Issue 49

D. Oshmarin et alii, Frattura ed Integrità Strutturale, 49 (2019) 800-813; DOI: 10.3221/IGF-ESIS.49.13 810 shell with an attached piezoelectric element without an electric circuit, the optimal parameters of the shunt circuit to damp vibrations at a single frequency and the corresponding values of complex eigenfrequencies for the shell with a shunted piezoelectric element. a b Figure 6 : Damping indices relationship surfaces depending on R and L parameters (a) and projections of the lines of their pairwise coincidence (b) for the frequencies under consideration. On the basis of the solution to the natural vibration problem for the shell with shunted piezoelectric element, we retrieved relationship surfaces of damping indices depending on variation of R and L parameters for these three frequencies (Fig. 6a). These relations allow us to define in the R-L space the lines of pairwise coincidence of the imaginary part of the frequencies under consideration. Fig. 6b shows the presented projections of these lines. According to the plots in Fig. 6b, there are points where the damping ratios for all three frequencies appear to coincide. Tab. 5 represents the values of the electric circuit parameters and corresponding values of comlex eigenfrequencies. The results show that option II has the highest values of damping indices. I R =5.38 [kOhm], L =6.33 [H] 549.84 + i 14.59 586.14 + i 13.08 647.18 + i 13.36 II R =4.92 [kOhm], L =6.30 [H] 547.17 + i 14.77 575.98 + i 15.71 648.74 + i 13.44 III R =4.42 [kOhm], L =6.33 [H] 573.07 + i 12.36 593.87 + i 13.86 650.18 + i 12.45 IV R =4.01 [kOhm], L =6.53 [H] 569.05 + i 10.23 593.96 + i 9.73 650.05 + i 9.67 Table 5 : Values of R and L at which imaginary parts of the first three complex eigenfrequencies within the frequency range up to 700 Hz are close to each other. Fig. 7 shows the frequency response amplitudes of displacement vector component r U at point A when the system is excited by harmonic force r P applied at point B. The results correspond to shunt circuit tuning for single-mode damping of the frequencies under study and to damp all the three modes within the frequency range up to 700 Hz.