# Analysis of Vibration Characteristics of the Grading Belt in Wolfberry Sorting Machines
## 2 Materials and Methods
### Mechanical Modeling of Transverse Vibration of a Flat Belt
#### 2.3.2 Transverse Vibration Analysis of Belt Drives  
The belt mainly demonstrates three vibration modes during its working process: the transverse vibration perpendicular to the centerline connection of the pulleys, the axial vibration parallel to the centerline, and the torsional vibration rotating around the centerline. In contrast to the impact of lateral vibration on belt vibration, the effects of longitudinal vibration and torsional vibration on belt vibration are relatively minor. Furthermore, longitudinal vibration and torsional vibration are more frequently employed in high-precision equipment. Therefore, this article primarily examines the lateral vibration characteristics of belts during the belt drive process and the impact of different working conditions on them.  
To simplify the mathematical model, hypotheses are formulated for the sorting device. In the belt transmission system, the lateral vibration is mainly concentrated on the straight portion between the two pulleys. Since the silicone rubber belt is a nonlinear viscoelastic material with non-negligible stiffness, the belt is simplified to an axially moving beam model [18; 19; 20; 21].  
Figure 4: Schematic diagram of belt drive system.  
The simplified diagram of the mathematical model for the transverse vibration of the flat belt is shown in Figure 5.  
By taking the contact point between the flat belt and the driving pulley as the coordinate origin, a differential element model of length \(dx\) located at a distance \(x\) from the origin along the belt movement direction is analyzed. This allows for the determination of the force in the y-direction and the moment equilibrium equation on the belt differential element as follows:  
\[[Q(x+dx)\cos\theta(x+dx)-Q(x)\cos\theta(x)]+[F\sin\theta(x+dx)-F\sin\theta(x)] =\rho Adxa_{y}-\beta\bigg{(}\frac{\partial y}{\partial t}\bigg{)}^{3}dx \tag{3}\]  
\[\sum M=0\frac{\partial M}{\partial x}=Q \tag{4}\]  
In the equations:  
\(\rho\): Belt density (kg/m\({}^{3}\)); \(A\): Cross-sectional area (m\({}^{2}\)); \(\rho A\): Linear density (kg/m); \(EI\): Bending stiffness (N-m\({}^{2}\)); \(T\): Axial tension (N); \(c\): Linear damping coefficient (N-s/m); \(\beta\): Nonlinear damping coefficient (N-s\({}^{3}\)/m\({}^{3}\)).  
Assuming that the lateral vibration is a small vibration (\(\theta\ll 1\)):  
\[\sin\theta\approx\tan\theta\approx\frac{\partial y}{\partial x},\cos\theta \approx 1,\theta(x+dx)\approx\theta(x)+\frac{\partial\theta}{\partial x}dx \tag{5}\]  
Based on the Euler-Bernoulli beam theory, the relationship between curvature and bending moment is formulated while incorporating the viscoelastic characteristics described by the Kelvin-Voigt model.  
\[M=-EI\frac{\partial^{2}y}{\partial x^{2}}-\eta I\frac{\partial^{3}y}{\partial x ^{2}\partial t} \tag{6}\]  
By substituting Formula (6) into the torque balance equation, the following result can be derived:  
\[\frac{\partial Q}{\partial x}=-EI\frac{\partial^{4}y}{\partial x^{4}}-\eta I \frac{\partial^{5}y}{\partial x^{4}\partial t} \tag{7}\]  
By simultaneously solving Equations (5) and (7) and substituting them into the transverse force equilibrium equation, the complete transverse vibration equation for the viscoelastic belt is derived.  
\[\rho A\bigg{(}\frac{\partial^{2}y}{\partial t^{2}}+2v\frac{\partial^{2}y}{ \partial x\partial t}+v^{2}\frac{\partial^{2}y}{\partial x^{2}}\bigg{)}+EI \frac{\partial^{4}y}{\partial x^{4}}+\eta I\frac{\partial^{5}y}{\partial x^{4 }\partial t}-T\frac{\partial^{2}y}{\partial x^{2}}+c\frac{\partial y}{ \partial t}+\beta\bigg{(}\frac{\partial y}{\partial t}\bigg{)}^{3}=0 \tag{8}\]  
From Equation (8), it can be observed that the lateral vibration amplitude y of the synchronous belt is related to the center distance \(L\) between the pulleys, the initial tension \(T\), and the operating speed \(v\)[22; 23; 24].  
Figure 5: Vibration mechanical model.