Active Damping of Transverse Vibrations of Console Beam by Piezoelectric Layer With Different Electrode Shapes of Damaged Media

Abstract

The damping efficiency is considered for a console beam described by a linear viscosity Bernoulli-Euler model. The article presents the methods of damping transverse vibrations implemented by a dynamic damper from a piezoelectric layer distributed symmetrically along the axis of symmetry of the beam. Piezoelectric layers with a triangular and rectangular shape of electrode plates are considered, which affects the nature of mechanical stresses upon application of electrical voltage. The electrode plates are thin layers made of nickel or silver several microns thick and located normal to the polarization axis, that is, along the length of the piezoceramic plate. The control of the piezoelectric layers is realized by changing the potential difference between the electrode plates, while the piezoelectric material uncoated by the electrode plate on both sides is useless to use as an active material. Mathematical models of the effect of piezoelectric elements on the cantilever beam are derived from the Hamilton principle. The Pareto-efficiency of quenching by piezoelectric plates with different electrode shapes is evaluated relative to two criteria: the level of control voltage and the maximum deflection of the beam. To compare the results with the best variant of vibration damping, in this formulation, the result of vibration damping for a beam with piezoelectric layer applied along the entire length is given. The damping efficiency was confirmed in an applied and particular example by means of vibrograms. The synthesis of Pareto-optimal controls is based on the Germeier convolution, and the search for optimal feedback is based on the application of the theory of linear matrix inequalities and effective algorithms for solving them.