# Design and Analysis of a Novel Floating Docking Mechanism for On-Orbit Refueling
## 4 Analysis and Discussion of Simulations and Experiments
### Static Analysis of a Refueling Pipe  
The static simulation and experiment on the mechanical properties of the refueling pipe are performed in this section to verify the accuracy of the analytical solution (Equations (20) and (26)). An ABAQUS simulation was performed for comparison with theoretical results. The parameters of the numerical validation model of the elliptical-helical pipe with bias loading are listed in Table 4.  
By substituting the parameters into Equations (20) and (26), the theoretical results of the numerical validation model can be obtained as follows:  
\[\left\{ \begin{array}{c}K_{\mathrm{t}}=0.2482\;\mathrm{N/mm}\;\Rightarrow \lambda_{\mathrm{t}}=40.3\;\mathrm{mm}\\ \sigma_{\mathrm{max}\;t}=147.0\;\mathrm{MPa}\end{array} \right. \tag{28}\]  
where the subscript 't' refers to the theoretical result. The axial deformation of the helical part is denoted as \(\lambda_{\mathrm{t}}\). However, in actual conditions, each end of the pipe is connected to the drogue and propellant tank. As shown in Figure 13, these connecting parts resemble cantilever beams whose deformation can be calculated by the mechanics of materials [32]. For the numerical validation model in Table 3, the deformation of the connecting part is 0.25 mm. The third strength theory is used as the strength-checking formula in this study, but the fourth strength theory is used in ABAQUS. To compare the strength-checking result with ABAQUS \(\sigma_{\mathrm{max}\;t}\) is calculated again by the fourth strength theory (0.866 times as much as the third strength theory), which is 127.3 MPa. Thus, Equation (28) is updated to create Equation (29):  
\[\left\{ \begin{array}{c}K_{\mathrm{t}}=0.2466\;\mathrm{N/mm}\;\Rightarrow \lambda_{\mathrm{t}}=40.3+0.25=40.55\;\mathrm{mm}\\ \sigma_{\mathrm{max}\;t}=127.3\;\mathrm{MPa}\end{array} \right. \tag{29}\]  
When ABAQUS is used to simulate the model above, the results slightly approach the theoretical results as the number of finite elements increases, as shown in Figure 16. Finally, the results converge when the number of finite elements is more than \(1.2\times 10^{5}\) and \begin{table}
\begin{tabular}{c c c} \hline \hline
**Definition** & **Symbol** & **Value** \\ \hline Semi-major axis of envelope/mm & \(a\) & 70 \\ Semi-minor axis of envelope/mm & \(b\) & 35 \\ External diameter of pipe/mm & \(D\) & 4 \\ Inner diameter of pipe/mm & \(d\) & 2.4 \\ Number of active coils of pipe & \(N\) & 4 \\ Pitch angle of pipe/\({}^{\circ}\) & \(\beta\) & 2 \\ Position of loading/mm & \(P\) (\(m\), \(n\)) & (\(-10\), \(-7\)) \\ Permissible stress of TCA/MPa & \([\sigma]\) & 825 \\ Elasticity modulus of TC4/GPa & \(E\) & 110 \\ Shear modulus of TC4/GPa & \(G\) & 41 \\ External force/N & \(F\) & 49 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of the numerical validation model of the elliptical-helical pipe with bias loading. the converged results of simulation are shown in Figure 17. The values of \(\lambda_{\rm A}\) and \(K_{\rm A}\) are obtained using the following equation:  
\[\left\{ \begin{array}{c}\lambda_{\rm A}=41.00\:\mathrm{mm}\;\Rightarrow K_{\rm A }=0.2439\:\mathrm{N/mm}\\ \sigma_{\rm max\:A}=126.5\:\mathrm{MPa}\end{array} \right. \tag{30}\]  
where subscript 'A' refers to the ABAQUS result. Compared with the theoretical results and simulation results--i.e., Equations (29) and (30)--the relative errors of the axial deformation and maximum stress are less than 1.2% and 0.6%, respectively, which verifies the validity and accuracy of the analytical solution. Because the effects of the pitch angle are neglected in the present study, the theoretical axial deformation result is lower than the ABAQUS result. Then, an experiment was performed to validate the theoretical results and the simulation of the axial stiffness of the pipe. As shown in Figure 18, the experimental system consists of foundation support, an air-bearing linear guideway, a weight, and a refueling pipe.