# Dynamics of Mechanisms with Superior Couplings
## 2 Materials and Methods
### Approximate Solution of the Lagrange Equation of Motion
\begin{cases}\dfrac{1}{2}\cdot J^{*I}=\dfrac{1}{2}\cdot\dfrac{dJ^{*}}{d\phi}=L \Rightarrow J^{*}\cdot\ddot{\phi}+L\cdot\dot{\phi}^{2}=M^{*}\Rightarrow\ddot{ \phi}+\dfrac{L}{J^{*}}\cdot\dot{\phi}^{2}=\dfrac{M^{*}}{J^{*}}\Rightarrow\ddot {\phi}=\dfrac{M^{*}}{J^{*}}-\dfrac{L}{J^{*}}\cdot\dot{\phi}^{2}\\ \text{with }\ddot{\phi}=\dfrac{d\dot{\phi}}{dt}=\dfrac{d\dot{\phi}}{d\phi} \cdot\dfrac{d\phi}{dt}=\dfrac{d\dot{\phi}}{d\phi}\cdot\dot{\phi}=\dfrac{d\omega }{d\phi}\cdot\omega\Rightarrow\omega\cdot\dfrac{d\omega}{d\phi}=\dfrac{M^{*}}{J^{ *}}-\dfrac{L}{J^{*}}\cdot\omega^{2}=\dfrac{M^{*}-L\cdot\omega^{2}}{J^{*}}\\ \text{with }\omega=\omega_{n}+d\omega\Rightarrow(\omega_{n}+d\omega)\cdot d \omega=\Big{[}\dfrac{M^{*}}{J^{*}}-\dfrac{L}{J^{*}}\cdot(\omega_{n}+d\omega) ^{2}\Big{]}\cdot d\phi\\ \omega_{n}\cdot d\omega+(d\omega)^{2}=\dfrac{M^{*}}{J^{*}}\cdot d\phi-\dfrac{L }{J^{*}}\cdot d\phi\cdot\Big{[}\omega_{n}^{2}+(d\omega)^{2}+2\cdot\omega_{n} \cdot d\omega\Big{]}\\ \omega_{n}\cdot d\omega+(d\omega)^{2}-\dfrac{M^{*}}{J^{*}}\cdot d\phi+\dfrac{L }{J^{*}}\cdot d\phi\cdot\omega_{n}^{2}+\dfrac{L}{J^{*}}\cdot d\phi\cdot(d \omega)^{2}+2\cdot\dfrac{L}{J^{*}}\cdot d\phi\cdot\omega_{n}\cdot d\omega=0\\ \Big{(}\dfrac{L}{J^{*}}\cdot d\phi+1\Big{)}\cdot(d\omega)^{2}+2\cdot\Big{(} \dfrac{L}{J^{*}}\cdot d\phi+\dfrac{1}{2}\Big{)}\cdot\omega_{n}\cdot d\omega- \Big{(}\dfrac{M^{*}}{J^{*}}\cdot d\phi-\dfrac{L}{J^{*}}\cdot d\phi\cdot\omega_ {n}^{2}\Big{)}=0\\ \Delta=\dfrac{L^{2}}{J^{*2}}\cdot(d\phi)^{2}\cdot\omega_{n}^{2}+\dfrac{\omega _{n}^{2}}{4}+\dfrac{L}{J^{*}}\cdot d\phi\cdot\omega_{n}^{2}+\dfrac{L\cdot M^{* }}{J^{*2}}\cdot(d\phi)^{2}+\dfrac{M^{*}}{J^{*}}\cdot d\phi-\dfrac{L^{2}}{J^{*2 }}\cdot(d\phi)^{2}\cdot\omega_{n}^{2}-\dfrac{L}{J^{*}}\cdot d\phi\cdot\omega_ {n}^{2}\\ \Delta=\dfrac{\omega_{n}^{2}}{4}+\dfrac{L\cdot M^{*}}{J^{*2}}\cdot(d\phi)^{2} +\dfrac{M^{*}}{J^{*}}\cdot d\phi\\ d\omega=\dfrac{-\frac{L}{J^{*}}\cdot d\phi\cdot\omega_{n}-\frac{\omega_{n}}{2} +\sqrt{\Delta}}{\frac{L}{J^{*}}\cdot d\phi+1}\end{cases}  \tag{19}\]  
Another novelty in the presented dynamic method is the way in which the dynamics of the follower is solved. It should be noted here that the Lagrange equation already presented can only determine in the end only the variable rotational speed \(\omega\) of the camshaft (cam). The complete dynamics of the cam and ratchet mechanism also requires the determination of the variable adept kinematics, i.e., its dynamic offset \(x\), different from the law of motion imposed on the ratchet \(s\), its dynamic speed \(x^{\prime}\) different from \(s^{\prime}\), and its dynamic acceleration \(x^{\prime\prime}\) different from \(s^{\prime\prime}\). The original Equations (20) and (21) used for this purpose were obtained by integrating the known classical Newtonian equation of motion. Those who want to know how to obtain the Equations (20) and (21) must study the paper in detail [1], because the calculations are long and tiring and there is no room here for their presentation. It is used to determine the dynamic displacement of the adept, the relations (20) and (21) [1]. \[ \begin{array}{c}\Delta X=(-1)\cdot\\ \frac{(k^{2}+2\cdot k\cdot K)\cdot\dot{s}^{2}+2\cdot k\cdot x_{0}\cdot(K+k)+ \lfloor\frac{k^{2}}{1\cdot K+k}\cdot m_{S}^{*}+(K+k)\cdot m_{T}^{*}\rfloor\cdot \omega^{2}\cdot(Dsq)^{2}}{2\cdot(s+\frac{k\cdot x_{0}}{k+k})\cdot(K+k)^{2}} \end{array}  \tag{20}\]  
\[X=s-\frac{\lfloor\frac{K^{2}}{K+k}\cdot m_{S}^{*}+(K+k)\cdot m_{T}^{*}\rfloor \cdot\omega^{2}\cdot(Dsq)^{2}}{2\cdot(s+\frac{k\cdot x_{0}}{K+k})\cdot(K+k)^{2 }}-\frac{(k^{2}+2\cdot k\cdot K)\cdot s^{2}+2\cdot k\cdot x_{0}\cdot(K+k) \cdot s}{2\cdot(s+\frac{k\cdot x_{0}}{K+k})\cdot(K+k)^{2}}\end{array} \tag{21}\]  
After obtaining the real, dynamic displacement \(x\) of the follower, with the help of the original Equation (21), thus it will be derived twice in relation to time and in this way, the exact equations for determining the dynamic tappet speed \(x^{\prime}\) and of its dynamic acceleration \(x^{\prime\prime}\) will be obtained (see [1]).