Abstract
The differential eigenvalue problem describing eigenvibrations of a bar with fixed ends and attached load at an interior point 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 formulate limit differential eigenvalue problems and prove 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. The original differential eigenvalue problem is approximated by the finite difference method on a uniform grid. Error estimates for approximate eigenvalues and eigenfunctions are established. Theoretical results are illustrated by numerical experiments for a model problem. 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 loads.
Highlights
Let us formulate the differential eigenvalue problem governing eigenvibrations of the barload system
The eigenvibrations of the bar-load mechanical system are characterized by the function w(x,t) of the following form w(x,t) = u(x) sin(ωt), x ∈[0,l], t > 0, where ω is a constant
For nonlinear differential spectral problems, the finite element method was studied in [16,17,18,19] based on the use general results in the linear case [20-23]
Summary
Let us formulate the differential eigenvalue problem governing eigenvibrations of the barload system. The longitudinal deflection w(x,t) of the bar at a point x at time t satisfies the following equations. The eigenvibrations of the bar-load mechanical system are characterized by the function w(x,t) of the following form w(x,t) = u(x) sin(ωt), x ∈[0,l], t > 0, where ω is a constant. The error of the finite difference method for solving differential eigenvalue problems with nonlinear dependence on the spectral parameter was investigated in [1, 15]. For nonlinear differential spectral problems, the finite element method was studied in [16,17,18,19] based on the use general results in the linear case [20-23]. Approximate methods for solving applied nonlinear boundary value problems and variational inequalities have been investigated in the papers [24-30]
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