Abstract
In this paper, the nonsmooth dynamics of two contacting rigid bodies is analysed in the presence of dry friction. In three dimensions, slipping can occur in continuously many directions. Then, the Coulomb friction model leads to a system of differential equations, which has a codimension-2 discontinuity set in the phase space. The new theory of extended Filippov systems is applied to analyse the dynamics of a rigid body moving on a fixed rigid plane to explore the possible transitions between the slipping and rolling behaviour. The paper focuses on finding the so-called limit directions of the slipping equations at the discontinuity. This leads to a complete qualitative description of the possible scenarios of the dynamics in the vicinity of the discontinuity. It is shown that the new approach consistently extends the information provided from the static friction force of the rolling behaviour. The methods are demonstrated on an application example.
Highlights
If dry friction is assumed between the surfaces of rigid bodies, the dynamical model of the bodies leads toM
We focus on the transitions between the slipping and rolling dynamics when applying the theory of extended Filippov systems
The slipping of the bodies in the presence of Coulomb friction leads to extended Filippov systems, and the rolling or sticking of the bodies corresponds to the sliding dynamics inside the discontinuity manifold
Summary
If dry friction is assumed between the surfaces of rigid bodies, the dynamical model of the bodies leads to. The direct substitution of the discontinuous friction models into the dynamical equations leads to discontinuous systems of differential equations. In the 2D case, the Coulomb friction leads to Filippov systems (for an overview and examples, see [7]). The generalization of the Filippov systems to codimension-2 discontinuity sets in the phase space leads to the concept of extended Filippov systems (see [1] and [4]) This type of differential equation can be used for modelling and analysis of 3D mechanical systems with dry friction, which was demonstrated in specific mechanical examples in [4]. We focus on the transitions between the slipping and rolling dynamics when applying the theory of extended Filippov systems. 4, where the theory of extended Filippov systems is applied to the dynamics of the moving body.
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