PSI - Issue 28

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

1444

7

electromechanical coupling coefficients was used (Oshmarin and Iurlov, 2017). This quantity characterizes the efficiency of transformation of mechanical energy of vibrations into electric one and is calculated according to (Hagood and von Flotow, 1991) as follows 2 2

2 O C S C S C    

,

(10)

K



where O C  , S C  are the natural vibration frequencies of a structure when piezoelectric element operates in open circuit and short circuit modes correspondingly. In order to reach maximal damping properties piezoelectric element must be located as such as the electromechanical coupling coefficient for the specified modes takes the maximal value. Within the frameworks of the current research, it was accepted that the center of masses of piezoelectric element was always located at the longitudinal axis of symmetry of the plate. Thus, the location of piezoelectric element was characterized by the only parameter – coordinate y of its center of masses. Parameters of a shunting circuit which provide the maximum damping properties of a structure at a given vibration mode were determined according to the formulas derived in (Hagood and von Flotow, 1991). In case when R- circuit is used the value of optimal resistance R opt R is determined as follows:

2

1 C  

i S C S K

(11)

R opt

R





 2 1 i

is the parameter which characterizes capacitive properties of piezoelectric element at i -th i K is the value of electromechanical coupling coefficient at i -th vibration mode, S С is the static

i S

Here

C

С K  

S

vibration mode,

capacitance of piezoelectric element. In case when series RL -circuit is used the values of optimal resistance RL

opt R and inductance

RL opt L are determined as

follows:

2

1

K

(12)

RL opt

RL opt

;

R

L







 2





3

2 

2

1 C K  i

2

1 C K  i



/ S C S

S C S

4. Results of numerical calculations In order to draw a comparison between different mechanisms of energy dissipation first we need to determine values of the first eight natural vibration frequencies of the elastic cantilever plate under study in case when no additional elements realizing mechanisms of vibration damping are present. The obtained values of natural vibration frequencies are presented in table 1. Apart of it the table indicates the character of vibration at the corresponding mode: b is for bending vibrations, t is for torsional vibrations, p is for in-plane vibrations (vibrations in plane XY). Next, we should determine values of natural vibration frequencies for the configuration I (the plate is completely covered with layer made of viscoelastic material). The obtained results are shown in table 2. Here Re Im i      is the complex eigenfrequency of the structural vibrations, in which Re  is the natural angular frequency of vibrations, and Im  is the damping index characterizing the rate of vibration damping;  is damping coefficient which is determined according to the formula Im Re     (Timoshenko et al., 1985). In case when complex dynamic moduli of viscoelastic material increased along with frequency they changed according to the following range for the bandwidth from 11 up to 900 Hz: Re 1500 12600 G   MPa, Im 300 2520 G   MPa, Re 7450 62600 B   MPa. For the case of decreasing complex dynamic moduli the ranges of change were as follows: Re 2000 642 G   MPa, Im 4000 128 G   MPa, Re 100000 32000 B   MPa. These ranges were chosen in order to provide approximately equal level of damping at individual modes for the both options of dependence on frequency of components of complex moduli under consideration.