This paper is concerned with mathematical aspects and numerical modeling of vibration of a circular plate with piezoelectric actuators. Particularly, a thin Kirchho-Love plate with arbitrary shaped actuators (e.g. pie-shaped, trapezoid, disc, and rectangular) is considered. In the theoretical model, the moments that act upon a structure and are induced by piezoelectric actuators are described by the generalized tensor product of a distribution and distribution-valued function. Numerical computations utilize the FEM approach supported by Ansys software. The possibility of active vibration control for minimization of acoustic energy radiated by vibrating surface elements was explored for over 50 years [1]. The advantage of using dierently shaped, distributed actuators for active control was demonstrated by a number of researchers, e.g. [25]. Many works deal with dierent problems like optimal placement of actuators [6,7], using specic congurations [8]. There are also works that try to use elements typically used in active methods in passive systems [9,10]. Among plates of various shapes, circular plates seem to have a particular importance [5] due to their axial symmetry. Circular geometries are used in a wide variety of applications and are often easily manufactured. In [11], a thin rectangular plate with arbitrary shaped actuators (e.g. triangles, parallelograms, discs) is considered. The theoretical model for a structure is given in a language of distribution-valued function. Also, the formula for the solution of the Cauchy problem in the class of absolutely continuous tempered distribution-valued functions is derived. This paper presents an analytical approach to modeling circular plates with piezoelectric actuators of arbitrary shape. In particular, pie-, trapezoid-, disc-, and rectangular-shaped actuators are considered. The natural tool to describe a structure that consists of a plate and a piezoelectric actuator is the theory of distributions. The moments that act upon a structure and are induced by piezoelectric actuators are expressed as the generalized tensor product of a distribution and distributionvalued function. For two actuator shapes (rectangular and disc), numerical models of a circular plate with two piezo elements attached are created. The analysis uses * corresponding author; e-mail: mwiciak@pk.edu.pl the FEM method for structural vibrations. One of the piezo elements acts as an exciter and the other as an actuator for vibration reduction. Mathematical formulation of the problem Equation of motionLet us consider a thin circular plate of thickness h 0 and radius R, and under the action of external forces and moments. The plate is also assumed to be made of linearly elastic, homogeneous and isotropic material of mass density ρ. The equation of motion for this plate is ([5]):where w = w(t, r, θ)is transverse displacements, ∇ 4 = ∇ 2 ∇ 2 , andis the Laplacian operator in polar coordinates r, θ. Moreover, F = F (t, r, θ) is the external excitation, µ is internal damping loss factor, D = Eh 3 0 /12(...

This paper is concerned with application of piezoelements with a step change of material properties occurring in the element's plane. Each part of an element may be controlled independently. Analytical models for dierent shape and location of the inner part of a two-part piezoactuator are developed. Finally, results of numerical simulations concerning eectiveness in vibration reduction of piezoactuators constructed this way are presented.

This paper is concerned with mathematical aspects of modelling vibration of a plate with piezoelectric actuators. Particularly, a thin KirchhoLove plate with arbitrary shaped actuators (e.g. triangles, parallelograms, discs) is considered. The moments that act upon a structure and are induced by piezoelectric actuators, are described by the generalized tensor product of a distribution and distribution-valued function. Finally, the formula for the solution of the Cauchy problem in the class of absolutely continuous tempered distribution-valued functions is derived.