# Determination of Kinematic and Dynamic Characteristics of Oscillating Conveyor Mechanism
###### Abstract  
This research focuses on the dynamic analysis of an oscillating conveyor mechanism using numerical methods to solve nonlinear differential equations that govern its motion. The system under study is modeled by a second-order differential equation of the form \(R(t)\frac{d\omega_{1}}{dt}+Q(t)\omega_{1}^{2}(t)=W(t)\), where \(R(t)\), \(Q(t)\), and \(W(t)\) are time-dependent functions representing system parameters such as resistance, damping, and external driving forces. To solve these equations, we employed a numerical approach based on [PERSON]'s method, which discretizes the time domain into small steps \(h\) and approximates the derivatives of angular velocity and angular displacement. The angular velocity \(\omega_{k+1}\) and angular displacement \(\varphi_{k+1}\) are updated iteratively using the formulas \(\omega_{k+1}=\omega_{k}+h\Big{(}\frac{W_{k}}{R_{k}}-\frac{Q_{k}}{R_{k}}\omega _{k}^{2}\Big{)}\) and \(\varphi_{k+1}=\varphi_{k}+h\omega_{k}\), respectively. Initial conditions, with \(\omega_{0}=0\) and \(\varphi_{0}=0\), were specified, and the system was simulated over a specified time range divided into \(N\) time steps. In the simulation, key 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 based on the system's geometry and the angular displacements. Due to the complexity of the system, analytical solutions were impractical, so the Runge-Kutta method was employed for higher accuracy in the integration process. The results from the numerical simulations were validated by comparing them with theoretical expectations, and the system's dynamic behavior was visualized using time-series and 3D plots. The simulation demonstrated that the system's stability and accuracy were highly dependent on the time step \(h\), with smaller values providing more precise results at the cost of increased computational time. The research confirms the applicability of numerical methods in solving complex nonlinear differential equations for dynamic systems and provides insights into the system's behavior under various operating conditions. m 2025 15 1676 1 2025 precise control of material flow [2]. The performance and reliability of oscillating conveyor systems depend heavily on their kinematic and dynamic characteristics, which govern their motion, stability, and energy consumption. Understanding these characteristics is crucial for optimizing system design, enhancing operational efficiency, and ensuring long-term durability [3]. In the study of dynamic systems, particularly for oscillating conveyor mechanisms, accurately modeling and predicting the system's behavior is critical for optimizing performance and ensuring stability. These systems are typically governed by nonlinear differential equations that describe the rotational motion of components such as angular velocities and displacements [4]. Due to the inherent complexity and nonlinearities, obtaining analytical solutions to these equations is often impractical. As a result, numerical methods have become essential tools for approximating the behavior of such systems and providing valuable insights into their dynamics. While numerous studies have utilized numerical methods for dynamic analysis, much of the existing work primarily focuses on simple systems or uses basic methods, often lacking higher-order accuracy or the ability to handle more complex system configurations [5]. The main contribution of this work lies in the development and application of a comprehensive numerical approach that enhances the accuracy and efficiency of dynamic simulations for complex oscillating systems. Specifically, we employ [PERSON]'s method for discretizing the differential equations and Runge-Kutta methods for higher-order integration. In addition to standard numerical techniques, this study introduces several innovative contributions [6], both in the formulation and implementation of the approach.