical load of the parabolic leaf spring [6]. Traditional approaches to vehicle suspension optimization require decomposing multiple performance objectives which is a tedious process. Tey et al. put forward a new optimization method to simplify the tedious process. The proposed multi-objective optimization method and the improved optimization engineering process are used to minimize the performance indicators of the whole vehicle, while the design parameter space of more than 100 dimensions is solved through the software-in-the-loop optimization process. Compared with traditional method, the method shows results in optimizing the suspension design of the whole vehicle based on ride comfort and handling stability [7]. Wang et al. designed a novel omnidirectional accelerometer by a method based on genetic optimization algorithm, which has higher sensitivity than the omnidirectional accelerometer. The strain-induced voltage output of curved piezoelectric fibers is analyzed by isogeometric analysis (IGA), which guarantees higher computational accuracy compared to the finite element method [8]. Liu et al. conducted a study of direct and iterative inversion methods for determining the spatial shear modulus distribution of elastic solids. Subsequently, the advantages and disadvantages of the two methods are compared. The results show that both methods can identify the nonhomogeneous shear modulus distribution of solids well and the direct inversion is much faster than the iterative inversion, whereas iterative inversion yields better shear modulus ratios even at higher noise [9]. According to the design of existing leaf spring and production process, Deng et al. used VC++ SolidWorks for secondary development. After the model was established, the characteristics of the leaf spring were analyzed, and the results were verified by prototype experiments [10]. In order to overcome the trade-off between handling stability and ride comfort in passenger cars equipped with traditional suspension, Qi et al. created a new suspension configuration and compared it with the traditional suspension. The test results show that vehicles equipped with hydraulic interconnected suspension and electronically controlled air springs can achieve higher performance in the light of the handling stability and ride comfort [11]. Shah et al. first established the leaf spring model through Creo 2.0 software and then used ANSYS 14.0 software to simulate stress, deflection, and fatigue through finite element analysis [12]. Liu chose the damping of shock absorber and spring stiffness of suspension as design variables and performed optimization of the suspension by virtue of the particle swarm algorithm. Through the comparison of simulation before and after optimization, the results show that, on the basis of taking into account the handling stability, the uncomfortable parameter, which is one of the indicators of ride comfort, is reduced at each vehicle speed and has been greatly improved. The dynamic deflection of the suspension is also optimized to a reasonable range, which proves the effectiveness of the optimization [13]. Zeng constructed a model of the active suspension based on analysis of the air spring and shock absorber structure and related parameter characteristics. The results show that the vehicles using the coordinated control method have improved ride comfort and handling stability of vehicle [14]. Quaglia and Sorli used the air suspension simulation model to simulate the vibration characteristics of the air spring with additional air chambers and analyzed the influence of the suspension parameters about the vibration characteristics [15].The above scholars have conducted research with respect to suspension springs from different perspectives. However, the influence of the change of spring stiffness on the suspension characteristics has not been studied systematically. Secondly, the influence mechanism of the change of the spring stiffness on the vehicle frequency characteristics is not discussed in depth, and the relationship between the frequency characteristic index and the frequency of the steering wheel angle under the change of the spring stiffness of the front and rear suspensions is not obtained. Finally, there is no research on the optimization of the suspension spring stiffness and the frequency response characteristics.For questions that have not been explored in depth by experts, this work mainly includes the following four aspects: (1) Kinematics analysis is carried out on the suspension, and the expressions of wheel alignment parameters of front and rear wheels are deduced. (2) The influence of the change of the spring stiffness on suspension characteristic index is systematically studied. (3) The influence mechanism of the spring stiffness change on the vehicle frequency characteristics index is revealed, and the relationship between the frequency characteristic index under spring stiffness change and frequency of steering wheel angle is gained. (4) The frequency characteristic index is determined. The suspension spring stiffness is the optimization variable, the frequency characteristic index at 0.5 Hz at which the people manipulates the steering wheel is the optimization objective, and genetic algorithm is performed for optimization to make the vehicle frequency response characteristics better.


## 2. The Establishment of Front and Rear Suspension Models

So as to discuss the suspension kinematics and influence of suspension springs on the suspension and the whole vehicle, the parameters of a real vehicle, the double wishbone suspension, the five-link suspension, and the multi-body dynamics model of vehicle are established in Figure1, and the modeling parameters of the vehicle are given in Table 1. The following analysis objects are the rubber springs of the front and rear suspensions. The characteristic curves of suspension spring are illustrated in Figure 2.Figure 1
The vehicle dynamics model. (a) Front double wishbone. (b) Rear five-link.
(a)(b)Table 1
Vehicle parameters.
ParametersValueSprung mass (kg)1097.00Unsprung mass (kg)551.98Center of mass to front wheelbase (m)0.71Center of mass to rear wheelbase (m)1.85Height from center of mass to ground (m)0.38Distance between front wheels (m)1.60Distance between rear wheels (m)1.52Wheelbase (m)2.56Figure 2
Characteristic curves of spring stiffness.


## 3. Kinematics Analysis of Front Double Wishbone Suspension

Since double wishbone mechanism consists of the RSSR link and the SSP link (R: revolute, S: spherical, P: prismatic) [16], two independent complete circuits of EDFGE and ABCDE can be used for complex motion analysis. So as to gain the wheel alignment angle, the vector (ek⟶ and ehub⟶) in the simplified suspension curve in Figure 3 needs to be solved first [17, 18]. So as to gain the vector ek⟶, the RSSR closed-loop link composed of EDFGE should be considered first. So as to analyze the RSSR closed loop, the parameters that need to be used are l1, lu, lk, ξ, and ϑ. If these parameters are defined and the lower wishbone angle θ is known, the RSSR closed-loop link can be analyzed. Then, the vector equation of the RSSR closed-loop link is DF⟶=EF⟶−ED⟶. The specific vector of equation is expressed as follows:(1)ED⟶=l1cosθi⟶+l1sink⟶,EF⟶=q3+lucosχcosξ+lusinχsinθsinξi⟶+q2+lucosχsinξ−lusinχsinθcosξj⟶+q1+lusinχcosθk⟶,where l1 is the length of lower wishbone (ED), lu is the length of upper wishbone (FG), ϑ is the anti-dive (or anti-lift) angle, ξ is the yaw angle of upper wishbone, and χ is the angular displacement of upper wishbone.Figure 3
Kinematic curve of double wishbone suspension.The displacement equation of the RSSR closed-loop link is expressed as follows:(2)c1+c2−c3cosθcosχ−c4+c5sinθ−c6cosθsin−c7cosθ−c8sinθ=0,where(3)c1=q22+q32+q12+l12+lu2−lk2,c2=2q2lusinξ+2q3lucosξ,c3=2l1lucosξ,c4=2q2lucosξsinϑ−2q1lusinξsinϑ,c5=2l1lucosϑ,c6=2l1lusinξsinϑ,c7=2q3l1,c8=2q1l1.When the lower wishbone angleθ is known, and from above equation (2), χ can be derived from the half tangent formula.(4)χ=2tan−1−F±F2−4DC2D,where(5)D=−c1+c2−c3cosθ+c7cosθ+c8sinθ,F=2c4+2c5sinθ+2c6cosθ,G=−c1−c2−c3cosθ+c7cosθ+c8sinθ.The position vector of wheel knuckle is determined asDF⟶=lkek⟶, lk is the length of wheel knuckle (DF), and the unit vector of wheel knuckle is determined as follows:(6)ek⟶=1lkq3+lucosχcosξ+lusinχsinθsinξ+l1cosθq2+lucosχsinξ−lusinχsinθcosξq1+lusinχcosθ−l1sinθ.The unit vector of lower wishbone arm is determined as follows:(7)e1→=cosθ0sinθT.Through (4)–(7), one can gain the position of wishbone and wheel knuckle for θ in closed-form equation set.So as to gain the vectorehub⟶, the SSP linkage system needs to be discussed, so the ABCDEA system is chosen. The required parameters are the original coordinates of ball joints E and F. The length of steering rod is determined as tr=AB⟶tr=AB⟶. The effective steering arm length is sa=BD⟶×BF⟶/FD⟶. The toe angle variation of the double wishbone caused by the input of the θ angle is defined by constraining the ABCDEA loop.The first constraint is that the length of tie rod remains the same, so(8)tr2=Bx−Ax02+By−Ay02+Bz−Az02.The second constraint is that the length of the steering arm remains the same, so(9)ta2=Cx−Bx2+Cy−By2+Cz−Bz2,where(10)EC⟶=l1e1⟶+laea⟶,la=DB⟶2−ta2.The third constraint is that the vector of steering arm and the wheel knuckle are vertical to each otherBC⟶⊥DC⟶; then, its scalar product is equal to zero; that is, BC⟶⋅DC⟶=0.(11)l1ek⟶+laek⟶−Bxi⟶+Byj⟶+Bzk⟶⋅laek⟶=0.In (11), we can get(12)a1Bx+a2By+a3Bz+a4=0,where a1=−laekx⟶,a2=−laeky⟶,a3=−laekz⟶,a4=la2+l1laekx⟶e1x⟶+eky⟶e1y⟶+ekz⟶e1z⟶.In (11) and (12), the unknown parameters Bx, By, and Bz indicate the new coordinates after the change of spherical point of steering arm B. Thus, all the positions of the joints can be gained. In addition, the unit vector of wheel hub GW is defined as follows:(13)ehub0⟶=cosε00−sinε0T,where ε0 is the positive static position of wheel in Figure 4(a). In addition, the static position of wheel center is defined as follows:(14)lhub0⟶=c00−10T+lhubehub0⟶.Figure 4
Vector transformations.
(a)(b)(c)So as to get any position of vector, rotation matrix is employed, and then the following order is employed. Starting with the original position of steering arm, the rotation matrix of the unit vector fromea0⟶ to ea⟶ is defined first. Because the unit vector ek⟶ is derived by (6), the rotation amount ek⟶ is defined from cross product as follows:(15)λ=sin−1ek0⟶×ek⟶⋅ekλ⟶,where ekλ⟶ is the vector about the axis of rotation formed perpendicular to the plane composed of ek⟶ and ek0⟶, as shown in Figure 4(b).(16)ekλ⟶=ek0⟶×ek⟶ek0⟶×ek⟶.The analysis of the second rotation needs to fulfill a rotation analysis of wheel knuckle in regard to the unit vectorek⟶, as shown in Figure 4(c).(17)λp=sin−1ea0λ⟶×ea⟶⋅ek⟶.The rotation matrix of the rotationθ angle around the axis along the direction of unit vector u⟶ is as follows:(18)Ru⟶,θ=cosθ+ux21−cosθuxuy1−cosθ+uzsinθuxuz1−cosθ+uysinθuyux1−cosθ+uzsinθcos+uy21−cosθuyuz1−cosθ−uxsinθuzux1−cosθ+uysinθuzuy1−cosθ+uxsinθcosθ+uz21−cosθ.To gain the relevant rotation matrix, the unit vectorehub⟶ from its original position ehub0⟶ to any recent position is as follows:(19)ehub⟶=Rek⟶,λpRekλ⟶,λpehub0⟶.


### 3.1. Derivation of Wheel Alignment Parameters

So as to obtain the wheel alignment angles expression of front suspension, the suspension is simplified in Figure5, and the wheel alignment angle can be calculated by vector.Figure 5
Simplified curve of wheel alignment parameters of double wishbone.In Figure5, ek⟶ refers to the unit vector of wheel hub, ehub⟶ refers to the unit vector of wheel axis, k⟶ refers to the unit vector in the z-axis, j⟶ refers to the unit vector in the y-axis, ehubp⟶ refers to the vector ehub⟶ being projected in the XY plane, ecaster⟶ is the vector vertical to vector ehub⟶ in the XY plane.The toe angle of front wheelαf is gained from the vectors ehubp⟶ and j⟶:(20)αf=π2−cos−1ehubp⟶⋅j⟶.The camber angle of front wheelσf is represented by vector ehub⟶ and vector k⟶:(21)σf=cos−1ehub⟶⋅k⟶−π2.The caster angle of front wheelψf is calculated by vector ek⟶ and vector ecaster⟶:(22)ψf=cos−1ecaster⟶⋅k⟶−π2,ecaster⟶=∓ehubyehubx1ehuby/ehubx2+1±1ehuby/ehubx2+10T.The kingpin inclination angle of front wheel is expressed by the vectorek⟶ and the vector ehubp⟶ehubp⟶,where(23)τf=cos−1ehub→⋅k→−π2,where (24)ehubp→=ehubpxehubpy0Tehubpx2+ehubpy2.


## 3.1. Derivation of Wheel Alignment Parameters

So as to obtain the wheel alignment angles expression of front suspension, the suspension is simplified in Figure5, and the wheel alignment angle can be calculated by vector.Figure 5
Simplified curve of wheel alignment parameters of double wishbone.In Figure5, ek⟶ refers to the unit vector of wheel hub, ehub⟶ refers to the unit vector of wheel axis, k⟶ refers to the unit vector in the z-axis, j⟶ refers to the unit vector in the y-axis, ehubp⟶ refers to the vector ehub⟶ being projected in the XY plane, ecaster⟶ is the vector vertical to vector ehub⟶ in the XY plane.The toe angle of front wheelαf is gained from the vectors ehubp⟶ and j⟶:(20)αf=π2−cos−1ehubp⟶⋅j⟶.The camber angle of front wheelσf is represented by vector ehub⟶ and vector k⟶:(21)σf=cos−1ehub⟶⋅k⟶−π2.The caster angle of front wheelψf is calculated by vector ek⟶ and vector ecaster⟶:(22)ψf=cos−1ecaster⟶⋅k⟶−π2,ecaster⟶=∓ehubyehubx1ehuby/ehubx2+1±1ehuby/ehubx2+10T.The kingpin inclination angle of front wheel is expressed by the vectorek⟶ and the vector ehubp⟶ehubp⟶,where(23)τf=cos−1ehub→⋅k→−π2,where (24)ehubp→=ehubpxehubpy0Tehubpx2+ehubpy2.


## 4. Analysis of Kinematics Characteristics of Five-Link Rear Suspension

The five-link rear suspension belongs to the spatial multi-link suspension [19]. In order to analyze its spatial mechanism, it is first critical to set a coordinate system on the rod. By using transformation of space coordinate, these spatial mechanisms are studied to gain the relationship between the coordinate systems about rods, thereby obtaining the position of rods and further researching the five-link suspension.


### 4.1. Rotation Matrix of Space Coordinates

In Figure6, o1−x1y1z1 is coordinate system 1, and o2−x2y2z2 is coordinate system 2. When coordinate system 2 is rotated by the γ angle around the z1 axis, the rotation matrix Uzγ between the two coordinate systems is as follows:(25)Uzγ=cosγ−sinγ0sinγcosγ0001.Figure 6
Coordinate plot rotated around thez-axis.Similarly, the resulting rotation matrixUyγ rotated around the y1 axis and the resulting rotation matrix Uxγ around the y1 axis are as follows:(26)Uyγ=cosγ−sinγsinγ001−sinγ0cosγ,Uxγ=1000cosγ−sinγ0sinγcosγ.Assuming that there is a point in coordinate system 1 whose coordinate isQ, the coordinate after rotating around an axis by γ angle is Q′ in coordinate system 1, so the relationship between Q and Q′ is as follows:(27)Q′=UQ.In Figure7, we assume that the axes of coordinate system o2−x2y2z2 and coordinate system o1−x1y1z1 are parallel to each other before the motion of the space body and C1 is a point on the space body; the coordinate is marked as C1x,C1y,C1z. After the motion, C1 point reaches C2 point; the coordinate of C2 point is marked as C2x,C2y,C2z. In the process of motion, the space body rotates by η1,η2,η3 angle around the z1y1x1 axis, which will make C1 and C2 coincide. Assuming there is an N point on the space body before motion, the original coordinate is N1x,N1y,N1z, and the coordinate after the motion is N2x,N2y,N2z. Accordingly, the coordinate of N point before and after the movement in the base coordinate is transformed into the following:(28)N2xN2yN2z1=n11n12n13n14n21n22n23n24n31n32n33n340000N1xN1yN1z1.Figure 7
Schematic curve of spatial body position change.The matrix on the right side of (28) is as follows:(29)D12=n11n12n13n14n21n22n23n24n31n32n33n340000=UY01,where(30)U=cosη1cosη2−sinη1sinη2sinη2sinη1cosη3+cosη1sinη2sinη3cosη1cosη3−sinη1sinη2sinη3−cosη2sinη3sinη1sinη3−cosη1sinη2cosη3cosη1sinη3+sinη1sinη2cosη3cosη2cosη3,Y=C2x−n11C1x+n12C1y+n13C1zC2y−n21C1x+n22C1y+n23C1zC2z−n31C1x+n32C1y+n33C1z.


### 4.2. Analysis of Kinematics Characteristics of Rear Suspension

Generally, there is no kinematic pair between the suspension rods. Therefore, before the kinematics analysis of the suspension, the connection between the rods needs to be simplified. Therefore, the following assumptions are first put forward: The chassis and the connecting rod are rigid, ignoring the influence of bushing elasticity. A simplified suspension model of the five-link suspension is established in Figure8. Supposing the original position coordinates of each point of the suspension are known, each point connected with the steering knuckle is Aii=1,2,3,4,5 and the connection point between the link and the body is Bii=1,2,3,4,5. From the degree of freedom, it is known that the spatial posture is defined by the z-direction displacement at the W point [20].Figure 8
Schematic curve of wheel traveling of the five-link suspension.(31)Wtz=W0z+dz.When the wheel travels up and down, thez coordinate change dz at the wheel center is used as the input; then, the coordinate of the W point before and after the motion is as follows:where W0z is the original z coordinate at W point. When the wheel is at Wt, from the relationship between the coordinate change matrix and Wt, the position of point Ait on the steering knuckle is obtained first. The change relationship is as follows:(32)Ait=DotAi0,Wt=DotW0.In order to determine the coordinateAit of each point on the steering knuckle after the motion, it is necessary to know η1,η2,η3,Wtx,Wty,Wtz. The above assumptions show that in the process of the wheel travel, the length and position of each rod remain unchanged.Assuming the length of the rod isLii=1,2,3,4,5, the motion constraint relationship of the connecting is as follows:(33)B1A1t=L1,B2A2t=L2,B3A3t=L3,B4A4t=L4,B5A5t=L5.The above is about the constraint equation ofη1,η2,η3,Wtx,Wtyη1,η2,η3,Wtx,Wty; the motion input Wtz is known; and there are five unknowns: η1,η2,η3,Wtx,Wty. After the unknowns are obtained, the homogeneous transformation matrix Dot is obtained by (29), and then the motion coordinates of each point on the steering knuckle is obtained by (28). After the position of steering knuckle is gained, the alignment parameters of rear wheel in the process of wheel travel can be obtained [21].In Figure8, W is the wheel center; supposing P is a point on the steering knuckle, the original position of the point is P0, and the coordinate of the point after suspension motion is Pt, so the relationship between P0 and Pt can be expressed as follows:(34)Pt=D0tP0.From the definition of the toe angle, the toe angle of rear wheel can be obtained:(35)αr=±tan−1Ptx−WtxPty−Wty.Similarly, the camber angle of the rear wheel can be obtained:(36)σr=±tan−1Ptz−WtzPty−Wty.


## 4.1. Rotation Matrix of Space Coordinates

In Figure6, o1−x1y1z1 is coordinate system 1, and o2−x2y2z2 is coordinate system 2. When coordinate system 2 is rotated by the γ angle around the z1 axis, the rotation matrix Uzγ between the two coordinate systems is as follows:(25)Uzγ=cosγ−sinγ0sinγcosγ0001.Figure 6
Coordinate plot rotated around thez-axis.Similarly, the resulting rotation matrixUyγ rotated around the y1 axis and the resulting rotation matrix Uxγ around the y1 axis are as follows:(26)Uyγ=cosγ−sinγsinγ001−sinγ0cosγ,Uxγ=1000cosγ−sinγ0sinγcosγ.Assuming that there is a point in coordinate system 1 whose coordinate isQ, the coordinate after rotating around an axis by γ angle is Q′ in coordinate system 1, so the relationship between Q and Q′ is as follows:(27)Q′=UQ.In Figure7, we assume that the axes of coordinate system o2−x2y2z2 and coordinate system o1−x1y1z1 are parallel to each other before the motion of the space body and C1 is a point on the space body; the coordinate is marked as C1x,C1y,C1z. After the motion, C1 point reaches C2 point; the coordinate of C2 point is marked as C2x,C2y,C2z. In the process of motion, the space body rotates by η1,η2,η3 angle around the z1y1x1 axis, which will make C1 and C2 coincide. Assuming there is an N point on the space body before motion, the original coordinate is N1x,N1y,N1z, and the coordinate after the motion is N2x,N2y,N2z. Accordingly, the coordinate of N point before and after the movement in the base coordinate is transformed into the following:(28)N2xN2yN2z1=n11n12n13n14n21n22n23n24n31n32n33n340000N1xN1yN1z1.Figure 7
Schematic curve of spatial body position change.The matrix on the right side of (28) is as follows:(29)D12=n11n12n13n14n21n22n23n24n31n32n33n340000=UY01,where(30)U=cosη1cosη2−sinη1sinη2sinη2sinη1cosη3+cosη1sinη2sinη3cosη1cosη3−sinη1sinη2sinη3−cosη2sinη3sinη1sinη3−cosη1sinη2cosη3cosη1sinη3+sinη1sinη2cosη3cosη2cosη3,Y=C2x−n11C1x+n12C1y+n13C1zC2y−n21C1x+n22C1y+n23C1zC2z−n31C1x+n32C1y+n33C1z.


## 4.2. Analysis of Kinematics Characteristics of Rear Suspension

Generally, there is no kinematic pair between the suspension rods. Therefore, before the kinematics analysis of the suspension, the connection between the rods needs to be simplified. Therefore, the following assumptions are first put forward: The chassis and the connecting rod are rigid, ignoring the influence of bushing elasticity. A simplified suspension model of the five-link suspension is established in Figure8. Supposing the original position coordinates of each point of the suspension are known, each point connected with the steering knuckle is Aii=1,2,3,4,5 and the connection point between the link and the body is Bii=1,2,3,4,5. From the degree of freedom, it is known that the spatial posture is defined by the z-direction displacement at the W point [20].Figure 8
Schematic curve of wheel traveling of the five-link suspension.(31)Wtz=W0z+dz.When the wheel travels up and down, thez coordinate change dz at the wheel center is used as the input; then, the coordinate of the W point before and after the motion is as follows:where W0z is the original z coordinate at W point. When the wheel is at Wt, from the relationship between the coordinate change matrix and Wt, the position of point Ait on the steering knuckle is obtained first. The change relationship is as follows:(32)Ait=DotAi0,Wt=DotW0.In order to determine the coordinateAit of each point on the steering knuckle after the motion, it is necessary to know η1,η2,η3,Wtx,Wty,Wtz. The above assumptions show that in the process of the wheel travel, the length and position of each rod remain unchanged.Assuming the length of the rod isLii=1,2,3,4,5, the motion constraint relationship of the connecting is as follows:(33)B1A1t=L1,B2A2t=L2,B3A3t=L3,B4A4t=L4,B5A5t=L5.The above is about the constraint equation ofη1,η2,η3,Wtx,Wtyη1,η2,η3,Wtx,Wty; the motion input Wtz is known; and there are five unknowns: η1,η2,η3,Wtx,Wty. After the unknowns are obtained, the homogeneous transformation matrix Dot is obtained by (29), and then the motion coordinates of each point on the steering knuckle is obtained by (28). After the position of steering knuckle is gained, the alignment parameters of rear wheel in the process of wheel travel can be obtained [21].In Figure8, W is the wheel center; supposing P is a point on the steering knuckle, the original position of the point is P0, and the coordinate of the point after suspension motion is Pt, so the relationship between P0 and Pt can be expressed as follows:(34)Pt=D0tP0.From the definition of the toe angle, the toe angle of rear wheel can be obtained:(35)αr=±tan−1Ptx−WtxPty−Wty.Similarly, the camber angle of the rear wheel can be obtained:(36)σr=±tan−1Ptz−WtzPty−Wty.


## 5. Roll and Load Transfer Caused by Body Roll

Generally, the car body tends to roll when turning, and the load of tire will be transferred due to the body roll. So as to make the suspension characteristic, vehicle handling stability, and vehicle roll stability better, suspension roll stiffness in a reasonable range must be designed, and thus the model of whole vehicle roll and load transfer are established in Figures9 and 10.Figure 9
The roll model of the whole vehicle.Figure 10
Front axle load transfer caused by body roll.In Figure9, Mx is the sprung mass, Mu is the unsprung mass, ay is lateral acceleration, hs is the distance from the center of sprung mass to the suspension roll axis, hc is the distance from the roll center to the ground, R is the radius of wheel, and ϕ is body roll angle. The moment caused by the centrifugal force of the suspension is MsAjhsMsAjhs. The rolling moment leading to the gravity of sprung mass is MsghsϕMsghsϕ. In an independent suspension, the roll moment led by mass centrifugal force of non-suspension is −Muayhc−R.According to the torque balance conditions, there is(37)Kϕϕ=Msayhs+Msghϕ−Muayhc−R.From (37), the expressions for the total roll stiffness and roll angle are as follows:(38)Kϕ=Msayhs+Muayhc−Rϕ+msghs,(39)ϕ=Msayhs+Muayhc−RKϕf+Kϕr−Msghs.The total roll stiffnessKϕ is approximately composed of the sum of the roll stiffness Kϕf of front axle and the roll stiffness Kϕr of rear axle; that is,(40)Kϕ=Kϕf+Kϕr.Among them, the roll stiffness of front or rear axle is composed of the roll stiffness of the spring and stabilizer bar; namely,(41)Kϕf=Kϕfs+Kϕfw.(42)Kϕr=Kϕrs+Kϕrw.In (41) and (42), Kϕfs,Kϕrs are the equivalent roll stiffness at the corresponding wheels; Kfs,Krs are the linear stiffness of spring; l1f,l1r are the arm length of front and rear suspension; l2f,l2r are the distance from the spring installation point to the hinge point of the arm; and Lf,Lr are the wheelbase of wheel.Incorporating (41) and (42) into (40), we can gain the total suspension roll stiffness; namely,(43)Kϕ=Kϕf+Kϕr=Kϕfs+Kϕrs+Kϕrs+Kϕrw.If the body rolls, it will cause load transfer between the left and right wheels in Figure10 for the load transfer of the front axle. It is assumed that the load transfer of front and rear axle is ΔFf,ΔFr. It is gained from the moment balance relation of the roll center:(44)Kϕfϕ=ΔFfLf−aymsdrdhcf.Assuming that the amount of load transfer between the wheels remained unchanged at this time and putting (39) into (44), we gain the vertical load transfer of front axle ΔFf as follows:(45)ΔFf=ayLfMshs−Muhc−R1+Kϕf/Kϕr−msghp/Kϕr+Msdfdhcf,where df is the distance from the front axle to center mass d=df+dr, and hcf is the distance from the roll center of front suspension to the ground.Similarly, the load transfer of rear axle is gained:(46)ΔFr=ayLrMshs−Muhc−R1+Kϕf/Kϕr−msghp/Kϕr+Msdrdhcr.The side slip stiffness of tire is related to its vertical load. Figure11 describes the relationship between the load transfer of tire and change of lateral force. The lateral force of tire changes in the form of a saturated curve with the transfer of the load. In the case of an axle, assuming that there is no lateral force acting on the car, the vertical loads on the left and right wheels are both F0. When lateral force acts on the car, the left and right wheels with load F0 produce a load transfer of ΔF0, so the lateral forces of the left and right wheels are, respectively, p1e1¯ and p2e2¯p2e2¯; among them, 2ke¯=p1e1¯+p2e2¯. When there is no load transfer between the left and right wheels, the sum of lateral force is 2pe¯, so the reduction amount of the side slip force is 2pk¯. When the load transfer of wheel occurs, the total of the lateral forces will reduce compared to when the load transfer is not taken into consideration. Moreover, the greater the load transfer of the left and right wheel, the greater the decrease of lateral force.Figure 11
The relationship between load transfer and lateral force.It is seen from the above analysis that the different spring stiffness will affect the change of the suspension roll center. From (45) and (46), it is known that the load transfer amount of the front and rear wheels will change. It is further known from Figure 11 that this will cause a change in the lateral force, so the side slip stiffness of wheel will vary.Equations (45) and (46) show that the distance from the sprung mass center to the suspension roll axis hshs; the roll center height hcf,hcrhcf,hcr; and the ratio Kϕf/Kϕr of suspension roll stiffness mainly determine the amount of load transfer during steering, which affects the understeer.


## 6. The Influence of Spring Stiffness Changes on Suspension Characteristics

According to engineering design experience, the spring stiffness values of the front and rear suspensions are reduced or increased to 20% and 50% of the original value, respectively (the original stiffness value is recorded as Origin±, and the ± sign indicates the percentage of decrease or increase). The suspension is simulated, and the relationship between Figures12 and 13 is obtained.Figure 12
The relationship between the suspension characteristic index and wheel travel after the change of spring stiffness of front suspension. (a) Toe angle. (b) Camber angle. (c) Roll stiffness. (d) Roll center height.
(a)(b)(c)(d)Figure 13
The relationship between the suspension characteristic index and wheel travel after the change of spring stiffness of rear suspension. (a) Toe angle. (b) Camber angle. (c) Roll stiffness. (d) Roll center height.
(a)(b)(c)(d)Generally, the toe angle of the front suspension should show a negative change; that is, the change gradient of the toe angle is negative, which indicates that when the body rolls, the toe angle of outer wheel decreases, the wheel turns outward, and the inner wheel is opposite, which is favorable to the understeer. The trend of toe angle of the rear suspension changes is the opposite.In Figure12(a), when the spring stiffness of front suspension changes from small to large or from large to small values, the change gradient of the toe angle of the front wheel is negative. Negative gradient indicates that when the wheel travels up, the toe angle changes from positive to negative values, which is favorable to the understeer.When the change gradient of the camber angle is positive, the camber angle of outer wheel increases, the inclination direction of wheel is opposite to the direction of the lateral force, and the side slip angle of the tire will increase. When the change gradient of the camber angle is negative, the effect is the opposite. For the front wheels, a positive gradient of camber changes favors understeer; that is, the camber of outer wheels will increase and the camber of inner wheels will decrease, while the opposite is true for the rear wheels.It is known from Figure12(b) that when the front suspension spring stiffness changes from small to large or from large to small values, the camber angle of front wheel changes in a positive direction, which is a positive gradient. As mentioned above, the slip angle of the wheel will increase, which is favorable to the understeer.In Figure12(c), when the front suspension spring stiffness changes from small to large values, the roll stiffness shows an increasing trend, which is favorable to the increase of the load transfer of wheel. The side slip angle of wheel will increase, which is favorable to understeer.In Figure12(d), when the front suspension spring stiffness changes from small to large values, the roll center height increases relative to the original value, and side slip stiffness of wheel will decrease, so side slip angle of wheel will increase, which is favorable to the understeer.In Figure13(a), when the spring stiffness of rear suspension changes from small to large or from large to small values, the change gradient of toe angle of rear wheel is positive. As mentioned above, the positive gradient is favorable to the understeer.It is known from Figure13(b) that when the spring stiffness of rear suspension changes from small to large or from large to small values, the gradient of the camber angle of rear wheels changes from a positive gradient to a negative gradient. As shown above, when the gradient is negative, the slip angle of the rear wheels will decrease, which is favorable for generating a positive returning torque and improving the side slip performance of vehicle.It is known from Figure13(c) that when the spring stiffness of rear suspension changes from small to large values, the roll angle stiffness decreases, which will increase the load transfer of the wheels, and the side slip angle will decrease, which is favorable to understeer. It is known from Figure 13(d) that when the spring stiffness value of rear suspension changes from small to large values, the roll center height decreases, so the side slip stiffness will be reduced, and the side slip angle will increase, which is favorable to the increase of the understeer.


## 7. Analysis of the Handling Stability Index under the Change of Suspension Spring Stiffness

Similarly, according to the above ratio, spring stiffness value of suspension is changed first, and then the swept-sine input simulation is performed. The relationship between the frequency characteristics index and the frequency of the steering wheel angle under different spring stiffness changes is gained as shown in Figures14 and 15.Figure 14
The relationship between the frequency characteristic index and the frequency of the steering wheel angle under the change of the spring stiffness of front suspension. (a) Yaw rate gain. (b) Roll angle gain. (c) Delay time of lateral acceleration. (d) Delay time of yaw rate.
(a)(b)(c)(d)Figure 15
The relationship between the frequency characteristic index and the frequency of the steering wheel angle under the change of the spring stiffness of rear suspension. (a) Yaw rate gain. (b) Roll angle gain. (c) Delay time of lateral acceleration. (d) Delay time of yaw rate.
(a)(b)(c)(d)It is known from Figure14 that when the spring stiffness of front suspension changes from small to large or from large to small values, with a low frequency up to 1 Hz, the effect on the yaw rate gain, the delay time of lateral acceleration, and the delay time of yaw rate is small, but the effect on the roll angle gain is large. When frequency is within 1 Hz to 2 Hz, the effect on the various indexes of the frequency response is the same as when frequency is up to 1 Hz. In the frequency within 2 Hz to 4 Hz, no matter whether the spring stiffness changes from small to large or from large to small values, it will have a significant effect on the yaw rate gain, the roll angle gain, the delay time of lateral acceleration, and the delay time of yaw rate. In frequency within 1 Hz to 4 Hz, the gain value and delay time do not show a regular change with the increase of frequency.It is known from Figure15 that in frequency up to 1 Hz, when the spring stiffness of the rear suspension changes from small to large or from large to small values, the effect on the yaw rate gain and the delay time of lateral acceleration is more obvious, while it has little effect on the roll angle gain and the delay time of yaw rate. In high frequency within 1 Hz to 3 Hz, no matter whether the spring stiffness changes from small to large or from large to small values, it has little effect on the yaw rate gain, the roll angle gain, the delay time of lateral acceleration, and the delay time of yaw rate. When frequency is within 3 Hz to 4 Hz, it has little effect on the yaw rate gain, the delay time of lateral acceleration, and the delay time of yaw rate, while the influence on the roll angle gain is more obvious. Furthermore, in frequency within 1 Hz to 4 Hz, the gain value and delay time do not show a regular change with the increase of frequency.


## 8. Multi-Objective Optimization

### 8.1. The Theoretical Model of Equivalent Handling Stability

Figure16 shows the established vehicle handling dynamics model. X′Y′Z is the absolute coordinate system. XYZ is the mass center coordinate.Figure 16
3-DOF manipulation model.The absolute accelerationO at the center of mass a is as follows:(47)a=φ˙×V,where the velocity of mass center is V, and φ is the angle between V and X′.The direction of the acceleration vectora is perpendicular to the speed vector V, and the component on the y-axis is as follows:(48)Y¨=φ˙Vcosβ,where β is the side slip angle of mass center.Sinceφ=ψ+β and cosβ≈1, (48) can be transformed into the following:(49)Y¨=Vβ˙+Vr,where r is the yaw rate.The centroid acceleration component of the sprung mass in they-axis is as follows:(50)Y¨s=Vβ˙+Vr−hsp˙,where p˙ is the roll acceleration.The balance equation of the lateral force is as follows:(51)M−Msr+β˙V+Msr+β˙V−hsp˙=∑i=1nFyi′+Fyi″.The moment balance equation around thez-axis is as follows:(52)Isr˙−Ixzp˙=∑i=1nxiFyi′+Fyi″,where Is is the yaw moment of inertia and Ixz is the product of inertia.The moment balance around thex-axis is as follows:(53)Iϕp˙−Ixzr˙−MshsVβ˙+r=−Kϕ+Msghsϕ−Cϕp,where Iϕ is the roll moment of inertia and Cϕ is the roll angle damping.The side slip anglesβi′,βi″ of the mass center of inner and outer wheels are as follows:(54)βi′=Vβ+dirV−Lir/2−δi=β+dir/V1−Lir/2V−δi≈β+dirV−δi,(55)βi″=Vβ+dirV+Lir/2V−δi=β+dir/V1+Lir/2V−δi≈β+dirV−δi,where δi is the steering wheel angle; di is the distance from the mass center to the front or rear axles; and Li is the wheel track, i=f+r.The relationship between roll steering angleαi′,αi″ and the roll angle of the wheel is as follows [22]:(56)αi′=αi″=∂αi∂ϕϕ,where ∂αi/∂ϕ is the roll steering angle corresponding to each unit of vehicle body roll angle, and it is positive when it is relative to the positive roll angle; that is, it is positive every time a counterclockwise steering angle is generated.Supposing that the camber angle led by the body roll is proportional to the body roll angle, the camber thrustYci′,Yci″ generated by the front and rear wheels is as follows:(57)Yci′=Yci″=−Kci∂σi∂ϕϕ,where Kcii=f,r is the camber lateral thrust coefficient of the front and rear tires and ∂γi/∂ϕ is the camber angle per unit body roll angle.Considering the roll steering and roll camber and combining (54)–(57), we obtain the lateral forces Fyi′ and Fyi″ of the tire acting on the wheels:(58)Fyi′=Fyi″=−Kiβ+dirV−δi−∂αi∂ϕϕ−Kci∂σi∂ϕϕ,where Ki is the side slip stiffness of tire. The side slip angle of wheel is calculated by (54) and (55), and the roll steering angle and camber thrust are calculated by (56) and (57). Equations (50)–(52) can be transformed into the following equations:(59)MVβ˙+2∑i=1nKiβ+MV+2∑i=1ndiKiVr−Mshsp˙−2Yϕϕ=2Kfδ,(60)2∑i=1ndiKiβ+Isr˙+2∑i=1ndi2KiVr−Ixzp˙−2Nϕϕ=2dfKfδ,(61)−MshsVβ˙−Ixzr˙−MshsVr+Iϕp˙+Cϕp+Kϕ−Msghs=0,where(62)Yϕ=∑i=1nKi∂αi∂ϕ−∑i=1nKci∂σi∂ϕ,Nϕ=∑i=1ndiKi∂αi∂ϕ−∑i=1ndiKci∂σi∂ϕ.By solving (59)–(61), the corresponding frequency characteristic indexes can be gained. By virtue of the above analysis and derivation, it can be concluded that the specific influence mechanism of the suspension spring stiffness on the frequency characteristics is as follows:(1)
It is known from Figures12(d) and 13(d) that different spring stiffness will affect the roll center height hcf,hcr of the front and rear suspensions, which will cause the parameter hs in (59)–(61) to change and cause the frequency response characteristic parameters to change.(2)
It is known from Figures12(c) and 13(c) that different spring stiffness will affect the roll stiffness, causing the total roll stiffness Kϕ in (59)–(61) to change, which in turn affects the change of frequency response characteristic parameters.(3)
It is from the analysis in Section4 that due to the change of roll stiffness, the load transfer will change correspondingly. This will affect the side slip stiffness Ki of tire in (59)–(61), which in turn affects the change of frequency response characteristic parameters.(4)
During the movement of the suspension, from the simulation results in Figures12(a), 12(b), 13(a), and 13(b), it is known that different spring stiffness will cause the toe angle and camber angle to change; that is to say, ∂αi/∂ϕ,∂σi/∂ϕ in (61) will change, which will affect the frequency response characteristic parameters.


### 8.2. Optimization Objectives and Optimization Algorithms

Since the frequency of people turning the steering wheel is about 0.5 Hz, the frequency response characteristic index at 0.5 Hz is used as the optimization objective function.f1 is the yaw rate gain at 0.5 Hz; f2 is the roll angle gain at 0.5 Hz; f3 is the resonance frequency at 0.5 Hz; f4 is the delay time of yaw rate at 0.5 Hz; f5 is the delay time of lateral acceleration at 0.5 Hz; and ζrω,ζϕω are the corresponding phase lag angle. The phase lag angle is transformed into delay time by the following equation:(63)t=ζ+180360⋅forζ−180360⋅f.Thus, the objective function is determined as follows:(64)f1=minr˙δωω=0.5Hz,f2=minϕayωω=0.5Hz,f3=ωr˙δωmax,f4=ζr+180360⋅forζr−180360⋅f,f5=ζϕ+180360⋅forζϕ−180360⋅f.Here, the genetic algorithm NSGA-II is used for multi-objective optimization of the front and rear suspension spring stiffness [23–25]. The algorithm is an improved multi-objective algorithm based on the NSGA algorithm. The algorithm introduces fast non-dominated sorting, crowding, and elite strategies. Through fast non-dominated sorting, the complexity of computing sorting is effectively reduced. The crowding degree and the comparison factor are introduced as the comparison criteria of the individuals in the population to ensure the diversity of the population. Compared with other genetic algorithms, NSGA-II has better convergence and robustness and has a good optimization effect on multi-objective optimization problems. The elite strategy is introduced to expand the sampling space and improve the global search ability and the accuracy of optimization results [26].


### 8.3. Optimization Results

After optimization, the iterative curve of each objective is obtained in Figure17. Figure 17(a) is an iterative curve of the yaw rate gain at 0.5 Hz. The value of the yaw rate gain at 0.5 Hz tends to decrease with the optimization. However, a smaller yaw rate gain within a certain range is favorable for making the handling stability better.Figure 17
Iterative curve of optimization objective. (a) Yaw rate gain at 0.5 Hz. (b) Resonance frequency at 0.5 Hz. (c) Roll angle gain at 0.5 Hz. (d) Delay time of lateral acceleration at 0.5 Hz. (e) Delay time of yaw rate at 0.5 Hz.
(a)(b)(c)(d)(e)Figure17(b) shows an iterative curve of the resonance frequency at 0.5 Hz. It is known from the figure that the resonance frequency tends to increase with the optimization. Generally speaking, the greater the resonance frequency within a certain range, the more stable the vehicle, which shows that its change trend satisfies the requirements of the optimization objective. Figure 17(c) indicates an iterative curve of the roll angle gain at 0.5 Hz. A small roll angle gain within a certain range is favorable to the roll stability. It is seen from the figure that the roll angle gain decreases with the progress of the optimization.Figures17(d) and 17(e) are the iterative curves of the delay time of lateral acceleration at 0.5 Hz and the delay time of yaw rate at 0.5 Hz. The smaller the delay time within a certain range is, the faster the vehicle reacts, and the better the performance of the vehicle is. It is seen from the figure that the absolute value of the delay time tends to decrease with the optimization. Meanwhile, the corresponding Pareto solution set is also gained in Figure 18.Figure 18
Pareto solution sets of frequency response index. (a) Roll angle gain at 0.5 Hz and resonance frequency at 0.5 Hz. (b) Yaw rate gain at 0.5 Hz and resonance frequency at 0.5 Hz. (c) Roll angle gain at 0.5 Hz and delay time 1 at 0.5 Hz. (d) Roll angle gain at 0.5 Hz and delay time 2 at 0.5 Hz.
(a)(b)(c)(d)Through optimization, the changes of parameters before and after optimization are presented in Table2.Table 2
Changes in parameters before and after optimization.
f1s−1f2s−1f3Hzf4sf5sKϕfsN/mmKϕrsN/mmOriginal car0.524.420.540.200.1442.0020.00Relative optimal solution0.474.310.5621.000.1352.4030.50Pareto solution0.484.350.580.200.1334.5024.50Percentage (Pareto solution) (%)−7.60−15.80+7.40−9.10−7.10−17.80+22.50


## 8.1. The Theoretical Model of Equivalent Handling Stability

Figure16 shows the established vehicle handling dynamics model. X′Y′Z is the absolute coordinate system. XYZ is the mass center coordinate.Figure 16
3-DOF manipulation model.The absolute accelerationO at the center of mass a is as follows:(47)a=φ˙×V,where the velocity of mass center is V, and φ is the angle between V and X′.The direction of the acceleration vectora is perpendicular to the speed vector V, and the component on the y-axis is as follows:(48)Y¨=φ˙Vcosβ,where β is the side slip angle of mass center.Sinceφ=ψ+β and cosβ≈1, (48) can be transformed into the following:(49)Y¨=Vβ˙+Vr,where r is the yaw rate.The centroid acceleration component of the sprung mass in they-axis is as follows:(50)Y¨s=Vβ˙+Vr−hsp˙,where p˙ is the roll acceleration.The balance equation of the lateral force is as follows:(51)M−Msr+β˙V+Msr+β˙V−hsp˙=∑i=1nFyi′+Fyi″.The moment balance equation around thez-axis is as follows:(52)Isr˙−Ixzp˙=∑i=1nxiFyi′+Fyi″,where Is is the yaw moment of inertia and Ixz is the product of inertia.The moment balance around thex-axis is as follows:(53)Iϕp˙−Ixzr˙−MshsVβ˙+r=−Kϕ+Msghsϕ−Cϕp,where Iϕ is the roll moment of inertia and Cϕ is the roll angle damping.The side slip anglesβi′,βi″ of the mass center of inner and outer wheels are as follows:(54)βi′=Vβ+dirV−Lir/2−δi=β+dir/V1−Lir/2V−δi≈β+dirV−δi,(55)βi″=Vβ+dirV+Lir/2V−δi=β+dir/V1+Lir/2V−δi≈β+dirV−δi,where δi is the steering wheel angle; di is the distance from the mass center to the front or rear axles; and Li is the wheel track, i=f+r.The relationship between roll steering angleαi′,αi″ and the roll angle of the wheel is as follows [22]:(56)αi′=αi″=∂αi∂ϕϕ,where ∂αi/∂ϕ is the roll steering angle corresponding to each unit of vehicle body roll angle, and it is positive when it is relative to the positive roll angle; that is, it is positive every time a counterclockwise steering angle is generated.Supposing that the camber angle led by the body roll is proportional to the body roll angle, the camber thrustYci′,Yci″ generated by the front and rear wheels is as follows:(57)Yci′=Yci″=−Kci∂σi∂ϕϕ,where Kcii=f,r is the camber lateral thrust coefficient of the front and rear tires and ∂γi/∂ϕ is the camber angle per unit body roll angle.Considering the roll steering and roll camber and combining (54)–(57), we obtain the lateral forces Fyi′ and Fyi″ of the tire acting on the wheels:(58)Fyi′=Fyi″=−Kiβ+dirV−δi−∂αi∂ϕϕ−Kci∂σi∂ϕϕ,where Ki is the side slip stiffness of tire. The side slip angle of wheel is calculated by (54) and (55), and the roll steering angle and camber thrust are calculated by (56) and (57). Equations (50)–(52) can be transformed into the following equations:(59)MVβ˙+2∑i=1nKiβ+MV+2∑i=1ndiKiVr−Mshsp˙−2Yϕϕ=2Kfδ,(60)2∑i=1ndiKiβ+Isr˙+2∑i=1ndi2KiVr−Ixzp˙−2Nϕϕ=2dfKfδ,(61)−MshsVβ˙−Ixzr˙−MshsVr+Iϕp˙+Cϕp+Kϕ−Msghs=0,where(62)Yϕ=∑i=1nKi∂αi∂ϕ−∑i=1nKci∂σi∂ϕ,Nϕ=∑i=1ndiKi∂αi∂ϕ−∑i=1ndiKci∂σi∂ϕ.By solving (59)–(61), the corresponding frequency characteristic indexes can be gained. By virtue of the above analysis and derivation, it can be concluded that the specific influence mechanism of the suspension spring stiffness on the frequency characteristics is as follows:(1)
It is known from Figures12(d) and 13(d) that different spring stiffness will affect the roll center height hcf,hcr of the front and rear suspensions, which will cause the parameter hs in (59)–(61) to change and cause the frequency response characteristic parameters to change.(2)
It is known from Figures12(c) and 13(c) that different spring stiffness will affect the roll stiffness, causing the total roll stiffness Kϕ in (59)–(61) to change, which in turn affects the change of frequency response characteristic parameters.(3)
It is from the analysis in Section4 that due to the change of roll stiffness, the load transfer will change correspondingly. This will affect the side slip stiffness Ki of tire in (59)–(61), which in turn affects the change of frequency response characteristic parameters.(4)
During the movement of the suspension, from the simulation results in Figures12(a), 12(b), 13(a), and 13(b), it is known that different spring stiffness will cause the toe angle and camber angle to change; that is to say, ∂αi/∂ϕ,∂σi/∂ϕ in (61) will change, which will affect the frequency response characteristic parameters.


## 8.2. Optimization Objectives and Optimization Algorithms

Since the frequency of people turning the steering wheel is about 0.5 Hz, the frequency response characteristic index at 0.5 Hz is used as the optimization objective function.f1 is the yaw rate gain at 0.5 Hz; f2 is the roll angle gain at 0.5 Hz; f3 is the resonance frequency at 0.5 Hz; f4 is the delay time of yaw rate at 0.5 Hz; f5 is the delay time of lateral acceleration at 0.5 Hz; and ζrω,ζϕω are the corresponding phase lag angle. The phase lag angle is transformed into delay time by the following equation:(63)t=ζ+180360⋅forζ−180360⋅f.Thus, the objective function is determined as follows:(64)f1=minr˙δωω=0.5Hz,f2=minϕayωω=0.5Hz,f3=ωr˙δωmax,f4=ζr+180360⋅forζr−180360⋅f,f5=ζϕ+180360⋅forζϕ−180360⋅f.Here, the genetic algorithm NSGA-II is used for multi-objective optimization of the front and rear suspension spring stiffness [23–25]. The algorithm is an improved multi-objective algorithm based on the NSGA algorithm. The algorithm introduces fast non-dominated sorting, crowding, and elite strategies. Through fast non-dominated sorting, the complexity of computing sorting is effectively reduced. The crowding degree and the comparison factor are introduced as the comparison criteria of the individuals in the population to ensure the diversity of the population. Compared with other genetic algorithms, NSGA-II has better convergence and robustness and has a good optimization effect on multi-objective optimization problems. The elite strategy is introduced to expand the sampling space and improve the global search ability and the accuracy of optimization results [26].


## 8.3. Optimization Results

After optimization, the iterative curve of each objective is obtained in Figure17. Figure 17(a) is an iterative curve of the yaw rate gain at 0.5 Hz. The value of the yaw rate gain at 0.5 Hz tends to decrease with the optimization. However, a smaller yaw rate gain within a certain range is favorable for making the handling stability better.Figure 17
Iterative curve of optimization objective. (a) Yaw rate gain at 0.5 Hz. (b) Resonance frequency at 0.5 Hz. (c) Roll angle gain at 0.5 Hz. (d) Delay time of lateral acceleration at 0.5 Hz. (e) Delay time of yaw rate at 0.5 Hz.
(a)(b)(c)(d)(e)Figure17(b) shows an iterative curve of the resonance frequency at 0.5 Hz. It is known from the figure that the resonance frequency tends to increase with the optimization. Generally speaking, the greater the resonance frequency within a certain range, the more stable the vehicle, which shows that its change trend satisfies the requirements of the optimization objective. Figure 17(c) indicates an iterative curve of the roll angle gain at 0.5 Hz. A small roll angle gain within a certain range is favorable to the roll stability. It is seen from the figure that the roll angle gain decreases with the progress of the optimization.Figures17(d) and 17(e) are the iterative curves of the delay time of lateral acceleration at 0.5 Hz and the delay time of yaw rate at 0.5 Hz. The smaller the delay time within a certain range is, the faster the vehicle reacts, and the better the performance of the vehicle is. It is seen from the figure that the absolute value of the delay time tends to decrease with the optimization. Meanwhile, the corresponding Pareto solution set is also gained in Figure 18.Figure 18
Pareto solution sets of frequency response index. (a) Roll angle gain at 0.5 Hz and resonance frequency at 0.5 Hz. (b) Yaw rate gain at 0.5 Hz and resonance frequency at 0.5 Hz. (c) Roll angle gain at 0.5 Hz and delay time 1 at 0.5 Hz. (d) Roll angle gain at 0.5 Hz and delay time 2 at 0.5 Hz.
(a)(b)(c)(d)Through optimization, the changes of parameters before and after optimization are presented in Table2.Table 2
Changes in parameters before and after optimization.
f1s−1f2s−1f3Hzf4sf5sKϕfsN/mmKϕrsN/mmOriginal car0.524.420.540.200.1442.0020.00Relative optimal solution0.474.310.5621.000.1352.4030.50Pareto solution0.484.350.580.200.1334.5024.50Percentage (Pareto solution) (%)−7.60−15.80+7.40−9.10−7.10−17.80+22.50


## 9. Comparative Analysis before and after Vehicle Optimization

After optimizing the entire vehicle, an optimal solution and the relative optimal solution are chosen in the Pareto solution set, and then they are, respectively, brought into the vehicle model to compare the transient characteristics of the simulation test with the original vehicle to judge whether the optimization results are reasonable.The curve in Figure19 shows the change of the yaw rate gain before and after optimization. When the frequency is within 0 Hz to 1 Hz, the relative optimal solution gained by the optimization and an optimal solution chosen both make the yaw rate gain smaller than the original vehicle in Figure 19, which shows that the requirements of the optimization objective are satisfied.Figure 19
Curve of the yaw rate gain.The curve in Figure20 shows the change of the roll angle gain before and after optimization. It is seen from the figure that when the frequency is around 0.5 Hz, the relative optimal solution gained by the optimization and the chosen optimal solution both make the value of the roll angle gain smaller than that of the original car, indicating that the value of roll angle gain has been improved after optimization.Figure 20
Curve of the roll angle gain.The curves in Figures21 and 22 show the change of the delay time of yaw rate and the delay time of lateral acceleration before and after optimization. It is seen from the figures that when the frequency is around 0.5 Hz, the relative optimal solution gained by the optimization and an optimal solution chosen both make the absolute value of the delay time of yaw rate smaller than that before optimization, indicating that the response speed of the car becomes faster after optimization. It is known that the delay time satisfies the requirements of the optimization objective.Figure 21
Curve of the delay time of lateral acceleration.Figure 22
Curve of the delay time of yaw rate.


## 10. Conclusion

The paper systematically analyzes the influence of the spring stiffness change on the suspension characteristics. It also explains the specific reasons why the roll stiffness of suspension and the changes in the load transfer of wheel affect the handling stability, and then it explains the influence mechanism of the change of spring stiffness on the vehicle frequency characteristic. The relationship between the frequency characteristic index and the frequency of steering wheel angle under the change of spring stiffness is gained. Then, multi-objective optimization of the spring stiffness is executed, and the concrete summaries as follows:(1)
The influence mechanism of the spring stiffness change on the vehicle frequency characteristic is revealed: The toe angle, camber angle, roll stiffness, and roll center will affect the vehicle frequency characteristics, and the difference in spring stiffness will cause them to change, so different spring stiffness will affect frequency characteristics. The roll stiffness and roll center will also affect the load transfer of wheel, which affects the side slip stiffness of tire, and then will affect the vehicle frequency characteristics, thereby affecting the handling stability.(2)
The relationship between the vehicle frequency characteristic index and the frequency of steering wheel angle is as follows: For the front suspension, when the frequency is up to 1 Hz, the influence on the delay time of lateral acceleration and yaw rate is small, and the influence on the gain of yaw rate and roll angle is more obvious. In the frequency range of 2 Hz to 4 Hz, it has obvious effects on the yaw rate gain, the roll angle gain, the delay time of lateral acceleration, and the yaw rate. For the rear suspension, when the frequency is up to 1 Hz, the effect on the yaw rate gain and the delay time of lateral acceleration is obvious, while the influence on the roll angle gain and the delay time of yaw rate is small. In the high frequency range of 3 Hz to 4 Hz, it has little effect on the yaw rate gain, the delay time of lateral acceleration, and the yaw rate, while the roll angle gain has a more obvious influence.(3)
The optimization objective of the frequency response characteristic index at 0.5 Hz is used to optimize the spring stiffness. After optimization, the optimal Pareto solution is selected and compared with the original vehicle. The results show that when the spring stiffness of front suspension is reduced by 17.5% and the spring stiffness of rear suspension is increased by 22.5%, the yaw rate gain, roll angle gain, delay time of lateral acceleration, and yaw rate at 0.5 Hz are all reduced, which is within an acceptable range and satisfies the requirements of the optimization objective, and the handling stability of the vehicle is improved.



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*Source: 9557483-2022-05-25.xml*