# Analytical Modeling and Application for Semi-Circular Notch Flexure Hinges
Currently, flexure hinges with non-constant cross-sections are extensively employed in the field of bio-medical engineering, such as in compliant artificial hands (Figure 1). The advantages of this type of flexure hinge lie in its ease of design and manufacturing. However, the aforementioned research findings are not applicable for analyzing the stiffness/compliance characteristics of semi-circular flexure hinges with constant or variable cross-sections. Furthermore, there are other mathematical approaches to analyzing the properties of flexure hinges. Topology optimization is also an approach that is utilized in the analysis and design of compliant mechanisms and flexure hinges. [PERSON] et al. surveyed the continuum topology optimization method from three levels [21; 22]. [PERSON] et al. proposed a topology optimization framework in the design of the compliant mechanism composed of flexure hinges and beams. A super flexure hinge element was proposed in the study and the stiffness matrix was derived with consideration of the stress concentration effects [23; 24]. However, the topology method is usually utilized in the design of a mechanism with an objective function, such as the requirement of compliance in the desired direction of the flexure hinges. In addition, [PERSON]'s second theorem is mainly utilized in the displacement calculation of an elastic body under strain energy. The theorem has been widely utilized in the stiffness calculation of the variable cross-section flexure hinge as it takes the effects  
Figure 1: A compliant artificial hand with semi-circular flexure hinges. of section structure and geometric dimension on stiffness into consideration. [PERSON] proposed several closed-form compliance equations based on [PERSON]'s second theorem [3]. Of course, the accuracy of the proposed closed-form compliance equations is limited under the small deflections. With the development of finite element analysis tools, the finite element analysis approach is utilized to amplify the application of [PERSON]'s second theorem. [PERSON] et al. proposed the characteristics of stiffness and stress of the elliptical and corner-filleded flexure hinges through the finite element method. The results show the accuracy of their proposed equations [25]. Similarly, [PERSON] et al. developed closed-form equations for symmetric single-axis semi-elliptical flexure hinges based on finite element analysis and experimental tests [26]. [PERSON] et al. proposed a multi-cavity flexure hinge with great compliance through the three-dimensional continuum topology optimization, the compliance equations of the hinge were obtained based on [PERSON]'s theorem [27]. [PERSON] et al. verified the stiffness of a variable thickness flexure pivot based on the FEA results [28]. [PERSON] et al. presented several new empirical stiffness equations for corner-filleded flexure hinges based on FEA results and the accuracy of the equations under large deformation was greatly improved [29]. It shows that the FEA approach could be utilized together with [PERSON]'s second theorem to analyze the characteristics of stiffness and stress of the flexure hinges with large deflection. Therefore, this paper analyzes the compliance characteristics and stress characteristics of the variable cross-section semi-circular notch flexure hinge under large deformation based on [PERSON]'s second theorem and the FEA approach. The proposed design equations in this paper can also be analyzed in bionic properties. The remainder of this paper is organized as follows: The model of the semi-circular flexure hinge is introduced, and the compliance and stress equations are presented in Section 2. Then, the finite element analysis is conducted, and the general design equations for the semi-circular notch flexure hinge are derived in Section 3. In Section 4, an improved optimization approach for the compliant mechanism is presented and a case study of compliant four-bar linkage mechanisms is conducted to verify the accuracy of the equations.