# Multi Body Dynamic Equations of Belt Conveyor and the Reasonable Starting Mode

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Rigid Finite Element Model of a Conveyor Belt

#### 2.1. Discrete Model of the Conveyor Belt

^{p}rigid body elements (RFEs) reflecting the rigid characteristics of the conveyor belt, and the rigid body elements are connected by spring damping elements (SDE) reflecting the flexible characteristics of the conveyor belt. Then, the whole conveyor system is simplified into a rigid flexible multi-body system composed of p-1 rigid bodies, m

^{p}RFEs and m

^{p}SDEs. The simplified dynamic model of conveyor belt is symmetrical in structure.

#### 2.2. Coordinate Description

#### 2.2.1. Rigid Finite Element

#### 2.2.2. Spring Damping Element (SDE)

#### 2.3. Spring Damping Element Parameters

#### 2.3.1. Potential Energy and Generalized Stiffness Matrix

#### 2.3.2. Coefficients of the Stiffness and Damping

- Each RFE can be simplified into an equal length beam element with a rectangular section shape;
- The deformation mode and velocity of the actual conveyor belt are the same as that of its equivalent SDE;
- The stress of each section in the conveyor belt section is equal, and its mechanical properties can be described by the Voigt model [20]:

#### 2.4. Rigid Finite Element Parameters

#### 2.4.1. Kinetic Energy and Lagrangian Operator

#### 2.4.2. Gravitational Potential Energy and Generalized Force

#### 2.4.3. Elastic Potential Energy

#### 2.5. Multibody Dynamic Equation

#### 2.5.1. Forward Recursive Formulation

#### 2.5.2. Dynamic Equation of the Rigid Flexible Multibody System

## 3. Verification of Belt Conveyor Simulation Model

#### 3.1. Field Test

_{d}= 2F

_{c}/sinθ, where F

_{d}is the dynamic tension in the cross section of the belt, F

_{c}is the output pressure value of a single pressure sensor, and θ is the angle between the belt pushed up by the idler and the horizontal plane.

#### Simulation Model

#### 3.2. Verification

## 4. Discussions

_{0}is the initial velocity (default 0), t is the time variable, T is the total acceleration time (staring time).

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

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**Figure 7.**Measured layout of belt conveyor: (

**a**) Principle of roller pressure test; (

**b**) Principle of speed measurement.

**Figure 14.**Acceleration under different starting speed (adding delay section). (

**a**) speed; (

**b**) acceleration.

Performance parameter | Transportation distance: 972 m Transportation capacity: 400 t/h Belt speed: 2 m/s |

Driving device | Rated power of motor: 150 KW Rated speed: 1000 rpm Diameter of driving drum: 0.8 m Drum mass: 800 kg |

Conveyor belt | Tensile strength: 1250 N/m Belt width: 1 m Mass per unit length of conveyor belt: 24.5 kg/m |

Conveying materials | 56 kg/m |

Model parameters and running resistance | Length of belt section: 314.23 mm Number of belt sections: 170 System degree of freedom: 309 |

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**MDPI and ACS Style**

Guo, Y.; Wang, F.
Multi Body Dynamic Equations of Belt Conveyor and the Reasonable Starting Mode. *Symmetry* **2020**, *12*, 1489.
https://doi.org/10.3390/sym12091489

**AMA Style**

Guo Y, Wang F.
Multi Body Dynamic Equations of Belt Conveyor and the Reasonable Starting Mode. *Symmetry*. 2020; 12(9):1489.
https://doi.org/10.3390/sym12091489

**Chicago/Turabian Style**

Guo, Yongbo, and Fansheng Wang.
2020. "Multi Body Dynamic Equations of Belt Conveyor and the Reasonable Starting Mode" *Symmetry* 12, no. 9: 1489.
https://doi.org/10.3390/sym12091489