Nonlinear free vibration of systems with inertia and static type cubic nonlinearities: An analytical approach

• A new application of Hamiltonian approach is proposed.
• We present a method for substituting of these nonlinear springs with a spring.
• The results of Hamiltonian approach compare with the exact solution and discussed.
• The effects of important parameters on the frequency of the system were considered.

The present paper deals with the new application of a powerful analytical method to a system with inertia and static type cubic nonlinearities. The free vibration of a mass grounded by linear and nonlinear springs in series is studied. A nonlinear ordinary differential equation with inertia and static type cubic nonlinearities represents the governing equation of the system. The new approach does not have the limitation of the traditional perturbation method which is valid for conservative systems with small perturbed parameters. An attempt has been made to provide simple approximate analytical expressions valid for small as well as large amplitudes of oscillation. The exact solution is also presented and discussed to validate the present analysis. In the paper the method for determining the equivalent rigidity for the serial connected nonlinear springs is developed.