Stabilization of beam parametric vibrations

A theoretical investigation of dynamic stability for linear elastic beams due to time dependent harmonic anal forces is presented. The concept of intelligent structure is used to insure the active damping. In the present paper the applicability of active vibration control is extended to linear continuous systems with parametric harmonic excitations. The study is based on the application of distributed sensors, actuators, and an appropriate feedback and is adopted for stability problems of system consisting of beam with control part governed by uniform partial differential equations with time, dependent coefficients. To estimate deviations of solutions from the equilibrium state (the distance between a solution with nontrivial initial conditions and the trivial solution) a scalar measure of distance equal to the square root of the functional is introduced. The tyapunoy method is used to derive a velocity feedback implying nonincreasing of the functional along an arbitrary beam motion and in conseqiaeftce to balance the supplied energy by the parametric excitation and the dissipated ejiergy by the inner and control damping. In order to calculate the energetic norm of disturbed solution as a function of time the partial differential equation is solved numencally. The numerical tests performed for the simply supported beam with surface bonded actuators and sensors show the influence of the fedback constant on the vibration decrease.