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The multicriteria problem of transverse vibrations damping of a console beam is colved by the active and passive damping methods. The mathematical model of beam is provided by Bernoulli-Euler's hypotheses with the linear viscosity. Perturbation acting on the beam belongs to a class L2 of functions. The beam mode is described by Krylov functions. The normal form method is used to convert to the main coordinates. A model of active vibration isolation applied along the entire length of the console beam and a model connected to a vertical base at one point were constructed. The task of transverse vibrations damping is a state feedback control problem with two controlled outputs. Two criteria are introduced: the level of the control force and the maximum deflection of the beam. The generalized H2-norm is used as a measure of functional evaluation in the synthesis of optimal regulators. The search for optimal feedback is based on the use of linear matrix inequalities and efficient algorithms for solving, implemented in the MATLAB package. Synthesis of Pareto optimal control is implemented on the basis of Germeyer convolution. The optimal values of the functional under distributed and concentrated vibration isolations are given with respect to two criteria for active and passive damping methods. The paper includes a comparison of vibration isolation for different damping methods.

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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.