# Determination of Kinematic and Dynamic Characteristics of Oscillating Conveyor Mechanism
## 3 Methodology  
The analysis of the vibration conveyor mechanism was conducted in three main stages: kinematic analysis, dynamic analysis [24], and numerical approximation [17]. The methodologies applied at each stage are as follows:  
1. Kinematic analysis. The kinematic analysis focused on describing the motion of the system's components without considering the forces or torques causing the motion: (a) System modeling--the conveyor mechanism was modeled as a system of interconnected rigid links with specified lengths (\(L_{1}\), \(L_{2}\), \(\ldots\)) and joints. Angular displacements (\(\varphi_{1}\), \(\varphi_{2}\), \(\ldots\)) were used to describe the positions of the links relative to reference points. (b) Geometric and trigonometric relationships--constraints between the links were expressed as equations involving sines and cosines of the angular variables. Relationships such as \(L_{1}cos(\varphi_{1})+L_{2}cos(\varphi_{2})=L_{3}cos(\varphi_{3})\) were used to ensure consistency in the system's geometry. (c) Velocities and accelerations--the angular velocities and angular accelerations were derived using time differentiation of the position equations. The motion constraints were used to compute velocities and accelerations for all components. (d) Visualization--3D plots of angular displacement, velocity, and acceleration were generated to illustrate the motion behavior over time. 2. Dynamic analysis. The dynamic analysis incorporated the forces and torques acting on the system to evaluate its response under load. (a) Derivation of equations of motion--[PERSON]'s Second Law and the principle of virtual work were applied to derive equations governing the system's dynamics. Inertia terms \(I(\varphi_{i})\) were computed using the mass distribution and geometry of the system. (b) Force and torque calculations--driving torques \(M_{d}\) and resisting torques \(M_{r}\) were modeled as functions of angular velocity \(\omega_{1}\) and angular displacement \(\varphi_{6}\). External factors, such as damping and load resistance, were incorporated into the dynamic equations. (c) Nonlinear effects--nonlinear terms, such as those involving products of angular velocities and displacements, were explicitly included to account for real-world interactions. (d) System stability--stability conditions were analyzed by evaluating the balance of forces and torques during operation. Parameters like stiffness K and base moments \(M_{r0}\) were studied to ensure system stability. 3. Numerical approximation. The numerical approximation was used to solve the nonlinear differential equations derived in the dynamic analysis. (a) Differential equation formulation--the equations were expressed in the standard form: \[R(t)\frac{d\omega_{1}}{dt}+Q(t)\omega^{2}(t)=W(t)\] where \(R(t)\), \(Q(t)\), and \(W(t)\) are time-dependent coefficients. (b) Discretization--[PERSON]'s method was employed to approximate the derivatives using finite time steps \(h\): \[\omega_{k+1}=\omega_{k}+h\left(\frac{W_{k}}{R_{k}}-\frac{Q_{k}}{R_{k}}\omega_{ k}^{2}\right)\] angular displacement was updated iteratively: \[\varphi_{k+1}=\varphi_{k}+h\omega_{k}\] (c) Initial conditions--Initial values for angular displacement \(\varphi_{1}=0\) and angular velocity \(\omega_{1}=0\) were specified at t = 0. (d) Simulation--Time was discretized into intervals, and the system's behavior was computed step by step. Parameters such as \(A(t)\), \(B(t)\), \(D(t)\), \(E(t)\), \(F(t)\), \(H(t)\), \(N(t)\), \(M(t)\), \(Q(t)\), \(R(t)\), and \(W(t)\) were evaluated at each time step. E) Validation and visualization--the numerical results were compared with theoretical expectations to ensure accuracy. Time-series and 3D plots of dynamic variables were generated for visualization. Due to the complexity of the equation, analytical solutions are impractical. Numerical integration (Runge-Kutta method) was employed, allowing for accurate computation of the time evolution of \(\varphi_{1}(t)\) and \(\omega_{1}(t)\).