Analysis of non-linear mode shapes and natural frequencies of continuous damped systems

In this paper, the aim is to find the non-linear mode shapes and natural frequencies for a class of one-dimensional continuous damped systems with weak cubic inertia, damping and stiffness non-linearities. This paper presents general formulations for natural frequencies and mode shapes with all non-linearity effects. Initially the non-linear system with general boundary conditions is discretized, and using a two-dimensional manifold, the model of cubic non-linearities is constructed and the general equation of motion which governs non-linear system is derived. The method of multiple scales is then used to extend the non-linear mode shapes and natural frequencies. During this analysis, it is realized that when the natural frequencies of the linear system become equal to the natural frequencies of the non-linear system a one-to-one internal resonance will appear. Also, there is a three-to-one internal resonance which is not dependent on the damping of the system. Finally, general formulations of amplitude for vibrations, natural frequencies and mode shapes of the non-linear system are obtained in parametric forms. Thus, a non-linear problem with some simple integration can be solved. The formulations are capable of handling any non-linearities in inertia, damping, stiffness, or any combination of them under any arbitrary boundary conditions.