An algorithm, based upon energy and force equilibrium, is proposed to evaluate the frequency response characteristics of nonlinear mechanical systems with symmetric and asymmetric dissipative and restoring elements. Dual stage dampers and hardening springs with symmetric or asymmetric characteristics are characterized by an array of local equivalent linear coefficients, using the methodology based upon principle of energy similarity. Frequency response characteristics of a dynamical system comprising nonlinear symmetric and asymmetric damping and hardening spring are evaluated, and compared to those determined from direct integration of the nonlinear system equations. An error analysis is performed to determine the magnitude of error between the nonlinear and local equivalent linear forces. The average force error is then integrated within the local equivalent algorithm to achieve the force similarity between the nonlinear and linear elements. The results revealed that the proposed algorithm can accurately estimate the frequency response characteristics of the nonlinear systems in a highly efficient manner. The algorithm based upon energy similarity alone yields slight error near the resonant frequencies, while the analysis based on the combined energy and force similarity can reduce this error considerably.