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Machines 2022, 10, 801. https://doi.org/10.3390/machines10090801 www.mdpi.com/journal/machines

Article

A Novel Numerical Method for Theoretical Tire

Model Simulation

Qianjin Liu, Dang Lu *, Yao Ma and Danhua Xia

State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun 130025, China

* Correspondence: ludang@jlu.edu.cn

Abstract: Theoretical tire models are often used in tire dynamics analysis and tire design. In the

past, scholars have carried out a lot of research on theoretical model modeling; however, little pro-

gress has been made on its solution. This paper focuses on the numerical solution of the theoretical

model. New force and moment calculation matrix equations are constructed, and different iterative

methods are compared. The results show that the modified Richardson iteration method proposed

in this paper has the best convergence-stability in the steady and unsteady state calculation, which

mathematically solves the problem of nonconvergence of discrete theoretical models in the pub-

lished reference. A novel discrete method for solving the total deformation of tires is established

based on the Euler method. The unsteady characteristics of tire models are only related to the path

frequency without changing its parameters, so the unsteady state ability of the tire model can be

judged based on this condition. It shows that the method in the references have significant differ-

ences at different speeds with the same path frequency under turn slip or load variations input, but

the method proposed in this paper has good results.

Keywords: tire model; theoretical; numerical solution; discrete solution; iteration method;

load variations; turn slip; Euler method

1. Introduction

For tire and vehicle dynamics simulations and analyses, different types of tire models

have been developed and categorized as empirical and theoretical models [1,2]. The em-

pirical model, or combined theoretical and empirical model, is used for vehicle dynamics,

and its model parameters are obtained by fitting measurements [2], including the

MF/PAC2002 [2–5], TMeasy [6–8], UniTire [9,10], Hankook-Tire [11], TameTire [12,13],

MF-Swift [14,15], FTire [16,17], CDTire [18,19] and RMOD-K [20,21] models, etc.

However, compared with the theoretical tire model, the empirical model is not suit-

able for the mechanical analysis of tire dynamics and tire design due to having too many

model parameters and being dependent on experimental data. Tire design engineers

should find the optimal design parameters that can satisfy the engineering requirements

simultaneously and, for more efficient product development, a reliable theoretical model

is essential because it enables systematic parameter study and design optimization. The

theoretical models have advantages over the finite-element models in terms of computa-

tional efficiency, and they can be incorporated into the parameter study and optimization

framework more easily [22]. Furthermore, combined with the static simulation data of the

finite-element, using the theoretical model to achieve the steady and unsteady simulation

should be a good prospect for future tire virtual development.

Citation: Liu, Q.; Lu, D.; Ma, Y.;

Xia, D. A Novel Numerical Method

for Theoretical Tire Model

Simulation. Machines 2022, 10, 801.

https://doi.org/10.3390/

machines10090801

Academic Editor: Dimitrios E.

Manolakos

Received: 17 August 2022

Accepted: 8 September 2022

Published: 11 September 2022

Publisher’s Note: MDPI stays neu-

tral with regard to jurisdictional

claims in published maps and institu-

tional affiliations.

Copyright: © 2022 by the authors. Li-

censee MDPI, Basel, Switzerland.

This article is an open access article

distributed under the terms and con-

Machines 2022, 10, 801 2 of 33

Fromm and Julien [3,23] proposed the brush tire model, which laid the foundation

for tread modeling and tire theoretical analysis. The beam [1,3,24–36] and string models

[3] are two classical belt/carcass modeling methods. Finally, the latest beam model in-

cludes belt tension, flexural rigidity, shear rigidity [1,22,25,26] and lateral shear force dis-

tribution [1], and the relationship between the parameters of the beam model and the de-

sign parameters of belts is established [1,22].

H. Sakai [28] pointed out that the previous practice of assuming the shape of the con-

tact-patch as rectangular, the pressure distribution as lateral uniform and the longitudinal

parabolic distribution does not reflect the reality; therefore, he studied the influence of

different load, inflation pressure, camber, side slip angle and lateral force on the contact

length in detail, and established a semi-empirical model of contact length with different

rib along the width. Through the study of the distribution of contact patch-pressure on

each rib, the empirical formula of contact patch pressure on each rib with load was estab-

lished [29]. Guo Konghui established the general function of pressure distribution based

on a rectangular contact patch in which the convexity, uniformity and fore-aft shift of

pressure-distribution were considered [37–39]. Patrick Gruber, Robin S. Sharp and An-

drew D studied normal and shear forces in the contact based on the finite-element model

and established a 2D semi-empirical contact patch, considering longitudinal grooves as a

function of vertical load and camber angle [40]. Ch. Oertel established the geometric rela-

tionship between the contact patch and vertical load, camber angle and tread profile cur-

vature [41]; however, the contact patch length calculated by this method will be larger

than the actual one, and the contact patch area caused by camber will be larger. Nakajima,

Y. and Hidano, S. proposed two-dimensional contact patch considering that shear force,

contact shape and contact pressure distribution are changed in the fore–aft and lateral

direction caused by the camber, fore–aft and lateral external force [1,42], and the defi-

ciency is that the contact patch shape is simply expressed as a rectangle or trapezoid.

Analytical [3,39,43–50] and numerical methods are usually used to solve theoretical

models. The advantage of the former is high efficiency, but it is only suitable for simple

theoretical models. With the increase of the complexity of the theoretical model, the nu-

merical method becomes more and more important. TreadSim [3] is a typical representa-

tive for the steady-state numerical solution; however, the iteration process turned out to

be unstable for certain combinations of vertical load, belt yaw and belt bend stiffness [51],

and the method proposed in [51] does not actually solve the problem essentially by itera-

tion. For unsteady-state numerical solutions, there are two different (but essentially the

same) methods that are both based on the Lagrange description. An element that leaves

the contact patch at one side is redefined in the model so that it will enter the contact patch

at the opposite side. After a new element has entered the contact patch, the elements are

also renumbered so that element number one is always the first element in the contact

patch, and when the traveling distance in an increment time is larger than the distance

between adjacent tread elements, a linear-interpolation method is used [52–55]. However,

this method has some problems under the condition of turn slip or load variations inputs,

which will be discussed in this paper. There are many methods for the iterative solution

of matrix equations, including the traditional [56,57] and the latest [58,59], and different

methods are suited for different application scenarios. For dynamic equations of tire mod-

els which usually consider mass, Newton’s method and the Newmark-beta method are

always used [60,61]. However, in this paper, we do not intend to focus on the develop-

ment of new iterative methods, but to reconstruct the matrix equations that can meet the

application requirements of different iterative methods and compare several common it-

erative methods to select the most suitable for the discrete theoretical model and make

appropriate optimizations.

ditions of the Creative Commons At-

tribution (CC BY) license (https://cre-

ativecommons.org/licenses/by/4.0/).

Machines 2022, 10, 801 3 of 33

In this paper, a novel matrix equation for the force and moment calculation is con-

structed and different iterative methods are compared under steady-state constant side

slip angle and unsteady-state conditions, including the step side slip, sine side slip, load

variations at constant side slip, sine side slip angle at constant turn slip and sine longitu-

dinal slip at constant side slip inputs. It shows that only the modified Richardson method

proposed in this paper can complete all of the simulations without any convergence. A

novel, discrete solution method for tire total deformation is proposed based on the Euler

method, and time derivatives are replaced by spatial gradients. Furthermore , based on

the condition of the unsteady characteristics of tire models only relating to the path fre-

quency without changing their parameters, the discrete solution method proposed is com-

pared with reference [52–55] under different unsteady-state conditions, including the step

side slip angle, sine side slip angle, load variations at constant side slip angle, step turn

slip simulation, sine longitudinal combined with a constant side slip, sine turn combined

with a constant side slip and load variations at constant longitudinal and side slip inputs.

It shows that the simulation results have little difference by using the proposed method,

but the reference method has a significant difference under turn slip or load variations

inputs at different speed or, more precisely, the ratio between the distance of the contact

patch center traveling in space within the time interval and distance between adjacent

tread elements in the contact patch. However, this paper only studies the numerical solu-

tion method of the theoretical model in detail, and the results of the comparison between

the model and the experiment show that a refined contact patch model considering cam-

ber and the influence of lateral force and longitudinal force on the contact patch model

demonstrate the need for more in-depth research on the dynamic friction model, consid-

ering that the influences of contact pressure and slip velocity are added to the theoretical

model.

The sections of this paper are arranged as follows. In Section 2, the tire total defor-

mation discrete solution method based on the Euler method will be given. In Section 3,

the modeling method of belt/carcass deformation, contact patch, stress direction in the

sliding region and anisotropic of tread stiffness is described. In Section 4, according to the

adhesion and sliding region, the matrix equations for the calculation of force and torque

are established, and the advantages and disadvantages of different iterative methods are

compared. In Section 5, the theoretical model is compared with the experiment to verify

the rationality of the theoretical model parameters, and based on the principle that the

unsteady state only depends on the path frequency, the discrete solution method pro-

posed in this paper is compared with reference [52,53] unsteady-state conditions. Finally,

the discussion and conclusions will be given in Section 6.

2. Tire Total Deformation Discrete Solution Method Based on the Euler Method

The total deformation of tires caused by friction consists of tread and carcass. When

the tire deformation is described, two coordinate systems are used; one is the road coor-

dinate system (, ,,)OXYZ and the other is the contact patch coordinate system

(,,,)Cxyz, named as

I

SO W−. When the tire slips relative to the road surface, the fric-

tion forces makes the tire deform (,)uv relative to the contact patch coordinates (, )

x

y,

as shown in Figure 1. P is the contact point at undeformed state and p is the vector of

P relative to the (, ,,)OXYZ system. C represents the contact patch center and c

is the vector C relative to the (, ,, )OXYZ system. (, )

x

y represents the contact point

vector at undeformed state, (,)uv represents the total tire deformation vector and q is

the vector of P relative to C.

α

represents the sideslip angle and

ψ

represents the

yaw angle of the

x

-axis with respect to the

X

-axis.

Machines 2022, 10, 801 4 of 33

Figure 1. Top view of tire contact area and its deformations

(,)uv

with respect to

(,,,)Cxyz

.

The position of the tread location relative to

(, ,,)OXYZ

can be expressed as the

equation:

OP OC CP=+

or

p=c+q

(1)

Herein,

()()

y

x

uyv=+ ++

x

qe e

. So, there is the slip velocity of the tread relative

to the road:

()() {()()}

g

xu yv xu yv

ψ

==+= + + + + + + − +

cxy yx

VpcqV e e e e

(2)

where

c

V is the contact patch center velocity and

x

e and

y

e

are the

(,,,)Cxyz

base

vector. When there is no relative slip between the tread and the road surface,

g

V=0

,

then

()

sx

uV yv

ψ

=− + ⋅ +

(3)

()

cy

vV xu

ψ

=− − ⋅ +

(4)

Machines 2022, 10, 801 5 of 33

Herein, er

dx

x

RV

dt

==−Ω=

, sx cx

VVx=+

, 0y=

, expression of Equations (3)

and (4) by the Euler method

()

sx

udx udy udS

uVyv

xdt ydt Sdt

ψ

∂∂∂

=⋅+⋅+⋅=−+⋅+

∂∂∂

(5)

()

cy

vdx vdy vdS

vVxu

xdt ydt S dt

ψ

∂∂∂

=⋅+⋅+⋅ =−−⋅+

∂∂∂

(6)

Then, there are:

()

c

x

r

V

uu Syv

xSV

ϕ

∂∂

+⋅=−+⋅+

∂∂ (7)

()

c

x

r

V

uu Syv

xSV

ϕ

∂∂

+⋅=−+⋅+

∂∂ (8)

Herein, c

dS V

dt =, 1

sx

x

r

V

SV

κ

κ

==

+, 11

tan

11

cy cy

y

rx

VV

SVV

α

κκ

==− ⋅=− ⋅

++

,

r

V

ψ

ϕ

=

, and

κ

is practical slip,

α

is side slip angle,

x

S is the longitudinal theoretical

slip, y

S is the lateral theoretical slip and

ϕ

is the turn theoretical slip. Equations (7) and

(8) are only related to the coordinates of the contact patch (, )

x

y and the space position

of the contact patch center. The expression of Equations (7) and (8) in a discrete method:

1 11 1 1 1 11 11

11

1111

{}(, ){}(, ){}(, ){}(, )

[{}(,)]

kij kij kij kij c

ii k k r

xjkij

uS x y uS x y uS x y uS x y V

x

xSSV

SyvSxy

ϕ

+ ++ + + + ++ ++

++

++++

−−

+⋅

−−

=− + ⋅ +

(9)

1 11 1 1 1 11 11

11

1111

{}(, ){}(, ){}(, ){}(, )

[{}(,)]

kij kij kij kij c

ii kk r

yjkij

vS xy vS xy vS xy vSxy V

x

xSSV

SxuSxy

ϕ

+ ++ + + + ++ ++

++

++++

−−

+⋅

−−

=− − ⋅ +

(10)

According to Equations (9) and (10), there are tire total deformation:

111 12 3

1

{}(, ) [ ]

11

kij uu u

c

r

uS x y p p p

V

ds V dx

ϕ

+++

=⋅ +⋅ ⋅

⋅−

(11)

111 12 3

1

{}(, ) [ ]

11

kij vv v

c

r

vS x y p p p

V

ds V dx

ϕ

+++

=⋅ −⋅ ⋅

⋅−

(12)

Machines 2022, 10, 801 6 of 33

Herein:

1

()

ii

dx x x

+

=− − represents the distance between adjacent tread elements in the

contact patch.

1kk

ds S S

+

=−

represents the distance of the contact patch center traveling in space

within the time interval dt .

11

2

11

11

1( )

11

uv

c

cr

r

pp V

Vds V dx

ds V dx

ϕ

== ⋅

+⋅−

⋅−

2111 11

11

{}(, ) {}(, ) c

ux j kij kij

r

V

pS yuSxy uSxy

dx ds V

ϕ

++ + ++

=− + ⋅ − ⋅ + ⋅ ⋅

3111 11

11

{}(, ) {}(, ) c

uyi kij kij

r

V

pS xvSxy vSxy

dx ds V

ϕ

++ + ++

=− − ⋅ − ⋅ + ⋅ ⋅

2111 11

11

{}(,) {}(, ) c

vyi kii kij

r

V

pS xvSxy vSxy

dx dx V

ϕ

+++ ++

=− − ⋅ − ⋅ + ⋅ ⋅

3111 11

11

{}(, ) {}(, ) c

vx j kij kij

r

V

pS yuSxy uSxy

dx ds V

ϕ

++ + ++

=− + ⋅ − ⋅ + ⋅ ⋅

If c

r

V

V is set to zero, it can be transformed into a steady state.

3. Model Description

3.1. Belt/Carcass Deformation [23,54]

Assuming that longitudinal translation deformation only occurs under longitudinal

force [54]:

0

x

c

cx

F

u

K

= (13)

where 0cx

K

is the longitudinal translational stiffness of carcass. Under lateral force, the

carcass is expressed as the deformation of an infinite beam under concentrated force. The

beam has in-plane flexural rigidity z

EI , where z

I is the moment of the inertia of the

area, E is the Young’s modulus of the belt in the circumferential direction,

s

k is the

lateral fundamental spring rate per unit length in the circumferential direction and the

tension 0

T is uniformly distributed in the belt width direction [23].

M

represents bend-

ing moment and Q represents shear force, as shown in Figure 2. The left diagram shows

the equilibrium of forces of the beam section and the right diagram shows model of the

beam on the elastic foundation.

Machines 2022, 10, 801 7 of 33

(a) (b)

Figure 2. Model of the beam on the elastic foundation. (a) The equilibrium of forces for the beam

section; (b) Belt/carcass deformation applied

y

F

.

The equilibrium of forces acting on the element of length

dx

along the belt width

results in the following equation:

2

00

2

() ( ) 0

s

yy y

wxdx Q dQ T dx Q T k y dx

xx

x

∂∂ ∂

++ +⋅ + ⋅ −−⋅ −⋅⋅ =

∂∂

∂

(14)

and for the Euler Bernoulli beam:

dM

Qdx

=

(15)

2

2

z

dy

MEI

dx

=− ⋅

(16)

Furthermore, according to (14), (15) and (16), the differential equation of the lateral defor-

mation of the beam can finally be expressed as:

42

0

42 ()

zs

yy

EI T k y w x

xx

∂∂

−+=

∂∂ (17)

According to boundary conditions, (1)

,() 0xyx

→∞ →

; (2)

0, ( ) 0xyx

==

; and (3)

0

2

ys

F

kydx

∞

=

, there is:

1

1

22

2

() (cos sin )

4

yx

s

F

yx e x x

k

λ

δλ

λλ

λ

−

=⋅ +

(18)

Herein:

0

4

1

(1 )

44

s

z

z

s

kT

EI EI k

λ

=+

⋅

,

0

4

2

(1 )

44

s

z

z

s

kT

EI EI k

λ

=−

⋅

22

12

1

λλ

δλ

+

=

,

0

4

zs

TEIk

ξ

=⋅ ⋅

Torsional deformation of carcass under aligning moment [54].

Machines 2022, 10, 801 8 of 33

() z

M

yx x

N

θ

=⋅

(19)

N

θ

is the carcass torsion stiffness.

Therefore, the lateral deformation of the carcass under lateral force and an aligning

moment can be expressed as

11

22

2

(cos sin )

4

xy z

c

s

FM

ve x xx

kN

λ

θ

δλ

λλ

λ

−

=⋅ + +⋅

(20)

3.2. Contact Patch

Reference [41] presents a method for calculating the shape of a contact patch based

on vertical force. Due to the fact that this method usually leads to a large contact patch

area following camber input, this paper does not consider camber.

22 2

(, ) ( )

y

n

j

ij l i

y

y

Gx y R R x R R

=− −− (21)

l

RRd=−

(22)

2

12zFz Fz

Fpdp d=⋅+⋅

(23)

where

R

is free radius, l

R is the loaded radius, d is the vertical deflection, y

R is the

lateral radius and

y

n is the lateral direction curvature exponent.

However, directly according to this method, the obtained contact length will be

larger. A correction method is proposed as follows:

2

lim 1 2

GGdGd=⋅+⋅

(24)

Then, the contact patch can be obtained when lim

(, )

ij

Gx y G<.

The pressure distribution of ground imprinting is expressed as follows [41,54]:

22

() (1())(1 ())(1 )

nn

ii i i

jj j j

x

xxx

AB

hh h h

ηλ

=⋅− ⋅+⋅ ⋅−⋅ (25)

26

011

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

jj

i

zij i j ij Fz Fz

ji i

yy

x

Fxy F a a

hb b

η

=⋅ ⋅+ − +⋅ (26)

assuming that 0ij

F is a constant, which can be obtained according to

(, )

zzijij

F

Fxy=.

j

h is the contact length at position

j

y, i

b is the contact width at position i

x

and

1Fz

a is the convexity factor along the contact width.

(2 1)(4 1)

2(4 1 )

nn

Ann

λ

++

=++ , 3(23)(43)(41)

(2 1)(4 1)(4 3 3 ) j

nnn

Bnnn h

λ

λ

++++Δ

=− ⋅

++++

Machines 2022, 10, 801 9 of 33

where

λ

is convexity factor of contact pressure along contact length,

j

h

Δ

is the contact

pressure center-offset factor and

n

is the contact pressure-uniformity factor.

When considering the tread pattern longitudinal groove, the

(, )

zij i j

Fxy

of the cor-

responding position is set to zero. The contact patch example is shown as in Figure 3

Figure 3. The contact patch model results.

3.3. Direction of Stress in the Sliding Regions

As shown in Figure 4,

,T,

iii

BP

are the upper (which is attached to the belt) and

lower end of the bristle and the bristle deformation, respectively, at previous times.

,T

111

,

iii

BP

+++

correspond to the current time.

dG

is the bristle slip displacement vector

relative to the road.

Figure 4. Tread element deformation and its slip displacement vector relative to the road.

Machines 2022, 10, 801 10 of 33

dt dt

== = = −

gBT i+1i

dG V (V + V ) dB + dT dB + T T

(27)

then

=−−=

i+1 i i i+1

dG T dB T P B

(28)

For sliding regions, the sliding direction

g

V

determines the direction of defor-

mation, so the vector

dG

is in the same direction as

i+1

T

and

i+1

T

is on vector

ii+1

PB

;

that is to say

ii+1

PB

determines the direction of

i+1

T

.

3.4. Stiffness of the Tread Element

The pattern block structure usually leads to the anisotropy of tread element stiffness.

It is assumed that the stiffness satisfies the elliptic equation, as shown in Figure 5:

22

cos sin

()()1

tt

tx ty

kk

kk

θθ

⋅⋅

+=

(29)

then:

22

1

cos sin

()()

t

tx ty

k

kk

θθ

=

+

(30)

where

θ

is the deformation direction angle of the tread element.

Figure 5. Tread element stiffness ellipse.

Machines 2022, 10, 801 11 of 33

4. Iterative Method for Force and Moment Calculation

4.1. Deformation and Stress Calculation of Tread Element

When the tread element is in the adhesion region, the deformation is calculated as

follows

Tc

uuu=− (31)

Tc

vvv=− (32)

Herein, u and v are obtained from Section 2, Equations (11) and (12); c

u and c

v

are obtained from Section 3.1, Equations (13) and (17). The deformation direction angle of

the tread element is expressed as

tan T

T

v

u

θ

= (33)

The longitudinal and lateral stresses of tread element are expressed as

x

atT

qku=⋅, ya t T

qkv=⋅, 22

axaya

qqq=+

(34)

If az

qq

μ

≤⋅

x

xa

qq=, yya

qq= (35)

else

cos

xa

xxs z z

a

q

qq q q

q

μμ

θ

== ⋅⋅=⋅⋅ (36)

sin

ya

yys z z

a

q

qq q q

q

μμ

θ

== ⋅⋅=⋅⋅ (37)

x

T

t

q

uk

=, y

T

t

q

vk

=; Tc

uu u=+

, Tc

vv v=+

(38)

Above are the stress and deformation of tread element in the sliding region.

4.2. Calculation of Force and Moment

4.2.1. Fx Calculation

0

xtTxs

x

t

tx xs

cx

F q dA k u dA q dA

k

k udA F dA q dA

K

== +

=−⋅ +

(39)

then:

(1 )

F

xx xr

pF F+=

(40)

Machines 2022, 10, 801 12 of 33

where:

0

t

Fx

cx

k

pdA

K

= , xra t

F

k udA= , xrs xs

F

qdA= ,

x

rxraxrs

FF F=+

4.2.2. Fy Calculation

ytTys

y

ty t zt ys

F q dA k v dA q dA

k vdA F k dA M k dA q dA

ης

== +

=−⋅− +

(41)

then:

(1 )

Fy y yr z MTF

pF F Mp+=+⋅

(42)

where:

11

22

2

[cos( ) sin( )]

4

x

s

exx

k

λ

λ

δ

ηλλ

λ

−

=⋅ ⋅ + ,

x

N

θ

ς

=

Fy t

p

kdA

η

= , MTF t

pkdA

ς

=−

, yra t

F

kvdA= , yrs ys

F

qdA=

yr yra yrs

F

FF=+

4.2.3. Mz Calculation

() ()

() () () ()

zyx

tT ys tT xs

MqxudAqyvdA

kv x u q x u ku y v q y v

=+−+

=⋅++⋅+−⋅+−⋅+

(43)

then:

(1 )

M

zz zr yFTM

pM M Fp+=+⋅

(44)

where:

()

Mz t

pkxudA

ς

=⋅+

,()

FTM t

pkxudA

η

=− ⋅ +

()

zrxa t T

M

ku y vdA=− ⋅ +

,()

zrxs xs

M

qyvdA=− ⋅ +

()

zrya t

M

kv x udA=⋅+

,()

zrys ys

M

qxu=⋅+

,

z

r zrxa zrxs zrya zrys

MM M M M=+++

According to Equations (33) and (35), there are

(1 )

11

FTM MTF MTF

F

yyyrzr

Mz Mz

pp p

pFFM

pp

⋅

+− =+ ⋅

++

(45)

(1 )

11

FTM MTF FTM

M

zzzryr

Fy Fy

pp p

pMMF

pp

⋅

+− =+ ⋅

++

(46)

Machines 2022, 10, 801 13 of 33

Make:

11 1

F

x

A

p=+ , 22 11

F

TM MTF

Fy

Mz

pp

Ap p

⋅

=+ − +, 33 11

F

TM MTF

Mz

Fy

pp

Ap p

⋅

=+ − +

1

x

r

B

F=, 21

MTF

yr zr

Mz

p

BF M

p

=+ ⋅

+, 31

FTM

z

ryr

Fy

p

B

MF

p

=+ ⋅

+

There are

1

11

22 2

33 3

00

00

00

x

y

z

F

B

A

A

FB

A

B

M

=

(47)

Make:

11

22

33

00

00

00

A

A

A

=

A,

1

2

3

B

B

B

=

B,

x

y

z

F

F

M

=

X (48)

Finally, there are

=

A

XB

(49)

A reasonable iterative method can be constructed according to this matrix equation.

4.3. Iterative Strategy

4.3.1. Matrix Splitting Method

Make:

−=

A

MN

(50)

then:

1m

−−

+=+

11

m

X

MNX MB

(51)

where ij

ij

NIR

M=, 1

I

R< and 1

ij

ij

A

M

I

R

=−,

1

ij ij

IR

NA

IR

=⋅

−

.

4.3.2. Steepest-Descent Method

=−

mm

rBAX

=

mm

pr

=

T

mm

mT

mm

rp

α

pAp

=+

m+1 m m m

XXαp

(52)

Machines 2022, 10, 801 14 of 33

4.3.3. Richrdson Method

()

ω

=+ −

m+1 m m

XX BAX

(53)

where

1

2

n

ωλλ

=+, 1

λ

and n

λ

are the maximum and minimum eigenvalues of A, re-

spectively.

4.3.4. Extrapolation Acceleration Method

According to matrixes M and N from Section 4.3.1, the extrapolation acceleration

method can be made:

(1 ) ( )

ωω

−−

=− + +

11

m+1 m m

XXMNXMB

(54)

where

1

2

2( )

n

ωλλ

=−+ , 1

λ

and n

λ

are the maximum and minimum eigenvalues of

−

MN, respectively.

4.3.5. Method in Reference [54,55]

This method obtains the solution directly from the matrix equation:

−

=1

m+1

XAB

(55)

This method is equivalent to Section 4.3.1 where

I

R is equal to zero.

4.3.6. Method in Reference [3,51]

This method is equivalent to that of the matrix M whose coefficients are equal to

one, and the method is the same as in Section 4.3.1.

4.3.7. Iterative Error

−= m+1

ERR B AX (56)

If TTol≤ERR ERR , the iterative reaches convergence, where Tol is the iterative

error tolerance.

4.3.8. Comparison of Iterative Methods

A comparison of the iterative methods at a steady state under a constant side slip

angle = three deg is shown in Figure 6, and method 4 and 5’s iterative processes coincide

completely.

Machines 2022, 10, 801 15 of 33

(a) (b)

Figure 6. Curves of lateral force and aligning moment with iteration times (a) Fy responses with

iteration times; (b) Mz responses with iteration times.

The unsteady comparison results of different iterative methods are shown in Table 1.

Table 1. The iterative CPU times of different methods at different conditions.

Iterative Method

Iterative CPU Time [s] and The Simulation Time is Set to 3 s

Step Side Slip Sine Side Slip Load Variations at

Constant Side Slip

Sine Side Slip Angle at

Constant Turn Slip

Sine Longitudinal Slip at

Constant Side Slip

Method 1 35.11 94.10 69.40 no convergence no convergence

Method 2 33.97 71.46 59.99 no convergence no convergence

Method 3 55.34 136.21 141.71 154.96 280.46

Method 4 33.61 65.07 59.67 no convergence no convergence

Method 5 32.50 66.26 60.14 no convergence no convergence

Method 6 no convergence no convergence no convergence no convergence no convergence

Figure 6 shows that method 6 has the fastest iteration speed at a steady state, followed

by method 3. It can be seen from Table 1 that although method 3 has the slowest iteration

speed in unsteady state, it has the most stable convergence. As such, method 3 is selected

as the iterative strategy of the discrete theoretical model. However, it should be noted that

in the case of unsteady large slip, due to the fact that the coefficients of the A matrix

parameters tend to one but the B matrix changes, the solution is sometimes not conver-

gent, so it is necessary to improve the iterative method. Through research, it is found that

the following equation can improve convergence:

1

1

2

n

n

p

λ

ω

λλ λλ

=

+++

(57)

The adjustment of p

λ

can better improve the convergence, but it will reduce a cer-

tain convergence speed. The greater p

λ

is, the better the convergence will be, but the

iteration speed will be reduced, and this parameter can be reasonably selected for specific

situations. This paper recomments 20p

λ

=, and the simulation in Table 1 uses this param-

eter.

Machines 2022, 10, 801 16 of 33

5. Simulation of the Discrete Theoretical Model

5.1. Model Parameters and Verification

The model parameters are listed in Table 2, Figures 7–11 show the comparison be-

tween the model and the test and the test information is shown in Table 3.

Table 2. The discrete tire model parameters.

Parameters Name Parameters Value Parameters Name Parameters Value

3

[/ ]

tx

kNm 1.3377×108 0[/]

cx

KNm

4.3735×105

3

[/ ]

ty

kNm 1.0332×108 2

[]

z

EI Nm 1×103

[]n− 2 2

[/ ]

s

kNm 1.25×105

[]

λ

− 0 []

ξ

− 0.2

[]Δ− 0 [/]NNmrad

θ

1.2994×104

1[]

z

aF − −0.05 []

μ

− 1.11

1[/]

Fz

pNm

2.01×105 0[]

R

m 0.3465

[]p

λ

− 20 []

y

n− 5.4

1[]G− 3.64 []

y

R

m 0.145

2[]G− −0.74 Tol 10

[]dx mm 2 []dy mm 2

Table 3. The test information.

Test No. Test Conditions Inflation Pressure Road Speed Steer Angle Steer Cycle Vertical Load

1 Contact Pressure 250 kPa 0 0 0 s 5415 N

2 Parking Maneuver 250 kPa 0 15°~−15° 32 s (Slope) 5386 N

3 Sideslip Angle Step 250 kPa 0 → 10 kph −1° - 5414 N

250 kPa 0 → 10 kph −4° - 5407 N

4 Sine Sideslip Angle 250 kPa 10 kph −15°~15° 20 s (0.05 Hz) 5406 N

The contact pressure is performed on the TekScan. The other three conditions are

performed on the MTS CT Flat-trac machine (MTS System Corporation). For the sideslip

angle step condition, the test procedure is to turn the steer angle first, then load and finally

apply the belt speed of the MTS to 10kph within one second. It should be noted that the

sign of steer angle in the table is the opposite to that of the slip angle which is determined

by the MTS Flat-trac machine.

Machines 2022, 10, 801 17 of 33

Figure 7. Contact patch shape comparison between model and test.

(a) (b)

Figure 8. Contact pressure comparison between model and test. (a) Contact pressure along x center

line; (b) Contact pressure along y center line.

-100 -80 -60 -40 -20 0 20 40 60 80 100

Contact Patch x Position [mm]

-80

-60

-40

-20

0

20

40

60

80

Contact Patch y Position [mm]

Fz=5415.12N

sim

test

Machines 2022, 10, 801 18 of 33

Figure 9. Parking comparison between model and test.

(a) (b)

Figure 10. Step side slip angle comparison between model and test. (a) Side slip angle = 1 deg; (b)

Side slip angle = 4 deg.

It should be noted that the response of normalized lateral force instead of lateral force

with travel distance is used, which does not affect the assessment of transient behavior.

Machines 2022, 10, 801 19 of 33

(a) (b)

Figure 11. Pure side slip comparison between model and test at 0.05 Hz at V = 10 km/h. (a) Side

force Fy; (b) Aligning moment Mz.

As shown in Figure 11, the aligning moment is quite different from the experimental

data, and a reasonable reason is whether the local camber of the carcass caused by the

lateral force is not considered, which can lead to greater aligning moment if taken into

consideration.

The unsteady characteristics of tire models are only related to the path frequency

without changing its parameters, so the unsteady state-ability of the tire model can be

judged based on this condition and changing the ratio between the distance of the contact

patch center traveling in space within the time interval and the distance between adjacent

tread elements in the contact patch by changing speed. The path frequency is defined as:

2

s

f

V

π

ω

= (58)

where

s

ω

[1

m−] represents the path frequency,

f

[1

s

−] represents the time frequency and

V[/ms

] represents the contact patch center velocity.

Next, the discrete method proposed for tire total-deformation calculation in this pa-

per (Section 2) will be compared with the references [52,53], and the comparison results

of two methods are given. The comparison error calculation method is as follows:

2

2

()

100%

pr

p

yy

err y

−

=×

(59)

where

p

y represents the proposed method in this paper and r

y represents the pro-

posed method in reference [52,53].

5.2. Pure Side Slip Simulation

5.2.1. Step Side Slip Angle

The simulation results of step side slip angle with the proposed method and the ref-

erence method are as shown in Figure 12.

Machines 2022, 10, 801 20 of 33

(a) (b)

(c) (d)

Figure 12. Response of lateral force and aligning moment under side slip angle input. (a) Response

of lateral force using the proposed method. (b) Response of lateral force using the reference method.

(c) Response of aligning moment using the proposed method. (d) Response of aligning moment

using the reference method.

5.2.2. Sine Side Slip Angle

The simulation results of sine side slip angle with the proposed method and the ref-

erence method are as shown in Figure 13.

(a) (b)

Machines 2022, 10, 801 21 of 33

(c) (d)

Figure 13. Response of lateral force and aligning moment under sine side slip angle input. (a) Re-

sponse of lateral force using the proposed method. (b) Response of lateral force using the reference

method. (c) Response of aligning moment using the proposed method. (d) Response of aligning

moment using the reference method.

5.2.3. Load Variations at Constant Side Slip Angle

The simulation results of load variations at constant side slip angle with the proposed

method and the reference method are as shown in Figure 14.

(a) (b)

(c) (d)

Figure 14. Response of lateral force and aligning moment under load variations at constant side slip

angle input. (a) Response of lateral force using the proposed method. (b) Response of lateral force

Machines 2022, 10, 801 22 of 33

using the reference method. (c) Response of aligning moment using the proposed method. (d) Re-

sponse of aligning moment using the reference method

5.2.4. Discussion

The above simulations have the same path frequency, and the simulation results for

different speeds should also be the same, in theory. The simulation results of the proposed

method are slightly different, but significant differences occur in reference methods under

the load variations condition at different speeds. The simulation errors of the two methods

at 3m/s are listed in Table 4.

Table 4. The simulation errors at pure side slip simulation.

Simulation Conditions Compare Items Error [%]

Step Side Slip Angle y

F

0.28

z

M

1.12

Sine Side Slip Angle y

F

1.37

z

M

2.18

Load Variations at Constant

Side Slip Angle

y

F

5.86

z

M

7.59

5.3. Step Turn Slip Simulation

The simulation results of step turn slip with the proposed method and the reference

method are as shown in Figure 15.

The method proposed in the references is obviously not suitable for the turn slip sim-

ulation because the steady-state simulation results at different speeds are quite different.

The reason is that the method proposed in the reference is based on the principle of linear

interpolation in the leading edge, while the tread deformation is nonlinear during turn

slip and the method fails when ds is much larger than dx . At the same time, it can be

seen that the aligning moment responds quickly in the initial position, which is explained

in Section 5.5.2. The simulation errors of the two methods at 3m/s are listed in Table 5.

(a) (b)

Machines 2022, 10, 801 23 of 33

(c) (d)

Figure 15. Response of the lateral force and aligning moment under step turn slip input. (a) Re-

sponse of lateral force using the proposed method. (b) Response of lateral force using the reference

method. (c) Response of aligning moment using the proposed method. (d) Response of aligning

moment using the reference method.

Table 5. The simulation errors at the step turn slip simulation.

Simulation Conditions Compare Items Error [%]

Step Turn Slip Simulation y

F

141.40

z

M

142.20

5.4. Combined Slip Simulation

5.4.1. Sine Longitudinal Combined with a Constant Side Slip

The simulation results of sine longitudinal combined with a constant side slip with

the proposed method and the reference method are as shown in Figure 16.

(a) (b)

(c) (d)

Machines 2022, 10, 801 24 of 33

(e) (f)

Figure 16. Response of lateral force and aligning moment under sine longitudinal combined with a

constant side slip. (a) Response of longitudinal force using the proposed method. (b) Response of

longitudinal force using the reference method. (c) Response of lateral force using the proposed

method. (d) Response of lateral force using the reference method. (e) Response of aligning moment

using the proposed method. (f) Response of aligning moment using the reference method.

5.4.2. Sine Turn Combined with a Constant Side Slip

The simulation results of sine turn combined with a constant side slip with the pro-

posed method and the reference method are as shown in Figure 17.

(a) (b)

(c) (d)

Figure 17. Response of lateral force and aligning moment under sine turn combined with a constant

side slip. (a) Response of lateral force using the proposed method. (b) Response of lateral force using

the reference method. (c) Response of aligning moment using the proposed method. (d) Response

of aligning moment using the reference method.

Machines 2022, 10, 801 25 of 33

5.4.3. Load Variations at Constant Longitudinal and Side Slip

The simulation results of load variations at constant longitudinal and side slip with

the proposed method and the reference method are as shown in Figure 18.

(a) (b)

(c) (d)

(e) (f)

Figure 18. Responses of longitudinal force and lateral force aligning moments under load variations

at constant longitudinal and side slip input. (a) Response of longitudinal force using the proposed

method. (b) Response of longitudinal force using the reference method. (c) Response of lateral force

using the proposed method. (d) Response of lateral force using the reference method. (e) Response

Machines 2022, 10, 801 26 of 33

of aligning moment using the proposed method. (f) Response of aligning moment using the refer-

ence method.

5.4.4. Discussion

In the case of load variations or turn slip input the reference method has obvious

problems, and the conclusions are consistent with Sections 5.2 and 5.3. The simulation

errors of the two methods at 3m/s are listed in Table 6.

Table 6. The simulation errors at combined slip simulation.

Simulation Conditions Compare Items Error [%]

Step Turn Slip Simulation

x

F

2.54

y

F

1.23

z

M

3.67

Sine Turn Combined with a

Constant Side Slip

y

F

61.01

z

M

6.43

Load Variations at Constant

Longitudinal and Side Slip

x

F

3.95

y

F

8.49

z

M

8.86

5.5. Deformation Analysis of Tread Element

5.5.1. Deformation of Tread Element under Step Side Slip Angle

In the side slip angle step condition, with the increase of the travel distance, the de-

formation of the tread element approximately changes shape from rectangular to trape-

zoidal and finally to triangular, as shown in Figure 19.

(a) (b)

(c) (d)

Machines 2022, 10, 801 27 of 33

(e) (f)

(g) (h)

Figure 19. Lateral deformation of the tread element at different traveling distances under the step

side slip angle input. (a) Response of lateral force and aligning moment. (b) Deformation of the tread

element at the contact patch center line under different travel distance. (c) Deformation of the tread

element at A position. (d) Deformation of the tread element at B position. (e) Deformation of the

tread element at C position. (f) Deformation of the tread element at D position. (g) Deformation of

the tread element at E position. (h) Deformation of the tread element at F position.

5.5.2. Deformation of the Tread Element under Step Turn Slip

In the turn slip step condition, the lateral deformation is first symmetrically distrib-

uted relative to the longitudinal center of contact patch, which is the reason why the slope

of lateral force with the travel distance is zero at the starting position and also explains

why aligning moment responds so quickly, as shown in Figure 20.

With the increase of the travel distance, the intersection point of the lateral defor-

mation curve with the contact patch longitudinal axis gradually shifts from the center to

the intersection point of the steady state. In this process, the lateral force increases contin-

uously, and the increment gradient of the aligning moment caused by the lateral defor-

mation decreases, which leads to a peak of the aligning moment which then decreases and

finally tends to the steady state value.

At the same time, the lateral deformation is approximate to the nonlinear change of

the parabolic shape, which also explains why the reference method will have problems in

the turn slip input simulation.

Machines 2022, 10, 801 28 of 33

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 20. Lateral deformation of the tread element at different traveling distances under the step

turn slip input. (a) Response of lateral force and aligning moment. (b) Deformation of the tread

element at the contact patch center line under different travel distance. (c) Deformation of the tread

element at A position. (d) Deformation of the tread element at B position. (e) Deformation of the

tread element at C position. (f) Deformation of the tread element at D position. (g) Deformation of

the tread element at F position. (h) Deformation of the tread element at E position.

5.5.3. Deformation of the Tread Element under Load Variations at a Constant Side Slip An-

gle

Machines 2022, 10, 801 29 of 33

Figure 21 shows the that the lateral deformation of the proposed method is more

regular and stable than reference. It explains why under load variations conditions at the

same path frequency, when the traveling distance in an increment time is larger than the

distance between adjacent tread elements, the lateral force and aligning moment of the

reference method will be different.

(a) (b)

(c) (d)

Figure 21. Lateral deformation of tread element at different traveling distance under load variations

at a constant side slip angle. (a) Response of vertical force and lateral force for proposed method. (b)

Response of vertical force and lateral force for the reference method. (c) Tread deformation at dif-

ferent positions for the proposed method. (d) Tread deformation at different positions for the refer-

ence method.

6. Discussion and Conclusions

6.1. Discussion

Figures 7 and 8 show that the contact patch parameters are reasonable, and the sim-

ulation results are consistent with the experiments. Figures 9, 10 and 11a show that the

belt/carcass parameters are reasonable, and simulation and test results are consistent as

well. However, Figure 11b shows that the aligning moment is quite different from the

experimental data, and a reasonable reason is that the local camber of the carcass caused

by the lateral force is not considered, which can lead to greater aligning moment if con-

sidered.

Figure 6 shows that different iterative methods converge to the same value and that

the iterative method in reference [54,55] has the fastest iteration speed, followed by the mod-

ified Richardson method proposed in this paper. Table 1 shows that the modified Richardson

has the best convergence and then method in reference [54,55] and the worst is the method in

reference [3,51].

Figures 12, 13 and 16 show that whether the discrete method proposed or the refer-

ences [52–55] are considered, the simulation results at different speeds with the same path

frequency are only slightly different under side slip angle or longitudinal slip input. Fig-

ures 14, 15, 17 and 18 show that for the discrete proposed method the simulation results

still have slight differences, but the results of the method in the references [52–55] have

Machines 2022, 10, 801 30 of 33

significant differences at different speeds with the same path frequency, and the greater

the speed difference, or more precisely the greater the ratio between the distance of the

contact patch center traveling in space within the time interval and distance between ad-

jacent tread elements in the contact patch, the greater the simulation difference. The cal-

culation errors in Tables 4–6 also confirm the above analysis. Figure 19 shows that the

tread deformation varies nearly linearly at the leading edge of the contact patch under

step side slip angle, but Figure 20 shows that the tread deformation varies nonlinearly in

an approximate parabolic under step turn slip. Figure 21 shows that the tread deformation

is more stable by using the proposed method under the load variations input. Tread de-

formation explains why the methods in the reference [52–55] have problems under the

turn slip or load variations inputs, since the method performs a linear interpolation process

at the leading edge which is not suitable for nonlinear deformation and deformation instabil-

ity. The above conclusions may be able to explain the reasons for the reference [3] “at each

time step in which the wheel is rolled further over a distance equal to the interval between two

successive tread elements” simulation settings.

6.2. Conclusions

In this paper, novel numerical methods for discrete theoretical tire model steady- and

unsteady-state simulations are proposed. Some conclusions are as follows.

The comparison between the theoretical model and the experiment shows that the

accuracy of the aligning moment needs further improvement, which may be affected by

tire local carcass camber caused by lateral force, but the model parameters are reasonable.

New basic matrix equations for force and moment calculation are constructed for the

convenient application of different iterative methods. Different iterative methods are

studied and compared. The results show that the new modified Richardson iteration

method proposed in this paper has the best convergence stability in the unsteady calcula-

tion, which mathematically solves the problem of nonconvergence of discrete theoretical

models in the published references [3,51,55].

A novel discrete method for solving the total deformation of tires is established based

on the Euler method. The unsteady characteristics of tire models are only related to the

path frequency without changing its parameters, so the unsteady-state ability of the tire

model can be judged based on this condition. It shows that the methods in the references

[52–55] have significant differences at different speeds with the same path frequency, and

the greater the speed difference, or more precisely the greater the ratio between the dis-

tance of the contact patch center traveling in space within the time interval and distance

between adjacent tread elements in the contact patch, the greater the simulation difference

under the turn slip or load variations inputs. However, the method proposed in this paper

has good results.

This paper only studies the numerical solution method of the theoretical model in

detail. In the future, the following work needs to be carried out. The establishment of a

refined contact patch model considering camber and the influence of lateral force and lon-

gitudinal force on the contact patch model and of a dynamic friction model considering

the influence of contact pressure and slip velocity is added to the theoretical model. Sys-

tem verification of theoretical models with experiments. Furthermore, it is hoped that the-

oretical models can be used in tire virtual development, since the computational efficiency

of the finite element model is low and virtual data simulation, especially for turn slip input

because the existing test machines cannot carry out this condition. Combined with the

static simulation data of the finite element, using the theoretical model to achieve the

steady and unsteady simulations should be a good prospect for the future of tire virtual

development.

Author Contributions: Conceptualization, Q.L. and D.L.; methodology, Q.L. and D.L.; software,

Q.L.; validation, Q.L., Y.M. and D.X.; formal analysis, Q.L., Y.M. and D.X.; investigation, Q.L.; re-

Machines 2022, 10, 801 31 of 33

sources, D.L.; data curation, Q.L., Y.M. and D.X.; writing—original draft preparation, Q.L.; writ-

ing—review and editing, D.L., Y.M. and D.X.; visualization, Q.L.; supervision, D.L.; project admin-

istration, D.L.; funding acquisition, D.L. All authors have read and agreed to the published version

of the manuscript.

Funding: This research was funded by the Strategic Priority Research Program of Chinese Academy

of Sciences, grant number XDC06060100.

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: Not applicable.

Conflicts of Interest: The authors declare no conflict of interest.

Abbreviations

Symbol Description

κ

Practical longitudinal slip

α

Side slip angle

x

S Theoretical longitudinal slip

y

S Theoretical lateral slip

ϕ

Theoretical turn slip

dx Distance between adjacent tread elements

ds Distance of the contact patch center traveling in space within the time intervaldt

0cx

K

Longitudinal translational stiffness of carcass

z

EI In-plane flexural rigidity of belt

s

k the lateral fundamental spring rate per unit length in the circumferential direction

N

θ

Carcass torsion stiffness

R

Tire free radius

l

R Tire loaded radius

d Tire vertical deflection

y

R

Tire lateral radius

y

n Lateral direction curvature exponent

1

G Contact patch shape correction quadratic coefficient

2

G Contact patch shape correction primary coefficient

λ

Convexity factor of contact pressure along contact length

n Contact pressure uniformity factor

Δ Contact pressure center offset factor

μ

Friction coefficient

Tol Iterative error tolerance

tx

k Tread element longitudinal stiffness

Machines 2022, 10, 801 32 of 33

ty

k Tread element lateral stiffness

t

k Tread element stiffness

1

F

z

p Convexity factor of contact pressure along the contact width

p

λ

Iterative correction coefficient

yss

F

Steady-state lateral force

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5. Pacejka, H.B.; Besselink, I.J.M. Magic formula tyre model with transient properties. Veh. Syst. Dyn. 1997, 27, 234–249.

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