# Analytical Modeling and Application for Semi-Circular Notch Flexure Hinges
## 4 Case Study: Compliant Four-Bar Linkage Mechanism

Table 4: Parameters of four-bar linkage mechanism designed by the general equations. Figure 7: Model of the compliant four-bar linkage mechanism. error between the \(M_{f}^{m}\) and \(M_{l}^{m}\) can be utilized to measure the accuracy of the proposed design equations while:  
\[\text{Error}_{\text{m}}=\left[\left(M_{l}^{m}-M_{f}^{m}\right)/M_{f}^{m}\right] \times 100\% \tag{13}\]  
As mentioned in Section 1, [PERSON] et al. analyzed the characteristics of the initially curved beam through the method of the PRB model. In this paper, two compliant four-bar linkage mechanisms are built on the basic of the stress and compliance equations proposed by [PERSON] and [PERSON]. The dimensions obtained from the two methods are shown in Tables 5 and 6, respectively. \(M_{l}^{H}\) and \(M_{l}^{L}\) denote the theoretical moment obtained from [PERSON] and [PERSON]'s equations, while \(M_{F}^{H}\) and \(M_{F}^{L}\) denote the corresponding FEA output results. The theoretical moment and the FEA moment are plotted in Figure 8. The stress distribution figure of the model obtained from the equations proposed by [PERSON] is shown in Figure 9 while the maximum stress is 23.291 MPa. The corresponding stress of the three models is shown in Figure 10. The errors \(\text{Error}_{\text{H}}\) and \(\text{Error}_{\text{L}}\) between \(M_{F}\) and \(M_{L}\) of the two mechanisms are compared with the \(\text{Error}_{\text{m}}\). These errors are printed in the Figure 11. \begin{table}
\begin{tabular}{c c c c c} \hline
**Dimension** & **Length \(L\) (mm)** & **Minimum Thickness \(t_{\text{min}}\) (mm)** & **Angle \(\theta_{\text{max}}\) (rad)** & **Width \(w\) (mm)** \\ \hline Hinge \#1 & 10 & 3.57 & 0.30 & 6 \\ Hinge \#2 & 10 & 2.38 & 0.47 & 6 \\ Hinge \#3 & 10 & 3.15 & 0.28 & 6 \\ Hinge \#4 & 10 & 4.66 & 0.11 & 6 \\ \hline \end{tabular}
\end{table} 
Table 6: Parameters of four-bar linkage mechanism designed by Lobontiu’s equations. \begin{table}
\begin{tabular}{c c c c} \hline
**Dimension** & **Length \(L\) (mm)** & **Minimum Thickness \(t_{\text{min}}\) (mm)** & **Angle \(\theta_{\text{max}}\) (rad)** & **Width \(w\) (mm)** \\ \hline Hinge \#1 & 10 & 3.32 & 0.30 & 6 \\ Hinge \#2 & 10 & 2.21 & 0.47 & 6 \\ Hinge \#3 & 10 & 3.56 & 0.28 & 6 \\ Hinge \#4 & 10 & 4.14 & 0.11 & 6 \\ \hline \end{tabular}
\end{table} 
Table 5: Parameter description of Hinges for the desired four-bar linkage mechanism. Figure 8: Schemes follow another format. Theoretical moment and FEA moment; **(a)** model designed by the general design equations; **(b)** model designed by the [PERSON]’s equations; **(c)** model designed by the [PERSON]’s equations. Figure 10: The relationship between the stress and the desired angle. Figure 9: Stress distribution. As is shown in Figure 11, for the nonlinear characteristics of hinge and the axis drift under large deformation, the maximum error which happened in [PERSON]'s equations reached 41.29%. The maximum error that happened in the [PERSON]'s equations is 25.42%. The emergence of large errors can be attributed to underlying assumptions. [PERSON] et al. conducted an analysis on the stiffness characteristics of numerous flexure hinges and compliant mechanisms using the PRB model method, which assumes small deformation without considering non-linear deformation. However, in the case of semi-circular flexure hinges investigated in this study, even very small deformations, cannot neglect non-linear deformation. Additionally, [PERSON] analyzed the characteristics of variable cross-section flexure hinges based on [PERSON]'s second theorem, which takes into account the influence of section structure and geometric dimensions on stiffness and proposes several equations. Nevertheless, it should be noted that these compliance equations have limited accuracy when applied to small deformations. Additionally, it can be figured out that with the increase of the rotation angle, the equations are more accurate.