# Determination of Kinematic and Dynamic Characteristics of Oscillating Conveyor Mechanism
## 4 Results  
To compute the exact derivatives of \(\vartheta_{S_{2}}\) and \(\vartheta_{S_{3}}\) with respect to \(\varphi_{1}\) over time, you need to express these variables in terms of \(\varphi_{1}\) and its time-dependent behavior shown in Figure 4. After that, we use the differentiation rules and chain rules to find time derivatives, including any relationships between angular velocity or other relevant parameters in the system. To plot the derivatives of \(u_{21}\) and \(u_{51}\) with respect to \(\varphi_{1}\) (i.e., \(\frac{du_{21}}{dp_{1}}\) and \(\frac{du_{51}}{dp_{1}}\)), we need the expressions for \(u_{21}\) and \(u_{51}\) in terms of \(\varphi_{1}\) shown in Figure 5. We can compute the derivatives of these expressions with respect to \(\varphi_{1}\) and then plot them. The programming language chosen to use (Python 3.13.0). \(M_{d}(t)=sin(\omega_{1})e^{-0.1t}\) A damped sinusoidal function. \(M_{r}(t)=cos(\varphi_{6})\times(1+0.5 sin(t))\) A modulated cosine function. Below are 3D plots for \(M_{d}(t)\) as a function of \(\omega_{1}\) and \(M_{r}(t)\) as a function of \(\varphi_{6}\). These assume some general functional forms for \(M_{d}(t)\) and \(M_{r}(t)\), but they can be replaced with specific equations or data shown in Figure 6. To analyze the 3D graphs below, it is first necessary to understand their physical meaning and how they change with time, angular velocity (\(\omega_{1}\)), and angular position (\(\varphi_{6}\)). In this graph, the torque \(M_{d}(t)\) is plotted as a function of time and angular velocity (\(\omega_{1}\)). The function \(\sin(\omega_{1})\) produces sinusoidal oscillations with respect to \(\omega_{1}\), indicating that the torque varies cyclically: \(\exp(-0.1t)\) as time \(t\) increases, the amplitude of the torque decreases, indicating the damping or damping effect. Initially, the amplitude of the \(M_{d}\) is maximum and decreases with time. This is explained by the energy loss in the kinematic system (e.g., due to friction). At high values of \(\omega_{1}\), the amplitude of the oscillations changes little, but the frequency increases. This graph is useful for predicting the change in torque, adjusting the damping parameters, and studying the stability of the system. The second graph shows the torque \(M_{r}(t)\) as a function of time and angular position (\(\varphi_{6}\)). \(\cos(\varphi_{6})\)\(M_{r}(t)\) varies sinusoidally with angular position. This indicates several complete cycles over a certain period of time. (\(1+0.5\) sin(\(t\))) The amplitude of the torque is modulated with time \(t\). That is, the vibration level changes with time. The dependence on \(\varphi_{6}\) indicates the  
Figure 4: Derivatives of \(\theta_{S_{2}}\) and \(\theta_{S_{3}}\) with respect to \(\varphi_{1}\) over time. Figure 5: Derivatives of \(u_{21}\) and \(u_{51}\) with respect to \(\varphi_{1}\) over time. cyclical nature of the vibration, which indicates the uniformity or repetition of the rotational motion in the mechanism. The dependence on \(t\) leads to an increase or decrease in the amplitude at a certain time, which indicates the sensitivity of the rotating system to external influences. This graph allows us to assess the stability of \(M_{r}(t)\) with respect to angular position and time. It is useful for studying the phase balance and dynamics of the device over time. The resulting 3D graphs from the Python provide insights into the oscillatory behaviors of the variables over time. Here is an analysis of each graph: \(A(t)\) exhibits sinusoidal oscillations whose amplitude gradually decreases over time due to exponential damping (\(\exp(-0.1t)\)) shown in Figure 7. \(B(t)\) is a cosine wave with slower exponential decay (\(\exp(-0.05t)\)) compared to \(A(t)\) shown in Figure 7. Useful for systems with slower energy dissipation, such as lightly damped mechanical systems. \(D(t)\) A sinusoidal wave superimposed on a linear increasing trend (\(2t\)) is shown in Figure 8. Represents oscillatory systems influenced by a steady growth, such as systems with a baseline drift.