Eigenoscillations of mechanical systems with boundary conditions containing the frequency

Abstract

Abstract.The problem of eigenoscillations of beam-mass systems is investigated and four examples are developed. For such systems the corresponding Sturm-Liouville problems contain the eigenvalue parameter in the boundary conditions. It is shown that the eigenfunctions for the systems considered form a basis of the appropriate Hilbert space. Rayleigh-Ritz formulas are also developed. Some lower bound estimations for eigenfrequencies are also found. Introduction.As is well known, the set of eigenfunctions of a positive selfadjoint second-order differential operator on L2[0, d] forms a basis of the space involved [1, p. 167]. There exists diverse literature [1] devoted to the problem of determining when a system of eigenfunctions forms a basis for the appropriate Hilbert space. The goal of this work is to contribute to this eigenfunction problem for certain mechanical systems that are defined by a Sturm-Liouville problem with an eigenvalue appearing in the boundary conditions. The intention of this paper is to show how known methods and results from the theory of operators on a Hilbert space can be applied to a class of eigenoscillation

Yeni Tipten Sturm-Liouville Problemlerinin Ürettiği Diferansiyel Operatörlerin Kendine Eşlenikliği ve Pozitivliği

“…The compactness of  is well-known fact (see, [34]). The proof of the compactness of 1  can be found by using the same arguments, as in [10,11] and [13].…”

“…Oscillation and comparison results have been obtained in [7]. The completeness of the eigenfunctions and eigenfunction expansions in various function spaces for the Sturm-Liouville problems with spectral parameter in the boundary conditions have been considered in [10][11][12][13][14][15][16]. Problems with various singularities have been analyzed in [17][18][19][20][21][22][23][24][25][26][27][28][29][30].…”

Energy stability for a class of two-dimensional interface linear parabolic problems

“…Theoretical results and applications of one-dimensional spectral problems with eigenvalue in the boundary conditions are presented in [2]. We begin with (18).…”

Some properties of eigenvalues and generalized eigenvectors of one boundary value problem

“…Basis properties and eigenfunction expansions for Sturm-Liouville problems involving eigenparameter in the boundary conditions have been considered in [1,5,9,10,14,[27][28][29]. Aliyev and Kerimov [1] studied basisness of root functions of Sturm-Liouville problems with a boundary condition depending quadratically on the spectral parameter.…”

“…Aliyev and Kerimov [1] studied basisness of root functions of Sturm-Liouville problems with a boundary condition depending quadratically on the spectral parameter. In [5], it is shown that the problem of eigenoscillations of various mechanical systems is formulated as the Sturm-Liouville problem with the eigenvalue parameter appearing in the boundary conditions. It is proven that the eigenfunctions form a Riesz basis of suitable Hilbert space.…”