# Determination of Kinematic and Dynamic Characteristics of Oscillating Conveyor Mechanism
## 2 Materials and Methods
### Geometric and Trigonometric Constraints
Angular velocity ratios:  
\[L_{1}sin\varphi_{1}+L_{2}sin\varphi_{2}+L_{k}+L_{3}sin\varphi_{3}=L_{7}sin \varphi_{7}+L_{5}sin\varphi_{5} \tag{4}\]3  
\[L_{1}sin\varphi_{1}+L_{2}sin\varphi_{2}+L_{k}+L_{3}sin\varphi_{3}=L_{7}sin \varphi_{7}+L_{5}sin\varphi_{5} \tag{4}\]4  
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:  
\[L_{1}sin\varphi_{1}+L_{2}sin\varphi_{2}+L_{k}+L_{3}sin\varphi_{3}=L_{7}sin \varphi_{7}+L_{5}sin\varphi_{5} \tag{4}\]5  
\[L_{1}sin\varphi_{1}+L_{2}sin\varphi_{2}+L_{k}+L_{3}sin\varphi_{3}=L_{7}sin \varphi_{7}+L_{5}sin\varphi_{5} \tag{4}\]6  
_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 (\(\varphi_{1}\), \(\varphi_{2}\)):  
\[L_{1}sin\varphi_{1}+L_{2}sin\varphi_{2}+L_{k}+L_{3}sin\varphi_{3}=L_{7}sin \varphi_{7}+L_{5}sin\varphi_{5} \tag{4}\]7  
\[L_{1}sin\varphi_{1}+L_{2}sin\varphi_{2}+L_{k}+L_{3}sin\varphi_{3}=L_{7}sin \varphi_{7}+L_{5}sin\varphi_{5} \tag{4}\]8  
These equations represent the coordinates of a point s2 that depend on the lengths \(L_{1}\) and \(L_{2}^{*}=\frac{L_{2}}{2}=As2\), as well as the angles \(\varphi_{1}\) and \(\varphi_{2}\). This is a purely kinematic representation, without any influence of oscillatory displacements. \[L_{1}sin\varphi_{1}+L_{2}sin\varphi_{2}+L_{k}+L_{3}sin\varphi_{3}=L_{7}sin \varphi_{7}+L_{5}sin\varphi_{5} \tag{4}\]9  
\[L_{1}cos\varphi_{1}+L_{2}cos\varphi_{2}=L_{8}cos\varphi_{8}+L_{6}cos\varphi_{ 6}+L_{4}cos\varphi_{4} \tag{5}\]0  
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(X_{S2}\right)^{2}+\left(Y_{S2}\right)^{2}}=\omega _{1}\sqrt{L_{1}^{2}+L_{