# Determination of Kinematic and Dynamic Characteristics of Oscillating Conveyor Mechanism
## 2 Materials and Methods
### Differential Equations of Motion of the Joints of an Oscillating Conveyor Mechanism  
Equation for joint 1 motion:  
\[I_{d}(\varphi_{1})\frac{d\omega_{1}}{dt}=\frac{\omega_{1}^{2}}{2}\frac{dI_{d}( \varphi_{1})}{d\varphi_{1}}=M_{d}-M_{r} \tag{45}\]  
where \(M_{d}\)--is the driving torque; \(M_{r}\)-- is the resisting torque. The driving torque on the joint is related to the torques from different components (such as \(M_{F}^{r}\), \(M_{F}^{r}\), \(M_{D}^{r}\), and \(M_{G}^{r}\)):  
Where \(M_{E}^{r}\), \(M_{F}^{r}\), \(M_{D}^{r}\), and \(M_{G}^{r}\) is the are the resisting torques from different parts of the system; \(\omega_{5}\) and \(\omega_{6}\) are the angular velocities of joints 5 and 6, respectively [18]. \[M_{d}\omega_{1}=M_{E}^{r}\omega_{5}+M_{D}^{r}\omega_{6}+M_{F}^{r}\omega_{5}+M_ {G}^{r}\omega_{6} \tag{46}\]  
By substituting and factoring the terms in the equation for \(M_{d}\), we get:  
\[M_{d}=M_{E}^{r}\frac{\omega_{5}}{\omega_{1}}+M_{D}^{r}\frac{\omega_{6}}{\omega _{1}}+M_{F}^{r}\frac{\omega_{5}}{\omega_{1}}+M_{G}^{r}\frac{\omega_{6}}{ \omega_{1}}\]  
Which can be simplified further to:  
\[M_{d}=M_{E}^{r}u_{51}+M_{D}^{r}u_{61}+M_{F}^{r}u_{51}+M_{G}^{r}u_{61}=4u_{51}M _{r} \tag{47}\]  
where \(u_{51}\) and \(u_{61}\) are terms relating to the motion of the system, possibly dependent on the geometry or kinematic relationships between different components. The motion of the oscillating conveyor mechanism is driven by the external torques \(M_{d}\), which depend on the angular velocities of different parts of the system. These equations couple the rotational dynamics of joint 1 with the other components in the system. \[M_{r}=M_{r0}+K\varphi_{6} \tag{48}\]  
\(M_{r0}\)--is the base moment of the applied resistance force (perhaps considering some other factors), \(M_{r0}=-2.98\) and this value represents the inherent resisting torque in the system at \(\varphi_{6}=0\); \(K\)--is the stiffness coefficient related to the change in \(M_{r}\) with respect to \(\varphi_{6}\). A higher \(K\) indicates a steeper increase in resistance torque as \(\varphi_{6}\) increases; \(\varphi_{6}\)--is the parameter (angle) representing the system's configuration or deformation, directly affecting \(M_{r}\). Expression for \(K\):  
\[K=tg(\alpha)=\frac{14.86-5.94}{2^{\circ}-1^{\circ}}=\frac{8.92}{0.017453}=511.84\]  
This constant \(K\) is calculated using the tangent of angle \(\alpha\), and it equals \(511.84\), which plays a key role in determining how the driving torque depends on \(\varphi_{6}\). The formula for \(M_{r}\) is given by:  
\[\varphi_{6}=\varphi_{6}(\varphi_{1}),\ M_{r}=M_{r}(\varphi_{1});\ M_{r}=-2.98 +511.87\varphi_{6}\]  
where \(M_{r0}\)--is the expressed as a linear function of \(\varphi_{6}\); the constant \(-2.98\) represents an offset, and \(511.84\) is the sensitivity factor, which amplifies the effect of \(\varphi_{6}\) on the driving torque. At \(\varphi_{6}=0\), the resisting torque \(M_{r}\) equals \(-2.98\), representing a baseline resistance. As \(\varphi_{6}\) increases, \(M_{r}\) grows linearly due to the stiffness coefficient \(K\). This behavior reflects how the resistance increases proportionally to the angular displacement \(\varphi_{6}\). Using the expression for \(M_{d0}\), the driving torque \(M_{d}\) becomes. The driving torque \(M_{d}\) is expressed as:  
\[M_{d}=M_{d_{o}}-\alpha_{0}\omega_{1} \tag{49}\]  
where \(\alpha_{0}\)--is the coefficient representing the rate at which driving torque decreases with increasing angular velocity. This accounts for damping effects, \(\omega_{1}\)--is the angular velocity of joint 1, \(M_{d_{o}}\)--is the base moment of the applied driving force \((M_{d_{o}}=2.92)\). This is the driving torque when the angular velocity \(\omega_{1}=0\). At \(\omega_{1}=0\) increases, \(M_{d}\) decreases linearly due to the damping effect represented by \(\alpha_{0}\). This reflects energy loss or reduced efficiency as the system accelerates.