Asymptotic and spectral analysis of nonselfadjoint operators generated by a coupled Euler-Bernoulli/Timoshenko beam model



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Texas Tech University


This dissertation is devoted to the asymptotic and spectral analysis of a coupled Euler-Bernoulli and Timoshenko beam model. The model is governed by a system of two coupled differential equations and a two parameter family of boundary conditions modeling the action of self-straining actuators. The aforementioned equations of motion together with a two-parameter family of boundary conditions form a coupled linear hyperbolic system, which is equivalent to a single operator evolution equation in the energy space. That equation defines a semigroup of bounded operators. The dynamics generator of the semigroup is our main object of interest in the dissertation. For each set of boundary parameters, the dynamics generator has a compact inverse. If both boundary parameters are not purely imaginary numbers, then the dynamics generator is a nonselfadjoint operator in the energy space. We calculate the spectral asymptotics of the dynamics generator. We find that the spectrum lies in a strip parallel to the horizontal axis, and is asymptotically close to the horizontal axis, thus the system is stable, but is not uniformly stable. The latter fact has been observed in numerical simulations and in the present dissertation, it has been rigorously proven.



Spectral analysis, Eulerian graph theory, Timoshenko beams