# Design and Performance Analysis of a Coal Bed Gas Drainage Machine Based on Incomplete Non-Circular Gears

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## Abstract

**:**

## 1. Introduction

## 2. Structure Design and Working Principle of Drainage Machines

#### 2.1. Overall Structure Design of Drainage Machines

#### 2.2. Transmission Scheme and Working Principle of Incomplete Non-Circle Gear Reversing Box

## 3. The Design of Non-Circular Gear Pair

#### 3.1. Transmission Principle of Non-Circular Gear Pair

_{1}and the instantaneous angular velocity as ω, setting the gear 2 angle as φ

_{2}and corresponding to the instantaneous angular velocity ω

_{2}. At the initial moment, φ

_{1}= 0, φ

_{2}= 0, the angular velocity ω

_{1}is described by ω

_{1}= dφ

_{1}/dt and angular velocity ω

_{2}= dφ

_{2}/dt. When gear 1 is engaged with gear 2, there is a certain rotation function relation:

_{1}, this is a finite and the positive smooth function. According to Equations (1)–(3), the angle function φ

_{2}of gear 2 can be obtained by:

_{1}, r

_{2}to be variable, that the instantaneous center P position is not fixed. At this time, the trajectory of instantaneous center P is a non-circular curve, which is the instantaneous line of two gears, called non-circular gear pitch curve.

_{1}and φ

_{2}is opposite to the corresponding rotation angle ω

_{1}and ω

_{2}. Therefore, when the center distance of a pair of non-circular gear pairs is determined, the pitch curve of the driving wheel 1 can be obtained by only knowing the transmission ratio function, and then the pitch curve of the driven wheel is obtained by Equation (7) [24,25].

#### 3.2. Design of the Non-Circular Gear Pair

_{1}as the origin of polar coordinates the curvature radius of the pitch curve varies counterclockwise increasing gradually in the Section 1 of 60 degrees before transmission, unchanged in Section 2 of 60~120 degrees, decreased in the Section 1 of 120~180 degrees, and it is unnecessary to calculate it in the Section 3 of 180~360 degrees.

_{2}as the origin of polar coordinates, the radius of curvature varies counterclockwise increased in Section 4, in does not need to be calculated in the no-tooth area 5, decreased again in Section 4, and unchanged in Section 6.

_{m}= π/6, T

_{3}= 2T

_{m}, K = (A − B)/2T

_{m}and J = (A − B)/T

_{m}

^{2}.

_{12}’ with the angle φ

_{1}change of the driving gear can be deduced as:

_{2}of the driven non-circular gear 4, 6 toothed part large in order to increase the non-circular gear’s meshing area and reduce the error, generally taking φ

_{2}as 300–330 degrees (300 degrees in this article). The design parameters of the non-circular gear are as follows: center distance of non-circular gear pair a = 360 mm, maximum transmission ratio i

_{12max}= 2.811, minimum transmission ratio i

_{12min}= 1/3, angle of driven non-circular gear 4, 6 = 300 degrees, so the pitch curve of driving non-circular gear 5 (Figure 12a) and non-circular curve section gears 4, 6 (Figure 12b) are calculated by a MATLAB program [27,28].

#### 3.3. Establishment of the Non-Circular Gear Pitch Curve

#### 3.4. Design of Non-Circular Gear Reversing Box

_{max}= 2.5 m, and roller diameter satisfies d = 300 mm. In a stroke, the speed of output shaft 10 is the same as that of the cylindrical spur gears 9 which is on output shaft 10 (see Figure 6):

_{max}= 2.5 m, the actual transmission ratio i

_{89}between standard cylindrical spur gears 8 and 9 will satisfy:

_{9}= 18, and large gears 8 and 7 satisfy Z

_{8}= Z

_{8}= 60, actual transmission ratio will satisfy:

_{max}= 2.6 m, diameter of the large cylindrical spur gears 8 and 7 are 720 mm, and the diameter of the small cylindrical spur gear 9 is 216 mm.

## 4. Efficiency Analysis of the New Drainage Machine and the Old

_{1}—motor working efficiency; η

_{2}—belt pulley working efficiency; η

_{3}—reducer transmission efficiency; η

_{4}—four-bar linkage transmission efficiency.

_{1}of the motor is only 0.8 at most.

_{3}is 0.91.

_{4}of the four-bar linkage is 0.95 when the efficiency of the wire rope is 0.98. Based on the above analysis, the ground efficiency of beam pumping unit can reach the maximum:

## 5. Kinematics Analysis Based on ADAMS

#### 5.1. Kinematics Simulation Analysis of the Gear System at Incomplete Non-Circular Gear Reversing Box

#### 5.2. Movement Analysis of Drainage Machine Polished Rod Displacement

## 6. Industrial Application

## 7. Conclusions

- (1)
- According to the characteristics of the inconvenient adjustment and low efficiency of the working parameters of the beam pumping unit, a new type of drainage machine is proposed to replace the conventional girder with the non-circular gear as the core for coalbed methane well pumping devices.
- (2)
- Through the transmission principle of the incomplete non-circular gear reversing box, a detailed design calculation of the incomplete non-circular gear is carried out, the non-uniform motion of non-circular gear pair is adopted to realize the variable speed movement in the stroke of the drainage machine, and the automatic reversing is realized by cooperating with other cylindrical gears in the gearbox, then the model of the incomplete non-circular gear and reversing gear system is established.
- (3)
- The paper analyzed the working performance of the drainage machine, and completed the theoretical calculation of the polished rod speed and acceleration; the efficiency of the new type of drainage machine and the conventional beam pumping unit is compared and analyzed simply, and the results show that the efficiency of the new type of drainage machine is about 11% higher than that of the traditional walking beam type pumping unit.
- (4)
- Whether the meshing process of one pair of non-circular gear pairs reaches the theoretical calculation value is verified by ADAMS; then the whole non-circular gear reversing gear train is simulated by ADAMS, and the output shaft speed and the simulation curve of the polished rod displacement are obtained. Based on the comparison with the theoretical values, the accuracy of the design is verified.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 5.**General structure of an automatic reversing vertical drainage machine for non-circular gears. 1—permanent magnet synchronous motor, 2—small belt pulley, 3—big belt pulley, 4—cycloid pinwheel reducer, 5—non-circular gear reversing box, 6—counterweight, 7—cylinder, 8—stroke reducer, 9—pumping rod, 10—transmission belt, 11—vertical frame, 12—crown sheave.

**Figure 6.**Sketch of a non-circular gear reversing box. 1—input shaft, 2, 3—transmission shaft, 4, 6—driven non-circular gear, 5—driving non-circular gear, 7, 8, 9—spur gear, 10—output shaft, 11—gear box.

**Figure 7.**Engagement condition of the non-circular gears when the input shaft rotates 180 degrees. 1—input shaft, 2, 3—transmission shaft, 4, 6-driven non-circular gear, 5—driving non-circular gear, 7, 8, 9—spur gear, 10—output shaft.

**Figure 12.**The pitch curve of non-circular gears. (

**a**) The pitch curve of gear 5; (

**b**) the pitch curve of gears 4 and 6.

**Figure 13.**Fitting curve of non-circular gears. (

**a**) Fitting curve of driving non-circular gear; (

**b**) fitting curve of driven non-circular gear.

**Figure 14.**Non-circular gear profile created by CAXA. (

**a**) Driving non-circular gear; (

**b**) driven non-circular gear.

**Figure 15.**Simulation video screenshot of the non-circular gear reversing box gear system’s movement.

**Figure 18.**Theoretical calculation value of output shaft speed of the non-circular gear reversing box.

Gear Name | Modulus | Number of Teeth | Angle of Pressure | Breadth of Tooth (mm) | Number |
---|---|---|---|---|---|

driving non-circular gear | 12 | 18 | 20 | 250 | 1 |

driven non-circular gear | 12 | 17 | 20 | 250 | 2 |

Standard cylindrical spur gear (big) | 12 | 60 | 20 | 190 | 2 |

Standard cylindrical spur gear (small) | 12 | 18 | 20 | 200 | 1 |

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**MDPI and ACS Style**

Xu, G.; Hua, D.; Dai, W.; Zhang, X.
Design and Performance Analysis of a Coal Bed Gas Drainage Machine Based on Incomplete Non-Circular Gears. *Energies* **2017**, *10*, 1933.
https://doi.org/10.3390/en10121933

**AMA Style**

Xu G, Hua D, Dai W, Zhang X.
Design and Performance Analysis of a Coal Bed Gas Drainage Machine Based on Incomplete Non-Circular Gears. *Energies*. 2017; 10(12):1933.
https://doi.org/10.3390/en10121933

**Chicago/Turabian Style**

Xu, Guiyun, Dezheng Hua, Weijun Dai, and Xiaoguang Zhang.
2017. "Design and Performance Analysis of a Coal Bed Gas Drainage Machine Based on Incomplete Non-Circular Gears" *Energies* 10, no. 12: 1933.
https://doi.org/10.3390/en10121933