Non-smooth dynamics and sensitivity analysis of multibody systems with clearances and friction in differential variational inequality framework

Abstract

Mathematical representations of multibody systems can be classified into two categories based on the type of constraints they carry. Ideal constraints in the multibody systems are the ones which enforce absolute alignment of bodies with respect to each other in the desired direction of motion. However, clearances in mechanical joints are integral to all real-world multibody systems. The clearances between the bodies allow the bodies to undergo a certain misalignment and the dynamics is governed by the contacts thus formed. The conventional formulations of joints as smooth algebraic constraints ignore the effect of clearances. To deal with contacts, and hence clearances, while the popular continuous dynamics approaches assume the Hertzian nature of the contact modeled by nonlinear unilateral spring-damper elements, the non-smooth dynamics approach results in a differential variational inequality (DVI) problem with complementarity constraints. This work provides a comprehensive re view of key contributions in the field of non-smooth dynamics and describes a framework to implement the non-smooth dynamics approach to fundamental mechanical joints with clearances with dry friction. Since clearances are small compared to the dimensions of me chanical components, contact is assumed to be inelastic. Based on this assumption and the general non-smooth dynamics framework, parametric formulations are derived for cylindri cal, prismatic, and revolute joints with clearances. The equations of motion of systems with clearances and friction, and their time-discretized counterparts are derived as second or der cone-programming (SOCP) problem. Although the SOCP framework can also simulate systems with ideal joints, eliminating friction carrying elements results in a condensed con vex quadratic programming (QP) form. Both formulations are compared for systems with frictionless joints with clearances. Additionally, special focus is given on rotation preserv ing Lie-group integration schemes for smooth as well as non-smooth systems, to circumvent the normalization constraint on quaternions. The problem formulations are assessed through multiple case-studies, demonstrating the versatility of the non-smooth formulation, and a de tailed analysis of the results is presented. With a unified framework available for simulating frictional contacts as well as ideal-joints with friction, sensitivity analysis equations are de rived forming a generalized sensitivity analysis framework for smooth as well as non-smooth systems. In addition, as another step towards making a robust framework, higher-order time integration is explored in differential algebraic equations on Lie-groups, using a time-finite element approach.