# Analytical Modeling and Application for Semi-Circular Notch Flexure Hinges
## 3 General Analysis Modeling  
### FEA Modeling of Flexure Hinge and Modified Coefficient  
FEA is an important approach to verify the nonlinear deformation and stress of a flexure hinge undergoing large deflection. The accuracy of these FEA models was verified by [PERSON] and [PERSON] et. al., and was with a maximum 8% error compared to three experimental results [4; 29]. This was completed in order to decrease the errors influenced by the materials and the geometrical parameters \(t/L\) and correct the theoretical compliance equations given by Equation (4). Eleven semi-circular beam flexure hinge models with different ratios of \(t/L\) are built in ANSYS 19.0 to obtain the reference principle compliance \(C_{1,\theta_{z}-M_{z}}^{F}\) and the principle stress \(\sigma^{F}\) of the hinges, the ratios of \(t/L\) are 0.025, 0.05, 0.1, 0.15,..., 0.5, respectively. It is noted that the model with the ratio \(t/L\) of 0.025 is chosen to improve the accuracy of the modified equations. In addition, the radius of the filled is chosen as \(a=0.5\)\(t_{\text{min}}\) to decrease the concentration of stress. The geometry of one of these analysis models is shown in Figure 5. The analytical models had a depth of 10 mm. Triangle element type is more suitable to model irregular shapes and is chosen in this paper to generate the model mesh. The meshing technique was refined five times and used to produce automatically refined meshing at parts that had high stress concentrations. The size of the mesh is only approximately 1/50 of the thickness. In this way, the analysis of flexure hinges is always assumed to be like a cantilever beam. One end of the flexure  
Figure 4: Relative errors with \(t/L\) = 0.10 and 0.25. Figure 3: Relationship between moment and rotation angles. hinge is fixed and the moment \(M_{z}\) is applied at Point \(P_{1}\) through the ANSYS 19.0, the triangles method and sizing of the mesh are utilized and set. In addition, the condition of the FEA is set as large deformations and the non-linear characteristic. In the meantime, the FEA investigation should follow the procedure described in the following to ensure the coincidence: The flexure hinges have the same deformation \(\theta_{z}{}^{F}\) under the moment \(M_{z}\). Therefore, the moment \(M_{z}\) will increase with the ratio of \(t/L\). Thus, the considered FEA simulations have been performed in ANSYS Workbench. In total, 220 data points are utilized to obtain modified equations. The rotation angle \(\theta_{z}{}^{F}\) can be obtained as described in Figure 5. The corresponding theoretical compliance \(C^{T}_{1,\theta_{z}-M_{z}}\) under the same load condition can be calculated through the Equations (4) and (5). The modified coefficient \(\mu\) is introduced in this paper while the \(\mu=\frac{C^{F}_{1,\theta_{z}-M_{z}}}{C^{T}_{1,\theta_{z}-M_{z}}}\) is used to describe the error between the theoretical compliance and FEA compliance quantitatively. In all these cases, the resulting difference for the parameter \(\mu\) is found to be always lower than 0.5 and 0.3%, respectively, allowing one to assume a negligible dependency on the material parameters. The compliance and deformation are usually determined by the load, material properties (Young Modulus \(E\)), rotation angles, and geometrical parameters (\(t/L\)) [29]. Therefore, the modified coefficient \(\mu\) and relative error may be determined by the Young Modulus \(E\), rotation angles, and geometrical parameters (\(t/L\)). In addition, the relative errors between the \(C^{T}_{1,\theta_{2}-M_{z}}\) and \(C^{F}_{1,\theta_{2}-M_{z}}\) with different parameters are obtained under the same FEA setting by following Equation (7):  
\[\text{Error}=\left[\frac{C^{F}_{1,\theta_{2}-M_{z}}-C^{T}_{1,\theta_{2}-M_{z} }}{C^{F}_{1,\theta_{2}-M_{z}}}\right]\times 100\% \tag{7}\]  
While the Young Modulus of 'Material A' is 345 MPa, and the Young Modulus of 'Material B' is 110 GA, the ratios of the hinge are chosen as 0.05, 0.1, and 0.25.