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

Abstract

The Euler–Bernoulli beam model with non-conservative feedback-type boundary conditions is investigated. Components of the two-dimensional input vector are shear and moment at the right end, and components of the observation vector are time derivative of displacement and slope at the right end. The boundary matrix containing four control parameters relates input and observation. The following results are presented: (i) if one and only one of the control parameters is positive and the rest of them are equal to zero, then the set of the eigenmodes is located in the open left half-plane of the complex plane, which means that all eigenmodes are stable; (ii) if the diagonal elements of the boundary matrix are positive and off-diagonal elements are zeros, then the set of the eigenmodes is located in the open left half-plane, which implies stability of all eigenmodes; (iii) specific combinations of the diagonal and off-diagonal elements have been found to ensure the stability results. To prove the results, two special relations between the eigenmodes and mode shapes of the non-self-adjoint problem and clamped–free self-adjoint problem have been established.