Keywords: Timoshenko, vibration spectrum, eigenvalues, eigenfunctions, controllability, controllable, stability.

The Timoshenko beam is the most complete linear model of a vibrating beam. Although it is an old problem (1920s), and linear, it still is of great interest, both mathematically in and of itself, and in the many and wide-ranging applications for which it is the most appropriate model. We consider the problem first with constant, then with x-dependent, coefficients, the most general model discussed being that given in [1]. This model is strongly clamped at one end, and subject to boundary controls in both shear and bending moment at the other. It is a rich model and is not restrictive, in that it is an easy step to generalize it to all cases of conservative and dissipative boundary conditions.

In order to deal with issues of stability and controllability and such, one needs first to have spectral results. We begin with a consideration of the eigenvalues and eigenfunctions for the constant-coefficient problem. In [2], asymptotic results for the vibration frequencies were derived for the dissipative problem, and it is here that we first see, to our knowledge, the signature double-branched asymptotically vertical spectrum for the Timoshenko beam. Reference [3], then, provides a comprehensive study of the eigenfunctions for the conservative problem. Also, we present new perturbation results by this author which improve the accuracy of asymptotic results for the low end of the spectrum. Lastly, we briefly discuss this author's numerical and perturbation experiments for the degenerate case EI/k = Irho/rho.

We follow with a discussion of the x-dependent coefficient problem, where Shubov (in [1] and other papers, in a very important series of articles on the Timoshenko beam) generalized the asymptotic results of [2] to this case. Further, the same author provides generalized eigenfunction formulas which reduce to those in [3] in that special case. Lastly, we mention briefly results by Shubov and others concerning stability of solutions and exact controllability of the general problem.

We conclude with a brief mention of numerical problems on which this author is working, and we hope that some preliminary results will be ready for presentation at the conference.

1

M.A. Shubov, "Asymptotic and spectral analysis of the spatially nonhomogeneous Timoshenko beam model", Mathematische Nachrichten, 241, 125-162, 2002. doi:10.1002/1522-2616(200207)241:1<125::AID-MANA125>3.0.CO;2-3

2

M.P. Coleman, H.K. Wang, "Analysis of the vibration spectrum of a Timoshenko beam, with boundary damping by the Wave Method", Wave Motion, 17, 223-239, 1993. doi:10.1016/0165-2125(93)90003-X

3

Q-P. Vu, J-M. Wang, G-Q. Xu, S-P. Yung, "Spectral analysis and system of fundamental solutions for Timoshenko beams", Applied Mathematics Letters, 18 (2), 127-134, 2005. doi:10.1016/j.aml.2004.09.001

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