A reducer design method based on multi-strategy improved dung beetle algorithm

Abstract

The present invention belongs to the technical field of reducer design methods, and specifically relates to a reducer design method based on a multi-strategy improved dung beetle algorithm. The method models the reducer design problem, determines the range of design variables and constraints, and obtains the objective function of the reducer design problem, namely, the minimum weight of the reducer under the constraints. The original dung beetle algorithm is improved through multi-strategy to obtain a multi-strategy improved dung beetle algorithm, denoted as MIDBO. The objective function of the reducer design is solved to obtain the optimal solution for the reducer design. The reducer design method based on the multi-strategy improved dung beetle algorithm (MIDBO) provided by the present invention overcomes the shortcoming of the original dung beetle algorithm (DBO), which is prone to falling into local optimality in the late optimization stage. The proposed algorithm is applied to the reducer design optimization problem, greatly reducing the design cost of the reducer.

Description

Technical Field

[0001] The present invention belongs to the technical field of reducer design methods, and in particular relates to a reducer design method based on a multi-strategy improved dung beetle algorithm. Background Art

[0002] Reducer design is a type of engineering design problem, essentially an optimization problem characterized by numerous local minima, multiple constraints, and nonlinearity. Existing algorithms for this problem include the particle swarm optimization (PSO) and the grey wolf optimization (GWO), but the design methods found by these algorithms require further optimization.

[0003] The dung beetle optimization algorithm is a novel swarm intelligence optimization algorithm proposed in 2022. It simulates the habits of dung beetles in nature and constructs a search framework based on the "rolling-reproduction-foraging-stealing" model. It has the characteristics of strong optimization ability and fast convergence speed. However, like other swarm intelligence optimization algorithms, it has the defect of easily falling into local optimality in the later stage of optimization, and it is difficult to adapt to various complex optimization problems. Summary of the Invention

[0004] In response to the technical problems of the above-mentioned reducer being heavy and having high production cost, the present invention provides a reducer design method based on a multi-strategy improved dung beetle algorithm, which can obtain better design parameters and effectively reduce the reducer design cost.

[0005] In order to solve the above technical problems, the technical solution adopted by the present invention is:

[0006] A reducer design method based on a multi-strategy improved dung beetle algorithm includes the following steps:

[0007] S1. Model the reducer design problem, determine the range of design variables and constraints, and obtain the objective function of the reducer design problem, that is, the minimum weight of the reducer under the constraints;

[0008] S2, perform multi-strategy improvements on the original dung beetle algorithm to obtain a multi-strategy improved dung beetle algorithm, denoted as MIDBO;

[0009] S3. Based on the multi-strategy improved dung beetle optimization algorithm of step S2, the objective function of the reducer design is solved to obtain the optimal solution of the reducer design.

[0010] The method for modeling the reducer design problem in S1 is:

[0011] The mathematical model of the reducer is defined as: X = [b, m, z, l1, l2, d1, d2] = [x1, x2, x3, x4, x5, x6, x7]. The reducer includes seven parameters: tooth face width b, gear module m, number of teeth z on the pinion, length l1 of the first shaft between bearings, length l2 of the second shaft between bearings, diameter d1 of the first shaft, and diameter d2 of the second shaft. These seven parameters are used as design variables for the reducer design and are denoted as x1, x2, x3, x4, x5, x6, and x7. X represents a set of design options consisting of the seven design variables.

[0012] The objective function of the reducer design is set as:

[0013]

[0014] The constraints of the mathematical model of the reducer are g1(x)~g 11 (x)

[0015]

[0016]

[0017]

[0018]

[0019]

[0020]

[0021]

[0022]

[0023]

[0024]

[0025]

[0026] The range of variable values ​​is: 2.6≤x1≤3.6, 0.7≤x2≤0.8, 17≤x3≤28, 7.3≤x4≤8.3, 7.8≤x5≤8.3, 2.9≤x6≤3.9, 5.0≤x7≤5.5.

[0027] The method for performing multi-strategy improvement on the original dung beetle algorithm in S2 is:

[0028] S2.1. Using Circle Chaos Mapping to enhance the randomness and diversity of particles, it is easier to search for the global optimal solution when there are many local optimal solutions.

[0029] S2.2. Replacing the rolling ball behavior of the dung beetle algorithm with the update mechanism of the sparrow algorithm's discoverer can effectively avoid the drawbacks of the dung beetle algorithm in its rolling ball behavior, which relies only on the worst value and has too many parameters.

[0030] S2.3. During the dung beetle breeding and foraging stages, the Sigmoid nonlinear convergence factor R is introduced. new Replace the linear convergence factor R in the original dung beetle algorithm, so that the algorithm has better global search capabilities in the early stage. As the number of iterations increases, the descent speed in the later stage accelerates, and has better local search capabilities.

[0031] S2.4. After the dung beetle position is updated, Levy flight is performed to update the individual position again, improving the algorithm's ability to find the best solution and escape from local extremes.

[0032] The formula for initializing the population of the Circle chaotic map in S2.1 is:

[0033]

[0034] Among them: a is 0.5, b is 0.2.

[0035] The formula for the location update mechanism of the Sparrow Algorithm discoverer in S2.2 is:

[0036]

[0037] Where: t represents the number of current iterations, j ranges from {1, 2, ..., d}, represents the value of the jth dimension of the i-th sparrow at the t-th iteration; α∈(0,1] is a random number, R2 and ST represent the alarm value and safety threshold respectively, R2∈[0,1], ST∈[0.5,1.0]; when R2≤ST, it indicates that there are no natural enemies around and the producer enters the wide-area search mode; when R2>ST, it indicates that there is danger nearby and all sparrows need to fly quickly to other safe areas; Q is a random number that satisfies the normal distribution, and L is a 1×d matrix with all elements being 1.

[0038] The convergence factor R in S2.3 new The nonlinear attenuation mode of is optimized using the following formula:

[0039]

[0040] Where: ω is the adjustment factor, ω=1.

[0041] The formula for Levy flight in S2.4 is as follows:

[0042]

[0043] Among them: α is the step size control factor, which is 1; is a vector operation; Levy(λ) is the step size that follows the Levy distribution.

[0044] The method for solving the objective function of the reducer design in S3 is:

[0045] S3.1. Using the objective function of the reducer design as the fitness function, determine the locations of the discoverer and breeding, foraging, and thieving dung beetles;

[0046] S3.2. Check whether the range of the updated solution exceeds the value range of the design variable. If so, replace the solution that exceeds the range with the boundary value of the design variable;

[0047] S3.3. Determine whether the current number of iterations is less than the maximum number of iterations. If so, return to step S3.1. Otherwise, output the minimum value of the objective function as the minimum weight of the reducer, and the currently found solution is used as the optimal reducer design solution.

[0048] The formula for updating the position of breeding dung beetles in S3.1 is as follows:

[0049] B i (t+1)=X * +b1×(B i (t)―Lb * )+b2×(B i (t)―Ub * )

[0050] Where b1 and b2 are independent random vectors of 1×D, and D is the dimension of the reducer design problem;

[0051] Dung beetles roll their dung balls to a safe location, providing a safe environment for female dung beetles to lay eggs. The safe zone formula is defined as:

[0052]

[0053] Among them, X * is the optimal position of the group in this iteration, that is, the local optimal position, R is the linear convergence factor in the original dung beetle algorithm (R = t / T), Lb and Ub represent the lower and upper bounds of the feasible region respectively; Lb * , Ub * are the lower and upper bounds of the safe area;

[0054] The formula for updating the position of a foraging dung beetle is:

[0055] X i (t+1)=X i (t)+C1×(X i (t)―Lb b )+C2×(X i (t)―Ub b )

[0056] Where C1 is a 1×D random number vector that obeys the normal distribution, and C2 is a random vector of (0,1);

[0057] The feeding location of the dung beetle is not random, it will choose an optimal feeding area. The range of the optimal feeding area is defined as:

[0058]

[0059] Among them, X b is the global optimal position, Lb b 、Ub b are the lower and upper bounds of the optimal foraging area, respectively;

[0060] The formula for updating the position of a stealing dung beetle is as follows:

[0061] X i (t+1)=X b +S×g×|X i (t)―X * |+|X i (t)―X b |

[0062] Where g is a random vector of size 1×D that follows a normal distribution, and S represents a constant.

[0063] Compared with the prior art, the present invention has the following beneficial effects:

[0064] The present invention provides a reducer design method based on the multi-strategy improved dung beetle algorithm (MIDBO), which makes up for the shortcoming of the original dung beetle algorithm (DBO) that it is easy to fall into local optimality in the later stage of optimization. The proposed algorithm is applied to the reducer design optimization problem, which greatly reduces the design cost of the reducer. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] To more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for the embodiments or the description of the prior art. Obviously, the drawings described below are merely exemplary, and those skilled in the art can, without inventive effort, derive other implementation drawings based on the provided drawings.

[0066] The structures, proportions, sizes, etc. illustrated in this specification are intended solely to complement the contents disclosed herein and to facilitate understanding and reading by persons skilled in the art. They are not intended to limit the conditions under which the present invention may be implemented and therefore have no substantive technical significance. Any structural modifications, changes in proportions, or adjustments in sizes, without affecting the efficacy and objectives of the present invention, shall remain within the scope of the technical contents disclosed herein.

[0067] Figure 1 This is a flow chart of the reducer design method based on the multi-strategy improved dung beetle algorithm of the present invention;

[0068] Figure 2 This is a schematic diagram of the reducer structure provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0069] In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below. Obviously, the described embodiments are only part of the embodiments of this application, not all the embodiments. These descriptions are only to further illustrate the features and advantages of the present invention, rather than to limit the claims of the present invention. Based on the embodiments in this application, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of this application.

[0070] The following embodiments of the present invention are described in further detail with reference to the accompanying drawings and examples. The following embodiments are used to illustrate the present invention but are not intended to limit the scope of the present invention.

[0071] In this embodiment, a reducer design method based on a multi-strategy improved dung beetle algorithm is provided. Figure 1 As shown, the following steps are included:

[0072] Step 1: Model the reducer design problem, determine the range and constraints of the design variables, and obtain the objective function of the reducer design problem, that is, the minimum weight of the reducer under the constraints.

[0073] The mathematical model of the reducer is defined as: X = [b, m, z, l1, l2, d1, d2] = [x1, x2, x3, x4, x5, x6, x7], where the reducer includes seven parameters: tooth face width b, gear module m, number of teeth in the pinion z, length l1 of the first shaft between bearings, length l2 of the second shaft between bearings, diameter d1 of the first shaft, and diameter d2 of the second shaft; these seven parameters are used as design variables for the reducer design, such as Figure 2 As shown, they are denoted as x1, x2, x3, x4, x5, x6, and x7; X represents a set of design schemes consisting of seven design variables.

[0074] The objective function of the reducer design is set as:

[0075]

[0076] The constraints of the mathematical model of the reducer are g1(x)~g 11 (x)

[0077]

[0078]

[0079]

[0080]

[0081]

[0082]

[0083]

[0084]

[0085]

[0086]

[0087]

[0088] The range of variable values ​​is: 2.6≤x1≤3.6, 0.7≤x2≤0.8, 17≤x3≤28, 7.3≤x4≤8.3, 7.8≤x5≤8.3, 2.9≤x6≤3.9, 5.0≤x7≤5.5.

[0089] Step 2: Perform multi-strategy improvements on the original dung beetle algorithm to obtain a multi-strategy improved dung beetle algorithm, denoted as MIDBO.

[0090] Step 2.1: Using Circle Chaos Mapping can enhance the randomness and diversity of particles, especially when there are many local optimal solutions, making it easier to search for the global optimal solution;

[0091] Circle chaotic map initialization population formula:

[0092]

[0093] Where a is 0.5 and b is 0.2

[0094] Step 2.2: The Sparrow Algorithm has the characteristics of high convergence accuracy, fast convergence speed, and strong robustness. Therefore, replacing the rolling ball behavior of the Dung Beetle Algorithm with the update mechanism of the Sparrow Algorithm's discoverer can effectively avoid the disadvantages of the Dung Beetle Algorithm in its rolling ball behavior, which only relies on the worst value and has too many parameters.

[0095] The position update mechanism of the Sparrow Algorithm discoverer is as follows:

[0096]

[0097] Where t represents the number of current iterations, j ranges from {1, 2, ..., d}, represents the value of the jth dimension of the i-th sparrow at the t-th iteration. α∈(0,1] is a random number, R2(R2∈[0,1]) and ST(ST∈[0.5,1.0]) represent the alarm value and safety threshold, respectively. When R2≤ST, it indicates that there are no natural enemies around and the producer enters a wide-area search mode. When R2>ST, it indicates that there is danger nearby and all sparrows need to quickly fly to other safe areas. Q is a random number that satisfies the normal distribution, and L is a 1×d matrix whose elements are all 1.

[0098] Step 2.3: Introduce the Sigmoid nonlinear convergence factor R during the dung beetle reproduction and foraging stages new Replace the linear convergence factor R in the original dung beetle algorithm, so that the algorithm has better global search capabilities in the early stage. As the number of iterations increases, the descent speed in the later stage accelerates, and has better local search capabilities.

[0099] Convergence factor R new The nonlinear attenuation mode of is optimized using the following formula:

[0100]

[0101] ω is the adjustment factor, ω=1.

[0102] Step 2.4: After the dung beetle position is updated, perform Levy flight to update the individual position again to improve the algorithm's ability to find the optimal solution and escape from local extremes.

[0103] The formula for Levy flight is as follows:

[0104]

[0105] α is the step size control factor, which takes a value of 1; is a vector operation; Levy(λ) is the step size that follows the Levy distribution.

[0106] Step 3: Based on the multi-strategy improved dung beetle optimization algorithm in step 2, the objective function of the reducer design is solved to obtain the optimal solution of the reducer design;

[0107] Step 3.1: Use the objective function of the reducer design as the fitness function to determine the locations of the discoverer and the breeding, foraging, and thieving dung beetles.

[0108] The formula for updating the breeding dung beetle position is as follows:

[0109] B i (t+1)=X * +b1×(B i (t)―Lb * )+b2×(B i (t)―Ub * )

[0110] Where b1 and b2 are independent random vectors of 1×D, and D is the dimension of the reducer design problem.

[0111] Dung beetles roll their dung balls to a safe location, providing a safe environment for female dung beetles to lay eggs. The safe area formula is defined as follows:

[0112]

[0113] Among them, X * is the optimal position of the group in this iteration, that is, the local optimal position, R is the linear convergence factor in the original dung beetle algorithm (R = t / T), Lb and Ub represent the lower and upper bounds of the feasible region respectively; Lb * 、Ub * are the lower and upper bounds of the safe area respectively.

[0114] The formula for updating the position of a foraging dung beetle is as follows:

[0115] X i (t+1)=X i (t)+C1×(X i (t)―Lb b )+C2×(X i (t)―Ub b )

[0116] Where C1 is a 1×D random number vector that follows a normal distribution, and C2 is a random vector of (0,1).

[0117] The foraging location of the dung beetle is not random, it will choose an optimal foraging area. The range of the optimal foraging area is defined as follows:

[0118]

[0119] Among them, X b is the global optimal position, Lb b , Ub bare the lower and upper bounds of the optimal foraging area.

[0120] The formula for updating the position of a stealing dung beetle is as follows:

[0121] X i (t+1)=X b +S×g×|X i (t)―X * |+|X i (t)―X b |

[0122] In the formula, g is a random vector of size 1×D and obeys the normal distribution, and S represents a constant.

[0123] Step 3.2: Check whether the range of the updated solution exceeds the value range of the design variable. If it exceeds, use the boundary value of the design variable to replace the solution that exceeds the range.

[0124] Step 3.3: Determine whether the current number of iterations is less than the maximum number of iterations. If so, return to step 3.1. Otherwise, output the minimum value of the objective function as the minimum weight of the reducer, and the currently found solution is used as the optimal reducer design solution.

[0125] In this embodiment, the pseudo code of the multi-strategy improved dung beetle optimization algorithm (MIDBO) is as follows:

[0126] Input: population size N, maximum number of iterations T

[0127] Output: optimal solution X best , the optimal fitness value f(X best )

[0128] 1. Randomly initialize the dung beetle population

[0129] 2.while t≤T do

[0130] 3.for i=1:N

[0131] 4.if i == discoverer

[0132] 5.α=rand(1)

[0133] 6.if α<0.9 then

[0134] 7. Update the finder's location

[0135] 8.else

[0136] 9. Update the finder's location

[0137] 10.end if

[0138] 11. If i = = breeding dung beetles

[0139] 12. Update the breeding location of dung beetles

[0140] 13.end if

[0141] 14.if i==foraging for dung beetles

[0142] 15. Updated the location of foraging dung beetles

[0143] 16.end if

[0144] 17. if i = = stealing dung beetle

[0145] 18. Update the location of the stealing dung beetle

[0146] 19.end if

[0147] 20.end for

[0148] 21. Use Levy flight to generate new solutions and use greedy selection strategy to select the optimal point

[0149] 22.end while

[0150] In this embodiment, a simulation experiment of the method is carried out with the help of MATLAB on an Intel(R) Core(TM) i5-8300H CPU 2.30GHz and 8.00GB memory in the Windows 10 operating environment. The experimental results show that the method of this embodiment is better than the experimental results of other algorithms.

[0151] In this embodiment, the multi-strategy improved dung beetle optimization algorithm (MIDBO) is used to design the minimum weight of the reducer, and the design effects are compared with the particle swarm optimization algorithm (PSO), the grey wolf algorithm (GWO), and the original dung beetle algorithm (DBO). The data results are shown in Table 1.

[0152] In this example, the parameters are set to a population size of 30, a maximum number of iterations of 500, and the program is repeated 30 times. The output results are shown in Table 1, where the best optimal solution, optimal value, average value, and standard deviation of all algorithms are bolded.

[0153] Comparative Analysis: The results in Table 1 show that the reducer design method based on the Multi-Strategy Improved Dung Beetle Optimization (MIDBO) algorithm achieves the best optimal value among all methods, and each design variable can reach the optimal solution. This shows that this embodiment can reduce the weight of the reducer in practical design, thereby reducing costs. Moreover, compared with other algorithms, MIDBO has the best mean and the smallest standard deviation, demonstrating the greater robustness of this embodiment.

[0154] Table 1 Optimization results and comparison table of reducer design problems

[0155]

[0156] Among them: Mean represents the mean, and Std represents the standard deviation.

[0157] The above only describes in detail the preferred embodiments of the present invention, but the present invention is not limited to the above embodiments. Various changes can be made within the knowledge of ordinary technicians in this field without departing from the purpose of the present invention, and various changes should be included in the scope of protection of the present invention.

Claims

1. A reducer design method based on a multi-strategy improved dung beetle algorithm, characterized by: The following steps are involved: S1. Model the reducer design problem, determine the range of design variables and constraints, and obtain the objective function of the reducer design problem, that is, the minimum weight of the reducer under the constraints; S2, perform multi-strategy improvements on the original dung beetle algorithm to obtain a multi-strategy improved dung beetle algorithm, denoted as MIDBO; The method for performing multi-strategy improvement on the original dung beetle algorithm in S2 is: S2.

1. Using Circle Chaos Mapping to enhance the randomness and diversity of particles, it is easier to search for the global optimal solution when there are many local optimal solutions. S2.

2. Replacing the rolling ball behavior of the dung beetle algorithm with the update mechanism of the sparrow algorithm's discoverer can effectively avoid the drawbacks of the dung beetle algorithm in its rolling ball behavior, which relies only on the worst value and has too many parameters. The formula for the location update mechanism of the Sparrow Algorithm discoverer in S2.2 is: ； Where: t represents the number of current iterations, j ranges from {1, 2, ..., d}, Indicates the value of the jth dimension of the i-th sparrow at the t+1th iteration; ∈(0,1] is a random number, R2 and ST represent the alarm value and safety threshold respectively, R2∈[0,1], ST∈[0.5,1.0]; when R2≤ST, it indicates that there are no natural enemies around and the producers enter the wide-area search mode; when R2>ST, it indicates that there is danger nearby and all sparrows need to fly quickly to other safe areas; Q is a random number that satisfies the normal distribution, L is a 1×d matrix with all elements set to 1; S2.

3. During the dung beetle breeding and foraging stages, the Sigmoid nonlinear convergence factor R is introduced. new Replace the linear convergence factor R in the original dung beetle algorithm, so that the algorithm has better global search capabilities in the early stage. As the number of iterations increases, the descent speed in the later stage accelerates, and has better local search capabilities. The convergence factor R in S2.3 new The nonlinear attenuation mode of is optimized using the following formula: ； Where: ω is the adjustment factor, ω = 1; S2.4, after the dung beetle position is updated, perform Levy flight to update the individual position again, improving the algorithm's ability to find the best solution and escape from local extreme values; The formula for Levy flight in S2.4 is as follows: ； Among them: α is the step size control factor, which is 1; is a vector operation; Levy(λ) is the step size that follows the Levy distribution; S3. Based on the multi-strategy improved dung beetle optimization algorithm of step S2, the objective function of the reducer design is solved to obtain the optimal solution of the reducer design.

2. The reducer design method based on the multi-strategy improved dung beetle algorithm according to claim 1 is characterized in that: The method for modeling the reducer design problem in S1 is: The mathematical model of the reducer is defined as: X = [b, m, z, l1, l2, d1, d2] = [x1, x2, x3, x4, x5, x6, x7]. The reducer includes seven parameters: tooth face width b, gear module m, number of teeth z on the pinion, length l1 of the first shaft between bearings, length l2 of the second shaft between bearings, diameter d1 of the first shaft, and diameter d2 of the second shaft. These seven parameters are used as design variables for the reducer design and are denoted as x1, x2, x3, x4, x5, x6, and x7. X represents a set of design options consisting of the seven design variables. The objective function of the reducer design is set as: ； The constraints of the reducer mathematical model are g1(x)~g11(x), ； ； ； ； ； ； ； ； ； ； ； The range of variable values ​​is: 2.6≤x1≤3.6, 0.7≤x2≤0.8, 17≤x3≤28, 7.3≤x4≤8.3, 7.8≤x5≤8.3, 2.9≤x6≤3.9, 5.0≤x7≤5.

5.

3. The reducer design method based on the multi-strategy improved dung beetle algorithm according to claim 1 is characterized in that: The formula for initializing the population of the Circle chaotic map in S2.1 is: ； Among them: a is 0.5, b is 0.

2.

4. The reducer design method based on the multi-strategy improved dung beetle algorithm according to claim 1 is characterized in that: The method for solving the objective function of the reducer design in S3 is: S3.

1. Using the objective function of the reducer design as the fitness function, determine the locations of the discoverer and breeding, foraging, and thieving dung beetles; S3.

2. Check whether the range of the updated solution exceeds the value range of the design variable. If so, replace the solution that exceeds the range with the boundary value of the design variable; S3.

3. Determine whether the current number of iterations is less than the maximum number of iterations. If so, return to step S3.

1. Otherwise, output the minimum value of the objective function as the minimum weight of the reducer, and the currently found solution is used as the optimal reducer design solution.

5. The reducer design method based on the multi-strategy improved dung beetle algorithm according to claim 4 is characterized in that: The formula for updating the position of breeding dung beetles in S3.1 is as follows: ； Where b1 and b2 are independent random vectors of 1×D, and D is the dimension of the reducer design problem; Dung beetles roll their dung balls to a safe location, providing a safe environment for female dung beetles to lay eggs. The safe zone formula is defined as: ； in, is the optimal position of the group in this iteration, that is, the local optimal position, R is the linear convergence factor in the original dung beetle algorithm, R = t / T, Lb, Ub represent the lower bound and upper bound of the feasible region respectively; , are the lower and upper bounds of the safe area; The formula for updating the position of a foraging dung beetle is: ； Where C1 is a 1×D random number vector that obeys the normal distribution, and C2 is a random vector of (0,1); The feeding location of the dung beetle is not random, it will choose an optimal feeding area. The range of the optimal feeding area is defined as: ； Among them, X b is the global optimal position, Lb b 、Ub b are the lower and upper bounds of the optimal foraging area, respectively; The formula for updating the position of a stealing dung beetle is as follows: ； Where g is a random vector of size 1×D that follows a normal distribution, and S represents a constant.

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