Resonance of multiple frequency excitated systems with quadratic, cubic and quartic non-linearity

This paper deals with certain forms of non-linear oscillations of a vibratory system with quadratic, cubic and quartic non-linearities subjected to a sinusoidal excitation that involves multiple frequencies. The theoretical results are obtained by the multiple scale method. Two first-order ordinary differential equations are derived for the variation of the amplitude and phase. The steady-state amplitude for each case is plotted, showing the influence of different parameters. Stability analysis is carried out for each case. Simulations are used to compare between the analytic and numerical solutions.