In this paper an active multimodal beam vibration reduction via one actuator is considered. The optimal actuator distribution is analyzed with two methods: an exact mathematical principles and the LQ problem idea. It turned out that the same mathematical expressions are derived. Thus, these methods are equivalent.

The article presents the new 2D asymmetrical PZT (a-PZT) and its effectiveness in the active reduction of triangular plate vibrations. The isosceles right triangular plate with simply supported edges was chosen as the research object. To determine the a-PZT asymmetry and its distribution on the plate, a maximum bending moment criterion for the beam was used. First of all, this criterion points out exact center location of the a-PZT. It was at the point, at which the plate bending moment has reached its maximum value. Next, at this point, it was assumed that the piezoelectric consists of active fibers located radially from the center. Each fiber acted on the plate as a separate actuator. Next, at each direction, the actuator asymmetry was found mathematically by minimizing the amplitude of the vibrations. By connecting the outer edges of individual fibers, the 2D a-PZT was obtained. It was quantitatively confirmed that the effectiveness of the new a-PZT was the best compared with the effectiveness of the standard square and the circular PZTs, adding the same exciting energy to the PZTs.

The article extended the idea of active vibration reduction of beams with symmetric modes to beams with asymmetric modes. In the case of symmetric modes, the symmetric PZT (s-PZT) was used, and the optimization of the problem led to the location of the s-PZT centre at the point with the greatest beam curvature. In the latter case, the asymmetric modes that occur due to the addition of the point mass cause an asymmetric distribution of the bending moment and transversal displacement of a beam. In this case, the optimal approach to the active vibration reduction requires both new asymmetric PZT (a-PZT) and its new particular distribution on the beam. It has been mathematically determined that the a-PZT asymmetry point (a-point), ought to be placed at the point of maximum beam bending moment. The a-PZT asymmetry was found mathematically by minimizing the amplitude of the vibrations. As a result, it was possible to formulate the criterion of the maximum bending moment of the beam. The numerical calculations confirmed theoretical considerations. So, it was shown that in the case of asymmetric vibrations, the a-PZTs reduced vibrations more efficiently than the s-PZT.