# Dynamics of Mechanisms with Superior Couplings
## 2 Materials and Methods
### Swiped Cam Mechanisms and Translated Adept with Roll  
Figure 5 shows a mechanism with rotating cam and adept with roller in translational movement. Turning to the mechanism with a rotating cam and a follower with roller in translational motion, the authors make important new contributions to mechanical systems with superior couples, among which could be listed a few: determining two angles of pressure for this type of mechanism; transmission of forces and speeds; determining the dynamic coefficients and the instantaneous mechanical efficiency; the exact theoretical realization of the dynamics of the mechanism, with the determination of the variable angular velocity of the cam \(w\) and of the variable dynamic displacement of the ratchet \(x\); dynamic synthesis of the mechanism.  
The pressure angle \(\delta\), which appears between the normal \(n\) taken through the contact point A and a vertical, has the known quantity given by the relations (27)-(29):  
\[\cos\delta=\frac{s_{0}+s}{\sqrt{\left(s_{0}+s\right)^{2}+\left(s^{\prime}-e \right)^{2}}} \tag{27}\]  
\[\sin\delta=\frac{s^{\prime}-e}{\sqrt{\left(s_{0}+s\right)^{2}+\left(s^{\prime }-e\right)^{2}}} \tag{28}\]  
\[tg\delta=\frac{s^{\prime}-e}{s_{0}+s} \tag{29}\]  
Figure 5: A mechanism with rotating cam and adept with roller in translational movement.  
We can also directly determine the angle \(\alpha_{A}\) (30) and (31):  
\[\cos\alpha_{A}=\frac{e+r_{b}\cdot\sin\delta}{r_{A}} \tag{30}\]  
\[\sin\alpha=\frac{s_{0}+s-r_{b}\cdot\cos\delta}{r_{A}} \tag{31}\]  
The cam profile can now be drawn directly using the polar coordinates \(r_{A}\) (known, see relation (9)) and \(\theta_{A}\) (which is determined by relations (32)-(37):  
\[\gamma=\alpha_{A}-\alpha_{0} \tag{32}\]  
\[\cos\gamma=\cos\alpha_{A}\cdot\cos\alpha_{0}+\sin\alpha_{A}\cdot\sin\alpha_{0} \tag{33}\]  
\[\sin\gamma=\sin\alpha_{A}\cdot\cos\alpha_{0}-\cos\alpha_{A}\cdot\sin\alpha_{0} \tag{34}\]  
\[\theta_{A}=\phi-\gamma \tag{35}\]  
\[\cos\theta_{A}=\cos\phi\cdot\cos\gamma+\sin\phi\cdot\sin\gamma \tag{36}\]  
\[\sin\theta_{A}=\sin\phi\cdot\cos\gamma-\sin\gamma\cdot\cos\phi \tag{37}\]  
One writes the following relations of forces and velocities (38)-(42) [1]:  
\[\left\{\begin{array}{l}v_{a}=v_{m}\cdot\sin(\alpha_{A}-\delta)\\ F_{a}=F_{m}\cdot\sin(\alpha_{A}-\delta)\end{array}\right. \tag{38}\]  
\[\left\{\begin{array}{l}v_{n}=v_{m}\cdot\cos(\alpha_{A}-\delta)\\ F_{n}=F_{m}\cdot\cos(\alpha_{A}-\delta)\end{array}\right. \tag{39}\]  
\[\left\{\begin{array}{l}v_{i}=v_{n}\cdot\sin\delta\\ F_{i}=F_{n}\cdot\sin\delta\end{array}\right. \tag{40}\]  
\[\left\{\begin{array}{l}v_{2}=v_{n}\cdot\cos\delta=v_{m}\cdot\cos(\alpha_{A }-\delta)\cdot\cos\delta\\ F_{u}=F_{n}\cdot\cos\delta=F_{m}\cdot\cos(\alpha_{A}-\delta)\cdot\cos\delta \end{array}\right. \tag{41}\]  
Instantaneous yield is determined (16):  
\[\begin{array}{l}\eta_{i}=\frac{p_{a}}{P_{c}}=\frac{F_{m}\cdot v_{n}\cdot\cos ^{2}(\alpha_{A}-\delta)\cdot\cos^{2}\delta}{P^{n}\cdot\gamma_{m}}=\cos^{2}( \alpha_{A}-\delta)\cdot\cos^{2}\delta=\\ \left[\cos(\alpha_{A}-\delta)\cdot\cos\delta\right]^{2}=\left[\frac{s\delta}{ r_{A}}\cdot\cos^{2}\delta\right]^{2}=\frac{s\delta^{2}}{r_{A}^{2}}\cdot\cos^{4} \delta\end{array} \tag{42}\]  
It can be seen that the instantaneous mechanical efficiency of this type of mechanism ultimately depends on the cosine of the classical pressure angle raised to the fourth power.  
The transmitted dynamic forces have more complex expressions for this type of mechanism, which makes the dynamics more difficult and more complex.