The solid small rocket housing is cylindrical. To facilitate torque loading and clamping, it is placed vertically during the tightening process, and the clamping mechanism is a three-jaw chuck. The main function of clamping is to prevent the solid small rocket body from rotating during the tightening process due to the tightening torque M.

Figure 3. Force analysis diagram of the clamped shell under torque loading.

As shown in Figure 3, to simplify the calculation, the force F on the three claws is approximately regarded as being of the same magnitude and distributed at 120˚ in space, and the radius of the solid small rocket shell is R. To facilitate automatic torque loading, the solid small rocket is set to be placed vertically. Therefore, when clamping the cylinder, the influence of its own gravity is not considered.

Therefore, taking point A as an example, the critical force that makes the rocket body rotate at each clamping point is:

The frictional force generated by the clamping force at each point is:

To prevent the rocket body from rotating, the frictional force must satisfy the following relationship:

In addition, considering the differences between theoretical calculations and the actual tightening process, as well as the influence of other factors, a safety factor K₃ should also be multiplied by the clamping force. Generally, K₃ is taken as 1.2 to 1.5. To distinguish from the axial preload F mentioned above, the clamping force is changed to F_j. The conditions that need to be met are:

Among them, M_max represents the maximum tightening torque during the tightening process.

In addition, during the clamping process, it is necessary to ensure that the rocket body does not move, rotate, or vibrate, and no plastic deformation or damage should occur to the surface coating of the small solid rocket. Therefore, the next step should be to consider the relationship between the clamping force and the deformation of the rocket shell, and compare it with the clamping force F that prevents the small solid rocket body from rotating. If F causes plastic deformation of the solid small rocket body, the clamping force needs to be adjusted, such as increasing the direct friction coefficient μ between the clamping mechanism and the rocket body, or increasing the direct contact area between the clamping mechanism and the rocket body. Or change the three-jaw clamping mechanism to four-jaw or six-jaw, etc.

Take a small solid rocket with a shell diameter of 90 mm as an example to calculate the minimum required clamping force. Meanwhile, this paper analyzes the stress distribution of the rocket shell under this clamping force, and conducts calculations and experimental measurements on the deformation at the shell end. To increase friction and protect the shell surface, graphite rubber pads are installed between the rocket shell and the clamping mechanism. The friction coefficient between steel and graphite rubber pad is found to range from 0.2 to 0.8, and a value of μ = 0.5 is adopted for calculation. The safety factor is set to 1.2. For the selected small solid rocket with a seal ring compression rate of 30%, the maximum required torque M_max is 390 N·m, which is also the maximum torque load for all types of small solid rockets. Substitute the above parameters into Equation (4):

Since hydraulic or pneumatic chucks generally express pressure in terms of pressure, pressure-pressure conversion is required. This subsequent verification test plans to use the hydraulic chuck of a certain machine tool for the test. After consulting relevant technical data, it is learned that the inner diameter of this hydraulic cylinder is 140 mm, the minimum pressure is 0.5 MPa, and the maximum pressure is 16 MPa.

So when using this hydraulic chuck and adjusting to a minimum pressure of 0.5 MPa, the small solid rocket remains stationary when tightened with a maximum torque of 390 N∙m.

The clamping force of each claw of the triangular chuck when P = 0.5 MPa：

As shown in Figure 4, the rocket casing is directly clamped with three claws, with a cross-section of approximately 20 mm × 60 mm. The clamping area is shown in Figure 4.

Figure 4. Illustration of the clamping area.

It can be calculated that:

That is, the pressure acting on the clamping part of the small rocket shell is 6.41 MPa.

The stress analysis was performed in Pro/E by inputting parameters such as the clamping cross-section of the three-jaw chuck, clamping pressure, and shell material characteristics, see Figures 5-7.

Figure 5. Displacement and stress cloud map.

Figure 6. The position with the maximum stress.

Figure 7.Displacement curve of the outer circle of the shell thread.

It can be obtained that the displacement of the part held by the three claws is the maximum, which is 0.16 mm. The part with the greatest stress is located at the ear branch, at 563 MPa, which is less than the yield strength of the shell at 785 MPa. The maximum displacement of the outer circle curve of the thread is 0.0637 mm, which has little impact on the thread assembly.

The above theoretical calculations and simulation analyses were conducted on the magnitude of the clamping force, stress distribution, and deformation of the solid small rocket. A typical small solid rocket with a 90 mm diameter shell and lugs was used for verification tests.

Test specimen preparation: 1) Twenty small solid rockets of the lug-mounted shell type with an outer diameter of 90 mm were selected and divided into two groups. The first group was tested by direct clamping with a three-jaw chuck, while the second group was tested with a 0.5 mm thick graphite rubber pad (Material Grade: 66-2, Standard: Ji Q/CSMQ6131-90) installed between the three-jaw chuck and the shell; 2) Six marking points were evenly arranged on the outer circular surface and the threaded end of each shell.

The main purpose of the test is: 1) To verify the differences between rubber gaskets with graphite and those without; 2) To verify the minimum clamping force that ensures the product does not rotate when a torque of 390 N∙m is applied to the solid small rocket nozzle; 3) Measure the magnitude of the deformation at the nozzle housing opening during clamping.

Test content: 1) Use the hydraulic three-jaw chuck of the CNC machine tool to hold the housing with a clamping force of 0.5 MPa (where the chuck and the product are divided into two states: direct clamping and wrapping with graphite rubber); 2) Measure the deformation of the six points marked at the shell opening; 3) Can it achieve no slippage when loading with a 390 N∙m torque after adopting three-jaw clamping? 4) After the torque loading is completed and the shell is removed, observe whether there is any damage or an appearance defect at the clamping part of the shell.

The process of the test part is shown in Figure 8.

Figure 8. Shows part of the test process.

The test results are shown in Table 2 and Table 3.

Table 2. Test records of direct clamping of the shell with three claws.

Table 3. Test results of adding 0.5 mm graphite rubber pads between the three claws and the shell.

By analyzing the above test data, it can be known that.

1) As shown in Table 1 and Table2, under a clamping pressure of 0.5 MPa, slippage occurs when a torque of 390 N·m is applied without the graphite rubber pad. After installing the graphite rubber pad, no slippage takes place under the same 390 N·m torque load.

2) It can be concluded from Table 2 that the maximum deformation is 0.1 mm and does not affect the torque loading.

3) It can be concluded from Table 2 that a clamping force of 0.5 MPa will not cause any damage or appearance defects to the product.