# Design Analysis of a Passive Buffer System for Space-Assembled Segmented Mirrors
## 2 Establishment of Double-Elastic Contact Model
This model is helpful in analyzing whether the maximum deformation and maximum compressive stress meet the assembly requirements of the on-orbit telescope. Under the double-elastic contact model, Figure 2 illustrates the details of transient docking collision where the elastic head of the protruding pin \(B\) collides with the elastic inner-wall of conical hole \(A\) with non-zero incident angle \(\varepsilon_{0}\) and velocity \(v_{0}\) in the position \(i\). The maximum deformation \(\delta\) is produced on the contact surface of \(A\) when the pin head of \(B\) moves to the position \(ii\). The pin head leaves the elastic wall at the position \(iii\) through the exit angle \(\varepsilon_{1}\) and exit velocity \(v_{k}\). At position \(ii\), centered on the collision point \(O\), the sectional view can be obtained in the plane perpendicular to the common normal of the collision point, as shown in the red region of Figure 2. Since the deformation \(\delta\), caused by the contact, is much smaller than the radius of the elastic inner-wall \(R_{a}\) and the pin head \(R_{b}\), it is assumed that the radius \(R_{1}\) is expressed by \(R_{1}=R_{a}/\cos\beta\) and the spherical cone head radius is \(R_{2}=R_{b}\). Therefore, the contact pressure and contact deformation will be discussed. Figure 1: The traditional study of the contact between the pin and the chamfered hole, both of which are considered rigid. As Figure 3 shows, during capture stage, it is assumed that the elastic pin head collides with the elastic inner-wall, at point \(O\) under the driving force \(F\). A local coordinate system \(OXY\) is established with the contact point \(O\) as the origin. The \(X\)-axis, and \(Y\)-axis, represent the common normal, and tangential direction of the collision location, respectively. It is assumed that two points \(C\) (\(x_{1}\), \(\xi\)) and \(D\) (\(x_{2}\), \(\xi\)) are taken arbitrarily and will coincide to anyone point within the contact surface during the collision. Then according to the geometric relationship, it can be obtained that  
\[\left\{ \begin{array}{l}x_{1}=\frac{\xi^{2}}{2R_{1}}\\ x_{2}=\frac{\xi^{2}}{2R_{2}}\end{array} \right.. \tag{3}\]  
Therefore, the deformation \(\delta\) of the collision can be represented as:  
\[\delta=(x_{2}-x_{1})+(e_{2}-e_{1}), \tag{4}\]  
where \(e_{1}\) and \(e_{2}\) represent the displacement of deformation generated by the point \(C\) and the point \(D\) during the collision, respectively. From the Equations (3) and (4), the deformation \(\delta\) can be written by  
\[\left\{ \begin{array}{l}\delta=\gamma\xi^{2}+k\big{[}\!\int fdsd\psi\\ \gamma=\frac{R_{1}-R_{2}}{2R_{1}R_{2}}\\ k=\frac{1-\gamma\xi^{2}}{\pi E_{2}}+\frac{1-\gamma^{2}}{\pi E_{1}}\end{array}  \right., \tag{5}\]  
Figure 3: The detailed contact model. And this figure shows the sectional view of the red region in Figure 2.  
Figure 2: Transient collision of docking connectors with an elastic pin head engaging with an elastic inner-wall of a chamfered hole. where \(E_{1}\), \(E_{2}\), and \(v_{1}\), \(v_{2}\) represent the Young's modulus and Poisson's ratio of the elastic inner-wall of \(A\), and the elastic head of \(B\), respectively. \(f\) is shown as a compressive stress function, whose integral area is the entire contact surface. According to the [PERSON] theory of elastic mechanics, the compressive stress function, \(f\) can be solved by the semi-inverse solution method. Assuming that the compressive stress function\(f\) on the contact circle of radius \(u\) is proportional to the ordinate of the hemispherical surface, so we can get:  
\[k\frac{\pi^{2}f_{0}}{4u}(2u^{2}-\xi^{2})=\delta-\gamma\xi^{2}. \tag{6}\]  
By comparing the coefficients on both sides of the Equation (6), it can be easily seen that Equation (6) can be rearranged as:  
\[\left\{ \begin{array}{l}\delta=k\frac{\pi^{2}uf_{0}}{Z}\\ \gamma=k\frac{\pi^{2}f_{0}}{4u}\end{array} \right., \tag{7}\]  
where \(f_{0}\) is represented as the maximum compressive stress on the contact circle.