# Optimized Design for Reliability of Pointer Irrigation Machine Components for Intelligent Computing

Authors: Chengwen Liu; Guoliang Liu; Haitao Zhang
Journal: International Transactions on Electrical Energy Systems
 (2022)

---

## Abstract

The application of intelligent technology has realized the transformation of people’s production and lifestyle, and also promoted the development and transformation of the agricultural field. At present, the application of agricultural intelligence is getting stronger and stronger; using its intelligent advanced methods and technologies, this paper aimed to achieve the optimization of sprinkler irrigation machine parts in the intelligent network environment to promote the rapid development of agriculture, and proposed the use of the NSGA-II algorithm in intelligent computing to guide the integration of artificial intelligence and pointer sprinkler parts, which helps to analyze and solve the objective problem of machine failure and parts damage in agriculture. In the study of the sprinkler gear system, from the perspective of gear efficiency, since it is optimized according to the minimum efficiency point of the fourth gear of the gear reducer, compared with the gear efficiency of 49.05% before this point, the efficiency of this point after optimization is 59.45%, and the minimum efficiency point will be increased by 21.2%. And because the energy loss unrelated to the power loss load will be greatly reduced, these energy losses have a greater relationship with the structure of the gearbox. In terms of each gear, compared with the previous period, the efficiency of the first gear was increased by 8.5% to 15.9%; the efficiency of the second gear increased by 8.7% to 17.4%; the efficiency of the third gear increased by 9.4% to 18.7%; and the efficiency of the fourth gear increased by 10.1% to 21.2%. Therefore, it is currently necessary to optimize the components of the sprinkler irrigation machine.


---

## Body

## 1. Introduction

At present, the area of new water-saving irrigation projects in China needs to exceed 300 million mu, and the effective utilization coefficient of farmland irrigation water exceeds 0.55. In agricultural production, traditional flood irrigation and other methods have large evaporation and low water resource utilization. In contrast, the efficiency of sprinkler irrigation can reach 40%–50%, and the spraying uniformity is high, which can increase the yield of crops. Therefore, it is of great significance to increase the use of agricultural sprinkler irrigation technology to alleviate the shortage of freshwater resources in China. The pointer sprinkler first appeared in Europe in the early 1960s, and compared with other types of sprinkler equipment, this type of sprinkler appeared relatively late. Due to the short life and high price of the hose at that time, it was not widely promoted. China’s per capita freshwater resources are only a quarter of the world’s level, so protecting and saving water resources is our inescapable responsibility. Agriculture is a major water user. In China’s existing 2.025 billion mu of arable land, the effective irrigation area of farmland is 952 million mu, and the utilization rate of irrigation water is only 52%. In order to ensure agricultural water use and promote water conservation, China will further expand the construction of modern farmland water conservancy, increase the high-efficiency water-saving irrigation area of 1.0 × 10°mu, and increase the effective utilization coefficient of agricultural irrigation water to more than 0.55.The pointer sprinkler has become one of the most promising agricultural irrigation equipment in China due to its advantages of strong mobility, convenient maintenance, and high spraying uniformity. However, due to the drawbacks of the traditional scheme design, the cylindrical gear reduction transmission efficiency of its matching core component is low. Especially under low-load and low-speed conditions, the transmission efficiency is much lower than that of the same type of gearbox, and the energy loss is relatively serious. At the same time, the transmission shaft of the reducer, especially the output shaft connected to the reel, has a low service life under high-load conditions and is prone to breakage. Therefore, in view of the above problems, this paper is based on the optimization simulation of the cylindrical gear reducer matched with the JP75 pointer-type sprinkler irrigation machine and the simulation calculation of the energy consumption of the whole machine. When the pointer sprinkler is working, first fix the frame at one end of the field, and drag the PE pipe wound on the reel together with the sprinkler truck to the other end of the field. At the same time, the speed of the drive device and the gear position of the transmission device are set according to the actual working conditions. The transmission device drives the large crankset on the reel, and then recycles the PE pipe, thereby realizing the automatic sprinkler irrigation process of recycling while spraying. One of the key components in the whole pointer sprinkler is the reduction box in the transmission device. The performance of the reduction box has a great impact on the work efficiency, energy consumption, and service life of the pointer sprinkler. Therefore, vigorously researching the reduction box has a more positive impact on improving the performance of the pointer sprinkler and developing the agricultural machinery irrigation equipment.Reliability optimization of pivot irrigation parts has profound implications for the agricultural sector. Among them, Vaux et al. resented its focus on the interaction between sprinkler systems and stratified smoke layers due to pool fires in mechanically ventilated enclosures in pivot system design [1]. Xu et al. proposed that the high-fidelity visualization of fires in urban areas is rarely mentioned in existing models. Therefore, the research on pointer-based sprinklers is the focus of current research [2]. Liu et al. used the digital weighting method to perform statistical analysis on the distribution of water droplet size and summed up the variation law of water droplet size frequency and cumulative frequency distribution under pressure at different distances from the sprinkler head of the sprinkler [3]. Soni et al.’s (2015) experiment was conducted with a randomized block design including 11 treatments. The results have shown that the water-saving rates of drip irrigation at 100% and 75% PE are 40.08% and 55.06%, respectively [4]. Naderianfar et al. experiments were carried out at the meteorological station of the Faculty of Agriculture, Ferdowsi University, Mashhad, using a line-source sprinkler irrigation system based on the single-sprinkler installation method [5]. However, due to the problems of equipment and data collection in the above research, the accuracy of the designed parts often only exists in the theoretical stage and has little practicality.It is a very novel idea to use an intelligent algorithm to analyze the reliability of the parts of the sprinkler irrigation machine, and many scholars have studied it. Among them: Xiang et al. conducted a series of experiments to study the ability of the intake structure to draw air and water into the impingement nozzle, and the suction ability was measured by the mass flow of water and the degree of vacuum [6]. Li et al. proposed to use the Matlab genetic algorithm toolbox to optimize the key parameters of the system such as the number of teeth and the tooth width coefficient of the spur gear drive and worm drive of the sprinkler [7]. Experiments by Bhupendra et al. observed, recorded, and compared the performance of a solar sprinkler with a conventional still, where the water flow on the glass cover and nanoparticles was modified in the basin water [8]. Zhu studied the irrigation uniformity under no wind conditions and obtained the radial water distribution of the 10-type full-jet sprinkler through experiments [9]. Efficient intercropping patterns, use of efficient irrigation systems, and proper irrigation scheduling in El-Mehy A A are one of the challenges currently facing the agricultural sector in terms of water conservation, crop yield, and economic maximization [10]. However, the current research on the optimization of parts between intelligent algorithms and sprinklers still does not get rid of the traditional thinking and definition of parts design, and lacks in-depth analysis and discussion on the functionality of intelligent algorithms, which also hinders the high integration and advantages of intelligent technology and agriculture.The innovations of this paper are as follows: (1) Analyzing the energy loss form of the gearbox supporting the pointer sprinkler irrigation machine and deriving the mathematical model of the gearbox’s transmission efficiency. Taking the transmission efficiency and gear volume of the reducer as the optimization objects, the multiobjective optimization is carried out through the fast nondominated genetic algorithm NSGA-II, and the decision is made through the objective weighted gray target model to select the optimal solution. The optimized reduction box not only improves the transmission efficiency of each gear to different extents but also the overall structure of the gear system becomes more compact. (2) Flexibility of the key gears in the gear system of the reduction box, and constraints between the components. At the same time, the contact force function between the gear pairs is set based on the impact function, and the load and speed under the maximum working condition are added to establish the rigid-flexible coupling dynamic model of the gear system. By analyzing the key gear speed, the time domain, and frequency domain diagrams of the meshing force, as well as its stress and strain, it is verified that the optimized geometric parameters of the gearbox gear meet the design requirements. (3) Based on the inverse simulation method, the block diagram of load, transmission, drive, and energy supply are established in Simulink and connected in series to obtain the energy consumption simulation model of the pointer sprinkler. By calculating the energy consumption value of the sprinkler truck under the same spraying length, different gears of the reduction box, and different driving speeds, the optimal form speed of the sprinkler truck with the lowest energy consumption under different gears of the reduction box is determined.


## 2. Optimization of the Parts of the Pointer Sprinkler

### 2.1. Transmission System Model

The pointer-type sprinkler gearbox is composed of two power input shafts and one power output shaft. Due to the different operating conditions of the pointer sprinkler, the large range of reel load and speed change, and to make the matching permanent magnet brushless DC motor and reduction box work within the high-efficiency area as much as possible, it is necessary to set a variety of transmission ratios to reduce energy usage. Therefore, the gearbox of the transmission system is a four-speed transmission, and its internal structure and transmission diagram are shown in Figure1.Figure 1
Transmission system diagram. (a) Internal structure diagram. (b) Transmission diagram.
(a)(b)


### 2.2. Transmission Ratio

As can be seen, when the motor drives the input shaft 1, the reduction box is an unfolded five-stage cylindrical gear reduction box. When the input shaft 2 is shaken by hand, the reduction box is an unfolding two-stage cylindrical gear reduction box, and the axis position of each gear is fixed on the reduction box housing, forming a fixed-axis gear train [11]. The transmission ratio of the fixed-axis gear train is the product of the gear ratios when each pair of gear pairs is engaged, which is also equal to the ratio of the product of the number of teeth of all driven gears in the gear train and the product of the number of teeth of the driving gear. Assuming that A represents the input shaft and B represents the output shaft, the formula for calculating the transmission ratio of each gear of the reducer is as follows:(1)iAB=wAwB.The transmission system of the pointer sprinkler is composed of the gear transmission of the reduction box and the chain transmission with a fixed transmission ratio. The parameters of each gear are shown in Table1. Combining formula (1), the transmission ratio of each gear of the entire transmission system is shown in Table 2.Table 1
Gear parameters.
DentitionNumber of teethModule (mm)Tooth width (mm)1153.5102413.5103391.564231.575131.586171.5217301.5238421.589151.57Table 2
Transmission ratio of each gear in the transmission system.
GearFirst gearSecond gearThird gearFourth gearQuick recovery fileGear ratio623.0291.4202.2102.29.2Chain drive ratio22.522.522.522.522.5Total gear ratio14017.56556.54549.52299.5207.0


### 2.3. Mathematical Model of Gearbox Transmission Efficiency

#### 2.3.1. Form of Power Loss

When the sprinkler truck is spraying normally, the gearbox is driven by a five-stage cylindrical gear. During the transmission process, in addition to lubrication, external temperature, and other factors, the geometric parameters of gears, bearings, and other components have a decisive impact on the efficiency of the gearbox [12, 13]. The power loss of the gearbox can be divided into load related and load-independent, and the specific structure is shown in Figure 2.Figure 2
Schematic diagram of the power loss composition of the gearbox.


#### 2.3.2. Mathematical Model of Transmission Efficiency

Because under the actual working conditions, clutch 2 will be closed only in emergencies, and the PE pipe will be recovered by shaking the input shaft 2 by hand. When the input shaft 1 is driven by the normal motor, the quick recovery gear under the input shaft 2 will not rotate due to the disconnection of the clutch 2, and the related gears, bearings, etc. will not produce power loss. The purpose of this paper is to establish the transmission efficiency model of the gearbox with the motor connected to the input shaft 1, so the input shaft 2 and its related components will not be considered in the establishment of the mathematical model of the gearbox transmission efficiency and the following gearbox optimization.As the largest part of power loss, gear meshing power loss is mainly composed of sliding friction power loss and rolling friction power loss [14–16]. The normal load of the tooth surface and the relative sliding speed of the tooth surface are the decisive factors of the meshing power loss. When the two gears are meshed, the magnitude and direction of the relative sliding speed of the two tooth surfaces at different meshing positions are also constantly changing. If a certain transient situation is analyzed, it is not practical to calculate the sliding and rolling friction power losses under the condition of elastohydrodynamic lubrication. Therefore, the average power loss of gear meshing must also be calculated. And GBISO/TR-14179-1 derives the semiempirical formula of gear meshing power loss P, which is expressed as follows:(2)PM=fmT1n1cos2β9.55M.In the formula,fm is the meshing friction factor; Ti is the torque of the driving wheel in the gear pair, N∗m:n is the rotational speed of the driving wheel in the gear pair, and r/min:β is the helix angle on the pitch circle.


#### 2.3.3. Establishment of Gear Flexible Body Model

Adams is based on the modal superposition method, which represents the elastic deformation of flexible body components by combining modal vectors and modal coordinates. The modal neutral file is a binary file containing information such as node coordinates, stress, strain, mode, mass, and inertia of flexible body components. Therefore, in Adams, the modal neutral file is the carrier of the flexible body.There are usually two ways to create a modal neutral file: one is to use the Flexiblebodies module that comes with Adams View to directly create a modal neutral file of a flexible body, which is suitable for components with low requirements on mesh accuracy and simple geometric shapes; the other is to mesh the components and perform modal analysis based on finite-element software, save the analysis results as modal neutral files, and import them into Adams to establish the flexible body model of the components. Because the gear structure is relatively complex and the mesh precision is relatively high, the second method is adopted in this paper to establish the modal neutral file of the flexible gear through ANSYS [17]. In order to improve the simulation efficiency, before the gear is flexible, it is necessary to simplify the gear reasonably, remove the chamfer, keyway, and other features that have little impact on the simulation calculation results, and the modal neutral file establishment steps are shown in Figure 3.Figure 3
Modal neutral file steps.For the steps of creating a modal neutral file, the key operations for creating a modal neutral file are as follows:Since the soli185 element is defined by eight nodes, and each node has 3 degrees of freedom in three directions, and has the advantages of large deformation and superelasticity, the gear model uses this solid element for mapping mesh division [18, 19]. Since the contact area of the gear teeth is relatively complex, in order to ensure the calculation accuracy, the gear teeth part is refined.Since the gear rotates, the rotation center of the gear is used as the interface point. During the rotation of the gear, compared with the tooth root and tooth surface, the stress and strain in the area near the rotation center are much smaller, so this area is regarded as a rigid area and the rotation center is connected with the nearby nodes through the beam188 element.The units of ANSYS and Adams need to be unified when outputting. At the same time, in order to ensure that each element contains component information such as stress and strain when extracting the modal order, the modal order of the modal neutral file must be equal to modal order in ANSYS +6 × interface point. In this paper, the first 14 modes of each gear are taken. The flexible gear is shown in Figure4:Figure 4
Flexible gear model diagram.


### 2.4. Multiobjective Optimization Design of Gearbox Based on Nsga-II Algorithm

In traditional multiobjective optimization design methods, multiobjective optimization is usually converted into single-objective optimization by weighting all objective functions, which can only lead to one solution. The fast nondominant sorting genetic algorithm is a genetic algorithm based on the concept of Pareto optimal solution. Compared with the previous NSGA, a fast nondominant sorting algorithm is proposed, and an elite strategy is introduced, as well as congestion comparison operators [20]. This ensures that the individuals in the population are more evenly distributed in the Pareto field, guaranteeing the diversity of the population. Its obvious advantage is the reduced complexity of inferior genetic algorithms, and an additional benefit is an excellent solution set convergence and fast execution speed, making it a benchmark for multiobjective optimization algorithms. Its flowchart is shown in Figure 5:Figure 5
Algorithm flow.As can be seen from Figure5, since the number of gears in the entire reduction box is relatively large, and the parameters corresponding to each gear structure are also relatively large, preprocessing is performed before the overall optimization of the reduction box.The original transmission ratio of each gear of the reduction box is not changed, and the shift gear modules on shaft 2 and shaft 3 are the same as gear 1, and other parameters remain unchanged.In order to avoid the pinching phenomenon caused by gear meshing, the radius of the transition fillet at the top of the tooth is 0.4 times the modulus of the gear, and the radius of the transition fillet at the root of the tooth is 0.38 times the modulus of the gear. At the same time, the gears used are standard spur gears [21, 22].In a pair of gear pairs, in order to ensure the length of the contact line, the tooth width of the pinion is 2 mm∼5 mm larger than that of the large gear. Based on the above preprocessing, the number of teeth of gears 1, 2, 7, 12, 13, 14, 15, and 16, as well as the corresponding module and pinion tooth width coefficient, these 16 parameters are selected as design variables, namely:(3)x=z1,z2,z7,z12,z13,z14,z15,z16,m1,m2,m3,m4,ϕ1,ϕ2,ϕ3,ϕ4.The transmission ratio error is generally controlled within the 4% error range. Combined with the original transmission ratio, except for the transmission ratio of the shift gear pair, the constraints on the number of teeth of the remaining gears are as follows:(4)134≤Z2Z1∗Z12Z7∗Z15Z13∗Z16Z14≤145.For single-stage spur gears, the transmission ratio needs to be less than 8. The reducer box is an unfolded five-stage cylindrical gearbox, and there are many transmission shafts and stages, so the agreed single-stage transmission ratio is less than 6. At the same time, according to the principle that the transmission ratio is small in front and large in the back, the following constraints are made:(5)1z2z1z12z7z15z13z16z146.For gears that transmit power from the gearbox, the minimum modulus is 1.5 mm. Due to the gradual increase in the torque of the gears at all levels, the modulus of the gears on each shaft should comply with the increasing rules, and the first series specified in the national standard is adopted, so the modulus constraints are as follows:(6)1.5≤m1≤m2≤m3≤m4.For standard gears, the minimum number of teeth without undercutting is 17. In order to avoid undercutting of gear meshing and ensure that the number of gear teeth is not too high, the constraints on the number of gear teeth are as follows:(7)17≤zi≤65,i=1,2,7,12,16.Since the gears in the reduction box are asymmetrically arranged and hard-toothed gears, the transmission capacity is reduced in order to ensure the bearing capacity of the gears and not cause eccentric loads. Based on the actual working conditions, the constraints on the tooth width of the pinion are as follows:(8)0.2≤ϕi≤0.6,i=1,7,3,14.Due to the nonstandard design of gear teeth and modules in the design process, the interference between the gear and the shaft is prone to occur, so this phenomenon should be avoided in the optimization design, therefore the following constraints are imposed on the gear and the shaft:(9)z2+2m816.5,z7m2+z13m2110,z13m3+z15m3−z12+2m325,z14m4+z16m4−z15+2m435.Therefore, the constraint on the bending fatigue strength of the tooth root is as follows:(10)σF=2KT1b1m12z1YF1YF2≤σFSF,YF1=0.169+0.006666z1−0.000854z12,YF2=0.2824+0.0003539iz1−0.000001576iz12.In the formula,K is the load factor. Since the reel sprinkler is a machine that runs smoothly, the value is 1.3. T1 is the nominal torque of the pinion in the gear pair, and N∗m;σF is the allowable bending fatigue strength of the material. All gears are made of 40Cr and are quenched and tempered. The allowable bending fatigue strength is 470 MPa and the allowable contact fatigue strength is 600 MPa. SF is the bending strength safety factor, according to GB3480-83, the value is 1.5; Y is the pinion tooth shape factor; and YF1 is the stress correction factor. z, m1, and bt are the number of teeth, module, and tooth width of the pinion (driving wheel) in the gear pair; i is the transmission ratio at all levels. The fatigue strength of the tooth surface is as follows:(11)σH=ZEZH2KT1i+1b1d12i≤σHSH.In the formula,ZE is the elastic coefficient; 40Cr steel material value is 189.8MPa; σH is the node area coefficient; the value 2.5 : 4 represents the allowable contact fatigue strength of the material; and Sr is the contact strength safety factor and according to GB3480-83, the value is 1.5.


## 2.1. Transmission System Model

The pointer-type sprinkler gearbox is composed of two power input shafts and one power output shaft. Due to the different operating conditions of the pointer sprinkler, the large range of reel load and speed change, and to make the matching permanent magnet brushless DC motor and reduction box work within the high-efficiency area as much as possible, it is necessary to set a variety of transmission ratios to reduce energy usage. Therefore, the gearbox of the transmission system is a four-speed transmission, and its internal structure and transmission diagram are shown in Figure1.Figure 1
Transmission system diagram. (a) Internal structure diagram. (b) Transmission diagram.
(a)(b)


## 2.2. Transmission Ratio

As can be seen, when the motor drives the input shaft 1, the reduction box is an unfolded five-stage cylindrical gear reduction box. When the input shaft 2 is shaken by hand, the reduction box is an unfolding two-stage cylindrical gear reduction box, and the axis position of each gear is fixed on the reduction box housing, forming a fixed-axis gear train [11]. The transmission ratio of the fixed-axis gear train is the product of the gear ratios when each pair of gear pairs is engaged, which is also equal to the ratio of the product of the number of teeth of all driven gears in the gear train and the product of the number of teeth of the driving gear. Assuming that A represents the input shaft and B represents the output shaft, the formula for calculating the transmission ratio of each gear of the reducer is as follows:(1)iAB=wAwB.The transmission system of the pointer sprinkler is composed of the gear transmission of the reduction box and the chain transmission with a fixed transmission ratio. The parameters of each gear are shown in Table1. Combining formula (1), the transmission ratio of each gear of the entire transmission system is shown in Table 2.Table 1
Gear parameters.
DentitionNumber of teethModule (mm)Tooth width (mm)1153.5102413.5103391.564231.575131.586171.5217301.5238421.589151.57Table 2
Transmission ratio of each gear in the transmission system.
GearFirst gearSecond gearThird gearFourth gearQuick recovery fileGear ratio623.0291.4202.2102.29.2Chain drive ratio22.522.522.522.522.5Total gear ratio14017.56556.54549.52299.5207.0


## 2.3. Mathematical Model of Gearbox Transmission Efficiency

### 2.3.1. Form of Power Loss

When the sprinkler truck is spraying normally, the gearbox is driven by a five-stage cylindrical gear. During the transmission process, in addition to lubrication, external temperature, and other factors, the geometric parameters of gears, bearings, and other components have a decisive impact on the efficiency of the gearbox [12, 13]. The power loss of the gearbox can be divided into load related and load-independent, and the specific structure is shown in Figure 2.Figure 2
Schematic diagram of the power loss composition of the gearbox.


### 2.3.2. Mathematical Model of Transmission Efficiency

Because under the actual working conditions, clutch 2 will be closed only in emergencies, and the PE pipe will be recovered by shaking the input shaft 2 by hand. When the input shaft 1 is driven by the normal motor, the quick recovery gear under the input shaft 2 will not rotate due to the disconnection of the clutch 2, and the related gears, bearings, etc. will not produce power loss. The purpose of this paper is to establish the transmission efficiency model of the gearbox with the motor connected to the input shaft 1, so the input shaft 2 and its related components will not be considered in the establishment of the mathematical model of the gearbox transmission efficiency and the following gearbox optimization.As the largest part of power loss, gear meshing power loss is mainly composed of sliding friction power loss and rolling friction power loss [14–16]. The normal load of the tooth surface and the relative sliding speed of the tooth surface are the decisive factors of the meshing power loss. When the two gears are meshed, the magnitude and direction of the relative sliding speed of the two tooth surfaces at different meshing positions are also constantly changing. If a certain transient situation is analyzed, it is not practical to calculate the sliding and rolling friction power losses under the condition of elastohydrodynamic lubrication. Therefore, the average power loss of gear meshing must also be calculated. And GBISO/TR-14179-1 derives the semiempirical formula of gear meshing power loss P, which is expressed as follows:(2)PM=fmT1n1cos2β9.55M.In the formula,fm is the meshing friction factor; Ti is the torque of the driving wheel in the gear pair, N∗m:n is the rotational speed of the driving wheel in the gear pair, and r/min:β is the helix angle on the pitch circle.


### 2.3.3. Establishment of Gear Flexible Body Model

Adams is based on the modal superposition method, which represents the elastic deformation of flexible body components by combining modal vectors and modal coordinates. The modal neutral file is a binary file containing information such as node coordinates, stress, strain, mode, mass, and inertia of flexible body components. Therefore, in Adams, the modal neutral file is the carrier of the flexible body.There are usually two ways to create a modal neutral file: one is to use the Flexiblebodies module that comes with Adams View to directly create a modal neutral file of a flexible body, which is suitable for components with low requirements on mesh accuracy and simple geometric shapes; the other is to mesh the components and perform modal analysis based on finite-element software, save the analysis results as modal neutral files, and import them into Adams to establish the flexible body model of the components. Because the gear structure is relatively complex and the mesh precision is relatively high, the second method is adopted in this paper to establish the modal neutral file of the flexible gear through ANSYS [17]. In order to improve the simulation efficiency, before the gear is flexible, it is necessary to simplify the gear reasonably, remove the chamfer, keyway, and other features that have little impact on the simulation calculation results, and the modal neutral file establishment steps are shown in Figure 3.Figure 3
Modal neutral file steps.For the steps of creating a modal neutral file, the key operations for creating a modal neutral file are as follows:Since the soli185 element is defined by eight nodes, and each node has 3 degrees of freedom in three directions, and has the advantages of large deformation and superelasticity, the gear model uses this solid element for mapping mesh division [18, 19]. Since the contact area of the gear teeth is relatively complex, in order to ensure the calculation accuracy, the gear teeth part is refined.Since the gear rotates, the rotation center of the gear is used as the interface point. During the rotation of the gear, compared with the tooth root and tooth surface, the stress and strain in the area near the rotation center are much smaller, so this area is regarded as a rigid area and the rotation center is connected with the nearby nodes through the beam188 element.The units of ANSYS and Adams need to be unified when outputting. At the same time, in order to ensure that each element contains component information such as stress and strain when extracting the modal order, the modal order of the modal neutral file must be equal to modal order in ANSYS +6 × interface point. In this paper, the first 14 modes of each gear are taken. The flexible gear is shown in Figure4:Figure 4
Flexible gear model diagram.


## 2.3.1. Form of Power Loss

When the sprinkler truck is spraying normally, the gearbox is driven by a five-stage cylindrical gear. During the transmission process, in addition to lubrication, external temperature, and other factors, the geometric parameters of gears, bearings, and other components have a decisive impact on the efficiency of the gearbox [12, 13]. The power loss of the gearbox can be divided into load related and load-independent, and the specific structure is shown in Figure 2.Figure 2
Schematic diagram of the power loss composition of the gearbox.


## 2.3.2. Mathematical Model of Transmission Efficiency

Because under the actual working conditions, clutch 2 will be closed only in emergencies, and the PE pipe will be recovered by shaking the input shaft 2 by hand. When the input shaft 1 is driven by the normal motor, the quick recovery gear under the input shaft 2 will not rotate due to the disconnection of the clutch 2, and the related gears, bearings, etc. will not produce power loss. The purpose of this paper is to establish the transmission efficiency model of the gearbox with the motor connected to the input shaft 1, so the input shaft 2 and its related components will not be considered in the establishment of the mathematical model of the gearbox transmission efficiency and the following gearbox optimization.As the largest part of power loss, gear meshing power loss is mainly composed of sliding friction power loss and rolling friction power loss [14–16]. The normal load of the tooth surface and the relative sliding speed of the tooth surface are the decisive factors of the meshing power loss. When the two gears are meshed, the magnitude and direction of the relative sliding speed of the two tooth surfaces at different meshing positions are also constantly changing. If a certain transient situation is analyzed, it is not practical to calculate the sliding and rolling friction power losses under the condition of elastohydrodynamic lubrication. Therefore, the average power loss of gear meshing must also be calculated. And GBISO/TR-14179-1 derives the semiempirical formula of gear meshing power loss P, which is expressed as follows:(2)PM=fmT1n1cos2β9.55M.In the formula,fm is the meshing friction factor; Ti is the torque of the driving wheel in the gear pair, N∗m:n is the rotational speed of the driving wheel in the gear pair, and r/min:β is the helix angle on the pitch circle.


## 2.3.3. Establishment of Gear Flexible Body Model

Adams is based on the modal superposition method, which represents the elastic deformation of flexible body components by combining modal vectors and modal coordinates. The modal neutral file is a binary file containing information such as node coordinates, stress, strain, mode, mass, and inertia of flexible body components. Therefore, in Adams, the modal neutral file is the carrier of the flexible body.There are usually two ways to create a modal neutral file: one is to use the Flexiblebodies module that comes with Adams View to directly create a modal neutral file of a flexible body, which is suitable for components with low requirements on mesh accuracy and simple geometric shapes; the other is to mesh the components and perform modal analysis based on finite-element software, save the analysis results as modal neutral files, and import them into Adams to establish the flexible body model of the components. Because the gear structure is relatively complex and the mesh precision is relatively high, the second method is adopted in this paper to establish the modal neutral file of the flexible gear through ANSYS [17]. In order to improve the simulation efficiency, before the gear is flexible, it is necessary to simplify the gear reasonably, remove the chamfer, keyway, and other features that have little impact on the simulation calculation results, and the modal neutral file establishment steps are shown in Figure 3.Figure 3
Modal neutral file steps.For the steps of creating a modal neutral file, the key operations for creating a modal neutral file are as follows:Since the soli185 element is defined by eight nodes, and each node has 3 degrees of freedom in three directions, and has the advantages of large deformation and superelasticity, the gear model uses this solid element for mapping mesh division [18, 19]. Since the contact area of the gear teeth is relatively complex, in order to ensure the calculation accuracy, the gear teeth part is refined.Since the gear rotates, the rotation center of the gear is used as the interface point. During the rotation of the gear, compared with the tooth root and tooth surface, the stress and strain in the area near the rotation center are much smaller, so this area is regarded as a rigid area and the rotation center is connected with the nearby nodes through the beam188 element.The units of ANSYS and Adams need to be unified when outputting. At the same time, in order to ensure that each element contains component information such as stress and strain when extracting the modal order, the modal order of the modal neutral file must be equal to modal order in ANSYS +6 × interface point. In this paper, the first 14 modes of each gear are taken. The flexible gear is shown in Figure4:Figure 4
Flexible gear model diagram.


## 2.4. Multiobjective Optimization Design of Gearbox Based on Nsga-II Algorithm

In traditional multiobjective optimization design methods, multiobjective optimization is usually converted into single-objective optimization by weighting all objective functions, which can only lead to one solution. The fast nondominant sorting genetic algorithm is a genetic algorithm based on the concept of Pareto optimal solution. Compared with the previous NSGA, a fast nondominant sorting algorithm is proposed, and an elite strategy is introduced, as well as congestion comparison operators [20]. This ensures that the individuals in the population are more evenly distributed in the Pareto field, guaranteeing the diversity of the population. Its obvious advantage is the reduced complexity of inferior genetic algorithms, and an additional benefit is an excellent solution set convergence and fast execution speed, making it a benchmark for multiobjective optimization algorithms. Its flowchart is shown in Figure 5:Figure 5
Algorithm flow.As can be seen from Figure5, since the number of gears in the entire reduction box is relatively large, and the parameters corresponding to each gear structure are also relatively large, preprocessing is performed before the overall optimization of the reduction box.The original transmission ratio of each gear of the reduction box is not changed, and the shift gear modules on shaft 2 and shaft 3 are the same as gear 1, and other parameters remain unchanged.In order to avoid the pinching phenomenon caused by gear meshing, the radius of the transition fillet at the top of the tooth is 0.4 times the modulus of the gear, and the radius of the transition fillet at the root of the tooth is 0.38 times the modulus of the gear. At the same time, the gears used are standard spur gears [21, 22].In a pair of gear pairs, in order to ensure the length of the contact line, the tooth width of the pinion is 2 mm∼5 mm larger than that of the large gear. Based on the above preprocessing, the number of teeth of gears 1, 2, 7, 12, 13, 14, 15, and 16, as well as the corresponding module and pinion tooth width coefficient, these 16 parameters are selected as design variables, namely:(3)x=z1,z2,z7,z12,z13,z14,z15,z16,m1,m2,m3,m4,ϕ1,ϕ2,ϕ3,ϕ4.The transmission ratio error is generally controlled within the 4% error range. Combined with the original transmission ratio, except for the transmission ratio of the shift gear pair, the constraints on the number of teeth of the remaining gears are as follows:(4)134≤Z2Z1∗Z12Z7∗Z15Z13∗Z16Z14≤145.For single-stage spur gears, the transmission ratio needs to be less than 8. The reducer box is an unfolded five-stage cylindrical gearbox, and there are many transmission shafts and stages, so the agreed single-stage transmission ratio is less than 6. At the same time, according to the principle that the transmission ratio is small in front and large in the back, the following constraints are made:(5)1z2z1z12z7z15z13z16z146.For gears that transmit power from the gearbox, the minimum modulus is 1.5 mm. Due to the gradual increase in the torque of the gears at all levels, the modulus of the gears on each shaft should comply with the increasing rules, and the first series specified in the national standard is adopted, so the modulus constraints are as follows:(6)1.5≤m1≤m2≤m3≤m4.For standard gears, the minimum number of teeth without undercutting is 17. In order to avoid undercutting of gear meshing and ensure that the number of gear teeth is not too high, the constraints on the number of gear teeth are as follows:(7)17≤zi≤65,i=1,2,7,12,16.Since the gears in the reduction box are asymmetrically arranged and hard-toothed gears, the transmission capacity is reduced in order to ensure the bearing capacity of the gears and not cause eccentric loads. Based on the actual working conditions, the constraints on the tooth width of the pinion are as follows:(8)0.2≤ϕi≤0.6,i=1,7,3,14.Due to the nonstandard design of gear teeth and modules in the design process, the interference between the gear and the shaft is prone to occur, so this phenomenon should be avoided in the optimization design, therefore the following constraints are imposed on the gear and the shaft:(9)z2+2m816.5,z7m2+z13m2110,z13m3+z15m3−z12+2m325,z14m4+z16m4−z15+2m435.Therefore, the constraint on the bending fatigue strength of the tooth root is as follows:(10)σF=2KT1b1m12z1YF1YF2≤σFSF,YF1=0.169+0.006666z1−0.000854z12,YF2=0.2824+0.0003539iz1−0.000001576iz12.In the formula,K is the load factor. Since the reel sprinkler is a machine that runs smoothly, the value is 1.3. T1 is the nominal torque of the pinion in the gear pair, and N∗m;σF is the allowable bending fatigue strength of the material. All gears are made of 40Cr and are quenched and tempered. The allowable bending fatigue strength is 470 MPa and the allowable contact fatigue strength is 600 MPa. SF is the bending strength safety factor, according to GB3480-83, the value is 1.5; Y is the pinion tooth shape factor; and YF1 is the stress correction factor. z, m1, and bt are the number of teeth, module, and tooth width of the pinion (driving wheel) in the gear pair; i is the transmission ratio at all levels. The fatigue strength of the tooth surface is as follows:(11)σH=ZEZH2KT1i+1b1d12i≤σHSH.In the formula,ZE is the elastic coefficient; 40Cr steel material value is 189.8MPa; σH is the node area coefficient; the value 2.5 : 4 represents the allowable contact fatigue strength of the material; and Sr is the contact strength safety factor and according to GB3480-83, the value is 1.5.


## 3. Experiment Design for Reliability Optimization of Sprinkler Parts

### 3.1. Algorithm Optimization Results

Taking the highest efficiency point of the fourth gearbox and the lowest total gear volume as the optimization goals, the multiobjective optimization is realized by programming in Matlab software. Since there are as many as 16 design variables, the number of populations is set to 250, the number of iterations is set to 500, the crossover probability is set to 0.9, and the distribution index of the crossover and mutation algorithms are both 20 [23, 24]. After 500 iterations, 42 groups of Pareto optimal solution sets of the objective function are obtained. The represented solution sets have high transmission efficiency, but the gear volume is too large, and the compact structure cannot be reflected in the optimal design. Therefore, six groups of schemes were selected from the blue dots at equal intervals, as shown in Table 3. The multiobjective weighted gray target model is used to make decisions to obtain the optimal solution. Its decision-making process is as follows:(12)S=1.161.211.251.331.391.4458.2258.4458.6258.3359.4459.99T.Table 3
Six groups of candidates.
IndexVolumeEfficiency/%Group 11.1558.02Group 21.2058.41Group 31.2758.60Group 41.3158.92Group 51.3659.45Group 61.4159.94The candidate program target matrixS is as follow:Among the two optimization goals, high transmission efficiency is more important than small gear volume, so a smaller weight is given to the gear volume, and a larger weight is given to the transmission efficiency. The objective function weight vector is as follows:(13)μ=μ1,μ2=0.3,0.7.Use the linear transformation operator [−1, 1] to linearly change the target matrixS of the candidate program to obtain the decision matrix K:(14)K=−2.199−0.09550.1866−0.4488−0.20031.0021−0.67711.0010−0.1575−0.43300.78870.0018T.From the above comparison, the fifth group of candidate schemes can be obtained as the final optimization scheme. The optimization was calculated and rounded, and the results are shown in Table4.Table 4
Gear parameters after optimization.
DentitionNumber of teethModulusTooth width1201.5112551.593421.544391.555221.576161.5207172168301.579441.5810451.5711561.57126221013183251418227156622016673181730218————It can be seen from Table4 that from the perspective of the transmission ratio, whether the distribution of transmission ratios at all levels of the reduction box is reasonable has a great impact on its service life and gear-bearing capacity. Before the optimization, except for the shifting gear pair, the other transmission ratios at all levels are smaller than the front and the rear, which does not conform to the principle. After optimization, the change in transmission ratios at all levels conforms to the principle that the front is small and the back is large, and the transmission ratios of all levels increase step by step, which is more reasonable [25].From the perspective of transmission efficiency, since this paper is optimized based on the lowest efficiency point of the fourth gear of the gearbox, compared with the transmission efficiency of 49.05% before this point, the efficiency at this point after optimization is 59.45% and the lowest point efficiency is increased by 21.2%. The optimized gear parameters are brought into the transmission efficiency mathematical model, the above calculation points are used as input conditions, and the optimized transmission efficiency curve of the gearbox is shown in Figure6.Figure 6
Theoretical transmission efficiency of each gear of the gearbox after optimization.
(a)(b)(c)(d)It can be seen from Figure6 that the transmission efficiency of the gearbox is not the same for different gears. Generally speaking, the lower the load, the higher the efficiency improvement. This is because the energy loss of this part of the power loss that is not related to the load is greatly reduced, and these energy losses are closely related to the gear structure. From the perspective of each gear, compared to before, the efficiency of the first gear has increased by 8.5%∼15.9%; the efficiency of the second gear has increased by 8.7%∼17.4%; the efficiency of the third gear has increased by 9.4%∼18.7%; and the fourth gear efficiency is increased by 10.1%∼21.2%.


### 3.2. Speed of the Gear System of the Reducer of the Sprinkler

The rigid-flexible coupling dynamics model of the gear system of the pointer sprinkler gearbox studied in this paper is applied by the motor through the input shaft 1, and the gear pairs on the transmission shafts at all levels are engaged in meshing transmission; finally, it is output by the output shaft to drive the chain drive and the reel. The speed curves of each flexible gear are shown in Figure7.Figure 7
Flexible body gear speed curve. (a) Gear 1 speed. (b) Gear 2 speed.
(a)(b)As can be seen from Figures7(a) and 7(b), since the constraint relationship between gear 1 and the input shaft is a fixed pair, the simulated and theoretical values of gear 1 fluctuate slightly when the PE pipe winding radius changes. The rest of the time is almost the same, and the speed of the remaining flexible body gears fluctuates around its theoretical value. And this is due to the continuous reduction of the gear load, which leads to the instability of the gear speed and the constant change in its meshing stiffness. However, in general, the comparison error between the simulated speed of the gear and the theoretical value is small, indicating that the speed of the gearbox meets the requirements, that is, the transmission ratio meets the design requirements.


#### 3.2.1. Analysis from the Perspective of Time Domain

Since the motor reaches the maximum required speed 0.1s after the motor starts, the load is also loaded to the maximum value at this time, and the load gradually decreases with the pullback of the PE pipe. Combining the above, it can be seen that the load of the PE pipe is the largest when it is wound for the first time, and the radius is the smallest at the same time. In order to clearly see the change in the gear meshing force under a large load, and based on the characteristics of the gradual reduction in the gear transmission load, the change in the meshing force between 0.1 s and 3.246 s of the two pairs of gear pairs that have been flexible is analyzed. Figure8 is a graph showing the change in the meshing force of two pairs of gear pairs with time at 0.1 s∼3.246 s.Figure 8
Curve of meshing force between gear pairs. (a) Meshing force between gear 1. (b) Meshing force between gear 2.
(a)(b)It can be seen from Figure8 that the change curves of the meshing force of the two pairs of gears with time are roughly the same. After the driving speed is stable within 0.1 s∼3.246 s, the meshing force of the two pairs of gears decreases with the decrease in the load, and the simulated value fluctuates around the theoretical value of the meshing force, which is because the meshing force of the gears has the characteristics of periodic meshing and meshing during the transmission process. At the same time, it can be found that the fluctuation center of the simulated value is smaller than the theoretical value. This is due to the elastic deformation of the flexible body gear during the meshing process. These deformations cause a certain loss of energy during the meshing process due to the existence of damping, and also buffer the rigid impact of the gear during the meshing process. The schematic diagram of the gear force is shown in Figure 9:Figure 9
Graph of gear axial force change. (a) Schematic diagram of the axial force of gear 1. (b) Schematic diagram of the axial force of gear 14.
(a)(b)Comparing the two graphs of Figures9(a) and 9(b) at the same time, it can be seen that the fluctuation range of the gear meshing force in Figure 8(a) is larger than that in Figure 8(b). This is due to when the load is transmitted from the output shaft all the way to the input shaft 1—because of the accumulated elastic deformation of the gear and the continuous accumulation and gradual increase in the gear machining error—the fluctuation range of the gear meshing force is larger than that in Figure 8(a). Figure 8 is a schematic diagram of the axial force of the driving wheel in the two pairs of gears, that is, gear 1 and gear 14. It can be seen from the figure that the axial force of the gear fluctuates near ON during the transmission process, and no obvious axial force is generated, which shows that the gear transmission process is relatively stable, and no obvious axial force is generated [26].


#### 3.2.2. Analysis from the Perspective of Frequency Domain

During the transmission process of the involute gear, the stiffness of each meshing point of the gear pair changes dynamically. The coincidence of gear 1 and gear 2, and gear 14 and gear 16 is between 1 and 2, and at the same time, the reduction box is under a continuously decreasing variable load condition, so the gear system will continuously emit excitation vibration like a spring system, especially under high load and high speed. Figure10 shows the frequency domain diagram of the meshing force between two pairs of flexible body gear pairs.Figure 10
Frequency domain curve of gear pair meshing force. (a) Frequency domain diagram of meshing force between gear 1 and gear 2. (b) Frequency domain diagram of meshing force between gear 14 and gear 16.
(a)(b)It can be seen from Figure10(a) that the vibration amplitude of the meshing force between gear 1 and gear 2 is relatively small, and decreases with the increase in the frequency domain, indicating that there is no obvious vibration when the pair of gear pairs meshes and the operation is relatively stable. In Figure 10(b), the maximum frequency domain of the meshing force between gear 14 and gear 16 is 5.8 Hz, and the corresponding meshing force is assigned a value of 18ON. The amplitude of the meshing force decreases gradually with the increase in frequency, the vibration decreases continuously, and the peak value appears at the integer multiple of its theoretical meshing frequency of 5.8 Hz.


## 3.1. Algorithm Optimization Results

Taking the highest efficiency point of the fourth gearbox and the lowest total gear volume as the optimization goals, the multiobjective optimization is realized by programming in Matlab software. Since there are as many as 16 design variables, the number of populations is set to 250, the number of iterations is set to 500, the crossover probability is set to 0.9, and the distribution index of the crossover and mutation algorithms are both 20 [23, 24]. After 500 iterations, 42 groups of Pareto optimal solution sets of the objective function are obtained. The represented solution sets have high transmission efficiency, but the gear volume is too large, and the compact structure cannot be reflected in the optimal design. Therefore, six groups of schemes were selected from the blue dots at equal intervals, as shown in Table 3. The multiobjective weighted gray target model is used to make decisions to obtain the optimal solution. Its decision-making process is as follows:(12)S=1.161.211.251.331.391.4458.2258.4458.6258.3359.4459.99T.Table 3
Six groups of candidates.
IndexVolumeEfficiency/%Group 11.1558.02Group 21.2058.41Group 31.2758.60Group 41.3158.92Group 51.3659.45Group 61.4159.94The candidate program target matrixS is as follow:Among the two optimization goals, high transmission efficiency is more important than small gear volume, so a smaller weight is given to the gear volume, and a larger weight is given to the transmission efficiency. The objective function weight vector is as follows:(13)μ=μ1,μ2=0.3,0.7.Use the linear transformation operator [−1, 1] to linearly change the target matrixS of the candidate program to obtain the decision matrix K:(14)K=−2.199−0.09550.1866−0.4488−0.20031.0021−0.67711.0010−0.1575−0.43300.78870.0018T.From the above comparison, the fifth group of candidate schemes can be obtained as the final optimization scheme. The optimization was calculated and rounded, and the results are shown in Table4.Table 4
Gear parameters after optimization.
DentitionNumber of teethModulusTooth width1201.5112551.593421.544391.555221.576161.5207172168301.579441.5810451.5711561.57126221013183251418227156622016673181730218————It can be seen from Table4 that from the perspective of the transmission ratio, whether the distribution of transmission ratios at all levels of the reduction box is reasonable has a great impact on its service life and gear-bearing capacity. Before the optimization, except for the shifting gear pair, the other transmission ratios at all levels are smaller than the front and the rear, which does not conform to the principle. After optimization, the change in transmission ratios at all levels conforms to the principle that the front is small and the back is large, and the transmission ratios of all levels increase step by step, which is more reasonable [25].From the perspective of transmission efficiency, since this paper is optimized based on the lowest efficiency point of the fourth gear of the gearbox, compared with the transmission efficiency of 49.05% before this point, the efficiency at this point after optimization is 59.45% and the lowest point efficiency is increased by 21.2%. The optimized gear parameters are brought into the transmission efficiency mathematical model, the above calculation points are used as input conditions, and the optimized transmission efficiency curve of the gearbox is shown in Figure6.Figure 6
Theoretical transmission efficiency of each gear of the gearbox after optimization.
(a)(b)(c)(d)It can be seen from Figure6 that the transmission efficiency of the gearbox is not the same for different gears. Generally speaking, the lower the load, the higher the efficiency improvement. This is because the energy loss of this part of the power loss that is not related to the load is greatly reduced, and these energy losses are closely related to the gear structure. From the perspective of each gear, compared to before, the efficiency of the first gear has increased by 8.5%∼15.9%; the efficiency of the second gear has increased by 8.7%∼17.4%; the efficiency of the third gear has increased by 9.4%∼18.7%; and the fourth gear efficiency is increased by 10.1%∼21.2%.


## 3.2. Speed of the Gear System of the Reducer of the Sprinkler

The rigid-flexible coupling dynamics model of the gear system of the pointer sprinkler gearbox studied in this paper is applied by the motor through the input shaft 1, and the gear pairs on the transmission shafts at all levels are engaged in meshing transmission; finally, it is output by the output shaft to drive the chain drive and the reel. The speed curves of each flexible gear are shown in Figure7.Figure 7
Flexible body gear speed curve. (a) Gear 1 speed. (b) Gear 2 speed.
(a)(b)As can be seen from Figures7(a) and 7(b), since the constraint relationship between gear 1 and the input shaft is a fixed pair, the simulated and theoretical values of gear 1 fluctuate slightly when the PE pipe winding radius changes. The rest of the time is almost the same, and the speed of the remaining flexible body gears fluctuates around its theoretical value. And this is due to the continuous reduction of the gear load, which leads to the instability of the gear speed and the constant change in its meshing stiffness. However, in general, the comparison error between the simulated speed of the gear and the theoretical value is small, indicating that the speed of the gearbox meets the requirements, that is, the transmission ratio meets the design requirements.


### 3.2.1. Analysis from the Perspective of Time Domain

Since the motor reaches the maximum required speed 0.1s after the motor starts, the load is also loaded to the maximum value at this time, and the load gradually decreases with the pullback of the PE pipe. Combining the above, it can be seen that the load of the PE pipe is the largest when it is wound for the first time, and the radius is the smallest at the same time. In order to clearly see the change in the gear meshing force under a large load, and based on the characteristics of the gradual reduction in the gear transmission load, the change in the meshing force between 0.1 s and 3.246 s of the two pairs of gear pairs that have been flexible is analyzed. Figure8 is a graph showing the change in the meshing force of two pairs of gear pairs with time at 0.1 s∼3.246 s.Figure 8
Curve of meshing force between gear pairs. (a) Meshing force between gear 1. (b) Meshing force between gear 2.
(a)(b)It can be seen from Figure8 that the change curves of the meshing force of the two pairs of gears with time are roughly the same. After the driving speed is stable within 0.1 s∼3.246 s, the meshing force of the two pairs of gears decreases with the decrease in the load, and the simulated value fluctuates around the theoretical value of the meshing force, which is because the meshing force of the gears has the characteristics of periodic meshing and meshing during the transmission process. At the same time, it can be found that the fluctuation center of the simulated value is smaller than the theoretical value. This is due to the elastic deformation of the flexible body gear during the meshing process. These deformations cause a certain loss of energy during the meshing process due to the existence of damping, and also buffer the rigid impact of the gear during the meshing process. The schematic diagram of the gear force is shown in Figure 9:Figure 9
Graph of gear axial force change. (a) Schematic diagram of the axial force of gear 1. (b) Schematic diagram of the axial force of gear 14.
(a)(b)Comparing the two graphs of Figures9(a) and 9(b) at the same time, it can be seen that the fluctuation range of the gear meshing force in Figure 8(a) is larger than that in Figure 8(b). This is due to when the load is transmitted from the output shaft all the way to the input shaft 1—because of the accumulated elastic deformation of the gear and the continuous accumulation and gradual increase in the gear machining error—the fluctuation range of the gear meshing force is larger than that in Figure 8(a). Figure 8 is a schematic diagram of the axial force of the driving wheel in the two pairs of gears, that is, gear 1 and gear 14. It can be seen from the figure that the axial force of the gear fluctuates near ON during the transmission process, and no obvious axial force is generated, which shows that the gear transmission process is relatively stable, and no obvious axial force is generated [26].


### 3.2.2. Analysis from the Perspective of Frequency Domain

During the transmission process of the involute gear, the stiffness of each meshing point of the gear pair changes dynamically. The coincidence of gear 1 and gear 2, and gear 14 and gear 16 is between 1 and 2, and at the same time, the reduction box is under a continuously decreasing variable load condition, so the gear system will continuously emit excitation vibration like a spring system, especially under high load and high speed. Figure10 shows the frequency domain diagram of the meshing force between two pairs of flexible body gear pairs.Figure 10
Frequency domain curve of gear pair meshing force. (a) Frequency domain diagram of meshing force between gear 1 and gear 2. (b) Frequency domain diagram of meshing force between gear 14 and gear 16.
(a)(b)It can be seen from Figure10(a) that the vibration amplitude of the meshing force between gear 1 and gear 2 is relatively small, and decreases with the increase in the frequency domain, indicating that there is no obvious vibration when the pair of gear pairs meshes and the operation is relatively stable. In Figure 10(b), the maximum frequency domain of the meshing force between gear 14 and gear 16 is 5.8 Hz, and the corresponding meshing force is assigned a value of 18ON. The amplitude of the meshing force decreases gradually with the increase in frequency, the vibration decreases continuously, and the peak value appears at the integer multiple of its theoretical meshing frequency of 5.8 Hz.


## 3.2.1. Analysis from the Perspective of Time Domain

Since the motor reaches the maximum required speed 0.1s after the motor starts, the load is also loaded to the maximum value at this time, and the load gradually decreases with the pullback of the PE pipe. Combining the above, it can be seen that the load of the PE pipe is the largest when it is wound for the first time, and the radius is the smallest at the same time. In order to clearly see the change in the gear meshing force under a large load, and based on the characteristics of the gradual reduction in the gear transmission load, the change in the meshing force between 0.1 s and 3.246 s of the two pairs of gear pairs that have been flexible is analyzed. Figure8 is a graph showing the change in the meshing force of two pairs of gear pairs with time at 0.1 s∼3.246 s.Figure 8
Curve of meshing force between gear pairs. (a) Meshing force between gear 1. (b) Meshing force between gear 2.
(a)(b)It can be seen from Figure8 that the change curves of the meshing force of the two pairs of gears with time are roughly the same. After the driving speed is stable within 0.1 s∼3.246 s, the meshing force of the two pairs of gears decreases with the decrease in the load, and the simulated value fluctuates around the theoretical value of the meshing force, which is because the meshing force of the gears has the characteristics of periodic meshing and meshing during the transmission process. At the same time, it can be found that the fluctuation center of the simulated value is smaller than the theoretical value. This is due to the elastic deformation of the flexible body gear during the meshing process. These deformations cause a certain loss of energy during the meshing process due to the existence of damping, and also buffer the rigid impact of the gear during the meshing process. The schematic diagram of the gear force is shown in Figure 9:Figure 9
Graph of gear axial force change. (a) Schematic diagram of the axial force of gear 1. (b) Schematic diagram of the axial force of gear 14.
(a)(b)Comparing the two graphs of Figures9(a) and 9(b) at the same time, it can be seen that the fluctuation range of the gear meshing force in Figure 8(a) is larger than that in Figure 8(b). This is due to when the load is transmitted from the output shaft all the way to the input shaft 1—because of the accumulated elastic deformation of the gear and the continuous accumulation and gradual increase in the gear machining error—the fluctuation range of the gear meshing force is larger than that in Figure 8(a). Figure 8 is a schematic diagram of the axial force of the driving wheel in the two pairs of gears, that is, gear 1 and gear 14. It can be seen from the figure that the axial force of the gear fluctuates near ON during the transmission process, and no obvious axial force is generated, which shows that the gear transmission process is relatively stable, and no obvious axial force is generated [26].


## 3.2.2. Analysis from the Perspective of Frequency Domain

During the transmission process of the involute gear, the stiffness of each meshing point of the gear pair changes dynamically. The coincidence of gear 1 and gear 2, and gear 14 and gear 16 is between 1 and 2, and at the same time, the reduction box is under a continuously decreasing variable load condition, so the gear system will continuously emit excitation vibration like a spring system, especially under high load and high speed. Figure10 shows the frequency domain diagram of the meshing force between two pairs of flexible body gear pairs.Figure 10
Frequency domain curve of gear pair meshing force. (a) Frequency domain diagram of meshing force between gear 1 and gear 2. (b) Frequency domain diagram of meshing force between gear 14 and gear 16.
(a)(b)It can be seen from Figure10(a) that the vibration amplitude of the meshing force between gear 1 and gear 2 is relatively small, and decreases with the increase in the frequency domain, indicating that there is no obvious vibration when the pair of gear pairs meshes and the operation is relatively stable. In Figure 10(b), the maximum frequency domain of the meshing force between gear 14 and gear 16 is 5.8 Hz, and the corresponding meshing force is assigned a value of 18ON. The amplitude of the meshing force decreases gradually with the increase in frequency, the vibration decreases continuously, and the peak value appears at the integer multiple of its theoretical meshing frequency of 5.8 Hz.


## 4. Conclusions

In this paper, as the core device of the model transmission system, the transmission efficiency of the reducer of the cylindrical gear is low. In the case of serious energy loss under low-load conditions, and the transmission shaft of the gearbox, especially the output shaft connected to the reel, has a low service life under high-load conditions, and is prone to breakage and other phenomena. Therefore, the optimization design and the simulation analysis are carried out. In order to improve the transmission efficiency of the pointer-type sprinkler reducer and improve the power performance of the whole machine, the gear system of the original four-speed cylindrical gear reducer was optimized. The optimization is carried out through the fast nondominated sorting genetic algorithm NSGA-II, and the decision is made through the target weighted gray target model so as to obtain the optimal solution. The simulation results show that in the gear system, the gear speed fluctuates around the theoretical value and the error is small. The meshing force of the flexible body gear in the time domain fluctuates around the theoretical value, the axial force is almost 0, and the fluctuation in the frequency domain is small, indicating that the gearbox runs relatively smoothly; gear 16 has the maximum stress at the root of the tooth, and the safety factor of tooth root bending fatigue strength is 1.6. At the same time, the minimum gear fatigue life in the entire gear system is 1.85 × 10h, indicating that the optimized reduction gearbox meets the design requirements. However, due to the limitation of resources and scientific research level, further research is needed in the future: (1) When the transmission efficiency of the reduction box is optimized, when deriving the mathematical model of the transmission efficiency of the reduction box, some energy loss formulas are calculated by empirical or semiempirical formulas, which may deviate from the transmission efficiency under actual working conditions. At the same time, due to the limitation of practical conditions, the optimized reduction gearbox has not been processed in kind and tested for efficiency. If conditions permit, it can be processed in the later stage for testing. (2) There is no actual verification of the energy consumption simulation value of the pointer-type sprinkler, and the actual energy consumption value needs to be obtained according to the energy consumption feedback through the user or a large number of experiments, which is also one of the basic tasks for the whole machine factory to improve the quality of its own products.



---

*Source: 7114934-2022-09-09.xml*