PSI - Issue 28

Liudmila Igusheva et al. / Procedia Structural Integrity 28 (2020) 1303–1309 L. Igusheva, Y. Petrov / Structural Integrity Procedia 00 (2019) 000–000

1305

3

2

2

3

u

u

u



 



2

2

2 b u  

0,

a

c





(2)

2

2

2

t

x

t x



 

E kP

 

,

,

.

a

b

c







where

S





In the special case when the coefficient c is equal to 0 , the equation of longitudinal vibrations of an elastic rod located in an elastic environment turns out to be the well – known Klein-Gordon equation (generalized wave equation). This equation finds its application in quantum mechanics and the mechanics of a deformable solid. The main interest is represented by rods of finite length, since they are found in almost all engineering structures. Therefore, we consider the problem of wave propagation in the rod of a finite wave under the action of pulsed loads. Fig. 1 shows the model of the final rod. It is assumed that all cross sections of the rod have the same shape and are subjected to the same environmental influences, so we can consider a one-dimensional problem of longitudinal vibrations.

Fig. 1. The model of a rod of finite length.

Direct the ox axis along the axis of the rod, take the left end of the rod as the origin of coordinates. Assume that an arbitrary dynamic force ( ) f t is applied to the left end of the rod, and the second end of the rod is stress-free. Mathematically this problem in displacements can be written in the following form (3).

2   

2

3

u

u

u

 



2

2

2 b u  

0

a

c





2

2

2

t

x

t x

 

0        0 t t u  

0

(3)

u t u x

0



    

1 ( ).

f t

 

0

x



2

a

The numerical solution of problem (3) yields the following distribution of deformations (Fig. 2) for the case when the rod is not affected by resistance forces. The viscoelastic model allows us to describe the attenuation of vibrations occurring in the rod. It can be seen that the calculation is similar to the experimental results. A number of experiments were performed to measure the deformation profile in a rod under dynamic influence (Smirnov I. V., SPbSU-NNSU, 2020). The rod is made of PMMA (plexiglass). Two cases were considered. The first is when the rod is in the air, and it is not affected by the forces of resistance from the environment. The second is when there is silicone around the rod. The size of the plexiglass rod in which the deformation profile was measured is 10x10x400 mm.