Optimal Vibration Feedback Control of an Euler-Bernoulli Beam: Toward Realization of the Active Sink Method

Abstract

This paper discusses the optimal vibration feedback control of an Euler-Bernoulli beam from a viewpoint of active wave control making all structural modes inactive (more than suppressed). Using a transfer matrix method, the paper derives two kinds of optimal control laws termed “active sink” which inactivates all structural modes; one obtained by eliminating reflected waves and the other by transmitted waves at a control point. Moreover, the characteristic equation of the active sink system is derived, the fundamental properties being investigated. Towards the goal of implementing the optimal control law that is likely to be non-causal, a “classical” velocity feedback control law (Balas, 1979) widely used in a vibration control engineering is applied, revealing a substantial shortcoming. Introduction of a “classical” displacement feedback to the velocity is found to realize the optimal control law in a restricted frequency range. Finally, two kinds of stability verification for closed feedback control systems are presented for distributed parameter structures.