Free vibration of structures with cubic non-linearity-remarks on amplitude equation and Rayleigh quotient Original Research Article

In the paper, several aspects concerning the formulation of and solution to amplitude equations for free vibration of systems with cubic non-linearity are discussed in detail. The use of several possible constraint equations to achieve the non-trivial solution to the amplitude equation is also presented. The well-known concept of the Rayleigh quotient is extended to the non-linear case of interest. Some general remarks on the formulation of the non-linear amplitude equation and the non-linear Rayleigh quotient are also presented. Moreover, the introduced Rayleigh quotient is used to develop a new and suitably efficient method to solve amplitude equations. The accuracy of solutions obtained by means of some widely used methods of analysis of free vibration problems is also presented. It is found that the Galerkin method gives the most accurate results.