Abstract
Nonsmooth mechanical systems, which are mechanical systems involving dry friction and rigid unilateral contact, are usually described as differential inclusions (DIs), that is, differential equations involving discontinuities. Those DIs may be approximated by ordinary differential equations (ODEs) by simply smoothing the discontinuities. Such approximations, however, can produce unrealistic behaviors because the discontinuous natures of the original DIs are lost. This paper presents a new algebraic procedure to approximate DIs describing nonsmooth mechanical systems by ODEs with preserving the discontinuities. The procedure is based on the fact that the DIs can be approximated by differential algebraic inclusions (DAIs), and thus they can be equivalently rewritten as ODEs. The procedure is illustrated by some examples of nonsmooth mechanical systems with simulation results obtained by the fourth-order Runge-Kutta method.
Highlights
Mechanical systems involving dry friction and rigid unilateral contact are usually described as differential inclusions (DIs)
Some previous friction models [5,6,7,8,9] and contact models [10,11,12,13] can be viewed as approximations of dry friction and rigid unilateral contact, respectively
This paper has introduced a new method to approximate DIs describing nonsmooth mechanical systems involving dry friction and rigid unilateral contact by ordinary differential equations (ODEs)
Summary
Mechanical systems involving dry friction and rigid unilateral contact are usually described as differential inclusions (DIs). Some previous friction models [5,6,7,8,9] and contact models [10,11,12,13] can be viewed as approximations of dry friction and rigid unilateral contact, respectively Physical meanings of such approximations can be usually interpreted as relaxation of constraints, that is, compliance that replaces rigid constraints between force and motion. The discretized equation is regarded as an algebraic equation, which is solved numerically [14,15,16,17,18,19,20,21,22] or analytically ([23, Section III.A], [24, Section 1.4.3.2], and [25]) Another type of approach (e.g., [26]) is to describe a system as an ODE in every period between discontinuous events such as transitions between static and kinetic friction states.
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