DESIGN OPTIMIZATION OF MECHANICAL SYSTEMS USING GENETIC ALGORITHMS

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Fen Bilin1leri Enstitüsü

Dergisi

7.Cilt,

2.Sayt (Tenımuz 2003)

Design Optimization Of Mechanical Systems

Using Genetic Algorithnıs H.Saruhan, İ. _Uygur

DESIGN OPTIMIZATION

_OF

MECHANICAL SYSTEMS USING GENETIC

ALGORITHMS

Harnit

SARUHAN,

İlyas

UYGUR

Abstract-

This papeı- prcsents an algorithnı foı· the design of minimum weight of speed reducer., gear train, subject to a specificd set of constraints. The study is primari1y aimed to expose the potential of genetic algorithms, to discuss their application capabilities, and to show the concept of these algorithms as optiınization techniques and their scope of application by implementing them to the

speed reducer. Results obtained for the minimum \Veight of speed reducer are prescnted to provide insight into the capabilities of these tecbniques. Genetic algorithms are efficient search techniqucs which are inspircd fronı natural genetics selection process to cxplore a given search space.

Keywords-

Genetic a

l

gorit

h

ms, design, optimization ••

Ozet-Bu

makalede sınır şartları verilen bir hız rcdüktörünün nıininıum ağırlığını hesaplayan bir algoritn1a tanıtılmaktadır. Bu çalışmanın asıl amacı genet

ik

algoritmaların potansiyellerini ve u

y

_gulanı_a kabiliyetlerini, bir optinnun hız redüktörü tasarımında göstermektir. Bu tasanın için elde edilen sonuçlar bu tekniklerin uygunluğunu göstermektedir. Genetik algoritmalar tabii seleksiyon (seçim) teknikleri kullanarak tanınılanmtş sınırlar içinde tarama yapan ve genetik fikrine da)'ah uygun araştırma teknikleridir.

A�ıalıtar k elim eler-

Genetik aJgoritnıala

r,

tasarım, optinıizasyon

I. INTRODUC1lON

Many nurner

i

cal

opti

miza

tion

a

l

g

o

ıi

t

hms

have been develope

_d

_{and used for}

_d

e_s

i

_g

_{n o}

_p

_tinnzat

i

_o

_{n of} e

_n

_g

_i

_n

eer

i

_n

g

_pr

o

_bl

_e

_m

_s

_{. Most}

of

_thcse

_{optiınization} algoritluns so]ve eng

i

nee

r

i

ng

probl

e

ms

_for

fı

n_d

i

_n

_g

optinıiın

design.

Solvin

g engineeıing

p

r

oblems _{can be} con1plex

and

a time

consuming

proc

ess

when there are large n

_u

m_be

_r

_s

_of

_des

i

_g

n _{variables and constraints.} Thus, there is a need for

n

ı_or

_e

_{efficient and reliable}

a

l

g

o

r

ithm

s

that solve such problen1S. The development

of faster

conıputers

has

a.l

l

o\v

e

d

developn1ent

of morc

1 l.Saruharı,

İ.

Uygur; Aba nt

İzzet

Baysal

Üniversitesi,

Teknik

Egitim

l"akültcsiMakine E

ği

li ın i Bölüınü, 14550, DOzce,TURKEY

robust and effıcient

o

p

timi

zation methods. One of these

methods

is the g

e

net

ic

algo

r

i

thrns

.

The genetic

algorithnıs are

search

procedure

s

based on

the

id

ea

of

natuı�aı

sel ec tion

and

g

en

etic

s [ 1]. Genetic

al

g

or

i

t

lu

ns

can

be a

pp

l

i

c_{d to conceptual}

_and

_prel

_i

_m

_i

_n

_ary

engineering de

s

i

g

n _{studies. Genetic a1gorithms have}

been increasingly

r

eco

g

n

i

zed and app

l

ied

in

many

app1ications.

Interested reader

c

an refer studies by

[2],

[3]. This

pa per shows h

o

w genetic algorithıns

s

e

a

r

c

h

tlu·

o

ugh a desi

g

n space to fırıd the

minimun1

value

of

the

objective

fu

n

c_t

_i

_on

for

_engineering

_des

i

_g

_{n problems.}

II.GENETIC ALGORITHMS

In thi_{s seetion of the paper, the}

_{fundamental intuition}

of

genet

i

c _al

_g

_or

i

_t

hms

_{and how}

_{they process are}

g

_i

v_en

_.

Genetic al

g

o_r

i

_{thms nıaintain a p}

_o

_pulat

io

_{n of en}c_od

_e

_d

solutions,

a

n

d guide the populat

i

on

to

w

a

r

_d

_{s t}

_h

_e o_{pti mum}

_{solution [ 4]. Thus,}

_{they sea}

_r

_{ch the}

_space

of

possible indiv

i

d

u_al

_s

_{and seek}

_{to fınd the best fitness}

s

tr_in

_g

_.

_{Rather than}

_s

_t

_art

_in

_g

_{from a}

_s

_in

_g

_{le point} s

_o

_l

_u

t

_i

_on within the search space as in traditional opti

m

iz

a

ti

o

n

n1ethods, genetic al

g

or_i

_t

_{hıns are}

i

n_itia

li

_z

_e

_{d with a} population of

solutions.

Viewing

the genetic

a

l

g

or

ithms

as

o

ptimization tcchniques, t hey bo

lo

n

g

to the

class of

zer_o-o_rde

_{r optinıizati}

_on

methods [ 5],

[ 6].

The description

of

a

s

i

mp

l

e genetic

al

g

oritlım

is

ou_tl

i

_ned

_in

Figure 1. An

_initial

_p

o

_pulat

_i

_on

_{is chosen}

randamly

at the begiıu1

in

g

. Then an iterative proce�·

starts until the termination criteri a have b

e

e

_n

_satisfıed. After the evaluation of each individual

fi

t_nes

_s

_{in the}

p

o

pu lati

o

n, the

g

enetic

operators, selection, crossover,

and

mutation, are applie

d

to

p

rodu

c

e

_a

_{new generation.}

Other

genetic operators

a

re appl

i

e

d as needed. The

newly

c

r

e_a

t

_ed

_indıvidua]s

_{replace t}

_h

_e

_existing

generation,

and reevaluation is started for

the

fitness of

new individuals.

T

h

e

loop is

repeated

unttl an

acc_eptab

_l

_e

_{solution is found. Genetic}

_a

l

_g

_or

i

_{tluns differ}

froın

t

r

_a

_dit

i

_o

n

_a

_l

_search

t_e

_c

h

n_i

q

_u

_e

_{s in the}

_following

ways [ 4]:

-Genetic algorithnıs

work

with

a c

o

d

i

n

g

of

de

si

g

n

variablcs and not the de

s

ig

n

variables thernselv_es.

-Genetic algo_r

i

_t

_h

rn_s

_use

_ob

j

ec

_t

_iv

_e

_{function or fitness} fuııction information. No derivatives are necessaıy as

SAU Fen

Bilimleri

Enstitüsü Dergisi 7.Cllt, 2.Sayı

(Temmuz

2003)

-Genetic algorithms search from a popula_{tion of points}

not

a

single point.

-Genetic algorithms ga

t

he

r

information fron1 current search points and direct them to the subsequent search. -Genetic algorithms can be used with discretc,

integer,

continuous, or a

mix of

t

h

e

s

e

three d esign variables.

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sah.siied NO

�neratiı::mc�:t\eratioıı+ 1

1

Figure 1 Flo\'V chart for _{a simple genetic} algorithms.

Figurc _{2 Seltematic Diagram of Speed Reducer.}

111.1.

Design Variables

�

ı

ı

I

X

ı !<.--'

ı

ı

_/

_Pinion

ı

ı

/'

ı

1 /

1

Design Optiınization _{Of l\1cchanical} Systems

l1sing Genetic Algoritbms

H.Saruhan,

İ.Uygur

DI. PROBLEM

STATEMENT

Figu re

2

shows the configuration of a compound gear

train which was takeı1 from Ra o [7].

lt is

desired to o btajn the lowest weight of the gear

t1'ain

subject to

a

set of constraints. The statement of the design optimization of _{the problem is forn1ulated as:}

Objective function

\1ininlize

F

(X)

Subject to

j

=

1,

· · · ·

NIC

(

n_umb

e

_{ı- of inequality constraints)}

X

�ower

<

X

. <

X

�tp per

ı - l - 1 where X;=

{X1,X2,

. • . . . .

,X,}

i= 1,

. .

.

. . . n

Fa�;ective

=

P(X)

=

0.7854Xı X i (3.3333XJ

+

14.9334X

3

-43 .0934)

-1.508X1

(X;+

Xi)+7.477(Xı +Xi)+

0.7854(X4X

;

+

X

5

Xf )

(1)

The design

variables

used for fınding the

minimum

N

ı ı

�

�

ı

ı

ı

ı

ı

,., Gear 2

ı

_/'

t _{_.r...,}

ı

� _-----r--1---,.._ ____________ ---.. -t--- -ı •• ı J _ı

1

_ı

ı

ı

ı

ı

....

_.......

1

1

'-....

ı

<-'" ı ı:-- ...

Shaft 2

ı

I

X

_!

�

Shaft 1 """

ı ı ı 1

"-

,

ı

1

Bearing

�

ı

_ı

ı

�

ı ı

)

t-" - _ı '

w

ei

g

h

t

of the

ge

a

r train include:

SAL Fen Bilinıleri Enstitüsü Dergisi /.Cilt,

2.Sayı (Temmuz 2003)

.�

.. 1 is the face width. 2.6 < X1

� 3.6

..

f., is

m_od

u

_{le of teeth.}

0.7

<X

2 < 0.8

X

3

is

n

u

n

ıb

er of teeth on pinion.

17

_<X

3 <

28

X

4

is

length of shaft

1

between bearings. 7.3<X4 <8.3

.. r

5 is length of

shaft 2

betwcen bearings.

7.3<X5<8.3

X

6 is da

i

meter of shaft

1.

2.9<X5

�3.9

.A...

₁ is daimeter of shaft 2.

5

.

0

_<

_X

₅

_{< 5.5}

III.2.

Constraints

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Constraints are conditions that must be met _in t_he

optiınum design _{and inc}

lu

_{de restrictions on the de}s_ign

variabfes value and optimum design of the function. These constraints define _{the boundaries of the fea}

s

_ibl

e

and infcasible

design space domain. The constraints

considered for the optin1un1 design of gear train

include

the

following:

-

2 7

V -1

v-2.x

_}

<

1

gl -

. /\ ı .. '1 2 3

-g,

J

2

745.X'

4 +

(16.9 )106

X2.r\'3

. l

2

go=

74SX5

+

(157.5)106

)[ıX

3

g7

=

x

2X1

< 40

t;T ₌

(ı

"

Y

+

ı

9 )x

_-ı

<

ı

o� ·�· ./\ 6

.

4

-(9)

( 1 O)

( J ı)

( 12)

0.5

o.ıx�

_<ı

_{( 13)}

0.5

o.ıx?

_<ı

_{( 14)}

(15)

(16)

( 1 7)

C

icnctic

algorithms are unconstrained optin1İzation

procedure. 'Therefore, th

e

constrained optimization

79

Design

Optiınization Of Mectıanical Systems

Using Genetic Algorithrns

H.Saruhan, İ.Uygur

pr_ob

l

_em _{has be}

e

_n transforn1ed into an unconstrained

optimization pr_{oblem by}

p

_en_{alizing the} o_bjective

function

v alue with the quadratic penal

ty function. In

case of any violation of a constraint boundaıy, the

fitness of corr_{esponding solutj on is pe}

n

_{alized, and th}

u

_s kept within feasible r_{egions of the design space by}

increas

i

ng the value of the ob

j

ective function when constraint violations are enco

un

tered. The penalty

coefficients, ri, for the j -th constraint have to be

judiciously selected. The fitness function provides a

measure of the performance

of

an individ

u

al, which is

use d t

o

bais the se leetion process in fa vor of the most

fıt me mb ers of

the cuıTeni

pop

u

la

tion.

Fitness0�jcctive

=

F-

(F(x)

+

P)

NCON

P

=

L

ri

(

nıax

f O,

g

i

]

)

₂ .i=l

(18)

(19)

where

F

is an arbit_rary

large

_eno

u

_{gh that is greater}

than

F(X)

+

P

to cxclude negative fitness function

values and

P

is the penalty function.

ITI.3 C

onstruction of Design

variab I

es and Genetic

Algoritlını Parameters

In optin

ıi

zation probleın, a design of variables, x(i) ,

represents a solut_{ion that} n

1i

n_i

rni

_{zes or maximizes an} ob

j

ective function. The first step

for

applying the genetic algor_{itluns to}

a

_ssigne_{d design pro}b_{lem is}

en_codin_{g of the}

de

_{sign variables.}

Genetic a

l

go_rit_{hms require the design variables of the}

opti

m.i

zation problem to be coded. Binary coding, as a

fınite l ength strings, is generally used although other codin_{g schemes hav}

e

_{bcen used. Th}es

e

_{sırings are}

represented as chron1osonıes. Each design variable has

a specified range

so

_that

x(i)ıower :5;x(i)<x(i)upper·

'The continuous design v_ar_iab

l

_{es can be representeu}

and discrctized to a precision of e . Genetic algoıithms

have abihty to deal with integer and disc_{rete desig�}

v_ar_iables. The _{nun1ber of the digits in the binary} s_trin

g

s_,

l

_{, is cstiı}n_ate

d

_{from the following relationship}

[8].

(

'

)

(

'

)

x z -x ı

1

upper

lower

2 >

₊₁

&

(20)

where

x(i)1

_{and x(i)} are _{the lower and}

ower

upper

upper

bo

u

nd for desig

n

vari ables respectively. The

design v_ariab

I

_{es are code d in to the binary eligit}

{O,

1

} .

The

physical value of the design _{variables. x(i), can be}

SAU Fen

Bilimleri Enstitüsü Dergısi

7.Cilt, 2.Sayı (Ten1muz 2003)

x(i)

up per

-

x(i) 1

OltVer _.

x(i)

= x(i)

lo1ver

+

₁

d(ı)

2

-1

(21)

where d(i)

represents t

he decin1al value of

s

tri

n

g

for

design v_ar

i

_a

b

_les

which is

_obtain

e

d

using

base-2

_forn1.

Tablc 1 Design variables mapping.

Design

Variables Lo"�er Limit

1

The face

width 2.6

rv1odule of

teeth

0.7

Number of

teetlı

on pini

a

n 17

L

en

g

th of

s

h

aft

1 between

b

ea

ri

ng

s

7.3

Length

of shaft

2

between bearings

_7.3

Daimeter of shaft

₁

2.9

Dainıeter or shaft

2

_5.0

To start the algoritlun, an injtial

p

o

p

u

l

a

ti

on set is

ra

n

_d

o

ml

y assigned. This

_{set of initialized popul}

ation

_is

a p

o

tent

i

_a

l

_{solution to the problen1. For}

exaınple,

the

b

i

nar

y

string

representation

for

the des

i

g

n va

ri

a

b

_les,

Table 2 The binary string repre.sentation of the vat·iablcs.

Dc.sign Optimization Of _{:vıechanical Systems}

l Ts ing Genetic Algorithms _•

H.Saruhan, I.Uygur

Design variabtes

are

represented in different level of

p

r

ec_{ision. Ta bl e 1 giv}

es

_{descriptions of}

these

m_ap

pi

_n

g

_.

Upper Limit

Precision

String Leııgth

3.6 0 .01

7

0.8

O.

1

2

28

ı

4

8.3 0.01

7

8.3 0.01

7

3.9 0.0

ı

7

5.5

0.01 6

xi,

in Tab

le 2 gives an exampl

e

of a chroınosome

that

re

p

r_es_e

nt

s _{design v}ar

i

_a

b

_le

s

_{accordıngly. This design}

string

is c

o

mposed of 40 o

n

es _{and zeros.}

Design Variables

-x(l)

x(2)

_x(3)

_x(4)

_x(5)

_x(6)

_x(7)

-, 1 000 1010 ' 1 1 0 0 0000 0 00011

o

0001100 1000000 100100 ' _{-- .}

Concatenated

Vaıiables Head-to-Tail

-0 -0 -01-01-0 -0 -0 -0 -0-0-0-0 -0 -0 -011-0-0-0-011-0-01-0-0-0-0 -0-01 -0 -0 1-0-0

In T

a

b

l

e

2, the

string

of 40-bit st

r

i

n

g length represents

one of 240

a_l

t

_eına

ti

_{ve individual s}o_l

uti

_on

s

_existing

in

the

design

space.

For ıunn

i

ng

genetic

algorithms, an

in

it

ial

population

is need to be

a

ss

ig

n

e

d

randomly at

the

begim1ing.

Population size

influences

the nun1ber

of search p

o

i

nts in

each

gener_at_ion.

_A

guideline

for

an

a

p

p

ro

p

r

_{iate population size}

is suggested by Goldberg

[ 1 O]. The guideJine

for optimal

population size

dep en ds

on

_the

i

n

d

iv

i

d

u

al chromosome le

n

g

t

h

, \Vhich is

valid up

to 60 c_xpresse_{d as follows:}

80

population size=

1.65 *

2°·21*1

(23)

For

a string lcngth of 40 bits, an optinıal population

size

of 558

nıay

be used [ 1 0]. Considering computation

1üne� a randomly selected

set,

1

O strings,

of potential

so] u tion is us cd in this

study sine

c t

h

e

r

e was seen to not

h:ıvc a significan1

iuıprovement

in

resu

l

t

s

. See

Table

3.

The genetic algoritlun

then

p

r

ocee_{ds by gene}

r

a

t

i

_ng

new

solutioııs with bit operations u

t

ili

zin

g

genetic algorithm

SA U Fen Bilimleri Enstitüsü

Dergisi

7 Ci lt, 2.Sayı (Temmuz 2003)

Ta Ille 3 Aset of starting popula tion.

Individual

n

um

b

er

Design Optimization Of Mechanical Systen1s

Using Genetic Algoa·ithıns

H.Sarulıan,

İ.

_Uygur

Randomized

bin

a

ı-y str

i

n

g

ı

1001010000100010001010011001000000101010

2

ooo1oıoıoooooooooııoıooıoooıooooooıooooo

3

1100010010010001011000011000101000111101

4

oooıııoooooooo1oooııoooıııo11ooıooı1ııoo

5

_. 010001 0000010000111010001001010010100101

6

1001011001100100011000011100010010100101 7 oooıoıooıooo1ooooıoooıooo1ııoo1ooooıoooo 8 0001001001000000011000111011101000100100 9 1010010100001000011000011001000010100101

lO

0001010000000010001000011001001000000000

111.4. Genetic

Algorithnı

Opcrators

In

a sin1

p

l

e

g

en

e

tic al

g

o

ri

t

hnı, there a

r

e

three basic

operators f

o

r

creating the

next generation. Each

of

these

o

pe

r_ato

_{rs is}

e_xp

_l

_a

_i

_n

_{ed and}

_demonstrated

_{in the} following:

th

e selection operator shown

in

this work is a touman1ent sel

e

cti

o

n

.

Toumament selectio n approach

vvorks as follows: a pair

of individuals

from

mating pool is ran

do

n

uy

picked and t

he best-fıt

hvo

indi

v

i

duals from this

pair will

be

chosen as a

parent.

Each

pair of the

p

a

ren

_t

_{creates t\vo Child as deseribed}

in

the met

h

od of uni

f

o

nn

crossover

shown

i

n

Table 4.

A

u

ni

f

omı

crossover

opc

r

a

to

r is

used

in

this

study. A

uniform

crossovcr

o

p

e

ra

t

o

r probability

of

O. S

is

recommended in ı

n

a

n

y works such as [ 11] and

[

_l

2].

Crossover

is

very

important

in

the

succes

s of g

e

n

e

ti

c

algorithıns. This

ope

ra

te

r

is _the

_primary

_{source of}

_the

new

candidate

s

o

luti

o

ns

and provides

the

search

nıechan1sm that

e

ffı

c

ie

n

tly

_g

u

i

d

e

s

the

evo

lu

t

ion through t

h

e

solution

space towa

r

d

s

_the

optimum.

In

unifoım

cros

s

ov

e

r

_,

_{every bit}

_of

_e

_ach

parent stTing

_{has a}

cbance

of

be ing exchanged

wi

t

_h

the

corre

s

pon

d

i

ng

bit

of the

other parent s

tr

i

n

g

.

Table 4 Oniform crossover.

The

pr

_oce

_d

_{ure is}

_{to ob}

_tain

any

_c

_o

_ın

_b

_i

_nati

_{on of}

t

_wo

parent string s

(

chronı.osomes)

from

the mating

poo

l

at

random and generate

n

e

w Child stri

ng

_s

from these

parcnt strings

by perfoıming bit-by-bit

crossover chosen according

to a randomly

g

en

e

rat

ed

crossover

mask [1 3]. Where

there is

a

1 in

the

crossover mas k, the _C

h

_i

ld bit is _co

_p

_ic

_{d froın the}

_{first parent string, and} \vhere _{there is a}

O

_{in the mask, the Child bit}

_{is co}

_p

_ie

_d

from the

s

e

c

o

n

d parent string. The second Child str

ing

uses the

opposite

nıl

e

to

the p

rev

i

ous

one as shown

in

Tab le 4 .

For

e

a

ch pair ofparent strings

_{a n}

e

w

crossover nıask is r

an

d

o

rnJy

ge

ncra

t

e

_d.

Prev

e

nti

n

g the genetic

algoıithm f

ro

m

the

premature

convergence to

a

non­

optimal

solution, which

may lose

diversity by r

e

pea

t

e

d

a

p

pl

i

cati

o

n of se

_le

cti

o

n and crossove

r

o

_per

ato

r

s, a

mutation operator is used.

M

ut

a

tio

n operator is

basically

a p

r

o

c

e

ss of

r

andamly

a

lt

e

ı·ing a paı1 of an

individua] to

p

r

o

d

uce a

new

individual

by S\Vitching

the bit

p

ositi

o

n from a

O

to

a

1 or

vice versa as s

een in

Table

5.

Crossover mask

.. . . '·

·

-

c.t

o

:

o:r

:

o

_

�(lO

o o o

o ooo.oo

ıı,o�o

o

·

·

o

�

ııto

·

oJ.o

·

o

·

q·o

·

o

·

o:ı

·

o

'·

ö

l

i

·

o·cf.:

�

;-

,

-'

��

·, !_ : • \ 1. ·i ı. •. . 'f ' 1' ' ... - . • t 1 .. ·� 1 · _••• ,, :" '· c ,

Parent

I

Parent

II

Chil d

I Chil

d

_II

T bl 5 M t t" a e. ı u a ıon opera t or.

Be fare

i\fter . . 1101010001000100001000001001000100110000 oıoıooooooıcıooooı ıoıooı1ooooo1oooıooıoı 010101000010100000101000100�001000100001 1101000001000100011000011000000100110100 1001010000100000011000001001000100100100 1001010000100010011000001001000100100100

81

SAU Fen Bi1imleri Enstitüsü Dergisi

7 .Cilt,

2.Say1 (Temnıuz 2003)

'Ine

mutatio n

rate

suggested by

_Back

(14)

is:

1/

_<"

_P

_.

<

1

/

/

_{population size}

... mutatron

/

c

h

ronıoso1n

e

length

(22)

A

specialized

mechanisnı,

elitis1n,

_{is added}

to

the

g

e

n

eti

c algorithın.

Elitism forces

_the

genetic

algorithm

to retain the best

individual

in

a gi ven

g

en

cr

a

tio

n and

proceed unch

ang

e

d

into the following

generatian [15].

The parameters

of

genetic

algorithm

for t his

_study

have

chosen as in

Ta

bl

e

6.

Table 6 Genetic search algorithm parameters.

Genetic

algorithm

pa rameters

Chromoson1e length

40

P

o

p

_ula

t

io

_n

s

i

z

_e

lO

Number of generatian

200

Crossover probabib

ty

0.5

Mutation pr obability

0.01

There

are

_{many different}

ways

to

deterınine

when

to

stop

ıunning

the

ge

n

e

t

ic algorithm.

One

ınethod is

to

De_,ign Optinıization Of _Mechanical Systems

lJsing Genetic AJgorithm!

H.Saruhan, i.Uygur

stop after a presct number of gencration

which is used

in this

study

or a

time lin-ıit. Anather is to stop

after

the

ge

n

e

tic algorithın has converged.

Convergence

is

the

progression towards

uniformity. A string is said

_to

have

converged vvhen

9 5

%

of the population

share the same

value

[16].

Thus, n1ost or all strings in

the

population

are identical or

si

mi

1

a

r when

population is co

nve

rg

ed

.

TV.

RESULTS

Figure

3

sho,vs

the plots of

the

n

o

r

m

a

li

ze

d

minimum,

average, and best

_fitness

function values

in each

g

ene

ra

t

ian

as

optiınization

proceeds.

As

can

be seen

from

Figure

3, the

normalized

fitness

function of

individuals

in a population

iınproves over

generations.

The o

v

er

a

ll

results sho\v that

the

best design rapidly

co nvergc

_over

the first several generations

and

refıne

the

_{design over remaining}

generations.

Thus, the

selectcd paramelers set has converged to

a stable

solutions

�rith

similar values. The

results and

their

comparison

witlı nurnerical nıethod used by

Rao [7]

are

shown in Tab l

e

7.

As

can be seen from

these

results,

the

genetic algorithn1S

produced much

better

results

than that the n

tınl

e

ri c

a

l

met

h

od.

Table 7 The problem design vaı-iable� and

objective

function results.

Speed Reducer Optinı.ization

Method

Design variables Nurnerical

Genetic

o

tiıniza tion

_Algorithm

The face width

3.5

_-

_2.6

:vfodule of teeth

_0.7

_0.7

r--

-Numbe r

ofteeth

on pinion

_{17 .O}

_ı

_7.0

Length of

shaft

1

between bearings _{7. 3 7. 3}

Length of

shaft 2 between

bearings

_{7.3 7. 3}

Diameter

of

shaft

1

3.35

_3.40

Dian1eter

of

shaft

₂

_ı

_'

5.29

_5.28

Design Objective

1\'linimum weight

of

ge ar

_train

_2985.22

_2654.19

.

1 •

,

c o ·--+-' u c :::.> u.. en C/) <D c -+-' ·-LL

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

o

• •

.

.

.

.

.

.

.

.

.

..

. . : . . . q

:

.

. . . <t;:>. w

.

. . . �.

.

. . <?. . . .: . �K��.

. �

. <?'?

.

. : . . . <Ç) . : Ç)_

• • •

�

• .

<Doo

· · · · _{· · · · ·}

O.

.

·O

·

O O

· · · · · • • • • • • • • • . . . .. . . .. . .

o

· . .. . . .. . . ,

.

.

. .

.

. .

. .

. .. .

. .

. .

.

.. . . • • • • • • • • • • • ·

O

•

((D . (()

aoo

• • •

o

·

(()

· • • • • • • . . · · · " · · · ·· · · ·· · · ·· · · · · · ' · · · ' · · · · · ·· · · • • • • 1 • • _• • • • • • • • • • • • • • • • • • • • • • • • • • • .

. .

.

.

. .

.

.

.

.

. . .

.

.

..

.

.

. .

..

.

.

. .

. '

. .

.

. .

.

. .

. . . .

.

. . . .

.

. .

.

. .

.

.

.

. .

• • • • • • • • • • • • • • • • • • • • • • _• • _• • • • • • • _• • • • • • • • • • • • • • • • • • • • • .. • .. • • • • .. • • • • • 1 • • • • • • • • • • • • • .. • • • • • 1 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • · · · • • tt • • • • · · · · · · . . . • • • • • • • • • • • • • _{• •} • • • • • • • • • • • •

.

'

.

.

.

.

. 00 .

.

.

• • • • t • • • _{• •} • • • • • • • _{• •} • _• • • • • • • 1 • • • • _• • • * • • _• • • _• • • • _• • • • • • • _{• •} • • • • • • • • • • _•

.

.

.

. "

.

.

.

.

. •

[)

arn

o.

• • • . , ..

.

.

.

. .

.

.

.

. .

.

.

.

• • • .

.

,

.

. .

..

.

.

. .

.

. . .

.

.

20

40

60

• • • • • •

([) G)D

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

.

.

.

. .

. .

'

.

. . . .

. ..

.

.

.

• • • • • • •

BO

100

1

20

Generatian

• •

tD

:o

•

co

• • • • • •

o

Minirnum Fitness

o

Average Fitness

• • •

•

Best Fitness

140

160

180

200

• •

Figure _{3 Co} n' crgence process of genetic _algoritlı ms for normalized minimum .. average, and best fitness function.

V. CONCLUSIONS

Systen1s :orithn1s

i.

Uygur

A genetic algori thın techııique was us ed to generate the

nıinin1uın

weiglıt of the gear train. The design variables

were sclected as those that influence the optimuın design. The results sho\v that genetic algoritlun provides goo d solutions when compared to a nurnerical

optinıization method. In this regard, the ef

fı

cac

y

of genetic a

l

gori

th

m optinrization techniques is

deınonstrated by employ

i

ng an engineering design

problen1. lt can be conc]uded that th_{e genetic}

algorithıns

can be successfully used for conceptual and pre

1

in1inary des ign optimization o

f

the engineering

problenıs.

3.

Saruhan, H., Rouch,

_K.E.,

and Roso, C.A., Design Optinuzation of Tilting-Pad Journal Bearing Using a Genetic Algorithm Approach, The

9th

of

International Symposium on Transport

Phcnomena and Dynaınics of Ro

t

at

in

g

Machinery, ISROMAC-9, Honolulu, Hawaii,

2002.

4.

Goldberg, D.

E.,

Gene

t

ic Algorit

hms

in Search,

Optiınization, and Machine Leaming, Addison­

Wesley, Readiııg,

1989.

5.

Dracopoulos, D.C., Evolutionary Leaming

Algorithms for �eural Adaptive Contro'

Sp ring er-Verlag, London, 1997.

6. Louis, S.J., Zhoo,

F.,

and Zeng,

X.,

Flaw

Dctection and Configuration with

G

enetic

Algorithmsl Evolutionary Algorithms in

Engineering Applications, Springer-

V

erlag,

1997.

Vl. REFERENCES

1.

Goldberg,

D.E.,

The

D

e

si

gn of

Innova

tion:

Lessons from Genetic Algorithms. Lessons for the

Re al World, Univer

_s

ity of

l

l

l

in

oi

s

at

U

rbana­

Champaign, IlliGAL Report:

_98004,

Urbana, IL

ı 998.

2.

Sanıhan,

H.,

Rouch,

K.E.,

and Roso, C.A., Design

Optirnization of Fixed Pad Journal Bearing for

Rotor Systen1 U s ing a Genetic Algorithın

Approach, The

1

st International S

ymp

osium on

Stability Control

of

Rotat

i

ng Machinery,

ISCORMA-1,

I.,ake Tahoe,

N

eva da,

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7.

Ra

o, S. S., Engineeıing Optiınization 11ıeory and

Practice, New Age international (P) Limited,

Pub., New Delhi,

1999.

8.

Lin, C.Y.

and Hajela,

P.,

Genetic Algorithms

in

Optinıization Problcms with Discrete and

Intege

r

Design Variables, Engineering Optiınization,

19,

309-327� 1992.

9.

Wu, S.J.

and Chow, P.l'., Genetic AJgorithms for

Nonlinear

Mixcd

Discrete-Integer Optinıization Problems _via Meta-Genetic Parameter

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Dergisı

7 .Cilt, 2.Sayı

(Temmuz 2003)

Optimization,

Engineeıing

Optimization, 24,

1 3 7-1 59, 7-1 99 5 .

1 O.

Goldberg,

_D

.E.,

Optinıal Initial Pop

u

l a

ti

o

n

Size

for

B

i

n

ar

_y

Coded

Genetic Algorithms, The

Clearinghouse for Genetic Algorithms,

University

of Alabama, TCGA

Rept. 8500 1 ,

Tuscaloosa,

1 985 .

1

l . Syswerda,

_G.,

Un

i

form Crossover

_in

G

_en

e

_tic

Algorithms, Proceedings of

_{the 3}

rd

International

Conference

on

Genetic Algorithnıs,

Morgan

Kaufman, 2-9, 1 989.

1 2.

_Spears,

W.M.,

_and

_De

_{Jong, K.A.,}

_{On the Virtues}

ofParameterized

Unıform

Crossover, Proceedings

of the

₄

th International

Conference on Genetic

A

lg

oıithms, Morgan

Kaufinan,

230-236, 1 99 1 .

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Design Optimization _Of Mechanical Systems Using Genetic Algorithms H.Saruhan, i.Uygur

1 3 .

B

_eas

l

y

_{, D., Bull,}

D.R. , and

Martin,

R.R., An

Overview of Genetic A lgorithms: Part2,

_Research

T

o

pic

s,

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Computing, 1 5

(

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1 70- 1 8 1 ,

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1 4.

Back, T., Optimal Mutation Rates

in

Genetic

Se

_a

r

c

h

,

Proceedings

of

the

5th

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Conference on Genetic Algorithms, Morgan

Kaufmann,

Los Angeles, 2-8, 1 993 .

1 5 .

Mitchell, M., An Introduction to

Genetic

Algorithms, The

_{MIT' P}

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e

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Massachusetts,

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16.

DeJong

_K.,

The Analysis and Behavior of Class

of

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t

e

ms,

Ph.D.

Thesis,

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f M

ich

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n

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