Issue 49

D. Oshmarin et alii, Frattura ed Integrità Strutturale, 49 (2019) 800-813; DOI: 10.3221/IGF-ESIS.49.13 804 damping index with the damping indices Im n  of the complex natural vibration frequencies related to the structure. However for each natural vibration frequency of the structure, there exists only one global extremum of the damping index over the entire space of possible values of the R-L parameters [35]. As a result, the condition providing a unambiguous determination of the optimal values of shunting circuit parameters of opt j R and opt j L , at which the highest damping rate of the appropriate vibration mode can be reached, can be was obtained in the form of (9) Im max n   (9) The results obtained during numerical calculations show that in the case of application of a resonant circuit for damping of vibrations at one mode there exist options when damping of vibrations is also observed at some other modes. This means it may be possible to implement a single piezoelectric element shunted with one single resonant RL circuit to damp vibrations on several modes. Imaginary parts of complex natural frequencies Im  were chosen as a parameter to formulate the criteria for shunt circuit parameters selection in order to damp vibrations on several modes. Taking into account the complex conjugation of the obtained eigenvalues Re Im i      , the ones are chosen are those that have a positive imaginary part. This choice provides the damped character of vibration processes under consideration. Representing eigenvectors (7) in trigonometric form (10), allows seeing that exactly the imaginary part of complex natural vibration frequency Im  determines the damping properties of a structure (in this case, the rate of decay of the vibration amplitude)           Im 0 Re Re cos sin t u u e t i t       (10) Analysis of the behaviour of the imaginary parts in the R-L parameters space shows the existence of variants for which damping indices related to different frequencies appeared to be equal, meaning that the damping rates for these frequencies is equal. In this regard, the following condition for selecting the parameters of a series resonant RL- circuit was chosen: in order to damp several vibration modes, the parameters of the shunt circuit should correspond to the point in the R-L space where absolute values damping ratios coincide and are maximal. D EFINITION OF SHUNT CIRCUIT PARAMETERS FOR DAMPING STRUCTURAL VIBRATIONS AT SEVERAL FREQUENCIES he applicability of the proposed approach can be demonstrated using a thin-walled shell in the form of a semi- cylinder that is clamped at the edges and simply supported along the generatrix (Fig. 1). a b Figure 1 : The design models of cylindrical shell with attached piezoelectric element (а) and cylindrical shell with attached piezoelectric element connected to series RL circuit (b). T A B