Issn 2181-1296 ilmiy axborotnoma научный вестник scientific journal

97

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ILMIY AXBOROTNOM A
M EXANIKA
2021-yil, 1-son
U = B  ex p [k z + —  ] 
(24)
Here k - number of waves, — - periodic frequency of oscillations.
Substituting this expression (24) into equation (23), we obtain the following frequency equation after 
some simple simplifications.
bl
—
- k 2 - -
k = 0
(25)
b 2 
( r + gz)
It can be seen that the z coordinate also actively participates in the frequency equation, that is, in different 
parts of the shell, the ratio between the vibration frequency and the number of waves changes depending on 
the distance from the origin.
Since the space velocity in equation (25) is c = — , we have the following space velocity equation.
k
- 1 - -
1
= o
(26)
b 
(ro + g z) k
We solve this equation numerically using the Maple 17 software package. Quantitative calculations 
were carried out for cases when the material of the rod was steel, aluminum and polymer. Moreover, their 
physical and mechanical characteristics are as follows:
Steel - 
£=2,0-10" Pa; v=0,25; p=7850 k g / m "  ;
aluminum - 
£=0,7-1011Pa; v=0,35; p=2750 k g j m ?  ;
polymer - 
£=5,5-1010 Pa; v=0,4; p=1700 k g / m *  ;
rod radius: r0 = 0.02 m ,
The calculation results are shown in Figures 2-3 in the form of graphs of the dependence of — - 
frequency and k  - the number of waves. Figure 2 shows the relationship between frequency and number of 
waves for aluminum, steel, and polymer rod materials. The graphs show that the relationship between the 
frequency of free vibrations of the rod and the number of waves is directly proportional to all three 
materials. In this case, at a fixed value of the number of waves, the frequency of the steel rod is greater than 
the frequency of the aluminum and polymer rods, and the frequency of the aluminum rod is greater than the 
frequency of the polymer rod. In other words, the greater the material's modulus of elasticity, the greater its 
frequency value. In fig. 3 shows the relationship between the frequency and the number of waves in the 
sections z = 0.2 and z = 0.8 of the aluminum rod. The graphs show that the farther from the origin, the 
lower the frequency.
У
e e l
---- - ■ - si
^ 
■
— a lu m in u m
s '
. 
**
s '
s
‘
■O
x ‘
XI
s
&
A S '
V
/S s
4L_
--- 2=0.2
=0.8
s'
S'
s'
S'"'
s'
S'
/
s~
/
V
a. j
1 
1 J
2 
2 3
и
3 
3.3
Figure 2. The relationship between frequency and 
number of waves for aluminum, steel and polymer.
0 J
1 
1.5 
2 
2.5 
3 
3.5 
4 
A 5  
5
к
Figure 3. Relationship between frequency and 
number of waves in sections z = 0.2 and 
z = 0.8 of an aluminum rod.
5. 
Conclusions. The obtained analytical and numerical results allow us to draw the following 
conclusions.:
- 
the equation of torsional vibrations of a round conical elastic rod is obtained. The coefficients of 
this equation are functions of the longitudinal coordinates;
2