Tyre and Vehicle Dynamics part 4 pps

contact line
ϕ
γ
tyre
peripheral
line
ϕ
γο
ϕ
contact line
contact line
wheel plane
peripheral line on
frictionless surface
γ
contact line
’large’
wheel plane
ϕ
(
as if
µ

)
ϕ
(contact line if
µ

)
ϕ
ϕ
(
as if
µ

)

(
γο
)
ϕ
ϕ
upward shifted

γ
=0
’large’
contact line
(| | > )
contact line
wheel plane
γο
γο
>
0
1
0
0.33

µ
µ
1
ϕ
1
θ

θ

= 5
asymptote
ϕ
’small’

(
γ

)
ϕ
α
ϕ
α
α
contact line
wheel plane
α
γ
contact line
0
tan

α
-1
1
-0.2
0.2
γ
α
1/
θ
*
+
1/
θ
*
-
1/
θ
α
α
α
,
µ
µ
α
γ
= 10
O
(1-
ε
)

γ
=
θ
= 5
1
3
-20 -10
20
0
γ
[deg]
α
[deg]
-1
1
-20
-10
20
010
-0.2
γ
[deg]
-20
-10
10
20
0
µ
µ
(1-

ε
)
γ
=
θ
= 5
1
3
0
-1
1
0.2
γ
[deg]
-20
-10
20
0
10
0
1
2
3
4
6
-1
-2
-3
-4
-5

-6
5
α
[deg]
µ
µ
(1-
ε
)
γ
=
θ
= 5
1
3
ϕ
’large’

(
γο
)
ϕ
α
ϕ
α
α
ϕ
upward shifted

contact line
wheel plane
α
γ
contact line
ϕ
to be ’large’
α
range for
ϕ
to be ’large’
α
range for
α
α
α
α
ϕ
to be ’large’
α
range for
’large’
γ
α

>
0
α

<

0

γο
ϕ
to be ’large’
α
range for
α
α
α
α
ϕ
to be ’large’
α
range for
’small’
γ

α

> 0
α

< 0
γο
-10
0
10
-0.3

2
1
0
0.1
0.2
0.30.4
0.5
0.6
0.7
0.8
ϕ
α
[deg]
µ
1
-1
θ


= 5
-10
0
10
α
[deg]
0.33
µ
-0.3
-0.2
-0.1

0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0
0.1
ϕ

θ


= 5
α
tread element
contact line
belt
carcass
ψ
wheel plane
wheel spin axis
∆

-3
0
3
α
-20-1001020
oo oo
[kN]
0
-0.2
γ

=
ϕ

=
0
κ

=
single row
rigid carcass
=0
µ
-3
0
3
α
-20-1001020

oo oo
[kN]
0
-0.2
1 row
rigid
=0.03
µ
γ

=
ϕ

=
0
κ

=
α
-40
0
40
-20 -10 0 10 20
oo oo
[Nm]
0
κ

=
-0.2

single row
rigid carcass
γ

=
ϕ

=

0
=
0
µ
α
-40
0
40
-20 -10 0 10 20
oo oo
[Nm]
0
κ

=
-0.2
1
row
rigid
=0.03
µ

γ

=
ϕ

=
0
-2
0
3
[kN]
-3 0 3
[kN]
α

=
0
-2
2
4
6
8
o
1 row
rigid
=0
µ
γ

=

ϕ

=

0
-50
0
50
[Nm]
-3
03
[kN]
0
-2
2
4
6
8
o
=0
µ
1 row, rigid
α

=
γ

=
ϕ

=
0
-2
0
3
[kN]
-3 0
3
[kN]
α

=
0
-2
2
4
6
8
o
1
row
rigid
=0.03
µ
γ

=
ϕ

=

0
-50
0
40
[Nm]
-3
0
3
[kN]
0
-2
2
4
6
8
o
=0.03
µ
1 row, rigid
γ

=
ϕ

=
0
α

=

α
-40
0
40
-20 -10 0 10 20
oo oo
[Nm]
0
κ

=
=
0.03
µ
1 row
flexible
γ

=
ϕ
=

0
-0.2
-3
0
3
α
-20 -10 0 10 20
oo oo

[kN]
-0.2
1 row
flexible
=
0.03
µ
γ

=10 ,
ϕ
=

0
o
0
γ

=
0
κ

=
-50
0
40
[Nm]
-3
0
3

[kN]
0
-2
α

= 2468
o
=0.03
µ
1 row
flexible
γ

=
ϕ

=
0
-3
0
3
κ
-0.8 -0.4 0 0.4 0.8
o
[kN]
8
1 row
flexible
=
0.03

µ
o
-2
α

=
γ

=
10
,
ϕ
=
0
o
-2
0
3
[kN]
-3 0
3
[kN]
0
2
4
8
=0.03
µ
γ

=
ϕ

=

0
1
row
flexible
α

=
-2
o
6
-2
0
3
[kN]
-3 0 3
[kN]
0
2
4
8
=
0.03
µ
1 row
flexible

α

=
-2
o
γ

=10 ,
ϕ
=
0
o
-150
0
200
[Nm]
-3
0
3
[kN]
0
-2
2
4
6
8
o
=
0.03
µ

1 row, flexible
γ

=10 ,
ϕ

=
0
o
α

=
-3
0
3
α
-20 -10 0 10 20
oo oo
[kN]
0
-0.2
κ

=
1
row
flexible
=0.03
µ
γ

=
ϕ

=

0
-3
0
3
α
-20 -10 0 10 20
oo oo
[kN]
2 rows
flexible
=
0.03
µ
-0.2
0
κ

=
γ

=
ϕ

0

-3
0
3
κ
-0.8 -0.4 0 0.4 0.8
o
[kN]
8
=
0.03
µ
o
-2
α

=
2 rows
flexible
γ

=
ϕ

0
-2
0
3
[kN]
-3 0
3

[kN]
0
2
4
8
=0.03
µ
2 rows
flexible
α

=
-2
o
γ

=
10
,
ϕ
=
0
o
α
-20-1001020
oo oo
2
rows
flexible
=

0.03
µ
0
γ

κ

=
0
,
ϕ
0
20
o
=-5
o
-100
0
150
[Nm]
-50
0
40
[Nm]
-3
0
3
[kN]
0
2

8
o
=0.03
µ
2 rows, flexible
-2
α

=
γ

=
ϕ


=

0
6
4
-2
0
3
[kN]
-3 0 3
[kN]
0
2
4
8

=
0.03
µ
2 rows
flexible
α

=
-2
o
γ

=
ϕ

0
-3
0
3
α
-20 -10 0 10 20
oo oo
[kN]
-5
2 rows
flexible
=
0.03
µ
0

γ

=
20
oo
κ

=
0
,
ϕ
0
-150
0
200
[Nm]
-3
03
[kN]
0
-2
2
8
o
=0.03
µ
2 rows, flexible
γ

=

10
,
ϕ

=

0
o
α

=
-3
0
3
α
-20 -10 0 10 20
oo oo
[kN]
2 rows
flexible
=
0.03
µ
-0.3
0.1
ϕ
=
-0.2
-0.1
γ

=
κ


=

0
0
0.8
0.2
γ

=
κ


=

0
α
-20 -10 0 10 20
oo oo
2
rows
flexible
=
0.03
µ
0

-200
0
200
[Nm]
-0.2
-0.1
0.8
0.1
-0.3
ϕ
=
-3
0
3
α
-20 -10 0 10 20
oo oo
[kN]
2
rows
flexible
=
0
µ
-0.3
-0.2
-0.1
γ
=
κ

=

0
0
0.8
0.2
0.1
ϕ
=
γ

=
κ


=
0
α
-20 -10 0 10 20
oo oo
0
-200
0
200
[Nm]
-0.2
-0.1
0.1

-0.3
2
rows
flexible
=
0
µ
0.8
ϕ
=
-1
0
3
α
0.1 0.200.30.5
[kN]
0
8
2
2
rows
flexible
=
0
µ
γ

=
κ

=

0
0.4
4
o
6
=-2
o
ϕ

0.1 0.200.30.5
0
2
2 rows
flexible
=
0
µ
γ

=
κ


=

0
0.4

4
8
o
6
α
=-2
o
ϕ

-50
0
200
[Nm]
-0.1
0
0.5
α
-20 -10 0 10 20
oo oo
[-]
1 row
rigid
=
0
µ
κ

=
γ

=
ϕ
0
-0.1
0
0.5
α
-20 -10 0 10 20
oo oo
[-]
2 rows
flexible
=
0
µ
κ

=
γ

=
ϕ
0