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 sy stem with cubic non-linearities and auto-parametric resonance omega(3) cong ruent to 3 omega(2) and omega(2) congruent to 3 omega(1) to a harmonic exci tation of the third mode, where the omega(n) are the linear natural frequen cies of the system. The method of multiple scale is applied to determine si x first-order non-linear ordinary differential equations that govern the ti me variation of the amplitudes and phases of the interacting modes. The fix ed points of the three equations are obtained and their stability are deter mined. Numerical solutions are conducted to obtain the response of the thre e modes. The effects of the different parameters on both response and stabi lity of the system are investigated. The obtained results are discussed fol lowed by the main conclusions.