Control of Vibrations of a Beam with Nonlocal Boundary Conditions

Abstract

In this article is considered the models of uniform Euler-Bernoulli beams with an arbitrary variable coefficient of foundation on a finite segment. The variable of foundation corresponds to the Winkler model. The control problem the first eigenvalues of the beam vibration is investigated. Two types of fastenings at the ends are considered: clamped-clamped and hinged-hinged. The control is based on the Kanguzhin algorithm through integral perturbations of one of the boundary conditions of the original problem. Conditions for the boundary parameters for controlling the first eigenvalues are found. First, a result is formulated regarding the control of the first eigenvalue of the oscillation of the Euler-Bernoulli beam with hinge fastening at both ends. The result is then extended to control with several eigenvalues for this beam, which are important from the point of view of the application. Such questions are especially relevant when studying the resonant natural frequencies of a mechanical system. A similar result was obtained for a Euler-Bernoulli beam with clamped fastening at both ends. Such results of eigenvalue control of a mechanical system contribute to the creation of various non-destructive testing devices that are widely used in technical acoustics.