Time-varying stiffness method for extracting the frequency–energy dependence in the nonlinear dynamical systems

In this work, a time-varying stiffness method is proposed for extracting highly accurate approximation for the fundamental backbone branches of the frequency–energy plot from the numerical simulation response of the nonlinear dynamical system. The purely nonlinear duffing oscillator with a nonnegative real power restoring force is firstly considered to develop the method, and later the method is applied to linear systems attached with a nonlinear energy sink for more demonstration. The systems of concern are numerically simulated at an arbitrary high level of initial input energy to apply the proposed method. Accordingly, the obtained responses of these systems are employed via the proposed time-varying stiffness method to extract an approximation for the fundamental backbone branches in the frequency–energy plot. The obtained backbones have been found in excellent agreement with the exact backbones of the considered systems. Even though these approximate backbones have been obtained for only one high energy level, they are valid for any other initial energy below that level. In addition, they are not affected by the damping variations in the considered systems. The proposed method is found to be applicable to well approximate the fundamental backbone branches of the large-scale nonlinear dynamical systems. The frequency–amplitude dependences have been also studied here for the considered systems.