1 Optimum design of helical gear transmission The mathematical model of the helical gear transmission can be optimized to establish a variety of single objective functions or multi-objective functions. In this paper, the mathematical model is established with the minimum center distance and the smallest volume of the transmission as the objective function, and the optimization design is carried out.
1.1 Establishing the objective function The minimum objective function is F1(X)=mina=mnz12cosβ(1 i)(1) The smallest objective function is F2(X)=minV=14π(mnz12cosβ)3(1 i2)φa( 2) where z1 is the number of pinion teeth; mn is the normal modulus; β is the helix angle; i gear ratio; φa is the tooth width coefficient.
1.2 Determining the design variables From equations (1) and (2), F1(X) and F2(X) are determined by four independent design parameters such as z1, mn, β, φa, so the design variables are: X=[x1 ,x2,x3,x4]T=[z,mn,β,φa](3)
1.3 Constraints 1) Tooth number constraint Normally, closed gear transmission z1 ≥ 20 ~ 40, from which g1 (x) = x1-20 ≥ 0 (4) g2 (x) = 40 - x1 ≥ 0 (5) 2 The modulus of the modulus-constrained power transmission is mn ≥ 2mm, so g3(x)=x2-2≥0(6)
3) The helix angle constraint can generally take β=8°～20°. If the β angle is too large, the axial force of the transmission will increase, and the gear processing is difficult. The optimum value of the β angle is between 8° and 15°. Therefore, g4(x)=15-x3≥0(7)g5(x)=x3-8≥0(8)
4) Tooth width coefficient constraint tooth width coefficient generally takes φa=0.1～1.2
Increase the center-to-center distance of the tooth width coefficient, but increase the tooth width to distribute the load evenly along the tooth width direction. Therefore, g6(x)=x4-0.1≥0(9)g7(x)=1.2-x4≥0(10)
5) Tooth surface contact fatigue strength Constrained tooth surface contact stress σH is less than or equal to the allowable contact stress [σH], and g8(x)=[σH]-σH≥0(11)6) Root root bending fatigue strength confines root bending The stress σF is less than or equal to the allowable bending stress [σF], and the calculation of the g9(x)=[σF]-σF≥0(12) tooth surface contact stress σH and the root bending stress σF is considered in the literature [1].
7) The longitudinal coincidence degree εβ constraint εβ can be calculated according to the following formula: εβ[3]=bsinβΠ(πmn)≥1, from which the constraint g10(x)=x1x41-cosx24-πcosx4≥0(13) can be obtained. The mathematical model for the optimal design of helical gear transmission is X=[x1,x2,x3,x4]T,minF1(X),minF2(X),s.tgu(x)≥0u=4,5,... , 13
2 Genetic algorithm-based optimization model solution 2.1 Gene coding uses floating-point number coding, each design variable as a chromosome gene. The chromosome code of the optimization problem is: X = [x1, x2, x3, x4] T = [z1, mn, β, φa] 2.2 fitness function and initial population generation GA generally does not require other external information during the search evolution process, It is only necessary to reasonably define the fitness function according to the objective function of the problem, to evaluate the individual's ability to adapt to the problem environment, that is, the advantages and disadvantages of the solution, and as the basis for future genetic operations.
Since the fitness function contains the influence of constraints, the N individual chromosomes in the initial population can be generated by a random method within the range of values ​​of each gene.
2.3 Genetic manipulation On the basis of the previous generation of chromosome groups, each individual of the population consisting of N chromosomes performs genetic operations such as selection, crossover and mutation according to the fitness function values, resulting in higher fitness function values. A new generation of groups that have evolved.
1) The selection operation is to select the better individuals from the previous generation to participate in the breeding of the next generation of populations. Generally, the method of fitness function value ratio is adopted, that is, the probability Pi of each individual is Pi=fiΠ∑ni=1fi, Pi—the probability that the i-th individual is selected; n—the size of the population; fi—the fitness function value of the i-th individual. 2) The cross-over operation is a chromosome randomly selected for cross-operation and two The two combinations constitute the parents who perform the crossover operation, exchanging the gene chain code information of the parents, so that each pair of parents produces two offspring to form a new generation of optimized individuals. In this paper, two points are crossed, and X and Y are used as parents, and the two intersections are located in the second and fourth genes, respectively.
X=[x1,x2,x3,x4]T
Y=[y1,y2,y3,y4]T The new chromosome X', Y' obtained after gene exchange is X'=[x1,x2,y3,y4]T
Y'=[y1,y2,x3,x4]T sets the crossover probability to Pc, then the PcN chromosomes in the population participate in the intersection, and the magnitude of the crossover probability Pc affects the convergence speed of the genetic algorithm.
3) Mutation For a selected chromosome, randomly select one or several genes and change their gene coding to generate a new chromosome with one or several genes different from the previous chromosome.
In this paper, two-point cross mutation was used and the mutation point was selected as the second gene. A new gene x2 was randomly selected in the range of the variation gene to replace x2, and a new chromosome X'=[x1,x2,x3,x4] was obtained. T.
4) Compose a new generation of groups to replace the previous generation group by a new generation of groups obtained by the aforementioned selection, intersection, and variation. Obviously, the average quality of the new group and the quality of the best individual are better than those of the previous generation. In this way, after several generations of iterative inheritance, when the fitness value of the group tends to be stable, the genetic operation is stopped, and the individual with the best fitness in the group is taken as the optimal solution to the optimization problem.
3 Optimization Example 3.1 Raw Data Design The helical gear transmission of the single-stage gear reducer for the belt conveyor. Known: transmission power P = 12KW, drive wheel speed n1 = 970, transmission ratio i = 4.8, one-way rotation, slight load shock, service life of 10 years, gear accuracy of 8 grades, material: pinion 45 Steel, quenching and tempering, HB240～270; large gear 45 steel, normalizing, HB180～210.
3.2GA algorithm implementation and results In the GA algorithm, the population size is N=100, the crossover probability is Pc=0.75, and the mutation probability is Pm=0.03. After 30 generations of genetic manipulation, the fitness value of the chromosome population tends to be stable, and the optimization problem is obtained. The optimal solution. This paper also uses conventional design methods and traditional optimization design methods to design, the results are more common.
By optimizing the design of the helical gear transmission with GA, the results obtained by the GA method are obviously superior to the conventional design and the traditional optimization design.
1Computation result comparison 4 Conclusions (1) Genetic algorithm as a non-numerical iterative global search algorithm has strong adaptability to objective function and constraint function, suitable for dealing with more complicated engineering optimization problems, and easy to get problems. Global optimal solution.
(2) This paper uses genetic algorithm to solve the optimization design of the helical gear transmission structural parameters is a new attempt, and obtained satisfactory results (center distance reduction of 26.6, volume reduction of 60.5), thus making the structure more compact The transmission stability is improved, the helix angle is increased, and the bearing capacity is further enhanced. Although the basic theory and application fields of genetic algorithms have yet to be further explored and discussed, its advantages over conventional methods will certainly make genetic algorithms more widely used.
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