Abstract
The nonlinear differential eigenvalue problem describing eigenvibrations of a simply supported beam with elastically attached load is investigated. The existence of an increasing sequence of positive simple eigenvalues with limit point at infinity is established. To the sequence of eigenvalues, there corresponds a system of normalized eigenfunctions. To illustrate the obtained theoretical results, the initial problem is approximated by the finite difference method on a uniform grid. The accuracy of approximate solutions is studied. Investigations of the present paper can be generalized for the cases of more complicated and important problems on eigenvibrations of plates and shells with elastically attached loads.
Highlights
Let us formulate the nonlinear differential eigenvalue problem governing eigenvibrations of the beam-spring-load system
Assume that the beam axis occupies in the equilibrium horizontal position the segment [0,l] on the Ox axis
To the sequence of eigenvalues, there corresponds a system of normalized eigenfunctions
Summary
Let us formulate the nonlinear differential eigenvalue problem governing eigenvibrations of the beam-spring-load system. If λ ≠ σ , from relation (7) we obtain =z σ u(x(0) ) (σ − λ), and equations (1)-(4) lead to the following nonlinear eigenvalue problem: find numbers λ and nonzero functions u(x), x ∈[0,l], such that (= p(x)u′′(x))′′ λr(x)u(x), x ∈ (0,l),. The error of the finite difference method for solving differential eigenvalue problems with nonlinear dependence on the spectral parameter was investigated in [1, 17]. For nonlinear differential spectral problems, the finite element method was studied in [18,19,20] based on the use general results in the linear case [21-23]. Approximate methods for solving applied nonlinear boundary value problems and variational inequalities have been investigated in the papers [24-30]
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