# A Single Motor-Driven Focusing Mechanism with Flexure Hinges for Small Satellite Optical Systems
## 2 Materials and Methods
### Design of Flexure Hinge (FlexHe)
#### 2.4.1 FEM Modeling of the FlexHe  
A finite element (FE) was created in ANSYS 19.2 (ANSYS Inc, Canonsburg, PA, USA) with static structure analysis, and the mechanical characteristics were analyzed (i.e., strain versus stress, strain energy of structure through cyclic loading/un-loading, cyclic fatigue, and fracture). The analysis identified titanium alloys (Ti-6 Al-4V) as having suitable material properties for the FlexHe due to its promising material properties for space use (i.e., specific strength, lightweight, low density, etc.) [38]. Each FlexHe type was defined by the number of slits: types 1 to 6 were defined according to slit number [n], which ranged from 2 to 7, respectively. In addition to static characteristics, the fatigue analysis for the FlexHe was carried out under a fully reversed loading condition (tension-compression loading, stress ratio R = \(-\)1). Here, the thermal and vibrational effects were neglected. The stress versus the number of cycles to failure (S-N curves) and endurance limits associated with high-cycle fatigue (HCF) were compiled as a reference in [39]. The simulation results showed different mechanical behaviors. First, the greater the number of slits, the less axial stiffness is observed, as depicted in Figure 4c,d. Such compliant characteristics cause the FlexHe to undergo undesired deformations (e.g., buckling or column squirm), since the correlation between the buckling pressure and the stiffness of the structure is proportional [40; 41]. Indeed, the axial stiffness of Type 6 is relatively more compliant than Type 1, which indicates that Type 6 could be too delicate for the compression load. Similarly, the plotted relation of the strain energy versus deformation showed that Type 1 has excessive axial stiffness (Reaction force of 35.6 N). Figure 3: Concept design of the SMFH, consisting of a. Secondary mirror dummy, b. Secondary mirror supporter, c. Flexure hinge, d. Rotation body, e. Motor screw, f. Linear screw, g. Motor, h. Roller, i. Rotation shaft. Comparing Type 1 with Type 6, the maximum reaction force of Type 6 was only 8 N, which is only 22% of Type 1. In other words, Type 1 needs a relatively strong force in order to achieve the desired deformation, and accordingly, this would consume a lot of energy in small satellite. In the cases of Types 1 through 3, the relationship between deformation and strain energy showed a non-linear behavior. On the other hand, in cases of Types 4 through 6, we identified that the relationship showed a linear response. Due to this, the strain energy was not dramatically increased even though deformation is imposed. In light of these analyses, we determined that FlexHe Type 4, consisting of five slits, was the optimal structure for the operation. Indeed, as the graphs in Figure 4 demonstrate, Type 4 showed not only relatively high axial stiffness compared to Types 5 and 6, but also relatively compliant rather than Types 1 through 3. With the optimized FlexHe (Type 4), the constant amplitude load fatigue analysis was carried out. By using the displacement of 25 um and the estimated reaction force (11.5 N) from the simulation study, various parameters (i.e., alternating equivalent stress, safety factor, biaxial indication, fatigue sensitivity, fatigue life) were evaluated. The local stress was determined by means of the biaxial stress. In general, the biaxiality indication refers to the ratio between the major and minor principal stresses. As a result of fatigue analysis, the biaxial indication ranged between \(-0.99996\) and \(0.99152\), as shown in Figure 5. Thus, the maximum stress of the FlexHe occurred at the inside of each slit. However, since the minimum safety factor was 6.239 (Not less than 1), the available fatigue life extends the design life to 100,000 cycles at 11.5 N and avoids any critical regions that could cause a material failure and/or fracture. The fatigue sensitivity was computed to identify the variations in the fatigue life according to the loads.