# Design Analysis of a Passive Buffer System for Space-Assembled Segmented Mirrors
## 2 Establishment of Double-Elastic Contact Model  
According to the mission requirements of the space assembled telescope, and the collision characteristics of the probe-cone connection mechanism, the docking process between PMAs should be divided as follows. 1. First collision. Within a reasonable capture range, two PMAs could achieve a capture operation of weak impact, preventing strong vibrations and interference of the mirrors. 2. Positioning. It is necessary to limit partial degrees of freedom (DOFs) between two PMAs to align and steady the combination. 3. Locking and Adjustment. A mechanically rigid connection between the optical elements is required. At the same time, the ability of mirror gap adjustment is a pre-requisite in improving optical performance. 4. Separation. The feasible undocking maneuver of the connection mechanism should be satisfied in case of unpredictable incidents, such as abnormal docking or irreversible collision. The buffer performance design mainly focuses on the first point mentioned above, because it produces the largest impact influence, which means the buffer mechanism should be required to absorb the impact energy. In our previous studies of the typical small-sized cooperative target collision system [20], the rigid protruding pin B made the first contact with the rigid chamfered entrance hole A, during the capture stage as shown in Figure 1. The conditions for successful capture by analysis are:  
\[\left\{ \begin{array}{c}\varepsilon_{1}<90^{\circ}-2\beta\\ V_{n}^{1}=SV_{n}^{0}\\ V_{\tau}^{1}=V_{\tau}^{0}+(1+5)TV_{n}^{0}\\ T=M\frac{2}{i}\bigg{(}\frac{\sin_{\mu}}{m_{i}}+\frac{r_{i}^{2}\cos\theta_{i}\sin( \theta_{i}-\mu)}{l_{iz}}\bigg{)}\quad(i=A,B)\end{array} \right., \tag{1}\]  
where \(\beta\) represents the half cone angle of the chamfered hole \(A\); the entrance angle, and the exit angle of the pin B, are represented by \(\varepsilon_{0},\ \varepsilon_{1}\), respectively. The normal and tangential velocities of the contact point before impact are shown as \(V_{n}^{0},\ V_{\tau}^{0}\), respectively. Similarly, \(V_{n}^{1},\ V_{\tau}^{1}\) represent the normal and tangential velocities of the contact point after impact. In addition, the velocity ratio before, and after, the collision is defined as the recovery factor \(S\); \(T\) stands for the equivalent stiffness; \(M\) is expressed by the equivalent mass of the connection mechanism; the coefficient of friction angle of conical surface is shown as \(\mu\); the angle between the vector of the two docking mechanisms centroid to collision point and the normal of the cone is expressed by \(\theta\); \(l_{iz}\) is the moment of inertia around the \(Z\) axis of the mechanism. According to Equation (1), in the case of successful capture, the relationship between the velocity of the rigid pin and the rigid inner-wall of the hole can be obtained by:  
\[\tan\varepsilon_{1}=\frac{S}{\frac{1}{\tan\varepsilon_{0}}+(1+S)T}<\tan(90^{ \circ}-2\beta). \tag{2}\]  
However, in the actual space on-orbit assembly environment, a rigid collision can cause irreversible damage to the mirror, such as vibration and interference damage. Therefore, a buffer system needs to be added to mitigate the effects of impact during the docking process. To minimize the energy requirements on such a large cluster of PMAs processing, a passive buffer system, rather than an electromagnet driving one, was developed in this paper. For this purpose, \"double-elastic contact model\" is established to satisfy this buffer system. To satisfy the optical specifications and ensure the impact force absorption, storage, and consumption, a passive buffer system was designed and analyzed. On the basis of the structural shape, the pin and the inner-wall of the entrance hole are re-designed flexibly in the probe-cone docking system. In this study, the \"double-elastic contact model\" was established to study the collision, between the elastic head of the protruding pin and the elastic inner-wall of the entrance hole, during the capture stage.