# Determination of Kinematic and Dynamic Characteristics of Oscillating Conveyor Mechanism
## 2 Materials and Methods  
### 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, Lk = 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:  
\[L_{1}+L_{2}+L_{k}+L_{3}=L_{7}+L_{5} \tag{1}\]  
And for closed contour OABCEGO:  
\[L_{1}+L_{2}+L_{k}=L_{8}+L_{6}+L_{4} \tag{2}\]  
These constraints maintain the equilibrium of the system during operation. In addition, trigonometric relationships are used to describe the motion angles \(\varphi_{1}=\varphi_{1}t\), 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:  
\[L_{1}cos\varphi_{1}+L_{2}cos\varphi_{2}+L_{3}cos\varphi_{3}=L_{7}cos\varphi_{ 7}+L_{5}cos\varphi_{5} \tag{3}\]  
\[L_{1}sin\varphi_{1}+L_{2}sin\varphi_{2}+L_{k}+L_{3}sin\varphi_{3}=L_{7}sin \varphi_{7}+L_{5}sin\varphi_{5} \tag{4}\]  
\[L_{1}cos\varphi_{1}+L_{2}cos\varphi_{2}=L_{8}cos\varphi_{8}+L_{6}cos\varphi_{ 6}+L_{4}cos\varphi_{4} \tag{5}\]  
\[L_{1}sin\varphi_{1}+L_{2}sin\varphi_{2}+L_{k}=L_{8}sin\varphi_{8}+L_{6}sin \varphi_{6}+L_{4}sin\varphi_{4} \tag{6}\]  
Additional constraints (assuming \(\varphi_{3}=\varphi_{4}=180^{\circ}\), \(\varphi_{k}=90^{\circ}\), \(L_{5}=L_{6}\), and \(\varphi_{5}=\varphi_{6}\)):  
\[L_{2}cos\varphi_{2}-L_{5}cos\varphi_{5}=L_{7}cos\varphi_{7}-L_{1}cos\varphi_{ 1}+L_{3} \tag{7}\]  
\[L_{2}sin\varphi_{2}-L_{5}sin\varphi_{5}=L_{7}sin\varphi_{7}-L_{1}sin\varphi_{ 1}-L_{k} \tag{8}\]  
Since \(L_{5}=L_{6}\) and \(\varphi_{5}=\varphi_{6}\), you can substitute those relationships to reduce the complexity:  
\[L_{2}cos\varphi_{2}-L_{5}cos\varphi_{5}=L_{8}cos\varphi_{8}-L_{1}cos\varphi_{ 1}-L_{4} \tag{9}\]  
\[L_{1}+L_{2}+L_{k}+L_{3}=L_{7}+L_{5} \tag{1}\]0  
\[L_{1}+L_{2}+L_{k}+L_{3}=L_{7}+L_{5} \tag{1}\]1 Kinematical schema of the three-class mechanism. Combine Equations (7) and (9):  
\[L_{1}+L_{2}+L_{k}+L_{3}=L_{7}+L_{5} \tag{1}\]2  
Combine Equations (8) and (10):  
\[L_{1}+L_{2}+L_{k}+L_{3}=L_{7}+L_{5} \tag{1}\]3  
Simplified System:  
\[L_{1}+L_{2}+L_{k}+L_{3}=L_{7}+L_{5} \tag{1}\]4  
\[L_{1}+L_{2}+L_{k}+L_{3}=L_{7}+L_{5} \tag{1}\]5  
Using trigonometric identities:  
\[L_{1}+L_{2}+L_{k}+L_{3}=L_{7}+L_{5} \tag{1}\]6  
\[L_{1}+L_{2}+L_{k}+L_{3}=L_{7}+L_{5} \tag{1}\]7  
Express \(\varphi_{8}\) in terms of \(\varphi_{7}\). From \(sin\varphi_{7}=\frac{L_{8}sin\varphi_{8}}{L_{7}}\) and \(cos\varphi_{7}=\frac{L_{8}cos\varphi_{8}}{L_{7}}\), for \(\varphi_{8}\) we solve using:  
\[L_{1}+L_{2}+L_{k}+L_{3}=L_{7}+L_{5} \tag{1}\]8  
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. \(tga=\frac{OM}{FM}=\frac{40}{1325}=0.03\), \(\alpha=1^{\circ}40^{\prime}\), \(\varphi_{7}=180^{\circ}+1^{\circ}40^{\prime}=181^{\circ}40^{\prime}\), \(tg\beta=\frac{MO}{MG}=\frac{40}{375}=0.1066\),  
\(\beta=6^{\circ}\), \(\varphi_{8}=360^{\circ}-6^{\circ}=354^{\circ}\), \(L_{3}=642\) mm, \(L_{4}=1058\) mm, \(L_{7}=1325.6\) mm, \(L_{8}=337.1\) mm.