# An Analytical Tire Model with Flexible Carcass for Combined Slips

Authors: Nan Xu; Konghui Guo; Xinjie Zhang; Hamid Reza Karimi
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

The tire mechanical characteristics under combined cornering and braking/driving situations have significant effects on vehicle directional controls. The objective of this paper is to present an analytical tire model with flexible carcass for combined slip situations, which can describe tire behavior well and can also be used for studying vehicle dynamics. The tire forces and moments come mainly from the shear stress and sliding friction at the tread-road interface. In order to describe complicated tire characteristics and tire-road friction, some key factors are considered in this model: arbitrary pressure distribution; translational, bending, and twisting compliance of the carcass; dynamic friction coefficient; anisotropic stiffness properties. The analytical tire model can describe tire forces and moments accurately under combined slip conditions. Some important properties induced by flexible carcass can also be reflected. The structural parameters of a tire can be identified from tire measurements and the computational results using the analytical model show good agreement with test data.


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## Body

## 1. Introduction

Tires are the only components of a road vehicle to directly contact with the road surface, and the forces and moments generated in the contact patch have significant effects on the vehicle performance [1–4]. Hence a huge number of tire models have been proposed for use in calculating the forces and moments at the tire-road interface [5–13].The empirical models are often employed for the vehicle simulation and control, which rely basically on curve-fitted experimental data and can provide a good representation of experimental data for longitudinal force, lateral force, and aligning moment. However, empirical models would become quite complicated without theoretical support while considering all kinds of operating variables, such as slip angle, inclination angle, slip ratio, vertical load, inflation pressure, road friction, and rolling speed. The analytical models, which usually include carcass model (beam, string or rigid) with elastic tread elements, such as Brush model, Fiala model and String model, establish the relationship between tire structure parameters and tire behavior [3]. So it can be employed to derive some useful qualitative conclusions for understanding tire properties, but most of these models are either relatively simple or more complicated, which limit their practical use. In this paper, we do not see the carcass as an actual beam or string, and the carcass deformation is described with relatively simple and general forms, composed by translational, bending, and twisting deformations. Besides, the arbitrary pressure distribution, dynamic friction coefficient, and anisotropic stiffness properties are also considered. Consequently, the analytical model would become more suitable for application and also appropriate for analyzing tire properties in detail.In this paper, the key factors for developing the analytical tire model are firstly discussed; then, considering all these factors, the analytical tire model with flexible carcass for combined slips is introduced. By employing the model, effects of carcass compliance on tire properties are discussed, which are valuable for understanding tire properties; at the end, the analytical model is validated by test data.


## 2. Tire Axis System and Slip Ratios

The tire axis system is shown in Figure1. The positive direction of the X-axis and the Y-axis is coincident with tire revolution direction (not wheel traveling direction); the Z-axis is perpendicular to road plane and upward. The wheel traveling speed is denoted as V, α is the slip angle, and γ is inclination angle. The figure shows all the forces and moments associated with a wheel.Figure 1
Tire axis system.The longitudinal and lateral slip ratios are defined in the unified form [11]
(1)Sx=-VsxΩRe=-Vcosα-ΩReΩReSx∈(-∞,+∞),Sy=-VsyΩRe=-VsinαΩReSy∈(-∞,+∞),
where Ω is the angular speed, Re is the effective rolling radius and Vsx and Vsy are the longitudinal and lateral sliding speeds of tire with respect to road surface.Usually, the longitudinal slip ratio used in tire force test is defined as(2)κ=-VsxVcosα.The relationship betweenSx and κ can be obtained easily:
(3)Sx=κ1+κ.


## 3. Key Factors for Analytical Tire Model

### 3.1. Arbitrary Pressure Distribution

Contact pressure distribution over contact patch strongly influences tire behaviors. Here, the contact patch is assumed to be rectangular in shape, and the contact pressure distribution is assumed to be uniform in lateral direction and of arbitrary form in circumferential direction to represent different kinds of pressure distributions.The contact pressureqz along the contact patch length 2a is expressed as an arbitrary form as
(4)qz(x)=Fz2a·η(u),
where Fz is the tire vertical load; u=x/a is the relative coordinate. η(u) is the normalized pressure distribution function and is expressed as
(5)η(u)=A·(1-u2n)·(1+λ·u2n)·(1-B·u)A=(2n+1)(4n+1)2n(4n+1+λ),B=-3(2n+3)(4n+3)(4n+1+λ)(2n+1)(4n+1)(4n+3+3λ)·Δa,
where n, λ, and Δ are the parameters which determine the shape of pressure distribution. With these three parameters, (5) can be employed to express arbitrary pressure distribution over contact patch, as shown in Figure 2.Pressure distribution.
(a)
(b)


### 3.2. Carcass Structure Parameters

The deformation of carcass has an important effect on the tire properties under combined slip conditions. In this paper, three tire carcass stiffness parameters for tire lateral carcass deformation are introduced, that is, tire carcass lateral translation stiffnessKcy, carcass bending stiffness Kcb, and carcass twisting stiffness Nθ. So, the carcass deformation includes lateral translating part yc0, bending part ycb, and twisting part yθ, as shown in Figure 3. The carcass longitudinal deformation is assumed to include longitudinal translating part xc0, and the longitudinal translation stiffness is Kcx.Figure 3
Tire lateral carcass deformation.The lateral translating deformation of carcass can be calculated as(6)yc0=FyKcy0,
where Kcy is the lateral translation stiffness of carcass; Fy is the lateral force.The lateral bending deformation of carcass can be calculated as(7)ycb(x)=FyKcb·ξ(xa),
where Kcb is the carcass bending stiffness; ξ(x/a) is the general function of carcass bending deformation.The zero-order moment and first-order moment ofξ(u) are expressed as
(8)D0(u)=∫u1ξ(u)du,D1(u)=∫u1u·ξ(u)du.The twisting deformation of carcass is calculated by(9)yθ(x)=θ·x,θ=MzNθ,
where θ is the carcass twisting angle; Nθ is the carcass twisting stiffness; Mz is the aligning moment.The longitudinal translating deformation of carcass is expressed as(10)xc0=FxKcx0,
where Kcx0 is the longitudinal translation stiffness of carcass; Fx is the longitudinal force.


### 3.3. Dynamic Friction Coefficient

Friction coefficient used in the analytical tire model is the dynamic friction coefficient, which considers the significant influence of slip speed. The expression is as follows [14]:
(11)μd=μs+(μm-μs)·exp((|VsVsm|+N·exp(-|VsVsm|)))-μh2·log2(|VsVsm|+N·exp(-|VsVsm|))),
where μ0, μs, μh, and Vsm are friction characteristic parameters; N (usually N=0.8) is a factor to make the friction coefficient increase slightly around the origin; Vs is the slip speed between the road and tire.


### 3.4. Anisotropic Stiffness Properties

The anisotropy of tire slip stiffness will arouse the difference of tire shear stress direction in adhesion region and sliding region, which are expressed as [15]
(12)adhesion:θad=arctan((KySy)(KxSx)),sliding:θs=arctan(SySx),
where Kx and Ky represent the longitudinal slip and cornering stiffness, respectively.


## 3.1. Arbitrary Pressure Distribution

Contact pressure distribution over contact patch strongly influences tire behaviors. Here, the contact patch is assumed to be rectangular in shape, and the contact pressure distribution is assumed to be uniform in lateral direction and of arbitrary form in circumferential direction to represent different kinds of pressure distributions.The contact pressureqz along the contact patch length 2a is expressed as an arbitrary form as
(4)qz(x)=Fz2a·η(u),
where Fz is the tire vertical load; u=x/a is the relative coordinate. η(u) is the normalized pressure distribution function and is expressed as
(5)η(u)=A·(1-u2n)·(1+λ·u2n)·(1-B·u)A=(2n+1)(4n+1)2n(4n+1+λ),B=-3(2n+3)(4n+3)(4n+1+λ)(2n+1)(4n+1)(4n+3+3λ)·Δa,
where n, λ, and Δ are the parameters which determine the shape of pressure distribution. With these three parameters, (5) can be employed to express arbitrary pressure distribution over contact patch, as shown in Figure 2.Pressure distribution.
(a)
(b)


## 3.2. Carcass Structure Parameters

The deformation of carcass has an important effect on the tire properties under combined slip conditions. In this paper, three tire carcass stiffness parameters for tire lateral carcass deformation are introduced, that is, tire carcass lateral translation stiffnessKcy, carcass bending stiffness Kcb, and carcass twisting stiffness Nθ. So, the carcass deformation includes lateral translating part yc0, bending part ycb, and twisting part yθ, as shown in Figure 3. The carcass longitudinal deformation is assumed to include longitudinal translating part xc0, and the longitudinal translation stiffness is Kcx.Figure 3
Tire lateral carcass deformation.The lateral translating deformation of carcass can be calculated as(6)yc0=FyKcy0,
where Kcy is the lateral translation stiffness of carcass; Fy is the lateral force.The lateral bending deformation of carcass can be calculated as(7)ycb(x)=FyKcb·ξ(xa),
where Kcb is the carcass bending stiffness; ξ(x/a) is the general function of carcass bending deformation.The zero-order moment and first-order moment ofξ(u) are expressed as
(8)D0(u)=∫u1ξ(u)du,D1(u)=∫u1u·ξ(u)du.The twisting deformation of carcass is calculated by(9)yθ(x)=θ·x,θ=MzNθ,
where θ is the carcass twisting angle; Nθ is the carcass twisting stiffness; Mz is the aligning moment.The longitudinal translating deformation of carcass is expressed as(10)xc0=FxKcx0,
where Kcx0 is the longitudinal translation stiffness of carcass; Fx is the longitudinal force.


## 3.3. Dynamic Friction Coefficient

Friction coefficient used in the analytical tire model is the dynamic friction coefficient, which considers the significant influence of slip speed. The expression is as follows [14]:
(11)μd=μs+(μm-μs)·exp((|VsVsm|+N·exp(-|VsVsm|)))-μh2·log2(|VsVsm|+N·exp(-|VsVsm|))),
where μ0, μs, μh, and Vsm are friction characteristic parameters; N (usually N=0.8) is a factor to make the friction coefficient increase slightly around the origin; Vs is the slip speed between the road and tire.


## 3.4. Anisotropic Stiffness Properties

The anisotropy of tire slip stiffness will arouse the difference of tire shear stress direction in adhesion region and sliding region, which are expressed as [15]
(12)adhesion:θad=arctan((KySy)(KxSx)),sliding:θs=arctan(SySx),
where Kx and Ky represent the longitudinal slip and cornering stiffness, respectively.


## 4. Analytical Tire Forces and Moments Model

### 4.1. Tire Forces and Moments without Sliding

In order to obtain the shear force in the contact patch, the deformations of the tread and carcass along theX and Y axes must be known firstly. The longitudinal and lateral deformations of the tread and carcass, under combined slip condition, are shown in Figure 4. In this figure, XOY is a coordinate system before the carcass is deformed, and xoy is a relative coordinate system for describing the tread deformation and bending and twisting deformation of carcass. xc0 and yc0 are the longitudinal and lateral translating deformation of carcass. “ABC” is the contact line of the contact patch under combined slip condition. V is the wheel traveling speed; α is the slip angle and θ is the carcass twisting angle. In general case, the whole length of contact patch, 2a, is divided into two parts, the adhesion region “AB” and the sliding region “BC,” by the initial sliding point “B.”Figure 4
Deformation of carcass and tread element under combined slip condition.P
c
P
t in the figure represents the tread element after rolling for a period of time t. The upper point of tread element, Pc, is attached to the belt of tire. Its coordinate could be written as
(13)xpc=x,ypc=yθ(x)+ycb(x).The lower point of tread element,Pt, is contacted with the ground. Its coordinate could be written as
(14)xpt=x-(Vtcosα-Vrt)=x+(a-x)Sx,ypt=-Vtsinα+aθ=aθ+(a-x)Sy.Therefore, the longitudinal and lateral deformations of tread element is(15)Δx=xpt-xpc=(a-x)Sx,Δy=ypt-ypc=(a-x)(MzNθ+Sy)-FyKcbξ(xa).The main source of anisotropy is due to different tire structural flexibility in lateral and longitudinal direction, whereas tread anisotropy is present but comparatively small. In this paper, the stiffness of tread, denoted askt, is considered to be isotropic. The shear stresses of tread element in X and Y directions can be expressed as
(16)qx=kt·Δx=kt·(a-x)Sx,qy=kt·Δy=kt·[(a-x)(MzNθ+Sy)-FyKcbξ(xa)].The shear force in the directionsX and Y can be determined as follows:
(17)Fx=∫-aaqxdx,Fy=∫-aaqydx,Mz=-∫-aaqx·ypcdx+∫-aaqy·xpcdx-Fx·yc0+Fy·xc0.Considering the previous equations (6)~(10), (13), and (16), the forces and moments could be written as
(18)Fx=2a2ktSx,Fy=3aεθMz+2a2ktSy-εbFy,Mz=-aεbSxFy+εθSxMz+12aεbSxFyD1(-1)-εθMz-23a3ktSy-12aεbFyD1(-1)+(1Kcx0-1Kcy0)FxFy,
where the bending characteristic ratio and twisting characteristic ratio have been introduced and defined by
(19)εb=2aktKcb,εθ=23a3ktNθ.Solving (18) to obtain the explicit expression of Fy and Mz, (18) becomes
(20)Fx=KxSx,Fy=KySy,Mz=KmSy,
with
(21)Kx=2a2kt,Ky=a(1+εθ-εθSx)Ky0-3εθKm0a(1+εb)(1+εθ-εθSx)+3aεbεθSx-3εθcFxKm=a3εθ[(1+εb)Ky-Ky0],
where Kx is the longitudinal slip stiffness, Ky is the cornering stiffness, and Km is the aligning stiffness with flexible carcass. Ky0 and Km0 are the cornering stiffness and aligning stiffness, respectively, when the carcass is assumed to be rigid, which are expressed as
(22)Ky0=2a2kt,Km0=23a3kt,wherec in (21) is the translating compliance coefficient that is defined by
(23)c=1Kcx0-1Kcy0.


### 4.2. Tire Forces and Moments in General Case with Sliding Region

Considering that the sliding region might exist, the shear force in this area should be expressed asq(x)=μqz(x), based on which the coordinate of initial sliding point, x=xc, or the relative coordinate uc=xc/a, can be solved. The coordinate of initial sliding point will satisfy the following equation:
(24)q=qx2+qy2=μFz2a·η(uc).Integrating through both adhesion and sliding regions, the tire forces and moments can be derived as(25)Fx=∫xcakt·Δxdx+∫-axcμFz2a·η(xa)·θsxdx,Fy=∫xcakt·Δydx+∫-axcμFz2a·η(xa)·θsydx,Mz=-∫xcakt·Δx·ypcdx+∫xcakt·Δy·xpcdx-∫-axcμFz2a·η(xa)·θsx·ypcdx+∫-axcμFz2a·η(xa)·θsy·xpcdx-Fx·yc0+Fy·xc0,
where θsx and θsy are, respectively, the longitudinal and lateral components of shear stress direction θs in sliding region.Substituted with (6)~(12), (13), and (15), (25) becomes
(26)Fx=B6·KxSx+B3·μFzθsx,Fy=[(P1B6·Ky0+P4B7·Km0)Sy+(P1B3+P4B4a)μFzθsy]×(P2P4+P1P3)-1,Mz=[(-P2B6·Ky0+P3B7·Km0)Sy+(-P2B3+P3B4a)μFzθsy]×(P2P4+P1P3)-1,
with
(27)B1=12εbD0(uc),B2=12εbD1(uc),B3=12m0(uc),B4=12m1(uc),B5=12mD(uc),B6=(1-uc)24,B7=14(1-3uc2+2uc3),P1=1-B7εθ+B7εθSx+B4μFzθsxa1Nθ,P2=(B2+B1Sx-B2Sx)a+B5μFzθsx1Kcb-cFx,P3=1+B1,P4=3aB6εθ,
where m0(uc), m1(uc), and mD(uc) are defined by
(28)m0(uc)=∫-1ucη(u)du,m1(uc)=∫-1ucuη(u)du,mD(uc)=∫-1ucξ(u)η(u)du.


## 4.1. Tire Forces and Moments without Sliding

In order to obtain the shear force in the contact patch, the deformations of the tread and carcass along theX and Y axes must be known firstly. The longitudinal and lateral deformations of the tread and carcass, under combined slip condition, are shown in Figure 4. In this figure, XOY is a coordinate system before the carcass is deformed, and xoy is a relative coordinate system for describing the tread deformation and bending and twisting deformation of carcass. xc0 and yc0 are the longitudinal and lateral translating deformation of carcass. “ABC” is the contact line of the contact patch under combined slip condition. V is the wheel traveling speed; α is the slip angle and θ is the carcass twisting angle. In general case, the whole length of contact patch, 2a, is divided into two parts, the adhesion region “AB” and the sliding region “BC,” by the initial sliding point “B.”Figure 4
Deformation of carcass and tread element under combined slip condition.P
c
P
t in the figure represents the tread element after rolling for a period of time t. The upper point of tread element, Pc, is attached to the belt of tire. Its coordinate could be written as
(13)xpc=x,ypc=yθ(x)+ycb(x).The lower point of tread element,Pt, is contacted with the ground. Its coordinate could be written as
(14)xpt=x-(Vtcosα-Vrt)=x+(a-x)Sx,ypt=-Vtsinα+aθ=aθ+(a-x)Sy.Therefore, the longitudinal and lateral deformations of tread element is(15)Δx=xpt-xpc=(a-x)Sx,Δy=ypt-ypc=(a-x)(MzNθ+Sy)-FyKcbξ(xa).The main source of anisotropy is due to different tire structural flexibility in lateral and longitudinal direction, whereas tread anisotropy is present but comparatively small. In this paper, the stiffness of tread, denoted askt, is considered to be isotropic. The shear stresses of tread element in X and Y directions can be expressed as
(16)qx=kt·Δx=kt·(a-x)Sx,qy=kt·Δy=kt·[(a-x)(MzNθ+Sy)-FyKcbξ(xa)].The shear force in the directionsX and Y can be determined as follows:
(17)Fx=∫-aaqxdx,Fy=∫-aaqydx,Mz=-∫-aaqx·ypcdx+∫-aaqy·xpcdx-Fx·yc0+Fy·xc0.Considering the previous equations (6)~(10), (13), and (16), the forces and moments could be written as
(18)Fx=2a2ktSx,Fy=3aεθMz+2a2ktSy-εbFy,Mz=-aεbSxFy+εθSxMz+12aεbSxFyD1(-1)-εθMz-23a3ktSy-12aεbFyD1(-1)+(1Kcx0-1Kcy0)FxFy,
where the bending characteristic ratio and twisting characteristic ratio have been introduced and defined by
(19)εb=2aktKcb,εθ=23a3ktNθ.Solving (18) to obtain the explicit expression of Fy and Mz, (18) becomes
(20)Fx=KxSx,Fy=KySy,Mz=KmSy,
with
(21)Kx=2a2kt,Ky=a(1+εθ-εθSx)Ky0-3εθKm0a(1+εb)(1+εθ-εθSx)+3aεbεθSx-3εθcFxKm=a3εθ[(1+εb)Ky-Ky0],
where Kx is the longitudinal slip stiffness, Ky is the cornering stiffness, and Km is the aligning stiffness with flexible carcass. Ky0 and Km0 are the cornering stiffness and aligning stiffness, respectively, when the carcass is assumed to be rigid, which are expressed as
(22)Ky0=2a2kt,Km0=23a3kt,wherec in (21) is the translating compliance coefficient that is defined by
(23)c=1Kcx0-1Kcy0.


## 4.2. Tire Forces and Moments in General Case with Sliding Region

Considering that the sliding region might exist, the shear force in this area should be expressed asq(x)=μqz(x), based on which the coordinate of initial sliding point, x=xc, or the relative coordinate uc=xc/a, can be solved. The coordinate of initial sliding point will satisfy the following equation:
(24)q=qx2+qy2=μFz2a·η(uc).Integrating through both adhesion and sliding regions, the tire forces and moments can be derived as(25)Fx=∫xcakt·Δxdx+∫-axcμFz2a·η(xa)·θsxdx,Fy=∫xcakt·Δydx+∫-axcμFz2a·η(xa)·θsydx,Mz=-∫xcakt·Δx·ypcdx+∫xcakt·Δy·xpcdx-∫-axcμFz2a·η(xa)·θsx·ypcdx+∫-axcμFz2a·η(xa)·θsy·xpcdx-Fx·yc0+Fy·xc0,
where θsx and θsy are, respectively, the longitudinal and lateral components of shear stress direction θs in sliding region.Substituted with (6)~(12), (13), and (15), (25) becomes
(26)Fx=B6·KxSx+B3·μFzθsx,Fy=[(P1B6·Ky0+P4B7·Km0)Sy+(P1B3+P4B4a)μFzθsy]×(P2P4+P1P3)-1,Mz=[(-P2B6·Ky0+P3B7·Km0)Sy+(-P2B3+P3B4a)μFzθsy]×(P2P4+P1P3)-1,
with
(27)B1=12εbD0(uc),B2=12εbD1(uc),B3=12m0(uc),B4=12m1(uc),B5=12mD(uc),B6=(1-uc)24,B7=14(1-3uc2+2uc3),P1=1-B7εθ+B7εθSx+B4μFzθsxa1Nθ,P2=(B2+B1Sx-B2Sx)a+B5μFzθsx1Kcb-cFx,P3=1+B1,P4=3aB6εθ,
where m0(uc), m1(uc), and mD(uc) are defined by
(28)m0(uc)=∫-1ucη(u)du,m1(uc)=∫-1ucuη(u)du,mD(uc)=∫-1ucξ(u)η(u)du.


## 5. Simulation Analysis and Experiment Validation

According to the analytical tire model, the tire forces and moments under combined slip conditions can be simulated and the effects of carcass structure parameters can be analyzed. Furthermore, the analytical tire model can also be used in vehicle dynamics simulation by identifying the model parameters with test data.


### 5.1. Effect of Carcass Flexible on Tire Cornering Stiffness under Combined Slip Condition

First of all, the cornering stiffness under pure side slip condition could be calculated by assumingSx=0 in (21),
(29)Kypure=Ky0·1(1+εb)(1+εθ).It can be seen that the bending characteristic ratioεb and the twisting characteristic ratio εθ have important influence on cornering stiffness. With the increase of εb or εθ, the cornering stiffness will decrease obviously.Then, the variation of cornering stiffness under combined slip condition is also our concern, which often has important influence on tire characteristics. Divided byKypure, the normalized cornering stiffness could be derived as
(30)KyKypure=(1-εθSx)(1+εb)(1+εθ)(1+εb)(1+εθ-εθSx)+3εbεθSx-(3/a)·εθcFx.It can be seen that the normalized cornering stiffness is a function ofSx, which means that the longitudinal slip ratio (or longitudinal force) will have influence on the tire cornering stiffness. Figure 5 shows the relationship between the normalized cornering stiffness and longitudinal slip ratio. It indicates that the cornering stiffness will increase with the action of braking force does not and decrease when acting driving force.Figure 5
Variation of cornering stiffness under combined slip condition.


### 5.2. Effect of Carcass Compliance on Aligning Moment under Combined Slip Condition

The carcass translating compliance has a significant influence on aligning moment for a braked or driven wheel. In Figure6, c is the translating compliance coefficient defined in (23). As shown in Figure 6, the curve becomes more and more asymmetric when c increases (absolute value) and Mz changes its sign in the braking half of the diagram. These phenomena correspond reasonably well with the experimental results given in the previous scholars’ researches [2].Figure 6
Influence of carcass compliance on aligning moment under combined slip condition.


### 5.3. Simulation Results of Tire Forces and Moments under Combined Slip Condition

In this part, a number of example results of using the analytical tire model have been presented, as shown in Figures7–11. Some important characteristics have been reflected.(1)
Dynamic friction coefficient: Figure7 shows the longitudinal force under combined slip conditions. With the increase of slip angle α, the longitudinal force will decrease because of the limitation of friction force between the tire and the road. More important, because we adopt the dynamic friction coefficient as shown in (11), the sliding velocity dependent friction coefficient can be seen obviously in Figure 7.(2)
The variation of cornering stiffness: Figure8 shows the lateral force for combined slip and Figure 9 shows the Fy-Fx characteristics, which indicate the decrease of lateral force with the increase of longitudinal slip ratio. Besides, it can be seen that the lateral force has an increase to some extent when the wheel is braked slightly, which is mainly due to the increase of cornering stiffness.(3)
The asymmetry of aligning moment: Figure10 shows the aligning moment for combined slip. It is obvious that the curve is asymmetric and Mz changes its sign in the braking half of the diagram.(4)
Anisotropic stiffness properties: the anisotropy of tire slip stiffness will cause the variation of resultant force direction under different combined slip conditions. Figure11 shows the relationship between the resultant force direction and the slip direction. It can be seen that the resultant force direction changes gradually from adhesion direction to slip direction, which is reasonably and coincides with the test data provided in [15].Figure 7
Longitudinal force characteristics.Figure 8
Lateral force characteristics.Figure 9
F
y versus Fx characteristics.Figure 10
Aligning moment characteristics (Mz versus Fx).Figure 11
Resultant force direction.Some simulated results using the Brush model are shown in Figures12 and 13. Figure 12 is the Fy-Fx characteristics with different slip angles under combined slip conditions, and Figure 13 shows the Mz-Fx characteristics. Comparing the results of analytical tire model with flexible carcass proposed in this paper, it is obvious that the Brush model  have the capacity to express the declining friction coefficient and the influence of carcass flexibility, which will lead to severe deviation for tire characteristics, especially for the aligning moments. The dynamic friction coefficient will cause the curve of Fy-Fx characteristics to turn inward, as shown in Figure 9. The flexible carcass will make the curve of Mz-Fx characteristics asymmetric, as shown in Figure 10.Figure 12
F
y versus Fx characteristics (Brush model).Figure 13
M
z versus Fx characteristics (Brush model).


### 5.4. Experiment Validation

Test data for the P245/65R17 specification at different slip angles are used to validate the analytical tire model. Figure14 shows the tire shear forces under combined slip conditions. The vertical load is 4000 N and both the longitudinal and lateral forces are tested when the tire is rolling under combined tire cornering and braking conditions. From Figure 14, it can be seen that the analytical tire model can describe the tire characteristics accurately under combined slip conditions.Figure 14
The comparison between the analytical tire model and test data.


## 5.1. Effect of Carcass Flexible on Tire Cornering Stiffness under Combined Slip Condition

First of all, the cornering stiffness under pure side slip condition could be calculated by assumingSx=0 in (21),
(29)Kypure=Ky0·1(1+εb)(1+εθ).It can be seen that the bending characteristic ratioεb and the twisting characteristic ratio εθ have important influence on cornering stiffness. With the increase of εb or εθ, the cornering stiffness will decrease obviously.Then, the variation of cornering stiffness under combined slip condition is also our concern, which often has important influence on tire characteristics. Divided byKypure, the normalized cornering stiffness could be derived as
(30)KyKypure=(1-εθSx)(1+εb)(1+εθ)(1+εb)(1+εθ-εθSx)+3εbεθSx-(3/a)·εθcFx.It can be seen that the normalized cornering stiffness is a function ofSx, which means that the longitudinal slip ratio (or longitudinal force) will have influence on the tire cornering stiffness. Figure 5 shows the relationship between the normalized cornering stiffness and longitudinal slip ratio. It indicates that the cornering stiffness will increase with the action of braking force does not and decrease when acting driving force.Figure 5
Variation of cornering stiffness under combined slip condition.


## 5.2. Effect of Carcass Compliance on Aligning Moment under Combined Slip Condition

The carcass translating compliance has a significant influence on aligning moment for a braked or driven wheel. In Figure6, c is the translating compliance coefficient defined in (23). As shown in Figure 6, the curve becomes more and more asymmetric when c increases (absolute value) and Mz changes its sign in the braking half of the diagram. These phenomena correspond reasonably well with the experimental results given in the previous scholars’ researches [2].Figure 6
Influence of carcass compliance on aligning moment under combined slip condition.


## 5.3. Simulation Results of Tire Forces and Moments under Combined Slip Condition

In this part, a number of example results of using the analytical tire model have been presented, as shown in Figures7–11. Some important characteristics have been reflected.(1)
Dynamic friction coefficient: Figure7 shows the longitudinal force under combined slip conditions. With the increase of slip angle α, the longitudinal force will decrease because of the limitation of friction force between the tire and the road. More important, because we adopt the dynamic friction coefficient as shown in (11), the sliding velocity dependent friction coefficient can be seen obviously in Figure 7.(2)
The variation of cornering stiffness: Figure8 shows the lateral force for combined slip and Figure 9 shows the Fy-Fx characteristics, which indicate the decrease of lateral force with the increase of longitudinal slip ratio. Besides, it can be seen that the lateral force has an increase to some extent when the wheel is braked slightly, which is mainly due to the increase of cornering stiffness.(3)
The asymmetry of aligning moment: Figure10 shows the aligning moment for combined slip. It is obvious that the curve is asymmetric and Mz changes its sign in the braking half of the diagram.(4)
Anisotropic stiffness properties: the anisotropy of tire slip stiffness will cause the variation of resultant force direction under different combined slip conditions. Figure11 shows the relationship between the resultant force direction and the slip direction. It can be seen that the resultant force direction changes gradually from adhesion direction to slip direction, which is reasonably and coincides with the test data provided in [15].Figure 7
Longitudinal force characteristics.Figure 8
Lateral force characteristics.Figure 9
F
y versus Fx characteristics.Figure 10
Aligning moment characteristics (Mz versus Fx).Figure 11
Resultant force direction.Some simulated results using the Brush model are shown in Figures12 and 13. Figure 12 is the Fy-Fx characteristics with different slip angles under combined slip conditions, and Figure 13 shows the Mz-Fx characteristics. Comparing the results of analytical tire model with flexible carcass proposed in this paper, it is obvious that the Brush model  have the capacity to express the declining friction coefficient and the influence of carcass flexibility, which will lead to severe deviation for tire characteristics, especially for the aligning moments. The dynamic friction coefficient will cause the curve of Fy-Fx characteristics to turn inward, as shown in Figure 9. The flexible carcass will make the curve of Mz-Fx characteristics asymmetric, as shown in Figure 10.Figure 12
F
y versus Fx characteristics (Brush model).Figure 13
M
z versus Fx characteristics (Brush model).


## 5.4. Experiment Validation

Test data for the P245/65R17 specification at different slip angles are used to validate the analytical tire model. Figure14 shows the tire shear forces under combined slip conditions. The vertical load is 4000 N and both the longitudinal and lateral forces are tested when the tire is rolling under combined tire cornering and braking conditions. From Figure 14, it can be seen that the analytical tire model can describe the tire characteristics accurately under combined slip conditions.Figure 14
The comparison between the analytical tire model and test data.


## 6. Conclusions

This paper presented an analytical tire model with flexible carcass for combined slips. With employing the model, the effects of carcass structure parameters on tire properties are discussed. The simulation results and experiment validation of longitudinal forces, lateral forces, and aligning moments under combined slip conditions are also provided. Some conclusions are summarized as follows.Firstly, arbitrary pressure distribution, translational, bending, and twisting compliance of the carcass, dynamic friction coefficient, and anisotropic stiffness properties are the key factors for developing the analytical tire model.Secondly, the carcass compliance has a great influence on tire cornering stiffness. With the increase of bending characteristic ratioεb or twisting characteristic ratio εθ, the cornering stiffness will decrease obviously. Moreover, the cornering stiffness is also influenced by the braking force or driving force and thus leads to the increase of lateral force when tire has a slight braking.Thirdly, the carcass translating compliance has a significant influence on aligning moment under combined slip conditions. The curve ofMz becomes more and more asymmetric when translating compliance coefficient c increases (absolute value) and Mz changes its sign in the braking half of the diagram.Fourthly, the dynamic friction coefficient can express the friction delay with sliding velocity effectively; the anisotropic stiffness properties can express the resultant force direction reasonably, which changes gradually from adhesion direction to slip direction.Finally, considering all these key factors, the analytical tire model is capable of describing all kinds of tire properties reasonably and accurately. The model parameters can also be identified from tire measurements and the computational results using the analytical model show good agreement with test data.



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*Source: 397538-2014-01-02.xml*