# Analytical Modeling and Application for Semi-Circular Notch Flexure Hinges
## 2 Compliance Modeling under Small Deformation  
The structure model of the variable cross-section semi-circular notch flexure hinge is shown in Figure 2, while the radii of the two concentric circles are \(R\) and \(r\), respectively. As is shown in Figure 1, \(L\), \(w\), and \(t\) denote the length, width, and minimum thickness of the hinge, respectively, and h represents the height of the rigid body. In general, the ratio of \(t/L\) is defined as the dimensionless design parameter to describe the geometry characteristics of the hinge. As a special variable cross-section flexure hinge, the inner circle of the hinge is always tangent to the two rigid bodies under the different value of \(t/L\), and it can be deduced that \(L=2r\). Two fillets are designed at both sides of the hinge to reduce stress concentration. The fillet is tangent to the outer circle and rigid body with a radius of a. The origin of the coordinate system is the midpoint of the line that connects the two tangent points of the fillets and the inner circle. For the human finger, the joints are driven by the muscle and the moment is generated at each joint. Therefore, one end of the flexure hinge is fixed, and a moment is given at point '1' of the other free end. Compared with many other semi-circular notch flexure hinges with a constant cross-section, both sides of the hinge proposed in this paper are thicker than the middle part. This kind of flexure hinge also shows good motion accuracy and better energy storage characteristics because of its geometrical characteristics. As is shown in Figure 2, the thickness \(t(x)\) of the variable cross-section semi-circular notch flexure hinge can be expressed as Equation (1):  
\[t(x)=\left\{ \begin{array}{cc}\sqrt{(R+a)^{2}-(r-a)^{2}}-\sqrt{a^{2}-(x-a)^{2}}- \sqrt{r^{2}-(x-r)^{2}}&x\in\left(0,r-\frac{R(r-a)}{K+a}\right)\\ \sqrt{R^{2}-(x-r)^{2}}-\sqrt{r^{2}-(x-r)^{2}}&x\in\left(r-\frac{R(r-a)}{R+a},r +\frac{R(r-a)}{R+a}\right)\\ \sqrt{(R+a)^{2}-(r-a)^{2}}-\sqrt{a^{2}-(x+a-2r)^{2}}-\sqrt{r^{2}-(x-r)^{2}}&x \in\left(r+\frac{R(r-a)}{K+a},2r\right)\end{array} \right. \tag{1}\]  
According to [PERSON]'s second theorem, the relationship between load and the deformation of the hinge can be expressed as:  
\[ \begin{bmatrix}\delta_{x}\\ \delta_{y}\\ \theta_{z}\end{bmatrix} = \begin{bmatrix}C_{1,x-F_{x}}&0&0\\ 0&C_{1,y-F_{y}}&C_{1,y-M_{z}}\\ 0&C_{1,\theta_{z}-M_{z}}&C_{1,\theta_{z}-M_{z}}\end{bmatrix}\begin{bmatrix}F_{ x}\\ F_{y}\\ M_{z}\end{bmatrix}  \tag{2}\]  
As the finger joint is mainly driven by the moment, for the moment \(M_{z}\) applied at the point '1' while the \(F_{x}\) and \(F_{y}\) are equal to zero, the Equation (2) can be simplified as:  
\[\theta_{z}=C_{1,\theta_{z}-M_{z}}M_{z} \tag{3}\]  
The \(\theta_{Z}\) is the rotation angle of the hinge. The theoretical compliance \(C_{1,\theta_{z}-M_{z}}^{T}\) of the flexure hinge under small deformation can be calculated by [PERSON]'s second theorem:  
\[C_{1,\theta_{z}-M_{z}}^{T}=\frac{12}{Ew}I_{1} \tag{4}\]  
where \(E\) is [PERSON]'s modulus of the material. Substitute Equation (1) and Equation (4) to Equation (3) and the \(\theta_{Z}\) can be expressed as:  
\[\theta_{z}=C_{1,\theta_{z}-M_{z}}^{T}M_{z}=\frac{12}{Ew}\int_{0}^{L}\frac{dx} {t(x)^{3}}M_{z} \tag{5}\]  
A given model is built to verify the accuracy of the compliance matrix based on [PERSON]'s second theorem under large deformation, and the results obtained from Equation (5) were compared with the finite element analysis (FEA) results. The different rotation angles obtained from the FEA results and Equation (5) under the same moment are shown in Figure 3.  
Figure 2: Structure model of semi-circular notch flexure hinge.