Periodic Solutions to Quasilinear Oscillation Equations for Cables and Beams
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Journal of Mathematical Sciences, Vol. 250, No. 1, October, 2020
PERIODIC SOLUTIONS TO QUASILINEAR OSCILLATION EQUATIONS FOR CABLES AND BEAMS I. A. Rudakov Bauman Moscow State Technical University 5/1, 2-ya Baumanskaya St., Moscow 105005, Russia Moscow Aviation Institute (National Research University 4, Volokolamskoe Shosse, Moscow 125993, Russia
UDC 517.956.35
rudakov [email protected]
We study periodic solutions to the problem for the quasilinear Euler–Bernoulli equation governed oscillations of an I-beam with the homogeneous boundary conditions corresponding to the hinged and fixed beam endpoints. We obtain an asymptotic formula for the eigenvalues of the Sturm–Liouville problem and prove the existence of infinitely many solutions provided that the nonlinear term has a power growth. Bibliography: 12 titles.
1
Introduction
We consider the problem utt + uxxxx − auxx + g(x, t, u) = 0,
0 < x < π,
t ∈ R,
(1.1)
u(0, t) = uxx (0, t) = 0,
t ∈ R,
(1.2)
u(π, t) = ux (π, t) = 0,
t ∈ R,
(1.3)
u(x, t + T ) = u(x, t),
0 < x < π,
t ∈ R,
(1.4)
where the function g is T -periodic with respect to t and the constant a is positive. Equation (1.1) is a mathematical model of oscillations of a rod capable of resisting bending as well as oscillations of cables and I-beams. The boundary condition (1.2) corresponds to the left hinged beam endpoint, whereas (1.3) means that the right beam endpoint is fixed. There are many works devoted to periodic and quasiperiodic solutions to the problems for nonlinear evolution equations, in particular, the wave equation and the beam oscillation equation. The existence of periodic solutions is established for the wave equation with constant and variable coefficients. We note that there are a lot of various methods developed for such Translated from Problemy Matematicheskogo Analiza 104, 2020, pp. 111-119. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2501-0123
123
problems. In this paper, to study the beam equation, we apply the variational method [1] developed for the wave equation. In the case a = 0, the problem on periodic oscillations of beams is studied, for example, in [2]–[11]. The case a = 0 is considered in [3, 11]. In particular, the existence of a periodic solution of small amplitude is proved in [3] if the right-hand side of (1.1) has the form εf (x, t), where ε is sufficiently small, and the existence of infinitely many solutions to the problem (1.1)– (1.4), where g has a power growth, is proved in [11]. Moreover, conditions of the form (1.2) are imposed on the left and right beam endpoints in [3, 11]. Under such conditions, the Sturm– Liouville problem is explicitly solved: n4 + an2 are eigenvalues and sin (nx) are eigenfunctions. If the right beam endpoint is fixed, whereas the left one is hinged, then one has to deal with a cumbersome transcendental equation for eigenvalues and derive asymptotic estimates. In this paper, we prove the existence of countably many solutions to the problem (1.1)–(1.4). Assume that the period T and coefficient a in (1.1) satisfy the condi
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