Finite difference approximation of eigenvibrations of a cantilever beam with elastically attached load

Abstract

© 2020 American Institute of Physics Inc.. All rights reserved. The nonlinear fourth-order differential eigenvalue problem governing eigenvibrations of a cantilever beam with elastically attached load is investigated. This problem has an increasing sequence of positive simple eigenvalues with a limit point at infinity. To the sequence of eigenvalues there corresponds a system of normalized eigenfunctions. The original nonlinear differential eigenvalue problem is approximated by the finite difference method on a uniform grid. The existence and error estimates for approximate eigenvalues and eigenfunctions are established. Theoretical results are illustrated by numencal expenments for a model problem. The studies reported in this paper can be extended to the cases of more complicated and important problems on eigenvibrations of plates and shells with elastically attached loads.