Parametric formulations of spatial joints with clearances: A non-smooth dynamics approach

The conventional approach of simulating multibody dynamic systems treats the joint interfaces as ideal, that means that the bodies are in absolute alignment with each other in the desired relative directions of motion. However, in real life systems the clearances between the bodies allow the bodies to undergo a certain misalignment and the dynamics is governed by the contacts thus formed. Contact detection and evaluation of contact forces is yet another problem that needs to be addressed. Popular approaches assume the Hertzian nature of the contact and thus evaluate contact forces using nonlinear unilateral spring-damper elements. This approach results in very stiff differential algebraic equations and hence make the numerical integration computationally expensive. Furthermore, the Hertzian approach does not address truly elastic or truly inelastic nature of the contact. This work describes the parametric formulations for fundamental spatial joints with clearances and the non-smooth dynamics approach to solve the resulting equations of motion. The sets of spatial joint expressions for cylindrical, prismatic and revolute joints, and the non-smooth dynamics formulations are derived, considering their interdependence with great care. Further, the nature of the contact with clearances is discussed. The formulation is demonstrated through three case-studies and a detailed analysis of the results is presented. Additionally, a differentiation with respect to the ideal joint counterpart of the revolute joint case study is presented using tangent space ordinary differential equation formulation.