Approximation of the frequency-energy dependence in the nonlinear dynamical systems

In this work, a method is introduced for extracting the approximate backbone branches of the frequency-energy plot from the numerical simulation response of the nonlinear dynamical system. The duffing oscillator is firstly considered to describe the method and later a linear oscillator (LO) coupled with a nonlinear energy sink (NES) is also considered for further demonstration. The systems of concern are numerically simulated at an arbitrary high level of initial input energy. Accordingly, the obtained responses of these systems are employed via the proposed method to extract an approximation for the fundamental backbone branches of 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 still valid for any other initial energy below that level. In addition, they are not affected by the damping variations in the considered systems. Unlike other existing methods, the proposed approach is applicable to well-approximate the backbone branches of the large-scale nonlinear dynamical systems.