# Insights into Bending Stiffness Modeling of Elementary Flexure Hinges
## 4 Bending Stiffness Model of a Flexure Hinge
### Conditions of Applicability of the Linear Model  
Once the profile of the flexure hinge is known, the applicability of the analytical model provided in Equation (14) can be verified by looking at the coefficient \(a(\widetilde{x})\). An analysis of this term means understanding under which conditions the assumptions of its negligibility can be considered valid, namely when:  
\[\max_{\widetilde{x}\in[-1;1]}|a(\widetilde{x})| \ll 1\] \[\max_{\widetilde{x}\in[-1;1]}|a(\widetilde{x})| \ll\min_{\widetilde{x}\in[-1;1]}\left|\frac{\widetilde{I}_{z}^{ \prime}(\widetilde{x})}{\widetilde{I}_{z}(\widetilde{x})}\right|\;. \tag{22}\]  
Figure 5: Elliptical flexure hinge with applied static loads. From the first of Equation (22) it is:  
\[\frac{a_{x}^{2}}{E\tilde{I}_{20}}\frac{|F|}{\min\limits_{\widetilde{x}\in[-1;1]} \widetilde{I}_{z}(\widetilde{x})}\ll 1. \tag{23}\]  
It is easy to verify that \(\min\limits_{\widetilde{x}\in[-1;1]}\widetilde{I}_{z}(\widetilde{x})= \widetilde{I}_{z}(0)={\beta_{y}}^{3}\) by looking at the expression of the terms involved. A convenient substitution in Equation (23) can be made by introducing the following dimensionless parameters, as in [15]:  
\[\rho=\frac{a_{x}}{a_{y}};\ \ \mu_{F}=\frac{F}{E\,I\,a_{x}}\ . \tag{24}\]  
The first indicates how far the elliptical profile deviates from a circular profile, represented by \(\rho=1\). The second allows the value of the axial force to be dimensionless. Taking advantage of expressions in Equations (20) and (24), from Equation (23) the following condition on \(\mu_{F}\) can be easily found:  
\[|\mu_{F}|\ll\frac{2}{3}\bigg{(}\frac{\beta_{y}}{\rho}\bigg{)}^{3}. \tag{25}\]  
The second condition in Equation (22) can be re-elaborated as follows:  
\[\frac{3}{2}\bigg{(}\frac{\rho}{\beta_{y}}\bigg{)}^{3}|\mu_{F}|\ll\min\limits_ {\widetilde{x}\in[-1;1]}\bigg{|}\frac{\widetilde{I}_{z}^{\prime}(\widetilde{ x})}{\widetilde{I}_{z}(\widetilde{x})}\bigg{|}. \tag{26}\]  
Said \(c(\widetilde{x})=\frac{\widetilde{I}_{z}^{\prime}(\widetilde{x})}{\widetilde {I}_{z}(\widetilde{x})}\), its trend with a single zero in \(\widetilde{x}=0\), due to the fact that it is an odd function as it will be shortly demonstrated, can be observed in Figure 6. It follows that:  
\[\min\limits_{\widetilde{x}\in[-1;1]}\bigg{|}\frac{\widetilde{I}_{z}^{\prime} (\widetilde{x})}{\widetilde{I}_{z}(\widetilde{x})}\bigg{|}=0. \tag{27}\]  
The result just obtained demonstrates that there is certainly a range of \(\widetilde{x}\) in which the condition in Equation (26) is not verified. What follows is dedicated to finding the limits of this range. A look at the expressions in Equation (21) reveals that \(\widetilde{I}_{z}(\widetilde{x})\) is positive for \(\widetilde{x}\in[-1;1]\) with \(\beta_{y}\in\left]0;1\right]\) and that it is an even function, namely \(\widetilde{I}_{z}(-\widetilde{x})=\widetilde{I}_{z}(\widetilde{x})\). Furthermore, its derivative is an odd function, namely \(-\widetilde{I}_{z}^{\prime}(-\widetilde{x})=\widetilde{I}_{z}^{\prime}( \widetilde{x})\) or also \(\widetilde{I}_{z}^{\prime}(-\widetilde{x})=-\widetilde{I}_{z}^{\prime}( \widetilde{x})\). Figure 6: Trend of function \(c(\widetilde{x})\). It follows that \(c(\widetilde{x})\) is an odd function, as mentioned above:  
\[\max_{\widetilde{x}\in[-1;1]}|a(\widetilde{x})| \ll\min_{\widetilde{x}\in[-1;1]}\left|\frac{\widetilde{I}_{z}^{ \prime}(\widetilde{x})}{\widetilde{I}_{z}(\widetilde{x})}\right|\;. \tag{22}\]0  
A similar analysis can be addressed for \(a(\widetilde{x})\). According to Equation (9), it is easy to verify that it is an even function, in fact:  
\[\max_{\widetilde{x}\in[-1;1]}|a(\widetilde{x})| \ll\min_{\widetilde{x}\in[-1;1]}\left|\frac{\widetilde{I}_{z}^{ \prime}(\widetilde{x})}{\widetilde{I}_{z}(\widetilde{x})}\right|\;. \tag{22}\]1  
Its sign depends on the sign of \(\cos\left(\varphi\right)\). Assuming that the latter is positive, it follows that \(a(\widetilde{x})<0\) for \(\widetilde{x}\in[-1;1]\), as shown in Figure 7, whereas \(c(\widetilde{x})\leq 0\) for \(\widetilde{x}\in[-1;0]\).