o> 0 < 0 , from where follows

Figure 8. The distribution of the radial displacement of the shell along the axis of the cylinder for two values of the cylinder rotation n.

Figure 9. Distribution of axial movement of the shell along the axis of the cylinder for two values of the cylinder rotation n.

k R m 0 ( R , k ) > 0 . (31)

Inequality (31) in the ( R , k ) determines the radius values R and coefficient k, where the shell will be in a compressed state. With values ( R , k ) , lying on the curve k R m 0 ( R , k ) = 0 , loss of stability of the shell occurs.

For the selected values of the task parameters, the value N ¯ 0 has order 10 15 ÷ 10 20 .

Therefore, in this case the second case is practically not realized.

In the second case, when boundary conditions are satisfied at the ends of the rubber layer, the movement is determined by formulas

u r = λ ( R 2 R 0 2 ) β c 1 2 b 1 R [ 1 λ l c h λ z 2 s h ( λ l / 2 ) ] R N ¯ 0 (32)

u z = ( β 1 + ν 0 β ) N ¯ 0 z + λ ν 0 ( R 2 R 0 2 ) β c 1 2 b 1 R 2 [ z l s h λ z 2 s h ( λ l / 2 ) ] (33)

It can be seen from formula (33) that the axial movement of the shell at the edges becomes zero only in the absence of an axial force N 0 . This case can be realized by contacting the ends of the rubber layer with a fixed flange, and the edges of the shell remain free. In this case, in the formulas (22), (23), (32) and (33) it is necessary to assume N ¯ 0 = 0 .

Figure 10 and Figure 11 show the distribution of radial and axial displacements of the shell for two values of the angular velocity rotation of the cylinder in rpm.

When N 0 0 , the axial displacement edges of shell do not vanish. It is possible that there is no movement of shell in the radial direction. Assuming in formula (28) u r = 0 , we find the value of the axial force N ¯ 0 in which the shell

will not mix in the radial direction N ¯ 0 = λ ( R 2 R 0 2 ) β c 1 2 b 1 R 2 [ λ l 2 c t h ( λ l / 2 ) 1 ]

From this equality, it follows the quantity: N ¯ 0 = m 1 c 1 k R m 1

Where m 1 = m 1 ( R , k ) = λ c β ( R 2 R 0 2 ) [ λ l / 2 t h ( λ l / 2 ) ] / 2 R 2 b 1 .

4. Conclusions

1) Theoretical distributions of the radial moving of layer have being studied for two values of turn cylinder in different distances from the axis to cylinder.

2) It is possible to improve process parameters feed in pneumomechanic spinning

Figure 10. The distribution of the radial displacement of the shell along the axis of the cylinder for two values of the cylinder rotation n.

Figure 11. The distribution of the axial displacement of the shell along the axis of the cylinder for two values of the cylinder rotation n.

machines.

3) The given diminishes such as damages of fibers in are feeding zone.

4) The graphs of the distribution for the radial displacement of the layer for two values of cylinder rotation with different distances from the axis to the cylinder are presented.

Cite this paper
Abdukarimovich, M. , Ibragimovich, A. and Sharipjanovich, S. (2018) Designing a New Design of a Loading Cylinder for Pneumomechanical Spinning Machines. Engineering, 10, 345-356. doi: 10.4236/eng.2018.106025.
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