# 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
Components:  
\[M_{d}\omega_{1}=M_{E}^{r}\omega_{5}+M_{D}^{r}\omega_{6}+M_{F}^{r}\omega_{5}+M_ {G}^{r}\omega_{6} \tag{46}\]7  
\[M_{d}\omega_{1}=M_{E}^{r}\omega_{5}+M_{D}^{r}\omega_{6}+M_{F}^{r}\omega_{5}+M_ {G}^{r}\omega_{6} \tag{46}\]8  
\[M_{d}\omega_{1}=M_{E}^{r}\omega_{5}+M_{D}^{r}\omega_{6}+M_{F}^{r}\omega_{5}+M_ {G}^{r}\omega_{6} \tag{46}\]9  
\[M_{d}=M_{E}^{r}\frac{\omega_{5}}{\omega_{1}}+M_{D}^{r}\frac{\omega_{6}}{\omega _{1}}+M_{F}^{r}\frac{\omega_{5}}{\omega_{1}}+M_{G}^{r}\frac{\omega_{6}}{ \omega_{1}}\]0  
The equation describes the differentiation of the moment of inertia \(I_{k}(\varphi_{1})\) with respect to the angular displacement \(\varphi_{1}\), accounting for various components of the system shown in Appendix B, such as link lengths, angular velocities, and the kinematic ratios between the joints. After simplifying all terms, we obtain the resultant derivative of the moment of inertia with respect to \(\varphi_{1}\). It is equal to:  
\[M_{d}=M_{E}^{r}\frac{\omega_{5}}{\omega_{1}}+M_{D}^{r}\frac{\omega_{6}}{\omega _{1}}+M_{F}^{r}\frac{\omega_{5}}{\omega_{1}}+M_{G}^{r}\frac{\omega_{6}}{ \omega_{1}}\]1Given the initial conditions t = 0, \(\omega_{1}=0\), and \(\varphi_{1}=0\), we aim to solve the equations that describe the system dynamics. Here is a step-by-step guide to proceed with the solution. Solving the Differential Equation of an Oscillating Conveyor Mechanism Using the Approximate Calculation Method  
The problem involves solving a complex nonlinear differential equation for the dynamics of a mechanical system [20], where angular velocity (\(\omega_{1}\)) and angle (\(\varphi_{1}\)) evolve over time under the influence of inertia, resistance, and driving forces. The equation is expressed as:  
\[M_{d}=M_{E}^{r}\frac{\omega_{5}}{\omega_{1}}+M_{D}^{r}\frac{\omega_{6}}{\omega _{1}}+M_{F}^{r}\frac{\omega_{5}}{\omega_{1}}+M_{G}^{r}\frac{\omega_{6}}{ \omega_{1}}\]2  
with \(R(t)\), \(Q(t)\), and \(W(t)\) defined as functions of system parameters, geometry, and time. The solution incorporates initial conditions (t = 0, \(\omega_{1}=0\), and \(\varphi_{1}=0\)) and relies on numerical approximation methods [21]. Where \(R(t)\)--is the represents the combined system inertia and dynamic effects; \(Q(t)\)--is the accounts for quadratic velocity-dependent effects; \(W(t)\)--is the Represents the driving force, including torques and damping effects. The solution requires numerically evaluating \(R(t)\), \(Q(t)\), and \(W(t)\) based on the provided relationships. \[M_{d}=M_{E}^{r}\frac{\omega_{5}}{\omega_{1}}+M_{D}^{r}\frac{\omega_{6}}{\omega _{1}}+M_{F}^{r}\frac{\omega_{5}}{\omega_{1}}+M_{G}^{r}\frac{\omega_{6}}{ \omega_{1}}\]3  
\(R(t)\) is the encapsulates the combined effects of system mass and dynamic configurations. Terms such as \(A(t)\) and \(H(t)\) involve trigonometric and geometric dependencies, linking lengths (\(L_{i}\)), angles (\(\varphi_{i}\)), and oscillatory terms. \(Q(t)\) 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. \(W(t)\) integrates external forces, damping, and oscillatory effects. It is directly influenced by torque (\(M_{r}\)) and damping coefficients (\(\alpha\)). This algorithm (Table 1) uses iterative numerical integration to solve the motion of the oscillating conveyor mechanism [22] over a series of discrete time steps. By calculating the system's dynamics step by step, it provides the angular velocity and displacement for each time step, ultimately leading to the desired system behavior over time. The key steps include updating the velocity and displacement using time-dependent parameters, followed by a check to continue the iteration or end the process based on the number of steps.