CN107577876B - Multi-objective optimization method for tooth surface loading performance of spiral bevel gear - Google Patents
Multi-objective optimization method for tooth surface loading performance of spiral bevel gear Download PDFInfo
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Abstract
The invention relates to a multi-objective optimization method for the tooth surface loading performance of a spiral bevel gear, which is characterized by comprising the following steps of: 1) establishing a mathematical model of a spiral bevel gear tooth surface loading performance multi-objective optimization problem, and obtaining a test design sample point; 2) establishing a tooth surface loading contact analysis method considering tooth root bending stress, and carrying out tooth surface loading contact analysis on each test design sample point to obtain response values of a target function and a constraint function corresponding to each test design sample point, so as to obtain an initial sample point set comprising each test design sample point and the corresponding response value thereof; 3) and fitting a Kriging agent model based on the initial sample point set, and solving the mathematical model of the spiral bevel gear tooth surface loading performance multi-objective optimization problem to obtain an optimal solution set of the spiral bevel gear tooth surface loading performance multi-objective optimization problem. The method has high calculation efficiency and high calculation accuracy, and can be widely applied to multi-target optimization of the tooth surface loading performance of the spiral bevel gear.
Description
Technical Field
The invention relates to a tooth surface loading performance optimization method, in particular to a spiral bevel gear tooth surface loading performance multi-objective optimization method.
Background
The spiral bevel gear has the advantages of stable transmission, high bearing capacity and the like, and is widely applied to the fields of automobiles and aviation. In order to obtain a tooth surface with good performance, a gear engineer needs to adjust and trial calculate tooth surface precontrol parameters for multiple times until a relatively ideal result is obtained, the process is more dependent on the experience of the engineer, and multiple tooth surface loading analysis is needed, so that a large amount of time is consumed. Therefore, the optimized design of the tooth surface loading performance has important engineering value.
The current research has the following problems for the optimization of the tooth surface loading performance of the spiral bevel gear: firstly, the research focuses on the single-target optimization problem, the processing of the multi-target optimization problem is also converted into the single-target problem, and compared with the single-target optimization, the tooth surface loading performance multi-target optimization can comprehensively evaluate each loading performance index, so that a more appropriate tooth surface design scheme is selected; secondly, the bending stress of the tooth root is not considered in the existing optimization model, and the excessive bending stress of the tooth root can cause tooth breakage, so that the optimization model is an important factor to be considered in the optimization design. The main reason for the above problems is that multiple times of tooth surface loading contact analysis (LTCA) is required when performing multi-objective optimization of tooth surface loading performance, and LTCA usually adopts a semi-analytic calculation method, calculates bending deformation by using a gear tooth finite element model (hereinafter, referred to as a coarse finite element model) with a large grid size, and calculates contact deformation and shear deformation by using a weber empirical formula. However, the tooth root bending stress cannot be accurately calculated by adopting the rough finite element model, and if the fine finite element model is adopted, the method is not suitable for multiple times of calculation in a multi-objective optimization problem due to large calculation amount.
Disclosure of Invention
Aiming at the problems, the invention aims to provide a multi-objective optimization method for the tooth surface loading performance of the spiral bevel gear, which has both calculation accuracy and calculation efficiency.
In order to achieve the purpose, the invention adopts the following technical scheme: a multi-objective optimization method for tooth surface loading performance of a spiral bevel gear is characterized by comprising the following steps: 1) establishing a mathematical model of a spiral bevel gear tooth surface loading performance multi-objective optimization problem, and obtaining a test design sample point; 2) establishing a tooth surface loading contact analysis method considering tooth root bending stress, and carrying out tooth surface loading contact analysis on each test design sample point to obtain response values of a target function and a constraint function corresponding to each test design sample point, so as to obtain an initial sample point set comprising each test design sample point and the corresponding response value thereof; 3) and fitting a Kriging agent model based on the initial sample point set, and solving the mathematical model of the spiral bevel gear tooth surface loading performance multi-objective optimization problem to obtain an optimal solution set of the spiral bevel gear tooth surface loading performance multi-objective optimization problem.
In the step 1), the method for acquiring the test design sample points comprises the following steps: 1.1) establishing a mathematical model of a spiral bevel gear tooth surface loading performance multi-objective optimization problem, and determining a design variable, a target function and a constraint function of the mathematical model; 1.2) determining the value range of each design variable in the step 1.1); 1.3) carrying out Latin hypercube test design according to the value range of the design variable, and establishing test design sample points.
In the step 1.1), the method for establishing the mathematical model of the spiral bevel gear tooth surface loading performance multi-objective optimization problem comprises the following steps: 1.1.1) determining design variables of a spiral bevel gear tooth surface loading performance multi-objective optimization problem, wherein the design variables comprise a pre-control transmission error curve coefficient, a contact trace slope and a contact half width; 1.1.2) determining a target function of a spiral bevel gear tooth surface loading performance multi-target optimization problem, wherein the target function comprises maximum contact stress of a tooth surface and a peak value of a loading transmission error; 1.1.3) determining a constraint function of a spiral bevel gear tooth surface loading performance multi-objective optimization problem, wherein the constraint function comprises the maximum bending stress of tooth roots of a small wheel and a large wheel and the area of an actual loading contact zone outside an ideal contact zone; 1.1.4) obtaining a mathematical model of the spiral bevel gear tooth surface loading performance multi-objective optimization problem according to the determined design variables, the objective function and the constraint function.
In the step 1.1.4), the mathematical model of the spiral bevel gear tooth surface loading performance multi-objective optimization problem is as follows:
in the formula, a is a pre-control transmission error curve coefficient; k is a radical ofsIs the contact trace slope; b is contact half width; sigmacMaximum contact stress of tooth surface, peak load transmission error value and peak value in LTE, △ S area of actual load contact area outside ideal contact area, △ SmaxTo set tolerances; sigmapAnd σgMaximum curvature of the tooth root of the small and large wheels respectivelyStress; sigmabmaxThe allowable value of the set maximum bending stress is obtained; k is a radical ofmin,kmax,amin,amax,bmin,bmaxRespectively the upper and lower value limits corresponding to each variable.
In the step 2), the step of establishing the tooth surface loading contact analysis method considering the tooth root bending stress comprises the following steps: 2.1) designing sample points based on the test, dividing a gear rough grid finite element model by adopting the existing tooth surface contact analysis method, and calculating to obtain tooth surface load distribution without considering tooth root bending stress; 2.2) refining the divided rough gear mesh finite element model to obtain a fine mesh finite element model for calculating tooth root bending stress; 2.3) applying the tooth flank load distribution obtained in step 2.1) to the mesh nodes of the fine mesh finite element model used for calculating the tooth root bending stress in step 2.2); 2.4) calculating the finite element model of the fine grid at the iteration time t, and returning the calculation result of the tooth root bending stress at the time; 2.5) judging whether the time is the last time: when the iteration time t does not reach the last time, updating the iteration time tnT +1, and at a new iteration time tnRepeating the steps 2.2) to 2.4); and when the iteration time t reaches the last time, completing the calculation to obtain the response values of the target function and the constraint function corresponding to each experimental design sample point.
In the step 3), the method for solving the bevel gear tooth surface loading performance multi-objective optimization problem comprises the following steps: 3.1) establishing a sample point updating method based on EI-RMSE, which comprises two stages of optimal point supplement based on EI indexes and maximum error point supplement based on RMSE indexes; 3.2) performing EI index-based design point supplementary optimization on the initial sample point set to obtain a first-stage optimized and updated sample point set; 3.3) carrying out maximum error point supplementary optimization based on RMSE indexes on the sample point set obtained in the step 3.2) after the first-stage optimization updating to obtain an optimal solution set of the spiral bevel gear tooth surface loading performance multi-objective optimization problem.
In the step 3.1), the calculation formula of the Kriging agent model is as follows:
y(x)＝f(x)β+z(x)；
wherein y (x) is unknown Kriging agent model, f (x) is known regression model, β is regression coefficient to be determined, z (x) is mean value 0 and variance sigma2The random process of (a);
the calculation formulas of EI and RMSE are respectively as follows:
RMSE＝s
wherein f isminIs the minimum of all the sample points and,
for the estimated value of the predicted point, s is the root mean square error of the predicted point, and phi is the mean value
A cumulative distribution function with a standard deviation of s, phi being the mean
The standard deviation is a probability density function of s.
In the step 3.2), the first-stage optimized and updated sample point set obtaining method includes the following steps:
3.2.1) fitting Kriging agent models of each objective function and constraint function in the tooth surface loading performance multi-objective optimization problem according to the initial sample point set;
3.2.2) establishing a single-target optimization problem of each objective function based on the established Kriging agent model, and calculating to obtain the maximum EI value of each objective function and the corresponding design point in the value range of each design variable; the agent model of the single-target optimization problem of each objective function is as follows:
constraint function gi(x) The proxy model of (2) is:
wherein the content of the first and second substances,
is an estimate of the ith constraint function, sgiFor the root mean square error of the constraint function, m is the control level that satisfies the constraint;
3.2.3) if the maximum EI value of each objective function is smaller than the set threshold, finishing the supplement of the maximum EI value design point in the first stage to obtain an optimized and updated sample point set; otherwise, entering step 3.2.4);
3.2.4) calculating the target function of the design point corresponding to each maximum EI value and the real response value of the constraint function by using the tooth surface loading contact analysis method which is established in the step 2) and takes the bending stress of the tooth root into consideration;
3.2.5) adding the response values of the objective function and the constraint function of each maximum EI value corresponding to the design point in the step 3.2.4) into the sample point set adopted by the current fitting Kriging proxy model, and returning to the step 3.2.1).
In the step 3.3), the method for obtaining the optimal solution set of the spiral bevel gear tooth surface loading performance multi-objective optimization problem comprises the following steps: 3.3.1) fitting a Kriging agent model of a target function and a constraint function in the tooth surface loading performance multi-target optimization problem based on the sample point set obtained in the step 3.2); 3.3.2) based on the Kriging agent model obtained in the step 3.3.1), carrying out optimization calculation on the tooth surface loading performance multi-objective optimization problem by adopting a multi-objective optimization algorithm to obtain a group of initial pareto solution sets; 3.3.3) calculating the RMSE value of each target function in the initial pareto solution set, and finding the RMSE maximum value of each target function and the corresponding design point; 3.3.4) if the maximum value of the RMSE of each objective function is greater than the respective set threshold, then go to step 3.3.7), otherwise go to step 3.3.5); 3.3.5) calculating the target function of the design point corresponding to each RMSE maximum value and the real response value of the constraint function by adopting the tooth surface loading contact analysis method which is established in the step 2) and takes the bending stress of the tooth root into consideration; 3.3.6) adding the design points corresponding to the maximum value of each RMSE in the step 3.3.5) and the response values of the target function and the constraint function thereof into a sample point set adopted by the current fitting Kriging agent model, and returning to the step 3.3.1); 3.3.7), completing multi-objective optimization, wherein the sample point set currently used for fitting the Kriging surrogate model is the optimal solution set of the spiral bevel gear tooth surface loading performance multi-objective optimization problem.
Due to the adoption of the technical scheme, the invention has the following advantages: 1. according to the invention, a multi-target optimization model of the tooth surface loading performance of the spiral bevel gear is established, each loading performance index of the tooth surface is comprehensively evaluated, and compared with the existing single-target optimization model, the optimal solution set of the tooth surface design with good comprehensive performance can be obtained. 2. According to the tooth surface loading analysis method, tooth root bending stress is added in the existing tooth surface loading analysis method, the phenomenon of tooth breakage caused by overlarge tooth root bending stress is avoided, and the consideration factor is more comprehensive. 3. According to the invention, due to the introduction of the Kriging surrogate model establishing method, a functional relation between a response and a design variable is established based on the test design sample points of the design variable and the response value corresponding to each test design sample point obtained by adopting the actual LTCA, and the actual LTCA calculation is converted into the surrogate model, so that the calculation efficiency is improved, and the problem of large calculation amount in the multi-objective optimization problem is solved. 4. Based on the established Kriging agent model, the invention carries out EI index and RMSE index-based sample point optimization on the initial design sample points, thereby ensuring that the obtained optimal solution set is close to the actual optimal solution set to the maximum extent and ensuring the accuracy of calculation. The method can be widely applied to optimization of the tooth surface loading performance of the spiral bevel gear.
Drawings
FIG. 1 is a flow chart of a tooth surface loading performance multi-objective optimization problem solution of the present invention;
FIG. 2 is a schematic illustration of the contact trace slope and contact half-width of the present invention;
FIG. 3 is a schematic view of a loaded drive error curve of the present invention;
FIG. 4 is a schematic view of a loaded contact footprint of the present invention;
FIG. 5(a) is a tooth surface loading contact analysis flow chart for the present invention that considers root bending stress calculations;
FIG. 5(b) is a schematic illustration of load distribution to a refined finite element model in a tooth surface loading contact analysis of the present invention taking into account root bending stress calculations;
FIG. 6 is a flowchart illustrating the variation of the maximum EI value of the objective function 1 during the proxy model building process according to the embodiment of the present invention;
FIG. 7 is a flowchart illustrating the variation of the maximum EI value of the objective function 2 during the proxy model building process according to the embodiment of the present invention;
FIG. 8 is a diagram illustrating a variation process of a maximum RMSE value of an objective function in a proxy model building process according to an embodiment of the present invention;
FIG. 9 is a diagram of pareto solution after optimization in an embodiment of the present invention;
FIGS. 10(a) -10 (d) are graphs of the results of the load-contact analysis of the original design without optimization in an embodiment of the present invention;
11(a) -11 (d) are graphs showing the results of the load contact analysis of the optimized solution G1 in the example of the present invention;
FIGS. 12(a) -12 (d) are graphs showing the results of the load contact analysis of the optimized solution G1 in the example of the present invention;
FIG. 13 is a graph of the results of comparing the load drive error curve of the present invention after optimization to that when not optimized.
Detailed Description
The invention is described in detail below with reference to the figures and examples.
As shown in FIG. 1, the invention provides a multi-objective optimization method for tooth surface loading performance of a spiral bevel gear, which comprises the following steps:
1) establishing a mathematical model of a spiral bevel gear tooth surface loading performance multi-objective optimization problem, and obtaining a test design sample point;
2) establishing a tooth surface loading contact analysis method considering tooth root bending stress, and carrying out tooth surface loading contact (LTCA) analysis on each test design sample point to obtain a corresponding response value of each test design sample point, so as to obtain an initial sample point set comprising each test design sample point and the corresponding response value thereof;
3) and fitting a Kriging agent model based on the initial sample point set, and solving the mathematical model of the spiral bevel gear tooth surface loading performance multi-objective optimization problem to obtain an optimal solution set (hereinafter referred to as pareto solution set) of the spiral bevel gear tooth surface loading performance multi-objective optimization problem.
In the step 1), the method for establishing the mathematical model of the spiral bevel gear tooth surface loading performance multi-objective optimization problem and obtaining the test design sample points comprises the following steps:
1.1) establishing a mathematical model of a spiral bevel gear tooth surface loading performance multi-objective optimization problem, and determining a design variable, a target function and a constraint function of the mathematical model;
1.2) determining the value range of each design variable in the step 1.1);
1.3) carrying out Latin hypercube test design according to the value range of the design variable to obtain test design sample points of each design variable.
In the step 1.1), the method for establishing the mathematical model of the spiral bevel gear tooth surface loading performance multi-objective optimization problem comprises the following steps:
1.1.1) determining design variables of the multi-objective optimization problem of the tooth surface loading performance of the spiral bevel gear, including a pre-control transmission error curve coefficient a and a contact trace slope ksAnd a contact half width b.
The design parameters of the tooth surface modification surface of a small wheel in the spiral bevel gear are used as the design variables of an optimization model, and the design variables comprise 3 design variables which are respectively a pre-control transmission error curve coefficient a and a contact trace slope ksAnd a contact half width b.
As shown in fig. 2, a schematic of contact trace slope and contact half-width is shown. Wherein the contact trace slope is ksRepresents the contact trace C in the coordinate system of the figure1C2The slope of (a); the contact half-width is denoted by b and represents the distance from a point on the profile surface to the contact trace; the contact center A is set at the center of the profile modification surface to ensure the symmetry of the no-load transmission error curve. The coefficient of the pre-control transmission error curve is represented by a, is used for controlling the peak-to-peak value of the no-load transmission error, and is the quadratic term coefficient of the no-load transmission error curve, wherein the calculation formula of the no-load transmission error curve is as follows:
in the formula, phi1,φ2Are respectively the rotating angles when the small wheel and the big wheel are meshed,
initial positions of the small and large wheels, respectively, i12Is the theoretical transmission ratio.
1.1.2) determining an objective function of a spiral bevel gear tooth surface loading performance multi-objective optimization problem, including maximum contact stress sigma of the tooth surfacecAnd loading a transmission error peak-to-peak value LTE.
As shown in fig. 3, a loaded transmission error curve diagram is shown, wherein a loaded transmission error peak-to-peak value LTE is an excitation source of vibration noise of the helical bevel gear, and generally, the smaller the value, the more ideal the performance is. Maximum contact stress sigma of tooth surfacecThe maximum value of the tooth surface contact stress is indicated, and in order to ensure that the tooth surface does not fail, the maximum contact stress of the tooth surface needs to be smaller as well as better under the condition that edge contact does not occur.
1.1.3) determining a constraint function of a spiral bevel gear tooth surface loading performance multi-objective optimization problem, including maximum bending stress sigma of tooth roots of a small wheel and a large wheelpAnd σgAnd an area △ S where the actual loaded contact zone is outside of the ideal contact zone.
As shown in FIG. 4, to accurately reflect the loaded contact footprint location, the loaded contact analysis results are placed in the large wheel projection plane, under full load, the dotted line is the contact zone obtained by LTCA (tooth surface loaded contact analysis), the dotted line is the ideal contact zone after loading, the positions of which are 5% of the flank width of each of the small end and the large end, 5% of the full tooth height of each of the addendum and the dedendum, and 90% of the full tooth width and height, and to ensure that no edge contact occurs, an area difference △ S is defined, as follows
△S＝SL-Sc(2)
Wherein S isLContact area, S, for LTCAcArea of the contact area common to the ideal contact area and the contact area obtained by LTCA, △ S being bothDifference, i.e., the area of the LTCA touch region that falls outside the ideal touch region (as shown by the shaded area in FIG. 4), by controlling this area to be less than the set tolerance △ SmaxIt is ensured that no edge contact occurs in the loading contact area.
Controlling the maximum root bending stress sigma of the small and large wheels while controlling the load contact zonepAnd σgNot exceeding the allowable value
1.1.4) obtaining a mathematical model of the spiral bevel gear tooth surface loading performance multi-objective optimization problem according to the determined design variables, the objective function and the constraint function, wherein the calculation formula is as follows:
in the formula, kmin,kmax,amin,amax,bmin,bmaxRespectively the upper and lower value limits corresponding to each variable.
In the above step 2), as shown in fig. 5(a) and 5(b), the tooth surface loading contact analysis method considering the tooth root bending stress includes the following steps:
2.1) designing sample points based on the test, dividing a gear rough grid finite element model by adopting the existing tooth surface contact analysis method, and calculating to obtain tooth surface load distribution without considering tooth root bending stress;
2.2) refining the gear rough grid finite element model to obtain a fine grid finite element model for calculating tooth root bending stress;
2.3) applying the tooth flank load distribution obtained in step 2.1) to the mesh nodes of the fine mesh finite element model used for calculating the tooth root bending stress in step 2.2);
2.4) using finite element calculation software (e.g.: nastran) calculates a finite element model of the fine grid at the iteration time t, and returns the calculation result of the tooth root bending stress at the time;
2.5) judgmentWhether the break is the last time: when the iteration time t does not reach the last time, updating the iteration time tnT +1, and at a new iteration time tnRepeating the steps 2.2) to 2.4); and when the iteration time t reaches the last time, completing the calculation to obtain the response values of the target function and the constraint function corresponding to each experimental design sample point.
In the step 3), the method for solving the bevel gear tooth surface loading performance multi-target optimization problem by fitting the Kriging agent model based on the initial sample point set comprises the following steps:
3.1) establishing a sample point updating method based on EI (Expected Improvement) -RMSE (Root Mean square error), wherein the method comprises two stages of optimal point supplementary updating based on EI indexes and maximum error point supplementary updating based on RMSE indexes;
3.2) performing EI index-based design point supplementary optimization on the initial sample point set to obtain a first-stage optimized and updated sample point set;
and 3.3) carrying out maximum error point supplement optimization based on RMSE indexes on the obtained first-stage optimized and updated sample point set to obtain an optimal solution set of the spiral bevel gear tooth surface loading performance multi-objective optimization problem.
In the step 3.1), the invention adopts a sample point updating method based on EI-RMSE to gradually and dynamically fill new sample points. The EI and RMSE are two measurement indexes respectively, the EI index represents the improvement expectation, and the index means that if the improvement expectation value is larger for a single target minimization problem, the better solution is possible at the point. RMSE represents the root mean square error value of the Kriging model at the predicted point, with smaller values indicating smaller prediction errors at that point.
The Kriging agent model is as follows:
y(x)＝f(x)β+z(x) (4)
wherein y (x) is unknown Kriging agent model, f (x) is known regression model, β is regression coefficient to be determined, z (x) is mean value 0 and variance sigma2The random process of (a).
EI and RMSE were calculated as follows:
wherein f isminIs the minimum of all the sample points and,
for the estimated value of the predicted point, s is the root mean square error of the predicted point, and phi is the mean value
A cumulative distribution function with a standard deviation of s, phi being the mean
The standard deviation is a probability density function of s.
In the step 3.2), the method for performing EI-index-based design point complementary optimization on the initial sample point set to obtain the first-stage optimized and updated sample point set includes the following steps:
3.2.1) fitting Kriging agent models of each objective function and constraint function in the tooth surface loading performance multi-objective optimization problem according to the initial sample point set.
3.2.2) establishing a single-target optimization problem of each objective function by using the established Kriging agent model, and calculating by adopting a downhill simplex method to obtain the maximum EI value of each objective function and the corresponding design point within the value range of each design variable.
The calculation formula of the single-target optimization problem of each objective function is as follows:
for the constraint function gi(x) Calculated using the following formula
Wherein the content of the first and second substances,
is an estimate of the ith constraint function, sgiFor the root mean square error of the constraint function, m is the control level that satisfies the constraint, and is taken to be 2.
3.2.3) if the maximum EI value of each objective function is smaller than the set threshold, finishing the supplement of the maximum EI value design point in the first stage to obtain an updated sample point set; otherwise, entering step 3.2.4);
3.2.4) calculating the target function of the design point corresponding to each maximum EI value and the real response value of the constraint function by using the tooth surface loading contact analysis method which is established in the step 2) and takes the bending stress of the tooth root into consideration;
3.2.5) adding the design points corresponding to the maximum EI values in the step 3.2.4) and the response values of the target function and the constraint function thereof into the sample point set adopted by the current fitting Kriging proxy model, and returning to the step 3.2.1).
In the step 3.3), the method for performing maximum error point supplementary optimization based on the RMSE index on the obtained first-stage optimized and updated sample point set to obtain an optimal solution set of the spiral bevel gear tooth surface loading performance multi-objective optimization problem includes the following steps:
3.3.1) fitting a Kriging agent model of a target function and a constraint function in the tooth surface loading performance multi-target optimization problem based on the sample point set obtained in the step 3.2);
3.3.2) based on the Kriging agent model obtained in the step 3.3.1), carrying out optimization calculation on the tooth surface loading performance multi-objective optimization problem by adopting a multi-objective optimization algorithm (NSGA-II) to obtain a group of initial pareto solution sets;
3.3.3) calculating the RMSE value of each target function in the initial pareto solution set, and finding the RMSE maximum value of each target function and the corresponding design point.
3.3.4) if the maximum value of RMSE of each objective function is greater than the respective set threshold epsilonR1～nIf so, the RMSE maximum optimization of the second stage is finished, and step 3.3.7) is entered, otherwise, step 3.3.5) is entered;
3.3.5) calculating the target function of the design point corresponding to each RMSE maximum value and the real response value of the constraint function by adopting the tooth surface loading contact analysis method which is established in the step 2) and takes the bending stress of the tooth root into consideration;
3.3.6) adding the design points corresponding to the maximum value of each RMSE in the step 3.3.5) and the response values of the target function and the constraint function thereof into the sample point set adopted by the current fitting Kriging agent model, and returning to the step 3.3.1).
3.3.7), completing multi-objective optimization, wherein the sample point set used for fitting the Kriging surrogate model at present is a pareto solution set of the spiral bevel gear tooth surface loading performance multi-objective optimization problem.
The multi-objective optimization method for the tooth surface loading performance of the spiral bevel gear is further described below by combining specific embodiments. In the invention, optimization calculation is carried out by taking the optimization problem of a pair of spiral bevel gear pairs as an example, and the design parameters of the spiral bevel gear pairs are shown in the following table 1.
TABLE 1 basic parameter table of helical bevel gear pair
1. Establishing a spiral bevel gear tooth surface loading performance multi-objective optimization model by using the method in the step 1), wherein the objective function 1 is the maximum contact stress sigma of the tooth surfacecAnd the target function 2 is a loading transmission error peak-to-peak value LTE:
in the formula, the maximum allowable value σ of bending stressbmaxA tolerance of △ S is set to 300MPamaxIs 3mm2。
2. And (3) carrying out test design by adopting a Latin hypercube method, selecting 33 test design sample points, obtaining response values of a target function and a constraint function corresponding to each test design sample point by utilizing the LTCA calculation method considering the tooth root bending stress in the step 2), and calculating the input torque 1500Nm under the working condition of full load of the gear pair.
3. And 3) solving the tooth surface loading performance multi-objective optimization problem by using the method in the step 3).
As shown in FIGS. 6 to 9, a Kriging multi-objective optimization method based on EI-RMSE update strategy is adopted. The first stage is as follows: maximum contact stress σcIs set to epsilonE1When the transmission error peak value LTE is loaded, the EI threshold value is set to be epsilonE2And 0.3, stopping when each objective function maximum EI value is less than the threshold. After 13 calculations, the first stage is completed, and the curve of the maximum EI value of the objective function 1 is shown in fig. 6, and the curve of the maximum EI value of the objective function 2 is shown in fig. 7. And a second stage: maximum contact stress σcIs set to εR1The RMSE threshold for loading the transmission error peak-to-peak LTE is set to ε at 5R2After 26 times of calculation, the maximum RMSE value of each objective function in the final pareto solution set is smaller than the set threshold, and the curve of the variation process of the maximum RMSE value is shown in fig. 8. When the maximum RMSE value of each objective function is smaller than the set threshold, the pareto solution set of the optimization results is obtained as shown in fig. 9. The obtained Kriging agent model parameters of the objective function and the constraint function are as follows:
TABLE 2 maximum tooth flank contact stress (objective function 1) proxy model parameters
TABLE 3 Loading Transmission error Peak to Peak (Objective function 2) proxy model parameters
TABLE 4 constraint function bull wheel maximum root bending stress proxy model parameters
TABLE 5 constraint function Small wheel maximum root bending stress proxy model parameters
TABLE 6 constraint function Δ S surrogate model parameters
4. In order to verify the precision of the agent model at the optimal solution set, two design points G1 and G2 (see fig. 9) at which the target function 1 and the target function 2 reach minimum values in the pareto solution set are first taken to perform LTCA analysis in step 2), table 7 shows that the optimal solution obtained by the agent model is compared with the real LTCA calculation, and the calculation result shows that the maximum error between the agent model and the real model is 1.69%, and the precision is high. Further performing LTCA analysis of step 2) on all solutions in the pareto solution set, wherein the objective function sigma is in all solutionscThe maximum error of the fitting value and LTCA analysis is 1.88%, the maximum error of the target function LTE fitting value and LTCA analysis is 5.63%, and the constraint function sigma ispThe maximum error between the fitting value and LTCA analysis is 1.00%, and the constraint function sigmagThe maximum error of the fitting value and the LTCA analysis is 1.11%, the maximum error of the fitting value and the LTCA analysis of the constraint function delta S is 7.27%, and the absolute error of the fitting value and the LTCA analysis is only 0.23mm although the delta S error is slightly larger2The above results show that the pareto solution set precision obtained by the method of the present invention meets the requirements.
The fit values of the objective function and the constraint function at the solutions of G1 and G2 in Table 7 are compared with the real values
The loading contact footprint, contact stress cloud, pinion root bending stress cloud and bull root bending stress cloud of the original design are shown in fig. 10(a) -10 (d). The load contact footprint, contact stress cloud, pinion root bending stress cloud and bull tooth root bending stress cloud corresponding to the optimized solution G1 are shown in fig. 11(a) -11 (d). The load contact footprint, the contact stress cloud, the pinion root bending stress cloud and the bull tooth root bending stress cloud corresponding to the optimized solution G2 are shown in FIGS. 12(a) -12 (d). The load drive error curves when not optimized and the load drive error curves of solution G1, G2 are shown in fig. 13.
To illustrate the optimized effect, solutions G1 and G2 in the pareto solution set were compared to the original design analysis results when the gear pair was not optimized, as shown in table 8.
TABLE 8 Pareto solutions G1, G2 in Pareto solution set versus original design calculation results
After optimization, the following conclusions can be drawn through calculation results:
(1) the optimized front tooth surface contact print has edge contact between the small end and the tooth top, the optimized rear tooth surface contact print has no edge contact, and the tooth top, the tooth root, the small end and the large end are not separated.
(2) The maximum contact stress is reduced compared with the original design, the G1 solution with the maximum reduction degree is reduced by 13.8 percent compared with the solution before optimization, and the G2 solution is reduced by 8 percent compared with the solution before optimization. For the maximum value of the contact stress near the center of the tooth surface, the solution G1 is slightly reduced by 3.8 percent compared with the original design, the solution G2 is slightly increased by 2.6 percent compared with the original design, and the maximum contact stress near the center is basically equal to the original design.
(3) Compared with the original design, the peak-to-peak loading transmission error is reduced, the solution G2 with the maximum reduction degree is reduced by 42.8 percent compared with the original design, and the solution G1 is reduced by 24.2 percent compared with the original design.
(4) In the optimized constraint function, the maximum tooth root bending stress of the large wheel and the small wheel is smaller than an allowable value and is reduced compared with the original design, the bending stress of the tooth root of the large wheel G1 and the bending stress of the tooth root of the large wheel G2 are reduced by 3.5 percent and 4.4 percent respectively, and the bending stress of the tooth root of the small wheel is reduced by 6.8 percent and 7.8 percent respectively. The area outside the ideal contact area is reduced by 85 percent compared with the original design and is kept atSet Δ SmaxTolerance boundaries.
(5) The two stages of the optimization process are respectively supplemented with 24 and 36 design points, and 93 times of LTCA analysis (containing 33 sample points in the experimental design) are carried out on the supply and demand of the optimization process. If the NSGA-II multi-target optimization algorithm is directly adopted, the calculation amount is huge, and the operation cannot be carried out;
(6) the method for solving the Kriging surrogate model provided by the invention obtains a pareto solution set of the tooth surface multi-objective optimization problem, and the solution set precision is higher than that of a true value.
The above embodiments are only used for illustrating the present invention, and the structure, connection mode, manufacturing process, etc. of the components may be changed, and all equivalent changes and modifications performed on the basis of the technical solution of the present invention should not be excluded from the protection scope of the present invention.
Claims (8)
1. A multi-objective optimization method for tooth surface loading performance of a spiral bevel gear is characterized by comprising the following steps:
1) establishing a mathematical model of a spiral bevel gear tooth surface loading performance multi-objective optimization problem, and obtaining a test design sample point;
2) establishing a tooth surface loading contact analysis method considering tooth root bending stress, and carrying out tooth surface loading contact analysis on each test design sample point to obtain response values of a target function and a constraint function corresponding to each test design sample point, so as to obtain an initial sample point set comprising each test design sample point and the corresponding response value thereof;
in the step 2), the step of establishing the tooth surface loading contact analysis method considering the tooth root bending stress comprises the following steps:
2.1) designing sample points based on the test, dividing a gear rough grid finite element model by adopting the existing tooth surface contact analysis method, and calculating to obtain tooth surface load distribution without considering tooth root bending stress;
2.2) refining the divided rough gear mesh finite element model to obtain a fine mesh finite element model for calculating tooth root bending stress;
2.3) applying the tooth flank load distribution obtained in step 2.1) to the mesh nodes of the fine mesh finite element model used for calculating the tooth root bending stress in step 2.2);
2.4) calculating the finite element model of the fine grid when the iteration time is t, and returning the calculation result of the tooth root bending stress at the time;
2.5) judging whether the time is the last time: when the iteration time t does not reach the last time, updating the iteration time tnT +1, and at a new iteration time tnRepeating the steps 2.2) to 2.4); when the iteration time t reaches the last time, the calculation is completed, and the response values of the target function and the constraint function corresponding to each test design sample point are obtained;
3) and fitting a Kriging agent model based on the initial sample point set, and solving the mathematical model of the spiral bevel gear tooth surface loading performance multi-objective optimization problem to obtain an optimal solution set of the spiral bevel gear tooth surface loading performance multi-objective optimization problem.
2. The multi-objective optimization method for the tooth surface loading performance of the spiral bevel gear according to claim 1, characterized in that: in the step 1), the method for acquiring the test design sample points comprises the following steps:
1.1) establishing a mathematical model of a spiral bevel gear tooth surface loading performance multi-objective optimization problem, and determining a design variable, a target function and a constraint function of the mathematical model;
1.2) determining the value range of each design variable in the step 1.1);
1.3) carrying out Latin hypercube test design according to the value range of the design variable, and establishing test design sample points.
3. The multi-objective optimization method for the tooth surface loading performance of the spiral bevel gear according to claim 2, wherein the method comprises the following steps: in the step 1.1), the method for establishing the mathematical model of the spiral bevel gear tooth surface loading performance multi-objective optimization problem comprises the following steps:
1.1.1) determining design variables of a spiral bevel gear tooth surface loading performance multi-objective optimization problem, wherein the design variables comprise a pre-control transmission error curve coefficient, a contact trace slope and a contact half width;
1.1.2) determining a target function of a spiral bevel gear tooth surface loading performance multi-target optimization problem, wherein the target function comprises maximum contact stress of a tooth surface and a peak value of a loading transmission error;
1.1.3) determining a constraint function of a spiral bevel gear tooth surface loading performance multi-objective optimization problem, wherein the constraint function comprises the maximum bending stress of tooth roots of a small wheel and a large wheel and the area of an actual loading contact zone outside an ideal contact zone;
1.1.4) obtaining a mathematical model of the spiral bevel gear tooth surface loading performance multi-objective optimization problem according to the determined design variables, the objective function and the constraint function.
4. The multi-objective optimization method for the tooth surface loading performance of the spiral bevel gear according to claim 3, wherein the method comprises the following steps: in the step 1.1.4), the mathematical model of the spiral bevel gear tooth surface loading performance multi-objective optimization problem is as follows:
in the formula, a is a pre-control transmission error curve coefficient; k is a radical ofsIs the contact trace slope; b is contact half width; sigmacMaximum contact stress of tooth surface, peak load transmission error value and peak value in LTE, △ S area of actual load contact area outside ideal contact area, △ SmaxTo set tolerances; sigmapAnd σgMaximum bending stress of tooth roots of the small wheel and the large wheel respectively; sigmabmaxThe allowable value of the set maximum bending stress is obtained; k is a radical ofmin,kmax,amin,amax,bmin,bmaxRespectively the upper and lower value limits corresponding to each variable.
5. The multi-objective optimization method for the tooth surface loading performance of the spiral bevel gear according to claim 1, characterized in that: in the step 3), the method for solving the bevel gear tooth surface loading performance multi-objective optimization problem comprises the following steps:
3.1) establishing a sample point updating method based on EI-RMSE, which comprises two stages of optimal point supplement based on EI indexes and maximum error point supplement based on RMSE indexes;
3.2) performing EI index-based design point supplementary optimization on the initial sample point set to obtain a first-stage optimized and updated sample point set;
3.3) carrying out maximum error point supplementary optimization based on RMSE indexes on the sample point set obtained in the step 3.2) after the first-stage optimization updating to obtain an optimal solution set of the spiral bevel gear tooth surface loading performance multi-objective optimization problem.
6. The multi-objective optimization method for the tooth surface loading performance of the spiral bevel gear according to claim 5, wherein the method comprises the following steps: in the step 3.1), the calculation formula of the Kriging agent model is as follows:
y(x)＝f(x)β+z(x)；
wherein y (x) is unknown Kriging agent model, f (x) is known regression model, β is regression coefficient to be determined, z (x) is mean value 0 and variance sigma2The random process of (a);
the calculation formulas of EI and RMSE are respectively as follows:
RMSE＝s
wherein f isminIs the minimum of all the sample points and,
for the estimated value of the predicted point, s is the root mean square error of the predicted point, and phi is the mean value
A cumulative distribution function with a standard deviation of s, phi being the mean
The standard deviation is a probability density function of s.
7. The multi-objective optimization method for the tooth surface loading performance of the spiral bevel gear according to claim 5, wherein the method comprises the following steps: in the step 3.2), the first-stage optimized and updated sample point set obtaining method includes the following steps:
3.2.1) fitting Kriging agent models of each objective function and constraint function in the tooth surface loading performance multi-objective optimization problem according to the initial sample point set;
3.2.2) establishing a single-target optimization problem of each objective function based on the established Kriging agent model, and calculating to obtain the maximum EI value of each objective function and the corresponding design point in the value range of each design variable;
the agent model of the single-target optimization problem of each objective function is as follows:
constraint function gi(x) The proxy model of (2) is:
wherein the content of the first and second substances,
is an estimate of the ith constraint function, sgiFor the root mean square error of the constraint function, m is the control level that satisfies the constraint;
3.2.3) if the maximum EI value of each objective function is smaller than the set threshold, finishing the supplement of the maximum EI value design point in the first stage to obtain an optimized and updated sample point set; otherwise, entering step 3.2.4);
3.2.4) calculating the target function of the design point corresponding to each maximum EI value and the real response value of the constraint function by using the tooth surface loading contact analysis method which is established in the step 2) and takes the bending stress of the tooth root into consideration;
3.2.5) adding the response values of the objective function and the constraint function of each maximum EI value corresponding to the design point in the step 3.2.4) into the sample point set adopted by the current fitting Kriging proxy model, and returning to the step 3.2.1).
8. The multi-objective optimization method for the tooth surface loading performance of the spiral bevel gear according to claim 5, wherein the method comprises the following steps: in the step 3.3), the method for obtaining the optimal solution set of the spiral bevel gear tooth surface loading performance multi-objective optimization problem comprises the following steps:
3.3.1) fitting a Kriging agent model of a target function and a constraint function in the tooth surface loading performance multi-target optimization problem based on the sample point set obtained in the step 3.2);
3.3.2) based on the Kriging agent model obtained in the step 3.3.1), carrying out optimization calculation on the tooth surface loading performance multi-objective optimization problem by adopting a multi-objective optimization algorithm to obtain a group of initial pareto solution sets;
3.3.3) calculating the RMSE value of each target function in the initial pareto solution set, and finding the RMSE maximum value of each target function and the corresponding design point;
3.3.4) if the maximum value of the RMSE of each objective function is greater than the respective set threshold, then go to step 3.3.7), otherwise go to step 3.3.5);
3.3.5) calculating the target function of the design point corresponding to each RMSE maximum value and the real response value of the constraint function by adopting the tooth surface loading contact analysis method which is established in the step 2) and takes the bending stress of the tooth root into consideration;
3.3.6) adding the design points corresponding to the maximum value of each RMSE in the step 3.3.5) and the response values of the target function and the constraint function thereof into a sample point set adopted by the current fitting Kriging agent model, and returning to the step 3.3.1);
3.3.7), completing multi-objective optimization, wherein the sample point set currently used for fitting the Kriging surrogate model is the optimal solution set of the spiral bevel gear tooth surface loading performance multi-objective optimization problem.
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