# Design and Analysis of a Novel Floating Docking Mechanism for On-Orbit Refueling
## 3 Mechanical Properties of an Elliptical-Helical Pipe
### Axial Stiffness of an Elliptical-Helical Pipe  
An energy-based method [32] is used to derive the axial stiffness of an elliptical-helical pipe with bias loading. According to the law of energy conservation, when a helical structure is telescoped, the work \(W\) of an external force \(F\) is transferred into the strain energy \(V_{\varepsilon}\) of the structure, as shown in Equation (6):  
\[W=\frac{1}{2}\cdot F\cdot\lambda=V_{\varepsilon}\Rightarrow F=\frac{2V_{ \varepsilon}}{\lambda} \tag{6}\]  
where \(\lambda\) is the axial deformation of the helical structure under the external force \(F\). By deriving the formula for the strain energy \(V_{\varepsilon}\) and substituting it into Equation (6), the equation relating the net external force \(F\) and the axial deformation \(\lambda\) can be  
Figure 12: Free-body diagram of the resetting conditions of the drogue. obtained; thus, the axial stiffness of the helical structure can be expressed as follows: As an inherent attribute, the value of the stiffness cannot be influenced by changing the value and position of the external force when the geometrical parameters and material of the structure are confirmed. The external force only influences the stress distribution of the helical structure. Thus, the study of the axial stiffness of an elliptical-helical pipe with bias loading is equivalent to the study of the axial stiffness of an elliptical-helical pipe with center loading. The condition of bias loading can be considered when determining the strength of the pipe. The design space for the refueling pipe is minimal because of the compactness required for an on-orbit refueling mechanism. The pitch angle of the pipe is generally less than \(5^{\circ}\), so its influence can be neglected [32]. The configuration of the refueling pipe proposed in this study is shown in Figure 13.  
For the helical pipe, the strain energies of each coil are the same. Therefore, the strain energy \(V_{\varepsilon}\) of the entire pipe can be derived by multiplying the number of coils and the strain energy of one coil (3D transformed into 2D). To obtain the strain energy \(V_{\varepsilon}\), the geometrical relationship of the elliptical envelope of the pipe is first analyzed. The relationship between \(O\) and \(Q\) from the top view of the elliptical-helical pipe and the coordinate system is shown in Figure 14. Point \(O\) is the center of the elliptical envelope, which is subjected to a force \(F\) along the positive direction of the \(z\)-axis. Point \(Q\) is an arbitrary point on an elliptical envelope. As shown in Figure 14, the elliptical envelope equation can be obtained as follows:  
\[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\text{ and }\left\{ \begin{array}{l}x=a \cdot\cos\varphi\\ y=b\cdot\sin\varphi\end{array} \right. \tag{7}\]  
where \(a\) is the semi-major axis of the ellipse, and \(b\) is the semi-minor axis of the ellipse. Figure 14: Relationship of \(O\)Q and \(P\)Q on the elliptical envelope. Figure 13: Configuration of the refueling pipe. The red point represents the connecting point of the drogue and the refueling pipe, which is subjected to the force \(F\) along the \(z\)-axis. The included angle \(\angle N_{2}OQ\) formed by line \(ON_{2}\), which connects point \(N_{2}\) on arbitrary point \(Q\)'s tangent line to the ellipse and the ellipse center \(O\). Line \(OQ\), which connects the ellipse center \(O\) and arbitrary point \(Q\), can be obtained, which is denoted as \(\alpha_{2}\):  
\[\alpha_{2}=\arctan\left(\frac{a^{2}}{b^{2}}\cdot\frac{y}{x}\right)-\arctan \left(\frac{y}{x}\right) \tag{8}\]  
Through the analysis above, the geometrical relationship of the ellipse center \(O\) to an arbitrary point \(Q\) can be obtained.