# Insights into Bending Stiffness Modeling of Elementary Flexure Hinges
## 7 Conclusions  
In this work, the nonlinear bending model derived from continuum mechanics, which describes the rotational behavior of a flexure hinge under static loads, is analyzed. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\mathbf{\alpha}_{Q}\) & \(\mathbf{\alpha}_{M}\) & \(\mathbf{\theta}_{f,A}^{-}[^{\circ}]\) & \(\mathbf{\theta}_{f,A}^{-}[^{\circ}]\) & \(\mathbf{\theta}_{f,A-}^{-}[^{\circ}]\) & \(\mathbf{\theta}_{f,A-}^{-}[^{\circ}]\) & \(\mathbf{\theta}_{f,A-}^{-}[^{\circ}]\) & \(\mathbf{\theta}_{f,FEM}^{-}[^{\circ}]\) \\ \hline
0.1 & 0.5 & \(-\)4.294 & \(-\)4.211 & \(-\)4.211 & \(-\)4.202 & \(-\)4.302 & \(-\)4.213 \\
0.5 & 0.5 & 4.085 & 3.935 & 3.935 & 3.982 & 4.034 & 3.935 \\
1.0 & 0.5 & 16.779 & 15.153 & 15.153 & 16.385 & 15.464 & 15.157 \\ \(-\)1.0 & 0.9 & \(-\)9.658 & \(-\)8.947 & \(-\)8.947 & \(-\)9.420 & \(-\)9.153 & \(-\)8.948 \\ \(-\)0.5 & 0.9 & \(-\)12.578 & \(-\)11.924 & \(-\)11.924 & \(-\)12.289 & \(-\)12.186 & \(-\)11.929 \\ \(-\)0.1 & 0.9 & \(-\)11.715 & \(-\)11.394 & \(-\)11.394 & \(-\)11.456 & \(-\)11.647 & \(-\)11.400 \\
0.1 & 0.9 & \(-\)10.918 & \(-\)10.737 & \(-\)10.737 & \(-\)10.681 & \(-\)10.975 & \(-\)10.743 \\
0.5 & 0.9 & \(-\)8.592 & \(-\)8.577 & \(-\)8.577 & \(-\)8.417 & \(-\)8.762 & \(-\)8.583 \\
1.0 & 0.9 & \(-\)1.686 & \(-\)1.690 & \(-\)1.690 & \(-\)1.676 & \(-\)1.700 & \(-\)1.693 \\ \hline \(\mathbf{\alpha}_{Q}\) & \(\mathbf{\alpha}_{M}\) & \(\mathbf{\Delta}_{\mathbf{\kappa}A}\mathbf{\%}\) & \(\mathbf{\delta}_{\mathbf{\tau}}\mathbf{\%}\) & \(\mathbf{\delta}_{\mathbf{\alpha}(\mathbf{\tau})}\mathbf{\%}\) & \(\mathbf{\Delta}_{\mathbf{\varepsilon}EM,A}\mathbf{\%}\) & \\ \hline \(-\)1.0 & 0.1 & 17.903 & \(2.65\times 10^{-7}\) & 15.235 & 1.699 & 17.857 & \\ \(-\)0.5 & 0.1 & 6.854 & \(1.11\times 10^{-7}\) & 4.445 & 2.085 & 6.806 & \\ \(-\)0.1 & 0.1 & 2.540 & \(3.35\times 10^{-8}\) & 0.233 & 2.286 & 2.496 & \\
0.1 & 0.1 & 2.463 & \(4.56\times 10^{-8}\) & 0.088 & 2.331 & 2.418 & \\
0.5 & 0.1 & 6.365 & \(1.11\times 10^{-7}\) & 3.954 & 2.126 & 6.318 & \\
1.0 & 0.1 & 17.476 & \(2.66\times 10^{-7}\) & 14.816 & 1.717 & 17.433 & \\ \(-\)1.0 & 0.5 & 13.142 & \(2.84\times 10^{-7}\) & 10.534 & 1.917 & 13.103 & \\ \(-\)0.5 & 0.5 & 6.178 & \(1.11\times 10^{-7}\) & 3.759 & 2.137 & 6.135 & \\ \(-\)0.1 & 0.5 & 2.666 & \(2.96\times 10^{-8}\) & 0.384 & 2.244 & 2.615 & \\
0.1 & 0.5 & 1.971 & \(5.18\times 10^{-9}\) & 0.214 & 2.161 & 1.923 & \\
0.5 & 0.5 & 3.812 & \(1.61\times 10^{-7}\) & 1.194 & 2.516 & 3.812 & \\
1.0 & 0.5 & 10.731 & \(3.00\times 10^{-7}\) & 8.130 & 2.052 & 10.701 & \\ \(-\)1.0 & 0.9 & 7.947 & \(3.41\times 10^{-7}\) & 5.287 & 2.302 & 7.935 & \\ \(-\)0.5 & 0.9 & 5.485 & \(1.14\times 10^{-7}\) & 3.061 & 2.197 & 5.441 & \\ \(-\)0.1 & 0.9 & 2.817 & \(2.86\times 10^{-8}\) & 0.544 & 2.220 & 2.763 & \\
0.1 & 0.9 & 1.686 & \(1.10\times 10^{-8}\) & 0.522 & 2.217 & 1.629 & \\
0.5 & 0.9 & 0.175 & \(2.06\times 10^{-8}\) & 1.865 & 2.157 & 0.105 & \\
1.0 & 0.9 & 0.237 & \(3.28\times 10^{-7}\) & 0.828 & 0.592 & 0.413 & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._  
Figure 16: CASE III—Trend of the tension \(\sigma\) evaluated by FE simulations varying \(\rho\) and \(\beta_{y}\) (**up**) and \(\alpha_{Q}\) and \(\alpha_{M}\) (**down**). The process of developing the model from nonlinear to linear through successive simplifications is presented in detail, showing the analytical solution of the latter and its conditions of applicability. The present work shows how the approximated analytical model may be used to describe the angular displacement of the free-loaded end of an elliptical flexure hinge, also showing its limits of applicability with respect to the conventional complete model, which can only be solved in numerical form. In particular, the axial force applied to the free end of the flexure hinge is shown to be the determining effect in choosing the reference stiffness model.