Unboundedness of Solutions of Timoshenko Beam Equations with Damping and Forcing Terms

Timoshenko beam equations with external damping and internal damping terms and forcing terms are investigated, and boundary conditions (end conditions) to be considered are hinged ends (pinned ends), hinged-sliding ends, and sliding ends. Unboundedness of solutions of boundary value problems for Timoshenko beam equations is studied, and it is shown that the magnitude of the displacement of the beam grows up to ∞ as under some assumptions on the forcing term. Our approach is to reduce the multidimensional problems to one-dimensional problems for fourth-order ordinary differential inequalities. 1. Introduction The most fundamental beam equations are of the following form: with the length , the mass density , the cross-sectional area , the modulus of elasticity , and the flexural rigidity (see [1, page 416]). Taking account of the rotary inertia and the deflection due to shear, we obtain the following fourth-order beam equation for the transverse vibrations of prismatic beams on elastic foundations: (see [1, page 433] and Wang and Stephens [2, page 150]). Dividing the above equation by , letting , and taking into account the nonlinear term , the external damping term and the internal damping terms we obtain the Timoshenko beam equation where , and are positive constants. Let and we assume throughout this paper that(H1) is a real-valued continuous function in ;(H2) and for ;(H3) is a nondecreasing function in ;(H4) is a real-valued continuous function on . Definition 1. By a solution of (5), one means a function such that the partial derivatives exist and are continuous on . Oscillations of beam equations have been treated by numerous authors; see, for example, Feireisl and Herrmann [3], Herrmann [4], Kopá？ková [5], Kusano and Yoshida [6], Yoshida [7–10], and the references therein. In particular, we mention the paper [4] by Herrmann which deals with the Euler-Bernoulli beam equations that is similar to (5). We note that the oscillation of (5) was studied by Yoshida [10]. We refer to Ball [11], Fitzgibbon [12], and Narazaki [13] for stability and existence results for beam equations. However, there appears to be no known unboundedness results for beam equations. The objective of this paper is to provide unboundedness results for (5) by reducing the multi-dimensional problems to one-dimensional problems for ordinary differential inequalities of fourth-order. In Section 2 we treat the hinged ends and reduce unboundedness problem for (5) to that for ordinary differential inequalities. Sections 3 and 4 are devoted to the hinged-sliding ends and sliding ends,