A solution method for dynamic analysis of elastic contact problems with rigid body motion under small deformation is presented. The contact surface is assumed unbonded and frictionless. The problem is first described by partial differential equations with inequalities denoting the geometric compatibility condition on the contact surface. A variational statement constrained by these side conditions is then proposed. The equivalence of the two descriptions is shown by considering the necessary conditions of the variational statement. An incremental from is obtained using the rigid body motion described by a moving coordinate system attached to the body. The geometric compatability conditions are accordingly linearized. The deforming body is discretized by the finite element method, while the nodal displacements are approximated by admissible basis functions over time increments. A possible discontinuity in the velocity due to contact is allowed in the representation. The Lagrange multiplier technique is employed to impose the geometric compatability condition of contact. For numerical implementation, several contact check points are selected conveniently at nodal points on the contact surface. In the time domain, the time point to check the contact condition is chosen at the final time of a time step. If the discontinuity in the velocity is negative, this discontinuity is considered occurred at the initial time in this time step. The resulting discretized system is formulated in the form of a linear complementarity problem, suitable for numerical calculation. Two examples are considered to show the implementations. The first one is about a longitudinal impact of two elastic rods for which an analytical solution is available. The obained contact force shows good agreement with the analytical solution. The second example is about the impact of an elastic sphere with a rigid plane. The results are discussed comparing with the quasi-static solutions by the Hertz model. It is observed that numerically obtained maximum contact forces are larger than that of the quasi-static solution and calculated duration of impact is shorter. Although the emphasis in the presentation has been on the analysis of two body contact, the method is directly applicable to multibody contact problems with arbitrarily selected check points. To apply the method to more practical problem, however, frictional conditions must be included with additional complexity.
