Eigenvibrations of a bar with two attached loads

Abstract

© 2020 American Institute of Physics Inc.. All rights reserved. The second-order differential eigenvalue problem governing eigenvibrations of a bar with fixed ends and with two rigidly attached loads at interior points is studied. This problem has an increasing sequence of positive simple eigenvalues with a limit point at infinity. To the sequence of eigenvalues, there corresponds a complete orthonormal system of eigenfunctions. The asymptotic properties of eigensolutions are investigated. The initial differential eigenvalue problem is approximated by the finite difference scheme on a uniform grid. Error estimates of this finite difference scheme are derived. The theoretical results are illustrated by numerical experiments for a model problem. The investigations reported in this paper can be extended to the cases of more complicated and important problems on eigenvibrations of beams, plates, and shells with attached loads.