Response of three-degree-of-freedom system with cubic non-linearities to harmonic excitations

An investigation is presented of the response of three-degree-of-freedom system with cubic non-linearities and auto-parametric resonance omega(3) congruent to 3 omega(2) and omega(2) congruent to 3 omega(1) to a harmonic excitation of the third mode, where the omega(n) are the linear natural frequencies of the system. The method of multiple scale is applied to determine six first-order non-linear ordinary differential equations that govern the time variation of the amplitudes and phases of the interacting modes. The fixed points of the three equations are obtained and their stability are determined. Numerical solutions are conducted to obtain the response of the three modes. The effects of the different parameters on both response and stability of the system are investigated. The obtained results are discussed followed by the main conclusions.