Elastomeric bushings are structural elements used in automotive suspension systems. A bushing is a hollow cylinder between an outer steel cylindrical sleeve and an inner steel cylindrical rod. The steel sleeve is connected to components of the suspension system and transfers forces from wheel to chassis. Force-Displacement relation for elastomeric bushings is critical for multi-body dynamics simulations. A boundary value issue for bushing response leads to force-displacement relation that requires extensive computation time to implement and therefore is unsuitable. Explicit force-displacement relation may be used in multi-body dynamics simulations. The relation is expressed in terms of a force relaxation function. Lianis model, Modified Lianis model, and Pipkin-Rogers model are introduced. Modified Pipkin-Rogers model was proposed and a boundary value issue was formulated for axial mode bushing response. Numerical solutions of the boundary value issue of Modified Pipkin-Rogers model were compared with results of the Pipkin-Rogers model. It is revealed that the method for determining bushing relaxation function and prediction of proposed force-displacement relation is in agreement with the original results.