An approach for accurate analytical solution of a two degree-of-freedom nonlinear dynamical system coupled with a strongly nonlinear restoring force is presented here. The approach is based on the application of the local equivalent linear stiffness method (LELSM) to linearize the nonlinear coupling stiffness in the system based on the nonlinear frequency calculation. Consequently, the system can be decoupled into two forced single degree-of-freedom subsystems by replacing the nonlinear coupling force with a forcing function where the solution can be analytically obtained. Different combinations of the positive and negative linear and cubic stiffness components are considered in the nonlinear coupling force. For all considered stiffness combinations, the obtained analytical solution strongly agrees with the numerical simulation of the system. In addition, the internal resonance is found not to significantly affect the accuracy of the analytical solution.
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