# Simulation Parameter Calibration and Test of Typical Pear Varieties Based on Discrete Element Method

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model Establishment

#### 2.1.1. Physical Model of Pears

#### 2.1.2. Selection of Contact Model

#### 2.2. Calibration of Contact Parameters between Typical Pears and Materials

#### 2.2.1. Calibration of Collision Recovery Coefficient between Typical Pears and Materials

_{x}is the ratio of normal instantaneous separation velocity v

_{1}to instantaneous contact velocity v

_{0}at the collision contact point before and after the collision between pears and materials, and it is as follows:

#### 2.2.2. Calibration of Static Friction Coefficient between Typical Pears and Materials

_{s}and the inclined plane angle α was as follows:

#### 2.2.3. Calibration of Rolling Friction Coefficient between Typical Pears and Materials

_{r}is rolling friction coefficient; m was the fruit mass of pear, g; g is gravity acceleration, 9.8 m/s

^{2}; S is the rolling distance of the inclined plane, mm; β is the inclination angle of the inclined plane, °; L is the rolling distance, cm.

#### 2.3. Calibration of Contact Parameters between Typical Pears

#### 2.3.1. Calibration of Collision Recovery Coefficient between Typical Pears

#### 2.3.2. Calibration of Pear Accumulation Angle

- (1)
- Experiment of Pear Accumulation Angle

- (2)
- Experiment of the steepest ascent

- (3)
- Experiment of horizontal rotation combinations

_{n}) and the coefficient of rolling friction (μ

_{f}) between pears, five-level simulation experiments of the pear accumulation angle were designed for each factor by the method of orthogonal rotation combination, whose codes were shown in Table 3.

- (4)
- Experiment of verification

## 3. Results and Discussion

#### 3.1. Contact Parameters between Typical Pears and Materials

#### 3.1.1. Collision Recovery Coefficient between Typical Pears and Materials

_{x}

_{1}was 0.542, and e

_{x}

_{2}was 0.652. The collision recovery coefficients were set as the above parameters in EDEM, the maximum simulated rebound height of Snow pear and contact materials were 143.71 mm and 145.87 mm, respectively, which were consistent with measured values. The errors between simulated values and measured value were 1.72% and 1.57%, respectively, which indicated that e

_{x}

_{1}and e

_{x}

_{2}were accurate and reliable.

#### 3.1.2. Static Friction Coefficient between Typical Pears and Materials

_{1}, α

_{2}is inclination angle of Snow pears and PVC and EVA foam materials, respectively; μ

_{s}

_{1}, μ

_{s}

_{2}is static friction coefficient between Snow pears and PVC and EVA foam materials, respectively.

_{s}

_{1}was 0.686, and μ

_{s}

_{2}was 0.472. The static friction coefficients were set as the above parameters in EDEM, the maximum simulated inclination angle of Snow pear and materials were 33.58° and 23.98°, respectively, which were consistent with measured values. The errors between simulated values and measured value were 1.98% and 2.28%, respectively, which indicated that μ

_{s}

_{1}and μ

_{s}

_{2}were accurate and reliable.

#### 3.1.3. Rolling Friction Coefficient between Typical Pears and Materials

_{r}

_{1}, μ

_{r}

_{2}is rolling friction coefficient between Snow pears and PVC and EVA foam materials, respectively; L

_{1}, L

_{2}is rolling distance of Snow pears and PVC and EVA foam materials, respectively, in cm.

_{r}

_{1}was 0.00597, and μ

_{r}

_{2}was 0.00706. The rolling friction coefficient were set as the above parameters in EDEM, the simulated inclination angle of Snow pear and materials were 130.24 cm and 108.97 cm, respectively, which were consistent with measured values. The errors between simulated values and measured value were 1.06% and 1.36%, respectively, which indicated that μ

_{r}

_{1}and μ

_{r}

_{2}were accurate and reliable.

#### 3.2. Calibration of Contact Parameters between Typical Pears

#### 3.2.1. Calibration of Collision Recovery Coefficient between Typical Pears

#### 3.2.2. Pear Accumulation Angle

#### 3.2.3. Optimal Value of Influencing Factors of the Pear Accumulation Angle

_{n}) and the coefficient of rolling friction between pears (μ

_{f}), the steepest ascent experiment was carried out to determine the optimal value of influencing factors of the pear accumulation angle. The collision recovery coefficient between pears was from 0.50 to 0.70 according to Table 10. The static friction coefficient of most agricultural materials is from 0.20 to 0.50 [34,35], and the rolling friction coefficient is from 0.01 to 0.05 [36,37]. Therefore, the design and the results of steepest ascent experiment were shown in Table 11. Among them, θ

^{,}was the simulated value of the pear accumulation angle, and σ was the error between the simulated value and the measured value.

#### 3.2.4. Error Optimization Model of the Pear Accumulation Angle

_{n}) and the coefficient of rolling friction between pears affected the pear accumulation angle. By the method of orthogonal rotation combination, five-level simulation experiments of the pear accumulation angle were designed for each factor. The codes were shown in Table 12, and the results were shown in Table 13.

#### 3.2.5. Verification of Simulation Parameters between Pears

## 4. Conclusions

- (1)
- Based on the intrinsic parameters of four kinds of pears (Snow pear, Crisp pear, Huangguan pear and Qiuyue pear), their simulation models were constructed by the method of DEM. The simulation parameters between pears and the contact material (PVC, EVA foam material) were calibrated by the methods of free fall collision, inclined sliding and rolling, respectively. The collision recovery coefficients of Snow pear, Crisp pear, Huangguan pear and Qiuyue pear with PVC material were 0.542, 0.516, 0.624 and 0.573, respectively; and the static friction coefficients were 0.686, 0.651, 0.627 and 0.661, respectively; and the rolling friction coefficients were 0.00597, 0.00602, 0.00323 and 0.00627, respectively. The collision recovery coefficients between Snow pear, Crisp pear, Huangguan pear and Qiuyue pear wih EVA foam material were 0.652, 0.608, 0.678 and 0.641, respectively; and the static friction coefficients were 0.472, 0.491, 0.394 and 0.574, respectively; and the rolling friction coefficients were 0.00706, 0.00735, 0.00638 and 0.00714, respectively.
- (2)
- The pear accumulation angle was obtained by experimental measurement. The steepest ascent experiment was carried out to determine the optimal value of influencing factors of the pear accumulation angle. Considering the coefficient of collision recovery, the coefficient of static friction and the coefficient of rolling friction between pears, five-level simulation experiments of the pear accumulation angle were designed for each factor by the method of orthogonal rotation combination. The regression model of the error between the measured value and the simulated value of the pear accumulation angle was established, and the influence of three factors on the pear accumulation angle was analyzed. The results showed that the collision recovery coefficients of Snow pear, Crisp pear, Huangguan pear and Qiuyue pear were 0.54, 0.44, 0.51, and 0.48, respectively; and the coefficients of static friction were 0.27, 0.24, 0.31, and 0.28, respectively; and the coefficients of rolling friction were 0.020, 0.024, 0.018, and 0.027, respectively.
- (3)
- The accumulation angle verification experiments were carried out by the method of bottomless barrel lifting. The results showed that the relative error between the simulated and measured accumulation angle of four kinds of pears were 1.42%, 1.68%, 2.19% and 1.83%, respectively, which indicated that the calibrated simulation parameters were reliable. The research can provide a basis for the design and parameter optimization of harvesting machinery of pears.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 2.**Geometric models of typical pears. Snow pear (

**a**); Crisp pear (

**b**); Qiuyue pear (

**c**); Huangguan pear (

**d**). Note: Blue is the origin of the coordinate system for the pear geometric model.

**Figure 3.**EDEM simulation models of typical pears. Snow pear (

**a**); Crisp pear (

**b**); Qiuyue pear (

**c**); Huangguan pear (

**d**).

**Figure 4.**Calibration experiment of collision recovery coefficient between pears and materials. Experiment principle (

**a**); Physical experiment (

**b**); Simulation experiment (

**c**).

**Figure 5.**Calibration experiment of static friction coefficient between pears and materials. Physical experiment (

**a**); Simulation experiment (

**b**).

**Figure 6.**Calibration experiment of rolling friction coefficient between pears and materials. Physical experiment (

**a**); Simulation experiment (

**b**).

**Figure 8.**Experiment of pear accumulation angle. Physical experiment (

**a**); Simulation experiment (

**b**).

**Figure 9.**Experiment of simulation parameters verification. Physical experiment (

**a**); Simulation experiment (

**b**).

**Figure 13.**Processing the image of pears with Matrix Laboratory software. Original image (

**a**); Grayscale processing (

**b**); Boundary outline (

**c**); Linear fitting (

**d**). Note: The black line is the contour boundary of the pear accumulation angle, and the red line is the fitting line.

Variety | Snow Pear | Crisp Pear | Huangguan Pear | Qiuyue Pear |
---|---|---|---|---|

Transverse diameter D/mm | 79.61 ± 4.8 | 76.81 ± 4.6 | 77.39 ± 3.9 | 70.69 ± 2.7 |

Longitudinal height H/mm | 88.38 ± 4.5 | 69.14 ± 4.4 | 73.74 ± 4.2 | 61.77 ± 2.5 |

Mass/g | 381.95 ± 50.11 | 272.87 ± 43.89 | 301.56 ± 37.52 | 207.59 ± 26.04 |

Density/(kg·m^{−3}) | 985.07 ± 17.35 | 944.64 ± 54.93 | 1049.30 ± 45.61 | 1012.80 ± 40.77 |

Moisture content/% | 89.28 ± 4.85 | 87.14 ± 2.78 | 87.53 ± 0.27 | 89.56 ± 0.15 |

Poisson’s ratio | 0.48 | 0.36 | 0.40 | 0.24 |

Young’s modulus/MPa | 3.16 | 2.69 | 2.18 | 0.84 |

Shear modulus/MPa | 1.07 | 0.99 | 0.78 | 0.34 |

Material | Parameter | Value |
---|---|---|

PVC | Poisson’s ratio | 0.47 |

Shear modulus/MPa | 2.00 | |

Density/(kg·m^{−3}) | 1282.00 | |

EVA | Poisson’s ratio | 0.30 |

Shear modulus/MPa | 0.46 | |

Density/(kg·m^{−3}) | 79.09 |

Code | Experiment Factors | ||
---|---|---|---|

e | μ_{n} | μ_{f} | |

−1.682 | 1 | 1 | 1 |

−1 | 2 | 2 | 2 |

0 | 3 | 3 | 3 |

1 | 4 | 4 | 4 |

1.682 | 5 | 5 | 5 |

Variety | Contact Material | Maximum Rebound Height/mm | Value of Collision Recovery Coefficient |
---|---|---|---|

Snow pear | PVC | 141.28 | 0.594 |

EVA | 143.62 | 0.599 | |

Crisp pear | PVC | 96.14 | 0.490 |

EVA | 120.80 | 0.550 | |

Huangguan pear | PVC | 138.58 | 0.589 |

EVA | 152.18 | 0.617 | |

Qiuyue pear | PVC | 105.05 | 0.512 |

EVA | 131.24 | 0.573 |

Variety | Contact Material | Collision Recovery Coefficient |
---|---|---|

Snow pear | PVC | 0.542 |

EVA | 0.652 | |

Crisp pear | PVC | 0.516 |

EVA | 0.608 | |

Huangguan pear | PVC | 0.624 |

EVA | 0.678 | |

Qiuyue pear | PVC | 0.573 |

EVA | 0.641 |

Variety | Contact Material | Inclination Angle | Static Friction Coefficient |
---|---|---|---|

Snow pear | PVC | 34.26° | 0.681 |

EVA | 24.54° | 0.457 | |

Crisp pear | PVC | 33.21° | 0.655 |

EVA | 22.78° | 0.420 | |

Huangguan pear | PVC | 32.06° | 0.626 |

EVA | 21.60° | 0.396 | |

Qiuyue pear | PVC | 33.43° | 0.660 |

EVA | 29.90° | 0.575 |

Variety | Contact Material | Static Friction Coefficient |
---|---|---|

Snow pear | PVC | 0.686 |

EVA | 0.472 | |

Crisp pear | PVC | 0.651 |

EVA | 0.491 | |

Huangguan pear | PVC | 0.627 |

EVA | 0.394 | |

Qiuyue pear | PVC | 0.661 |

EVA | 0.574 |

Variety | Contact Material | Rolling Distance/cm | Rolling Friction Coefficient |
---|---|---|---|

Snow pear | PVC | 128.88 | 0.00629 |

EVA | 107.51 | 0.00742 | |

Crisp pear | PVC | 128.27 | 0.00631 |

EVA | 103.91 | 0.00766 | |

Huangguan pear | PVC | 125.09 | 0.00336 |

EVA | 119.79 | 0.00674 | |

Qiuyue pear | PVC | 122.29 | 0.00666 |

EVA | 106.66 | 0.00748 |

**Table 9.**Simulated values of static friction coefficient between other pear varieties and contact materials.

Variety | Contact Material | Rolling Friction Coefficient |
---|---|---|

Snow pear | PVC | 0.00597 |

EVA | 0.00706 | |

Crisp pear | PVC | 0.00602 |

EVA | 0.00735 | |

Huangguan pear | PVC | 0.00323 |

EVA | 0.00638 | |

Qiuyue pear | PVC | 0.00627 |

EVA | 0.00714 |

Variety | Value of Maximum Rebound Height/mm | Value of Collision Recovery Coefficient |
---|---|---|

Snow pear | 100.40~194.88 | 0.501~0.698 |

Crisp pear | 71.57~106.50 | 0.423~0.516 |

Huangguan pear | 93.70~132.25 | 0.484~0.575 |

Qiuyue pear | 76.39~119.25 | 0.437~0.546 |

Number | Experiment Factors | Experiment Results | |||
---|---|---|---|---|---|

e | μ_{n} | μ_{f} | θ^{,}/(°) | σ/% | |

1 | 0.50 | 0.10 | 0.01 | 16.98 | 7.97 |

2 | 0.54 | 0.20 | 0.02 | 18.14 | 1.69 |

3 | 0.58 | 0.30 | 0.03 | 19.35 | 4.88 |

4 | 0.62 | 0.40 | 0.04 | 20.52 | 11.22 |

5 | 0.66 | 0.50 | 0.05 | 22.75 | 23.31 |

6 | 0.70 | 0.60 | 0.06 | 23.43 | 26.99 |

Code | Experiment Factors | ||
---|---|---|---|

e | μ_{n} | μ_{f} | |

−1.682 | 0.47 | 0.03 | 0.003 |

−1 | 0.50 | 0.1 | 0.01 |

0 | 0.54 | 0.2 | 0.02 |

1 | 0.58 | 0.3 | 0.03 |

1.682 | 0.61 | 0.37 | 0.037 |

Number | Parameter | θ^{,}/(°) | Y(σ)/% | ||
---|---|---|---|---|---|

A (e) | B(μ_{n}) | C(μ_{f}) | |||

1 | 1 | −1 | 1 | 17.46 | 5.37 |

2 | −1 | 1 | 1 | 18.95 | 2.71 |

3 | 0 | 0 | 0 | 18.13 | 1.73 |

4 | 0 | 1.682 | 0 | 18.65 | 1.08 |

5 | 0 | 0 | 0 | 18.00 | 2.44 |

6 | 1.682 | 0 | 0 | 17.94 | 2.76 |

7 | −1 | 1 | −1 | 18.23 | 1.19 |

8 | 1 | 1 | 1 | 19.09 | 3.47 |

9 | 1 | 1 | −1 | 17.91 | 2.93 |

10 | 0 | 0 | 0 | 18.26 | 1.03 |

11 | −1.682 | 0 | 0 | 18.19 | 1.41 |

12 | 0 | 0 | 0 | 18.12 | 1.79 |

13 | 1 | −1 | −1 | 16.94 | 8.18 |

14 | 0 | 0 | 1.682 | 18.35 | 0.54 |

15 | −1 | −1 | 1 | 17.79 | 3.58 |

16 | 0 | 0 | 0 | 18.24 | 1.14 |

17 | 0 | 0 | 0 | 18.29 | 0.87 |

18 | 0 | −1.682 | 0 | 16.79 | 9.00 |

19 | 0 | 0 | −1.682 | 17.72 | 3.96 |

20 | −1 | −1 | −1 | 16.88 | 8.51 |

Source | Sum of Squares | Degree of Freedom | Mean Square | F-Value | p-Value |
---|---|---|---|---|---|

Model | 121.46 | 9 | 13.50 | 14.90 | 0.0001 ** |

A | 2.85 | 1 | 2.85 | 3.14 | 0.1067 |

B | 60.09 | 1 | 60.09 | 66.33 | <0.0001 ** |

C | 9.57 | 1 | 9.57 | 10.57 | 0.0087 ** |

AB | 0.13 | 1 | 0.13 | 0.15 | 0.7101 |

AC | 0.16 | 1 | 0.16 | 0.18 | 0.6814 |

BC | 12.03 | 1 | 12.03 | 13.28 | 0.0045 ** |

A^{2} | 3.25 | 1 | 3.25 | 3.59 | 0.0873 |

B^{2} | 33.27 | 1 | 33.27 | 36.73 | 0.0001 ** |

C^{2} | 4.09 | 1 | 4.09 | 4.51 | 0.0596 |

Residual | 9.06 | 10 | 0.91 | ||

Lack of Fit | 7.29 | 5 | 1.46 | 4.11 | 0.0734 |

Pure error | 1.77 | 5 | 0.35 | ||

Sum | 130.52 | 19 |

Variety | Collision Recovery Coefficient | Static Friction Coefficient | Rolling Friction Coefficient |
---|---|---|---|

Snow pear | 0.54 | 0.27 | 0.020 |

Crisp pear | 0.44 | 0.24 | 0.024 |

Huangguan pear | 0.51 | 0.31 | 0.018 |

Qiuyue pear | 0.48 | 0.28 | 0.027 |

Variety | Measured Value | Simulated Value | Error/% |
---|---|---|---|

Snow pear | 15.53 | 15.75 | 1.42 |

Crisp pear | 15.04 | 15.29 | 1.68 |

Huangguan pear | 15.31 | 15.65 | 2.19 |

Qiuyue pear | 15.49 | 15.77 | 1.83 |

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Fan, G.; Wang, S.; Shi, W.; Gong, Z.; Gao, M.
Simulation Parameter Calibration and Test of Typical Pear Varieties Based on Discrete Element Method. *Agronomy* **2022**, *12*, 1720.
https://doi.org/10.3390/agronomy12071720

**AMA Style**

Fan G, Wang S, Shi W, Gong Z, Gao M.
Simulation Parameter Calibration and Test of Typical Pear Varieties Based on Discrete Element Method. *Agronomy*. 2022; 12(7):1720.
https://doi.org/10.3390/agronomy12071720

**Chicago/Turabian Style**

Fan, Guiju, Siyu Wang, Wenjie Shi, Zhenfeng Gong, and Ming Gao.
2022. "Simulation Parameter Calibration and Test of Typical Pear Varieties Based on Discrete Element Method" *Agronomy* 12, no. 7: 1720.
https://doi.org/10.3390/agronomy12071720