On the Use of the Momentum Balance in the Impact Analysis of Constrained Elastic Systems

In elastic systems, impulsive forces that act at a point on a deformable body produce stress waves that travel with finite speeds. This paper examines, both theoretically and numerically, the validity of using the generalized impulse momentum approach in modeling impact or collisions in the constrained motion of deformable bodies. The generalized impulse momentum equations that involve the coefficient of restitution and the kinematic constraint Jacobian matrix are used to predict the jump discontinuity in the velocity vector as well as the joint reaction forces. The series solutions obtained by solving these algebraic equations are used to establish a closed form relationship between the jump discontinuity in velocities and joint reactions due to impact and the number of elastic degrees of freedom. It is shown that by increasing the number of elastic coordinates these series converge to their limits. The convergence of these series is used to prove that the generalized impulse momentum equations with the coefficient of restitution can be used with confidence to study impact problems in constrained multibody systems consisting of interconnected rigid and deformable bodies. The results obtained are compared with the classical treatment of the impact problems in the theory of elasticity wherein the case of perfectly plastic impact is assumed.