Asymptotic properties of the problem on eigenvibrations of the bar with attached load

The ordinary second-order differential eigenvalue problem describing eigenvibrations of an elastic bar with a load attached to its end was investigated. The 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. In the paper, the behavior of the solutions in dependence on the load mass was studied. More precisely, limit differential eigenvalue problems were formulated and the convergence of the eigenvalues and eigenfunctions of the initial problem to the corresponding eigenvalues and eigenfunctions of the limit problems as load mass tending to infinity were proved. The original differential eigenvalue problem was approximated by the mesh scheme of the finite element method on a uniform grid. Error estimates for approximate eigenvalues and eigenfunctions were established. The results of this paper can be generalized for the cases of more complicated and important applied problems on eigenvibrations of beams, plates, and shells with attached loads