Vibration analysis on a thin plate with the aid of computation of normal forms

Abstract

Through the dynamical analysis of a thin plate, normal forms have been explicitly derived for a non-autonomous system associated with 1 : 1 semisimple internal resonance as well as external resonances. The formulas are given for general n-dimensional systems rather than the particular example of the thin plate. Computation is divided into two steps: first a perturbation approach combined with the multiple scales is applied to find averaged equations. Then based on the autonomous system (averaged equations), normal form theory is used to obtain the explicit expressions of the normal form associated with a double zero and a pair of pure imaginary eigenvalues. Maple has been used to develop symbolic computer programs for the two procedures: one for non-autonomous system and one for autonomous system. The normal form for the example of the thin plate is obtained by executing the two Maple programs. On the basis of the normal form, bifurcation analysis of the thin plate is given. In particular, heteroclinic bifurcation is discussed in detail.