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

摘要

In the current paper, we present a series of results on the asymptotic and spectral analysis of coupled Euler-Bernoulli and Timoshenko beam model. The model is well-known in the different branches of the engineering sciences, such as in mechanical and civil engineering (in modelling of responses of the suspended bridges to a strong wind), in aeronautical engineering (in predicting and suppressing flutter in aircraft wings, tails, and control surfaces), in engineering and practical aspects of the computer science (in suppressing bending-torsional flutter of a new generation of hard disk drives, which is expected to pack high track densities (20,000+TPI) and rotate at very high speeds (25,000+RPM)), in medical science (in bio mechanical modelling of blood-carrying vessels in the body, which are elastic and collapsible). The aforementioned mathematical model is governed by a system of two coupled differential equations and a two parameter family of boundary conditions representing the action of the self-straining actuators. This linear hyperbolic system is equivalent to a single operator evolution equation in the energy space. That equation defines a semigroup of bounded operators and a dynamics generator of the semigroup is our main object of interest. We formulate and proof the following results: (a) the dynamics generator is a nonselfadjoint operator with compact resolvent from the class G{sub}p with p > 1; (b) precise spectral asymptotics for the two-branch discrete spectrum; (c) a nonselfadjoint operator, which is the inverse of the dynamics generator, is a finite-rank perturbation of a selfadjoint operator. The latter fact is crucial for the proof that the root vectors of the dynamics generator form a complete and minimal set. In our forthcoming paper, we will use the spectral results to prove that the dynamics generator is Riesz spectral, which will allow us to solve several boundary and distributed controllability problems via the spectral decomposition method.