PSI - Issue 32

D.A. Oshmarin et al. / Procedia Structural Integrity 32 (2021) 158–165 Oshmarin D.A., Iurlova N.A., Sevodina N.V., Iurlov M.A. / StructuralIntegrity Procedia 00 (2021) 000 – 000

161

4

where 3

nel S is the part of the surface of volume

3 V not covered with electrodes, and 3

el S is the part of the surface of

volume 3 V which has electrode coating (Karnaukhov and Kirichok, 1988; Parton and Kudryavtsev, 1988). The solution to the problem of natural vibrations will be searched for in the following form:     0 , e i t u x t u x   (8) ( ) ( , , ), ( , , ), ( , , ), ( , , ) u x u x x x u x x x u x x x x x x   is the generalized vector of state containing the components of mechanical displacements 1 2 3 , , u u u and the electric potential  ; Re Im i      is the complex eigenfrequency of the structure vibrations, in which Re  is the natural angular frequency of vibrations, and Im  is the damping coefficient characterizing the rate of vibration damping. The solution to the problem is found using the finite element method. The mathematical formulation of the problem on natural vibrations of electro-viscoelastic bodies and the algorithm of its numerical implementation based on the finite element method are given in detail in (Iurlova et al., 2019). This approach is based on capabilities of the ANSYS package which allow writing global assembled stiffness and mass matrices. The solution to the eigenvalue problem was found using the Mueller’s method realized in program written in Fortran language (Iurlova et al., 2019). 3. An influence of an external electric R-circuit on dissipative properties of an electro-viscoelastic structure It is a well-known fact that application of viscoelastic elements allows damping bending modes of vibrations in the most effective manner. Alongsidethisplanarandtwistingmodesare damped much less effective (Mead, 1999; Nashif et.al., 1985). Herewith one also knows that application of piezoelectric elements shunted with external electric circuits exhibits the highest efficiency also in case of damping of bending vibrations (Hagood and Von Flotow, 1991; Matveenko et al., 2018). It is explained by the fact that in this case strains in the direction of polarization of a piezoelectric element are maximal. This effect provides the highest level of mechanical energy converted into the electric one. Consideringthesepeculiaritiesofdynamicbehavioroftheelements under study all further investigation are conducted on the example of only bending modes of vibrations. Theseinvestigations are aimed on establishing the characteristic features of a joint performance of shunted piezoelectric and viscoelastic elements. Here,   0 1 1 2 3 2 1 2 3 3 1 2 3 1 2 3

Fig.1.Computationalscheme of a plate with a viscoelastic layer and a piezoelectric element

Acantileverplatemadeofisotropicelasticmaterialwaschosenastheobjectunderstudy. Thecharacteristicsoftheplatematerialareasfollows: Youngmodulus

11 2 10 E   Pa, Poissoncoefficient

0.3   ,

7800   kg/m 3 . An element made of a viscoelastic material is located at the surface of the plate.

specific density