The stability analysis of a slider-crank mechanism due to the existence of two-component parametric resonance

The objective of this paper is an analytical and numerical study of the dynamics and dynamic instability of a slider!crank mechanism with an inextensible elastic coupler[ Special attention is given to the phenomena arising due to modal interactions produced by the existence of multi!component\ speci_cally two!componentparametric resonance[ Such modal couplings are very common in the bending!bending motions of _xed: rotating beams[ The two!component parametric resonance occurs when one of the natural frequencies of ~exible parts of the mechanism is one!half or twice of the excitation frequency and simultaneously the sums or the di}erences among the internal frequencies are the same\ or neighboring\ as the frequency of excitation[ The e}ects of two!component parametric resonance post on instability condition are also investigated[ Resonance generated by more than two component modes are neglected due to its remote probability of occurrence in nature[ The mechanics of the problem is Newtonian[ Methods of analysis will consist of the dynamics of small deformations superimposed on the undeformed state[ Without loss of generality and based on the Euler Bernoulli beam theory\ the coupled nonlinear equations of motion of a slider!crank mechanism with an inextensible ~exible linkage are derived[ The Newton|s second law is used to obtain the boundary constraints at the piston end[ Galerkin|s procedure was used to remove the dependence of spatial coordinates in the partial di}erential equations[ The method of multiple time scales is applied to consider the steady state solutions and the occurrence of dynamic instability of the resulting multidegree!of!freedom dynamical system with time!periodic coe.cients