Closed-loop non-linear control of an initially imperfect beam with non-collocated input

A closed-loop non-linear control strategy to reduce the flexural vibrations of a hinged–hinged initially imperfect beam is investigated. The beam is subjected to a harmonic transverse excitation involved in a primary resonance of the first antisymmetric mode. A closed-loop symmetric control action—bending moments imparted by two piezoceramic actuators—although non-collocated, is designed to be non-orthogonal, in a non-linear sense, to the excited mode and be capable of exerting resonant beneficial damping effects onto it. The approximate responses of the controlled and uncontrolled beam are constructed by applying the method of multiple scales directly to the integral–partial differential equations of motion and boundary conditions. The frequency response curve governing the primary resonance of the uncontrolled system is compared with that obtained when the controller is in action. It is shown that, by exerting feasible control efforts, the response of the beam may be reduced by an order of magnitude and is stable in the overall frequency range in contrast with the uncontrolled large-amplitude responses which undergo jumps at the saddle-node bifurcations.