Optimization method of tooth surface modification for two-stage reduction gear system based on NSGA-Ⅱ algorithm

Abstract

The application discloses a tooth surface modification optimization method of a two-stage reduction gear system based on an NSGA-II algorithm, and comprises the following steps: establishing a two-stage reducer gear system model according to an actual structure; determining multiple typical working conditions of the two-stage reducer gear system; determining a modification method; establishing a multi-objective optimization mathematical model with the minimum peak-peak value of load transmission error under multiple working conditions as an objective function; based on the NSGA-II algorithm, using a secondary development interface of professional gear analysis software, using a Python language to compile an optimization program, respectively optimizing tooth surface modification parameters of a first-stage reduction gear pair and a second-stage reduction gear pair, and obtaining respective Pareto optimal solution sets; selecting a tooth surface modification optimization scheme in the Pareto optimal solution set, and completing tooth surface modification optimization work. The application combines the NSGA-II multi-objective genetic algorithm with professional gear analysis software, and provides an optimization method with stronger optimization ability and more rapid and accurate optimization for tooth surface modification optimization work under multiple working conditions.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of tooth surface modification optimization of two-stage reducer gear system, and particularly relates to a tooth surface modification optimization method for two-stage reduction gear system based on NSGA-II multi-objective genetic algorithm. BACKGROUND

[0002] An electric vehicle is powered by an electric motor. Since the electric motor can realize stepless speed regulation, forward rotation and reverse rotation, a complex multi-gear transmission in a fuel vehicle is no longer needed. However, from the aspects of power performance and economy, it is still necessary to equip the electric vehicle with a reducer. A two-stage reducer is the most common reduction mechanism on the current electric vehicle, which mainly consists of a first-stage gear pair, a second-stage gear pair, an input shaft, an intermediate shaft, an output shaft, bearings and a housing and the like. The power from the electric motor is transmitted to the intermediate shaft after being reduced by the first-stage gear pair, and then transmitted to the output shaft after being reduced by the second-stage gear pair, so as to realize power transmission.

[0003] During the transmission process of the gear system, due to the load deformation of the gears, shafts, bearings, housings and the like, manufacturing errors and assembly errors, transmission error will inevitably occur, that is, the difference between the actual rotation angle of the driven gear and the theoretical rotation angle when the driving gear rotates through a certain angle. Research results show that there is a corresponding relationship between gear noise and transmission error. Through the way of tooth surface modification, the peak value of transmission error can be effectively reduced, thereby reducing the vibration noise generated in the process of gear meshing. At present, the method of manual modification or orthogonal table is often used to determine the tooth surface modification parameters in engineering. Since the torque range of the electric motor is relatively wide and the working conditions are more, the tooth surface modification parameters are more, the traditional method of determining the tooth surface modification parameters is low in efficiency and easy to get local optimal result. Therefore, how to quickly and accurately determine the tooth surface modification scheme for multiple working conditions is a problem to be solved.

[0004] The Chinese invention patent CN114880813A provides a "three-way reduction reducer noise reduction method based on step-by-step modification and step-by-step optimization", which establishes a helical gear model of the three-way reduction reducer in ROMAX, and optimizes the tooth surface micro-modification parameters by using a genetic algorithm. The optimized tooth surface micro-modification parameters have good effect. However, the optimization algorithm used in this scheme is the ordinary genetic algorithm, which is easy to fall into local optimum when performing multi-objective optimization, and the optimization model used in this scheme will lead to the situation that the tooth surface contact spots are biased during the optimization process in order to reduce the transmission error as much as possible. SUMMARY

[0005] The application provides a tooth surface modification optimization method of a two-stage reduction gear system based on a NSGA-Ⅱ (Nondominated Sorting Genetic Algorithm-Ⅱ) multi-objective genetic algorithm, which can quickly and accurately reduce transmission errors under multiple working conditions and effectively improve the NVH (Noise, Vibration, Harshness) performance of the two-stage reduction gear system under the condition of ensuring good tooth surface contact state.

[0006] In order to achieve the object of the application, the application provides a tooth surface modification optimization method of a two-stage reduction gear system based on a NSGA-Ⅱ algorithm, which comprises the following steps:

[0007] S1. According to the actual structure, a two-stage reducer gear system model is established in professional gear analysis software;

[0008] S2. Multiple typical working conditions of the two-stage reducer gear system are determined;

[0009] S3. A modification method is determined;

[0010] S4. Optimization variables, optimization objectives and constraint conditions are determined, and a multi-objective optimization mathematical model with the minimum peak-to-peak value of the bearing transmission error under multiple working conditions as an objective function is established;

[0011] S5. Based on the NSGA-Ⅱ (Nondominated Sorting Genematic Algorithm-Ⅱ) algorithm, an automatic optimization program is compiled by using a secondary development interface of professional gear analysis software and a Python language, and the tooth surface modification parameters of the first-stage reduction gear pair and the second-stage reduction gear pair are optimized respectively to obtain respective Pareto optimal solution sets;

[0012] S6. A tooth surface modification optimization scheme is selected from the Pareto optimal solution set, and the tooth surface modification optimization work is completed;

[0013] Further, in step S1, an input shaft, an intermediate shaft, an output shaft, a first-stage reduction gear pair, a second-stage reduction gear pair, a bearing and a reducer housing are included; except the reducer housing, other components are completed by parameterized modeling, and the reducer housing is imported by an external entity grid file;

[0014] Further, in step S2, the working torque range of the motor is discretized according to the characteristics of the motor to obtain multiple typical working conditions;

[0015] Further, in step S3, the tooth surface modification method comprises pressure angle modification, tooth profile drum modification, helix angle modification and tooth direction drum modification;

[0016] Further, the optimization variables in step S4 are the pressure angle modification amount x1, the profile drum modification amount x2, the helical angle modification amount x3, and the tooth trace drum modification amount x4 of the pinion in the meshing gear pair.

[0017] Further, in step S4, since the torque range of the motor is relatively wide, in order to make the optimization result effective for multiple working conditions, the minimum peak-to-peak value of the transmission error under three different load working conditions is selected as three objective functions, and the expressions are as follows:

[0018]

[0019] In the formula, T ej represents the transmission error of the gear pair in one meshing period under working condition j.

[0020] Further, in step S4, the constraint range of the optimization variable is determined according to the actual machining capability, wherein:

[0021]

[0022] Further, in step S4, in addition to the constraints on the optimization variables in claim 4, considering the precision limitation of gear machining, integer constraints on the optimization variables are also included:

[0023] x i ∈integers,i=1,2,3,4 (3)

[0024] Further, in order to effectively ensure that the tooth surface load distribution does not appear to be biased, the constraint in step S4 also includes:

[0025] P edgemaxj / P maxj <k (4)

[0026] In the formula, k is a number less than 1, P edgemaxj is the maximum contact stress of the tooth surface edge under working condition j, and P maxj is the maximum contact stress of the tooth surface under working condition j.

[0027] Further, step S5 includes the following steps:

[0028] S501, setting the population size, the maximum evolution generation number, the crossover probability, and the mutation probability;

[0029] S502, initializing the population;

[0030] S503, calculating the objective function and performing Pareto classification on the population;

[0031] S504, calculating the crowding degree of each individual under each Pareto level.

[0032] S505, performing selection, crossover and mutation operations;

[0033] S506, generating a new population using an elite selection strategy;

[0034] S507, setting the maximum evolution generation Gen as a termination condition, if the maximum evolution generation is not met, the evolution number count is added by 1 and the process returns to step S503; if the maximum evolution number is met, the optimization ends and the final Pareto solution set is obtained.

[0035] Further, in step S5, the secondary development interface of the professional gear analysis software plays the following two roles: 1) using the secondary development interface, each individual in the population generated by the NSGA-II algorithm, including the pressure angle modification amount x1, the tooth profile drum modification amount x2, the helical angle modification amount x3 and the tooth direction drum modification amount x4, are all transmitted to the professional gear analysis software and subjected to bearing contact analysis calculation; 2) using the secondary development interface, the results obtained by the bearing contact analysis, i.e. the bearing transmission error T ej , the maximum contact stress P edgemaxj and the maximum contact stress P maxj of the tooth surface edge under the working condition are transmitted back to the NSGA-II algorithm for fast non-dominated sorting and crowding calculation.

[0036] Further, a bearing tooth surface contact analysis method based on the hybrid method is adopted, which is very close to the result of using full finite element for contact analysis, and has great advantages in efficiency. The basic theory is as follows:

[0037] First, the finite element grid of the gear is generated according to the macro parameters of the gear, and the flexibility matrix of the tooth surface grid node is obtained by the Guyan reduction method:

[0038]

[0039] In the formula, f bij(1) represents the bending-shear flexibility coefficient of the driving gear contact point j to the contact point i, f bij(2) represents the bending-shear flexibility coefficient of the driven gear contact point j to the contact point i, and n represents the number of contact points at the meshing position.

[0040] Secondly, according to the Hertz contact theory, the local contact deformation vector u of each segment contact line is obtained:

[0041] u = [u1 u2 u3 … u n ] T (6)

[0042] In the formula, u ilocal contact deformation of the i th contact point;

[0043] The n-order bearing tooth surface contact equation set is listed according to the deformation coordination condition of the contact point:

[0044]

[0045] In the formula, p is a load distribution vector of the meshing position, x is a bearing transmission error of the meshing position, d is a residual gap column vector at the contact point, w is an initial tooth side gap column vector including factors such as gear manufacturing error, assembly error, meshing misalignment amount and gear modification amount, e is a unit row vector, and P is a total normal load transmitted by the gear.

[0046] Finally, the bearing transmission error and the tooth surface load distribution at the meshing position can be finally obtained by iteratively solving the nonlinear equation set. Finally, the bearing transmission error and the contact stress distribution of the entire tooth surface of the gear pair in an engagement period can be obtained by performing the above analysis on different meshing positions in an engagement period.

[0047] Further, in step S6, the relationship between each optimization scheme in the obtained Pareto solution set is non-dominated, that is, there is no clear superior-inferior relationship between the optimization results, and the selection can be performed according to the demand.

[0048] Compared with the related art, the present application has at least the following advantages:

[0049] 1) The NSGA-II algorithm is a multi-objective genetic algorithm, which introduces fast non-dominated sorting technology, crowding sorting technology and elite selection strategy, reduces the time complexity of the algorithm, and at the same time, expands the distribution space of the optimal solution set. By using the non-dominated sorting method to save good individuals and using the elite selection strategy to ensure the diversity of the population, the algorithm has good robustness, strong optimization ability and other characteristics.

[0050] 2) The present application is based on the NSGA-II multi-objective genetic algorithm, uses the secondary development interface of professional gear analysis software, and uses Python language to compile an automatic optimization program. It has strong universality and can be applied to different models of secondary reducers, and can quickly and accurately obtain the corresponding tooth surface modification scheme.

[0051] 3) In the optimization model, the constraint condition of avoiding tooth surface contact spot partial load is added, which can effectively ensure that the phenomenon of tooth surface contact spot partial load caused by reducing the bearing transmission error does not occur in the optimization process, and the NSGA-II genetic algorithm has strong ability in multi-objective optimization, which can quickly and accurately reduce the transmission error under multiple working conditions while ensuring good tooth surface contact state.

[0052] 4) The NSGA-Ⅱ multi-objective genetic algorithm is combined with professional gear analysis software in the application, a more powerful and more rapid and accurate optimization method is provided for the tooth surface modification optimization work for multiple working conditions, and the time cost of enterprises in the tooth surface modification optimization work is reduced. BRIEF DESCRIPTION OF DRAWINGS

[0053] Figure 1 is a tooth surface modification optimization method flowchart of a two-stage reducer gear system based on the NSGA-Ⅱ algorithm provided by the embodiment of the application.

[0054] Figure 2 is a structural schematic diagram of a two-stage reduction gear system in the embodiment of the application.

[0055] Figure 3 is a pressure angle modification principle diagram in the embodiment of the application.

[0056] Figure 4 is a tooth profile drum modification principle diagram in the embodiment of the application.

[0057] Figure 5 is a helix angle modification principle diagram in the embodiment of the application.

[0058] Figure 6 is a tooth direction drum modification principle diagram in the embodiment of the application.

[0059] Figure 7 is a comparison diagram of bearing transmission error peak-to-peak values under different working conditions before and after optimization of a first-stage reduction gear pair in the embodiment of the application.

[0060] Figure 8 is a comparison diagram of bearing transmission error peak-to-peak values under different working conditions before and after optimization of a second-stage reduction gear pair in the embodiment of the application.

[0061] Figure 9 is a tooth surface contact spot diagram of a first-stage reduction gear pair under working condition 3 without modification of a driving wheel in the embodiment of the application.

[0062] Figure 10 is a tooth surface contact spot diagram of a first-stage reduction gear pair under working condition 3 after modification of a driving wheel in the embodiment of the application. DETAILED DESCRIPTION

[0063] The specific embodiments of the application are described in further detail below in combination with the drawings and examples. The following examples are used to illustrate the application, but are not used to limit the scope of the application.

[0064] As Figure 1As shown, the application provides a tooth surface modification optimization method of a two-stage reduction gear system based on a NSGA-Ⅱ (Nondominated Sorting Genematic Algorithm-Ⅱ) algorithm, which comprises the following steps:

[0065] Step S1, according to the actual structure, a two-stage reducer gear system model is established in professional gear analysis software.

[0066] In some embodiments of the application, the professional gear analysis software adopts MASTA.

[0067] The two-stage reducer gear system model includes an input shaft, an intermediate shaft, an output shaft, a first-stage reduction gear pair, a second-stage reduction gear pair, a bearing, a reducer housing and the like. Except the reducer housing, other components are completed by parameterized modeling, and the reducer housing is imported by an external entity grid file. In some embodiments of the application, the established two-stage reducer gear system model is as shown in Figure 2 The helix angle of the first-stage reduction gear pair is 26.5°, the normal modulus is 2 mm, the normal pressure angle is 17°, the pinion gear has 26 teeth and a tooth width of 18 mm, the gear has 59 teeth and a tooth width of 18 mm, the helix angle of the second-stage reduction gear pair is 26.5°, the normal modulus is 2.29 mm, the normal pressure angle is 17°, the pinion gear has 25 teeth and a tooth width of 30 mm, the gear has 72 teeth and a tooth width of 26 mm, and the gear material is 20CrMnTi.

[0068] Step S2, a plurality of typical working conditions of the two-stage reducer gear system are determined.

[0069] According to the characteristics of the used motor, the working torque range of the motor is discretized to obtain a plurality of typical working conditions. In some embodiments of the application, the two-stage reducer gear system includes working condition 1, working condition 2, working condition 3, working condition 4, working condition 5 and working condition 6, and the specific parameters are shown in Table 1:

[0070] Table 1

[0071]

[0072] S3, a modification method is determined.

[0073] The selected tooth surface modification method includes pressure angle modification, tooth profile drum modification, helix angle modification and tooth direction drum modification, as shown in Figures 3 to 6

[0074] ​S4, determine optimization variables, optimization target and constraint conditions, since the torque range of the motor is relatively wide, in order to make the optimization result effective for various working conditions, a multi-objective optimization mathematical model is established, which takes the minimum peak-to-peak value of load transmission error under multiple working conditions as the objective function.

[0075] In some embodiments of the application, a multi-objective optimization mathematical model is established, which takes the minimum peak-to-peak value of load transmission error under working condition 1, working condition 3 and working condition 5 (in other embodiments, other working conditions or other number of working conditions can be selected, and it is not limited to three) as the objective function:

[0076] Optimization variables: x1, x2, x3, x4;

[0077] Optimization target:

[0078] Constraint conditions:

[0079] x i ∈integers,i=1,2,3,4

[0080] In the formula, x 1min is the minimum value of the pressure angle modification amount, x 1max is the maximum value of the pressure angle modification amount, x 2min is the minimum value of the tooth profile drum modification amount, x 2max is the minimum value of the tooth profile drum modification amount, x 3min is the minimum value of the helix angle modification amount, x 3max is the minimum value of the helix angle modification amount, x 4min is the minimum value of the tooth direction drum modification amount, x 4max is the minimum value of the tooth direction drum modification amount, k is a number less than 1, P edgemaxj is the maximum contact stress of the tooth surface edge under working condition j, P maxj is the maximum contact stress of the tooth surface under working condition j.

[0081] The constraint condition of avoiding tooth surface contact spot bias load (P edgemaxj / P maxj <k) is added, which can effectively ensure that the phenomenon of tooth surface contact spot bias load caused by reducing the load transmission error during the optimization process.

[0082] In some embodiments of the application, after specific values of each parameter in the constraint condition are obtained:

[0083]

[0084] Wherein, x1-x4 are respectively the pressure angle modification amount, the tooth profile drum modification amount, the helix angle modification amount and the tooth trace drum modification amount (μm) of the right tooth surface of the pinion in the meshing gear pair, T ej T is the load transmission error (μm) of the meshing process of the crankshaft ring gear and the shear pair in the working condition j in Table 1, P maxj and P edgemaxj are respectively the maximum contact stress of the tooth surface and the maximum contact stress of the tooth surface edge (MPa) in the meshing process of the crankshaft ring gear and the shear pair in the working condition j in Table 1. f1(x1, x2, x3, x4) is the peak-to-peak value of the gear load transmission error in working condition 1, f2(x1, x2, x3, x4) is the peak-to-peak value of the gear load transmission error in working condition 3, f3(x1, x2, x3, x4) is the peak-to-peak value of the gear load transmission error in working condition 5, and integers represent integers.

[0085] Wherein, in some embodiments of the application, the load transmission error T ej , the maximum contact stress of the tooth surface edge P edgemaxj and the maximum contact stress of the tooth surface P maxj are all calculated by running the load tooth surface contact analysis program in the professional gear analysis software, and the classic gear slice model is used for calculation in the load tooth surface contact analysis.

[0086] S5, based on the NSGA-II algorithm, an automatic optimization program is compiled using Python language by using the secondary development interface of the professional gear analysis software, and the tooth surface modification parameters of the first-stage reduction gear pair and the second-stage reduction gear pair are optimized, and the specific steps are as follows:

[0087] S501, setting the population size to 25, the maximum evolution generation to 100, the crossover probability to 0.9, and the mutation probability to 0.2;

[0088] S502, initializing the population;

[0089] S503, calculating the objective function and performing Pareto classification on the population;

[0090] S504, calculating the crowding degree of each individual in each Pareto level;

[0091] S505, performing selection, crossover and mutation operations;

[0092] S506, using the elite selection strategy to generate a new population;

[0093] S507, set the maximum evolution number Gen as a termination condition, if the maximum evolution number is not met, the evolution number count is added by 1 and go back to step S503 to continue running, if the maximum evolution number is met, the optimization ends and the final Pareto solution set is obtained.

[0094] In some embodiments of the present application, Table 2 shows the obtained Pareto solution set of the first-stage reduction gear pair, and Table 3 shows the obtained Pareto solution set of the second-stage reduction gear pair.

[0095] Table 2

[0096]

[0097] Table 3

[0098]

[0099]

[0100] Finally, scheme 5 in Table 2 is selected as the modification scheme of the first-stage reduction gear pair, and scheme 3 in Table 3 is selected as the modification scheme of the second-stage reduction gear pair. (The solutions in the table are non-dominated, and each solution has its own advantages. The two modification schemes are selected because their performances in the three working conditions are relatively average.)

[0101] After modification and optimization by the method, the peak-peak value of the load transmission error of the first-stage reduction gear pair is reduced in all working conditions, with an average reduction of 40%, as shown in Table 4. Figure 7 The peak-peak value of the load transmission error of the second-stage reduction gear pair is reduced in all working conditions, with an average reduction of 38%, as shown in Table 5. Figure 8 The above results show that the optimization method can effectively reduce the peak-peak value of the load transmission error of each reduction gear pair in the entire torque range, thereby effectively improving the NVH (Noise, Vibration and Harshness) performance of the two-stage reduction gear system.

[0102] After modification and optimization by the method, the tooth surface contact state in each working condition is also improved, and the tooth surface contact spot is in central contact. Taking the first-stage reduction gear pair in working condition 3 as an example, the tooth surface contact spots before and after optimization are shown in Figs. 6 and 7, respectively. Figure 9 10

[0103] ​​The above examples are only used to illustrate the calculation process of the present application, and are not intended to limit the present application. Although the present application is described in detail with reference to the foregoing examples, those skilled in the art should understand that the calculation process described in the foregoing examples can be modified, or some of the hyperparameters can be replaced by equivalent parameters, and these modifications or replacements do not make the essence of the corresponding calculation method deviate from the spirit and scope of the calculation method of the present application.

Claims

1. A method for optimizing the tooth surface modification of a two-stage reduction gear system based on the NSGA-II algorithm, characterized in that, The method comprises the following steps: S1. According to the actual structure, a two-stage reducer gear system model is established in professional gear analysis software; S2. A plurality of working conditions of the two-stage reducer gear system are determined; S3. A modification method is determined; S4. Optimization variables, optimization objectives and constraint conditions are determined, a multi-objective optimization mathematical model with the minimum peak-to-peak value of load transmission error under multiple working conditions as an objective function is established, and the constraint condition of avoiding load deviation of tooth surface contact spots is added to the multi-objective optimization mathematical model; S5. Based on the NSGA-II algorithm, an optimization program is compiled by using a secondary development interface of professional gear analysis software and a Python language to optimize the tooth surface modification parameters of the first-stage reduction gear pair and the second-stage reduction gear pair respectively, and a respective Pareto optimal solution set is obtained; S6. A tooth surface modification optimization scheme is selected from the Pareto optimal solution set, and tooth surface modification optimization work is completed; In step S4, the minimum peak-to-peak value of load transmission error under multiple different load working conditions is selected as the objective function, and when the load working condition is three, the expression of the objective function is as follows: wherein The gear pair's load transmission error in a meshing cycle under the working condition The gear pair's load transmission error in a meshing cycle under the working condition , , The gear pair's load transmission error peak-to-peak value under three different working conditions The constraint range of the optimization variable is determined according to the actual machining capacity, wherein: wherein is a pressure angle modification, is a profile drum modification, is a helix angle modification, is a profile drum modification, is a minimum value of the pressure angle modification, is a maximum value of the pressure angle modification, is a minimum value of the profile drum modification, is a maximum value of the profile drum modification, is a minimum value of the helix angle modification, is a maximum value of the helix angle modification, is a minimum value of the profile drum modification, is a maximum value of the profile drum modification; The expression of the constraint condition of avoiding load deviation of tooth surface contact spots is as follows: wherein is a number less than 1, is the maximum contact stress of the tooth flanks under the operating condition is the maximum contact stress of the tooth flanks under the operating condition is the maximum contact stress of the tooth flanks under the operating condition is the maximum contact stress of the tooth flanks under the operating condition 2. The method for optimization of tooth profile modification of two-stage reduction gear system based on NSGA-II algorithm according to claim 1, wherein, The two-stage reducer gear system model comprises an input shaft, an intermediate shaft, an output shaft, a first-stage reduction gear pair, a second-stage reduction gear pair, a bearing and a reducer housing. Except the reducer housing, other components are completed by parameterized modeling, and the reducer housing is imported by an external entity grid file.

3. The method for optimization of tooth profile modification of two-stage reduction gear system based on NSGA-II algorithm as claimed in claim 1 wherein, In step S2, the working torque range of the motor is discretized according to the characteristics of the motor used, and a plurality of different working conditions are obtained.

4. The method for optimization of tooth profile modification of two-stage reduction gear system based on NSGA-II algorithm according to claim 1, wherein, In step S3, the tooth surface modification method used includes pressure angle modification, tooth profile drum modification, helix angle modification and tooth direction drum modification.

5. The method for optimization of tooth profile modification of two-stage reduction gear system based on NSGA-II algorithm as claimed in claim 1 wherein, In step S4, the optimization variables are the pressure angle modification amount, the tooth profile drum modification amount, the helix angle modification amount and the tooth direction drum modification amount of the pinion in the meshing gear pair.

6. The method for optimization of tooth profile modification of two-stage reduction gear system based on NSGA-II algorithm according to claim 1, wherein, Considering the precision limitation of gear machining, the constraint condition in step S4 further includes an integer constraint for the optimization variables: In the formulae, denotes an integer.

7. The method for optimization of tooth profile modification of two-stage reduction gear system based on NSGA-II algorithm according to any one of claims 1-6, characterized in that, Step S5 comprises the following steps: S501. A population size, a maximum evolution generation number, a crossover probability and a mutation probability are set; S502. A population is initialized; S503. The objective function is calculated, and the population is Pareto graded; S504. The crowding degree of each individual under each Pareto grade is calculated; S505. Selection, crossover and mutation operations are performed; S506. A new population is generated by using an elite selection strategy; S507. The maximum evolution generation number Gen is set as a termination condition. If the maximum evolution generation number is not met, the evolution count is increased by 1 and the step S503 is returned to continue running. If the maximum evolution number is met, the optimization is ended, and a final Pareto solution set is obtained.

8. The method for optimization of tooth profile modification of two-stage reduction gear system based on NSGA-II algorithm according to claim 7, wherein, In step S5, the secondary development interface of the professional gear analysis software has the following two functions: 1) Using the secondary development interface, each individual in the population generated by the NSGA-II algorithm is passed to the professional gear analysis software, including the pressure angle modification amount , the tooth profile drum modification amount , the helical angle modification amount , and the tooth direction drum modification amount , and the bearing tooth surface contact analysis calculation is performed; 2) The results of the contact analysis, i.e. the load transmission errors under the working conditions, are transferred back to the NSGA-II algorithm for fast non-dominated sorting and crowdedness calculation using the secondary development interface. the maximum contact stress on the edge of the tooth surface and the maximum contact stress on the tooth surface are transferred back to the NSGA-II algorithm for fast non-dominated sorting and crowdedness calculation using the secondary development interface.

9. The method for optimization of tooth profile modification of two-stage reduction gear system based on NSGA-II algorithm according to claim 8, wherein, A load tooth surface contact analysis method based on a hybrid method is adopted, and the basic theory is as follows: Firstly, the finite element mesh of gear is generated according to the macro parameters of gear, and the flexibility matrix of the gear mesh nodes is obtained by Guyan reduction method: wherein denotes the pinion contact point for the contact point the bending-shearing flexibility coefficient of the contact point, denotes the driven gear contact point for the contact point the bending-shearing flexibility coefficient of the contact point, denotes the number of meshing position contact points; Secondly, according to the Hertz contact theory, the local contact deformation vectors of each segment contact line are obtained : In the formulae, denotes the local contact deformation at the contact point. According to the deformation compatibility conditions of the contact points Step 2: Derive the bearing tooth surface contact equations wherein, is the load distribution vector for the engaged position, is the load transmission error for the engaged position, is the residual gap column vector at the contact point, is the initial flank gap column vector consisting of gear manufacturing error, assembly error, amount of meshing misalignment, and amount of gear modification factors, is the unit row vector, is the total normal load transmitted by the gear; Finally, the load transmission error and the load distribution of the gear mesh are obtained by solving the nonlinear equations iteratively. Finally, the load transmission error and the load distribution of the gear mesh are obtained by solving the nonlinear equations iteratively.

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