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 $R\left(t\right){\displaystyle \frac{d{\omega }_1}{dt}}+Q\left(t\right){\omega }_1^2\left(t\right)=W\left(t\right)$, 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 Euler’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 ${\phi }_{k+1}$ are updated iteratively using the formulas ${\omega }_{k+1}={\omega }_k+h\left({\displaystyle \frac{W_k}{R_k}}-{\displaystyle \frac{Q_k}{R_k}}{\omega }_k^2\right)$ and ${\phi }_{k+1}={\phi }_k+h{\omega }_k$, respectively. Initial conditions, with ${\omega }_0=0$ and ${\phi }_0=0$, were specified, and the system was simulated over a specified time range divided into N time steps. In the simulation, key parameters such as $A\left(t\right)$, B(t), D(t), E(t), F(t), $H\left(t\right)$, N(t), M(t), $Q\left(t\right)$, $R\left(t\right)$, and $W\left(t\right)$ 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.

1. Introduction

Oscillating conveyor mechanisms are widely used in various industries, such as mining, food processing, and manufacturing, for the transportation and handling of bulk materials. These systems are essential for ensuring efficient [1], continuous movement of materials along a predefined path, often over long distances or through complex processes. The key advantage of oscillating conveyors lies in their ability to minimize material degradation while maintaining high throughput, especially in applications that require 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 Euler’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. In the field of mechanical systems, particularly for oscillating conveyor mechanisms, understanding and predicting the system’s dynamic behavior is essential for optimizing performance and ensuring stability. These systems are typically governed by nonlinear differential equations that describe the rotational motion of various components, such as angular velocities and displacements [7]. Analytical solutions to these equations are often difficult or impractical to obtain, especially when the system is complex and involves time-dependent forces. Therefore, numerical methods become indispensable for approximating the behavior of such systems. This study aims to analyze the dynamic behavior of an oscillating conveyor mechanism by solving the nonlinear differential equations that govern its motion. The system is modeled by a second-order differential equation [8]. To solve this equation, we employ numerical methods, specifically the Euler method for discretization. In this approach, the system’s time domain is divided into small intervals, and the derivatives are approximated using finite differences. The simulation begins with the initial conditions ${\omega }_0=0$ and ${\phi }_0=0$ and proceeds over a specified time interval divided into N steps. The time-dependent coefficients R_k, Q_k, and W_k are computed at each step based on the system parameters. Given the complexity of the system, analytical solutions are often impractical, so we also employ the Runge–Kutta method to achieve higher-order accuracy [9]. This method allows for more precise numerical integration, improving the overall stability and convergence of the solution. The primary goal of this study is to investigate the dynamic behavior of the oscillating conveyor mechanism by calculating the angular velocity ${\omega }_1\left(t\right)$ and angular displacement ${\phi }_1\left(t\right)$ over time [10]. The results of the simulation are validated by comparing the numerical predictions with theoretical expectations, and the system’s performance is visualized through time-series plots and 3D graphs. Through this approach, we demonstrate the applicability of numerical methods in solving complex nonlinear differential equations for dynamic systems. The results not only provide insights into the behavior of oscillating conveyors under various operating conditions but also illustrate the trade-offs between computational accuracy and efficiency in such simulations. In the introduction, we thought it appropriate to briefly explain the innovative contributions of the authors. While many studies rely solely on Euler’s method, which is first-order accurate, this work introduces the use of the Runge–Kutta method alongside Euler’s method for better accuracy. The combination of these methods ensures that the trade-off between computational efficiency and solution precision is optimized, especially for stiff systems. The Runge–Kutta method allows for the efficient simulation of complex, nonlinear dynamics while minimizing errors associated with simple Euler-based integration. The research introduces a novel way of modeling time-dependent system parameters, such as resistance, damping, and external forces. The terms R(t), Q(t), and W(t) are dynamically updated at each time step based on the evolving state of the system, enabling more realistic simulations. This dynamic parameterization ensures that the numerical simulation reflects the time-varying nature of the forces and interactions within the system. One of the key innovations of this work is the handling of nonlinear interactions within the system. The nonlinear terms, such as ${\omega }_k^2$, are carefully integrated using advanced numerical methods, allowing the model to account for complex feedback mechanisms and their effects on system behavior [11]. This level of detail in simulating nonlinearities provides more accurate and reliable predictions for real-world systems. The work incorporates an iterative time-step approach that adjusts for computational stability. By iterating over small time steps and adjusting parameters dynamically, the method ensures that the system remains stable under various operating conditions. The careful selection of the time step h, along with stability considerations, marks an important advancement in handling large-scale dynamic systems with multiple interacting components. To validate and present the simulation results, this study emphasizes the generation of both time-series plots and 3D visualizations of dynamic variables, such as angular velocity and angular displacement. This approach not only aids in understanding the evolution of the system over time but also provides an intuitive way to communicate the complex behavior of the system to both researchers and engineers. Another innovative aspect of this research is the systematic validation of numerical results against theoretical models [12]. The authors have gone beyond simply generating simulations; they have rigorously compared the results with existing analytical solutions (when possible) and theoretical expectations. This validation process ensures that the proposed method is not only accurate but also applicable to real-world systems. The numerical simulations yielded accurate predictions of the system’s angular velocity ${\omega }_1\left(t\right)$ and angular displacement ${\phi }_1\left(t\right)$ over time. By adjusting the time step h, the simulations were able to capture the system’s dynamic response with high precision. The Runge–Kutta method, particularly for systems involving nonlinear terms, significantly improved the accuracy compared to simple Euler integration [13]. The results were validated against theoretical expectations, confirming the correctness and reliability of the numerical model. Moreover, the time-series and 3D plots of the system’s dynamic behavior provided valuable insights into the oscillatory nature of the conveyor mechanism. This work makes significant contributions to the numerical analysis of oscillating conveyor mechanisms. By combining advanced numerical methods such as Euler’s and Runge–Kutta, along with a careful parameterization of time-dependent forces, the authors have enhanced the accuracy, stability, and computational efficiency of dynamic simulations for nonlinear systems. The results not only contribute to a deeper understanding of the system’s behavior but also provide a robust framework for simulating complex dynamic systems in mechanical engineering applications.

2. Materials and Methods

2.1. Geometric and Trigonometric Constraints

The following geometric constraints are applied to ensure proper motion of the mechanism [14]. These equations describe the geometric relationships between the positions and angles of the mechanism components shown in Figure 1.

Parameters of the six-link oscillating conveyor mechanism:

L₁ = OA = 60 mm, L₂ = AB = 430 mm, L_k = BC = 82 mm, L₃ = CE = 642 mm, L₄ = CD = 1058 mm, L₅ = L₆ = EF = DG = 440 mm, ED = FG = 1700 mm, m₁ = 3.0 kg, m₂ = 6.5 kg, m₃ = 11.60 kg, m₅ = m₆ = 5.9 kg.

For closed contour OABCEFO:

$$L_1+L_2+L_k+L_3=L_7+L_5$$

(1)

And for closed contour OABCEGO:

$$L_1+L_2+L_k=L_8+L_6+L_4$$

(2)

These constraints maintain the equilibrium of the system during operation. In addition, trigonometric relationships are used to describe the motion angles ${\phi }_1={\phi }_{1}t$, which vary as functions of time. The determination of the kinematic and dynamic parameters for the oscillating conveyor mechanism involves solving a system of nonlinear equations that describe the motion of the mechanism’s components based on given constraints, link lengths, masses, and angular velocities [15].

The vector Equations (1) and (2) are projected on the Cartesian reference system. It is obtained:

$$L_{1}cos{\phi }_1+L_{2}cos{\phi }_2+L_{3}cos{\phi }_3=L_{7}cos{\phi }_7+L_{5}cos{\phi }_5$$

(3)

$$L_{1}sin{\phi }_1+L_{2}sin{\phi }_2+L_k+L_{3}sin{\phi }_3=L_{7}sin{\phi }_7+L_{5}sin{\phi }_5$$

(4)

$$L_{1}cos{\phi }_1+L_{2}cos{\phi }_2=L_{8}cos{\phi }_8+L_{6}cos{\phi }_6+L_{4}cos{\phi }_4$$

(5)

$$L_{1}sin{\phi }_1+L_{2}sin{\phi }_2+L_k=L_{8}sin{\phi }_8+L_{6}sin{\phi }_6+L_{4}sin{\phi }_4$$

(6)

Additional constraints (assuming ${\phi }_3={\phi }_4=180°, {\phi }_k=90°, L_5=L_6,$ and ${\phi }_5={\phi }_6$):

$$L_{2}cos{\phi }_2-L_{5}cos{\phi }_5=L_{7}cos{\phi }_7-L_{1}cos{\phi }_1+L_3$$

(7)

$$L_{2}sin{\phi }_2-L_{5}sin{\phi }_5=L_{7}sin{\phi }_7-L_{1}sin{\phi }_1-L_k$$

(8)

Since $L_5=L_6$ and ${\phi }_5={\phi }_6$, you can substitute those relationships to reduce the complexity:

$$L_{2}cos{\phi }_2-L_{5}cos{\phi }_5=L_{8}cos{\phi }_8-L_{1}cos{\phi }_1-L_4$$

(9)

$$L_{2}sin{\phi }_2-L_{5}sin{\phi }_5=L_{8}sin{\phi }_8-L_{1}sin{\phi }_1-L_k$$

(10)

Combine Equations (7) and (9):

$$L_{7}cos{\phi }_7-L_{1}cos{\phi }_1+L_3=L_{8}cos{\phi }_8-L_{1}cos{\phi }_1-L_4$$

(11)

Combine Equations (8) and (10):

$$L_{7}sin{\phi }_7-L_{1}sin{\phi }_1-L_k=L_{8}sin{\phi }_8-L_{1}sin{\phi }_1-L_k$$

(12)

Simplified System:

$$L_{7}cos{\phi }_7+L_3=L_{8}cos{\phi }_8-L_4$$

(13)

$$L_{7}sin{\phi }_7=L_{8}sin{\phi }_8$$

(14)

Using trigonometric identities:

$$L_7^2={\left(L_{7}cos{\phi }_7\right)}^2+{\left(L_{7}sin{\phi }_7\right)}^2$$

(15)

$$L_8^2={\left(L_{8}cos{\phi }_8\right)}^2+{\left(L_{8}sin{\phi }_8\right)}^2$$

(16)

Express ${\phi }_8$ in terms of ${\phi }_7$. From $sin{\phi }_7={\displaystyle \frac{L_{8}sin{\phi }_8}{L_7}}$ and $cos{\phi }_7={\displaystyle \frac{L_{8}cos{\phi }_8}{L_7}}$, for ${\phi }_8$ we solve using:

$$tan{\phi }_7={\displaystyle \frac{sin{\phi }_7}{cos{\phi }_7}}\mathrm{and} tan{\phi }_8={\displaystyle \frac{sin{\phi }_8}{cos{\phi }_8}}$$

(17)

The equations describe the geometric and kinematic constraints of the oscillating conveyor mechanism [16], now grouped into simplified forms. Let us organize them and analyze for clarity and solvability.

$$\begin{gathered} tg\alpha ={\displaystyle \frac{OM}{FM}}={\displaystyle \frac{40}{1325}}=0.03, \alpha =1°{40}^{\prime }, {\phi }_7=180°+1°{40}^{\prime }=181°{40}^{\prime }, tg\beta ={\displaystyle \frac{MO}{MG}}={\displaystyle \frac{40}{375}}=0.1066, \\ \beta =6°, {\phi }_8=360°-6°=354°, L_3=642 \mathrm{mm}, L_4=1058 \mathrm{m}\mathrm{m}, L_7=1325.6 \mathrm{mm}, L_8=337.1 \mathrm{mm}. \end{gathered}$$

A and B are determined Equations (11) and (12) directly by ${\phi }_7$:

$$A=L_{7}cos{\phi }_7-L_{1}cos{\phi }_1+L_3$$

(18)

$$B=L_{8}sin{\phi }_8-L_{1}sin{\phi }_1-L_k$$

(19)

These equation link ${\phi }_2$ and ${\phi }_5$ using A and B Equations (9) and (10):

$$L_{2}cos{\phi }_2-L_{5}cos{\phi }_5=A$$

(20)

$$L_{2}sin{\phi }_2-L_{5}sin{\phi }_5=B$$

(21)

From the provided constraints:

$$sin{\phi }_2={\displaystyle \frac{B+L_{5}sin{\phi }_5}{L_2}}$$

(22)

Using $si{n}^2{\phi }_2+co{s}^2{\phi }_2$ = 1, substitute $sin{\phi }_2$:

$$1-co{s}^2{\phi }_2={\left({\displaystyle \frac{B+L_{5}sin{\phi }_5}{L_2}}\right)}^2$$

(23)

Rearranging for $co{s}^2{\phi }_2$:

$$co{s}^2{\phi }_2=1-{\displaystyle \frac{{\left(B+L_{5}sin{\phi }_5\right)}^2}{L_2^2}}$$

(24)

Now, $sin{\phi }_2$ becomes:

$$cos{\phi }_2=\sqrt{{\displaystyle \frac{L_2^2-{\left(B+L_{5}sin{\phi }_5\right)}^2}{L_2^2}}}$$

(25)

The constraint linking ${\phi }_2$ and ${\phi }_5$ is:

$$L_{2}cos{\phi }_2-L_{5}cos{\phi }_5=A$$

Substitute $cos{\phi }_2$ into this equation:

$$L_2\sqrt{{\displaystyle \frac{L_2^2-{\left(B+L_{5}sin{\phi }_5\right)}^2}{L_2^2}}}-L_{5}cos{\phi }_5=A$$

(26)

Rearranging for $cos{\phi }_5$:

$$cos{\phi }_5={\displaystyle \frac{L_2\sqrt{\frac{L_2^2-{\left(B+L_{5}sin{\phi }_5\right)}^2}{L_2^2}}-A}{L_5}}$$

(27)

Using $si{n}^2{\phi }_5+co{s}^2{\phi }_5$ = 1, and $co{s}^2{\phi }_5$ = 1 − $si{n}^2{\phi }_5$:

Squaring both sides of the expression for $cos{\phi }_5$ gives:

$${\left({\displaystyle \frac{L_2\sqrt{\frac{L_2^2-{\left(B+L_{5}sin{\phi }_5\right)}^2}{L_2^2}}-A}{L_5}}\right)}^2=1-si{n}^2{\phi }_5$$

(28)

Simplifying:

Let $C={\displaystyle \frac{L_2^2}{2A}}-{\displaystyle \frac{B^2}{2A}}+{\displaystyle \frac{A}{2}}-{\displaystyle \frac{L_5^2}{2A}}$. Expanding the terms results in:

$$L_2^2-{\left(B+L_{5}sin{\phi }_5\right)}^2=C-{\displaystyle \frac{B}{A}}{L}_{5}sin{\phi }_5$$

(29)

Substitute back to form a quadratic Equation (29) in $sin{\phi }_5$:

$$Msi{n}^2{\phi }_5+Nsin{\phi }_5+P=0$$

(30)

where $M=L_5^2\left(1+{\displaystyle \frac{B^2}{A^2}}\right)$, $N=2B{L}_5\left(1-{\displaystyle \frac{C}{A}}\right)$, $P=B^2+C^2-L_2^2$

The quadratic formula gives:

$$sin{\phi }_5={\displaystyle \frac{-N\pm \sqrt{N^2-4MP}}{2M}}=W, W=W\left({\phi }_1\right)$$

(31)

Once $sin{\phi }_5$ is determined, ${\phi }_5$ can be expressed as:

$${\phi }_5=arcsinW+2\pi n, {\phi }_5=f\left({\phi }_1\right)$$

Phase angles relationships:

$$L_{2}sin{\phi }_2-L_{5}sin{\phi }_5=B$$

$$L_{2}sin{\phi }_2-L_{5}W=B$$

$$sin{\phi }_2={\displaystyle \frac{\left(B+L_{5}W\right)}{L_2}}$$

(32)

$${\phi }_2=arcsin\left({\displaystyle \frac{B+L_{5}W}{L_2}}+2\pi n\right), {\phi }_2=f\left({\phi }_1\right)$$

where ${\phi }_5$ = f(${\phi }_1$). Substituting ${\phi }_5$(${\phi }_1$) into this equation gives ${\phi }_2$ as a function of ${\phi }_1$.

Derivatives, to compute ${\displaystyle \frac{d{\phi }_2}{d{\phi }_1}}$:

$${\displaystyle \frac{d{\phi }_2}{d{\phi }_1}}={\displaystyle \frac{\frac{d{\phi }_2}{dt}}{\frac{d{\phi }_1}{dt}}}={\displaystyle \frac{{\omega }_2}{{\omega }_1}}=u_{21}$$

(33)

Using the chain rule and differentiating ${\phi }_2$(${\phi }_1$) explicitly will yield the angular velocity ratio $u_{21}$.

Derivatives, to compute ${\displaystyle \frac{d{\phi }_5}{d{\phi }_1}}$:

$${\displaystyle \frac{d{\phi }_5}{d{\phi }_1}}={\displaystyle \frac{\frac{d{\phi }_5}{dt}}{\frac{d{\phi }_1}{dt}}}={\displaystyle \frac{{\omega }_5}{{\omega }_1}}=u_{51}$$

(34)

For the given ratios:

$$u_{21}={\displaystyle \frac{L_{1}sin\left({\phi }_5-{\phi }_1\right)}{L_{2}sin\left({\phi }_2-{\phi }_5\right)}}$$

(35)

$$u_{51}={\displaystyle \frac{L_{1}sin\left({\phi }_2-{\phi }_1\right)}{L_{5}sin\left({\phi }_2-{\phi }_5\right)}}$$

(36)

Each $u_{ij}$ depends on the geometric and trigonometric relationships between the angles ${\phi }_1$, ${\phi }_2$, ${\phi }_5$. These relationships can be used to calculate the respective angular velocities.

Angular velocity ratios:

$${\displaystyle \frac{{\omega }_2}{{\omega }_1}}=u_{21}={\displaystyle \frac{L_{1}sin\left({\phi }_5-{\phi }_1\right)}{L_{2}sin\left({\phi }_2-{\phi }_5\right)}}$$

$${\displaystyle \frac{{\omega }_5}{{\omega }_1}}=u_{51}={\displaystyle \frac{L_{1}sin\left({\phi }_2-{\phi }_1\right)}{L_{5}sin\left({\phi }_2-{\phi }_5\right)}}$$

Similarly, we substitute the corresponding expressions for $u_{31}$, $u_{41}$, $u_{61}$.

Noting that the center of gravity of the third-generation changes according to the law of ellipticity shown in Figure 2, the equations for the elliptical motion can be expressed as:

$$RC=a_0+asin{\omega }_5$$

(37)

$$RS=b_0+bcos{\omega }_5$$

(38)

RC—is the horizontal position of the center of gravity, varying sinusoidally over time; RS—is the vertical position of the center of gravity, varying cosinusoidally over time; $a_0$ and $b_0$—is the initial positions (offsets) of the center of gravity in the horizontal and vertical directions, respectively; $a$ and b—is the amplitudes of the elliptical motion in the horizontal and vertical directions, respectively; ${\omega }_5$—is the angular frequency of the motion [17], determining how quickly the center of gravity traverses the elliptical path; t—is the time variable.

The trajectory described by RC and RS forms an ellipse with:

-

Semi-major axis $a$ (horizontal extent);

-

Semi-minor axis b (vertical extent);

-

Offset determined by $a_0$ and $b_0$, which places the ellipse at a specific location in the coordinate system.

This formulation is often used to describe oscillatory systems with elliptical trajectories, such as oscillating conveyors or mechanisms with rotational and translational motions.

The equations you provided for $X_{S2}$, $Y_{S2}$, $X_{S3}$, and $Y_{S3}$ define the coordinates of two points (s2 and s3) in terms of the oscillatory motion variables (RC and RS) and the angular parameters (${\phi }_1$, ${\phi }_2$):

$$X_{S2}=-L_{1}cos{\phi }_1-L_2^{*}cos{\phi }_2$$

(39)

$$Y_{S2}=L_{1}sin{\phi }_1+L_2^{*}sin{\phi }_2$$

(40)

These equations represent the coordinates of a point s2 that depend on the lengths $L_1$ and $L_2^{*}={\displaystyle \frac{L_2}{2}}=As2$, as well as the angles ${\phi }_1$ and ${\phi }_2$. This is a purely kinematic representation, without any influence of oscillatory displacements.

$$\dot{X_{S2}} = L_{1}sin{\phi }_1\frac{d{\phi }_1}{dt}+L_2^{*}sin{\phi }_2\frac{d{\phi }_2}{dt} = L_{1}sin{\phi }_1\frac{d{\phi }_1}{dt}+L_2^{*}sin{\phi }_2\frac{d{\phi }_2}{dt}\frac{d{\phi }_1}{dt} = \left(L_{1}sin{\phi }_1+L_2^{*}sin{\phi }_2{u}_{21}\right){\omega }_1$$

(41)

$$\dot{Y_{S2}}=L_{1}cos{\phi }_1{\displaystyle \frac{d{\phi }_1}{dt}}+L_2^{*}cos{\phi }_2{\displaystyle \frac{d{\phi }_2}{dt}}=L_{1}cos{\phi }_1{\displaystyle \frac{d{\phi }_1}{dt}}+L_2^{*}cos{\phi }_2{\displaystyle \frac{d{\phi }_2}{dt}}{\displaystyle \frac{d{\phi }_1}{dt}}=\left(L_{1}cos{\phi }_1+L_2^{*}cos{\phi }_2{u}_{21}\right){\omega }_1$$

(42)

To determine the magnitude of velocity ${\vartheta }_{S2}$, we use the formula for the resultant velocity from its components along the X- and Y-directions:

$${\vartheta }_{S2}=\sqrt{{\left(\dot{X_{S2}}\right)}^2+{\left(\dot{Y_{S2}}\right)}^2}={\omega }_1\sqrt{L_1^2+L_1{L}_2^{*}{u}_{21}cos\left({\phi }_1-{\phi }_2\right)+2L_2^{*2}{u}_{21}^2}$$

(43)

To determine ${\vartheta }_{S3}$, the velocity magnitude of point s3, we follow a similar approach to earlier calculations. The velocity components in the X- and Y-directions are given by. To calculate ${\vartheta }_{S3}$, the magnitude of the velocity at point s3, given the components $\dot{X_{S3}}$ and $\dot{Y_{S3}}$, we follow these steps:

$$\begin{array}{l}{\vartheta }_{S3} = \sqrt{{\left(\dot{X_{S3}}\right)}^2+{\left(\dot{Y_{S3}}\right)}^2} \\ = {\omega }_1^2\{{\left(L_{1}sin{\phi }_1+L_2^{*}sin{\phi }_2{u}_{21}\right)}^2+{\left(L_{1}cos{\phi }_1+L_2^{*}cos{\phi }_2{u}_{21}\right)}^2\\ -2\left(L_{1}sin{\phi }_1+L_2^{*}sin{\phi }_2{u}_{21}\right)a{u}_{51}cos\left({\omega }_{5}t\right)-2\left(L_{1}cos{\phi }_1+L_2^{*}cos{\phi }_2{u}_{21}\right)b{u}_{51}sin\left({\omega }_{5}t\right)\\ +{\left(a{u}_{51}cos\left({\omega }_{5}t\right)\right)}^2+{\left(b{u}_{51}sin\left({\omega }_{5}t\right)\right)}^2\}\end{array}$$

(44)

Equations gives the rate of change of the magnitude of the velocity ${\vartheta }_{S2}$ and ${\vartheta }_{S3}$ with respect to ${\phi }_1$, considering both the geometric terms and the influence of $u_{21}$ and $u_{51}$ and its derivative shown in Appendix A.

2.2. Differential Equations of Motion of the Joints of an Oscillating Conveyor Mechanism

Equation for joint 1 motion:

$$I_d\left({\phi }_1\right){\displaystyle \frac{d{\omega }_1}{dt}}={\displaystyle \frac{{\omega }_1^2}{2}}{\displaystyle \frac{d{{\rm I}}_d\left({\phi }_1\right)}{d{\phi }_1}}=M_d-M_r$$

(45)

where $M_d$—is the driving torque; $M_r$—is the resisting torque.

The driving torque on the joint is related to the torques from different components (such as $M_E^r$, $M_F^r$, $M_D^r$, and $M_G^r$):

Where $M_E^r$, $M_F^r$, $M_D^r$, and $M_G^r$ is the are the resisting torques from different parts of the system; ${\omega }_5$ and ${\omega }_6$ are the angular velocities of joints 5 and 6, respectively [18].

$$M_d{\omega }_1=M_E^r{\omega }_5+M_D^r{\omega }_6+M_F^r{\omega }_5+M_G^r{\omega }_6$$

(46)

By substituting and factoring the terms in the equation for $M_d$, we get:

$$M_d=M_E^r{\displaystyle \frac{{\omega }_5}{{\omega }_1}}+M_D^r{\displaystyle \frac{{\omega }_6}{{\omega }_1}}+M_F^r{\displaystyle \frac{{\omega }_5}{{\omega }_1}}+M_G^r{\displaystyle \frac{{\omega }_6}{{\omega }_1}}$$

Which can be simplified further to:

$$M_d=M_E^r{u}_{51}+M_D^r{u}_{61}+M_F^r{u}_{51}+M_G^r{u}_{61}=4u_{51}{M}_r$$

(47)

where $u_{51}$ and $u_{61}$ are terms relating to the motion of the system, possibly dependent on the geometry or kinematic relationships between different components.

The motion of the oscillating conveyor mechanism is driven by the external torques $M_d$, which depend on the angular velocities of different parts of the system. These equations couple the rotational dynamics of joint 1 with the other components in the system.

$$M_r=M_{r0}+K{\phi }_6$$

(48)

$M_{r0}$—is the base moment of the applied resistance force (perhaps considering some other factors), $M_{r0}=-2.98$ and this value represents the inherent resisting torque in the system at ${\phi }_6=0$; K—is the stiffness coefficient related to the change in $M_r$ with respect to ${\phi }_6$. A higher K indicates a steeper increase in resistance torque as ${\phi }_6$ increases; ${\phi }_6$—is the parameter (angle) representing the system’s configuration or deformation, directly affecting $M_r$.

Expression for K:

$$K=tg\left(\alpha \right)={\displaystyle \frac{14.86-5.94}{2°-1°}}={\displaystyle \frac{8.92}{0.017453}}=511.84$$

This constant K is calculated using the tangent of angle α, and it equals 511.84, which plays a key role in determining how the driving torque depends on ${\phi }_6$.

The formula for $M_r$ is given by:

$${\phi }_6={\phi }_6\left({\phi }_1\right), M_r=M_r\left({\phi }_1\right); M_r=-2.98+511.87{\phi }_6$$

where $M_{r0}$—is the expressed as a linear function of ${\phi }_6$; the constant −2.98 represents an offset, and 511.84 is the sensitivity factor, which amplifies the effect of ${\phi }_6$ on the driving torque. At ${\phi }_6=0$, the resisting torque $M_r$ equals −2.98, representing a baseline resistance. As ${\phi }_6$ increases, $M_r$ grows linearly due to the stiffness coefficient K. This behavior reflects how the resistance increases proportionally to the angular displacement ${\phi }_6$.

Using the expression for $M_{d0}$, the driving torque $M_d$ becomes. The driving torque $M_d$ is expressed as:

$$M_d=M_{d_o}-{\alpha }_0{\omega }_1$$

(49)

where ${\alpha }_0$—is the coefficient representing the rate at which driving torque decreases with increasing angular velocity. This accounts for damping effects, ${\omega }_1$—is the angular velocity of joint 1, $M_{d_o}$—is the base moment of the applied driving force $\left(M_{d_o}=2.92\right)$. This is the driving torque when the angular velocity ${\omega }_1=0$. At ${\omega }_1=0$ increases, $M_d$ decreases linearly due to the damping effect represented by ${\alpha }_0$. This reflects energy loss or reduced efficiency as the system accelerates. The final equation also includes a damping or resisting term proportional to the angular velocity ${\omega }_1$, with a constant ${\alpha }_0$. This structure allows you to compute the driving torque in terms of the angular position and velocity of the system components, which are key to understanding the dynamics of the oscillating conveyor mechanism [19]. The first subplot shows the relationship between $M_r$ and ${\phi }_6$. The second subplot shows the relationship between $M_d$ and ${\omega }_1$ show in Figure 3. Resisting Torque ($M_r$) Calculated for a range of ${\phi }_6$ values (0 to 0.02 rad) based on the formula $M_r=-2.98+511.87\cdot {\phi }_6$. Driving Torque ($M_d$) Calculated for a range of ${\omega }_1$ values (0 to 2000 rad/s) based on the formula $M_d=2.92-0.00074\cdot {\omega }_1$.

Let us break down the equations step by step:

$${\alpha }_0=tg\alpha ={\displaystyle \frac{2.8-1.8}{1500-150}}={\displaystyle \frac{1}{1350}}=0.00074$$

Given values:

$Y$ = 2.92, $Y_1$ = 2.8, $Y_2$ = 1.8, $X_1$ = 150, $X_2$ = 1500, $Mr$ = 2.92

Linear relationship equation:

$${\displaystyle \frac{Y-Y_1}{Y_2-Y_1}}={\displaystyle \frac{{\rm X}-{{\rm X}}_1}{{{\rm X}}_2-{{\rm X}}_1}}$$

(50)

This is a form of linear interpolation or a proportionality equation, where X and Y are related within the intervals [X₁, X₂] and [Y₁, Y₂]. Substituting the values:

$${\displaystyle \frac{Y-2.8}{1.8-2.8}}={\displaystyle \frac{{\rm X}-150}{1500-150}}$$

Simplifying the left-hand side:

$${\displaystyle \frac{0.12}{-1}}={\displaystyle \frac{{\rm X}-150}{1350}}$$

$$-0.12={\displaystyle \frac{{\rm X}-150}{1350}}$$

So, X = −12.

Now, based on we last expression:

$$M_d=2.92-0.00074{\omega }_1$$

This shows that the driving torque $M_d$ depends on the angular velocity ${\omega }_1$, with a small damping effect due to the factor ${\alpha }_0$.

To ensure the system operates stably:

The driving torque $M_d$ must overcome the resisting torque $M_r$. This establishes the condition for motion.

The equations show that $M_r$ grows with ${\phi }_6$, while $M_d$ decreases with ${\omega }_1$. A balance is achieved when the two torques are equal.

This analysis provides a foundation for understanding and optimizing the torques in the oscillating conveyor mechanism. The interaction between $M_r$ and $M_d$ governs the system’s behavior. If $M_d$< $M_r$, the system will decelerate or stop. Conversely, if $M_d$>$M_r$, the system will accelerate.

The moment of inertia equation is:

$${{\rm I}}_d\left({\phi }_1\right)={{\rm I}}_{S1}+m_2{\vartheta }_{S2{\phi }_1}+{{\rm I}}_{S2}{u}_{21}^2+m_3{\vartheta }_{S3{\phi }_1}^2+\left({{\rm I}}_{5F}+{{\rm I}}_{6G}\right)L_5^2$$

(51)

where ${{\rm I}}_{S1}$—is the moment of inertia of the first component; ${{\rm I}}_{S2}$—is the moment of inertia of the second component; ${{\rm I}}_{5F}$—is the combined moment of inertia of the fifth component; ${{\rm I}}_{6G}$ is the combined moment of inertia of the sixth component; $m_2{\vartheta }_{S2}$ and $m_3{\vartheta }_{S3}$—is the terms accounting for the contributions of specific masses $m_1$ and $m_3$ at velocities ${\vartheta }_{S2}$ and ${\vartheta }_{S3}$ respectively; $u_{21}$ and $u_{51}$—is the ratios representing relationships between components in the system.

Components:

$${{\rm I}}_{S1}=m_1\cdot {\displaystyle \frac{L_1^2}{2}}$$

(52)

$${{\rm I}}_{S2}=m_2\cdot {\displaystyle \frac{L_2^2}{2}}$$

(53)

$${{\rm I}}_{5F}={{\rm I}}_{S5}+m_5{L}_5^{*}=m_5{\displaystyle \frac{L_5^2}{2}}+m_5{\displaystyle \frac{L_5^2}{4}}$$

(54)

$${{\rm I}}_{6G}={{\rm I}}_{S6}+m_6{L}_6^{*}=m_6{\displaystyle \frac{L_6^2}{2}}+m_6{\displaystyle \frac{L_6^2}{4}}$$

(55)

The equation describes the differentiation of the moment of inertia ${{\rm I}}_k\left({\phi }_1\right)$ with respect to the angular displacement ${\phi }_1$, 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 ${\phi }_1$. It is equal to:

$$M_{d_o}-\alpha {\omega }_1-4u_{51}\left(M_{r0}+K{\phi }_6\right)$$

(56)

Given the initial conditions t = 0, ${\omega }_1=0$, and ${\phi }_1=0$, we aim to solve the equations that describe the system dynamics. Here is a step-by-step guide to proceed with the solution.

2.3. 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 (${\phi }_1$) evolve over time under the influence of inertia, resistance, and driving forces. The equation is expressed as:

$$R\left(t\right){\displaystyle \frac{d{\omega }_1}{dt}}+Q\left(t\right){\omega }^2\left(t\right)=W\left(t\right)$$

(57)

with $R\left(t\right), Q\left(t\right)$, and $W\left(t\right)$ defined as functions of system parameters, geometry, and time.

The solution incorporates initial conditions (t = 0, ${\omega }_1=0$, and ${\phi }_1=0$) and relies on numerical approximation methods [21].

Where $R\left(t\right)$—is the represents the combined system inertia and dynamic effects; $Q\left(t\right)$—is the accounts for quadratic velocity-dependent effects; $W\left(t\right)$—is the Represents the driving force, including torques and damping effects. The solution requires numerically evaluating $R\left(t\right)$, $Q\left(t\right)$, and $W\left(t\right)$ based on the provided relationships.

$$R\left(t\right)={{\rm I}}_{S_1}+(m_{2}A\left(t\right)+m_{3}H\left(t\right){\omega }_1^2+B\left(t\right)+D\left(t\right)$$

(58)

$R\left(t\right)$ is the encapsulates the combined effects of system mass and dynamic configurations. Terms such as $A\left(t\right)$ and $H\left(t\right)$ involve trigonometric and geometric dependencies, linking lengths $\left(L_i\right),$ angles $\left({\phi }_i\right),$ and oscillatory terms. $Q\left(t\right)$ represents quadratic velocity-dependent effects and includes contributions from terms $E\left(t\right)$, $F\left(t\right)$,$N\left(t\right)$ and $M\left(t\right)$, which are influenced by angular relationships and material properties. $W\left(t\right)$ integrates external forces, damping, and oscillatory effects. It is directly influenced by torque $\left(M_r\right)$ and damping coefficients $\left(\alpha \right)$.

This algorithm (

Table 1. A tabular representation of the steps and their descriptions.

Step	Description
Inputs	$\mathrm{Inputs}: {\omega }_0$$, {\phi }_0$, N, h
Increment n	Set n: = n + 1
Compute A(tn)	Compute A(tn)
Compute B(tn)	Compute B(tn)
Compute D(tn)	Compute D(tn)
Compute E(tn)	Compute E(tn)
Compute F(tn)	Compute F(tn)
Compute H(tn)	Compute H(tn)
Compute N(tn)	Compute N(tn)
Compute M(tn)	Compute M(tn)
Compute Q(tn)	Compute Q(tn)
Compute R(tn)	Compute R(tn)
Compute W(tn)	Compute W(tn)
$\mathrm{Update} {\omega }_{n+1}$	${\omega }_{n+1}={\omega }_n+h$$(\mathrm{Wn}/\mathrm{Rn}-\mathrm{Qn}/\mathrm{Rn}{\omega }_n$)
$\mathrm{Update} {\phi }_{n+1}$	${\phi }_{n+1}={\phi }_n$$+\mathrm{h}{\omega }_n$
Check if n < N	Check if n < N
$\mathrm{Yes}: \mathrm{Update} {\omega }_n, {\phi }_n$ and Repeat	$\mathrm{If} ‘\mathrm{Yes}’, \mathrm{Update} {\omega }_n\to {\omega }_{n+1}$$, {\phi }_n\to {\phi }_{n+1}$ and Repeat
$\mathrm{No}: \mathrm{Output} {\omega }_{n+1}$$, {\phi }_{n+1}$	$\mathrm{If} ‘\mathrm{No}’, \mathrm{Output} \mathrm{final} {\omega }_{n+1}$$, {\phi }_{n+1}$
End	End of process

) 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.

The expression for $A\left(t\right)$ is given as:

$$A\left(t\right)=L_1^2+2L_1{L}_2^{*}\left(\sin {\phi }_1\sin {\phi }_2+\cos {\phi }_1\cos {\phi }_2\right)u_{21}+{L_2^{*}}^2{u}_{21}^2$$

(59)

$A\left(t\right)$—is the contributes to the inertia and energy terms of the system, making it crucial in describing dynamic behaviors.

$$\begin{array}{l}H\left(t\right)=L_1^2-L_2^2{u}_{21}^2+\left(a^2{\cos }^2{\omega }_{5}t+b^2{\sin }^2{\omega }_{5}t\right)u_{51}^2+2L_1{L}_2{u}_{21}\left(\sin {\phi }_1\sin {\phi }_2+\cos {\phi }_1\cos {\phi }_2\right)\\ -2L_1{u}_{51}\left(a\sin {\phi }_1\cos {\omega }_{5}t+b\cos {\phi }_2\sin {\omega }_{5}t\right)-2L_2{u}_{21}\left(a\cos {\omega }_{5}t\sin {\phi }_2+b\sin {\omega }_{5}t\cos {\phi }_2\right)\end{array}$$

(60)

$H\left(t\right)$—is the governs dynamic inertia contributions from the links, angles, and elliptical motion components.

$$u_{51}=u_{61}={\displaystyle \frac{L_1\sin ({\phi }_2-{\phi }_1)}{L_5\sin ({\phi }_2-{\phi }_5)}}$$

$$u_{21}={\displaystyle \frac{L_1\sin ({\phi }_5-{\phi }_1)}{L_2\sin ({\phi }_2-{\phi }_5)}}$$

$$B\left(t\right)={{\rm I}}_{S_2}{\displaystyle \frac{L_1^2{\sin }^2({\phi }_5-{\phi }_1)}{L_2^2{\sin }^2({\phi }_2-{\phi }_5)}}$$

(61)

B(t)—is the representation of the rotational effect of link 2 influenced by angular positions ${\phi }_5$, ${\phi }_1$, ${\phi }_2$.

$$D\left(t\right)=\left({{\rm I}}_{5F}+{{\rm I}}_{6G}\right){\displaystyle \frac{L_1^2{\sin }^2({\phi }_2-{\phi }_1)}{L_5^2{\sin }^2({\phi }_2-{\phi }_5)}}$$

(62)

D(t)—is the account for the combined inertial effects of links 5 and 6 modulated by relative angles ${\phi }_1$, ${\phi }_2$, ${\phi }_5$.

$$Q\left(t\right)=\left(m_{2}E\left(t\right)+m_{3}F\left(t\right)\right)2{\omega }_1^2+2{{\rm I}}_{S_2}{\displaystyle \frac{L_1^2}{L_2^2}}N\left(t\right)+M\left(t\right)$$

(63)

$Q\left(t\right)$—are the accounts for quadratic velocity-dependent effects. $Q\left(t\right)$ represents quadratic velocity-dependent effects and includes contributions from terms $E\left(t\right)$, $F\left(t\right)$,$N\left(t\right)$ and $M\left(t\right)$, which are influenced by angular relationships and material properties.

$$\begin{array}{l}E\left(t\right)=\left(L_1\sin {\phi }_1+L_2^{*}\sin {\phi }_2{u}_{21}\right)\left(L_1\cos {\phi }_1+L_2^{*}\cos {\phi }_2{u}_{21}^2+L_2^{*}\sin {\phi }_2{\displaystyle \frac{d{u}_{21}}{d{\phi }_1}}\right)\\ +\left(L_1\cos {\phi }_1+L_2^{*}\cos {\phi }_2{u}_{21}\right)\left(-L_1\sin {\phi }_1-L_2^{*}\sin {\phi }_2{u}_{21}^2+L_2^{*}\cos {\phi }_2{\displaystyle \frac{d{u}_{21}}{d{\phi }_1}}\right)\end{array}$$

(64)

E(t)—is the energy term tied to angular positions and coupling ratios.

$$\begin{gathered} F\left(t\right) = \left(-L_1\sin {\phi }_1-L_2\sin {\phi }_2{u}_{21}+a{u}_{51}\cos {\omega }_{5}t\right)(-L_1\cos {\phi }_1-L_2\cos {\phi }_2{u}_{21}^2-L_1\sin {\phi }_1\frac{d{u}_{21}}{d{\phi }_1}+a\cos {\omega }_{5}t\frac{d{u}_{51}}{d{\phi }_1})+(L_1\cos {\phi }_1 \\ +L_2\cos {\phi }_2{u}_{21}-b{u}_{51}\sin {\omega }_{5}t)(-L_1\sin {\phi }_1-L_2\sin {\phi }_2{u}_{21}^2+L_2\cos {\phi }_2\frac{d{u}_{21}}{d{\phi }_1}-b\sin {\omega }_{5}t\frac{d{u}_{51}}{d{\phi }_1}) \end{gathered}$$

(65)

F(t)—is the damping and inertial effects with oscillatory external inputs.

$$N\left(t\right)={\displaystyle \frac{\left(\cos ({\phi }_1+{\phi }_5\right)-\cos ({\phi }_1-{\phi }_1))\sin ({\phi }_1-{\phi }_5)}{{\sin }^2({\phi }_2-{\phi }_5)}}$$

(66)

N(t)—is the coupling strength based on angular misalignments.

$$M\left(t\right)=\left({{\rm I}}_{5F}+{{\rm I}}_{6G}\right)\left[2{\displaystyle \frac{L_1^2}{L_2^2}}{\displaystyle \frac{\left(\cos ({\phi }_1+{\phi }_2\right)-\cos ({\phi }_1-{\phi }_2))\sin ({\phi }_2-{\phi }_1)}{{\sin }^2({\phi }_2-{\phi }_5)}}\right]$$

(67)

M(t)—is the torque contribution influenced by inertia and geometry.

$$W\left(t\right)=M_{d_o}-\alpha {\omega }_1-4u_{51}\left(M_{r0}+K{\phi }_6\right)$$

(68)

Given your detailed numerical approximation method for solving the differential equation:

$$\omega \left(0\right)=0, {\phi }_0=60°, h=0.01, k=0,1,2,\dots ,N-1, t_k=kh, {\omega }_k=\omega \left(t_k\right)\text{for} k=0,1,\dots ,N.$$

Approximation of the differential equation:

$$R_k{\displaystyle \frac{{\omega }_{k+1}-{\omega }_k}{h}}+Q_k{\omega }_k^2=W_k$$

(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:

$${\omega }_{k+1}={\omega }_k+h\left({\displaystyle \frac{W_k}{R_k}}-{\displaystyle \frac{Q_k}{R_k}}{\omega }_k^2\right)$$

(70)

Angular position update for ${\phi }_k.$ The angular position $\phi \left(t\right)$ is related to the angular velocity $\omega \left(t\right)$. As per the equation:

$${\phi }_{k+1}={\phi }_k+h{\omega }_k$$

(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 ${\phi }_k$. 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 ${\phi }_0=0$. The equation of motion is discretized using the Euler method, leading to the iterative formulas for ${\omega }_{k+1}$ and ${\phi }_{k+1}$. 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 decisions about system design, energy efficiency, and control strategies. The study highlights the sensitivity of the oscillating conveyor mechanism to initial conditions, but a more detailed sensitivity analysis of key parameters such as driving torque (M_d), resistance coefficient (K), and moment of inertia ${{\rm I}}_d\left({\phi }_1\right)$ is necessary to fully understand the system’s oscillatory behavior and stability margins. This section presents an in-depth investigation of how variations in these parameters affect the oscillation frequency, amplitude, and energy dissipation, which are critical for ensuring the system operates efficiently under different conditions.

• Influence of driving torque (M_d) on system behavior

Driving torque plays a crucial role in determining the oscillation intensity and stability of the conveyor system. Changes in M_d directly influence:

-

higher M_d values lead to increased energy input, causing larger oscillation amplitudes and potential instability;

-

higher torque may shift the system’s natural frequency, leading to resonance risks.

-

if M_d is too low, the conveyor may experience damped oscillations, limiting material transport efficiency.

2.

Effect of resistance coefficient (K) on damping and stability

The resistance coefficient governs energy dissipation and influences how quickly oscillations decay. It primarily affects:

-

a higher K results in faster energy dissipation, reducing oscillation persistence;

-

if K is too low, oscillations may persist longer than desired, potentially affecting material transport consistency;

-

excessively high K values could lead to overdamping, preventing efficient oscillatory motion.

3.

Influence of moment of inertia ${{\rm I}}_d\left({\phi }_1\right)$ on system response

Moment of inertia determines how the system reacts to applied forces. It affects:

-

higher ${{\rm I}}_d\left({\phi }_1\right)$ reduces acceleration, slowing oscillation growth;

-

lower ${{\rm I}}_d\left({\phi }_1\right)$ increases system reactivity, which may lead to excessive motion variations;

-

higher inertia requires more energy to maintain oscillations.

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:

• 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 ($L_1$, $L_2$, …) and joints. Angular displacements (${\phi }_1, {\phi }_2, \dots$) 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 $L_{1}cos\left({\phi }_1\right)+L_{2}cos\left({\phi }_2\right)=L_{3}cos\left({\phi }_3\right)$ 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.
• 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—Newton’s Second Law and the principle of virtual work were applied to derive equations governing the system’s dynamics. Inertia terms $I\left({\phi }_i\right)$ 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 ${\phi }_6$. 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.
• 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:

  $$R\left(t\right){\displaystyle \frac{d{\omega }_1}{dt}}+Q\left(t\right){\omega }^2\left(t\right)=W\left(t\right)$$

  where $R\left(t\right), Q\left(t\right)$, and $W\left(t\right)$ are time-dependent coefficients. (b) Discretization—Euler’s method was employed to approximate the derivatives using finite time steps h:

  $${\omega }_{k+1}={\omega }_k+h\left({\displaystyle \frac{W_k}{R_k}}-{\displaystyle \frac{Q_k}{R_k}}{\omega }_k^2\right)$$

  angular displacement was updated iteratively:

  $${\phi }_{k+1}={\phi }_k+h{\omega }_k$$

(c) Initial conditions—Initial values for angular displacement ${\phi }_1=0$ 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 $A\left(t\right)$, B(t), D(t), E(t), F(t), $H\left(t\right)$, N(t), M(t), $Q\left(t\right)$, $R\left(t\right)$, and $W\left(t\right)$ 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 ${\phi }_1\left(t\right)$ and ${\omega }_1\left(t\right)$. For each time step, we iteratively calculate the angular velocity ${\omega }_k$ and the angular position ${\phi }_k$. 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 ${\phi }_0=0$. The equation of motion is discretized using the Euler method, leading to the iterative formulas for ${\omega }_{k+1}$ and ${\phi }_{k+1}$. 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.

4. Results

To compute the exact derivatives of ${\vartheta }_{S_2}$ and ${\vartheta }_{S_3}$ with respect to ${\phi }_1$ over time, you need to express these variables in terms of ${\phi }_1$ and its time-dependent behavior shown in Figure 4. After that, we use the differentiation rules and chain rules to find time derivatives, including any relationships between angular velocity or other relevant parameters in the system. To plot the derivatives of $u_{21}$ and $u_{51}$ with respect to ${\phi }_1$ (i.e., ${\displaystyle \frac{d{u}_{21}}{d{\phi }_1}}$ and ${\displaystyle \frac{d{u}_{51}}{d{\phi }_1}}$), we need the expressions for $u_{21}$ and $u_{51}$ in terms of ${\phi }_1$ shown in Figure 5. We can compute the derivatives of these expressions with respect to ${\phi }_1$ and then plot them. The programming language chosen to use (Python 3.13.0).

$M_d\left(t\right)=sin\left({\omega }_1\right)e^{-0.1t}$ A damped sinusoidal function. $M_r\left(t\right)=cos\left({\phi }_6\right)\times \left(1+0.5sin\left(t\right)\right)$ A modulated cosine function. Below are 3D plots for $M_d\left(t\right)$ as a function of ${\omega }_1$ and $M_r\left(t\right)$ as a function of ${\phi }_6$. These assume some general functional forms for $M_d\left(t\right)$ and $M_r\left(t\right)$, but they can be replaced with specific equations or data shown in Figure 6. To analyze the 3D graphs below, it is first necessary to understand their physical meaning and how they change with time, angular velocity (${\omega }_1$), and angular position (${\phi }_6$). In this graph, the torque $M_d\left(t\right)$ is plotted as a function of time and angular velocity (ω₁). The function sin(${\omega }_1$) produces sinusoidal oscillations with respect to ${\omega }_1$, indicating that the torque varies cyclically. exp(−0.1t) as time t increases, the amplitude of the torque decreases, indicating the damping or damping effect. Initially, the amplitude of the $M_d$ is maximum and decreases with time. This is explained by the energy loss in the kinematic system (e.g., due to friction). At high values of ${\omega }_1$, the amplitude of the oscillations changes little, but the frequency increases. This graph is useful for predicting the change in torque, adjusting the damping parameters, and studying the stability of the system. The second graph shows the torque $M_r\left(t\right)$ as a function of time and angular position (${\phi }_6$). cos(${\phi }_6$) $M_r\left(t\right)$ varies sinusoidally with angular position. This indicates several complete cycles over a certain period of time. (1 + 0.5 sin(t)) The amplitude of the torque is modulated with time t. That is, the vibration level changes with time. The dependence on ${\phi }_6$ indicates the cyclical nature of the vibration, which indicates the uniformity or repetition of the rotational motion in the mechanism. The dependence on t leads to an increase or decrease in the amplitude at a certain time, which indicates the sensitivity of the rotating system to external influences. This graph allows us to assess the stability of $M_r\left(t\right)$ with respect to angular position and time. It is useful for studying the phase balance and dynamics of the device over time.

The resulting 3D graphs from the Python provide insights into the oscillatory behaviors of the variables over time. Here is an analysis of each graph: A(t) exhibits sinusoidal oscillations whose amplitude gradually decreases over time due to exponential damping (exp(−0.1t)) shown in Figure 7. B(t) is a cosine wave with slower exponential decay (exp(−0.05t)) compared to A(t) shown in Figure 7. Useful for systems with slower energy dissipation, such as lightly damped mechanical systems. D(t) A sinusoidal wave superimposed on a linear increasing trend (2t) is shown in Figure 8. Represents oscillatory systems influenced by a steady growth, such as systems with a baseline drift. E(t) Superposition of two sinusoidal waves with different frequencies leads to a complex pattern, showing periodic reinforcement and cancellation (beat phenomena). Common in wave interference, acoustics, or alternating currents with multiple harmonics are shown in Figure 8. F(t) The product creates a modulated waveform with periodic variations in amplitude shown in Figure 9. Found in amplitude-modulated (AM) signals in communication systems. H(t) Sinusoidal oscillations where the amplitude grows quadratically with time (0.5t²). Useful in modeling systems where oscillatory behavior becomes increasingly intense, such as resonant systems shown in Figure 9 (average around 1.0215 with oscillations ±0.1). N(t) A simple sine wave with a fixed frequency and amplitude is shown in Figure 10. Idealized representation of pure harmonic motion. M(t) Similar to N(t), but with a phase shift of π/2. Complements N(t) in systems requiring orthogonal oscillations shown in Figure 10, such as in-phase and quadrature components. Q(t) High-frequency oscillation with a small amplitude is shown in Figure 11. Models small perturbations or noise-like oscillatory components in systems. R(t) Oscillatory behavior with a baseline offset (5+…) is shown in Figure 11. Represents resistive fluctuations in electrical or mechanical systems with a steady mean resistance. W(t) A pure sine wave driving force (ranges from 1.8 to 2.2; periodic fluctuations add variability). This type of behavior is typical for systems subject to periodic forcing, such as driven pendulums or AC circuits shown in Figure 12.

5. Discussion

The presented differential equation governing the movement of the oscillating conveyor mechanism [25] provides a comprehensive model for analyzing the system’s dynamic behavior over time. This model incorporates various parameters related to the mechanical system, such as resisting torque ($M_r$), driving torque ($M_d$), inertial moments ($I_{S1}$, $I_{S2}$, $I_{S}F$, $I_{6}G$), and dynamic terms involving angular displacements (${\phi }_1$, ${\phi }_2$, etc.), angular velocities (${\omega }_1$, ${\omega }_2$, ${\omega }_6$), and their corresponding time derivatives. The equation captures the interactions between the resistive and driving forces in the system and how these forces influence the motion of the mechanism. The system is subject to a variety of forces that are modeled through functions like A(t), H(t), Q(t), and W(t), which depend on the time-varying nature of the system’s geometry, velocity, and other dynamic variables. Notably, the resistive torque $M_r$ is modeled as a linear function of ${\phi }_6$, reflecting the physical behavior of the resisting torque as the angular position changes. Similarly, the driving torque $M_d$ is affected by a damping term proportional to ${\omega }_1$, which accounts for frictional or resistive losses in the system. These dynamic interactions influence the evolution of angular velocities and positions over time. The equation is solved using an approximate method, specifically an explicit finite difference approach, where the angular velocity $\omega$ and angular displacement $\phi$ are computed at discrete time steps. This numerical integration method enables the prediction of system behavior over time based on initial conditions. The time step h and total simulation time T are key factors that influence the accuracy of the numerical solution. In this case, an initial angular displacement of ${\phi }_0=0$ and initial angular velocity ${\omega }_0=0$ are used as starting conditions, and the system is modeled over the interval [0, T]. By iterating through the algorithm at each time step, we can track the evolution of angular velocities and displacements, providing insight into the system’s stability and response to various forces. This approach is particularly useful for studying complex systems like oscillating conveyors [17], where analytical solutions are difficult to derive.

Key observations:

System damping—the model’s inclusion of damping in the form of α (a damping coefficient) highlights the importance of frictional forces in the system’s behavior. The damping term, which is proportional to angular velocity ${\omega }_1$, plays a significant role in regulating the amplitude of oscillations over time. This is crucial in understanding how the conveyor mechanism stabilizes and the rate at which it dissipates energy.

Nonlinear dynamics—the nonlinear relationship between the angular velocities and torques, particularly in the form of ω₂ terms and the dependency of R(t) and Q(t) on time-varying functions, reflects the complex nature of the system’s dynamics. These nonlinearities must be carefully accounted for when analyzing the system’s behavior, especially in cases where large displacements or velocities occur.

Sensitivity to initial conditions—the system’s sensitivity to initial conditions, such as the starting angular displacement and velocity, indicates the importance of accurate measurements and control in real-world applications. The behavior of the system may vary significantly depending on the initial conditions, which highlights the need for precise calibration in practical implementations.

Numerical stability and accuracy—the choice of time step h is crucial for the stability and accuracy of the numerical solution. A smaller h improves accuracy but increases computational effort, while a larger h may lead to numerical instability or inaccuracies. Thus, balancing computational efficiency with accuracy is an essential consideration when solving this type of differential equation.

Future work—to improve the model’s realism and applicability, further work could involve incorporating more detailed effects such as non-constant damping coefficients, elastic deformations of the conveyor components, or external disturbances like loading variations. Additionally, real-time measurements and feedback control strategies could be integrated to optimize the conveyor’s performance and mitigate issues such as resonance or instability. By extending this model to include more components or refining the numerical methods used, the system could be better suited for practical implementation in vibration conveyors or similar systems.

6. Conclusions

The kinematic analysis of the vibration conveyor mechanism provided insights into the motion of its components, including their positions, velocities, and accelerations. By modeling the geometric and trigonometric relationships between the links, the study established how the angular displacements, velocities, and accelerations evolve over time. The analysis highlighted the importance of understanding the interplay between link lengths, joint angles, and motion constraints for optimizing the mechanism’s design and ensuring smooth operation. The results can be used to predict motion trajectories and to ensure that the conveyor operates within its intended range of motion, minimizing the risk of mechanical interference or instability. Dynamic analysis revealed the forces and torques acting on the vibration conveyor system, accounting for inertia, damping, and external forces. The study demonstrated how the system responds to dynamic loads and how these forces influence the stability and efficiency of the mechanism. The derived equations of motion captured the nonlinear interactions between the system’s components, including the effects of centrifugal and inertial forces. By quantifying these forces, the analysis enabled the identification of optimal operating conditions and highlighted the need for proper force balancing and damping to minimize vibrations and wear. The numerical approximation provided a practical method for solving the nonlinear differential equations governing the vibration conveyor mechanism. Methods such as Euler’s method and the Runge–Kutta approach allowed for accurate simulations of the system’s behavior over time. These techniques enabled the visualization of time-dependent variables, such as angular velocity and system forces, under varying conditions. The numerical results closely matched the expected theoretical outcomes, validating the mathematical model and the approximations used. This approach is particularly valuable when exact analytical solutions are intractable, offering a reliable way to simulate and analyze complex dynamic systems. The combination of kinematic analysis, dynamic analysis, and numerical approximation provides a comprehensive understanding of the vibration conveyor mechanism. This multi-faceted approach facilitates: Insights into motion and force interactions, enabling improved designs for efficiency and reliability. By simulating system responses under various conditions, potential failure modes can be identified early. Accurate modeling supports fine-tuning of operational parameters to achieve desired performance. The methodologies and results from this study can be extended to similar mechanisms, laying the foundation for robust mechanical system analysis and design in industrial applications.