# Lightweight Design and Dynamics Analysis of ZYL-15000D Directional Drill Reducer

Authors: Xiaohong Zhao; Xianguo Yan; Zhi Chen; Hang Su
Journal: Advances in Materials Science and Engineering
 (2021)

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## Abstract

Since the output torque of the rotary reducer of the ZYL-15000D-kilometer directional drill is proportional to the gear train transmission ratio, the output torque is large enough when the input speed and output speed meet the design goal. This paper selects the method of reducing the transmission ratio to optimize the size parameters of the reducer. MATLAB genetic algorithm is used in the optimization design, and the minimum volume of the rotary part reducer is taken as the objective of optimization design, while the design variables and constraints are determined. The optimal value of design variables was obtained through optimization, and the parameter values of each gear were determined accordingly. Through analysis, the total volume of the optimized gear reducer was reduced by 49.6%. Then, the 3D model of the optimized gear was created, and the analysis of the transient dynamics of the optimized gear was carried out with ANSYS Workbench software. According to the analysis results, the optimized gear met the strength requirements and provided a reference for the subsequent optimization design of other types of gear reducers.


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## Body

## 1. Introduction

Horizontal directional drilling is a trenchless construction method, which is usually completed by directional drilling RIGS, steering bits, and steering instruments [1]. The steps of the directional drill include drilling along the design path, reaming with a reamer, and pulling back of the product or casing [2, 3].Directional drilling technology is not only used in pipeline laying and geological exploration but more importantly in coal mine gas drainage and discharge operations. It is the key technology of coal mine gas control and can ensure efficient extraction and safe production of coal, so it has been listed as a key development direction in the national coal industry “The Thirteenth Five-Year” development plan [2, 4–7].The horizontal directional drilling machine is a piece of special equipment for underground gas extraction. It is a directional drilling technology with a screw motor with a bent joint. A screw motor is a kind of bottom-hole dynamic drilling tool with high-pressure washing fluid as the transmission power medium. It adopts screw drilling tools with different forms of deflecting parts and is equipped with measuring instruments while drilling, electrical control system, etc., which can flexibly meet the directional drilling needs of different design requirements [1, 8]. The reducer drives the power head to realize the all-round operation of the roadway. Due to the environmental characteristics of narrow roadway in coal mine, the drilling rig must be convenient for installation and disassembly. Therefore, under the condition of meeting the requirements of working conditions, it is very important to optimize the design of reducer [9]. The object of this paper is the reducer of the rotary part of ZYL-15000D horizontal directional drill. Taking the minimum volume of reducer as the optimization design goal, the mathematical model of the optimization design of directional drill reducer was established, including the objective function, design variables, and constraints. MATLAB software is used to the optimization design based on genetic algorithm, and the optimization results were obtained [10–12]. Creo was used to model the optimized gear to obtain the 3D model of the gear. Then, the 3D gear model was imported into ANSYS Workbench to perform transient dynamics analysis on the gear to verify the reliability of the optimized model [13–15].


## 2. Structural Analysis of Reducer

Figure1 shows the simplified transmission diagram of the reduction box of the rotary part of the drill. The overall structure is divided into two stages of reduction. The first stage of reduction involves the planetary gear system, and the second stage of reduction involves the helical gear group driven by the planetary frame. The gear parameters of the original gearbox are shown in Table 1.Figure 1
Transmission diagram of reduction gearbox.Table 1
Original design parameters of reducer.
ParameterGearSun gear aPlanetary gear cRing gear bHelical gear dDrive gear emn3.53.53.577Z1725671969B4141418585Material20CrNi420CrNi442CrMo20CrNi420CrNi4To facilitate the analysis of this optimization calculation, the whole gear train is divided into three zones: planetary gears A and B and helical gears.Because the transmission structure and transmission mode of planetary gears A and B are consistent, the calculation is mainly carried out by planetary gear train A and helical gear set.


## 3. Mathematical Model of Reducer Optimization Design

Genetic algorithm (GA) is an algorithm that simulates the evolution of organisms by computer based on the genetic characteristics of organisms; the general steps of genetic algorithm are as follows: first replace the feasible solution with chromosomes, and these produced chromosomes are called feasible solution set. Then, the chromosome is put into “nature” according to the law of biological evolution in nature, excellent individuals are selected for crossover and mutation, and new chromosome individuals are produced. After several times of screening, the optimal individual is finally converged according to the conditions, and the genetic algorithm completes the optimization [16–18].MATLAB genetic algorithm toolbox provides two methods; the two methods are command line call GA function and graphical user interface. The GUI method only needs to input the number of variables, the upper and lower limits of constraint values, theM file name of the prepared fitness function, and the M file name of the constraint function on the interface and set the population size, crossover probability, and mutation probability of the optional parameters to carry out optimization calculation [19–23]. The flowchart of the GA is shown in Figure 2.Figure 2
The flowchart of the GA.


### 3.1. Determining Design Variables

By analyzing the structure of the reducer, we can know that the size of the planet wheel, the center wheel, and the driving wheel can greatly change the volume of the reducer. Therefore, the tooth width, modulus, and tooth number of each gear are selected as design variables, and the number of planetary gears is 4. The design variables taken in this paper are as follows:(1)X=x1,x2,x3,x4,x5,x6,x7=b1,m1,za,zc,b2,m2,z1,where b1 and b2 are the tooth widths of high-speed and low-speed stages, m1 and m2 are the moduli of high-speed and low-speed stages, za is the number of teeth of the sun gear, zc is the number of planetary gears, and z1 is the number of helical gears.


### 3.2. The Objective Function of the Reducer

According to the symmetrical installation condition of planet gear, concentric condition of sun gear and planet gear, and the transmission ratio formula of planetary gear train, the volume function of the reducer is as follows [19, 24]:(2)fx=2f1x+f2x=2πm12b114za2+zc2+π4m22z12b21+i2.


### 3.3. Constraint Condition

(1)
Adjacency conditions: to ensure that the planet gear does not collide, make sure there is a gap between the tips of the two planetary gears on the center line, that is, the sum of the radius of the top circle of the two adjacent planets gear should be less than the center distance.(3)dac<2aac′sinπ4,wheredac is the diameter of the apical circle of the planetary wheel and aac′ is the center distance between the sun gear and the planet gear.(2)
The minimum number of teeth:(4)z≥zmin.The minimum number of teeth of the sun gear in the high-speed stage of the reducer is 14–18.(3)
The minimum modulus:(5)m1≥2.86,m2≥5,wherem1 is the modulus of the planetary gear train and m2 is the modulus of the helical gear set.(4)
Minimum tooth width:(6)b≥10.(5)
Coefficient of face width:For planetary gear sets:0.8≤b1/da≤1.4;For the helical gear set:1≤b2/d2≤1.4;whereb1 is the tooth width of the planetary gear train; da is the diameter of the sun gear’s indexing circle; b2 is the tooth width of the helical gear set; and m2 is the modulus of the helical gear set.(6)
Contact fatigue strength of tooth surface:(7)da≥KdT1KAKHΣKHpϕdσHlim2×u+1u3,whereKd is the coefficient of the formula. If a pair of gears is spur, then Kd=768; if a pair of gears is helical, then Kd=720; T1 is the nominal torque of the pinion, N•m; KA is the use coefficient; u is the ratio of teeth; KHΣ is the comprehensive coefficient; KHp is the non-uniform coefficient of the load distribution of the planetary wheel when calculating the contact strength; ϕd is the pinion tooth width coefficient; and σHlim is the contact fatigue strength of the gear, N/mm2.“In the formula, “+” is for external engagement, and “-” is for internal engagement.(7)
Bending fatigue strength of tooth root:(8)m≥KmT1KAKFΣKFpYFa1ϕdz12σFlim3,whereKm is the coefficient of calculation (in spur gear transmission, Km=12.1; in helical gear transmission, Km=11.5); KFΣ is the comprehensive coefficient; KFp is the non-uniform coefficient of load distribution between planetary wheels when calculating bending strength; z1 is the number of pinion teeth; and σFlim is the bending fatigue strength of the test gear, N/mm2.(8)
Assembly conditions:(9)za+zbnp=CInteger.Since assembly conditions are difficult to be directly substituted in the constraint formula, they can be used as the constraint conditions to determine the number of teeth of the inner ring gear after optimization.za is the number of sun gear teeth; zb is the number of teeth of the inner ring gear; and np is the number of planet gears, and the value in this paper is 4.(9)
The transmission ratio error of planetary gear:(10)1−2za+2zczaip−4≤0,whereip is the rated transmission ratio.(10)
Interference problem of the inner ring gear. As shown in Figure3, in the process of optimizing the size, the two inner gear rings may interfere with each other because the driving gear C shrinks too much. The interference constraints of inner ring gear can be obtained as follows:(11)1.2≤1/2m2z11+iinclinem1za+2zc≤1.4,whereiincline is the transmission ratio of the helical gear set.


### 3.4. Optimization Case Study and Optimization Result Analysis

#### 3.4.1. Reducer Case Study

In this part, the relevant parameters of the reducer optimization design of directional drilling machine are given, the input torque isTout=380N•M, the output torque is Tenter=7200N•M, the output speed is nout=150r/min, the transmission efficiency of the planetary gear is ηRow=0.97, the transmission efficiency of the helical gear is ηincline=0.98, and the design life of the reducer is 5∼7 years.Figure 3
Gear installation position diagram.The material used for planetary gear train is 20CrMnTi, and the hardness of the material is 217HB after carburizing and quenching treatment. The material used for the helical gear group is 20CrNi4, and the hardness of the material is 296HB after carburizing and quenching treatment. The accuracy of all the gears is 7 grade.(1)
Determination of transmission ratio at each stage: according to the original parameters of the reducer, before the optimization design, the transmission ratio of the planetary gear train is 5, and the transmission ratio of the helical gear is 3.6. Because the output torque is proportional to the transmission ratio, the output torque is sufficient while the input and output speeds meet the design objectives. Under the condition of meeting the overall layout of the reducer, the purpose of reducing the weight and saving the cost is achieved by optimizing the parameters of the reducer. According to the distribution principle of the transmission ratio of the second-level reducer, it can be known that after the transmission ratio is redistributed, the planetary gear train isiRow=3.55, the helical gear set is iincline=2.8, and the spiral angle is 10°.(2)
Determination of input torque at each stage: in the planetary gear train, the input torque of the sun gear at each power split is as follows:(12)T1=Tenternp=3804N•M=95N•M.The input torque of the helical gear set is as follows:(13)T2=Tenter×iRow×ηRow=1308.53N•M.(3)
The allowable contact stress for high-speed gear isσH1=1591MPa, allowable bending stress of tooth root is σF1=346.5MPa, the allowable contact stress of the tooth surface of the low-speed gear is σH2=1295MPa, and the allowable bending stress of the tooth root is σF2=443.52MPa.Put the above parameters into the constraint condition, and the constraint function is as follows [25]:(14)c1x=−22za+2−22zc<−2,c2x=0.8m1za−b1≤0,c3x=b1−1.4m1za≤0,c4x=m2z1−b2≤0,c5x=b2−1.4m2z1≤0,c6x=88382.7765×za+zcb1zc−m12za2≤0,c7x=549047.233×1b2−m22z12≤0,c8x=7335.3088×1b1za−m12≤0,c9x=27675.4×1b2z1−m22≤0,c10x=1−2za+2zczaip−4%≤0,c11x=1.2m1za+2zc−12m2z11+iincline≤0,c12x=12m2z11+iincline−1.4m1za+2zc≤0.


#### 3.4.2. Optimization Design and Result Analysis

In this study, the optimization problem belongs to the optimization solution of the single-objective function nonlinear constraint minimization problem [26]. By using the MATLAB genetic algorithm toolbox, users only need to input the number of variables, the upper and lower limits of constraint values, the file name of fitness function M, and file name of constraint condition function M on the interface and set the population size, crossover probability, and mutation probability to calculate the optimal value [27, 28]. The specific steps are as follows:(1)
Write theM file for the fitness function, as shown in Figure 4.(2)
Write theM file of the constraints, as shown in Figure 5.(3)
Use the genetic algorithm toolbox of MATLAB to call the written fitness functionM file and constraint condition function M file and input the number of variables, the corresponding linear constraint value, and boundary, as shown in Figure 6. The crossover probability is 0.6, the mutation probability is 0.08, and the other parameters are the default parameters of the system.Figure 4
M file of the objective function.Figure 5
M files for constraints.Figure 6
Genetic algorithm toolbox interface.After the setting is complete, click Start to run, and the running result is shown in Figure7.Figure 7
Optimization results of genetic algorithm.Since the number of teeth and moduli are discrete variables, the results should be rounded, and the rounded design variables should be substituted into the constraint conditions to verify that they meet the requirements [29]. The comparison between the final parameters and the original data is as follows.Table2 shows that the volume of the reducer optimized by the genetic algorithm is reduced by 49.6% compared with the original volume. The optimized reducer has a smaller volume, a more compact structure, and saves materials. Finally, the optimization goal is achieved.Table 2
Comparison of reducer parameters before and after optimization.
ParametersResults before optimizationResults after optimizationSun gearRing gearPlanetary gearHelical gearDrive gearSun gearRing gearPlanetary gearHelical gearDrive gearm (mm)3.53.53.57733355z17672519692357172159b (mm)4141418585686868103103Material20CrNi442CrMo20CrNi420CrNi420CrNi420CrMnTi42CrMo20CrMnTi20CrNi420CrNi4Volume (mm3)18945724.29546848.15Volume decrease ratio49.6%


## 3.1. Determining Design Variables

By analyzing the structure of the reducer, we can know that the size of the planet wheel, the center wheel, and the driving wheel can greatly change the volume of the reducer. Therefore, the tooth width, modulus, and tooth number of each gear are selected as design variables, and the number of planetary gears is 4. The design variables taken in this paper are as follows:(1)X=x1,x2,x3,x4,x5,x6,x7=b1,m1,za,zc,b2,m2,z1,where b1 and b2 are the tooth widths of high-speed and low-speed stages, m1 and m2 are the moduli of high-speed and low-speed stages, za is the number of teeth of the sun gear, zc is the number of planetary gears, and z1 is the number of helical gears.


## 3.2. The Objective Function of the Reducer

According to the symmetrical installation condition of planet gear, concentric condition of sun gear and planet gear, and the transmission ratio formula of planetary gear train, the volume function of the reducer is as follows [19, 24]:(2)fx=2f1x+f2x=2πm12b114za2+zc2+π4m22z12b21+i2.


## 3.3. Constraint Condition

(1)
Adjacency conditions: to ensure that the planet gear does not collide, make sure there is a gap between the tips of the two planetary gears on the center line, that is, the sum of the radius of the top circle of the two adjacent planets gear should be less than the center distance.(3)dac<2aac′sinπ4,wheredac is the diameter of the apical circle of the planetary wheel and aac′ is the center distance between the sun gear and the planet gear.(2)
The minimum number of teeth:(4)z≥zmin.The minimum number of teeth of the sun gear in the high-speed stage of the reducer is 14–18.(3)
The minimum modulus:(5)m1≥2.86,m2≥5,wherem1 is the modulus of the planetary gear train and m2 is the modulus of the helical gear set.(4)
Minimum tooth width:(6)b≥10.(5)
Coefficient of face width:For planetary gear sets:0.8≤b1/da≤1.4;For the helical gear set:1≤b2/d2≤1.4;whereb1 is the tooth width of the planetary gear train; da is the diameter of the sun gear’s indexing circle; b2 is the tooth width of the helical gear set; and m2 is the modulus of the helical gear set.(6)
Contact fatigue strength of tooth surface:(7)da≥KdT1KAKHΣKHpϕdσHlim2×u+1u3,whereKd is the coefficient of the formula. If a pair of gears is spur, then Kd=768; if a pair of gears is helical, then Kd=720; T1 is the nominal torque of the pinion, N•m; KA is the use coefficient; u is the ratio of teeth; KHΣ is the comprehensive coefficient; KHp is the non-uniform coefficient of the load distribution of the planetary wheel when calculating the contact strength; ϕd is the pinion tooth width coefficient; and σHlim is the contact fatigue strength of the gear, N/mm2.“In the formula, “+” is for external engagement, and “-” is for internal engagement.(7)
Bending fatigue strength of tooth root:(8)m≥KmT1KAKFΣKFpYFa1ϕdz12σFlim3,whereKm is the coefficient of calculation (in spur gear transmission, Km=12.1; in helical gear transmission, Km=11.5); KFΣ is the comprehensive coefficient; KFp is the non-uniform coefficient of load distribution between planetary wheels when calculating bending strength; z1 is the number of pinion teeth; and σFlim is the bending fatigue strength of the test gear, N/mm2.(8)
Assembly conditions:(9)za+zbnp=CInteger.Since assembly conditions are difficult to be directly substituted in the constraint formula, they can be used as the constraint conditions to determine the number of teeth of the inner ring gear after optimization.za is the number of sun gear teeth; zb is the number of teeth of the inner ring gear; and np is the number of planet gears, and the value in this paper is 4.(9)
The transmission ratio error of planetary gear:(10)1−2za+2zczaip−4≤0,whereip is the rated transmission ratio.(10)
Interference problem of the inner ring gear. As shown in Figure3, in the process of optimizing the size, the two inner gear rings may interfere with each other because the driving gear C shrinks too much. The interference constraints of inner ring gear can be obtained as follows:(11)1.2≤1/2m2z11+iinclinem1za+2zc≤1.4,whereiincline is the transmission ratio of the helical gear set.


## 3.4. Optimization Case Study and Optimization Result Analysis

### 3.4.1. Reducer Case Study

In this part, the relevant parameters of the reducer optimization design of directional drilling machine are given, the input torque isTout=380N•M, the output torque is Tenter=7200N•M, the output speed is nout=150r/min, the transmission efficiency of the planetary gear is ηRow=0.97, the transmission efficiency of the helical gear is ηincline=0.98, and the design life of the reducer is 5∼7 years.Figure 3
Gear installation position diagram.The material used for planetary gear train is 20CrMnTi, and the hardness of the material is 217HB after carburizing and quenching treatment. The material used for the helical gear group is 20CrNi4, and the hardness of the material is 296HB after carburizing and quenching treatment. The accuracy of all the gears is 7 grade.(1)
Determination of transmission ratio at each stage: according to the original parameters of the reducer, before the optimization design, the transmission ratio of the planetary gear train is 5, and the transmission ratio of the helical gear is 3.6. Because the output torque is proportional to the transmission ratio, the output torque is sufficient while the input and output speeds meet the design objectives. Under the condition of meeting the overall layout of the reducer, the purpose of reducing the weight and saving the cost is achieved by optimizing the parameters of the reducer. According to the distribution principle of the transmission ratio of the second-level reducer, it can be known that after the transmission ratio is redistributed, the planetary gear train isiRow=3.55, the helical gear set is iincline=2.8, and the spiral angle is 10°.(2)
Determination of input torque at each stage: in the planetary gear train, the input torque of the sun gear at each power split is as follows:(12)T1=Tenternp=3804N•M=95N•M.The input torque of the helical gear set is as follows:(13)T2=Tenter×iRow×ηRow=1308.53N•M.(3)
The allowable contact stress for high-speed gear isσH1=1591MPa, allowable bending stress of tooth root is σF1=346.5MPa, the allowable contact stress of the tooth surface of the low-speed gear is σH2=1295MPa, and the allowable bending stress of the tooth root is σF2=443.52MPa.Put the above parameters into the constraint condition, and the constraint function is as follows [25]:(14)c1x=−22za+2−22zc<−2,c2x=0.8m1za−b1≤0,c3x=b1−1.4m1za≤0,c4x=m2z1−b2≤0,c5x=b2−1.4m2z1≤0,c6x=88382.7765×za+zcb1zc−m12za2≤0,c7x=549047.233×1b2−m22z12≤0,c8x=7335.3088×1b1za−m12≤0,c9x=27675.4×1b2z1−m22≤0,c10x=1−2za+2zczaip−4%≤0,c11x=1.2m1za+2zc−12m2z11+iincline≤0,c12x=12m2z11+iincline−1.4m1za+2zc≤0.


### 3.4.2. Optimization Design and Result Analysis

In this study, the optimization problem belongs to the optimization solution of the single-objective function nonlinear constraint minimization problem [26]. By using the MATLAB genetic algorithm toolbox, users only need to input the number of variables, the upper and lower limits of constraint values, the file name of fitness function M, and file name of constraint condition function M on the interface and set the population size, crossover probability, and mutation probability to calculate the optimal value [27, 28]. The specific steps are as follows:(1)
Write theM file for the fitness function, as shown in Figure 4.(2)
Write theM file of the constraints, as shown in Figure 5.(3)
Use the genetic algorithm toolbox of MATLAB to call the written fitness functionM file and constraint condition function M file and input the number of variables, the corresponding linear constraint value, and boundary, as shown in Figure 6. The crossover probability is 0.6, the mutation probability is 0.08, and the other parameters are the default parameters of the system.Figure 4
M file of the objective function.Figure 5
M files for constraints.Figure 6
Genetic algorithm toolbox interface.After the setting is complete, click Start to run, and the running result is shown in Figure7.Figure 7
Optimization results of genetic algorithm.Since the number of teeth and moduli are discrete variables, the results should be rounded, and the rounded design variables should be substituted into the constraint conditions to verify that they meet the requirements [29]. The comparison between the final parameters and the original data is as follows.Table2 shows that the volume of the reducer optimized by the genetic algorithm is reduced by 49.6% compared with the original volume. The optimized reducer has a smaller volume, a more compact structure, and saves materials. Finally, the optimization goal is achieved.Table 2
Comparison of reducer parameters before and after optimization.
ParametersResults before optimizationResults after optimizationSun gearRing gearPlanetary gearHelical gearDrive gearSun gearRing gearPlanetary gearHelical gearDrive gearm (mm)3.53.53.57733355z17672519692357172159b (mm)4141418585686868103103Material20CrNi442CrMo20CrNi420CrNi420CrNi420CrMnTi42CrMo20CrMnTi20CrNi420CrNi4Volume (mm3)18945724.29546848.15Volume decrease ratio49.6%


## 3.4.1. Reducer Case Study

In this part, the relevant parameters of the reducer optimization design of directional drilling machine are given, the input torque isTout=380N•M, the output torque is Tenter=7200N•M, the output speed is nout=150r/min, the transmission efficiency of the planetary gear is ηRow=0.97, the transmission efficiency of the helical gear is ηincline=0.98, and the design life of the reducer is 5∼7 years.Figure 3
Gear installation position diagram.The material used for planetary gear train is 20CrMnTi, and the hardness of the material is 217HB after carburizing and quenching treatment. The material used for the helical gear group is 20CrNi4, and the hardness of the material is 296HB after carburizing and quenching treatment. The accuracy of all the gears is 7 grade.(1)
Determination of transmission ratio at each stage: according to the original parameters of the reducer, before the optimization design, the transmission ratio of the planetary gear train is 5, and the transmission ratio of the helical gear is 3.6. Because the output torque is proportional to the transmission ratio, the output torque is sufficient while the input and output speeds meet the design objectives. Under the condition of meeting the overall layout of the reducer, the purpose of reducing the weight and saving the cost is achieved by optimizing the parameters of the reducer. According to the distribution principle of the transmission ratio of the second-level reducer, it can be known that after the transmission ratio is redistributed, the planetary gear train isiRow=3.55, the helical gear set is iincline=2.8, and the spiral angle is 10°.(2)
Determination of input torque at each stage: in the planetary gear train, the input torque of the sun gear at each power split is as follows:(12)T1=Tenternp=3804N•M=95N•M.The input torque of the helical gear set is as follows:(13)T2=Tenter×iRow×ηRow=1308.53N•M.(3)
The allowable contact stress for high-speed gear isσH1=1591MPa, allowable bending stress of tooth root is σF1=346.5MPa, the allowable contact stress of the tooth surface of the low-speed gear is σH2=1295MPa, and the allowable bending stress of the tooth root is σF2=443.52MPa.Put the above parameters into the constraint condition, and the constraint function is as follows [25]:(14)c1x=−22za+2−22zc<−2,c2x=0.8m1za−b1≤0,c3x=b1−1.4m1za≤0,c4x=m2z1−b2≤0,c5x=b2−1.4m2z1≤0,c6x=88382.7765×za+zcb1zc−m12za2≤0,c7x=549047.233×1b2−m22z12≤0,c8x=7335.3088×1b1za−m12≤0,c9x=27675.4×1b2z1−m22≤0,c10x=1−2za+2zczaip−4%≤0,c11x=1.2m1za+2zc−12m2z11+iincline≤0,c12x=12m2z11+iincline−1.4m1za+2zc≤0.


## 3.4.2. Optimization Design and Result Analysis

In this study, the optimization problem belongs to the optimization solution of the single-objective function nonlinear constraint minimization problem [26]. By using the MATLAB genetic algorithm toolbox, users only need to input the number of variables, the upper and lower limits of constraint values, the file name of fitness function M, and file name of constraint condition function M on the interface and set the population size, crossover probability, and mutation probability to calculate the optimal value [27, 28]. The specific steps are as follows:(1)
Write theM file for the fitness function, as shown in Figure 4.(2)
Write theM file of the constraints, as shown in Figure 5.(3)
Use the genetic algorithm toolbox of MATLAB to call the written fitness functionM file and constraint condition function M file and input the number of variables, the corresponding linear constraint value, and boundary, as shown in Figure 6. The crossover probability is 0.6, the mutation probability is 0.08, and the other parameters are the default parameters of the system.Figure 4
M file of the objective function.Figure 5
M files for constraints.Figure 6
Genetic algorithm toolbox interface.After the setting is complete, click Start to run, and the running result is shown in Figure7.Figure 7
Optimization results of genetic algorithm.Since the number of teeth and moduli are discrete variables, the results should be rounded, and the rounded design variables should be substituted into the constraint conditions to verify that they meet the requirements [29]. The comparison between the final parameters and the original data is as follows.Table2 shows that the volume of the reducer optimized by the genetic algorithm is reduced by 49.6% compared with the original volume. The optimized reducer has a smaller volume, a more compact structure, and saves materials. Finally, the optimization goal is achieved.Table 2
Comparison of reducer parameters before and after optimization.
ParametersResults before optimizationResults after optimizationSun gearRing gearPlanetary gearHelical gearDrive gearSun gearRing gearPlanetary gearHelical gearDrive gearm (mm)3.53.53.57733355z17672519692357172159b (mm)4141418585686868103103Material20CrNi442CrMo20CrNi420CrNi420CrNi420CrMnTi42CrMo20CrMnTi20CrNi420CrNi4Volume (mm3)18945724.29546848.15Volume decrease ratio49.6%


## 4. Transient Dynamics Analysis of the Reducer

In this part, to verify the reliability of the optimized reducer, transient dynamics analysis of the planetary gear set and helical gear set in the reducer was carried out by using ANSYS Workbench.


### 4.1. Transient Dynamics Equation

Transient dynamics analysis is a method used to determine the dynamic response of a structure under any time-varying load. By transient dynamics analysis, the time-varying displacements, forces, stresses, and strains of the structure under the random combination of transient, steady, and harmonic loads can be determined [30].The basic equation of motion for transient dynamics is as follows:(15)Mu¨+Cu˙+Ku=Ft,where M is the mass matrix; C is the damping matrix; K is stiffness matrix; u¨ is nodal acceleration vector; u˙ is node velocity vector; and u is the node displacement vector.


### 4.2. The Establishment of Finite Element Model

The optimized gear was modeled by Creo, and then the finite element model of planetary gear and helical gear was obtained, as shown in Figures8(a) and 8(b). Save the 3D model as .STP format.Figure 8
.3D model of gears. (a) 3D model of planetary gear. (b) 3D model of helical gear.
(a)(b)


### 4.3. Establishment of the Transient Dynamics Model

#### 4.3.1. Definition of Gear Material

The transient structure was opened in ANSYS Workbench, and the 3D model of the gears was imported into ANSYS Workbench for analysis, and its material properties were defined. 20CrMnTi is used for both the sun wheel and the planet wheel, 42CrMo is used for the inner gear ring, and 20CrNi4 is used for the helical gear set. The material properties are shown in Table3 [31].Table 3
Material properties of gears.
No.Property20CrMnTi42CrMo20CrNi41Density （Kg/m³）7800785078002Poisson’s ratio0.250.260.293Young’s modulus （Pa）2.07 E+112.21 E+112.07 E+11


#### 4.3.2. Finite Element Mesh Determination of Gears

The gear model was imported, tetrahedral mesh was adopted, 1/3 of the minimum side length of the gear was taken as the minimum mesh size to divide, and the medium smoothing method was considered. Sizing command was used to control the mesh size of 3 mm in planetary gear train, and contact sizing command was added to encrypt the mesh of the contact region of the gear, and the mesh size was 1 mm. Sizing command was used to control the mesh unit size of 5 mm in the helical gear group, and contact sizing command was added to encrypt the mesh in the contact region of the gear, and the mesh size was 2 mm. After setting, the mesh will be generated, and the average quality of the mesh should be controlled above 0.7 [32]. The contact problem of gears is a highly nonlinear behavior, which requires a lot of computing resources. In order to improve the operation efficiency, reduce the amount of computer calculation, minimize the size of the model on the premise of satisfying the calculation accuracy, and ensure that it is close to the actual working conditions, and the model in Creo needs to be simplified. In the planetary gear train, because there are four planetary gears in contact with the sun gear, the load on each planetary gear is uniform. Therefore, you only need to analyze and calculate one group. In the helical gear set, only the gear meshing part is kept for analysis and calculation. Generated finite element meshes are presented in Figure 9 [33, 34].Figure 9
Finite element mesh of gears. (a) Planetary gear. (b) Helical gears.
(a)(b)


#### 4.3.3. Boundary Conditions and Load Conditions

Boundary conditions in planetary gear train include frictionless contact between solar gear and planetary gear, as shown in Figure10(a); frictionless contact between planetary gear and inner ring gear, as shown in Figure10(b); the rotation pair between the planetary frame and the planetary wheel, as shown in Figure 11(a); the rotation pair of the solar wheel to the earth, as shown in Figure 11(b); the rotation pair of the planetary frame to the earth, as shown in Figure 11(c); and the fixed pair of the inner ring to the ground, as shown in Figure 12.Figure 10
Friction contact pairs in a planetary gear train. (a) The contact pair between the sun gear and the planetary gear. (b) Contact pair of planetary gear and inner ring gear.
(a)(b)Figure 11
Rotation pairs in planetary gear system. (a) Rotation pairs between planet carrier and planetary gear. (b) Rotation pairs between sun gear and earth. (c) Rotation pairs between planet carrier and earth.
(a)(b)(c)Figure 12
Fixed pair of inner gear ring to ground.In the helical gear set, the boundary conditions include frictionless contact pairs of the pinion and the large gear and rotation pairs of the two gears against the ground, as shown in Figure13 [35].Figure 13
Boundary conditions of helical gears. (a) Helical gear contact pair. (b) The ground pair of helical gears.
(a)(b)In the planetary gear train, the sun gear is chosen as the driving wheel, and the load is applied to the sun gear in the form of torque, and the planet rack is the output. In the helical gear, the pinion is selected as the driving gear, and the load is also applied to the pinion in the form of torque.


## 4.1. Transient Dynamics Equation

Transient dynamics analysis is a method used to determine the dynamic response of a structure under any time-varying load. By transient dynamics analysis, the time-varying displacements, forces, stresses, and strains of the structure under the random combination of transient, steady, and harmonic loads can be determined [30].The basic equation of motion for transient dynamics is as follows:(15)Mu¨+Cu˙+Ku=Ft,where M is the mass matrix; C is the damping matrix; K is stiffness matrix; u¨ is nodal acceleration vector; u˙ is node velocity vector; and u is the node displacement vector.


## 4.2. The Establishment of Finite Element Model

The optimized gear was modeled by Creo, and then the finite element model of planetary gear and helical gear was obtained, as shown in Figures8(a) and 8(b). Save the 3D model as .STP format.Figure 8
.3D model of gears. (a) 3D model of planetary gear. (b) 3D model of helical gear.
(a)(b)


## 4.3. Establishment of the Transient Dynamics Model

### 4.3.1. Definition of Gear Material

The transient structure was opened in ANSYS Workbench, and the 3D model of the gears was imported into ANSYS Workbench for analysis, and its material properties were defined. 20CrMnTi is used for both the sun wheel and the planet wheel, 42CrMo is used for the inner gear ring, and 20CrNi4 is used for the helical gear set. The material properties are shown in Table3 [31].Table 3
Material properties of gears.
No.Property20CrMnTi42CrMo20CrNi41Density （Kg/m³）7800785078002Poisson’s ratio0.250.260.293Young’s modulus （Pa）2.07 E+112.21 E+112.07 E+11


### 4.3.2. Finite Element Mesh Determination of Gears

The gear model was imported, tetrahedral mesh was adopted, 1/3 of the minimum side length of the gear was taken as the minimum mesh size to divide, and the medium smoothing method was considered. Sizing command was used to control the mesh size of 3 mm in planetary gear train, and contact sizing command was added to encrypt the mesh of the contact region of the gear, and the mesh size was 1 mm. Sizing command was used to control the mesh unit size of 5 mm in the helical gear group, and contact sizing command was added to encrypt the mesh in the contact region of the gear, and the mesh size was 2 mm. After setting, the mesh will be generated, and the average quality of the mesh should be controlled above 0.7 [32]. The contact problem of gears is a highly nonlinear behavior, which requires a lot of computing resources. In order to improve the operation efficiency, reduce the amount of computer calculation, minimize the size of the model on the premise of satisfying the calculation accuracy, and ensure that it is close to the actual working conditions, and the model in Creo needs to be simplified. In the planetary gear train, because there are four planetary gears in contact with the sun gear, the load on each planetary gear is uniform. Therefore, you only need to analyze and calculate one group. In the helical gear set, only the gear meshing part is kept for analysis and calculation. Generated finite element meshes are presented in Figure 9 [33, 34].Figure 9
Finite element mesh of gears. (a) Planetary gear. (b) Helical gears.
(a)(b)


### 4.3.3. Boundary Conditions and Load Conditions

Boundary conditions in planetary gear train include frictionless contact between solar gear and planetary gear, as shown in Figure10(a); frictionless contact between planetary gear and inner ring gear, as shown in Figure10(b); the rotation pair between the planetary frame and the planetary wheel, as shown in Figure 11(a); the rotation pair of the solar wheel to the earth, as shown in Figure 11(b); the rotation pair of the planetary frame to the earth, as shown in Figure 11(c); and the fixed pair of the inner ring to the ground, as shown in Figure 12.Figure 10
Friction contact pairs in a planetary gear train. (a) The contact pair between the sun gear and the planetary gear. (b) Contact pair of planetary gear and inner ring gear.
(a)(b)Figure 11
Rotation pairs in planetary gear system. (a) Rotation pairs between planet carrier and planetary gear. (b) Rotation pairs between sun gear and earth. (c) Rotation pairs between planet carrier and earth.
(a)(b)(c)Figure 12
Fixed pair of inner gear ring to ground.In the helical gear set, the boundary conditions include frictionless contact pairs of the pinion and the large gear and rotation pairs of the two gears against the ground, as shown in Figure13 [35].Figure 13
Boundary conditions of helical gears. (a) Helical gear contact pair. (b) The ground pair of helical gears.
(a)(b)In the planetary gear train, the sun gear is chosen as the driving wheel, and the load is applied to the sun gear in the form of torque, and the planet rack is the output. In the helical gear, the pinion is selected as the driving gear, and the load is also applied to the pinion in the form of torque.


## 4.3.1. Definition of Gear Material

The transient structure was opened in ANSYS Workbench, and the 3D model of the gears was imported into ANSYS Workbench for analysis, and its material properties were defined. 20CrMnTi is used for both the sun wheel and the planet wheel, 42CrMo is used for the inner gear ring, and 20CrNi4 is used for the helical gear set. The material properties are shown in Table3 [31].Table 3
Material properties of gears.
No.Property20CrMnTi42CrMo20CrNi41Density （Kg/m³）7800785078002Poisson’s ratio0.250.260.293Young’s modulus （Pa）2.07 E+112.21 E+112.07 E+11


## 4.3.2. Finite Element Mesh Determination of Gears

The gear model was imported, tetrahedral mesh was adopted, 1/3 of the minimum side length of the gear was taken as the minimum mesh size to divide, and the medium smoothing method was considered. Sizing command was used to control the mesh size of 3 mm in planetary gear train, and contact sizing command was added to encrypt the mesh of the contact region of the gear, and the mesh size was 1 mm. Sizing command was used to control the mesh unit size of 5 mm in the helical gear group, and contact sizing command was added to encrypt the mesh in the contact region of the gear, and the mesh size was 2 mm. After setting, the mesh will be generated, and the average quality of the mesh should be controlled above 0.7 [32]. The contact problem of gears is a highly nonlinear behavior, which requires a lot of computing resources. In order to improve the operation efficiency, reduce the amount of computer calculation, minimize the size of the model on the premise of satisfying the calculation accuracy, and ensure that it is close to the actual working conditions, and the model in Creo needs to be simplified. In the planetary gear train, because there are four planetary gears in contact with the sun gear, the load on each planetary gear is uniform. Therefore, you only need to analyze and calculate one group. In the helical gear set, only the gear meshing part is kept for analysis and calculation. Generated finite element meshes are presented in Figure 9 [33, 34].Figure 9
Finite element mesh of gears. (a) Planetary gear. (b) Helical gears.
(a)(b)


## 4.3.3. Boundary Conditions and Load Conditions

Boundary conditions in planetary gear train include frictionless contact between solar gear and planetary gear, as shown in Figure10(a); frictionless contact between planetary gear and inner ring gear, as shown in Figure10(b); the rotation pair between the planetary frame and the planetary wheel, as shown in Figure 11(a); the rotation pair of the solar wheel to the earth, as shown in Figure 11(b); the rotation pair of the planetary frame to the earth, as shown in Figure 11(c); and the fixed pair of the inner ring to the ground, as shown in Figure 12.Figure 10
Friction contact pairs in a planetary gear train. (a) The contact pair between the sun gear and the planetary gear. (b) Contact pair of planetary gear and inner ring gear.
(a)(b)Figure 11
Rotation pairs in planetary gear system. (a) Rotation pairs between planet carrier and planetary gear. (b) Rotation pairs between sun gear and earth. (c) Rotation pairs between planet carrier and earth.
(a)(b)(c)Figure 12
Fixed pair of inner gear ring to ground.In the helical gear set, the boundary conditions include frictionless contact pairs of the pinion and the large gear and rotation pairs of the two gears against the ground, as shown in Figure13 [35].Figure 13
Boundary conditions of helical gears. (a) Helical gear contact pair. (b) The ground pair of helical gears.
(a)(b)In the planetary gear train, the sun gear is chosen as the driving wheel, and the load is applied to the sun gear in the form of torque, and the planet rack is the output. In the helical gear, the pinion is selected as the driving gear, and the load is also applied to the pinion in the form of torque.


## 5. Transient Dynamics Analysis Results and Discussion

According to the above settings, the total deformation program, equivalent stress program, and equivalent strain program of the gear train are obtained after transient dynamics analysis of the gear train, as shown in Figures14–21 [36, 37]. Since the maximum strain and stress in the planetary gear train occur in the sun gear, only the stress and strain on the sun gear are analyzed.Figure 14
Deformation program of the planetary gear train.Figure 15
Equivalent strain program of the planetary gear train. (a) Equivalent strain program. (b) Program of sun gear strain.
(a)(b)Figure 16
Program of equivalent stress of the planetary gear train. (a) Equivalent stress program. (b) Program of sun gear stress.
(a)(b)Figure 17
Root stress of the planetary gear train. (a) Root stress cloud of the sun gear. (b) Root stress cloud of the planetary gear.
(a)(b)Figure 18
Deformation nephogram of helical gear set.Figure 19
Equivalent strain program of the helical gear group. (a) Equivalent strain program. (b) Strain program of helical gear.
(a)(b)Figure 20
Equivalent stress nephogram of the helical gear group. (a) Equivalent stress nephogram. (b) Stress nephogram of helical gear.
(a)(b)Figure 21
Root stress of the helical gear group. (a) Root stress cloud of the helical gear. (b) Root stress cloud of the drive gear.
(a)(b)Table4 shows the output response of transient dynamics analysis, and Table 5 shows the maximum output response of each contact pair in the reducer gear train.Table 4
Output response of finite element analysis.
No.Output characteristicsSymbolSI unit1Total deformationR1mm2Equivalent elastic strainR2—3Equivalent von Mises stressR3MPaTable 5
Maximum results of finite element analysis.
Output variableSun gear-planetary gearPlanetary gear-internal gearHelical gearR10.13640.09090.79741R20.011610.005160.0024692R3705.67306.82478.23As shown in Figures13–15, the maximum deformation in the planetary gear train occurs at the edge of the sun gear tooth, and the maximum deformation is 0.1364 mm. Both the maximum strain and the maximum stress appear near the node line of the sun gear. The maximum strain is 0.01161, and the maximum stress is 705.67 MPa. As shown in Figures 17 and 18, the maximum deformation of the helical gear group appears at the edge of the helical gear, and the maximum deformation is 0.7974 mm. The maximum strain and maximum stress both appear near the pitch line of the helical gear, and the maximum strain is 2.4692 and the maximum stress is 478.23 MPa [33, 38].Figures16 and 20 show the stress diagrams of the planetary gear train and the helical gear group. The root stress on the tensile side is the key factor leading to tooth failure, so the stress on the tensile side is mainly considered [39]. In the planetary gear train, the stress value of the teeth root radius on the tensile side of the sun gear and the planet gear is 393.84MPa and 472.61MPa, respectively, which is less than the yield limit of the materials. In the helical gear set, the root stress on the tensile side of the two gears is about 288.3MPa. It is also less than the yield limit of the materials.


## 6. Conclusion

In this paper, MATLAB genetic algorithm is used to carry out a lightweight design for the directional drilling gear reducer, and the optimized gear parameters are obtained. Through calculation, the volume of the optimized model is reduced by 49.6% compared with the original gear model, achieving the purpose of being lightweight (Figure21).Three-dimensional modeling software was used to model the optimized gear, and transient dynamics analysis of planetary gear set and helical gear set was carried out with ANSYS Workbench. Through analysis, we can obtain the convergent force curve the convergent displacement curve and the response cloud graph. Through analysis, the maximum stress and strain appear on the contact line of gear meshing. The maximum stress is less than the yield limit of the sun gear material, and the maximum strain is less than the elongation and section shrinkage of the sun gear material. In the helical gear group, the maximum stress and strain also appear on the contact line of gear meshing, and the maximum stress value is less than the yield limit of the helical gear material, and the maximum strain value is less than the elongation and section shrinkage of the helical gear material. Therefore, the optimized gear meets the strength requirements and achieves the purpose of being lightweight.



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*Source: 7164276-2021-12-20.xml*