PSI - Issue 28

D.A. Oshmarin et al. / Procedia Structural Integrity 28 (2020) 1438–1448 Author name / Structural Integrity Procedia 00 (2019) 000–000

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where S  is the surface, where the stresses ij  are prescribed; u S is the surface, where the components of the displacement vector i u are given and j n are the components of the unit normal vector for the S  surface. For the non-electrode parts of the surface 0 S of a piezoelectric body of volume 2 V there are no electric boundary conditions due to the fact that these parts of the surface are non-conductive and hence there are no free electric charges on them. Taking into account the used form of Maxwell’s equations, this condition can be written as:   0 S n D dS    or 0 divD  (7) Supposing one of the electrode surfaces of the piezoelectric element is grounded, i.e. it has a zero-value electric potential, the electric boundary condition for this part of the surface has the following form: where el S el S of volume 2 V , where the electric potential is prescribed. If there is no an external power supply, the other parts of the electrode surface el S can be considered free (in this case, the open circuit mode is realized) or it can be considered to have zero-value electric potential (8) (here, the short circuit mode is realized). By using the electrode surfaces, the external electric circuits can be attached with an arbitrary configuration to the system under consideration, which could include resistive elements ( R ), capacitive elements ( C ) or inductive elements ( L ). 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   (9) ( ) ( , , ), ( , , ), ( , , ), ( , , ) 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 index characterizing the rate of vibration damping. The solution of 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). 3. An object under study A cantilever plate having the following dimensions 210 l  mm, 26 b  mm, 0.6 h  mm was chosen as the object under study. The plate was made of the material with the following physical and mechanical properties: Young’s modulus 11 2 10 E   Pa, Poisson’s ratio 0.3   , specific density 7800   kg/m 3 . Two configurations of the cantilever plate were considered in order to carry out a comparative analysis of efficiency of damping of vibrations by two different mechanisms. The first mechanism was the internal mechanism of energy dissipation (when elements made of viscoelastic materials were used) and the second one was related to transformation of mechanical vibration energy into an electric one and its further dissipation as a heat and an electromagnetic radiation (when piezoelectric elements electrodes of which connected to external electric circuits were used). A special feature of the configuration I is damping of vibrations by a mechanism of internal energy dissipation in structural elements made of viscoelastic materials. In this case the surface of the plate is completely covered by a layer  is the part of the electrode surface Here,   0 1 1 2 3 2 1 2 3 3 1 2 3 1 2 3 0 el S    (8)