Issue 49

D. Oshmarin et alii, Frattura ed Integrità Strutturale, 49 (2019) 800-813; DOI: 10.3221/IGF-ESIS.49.13 805 The geometrical dimensions of the shell are as follows: 1 76 r  mm 1 300 l  mm, 1 0.25 h  mm. The shell is made of elastic isotropic material having the following characteristics: 11 1.96 10 E   Pa, 0.3   , 7700   kg/m 3 . The piezoelectric element, which is attached to the surface of the shell has the shape of the ring segment with the following dimensions: 76.25 p r  mm, 15.08 p    , 0.36 p h  mm. The centre of mass of the piezoelectric element is 15 mm away from the clamped edges and is shifted by 90 о from the simply supported generatrix along the angular coordinate. The piezoelectric element is made of peizoceramics PZT-4 polarised along the r -axis and having the following characteristics (in the cylindrical coordinate system): 10 11 22 13.9 10 C C    Pa, 10 12 7.78 10 C   Pa, 10 13 23 7.43 10 C C    Pa, 10 33 11.5 10 C   Pa, 10 44 3.06 10 C   Pa, 10 55 66 2.56 10 C C    Pa, 31 32 5.2      C/m 2 , 33 15.1   C/m 2 , 52 61 12.7     C/m 2 , 9 11 22 6.45 10 e e     F/m, 9 33 5.62 10 e    F/m, 7700   kg/m 3 . The upper and lower surfaces of the piezoelectric element are covered with electrodes. The external series RL circuit is connected to piezoelectric element electrodes (Fig. 1b). The solution of the natural vibration problem for the structure with an attached piezoelectric element and without an external circuit shows that in the frequency range 0–1500 Hz there are 15 natural vibration frequencies 2 f   . For the specified piezoelectric element location, the electric potential generated on its electrodes appears to be sufficient for registration and further operation only at five frequencies from the specified frequency range: 1st, 4th, 5th, 12th and 15th (coupled modes of vibration). An almost negligible electric potential at other frequencies indicates that the mode shapes for these frequencies are such that the piezoelectric element either remains undeformed or it deforms symmetrically, which leads to zero-value total electric charge (uncoupled modes of vibration) [36]. On the basis of the solution to the natural vibration problem (1)–(7), values of natural vibration frequencies were obtained for the shell without an external circuit (real) and with a series RL- circuit (complex), with the parameters chosen to reach maximal damping of corresponding frequencies from the specified frequency range. Tab. 1 presents the five frequencies for which generated electric potential is sufficient for the efficient operation of the piezoelectric element (only for coupled modes). Tab. 1 also shows the optimal shunt circuit parameters with corresponding complex eigenfrequency values to damp vibrations at these frequencies separately. Number of frequency Shell eigenfrequencies without electric circuit f [Hz] Shunt circuit optimal parameters for single-mode damping L [H], R [kOhm] Complex eigenfrequencies of the shell with tuned shunt circuit Re Im f f if   1 557.41 L =6.93, R =5.3 553.60 + i 28.30 4 759.57 L =3.66, R =4.1 750.59 + i 32.06 5 803.79 L =3.67, R =4.02 813.58 + i 31.12 12 1293.42 L =1.3, R =2.18 1308.34 + i 48.24 15 1482.21 L =1.04, R =2.57 1493.29 + i 81.48 Table 1 : Natural vibration frequencies for the shell with piezoelectric element and no external circuit and with external circuit tuned to a single frequency. On the basis of the solution to the natural vibration problem for the system (shell with shunted piezoelectric element), there were retrieved relationship surfaces of imaginary parts of complex eigenfrequiencies depending on variation of R and L parameters for the this five frequencies under consideration (Fig. 2) When analysing relationship surfaces in Fig. 2, one can identify frequencies for which the surfaces intersection exists. These are at the 4th and 5th frequencies (Fig. 3a) and the 12th and 15th frequencies (Fig. 3c). Let us consider these pairs separately. Figs. 3b and 3d show the lines of Im  coincidence in the R-L space. Along these lines, damping indices take different values. To define the best damping of structural vibrations for the corresponding pairs, it is necessary to find the values of the external circuit parameters that guarantee the maximal decay rate of vibrations.