Modeling and analysis of multiple impacts in multibody systems under unilateral and bilateral constrains based on linear projection operators

This paper presents a unifying dynamics formulation for nonsmooth multibody systems (MBSs) subject to changing topology and multiple impacts based on a linear projection operator. An oblique projection matrix ubiquitously yields all characteristic variables of such systems as follows: (i) the constrained acceleration before jump discontinuity from the projection of unconstrained acceleration, (ii) post-impact velocity from the projection of pre-impact velocity, (iii) impulse during impact from the projection of pre-impact momentum, (iv) generalized constraint force from the projection of generalized input force, and (v) post-impact kinetic energy from pre-impact kinetic energy based on projected inertia matrix. All solutions are presented in closed-form with elegant geometrical interpretations. The formulation is general enough to be applicable to MBSs subject to simultaneous multiple impacts with nonidentical restitution coefficients, changing topology, i.e., unilateral constraints becoming inactive or vice versa, or even when the overall constraint Jacobian becomes singular. Not only do the solutions always exist regardless of the constraint condition, but also the condition number for a generalized constraint inertia matrix is minimized in order to reduce numerical sensitivity in computation of the projection matrix to roundoff errors. The model is proven to be energetically consistent if a global restitution coefficient is assumed. In the case of nonidentical restitution coefficients, the set of energetically consistent restitution matrices is characterized by using a linear matrix inequality (LMI).