## 3 Proposed Mechanism
### Final Mechanism  
The mechanism proposed in Section 3.1, although theoretically suitable for the target application, is difficult to build. In particular, it would be quite impractical to include sliders at the at the anchor points with the frame, in order to keep the springs aligned with the reference axes. Therefore, a new mechanism is proposed with the springs fixed at some distance from the crank fixed hinge (Figure 5). Even with this modification, the basic principles described in Section 3.1 hold valid.  
Additionally, in this modified design, the springs are unloaded when the mechanism is close to the TDC (\(x\to 0\)). The principle of virtual work is to be applied in the same way as in (7), but the virtual displacements of \(k_{1}\) and \(k_{2}\) (\(s_{1}\) and \(s_{2}\), respectively) have to be  
Figure 4: Variation of the force necessary to maintain the mechanism in static equilibrium as a function of \(x\). (**a**) Only \(k_{1}\) and \(k_{2}\) are used (\(k_{1}=\)1500 N/m, \(k_{2}=\) 3000 N/m). (**b**) Only \(k_{1}\) and \(k_{3}\) are used (\(k_{1}=\) 1500 N/m, \(k_{3}=\) 150 N/m).  
Figure 5: Slider-crank mechanism with translational springs connected to fixed ends (**a**). Two particulars of the mechanism are shown in the right figure (**b**).  
modified according to the new mechanism. Starting from the dimensions of Figure 5, the spring displacements are:  
\[\begin{array}{c}s_{1}=\sqrt{l_{1}^{2}+d^{2}-2l_{1}d\cos(\gamma_{1}+\theta)}- \sqrt{\left(a-l_{1}\right)^{2}+c^{2}}\\ s_{2}=\sqrt{l_{1}^{2}+b^{2}}-\sqrt{l_{1}^{2}+b^{2}}-2l_{1}b\sin(\theta)\end{array} \tag{15}\]  
where \(d\) is the distance between the anchor point of \(k_{1}\) and the crank hinge, and \(\gamma_{1}\) is the angle of the vector crank hinge-anchor point with respect to the \(x\)-axis:  
\[d=\sqrt{a^{2}+c^{2}},\ \ \ \gamma_{1}=\tan\left(\frac{c}{a}\right) \tag{16}\]  
The differentials of \(s_{1}\) and \(s_{2}\) can be calculated as:  
\[\begin{array}{c}\delta s_{1}=\frac{\partial s_{1}}{\partial\theta}\cdot \delta\theta=\frac{l_{1}d\sin(\gamma_{1}+\theta)}{\sqrt{l_{1}^{2}+d^{2}-2l_{1 }d\cos(\gamma_{1}+\theta)}}\cdot\delta\theta\\ \delta s_{2}=\frac{\partial s_{2}}{\partial\theta}\cdot\delta\theta=\frac{l _{1}b\cos(\theta)}{\sqrt{l_{1}^{2}+b^{2}-2l_{1}b\sin(\theta)}}\cdot\delta \theta\end{array} \tag{17}\]  
As a result, the force \(F\) at the slider becomes:  
\[F=k_{3}x+\left(k_{1}s_{1}\frac{l_{1}d\sin(\gamma_{1}+\theta)}{\sqrt{l_{1}^{2}+ d^{2}-2l_{1}d\cos(\gamma_{1}+\theta)}}+k_{2}s_{2}\frac{l_{1}b\cos(\theta)}{ \sqrt{l_{1}^{2}+b^{2}-2l_{1}b\sin(\theta)}}\right)\tau_{\theta x} \tag{18}\]  
The expression (18) for \(x\to 0\) simplifies to:  
\[\underset{x\to 0}{\lim}F=\frac{k_{1}l_{1}l_{2}c^{2}\left(b^{2}+l_{1}^{2} \right)+k_{2}l_{1}l_{2}b^{2}\left(\left(a-l_{1}\right)^{2}+c^{2}\right)}{ \left(l_{1}+l_{2}\right)\left(\left(a-l_{1}\right)^{2}+c^{2}\right)\left(b^{ 2}+l_{1}^{2}\right)} \tag{19}\]  
Differently from (14), which depends on \(k_{1}\) only, (19) depends both on \(k_{1}\) and \(k_{2}\). Similarly to the mechanism of Figure 3, the new mechanism behaviour depends on its main dimensions. Nevertheless, in this case also the dimensions \(a\), \(b\), and \(c\) have a great impact on \(F\). As an example, two different mechanisms are shown in Figure 6, whose main parameters are listed in Table 1. Similarly to the mechanism presented in Section 3.1, both progressive and regressive forces as a function of the \(x\) displacement can be obtained, with a finite value of the force around the TDC. The regressive curve is preferred for the application at hand since it is related to lower cutting forces.  
Figure 6: Variation of the force necessary to maintain the mechanism in static equilibrium as a function of \(x\). (**(a)** Mechanism #1, **(b)** Mechanism #2).