# Analysis on the Isostatic Bipod Mounts for the HERA Mission LIDAR
## 5 Thermoelastic Simulations
### Thermoelastic Results  
The aim of these simulations was to assess the isostatic bipod mount's behavior in the extreme temperatures. Two topics were studied, namely the survivability of the mirror (i.e., assessment of the stress in the mirror) and the analysis of the mirror displacement regarding its nominal position (to evaluate if the flexures were optimized and working correctly). Figures 9 and 10 represent the XZ displacement results for the first isostatic bipod mount design for the hot and cold non-operational scenarios, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Component** & & **Value [mm]** & **Comments** \\ \hline \multirow{6}{*}{Primary Mirror} & \multirow{6}{*}{\(\pm 1\)} & XZ displacement & \(\pm 0.01\) & \\ \cline{2-3}  & Distance of Mirror to the Base & \(\pm 0.09\) & \\ \cline{2-3}  & & & \\ \cline{2-3}  & & & \(\pm 1\) mm radius & variation is \\ \cline{2-3}  & & & equivalent of a \\ \cline{2-3}  & & & maximum surface \\ \cline{2-3}  & & & displacement \\ \cline{2-3}  & & & of \(\pm 2\) μm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Thermal acceptable tolerances. Figure 8: The boundary conditions of the final design. From the results, it is evident that there is a displacement of the mirror relative to its nominal position. From the hot scenario, the extreme values of XZ displacement are in range of \(\left[-3.461\times\ 10^{-4};\,3.027\times\ 10^{-4}\right]\) mm, and for the cold scenario, the extreme values of XZ displacement are in range of \(\left[-4.280\times\ 10^{-4},\ 4.894\times\ 10^{-4}\right]\) mm, which is fully within the range of acceptable tolerances from Table 2. An animation of the simulations can be found in ([[https://www.youtube.com/watch?v=M6](https://www.youtube.com/watch?v=M6) bY2 InInNoE]([https://www.youtube.com/watch?v=M6](https://www.youtube.com/watch?v=M6) bY2 InInNoE) accessed on 23 March 2020)  
Regarding the stress, the critical components were analyzed, namely the mirror and the isostatic bipod mounts. The thermal conductance along the mirror and the adhesives, and between the adhesives and the isostatic bipod mounts is established as 2500 W/(m\({}^{2}\)K). The contacts between the other materials (aluminum and titanium) are established as 150 W/(m\({}^{2}\)K) (as defined in [11]). The results of the von-Mises stress for the hot non-operational case and for the cold non-operational case are presented in Figures 11 and 12, respectively. Figure 10: Result of cold (\(-60\ ^{\circ}\)C) non-operational scenario—First design: (**a**) \(X\)-axis displacement, (**b**) \(Z\)-axis displacement. Figure 9: Result of hot (+80 \({}^{\circ}\)C) non-operational scenario—First design: (**a**) \(X\)-axis displacement, (**b**) \(Z\)-axis displacement. The mirror maximum stress is sited on the bonding areas, with a safety factor of 3.29 for the hot scenario, and 2.33 for the cold scenario, which means that the mirror will withstand the non-operational temperatures. The isostatic bipod mounts have a good overall performance, since their displacement is located on the radial blades. The tangential blade does not present a considerable stress since the displacement condition is essentially radial. The maximum stress is located on the base of the isostatic bipod mounts, but since the critical stress is on the blades and in the bonding area, that maximum is not considered. When the maximum stress range is decreased, the value of stress starts to appear on the radial blade, which is coincident with the radial deformation of the base (see Figures 11b and 12b). The safety factor for the isostatic bipod mount considers the maximum stress value, which converts into a safety factor of 2.82 for the hot scenario, and 1.99 for the cold scenario. Making a similar study for the final design of the isostatic bipod mounts, the XZ displacement results are depicted in Figures 13 and 14 for the hot and cold non-operational scenarios, respectively.