NGW planetary reducer multi-objective optimization method based on Kriging proxy model and NSGA-II

Abstract

The invention is suitable for the technical field of reducer design, and particularly relates to an NGW planetary reducer multi-objective optimization method based on a Kriging proxy model and NSGA-II, and the method comprises the steps: S1, building an NGW type planetary reducer model, and carrying out the simulation; s2, establishing a multi-objective optimization model of the NGW type planetary reducer; s3, an optimal Latin hypercube sampling method is selected for sampling the NGW type planetary reducer; s4, training the multi-objective optimization proxy model of the NGW type planetary reducer; s5, performing multi-objective optimization on the NGW type planetary reducer; and S6, analyzing an optimization result. According to the method, sampling points are generated by adopting an optimal Latin hypercube experimental design method, an approximate model of input and output is constructed by using a Kriging agent model, and an optimal solution set is obtained through an NSGA-II optimization algorithm, so that the optimization efficiency is greatly improved, and the optimization effect is greatly improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of reducer design, and in particular relates to a multi-objective optimization method for an NGW planetary reducer based on a Kriging surrogate model and NSGA-II. Background Art

[0002] The rocket launch platform is used to support and secure the carrier and is an essential component of the launch tower. As the core component of the vertical adjustment mechanism of the rocket launch platform, the NGW planetary reducer features high transmission accuracy, strong load-bearing capacity, compact structure, and fast response speed. The NGW planetary reducer can change the angle of the launch platform through coordinated movement, thereby achieving azimuth aiming during rocket launch. Its load-bearing stiffness, adjustment accuracy, and impact resistance play an important role in the initial stage of rocket launch motion performance. Therefore, optimizing the performance of the NGW planetary reducer is of great significance, and scholars at home and abroad have conducted extensive research.

[0003] Current research on multi-objective optimization of planetary reducers focuses primarily on volume, efficiency, noise, and gear strength, with limited research on reducer torsional stiffness. To meet the requirements of rocket launch platforms for vertical adjustment mechanisms with higher load-bearing capacity, greater torsional stiffness, and lightweight design, this paper, based on the ISIGHT platform and combining parametric modeling with finite element statics simulation, investigates the volume and torsional stiffness trends of planetary reducers under different structural parameter combinations. Using the DOE method and Kriging model, a mapping relationship between the structural parameters and the target parameters of the planetary reducer is constructed, resulting in the optimal structural parameters for the NGW-type planetary reducer that meet engineering applications. Summary of the Invention

[0004] The purpose of the present invention is to provide a multi-objective optimization method for NGW planetary reducers based on Kriging surrogate model and NSGA-II, aiming to solve the problem that the existing optimization method requires multiple re-import of the model and has low optimization efficiency.

[0005] The present invention is implemented as follows: a multi-objective optimization method for NGW planetary reducer based on Kriging surrogate model and NSGA-II, the method comprising:

[0006] S1. Establish an NGW planetary reducer model and perform simulation;

[0007] S2. Establish a multi-objective optimization model for NGW planetary reducer;

[0008] S3. Select the optimal Latin hypercube sampling method to sample the NGW planetary reducer;

[0009] S4. Training the multi-objective optimization agent model for NGW planetary reducer;

[0010] S5. Multi-objective optimization of NGW type planetary reducer;

[0011] S6. Analysis of optimization results.

[0012] Furthermore, step S1 is specifically as follows:

[0013] S1.1. Design of NGW planetary reducer

[0014] S1.2. Establish the finite element model of NGW planetary reducer

[0015] S1.3. Verify the NGW planetary reducer model

[0016] Furthermore, step S1.3 is specifically as follows:

[0017] 1.3.1 Verifying Grid Independence

[0018] The tooth profile mesh size range was selected from 0.1 to 0.7 mm. Simulated stress values ​​were closest to theoretically calculated stress values ​​when the mesh lengths were 0.1 and 0.2 mm, but the calculation time increased significantly compared to a mesh size of 0.3 mm. The error between the simulated and theoretically calculated stress values ​​was only 1.2% for a mesh size of 0.3 mm, significantly better than the accuracy achieved with a mesh size of 0.4 mm. To balance accuracy and efficiency, a mesh size of 0.3 mm was ultimately selected for the tooth profile in the finite element simulation.

[0019] 1.3.2 Verify the correctness of torsional stiffness simulation results

[0020] In order to ensure the accuracy, reliability and dynamic performance of the transmission system, it is necessary to calculate the torsional stiffness of the NGW planetary reducer. The expression of torsional stiffness K is:

[0021]

[0022] Where T is the torque; θ is the torsion angle of the mechanical structure.

[0023] The torsional stiffness of the NGW planetary reducer can be calculated by applying a torsional load to the primary sun gear using a finite element model. Furthermore, when optimizing the design of the NGW planetary reducer, torsional stiffness is also a primary optimization objective, and the correctness of the simulation results needs to be verified.

[0024] The test bench features a symmetrical structure, consisting of an electric motor, a torque and speed sensor, two NGW-type planetary reducers, and a loader. The two reducers under test use identical configurations. During the experiment, the torque output by the electric motor, after being reduced in speed by the gearbox, is first transmitted to reducer No. 1 under test. This torque is then connected to reducer No. 2 through a drive shaft, and finally absorbed by the loader to maintain system stability.

[0025] During the test, the loader used maximum power to fix the output of test reducer No. 2. The motor then applied loads to the NGW planetary reducer in stages, at a certain percentage of the rated load. Torque and speed measurements were taken using a JZ2000 torque and speed sensor. The sensor measured torque with an accuracy of ±0.2% FS and a speed accuracy of ±0.1% RPM. The torque range was 0-2000 Nm, and the speed range was 0-6000 RPM.

[0026] Furthermore, when establishing the optimization model in step S2, the objective function is to minimize the volume of the reducer and maximize the torsional stiffness, and to satisfy the bending strength as a constraint condition. The overall optimization equation is as follows:

[0027]

[0028] The objective function f1(x) of the volume is expressed as follows:

[0029]

[0030] Where: V s is the volume of the sun gear; V n is the volume of the inner gear ring; V p is the volume of the planetary gear; C is the number of planetary gears; b is the width of the gear; Z s is the number of teeth of the center wheel; m is the module of the gear; i is the transmission ratio of the planetary gear.

[0031] The objective function f2(x) of torsional stiffness is expressed as follows:

[0032]

[0033] The bending strength as a function of the constraint condition f3(x) is expressed as follows:

[0034]

[0035] Where F t Gear tangential force; K A Service factor; K V Dynamic load coefficient; K Fβ Tooth load distribution coefficient; K Fα tooth load distribution coefficient; b tooth width; m normal module; Y FTooth form factor; Y S Stress correction factor.

[0036] Furthermore, step S3 is specifically as follows:

[0037] The number of sun and planetary gear teeth remains unchanged during sampling. For each transmission stage, only four factors are sampled: the gear module m, tooth width b, sun gear displacement coefficient x, and center distance variation coefficient y. To ensure sampling accuracy, the recommended number of sampling points is 2 × (n + 1) × (n + 2) (n is the number of sampling factors), resulting in 60 sampling points for each gear transmission stage. The sampling point parameters are rounded and standardized.

[0038] Based on the rounded sampling point parameters, a three-dimensional model of the NGW planetary reducer was established and finite element analysis was performed. The gear module m, tooth width b, sun gear modification coefficient x, and center distance variation coefficient y were used as inputs, and torsional stiffness and volume were used as outputs. DOE analysis was then performed on the ISISHT platform. Main effect plots of the gear module m, tooth width b, sun gear modification coefficient x, and center distance variation coefficient y on the reducer volume and sun gear torsion angle were obtained. The volume of the NGW planetary reducer showed a linear growth trend with increasing tooth width b, and an exponential growth trend with increasing module m when the number of teeth remained unchanged. The modification coefficient x and center distance variation coefficient y had little effect on the volume of the planetary reducer. The sun gear torsion angle of the NGW planetary reducer first increased and then decreased with increasing tooth width b, and decreased with increasing module m when the number of teeth remained unchanged. The modification coefficient x and center distance variation coefficient y had some influence on the torsion angle, but the overall effect was small.

[0039] Furthermore, step S4 is specifically as follows:

[0040] The Kriging model was used as a surrogate model for the multi-objective optimization of an NGW planetary reducer. After model training, 15 randomly selected sample data sets were used to validate the constructed surrogate model. The model's accuracy was evaluated by comparing the model's predictions with actual simulation data. To verify the accuracy of the surrogate model constructed for the test sample points, a multidimensional error analysis method was used to quantitatively evaluate the surrogate model's fitting performance.

[0041] A cross-validation strategy was used to further analyze the reliability of the model. Six groups of sample parameters were randomly selected to construct an independent validation set, and the estimated results of the NGW planetary reducer proxy model were quantitatively compared with the finite element simulation results.

[0042] Furthermore, step S5 is specifically as follows:

[0043] The NSGA-II algorithm is used to optimize the primary gear train. To verify whether the optimal solution specified by the NSGA-II algorithm has the best effect, different weight ratios are set for the torsional stiffness and volume of the NGW planetary reducer objective functions, thereby obtaining multiple optimal solutions. The optimization process of the secondary gear train of the NGW planetary reducer is similar to that of the primary gear train.

[0044] Furthermore, step S6 is specifically as follows:

[0045] To verify the optimization effect of the dynamic characteristics of an NGW planetary reducer, the speed fluctuations of the reducer before and after optimization were compared. This NGW planetary reducer uses a sun gear as input, a fixed ring gear, and a planetary carrier as output. Therefore, the speed fluctuations of the secondary planetary carrier before and after optimization were compared. The three-dimensional models of the reducer before and after optimization were imported into the ADAMS platform to establish a dynamic model of the NGW planetary reducer, and the results were analyzed.

[0046] Compared with the prior art, this patent application has the following beneficial effects:

[0047] 1) Using the optimal Latin hypercube as the sampling method for DOE and the Kriging model as the training method has a good fitting effect and shows very good accuracy in predicting the objective function of the NGW type planetary reducer.

[0048] 2) The Pareto frontier and comprehensive optimal solution for the NGW planetary reducer were obtained using the multi-objective NSGA-II algorithm. The optimization results show that the torsional stiffness of the planetary reducer increased by 11.20% and the volume decreased by 4.23%. This effectively suppresses the elastic deformation during torque transmission and optimizes space utilization while ensuring load-bearing capacity.

[0049] 3) A dynamic model of an NGW-type planetary reducer was established, and the speed fluctuation amplitude of the secondary planetary carrier was compared before and after optimization. The results showed that the speed fluctuation amplitude was reduced by 7.4% compared to the original speed, effectively suppressing the periodic vibration of the transmission system and providing key support for reliable operation under heavy load conditions.

[0050] 4) Based on the ISIGHT platform, combined with parametric modeling and finite element statics simulation methods, the changing trends of the volume and torsional stiffness of the planetary reducer under different structural parameter combinations are studied. The DOE method and Kriging model are used to construct the mapping relationship between the structural parameters of the planetary reducer and the target parameters, and the optimal structural parameters of the NGW-type planetary reducer that meets engineering applications are obtained. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] Figure 1 This is a simplified structural diagram of the NGW type planetary reducer;

[0052] Figure 2 is the stress and strain result of the planetary reducer gear train;

[0053] Figure 3 is the torsion angle of the input sun gear;

[0054] Figure 4 This is the relationship between the tooth profile mesh size and the number of meshes;

[0055] Figure 5 This is the relationship between the tooth profile mesh size and calculation time;

[0056] Figure 6 This is the relationship diagram between the tooth profile mesh size and the simulation stress;

[0057] Figure 7 This is the test bench test principle diagram;

[0058] Figure 8 It is the site layout diagram;

[0059] Figure 9 The torsional stiffness test results and simulation results of NGW type planetary speed reducer are shown below;

[0060] Figure 10 Schematic diagram of the main effect of reducer volume;

[0061] Figure 11 Schematic diagram of the main effect of the reducer torsion angle;

[0062] Figure 12 This is the effect diagram of the surrogate model volume response fitting;

[0063] Figure 13 This is the fitting effect diagram of the torsional angle response of the surrogate model;

[0064] Figure 14 Optimize the solution set for the NSGA-II algorithm;

[0065] Figure 15 To optimize the front two-stage planetary carrier speed curve;

[0066] Figure 16 This is the optimized secondary planet carrier speed curve;

[0067] Figure 17 This is the flow chart of this application. DETAILED DESCRIPTION

[0068] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0069] like Figure 1-17As shown, the present invention provides a multi-objective optimization method for NGW planetary reducer based on Kriging surrogate model and NSGA-II, the method comprising:

[0070] S1. Establish NGW planetary reducer model and perform simulation

[0071] S1.1. Design of NGW planetary reducer

[0072] In the embodiment of the present invention, the technical requirements for the planetary reducer of the vertical adjustment mechanism of the rocket launch platform must meet the comprehensive indicators of high load-bearing performance, fast dynamic response and high operational reliability, and at the same time meet the requirements of production process economy. The present invention adopts a two-stage NGW type planetary gear reduction mechanism as the core transmission solution. The mechanism has high-efficiency transmission characteristics and high compactness. Its power density, motion smoothness and transmission accuracy and other indicators meet the engineering application standards, so it has wide application value in the field of heavy equipment. Figure 1 As shown in the figure, the NGW type planetary reducer includes a primary transmission unit and a secondary transmission unit. The primary transmission unit includes a primary sun gear, a planetary carrier assembly and three evenly distributed planetary gear assemblies; the secondary transmission unit adopts a symmetrical layout of four planetary gears, and a composite transmission system composed of a secondary sun gear, a planetary gear set, a planetary carrier and an inner ring gear.

[0073] like Figure 1 As shown, s1 represents the primary sun gear, p1 represents the primary planetary gear assembly, n1 represents the primary inner ring gear, and x1 represents the primary planet carrier; s2 represents the secondary sun gear, p2 represents the secondary planetary gear assembly, n2 represents the secondary inner ring gear, and x2 represents the secondary planet carrier.

[0074] Combined with the design requirements of the reducer in Table 1, the material of each gear of the NGW planetary reducer is 18CrNiMo7-6. The design basis is to meet the load-bearing capacity requirements, verify the gear strength, tooth matching conditions, coaxial conditions, adjacent conditions, etc., determine the gear pair tooth number matching scheme and meshing parameter configuration, and the design reducer parameters are shown in Table 2.

[0075] Table 1 Technical indicators of reducer

[0076]

[0077] Table 2 Reducer parameters

[0078]

[0079] As the core component of a planetary reducer, gears have complex involute tooth surfaces and include a series of variable parameters such as the number of teeth, module, and displacement coefficient. When obtaining planetary reducers with different transmission ratios and multi-stage series connections, it is necessary to change each relevant parameter corresponding to the gear. Each change is a large task, so SOLIDWORKS was chosen to perform parametric modeling of the planetary reducer gears and perform parametric assembly to obtain a three-dimensional model.

[0080] S1.2. Establish the finite element model of NGW planetary reducer

[0081] The three-dimensional model of the reducer was imported into ANSYS for static simulation. In order to balance the calculation efficiency and calculation accuracy, the meshing strategy selected was local encryption for the teeth that are not involved in the meshing and have no effect on the simulation results. A finer mesh was used in the stress concentration area, and a coarser mesh was used in other areas. The gear tooth profile, tooth top fillet, and tooth root transition curve were encrypted, and the initial mesh size was selected as 0.3mm. The stress and strain results of the planetary reducer gear train and the torsion angle of the input sun gear are shown in Figure 2. Figure 2 and Figure 3 .

[0082] S1.3. Verify the NGW planetary reducer model

[0083] 1.3.1 Verifying Grid Independence

[0084] In order to avoid inaccurate simulation results due to improper meshing of the NGW reducer finite element model, mesh independence verification is performed. The core purpose is to ensure the accuracy and reliability of the simulation results. The mesh size range of the tooth profile is selected as 0.1-0.7mm. The mesh independence verification results are shown in Figure 4-6 .Depend on Figure 4-6 As can be seen, the simulated stress values ​​are closest to the theoretically calculated stress values ​​when the mesh lengths are 0.1mm and 0.2mm, but the calculation time is significantly increased compared to the mesh length of 0.3mm. Because the error between the simulated stress values ​​and the theoretically calculated stress values ​​is only 1.2% when the mesh length is 0.3mm, it is much more accurate than the mesh length of 0.4mm. To balance calculation accuracy and efficiency, a mesh length of 0.3mm was ultimately selected for the tooth profile in the finite element simulation.

[0085] 1.3.2 Verification of the correctness of torsional stiffness simulation results

[0086] Torsional stiffness reflects the ability of the NGW planetary reducer to resist elastic deformation when subjected to torque. The higher the stiffness, the smaller the torsional deformation under the same torque, and the higher the rocket launch quality. Sufficient torsional stiffness can ensure that the planetary reducer can maintain high transmission accuracy when the load changes, avoid position errors caused by deformation, and at the same time reduce the backlash caused by insufficient stiffness, which can improve the dynamic performance of the planetary reducer and the stability of motion control. In order to ensure the accuracy, reliability and dynamic performance of the transmission system, it is necessary to calculate the torsional stiffness of the NGW planetary reducer. The expression of torsional stiffness K is:

[0087]

[0088] Where T is the torque; θ is the torsion angle of the mechanical structure.

[0089] The torsional stiffness of the NGW planetary reducer can be calculated by applying a torsional load to the primary sun gear using a finite element model. Furthermore, when optimizing the design of the NGW planetary reducer, torsional stiffness is also a primary optimization objective, and the correctness of the simulation results needs to be verified.

[0090] The test bench schematic and physical diagram are as follows Figure 7-8 The test bench features a symmetrical structure, primarily consisting of an electric motor, a torque and speed sensor, two NGW-type planetary reducers, and a loader. The two reducers under test used identical models and configurations; their specific structural parameters are shown in Table 2. During the experiment, the torque output by the electric motor, after being reduced in speed by the gearbox, was first transmitted to reducer No. 1 under test. This torque was then connected to reducer No. 2 through a drive shaft, and finally absorbed by the loader to maintain system stability.

[0091] During the test, the loader used maximum power to fix the output of test reducer No. 2. The motor then applied loads to the NGW planetary reducer in stages, at a certain percentage of the rated load. Torque and speed measurements were taken using a JZ2000 torque and speed sensor. The sensor measured torque with an accuracy of ±0.2% FS and a speed accuracy of ±0.1% RPM. The torque range was 0-2000 Nm, and the speed range was 0-6000 RPM.

[0092] The torsional stiffness test results and simulation results of NGW planetary speed reducer are shown in Figure 9. The simulated torsional stiffness of the designed planetary reducer is 1265.5N·m / °, which has a strong ability to resist torsional deformation and can avoid position errors caused by deformation. The torsional stiffness of the reducer obtained by experiment is 1074.8N·m / °, and the relative error of the simulation result is 15.07%. Those skilled in the art know that it is reasonable for the torsional stiffness of the planetary reducer calculated by finite element method to be within 20% greater than the measured torsional stiffness. The reason for this error is that the bearing model is not established during the simulation, and the bearing constraint is used as a rigid constraint, which leads to an increase in torsional stiffness. The above results show that the established NGW planetary reducer finite element model and the simulation analysis method of torsional stiffness are feasible, and its torsional stiffness simulation results can serve as the basis for subsequent multi-objective optimization.

[0093] S2. Establish a multi-objective optimization model for NGW planetary reducers

[0094] According to the technical requirements of the NGW planetary reducer, the device must meet the comprehensive indicators of high load-bearing performance, fast dynamic response, and high operational reliability, and is also limited by the installation space. Therefore, when establishing the optimization model, the objective function is to minimize the reducer's volume and maximize the torsional stiffness, and to meet the bending strength as a constraint condition. The overall optimization equation is as follows:

[0095]

[0096] The objective function f1(x) of the volume is expressed as follows:

[0097]

[0098] Where: V s is the volume of the sun gear; V n is the volume of the inner gear ring; V p is the volume of the planetary gear; C is the number of planetary gears; b is the width of the gear; Z s is the number of teeth of the center wheel; m is the module of the gear; i is the transmission ratio of the planetary gear.

[0099] The objective function f2(x) of torsional stiffness is expressed as follows:

[0100]

[0101] The bending strength as a function of the constraint condition f3(x) is expressed as follows:

[0102]

[0103] Where F t Gear tangential force; K A Service factor; K V Dynamic load coefficient; K FβTooth load distribution coefficient; K Fα tooth load distribution coefficient; b tooth width; m normal module; Y F Tooth form factor; Y S Stress correction factor.

[0104] S3. Select the optimal Latin hypercube sampling method to sample the NGW planetary reducer

[0105] Considering that this optimization is based on a complex, two-stage NGW planetary reducer, a hierarchical optimization approach was chosen to reduce computational time and improve efficiency. Because NGW planetary reducers have clear requirements for output torque and transmission ratio, the number of sun and planetary gear teeth remains unchanged during sampling. Only four factors—gear module m, tooth width b, sun gear displacement coefficient x, and center distance variation coefficient y—are sampled for each transmission stage. The parameter variation range for the NGW planetary reducer is shown in Table 3. To ensure sampling accuracy, the recommended number of sampling points is 2 × (n + 1) × (n + 2) (n is the number of sampling factors), resulting in 60 sampling points for each gear transmission stage. The sampling point parameters are rounded and standardized.

[0106] Table 3 NGW planetary reducer parameter variation range

[0107] Factor Name Level range Modulus m / mm 1<m<3 Tooth width b / mm 30<b<50 Sun gear displacement coefficient x 0.2<x<0.5 Center distance variation coefficient y 0.3<y<0.6

[0108] According to the rounded sampling point parameters, the three-dimensional model of the NGW planetary reducer is established in sequence, and the finite element analysis is performed. The gear module m, tooth width b, sun gear modification coefficient x and center distance variation coefficient y are used as input, and the torsional stiffness and volume are used as output. The DOE analysis is carried out on the ISISHT platform to obtain the main effect diagram of the gear module m, tooth width b, sun gear modification coefficient x, and center distance variation coefficient y of the NGW planetary reducer on the reducer volume and sun gear torsion angle, as shown in the figure. Figure 10 and 11 . Figure 10 It shows that the volume of the NGW planetary reducer shows a linear growth trend with the increase of tooth width b, and the volume increases exponentially with the increase of module m under the condition of unchanged number of teeth, while the displacement coefficient x and center distance variation coefficient y have almost no effect on the volume of the planetary reducer. Figure 11 It shows that the torsion angle of the sun gear of the NGW type planetary reducer shows a trend of first increasing and then decreasing with the increase of tooth width b. Under the condition that the number of teeth remains unchanged, the torsion angle of the sun gear shows a trend of decreasing with the increase of module m. The displacement coefficient x and the center distance variation coefficient y have a certain influence on the torsion angle, but the overall influence is small.

[0109] S4. Training the multi-objective optimization agent model for NGW planetary reducer

[0110] The Kriging model is used as the proxy model for the multi-objective optimization of the NGW planetary reducer. After the model training, 15 groups of sample data are randomly selected to verify the constructed proxy model. The model accuracy is evaluated by comparing the model prediction results with the actual simulation data. The proxy model response fitting effect is shown in Figure 12-13 In order to meet the accuracy verification requirements of the proxy model constructed by the sample points of the experimental group, the fitting performance of the proxy model was quantitatively evaluated based on the multidimensional error analysis method. Table 4 shows the fitting evaluation of the volume and torsion angle proxy models. The four evaluation indicators can all represent the fitting effect of the proxy model, among which R 2 The evaluation index intuitively reflects the model's explanation of data volatility through normalized form. Generally speaking, R 2 The larger the value, the better. The default lower limit is 0.9. When the index approaches 1, it indicates that the model reliability is excellent. Generally speaking, R 2 The value of is greater than 0.9, indicating that the model is credible. Table 4 shows that the R 2 The values ​​reached 0.95659 and 0.97312 respectively, which not only significantly exceeded the baseline of 0.9, but were also closer to the theoretical optimal value of 1, indicating that the training effect of the Kriging model was better and the surrogate model for the multi-objective optimization of the NGW planetary reducer was more credible.

[0111] Table 4 Goodness of fit evaluation of volume and torsion angle surrogate models

[0112] Fitting accuracy evaluation index volume Twist angle Average value (<0.2) 0.05244 0.03422 Maximum value (<0.3) 0.15988 0.08558 RMS (<0.3) 0.0677 0.04205 R-square root (>0.9) 0.95695 0.97312

[0113] A cross-validation strategy was used to further analyze the model's reliability. Six sets of sample parameters were randomly selected to construct an independent validation set. The estimated results of the NGW planetary reducer surrogate model were quantitatively compared with the finite element simulation results. Table 5 shows a comparison of the estimated values ​​of the fitted model and the simulation results. As shown in Table 5, the prediction deviations of the volume and torsion angle surrogate models established based on the Kriging model are within acceptable engineering standards, demonstrating that this surrogate model can effectively replace the NGW planetary reducer simulation model for optimization design.

[0114] Table 5 Comparison between proxy model estimates and simulation results

[0115]

[0116] S5. Multi-objective optimization of NGW planetary reducer

[0117] The NSGA-II algorithm is used to optimize the primary gear train. The NSGA-II algorithm optimization solution set is shown in Figure 14 . Figure 14The paper reveals the algorithm's convergence behavior during the iteration process, where the initial feasible solution gradually converges to the Pareto frontier, which is composed of non-dominated solutions. In multi-objective optimization, the weighting of the objective function directly determines the degree to which the optimization result favors each objective. By adjusting the weights, different compromise solutions can be obtained on the Pareto frontier. The changes in weights guide the solution along the Pareto frontier.

[0118] The NSGA-II algorithm can directly find all non-dominated solutions without relying on weights. To verify the effectiveness of the optimal solution specified by the NSGA-II algorithm, different weight ratios were set for the objective functions of torsional stiffness and volume for the NGW planetary reducer: 1:1, 1:2, and 2:1. Optimal Solutions 1, 2, and 3 were obtained, respectively. The optimization results for different objective function weights are shown in Table 6. Analysis shows that while the optimized solution 2 achieves a reduced volume compared to the pre-optimization solution, the torsional stiffness optimization effect is insufficient, failing to address the torsional stiffness requirement. The optimized solution 3 achieves an increased torsional stiffness compared to the pre-optimization solution, resulting in an insufficient volume optimization effect and failing to address the lightweighting requirement. The optimized solution 1 achieves a 3.9% reduction in volume and a 10.8% increase in torsional stiffness compared to the pre-optimization solution, effectively balancing both volume and torsional stiffness. Overall, the NSGA-II algorithm's optimal solution 1 achieves the best optimization results.

[0119] Table 6 Optimization effects of different weights of objective function

[0120] Before optimization Optimal Solution 1 Optimal Solution 2 Optimal Solution 3 Volume V / mm3 3069190 2950183 2627136 3036547 Torsion angle θ / ° 0.203 0.183 0.198 0.172

[0121] The optimization process for the secondary gear train of the NGW planetary reducer is similar to that for the primary gear train and will not be repeated here. The optimization results for the secondary gear train are shown in Table 7. Summarizing the optimization results for the primary and secondary gear trains, the torsional stiffness of the NGW planetary reducer increased by 11.20% and its volume decreased by 4.23%, demonstrating significant optimization results. The optimization results for the NGW planetary reducer are shown in Table 8.

[0122] Table 7 Optimization results of the secondary gear train of the reducer

[0123] parameter Before optimization After optimization Secondary gear tooth width b2 / mm 65 70 Secondary gear train module m2 / mm 3 2.5 Secondary sun gear displacement coefficient x2 0.285 0.322 Secondary gear train center distance variation coefficient y2 0.46 0.52 Volume V / mm3 7.951×106 7.620×106 Torsion angle θ / ° 0.198 0.193

[0124] Table 8 Optimization results of NGW planetary reducer

[0125] parameter Before optimization After optimization First gear tooth width b1 / mm 30 35 Primary gear train module m1 / mm 2 2 First-stage sun gear displacement coefficient x1 0.353 0.272 Primary gear train center distance variation coefficient y1 0.45 0.55 Secondary gear tooth width b2 / mm 65 70 Secondary gear train module m2 / mm 3 2.5 Secondary sun gear displacement coefficient x2 0.285 0.322 Secondary gear train center distance variation coefficient y2 0.46 0.52 Volume V / mm3 1.102×107 1.057×107 Torsional stiffness K / (Nm / °) 1265.5 1407.2

[0126] S6. Optimization results analysis

[0127] In order to verify the optimization effect of the dynamic characteristics of the NGW planetary reducer, the speed fluctuation of the reducer before and after optimization is compared; this NGW planetary reducer uses the sun gear as input, the inner ring gear is fixed, and the planetary carrier is output, so the speed fluctuation of the secondary planetary carrier before and after optimization is compared. The three-dimensional models of the reducer before and after optimization are imported into the ADAMS platform to establish the dynamic model of the NGW planetary reducer. The speed curves of the secondary planetary carrier before and after optimization are shown in Figure 2. Figure 15 and Figure 16 Table 9 shows a comparison of the secondary planetary carrier speed fluctuations before and after optimization. As shown in Table 9, the optimized secondary planetary carrier speed fluctuations for the NGW planetary reducer are reduced by 7.4%, demonstrating the effectiveness of the ISIGHT platform-based multi-objective optimization of the torsional stiffness and volume of the NGW planetary reducer. While improving the reducer's torsional stiffness, it also reduces the secondary planetary carrier speed fluctuations under rated operating conditions, weakening the periodic vibration during transmission and further improving transmission reliability.

[0128] Table 9 Comparison of secondary planet carrier rotation speed fluctuation before and after optimization

[0129] Before optimization After optimization Secondary planet carrier speed fluctuation (deg / sec) 2.7 2.5

[0130] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A multi-objective optimization method for NGW planetary reducer based on Kriging surrogate model and NSGA-II, characterized by: The method comprises: S1. Establish an NGW planetary reducer model and perform simulation; S2. Establish a multi-objective optimization model for NGW planetary reducer; S3. Select the optimal Latin hypercube sampling method to sample the NGW planetary reducer; S4. Training the multi-objective optimization agent model for NGW planetary reducer; S5. Multi-objective optimization of NGW type planetary reducer; S6. Analysis of optimization results.

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