© 2017, Pleiades Publishing, Ltd. The eigenvalue problem describing eigenvibrations of a beam with load is investigated. The problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. To the sequence of eigenvalues, there corresponds a system of normalized eigenfunctions. Limit properties of eigenvalues and eigenfunctions are studied

Samsonov A.

Solov’ev S.

Lobachevskii Journal of Mathematics

Kazan Federal University Digital Repository

ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2017, Vol. 38, No. 5, pp. 849–855. c© Pleiades Publishing, Ltd., 2017.Eigenvibrations of a Beam with LoadA. A. Samsonov* and S. I. Solov’ev**(Submitted by A. V. Lapin)1Department of Computational Mathematics,Institute of Computational Mathematics and Information Technologies,Kazan (Volga Region) Federal University, Kazan, 420008 RussiaReceived September 27, 2016Abstract—The eigenvalue problem describing eigenvibrations of a beam with load is investigated.The problem has an increasing sequence of positive simple eigenvalues with limit point at inﬁnity.To the sequence of eigenvalues, there corresponds a system of normalized eigenfunctions. Limitproperties of eigenvalues and eigenfunctions are studied.DOI: 10.1134/S1995080217050298Keywords and phrases: Eigenvalue, eigenfunction, eigenvalue problem, beam, eigenvibration.1. INTRODUCTIONLet us formulate the diﬀerential eigenvalue problem governing eigenvibrations of the beam-loadsystem. Assume that the beam axis occupies in the equilibrium horizontal position the segment [0, l]on the Ox axis. Denote by ρ(x) the volume mass density, E(x) the elasticity Young modulus, S(x) thesquare of transversal cut of the beam, J(x) the inertia moment of the cut with respect to its horizontalaxis at the point x ∈ [0, l]. Suppose that the end x = 0 of the beam is rigidly ﬁxed while the end x = l isfree, at the point x = l of the beam a load of mass M is rigidly joined. Then the vertical deﬂection w(x, t)of the beam at a point x at time t satisﬁes the following equations∂2∂x2(p(x)∂2∂x2w(x, t))+ r(x)∂2∂t2w(x, t) = 0, x ∈ (0, l), t > 0, (1)w(0, t) =∂∂xw(0, t) = 0, t > 0, (2)∂2∂x2w(l, t) =∂∂x(p(l)∂2∂x2w(l, t))−M ∂2∂t2w(l, t) = 0, t > 0, (3)where p(x) = E(x)J(x), r(x) = ρ(x)S(x), x ∈ [0, l].The eigenvibrations of the beam-load system are characterized by the function w(x, t) of the formw(x, t) = u(x)v(t), x ∈ [0, l], where v(t) = a0cos√λt+ b0sin√λt, t > 0; a0, b0, and λ are constants.Then equations (1)–(3) lead to the following eigenvalue problem: ﬁnd values λ and nontrivial functionsu(x), x ∈ [0, l], such that(p(x)u′′(x))′′ = λr(x)u(x), x ∈ (0, l), u(0) = u′(0) = 0, u′′(l) = (p(l)u′′(l))′ + λMu(l) = 0. (4)Problem (4) has an increasing sequence of positive simple eigenvalues with limit point at inﬁnity. Tothe sequence of eigenvalues, there corresponds a system of normalized eigenfunctions. In the present*E-mail: anton.samsonov.kpfu@mail.ru**E-mail: sergei.solovyev@kpfu.ru849

Eigenvibrations of a beam with load

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Eigenvibrations of a beam with load

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