Issue 49

D. Oshmarin et alii, Frattura ed Integrità Strutturale, 49 (2019) 800-813; DOI: 10.3221/IGF-ESIS.49.13 803 ij ijkl kl ijk k k ijk ij ki i C E D e E              (4) where ijkl C and ijkl C are the components of the tensors of elastic constants, ijk  and ki e are the components of the tensors of piezoelectric and dielectric coefficients. For the examined system, the restrictions are set in the form of mechanical and electrical boundary conditions. The mechanical boundary conditions are given as: : 0, : 0 ij j u i S n S u     (5) where u S S S    is the surface of full volume V . The expressions for the electrical boundary condition on the piezoelectric parts confined in volume 2 V read as follows: 0, el     (6) where el   is the part of the electrode surface of the volume 2 V , with the prescribed electric potential. In the absence of the electrode covering all electrical boundary conditions take zero values. The eigenvectors of vibrations at zero-valued boundary conditions (5-6) are sought in the following form:     , i t i i u x t u x e    ,     , i t x t x e      (7) where Re Im i      is the complex eigenfrequency, in which Re  corresponds to the vibration eigenfrequency itself, Im  corresponds to the decay rate of vibration (damping index) and     , i u x x    are eigenmodes of vibrations. A complete mathematical formulation of the problem is given in [33-34]. The stated problem is solved by the finite element method, using the algorithm elaborated by the authors of this paper using ANSYS package possibilities and program written in FORTRAN language. As shown in [2], the series resonant electric circuit, shunting the piezoelectric element can decrease oscillations for only one selected frequency, to which the external circuit is tuned by choosing the appropriate parameters R and L . Here will be given explanations related to the process of tuning. A piezoelectric element, by virtue of its nature, has capacitive properties and in the presence of the external RL -circuit forms a series resonance oscillatory RLC -circuit. This leads to the appearance of additional eigenfrequency e  in the natural frequencies spectrum of the base structure, which is due to the interaction of the inductive element and the inherent capacitance of the piezoelectric element. As a result, the spectrum of the eigenfrequencies of the structure with a piezoelectric element and an external circuit is formed. This additional eigenfrequency e  can vary over a wide range due to changing the parameters of the external circuit. During the process of tuning this additional frequency is brought closer and closer to the eigenfrequency of the structure n  until they coincide. Since in the complex eigenfrequency (according to the mathematical formulation of the problem) the real part characterizes the frequency of vibrations, one of the conditions for selecting parameters of the shunting circuit can be given as Re Re n e    (8) From the one hand the performed numerical investigations [35] showed, that in the space of R-L parameters there is a set of points corresponding to different values of resistance and inductance, at which condition (8) is satisfied. From the other hand the performed numerical investigations have revealed that the damping index Im e  related to the additional complex frequency of the electric oscillatory circuit varies within very wide range and has, in the space of electric circuit parameters R and L , a great number of local extrema, which appear at the instant of convergence of electric circuit