Spectral Analysis and Numerical Investigation of a Flexible Structure with Nonconservative Boundary Data

Abstract

Analytic and numerical results of the Euler-Bernoulli beam model with a twoparameter family of boundary conditions have been presented. The co-diagonal matrix depending on two control parameters (k 1 and k 2 ) relates a two-dimensional input vector (the shear and the moment at the right end) and the observation vector (the time derivatives of displacement and the slope at the right end). The following results are contained in the paper. First, high accuracy numerical approximations for the eigenvalues of the discretized differential operator (the dynamics generator of the model) have been obtained. Second, the formula for the number of the deadbeat modes has been derived for the case when one control parameter, k 1 , is positive and another one, k 2 , is zero. It has been shown that the number of the deadbeat modes tends to infinity, as k 1 ! 1 þ and k 2 ¼ 0. Third, the existence of double deadbeat modes and the asymptotic formula for such modes have been proven. Fourth, numerical results corroborating all analytic findings have been produced by using Chebyshev polynomial approximations for the continuous problem.

Location of eigenmodes of Euler–Bernoulli beam model under fully non-dissipative boundary conditions

“…Finally, we represent the spectral problem for the dynamics generator in the form of the spectral problem for a polynomial operator pencil. In the same section, we provide formulation of the results on the eigenvalue distribution in the model from Shubov & Kindrat [29,30]. From these results (see theorems 2.2 and 2.3), it follows that, using the spectral asymptotics, one can claim that, for specific combinations of control parameters, all distant eigenvalues are located in the upper half-plane.…”