Perturbation Method for Analysis of Strongly Non-Linear Free Vibration of Beams

The perturbation method is one of the oldest methods used to analyse the dynamic behaviour of nonlinear systems. There are many versions of the perturbation method but most of them apply to weakly non-linear cases only. To overcome this limitation, new techniques have been proposed. Cheung et al. [

1

], Lim et al. [

2

] and Hu [

3

] proposed some modifications which make possible the analysis of strongly non-linear systems but with one degree of freedom only.

In this paper, the possibility of application of the perturbation method to the dynamic analysis of strongly non-linear free vibrations of beams is discussed. Beams are treated as geometrically nonlinear systems. The von Karman theory is used to describe non-linear effects. The finite element method is adopted to discetize the beam and the motion equation is written in a matrix form.

The first order perturbation equation is solved and the obtained solution is compared with the solution found with the help of the harmonic balance method which is widely used and applicable to the analysis of strongly non-linear dynamic systems. It was proved that both solutions are almost identical and differences are negligibly small. On the basis of similarities discovered in the both solutions, it was concluded that, for the value of small parameter 1 = e, the solution obtained by means of the perturbation method is almost identical to the one given by the harmonic balance method. The results of typical calculations confirm these observations. Finally, it is concluded that the perturbation solution has also enough accuracy when the strongly non-linear systems are considered. It is believed that the reason of success of the presented perturbation method comes from a feedback which is taken into account when the tangent stifness matrix is introduced. The numerical procedure enabling determination of backbone curves is also briefly described. Theoretical results are supplemented by a description of the results of typical calculations.