Issue 49

D. Oshmarin et alii, Frattura ed Integrità Strutturale, 49 (2019) 800-813; DOI: 10.3221/IGF-ESIS.49.13 802 natural vibration frequencies where the real part defines the circular natural vibration frequency of the structure and the imaginary part defines its vibration damping rate (i.e. appears to be the parameter of structural vibration damping). A previous analysis was performed of the behaviour of imaginary parts of complex eigenfrequencies in the space of external circuit parameters (resistance – inductance) while tuning the external circuit to a given frequency of vibrations of the structure. This analysis revealed some peculiarities of the behaviour of damping ratios at certain frequencies, which also allowed for a sufficiently high damping rate at some other frequencies. From this, it may be possible to use one single piezoelectric element and one single resonant RL circuit to damp vibrations at several frequencies with an appropriate choice of parameters. This work is devoted to the demonstration of such a possibility. M ATHEMATICAL STATEMENT OF THE PROBLEM onsider a piecewise homogeneous body of volume 1 2 V V V   , where volume 1 V is composed of homogeneous elastic parts and volume 2 V of piezoelectric elements. Part of the surface of volume 2 V is covered with electrodes. To one of the electrode surfaces of volume 2 V is attached an external passive electrical circuit of arbitrary configuration, consisting of resistive (resistance R ), inductive (inductance L ) and capacitive (capacitance C ) elements. One of the quite efficient problems in mechanics that is used for optimization of dissipative properties of deformable solids is problem on natural vibrations of bodies in which energy dissipation occurs during vibrations. Mathematical statement of this problem and its applications for studying natural vibrations of viscoelastic bodies were proposed in [29]. Algorithm of numerical implementation of the problem with the aid of capabilities of the ANSYS software package was presented in [30]. The approach proposed in these papers has been applied for investigating natural vibrations of electroelastic bodies with external electric circuits. The variational equation for natural vibrations of the body consisting of elastic and piezoelectric elements is constructed based on the relations of linear theory of elasticity and quasi-static Maxwell equations [31-34]:           1 2 1 2 1 2 1 2 1 1 1 1 1 0 i C L R p p q q r r ij ij i ij ij i i i i V V n n n L L R R C C r p q p q r u u dV D E u u dV dtdt dt C L R                                           (1) Here , i i D E are the components of electric flux density and electric field intensity vectors; ij  are the components of the symmetric Cauchy stress tensor, ij  are the components of the linear strain tensor, i u are the components of the displacement vector,  is electric potential, 1 2    is the potential difference on external circuit element, , , L R C n n n are the numbers of inductive, resistive and capacitive elements respectively, p L , q R , r C are inductance, resistance and capacitance values for corresponding circuit element. For the electric field, the potentiality condition is fulfilled: , i i E    (2) For elastic parts ( 1 V ), the following physical equations of the linear theory of elasticity hold true: ij ijkl kl C    (3) For isothermal processes, the electroelastic parts ( 2 V ) obey the following physical relations: C