Extended Abstract
The 4th Joint International Conference on Multibody System Dynamics
May 29 – June 2, 2016, Montréal, Canada
Modeling and numerical simulation of anisotropic dry friction with
non-convex friction force reservoir
Simon V. Walker1 and Remco I. Leine1
1 Institute for Nonlinear Mechanics, University of Stuttgart, {walker,leine}@inm.uni-stuttgart.de
In this work an anisotropic dry friction model for non-convex friction force reservoirs is developed in the
framework of convex analysis. The model is implemented in a time-stepping scheme allowing for numerical
simulation of systems undergoing anisotropic dry friction during stick and slip.
Anisotropic dry friction occurs in various technical applications. Although it might be exploited for specific
applications like snake robots [1], it is often an unwanted side effect, which can not be neglected. For example,
machining or finishing of a surface can lead to an anisotropic surface roughness which in turn causes anisotropic
frictional behavior. Anisotropy may also originate from the crystal structure of a material and occurs on the
surface of composite, textile and biological materials. The classical Coulomb friction law describes the constitutive
behavior of stick and slip. Regularizations of the Coulomb friction law lack the ability to describe stiction, which
motivates the use of set-valued force laws. This non-smooth approach makes use of methods of convex analysis
such as normal cone inclusions. The normal cone NC (x) to a convex set C is zero for any argument x ∈ C and
returns the set of outward normal rays for an argument at the boundary of C . Using γ T as the relative sliding
velocity and λT as the friction force between two bodies, the spatial Coulomb friction law can be written as
γ T ∈ NC (−λT ).
(1)
In the isotropic case, the force reservoir C containing
all admissible friction
forces is a disc defined by the friction
2
coefficient µ and the normal force λN , C = −λT ∈ R | kλT k ≤ µλN . Assuming the principle of maximal
dissipation to hold, the Coulomb friction law is readily extended to the anisotropic case by choosing a non-circular
force reservoir C . Again, a normal cone inclusion force law in the form of Eq. (1) is obtained [2]. For orthotropic
friction, an ellipsoidal force reservoir is often assumed. Figure 1 depicts a body sliding over an anisotropic surface
with the sliding velocity γ T . On the right, an ellipsoidal force reservoir and the relationship between the sliding
direction and the friction force according to Eq. (1) are shown. The sliding direction and the friction force are not
collinear, which causes the sliding body to deflect.
Fig. 1: Left: Body sliding over anisotropic surface. Right: Normal cone inclusion force law with elliptical force reservoir.
The ellipsoidal force reservoir and the principle of maximal dissipation are constitutive assumptions and therefore not necessarily valid for measured anisotropic frictional behavior. The asperity model introduced in [3], where
frictional anisotropy is imposed through uniformly distributed wedge-shaped asperities having isotropic friction
properties, motivates the use of non-convex friction force reservoirs. Here, the sliding direction no longer lies in
the normal cone of the force reservoir C , but is defined by normality to a convex set D. In this work an anisotropic
dry friction model, based on the asperity model developed in [3, 4], is formulated within convex analysis. Our
work thereby extends the model of [4], which is limited to ellipsoidal sets C and D, to non-convex star-shaped
force reservoirs C and arbitrary convex sets D. The normal cone inclusion force law
γ T ∈ ND (−αλT )
(2)
is introduced, where the parameter α is used to scale the friction force such that it lies on the boundary of D in
the sliding case. Convex and non-convex but star-shaped sets can be described as level sets of nonnegative, lower
semicontinuous and positively homogeneous gauge functions, e.g. C = {−λT | kC (−λT ) ≤ 1}. Using the gauge
functions kC and kD , the scaling parameter α is determined by
α=
1
.
kD (−λT ) − kC (−λT ) + 1
(3)
Force laws given as normal cone inclusions can be transformed into implicit equations using the proximal
point function and be used for numerical simulations. The resulting contact problem is solved iteratively. In each
iteration, the parameter α is adapted. For time integration, the time-stepping method of Moreau [5] is used.
Fig. 2: Left: Force law with generic non-convex force reservoir. Right: Sliding paths for different force laws.
The graph of a force law with generic non-convex force reservoir and the corresponding sliding directions is
shown on the left of Fig. 2. In order to compare various force laws, a body with initial velocity u0 sliding on a
horizontal orthotropic surface is considered. The friction coefficients µ1 and µ2 along the semi-axes are the same
for all cases, with µ1 > µ2 . On the right, Fig. 2 depicts the numerical results of the sliding paths until stick for
different force laws. Naturally, the collinear force law leads to a linear sliding path, whereas all other force laws
cause a deflection of the body in the direction of µ2 .
The capability of the proposed normal cone inclusion force law to handle non-convex star-shaped force reservoirs is shown. The set-valued force law accounts for stiction and enables the description of a large class of
anisotropic dry friction models.
References
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Archiv. Mech., vol. 30, pp. 259–276, 1978.
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