Abstract
AbstractIn this paper we use step size adjustment and extrapolation methods to improve Moreau's time‐stepping scheme for the numerical integration of non‐smooth mechanical systems, i.e. systems with impact and friction. The scheme yields a system of inclusions, which is transformed into a system of projective equations. These equations are solved iteratively. Switching points are time instants for which the structure of the mechanical system changes, for example, time instants for which a sticking friction element begins to slide. We show how switching points can be localized and how these points can be resolved by choosing a minimal step size. In order to improve the integration of non‐smooth systems in the smooth parts, we show how the time‐stepping method can be used as a base integration scheme for extrapolation methods, which allow for an increase in the integration order. Switching points are processed by a small time step, while time intervals during which the structure of the system does not change are computed with a larger step size and improved integration order. The overall algorithm, which consists of a time‐stepping module, an extrapolation module and a step size adjustment module, is discussed in detail and some examples are given. Copyright © 2008 John Wiley & Sons, Ltd.
Highlights
The modelling of mechanical systems with unilateral contacts and friction is a challenging task, which is studied in the field of elastic as well as rigid body mechanics
One possibility is to regularize the non-smooth force laws that are associated with the unilateral contact and friction, i.e. the contact and the friction forces are defined as a function of the penetration depth or volume and the sliding velocity, respectively, see [8,9,10,11]
The presented algorithm provides switching point search and higher integration order in the smooth parts of the motion, but because the base scheme is still the time-stepping scheme, it is suited for systems with more than a few contacts
Summary
The modelling of mechanical systems with unilateral contacts and friction is a challenging task, which is studied in the field of elastic as well as rigid body mechanics. One possibility is to regularize the non-smooth force laws that are associated with the unilateral contact and friction, i.e. the contact and the friction forces are defined as a function of the penetration depth or volume and the sliding velocity, respectively, see [8,9,10,11] Such an approach leads to an ordinary differential equation system, which can be simulated straightforwardly using a classical numerical scheme for stiff systems, for example, an implicit Runge–Kutta scheme. The determination of suitable regularization parameters can pose a problem Another approach to handle mechanical systems with unilateral contacts and friction is the non-smooth approach, which was originated by Moreau (see [12,13,14]). The aim of this paper is to improve the accuracy of Moreau’s integration scheme by using step size adjustment and extrapolation [27, 28]
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