机械外文翻译

Optimization design of spur gear reducer based on

genetic algorithm

Zhang Xiao-qin1 2

1 College of Mechanical Engineering

Yan Shan University

2 College of Mechanical and Electrical Engineering Hebei Normal University of Science & Technology

Qinhuangdao China

Hu Zhan-qi

College of Mechanical Engineering Yan Shan University

Qinhuangdao China

Lun Cui-fen Yu Jing-jing

College of Mechanical and Electrical Engineering Hebei Normal University of Science & Technology

Qinhuangdao China

Abstract—Gear reducer is one of the most widely used methods in m echanical transm ission, optim ization of which is of great significance in im proving the bearing capacity, prolonging service life and reducing its size and quality. By Visual Basic program m ing m ixed with MATLAB, autom atic optim ization design for gear reducer is realized in the paper, design efficiency and quality greatly im proved. Genetic algorithm and genetic toolbox of MATLAB is used when calculating, with the advantages of sim ple program m ing, good reliability and high efficiency.

Keywords-reducer; optimiz ation of design; genetic algrithm; MATLAB

I.I NTRODUCTION

Gear reducer is one of the most widely used methods in mechanical transmission, which has many advantages, such as accuracy ratio, high efficiency, stable and accurate. With the rapid development of science and technology it will make continuous progress. So it is of great economic significance to improve the bearing capacity, prolong the service life and decrease its size and quality. Optimized design of gear reducer, choosing the best parameters is a kind of important ways to improve the bearing capacity, reduce weight and cost[1][2][3].

Optimization design of gear reducer includes the continuous variables and discrete variables. The traditional method is to round the optimal design to the adjacent discrete points. Thus, design point might run out of the feasible region, besides traditional optimization method is mostly based on gradient algorithm, which is likely trapped into local minimum search. Many studies indicate that the genetic algorithm has strong ability of general optimization, which is very effective in treating optimization problem containing continuous and discrete variables[4][5].

Based on single spur gear reducer, the paper establishes the mathematical model for optimization design of gear reducer, with the purpose of minimum volume or quality. By using genetic algorithm and genetic toolbox of MATLAB to get optimum solution quickly and accurately, the efficiency and quality of gear design is greatly improved and the production cycle is shorten.

II.M ATHEMATICAL M ODEL FOR O PTIMIZATION D ESIGN Optimization design of gear reducer is usually given power P, gear ratio i, input speed n1 and other technical conditions and requirements, to seek a group of design parameters that achieve technical and economic indexes of a reducer optimum.

A single spur gears reducer transmission diagram and gear structure is shown in Fig 1. Assume that the gear ratio is known as i, the input power P(kW), active gear speed n1 (r/min), it’s required to figure out various design parameters to make the reducer as light as possible under the condition that

the strength and stiffness are guaranteed.

Figure 1. Single spur gear reducer and gear structure Since the size of the gear and axis is to determine the size and quality of reducer assembly, the objective function is the sum of their value, without considering cabinet and bearing volume or quality. According to the fig 1, the sum of the

978-1-4244-7161-4/10/$26.00 ?2010 IEEE

volume of gears and shafts can be approximately expressed as follows:

()

)

1(42.14)(3.04)(421

2

2122

2222j j j j j nj nj wj wj oj l d b d d b d d b d d V ×+??o??a×?+×?+?=||==ππππWhere d 01=mz 1;d 02=mz 1i ;d w1=mz 1-8m ;d w2=mz 1i -8m ;d n1=1.6d 1;d n2=1.6d 2.

From above equation, it can be seen that if i have been given beforehand, the volume of gear and shaft is determined only by face width b , the number of pinion teeth z 1, modulus

m , shaft diameter d 1 and d 2 and distance l between two bearings, i.e. V is the function of these parameters: V=f (b,z 1,m ,d 1,d 2,l 1,l 2). Take these variables as design variables and expressed as: X =[x1,x2,x3,x4,x5,x6]= [b,z1,m,d1,d2,l] (2) Thus the objective function can be expressed as: f(X)=V=f(x1,x2,x3,x4,x5,x6) (3)The constraint conditions can be given by follows [1][2]: ?

Avoiding undercut condition z1 should be greater than

the smallest number 17, i.e.

g 1(X )=-x 2 -17 (4) ?

Face width condition face width should meet ?dmin ?d=b /d 1 ?dma , i.e.

)6(0

)()5(0

)(max 32133

21min 2≤?=≤?=d d x x x

X g x x x

X g ???Total size condition If given d1+d2 F, then

g 4(X )=x 2x 3(1+i )-F 0 (7) ?

Module condition Module of Power transmission gear should be not less than 2, i.e.

g 5(X )=-x 3 -2 (8) ?

Minimum shaft diameter condition If the least sizes of d1,d2 are d1min,d2min respectively, then

g 6(X )=d 1m i n -x 4 0; (9) g 7(X )=d 2m i n -x 5 0; (10) where 3min 11/n P A d =,3min 21/n iP A d =, where A=108. ?

According to the structure, support distance of shaft should satisfy: l b+2¨min+0.5d2(¨min can be identified as 20mm), e.i.

g 8(X )=x 1+0.5x 5+40-x 6 0; (11) ?

The shaft’s bending condition The stress of gear axis ?j should be not more than the allowable value [?j ], i.e. g 9(X )=?1-[?1] 0; (12) g 10(X )=?2-[?2] 0; (13) Where

j

nj

wj

j W M M 2

275.0+

=

σ (j=1,2),

?

Gear strength conditions the bending stress ?H and

contact stress ?F of gears should be not more than the allowable value, i.e.

g 13(X )=?H -[?]H 0 (14) g 14(X )=?F 1-[?]F 1 0 (15) g 15(X )=?F 2-[?]F 2 0 (16)

Where i

x x x i KT Z Z Z H E H 232211)

1(2+=σε

,Z E is elastic coefficient, Z H

is regional coefficient of pitch point, Z ? is contact ratio coefficient, K is load coefficient, T1=9550*P /n1.)

2,1(2

22311

==

σεi Y Y Y x x x KT Sai Fai Fi , YFa is tooth shape coefficient,

Ysa is correction coefficient, Y ? is coincidence degree coefficient for bending strength.

III.

D ATA PROCESSING

A.Input/output of data

In order to increase the versatility of optimization design program, the friendly human-computer interface is designed by using Visual Basic language in order that user can select or input related parameters in the interface. After all known parameter selected or input, click "optimization design" button, the optimal result will be displayed, simply and visually. B.The graph data accessing and programming

In gear design manuals, many coefficients and other parameters are estimated by table (table 1) or line graph (fig. 2). In the traditional design, we can determine the values of relevant coefficients by consulting manual, while in numerical optimization design, all designs and calculations are fulfilled through the computer automatically, and therefore programming chart data is needed.

1)Table programming.

Tables used in the mechanical design are divided into simple lists and list functions according to having function relations or not between data. In simple lists various data are independent, having no clear relationship, which can be stored in one-dimensional array, two-dimensional array or three-dimensional array, retrieved by using look-up, interpolation method, and so on. Function relation is existed between function data and variables, but cannot be expressed by clear expression in list function, which is usually treated by interpolation or curve fitting methods [7].

TABLE I.

USING COEFFICIENT K A

Prime mover conditions

Work Machine Conditions

Steady

Slight impact

Moderate impact

Steady 1.00 1.25 1.50 Slight impact 1.10 1.35 1.60 Moderate impact

1.25

1.50

1.75

The tables involved in the paper are all simple, such as using coefficient KA, treated by seeking array of

table. MATLAB function of determining using

coefficient KA (table 1) is as follows:

function ka=Ka(i,j)

k=[1.00 1.25 1.50;1.10 1.35 1.60;1.25 1.50 1.75] ka=k(i,j);

2)Graph programming.

According to sources, line graph can be divided into two kinds: one kind is represented by the parameters of the graph between which have calculation formula, just because the formula is so complex that drawn into graph to manually search; one is that the parameters of the graph can’t be found calculation formula. For the first type, it’s needed to find the original formula, and incorporated into the program; For the second, it’s needed to discrete the graph into table, and treat it as table or seek fitting formulae by curve fitting method and incorporated into the program.

The following is programming process of tooth shape coefficient Y F a (fig.2) when modification coefficient x=0. a)Separate the graph into table shown as table 2.

The basic principle selecting node is that the differences between two adjacent points are of uniform. In figure 3, z value is smaller, the greater the effect on the profile coefficients, node interval should be made smaller; the number of teeth is higher, less impact on the profile coefficient, node interval should be made bigger, to improve the accuracy of list

functions and reduce the interpolation error.

Figure 2. Tooth shape coefficient Yfa

TABLE II.

DISCRETE TABLE OF TOOTH SHAPE COEFFICIENTS

z

Yfa z Yfa z Yfa

17 2.95 23 2.68 29 2.54 18 2.9 24 2.65 30 2.52 19 2.85 25 2.625 35 2.5 20 2.8 26 2.6 40 2.48 21 2.76 27 2.575 45 2.44 22 2.72 28 2.56 50 2.32

b)Programming with interpolation method.

function [yfa]= Yfa(z)

x=[30 35 40 45 50]; y1=[2.95 2.9 2.85 2.80 2.76 2.72 2.68 2.65 2.625 2.60 2.575 2.56 2.54 2.525]; y2=[2.525 2.45 2.40 2.32]; if z>=17&&z<=30 yfa=y1(z-16) elseif z>30&&z<=50 yfa=interp1(x,y2,z,'cubic') end

IV.

PRINCIPLE OF GENETIC ALGORITHM

Genetic Algorithm (GA) is referred to as a search method of optimal solution to simulating Darwin's genetic selection and biological evolution process. Genetic algorithm is a series of random iterations and evolutionary computations simulating the process of selection, crossover and mutation occurred in natural selection and population genetic, in according to the survival of the fittest, through crossover and mutation, good quality gradually maintained and combined, while continually producing better individuals and out of bad individuals. Through the generational produce and optimizing the individual, the whole group evolves forward and constantly approaches to the optimal solution.

Genetic algorithm, not requiring gradient information and continuous function, optimization results being global, applied to mechanical design optimization problems, can effectively avoid local optimal solutions, and get the global optimal solution [6]. So genetic algorithm is selected for gear optimization, and achieved through the MATLAB GA toolbox, optimizing process simple and efficient.

V.

S OLVING THROUGH MATLAB GA TOOLBOX

MATLAB is advanced mathematics software launched by the MathWorks Company since the mid 1980s, which faces to science and engineering. His ability of excellent numerical computation and data visualization makes it fast eminent in mathematics software. MATLAB includes two parts: the core part and various optional toolboxes, and these distinctive toolboxes provide quick solution for different researchers. MATLAB genetic toolbox is customized toolbox for genetic algorithm, with which various problems to optimize using genetic algorithm can be easily figured out.

Below is the detailed process using MATLAB 7.1 genetic toolbox to the optimal solution to cylindrical gear reducer. A.Convert the optimization mathematical model of I into the following forms applied to MATLAB: Seek X =[x 1,x 2,x 3,x 4,x 5,x 6,x 7]T

To make

(

)+

π+??=4

2094.1)8(7.0)(1

2

423322

2

23x x x x x x x X F

())44(

4

2094.1)8(7.06

256241

2

5

233222223x x x x x

x x i x x i x x

+

π+π+?? min

And subjected to

A*X<=b (linear inequality constraints); Aeq*X=beq (linear equality constraints); LB<=X<=UB(bound constraints); C (X)<=0 (nonlinear inequality constraints); C eq(X)=0 (non

linear equality constraints).

B.Establish .m file for objecitve function:

function [f]=FitnessFcn(x) global u f=0.785*x(1)*(x(3)^2*x(2)^2-0.7*(x(2)*x(3)-8*x(3))^2+ 1.20*x(4)^2)+0.785*x(1)*(x(3)^2*x(2)^2*u^2-0.7*(x(2)* x(3)*u-8*x(3))^2+1.20*x(4)^2)+0.785*x(6)*(x(4)^2+x(5)^2)C.Establish .m file for nonlinear constraints:

function [c,ceq]= nonlconfun(x)%define global variables global wcon, pcon, fdmin, fdmax, u, f, P ,n ,sigma1, sigma2, cl1, cl2

ka=gearka(wcon,pcon)%calculating load coefficient KA ...

c=[fdmin-x(1)/x(2)/x(3);x(1)/x(2)/x(3)-fdmax;x(2)*x(3) *(1+u)-f;P*(x(6)^2+1094.43*19^12/1.57/x(4))^0.5/n/x(4)-45;P*(x(7)^2+1094.43*19^12/1.57/x(5))^0.5/n/x(5)-75; ze*zh*zs*(2*kh*t1*(u+1)/x(1)/x(2)/x(2)/x(3)/x(3)/u)^0.5-sigmah;2*kf*t1*yfa1*ysa1*ysi/x(1)/x(3)/x(3)/x(2)-sigmaf1; 2*kf*t1*yfa2*ysa2*ysi/x(1)/x(3)/x(3)/x(2)-sigmaf2] ceq=[]

D.A, b, Aeq, beq, LB and UB are assigned by linear constraint and bound constraints respectively

A=[0 -1 0 0 0 0 ;0 0 -1 0 0 0 ;0 0 0 -1 0 0 ;0 0 0 0 -1 0 ;1 0 0 0 0.5 -1];

b=[-17;-2;-29.3;-42.3;-40]; Aeq=[];beq=[];

LB=[0;17;2;29.3;42.3;0];UB=[];

E.Set optimization options:

options=gaoptimset('MutationFcn',@mutationadaptfeasible )

F.Call ga() function:

[X,FVAL]=ga(@FitnessFcn,nvars,A,b,Aeq,beq,LB,UB,@n onlconfun,options)%X is optimum point; FAVAL is optimum value;nvars is number of design variable .

VI.A PPLICATION EXAMPLE

It is known that input power P=22kw, small gear speed n1=960r/min, gear ratio u=3.5; load stability, expected life is 10 years, 300 days a year, accounting for 20% of working time.

Power machine is the motor, moderate vibration exists, transmission is not reversed, the gear symmetrical layout. Enter known parameters and design variables initial values in the user interface, click on the "Optimization" button, you can output optimization results, shown in Figure 4. Result

compared with the results of traditional optimization design

and conventional design is shown in Table 2. TABLE III.

R ESULTS COMPARED WITH THE TRADITIONAL OPTIMIZATION

DESIGN AND CONVENTIONAL DESIGN

Design method Face width Tooth number

module Shaft diameter 1 Genetic

algrithm 49.632 24 2 15.463 Traditional optimization method

49.654 24

2

15.465

Conventional design 85 34 2.5 78.4 Design method Shaft diameter

2

Support distance

Objective function mm2

Genetic algrithm 45.125 115.359 648480 Traditional optimization method

45.133 115.360 649610 Conventional design

12.60 130

3584900

VII.C ONCLUSION

Using VB and MATLAB as development tool, selecting the genetic algorithm in artificial intelligence, the paper developed optimization design software for spur gear reducer to achieve the automatic optimization and obtain the global optimal solution, greatly improved the design efficiency and quality of gear reducer, shorten the production cycle.

R EFERENCES

[1]Zhu Xiaolu, “Gear drive design manual,” C hemical Industry Press, Beijing, 2005, pp. 114 -185

[2]Liu Weixin, “Mechanical design optimization,” Tsinghua University Press, Beijing, 1994, pp. 287-289

[3]Xie Qingsheng, “Mechanical Engineering Neural Network,” Mechanical Industry Press, Beijing, 2003, pp.100-104

[4]

Song C hongzhi, Xie Zhigang, “Application of intelligent algorithm to optimize the design of gear transmission,” Instrumentation and Automation, vol. 6, 2006, pp.1-3

[5]

Zhou Tingmei, Lan Yueming, “Optimization design modeling and applications in mechanical parts and system,” Chemical Industry Press, Beijing, 2005, pp. 204-230

[6]

Bi C hangchun, Shi Lei, Ding Yuzhan, “Artificial intelligence gear transmission CAD optimum Design,” Mechanical Design, vol. 2, 2000, pp.35-37

[7]

C heng Kai, Li Renjiang, Li Jing, “C omputer aided design technology base,” Chemical Industry Press, Beijing, 2005, pp.270-140