a comparison of dynamic tyre models for vehicle shimmy ... · with the derivation of these tyre...

MSc Thesis

Supervisors: Prof. Dr. H. Nijmeijer (TU/e)

Dr. Ir. I.J.M. Besselink (TU/e) Ir. S.G.J. de Cock (DAF Trucks) Ir. R.M.J. Liebregts (DAF Trucks)

Master Thesis Committee: Prof. Dr. H. Nijmeijer (TU/e) Dr. Ir. I.J.M. Besselink (TU/e) Dr. Ir. T. Hofman (TU/e) Ir. R.M.J. Liebregts (DAF Trucks) Prof. Dr. Ir. H.B. Pacejka

Eindhoven University of Technology Department of Mechanical Engineering Dynamics and Control Group

Eindhoven, October 2009

A Comparison of Dynamic Tyre Models for Vehicle Shimmy

Stability Analysis

J.W.L.H. Maas

DCT 2009.101

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Abstract In this thesis, a comparison is made between different dynamic tyre models for the analysis of shimmy instability. First, three different tyre models which include the straight tangent tyre model, the Von Schlippe tyre model and the rigid ring tyre model are discussed. The equations of motion for these tyre models are derived and are validated using measurements. For this validation, both step responses at a low forward velocity are used as well as frequency responses at high forward velocities. The tyre models are then coupled to a basic front axle model and the resulting equations of motion are derived. For the system with the relatively simple straight tangent tyre model, analytical expressions for the stability boundaries are derived as a function of system parameters. Finally, a numerical analysis of the eigenvalues and eigenvectors of the system with the different tyre models is carried out.

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Table of Contents List of Symbols ............................................................................................................................... 3

1. Introduction ................................................................................................................................. 6

1.1 Background and motivation .................................................................................................. 6 1.2 Problem statement................................................................................................................. 6 1.3 Outline of the thesis .............................................................................................................. 6

2. Tyre Modelling............................................................................................................................ 8 2.1 Survey of literature on dynamic tyre modelling.................................................................... 8 2.2 Von Schlippe......................................................................................................................... 9 2.3 Straight tangent ................................................................................................................... 12 2.4 Kluiters................................................................................................................................ 13 2.5 Rigid ring ............................................................................................................................ 14 2.6 State space representation and rim interface....................................................................... 23

3. Tyre Model Comparison and Validation................................................................................... 26 3.1 Parameter values ................................................................................................................. 26 3.2 Step responses ..................................................................................................................... 27 3.3 Frequency responses ........................................................................................................... 31 3.4 Conclusions......................................................................................................................... 37

4. Basic Front Axle........................................................................................................................ 41 4.1 Survey of literature on shimmy stability analysis ............................................................... 41 4.2 Font axle model................................................................................................................... 42 4.3 Analytical computation of the stability boundaries............................................................. 51 4.4 Parameter influence on the analytical expressions of the stability boundaries ................... 53 4.5 Numerical computation of the stability boundaries ............................................................ 55 4.6 Stability boundary sensitivity to parameters adjustments ................................................... 58 4.7 Conclusions......................................................................................................................... 61

5. Truck Front Axle ....................................................................................................................... 63 5.1 Parameter values ................................................................................................................. 63 5.2 Eigenvalue and eigenvector analysis .................................................................................. 64 5.3 Conclusions......................................................................................................................... 74

6. Conclusions and Recommendations.......................................................................................... 76 6.1 Conclusions......................................................................................................................... 76 6.2 Recommendations............................................................................................................... 77

References ..................................................................................................................................... 79 A. Padé versus Taylor Approximation .......................................................................................... 81 B. Equations of Motion for the Rigid Ring Tyre Model ............................................................... 83 C. Optimization of Parameters for the Rigid Ring Tyre Model .................................................... 88 D. System Matrix of the Front Axle with the Straight Tangent Tyre Model ................................ 90 E. System Matrix of the Front Axle with the Rigid Ring Tyre Model.......................................... 91

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List of Symbols a half of the tyre contact length ai coefficient of the characteristic equation c stiffness

CcFϕ turn slip stiffness for the lateral force

CcMϕ turn slip stiffness for the self aligning moment

CFα cornering stiffness

CMα self aligning stiffness Fy tyre lateral force Hi i-th Hurwitz determinant Hy,x(s) transfer function: input x, output y HCM angular momentum with respect to the centre of mass I moment of inertia

k damping m mass Mx overturning moment Mz tyre self aligning moment n steering axis offset with respect to the wheel centre N distance between centre of gravity and swivel axis, N = q + n q centre of gravity offset with respect to the wheel centre qi i-th generalised coordinate Qi i-th generalised force ri i-th stability boundary expression R tyre radius Re effective rolling radius s Laplace variable st travelled distance t time tp tyre pneumatic trail T kinetic energy U potential energy v1 deflection of the tyre string at the leading contact point v2 deflection of the tyre string at the trailing contact point V forward velocity w1, w2 weighting factors x longitudinal direction y lateral position, lateral direction y1 lateral position of the tyre string at the leading contact point y2 lateral position of the tyre string at the trailing contact point z intermediate variable z vertical direction

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α tyre side slip angle

α' tyre deformation angle

γ camber angle

δqi virtual change in generalised coordinates (Lagrange equations)

δW virtual work (Lagrange equations)

ε caster angle

η amplitude ratio

λ root of the characteristic equation

ξ relative phase angle

σ tyre relaxation length

τ time delay

ϕ turn slip

ϕ' transient turn slip

ψ yaw angle

ω angular velocity

Ω angular velocity of the wheel

A12 transformation matrix between frame 1 and 2 A system matrix (state space) B input matrix (state space) C output matrix (state space) C stiffness matrix D feedthrough matrix (state space) F matrix related to the tyre forces I unity matrix ICM inertia tensor with respect to the centre of mass K damping matrix M mass matrix q vector containing generalised coordinates

Q vector containing generalised forces u input vector Wv, Wp matrices related to the tyre deformation angle x state vector y output vector Z1, Z2, Z3 component matrices

xɺ first time derivative of x

xɺɺ second time derivative of x

x

vector x

x magnitude of x

A-1 inverse of matrix A AT transpose of matrix A

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Subscripts and superscripts a axle b belt c contact patch r rim rr rigid ring st straight tangent t total (axle and wheel) vs Von Schlippe w wheel x x-direction y y-direction z z-direction

α side slip

γ rotational component around the x-axis

ψ rotational component around the z-axis

ϕ turn slip

ω rotational component around the y-axis

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1. Introduction

1.1 Background and motivation

Occasionally, vehicles may show an unstable oscillation of the steered wheels. This phenomenon can occur on motorcycles, automobiles and airplanes. This oscillation is referred to as “shimmy” and is caused by a variety of flexibilities in the design of the suspension of the vehicle. This instability is not only an uncomfortable phenomenon for occupants, but can cause more dangerous effects such as loss of control, excessive tyre wear and even failure of mechanical components. Pacejka [Pacejka; 1966, Pacejka; 2004] points out that there are two types of shimmy instability, each caused by different factors. One of the first investigators who developed a theory for the shimmy motion of automobiles is Fromm [Becker, Fromm & Maruhn; 1931]. He investigates the first type of shimmy where the main factor causing this instability is the gyroscopic coupling between the angular motions of the wheel about the longitudinal and steering axis. The main factors causing the second type of shimmy instability are the lateral compliances in tyres and suspension. This type of instability is discussed in this thesis. At DAF Trucks N.V. in Eindhoven, the shimmy phenomenon is studied. However, analysis methods and models are required to assess the sensitivity of certain design modifications with respect to stability. The shimmy sensitivity is generally addressed by the natural damping or friction around the steering axis. It is, however, desirable to design a front axle that is stable regardless of the damping in the system.

1.2 Problem statement

This project focuses on the shimmy stability of steered front axle tyres. The problem presented in this thesis can be summarised as follows: Provide a detailed parameter study regarding shimmy stability in a basic front axle model. The

front axle model is split into a tyre model and a suspension model to analyse and validate the

dynamic behaviour of the separated tyre models.

1.3 Outline of the thesis

As stated in the problem statement, a distinction is made between the tyre model and suspension model. A schematic representation of this and of the nomenclature used is displayed in Figure 1.1.

Figure 1.1. Schematic representation and nomenclature of the two parts of the front axle.

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The initial focus in this thesis is on the tyre modelling. The tyre models discussed here have the restriction that they are only capable of showing linear tyre behaviour, thus limited to small side slip angles. At first, the simple straight tangent tyre model developed by Pacejka [Pacejka; 1966] is used, continuing with Kluiters’ approach to the tyre model developed by Von Schlippe [Von Schlippe & Dietrich; 1954] and finally ends with the recently developed rigid ring tyre model, based on the work of Maurice [Maurice; 2000] and Zegelaar [Zegelaar; 1998]. Chapter 2 deals with the derivation of these tyre models. In Chapter 3, the dynamic behaviour of these tyre models is analysed using both step responses at low forward velocity and frequency responses at higher forward velocities. The frequency responses are analysed at three different forward velocities of 25, 59 and 92 km/h. Both step and frequency responses consist of three different inputs which are a pure yaw input, a side slip input and a turn slip input. Subsequently, the responses are validated using measurements on a passenger car tyre where available.

The focus in the second part is on the stability analysis of a basic front axle model with the tyre models used in the first part of this thesis. In the first part of Chapter 4, equations of motion are derived for the suspension model with two degrees of freedom and are coupled to the tyre models. With the equations of motion being derived, the focus in remaining part of Chapter 4 is on the stability boundaries. First, analytical expressions for the stability boundaries are derived for the front axle with the straight tangent tyre model. Second, a numerical computation of the stability boundaries is made using all three models for parameter values of a passenger car. Chapter 4 ends with an overview of the sensitivity of the stability of the front axle with all three models to the most influential parameter changes. In Chapter 5, the same front axle model is used with parameter values of a truck for both the tyre and the suspension. The analysis in this chapter is based on an eigenvalue and eigenvector analysis of the front axle model with all three tyre models. The damping and eigenfrequencies of the two modes representing the two degrees of freedom of the suspension are calculated as well as the amplitude ratio and the relative phase angle of the motions of these two degrees of freedom. Finally, conclusions are drawn and recommendations for future research are given in Chapter 6.

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2. Tyre Modelling

In this chapter, the equations of motion of three dynamic tyre models are derived. These tyre models are the straight tangent tyre model, the Von Schlippe tyre model and the rigid ring tyre model. These tyre models are analysed and validated using step and frequency responses further on in this thesis. The tyre models presented in this chapter have the restriction that they can only show linear tyre behaviour in a straight line motion and are restricted to lateral dynamics. Section 2.1 starts with a short survey of literature, followed by the explanation of the Von Schlippe tyre model in Section 2.2. In Sections 2.3 and 2.4, two of the derivatives of the Von Schlippe tyre model are discussed. These are the straight tangent tyre model and Kluiters’ approach respectively. In Section 2.5, the rigid ring tyre model is discussed. Finally, the models are transformed into state space representation in Section 2.6.

2.1 Survey of literature on dynamic tyre modelling

Over the years, many tyre models have been developed. The first models that are used in the analysis of shimmy are based on an approach where the road-tyre interface is reduced to a single contact point. However, in 1942 Von Schlippe [Von Schlippe; 1954] introduced the concept of a stretched string with a finite contact length. Many tyre models based on this concept have since then been developed and some of them are discussed in this chapter. Besselink [Besselink; 2000] made a detailed overview of several of the derivatives of the stretched string approach on which the majority of this section is based. More recently, the dynamic behaviour of the tyre has been studied and has resulted in a rigid ring approach. This rigid ring approach is based on the work of Maurice [Maurice; 2000] and Zegelaar [Zegelaar; 1998].

The straight tangent tyre model developed by Pacejka [Pacejka; 1966] is a simple linear approximation of the stretched string concept. It uses only the deflection in the leading contact point of the tyre to calculate the lateral force and self aligning moment generated by the tyre. Due to the simplicity of this model, Besselink [Besselink; 2000] proves that this model becomes less accurate at low forward velocities. More elaborate models have, amongst others, been developed in [Segel; 1966], [Von Schlippe; 1954], [Rogers; 1972], [Pacejka; 1966] and [Smiley; 1957]. Unlike the straight tangent approach, these models consider both the deflections of leading and trailing contact points of the tyre. The stretched string approach assumes that no sliding occurs in the contact area and as a result, the trailing contact point follows the same path as the leading contact point with a time delay. Due to this time delay, an exponential function occurs in the analytical expressions for the tyre transfer functions of the stretched string approach. This can be seen from the expressions of Segel [Segel; 1966] and Von Schlippe [Von Schlippe; 1954]. Segel includes the deflection of the entire string and this is therefore exact in his expressions. However, Pacejka [Pacejka; 2004] points out that Segel forgets to include the radial forces acting on the circular shape of the string due to the string tension. Because of the lateral deflection of the string, these radial forces cause a moment around the vertical axis. Von Schlippe approximates the contact line by forming a straight connection between the exact leading and trailing contact point.

Getting rid of the exponential functions in order to obtain a set of differential equations with constant coefficients, several approximations have been made. These approximations reduce the complexity of the mathematics involved. For example, Smiley [Smiley; 1957] uses a Taylor series to approximate the lateral position of the centre of the contact patch. Several derivatives of both Segel’s and Von Schlippe’s expressions have been made as well. These derivatives use Taylor expansions and are made by Rogers [Rogers; 1972] and Pacejka [Pacejka; 1966]. Rogers uses a Taylor series expansion to approximate the deflection of the trailing contact point with

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respect to the leading contact point and this is therefore an approximation to the Von Schlippe tyre model. Pacejka develops the exponential function occurring in the transfer functions of Segel in a Taylor series of second order (parabolic) and first order (straight tangent). Instead of the Taylor approximations, Kluiters replaces the time delay in the Von Schlippe approximation between leading and trailing contact point with a Padé filter [Kluiters; 1969]. More recently, the rigid ring tyre model has been developed by Zegelaar [Zegelaar; 1998] and Maurice [Maurice; 2000]. Unlike the stretched string approach, it includes the dynamics of the belt and contact patch and is therefore capable of showing for example gyroscopic effects and resonances of the belt.

2.2 Von Schlippe

In the stretched string approach [Von Schlippe; 1954], the tyre is considered as a massless string of infinite length under a constant pre-tension force and it is uniformly supported elastically in the lateral direction, see Figure 2.1.

Figure 2.1. Stretched string model.

Here, a represents half of the contact area and σ represents the relaxation length. The lateral stiffness per unit length between the string and the wheel plane is represented by cc. In [Pacejka; 2004], the boundary condition is used that through the rolling process of the tyre, the string forms a continuously varying slope around the leading contact point. At the rear however, the absence of bending stiffness may cause a discontinuity in slope. This boundary condition yields the following differential equation for the string deflection at the leading contact point v1 with respect to the travelled distance st [Besselink; 2000, Pacejka; 2004]:

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1

t

dvv a

dsα ϕ

σ+ = + (2.1)

Here, α represents the tyre side slip angle and ϕ represents turn slip and they are given by:

cc

t

dy

dsα ψ= − (2.2)

c

t

d

ds

ψϕ = − (2.3)

where yc and ψc represent the lateral position and the yaw angle of the contact patch respectively. A more detailed description of turn slip is given further on in this chapter. Figure 2.2 shows a schematic representation of the situation. In this figure, v1 and v2 are the string deflections of the

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leading and trailing contact point respectively and y1 and y2 are the lateral positions of the leading and trailing contact point respectively.

Figure 2.2. Schematic representation of the string deflection. When the forward velocity V is assumed to be constant, it may be assumed that st = Vt, where t is the time. As a result, (2.1) can be rewritten as:

1 1 c c c

Vv v V y aψ ψ

σ+ = − − ɺɺ ɺ (2.4)

From Figure 2.2 it can be seen that the lateral position of the leading contact point of the string y1 is given by:

1 1c cy y a vψ= + + (2.5)

By differentiating (2.5) with respect to time (a is constant) and substituting the obtained equation in (2.4), the string deflection can be eliminated:

( )1 1 c cy y y a

V

σσ ψ+ = + +ɺ (2.6)

Furthermore, Von Schlippe made the assumption that no sliding occurs with respect to the road in the contact region. Therefore, the trailing contact point follows the same path as the leading

contact point with a time delay τ. The lateral position of the tyre string at the trailing contact point y2 can be described by:

( ) ( )2 1y t y t τ= − (2.7)

This time delay is equal to the time between the leading contact point and the trailing contact point following the same path and can be described with:

2a

Vτ = (2.8)

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Von Schlippe assumes the contact line to be a straight line between y1 and y2. This is schematically displayed in Figure 2.3 where the interpretation of the straight tangent tyre model, discussed in the next section, is also displayed.

Figure 2.3. Physical interpretations of the Von Schlippe tyre model and straight tangent tyre model.

Consequently, the lateral force Fy and self aligning moment Mz for the Von Schlippe tyre model are given by:

( ) ( )

( ) ( )

1 2 1 2

2 21 2 1 2

2 22 2

1 12 2

3 2 3 2

y c c c

z c c c

v v y yF c a c a y

v v y yM c a a a c a a a

a a

σ σ

σ σ σ σ ψ

+ + = + = + −

− − = + + = + + −

(2.9)

The tyre cornering stiffness CFα and the tyre self aligning stiffness CMα, assuming a straight line between y1 and y2, are given by:

( )

( )

2

2

2

12

3

F c

M c

C c a

C c a a a

α

α

σ

σ σ

= +

= − + +

(2.10)

The cornering stiffness and self aligning stiffness are tyre parameters that are given by:

0

0

y

F

zM

FC

MC

α

α

α

α

α

α

=

=

∂=

∂

∂ = ∂

(2.11)

Given the sign convention displayed in Figure 2.4, the self aligning stiffness is a negative parameter. Finally, combining (2.9) and (2.10) yields the following expression for the lateral force and self aligning moment:

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1 2

1 2

2

2

Fy c

z M c

C y yF y

a

y yM C

a

α

α

σ

ψ

+ = − +

− = − −

(2.12)

The sign convention used throughout this thesis is displayed in Figure 2.4. In this figure, x, y and z represent the longitudinal, lateral and vertical direction respectively.

Figure 2.4. Sign convention used in this thesis with top view (left) and rear view (right).

2.3 Straight tangent

As can be seen from Figure 2.3, the contact line of the straight tangent tyre model is solely

governed by the deflection v1 at the leading contact point. It uses the tyre deformation angle α' to

compute the forces generated by the tyre. This deformation angle α' is given by:

1vα

σ′ = (2.13)

Combining (2.13) with (2.4) yields the expression for the tyre deformation angle:

c c c

y F

z M

V V y a

F C

M C

α

α

σα α ψ ψ

α

α

′ ′ + = − −

′= ′=

ɺ ɺɺ

(2.14)

The tyre self aligning moment Mz for the straight tangent tyre model can be replaced by giving the tyre lateral force Fy an offset tp in longitudinal direction with respect to the wheel centre. This offset is referred to as the pneumatic trail of a tyre and can be calculated by:

M zp

F y

C Mt

C F

α

α

= − = − (2.15)

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2.4 Kluiters

Kluiters uses the Von Schlippe approach and replaces the time delay between leading and trailing contact point with a Padé filter [Kluiters; 1969]. This tyre model is referred to as the Von Schlippe tyre model in the remainder of this thesis. Kluiters observes that a Padé filter has better convergence properties when approximating the time delay compared to the Taylor expansion as can be seen from Appendix A. He points out that a Padé filter of order 2 is sufficient and increasing the order of the filter does not change the results significantly. The transfer function of the second order Padé filter that he uses can be described by:

( )

2

2, 1 2

11

3

11

3

y y

as as

V VH s

as as

V V

− +

=

+ +

(2.16)

where s is the Laplace variable. Because (2.16) defines the relationship between y1 and y2, the tyre forces can be computed using an intermediate variable z. This variable has no physical meaning but has to be introduced in order to be able to calculate the tyre forces. This yields the following set of equations:

2

1

2

2

11

3

11

3

as asy z

V V

as asy z

V V

= + +

= − +

(2.17)

Substituting (2.17) into (2.6) and (2.12) yields the following expressions describing the tyre model:

( )2 2

2

1 11 1

3 3

11

3

c c

Fy c

z M c

as as as asz z y a

V V V V V

C asF z y

a V

sM C z

V

α

α

σσ ψ

σ

ψ

+ + + + + = + +

= + − +

= − −

ɺ

(2.18)

After rearranging (2.18), the equations can be transformed into:

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( )22 1

3

3 2

2

3

1

3

1

c c

F

y c

z M c

a aa az z z z y a

V V V

C aF z z y

a V

M C zV

α

α

σσ σσ ψ

σ

ψ

+ ++ + + = + +

= + − +

= − −

ɺɺɺ ɺɺ ɺ

ɺɺ

ɺ

(2.19)

As is pointed out earlier, the pneumatic trail (2.15) is equal to minus the tyre self aligning moment divided by the tyre lateral force. As can be seen from (2.19), unlike the straight tangent tyre model, the pneumatic trail for this tyre model is not constant, but a dynamic variable.

2.5 Rigid ring

The rigid ring tyre model, based on the work of Maurice [Maurice; 2000] and Zegelaar [Zegelaar; 1998], consists of three masses which correspond to a rim, belt and contact patch. The belt is elastically suspended with respect to the rim which represents the flexible carcass. The out-of-plane relative motion between the rim and belt has three degrees of freedom which are a lateral translation and a rotation about the vertical axis (yaw) and longitudinal axis (roll/camber). The interface between the belt and the road surface is modelled with residual stiffnesses and the actual slip model. The residual stiffnesses, which consist of a yaw and lateral degree of freedom, represent the local deformations of the tyre contact patch. The residual stiffnesses have to be included to obtain correct static tyre deformations in the corresponding directions. The contact patch mass has to be included to ensure that no numerical problems occur. Figure 2.5 shows a

schematic representation of the tyre model. In this figure, ψ represents the yaw angle, γ represents

the camber angle and Ω represents the angular velocity of the wheel.

Figure 2.5. Schematic representation of the rigid ring tyre model.

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Rim and fixed part of the tyre The rim has three degrees of freedom (Figure 2.6). These degrees of freedom can be described with the following equations of motion:

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

r r by b r by b r yr

r r r r b b r b b r b b r zr

r r r r b b r b b r b b r xr

m y k y y c y y F

I I k k c M

I I k k c M

ψ ω ψ ψ ψ

γ ω γ γ γ

ψ γ ψ ψ γ γ ψ ψ

γ ψ γ γ ψ ψ γ γ

= − + − +

= − Ω + − + Ω − + − +

= Ω + − − Ω − + − +

ɺɺ ɺ ɺ

ɺɺ ɺ ɺ ɺ

ɺɺ ɺ ɺ ɺ

(2.20)

In this equation, mr, Irγ, Irψ and Irω represent the mass of the rim and the moments of inertia of the rim around the longitudinal axis, vertical axis and lateral axis respectively. In these inertia parameters, part of the tyre is included that moves along with the rim. Rules of thumb are pointed out in [Pacejka; 2004] which state that 25% of the tyre mass is fixed to the rim and 75% of the tyre mass represents the belt. For the moments of inertia of the tyre, 15% and 85% are used for the fixed part and the belt respectively. In the remainder of this thesis, the subscript r is used for the inertia parameters of the rim including the fixed part of the tyre. The degrees of freedom of

the rim indicated with yr, ψr and γr are the lateral position, yaw angle and camber angle

respectively and yb, ψb and γb are the lateral position, yaw angle and camber angle of the belt respectively. Finally, kby and cby represent the lateral damping and stiffness between the rim and

belt respectively, kbγ and cbγ represent the camber damping and stiffness between the rim and belt

respectively and kbψ and cbψ represent the yaw damping and stiffness between the rim and belt respectively. Because the motions of the rim are defined as inputs in this chapter, these dynamics are only used for the calculation of the output force and moments later on this chapter, which are measured on the mounting point of the rim. Fyr, Mzr and Mxr are the lateral force, self aligning moment and overturning moment acting on the rim respectively. The output force and moments that are analysed further on in this thesis are the force and moments acting on the mounting point (axle) of the rim, represented by Fya, Mza and Mxa, and are therefore equal to minus the force and moments acting on the rim:

yr ya

zr za

xr xa

F F

M M

M M

= −

= − = −

(2.21)

Belt and contact patch

The differential equations that describe the three degrees of freedom of the belt are derived in Appendix B and are given by:

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( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( )

+

b b by b r cy c b b

by b r cy c b b

b b b b b b r c c b

b b r b b r c c b

b b b b b b r cy c b b

m y k y y k y y R

c y y c y y R

I I k k

k c c

I I k Rk y y R

k

ψ ω ψ ψ

ψ ψ ψ

γ ω γ

γ

γ

ψ γ ψ ψ ψ ψ

γ γ ψ ψ ψ ψ

γ ψ γ γ γ

= − − + − −

− − + − −

= − Ω − − + −

− Ω − − − + −

= Ω − − + − −

ɺɺɺ ɺ ɺ ɺ ɺ

ɺɺ ɺ ɺ ɺ ɺ ɺ

ɺɺ ɺ ɺ ɺ ɺɺ ɺ

( ) ( ) ( )b b r b b r cy c b bc Rc y y Rγ γψ ψ γ γ γ

Ω − − − + − −

(2.22)

Here, mb, Ibγ, Ibψ and Ibω represent the mass of the belt and the moments of inertia of the belt

around the longitudinal axis, vertical axis and lateral axis respectively. kcy and kcψ represents the

lateral and yaw damping between the belt and the contact patch respectively and ccy and ccψ represent the lateral and yaw stiffness between the belt and the contact patch respectively. Finally,

R is the tyre radius. Ω is the angular velocity of the wheel and is given by Ω = V/Re, where V is the forward velocity and Re is the effective rolling radius.

Figure 2.6. Rear view (left) and top view (right) of the degrees of freedom of the rigid ring tyre model.

The contact patch has two degrees of freedom with respect to the belt which are a lateral translation and a rotation around the vertical (yaw) axis. These degrees of freedom can be described with the following differential equations:

( ) ( )

( ) ( )c c cy c b b cy c b b y

c c c c b c c b z

m y k y y R c y y R F

I k c Mψ ψ ψ

γ γ

ψ ψ ψ ψ ψ

= − − − − − − +

= − − − − +

ɺɺɺ ɺ ɺ

ɺɺ ɺ ɺ (2.23)

Here, mc and Icψ represent the mass of the contact patch and the moment of inertia of the contact patch around the vertical axis respectively. The six degrees of freedom described in (2.22) and (2.23) along with the degrees of freedom of the rim are displayed in Figure 2.6.

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Slip model The lateral force and self aligning moment originating from a relative movement of the contact patch with respect to the road consists of two contributions: one due to side slip and one due to

turn slip. The contribution of the side slip is calculated using the deformation angle α' according to a first-order equation that takes the relaxation effect into account. It is similar to the equation describing the deformation angle of the straight tangent tyre model (2.14) without the turn slip component:

c c c

V V yσ α α α ψ′ ′+ = = −ɺ ɺ (2.24)

Here, σc indicates the relaxation length of the contact patch which is equal to half of the contact length a. Furthermore, measurements show an initial slope of the self aligning moment equal to zero upon applying a step in side slip angle [Higuchi; 1997]. This step response is schematically displayed in Figure 2.7a. Note that the travelled distance in this figure is defined as st = Vt. Pacejka [Pacejka; 2004] describes a similar response for the lateral force for a step in turn slip. He

suggests that this initial slope can be achieved by using the first-order equation with σc as relaxation length (Figure 2.7b), and subtracting a response curve that starts with the same slope but dies out after having reached its peak (Figure 2.7c). This latter response curve can be obtained by taking the difference of two responses, each leading to the same level but starting at different slopes. These different slopes can be achieved by two first-order equations with relaxation lengths

equal to σc (Figure 2.7b) and σc/2 (Figure 2.7d).

Figure 2.7. Schematic representations of step responses in self aligning moment to side slip. Consequently, an additional first order differential equation is needed for the calculation of the self aligning moment:

t t t c c

V V yσ α α α ψ′ ′+ = = −ɺ ɺ (2.25)

where σt is given by:

/ 2t c

σ σ= (2.26)

Summarising this extra effect results in the following equation for the deformation angle for the

self aligning moment α'M:

2M t

α α α′ ′ ′= − (2.27)

where α' and α't are given in (2.24) and (2.25) respectively. The transfer function between the

deformation angle for the self aligning moment α'M and the side slip angle α becomes:

18

( ) ( ) ( ), , , 22

2

2 1 12

31 1 1

2 2 2

M t

c c c c

H s H s H s

s s s sV V V V

α α α α α α σ σ σ σ′ ′ ′= − = − =

+ + + +

(2.28)

Figure 2.8 shows the response of the deformation angle to a step in side slip using the transfer function described in (2.28). The forward velocity is equal to 5 cm/s which is a typical velocity for step response measurements made on the flat plank tyre tester. As expected, the initial slope of the deformation angle is 0.

Because the initial slope of the lateral force in this step response is not equal to zero, the effect discussed above is not applied to the lateral tyre force and as a result, only the deformation angle described in (2.24) is used for the computation of the lateral force. This results in the

following expressions for the lateral force Fyα and self aligning moment Mzα regarding side slip:

y cF

z p cF M

F C

M t C

α α

α α

α

α

′=

′= − (2.29)

where CcFα is the cornering stiffness for the contact patch and α' and α'M are described in (2.24) and (2.27) respectively.

Figure 2.8. Response of the deformation angle for the self aligning moment to a step in side slip (V = 5 cm/s).

The second contribution to the lateral force and self aligning moment is made by turn slip. This typically occurs when the tyre is travelling in a circular path. A schematic representation of this situation is given in Figure 2.9. It can also be seen from this figure that the velocity vector is tangent to the travelled path and the side slip angle is equal to zero in this situation consequently.

Figure 2.9. Schematic representation of a typical situation where turn slip occurs.

19

The turn slip ϕ is given by:

c

V

ψϕ = −

ɺ (2.30)

Some tyre models have an inaccurate response to a step in turn slip because the circular path causes the side slip angle described in (2.2) to remain 0. Turn slip dynamics can be described using four first-order equations as has been pointed out by Pacejka [Pacejka; 2004]. Measurements show an initial slope of the lateral force equal to zero in a step response in turn slip. Similar as is done for the self aligning moment for a step in side slip, this can be achieved by

using two first-order equations with σc and σc/2 as relaxation lengths:

2 2 2

c c c c

F F F c

V V

V V

σ ϕ ϕ ϕ ψ

σ ϕ ϕ ϕ ψ

′ ′+ = = −

′ ′+ = = −

ɺ ɺ

ɺ ɺ (2.31)

where σF2 is given by:

2 / 2F c

σ σ= (2.32)

Here, ϕ'c and ϕ'F2 are the two components describing the transient turn slip for the lateral force.

The transient turn slip for the lateral force F

ϕ′ used for the calculation of the lateral force equals:

22F c F

ϕ ϕ ϕ′ ′ ′= − (2.33)

where ϕ'c and ϕ'F2 are given in (2.31). The tyre self aligning moment response to a step in turn slip is a curve that after having reached a peak tends to a lower stationary value, which is schematically displayed in Figure 2.10a. This curve can be achieved by using the first-order

equation with σc as relaxation length (Figure 2.7b), and adding a response curve that dies out after having reached its peak (Figure 2.10b). This latter response curve can be obtained by taking the difference of two responses, each leading to the same level but starting at different slopes. Values

for the relaxation lengths are taken as σc/2 (Figure 2.10c) and σc/1.5 (Figure 2.10d) and are obtained from [Pacejka; 2004].

Figure 2.10. Schematic representations of step responses in self aligning moment to turn slip. The two additional equations that describe the turn slip influencing the self aligning moment are:

1 1 1

2 2 2

c

c

V V

V V

ϕ

ϕ

σ ϕ ϕ ϕ ψ

σ ϕ ϕ ϕ ψ

′ ′+ = = −

′ ′+ = = −

ɺ ɺ ɺ

ɺ ɺ ɺ (2.34)

20

where σϕ1 and σϕ2 are given by:

1

2

/ 2

/1.5

c

c

ϕ

ϕ

σ σ

σ σ

=

= (2.35)

The transient turn slip for the self aligning moment ϕ'M used for the calculation of the self aligning moment equals:

( )1 2 1 2M cw wϕ ϕ ϕ ϕ′ ′ ′ ′= + − (2.36)

where ϕ'c, ϕ'1 and ϕ'2 are described in (2.31) and (2.34). In (2.36), w1 and w2 are factors that govern the characteristics of the tyre moment response to turn slip. From a physical point of view, w1 influences the effect of tread width and w2 influences the magnitude of the overshoot visible in Figure 2.10a. Values for these factors are taken equal to 1 and 4 respectively and they are obtained from [Pacejka; 2004]. The transfer function between transient turn slip for the self

aligning moment ϕ'M and the turn slip ϕc becomes:

( ) ( ) ( ) ( )( )1 2, , , ,

22

2

3 23 2

3 2

1 4 44

1 1 12 1.5

111

6

1.5 131

3 6

M c c

c c c

c c

c c c

H s H s H s H s

s s sV V V

s sV V

s sV V V

ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ σ σ σ

σ σ

σ σ σ

′ ′ ′ ′

= + − = + −

+ + +

+ + =

+ + +

(2.37)

Figure 2.11 shows the response of the transient turn slip for the self aligning moment to a step in turn slip using the transfer function described in (2.37). Also included are the separate components of which the total transfer function is composed of. The parameter values are chosen to correspond to the values used in Chapter 3.

Figure 2.11. Response of the deformation angle for the self aligning moment to a step in side slip (V = 5 cm/s).

21

As can be seen from Figure 2.11, the value of 4 for w2 is not sufficient to have an overshoot in the

response curve for ϕ'M. However, as can be seen from the next chapter, the dynamics between the rim and the contact patch cause the response of the self aligning moment at the wheel centre to have an overshoot similar to the schematic representation in Figure 2.10a. It can be noted that the

response curve of ϕ'M is similar to a first-order response curve with a small relaxation length and the question arises if the four differential equations describing the turn slip response of the self aligning moment are necessary. This is however beyond the scope of this research.

Assuming the turn slip remains relatively small, this results in the following expression

for the lateral force Fyϕ and self aligning moment Mzϕ regarding turn slip:

y cF F

z cM M

F C

M C

ϕ ϕ

ϕ ϕ

ϕ

ϕ

′=

′= (2.38)

where CcFϕ and CcMϕ are the turn slip stiffnesses for the lateral force and self aligning moment

respectively and ϕ'F and ϕ'M are described in (2.33) and (2.36) respectively. Because the moment

generated by turn slip Mzϕ causes a small difference in angle between the rim and belt and therefore also a small side slip angle and side force, the turn slip stiffnesses cannot be obtained from the steady-state values directly. Instead, linear simulations are made to obtain the same steady-state values as the measurements further on in this thesis. Finally, the total lateral force and self aligning moment including both side slip and turn slip becomes:

y y y

z z z

F F F

M M M

α ϕ

α ϕ

= +

= + (2.39)

Figure 2.12. Schematic representation of the slip model of the rigid ring tyre model.

22

A schematic representation of the different components of the slip model described in this section is displayed in Figure 2.12.

In this section, there is a difference in parameters that are used for the relaxation length and cornering stiffness for the rigid ring tyre model with respect to the other two tyre models. This is a result of the dynamics involved between rim and contact patch. First, an adjustment has to be made to the cornering stiffness. The yaw stiffnesses in the rigid ring tyre model between rim and contact patch causes a difference in the yaw angle applied to the rim and the yaw angle of the contact patch. Because this is not the case for the straight tangent tyre model and the Von Schlippe tyre model, the slip angle of the contact patch is different for the tyre models upon

applying the same yaw angle to the rim. The difference in side slip angle δα between the straight tangent tyre model and the rigid ring tyre model can be described as:

,

zrr st st

total

M

cψ

α α δα α= − = − (2.40)

Here, the subscripts rr and st refer to the rigid ring tyre model and straight tangent tyre model

respectively. The total yaw stiffness between rim and contact patch cψ,total is given by:

,

1

1 1total

b c

c

c c

ψ

ψ ψ

=

+

(2.41)

To obtain the same steady-state characteristics in spite of the different side slip angles, the parameter value of the cornering stiffness has to be adjusted according to the difference in side

slip angle δα [Pacejka; 2004]:

1

,

1F p

cF F

total

C tC C

c

α

α α

ψ

−

= −

(2.42)

Here, CcFα and CFα indicate the cornering stiffness of the rigid ring tyre model and the straight tangent tyre model respectively. A similar adjustment has to be made for the relaxation length of the tyre. For the straight tangent tyre model and the Von Schlippe tyre model, the relaxation effect of the tyre is caused only by the parameter value of the relaxation length. The rigid ring tyre model has an additional relaxation effect caused by the lateral stiffness between rim and

contact patch. The relaxation length of the rigid ring tyre model σc is equal to half of the contact length a. Pacejka [Pacejka; 2004] states that this difference in relaxation lengths can be described by:

, ,

, , , ,

total totalcF F

c c

total cF p y total total cF p y total

c cC C

c C t c c C t c

ψ ψα α

ψ α ψ α

σ σ σ

= + = + + + (2.43)

Here, σc and σ are the relaxation length of the rigid ring tyre model and straight tangent tyre model respectively and cy,total is the total lateral stiffness between rim and contact patch and can be calculated by:

23

, 2

1

1 1y total

by cy b

cR

c c c γ

=

+ +

(2.44)

2.6 State space representation and rim interface

In order to compare the models to measurements, the systems can be transformed into a state space representation:

( ) ( ) ( )

( ) ( ) ( )

t t t

t t t

x Ax Bu

y Cx Du

= +

= +

ɺ

(2.45)

Here, x(t) represents the state vector, y(t) the output vector and u(t) the input vector. The matrices A, B, C and D represent the system matrices. Because the straight tangent tyre model and the Von Schlippe tyre model assume the wheel to be a rigid mass, the states of the contact patch indicated with the subscript c are given by:

c r r

c r

c r

y y Rγ

ψ ψ

γ γ

= +

= =

(2.46)

These formulas can be substituted in (2.14) and (2.19) so that yc, ψc and γc do not have to be included in the state vector of the straight tangent tyre model and the Von Schlippe tyre model.

In the next chapter, the tyre models are analysed and validated using measurements on a tyre. In these measurements carried out by Higuchi [Higuchi; 1997] and Maurice [Maurice; 2000], the input consists of applied motions to the rim and the output is measured at the mounting point of the wheel at the wheel centre. To imitate this situation, the force and moment acting on the mounting point of the rim are calculated in the simulations rather than the actual tyre force and moment in the contact area. Moreover, calculating the outputs at the wheel centre includes the inertia effects of the wheel in the analysis of the force and moments acting on the mounting point. Consequently, the output force and moments in the simulations are given by:

ya w r F

st xa w r w r F

za w r w r F p

F m y C

M I I C R

M I I C t

y

α

γ ω α

ψ ω α

α

γ ψ α

ψ γ α

′ − + ′= = − + Ω +

′− − Ω −

ɺɺ

ɺɺ ɺ

ɺɺ ɺ

(2.47)

( )

( )

2

2

2

2

3

3

F F F

w r r

ya

F F F

vs xa w r w r r

za

Mw r w r M r

C C a Cm y y z z

a a V aF

C R C a R C RM I I y z z

a a V aM

CI I C z

V

y

α α α

α α αγ ω

αψ ω α

σ σ σ

γ ψσ σ σ

ψ γ ψ

− − + +

+ + + = = − + Ω − + + + + +

− − Ω + −

ɺɺ ɺɺ

ɺɺ ɺ ɺɺ

ɺɺ ɺ ɺ

(2.48)

24

( ) ( )( ) ( ) ( )

( ) ( ) ( )

ya r r by b r by b r

rr xa r r b b r r r b b r b b r

za r r r r b b r b b r b b r

F m y k y y c y y

M I k I c k

M I I k k c

γ γ ω γ γ

ψ ω ψ ψ ψ

γ γ γ ψ γ γ ψ ψ

ψ γ ψ ψ γ γ ψ ψ

− + − + −

= = − + − + Ω + − − Ω − − − Ω + − + Ω − + −

y

ɺɺ ɺ ɺ

ɺɺ ɺ ɺ ɺ

ɺɺ ɺ ɺ ɺ

(2.49)

Here, the subscripts st, vs and rr refer to straight tangent, Von Schlippe and rigid ring

respectively. The inertia parameters mw and Iwψ are equal to the mass and moment of inertia of the wheel respectively. They are equal to the inertia values of the rim plus the tyre. A distinction is made between the outputs of the step and frequency response simulations. This is done because the step response measurements [Higuchi; 1997] have been made with initial conditions instead of a step motion applied to the rim. Consequently, the terms in (2.47), (2.48) and (2.49)

depending on derivatives of yr, ψr and γr are neglected for the step responses. The inputs in the step response measurements consist of a side slip input dyr/dt, a pure

yaw input ψr and a turn slip input. The turn slip input is a combination of the side slip and pure yaw input, such that the side slip angle described in (2.2) equals 0. Because the equations of motion of the tyre models contain derivatives of both the side slip input and the pure yaw input from the measurements, the inputs for the simulations are taken as the derivatives of the measurement inputs. Consequently, an impulse is applied to the simulation inputs to represent the step responses from the measurements. The input u is given by:

( )( )T

impulse r r r rt Vy Vyψ ψ= − − −u ɺ ɺɺɺ ɺɺ (2.50)

Here, the first input is used for the side slip input. This input is negative and multiplied with the forward velocity to obtain the same side slip angle as for the pure yaw input. Applying an impulse to this input represents the step in side slip angle in the measurements. The second input is used for the pure yaw input. Applying an impulse to this input represents the step in pure yaw angle in the measurements. Finally, the third input is used for the turn slip input. Applying an impulse to this input represents an impulse in turn slip. Consequently, this latter curve has to be integrated over the travelled distance to obtain a step in turn slip.

In the frequency response measurements from [Maurice; 2000], the inertia terms for the rim cannot be neglected. Therefore, the second derivatives of both the lateral position and the yaw angle have to be included in the inputs. The inputs for the frequency response of the side slip and yaw input are therefore given by:

( )( )T

frequency r rt Vyu ψ= − ɺɺɺɺ (2.51)

Because the inputs are defined as the second derivatives of the lateral position and yaw angle, the inputs have to be integrated to obtain the proper transfer functions and be able to make a comparison with measurements. This can be achieved by multiplying the transfer function for side slip with s and for pure yaw with s2.

The third input that is used in this chapter is the turn slip input. As is pointed out already, it is a combination of the side slip and yaw input, such that the side slip angle described in (2.2) equals 0. The relation between the turn slip input and the combination of side slip and yaw input is defined as [Besselink; 2000]:

25

( ) ( ) ( )( )

( ) ( ) ( )( )

, , ,

, , ,

y y y

z z z

F F F

M M M

VH s H s H s

s

VH s H s H s

s

ϕ α ψ

ϕ α ψ

= −

= −

(2.52)

Here, H indicates the transfer function and the subscripts ϕ, α and ψ refer to turn slip, side slip and pure yaw respectively. Because of the different number of degrees of freedom, each tyre model has its own state vector. The state vectors are therefore given by:

( )

( )

(

)2 1 2

...

T

st r r r r r r

T

vs r r r r r r

rr r r r r r r b b b b b b

T

c c c c t c F

y y

y y z z z

y y y y

y y

γ ψ γ ψ α

γ ψ γ ψ

γ ψ γ ψ γ ψ γ ψ

ψ ψ α α ϕ ϕ ϕ ϕ

′==

=

′ ′ ′ ′ ′ ′

x

x

x

ɺ ɺɺ

ɺ ɺɺ ɺɺ ɺ

ɺ ɺ ɺ ɺɺ ɺ

ɺɺ

(2.53)

Note that for the step responses, dψr/dt is defined as an input. As a result, this state is redundant for the step responses.

26

3. Tyre Model Comparison and Validation The three tyre models from Chapter 2 can be compared with each other and with measurements on tyres. This is done by using step responses and frequency responses. The step response measurements are carried out by Higuchi [Higuchi; 1997] and are available for three inputs. These inputs include a step in side slip angle, a step in yaw angle and a step in turn slip. The frequency response measurements are carried out by Maurice [Maurice; 2000] and are only available for the pure yaw input. In Section 3.1, the tyre model parameter values are given. Step responses and frequency responses are given in Sections 3.2 and 3.3 respectively. Finally, Section 3.4 gives the main conclusions that can be drawn from this chapter.

3.1 Parameter values

The parameters that are used are displayed in Table 3.1 and are based on parameters used in [Maurice; 2000]. However, the parameters that are used by Maurice are based on a model without turn slip and are therefore inaccurate for the rigid ring tyre model described in this thesis. Therefore, an optimization of parameters is carried out as described in Appendix C. This is done using the fmincon function of the optimization toolbox of Matlab and is based on an objective function that describes the relative error between the frequency responses of the measurements and the model. The parameters used in this chapter are applicable for a passenger car tyre. This is done because, unlike for truck tyres, measurements on passenger car tyres are available. Consequently, the experimental results can be used to obtain model parameters that represent the measurements accurately. Table 3.1. Tyre model parameters for a passenger car tyre.

Parameter Value Unit Parameter Value Unit

a 4.88e-2 m Icψ 1.00e-2 kgm2 cby 6.39e5 N/m kby 7.68e1 Ns/m cbγ, cbψ 2.02e4 Nm/rad kbγ, kbψ 4.00 Nms/rad ccy 8.00e5 N/m kcy 1.00e2 Ns/m ccψ 1.00e4 Nm/rad kcψ 3.00 Nms/rad CcFφ 8.80e2 Ns/rad mb 7.64 kg CcMφ 1.77e2 Nms/rad mc 1.00 kg CFα 6.70e4 N/rad mr 1.66 kg Ibγ, Ibψ 3.21e-1 kgm2 mw 9.30 kg Ibω 6.61e-1 kgm2 R 2.80e-1 m Irγ, Irψ 6.98e-2 kgm2 Re 3.00e-1 m Irω 7.50e-2 kgm2 tp 3.00e-2 m Iwγ, Iwψ 3.91e-1 kgm2 σc 4.88e-2 m Iwω 7.36e-1 kgm2

The turn slip stiffnesses CcFφ and CcMφ for respectively the lateral force and self aligning moment are obtained from step response measurements [Higuchi; 1997]. The correction of the cornering stiffness of the rigid ring tyre model (2.42), results in a value for CcFα equal to 94560 N/rad. The total relaxation length σ (2.43) used for the straight tangent tyre model and the Von Schlippe tyre model is equal to 0.4833 m. Note that in Table 3.1, the total inertia of the wheel is indicated with

27

parameters with the subscript w. In this chapter however, these values do not include the inertia parameters for the rim, because the experimentally obtained response functions presented in this chapter are corrected for these rim inertia effects. Consequently, the inertia parameters of the wheel, indicated with subscript w, only include the inertia of the tyre and the inertia parameters of the rim indicated with subscript r only include the inertia of the part of the tyre that moves along with the rim.

3.2 Step responses

Step responses can be calculated with the tyre models in state space format (Section 2.6) using the lsim command in Matlab. In this section, the simulated step responses are compared to real measurements and differences are identified. Parameters are taken as in Table 3.1. The forward velocity is taken equal to 5 cm/s which is similar to the forward velocity used in the measurements. The steps are applied when the travelled distance equals 0. As is already pointed out in Section 2.6, the derivatives of the motions of the rim are neglected here.

Figure 3.1. Step response of the lateral force and self aligning moment to side slip.

28

Figure 3.1 shows the step responses to a step of 1 degree in side slip. Zoomed in plots are provided where differences are too small to be distinguished in the regular plots. The models show very small differences. The largest difference can be found in the initial slope of the self aligning moment equal to zero for the Von Schlippe tyre model and the rigid ring tyre model, which can also be seen from the measurements. The straight tangent tyre model does not show this response and has therefore a small discrepancy. Figure 3.2 shows the string model in a step in side slip angle for the initial situation when the step is applied at the right-hand side and the steady-state situation on the left-hand side. The path of the contact points has a small delay with respect to the path of the wheel centre and causes the zero initial slope of the self aligning moment.

Figure 3.2. The string model in a step in side slip angle for the initial situation (right) and steady-state situation (left).

Figure 3.3 shows the step responses to a step of 1 degree in yaw angle. Differences between the tyre models are larger compared to the step in side slip. Especially at the initial stage, differences can be seen from both the magnitude and the sign of the tyre forces. For the lateral force, the straight tangent tyre model has a small negative initial value which cannot be seen from the measurements. The Von Schlippe tyre model shows a small delay in lateral force build-up. For the self aligning moment, the straight tangent tyre model has a positive initial value, whereas the Von Schlippe tyre model and the rigid ring tyre model have a negative initial value similar to the measurements. However, the initial peak value of the Von Schlippe tyre model is equal to its steady-state value and the peak value is therefore too small. The rigid ring tyre model shows an initial peak that is too large compared to the measurements.

29

Figure 3.3. Step response of the lateral force and self aligning moment to yaw.

30

Figure 3.4. Step response of the lateral force and self aligning moment to turn slip.

Finally, Figure 3.4 shows the step responses to a step in turn slip. This response shows much larger differences than the other step responses. The initial slope of the lateral force is equal to zero for the Von Schlippe tyre model and the rigid ring tyre model, which is not the case for the straight tangent tyre model. Differences can also be found in the stationary values of the lateral force. For the self aligning moment, the largest differences can be found in the stationary values. The straight tangent tyre model has a negative value, whereas the Von Schlippe tyre model and the rigid ring tyre model have positive values as is the case for the measurements. However, the Von Schlippe tyre model has a zero steady-state response to turn slip. A schematic representation of the steady-state situation for a step in turn slip is displayed in Figure 3.5 for the straight tangent tyre model and the Von Schlippe tyre model. This figure clarifies what happens in the steady-state situation: the lateral force of the straight tangent tyre model has a larger positive value than the Von Schlippe tyre model and the self aligning moments for the straight tangent tyre model and the Von Schlippe tyre model are negative and zero respectively.

31

Figure 3.5. Schematic representation of the straight tangent tyre model and the Von Schlippe tyre model in a step in turn slip.

3.3 Frequency responses

Frequency response measurements are carried out by Maurice [Maurice; 2000] and consist of a pure yaw input motion on the rim. The frequency responses for side slip and turn slip are not available and are therefore only compared between the tyre models. For the input in the measurements, a white noise input signal is applied to the rim with a 0 – 64 Hz bandwidth.

Figure 3.6 shows the Bode plots for the lateral force and self aligning moment for the three tyre models with a side slip input. Several effects can be recognised in the Bode plots. The tyre models are similar to each other for low frequencies. For a steady-state situation, the magnitude of the Bode plot between the side force and side slip input is equal to the cornering stiffness. The phase in this situation is equal to zero, meaning that a positive slip angle results in a positive tyre lateral force. As the frequency increases, the relaxation effect becomes visible. This results in a decreasing magnitude and increasing phase lag. All three tyre models are capable of showing these effects. Above approximately 15 Hz, the magnitude of the lateral force increases because of the mass of the tyre. As the frequency of excitation is increasing, the necessary force to excite the wheel in lateral direction increases as well. The differences between tyre models become larger towards the resonance frequencies of the tyre belt. These are visible by the peaks in magnitude and the two peaks visible represent the yaw and camber degrees of freedom of the belt. Because of the gyroscopic coupling of these degrees of freedom, there is no clear yaw and camber mode however. The straight tangent tyre model and the Von Schlippe tyre model are not capable of showing these resonance peaks.

The self aligning moment shows similar effects as the lateral force, but there are some differences however. The steady-state magnitude for the tyre models is equal to the self aligning stiffness instead of the cornering stiffness. The phase in this situation is 180 degrees because a positive side slip angle results in a negative tyre self aligning moment. For increasing frequencies, the straight tangent tyre model and the Von Schlippe tyre model display the same relaxation effect as in the lateral force Bode plot. This results in a decreasing magnitude and increasing phase lag. The increase in phase lag of the rigid ring tyre model is compensated by the phase leading nature of the gyroscopic effects of the belt. The straight tangent tyre model and the Von Schlippe tyre model are not capable of showing this phase lead because gyroscopic effects are not present for these tyre models.

32

Figure 3.6. Bode plots for the lateral force and self aligning moment for side slip (V = 92 km/h).

Figures 3.7 through 3.9 show the Bode plots for the lateral force and self aligning moment with a pure yaw input at different forward velocities. For this input, measurements are available that are carried out by Maurice [Maurice; 2000]. Similar effects for a pure yaw input that can also be observed in a side slip input, include the steady-state magnitude, the relaxation effect and the resonance frequencies of the rigid ring tyre model. The gyroscopic effects that cause a phase lead for the rigid ring tyre model with respect to the other tyre models can again be found in the self aligning moment at frequencies just below the resonance frequencies. A difference between the side slip and pure yaw input can be found in the inertia effects which cause an increase in magnitude at higher frequencies. For the pure yaw input, this increase can be found for the self aligning moment instead of the lateral force because of the moment of inertia of the tyre. Another difference is the dip in magnitude of the lateral force for the Von Schlippe tyre model. This dip occurs when the wavelength of the travelled path equals the contact length of the string model. Because exactly one wavelength fits in the contact area in this point, the resulting lateral force equals zero and a dip occurs in the magnitude of the Bode plot. Because the wavelength of the

33

travelled path is dependent on the forward velocity, this dip is also dependent on the forward velocity. Therefore, this dip disappears from Figure 3.9 because it exceeds the frequency range considered. As can be seen from Figures 3.7 through 3.9, the velocity has several effects on the plots. First, it can be noted that the relaxation effect shifts to higher frequencies for higher velocities. For the rigid ring tyre model, the shifting of the relaxation effect to higher frequencies causes the relaxation effect to become less visible in the Bode plots because of the resonance frequencies of the rigid ring tyre model. The decrease in magnitude because of the relaxation effect is compensated by the increase in magnitude because of the resonance frequencies. For the other tyre models, the relaxation effect does not become smaller but shifts to higher frequencies. Secondly, the two lowest resonance frequencies for the rigid ring tyre model represented by the yaw and camber modes tend to shift from each other for an increasing velocity. This is because of the gyroscopic effects in the belt.

Figure 3.7. Bode plots for the lateral force and self aligning moment for pure yaw (V = 25 km/h).

34

Figure 3.8. Bode plots for the lateral force and self aligning moment for pure yaw (V = 59 km/h).

35

Figure 3.9. Bode plots for the lateral force and self aligning moment for pure yaw (V = 92 km/h). When comparing the models with the measurements, it is obvious that the rigid ring tyre model has the most accurate frequency response. Effects such as resonance frequencies and gyroscopic effects are not present for the straight tangent tyre model and the Von Schlippe tyre model. However, the rigid ring tyre model has some discrepancies as well. One of these discrepancies can be found between 10 and 30 Hz for the lateral force at low forward velocities. The lateral force shows an increase in phase in this region which can not be seen from the measurements. Furthermore, the dip in magnitude for the self aligning moment visible for all velocities and the magnitude at the resonance frequencies show differences as well. The third input that is discussed in this thesis is turn slip. Figure 3.10 shows the Bode plots for the lateral force and self aligning moment with a turn slip input at a forward velocity of 92 km/h. For a steady-state situation, the same effects can be seen as the steady-state values for the step response. The magnitudes in lateral force are slightly different but have the same sign. As a result, the phase in this situation is equal for all tyre models. For the self aligning moment however, the straight tangent tyre model shows a negative steady-state value in the step

36

responses. This can also be seen from the Bode plots where there is a phase difference of 180 degrees between the straight tangent tyre model and the rigid ring tyre model for an infinitely low frequency. The zero response in self aligning moment of the Von Schlippe tyre model to a step in turn slip can also be seen from the Bode plot: the magnitude goes to 0 for an infinitely low frequency. For increasing frequencies, the relaxation effect, gyroscopic effect and the resonance frequencies can be recognised in the lateral force bode plot. For the self aligning moment, the gyroscopic effect, the increase in magnitude because of the moment of inertia of the tyre and the resonance frequencies can be recognised.

Figure 3.10. Bode plots for the lateral force and self aligning moment for turn slip (V = 92 km/h). Until now, the comparison between models and measurements are made using the lateral force and self aligning moment. However, the measurements that have been made for the frequency response to pure yaw also include the overturning moment around the x-axis that acts on the rim. This moment is less relevant for the scope of this thesis because the rim does not have a degree of freedom around the x-axis when attached to the suspension model for the systems studied. This is

37

explained in the next chapter. However, it is an additional validation output, so the measurements are compared to the second output Mxa defined in (2.47) of the three tyre models. The results are shown in Figure 3.11 for three different forward velocities. These Bode plots show effects that can be found in the other outputs as well. These include the relaxation effect, phase lead because of gyroscopic effects around 10 Hz and the resonance frequencies. However, the main discrepancies as are pointed out earlier are the dip in magnitude between approximately 20 and 30 Hz and the magnitude at resonance frequencies.

Figure 3.11. Bode plots for the overturning moment for pure yaw.

3.4 Conclusions

The tyre models in this chapter are validated with measurements. First, this validation is made regarding step responses. Overall, the rigid ring tyre model has the most accurate response for the step responses, although the relaxation effect for all three tyres appears to be too small. However,

38

it has to be noted that the tyre parameters are optimised using the frequency response functions. Because different passenger car tyres have been used for the frequency- and step response measurements, the discrepancies of the tyre models in the step responses may be caused by inaccurate parameters. The results for the step responses can be summarised as is done in Table 3.2. In this table, an error function is added that indicates the relative error between models and measurements. The error function for the lateral force and self aligning moment are respectively given by:

( ) ( )( ) ( )

( ) ( )( ) ( )

340,model ,measurement

1 ,measurement ,measurement

340z,model z,measurement

1 z,measurement z,measurement

1Error

340 0.05max

1Error M

340 0.05max

y y

y

n y y

z

n

F n F nF

F n F

M n M n

M n M

=

=

−=

+ −

= +

∑

∑

(3.1)

Here, n is the number of measurement points. This error function computes the relative error between model and measurement. However, there is an extra term in de denominator of the function which is equal to 5% of the maximum value of the measurement. This is done in order to avoid having a very large error value when dividing by a very small value in the initial stage of the step responses. The error values show what can be seen from the step response plots: the largest errors can be found for the straight tangent tyre model and the Von Schlippe tyre model in the turn slip responses. Table 3.2. Results step response.

Input Output Tyre model Error Remarks

Straight tangent 0.0419 Correct response

Von Schlippe 0.0400 Correct response

Fy

Rigid ring 0.0559 Correct response

Straight tangent 0.1344 Initial slope is not equal to zero

Von Schlippe 0.0630 Correct response

Side slip

Mz


Straight tangent 0.0790 Negative initial value

Von Schlippe 0.0461 Small delay in force build-up

Fy


Straight tangent 0.1483 Positive initial value

Von Schlippe 0.1066 Negative initial value is too small

Pure yaw

Mz

Rigid ring 0.0810 Negative initial value is too large

Straight tangent 0.5654 Initial slope is not equal to zero Steady-state value is too large

Von Schlippe 0.2885 Steady-state value is too large

Fy

Rigid ring 0.1810 Relaxation effect is too small

Straight tangent 1.3974 Negative steady-state value

Von Schlippe 0.7562 Steady-state value is equal to zero

Turn slip

Mz

Rigid ring 0.0419 Overshoot is too large

The second validation that is made in this chapter includes frequency responses. As well as for the step responses, the rigid ring tyre model is shown to be the most accurate tyre model. The results for the frequency responses can be summarised as is done in Table 3.3.

39

Table 3.3. Results frequency response.

Input Output Remarks

Fy • Steady-state values are equal to the cornering stiffness CFα for all tyre models

• Relaxation effect is visible at frequencies above approximately 2 Hz for all tyre models

• Increase in magnitude because of the mass of the wheel is visible at frequencies above approximately 10 Hz for all tyre models

• Only the rigid ring tyre model shows resonance peaks of the belt between approximately 30 Hz and 60 Hz

Side slip

Mz • Steady-state values are equal to the self aligning stiffness CMα for all tyre models

• Relaxation effect is visible at frequencies above approximately 2 Hz for the straight tangent tyre model and the Von Schlippe tyre model

• The relaxation effect for the rigid ring tyre model is compensated by the increase in magnitude and phase because of the gyroscopic effects of the belt


Fy • Steady-state values are equal to the cornering stiffness CFα for all tyre models

• Relaxation effect is visible at frequencies above approximately 2 Hz for the all tyre models

• The Von Schlippe tyre model shows a dip in magnitude where exactly one wavelength fits in the contact area


Pure yaw

Mz • Steady-state values are equal to the self aligning stiffness CMα for all tyre models

• Relaxation effect is visible at frequencies above approximately 2 Hz for the straight tangent tyre model and the Von Schlippe tyre model

• The relaxation effect for the rigid ring tyre model is compensated by the increase in magnitude and phase because of the gyroscopic effects of the belt

• Increase in magnitude because of the moment of inertia of the wheel is visible at frequencies above approximately 10 Hz for all tyre models


Fy • Small differences exist between tyre models for steady-state values


Turn slip

Mz • Large differences exist between tyre models in both magnitude and phase at steady-state values


40

As well as for the step responses, an error value can be calculated for the frequency responses. For the lateral force and self aligning moment, the error functions are respectively given by:

( ) ( )( )

( ) ( )

( ) ( )( )

472,model ,measurement

1 ,measurement

472

,model ,measurement

1

472z,model z,measurement

1 z,measurement

1Magnitude error

472

1Phase error

472

1Magnitude error

472

Phase e

y y

y

n y

y y y

n

z

n

F n F nF

F n

F F n F n

M n M nM

M n

=

=

=

−=

= ∠ − ∠

−=

∑

∑

∑

( ) ( )472

z,model z,measurement

1

1rror

472z

n

M M n M n=

= ∠ − ∠

∑

(3.2)

Here, n is the number of measurement points. For the magnitude, the error is defined as a relative error between the model and the measurement, because of the large range of the magnitude values. The phase error is defined as the absolute value of the phase difference. Here, the phase difference in degrees is wrapped to the interval [-180, 180]. Results for these error functions are given in Table 3.4. The error values show what can be seen from the frequency response plots: the rigid ring tyre model has the smallest errors with respect to the other two tyre models. Table 3.4 Results for the error function for the frequency response with yaw input.

Output Magnitude

/phase

Tyre model Error

V=25 km/h

Error

V=59 km/h

Error

V=92 km/h

Straight tangent 0.6191 0.6674 0.6715

Von Schlippe 0.7439 0.6982 0.6847

Magnitude

Rigid ring 0.2491 0.1753 0.1807


Von Schlippe 34.6428 47.6800 53.3547

Fy

Phase

Rigid ring 13.7721 6.0997 4.3681


Von Schlippe 0.5987 0.6453 0.5605

Magnitude

Rigid ring 0.4082 0.3853 0.3953


Von Schlippe 62.1076 69.0303 82.2399

Mz

Phase

Rigid ring 11.5634 10.0253 9.2575

41

4. Basic Front Axle

After the tyre models are analysed and validated in the previous chapter, the step towards the front axle model is made in this chapter. This chapter focuses on the derivation of the equations of motion of a basic suspension model attached to tyre models as discussed in the previous chapters. In Section 4.1, a literature survey on shimmy stability analysis is given. The equations of motion of the front axle model including the three tyre models are derived in Section 4.2. Next, Section 4.3 focuses on the shimmy stability of the front axle for a passenger car. Because of the simplicity of the straight tangent tyre model, this model is used to derive analytical expressions for the stability boundaries as a function of parameter values. In Section 4.4, a short overview of the effects of parameter adjustments are given with respect to the analytical expressions for the stability boundaries derived in Section 4.3. Next, all three tyre models are compared to each other by a numerical eigenvalue analysis in Section 4.5 and a sensitivity analysis of the stability boundaries of all three tyre models to parameter changes is made in Section 4.6. Finally, a short overview of the main conclusions is given in Section 4.7.

4.1 Survey of literature on shimmy stability analysis

As pointed out in [Pritchard; 1999], the first fundamental contributions toward understanding the shimmy phenomenon emerged from the automobile industry in France around 1920. Broulhiet [Broulheit; 1925] published his work on the effect of tyre mechanics on shimmy in 1925. He first described the concept of side slip and suggested that the energy for the self-sustainable shimmy oscillations comes from the tyre mechanics via the side slip. Fromm [Becker, Fromm & Maruhn; 1931] also recognised the important role that lateral slip of tyres plays in automobile motions.

Initially, theories that explain shimmy instability problems have been based on linear models. These models consisted typically of a wheel with an elastic tyre capable of swivelling about a king-pin that moves along a straight line. Amongst others, these models have been developed and theoretically investigated in [Kantrowitz; 1937], [Von Schlippe & Dietrich; 1954], [Moreland; 1951] and [Smiley; 1957].

Pacejka [Pacejka; 1966] is one of the first investigators to include non-linear effects in his shimmy stability analysis. He included non-linear effects in the tyre and in the suspension. These non-linear effects include degressive tyre characteristics, dry friction in the king-pin bearings and rotational clearance in the wheel bearings. Furthermore, a model for non-linearities in yaw stiffness, which include friction forces and free-play, has been developed by Black [Black; 1976] and later on by Baumann [Baumann; 1991] and Li [Li; 1993]. Black also points out that it is advisable to include the movements of the supporting structure in the shimmy analysis.

More recently, the shimmy phenomenon has been studied by several researchers using experimental, analytical and numerical techniques in [Krabacher; 1996], [Glaser & Hrycko; 1996] and [Besselink; 2000]. Besselink focuses his research on the shimmy stability of twin-wheeled aircraft main landing gears and uses several tyre models in his analysis. Nonlinear analysis is used not only to study the qualitative behaviour of the Hopf bifurcation but also to analyze the system beyond the Hopf bifurcation in [Thota, Krauskopf & Lowenberg; 2008]. Also, Somieski [Somieski; 1997] studied shimmy as a non-linear dynamics phenomenon for a non-linear set of ODEs. Here, time domain analysis showed a case of supercritical Hopf bifurcation leading to a stable limit cycle past the bifurcation point.

42

4.2 Font axle model

In order to derive equations of motion for the front axle model, some assumptions have to be made:

• The front axle consists of one body which has one lateral degree of freedom.

• The mass of the axle contains the mass of the axle itself including moving parts of the suspension.

• The inertia parameters of the rim are including additional components such as the brakes, hub and steering rod.

• Because of the assumed symmetry of the front axle, only half of the front axle of the truck is considered to reduce the complexity of the model.

• The rim has one rotational degree of freedom around the steering axis with respect to the axle.

• The forward velocity V does not vary in time.

• The system parameters are constant.

• No longitudinal forces act on the tyres

• Vertical forces acting on the tyres are neglected because of the small caster- and steering

angle involved. The component of the vertical force perpendicular to the steering axis Fzψ is given by (see also Figure 4.1):

sinz zF Fψ ε= (4.1)

Here, Fz is the vertical force acting on the tyre because of the mass of the vehicle and ε is the caster angle. Assuming the steering angle remains small, the resulting moment around

the steering axis Mzψ caused by this component is given by:

( )sin tan cosz z aM F R nψ ε ε ψ ε= + (4.2)

Here, R is the tyre loaded radius, n is the steering axis offset with respect to the wheel

centre and ψa is the steering angle. As can be seen from (4.2), the moment around the

steering axis caused by the vertical force remains small when the caster angle ε remains small. Moreover, the tyre models presented in this thesis are accurate for a small side slip angle and therefore also for a small steering angle assuming a straight-line motion. Consequently, the additional term for the self aligning moment described in (4.2) is neglected.

The front axle model developed in this chapter is schematically displayed in Figure 4.1. In this model, two degrees of freedom of the suspension are shown which are a lateral translation ya and

a rotation around the steering axis ψa. Both degrees of freedom have a spring and damper acting

on them, indicated with cay and kay respectively for the lateral translation and caψ and kaψ respectively for the rotation around the steering axis. The axle itself has a mass ma which is equal to half of the total axle mass. The steering axis is rotated in the plane of the wheel with caster

angle ε and has an offset n with respect to the wheel centre. The centre of gravity of the wheel has an offset q with respect to the wheel centre and has a mass mw and moment of inertia around the

steering axis Iwψ. Finally, the tyre loaded radius is indicated with R.

43

Figure 4.1. Side view (left) and top view (right) of the model. The steering axis is rotated by a caster angle ε in the plane of the wheel with respect to the vertical axis. As a result, the steering angle has a component in both the yaw and camber angle of the rim. For small steering angles, this can be described with:

cos

cos

sin

r a a

r a

r a

y y nψ ε

ψ ψ ε

γ ψ ε

= −

= = −

(4.3)

Here, symbols with the subscript r are defined in the centre of the rim. The derivation of a force and moment equilibrium can be made using the Lagrange method [De Kraker; 2001]. For this method, the kinetic energy T and potential energy U of the system as well as the generalised forces Q are needed. The Langrangian equation that is used for the derivation of the equations of motion can be described by:

i

i i i

d T T UQ

dt q q q

∂ ∂ ∂− + =

∂ ∂ ∂ɺ (4.4)

where qi indicates the i-th generalised coordinate:

1

2

a

a

yq

qq

ψ

= =

(4.5)

The generalised forces Qi can be determined by the equation:

2

1

i i

i

W Q qδ δ=

=∑ (4.6)

where δW indicates the virtual work done by the generalised forces and δqi indicates a vertical change in the generalised coordinate. The potential energy and the generalised forces are dependent on the used tyre model and are calculated in the next sections.

44

Straight tangent The system with two degrees of freedom described in the previous section can be analysed in combination with the simple straight tangent tyre model. The slip model for the straight tangent tyre model is derived in Chapter 2 and can be described by (2.14). However, because of the geometry of the system, (2.14) has to be rewritten in terms of the generalised coordinates given in (4.5). The straight tangent tyre model assumes the wheel to be a rigid mass and because the parameters are assumed to be constant, the following set of equations holds:

sin cos

sin cos

cos

cos

c a a a

c a a a

c a

c a

y y R n

y y R n

ψ ε ψ ε

ψ ε ψ ε

ψ ψ ε

ψ ψ ε

= − −

= − −

= =

ɺ ɺɺ ɺ

ɺ ɺ

(4.7)

Here, symbols with the subscript c are defined in the contact centre. Combining (2.14) and (4.7) yields:

( )' ' cos tan cosa a a

y F

z M

V V a R n y

F C

M C

α

α

σα α ψ ε ε ψ ε

α

α

+ = − − − −

′= ′=

ɺ ɺ ɺ

(4.8)

For the remaining equations of motion, the kinetic energy T, the potential energy U and the generalised forces Qi are needed. The expression for the kinetic energy for the two degrees of freedom of the front axle with the straight tangent tyre model reads as:

( )22 21 1 1

cos2 2 2

a a w a a w aT m y m y N I ψψ ε ψ= + − +ɺ ɺɺ ɺ (4.9)

In this equation, N is the distance between centre of gravity and swivel axis and can be calculated by:

N q n= + (4.10)

where q and n are the centre of gravity offset and the steering axis offset respectively. The potential energy U of the system with the straight tangent tyre model can be described by:

2 21 1

2 2ay a a a

U c y c ψψ= + (4.11)

The virtual work δW in the system is done by the external tyre forces and damping forces. The lateral tyre force acts on the tyre with a pneumatic trail. Figure 4.2 shows this effect.

45

Figure 4.2. Relevant lengths and angles for equations of motion. The virtual work done by the generalised forces can be described by:

( )( )tan cosy a p a ay a a a a aW F y R n t k y y k ψδ δ ε δψ ε δ ψ δψ= − + + − − ɺɺ (4.12)

In this equation, δya is the virtual lateral displacement of the axle and δψa is the virtual angular displacement of the wheel around the steering axis. Combining (4.12) and (4.6) yields expressions for the generalised forces:

( )

1

2 tan cos

y ay a

y p a a

Q F k y

Q F R n t k ψε ε ψ

= −

= − + + −

ɺ

ɺ (4.13)

Solving the Lagrange equation for the first variable (ya) and using the expression for Fy from (4.8) yields a force equilibrium in the y-direction:

( ) cosa w a w a ay a f ay am m y m N c y C k yαψ ε α ′+ − + = −ɺɺɺɺ ɺ (4.14)

Solving the Lagrange equation for the second variable (ψa) and using the expression for Mz from (4.8) and (2.15) yields a moment equilibrium around the swivel axis:

( ) ( )2 2cos cos tan cosw a w w a a a f p a a

m Ny m N I c C R n t kψ ψ α ψε ε ψ ψ α ε ε ψ′− + + + = − + + −ɺɺ ɺɺɺ (4.15)

To analyse the stability of the front axle with the straight tangent tyre model, the equations of motion have to be transformed into a state space form. In order to do this, the system can be transformed into a second-order system in matrix form. This form can be described as:

Mq Kq Cq Fα ′+ + =ɺɺ ɺ (4.16)

46

The 2x2 mass matrix M reads as:

2 2

cos

cos cos

t w

w w w

m m N

m N m N IM

ψ

ε

ε ε

− =

− + (4.17)

Here, mt is the total mass of the system and can be calculated by:

t a w

m m m= + (4.18)

The 2x2 damping matrix K reads as:

0

0

ay

a

k

kK

ψ

=

(4.19)

The 2x2 stiffness matrix C reads as:

0

0

ay

a

c

cC

ψ

=

(4.20)

The 2x1 matrix F is related to the tyre forces and reads as:

( )tan cos

F

F p

C

C R n tF

α

α ε ε

= − + +

(4.21)

The state-space form ( )x Ax=ɺ of the system can now be derived. Combining the equation

describing the slip model (4.8) and the second-order system in matrix form (4.16) yields:

1 1 1

' 'V

d

dtσα α

− − −

= − − − p v

q q0 I 0

q M C M K M F q

W W

ɺ ɺ (4.22)

where I is a 2x2 unity matrix and Wv and Wp read as:

( )

( )

sin cos cos

cos0

V R n a

V

v

p

W

W

ε ε εσ σ

εσ

− + − =

= (4.23)

The system matrix A is given in Appendix D.

Von Schlippe As well as the straight tangent tyre model, the Von Schlippe tyre model assumes the wheel to be a rigid mass. As is done for the system with the straight tangent tyre model, the slip model has to be rewritten in terms of the generalised coordinates (4.5). Substituting (4.7) in (2.19) yields:

47

( )

( )

22 13

3 2

2

tan cos3

1tan cos

3

1cos

a a

F

y a a

z M a

a aa az z z z y a R n

V V V

C aF z z y R n

a V

M C zV

α

α

σσ σσ ε ψ ε

ε ψ εσ

ψ ε

+ ++ + + = + + − −

= + − + + +

= − −

ɺɺɺ ɺɺ ɺ

ɺɺ

ɺ

(4.24)

The same expressions for the kinetic energy (4.10) and potential energy (4.11) can be used for the system with the Von Schlippe tyre model. However, as can be seen from (4.24), the self aligning moment can no longer be calculated by giving the lateral tyre force a constant pneumatic trail. As a result, the generalised forces and therefore the virtual work of the system change due to the different tyre model. The virtual work done by the external tyre forces and friction can be described by:

( )( )tan cos cosy a a z a ay a a a a aW F y R n M k y y k ψδ δ ε δψ ε δψ ε δ ψ δψ= − + + − − ɺɺ (4.25)

Combining (4.25) and (4.6) yields expressions for the generalised forces:

( )

1

2 tan cos cos

y ay a

y z a a

Q F k y

Q F R n M k ψε ε ε ψ

= −

= − + + −

ɺ

ɺ (4.26)

Substituting (4.9), (4.11) and (4.26) in the Lagrange equation (4.4) and solving for the first variable (ya) yields a force equilibrium in the y-direction:

( ) cosa w a w a ay a y ay am m y m N c y F k yψ ε+ − + = −ɺɺɺɺ ɺ (4.27)

Substituting the expression for the lateral tyre force Fy from equation (4.24) into (4.27) and rearranging yields an equation that can be transformed into the state-space form:

( )

( )( ) ( )

2

2

cos

tan cos3

F

a w a w a ay a ay a

F F F

a

Cm m y m N k y c y

a

C C a CR n z z

a a V a

α

α α α

ψ εσ

ε ψ εσ σ σ

+ − + + + +

− + = ++ + +

ɺɺɺɺ ɺ

ɺɺ

(4.28)

Solving the Lagrange equation for the second variable (ψa) yields a moment equilibrium around the swivel axis:

( ) ( )2 2cos cos tan cos cosw a w w a a y z a

m Ny m N I c F R n M kψ ψ ψε ψ ε ψ ε ε ε ψ− + + + = − + + −ɺɺ ɺɺɺ (4.29)

Substituting the expression for the lateral tyre force Fy and the self aligning moment Mz from equation (4.24) into (4.29) and rearranging yields an equation that can be transformed into the state-space form:

48

( ) ( )

( )

( )( )

( )( )

2 2

2 2 2

2

2

cos cos tan cos

tan cos cos

tan cos cos tan cos3

Fw a w w a a a a

Fa M a

F M F

Cm Ny m N I k R n y

a

Cc R n C

a

C a C CR n z z R n z

a V V a

αψ ψ

αψ α

α α α

ε ψ ε ψ ε εσ

ε ε ε ψσ

ε ε ε ε εσ σ

− + + + − +

+

+ + + − = + − + − − +

+ +

ɺɺ ɺɺɺ

ɺɺ ɺ

(4.30)

To analyse the stability of the front axle with the Von Schlippe tyre model, the equations of motion have to be transformed into a state space form. As is done for the system with the straight tangent tyre model, the second-order system in matrix form can be used for this transformation. Because only the slip model changes with respect to the system with the straight tangent tyre model, the mass matrix M (4.17) and damping matrix K (4.19) are unchanged. The stiffness

matrix C changes however due to the dependency of the tyre forces on ya and ψa. The 2x2 stiffness matrix C becomes:

( )

( ) ( )2 2 2

tan cos

tan cos tan cos cos

F F

F F

C C

ay a a

C C

a Ma a

c R n

R n c R n CC

α α

α α

σ σ

ψ ασ σ

ε ε

ε ε ε ε ε

+ +

+ +

+ − + = − + + + −

(4.31)

Because Kluiters uses a third order equation that describes the slip model, the 2x3 matrix F becomes:

( ) ( )

( )( ) ( ) ( )

2

2

2

2

3

cos

3

0

tan cos tan cos

F F

F M F

C a C

aa V

C a C C

V aa VR n R n

F

α α

α α α

σσ

ε

σσε ε ε ε

++

++

= − + − − +

(4.32)

The state-space form ( xɺ = Ax) of the system can now be derived. Combining the equation

describing the slip model (4.24) and the second-order system in matrix form (4.16) yields:

1 1 1

dz z

dtz z

z z

− − −

− − =

1

2

p 3

0 I 0q q

M C M K M Fq q

0 0 Z

0 0 Z

W 0 Z

ɺ ɺ

ɺ ɺ

ɺɺ ɺɺ

(4.33)

where I is a 2x2 unity matrix and Wp, Z1, Z2 and Z3 read as:

( )( )( )

( )( ) ( )( )

33

2 2

23

2 2

3 tan cos3

3 33

0 1 0

0 0 1

V a R nVp a a

a V V aV

aa a

σ ε ε

σ σ

σ σ

σσ σ

+ − +

+ +

=

=

=

= − −

1

2

3

W

Z

Z

Z

(4.34)

49

Rigid ring For the equations of motion for the rigid ring tyre model itself, the equations of motion derived in Section 2.5 can be used. As well as for the other two tyre models, these equations have to be rewritten in terms of the generalised coordinates (4.5). Substituting (4.3) in the equations derived for the belt (2.22) yields:

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( )

cos

cos

cos

sin cos

sin

b b by b a a cy c b b

by b a a cy c b b

b b b b b b a c c b

b b a b b a c c b

b b b b b b a

m y k y y n k y y R

c y y n c y y R

I I k k

k c c

I I k

ψ ω ψ ψ

ψ ψ ψ

γ ω γ

ψ ε γ

ψ ε γ

ψ γ ψ ψ ε ψ ψ

γ ψ ε ψ ψ ε ψ ψ

γ ψ γ ψ

= − − + + − −

− − + + − −

= − Ω − − + −

− Ω + − − + −

= Ω − +

ɺ ɺɺɺ ɺ ɺ ɺ ɺ

ɺɺ ɺ ɺ ɺ ɺ ɺ

ɺɺ ɺ ɺ ɺ( ) ( )

( ) ( ) ( ) + cos sin

cy c b b

b b a b b a cy c b b

Rk y y R

k c Rc y y Rγ γ

ε γ

ψ ψ ε γ ψ ε γ

+ − − Ω − − + + − −

ɺɺ ɺ

(4.35)

Equations for the contact patch (2.23) as well as the equations for the slip model (2.24) through (2.39) remain unchanged. The rigid ring tyre model assumes that the wheel consists of more than one rigid body. Between these bodies, additional degrees of freedom exist. A schematic representation of this tyre model attached to the front axle model is displayed in Figure 4.3. In this figure, the offsets n and q are not displayed to improve the readability. Because the equations of motion of the belt and contact patch as well as the slip model are derived already in Chapter 2 and Appendix B, only the equations of motion of the two degrees of freedom of the axle are considered here.

Figure 4.3. The rigid ring tyre model attached to a suspension model. When compared to the systems with the other two tyre models, a similar equation for the kinetic

energy (4.9) can be used. However, instead of the inertia parameters of the wheel mw and Iwψ, the

inertia parameters of the rim mr and Irψ have to be used for the two degrees of freedom of the front axle:

50

( )22 21 1 1

cos2 2 2

a a r a a r aT m y m y N I ψψ ε ψ= + − +ɺ ɺɺ ɺ (4.36)

The potential energy U changes because the tyre forces do not act on the rim itself anymore. Instead, additional spring forces act on the rim:

( )

( ) ( )

22 2

2 2

1 1 1cos

2 2 2

1 1 cos sin

2 2

ay a by a a b a a

b a b b a b

U c y c y n y c

c c

ψ

ψ γ

ψ ε ψ

ψ ε ψ ψ ε γ

= + − − +

+ − + − −

(4.37)

The virtual work δW in the axle is done by the damping forces:

( )

( ) ( )

( ) ( )

cos

+ cos cos cos sin

+ sin cos cos sin

ay a a a a a by b a a a

by b a a a b b b a a

b b a a b b a a

W k y y k k y y n y

k n y y n k

k k

ψ

ψ

ψ γ

δ δ ψ δψ ψ ε δ

ε ψ ε δψ ψ ε γ ε ψ δψ

γ ψ ε δψ ε ψ ψ ε δψ ε

= − − + − +

− + + − −

Ω + + Ω −

ɺ ɺɺ ɺ ɺ

ɺɺ ɺ (4.38)

Combining (4.38) and (4.6) yields expressions for the generalised forces:

( )

( ) ( )

( ) ( )

1

2

cos

cos cos cos sin

+ sin cos cos sin

ay a by b a a

a a by b a a b b b a

b b a b b a

Q k y k y y N

Q k k n y y n k

k k

ψ ψ

ψ γ

ψ ε

ψ ε ψ ε ψ ε γ ε ψ

γ ψ ε ε ψ ψ ε ε

= − + − +

= − − − + + − −

Ω + + Ω −

ɺɺ ɺ ɺ

ɺ ɺ ɺ ɺ ɺɺ ɺ (4.39)

Substituting (4.9), (4.37) and (4.39) in the Lagrange equation (4.4) and solving for the first variable (ya) yields a force equilibrium in the y-direction:

( ) ( )

( )

cos cos

cos

a a r a a ay a by a b a

ay a by b a a

m y m y N c y c y y n

k y k y y n

ψ ε ψ ε

ψ ε

+ − + + − − =

− + − +

ɺɺɺɺ ɺɺ

ɺɺ ɺ ɺ (4.40)

Solving the Lagrange equation for the second variable (ψa) yields a moment equilibrium around

the swivel axis. Furthermore, if the rotational stiffness constants cbψ and cbγ are taken equal to

each other as well as the rotational damping constants kbψ and kbγ, the moment equilibrium can be simplified to:

( ) ( )

( ) ( )

( ) ( )

2 2cos cos cos cos

+ cos sin cos cos

+ cos sin sin cos

r a a r a by a a b

a a b a b b a a by b a a

b b b a b b b

m Ny N I c n n y y

c c k k n y y n

k k

ψ

ψ ψ ψ

ψ ψ

ε ψ ε ψ ε ψ ε

ψ ψ ψ ε γ ε ψ ε ψ ε

ψ ε γ ε ψ ψ ε γ ε

− + + + − +

+ − + = − − − +

− − + Ω +

ɺɺ ɺɺɺɺ

ɺ ɺɺ ɺ

ɺ ɺ ɺ

(4.41)

To analyse the stability of the front axle with the rigid ring tyre model, the equations of motion have to be transformed into a state space form. This state-space form ( xɺ = Ax) has

become larger and is therefore given in Appendix E.

51

4.3 Analytical computation of the stability boundaries

In order to find analytic expressions for the boundaries between stability and instability, the Hurwitz criterion [Hagedorn; 1988, Pacejka; 1966] can be applied to the characteristic equation

of the system. The damping coefficients kay and kaψ as well as the steering axis offset n and centre of gravity offset q are neglected. The characteristic equation of the system described by (4.22) reads as:

5 4 3 2

0 1 2 3 4 5 0a a a a a aλ λ λ λ λ+ + + + + = (4.42)

where:

( )( )

( )

( )( ) ( )

( )

0

1

2

2

2

3

2

4

2

5

1

cos tan tan

cos tan

cos tan tan

cos tan

F p ayaF

w t w t

F p ay a

w t w

F ay p a F ay

t w t w

F ay p ay a

t w t w

a

Va

C R t R a ccCa

I m I m

C V R t c V c Va

I m I

C c R a R t c C ca

m I m I

C c V R t c c Va

m I m I

α ψα

ψ ψ

α ψ

ψ ψ

α ψ α

ψ ψ

α ψ

ψ ψ

σ

ε ε ε

σ σ

ε ε

σ σ σ

ε ε ε σ

σ σ

ε ε

σ σ

= =

+ −= + + +

+= + +

− + += +

+= +

(4.43)

The Hurwitz criterion can be applied to the characteristic equation of the system in order to determine stability. The system is asymptotically stable if all of the following conditions are met (a0 = 1):

( )( )( )

( ) ( )

1

2

5

2

2

1

1

3

1

1

1

2 3 2 3

4

0 : 0

0 : cos tan

0 : cos tan tan

for tan

0 :

for tan

min , , max , for tan

0 :

a F p

w t p

p

a

p

a

p

a a

Va

a c C R t

H I m R a R t

tc r

RH

tc r

R

tc r r c r r

H

ψ α

ψ

ψ

ψ

ψ ψ

σ

ε ε

ε ε σ ε

ε

ε

ε

−

−

−

> >

> > − +

> > − + + +

− > <

>

− < >

−< > <

>

( ) ( ) 1

2 3 2 3min , max , for tanp

a

R

tr r c r r

Rψ ε −

− < < >

(4.44)

52

where r1, r2 and r3 are given by:

( )( ) ( )( )

( )( )

2

2

1 2 2

2

2

3

cos tantancos tan tan

cos tantan

w ay F w

F p

tt p

ay w

t

ay wF w

F p

t t

I c C Ir C R t

m R am R a R t

c Ir

m

c IC Ir C R t

m R a m

ψ α ψ

α

ψ

ψα ψα

ε εε σε ε σ ε

ε εε σ

= + − +

− + +− + + +

= = + − +

− + +

(4.45)

The first condition of the Hurwitz criteria is trivially fulfilled: the forward velocity and relaxation length are both positive. Furthermore, the condition for H3 is always covered by a combination of the conditions for H4 and a5. The stability criteria are schematically displayed as function of the yaw stiffness and caster angle in Figure 4.4. The choice of these parameters is made because this plot shows all stability boundaries except for H2. However, this criterion is shown to be redundant later on in this section. The shaded grey area indicates a negative yaw stiffness which is a physically impossible situation.

Figure 4.4. Schematic representation of the stability criteria. As can be seen from Figure 4.4, there are two characteristic values of the caster angle which are indicated by the black dots numbered with I and II. The first characteristic value for the caster angle, indicated with I, describes the point in the given caster angle range where the stability boundary described by a5 in (4.44) and the horizontal line describing a yaw stiffness equal to zero intersect. Computing the intersection of the latter two lines results in the following caster angle value:

53

1tan 0 tanp

p

tR t

Rε ε −

− + = ⇒ =

(4.46)

This point can also be seen from the Hurwitz criteria H3 and H4. This value for the caster angle describes the situation where the lateral tyre force acts exactly on the axis of rotation of the wheel. Consequently, this is the point where the self aligning moment switches its sign.

The second characteristic value indicates the maximum value for the caster angle for a stable system in the given caster angle range. This value can be found by solving the equation:

( )

( )2cos tantan

ay w ay wF w

F p

t t t

c I c IC IC R t

m m R a m

ψ ψα ψ

α ε εε σ

= + − +− + +

(4.47)

This equation can be reduced to:

( )( )2cos tan tanw t p

I m R a R tψ ε ε σ ε= − + + + (4.48)

The solution is chosen as an expression for the moment of inertia rather than the caster angle to reduce complexity. When comparing (4.48) with H2 of the Hurwitz criteria, it appears that the expressions are equal to each other. Consequently, H2 implies that the system is unstable for a caster angle larger than the value indicated with II in Figure 4.4. Therefore, H2 is covered by H4 and does not impose an additional stability restriction.

4.4 Parameter influence on the analytical expressions of the stability

boundaries

The vertical line indicated with I (4.46) separates the three stable areas in Figure 4.4: there are two stable areas on the left-hand side of this line and one on the right-hand side of this line. Because both the pneumatic trail tp and the radius of the tyre R are always positive, this line is always described by a negative value. As a result, there is only one stable area in the positive caster angle domain. In practice, the caster angle is usually positive in order to improve the straight line stability of the system and therefore the stable area on the right-hand side of the vertical line is the most interesting area in practical point of view and is therefore considered in this section.

The stable area on the right-hand side of I is bound by three lines which can be described by (4.46) and by r2 and r3. These three boundaries can also be seen from Figure 4.5. However, as pointed out already, the vertical line described by (4.46) is always in the negative caster angle domain and is therefore neglected. Consequently, only r2 and r3 are considered here.

54

Figure 4.5. Schematic overview of the stability boundaries. First, the boundary on the right-hand side of the stable area is the point where r2 and r3 intersect. Because the caster angle is usually positive, this point should be shifted towards a large positive caster angle where (4.48) has to be fulfilled. This can be achieved by decreasing the mass of both the wheel mw and the axle ma. Other changes that have a similar effect are decreasing the relaxation length σ, half of the contact length a and the pneumatic trail tp and increasing the

moment of inertia Iwψ. Adjusting the radius of the tyre R has a different effect. When the maximum stable caster angle value described by (4.48) has a negative value, decreasing R results in a decrease of the maximum stable caster angle value. For a positive value, decreasing R results in an increase of the maximum stable caster angle value.

Secondly, the difference between r2 and r3 should be as large as possible in order to have a stable system over a large yaw stiffness range. This difference can be described by:

( )

( ) 2

3 2 tan costan

w

F p

t

Ir r C R t

m R a

ψα ε ε

ε σ

− = − + − + +

(4.49)

As can be seen from (4.49), the difference can be increased by decreasing the mass of both the wheel mw and the axle ma, the relaxation length σ, half of the contact length a and the pneumatic

trail tp and increasing the moment of inertia Iwψ. This is in accordance with the changes that have

to be made to increase the maximum stable caster angle. The remaining two parameters CFα and R have a different influence on the stability: they both affect the gradient of r3. This influence is shown in Figure 4.6. In this figure, the solid lines display the original stability boundaries and the

dashed lines display the boundaries with increasing CFα and decreasing R. Increasing CFα results in an increased negative gradient of r3. Because the value for the caster angle indicated with II in

Figure 4.4 is not dependent on CFα, r3 rotates around this point. Consequently, an increase of CFα is desirable for stability considerations. Secondly, decreasing R results in a decreased negative gradient of r3. This may seem undesirable for an increase of the stable area, but it seems that r3

rotates around the point where it intersects with the vertical line where ε is equal to zero.

55

Consequently, decreasing R results in an increase of the difference described with (4.49) for a positive caster angle and a decrease of the difference in (4.49) for a negative caster angle value.

The effects of a variation of R and CFα are schematically displayed in Figure 4.6. However, it has to be taken into account that these are variations made to only one parameter. In reality, adjusting one parameter often results in the adjustment of several parameters because they can be physically linked. A final remark can be made about the lateral stiffness cay. This parameter does not influence the difference between r2 and r3 nor does it influence the maximum stable caster angle. However, when increasing cay, r2 and r3 both shift upwards equally.

Figure 4.6. Effect on stability boundaries when changing CFα (left) and R (right).

4.5 Numerical computation of the stability boundaries

In this section, a comparison is made between the influences of the tyre models on the stability regions. Due to the increased complexity of the Von Schlippe tyre model and the rigid ring tyre model with respect to the straight tangent tyre model, a numerical computation of the eigenvalues of the system matrix is made rather than deriving analytical expressions of the stability boundaries as is done in the previous section. Table 4.1. Parameter values for the front axle of a passenger car.

Parameter Value Unit Description

cay 1.40e6 N/m Lateral stiffness of the front axle

caψ 1.91e4 Nm/rad Rotational stiffness around the steering axis

Irψ 3.75e-1 kgm2 Moment of inertia of the rim including part of the tyre that is fixed to the rim around the x- and z-axis

Irω 7.5e-1 kgm2 Moment of inertia of the rim including part of the tyre that is fixed to the rim around the y-axis

kay 0 Ns/m Lateral damping of the front axle

kaψ 0 Nms/rad Rotational damping around the steering axis

ma 2.00e1 kg Half of the mass of the front axle mr 1.50e1 kg Mass of the rim including part of the tyre that is fixed to

the rim n 0 m Steering axis offset with respect to the wheel centre q 0 m Centre of gravity offset with respect to the wheel centre

ε 6.46 ˚ Caster angle

Parameters that are used in this section are representative for a steering system of a passenger car. The parameter values for a tyre have been discussed already and are displayed in Table 3.1. However, because other components such as the braking system and hub are included in this

chapter, inertia parameters mr, Irψ and Irω are increased to account for these components. Parameters that are used for the steering system are taken from a multibody model developed at the University of Technology Eindhoven and are displayed in Table 4.1. In order to reduce the

56

complexity of the stability analysis in this chapter, the damping coefficients kay and kaψ and the two offsets q and n are taken equal to zero. The parameters are taken as given in Table 3.1 and Table 4.1, unless indicated otherwise.

Figure 4.7 shows a contour plot of the maximum value of the real parts of the eigenvalues, and therefore the level of stability, for varying yaw stiffness and caster angle. As can be seen from this figure, there are three stable areas and three unstable areas. Furthermore, the stability boundaries derived in the previous section can be seen from this figure. The largest level of instability occurs for a large negative caster angle in combination with low yaw stiffness. In this area, the system is monotonically unstable. In this situation, the eigenvalues have a positive real part and no imaginary part. This means that the states of the system go to infinity without any form of vibration: the wheel simply steers to one side. Furthermore, a positive caster angle causes the system to be completely unstable with an exception for a small area at a mediocre value of the yaw stiffness. Finally, the third unstable area is located at a negative caster angle in combination with a yaw stiffness of approximately 2.5·104 Nm/rad.

Figure 4.7. Contour plot of the maximum real part of the eigenvalues with varying caster angle

and yaw stiffness for the system with the straight tangent tyre model (V = 25 m/s). Unlike the system with the straight tangent tyre model, the stability boundaries of the system with the Von Schlippe tyre model are dependent on the forward velocity. This is because the Von Schlippe tyre model includes the trailing contact point of the tyre in the calculations of the tyre forces. As already mentioned, the trailing contact point has a velocity dependent delay with respect to the leading contact point. Figure 4.8 shows the stability results for the front axle model with the Von Schlippe tyre model as a function of caster angle and yaw stiffness. Because the stability boundaries are dependent on the forward velocity, various forward velocities are used.

57

The grey areas indicate an unstable system. At low forward velocities, the stable regions change drastically when changing the forward velocity and there are very few configurations that are stable for every forward velocity. For higher forward velocities, the results become more similar to the system with the straight tangent tyre model (Figure 4.7). Although the stability regions are dependent on the forward velocity, there are two stability boundaries that do not change when the forward velocity changes. These are also equal to the corresponding stability boundaries of the system with the straight tangent tyre model which are the stability boundaries indicating the monotonically unstable area in the lower left-hand corner and the boundary indicated with r3 in Figure 4.5.

Figure 4.8. Unstable areas (grey) and stable areas (white) for various forward velocities for the

system with the Von Schlippe tyre model. Figure 4.9 shows the stability results for the system with the rigid ring tyre model as a function of caster angle and yaw stiffness for various forward velocities. In this figure, the grey areas indicate an unstable system. The system with the rigid ring tyre model shows very few similarities with the systems with the other two tyre models. The monotonically unstable area in the lower left hand corner is also present for the system with the rigid ring tyre model as well as the unstable area in the lower right-hand corner. As can be expected for the system with the rigid ring tyre model, the system has a larger stable area compared to the systems with the other two tyre models. This is caused by the turn slip effect and the damping forces between the rim and belt. Because of the turn slip effect, the system is stable almost everywhere at low forward velocities, except for the monotonically unstable area. It is interesting to see is that at extremely high forward velocities, the unstable area in the lower right-hand corner disappears again.

58

Figure 4.9. Unstable areas (grey) and stable areas (white) for various forward velocities for the

system with the rigid ring tyre model.

4.6 Stability boundary sensitivity to parameters adjustments

In this section, the dependency of the stability boundaries on the most influential parameters is

investigated. So far, the damping coefficients kay and kaψ and the offsets q and n are neglected in this chapter. However, it turns out that these parameters have a large influence on the stability

boundaries. Figures 4.10 and 4.11 show the influence of the parameters kay and kaψ with respect to the stability boundaries. The values for these parameters are based on 3% damping of the corresponding eigenmode when the tyre is not touching the road. The values are taken equal to

460 Ns/m for kay and 6.8 Nsm/rad for kaψ. The forward velocity is taken equal to 25 m/s throughout this entire section. In Figures 4.10 and 4.11, the grey areas indicate the unstable areas where the damping is taken equal to zero and the shaded areas with the dashed stability boundaries indicate the unstable areas with adjusted damping. From Figures 4.10 and 4.11 it can be seen that introducing damping in the system causes the unstable areas to become smaller. This effect is larger for adjusting the rotational damping instead of the lateral damping. Moreover, the effect is also larger for the systems with the straight tangent tyre model and the Von Schlippe tyre model compared to the system with the rigid ring tyre model. However, the stability boundary indicating the monotonically unstable area is not affected by adjusting the damping constants because no vibrations are involved in this situation.

59

Figure 4.10. Unstable areas for kay = 0 (grey) and kay = 460 (shaded) for the system with the three tyre models at 25 m/s.

Figure 4.11. Unstable areas for kaψ = 0 (grey) and kaψ = 6.8 (shaded) for the system with the three tyre models at 25 m/s.

Figures 4.12 and 4.13 show the influence of the offsets n and q with respect to the stability boundaries. The parameter values are taken equal to 3.11 cm and 3.00 cm for n and q respectively. The value for n is based on a multibody model developed at the University of Technology Eindhoven and the value for q is based on the assumption that the brake calliper is positioned at the rear side of the brake disk. The grey areas indicate an unstable system for the offsets equal to zero and the shaded areas with dashed stability boundaries indicate an unstable system for adjusted offsets.

As can be seen from Figures 4.12 and 4.13, both adjustments cause the unstable areas to become larger for all three models. The monotonically unstable area in Figure 4.12 becomes smaller. For a positive caster angle however, positive values for the offsets n and q are desirable for a large value of the yaw stiffness and undesirable for low values of the yaw stiffness.

Figure 4.12. Unstable areas for n = 0 (grey) and n = 0.0311 (shaded) for the system with the three tyre models at 25 m/s.

60

Figure 4.13. Unstable areas for q = 0 (grey) and q = 0.03 (shaded) for the system with the three tyre models at 25 m/s.

As well as for the suspension parameters that are analysed so far, the sensitivity of the stability boundaries to the tyre parameters can be investigated. As can be seen from previous chapters, there are numerous tyre parameters that can be varied. However, most of these parameters only influence the stability boundaries slightly when realistic adjustments are made. It seems that the parameters that influence the stability areas the most are the parameters used for the calculation

of tyre forces. These are the pneumatic trail tp, the cornering stiffness CFα and the turn slip

stiffness for the self aligning moment CcMϕ. The parameters used in the calculation of the tyre forces are also the parameters that are most likely to change during load changes on the tyre for example while driving around. Other parameters such as the contact length a and relaxation

length σ are only influencing the stability boundaries of the system with the straight tangent tyre model and the Von Schlippe tyre model rather than the system with the rigid ring tyre model. Because the rigid ring tyre model is shown to be the most accurate tyre model, these parameters are left out in the sensitivity analysis in this section.

Figure 4.14. Unstable areas for CFα = 67000 (grey) and CFα = 134000 (shaded) for the system with the three tyre models at 25 m/s.

Figure 4.15. Unstable areas for tp = 0.03 (grey) and tp = 0.06 (shaded) for the system with the three tyre models at 25 m/s.

61

Figures 4.14 and 4.15 show the influence of a variation of the cornering stiffness CFα and the pneumatic trail tp to the stability boundaries respectively. Here, the grey areas show the unstable areas for the original situation and the shaded areas with dashed stability boundaries show the unstable areas when the cornering stiffness and the pneumatic trail are increased by a factor 2. As is pointed out already in Section 4.5, it has to be taken into account that adjusting one parameter can result in the adjustment of a group of parameters in reality. The changes for the system with the straight tangent tyre model in Figures 4.14 and 4.15 are in accordance to what is pointed out in Section 4.5. The most interesting change can be found for the system with the rigid ring tyre model for both parameters where an additional unstable area arises for large positive caster angles in combination with a large yaw stiffness.

Finally, the turn slip stiffness for the self aligning moment CcMϕ only influences the stability boundaries of the system with the rigid ring tyre model. This is shown in Figure 4.16 where the grey areas indicate an unstable system for the original situation and the shaded area with dashed stability boundaries indicate the unstable areas for a turn slip stiffness that is increased by a factor 2. As can be expected, the unstable area decreases in size because of the damping effect of turn slip. The monotonically unstable area in the lower left-hand corner is however not affected by this change.

Figure 4.16. Unstable areas for CcMϕ = 177 (grey) and CcMϕ = 354 (shaded) for the system with the rigid ring tyre model at 25 m/s.

4.7 Conclusions

In this chapter, the model derived in Section 4.2 is used to analyse the stability of the system as a function of parameters. The analytical results from the front axle model with the straight tangent tyre model show that there are three stability boundaries present that are independent on the forward velocity. The first stability boundary indicates the boundary where the system becomes monotonically unstable. This is instability where the wheel steers to one side without vibrations.

This situation is present when a small value for the yaw stiffness caψ is used in combination with

a large negative caster angle ε. The other two stability boundaries determine the areas where unstable modes exist. The stability boundaries indicate that there is only a small stable area in the positive caster angle domain. This is the most interesting domain because the caster angle is usually made positive in order to improve the straight line stability. Next, the stability boundaries for all three tyre models are analysed using a numerical computation of the eigenvalues of the systems. For this analysis, parameter values of a passenger

62

car are used. Unlike the stability boundaries for the front axle model with the straight tangent tyre model, the stability boundaries for the other two models are velocity dependent. The model with the Von Schlippe tyre model shows different unstable areas with respect to the model with the straight tangent tyre model. These differences are larger at a low forward velocity and become smaller towards high forward velocities. The model with the rigid ring tyre model shows only two unstable areas that are present for a low yaw stiffness. At low forward velocities, only the monotonically unstable area exists for a large negative caster angle. At increasing velocities, an unstable area appears for a positive caster angle. This latter area disappears again for an extremely high forward velocity. Only the monotonically unstable area for a low yaw stiffness and a large negative caster angle is independent of forward velocity and tyre model. Finally, a sensitivity analysis is made for the stability boundaries. Several influential parameters are adjusted and the corresponding boundaries are plotted. The most influential

parameters for the suspension model appear to be the two damping constants kay and kaψ and the two offsets n and q. For the tyre model, the most influential parameters are the cornering stiffness

CFα, the pneumatic trail tp and the turn slip stiffness for the self aligning moment CcMϕ. It is shown

that increasing kay. kaψ and CcMϕ leads to decreasing the unstable areas because of the additional

damping. Furthermore, increasing CFα, and tp leads to an increase of the unstable areas. Finally, adjusting the offsets n and q have a more complicated effect where it is hard to draw conclusions whether or not an adjustment in these parameters is desirable. For a positive caster angle however, positive values for the offsets n and q are desirable for a large value of the yaw stiffness and undesirable for low values of the yaw stiffness.

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5. Truck Front Axle So far, the emphasis of this thesis has been on the influence of tyre models on the stability of a basic front axle model using parameters for a passenger car. In this chapter, the step is made to a truck front axle using the same model that is derived in Chapter 4. The analysis in this chapter is augmented with the calculation of the damping and frequency of the modes representing the degrees of freedom of the suspension as well as the calculation of eigenvectors. Parameter values used in this chapter are given in Section 5.1. Next, the eigenvalue and eigenvector analysis is given in Section 5.2 and the chapter ends with the main conclusions in Section 5.3.

5.1 Parameter values

Parameters that are used for a truck tyre are displayed in Table 5.1. They are based on simulations with a tyre property file. A vertical load of 3550 kg is applied which is a typical value for a load for a truck front tyre. Table 5.1. Parameter values for a truck tyre.


a, σc 1.16e-1 m Half of the contact length Relaxation length of the contact patch

cby 2.68e6 N/m Lateral stiffness between rim and belt cbγ, cbψ 3.24e5 Nm/rad Rotational stiffness between rim and belt ccy 3.36e6 N/m Lateral stiffness between belt and contact patch ccψ 4.10e4 Nm/rad Rotational stiffness between belt and contact patch CFα 2.12e5 N/rad Cornering stiffness CcFφ 8.00e3 Ns/rad Turn slip stiffness for the lateral force CcMφ 6.25e2 Nms/rad Turn slip stiffness for the self aligning moment Ibγ, Ibψ 5.61 kgm2 Moment of inertia of the belt around the x- and z-

axis Ibω 1.01e1 kgm2 Moment of inertia of the belt around the y-axis Icψ 1.00e-2 kgm2 Moment of inertia of the contact patch around the z-

axis kby 6.43e2 Ns/m Lateral damping between rim and belt kbγ, kbψ 7.24e1 Nms/rad Rotational damping between rim and belt kcy 2.06e2 Ns/m Lateral damping between belt and contact patch kcψ 5.43e1 Nms/rad Rotational damping between belt and contact patch mb 4.70e1 kg Mass of the belt mc 1.00 kg Mass of the contact patch Re 5.33e-1 m Effective rolling radius R 5.16e-1 m Tyre loaded radius tp 4.02e-2 m Pneumatic trail

Note that for the straight tangent tyre model and the Von Schlippe tyre model, the inertia parameters of the entire wheel are used and they are equal to the values for the rim plus the belt. Parameters that are used for the front axle model are displayed in Table 5.2. The damping

constants kay and kaψ are taken equal to zero, because the introduction of damping in the suspension causes most of the stability boundaries to disappear and make the system almost

64

completely stable. However, the scope of this thesis is to identify trends caused by parameter adjustments and therefore the damping in the suspension is neglected. Parameters in the remainder of this chapter are taken as given in Table 5.1 and Table 5.2, unless indicated otherwise. Table 5.2. Parameter values for the front axle of a truck.


cay 3.57e6 N/m Lateral stiffness of the front axle

caψ 1.20e5 Nm/rad Rotational stiffness around the steering axis

Irψ 8.69 kgm2 Moment of inertia of the rim including part of the tyre that is fixed to the rim around the x- and z-axis

Irω 9.10 kgm2 Moment of inertia of the rim including part of the tyre that is fixed to the rim around the y-axis

kay 0 Ns/m Lateral damping of the front axle

kaψ 0 Nms/rad Rotational damping around the steering axis

ma 91 kg Half of the mass of the front axle mr 232 kg Mass of the rim including part of the tyre that is fixed to

the rim n -1.90e-2 m Steering axis offset with respect to the wheel centre q -1.31e-2 m Centre of gravity offset with respect to the wheel centre

ε 3.50 ˚ Caster angle

5.2 Eigenvalue and eigenvector analysis

For the analysis in this chapter, eigenvalues and eigenvectors of the system matrix are calculated as a function of varying parameter values. The forward velocity is taken equal to 25 m/s throughout this chapter unless indicated otherwise. This velocity is taken because shimmy instability is most likely to occur at high velocities and is a on average the maximum velocity a truck is allowed to reach. Except for the varying parameter, the other parameter values are taken from Table 5.1 and Table 5.2. Figures 5.1 through 5.4 show the effects on the eigenvalues of the system for the four most influential parameters. These are the caster angle, yaw- and lateral stiffness and the forward velocity which are displayed in Figures 5.1 through 5.4 respectively. For these figures, both the frequency and damping are calculated for the two modes that correspond to the two degrees of freedom of the front axle. The other modes such as the tyre modes for example are neglected because they cannot become unstable. Besides the eigenvalues, the eigenvectors are

analysed as well with a calculation of the amplitude ratio η and the relative phase angle ξ. The

amplitude ratio is defined as the amplitude of the steering angle ψa in radians divided by the amplitude of the lateral position of the axle ya in metres. Consequently, a large value for the amplitude ratio indicates a large contribution of the steering angle to the shimmy motion. The relative phase angle is defined as the phase difference between the yaw motion and the lateral motion. Here, a positive relative phase angle indicates a phase lead of the yaw motion with respect to the lateral motion.

65

Figure 5.1. Frequency, damping, amplitude ratio and relative phase angle as a function of the

caster angle for the system with the three tyre models at V = 25 m/s. In Figure 5.1, the four rows show the frequency, the damping, the amplitude ratio and the relative phase angle of the system with the three tyre models. The two columns display these values for the two different modes representing the degrees of freedom of the suspension. The caster angle is varied between -20 and 20 degrees. Overall, it can be seen that the system with the straight tangent tyre model and the Von Schlippe tyre model are similar to each other, whereas the system with the rigid ring tyre model shows more differences. The system with the rigid ring tyre model shows the least unstable behaviour only at a large positive caster angle. This can be explained by the additional damping between the rim and belt and the turn slip effect of the rigid ring tyre model. The results for the eigenvectors, displayed in the last two rows, indicate that it is not possible to identify one ‘yaw mode’ and one ‘lateral mode’.

66

Figure 5.2. Frequency, damping, amplitude ratio and relative phase angle as a function of the yaw

stiffness for the system with the three tyre models at V = 25 m/s. Figures 5.2 and 5.3 are similar to Figure 5.1 where instead of the caster angle, both the yaw- and lateral stiffness are varied respectively. As expected, the frequencies for both modes increase when increasing the stiffness. Again, the systems with the straight tangent tyre model and the Von Schlippe tyre model are similar and their differences with the system with the rigid ring tyre model are much larger. As well as for the varying caster angle, the system with the rigid ring tyre model shows the most stable behaviour. The system with the rigid ring tyre model is only unstable for small values of both the yaw- and lateral stiffness.

67


lateral stiffness for the system with the three tyre models at V = 25 m/s.

68


forward velocity for the system with the three tyre models. Finally, the influence of a varying forward velocity is shown in Figure 5.4. Differences between the systems with the tyre models are large and the models with the straight tangent tyre model and the Von Schlippe tyre model show unstable behaviour for some velocities in the full range of the given forward velocity range. The system with the straight tangent tyre model is even completely unstable.

Figures 5.1 through 5.4 give insight in the effect of varying one parameter to the eigenvalues and eigenvectors of the system. However, it may be useful to study the effect of a combination of varying parameters as is done in the previous chapter. Different combinations of the varying parameters used in Figures 5.1 through 5.4 are displayed in Figures 5.5 through 5.7. In these figures, the first, second and third row display the results for the systems with the straight tangent tyre model, the Von Schlippe tyre model and the rigid ring tyre model respectively and the first and second column dispay the results for a low and high forward velocity of 5 and 25 m/s.

69

Figure 5.5. Unstable areas (grey) and stable areas (white) as function of yaw stiffness and caster

angle for the system with the three tyre models at two different forward velocities. Figure 5.5 shows a contour plot where the grey areas indicate an unstable system as a function of yaw stiffness and caster angle. Trends that can be seen from this figure are the larger stable areas of the system with the rigid ring tyre model as is already pointed out in this section and the increasing instability at higher velocities mostly for positive caster angles. As is derived in Chapter 4, the stability boundaries of the system with the straight tangent tyre model are independent of the forward velocity. Finally, the system with the Von Schlippe tyre model becomes more similar to the system with the straight tangent tyre model for higher forward velocities.

70

Figure 5.6. Unstable areas (grey) and stable areas (white) as function of lateral stiffness and caster

angle for the system with the three tyre models at two different forward velocities. The same trends can also be seen from Figure 5.6 where the stability is shown as a function of lateral stiffness and caster angle. First, the stability boundaries of the system with the straight tangent tyre model are independent on the forward velocity, unlike the other two models. Next, the system with the Von Schlippe tyre model becomes more similar to the system with the straight tangent tyre model at higher forward velocities. Finally, the majority of the unstable areas are located in the positive caster angle domain.

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Figure 5.7. Unstable areas (grey) and stable areas (white) as function of yaw- and lateral stiffness

for the system with the three tyre models at two different forward velocities. Finally, Figure 5.7 shows the stable and unstable areas as a function of the yaw- and lateral stiffness. From this figure, it can be seen that the system with the straight tangent tyre model is unstable for the majority of the two stiffness domains whereas the system with the rigid ring tyre model is stable for the majority of the two stiffness domains. The system with the Von Schlippe tyre model becomes more similar to the system with the straight tangent tyre model at higher forward velocities. Besides calculating the stability of the system, it can be useful to calculate the eigenfrequencies as a function of varying parameters. The most interesting plot is obtained when these parameters are the caster angle and yaw stiffness. Because there are two oscillatory modes representing the two degrees of freedom of the suspension, there exist two eigenfrequencies in any working point. With respect to the shimmy analysis in this report, the eigenfrequency with the least damping is the most interesting one.

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Figure 5.8. Contour plot of the eigenfrequency of the mode with the least damping for the system

with the straight tangent tyre model (V = 25 m/s).

Figure 5.9. Contour plot of the eigenfrequency of the mode with the least damping for the system

with the Von Schlippe tyre model (V = 25 m/s).

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Figure 5.10. Contour plot of the eigenfrequency of the mode with the least damping for the

system with the rigid ring tyre model (V = 25 m/s). Figures 5.8 through 5.10 show the eigenfrequencies of the mode with the least damping for the system with the straight tangent tyre model, the Von Schlippe tyre model and the rigid ring tyre model respectively. Here, the contour lines are plotted with a 0.5 Hz interval and the forward velocity is taken equal to 25 m/s. It can clearly be seen that there are switches in the modes where a sudden change of more than one level occurs. These switches are indicated with thick black lines. Roughly three areas can be distinguished in the contour plot where the frequency of different modes is plotted. The area in the lower right-hand corner and the area on the left-hand side for intermediate yaw stiffness display a slightly lower frequency with respect to the area in between. Another effect that can be seen from the contour plot is that the modes are not separated from each other by a continuous line for the system with every tyre model. The lines separating the areas are interrupted in some places where there is a smooth change in frequency. A closer look to what happens in these areas is displayed in Figures 5.11 and 5.12.

Figure 5.11 shows the frequency and damping of the two oscillatory modes of the system with the rigid ring tyre model with a yaw stiffness caψ equal to 1.30·105 Nm/rad. As can be seen from Figure 5.11, there is only one switch in mode when going through the entire caster angle range. This is confirmed by Figure 5.10 where mode 2 is the mode with the least damping for a negative caster angle.

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Figure 5.11. Frequency and damping of the oscillatory modes as a function of caster angle for the system with the rigid ring tyre model (V = 25 m/s, caψ = 1.30·105 Nm/rad).

When the yaw stiffness is increased slightly to 1.35·105 Nm/rad, the contour plot in Figure 5.10 shows two switches in modes when going through the caster angle range. This is confirmed by Figure 5.12 where the frequency and damping of the two oscillatory modes with a yaw stiffness caψ equal to 1.35·105 Nm/rad are plotted. For a value of the caster angle of approximately -16.5 degrees, there is an additional switch between modes with respect to Figure 5.11.

Figure 5.12. Frequency and damping of the oscillatory modes as a function of caster angle for the system with the rigid ring tyre model (V = 25 m/s, caψ = 1.35·104 Nm/rad).

When comparing Figures 5.11 and 5.12 with each other, the parts of the two lines at a caster angle value lower than approximately -16.5 degrees are switching from one mode to the other. This effect explains the interruptions in the lines that separate the modes in Figures 5.10 and 5.11.

5.3 Conclusions

This chapter focuses on a more extensive eigenvalue and eigenvector analysis of the front axle model derived in Chapter 4. This analysis includes the calculation of both the damping and frequencies of the two modes representing the two degrees of freedom of the suspension as a function of different varying parameters. The amplitude ratio and the relative phase angle between the yaw angle and lateral displacement of the wheel are also calculated from the

eigenvectors. The parameters used for variation are the caster angle ε, the yaw stiffness caψ, the

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lateral stiffness cay and the forward velocity V. Furthermore, the eigenvectors are calculated in order to be able to analyse the mode shapes. In this chapter, parameter values of a truck are used. From the differences between the tyre models, it can be seen that the front axle model with the rigid ring tyre model has the most stable areas. Furthermore, shimmy instability is most likely to occur at high forward velocities. It is also shown that almost no conclusions can be drawn with respect to the desirability of adjusting the parameters. The plots in this chapter show several maxima and minima in both the frequency and damping in the domains where the parameters are varied. Consequently, the stability of the system is dependent on the baseline set of parameter values when adjusting one or more parameters.

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6. Conclusions and Recommendations

6.1 Conclusions

The study presented in this thesis can be divided into two parts. The first part includes the comparison and validation of three different linear tyre models. These tyre models are the straight tangent tyre model, the Von Schlippe tyre model and the rigid ring tyre model. It is shown that the rigid ring tyre model has the most accurate responses to a variety of different inputs.

The first validation of the tyre models is done with step responses and they consist of a step in side slip angle, yaw angle and turn slip. For all three responses, measurements are available and are used to validate the tyre models. For the side slip input, differences between models and differences with the measurements are minimal except for an erroneous initial slope of the straight tangent tyre model in the self aligning moment response. Differences for the yaw input are larger and can be found in the initial values of both the lateral force and self aligning moment. For the lateral force, the straight tangent tyre model shows a negative initial value whereas the measurements and the other two tyre models do not show an initial peak. For the self aligning moment, the straight tangent tyre model shows a positive initial value and the measurements and the other two tyre models show a negative initial value. However, the initial value for the Von Schlippe tyre model, which is equal to the steady-state value, is too small and the initial value for the rigid ring tyre model is too large. The largest differences however can be found for the turn slip input. Here, differences occur in both the initial stage as well as the steady-state values. For the lateral force, the straight tangent tyre model is the only tyre model that does not show an initial slope equal to zero which can also be seen in the measurements. The steady-state values for the lateral force for all tyre models is too large where the straight tangent tyre models shows the largest difference with the measurements and the rigid ring tyre model the smallest difference. For the self aligning moment, the straight tangent tyre model shows a negative steady-state value and the Von Schlippe tyre model shows a steady-state value equal to zero. The measurement, as well as the rigid ring tyre model, shows a positive steady-state value.

The second validation of the tyre models is done with frequency responses and consists of the same inputs as are used for the step responses. However, measurements are only carried out for the yaw angle input and this input is therefore the only input that can be used to validate the tyre models. The frequency response functions are derived using three different forward velocities. From the validation, it is shown that the rigid ring tyre model is the only model capable of showing effects such as resonances within the tyre, gyroscopic effects and the turn slip effect which are visible in measurements. However, the rigid ring tyre model shows discrepancies at the dip in magnitude of the self aligning moment between 5 and 10 Hz and the magnitude at resonance frequencies.

The second part of this thesis focuses on the stability of the three tyre models attached to a simplified front axle system with a lateral and a yaw degree of freedom. In the first place, this is done for a passenger car. For the system with the straight tangent tyre model, the Hurwitz criterion is used to derive analytical expressions for the stability boundaries as function of system parameters. In these expressions, different offsets and damping constants are neglected in order to reduce the complexity of the expressions. From the Hurwitz criterion, it is shown that three stable areas exist for the front axle model with the straight tangent tyre model. Two of these stable areas are located in the negative caster angle domain. Because the caster angle is usually positive to increase the straight line stability of the vehicle, these stable areas are less interesting. Nevertheless, there exists one stable area in the positive caster angle domain. From the eigenvalues of the system with the straight tangent tyre model, it is shown that a low yaw

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stiffness should be avoided in the complete caster angle domain. Because of the increasing complexity of the Von Schlippe tyre model and the rigid ring tyre model, the analytical expressions for the system with these tyre models become too complex and therefore the eigenvalues are calculated numerically to analyse the stability of these systems as function of the system parameters. It is shown that the latter two systems show stability boundaries that are dependent on the forward velocity, unlike the system with the straight tangent tyre model. Differences between the systems with the Von Schlippe tyre model and the straight tangent tyre model disappear at high forward velocities. For low forward velocities however, the stability boundaries for the system with the Von Schlippe tyre model change drastically. Because of the turn slip effect, the system with the rigid ring tyre model is stable in larger areas than the other two systems. Secondly, the step is made to a truck front axle using the same models. The analysis for the truck consists of a more extensive analysis of the eigenvalues such as an analysis of the eigenfrequencies and damping of the system. Furthermore, the amplitude ratios and the relative phase angles of the two degrees of freedom of the suspension are calculated from the eigenvectors of the system. It is shown that these variables are not linearly dependent on the variation of parameters and several minima and maxima can be found in the plots.

It is shown in this thesis that the shimmy phenomenon is a complicated dynamic problem which is dependent on the complete set of parameters. As a result, conclusions with respect to an optimal value for one parameter with respect to stability cannot be given.

6.2 Recommendations

With respect to the shimmy research presented in this thesis, the following recommendations are made:

• Tyre parameters

Some of the tyre parameters are easier to obtain than others. The cornering stiffness and pneumatic trail for example are obtained from steady-state measurements. Much harder to asses are for example the stiffness and damping constants in the tyres. Another problem can arise when a parameter that can easily be measured has to be divided into several components. For example, the mass of the tyre has to be divided into a part that moves separately from the rim and a part that moves along with the rim. In this thesis, especially the parameters for a truck tyre can be improved because they are not validated using step and frequency response tests. Step and frequency response measurements and static tests provide a more accurate source for obtaining these parameters. The parameters for a passenger car are however validated and are likely to be much more accurate.

• Suspension parameters

As well as for the tyre parameters, some parameters describing the suspension are hard to obtain. Modal tests on a real truck are most likely to be more accurate regarding the shimmy analysis in this thesis.

• String-based tyre models

As has been pointed out by Pacejka [Pacejka; 2004], turn slip and gyroscopic effects can also be implemented for the string-based tyre models. He suggests that the yaw moment

because of turn slip can be calculated by multiplying the turn slip ϕ by a turn slip factor

78

κ*. Because the string is assumed to be massless, the yaw moment because of gyroscopic effects can be based on the time rate of change of the camber angle of the string and therefore on the time rate of change of the lateral force.

• Camber effects The vertical force acting on the tyre because of the mass of the tyre has a small component around the steering axis. However, because the steering angle is assumed to remain small, this vertical force is neglected. Furthermore, this effect is partly compensated by camber force and moments which are dependent on the camber angle of the tyre.

• Stability measurements

Unlike the validation of the tyre models, the stability results that are derived in Chapters 4 and 5 are not validated using measurements. An experimental set up where different parameters can be adjusted to obtain stability criteria is very costly and difficult to achieve. It does however allow a validation of the used front axle model.

• Augmenting the front axle model

This thesis provides a good insight of the accuracy of the tyre models. The front axle model is kept relatively simple in order to give insight in the effects of basic designing criteria to the stability. Augmenting the front axle model most likely gives more accurate results regarding shimmy stability. Additional features that can be included are for example:

o Adding an additional offset between the steering axis and the wheel centre in lateral direction.

o Adding a kingpin inclination angle. This inclination angle is defined as the angle between the steering axis and the wheel around the longitudinal axis.

o Make the parameters dependent on for example vertical load or forward velocity. o Adding an additional degree of freedom of the tyre in longitudinal direction. o Including the steering system. Stiffness and damping within the steering box can

be included. o Adding a second tyre with interaction between the tyres on the left and right-

hand side. o Adding non-linear effects such as free-play and friction of the kingpin.

• Finite element software and multibody software The suspension with two degrees of freedom is relatively simple in this research. Coupling the tyre models presented in this thesis to finite element- or multibody software possibly makes the complete front axle model more accurate. When using finite element- or multibody software however, the eigenvalues on which a large part of this research is based cannot be computed from the model. As a result, a parameter study is hard to carry out. Alternatively, the flexibility data representing the truck and chassis could be exported from the finite element model and included in a linear MATLAB or non-linear Simulink model.

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References J. Baumann, C.R. Barker and L.R. Koval, A nonlinear model for landing gear shimmy, ASME 91-WA-DSC-14, 1991 G. Becker, H. Fromm and H. Maruhn, Schwingungen in Automobillenkungen (“Shimmy”), M. Krayn, Berlin, 1931 I.J.M. Besselink, Shimmy of aircraft main landing gears, PhD thesis, Delft University of Technology, 2000 R.J. Black, Realistic evaluation of landing gear shimmy stabilization by test and analysis, SAE technical paper 760496, 1976 G. Broulhiet, The Suspension of the Automobile Steering Mechanism: Shimmy and Tramp, Bulletin 78, Societe des Ingenieurs Civils de France, July 1925 J. Glaser and G. Hrycko, Landing Gear Shimmy – De Havilland’s Experience, AGARD-R-800, March 1996 P. Hagedorn, Non-Linear Oscillations, second edition, Oxford Science Publications, Clarendon Press, Oxford, 1988, pp. 90-92 A. Higuchi, Transient response of tyres at large wheel slip and camber, dissertation, TU Delft, 1997 A. Kantrowitz, Stability of castering wheels for aircraft landing gears, NACA Rep. 686, 1937 M.A.M. Kluiters, An investigation into F-28 main gear vibrations, Fokker report X-28-430, September 1969 W. E. Krabacher, A Review of Aircraft Landing Gear Dynamics, AGARD-R-800, March 1996 A. de Kraker and D.H. van Campen, Mechanical Vibrations, Shaker Publishing, Maastricht, 2001 G.X. Li, Modelling and analysis of a dual-wheel nose gear: shimmy instability and impact motions, SAE technical paper 931402, 1993 J.P. Maurice, Short wavelength and dynamic tyre behaviour under lateral and combined slip conditions, dissertation, TU Delft, 2000 W. J. Moreland, Landing Gear Vibration, AF Technical Report No. 6590, October 1951 H.B. Pacejka, The wheel shimmy phenomenon, Doctoral Thesis, Delft University of Technology, December 1966 H.B. Pacejka, Tyre and vehicle dynamics, Butterworth-Heinemann, Oxford, 2004

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I. J. Pritchard, An overview of landing gear dynamics, NASA Technical Reports, NASA/TM-1999-209143, Hampton, Virginia, 1999 L.C. Rogers, Theoretical tire equations for shimmy and other dynamic studies, AIAA Journal of Aircraft, Vol.9, No.8, August 1972, pp. 585-589 B. von Schlippe and R. Dietrich, Das Flattern eines bepneuten Rades, Bericht 140 der Lilienthal Gesellschaft, 1941, English translation: NACA TM 1365, 1954 L. Segel, Force and moment response of pneumatic tyres to lateral motion inputs, Transactions of ASME, Journal of Engineering for Industry, Paper No. 65-AV-2, 1966 R.F. Smiley, Correlation and extension of linearised theories for tire motion and wheel shimmy, NACA report 1299, 1957 G. Somieski, Shimmy analysis of a simple aircraft nose landing gear model using different mathematical methods, Aerospace Science and Technology, 1270-9638 (8), 1997 P. Thota, B. Krauskopf and M. Lowenberg, Shimmy in a nonlinear model of an aircraft nose landing gear with non-zero rake angle, paper in the 6th European Nonlinear Dynamics Conference (ENOC 2008), Saint Petersburg, 2008 P.W.A. Zegelaar, The dynamic response of tyres to brake torque variations and road unevennesses, dissertation, TU Delft, 1998

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A. Padé versus Taylor Approximation

The time delay between the lateral position of the tyre string at the leading contact point y1 and trailing contact point y2 of the tyre can be described by:

( ) ( )2 1y t y t τ= − (A.1)

In this equation, t represents the time and τ represents the time delay. When a Laplace transformation is performed, the following equation is obtained:

( ) ( )2 1

sY s e Y s

τ−= (A.2)

Therefore, the transfer function between the lateral position of the leading and trailing contact point of the tyre is:

( )( )( )

2

2, 1

1

s

y y

Y sH s e

Y s

τ−= = (A.3)

An approximation of (A.3) using a second order Taylor series can be described by:

2 212

1se s s

τ τ τ− = − + (A.4)

An approximation of (A.3) using a second order Padé approximant can be described by:

2 21 1

12 2

2 21 112 2

1

1

s s se

s s

τ τ τ

τ τ− − +

=+ +

(A.5)

The transfer functions describing a Taylor and Padé approximation can be used for comparison using different time delays. Given the delay described in (2.7) and half of the contact length equal to 0.0488 m (Table 3.1), several time delays can be computed depending on the forward velocity. Figures A.1 and A.2 show Bode plots of the exact transfer function and the two approximations at low and high forward velocity respectively. Figures A.1 and A.2 confirm the conclusions that are drawn by Kluiters [Kluiters; 1969]: the Padé approximation yields a better result at low forward velocities.

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Figure A.1. Approximations of the time delay (V = 5 km/h).

Figure A.2. Approximations of the time delay (V = 50 km/h).

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B. Equations of Motion for the Rigid Ring Tyre Model

The derivation of the equations of motion for the rigid ring tyre model can be made using Lagrange’s theorem. Because of the gyroscopic effects, the non-linear effects have to be taken into account when computing the kinetic energy. In order to compute the kinetic energy of the

belt, the angular velocity vector b

ω

is needed and can be described by:

( ) ( ) ( )

1 2 3

1 2 3 0 0 0 0 0 0b b b b b

x x x

z x y y y y

z z z

ω ψ γ ω ψ γ ω

= + + = + +

ɺ ɺ ɺ ɺ ɺ ɺ

(B.1)

where vectors x

, y

and z

and angles ψ, γ and ω are displayed in Figure 2.5. The subscript b

indicates the belt and the superscript 1 is the notation used for the base frame as indicated in Figure 2.5. This base frame moves along with the wheel centre with velocity V and does not rotate in any direction. The second frame indicated with the superscript 2 can be obtained by rotating the base frame over an angle ψ, the third frame indicated with the superscript 3 can be obtained by rotating the second frame over and angle γ and the fourth frame indicated with the

superscript 4 can be obtained by rotating the third frame over an angle ω, where ω = Ωɺ . As a

result, the angular velocity vector can be written as:

( ) ( ) ( )( )

3

12 23 230 0 0 0 0 0b b b

x

y

z

A A Aω ψ γ

= + + Ω

ɺ ɺ

(B.2)

with

1 2 3

12 12 23

x x x

y y y

z z z

A A A

= =

(B.3)

and the transformation matrices A12 and A23 equal to:

12 23

cos sin 0 1 0 0

sin cos 0 , 0 cos sin

0 0 1 0 sin cos

b b

b b b b

b b

ψ ψ

ψ ψ γ γ

γ γ

−

= = −

A A (B.4)

The expression for the angular velocity vector can therefore be rearranged:

( )

3

sin cosb b b b b b

x

y

z

ω γ ψ γ ψ γ

= + Ω

ɺ ɺ ɺ

(B.5)

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Next, the angular momentumbCM

H

of the belt with respect to the centre of mass can be computed

according to:

bCM bCM b

H I ω=

(B.6)

where IbCM is the inertia tensor of the belt and is equal to:

3 3 3 3 3 3

bCM b b bI x x I y y I z zI γ ω ψ= + +

(B.7)

Here, I represents the moment of inertia and the subscripts γ, ω and ψ indicate the rotational components around the x-, y- and z-axes respectively. Substituting (B.5) and (B.7) in (B.6) yields the following expression for the angular momentum of the belt:

( )( )

3

sin cosbCM b b b b b b b b

x

H I I I y

z

γ ω ψγ ψ γ ψ γ

= + Ω

ɺ ɺ ɺ

(B.8)

The kinetic energy T consists of a translation and rotational component and can be computed according to:

( ) ( )

( )

2

2 22 2

2 2 2 2 2 2 2

1 1

2 2

1 1 1 1sin cos

2 2 2 2

1 1 1 1sin 2 sin cos

2 2 2 2

b b b CM

b b b b b b b b b b

b b b b b b b b b b b b

T m y H

m y I I I

m y I I I

γ ω ψ

γ ω ψ

ω

γ ψ γ ψ γ

γ ψ γ ψ γ ψ γ

= + ⋅

= + + + Ω +

= + + + Ω + Ω +

ɺ

ɺ ɺ ɺɺ

ɺ ɺ ɺ ɺɺ

(B.9)

Here, y is equal to the lateral position and m is equal to the mass. For the application in Lagrange’s equations, several derivatives of the kinetic energy should be computed. Because the belt has only three degrees of freedom with respect to the rim, the generalised coordinates q can be described with:

b

b

b

y

ψ

γ

=

q (B.10)

The necessary derivatives of the kinetic energy for Lagrange’s equations therefore become:

( )( )( )( )

( ) ( )2

2 2

,

2 2

,

, ,

0 0 sin cos cos sin cos

sin sin cos

b b b b b b b b b b

b b b b b b b b b b b

b q b b

T I I

T m y I I I

d dT my T I

dt dt

ω ψ

ω ψ γ

γ

ψ γ γ ψ γ ψ γ γ

ψ γ γ ψ γ γ

γ

= + Ω −

= + Ω +

=

q

q

q

ɺ

ɺ ɺ

ɺ ɺ ɺ

ɺ ɺ ɺɺ

ɺɺɺɺ

(B.11)

85

with:

( ) ( )

( )

2

2

,

2

sin 2 sin cos sin cos

cos 2 sin cos

q b b b b b b b b b b

b b b b b b b

dT I

dt

I

ω

ψ

ψ γ ψ γ γ γ γ γ γ

ψ γ ψ γ γ γ

= + + Ω + Ω

+ −

ɺɺɺɺ ɺ ɺ ɺ

ɺɺ ɺ ɺ

(B.12)

Because the system is eventually transformed into the state space form, it is assumed that the camber angle is small. Furthermore, the forward velocity and therefore the rotational speed of the wheel are assumed to be constant. This leads to the following assumptions:

sin

cos 1

0

b b

b

γ γ

γ

=

=

Ω = ɺ

(B.13)

When substituting (B.12) and (B.13) in (B.11) and neglecting the third order components, the derivatives can be simplified to:

( )

( ) ( )

,

,

0 0 b b

b b b b b b b b

T I

dT m y I I I

dt

ω

ω ψ γ

ψ

γ ψ γ

= Ω

= Ω +

q

qɺ

ɺ

ɺ ɺɺ ɺɺɺɺ (B.14)

Next, the potential energy of the belt with respect to the rim can be computed according to:

( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( )

2 2 2

2 2

2 2 2 2

2 2 2 2

2 2 2 2

1 1 1

2 2 2

1 1

2 2

1 12 2

2 2

1 1 2 2

2 2

1

2

by b r b b r b b r

cy c b b c c b

by b r b r b b r b r

b b r b r c c b b c

cy c b b

U c y y c c

c y y R c

c y y y y c

c c

c y y R

ψ γ

ψ

ψ

γ ψ

ψ ψ γ γ

γ ψ ψ

ψ ψ ψ ψ

γ γ γ γ ψ ψ ψ ψ

γ

= − + − + −

+ − − + −

= + − + + −

+ + − + + −

+ + +( )2 2 2c b c b b b

y y y R y Rγ γ

− − +

(B.15)

Here, cb and cc indicate the stiffness constant between rim-belt and belt-contact patch respectively. The subscript y indicates the translational component in the y-direction and the subscripts r and c indicate the rim and contact patch respectively. The potential energy with respect to the height of the centre of gravity of the belt is neglected because of the small camber angle involved. The necessary derivative of the potential energy for Langrange’s equation is:

( ) ( )( ) ( )

( ) ( ),

by b r cy c b b

b b r c c b

b b r cy c b b

c y y c y y R

U c c

c c R y y R

ψ ψ

γ

γ

ψ ψ ψ ψ

γ γ γ

− − − −

= − − − − + − + +

q (B.16)

86

Finally, the virtual work δW done by the external tyre forces and damping can be described by:

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

by b r b cy c b b b

b b r b c c b b b b r b

b b r b cy c b b b b b r b

W k y y y k y y R y

k k k

k Rk y y R k

ψ ψ ψ

γ γ

δ δ γ δ

ψ ψ δψ ψ ψ δψ γ γ δψ

γ γ δγ γ δγ ψ ψ δγ

= − − + − −

− − + − − Ω −

− − + − − + Ω −

ɺɺ ɺ ɺ ɺ

ɺ ɺ ɺ ɺ

ɺ ɺ ɺɺ ɺ

(B.17)

In this equation, kb indicates the damping constant between rim and belt and R represents the tyre radius. δ indicates a virtual displacement or rotation. The two damping terms that are dependent

on the angular velocity of the wheel Ω are caused by the relative angle between the rim and belt. A schematic representation of this is given in Figure B.1. In this figure a quarter of both the rim

and belt are displayed. When there exists a difference in yaw angle between the rim and belt δψ,

a damper between the rim and belt that moves in the direction of the angular velocity Ω is constantly dissipating energy. A difference in yaw angle between rim and belt results in a damping moment around the longitudinal axis. The same applies to a difference in camber angles between the rim and belt which causes a damping moment around the vertical axis.

Figure B.1. Damping forces as a result of a difference in yaw angle between rim and belt. In order to derive the equations of motion, the Lagrangian equation can be used and can be described by:

( )1,2,3i

i i i

d T T UQ i

dt q q q

∂ ∂ ∂− + = =

∂ ∂ ∂ɺ (B.18)

where Qi denotes the generalised force determined by the equation:

2

1

i i

i

W Q qδ δ=

=∑ (B.19)

Combining (B.17) and (B.19) yields expressions for the generalised forces:

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1

2

3

by b r cy c b b

b b r c c b b b r

b b r cy c b b b b r

Q k y y k y y R

Q k k k

Q k Rk y y R k

ψ ψ ψ

γ γ

γ

ψ ψ ψ ψ γ γ

γ γ γ ψ ψ

= − − + − −

= − − + − − Ω −

= − − + − − + Ω −

ɺɺ ɺ ɺ ɺ

ɺ ɺ ɺ ɺ

ɺ ɺ ɺɺ ɺ

(B.20)

87

The equations of motion for the belt therefore become:

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( )

b b by b r cy c b b

by b r cy c b b

b b b b b b r c c b

b b r b b r c c b

b b b b b b r cy c b b

b

m y k y y k y y R

c y y c y y R

I I k k

k c c

I I k Rk y y R

k

ψ ω ψ ψ

ψ ψ ψ

γ ω γ

γ

γ

γ

ψ γ ψ ψ ψ ψ

γ γ ψ ψ ψ ψ

γ ψ γ γ γ

= − − + − −

− − + − −

= − Ω − − + −

− Ω − − − + −

= Ω − − + − −

+ Ω

ɺɺɺ ɺ ɺ ɺ ɺ

ɺɺ ɺ ɺ ɺ ɺ ɺ

ɺɺ ɺ ɺ ɺ ɺɺ ɺ

( ) ( ) ( )b r b b r cy c b bc Rc y y Rγψ ψ γ γ γ

− − − + − −

(B.21)

88

C. Optimization of Parameters for the Rigid Ring Tyre Model

Parameters that can be used for the rigid ring tyre model are obtained from [Maurice; 2000]. Because these parameters are based on a rigid ring tyre model without turn slip, the parameters that are used may be inaccurate for the rigid ring tyre model described in this thesis. As a result, the parameters can be improved using the optimization toolbox of Matlab. This toolbox contains a function fmincon that finds a constrained minimum of an objective function of several variables. The objective function chosen for this optimization is based on the complex values obtained from the frequency responses with pure yaw input. An optimization based on the magnitude and phase is not suitable here because of the of the phase wrap in degrees to the interval [-180, 180]. In total, twelve vectors are optimised which consist of the real and imaginary values of both the lateral force and self aligning moment at three forward velocities. The objective function f used for each of the twelve vectors is given by:

( ) ( )( )

( )( ) ( )( )

( )

2 2

n,measurement n,model n,measurement n,model

1 1n,measurement n,measurement

Re Re Im Im

Re 10 Im 10

k k

n n

fλ λ λ λ

λ λ= =

− − = + + +

∑ ∑ (C.1)

where λn is the nth eigenvalue, k is the number of frequency points that are used and Re and Im indicate the real and imaginary part respectively of the eigenvalues. As can be seen from (C.1), a combination of the relative and absolute difference is used. Furthermore, an extra constraint is added to the mass of the wheel. Because the mass of the wheel can be determined accurately, this value is kept constant. However, the division of the total mass and moment of inertia between the belt and rim is based on a rule of thumb and may therefore be inaccurate. The constraint used here is given as:

9.3

0.391

0.391

0.736

r b

r b

r b

r b

m m

I I

I I

I I

ψ ψ

γ γ

ω ω

+ =

+ =

+ = + =

(C.2)

The range in which most of the parameters are varied is between 75% and 125% of the original value. As said before, especially the damping constants between rim and belt may be inaccurate due to the introduction of turn slip. The range for these parameters is therefore increased to between 1% and 100%. Finally, some parameters are fixed in order to reduce computation times. These include the parameters for the contact patch indicated with the subscript c, the cornering stiffness, the pneumatic trail, the turn slip stiffnesses, the radius of the tyre and the effective rolling radius of the tyre. The parameters for the contact patch are fixed because they only influence the Bode plots for higher frequencies which are not relevant for the scope of this thesis. The cornering stiffness, pneumatic trail and turn slip stiffnesses do not have to be optimised because they can be obtained from steady-state values in the measurements. The radius and effective rolling radius of the tyre finally can be measured accurately. The original parameter values and the adjusted values are shown in Table C.1. It can be seen from this table that many parameter values are on the upper or lower limit of the variation range. An increase in this range results in more accurate Bode plots, but can result in unrealistic parameter values. Only the variation range for the damping constants is taken larger because unlike the other parameters, damping is very hard to identify accurately.

89

Table C.1 Optimization of parameters

Parameter Original value

Adjusted value

Unit Variation range

% of original value

a, σc 6.50e-2 4.88e-2 m 75%-125% 75% cby 5.11e5 6.39e5 N/m 75%-125% 125% cbγ, cbψ 2.55e4 2.02e4 Nm/rad 75%-125% 79% ccy 8.00e5 8.00e5 N/m fixed 100% ccψ 1.00e4 1.00e4 Nm/rad fixed 100% CcFα 9.30e4 9.30e4 N/rad fixed 100% CcFφ 8.80e2 8.80e2 N/rad fixed 100% CcMφ 1.77e2 1.77e2 Nm/rad fixed 100% Ibγ, Ibψ 3.26e-1 3.21e-1 kgm2 75%-125% 99% Ibω 6.36e-1 6.61e-1 kgm2 75%-125% 104% Icψ 1.00e-2 1.00e-2 kgm2 fixed 100% Irγ, Irψ 6.50e-2 6.98e-2 kgm2 75%-125% 107% Irω 1.00e-1 7.50e-2 kgm2 75%-125% 75% kby 3.12e2 7.68e1 Ns/m 1%-100% 25% kbγ, kbψ 4.00 4.00 Nms/rad 1%-100% 100% kcy 1.00e2 1.00e2 Ns/m fixed 100% kcψ 3.00 3.00 Nms/rad fixed 100% mb 7.10 7.64 kg 75%-125% 108% mc 1.00 1.00 kg fixed 100% mr 2.20 1.66 kg 75%-125% 75% R 2.80e-1 2.80e-1 m fixed 100% Re 3.00e-1 3.00e-1 m fixed 100% tp 3.00e-2 3.00e-2 m fixed 100%

90

D. System Matrix of the Front Axle with the Straight Tangent

Tyre Model The system matrix of the front axle model with the straight tangent tyre model is given in (A.1) through (A.3).

( ) ( )

2 2

2 2

2 2

2 2

2 2

00

00

coscos

coscos:,1 ; :,2

coscos

coscos

0

ww w a

ay t w a w

t w a w

t

away t w a w

t w a w

m NI m N c

c m I m m Nm I m m N

mcm N

c m I m m Nm I m m N

V

ψ ψ

ψψ

ψ

ψψ

εε

εε

εε

εε

σ

+ − − + += = − − + +

A A (D.1)

( ) ( )

( )

2 2

2 22 2

2 22 2

1 0

0 1

coscos

coscos:,3 ; :,4

cos

coscos

tan cos1

ww waay

t w a wt w a w

twaay

t w a wt w a w

m NI m Nkk

m I m m Nm I m m N

mm Nkk

m I m m Nm I m m N

R n a

ψψ

ψψ

ψ

ψψ

εε

εε

εεε

ε ε

σσ

+

− −++ = =

−−+ +

+ − −

A A (D.2)

( )

( )

( )( )

2 2 2

2 2

2 2

0

0

cos cos tan

cos:,5

tan cos

cos

w w w p

F

t w a w

w t p

F

t w a w

I m N m N R n tC

m I m m N

m N m R n tC

m I m m N

V

ψ

α

ψ

α

ψ

ε ε ε

ε

ε ε

ε

σ

+ − + + +

= − + +

+ −

A (D.3)

91

E. System Matrix of the Front Axle with the Rigid Ring Tyre

Model The state-space form ( xɺ = Ax) of the system with the rigid ring tyre model is given as:

1 2

3 4 5

6 7

d

dt α α

ϕ ϕ

=

-1 -1

q q0 I 0 0

q q-M C -M K A A

q qA A A 0

q q0 A 0 A

ɺ ɺ

(E.1)

where the state vectors q, qα and qϕ are given by:

2

1

2

; ;

a

a

c

b

F

b

t

b

c

c

y

y

y

α ϕ

ψϕ

ϕαγ

α ϕψ

ϕ

ψ

′ ′′ = = = ′ ′ ′

q q q (E.2)

The mass matrix M, the damping matrix K and the stiffness matrix C are given in (E.3) through (E.8).

( )

( )2 2

cos 0 0 0 0 0

cos cos 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

a r r

r r r

b

b

b

c

c

m m m N

m N m N I

m

I

I

m

I

ψ

γ

ψ

ψ

ε

ε ε

+ −

− + =

M (E.3)

( )( )

( )

2 2

cos

coscoscos

cos

(:,1) ; (:,2) ; (:,3)0 sin

0 0cos

0 0

0 00

by byay by

bya by bby

by cyby by

cyb

b

cy

k n kk k

k nk k n kk n

k kk k n

k Rk

k

k

ψ ψ

γ

ψ

ε

εεε

ε

ε

ε

− − +

+ +− +−

= = =

− −

K K K (E.4)

92

( )( )

2

00 00

cossin 00

0 0

0(:,4) ; (:,5) ; (:,6) ; (:,7)

0

00

00

bb

cy cy

bb cy cy

b c cb

cycy

cc

kk

k R k

Ik k R k R

k k kI

kk R

kk

ψψ

ωγ

ψ ψ ψω

ψψ

εε

− − − Ω= = = =+ − + −Ω − −

K K K K

(E.5)

( )( )

( )( )

( )

2 2

cos

cos coscoscos

(:,1) ; (:,2) ; (:,3)cos sin0

0 0sin cos

0 00 00

bybyay by

a by b byby

by by cyby

b b cy

b b

cy

c n cc cc c n c c nc n

c n c cc

k c c R

k c

c

ψ ψ

γ γ

ψ ψ

ε

ε εεε

ε ε

ε ε

− − + + + − +− = = =Ω + Ω − −

C C C

(E.6)

( )

( )

( )

( )

2

00

cos cossin cos

0

(:,4) ; (:,5)

0

0

b bb b

cy

bb cy

b cb

cy

c

c kc k

c R

kc c R

c ck

c R

c

ψ ψψ ψ

γγ

ψ ψψ

ψ

ε εε ε

− + Ω− Ω − Ω= =+

+ Ω

− −

C C (E.7)

00

00

0

0(:,6) ; (:,7)

0

0

0

cy

cy

c

cy

c

c

c R

c

c

c

ψ

ψ

−

= =− −

C C (E.8)

Finally, the matrices A1 through A7 are displayed in (E.9) through (E.14).

93

( )1 2

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

2 20 0 0

2 4 40

cF cFcF

c c c

cF p cF p cM cM cM

c c c c c

C CC

m m m

C t C t C C C

I I I I I

ϕ ϕα

α α ϕ ϕ ϕ

ψ ψ ψ ψ ψ

=

−

− − −

A A (E.9)

3

0 0 0 0 0 0

0 0 0 0 0 0

c

t

V

V

σ

σ

=

A (E.10)

4

10 0 0 0 0 0

10 0 0 0 0 0

c

t

σ

σ

−

=

−

A (E.11)

5

0

0

c

t

V

V

σ

σ

− =

−

A (E.12)

2

6

1

2

10 0 0 0 0 0

10 0 0 0 0 0

10 0 0 0 0 0

10 0 0 0 0 0

c

F

ϕ

ϕ

σ

σ

σ

σ

−

− =

− −

A (E.13)

94

2

7

1

2

0 0 0

0 0 0

0 0 0

0 0 0

c

F

V

V

V

V

ϕ

ϕ

σ

σ

σ

σ

−

− =

− −

A (E.14)

a comparison of dynamic tyre models for vehicle shimmy ... · with the derivation of these tyre...

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