Af. El-bassiouny et M. Eissa, Response of three-degree-of-freedom system with cubic non-linearities to harmonic excitations, PHYS SCR, 59(3), 1999, pp. 183-194
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.