# Design and Analysis of a Novel Floating Docking Mechanism for On-Orbit Refueling
## 3 Mechanical Properties of an Elliptical-Helical Pipe
### Strength Checking of an Elliptical-Helical Pipe  
Except for the axial stiffness, the maximum stress of the refueling pipe, which cannot exceed the allowable stress on the material, is another important mechanical property of the pipe to be analyzed. In this section, the bending moment \(M^{\prime}\) and torque \(\mathrm{T^{\prime}}\) acting on point \(Q\) of the pipe are derived. Using the third strength theory as the strength-checking formula and substituting \(M^{\prime}\) and \(\mathrm{T^{\prime}}\) into it, the formula can be obtained.  
The relationship between \(P\) and \(Q\) from the top view of the elliptical-helical pipe is shown in Figure 14. Point \(P\) is the loading position, 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.  
According to Figure 14, Equations (22)-(25) can be obtained as follows:  
The included angle \(\angle N_{1}PQ\) formed by line \(PN_{1}\), which connects point \(N_{1}\) on the arbitrary point \(Q\)'s tangent line to the ellipse to the loading position \(P\). Line \(PQ\), which connects \(P\) to an arbitrary point \(Q\), can be obtained, which is denoted as \(\alpha_{1}\):  
\[\alpha_{1}=\arctan\biggl{(}\frac{a^{2}}{b^{2}}\cdot\frac{y}{x}\biggr{)}- \arctan\biggl{(}\frac{y-n}{x-m}\biggr{)} \tag{22}\]  
where (\(m\), \(n\)) is the coordinate of the loading position \(P\).  
The distance between the loading position \(P\) and arbitrary point \(Q\) on the ellipse is denoted as \(L_{1}\).  
\[L_{1}=\sqrt{\left(x-m\right)^{2}+\left(y-n\right)^{2}} \tag{23}\]  
The formula of torque \(\mathrm{T^{\prime}}\) and bending moment \(M^{\prime}\)'s action on the arbitrary point \(Q\) of the ellipse by the force \(F\) in the loading position \(P\) can be expressed as follows:  
\[T^{\prime}=F\cdot L_{1}\cdot\left|\cos\alpha_{1}\right| \tag{24}\]  
\[M^{\prime}=F\cdot L_{1}\cdot\left|\sin\alpha_{1}\right| \tag{25}\]  
The third strength theory [32] is used as the strength-checking formula by combining the formula with Equations (24) and (25), the expression of strength checking can be obtained using the equation below:  
\[\sigma_{\mathrm{max}}=\frac{32 FL_{\mathrm{1 max}}}{\pi D^{3}(1-a^{4})}\leq[ \sigma],\ \mathrm{where}\ \alpha=\frac{d}{D} \tag{26}\]  
where \([\sigma]\) is the allowable stress of the material.  
The expression of \(L_{\mathrm{1 max}}\) is obtained by combining Equations (7) and (23) to find the point \(Q_{\mathrm{max}}\) corresponding to \(L_{\mathrm{1 max}}\). The derivative of \((L_{1})^{2}\) is calculated by ordering the derivative of \((L_{1})^{2}\) equal to zero and calculating its square to obtain the \(x\)-coordinates \(x\) of the point \(Q_{\mathrm{max}}\), as shown in the following equation:  
\[\left(a^{2}-b^{2}\right)^{2}x^{4}-2a^{2}m\left(a^{2}-b^{2}\right)x^{3}-a^{2} \biggl{[}\left(a^{2}-b^{2}\right)^{2}-a^{2}m^{2}-b^{2}n^{2}\biggr{]}x^{2}+2a^{ 4}m\left(a^{2}-b^{2}\right)x-a^{6}m^{2}=0 \tag{27}\]  
Equation (27) is a quartic equation with one variable, \(x\). The results of Equation (27) consist of two positive real roots and two negative real roots. The selection of \(x\) depends on \(m\), in that their signs are opposite, and \(x\) then chooses the larger value in the left two values, which is the correct result. Then, by substituting \(x\) into Equation (7). Similarly, the signs of \(y\) and \(n\) are opposite, and the \(y\)-coordinates \(y\) of the point \(Q_{\mathrm{max}}\) can be obtained. Subsequently, \(L_{\mathrm{1 max}}\) can be obtained by substituting \(x\) and \(y\) into Equation (23). Finally, by substituting \(L_{\mathrm{1 max}}\) in Equation (26), the strength of the pipe is checked. This completes the analysis of the mechanical properties of the elliptical-helical refueling pipe, which can guide the design of such pipes.