PSI - Issue 28

9

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

1446

Table 3. Optimal values of parameters which provide maximal damping of vibrations for configuration II

Resistive R -circuit

Resonant RL -circuit

Number of vibration mode

Optimal location y (mm) of piezoelectric element

Optimal resistance R

RL opt L , H

RL opt R , kΩ

opt R , kΩ

Optimal inductance

Optimal resistance

1 2 4 5 8

28

1002.43 180.16

10446.70

459.16 66.59 25.25 12.98

108 143 157 170

356.14 42.13 11.24

62.43 32.23 19.56

4.18

7.58

Table 4. Results of calculation of dynamic properties of the system for configuration II

Number of vibration mode

Resistive R -circuit

Resonant RL -circuit

Complex natural vibration frequencies Re Im i     

Complex natural vibration frequencies Re Im i     

Damping coefficient 

Damping coefficient 

1 2 4 5 8

14.2486 + i 0.2034 77.0225 + i 0.7004 223.5774 + i 2.4626 432.7828 + i 4.7245 710.9157 + i 7.1479

0.0143 0.0091 0.0110 0.0110 0.0101

14.4956 + i 1.5726 76.8158 + i 6.9805 220.8405 + i 20.3071 423.2150 + i 32.4653 693.2046 + i 47.9109

0.1085 0.0909 0.0920 0.0767 0.0691

The results obtained on the basis of the considered design schemes with corresponding dimensions of structural elements and material properties and presented in tables 2 and 4 allows resuming the following. Due to the fact that the parameters of the configurations under study were chosen properly, applying piezoelectric element shunted with R -circuit allowed one to achieve the level of damping comparable to that achieved when viscoelastic material with frequency-independent properties was used (both cases showed that damping coefficients change from 0.01 up to 0.014). The numerically obtained results confirmed the conclusions of (Hagood and von Flotow, 1991) that the dynamic behavior of a structure with a piezoelectric element shunted by an R- circuit is similar to the behavior of a structure made of a viscoelastic material (in this work, the authors did not take into account the frequency dependence of the mechanical characteristics viscoelastic material, therefore the comparison was made for a viscoelastic material with frequency-independent characteristics).

Fig. 4. Graphical curves describing behavior of damping coefficients for bending modes of vibrations

Application of piezoelectric element shunted with resonant RL -circuit (fig.4, red dashed line) shows that the higher damping properties can be reached in comparison as to application resistively shunted piezoelectric element (fig.4, blue dashed line) as so to applications of viscoelastic material either with frequency-independent properties (fig.4, blue solid line) or frequency-dependent increasing (fig.4, black solid line) or decreasing (fig.4, red solid line) complex dynamic moduli.