# Dynamics of Mechanisms with Superior Couplings
## 2 Materials and Methods  
### Geometric Synthesis  
A rapid method of geometric synthesis is that of Cartesian coordinates (see Figure 1). In the fixed system \(xOy\), the Cartesian coordinates of the contact point A (belonging to the block 2) are given by the projections of the position vector \(r_{A}\) on the \(\mathsf{O}_{\mathrm{x}}\) and \(\mathsf{O}_{\mathrm{y}}\) axes respectively, and have the analytical expressions expressed by the relational system (1). \[\left\{ \begin{array}{l}x_{T}=r_{A}\cdot\cos\bigl{(}\phi+\tau+\frac{\pi}{2}- \phi\bigr{)}=r_{A}\cdot\cos\bigl{(}\frac{\pi}{2}+\tau\bigr{)}=-r_{A}\cdot\sin \tau=-r_{A}\cdot\frac{\pi\prime}{r_{A}}=-st\\ \\ y_{T}=r_{A}\cdot\sin\bigl{(}\phi+\tau+\frac{\pi}{2}-\phi\bigr{)}=r_{A}\cdot\sin \bigl{(}\frac{\pi}{2}+\tau\bigr{)}=r_{A}\cdot\cos\tau=r_{A}\cdot\frac{r_{0}+s }{r_{A}}=r_{0}+s\end{array} \right. \tag{1}\]  
In the mobile system \(x^{\prime}Oy^{\prime}\), the Cartesian coordinates of the contact point A (belonging to the profile of the cam 1, which rotated clockwise with the angle \(\varphi\)), are given by the relations of the systems (2) and (3) [1]. \[\left\{ \begin{array}{l}x_{C}=r_{A}\cdot\cos\bigl{(}\phi+\tau+\frac{\pi}{2}- \phi+\phi\bigr{)}=r_{A}\cdot\cos\bigl{(}\frac{\pi}{2}+\tau+\phi\bigr{)}=r_{A} \cdot\sin(-\phi-\tau)=-r_{A}\cdot\sin(\phi+\tau)=\\ =-r_{A}\cdot(\sin\phi\cdot\cos\tau+\sin\tau\cdot\cos\phi)=-r_{A}\cdot\frac{ \pi\prime}{r_{A}}\cdot\sin\phi-r_{A}\cdot\frac{\pi\prime}{r_{A}}\cdot\cos \phi=-(r_{0}+s)\cdot\sin\phi-st\cdot\cos\phi\end{array} \right. \tag{2}\]  
\[\left\{ \begin{array}{l}y_{C}=r_{A}\cdot\sin\bigl{(}\phi+\tau+\frac{\pi}{2}- \phi+\phi\bigr{)}=r_{A}\cdot\sin\bigl{(}\frac{\pi}{2}+\tau+\phi\bigr{)}=r_{A} \cdot\cos(-\phi-\tau)=r_{A}\cdot\cos(\phi+\tau)=\\ =r_{A}\cdot(\cos\phi\cdot\cos\tau-\sin\tau\cdot\sin\phi)=r_{A}\cdot\frac{\pi \prime_{0}+s}{r_{A}}\cdot\cos\phi-r_{A}\cdot\frac{\pi\prime}{r_{A}}\cdot\sin \phi=(r_{0}+s)\cdot\cos\phi-st\cdot\sin\phi\end{array} \right. \tag{3}\]  
Distribution of Forces and Determination of Efficiency a Classic Mechanism with Rotating Cam and Translucent Flat Ratchet  
The distribution of forces was first introduced by the author [1] and it does not refer to the determination of forces acting on a mechanism. The reactions in the mechanism known from the classical methods determine the forces acting in the torques of the mechanism and show the maximum loads in the mechanism, as well as the variation of the inertia forces in the respective mechanism. The method of determining the distribution of forces shows the way in which the forces act in couples, on elements, and especially the way in which they are transmitted from one element to another within the respective kinematic coupling. It is an original method that allowed, in time, the author to deduce the exact mechanical efficiency of the mechanisms, the dynamic coefficients in the mechanism, the variation of the forces acting in the mechanism, and new methods for deducing the dynamics of the mechanism in question. Figure 3 analyzes the distribution of forces and velocities within a mechanism with superior couple with a rotating cam, ratchet, and classic translational movement, plane. The main angles that appear in the mechanism are the pressure angle (delta) and the transmission angle (tau) both being in a relationship of complementarity. The consumed motor force, \(F_{c}\), perpendicular in A on the vector \(r_{A}\), is divided into two components: (a) \(F_{m}\), which are the use force, or the reduced motor force at the dowel; (b) \(F_{y}\), which is the sliding force between the two cam and dowel profiles, (see Figure 3) and relations (4)-(13) [1]. The efficiency of the mechanism (8), originally determined by the distribution of forces in the mechanism, can be expressed simply by the relation (8) either according to the transmission angle (tau) or according to the pressure angle (delta). The relationship indicates practically instantaneous mechanical efficiency.