PSI - Issue 50

D.A. Oshmarin et al. / Procedia Structural Integrity 50 (2023) 212–219 Oshmarin D.A., Iurlova N.A., Sevodina N.V. / Structural Integrity Procedia 00 (2019) 000 – 000

217

6

in Fig.3 refers to the first mode of vibrations, red – to the second one, green – to the third mode, correspondingly. Due to the linearity of the chosen mathematical model, the resulting dependencies are also linear.

a

b

Fig. 3. Dependencies of optimal values of control electric impact on a magnitude of mechanical response foe the three vibration modes under study: (a) potential difference, (b) electric current

  cont U u and

  cont I u using the equation

Based on the plots from fig.3 we obtained explicit functional relations

  cont U u and

  cont I u take

of a straight line passing through two points. After some ease transforms the relations

the following form:

 



  u u I   1







2 u u U U    cont 1

1

2

1

I



cont

cont

cont

1 U I cont ;

1

(8)

U

I









cont

cont

cont

u u 

u u 

2 1

2 1

The data for obtaining functional relations (8) were chosen in such a way that we could have a possibility of using one and the same values of 1 u and 2 u for deriving the relations   cont U u and   cont I u . Thus, we took values     cont 1 cont 2 , U u U u and     cont 1 cont 2 , I u I u directly from the plots from fig.3. As a result, we obtain functions describing the relationship between the required mechanical response and the optimal parameters of the electrical signal for the three vibration modes under consideration, represented in Table 1. Thus, having the obtained dependences, we can calculate the value of the potential difference or current amperage required to achieve a given level of amplitude of forced vibrations of an electro-viscoelastic system.

  cont U u and

  cont I u for the three vibration modes under study.

Table 1. Dependencies

  cont I u

  cont U u

№ of mode









4

3 3 10 37.040   3 10 22.387   3 10 6.466     4 5

3

3 3 10 1.968 10    3 10 7.267 10    4

5

1 2 4

cont 1.232 10  

6.543 10 2.431 10





U U U

u            u u

I I

u       u 

   

cont



4

1

5

7.444 10 2.084 10





   

cont

cont



 5

5

5 cont 1.931 3 10 5.667 10 I u        

cont

The possibilities of controlling the amplitude of forced vibrations will be demonstrated by the example of the considered cantilever plate with a piezoelectric element located on its surface. Let's consider two variants of the force loading of this system, presented in Fig. 4:   ,0,0, x F F  , where 3 5 10 x F    N (fig.4a) and   0, ,0, y F F  , where 3 1 10 y F    N (fig.4b). Based on the solution to the problem of forced steady-state vibrations for the loading schemes represented at fig.4 we obtained frequency response plots (FRP) of the mechanical response / z stat u u u  for the first vibration mode.