Two-to-one internal resonances in nonlinear two degree of freedom system with parametric and external excitations

A system of two degrees of freedom with quadratic and cubic nonlinearities and subjected to parametric and external excitations is studied and solved. The method of multiple scale perturbation technique (MSPT) is used to analyze the response of this system. Four ordinary differential equations are derived to describe the modulation of the amplitudes and phases of the two modes of vibrations for the principal parametric resonances. The steady-state solutions and their stability are determined and representative numerical results are included. The theoretical resonance cases of this system have been obtained from the first approximation differential equations and some of them are confirmed by applying well-known numerical technique. The stability of the obtained numerical solution is studied using phase–plane method. The effects of different parameters on the vibration of this system are investigated.