# 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
The final equation also includes a damping or resisting term proportional to the angular velocity \(\omega_{1}\), with a constant \(\alpha_{0}\). This structure allows you to compute the driving torque in terms of the angular position and velocity of the system components, which are key to understanding the dynamics of the oscillating conveyor mechanism [19]. The first subplot shows the relationship between \(M_{r}\) and \(\varphi_{6}\). The second subplot shows the relationship between \(M_{d}\) and \(\omega_{1}\) show in Figure 3. Resisting Torque (\(M_{r}\)) Calculated for a range of \(\varphi_{6}\) values (0 to 0.02 rad) based on the formula \(M_{r}=-2.98+511.87\cdot\varphi_{6}\). Driving Torque (\(M_{d}\)) Calculated for a range of \(\omega_{1}\) values (0 to 2000 rad/s) based on the formula \(M_{d}=2.92-0.00074\cdot\omega_{1}\). Let us break down the equations step by step:  
\[\alpha_{0}=tg\alpha=\frac{2.8-1.8}{1500-150}=\frac{1}{1350}=0.00074\]  
Given values:  
\(Y=2.92\), \(Y_{1}=2.8\), \(Y_{2}=1.8\), \(X_{1}=150\), \(X_{2}=1500\), \(Mr=2.92\)  
Linear relationship equation:  
\[M_{d}\omega_{1}=M_{E}^{r}\omega_{5}+M_{D}^{r}\omega_{6}+M_{F}^{r}\omega_{5}+M_ {G}^{r}\omega_{6} \tag{46}\]0  
This is a form of linear interpolation or a proportionality equation, where \(X\) and \(Y\) are related within the intervals \([X_{1},\,X_{2}]\) and \([Y_{1},\,Y_{2}]\). Substituting the values:  
\[M_{d}\omega_{1}=M_{E}^{r}\omega_{5}+M_{D}^{r}\omega_{6}+M_{F}^{r}\omega_{5}+M_ {G}^{r}\omega_{6} \tag{46}\]1  
\[M_{d}\omega_{1}=M_{E}^{r}\omega_{5}+M_{D}^{r}\omega_{6}+M_{F}^{r}\omega_{5}+M_ {G}^{r}\omega_{6} \tag{46}\]2 The first subplot shows the relationship between \(M_{r}\) and \(\varphi_{6}\). The second subplot shows the relationship between \(M_{d}\) and \(\omega_{1}\). Simplifying the left-hand side:  
\[M_{d}\omega_{1}=M_{E}^{r}\omega_{5}+M_{D}^{r}\omega_{6}+M_{F}^{r}\omega_{5}+M_ {G}^{r}\omega_{6} \tag{46}\]3  
\[M_{d}\omega_{1}=M_{E}^{r}\omega_{5}+M_{D}^{r}\omega_{6}+M_{F}^{r}\omega_{5}+M_ {G}^{r}\omega_{6} \tag{46}\]4  
So, \(X=-12\). Now, based on we last expression:  
\[M_{d}\omega_{1}=M_{E}^{r}\omega_{5}+M_{D}^{r}\omega_{6}+M_{F}^{r}\omega_{5}+M_ {G}^{r}\omega_{6} \tag{46}\]5  
This shows that the driving torque \(M_{d}\) depends on the angular velocity \(\omega_{1}\), with a small damping effect due to the factor \(\alpha_{0}\). To ensure the system operates stably:  
The driving torque \(M_{d}\) must overcome the resisting torque \(M_{r}\). This establishes the condition for motion. The equations show that \(M_{r}\) grows with \(\varphi_{6}\), while \(M_{d}\) decreases with \(\omega_{1}\). A balance is achieved when the two torques are equal. This analysis provides a foundation for understanding and optimizing the torques in the oscillating conveyor mechanism. The interaction between \(M_{r}\) and \(M_{d}\) governs the system's behavior. If \(M_{d}<M_{r}\), the system will decelerate or stop. Conversely, if \(M_{d}>M_{r}\), the system will accelerate. The moment of inertia equation is:  
\[M_{d}\omega_{1}=M_{E}^{r}\omega_{5}+M_{D}^{r}\omega_{6}+M_{F}^{r}\omega_{5}+M_ {G}^{r}\omega_{6} \tag{46}\]6  
where \(I_{S1}\)--is the moment of inertia of the first component; \(I_{S2}\)--is the moment of inertia of the second component; \(I_{5F}\)--is the combined moment of inertia of the fifth component; \(I_{6G}\) is the combined moment of inertia of the sixth component; \(m_{2}\theta_{S2}\) and \(m_{3}\theta_{S3}\)--is the terms accounting for the contributions of specific masses \(m_{1}\) and \(m_{3}\) at velocities \(\theta_{S2}\) and \(\theta_{S3}\) respectively; \(u_{21}\) and \(u_{51}\)--is the ratios representing relationships between components in the system.