# 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
\begin{array}{c}F(t)=(-L_{1}\sin\varphi_{1}-L_{2}\sin\varphi_{2}u_{21}+au_{ 1}\cos\omega_{5}t)\left(-L_{1}\cos\varphi_{1}-L_{2}\cos\varphi_{2}u_{21}^{2}-L _{1}\sin\varphi_{1}\frac{du_{21}}{d\varphi_{1}}+a\cos\varphi_{5}\frac{du_{1} }{d\varphi_{1}}\right)+(L_{1}\cos\varphi_{1}\\ \hskip 142.26378 pt+L_{2}\cos\varphi_{2}u_{21}-bu_{1}\sin\omega_{5}t)\left(-L _{1}\sin\varphi_{1}-L_{2}\sin\varphi_{2}u_{21}^{2}+L_{2}\cos\varphi_{2}\frac{ du_{21}}{d\varphi_{1}}-b\sin\omega_{5}t\frac{du_{21}}{d\varphi_{1}}\right)\end{array}  \tag{65}\]  
\(F(t)\)--is the damping and inertial effects with oscillatory external inputs. \[N(t)=\frac{(\cos(\varphi_{1}+\varphi_{5})-\cos(\varphi_{1}-\varphi_{1}))\sin( \varphi_{1}-\varphi_{5})}{\sin^{2}(\varphi_{2}-\varphi_{5})} \tag{66}\]  
\(N(t)\)--is the coupling strength based on angular misalignments. \[M(t)=(l_{\rm SF}+l_{\rm 6G})\left[2\frac{l_{1}^{2}}{l_{2}^{2}}\frac{(\cos( \varphi_{1}+\varphi_{2})-\cos(\varphi_{1}-\varphi_{2}))\sin(\varphi_{2}- \varphi_{1})}{\sin^{2}(\varphi_{2}-\varphi_{5})}\right] \tag{67}\]  
\(M(t)\)--is the torque contribution influenced by inertia and geometry. \[W(t)=M_{d_{o}}-\alpha\omega_{1}-4u_{51}(M_{\rm 9}+K\varphi_{6}) \tag{68}\]  
Given your detailed numerical approximation method for solving the differential equation:  
\[\omega(0)=0,\ \varphi_{0}=60^{o},\ h=0.01,\ k=0,1,2,\ldots,N-1,\ t_{k}=kl,\ \omega_{k}=\omega(t_{k})\mbox{for}\ k=0,1, \ldots,N.\]  
Approximation of the differential equation:  
\[R_{k}\frac{\omega_{k+1}-\omega_{k}}{h}+Q_{k}\omega_{k}^{2}=W_{k} \tag{69}\]  
where \(h\)--is the time step size; \(\omega\)--is the angular velocity at the time step \(t_{k}\); \(W_{k}\), \(R_{k}\), and \(Q_{k}\) are the corresponding values at \(t_{k}\). Simplifies to:  
\[\omega_{k+1}=\omega_{k}+h\left(\frac{W_{k}}{R_{k}}-\frac{Q_{k}}{R_{k}}\omega_{ k}^{2}\right) \tag{70}\]  
Angular position update for \(\varphi_{k}\). The angular position \(\varphi(t)\) is related to the angular velocity \(\omega(t)\). As per the equation:  
\[\varphi_{k+1}=\varphi_{k}+h\omega_{k} \tag{71}\]  
This is a simple update based on the angular velocity at each step. For each time step, we iteratively calculate the angular velocity \(\omega_{k}\) and the angular position \(\varphi_{k}\). The total simulation time is divided into \(N\) steps, and the time step \(h\) determines the accuracy of the approximation. Initial conditions are specified at t = 0: \(\omega_{0}=0\) and \(\varphi_{0}=0\). The equation of motion is discretized using the Euler method, leading to the iterative formulas for \(\omega_{k+1}\) and \(\varphi_{k+1}\). Stability of the solution depends on the choice of step size \(h\), with smaller values yielding more accurate results at the cost of increased computational time. The accuracy of the method is first-order, meaning the error decreases quadratically with the step size \(h\). The computational efficiency is an important consideration, especially if the system is large or requires many time steps [2]. This analysis demonstrates a comprehensive approach to solving and interpreting the dynamics of a nonlinear mechanical system using numerical methods [23]. The results highlight the interplay between inertia, resistance, and driving forces, providing valuable insights into the system's behavior and stability under different conditions. The comparative analysis of the driving and resisting moments provides a detailed understanding of the system's dynamics. Through this comparison, one can gain insights into the forces governing system motion, identify potential stability concerns, and optimize the system for performance and efficiency. Numerical simulations offer a powerful tool for visualizing these interactions, and through careful analysis of the results, engineers can make informed decisionsabout system design, energy efficiency, and control strategies.