Numerical solution of the plate bending and free vibrations problems

Algazin, S. D.; Babenko, K. I.

doi:10.1016/0021-8928(82)90065-x

Cited by 5 publications

(3 citation statements)

References 2 publications

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“…Iterative methods for solving spectral problems with nonlinear parameter are proposed and investigated in the papers [15][16][17][18][19][20][21][22][23][24][25][26]. Numerical algorithm without saturation for solving problems of mathematical physics and mechanics were constructed and investigated in [27][28][29][30][31][32][33][34][35][36][37][38]. This paper develops and generalizes results of the papers [1][2][3][4][5][6].…”

Section: Introductionmentioning

confidence: 99%

Finite difference approximation of eigenvibrations of a bar with oscillator

et al. 2020

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The second-order ordinary differential spectral problem governing eigenvibrations of a bar with attached harmonic oscillator is investigated. We study existence and properties of eigensolutions of formulated bar-oscillator spectral problem. The original second-order ordinary differential spectral problem is approximated by the finite difference mesh scheme. Theoretical error estimates for approximate eigenvalues and eigenfunctions of this mesh scheme are established. Obtained theoretical results are illustrated by computations for a model problem with constant coefficients. Theoretical and experimental results of this paper can be developed and generalized for the problems on eigenvibrations of complex mechanical constructions with systems of harmonic oscillators.

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Section: Introductionmentioning

confidence: 99%

Finite difference approximation of eigenvibrations of a bar with oscillator

et al. 2020

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Section: Introductionmentioning

confidence: 99%

Eigenvibrations of a beam with two mechanical resonators attached to the ends

et al. 2020

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The fourth-order ordinary differential spectral problem describing vertical eigenvibrations of a beam with two mechanical resonators attached to the ends is studied. This problem has positive simple eigenvalues and corresponding eigenfunctions. We define limit differential spectral problem and establish the convergence of the eigenvalues and eigenfunctions of the original spectral problem to the eigenvalues and eigenfunctions of the limit spectral problem as parameters of the attached resonators tending to infinity. The initial fourth-order ordinary differential spectral problem is approximated by the finite difference method. Theoretical error estimates for approximate eigenvalues and eigenfunctions are derived. Obtained theoretical results are illustrated by computations for model problem with constant coefficients. Theoretical and experimental results of this paper can be developed for the problems on eigenvibrations of complex mechanical constructions with systems of resonators.

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“…Moreover, there are reasons to believe that the suggested numerical procedures may be advantageous for the solution of applied problems. Results reported in [1,25] indicate what can be gained in some essentially less general classes of problems from the standpoint of further reducing W(s) or of constructing real high-accuracy approximations.…”

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confidence: 99%

Effective methods for solving eigenvalue problems with fourth-order elliptic operators

D'yakonov¹

1986

Russian Journal of Numerical Analysis and Mathematical Modelling

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A generalized eigenvalue problem with an elliptic fourth-order operator in a domain with a piecewise-smooth boundary is approximated by a finite element scheme. An efficient iterative method is suggested for evaluating the smallest eigenvalues of the constructed generalized algebraic eigenvalue problem, and an estimate is found for the complexity of the iterative method. The results are illustrated by using as examples three main types of problems arising in the theory of elastic plates. This paper is connected with the author's previous publications [6][7][8][9][10][11][12][13][14] and is devoted to effective numerical procedures of finding the smallest eigenvalue λ ί of eigenvalue problems with linear elliptic fourth-order operators in a bounded domain Ω on the plane with a piecewise-smooth boundary Γ. Such problems can be written in the operator form Lw = AMw, 0<λ ί <λ 2 <·-(0.1) with bounded symmetric linear operators L and M in Hubert space H, where L and M are self-adjoint and positive definite; L~1M is a compact operator. The Hubert space//can be of the form//! χ H 2 χ ··· χ // p ,where// r isasubspaceof the Sobolev space W\(£l) if 1 < r ^ k^ ^ p, and a subspace of W\(ty ifk 1

k( see [3,4,24]). The elements of H are denoted by w = {w { ,..., w p }. Let the eigenspace corresponding to λ ί consist of functions w such that Wr eW m(r) + y (Q), y>0 , l^r^p (0.2) where w r is the rth component of any eigenfunction w corresponding to m(r) = 2 if I k 1 . The main result of the paper is the construction, under proper conditions on the types of the boundary value problems, of some variants of the finite element method (FEM) and iterative methods (IM) for finding the smallest eigenvalues of the arising algebraic problems such that an approximation of λ± with 0(e 2 )-accuracy may be obtained at the computational cost of W(s) arithmetic operations, where W(s) = 0(s-2/ ?\lns\ l \ 1=12. (0.3)A generalization is possible if the condition Μ > 0 is not satisfied and λ 1 is either the smallest positive or the largest negative eigenvalue. For such problems, the estimates (0.3) are still valid. The admissible boundary conditions can be of a rather general nature. If, for example, either the equation Δ 2 νν> = λ\ν or Δ 2 νν = -ΑΔνν is considered, then the boundary conditions can be any of three main types known in the theory of elastic plates that correspond to the cases of clamped, simply supported, and free plates. Brought to you by | University of Arizona Authenticated Download Date | 7/5/15 3:09 PM Brought to you by | University of Arizona Authenticated Download Date | 7/5/15 3:09 PM

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Numerical solution of the plate bending and free vibrations problems

Cited by 5 publications

References 2 publications

Finite difference approximation of eigenvibrations of a bar with oscillator

Finite difference approximation of eigenvibrations of a bar with oscillator

Eigenvibrations of a beam with two mechanical resonators attached to the ends

Effective methods for solving eigenvalue problems with fourth-order elliptic operators

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