Analytical treatment for bistable nonlinearly coupled Oscillators

Here, we introduce an analytical approximation to the exact solution of a bistable nonlinearly coupled oscillators (NLC-LOs) to study the internal resonance at the nonlinear normal modes (NNMs). The considered system is composed of two symmetrical linear oscillators coupled by a bistable nonlinear coupling restoring force. The coupling restoring force includes negative and nonnegative linear and nonlinear stiffness components. The introduced approximate analytical solution for the considered bistable NLC-LOs system is mainly proposed for the cases of which the exact frequency and the exact solution are neither available nor valid. The proposed solution depends on the application of the local equivalent linear stiffness method (LELSM) to linearize the nonlinear coupling force according to the non-linear frequency content in the original system. Accordingly, the bistable nonlinear coupling force in the NLC-LOs is replaced by an equivalent periodic forcing function of which the frequency is equal to that of the original NLC-LOs system. Therefore, the original NLC-LOs system is decoupled into two forced single degree-of-freedom subsystems where the analytical solution can be directly obtained. This obtained analytical solution is found to be highly accurate approximation for the exact solution, especially at internal resonances that occur on some NNMs of the system.