Periodic Solutions of a Modified Duffing Equation Subjected to a Bi-Harmonic Parametric and External Excitations

In this paper, we investigated the periodic solutions of type superharmonic and subsuperharmonic of modified Duffing equation subjected to a bi-harmonic parametric and external excitations. The method of multiple scales is used to construct a first order uniform expansion of approximate solutions. Two first-order nonlinear ordinary differential equations(Modulation Equation) are derived from the evolution of the amplitude and the phase. Steady state solutions and their stability are given for selected values of the system parameters. The consequences of these (quadratic and cubic) nonlinearities on these the vibrations are particularly examined. With this research, it has been confirmed that the qualitative effects of these nonlinearities are different. Regions of the hard (soft) nonlinearity of the system exist for the case of subsuperharmonic oscillation. Numerical solutions are presented in a group of figures which demonstrate the actions of the steady-state reaction plenitude as the purpose of the detuning parameter.