Dynamics of a single-degree-of-freedom structure with quadratic, cubic and quartic non-linearities to a harmonic resonance

摘要

The solution of a non-linear equation having often encountered in the study of structural dynamically and in which both parametric and external excitations are present, have been obtained analytically and numerically. The only considered case is when both the parametric and external excitations have the same frequency. The method of multiple time scale is applied to determine the equations that describe up to and including the second order both the modulation of the amplitude and phase. These equations are used to determine the fixed points and their stability. Frequency response curves are presented graphically. Discussion of the results is given. Numerical solutions are presented for the effects of the different equation parameters on system stability, response and chaos. Different cases of resonance are investigated and reported.