Using the Soft-Soil tire model

Using the Soft-Soil tire model
The Adams/Tire Soft Soil tire model offers a basic model to describe the tire-soil interaction forces for
any tire on elastic/plastic grounds, such as sand, clay, loam and snow.
The model requires a tire property file with keyword SOFT-SOIL and a road data file (one of the existing
formats) with additional soil properties. Two tire-soil contact models are offered:
• Elastic-plastic soil deformation model, USE_MODE = 1
• Visco-elastic soil deformation model, USE_MODE = 2
Definition of Tire Slip Quantities
Figure 1
Definition of the slip velocities in the tire-road contact point
The longitudinal slip velocity V sx in the SAE-axis system is defined using the longitudinal speed V x , the
wheel rotational velocity and the the effective rolling radius R e :
V sx = V z – R e
(1)
The lateral slip velocity is equal to the lateral speed in the contact point with respect to the road plane:
V sy
=
V y
(2)
The slip quantities (longitudinal slip) and (slip angle) are calculated with these slip velocities in the
contact point, for negative V sx they are defined as:
V sx
V x
= –------- and tan
=
V
-------- sy
V x
(3)

2
Adams/Tire
Using the Soft-Soil tire model
and for positive V sx (driving) as:
V sx
V r
= –------- and tan
=
V
------- sy
V r
(4)
V r is the rolling speed V r is determined using the effective rolling radius R e : (5)
Note that for realistic tire forces the slip angle is limited to degrees and the longitudinal slip
1
to .
90
V r R e
Loaded and Effective Tire Rolling Radius
The loaded rolling tire radius R l is defined as the unloaded tire radius R 0 minus the tire deflection f 0 due
to the vertical load:
R l
R 0 f 0
(6)
The effective rolling radius R e (at free rolling of the tire), which is used to calculate the rotational speed
of the tire, is defined by:
V
Re
x
(7)

Using the Soft-Soil tire model
Definition of Tire Slip Quantities
3
For radial tires, the effective rolling radius is rather independent of load in its load range of operation
because of the high stiffness of the tire belt circumference. Only at low loads does the effective tire radius
decrease with increasing vertical load due to the tire tread thickness, see the Figure 2.
Figure 2
Effective and loaded tire radius as a function of the vertical load

4
Adams/Tire
Using the Soft-Soil tire model
Effective Rolling Radius and Longitudinal Slip
Figure 3
Side view of a rolling tire
To represent the effective rolling radius R e , a PAC2002 compatible equation is used:
R
d
d
e R f0,F
( Deff
atan( Beff
f ) Feff
f
z 0 0
0 0
)
(8)
f
in which ,F is the nominal tire deflection at the nominal tire load F z0 :
0 z0
f
0,F
z0 F
C
z0
z
(9)
and
d
f 0
is called the dimensionless radial tire deflection, defined by:

Using the Soft-Soil tire model
Elastic-plastic tire-soil contact
5
f
d
0
f
f
0
0,F z 0
(10)
Elastic-plastic tire-soil contact
The interaction forces for a rigid wheel
The static sinkage of a rigid object into a soft soil depends on the load on that object: Bekker [1]
formulated the sinkage h of a flat plate with width b as follows:
p(
h )
( k
c
/ b k
) h
n
(11)
k
k
in which c and are the cohesive and frictional moduli respectively, n the sinkage exponent. The
static stress p is in equilibrium with the vertical force Fz.
Figure 4
Pressure distribution under a flat plate

6
Adams/Tire
Using the Soft-Soil tire model
When applying this approach to a non-rolling wheel the static stress distribution can be estimated as
shown in the Figure 5
Figure 5
Static stress distribution under a non-rolling rigid wheel
For the dynamic sinkage the wheel rotational speed must be taken into account.

Using the Soft-Soil tire model
Elastic-plastic tire-soil contact
7
Figure 6
Wheel entry and exit angle when rolling on soil
Assume a wheel soil contact with entry angle f and exit angle r , see also [2], then these angles
can be written as a function of the total sinkage h and the exit penetration h e as follows:
f
r
acos( 1
h / R )
acos( 1
h
e
/ R )
The exit penetration h e depends on the elastic stiffness C s of the soil.
(12)

8
Adams/Tire
Using the Soft-Soil tire model
Based on the terramechanical approach as described in [2] the normal and shear stresses can be modeled
as shown in the Figure 7.
Figure 7
Normal and shear stress modelling of a rotating wheel
The wheel normal stress distribution can be defined as function of the wheel angle [2,3]:
for m f :

Using the Soft-Soil tire model
Elastic-plastic tire-soil contact
9
n
kc
( ) R0
k
cos( ) cos(
b
for
r m
:
f
)
n
(13)
(
) R
n
0
kc
b
k
cos
f
r
m
r
(
f
m
)
cos(
f
)
n
with b the wheel width and R 0 the wheel radius.
The angle m is the angle at which the maximum normal stress occurs [4]:
m a0
a1
)
(
f
(14)
The shear stress [5,6] in longitudinal direction is:
( ) ( c (
)tan( ))( 1
e
x x
x
and in lateral direction yields:
j ( ) / k
)
(15)
( ) ( c (
)tan( ))( 1
e
y
jy(
) / k y
)
(16)
In equations 15 and 16 c represents the cohesion stress of the soil, the friction angle of the soil and k x
and k y the shear deformation moduli. Assuming that the wheel has a longitudinal slip , the longitudinal
shear displacement along the contact area j x in equation 16 can be estimated [5,6] by using the
longitudinal slip and wheel radius R 0 :
j ( ) R [ (
1 )(sin(
x
0
f
f
) sin( ))]
(17)
Similar the lateral shear displacement j y will depend on the slip angle and the wheel radius R 0 :
j ( ) R ( 1 )(
y
0
f
)tan( )
(18)

10
Adams/Tire
Using the Soft-Soil tire model
Figure 8 illustrates the shear stress as a function of soil deformation.
Figure 8 Measured shear stress compared to fitted stress (equation 15)
The longitudinal shear deformation modulus k x is defined as:
k
x
k k 1
x0 x
(19)
and the lateral shear deformation modulus k y :
k
y
k k 1
y0 y
Having the normal and shear stress for the rotating wheel, the tire-soil interaction contact forces and
moments can be calculated:
• Longitudinal force:

Using the Soft-Soil tire model
Elastic-plastic tire-soil contact
11
F
x
bR
f
r
{ ( )cos( ) (
)sin( )}d
x
(20)
• Lateral force:
F
y
f
bR ( )d
r
y
(21)
• Vertical load:
F
z
bR
f
r
{ ( )sin( ) (
)sin( )}d
x
(22)
• Overturning moment:
M x
0
(23)
• Rolling resistance moment:
M
y
2
bR
f
r
0
( )sin( )d
c
rol
F
z
(24)
with c rol the tire (internal) rolling resistance coefficient.
• Aligning moment:
M
z
f
2
bR ( )sin( )d
r
y
(25)
Tire deformation
In order to take the tire deflection into account the substitution circle approach is taken as was suggested
by Bekker [7]:

12
Adams/Tire
Using the Soft-Soil tire model
Figure 9
Substitution circle to account for tire deflection
At a certain penetration of the tire into the soil the tire deflection and sinkage can be determined by an
iteration process based on the fact that the vertical tire force and the force due to the sinkage must be
equal.
The tire force can be calculated with the tire stiffness C z and tire deflection f 0 by:
F
z,tire
C
z f 0
(26)
while the tire sinkage force is defined by equation (22), however, replacing the unloaded tire radius R 0
by the radius of the substitution circle R*.
Bekker [7] derived following relation in between the tire deflection f 0 and tire sinkage h:

Using the Soft-Soil tire model
Elastic-plastic tire-soil contact
13
*
R
f0
1
h
f
h
R 0
0
(27)
Elastic and Plastic deformation
Depending on the soil properties one part of the deformation is elastic and the remaining part is nonirreversible
(plastic deformation). The elastic deformation is calculated with by the soil stiffness C s at the
maximum normal stress max :
(
he
C
m
s
)
(28)
Multi-pass effect
When a tire has passed a certain spot of soil, a second tire will experience different soil properties when
rolling over that spot due to the plastic deformation of the soil by the first tire.
Therefore this Soft Soil tire model stores the elastic and plastic deformation of each tire as a function of
the contact point x,y coordinates. When a tire passes a point with plastic deformation caused by a
previous tire, the normal pressure calculation will account for the plastic deformation history.
Figure 10 explains the mechanism applied in this tire model [8]:
Assume two tires rolling after each other over the same spot of soil. The first tire will have a total
deformation h 1 existing of a plastic part h p1 and an elastic part h e1 . When a second tire passes the same
spot, the soil will first have an elastic deformation from A to B (= h e1 ) and then continue to follow the
normal pressure characteristic to point C. The plastic deformation of the second tire h p2 will be equal to
the total deformation h 2 subtracted with the elastic deformation h e2 .

14
Adams/Tire
Using the Soft-Soil tire model
Figure 10
Normal pressure characteristic for multi-pass approach
Note:
The tire model stores the x, y coordinates, the elastic and plastic deformation and tire width
of each tire. Because of the one-point of contact approach used in this Soft-Soil tire model,
the total stored plastic deformation will be applied for a next tire when its contact point
comes into the rut of a previous tire.
Visco-elastic tire-soil contact
Next to elastic-plastic deformation models for soft soil, also visco-elastic modeling approaches exist.
Wanjii e.o. [9] derived a visco-elastic model for the normal stress along the contact line in between the
tire and the soil. A three element Maxwell approach is used for a rigid wheel, see Figure 11.

Using the Soft-Soil tire model
Visco-elastic tire-soil contact
15
Figure 11
Three element Maxwell model for a rigid wheel on visco-elastic soil
For this model the normal stress in the contact in between tire and ground is:
x
A x
G1
2 2 G2VxTr
V T
( x ) ( xA
x ) (
xA
VxTr
) 1
e
x r
xA
2R0
R0
x
(29)
With
x
a
R
0
)
sin(
f
)
x
R
sin(
0
2
T r /
G
In which
T r is the relaxation time
is the viscosity of the soil

16
Adams/Tire
Using the Soft-Soil tire model
V x is the forward velocity of the tire
G 1 is the first elastic modulus
G 2 is the second elastic modulus
The longitudinal and lateral shear stresses are calculated using the equations 15 until and including 19 as
used for the elastic-plastic tire-soil model. Similar for the tire-ground interaction forces equation (20 -
24) are used.
For the multi-pass effect, the road deformation at the exit of the tire-soil contact (point B) and the time
of deformation occurrence is stored. When a second tire passes the same spot, the road deformation
corrected with the relaxation effect is taken to correct the road height input.
References:
1. Bekker, M.G., Off-the-road-locomotion, Ann Arbor, The University of Michigan Press, 1960.
2. G. Ishigami, A. Miwa, K. Nagatani, K. Yoshida, Terramechanics - Based Model for Steering
Maneuver of Planetary Exploration Rovers on Loose Soil, Journal of Field robotics 24(3), 233-
250 (2007), Wiley Periodicals, Inc.
3. Yoshida, K., Watanabe, T., Mizuna, N., Ishigami, G., Terramechanics - based analysis and
traction control of a lunar/planetary rover. In Proceedings of the Int. Conf. Of Field and Service
Robotics (FSR '03), Yamanashi, Japan.
4. Wong, J.Y., Reece, A., Prediction of rigid wheel performance based on the analysis of soil-wheel
stresses part I, performance driving rigid wheels, Journal of Terramechanics, 4, 81-98.
5. Janosi, Z. Hanamoto, B., The analytical determination of drawbar pull as a function of slip for
tracked vehicle in deformable soils, In proceedings of the 1 st Int. conf. on Terrain-Vehicle
systems, Torino, Italy.
6. Wong, J.Y., Theory of Ground Vehicles, John Wiley & Sons, Inc., second edition, 1993.
7. Bekker, M.G., Introduction to terrain-vehicle systems, Ann Arbor, The University of Michigan
Press, 1969.
8. AS 2 TM User's Guide, version 1.12, AESCO GbR, Hamburg.
9. S. Wanjii, T. Hiroma, Y. Ota, T. Kataoka, Predicition of Wheel Performance by Analysis of
Normal and Tangential Stress Distributions under the Wheel-Soil Interface, Journal of
Terramechanics, Vol. 34, No. 3, pp. 165-186, 1997.
10. Schmid, I.C., Interaction of Vehicle and Terrain Results from 10 Years Research at IKK, Journal
of Terramechanics, Vol. 32, No. 1, pp. 3-26, 1995.
11. Schmid, I.C., Aubel, Th., Der elastische Reifen auf nachgiebiger Fahrbahn - Rechenmodell im
Hinblick auf Reifendruckregelung, VDI Berichte nr. 916, 1991.
12. Faßbender, F., Simulation der Vertikaldynamik von Fahrzeugen auf Geländeböden mit STINA -
SOIL TIRE INTERFACE TO ADAMS einem Zusatzmodul für das Mehrkörperprogramm
ADAMS. Number 521 in Fortschritt-Berichte VDI Reihe 12. VDI Verlag, Düsseldorf, 2002.
Dissertation Universität der Bundeswehr Hamburg.

Using the Soft-Soil tire model
Feature and property overview of the Adams/Tire Soft Soil Tire model
17
Feature and property overview of the Adams/Tire Soft Soil Tire
model
• Two tire-road contact models:
• Elastic-plastic contact model
elastic tire: tire deflection is taken into account
multi-pass effect: road plastic deformation history is stored and taken into account when
another tire passes the same spot
• Visco-elastic contact model
rigid tire: no tire deflection
multi-pass effect: road viscous deformation is stored. The stored deformation reduced by the
relaxation effect is taken into account when another tire passes the same spot
• Tire effective rolling radius is defined similar to pac2002 tire model
• Tire properties are very basic (tire vertical stiffness and damping, unloaded radius, width and
effective rolling radius parameters)
• The existing Adams/Tire roads can be used, just an additional section with the soil properties is
required. These soil properties are valid for the whole road.
• Linearization of - F x characteristic during q-statics to ensure robust q-statics
• Linear vertical tire stiffness can be replaced by a (non-linear) deflection-load curve
• Scaling factors of road friction, tire cornering and longitudinal stiffness' are supported
• SMP (multi-thread, C++ solver) is supported
• Tire-road contact is a one-point contact
• Camber effects are not taken into account
• Overturning moment is not calculated
• No bulldozing effects
Example of the tire property file for the Soft-Soil Tire model:
$----------------------------------------------------------MDI_HEADER
[MDI_HEADER]
FILE_TYPE = 'tir'
FILE_VERSION = 2.0
FILE_FORMAT = 'ASCII'
(COMMENTS)
{comment_string}
'Tire - XXXXXX'
'Pressure - XXXXXX'
'Test Date - XXXXXX'
'Test tire'
'New File Format v2.1'
$---------------------------------------------------------------units

18
Adams/Tire
Using the Soft-Soil tire model
[UNITS]
LENGTH
= 'mm'
FORCE
= 'newton'
ANGLE
= 'degree'
MASS
= 'kg'
TIME
= 'sec'
$---------------------------------------------------------------model
! use mode 1 2
! --------------------------------------------------------------
! flexible wheel/tire with elastic-plastic road X
! rigid wheel/tire with visco-elastic road X
!
PROPERTY_FILE_FORMAT = 'SOFT-SOIL'
USE_MODE = 1.0
$-----------------------------------------------------------dimension
[DIMENSION]
UNLOADED_RADIUS = 309.9
WIDTH = 235.0
ASPECT_RATIO = 0.45
$-----------------------------------------------------------parameter
[PARAMETER]
NOMINAL_TIRE_LOAD = 4000
VERTICAL_STIFFNESS = 310.0
VERTICAL_DAMPING = 0.5
ROLLING_RESISTANCE = 0.01
BREFF = 8.4
DREFF = 0.27
FREFF = 0.07
$---------------------------------------------------------------shape
[SHAPE]
{radial width}
1.0 0.0
1.0 0.2
1.0 0.4
1.0 0.5
1.0 0.6
1.0 0.7
1.0 0.8
1.0 0.85
1.0 0.9
0.9 1.0
$----------------------------------------------------------load_curve
$ For a non-linear tire vertical stiffness
$ Maximum of 100 points
[DEFLECTION_LOAD_CURVE]
{pen fz}
0 0.0
1.0 212.0
2.0 428.0
3.0 648.0
5.0 1100.0
10.0 2300.0
20.0 5000.0
30.0 8100.0

Using the Soft-Soil tire model
Example of the required Soil properties in the Road Data File:
19
Example of the required Soil properties in the Road Data File:
Existing road data files can be used, but a 'SOIL_PROPERTIES' section has to be added:
$-----------------------------------------------------SOIL_PROPERTIES
[SOIL_PROPERTIES]
FRICTION_ANGLE = 37.2 $units: degree
COHESION_STRESS = 8.0E-4 $units: N/mm**2
SOIL_DEFORM_MOD_KX0 = 43.0 $units: mm
SOIL_DEFORM_MOD_KX1 = 0.6283 $units: mm/deg
SOIL_DEFORM_MOD_KY0 = 20.0 $units: mm
SOIL_DEFORM_MOD_KY1 = 0.2269 $units: mm/deg
!visco-elastic tire:
ELASTIC_MODULUS_G1 = 0.071E-3 $units: N/mm**3
ELASTIC_MODULUS_G2 = 1.072E-3 $units: N/mm**3
SOIL_VISCOSITY = 7.14E-3 $units: Ns/mm**3
!plastic-elastic tire:
PRESSURE_SINKAGE_KC = 1.37E-3 $units: N/mm**(n+1)
PRESSURE_SINKAGE_KFI = 8.14E-4 $units: N/mm**(n+2)
SINKAGE_EXPONENT = 1 $units: = n
SOIL_INTERACTION_A0 = 0.4 $units: -
SOIL_INTERACTION_A1 = 0.15 $units: -
SOIL_STIFFNESS = 8.14E-3 $units: N/mm**3
Symbols
B eff
b
c
c rol
D eff
C z
C s
f 0
effective rolling radius factor
tire/wheel width
cohesion
tire rolling resistance coefficient
effective rolling radius factor
tire vertical stiffness
soil stiffness
tire deflection
d
dimensionless tire deflection
f 0
f0,F
z 0
G 1
G 2
F eff
F x
nominal tire deflection
elastic modulus
elastic modulus
effective rolling radius factor
longitudinal force

20
Adams/Tire
Using the Soft-Soil tire model
F y
F z
F z0
h
h e
h p
k c
k
lateral force
vertical load
nominal tire load
sinkage
elastic deformation
plastic deformation
cohesive modulus
frictional modulus
k x soil deformation modulus
k y soil deformation modulus
M x overturning moment
M y rolling resistance moment
M z aligning moment
n sinkage component
p static stress
R e effective rolling radius
R 0 unloaded (free) tire/wheel radius
R l tire loaded radius
R* radius of substitution circle
T r relaxation time
V total tire/wheel speed
V r tire rolling velocity
V x tire/wheel forward speed (parallel to wheel plane)
V sx longitudinal slip speed
V sy lateral slip speed
slip angle
longitudinal slip
friction angle
normal stress
tire/wheel rotational speed

Using the Soft-Soil tire model
Symbols
21
x
y
f
r
longitudinal shear stress
lateral shear stress
wheel angle
wheel soil entry angle
wheel soil entry angle
viscosity of the soil

22
Adams/Tire
Using the Soft-Soil tire model