Nonlinear asymptotic analysis of a system of two free coupled oscillators with cubic nonlinearities

In this paper a two degrees of freedom undamped nonlinear system of two unforced coupled oscillators with cubic nonlinearities is analyzed. Through a decoupling procedure and using admissible functional transformations it is proved that this system can be reduced to an intermediate second order nonlinear ordinary differential equation (ODE) connecting both displacements to each other. By nonlinear asymptotic approximations the above equation can be further reduced to new nonlinear ODE that can be analytically solved. The solutions in the physical plane are extracted in parametric form. As generalization, the model of a damped system of two masses connected with stiffness with linear and nonlinear coefficient of rigidities respectively is analyzed and exact analytical solutions are extracted. Finally an application has been given in the case of a two mass system with cubic strong non-linearity.