Active vibration control of mechanical systems via maximum principle

Abstract

In the present work, active vibration control of mechanical components is studied using smart materials technology and within the framework of optimal control of distributed parameter systems. In particular, beams with internal damping and plates with timedependent boundary excitation are studied with the mechanical systems modeled as distributed parameter systems. Vibration control is exercised using discretely placed piezoelectric patch actuators which are bonded on the surface of the specific component. The piezoelectric patches are actuated by applying a transverse electric field the voltage of which is the control variable of the optimal control problem aimed at damping out the vibrations in a given interval of time. For this purpose a performance index is formulated consisting of the dynamical response of the mechanical system and a penalty function involving the expenditure of the control voltage over a given time interval. In particular the dynamical response is defined as a convex functional of the displacement and velocity integrated over the domain of the component. The objective of the optimal control is to minimize the performance index and thereby to minimize the dynamical response and the expenditure of control energy. By introducing an adjoint variable into the problem formulation, a Hamiltonian functional is formulated in terms of this adjoint variable which, in turn, leads to the derivation of the necessary and sufficient conditions of optimality. Present formulation leads to a boundary-initial-terminal value problem in terms of state and adjoint variables for the systems under consideration. Analytical solutions to these problems are developed by means of Galerkin method and in terms of eigenfuction expansions. Numerical simulations are presented to assess the effectiveness and the capabilities of the piezoelectric control to damp out excessive vibrations. It is demonstrated that the present approach to the optimal control of vibrating systems provides an efficient framework for vibration damping.