14. Active Damping of Transverse Vibrations of Console Beam by Piezoelectric Layer with Different Electrode Shapes

In the problems of damping vibration, the question often arises on the practical implementation of damping actuators. 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 affect 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. In turn, 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. Also, for a more general analysis, the quenching efficiency is also given for a beam with a piezoelectric plate applied along the entire length and an electrode layer. In addition to Pareto sets, efficiency is also considered in a more applied and particular example—time history. It is worth noting that 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.