Gear transmission system optimization method adopting improved MOEA / D algorithm

Abstract

The invention provides a gear transmission system optimization method adopting an improved MOEA / D algorithm. According to the method, constraint condition processing and a weight vector adaptive strategy of an MOEA / D algorithm are improved, a traditional MOEA / D algorithm is taken as a basic framework, simulated annealing penalty function processing is carried out on constraint conditions, an initial weight vector is generated by adopting a uniform stochastic method, and adaptive adjustment is carried out on the weight vector; the Pareto leading edge can keep uniform distribution and is prevented from falling into local optimum; the minimum acceleration, the highest reliability and the maximum power density of the gear transmission system serve as optimization objectives, design variables are screened in a partial derivative solving mode, and the system size, the material fatigue strength and the like serve as constraint conditions to establish a gear transmission system multi-objective optimization model. The problems that an NSGA-II algorithm, a genetic algorithm and the like are low in calculation efficiency and a traditional MOEA / D algorithm is prone to falling into local optimum under complex constraint conditions are solved, and better gear transmission system optimization performance is achieved.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of mechanical transmission component design, and particularly relates to a gear transmission system optimization method using an improved MOEA / D algorithm. BACKGROUND

[0002] In the key fields of modern industrial equipment, transportation tools and aerospace vehicles, gear transmission systems play a core role in power transmission. Their performance not only deeply affects the efficiency, maintenance period and economy of the equipment, but also directly determines the stability and safety of the overall system. If the core performance indicators are not fully optimized in the design stage, it may lead to low equipment efficiency, increased energy consumption, frequent equipment failures, high maintenance costs, and even serious safety hazards. Optimizing the gear system in the design stage for multiple performance objectives is of great significance to improving the overall performance of the equipment, reducing the life cycle cost, and ensuring the safety of personnel and assets.

[0003] In order to deal with the complexity of multi-objective optimization of gear transmission systems, MOEA / D (Multi-objective Evolutionary Algorithm Based on Decomposition) is often considered as an effective solution tool. MOEA / D decomposes the complex multi-objective problem into a set of relatively simple scalar sub-problems through weight vectors, and guides the evolution population to approach the Pareto front along different weight directions. Compared with meta-heuristic algorithms (such as standard genetic algorithm), MOEA / D has higher computational efficiency. However, the optimization design of gear transmission systems often involves high-dimensional design variables and a large number of nonlinear constraint conditions. The inherent mechanism of traditional MOEA / D algorithm easily leads to premature convergence to local Pareto optimal front when facing such complex optimization problems, making it difficult to obtain truly global optimal solutions. In addition, due to the initial setting of weight vectors or their aggregation method, traditional MOEA / D is usually difficult to ensure that the final obtained Pareto solution set has good distribution uniformity and wide coverage in the target space when dealing with high-dimensional and complex constraint problems. This uneven distribution greatly reduces the diversity and selectivity of the design scheme, making it difficult to fully demonstrate the trade-off relationship between performance indicators and restricting the ability of designers to obtain optimal solutions.

[0004] Therefore, it is of great significance to develop a gear transmission system optimization method using an improved MOEA / D algorithm. SUMMARY

[0005] The purpose of the present application is to provide a gear transmission system optimization method using an improved MOEA / D algorithm to solve the problems in the prior art.

[0006] The technical solution adopted to achieve the purpose of the present application is as follows: a gear transmission system optimization method using an improved MOEA / D algorithm, comprising the following steps:

[0007] 1) Collect and input the initial design parameters of the gear transmission system. The initial gear design parameters include the modulus, the number of teeth, the pressure angle, the helix angle, and the tooth width.

[0008] 2) Establish a performance analysis model of the gear transmission system to provide calculation basis for subsequent optimization. The performance analysis model includes a dynamics model, a time-varying reliability model, and a system volume representation function. The dynamics model calculates the vibration acceleration of the gear transmission system based on the node finite element method. The time-varying reliability model calculates the reliability through a stress-strength interference model. The volume representation function calculates the volume of the gear box.

[0009] 3) Determine the influence degree of each gear parameter on the system performance through sensitivity calculation, and select the gear parameters that have a significant impact on the performance as design variables.

[0010] 4) Establish a multi-objective optimization model with the minimum vibration acceleration, the highest reliability, and the minimum volume as optimization objectives, combined with size constraints and material fatigue strength constraints.

[0011] 5) Initialize the improved MOEA / D algorithm. Generate uniformly distributed initial weight vectors using the uniform random method, and set the algorithm parameters.

[0012] 6) Run the improved MOEA / D algorithm to complete the optimization calculation. Decompose the multi-objective problem into multiple single-objective sub-problems, and gradually approach the Pareto frontier along the weight vector direction until the convergence condition is met.

[0013] 7) After the algorithm converges, output the optimized gear design parameters.

[0014] Further, in step 3), the sensitivity of a certain gear parameter x to the system performance G is

[0015]

[0016] where x0 and G|x0 represent the initial value of the gear parameter and the system performance value under the initial value, respectively.

[0017] Through sensitivity analysis, it can be known that the vibration acceleration is most sensitive to the number of teeth Z, and the reliability is most sensitive to the modulus m. n Finally, the modulus m n and the number of teeth Z of the driving wheel are selected.p , number of driven gear teeth Z g , pressure angle a, helix angle β and tooth width B as design variables.

[0018] Further, in step 4), define the gear system vibration acceleration a ∑ The minimum is the first objective function f1. Define the total time-varying reliability R ∑ (t) of the gear system. The maximum is the second objective function f2. Define the volume V ∑ The minimum is the third objective function f3.

[0019] f1 = min(a ∑ ) (2)

[0020] f2 = max(rms(R ∑ (t))) (3)

[0021] f3 = min(V ∑ ) (4)

[0022] Wherein, the gear system acceleration is based on the node finite element method to establish the dynamic model. Gear reliability can be obtained by stress-strength interference model. Gear system power density refers to the power transmitted per unit volume.

[0023] Further, in step 4), the size constraints include width-diameter ratio constraints, end face coincidence constraints, non-interference constraints and center distance constraints.

[0024] Further, in step 4), the material fatigue strength constraints include that the gear contact fatigue stress and bending fatigue stress should not exceed the strength requirement value.

[0025] Further, in step 5), the simulated annealing penalty function is used to handle the constraint conditions, so that the solution gradually converges to the feasible region. The simulated annealing penalty function is:

[0026] M = 1 / T(t) (5)

[0027] Wherein, T(t) is the temperature cooling table in simulated annealing algorithm, that is,

[0028] T(t) = c t T(t-1) (6)

[0029] Wherein, c t represents the temperature coefficient, c t ∈(0, 1].

[0030] Further, in step 6), the weight vector is dynamically adjusted according to the congestion degree to avoid local concentration of the Pareto frontier. The adaptive weight vector formula is as follows:

[0031]

[0032] wherein, denotes the jth weight vector at the tth iteration. denotes the weight vector the individual corresponding to the ith weight vector in the field. denotes the individual the next generation individual the increment in each dimension. γ i denotes the crowding degree weighting factor. α, β denote two hyperparameters for controlling the weight vector offset.

[0033] Further, after step 7), there is also a step of verifying and comparing the optimization result obtained in step 7) with the traditional algorithm.

[0034] The technical effects of the present application are self-evident:

[0035] A. The problem of long time consumption of traditional meta-heuristic algorithms on complex multi-objective problems is solved.

[0036] B. Through the combined action of the simulated annealing penalty function and the adaptive weight vector, the Pareto front distribution is more uniform, and local optimum is avoided.

[0037] C. The vibration acceleration of the optimized system is reduced, directly reducing the operating noise and dynamic load impact. Under the premise of ensuring reliability, the system volume is reduced, realizing higher power density design. A good balance between performance and reliability is achieved.

[0038] D. The optimization algorithm is based on gear transmission system parameters for calculation, suitable for various gear optimization design, good practicability, and worth popularizing. BRIEF DESCRIPTION OF DRAWINGS

[0039] Figure 1 It is an improved MOEA / D algorithm-based gear transmission system optimization flowchart;

[0040] Figure 2 It is an improved MOEA / D algorithm initialization weight vector flowchart;

[0041] Figure 3 It is an improved MOEA / D algorithm weight vector adaptive process chart;

[0042] Figure 4 It is an improved MOEA / D algorithm flowchart;

[0043] Figure 5 It is a schematic diagram of the gear transmission system in the embodiment of the present application;

[0044] Figure 6 Sensitivity of vibration acceleration of gear transmission system to gear parameters;

[0045] Figure 7 Sensitivity of time-varying reliability of gear transmission system to gear parameters;

[0046] Figure 8 Pareto front comparison chart of different algorithms. DETAILED DESCRIPTION

[0047] The application will be further described below in conjunction with the embodiments, but should not be understood as limiting the above-mentioned subject matter of the application to the following embodiments. According to ordinary technical knowledge and common means in the art, various substitutions and modifications can be made without departing from the above-mentioned technical idea of the application, and all the substitutions and modifications shall be included in the protection scope of the application.

[0048] Example 1

[0049] Referring to Figure 1 The embodiment provides a gear transmission system optimization method using an improved MOEA / D algorithm, including the following steps:

[0050] 1) Collect and input initial design parameters of the gear transmission system. The initial gear design parameters include modulus, number of teeth, pressure angle, helix angle and tooth width.

[0051] 2) Establish a performance analysis model of the gear transmission system to provide calculation basis for subsequent optimization. The performance analysis model includes a dynamics model, a time-varying reliability model and a system volume representation function. The dynamics model calculates the vibration acceleration of the gear transmission system based on the node finite element method. The time-varying reliability model calculates the reliability through a stress-strength interference model. The volume representation function calculates the volume of the gear box.

[0052] 3) Determine the influence degree of each gear parameter on the system performance through sensitivity calculation, and select the gear parameters having significant influence on the performance as design variables.

[0053] 4) Establish a multi-objective optimization model with the minimum vibration acceleration, the highest reliability and the minimum volume as optimization objectives, combined with size constraints and material fatigue strength constraints.

[0054] 5) Initialize the improved MOEA / D algorithm. A uniform random method is used to generate an initial weight vector with uniform distribution, and algorithm parameters are set.

[0055] 6) Run the improved MOEA / D algorithm to complete optimization calculation. The multi-objective problem is decomposed into multiple single-objective sub-problems, and the Pareto front is gradually approached along the weight vector direction until the convergence condition is met.

[0056] 7) After the algorithm converges, the optimized gear design parameters are output.

[0057] Example 2:

[0058] The main content of this embodiment is the same as that of Example 1, wherein in step 3), the sensitivity of a certain gear parameter x to the system performance G is

[0059]

[0060] wherein x0 and G|x0 represent the initial value of the gear parameter and the system performance value at the initial value, respectively.

[0061] Through sensitivity analysis, the sensitivity of the gear transmission system vibration acceleration to different gear parameters is obtained, and it is found that the sensitivity of the displacement coefficient X and the gear slot width b is obviously lower than that of other parameters, and the number of teeth Z has the greatest influence on the gear transmission system vibration acceleration, such as Figure 5 The sensitivity of the gear transmission system time-varying reliability to different gear parameters is analyzed. It is found that the modulus m n and the number of teeth Z have the greatest influence, while the displacement coefficient X has the least influence, such as Figure 6 The finally selected design variables are the modulus m n , the number of teeth of the driving wheel Z p , the number of teeth of the driven wheel Z g , the pressure angle a, the helix angle b and the tooth width B.

[0062] Example 3:

[0063] The main content of this embodiment is the same as that of Example 1 or 2, wherein in the dynamic response of the gear system, the vibration acceleration can effectively reflect the change of the system under load, and it is also a vibration noise measurement standard, and its sensitivity is better than that of vibration displacement and vibration speed; the reliability of the gear transmission system needs to establish a gear time-varying reliability model, collect the stress and strength of the transmission system, and solve the reliability to avoid gear failure; the power density of the gear transmission system refers to the power transmitted per unit volume, and the high power density design is to improve the transmission power and gear bearing capacity of the gear box as much as possible under the condition of a certain volume, so as to ensure the optimization target of the gear transmission system. Therefore, the minimum vibration acceleration a ∑ of the gear transmission system is defined as the first objective function f1. The effective value of the total time-varying reliability R ∑ (t) of the gear transmission system is defined as the second objective function f2. The minimum volume V ∑ of the gear transmission system is defined as the third objective function f3.

[0064] The gear transmission system vibration acceleration is obtained based on the node finite element method to establish a dynamic model. The gear transmission system vibration acceleration a Σ is equal to the driving wheel vibration acceleration a psuperimposed on the driven wheel vibration acceleration a g :

[0065] a ∑ = w1a p + w2a g (2)

[0066]

[0067] where w1 and w2 represent weights, which can be obtained according to sensitivity; subscript "i" is p, g, representing the driving wheel and the driven wheel respectively; r represents the gear pitch circle radius.

[0068] The minimum vibration acceleration of the gear transmission system is defined as the first objective function f1:

[0069] f1 = min(a ∑ ) (4)

[0070] The gear reliability can be obtained by the stress-strength interference model. The maximum effective value of the total time-varying reliability R ∑ (t) of the gear transmission system is defined as the second objective function f2:

[0071] R j (t) = P{s j (t) ≥ σ j (t)} (t ∈ [0, T]) (5)

[0072] R ∑ (t) = ΠR ij (t) (6)

[0073] f2 = max(rms(R ∑ (t))) (7)

[0074] where subscript "i" is p, g, representing the driving wheel and the driven wheel respectively; subscript "j" is the tooth surface contact fatigue and the tooth root bending fatigue, s is the fatigue strength, and σ is the stress.

[0075] The power density of the gear transmission system refers to the power transmitted per unit volume. The minimum volume V Σ of the gear transmission system is defined as the third objective function f3:

[0076]

[0077] f3 = min(V ∑ ) (9)

[0078] where d p and d g represent the diameters of the driving wheel and the driven wheel respectively; B and b represent the single-sided helical tooth width and the tooth groove width respectively.

[0079] Example 4:

[0080] The main content of this embodiment is the same as any one of Examples 1-3, wherein in step 4), the size constraints include a width-to-diameter ratio constraint, an end face coincidence degree constraint, a non-interference constraint, and a center distance constraint. The material fatigue strength constraint includes that the gear contact fatigue stress and the bending fatigue stress should not exceed the strength requirement value. The following constraint conditions are usually required to be met during the optimization design of the gear transmission system:

[0081] 1) The width-to-diameter ratio constraint, according to the limiting conditions of the upper and lower boundaries of the tooth width coefficient, the tooth width-to-diameter ratio range should satisfy the following equation:

[0082]

[0083] wherein B represents the tooth width; d p and d g respectively represent the diameters of the pitch circles of the driving gear and the driven gear.

[0084] 2) The end face coincidence degree constraint, in order to ensure continuous transmission of the gear, the end face coincidence degree of the gear should satisfy the following equation:

[0085]

[0086] wherein α apt and α agt respectively represent the end face addendum circle pressure angles of the driving gear and the driven gear; and α' t represents the end face engagement angle.

[0087] 3) The non-interference constraint, in order to ensure that the gears can be engaged with each other during the meshing process, the parameters should satisfy the following equation:

[0088]

[0089] wherein α' represents the engagement angle; α ap and α ag respectively represent the addendum circle pressure angles of the driving gear and the driven gear; represents the normal addendum coefficient; X np and X ng respectively represent the normal displacement coefficients of the driving gear and the driven gear; β represents the helix angle; and α represents the pressure angle.

[0090] 4) The center distance constraint, during the optimization design, the center distance s is determined to satisfy the following equation:

[0091]

[0092] wherein m n represents the normal modulus.

[0093] 5) Fatigue strength constraint, gear contact fatigue stress and bending fatigue stress shall not exceed the strength requirement value, satisfying the following equation:

[0094]

[0095] wherein F t represents gear circumferential force; u represents transmission ratio; [σ] H , [σ] F respectively represent gear face contact fatigue strength allowable value, gear root bending fatigue strength allowable value.

[0096] Example 5:

[0097] The main content of the embodiment is the same as any one of examples 1-3, wherein in step 5), the core idea of improving the MOEA / D algorithm is to decompose the multi-objective problem into several single-objective problems, and to approach the Pareto front along the weight vector. Among them, taking the Tchebycheff method as an example to decompose the weight vector as the basis of the MOEA / D algorithm, the expression form of the single-objective optimization function is:

[0098]

[0099] wherein, represents ideal reference point, i.e. is a reference vector composed of the minimum values of all objective functions; λ represents weight vector; f i (x) represents the ith single-objective function.

[0100] wherein, represents the distance between the solution of the ith single-objective function and the ideal reference point , λ i represents the weight of the ith single-objective function, and the greater the value, the farther the distance between the objective function and the ideal point. Assuming is the maximum value of all , changing x will gradually reduce , i.e. approaching the ideal point, until reaching the corresponding point on the Pareto front, i.e. completing the minimization task, so other will also be minimized.

[0101] The penalty function is used to process the constraint condition, and the multi-objective optimization problem is assumed as follows:

[0102]

[0103] Wherein: x represents n-dimensional design variables; subscript "m" represents m objective functions; Ω represents the range of x values; g and h represent the constraints that the design variable x needs to meet, including k inequality constraints and q equality constraints, and M represents the penalty factor.

[0104] The penalty function can convert the constrained optimization problem into an unconstrained optimization problem

[0105]

[0106] Wherein: M represents the penalty factor; φ i represents the penalty function, when x meets the constraint condition, φ i = 0, when x does not meet the constraint condition, φ i > 0.

[0107] The penalty factor M is often treated as a constant, and its function is:

[0108] M = 1 / T(t) (4)

[0109] Wherein: T(t) is the temperature cooling table in the simulated annealing algorithm, that is,

[0110] T(t) = c t T(t-1) (5)

[0111] Wherein: c t represents the temperature coefficient, c t ∈(0, 1].

[0112] The penalty factor gradually increases as the temperature decreases, so that the solution gradually converges to the feasible region.

[0113] Example 6:

[0114] The main content of this embodiment is the same as any one of examples 1-5, see Figure 2 , the improved MOEA / D algorithm needs to create a weight vector to generate a subproblem, assuming that the number of multi-objective problems is m, and the number of weight vectors required is l. First, use the uniform random method to generate N weight vectors, wherein each vector is a unit vector, and N > l, denoted as V1 ∈ N × m; generate an m × m unit matrix I m . Second, mark the vector in V1 to the vector in I m The maximum value of the Euclidean distance, add the vector to I m , and delete the vector in V1. Repeat the above operation until the dimension of the unit matrix changes to l × m, at which time the initialization of the weight vector is completed, that is, W ∈ l × m.

[0115] Example 7:

[0116] The main content of this embodiment is the same as any one of embodiments 1-6, wherein in step 6), the improved MOEA / D algorithm needs to adjust the weight vector in each evolution process, avoid local concentration of the pareto front, and make the weight vector deviate to uniform distribution.

[0117] The weight vector should meet the following requirements:

[0118] 1)δ t Restrict the maximum change step to avoid individual jumping across the field;

[0119] 2)γ i Weighted adjustment to deviate to the low congestion direction;

[0120] 3) Exponential decay term e -kt With δ t Ensure that the adjustment amplitude decreases with iteration.

[0121] The weight vector adaptive process is as follows: Figure 3 The black dots represent the current evolution population individuals, the yellow box represents the field corresponding to the weight vector, and the arrow represents the individual evolution direction. The individual G0 of the field of the weight vector w2 evolves in the direction of G1 initially, but it is more crowded, so the weight vector w2 is changed to w2' downward, so that the individual G0 evolves in the direction of G1'. In this way, the Pareto front of the multi-objective problem is more evenly distributed, and the search ability of the algorithm is improved.

[0122] The adaptive weight vector formula is defined as follows:

[0123]

[0124] Wherein, represents the jth weight vector at the tth iteration; represents the weight vector The individual corresponding to the ith weight vector in the field; represents the individual and the next generation individual The increment in each dimension; γ i represents the congestion weighting factor; α, β represent two hyperparameters for controlling the weight vector offset.

[0125] In order to limit the change amplitude of the weight vector, the following constraint condition is added to the weight vector:

[0126]

[0127] Wherein, if The weight vector needs to be adjusted to deviate, and does not exceed the threshold δ t , δt As the number of iterations gradually decreases.

[0128] For the congestion degree weighting factor γ i The following definitions are made:

[0129]

[0130] Where CrowdingDegree represents the normalized field congestion degree, the closer to 1 indicates the more crowded, the calculation formula of the field congestion degree is as follows:

[0131]

[0132] Where, The maximum Euclidean distance of the jth sub-problem is represented by The average Euclidean distance of the jth sub-problem is represented by The maximum Euclidean distance of the jth sub-problem is represented by

[0133] For two hyperparameters, the following constraint conditions are given:

[0134]

[0135] Where the parameter kk can adjust the decay rate, the search step is rapidly reduced at the beginning, and it is gently adjusted at the later stage to accurately adjust.

[0136] Embodiment 8:

[0137] The main content of this embodiment is the same as any one of embodiments 1-7, wherein the improved MOEA / D algorithm takes the traditional MOEA / D algorithm as the basic framework, and the initial weight vector and the iteration process weight vector are improved respectively. The constraint condition is added to the simulated annealing penalty function to improve the population evolution efficiency. The basic process of the improved MOEA / D algorithm is as follows, Figure 4 .

[0138] Embodiment 9:

[0139] The main content of this embodiment is the same as any one of embodiments 1-8, wherein after step 7), there is also a step of verifying and comparing the optimization results obtained in step 7) with traditional algorithms. This embodiment takes a single-stage double helical gear transmission system as an example to perform multi-performance collaborative optimization on gear design parameters. The principle of the gear transmission system is that the motor shaft connects the driving wheel and the small bearing, and then connects the driven wheel to the large bearing and the output shaft, as shown in Figure 7 , the initial design parameters of the gear are shown in Table 1.

[0140] Table 1

[0141]

[0142] The initial design variables are x = [x1, x2,..., x6] T = [m n , Z p , Z g , a, b, B] T The multi-objective optimization model of the double helical gear transmission system is as follows:

[0143] Design variables: x = [m n , Z p , Z g , a, b, B] T

[0144] Objective function:

[0145]

[0146] Constraint condition design variable range:

[0147]

[0148] For the improved MOEA / D algorithm, the temperature coefficient c t is 0.99; the hyperparameters a and b are 0.1 and 0.15 respectively, and the parameter kk is 0.2; the weight interval is 100; the iteration number is 50; the scaling factor is 0.5; and the mutation rate is 0.5.

[0149] To verify the effectiveness of the improved MOEA / D algorithm, the traditional MOEA / D algorithm and the NSGA-II algorithm are used for multi-objective optimization of the double helical gear transmission system, where the initial parameters of the traditional MOEA / D algorithm are the same as those of the improved MOEA / D algorithm, and the population size of the NSGA-II algorithm is 100, the iteration number is 50, the mutation probability is 0.1, and the crossover probability is 0.9. To facilitate observation of the iteration process, the fitness is defined as:

[0150]

[0151] where subscript i represents the ith Pareto solution, subscript 0 represents the initial value, and n represents a total of n Pareto solutions.

[0152] Through comprehensive evaluation of the Pareto solutions of different algorithms, the design variable optimization results of different algorithms are calculated as shown in Table 2.

[0153] Table 2

[0154]

[0155]

[0156] The improved MOEA / D algorithm is slightly faster than the traditional MOEA / D algorithm and the NSGA-II algorithm in the iteration process, but in terms of computational efficiency, the MOEA / D algorithm shows a significant advantage. The NSGA-II solution takes 803 minutes, while the traditional MOEA / D and the improved MOEA / D reduce the calculation time by 36.86% and 38.23%, respectively, because NSGA-II needs frequent non-dominated sorting operations, while MOEA / D only needs to converge to the Pareto front along the weight vector direction. As shown in Table 3.

[0157] Table 3

[0158]

[0159] As Figure 8 , the Pareto front of the NSGA-II algorithm is more evenly distributed, but the optimization result is not as good as that of the MOEA / D algorithm. The Pareto front of the traditional MOEA / D algorithm is more concentrated, while the Pareto front of the improved MOEA / D algorithm is more evenly distributed.

[0160] In terms of vibration performance, NSGA-II is only 9.60%, while the MOEA / D series algorithm is more than 10%. In terms of reliability, NSGA-II improves by 0.14%, but the MOEA-D series algorithm only decreases by 0.3%-0.5%. In terms of volume, NSGA-II has the smallest improvement rate. As shown in Table 4.

[0161] Table 4

[0162]

[0163] Among them, the improved MOEA / D, with the adaptive weight vector strategy, has a better Pareto front in terms of vibration and volume than the traditional MOEA / D algorithm, and the calculation time is reduced by 38% compared with NSGA-II. The results show that the improved MOEA / D has a more balanced multi-objective optimization capability of the gear transmission system while ensuring reliability.

Claims

1. A method for optimizing a gear transmission system using an improved MOEA / D algorithm, characterized in that, Includes the following steps: 1) Collect and input the initial design parameters of the gear transmission system; the initial gear design parameters include module, number of teeth, pressure angle, helix angle, and tooth width; 2) Establish a performance analysis model for the gear transmission system to provide a calculation basis for subsequent optimization; the performance analysis model includes a dynamic model, a time-varying reliability model, and a system volume characterization function; the dynamic model calculates the vibration acceleration of the gear transmission system based on the nodal finite element method; the time-varying reliability model calculates the reliability through a stress-intensity interference model; the volume characterization function calculates the volume of the gearbox; 3) Determine the degree of influence of each gear parameter on the system performance through sensitivity calculation, and select the gear parameters that have a significant impact on performance as design variables; 4) A multi-objective optimization model is established with the optimization objectives of minimizing vibration acceleration, maximizing reliability, and minimizing volume, combined with size constraints and material fatigue strength constraints; 5) Initialize the improved MOEA / D algorithm; generate a uniformly distributed initial weight vector using the uniform random method, and set the algorithm parameters; 6) Run the improved MOEA / D algorithm to complete the optimization calculation; decompose the multi-objective problem into multiple single-objective sub-problems, and gradually approach the Pareto front along the weight vector direction until the convergence condition is met; 7) After the algorithm converges, output the optimized gear design parameters.

2. The gear transmission system optimization method using the improved MOEA / D algorithm according to claim 1, characterized in that, In step 3), the sensitivity of a certain gear parameter x to the system performance G is: Where x0 and G|x0 represent the initial values ​​of the gear parameters and the system performance values ​​under the initial values, respectively; Sensitivity analysis shows that vibration acceleration is most sensitive to the number of teeth Z, while reliability is most sensitive to the module m. n Most sensitive; ultimately choose the modulus m n Number of teeth on the driving gear Z p Number of teeth Z of driven gear g Pressure angle α, helix angle β, and tooth width B are used as design variables.

3. The gear transmission system optimization method using the improved MOEA / D algorithm according to claim 1, characterized in that: In step 4), the vibration acceleration a of the gear transmission system is defined. ∑ The lowest value is f1, which is the first objective function; the total time-varying reliability R of the gear transmission system is defined. ∑ The highest effective value of (t) is the second objective function f2; the volume V of the gear transmission system is defined. ∑ The minimum is the third objective function f3; f1=min(a ∑ ) (2) f2=max(rms(R ∑ (t))) (3) f3=min(V ∑ ) (4) Among them, the acceleration of the gear transmission system is obtained by establishing a dynamic model based on the nodal finite element method; the reliability of the gear can be obtained through the stress-strength interference model; the power density of the gear transmission system refers to the power transmitted per unit volume.

4. The gear transmission system optimization method using the improved MOEA / D algorithm according to claim 3, characterized in that: In step 4), the dimensional constraints include aspect ratio constraints, end face overlap constraints, non-interference constraints, and center distance constraints.

5. The gear transmission system optimization method using the improved MOEA / D algorithm according to claim 1, characterized in that: In step 4), the material fatigue strength constraint includes ensuring that the gear contact fatigue stress and bending fatigue stress do not exceed the required strength values.

6. The gear transmission system optimization method using the improved MOEA / D algorithm according to claim 1, characterized in that: In step 5), a simulated annealing penalty function is used to process the constraints, causing the solution to gradually converge to the feasible region; the simulated annealing penalty function is: M = 1 / T(t) (5) Where T(t) is the temperature cooling table in the simulated annealing algorithm, that is, T(t)=c t T(t-1) (6) Among them, c t c represents the temperature coefficient. t ∈(0,1).

7. The gear transmission system optimization method using the improved MOEA / D algorithm according to claim 1, characterized in that: In step 6), the weight vector is dynamically adjusted according to the crowding level to avoid local concentration in the Pareto front; the adaptive weight vector formula is defined as follows: in, This represents the j-th weight vector in the t-th iteration; Represents the weight vector The individual corresponding to the i-th weight vector in the domain; Represents an individual With the next generation of individuals Increment in each dimension; γ i α represents the crowding weighting factor; α and β represent two hyperparameters used to control the offset of the weight vector.

8. The gear transmission system optimization method using the improved MOEA / D algorithm according to claim 1, characterized in that: After step 7), there is also a step to verify and compare the optimization results obtained in step 7) with the traditional algorithm.

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