# 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
The formula of torque T and bending moment \(M\)'s action on the arbitrary point \(Q\) of the ellipse can therefore be expressed as follows:  
\[T=F\cdot\sqrt{x^{2}+y^{2}}\cdot|\cos\alpha_{2}| \tag{9}\]  
\[M=F\cdot\sqrt{x^{2}+y^{2}}\cdot|\sin\alpha_{2}| \tag{10}\]  
The work done by an external force on an elliptical-helical pipe is converted into strain energy, which can be derived using the infinitesimal method (IM):  
\[V_{\epsilon}=V_{\epsilon\mathrm{T}}+V_{\epsilon\mathrm{B}} \tag{11}\]  
where \(V_{\epsilon\mathrm{T}}\) and \(V_{\epsilon\mathrm{B}}\) are the strain energies caused by torque T and bending moment \(M\), respectively. As shown in Figure 15, the \(V_{\epsilon\mathrm{T}}\) can be obtained by IM:  
\[V_{\epsilon\mathrm{T}}=N\cdot\int\limits_{V}v_{\mathrm{T}}\mathrm{d}V_{ \mathrm{T}} \tag{12}\]  
where \(N\) and \(V\) are the number of active coils and the pipe volume, respectively, \(\mathrm{d}V_{\mathrm{T}}\) is the volume of the infinitesimal influenced by torque T, and \(v_{\mathrm{T}}\) is the density of strain energy caused by torque T.  
According to the mechanics of the materials [32], \(v_{\mathrm{T}}\) can be obtained using the following equation:  
\[v_{\mathrm{T}}=\frac{512}{G}\cdot\left[\frac{T\rho}{\pi(D^{4}-d^{4})}\right]^{2} \tag{13}\]  
Figure 15: Infinitesimal area \(\mathrm{d}A_{\mathrm{T}}\) (the curved shadow) and \(\mathrm{d}A_{\mathrm{B}}\) (the straight shadow) of the refueling pipe section subjected to torque T and bending moment \(M\), respectively. where \(G\) is the shear modulus of the material, \(\rho\) is the distance from the infinitesimal to the original point \(O\), and \(d\) and \(D\) are the internal and external diameters of the refueling pipe, respectively. According to Figures 14 and 15, \(\mathrm{d}V_{\mathrm{T}}\) can be obtained as follows:  
\[\mathrm{d}V_{\mathrm{T}}=\rho\cdot\mathrm{d}\theta\cdot\mathrm{d}\rho\cdot \sqrt{\left(a\sin\varphi\right)^{2}+\left(b\cos\varphi\right)^{2}}\cdot \mathrm{d}\varphi \tag{14}\]  
where \(\mathrm{d}\theta\) is the included angle of the infinitesimal. Substituting Equations (13) and (14) into Equation (12), \(V_{\varepsilon\mathrm{T}}\) can be obtained as:  
Figure 13:0  
where EllipticK is the first kind of complete elliptic integral. Similarly, as shown in Figure 15, \(V_{\varepsilon\mathrm{B}}\) can be obtained by the IM:  
Figure 13:1  
where \(\mathrm{d}V_{\mathrm{B}}\) is the volume of the infinitesimal influenced by \(M\), and \(v_{\mathrm{B}}\) is the strain energy density caused by the bending moment \(M\). According to the mechanics of materials [32], \(v_{\mathrm{B}}\) can be obtained as follows:  
Figure 13:2  
where \(E\) is the elastic modulus of the material, and \(z\) is the distance of the infinitesimal from the center. According to Figures 14 and 15, \(\mathrm{d}V_{\mathrm{B}}\) can be obtained using the following equation:  
Figure 13:3  
where \(\mathrm{d}z\) is the length of the infinitesimal along the \(z\)-axis, and \(r\) is the radius of the pipe, including the internal radius \(r_{1}\) and external radius \(r_{2}\). Substituting Equations (17) and (18) into Equation (16), \(V_{\varepsilon\mathrm{B}}\) can be obtained as:  
Figure 13:4  
where EllipticE is the second kind of complete elliptic integral. Substituting Equations (15) and (19) into Equations (6) and (11), the axial stiffness of the elliptical-helical pipe \(K\) can be obtained as:  
Figure 13:5  
When \(a=b=R\), the elliptical pipe degrades into a circular pipe. Equation (20) can be rewritten as follows:  
Figure 13:6  
where \(R\) is the radius of the circular pipe envelope.