# Analytical Modeling and Application for Semi-Circular Notch Flexure Hinges
## 4 Case Study: Compliant Four-Bar Linkage Mechanism  
The compliant four-bar linkage mechanism is the most common compliant mechanism which can meet the accuracy requirements of motion in the general case. A compliant four-bar linkage mechanism based on the semi-circular notch flexure hinge is built to verify the accuracy of the general design equations in this part of the paper. In addition, the comparison of the given moment and the moment in the finite element analysis under the same desired rotation angle are conducted. The PRB model of the mechanism is shown in Figure 6. The revolute is simplified as the joint with a torsion spring. Since the accuracy of the hinge is related to the motion direction for the hinge of transverse symmetry, the semi-circular flexure hinge can be applied in the unidirectional rotation. When a moment is loaded on the Rocker \(r_{2}\) as described in Figure 5, the direction of rotation for \(r_{2}\) will be clockwise. The flexure hinges 1 and 4 will also rotate in this same direction, while the flexure hinges 2 and 3 will rotate counterclockwise. Therefore, the model of the compliant mechanism based on the semi-circular notch flexure hinge is shown \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Coefficients** & **Value** & **Coefficients** & **Value** & **Coefficients** & **Value** \\ \hline \(b_{00}\) & \(0.0007568\) & \(b_{02}\) & \(0.008669\) & \(b_{40}\) & \(0.3867\) \\ \(b_{10}\) & \(-0.02399\) & \(b_{30}\) & \(-0.4565\) & \(b_{31}\) & \(0.6169\) \\ \(b_{01}\) & \(-0.01059\) & \(b_{21}\) & \(-0.6463\) & \(b_{22}\) & \(0.4631\) \\ \(b_{20}\) & \(0.1761\) & \(b_{12}\) & \(-0.2095\) & \(b_{13}\) & \(0.20297\) \\ \(b_{11}\) & \(0.8788\) & \(b_{03}\) & \(-0.0005916\) & \(b_{04}\) & \(-0.0003766\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Coefficient for Equations (11) and (12). Figure 6: Pseudo-rigid body model of the mechanism. in Figure 7. The mechanism is composed of the rockers \(r_{2}\) and \(r_{4}\), coupler \(r_{3}\), base \(r_{1}\) and four semi-circular notch flexure hinges 1 to 4. A pure moment \(M\) is applied at the rocker \(r_{2}\) and the corresponding angle of \(r_{2}\) is \(\gamma_{2}\) while the initial angle is \(\gamma_{20}\). As is listed in Table 3, the lengths of base \(r_{1}\), rocker \(r_{2}\), coupler \(r_{3}\), rocker \(r_{4}\), and the width of the flexure hinge are given in advance. For the given desired rotation angles \(\theta_{d}\) of \(r_{2}\), the corresponding rotation angles \(\theta_{2}\), \(\theta_{3}\), and \(\theta_{4}\) of hinges 2 to 4 can be obtained from the closed-form equations of the compliant four-bar linkage mechanism while the initial angle \(\gamma_{20}\) is 65 deg. For the maximum rotation angles \(\theta_{\max}\) of \(\theta_{2}\) to \(\theta_{4}\), the ratio \(t/L\) can be figured out through the stress design equation in Equation (12) while the Teflon (\(E=345\) MPa, \(\sigma_{\rm s}=23\) MPa) is chosen as the material of the mechanism. The remaining dimensions of the mechanism are shown in Table 4 considering the lengths of \(r_{1}\), \(r_{4}\), and the compliant equation. For the given flexure hinges, the corresponding compliance \(C_{1}\) to \(C_{4}\) can be calculated by Equation (10) with the width of 6 mm. The theoretical moment \(M_{l}^{m}\) can be figured out utilizing the principle of virtual work with the compliance and rotation angles. Furthermore, the finite element analysis with same desired rotation angles \(\theta_{d}\) of hinge 2 of the mechanism is conducted and the moment \(M_{f}^{m}\) obtained from the FEA is compared with the \(M_{l}^{m}\). The \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dimension** & **Length \(L\) (mm)** & **Minimum Thickness \(t_{\min}\) (mm)** & **Angle \(\theta_{\max}\) (rad)** & **Width \(w\) (mm)** \\ \hline Hinge\#1 & 10 & 3.19 & 0.30 & 6 \\ Hinge \#2 & 10 & 2.07 & 0.47 & 6 \\ Hinge \#3 & 10 & 3.45 & 0.28 & 6 \\ Hinge \#4 & 10 & 3.57 & 0.11 & 6 \\ \hline \hline \end{tabular}
\end{table}