Friction
The simulation of driving and braking traction
would be an important extension of this research.
Forces parallel to the interface are transmitted
More-elaborate friction models could be generated
based on a friction law. Friction is a complex phe-
based on the multitude of experimental traction data
nomenon involving stress and strain distributions;
obtained by CRREL. However, this need for a trac-
heat transfer; hydrodynamic and elastohydrodynamic
tion curve a priori seems to negate the usefulness of
fluid flow; material structure; chemical interactions
the model for predicting traction (i.e. the traction
between surfaces, surface coatings, and lubricants;
input is needed to get a valid traction output). Thus,
and phase change. Current models of friction are em-
the current modeling effort focused on the tire forces
pirical or semi-empirical formulations in which sev-
developed due to deformation of the snow rather than
eral variables and combinations of variables are
due to the interaction at the interface (i.e. a model of
modified with coefficients and exponents based on
rolling resistance rather than traction). Therefore, for
experimental observations. These equations apply
this project the simple Coulomb friction model was
only to the system for which they were developed,
used:
though they may be used to estimate the behavior of
similar systems. They are not applicable to other ex-
= τ .
perimental configurations or test conditions and can-
(27)
σ
not be extrapolated outside of their bounds with any
degree of confidence. Thus, an accurate model of
Rigid wheel in snow
friction is currently non-existent and is not likely to
be developed in the immediate future. The only rea-
Model construction
sonable way to predict the behavior of most frictional
The snowwheel model simulates one wheel of
systems today is to test them (Ludema 1996a, 1996b,
the CRREL Instrumented Vehicle moving through a
Barber 1991).
range of shallow (20 cm) to deep (50 cm) fresh snow
The experimental measurement of tireterrain fric-
with a density of 200 kg/m3. The wheel is modeled as
tion (including pavement) is typically performed as a
a rigid surface 0.74 m in diameter and 0.272 m wide
vehicle traction or braking test [a discussion of winter
(for a full wheel) with a 0.051-m radius of curvature
traction test methods is available in Shoop et al.
on the tire shoulder. An unsprung mass of 636 kg is
(1994)]. Test data are generally reported as a traction
placed at the hub to simulate the weight of the vehi-
(or friction) coefficient (a peak value or an averaged
cle and tire; the rotational inertia of the wheel was set
value over a specified range) or as a traction curve
to 2.15 kg m2. The snow was modeled with eight-
with respect to wheel slip. Traction coefficients and
node continuum elements near the wheel and four-
curve shapes are functions of both the tire and the
node infinite elements for the far-field snow cover.
surface. Curves of traction data with wheel slip can
The DruckerPrager cap material model with linear
be implemented in ABAQUS or other finite element
elasticity was used for the snow, and a linear elastic
analysis codes by specifying friction as a function of
model was used for the infinite elements. Coulomb
relative velocity. Liu and Wong (1996) took this ap-
friction was applied at the tiresnow interface ( =
proach using a friction law of the following form and
0.3). The deep snow was modeled using Adaptive
implementing it in a tireterrain model using the gen-
LagrangianEularian (ALE) meshing to accommo-
eral-purpose finite element program called MARC:
date the large displacements, so slide planes were not
needed along the sides of the tire. The shallow snow
-j k
τ = σ(1 - e
(26)
)
model had 11,520 continuum elements and 1,418
infinite elements; the deep snow model had 14,400
where
=
friction coefficient
continuum and 1,962 infinite elements. In all models
τ
=
shear stress
the snow rested on a rigid surface. The shallow snow
σ
=
normal stress
model is shown in Figure 56.
j
=
relative slip distance between the
The rolling resistance test is simulated by first
wheel and the terrain
lowering the wheel into the snow by gravity, accelerat-
k = constant having the same units as j.
ing it to the desired speed, and then translating it at a
This equation is of the type commonly used in
constant velocity. The wheel is moved longitudinally
conjunction with soil shear data. However, Liu and
by displacing the axle node. This simulates a towed
Wong were not convinced that this equation ade-
wheel and duplicates the procedure of a rolling resis-
quately described traction.
tance test using the CRREL Instrumented Vehicle.
42
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