# Analysis on the Isostatic Bipod Mounts for the HERA Mission LIDAR
## 5 Thermoelastic Simulations
### Thermoelastic Results
Figure 11: Result of hot (+80 \({}^{\circ}\)C) non-operational scenario—First design: (**a**) Mirror stress, (**b**) Isostatic bipod mount stress (deformation scale factor: 76). Figure 12: Result of cold (\(-\)60 \({}^{\circ}\)C) non-operational scenario—First design: (**a**) Mirror stress, (**b**) Isostatic bipod mount stress (deformation scale factor: 76). From the results, it is evident that there is a displacement of the mirror relative to its nominal position. Nevertheless, the results are within the established values for XZ displacement tolerances from Table 2. The extreme values of the hot non-operational temperatures for the final design of the isostatic bipod mounts are within the range of \(\left[-1.159\times\,10^{-4};\,1.133\times\,10^{-4}\right]\) mm, and for the cold non-operational temperatures are in range of \(\left[-1.602\times\,10^{-4};\,1.683\times\,10^{-3}\right]\) mm. Regarding the stress in the mirror and the isostatic bipod mounts, the same assessment previously mentioned was made. The results are presented in Figures 15 and 16 for the hot and cold non-operational scenario, respectively. Figure 14: Result of cold (\(-60\,^{\circ}\)C) non-operational scenario—Final design: (**a**) \(X\)-axis displacement, (**b**) \(Z\)-axis displacement. Figure 13: Result of hot (+80 \({}^{\circ}\)C) non-operational scenario—Final design: (**a**) \(X\)-axis displacement, (**b**) \(Z\)-axis displacement. The mirror's maximum stress is located on the bonding areas, with a safety factor of 1.75 for the hot scenario, and 1.24 for the cold scenario. Since the safety factor of the mirror is higher than unity, the mirror can withstand the hot and cold non-operational scenarios. Regarding the isostatic bipod mounts stress distribution, it can be concluded that they 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 safety factor for the isostatic bipod mount converts into a safety factor of 3.66 for the hot scenario, and 2.59 for the cold scenario. Since the safety factor is higher than unity, the isostatic bipod mounts can withstand the hot and cold non-operational scenarios. The maximum stress is found on the designated area of the radial blades (see Figures 15b and 16b), which is expected. The distance between the primary mirror and the base is given in Figures 17 and 18. Figure 8:0 Result of cold (\(-60\)\({}^{\circ}\)C) non-operational scenario—Final design: (**a**) Mirror stress, (**b**) Isostatic bipod mount stress (deformation scale factor: 73). Figure 8:1 Result of hot (\(+80\)\({}^{\circ}\)C) non-operational scenario—Final design: (**a**) Mirror stress, (**b**) Isostatic bipod mount stress (deformation scale factor: 73). The distance of the mirror towards the base for the first design is between \([-0.0248,\ 0.035]\) mm, and for the final design is between \([-0.0340,\ 0.0480]\) mm. These values are within the requirement of \(\pm 0.09\) mm defined in Figure 8:2  
For the analysis of curvature radius variation, the following result processing approach was implemented: (1) the displacement offset of the mirror was removed; (2) the difference between the mirror outer edge and mirror inner edge was calculated (see Figure 19). To this difference we named the radius curvature variation of the mirror, and its results are presented in Figure 8:3  
Figure 8:4 Result of \(Y\)-axis displacement for the first design: (**a**) hot (+80 \({}^{\circ}\)C) non-operational scenario, (**b**) cold (\(-60\)\({}^{\circ}\)C) non-operational scenario. Figure 8:5 Result of \(Y\)-axis displacement for the final design: (**a**) hot (+80 \({}^{\circ}\)C) non-operational scenario, (**b**) cold (\(-60\)\({}^{\circ}\)C) non-operational scenario. According to Table 2, a 1 mm curvature radius corresponds to a surface displacement of \(\pm 2\) um. The cold scenario does not fulfil this requirement; and for this case, the LIDAR performance has a loss of energy.