# Design and Evaluation of Flexible Support Based on Space Mirror
## 3 Design of Support Structure for Space Mirror Component
### Design of Flexible Hinges  
Flexible hinges play a role in the whole support structure to protect the top and stabilize the bottom. It is necessary to have a certain degree of rigidity to ensure that the components can withstand the harsh mechanical environment, but it also needs to have a certain degree of flexibility to adapt to a wide range of temperature change environments. In order to find a point of agreement in the contradictory conditions, we need to find a compromise value through the formula calculation and simulation analysis.  
The new biaxial circular arc flexible hinges support structure designed in this paper is such that each supporting point provides two degrees of freedom in the optical axis direction and tangential direction, and the three supporting points provide exactly six degrees of freedom (three axial supports constrain axial movement, radial rotation, and tangential rotation; three tangential supports constrain tangential movement, radial movement, and axial rotation), which realize decoupling of the degrees of freedom, and this is the typical Boise kinematic localization support. This support structure restricts the rigid motion of the mirror, and it can isolate the mirror from the elastic deformation of the surroundings, thus effectively protecting the mirror.  
Analyze and calculate the flexible hinge deflection and corner of the flexible hinge support structure [15]. The empirical equations for the flexible hinge deflection \(y\) and corner \(\theta\) are shown below:  
\[y=\int[\int\frac{12 RsinaM(X)}{Eb(2R+\tau-2 Rsina)^{3}}dx]da \tag{2}\]  
\[\theta=\int\frac{12 RsinaM(X)}{Eb(2R+\tau-2 Rsina)^{3}}dx \tag{3}\]  
In the equations, \(R\) is the cutting radius of the hinge, \(\tau\) is the thinnest part of the thickness of the hinge, \(b\) is the width of the hinge, and \(M(x)\) is the torque applied to the end of the hinge, which can be considered as a constant.  
The numerical integration method can be used to obtain the value of the corner stiffness of the flexible hinge under different values of \(R\) and \(\tau\), and then carry out curve  
Figure 5: Schematic diagram of mirror component.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Material**} & \multirow{2}{*}{\begin{tabular}{c} **Density** \\ **g/cm\({}^{3}\)** \\ \end{tabular} } & \multicolumn{2}{c}{\
\begin{tabular}{c} **Coefficient of** \\ **Linear** \\ **Expansion** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Modulus of** \\ **Elasticity** \\ **GPa** \\ \end{tabular} } & \multirow{2}{*}{**Poisson Ratio**} \\  & & \(\alpha\times 10^{-6}\)/K & & \\ \hline ULE & 2.21 & 0.03 & 67.6 & 0.17 \\ TC4R & 4.45 & 9.1 & 110 & 0.34 \\ Invar Steel (4J32) & 8.18 & 0.05 & 145 & 0.25 \\ AISiC & 2.9 & 9.8 & 145 & 0.32 \\ Optical & 1.7 & 110 & 600 & 0.49 \\ Adhesive & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Mirror material parameters.  
fitting; the results find that the corner stiffness increases with the increase in the \(\tau\) value, and the corner stiffness is not sensitive to the change of \(R\) value, so in the process of flexible hinge parameter selecting, the value should be set according to the structural dimensions and technological requirements, then adjust the value of t until the mirror ultimately meets the requirements of stiffness, strength, and thermal dimensional stability requirements. According to the above equations, the \(R\) value of the flexible hinge in this paper is 4 mm, and the \(\tau\) value is 1 mm. The design model of the flexible hinge is shown in Figure 6.