# Application of Multibody Dynamics and Bonded-Particle GPU Discrete Element Method in Modelling of a Gyratory Crusher

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## Abstract

**:**

## 1. Introduction

**Figure 1.**Schematic of the gyratory crusher system; (

**a**) major mechanical component; (

**b**) typical cavity study for optimising the concave and mantle profiles for gyratory crushers [31].

## 2. Multibody Dynamics Modelling

## 3. GPU-Based Discrete Element Modelling

#### 3.1. DEM Governing Equations and GPU Implementation

_{i}, mass m

_{i}, and moment of inertia I

_{i}can be written as

**v**and

_{i}**ω**are the translational and angular velocities of the particle, respectively, and k

_{i}_{c}is the number of particles in interaction. The forces involved are as follows: the gravitational force m

_{i}

**g**, and interparticle forces between particles, which include elastic force

**f**

_{c,ij}, and viscous damping force

**f**

_{d,ij}. These interparticle forces can be resolved into normal and tangential components at a contact point. The torque acting on particle i by particle j includes two components:

**M**

_{t,ij}, which is generated by tangential force and causes particle i to rotate, and

**M**

_{r,ij}, which is commonly known as the rolling friction torque. The rolling friction torque is generated by asymmetric normal forces, and slows down the relative rotation between particles.

**M**

_{n,ij}is the torque generated by the non-spherical particles as the contact force between two particles may not go through the centre. A particle may undergo multiple interactions, so the individual interaction forces and torques are summed over the k

_{c}particles interacting with particle i.

#### 3.2. DEM Particle Breakage Model

^{n}and k

^{s}are the bond normal and tangential stiffness. ∆δ

^{n}, ∆δ

^{s}, ∆θ

^{n}, and ∆θ

^{s}are the incremental displacements and rotation in the normal and tangential directions. For a bond with radius R

_{b}and length L

_{b}, its bonding area is $A=\pi {R}_{b}^{2}$, its moment of inertia is $I=\pi {R}_{b}^{2}/4$, and its polar moment of inertia is $J=\pi {R}_{b}^{2}/2$. Bonds can break either by tensile or shear stress and the criteria for bond failure are given by

_{b}is the strength of the bonds. Once broken, these bonds can no longer be restored. In this work, the bonding area is set to be proportional to the contact area between particles. Within a group of bonded particles, bonds are generated with the bond stress following a normal distribution as shown in Equation (8).

- ${\sigma}_{b}$ is the bonding strength;
- N is a random number following the standard normal distribution;
- S
_{c}is the critical bonding strength.

#### 3.3. Particle Breakage Model Setting

## 4. Cosimulation Framework

^{−4}s time step for MBD and 5 × 10

^{−7}s time step for DEM) was used for the crusher cosimulation [41].

## 5. Results and Discussion

#### 5.1. Rock Flow Pattern and Drivetrain Performance

- Bond radius of 5 mm
- 0.4~1.6 times the critical bond stress

_{bn}) experienced by the meta-particles during crushing. It was observed that higher bonding forces were predominantly induced to the meta-particles when they were in close contact with the concave and mantle surfaces. When the crushing force experienced by the meta-particles was larger than the bonding stress, meta-particles broke into smaller meta-particles for further interactions with the crusher geometry.

#### 5.2. Effect of Meta Filling Methods

- Bond radius of 5 mm
- 0.4~1.6 times the critical bond stress

_{Meta}/d

_{fractional}ratios, 10, 6, and 2, were utilised for comparison purposes, and the corresponding number of fractional particles in a meta-particle was calculated. As shown in Figure 12, when d

_{Meta}/d

_{fractional}was selected as 10, 6, and 2, the total fractional particles in a meta-particle reduced from 640 to 140, and then 40, which greatly assisted in GPU global memory reduction.

_{Meta}/d

_{fractional}ratio resulted in a significant reduction in the total bonds in the simulation, which would improve the computational efficiency. Comparing the ore flow and crushing behaviours of different d

_{Meta}/d

_{fractional}ratios shown in Figure 13b–d, a minor difference was observed across three settings. For the purpose of better tracking of fine particle size distribution in the product stream, as well as considering computational efficiency, d

_{Meta}/d

_{fractional}= 6 was selected for further modelling cases conducted below.

#### 5.3. Effect of Bond Distribution

_{Meta}/d

_{fractional}ratio = 6. An alternative bond stress range of 0.2–1.8 was utilised to compare the total bonds generation in each case and the results are shown in Figure 14. They indicated that the total number of bonds increased marginally when the bond stress range of 0.2–1.8 was selected. No visible difference was observed in the ore flow and crushing performance across the two different bond stress ranges. Therefore, it was suggested that the effect of bond stress range can be neglected, and a nominal range of 0.4–1.6 can be selected.

#### 5.4. Effect of Bond Radius

_{b}= [0.1R

_{fractional}, 0.5R

_{fractional}, R

_{fractional},]:

- Meta-particle size distribution: 50–500 mm
- Fractional particle size distribution: mono-sized with d
_{Meta}/d_{fractional}= 6 - Bond stress range: 0.4–1.6

_{b}= R

_{fractional}), the resulting bond stress was too high, rocks were more difficult to break, and a minimal discharge mass flow rate was observed in Figure 15c. This was further validated by the mantle speed and torque obtained from MBD modelling as shown in Figure 16a–c. When the material was fed into the crusher, the mantle torque increased substantially while the mantle rotational speed dropped to zero rapidly, indicating that the drivetrain was overloaded. In comparison, when an intermediate bond radius (R

_{b}= 0.5R

_{fractional}) was selected, normal crushing and product stream discharge were observed; however, the mantle torque, speed, and the discharge mass flow rate were observed to be reduced compared with the modelling results from the bond radius of R

_{b}= 0.1R

_{fractional}.

^{3}. Such a performance is within the range of the throughput obtained above with R

_{b}= 0.1R

_{fractional}.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**(

**a**) Multibody system diagram for the selected gyratory crusher; (

**b**) mechanical and drivetrain system for the selected gyratory crusher.

**Figure 3.**Crusher rotational speed for main shaft and pinion (motor nominal power 450 kW, maximum pinion rotation speed 600 rpm, gear ratio = 3.75, eccentric distance = 21.7 mm, simulation time step = 0.001 s).

**Figure 4.**Modelling principle of the bonded-particle model; (

**a**) Stresses formed by bonds between two particles; (

**b**) Force and bonding behaviours of bonded particles [17].

**Figure 6.**(

**a**) Initial meta- and fractional particle size distribution used in the modelling; (

**b**) Fractional particle factory used in constituting the meta-particles.

**Figure 10.**Results of particle breakage in crusher represented by (

**a**) Meta Id and (

**b**) normal bond force F

_{bn}; (

**c**) Transient mantle speed from the MBD system; (

**d**) Transient mantle torque from the MBD system.

**Figure 11.**(

**a**) Average number of fractional particles inside a meta-particle (when fractional particles inside have a size distribution between 8 and 45 mm); (

**b**) Reduction of fractional particles when increasing the population of meta particles.

**Figure 12.**Average number of fractional particles inside a meta-particle when fractional particles inside are mono-sized with d

_{Meta}/d

_{fractional}of (

**a**) 6–10 and (

**b**) 2–6.

**Figure 13.**(

**a**) Comparison of the total number of bonds for meta-particles filled with mono-sized fractional particles with d

_{Meta}/d

_{fractional}of 10, 6, and 2; (

**b**) Crusher performance of d

_{Meta}/d

_{fractional}of 2; (

**c**) Crusher performance of d

_{Meta}/d

_{fractional}of 6; (

**d**) Crusher performance of d

_{Meta}/d

_{fractional}of 10.

**Figure 14.**Comparison between the total number of bonds generated in the simulation for different bond stress distribution ranges selected.

**Figure 15.**Crushing behaviours of the rock particles with increasing bond radius settings; (

**a**) R

_{b}= 0.1R

_{fractional}; (

**b**) R

_{b}= 0.5R

_{fractional}; (

**c**) R

_{b}= R

_{fractional}.

**Figure 16.**(

**a**) Mantle speed, (

**b**) Mantle torque, and (

**c**) Mass flow rate results obtained from the numerical framework when different bond radius settings were utilised.

Parameter | Value | Units | |
---|---|---|---|

DEM Material Properties | Rock | Steel | |

Solid Density | 2600 | 7800 | kg/m^{3} |

Shear Stiffness | 5.57 × 10^{8} | 7.0 × 10^{10} | Pa |

Poisson’s Ratio | 0.35 Rock-rock | 0.3 Rock-steel | |

Coefficient of Static Friction | 0.5 | 0.7 | |

Coefficient of Restitution | 0.2 | 0.25 | |

Coefficient of Rolling Friction | 0.001 | 0.001 | |

Feed | |||

Feed Fate | 3500 | t/h | |

BPM Parameters | |||

Normal Stiffness | 1670 | N/m | |

Shear Stiffness | 667 | N/m | |

Normal Critical Stress | 36 | MPa | |

Shear Critical Stress | 24 | MPa | |

Drivetrain | |||

Eccentric Throw | 21.7 | mm | |

Eccentric Speed | 160 | rpm | |

Close Side Setting | 120 | mm | |

Liner Geometry | 3D model | ||

GPU | GPU V100 | ||

Time step | 3 × 10^{−7} | s |

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**MDPI and ACS Style**

Xiong, Y.; Chen, W.; Ou, T.; Zhao, G.; Wu, D.
Application of Multibody Dynamics and Bonded-Particle GPU Discrete Element Method in Modelling of a Gyratory Crusher. *Minerals* **2024**, *14*, 774.
https://doi.org/10.3390/min14080774

**AMA Style**

Xiong Y, Chen W, Ou T, Zhao G, Wu D.
Application of Multibody Dynamics and Bonded-Particle GPU Discrete Element Method in Modelling of a Gyratory Crusher. *Minerals*. 2024; 14(8):774.
https://doi.org/10.3390/min14080774

**Chicago/Turabian Style**

Xiong, Youwei, Wei Chen, Tao Ou, Guoyan Zhao, and Dongling Wu.
2024. "Application of Multibody Dynamics and Bonded-Particle GPU Discrete Element Method in Modelling of a Gyratory Crusher" *Minerals* 14, no. 8: 774.
https://doi.org/10.3390/min14080774