Use of the Generalized Impulse Momentum Equations in Analysis of Wave Propagation

A procedure is developed in this paper to study the propagation of impact-induced axial waves in constrained beams that undergo large rigid body displacements. The solution of the wave equations is obtained using the Fourier method. Kinematic conditions that describe mechanical joints in the system are formulated using a set of nonlinear algebraic constraint equations that are introduced to the dynamic formulation using the vector of Lagrange multipliers. The initial conditions which represent the jump discontinuity in the elastic coordinates as the result of impact are predicted using the generalized impulse momentum equations that involve the coefficient of restitution as well as the Jacobian matrix of the kinematic constraints. The convergence of the series solutions presented in this paper is examined and the analytical and numerical results are found to be consistent with the solutions obtained by the use of the classical theory of elasticity in the case of plastic impact. The cases in which the coefficient of restitution is different from zero are also examined and it is shown that the generalized impulse momentum equations can be used with confidence to study the propagation of elastic waves in applications related to multibody dynamics.