# 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{table}
\begin{tabular}{c c} \hline
**Step** & **Description** \\ \hline Inputs & Inputs : \(\omega_{0}\), \(\varphi_{0}\), \(N\), \(h\) \\ \hline Increment n & Set n: = n + 1 \\ \hline Compute A(tn) & Compute A(tn) \\ \hline Compute B(tn) & Compute B(tn) \\ \hline Compute D(tn) & Compute D(tn) \\ \hline Compute E(tn) & Compute E(tn) \\ \hline Compute F(tn) & Compute F(tn) \\ \hline Compute H(tn) & Compute H(tn) \\ \hline Compute N(tn) & Compute N(tn) \\ \hline \end{tabular}
\end{table} 
Table 1: A tabular representation of the steps and their descriptions. The expression for \(A(t)\) is given as:  
\[A(t)=L_{1}^{2}+2L_{1}L_{2}^{s}(\sin\varphi_{1}\sin\varphi_{2}+\cos\varphi_{1}\cos \varphi_{2})u_{21}+{L_{2}^{s}}^{2}u_{21}^{2} \tag{59}\]  
\(A(t)\)--is the contributes to the inertia and energy terms of the system, making it crucial in describing dynamic behaviors  
\[ \begin{array}{c}H(t)=L_{1}^{2}-L_{2}^{2}u_{21}^{2}+\left(a^{2}\cos^{2}\omega_ {5}t+b^{2}\sin^{2}\omega_{5}t\right)u_{51}^{2}+2L_{1}L_{2}u_{21}(\sin\varphi_{1 }\sin\varphi_{2}+\cos\varphi_{1}\cos\varphi_{2})\\ -2L_{1}u_{51}(a\sin\varphi_{1}\cos\omega_{5}t+b\cos\varphi_{2}\sin\omega_{5}t) -2L_{2}u_{21}(a\cos\omega_{5}t\sin\varphi_{2}+b\sin\omega_{5}t\cos\varphi_{2}) \end{array}  \tag{60}\]  
\(H(t)\)--is the governs dynamic inertia contributions from the links, angles, and elliptical motion components. \[u_{51}=u_{61}=\frac{L_{1}\sin(\varphi_{2}-\varphi_{1})}{L_{5}\sin(\varphi_{2}- \varphi_{5})}\]  
\[u_{21}=\frac{L_{1}\sin(\varphi_{5}-\varphi_{1})}{L_{2}\sin(\varphi_{2}- \varphi_{5})}\]  
\[B(t)=I_{S_{2}}\frac{L_{1}^{2}\sin^{2}(\varphi_{5}-\varphi_{1})}{L_{2}^{2}\sin ^{2}(\varphi_{2}-\varphi_{5})} \tag{61}\]  
\(B(t)\)--is the representation of the rotational effect of link 2 influenced by angular positions \(\varphi_{5}\), \(\varphi_{1}\), \(\varphi_{2}\). \[D(t)=(I_{\rm SF}+I_{\rm 6G})\frac{L_{1}^{2}\sin^{2}(\varphi_{2}-\varphi_{1})}{L_{ 5}^{2}\sin^{2}(\varphi_{2}-\varphi_{5})} \tag{62}\]  
\(D(t)\)--is the account for the combined inertial effects of links 5 and 6 modulated by relative angles \(\varphi_{1}\), \(\varphi_{2}\), \(\varphi_{5}\). \[Q(t)=(m_{2}E(t)+m_{3}F(t))2\omega_{1}^{2}+2I_{S_{2}}\frac{L_{1}^{2}}{L_{2}^{2} }N(t)+M(t) \tag{63}\]  
\(Q(t)\)--are the accounts for quadratic velocity-dependent effects. \(Q(t)\) represents quadratic velocity-dependent effects and includes contributions from terms \(E(t)\), \(F(t)\), \(N(t)\) and \(M(t)\), which are influenced by angular relationships and material properties. \begin{table}
\begin{tabular}{c c} \hline
**Step** & **Description** \\ \hline Compute M(tn) & Compute M(tn) \\ \hline Compute Q(tn) & Compute Q(tn) \\ \hline Compute R(tn) & Compute R(tn) \\ \hline Compute W(tn) & Compute W(tn) \\ \hline Update \(\omega_{n+1}\) & \(\omega_{n+1}=\omega_{n}+h(\rm{Wn/Rn-Qn/Rn\omega_{n}})\) \\ \hline Update \(\varphi_{n+1}\) & \(\varphi_{n+1}=\varphi_{n}+\rm{h}\omega_{n}\) \\ \hline Check if n \(<\) N & Check if n \(<\) N \\ \hline Yes : Update \(\omega_{n}\), \(\varphi_{n}\) and Repeat & If \(\rm{\gamma}\rm{Yes}^{\prime}\), Update \(\omega_{n}\rightarrow\omega_{n+1}\), \(\varphi_{n}\rightarrow\varphi_{n+1}\) \\  & and Repeat \\ \hline No : Output \(\omega_{n+1}\), \(\varphi_{n+1}\) & If \(\rm{\gamma}\rm{No}^{\prime}\), Output final \(\omega_{n+1}\), \(\varphi_{n+1}\) \\ \hline End & End of process \\ \hline \end{tabular}
\end{table} 
Table 1: Cont. \[ \begin{array}{c}E(t)=(L_{1}\sin\varphi_{1}+L_{2}^{*}\sin\varphi_{2}u_{21})\left(L_ {1}\cos\varphi_{1}+L_{2}^{*}\cos\varphi_{2}u_{21}^{2}+L_{2}^{*}\sin\varphi_{2} \frac{du_{21}}{d\varphi_{1}}\right)\\ \hskip 142.26378 pt+(L_{1}\cos\varphi_{1}+L_{2}^{*}\cos\varphi_{2}u_{21})\left(-L _{1}\sin\varphi_{1}-L_{2}^{*}\sin\varphi_{2}u_{21}^{2}+L_{2}^{*}\cos\varphi_{2 }\frac{du_{21}}{d\varphi_{1}}\right)\end{array}  \tag{64}\]  
\(E(t)\)--is the energy term tied to angular positions and coupling ratios. \[