Perturbation methods for nonlinear eigenvalue problems

Abstract

In this paper, the dynamics of a microbeam is investigated from the point of view of nonlinear oscillations. Because it regards a non-linear problem, the natural frequency is more complex to obtain. Phenomena such as bifurcations and doubling periods, common in nonlinear systems, may appear. To carry out the ongoing analysis, two components are necessary: first, the equations of motion and, second, the techniques for investigating the behavior of the system. With respect to the equations of motion, deformation gradient theory is used. Concerning the second component, the following approach is employed: techniques of perturbation methods due to the non-linearities present in the model, with the objective of analyzing its oscillations. The important contribution of this investigation resides in a new approach to the equations of motion originated from the formulation of deformation gradient for the context of beams. For future research, it is intended to propose a new stiffness matrix. In the section of computational experiments, results that simulate the behavior of the eigenvalues, eigenfunctions, and solutions of the equation of motion are presented.