Elastomeric bushings are structural elements and used in automotive suspension systems. An idealized bushing is an elastomeric hollow cylinder between an outer steel cylindrical sleeve and an inner steel cylindrical rod. The outer sleeve is connected with the components of the suspension system and transfer forces and moments from the wheel to the chassis. The accurate determination of the transmitted forces and moments among autocomponents, the motion of the components, the stresses in the components, and energy dissipation are affected by the quality of the elastomeric bushing model. Force- Displacement relation, moment-rotational angle relation, and coupled relations for elastomeric bushings are imperative for multi-body dynamics simulations. The boundary value problem in the bushing response leads to force-displacement relation and moment-rotational angle relations which require extensive computation time for implementation. Herein, an explicit moment-rotational angle relation has been introduced for use in multi-body dynamics simulations, a modified Pipkin-Rogers model is proposed and a boundary value problem is formulated for torsional mode elastomeric bushing response. Lianis" experimental equation and Pipkin-Rogers model are used for numerical experimental research. The proposed method and the prediction of the proposed moment-rotational angle relation are observed to exhibit excellent agreement with the original results.