PSI - Issue 28

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

1445

8

Table 1. Values of natural vibration frequencies of the original cantilever plate with no additional elements

Natural vibration frequencies Re   

Natural vibration

Number of vibration mode

Number of vibration mode

Type of mode

Type of mode

frequencies

Re   

1 2 3 4

11.233 70.359 177.096 197.193

5 6 7 8

387.174 477.349 535.593 641.627

b b

b p

t

t

b

b

Table 2. Dynamic characteristics of the structure with viscoelastic layer (Configuration I)

Complex dynamic moduli with constant values over the entire frequency range

Frequency-dependent complex dynamic moduli (Increasing moduli)

Frequency-dependent complex dynamic moduli (Decreasing moduli)

Number of vibration mode

Complex natural vibration frequencies

Complex natural vibration frequencies

Complex natural vibration frequencies

Damping coefficient 

Damping coefficient 

Damping coefficient 

Re Im i     

Re Im i     

Re Im i     

1 2 3 4 5 6 7 8

11.14653 + i 0.13014 69.78653 + i 0.81102 174.28094 + i 1.72113 195.58086 + i 2.27663 384.03496 + i 4.48638 446.86240 + i 0.44975 572.20850+i5.52291 636.52880+i7.47042

0.0117 0.0116 0.0010 0.0116 0.0117 0.0010 0.0010 0.0117

11.3371+i0.16114 77.0898+i2.03399 206.260+i7.02271 239.985+i9.19806 460.203+ i3.00853 499.362+i21.7310 684.413+i29.6077 876.184+i41.8213

0.0142 0.0264 0.0341 0.0383 0.0065 0.0435 0.0433 0.0477

16.8831+i 0.88578 91.8046+i 4.07921 208.825+i 7.39871 236.884+i 8.77354 423.255+i 11.0613 450.0323+i 11.1965 540.065+i 7.57588 617.266+i 3.85222

0.0525 0.0444 0.0354 0.0370 0.0261 0.0249 0.0140 0.0062

The results presented in table 2 lead us to the following conclusions. Application of the model of viscoelastic material with frequency-independent complex dynamic moduli (this assumption is valid for specific parts of a frequency range) allows providing equal level of vibration damping at all modes related to one and the same group (bending, torsional or in-plane). Herewith the highest damping coefficients correspond to the bending vibration modes. It should be mentioned that adding the layer of frequency-independent material leads to the change in values of natural vibration frequencies which are less than 7%. Taking into account frequency-dependent properties of viscoelastic material allows reaching higher values of damping coefficients at the specified modes. According to the obtained results these vibration modes realized at frequencies for which complex dynamic moduli takes higher values. Herewith distribution of level of damping along the spectrum is determined by the character of dependence of mechanical parameters of viscoelastic material on frequency. Beside this application of the material having the frequency-dependent properties can lead to a considerable shift of values of natural vibration frequencies from their original values. It was found that maximal shift of the first natural vibration frequency in case of using the material with decreasing values of complex moduli reached 50% in comparison to the original value represented in table 1. Next, we proceed to the configuration II of the plate. In this case piezoelectric element electrodes of which can be connected to resistive or resonant electric circuits was attached to the surface of the original elastic structure. It is the well-known fact that the efficiency of performance of piezoelectric element depends on a character of its deformation. Due to this one can face the situations when piezoelectric element, located in such a way that it either does not deform at the specified modes (in-plane modes, for example) or it deforms in a way that the total charge generated when piezoelectric element deformed is equal to zero, becomes unable to damp vibrations. In the current research we accepted that center of masses of piezoelectric element was located only at the longitudinal axis of symmetry of the plate. Thus, piezoelectric element could not be used for damping of vibrations at torsional and in plane modes due to its character of deformation at these modes. According to the table 1 these modes are 3, 6 and 7. Table 3 represents the following data for the bending modes of vibration: coordinates of the location of piezoelectric element which is optimal for damping of vibrations at the specified mode and where electromechanical coupling coefficient takes the highest values; values of the parameters of the elements of shunting circuits which provide the best vibration damping at a given mode. Table 4 shows the results of calculations of dynamic properties of the plate with piezoelectric element shunted with resistive or resonant electric circuit with corresponding values of parameters represented in table 3.