Eigenvibrations of an elastic bar with mechanical resonator

Abstract

The differential eigenvalue problem governing eigenvibrations of an elastic bar with fixed first end and mechanical resonator attached to second end is investigated. This problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. To the sequence of eigenvalues, there corresponds a complete orthonormal system of eigenfunctions. We introduce limit differential eigenvalue problems and derive the convergence of the eigenvalues and eigenfunctions of the initial problem to the corresponding eigenvalues and eigenfunctions of the limit problems as a resonator parameter tending to infinity. The original differential eigenvalue problem is approximated by the finite difference method on a uniform mesh. Error estimates for approximate eigenvalues and eigenfunctions are established. Theoretical results are illustrated by numerical experiments for model problems. Investigations of this paper can be generalized for the cases of more complicated and important problems on eigenvibrations of beams, plates and shells with attached resonators.