TELKOM
NIKA
, Vol. 11, No. 10, Octobe
r 2013, pp. 5
725 ~ 5
733
ISSN: 2302-4
046
5725
Re
cei
v
ed Fe
brua
ry 17, 20
13; Re
vised Ju
ly 1, 201
3; Accepted
Jul
y
15, 2013
Analysis on the Key Parameters of Aerospace
Microminiaturization Decelerat
or
Xiang
y
ang J
i
n*, Jian
y
u
an Feng, Lili
Zhao
Schoo
l of Lig
h
t Industr
y
,
Har
b
i
n
Univ
ersit
y
of
Commerce, H
a
rbin 1
500
28, H
e
ilo
ng
jia
ng, Ch
ina
*Corres
p
o
ndi
n
g
author, e-ma
i
l
:
jinxiangyan
g@1
26.com
A
b
st
r
a
ct
A hi
gh
efficie
n
cy sp
ecia
l tr
ans
missi
on
d
e
c
eler
ator is
d
e
sig
ned
by
sc
he
me
des
ig
n, structura
l
desi
gn, o
p
ti
mi
zation
des
ign,
a
nd retur
n
d
i
fference
an
alysis
and c
o
mp
uter
simulati
on. S
m
all te
eth d
i
ffere
nce
transmissio
n
w
i
th the first lev
e
l bev
el g
ear
p
a
ir an
d the sec
ond l
e
ve
l bev
el
oid g
ear
pair is
ado
pted as th
e
form of trans
mission
in the
d
e
cel
e
ratin
g
sys
tem. Gr
eat tor
que a
nd
big
b
end
ing
mo
men
t
are avai
lab
l
e
by
transmissio
n
ra
tio formula
de
d
u
ction, force a
nalysis
an
d strength
ana
lysis.
T
he particl
e sw
arm opti
m
i
z
a
t
io
n
and the genetic algorithm
have be
en c
o
m
b
ined
and the
muta
tion
operator optim
i
z
at
ion
m
o
del has
been
prop
osed
to o
p
timi
z
e
th
e d
e
s
ign
of d
e
cel
e
r
a
tor. T
hen t
he
transmissio
n
w
i
th lar
ge tra
n
s
m
iss
i
on
ratio,
hig
h
torque, hi
gh p
o
w
er and hig
h
p
r
ecisio
n w
ill be
reali
z
e
d
in s
m
all sp
ace. This mo
de
l has solv
ed the pro
b
l
e
ms
of particl
e sw
arm
opti
m
i
z
a
t
i
on i
n
mec
h
a
n
i
cal
desi
gn.
F
o
r exa
m
ple, t
here
are
mor
e
vari
abl
es, it
is
constrai
ned
a
n
d
it is
eas
ily
p
r
ecoci
ous. T
h
i
s
mode
l
has
p
r
ovid
ed th
e th
eoretic
al
basis
for the
opti
m
u
m
desi
gn of micro
m
i
n
iat
u
ri
z
a
ti
on
gear syste
m
.
Ke
y
w
ords
: Re
turn Differenc
e
Analysis, Parti
c
le Sw
arm, Mu
tation Operat
or
Copy
right
©
2013 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
Microminiatu
rization d
e
cel
e
rato
r for a
e
rosp
ace pu
rpo
s
e ad
opts th
e
first level be
vel gear
transmissio
n
and the seco
nd level
small teet
h
difference inner g
e
a
r
ing
beveloid g
ear
transmissio
n to succe
s
sfull
y
apply beveloid gea
rs
to
decelerators.
It repl
ace
s
the cycl
oidal
pin
wheel in
RV
transmission, whic
h h
e
lp
s the
de
cele
rator to
not
only inhe
rit
RV de
cel
e
rat
o
r’s
advantages of small volume, great transmissi
on rati
o, great carryin
g capabilities, large rigidity
,
high kin
e
mat
i
c
a
c
cu
ra
cy and high
tra
n
smi
ssi
on
ef
ficien
cy etc.
But al
so
m
a
ke
u
s
e
of
the
stru
ctural feature
s
of the b
e
veloid ge
ar to c
onve
n
ientl
y
adjust the g
earin
g ba
ckl
a
s
h, re
du
ce th
e
return differe
nce for p
u
rpo
s
e of pre
c
i
s
io
n transmissio
n [1, 2]. The
stru
ctural dia
g
ram i
s
sh
own in
Figure 1, the first level ad
opting bevel
gear tran
smi
ssi
on an
d the se
con
d
level con
s
i
s
ting
of a
parall
e
log
r
am
linka
ge an
d
gear
me
chan
isms. Excent
ric layo
ut is
adopte
d
for t
he outp
u
t sh
aft
and su
ppo
rti
ng axle se
rving as t
he crank of the p
a
rallel
ogram
linka
ge; the internal g
ear
and
output sh
aft are fixedly co
nne
cted or m
ade a
s
a w
h
o
l
e [3, 4]. During the ope
rat
i
on, the external
gear is
driven
by the input
shaft an
d sup
porting
axle t
o
do tran
slati
onal motio
n
.
Then th
e mot
i
ve
power is o
u
tp
ut through th
e output sh
aft by
gearing of
internal an
d external ge
ars.
The si
ze of
d
e
cel
e
rato
r sh
ould not
b
e
much
la
rge
r
due to
the li
mitation. The
r
efore, it i
s
critical to em
ploy a good
optimizatio
n
algorith
m
to optimize th
e para
m
eters o
f
decele
r
ato
r
for
this
mic
r
ominiaturization trans
miss
ion
devic
e
. As
th
e
modificatio
n
coeffici
ents
o
f
the thicken
e
d
gear
are diffe
rent in diffe
re
nt se
ction
s
, if the
modifica
tion co
efficie
n
ts a
r
e de
sig
ned to p
r
e
s
e
n
t
the linea
r
ch
ange
s al
ong
the axis, th
e tooth
cres
t
will h
a
ve a
certai
n tap
e
r along
the
a
x
ial
dire
ction [5,
6
]. Afterwards,
the g
ear me
shin
g
ba
ckla
sh can
be a
d
j
u
sted
thro
ug
h controlli
ng t
h
e
axial displa
cement
so
as to achieve t
he pu
rp
os
e o
f
adjustin
g
th
e ba
ckl
ash o
r
elimin
ating
the
backla
s
h. At the sa
me time
, this tran
smi
ssi
on ha
s the
small ge
ar b
a
ckla
sh, the
high ri
gidity, the
la
r
g
e
c
a
r
r
y
ing c
a
pa
c
i
ty a
nd th
e
s
m
ooth o
peratio
n, whi
c
h i
s
suitable
for the me
ch
anical drive t
hat
has hig
h
pre
c
isi
on, high rotation spe
e
d
and that ca
n be accele
rated and de
celerate
d in short
time. This is the driving body that is greatly
nee
d
ed by the aero
s
pa
ce mi
cro
m
iniatu
rization
decelerator a
nd other p
r
e
c
i
s
e ma
chin
es.
Based
on t
h
is d
e
man
d
, the thicke
ned g
ear
h
a
s b
een firstly applie
d
in the
microminiatu
rization d
e
cel
e
rato
r, whi
c
h
not only
inh
e
rits the va
rious m
e
rits
without the teeth
differen
c
e d
r
i
v
ing, but also
adjust
s
the meshi
ng
ba
ckla
sh an
d red
u
ce
s the ge
ar backla
s
h a
s
well
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 10, Octobe
r 2013 : 572
5 –
5733
5726
as
achieve
s
the preci
s
io
n
driving. A
ccordin
g
to th
e
ch
ara
c
te
risti
c
s of me
ch
a
n
ical
optimum
desi
gn, it has bee
n impro
v
ed based o
n
the parti
cle
swa
r
m opti
m
ization. Th
e particl
e swarm
optimizatio
n
and th
e g
e
n
e
tic al
gorith
m
have
be
e
n
al
so
su
cce
ssfully
appli
e
d in th
e o
p
timum
desi
gn of the para
m
eters o
f
micromini
a
turization
de
celerato
r [7].
The practi
ce
has
sho
w
n th
at
Figure 1. Driv
ing diag
ram o
f
micromini
a
turization de
celerato
r
the improve
d
particl
e swa
r
m optimi
z
ati
on in this
p
a
per i
s
effecti
v
e in the optimum de
sign
of
decelerators.
The cran
k
sh
aft with
powe
r
input in the
mech
ani
sm
is call
ed inp
u
t
shaft and t
h
e
cra
n
k
sh
aft without p
o
we
r input i
s
cal
l
ed suppo
rt
in
g axle. In this ki
nd of transmi
ssion, t
he
external
gea
r makes tra
n
s
lation
al moti
on in
stead
o
f
planeta
r
y
motion a
s
i
n
tran
smi
ssi
on
o
f
gene
ral
plan
etary ge
ar
and th
e inte
rnal
gea
r m
a
ke
s fixed
-
a
x
is rotatio
n
.T
he
stru
cture
of
microminiatu
rization d
e
celerato
r is con
s
ist of t
he first level bevel drive transmi
ssi
on and
se
con
d
level small te
eth intern
al g
earin
g bevel
o
i
d gea
r tran
smissi
on [8]. In the tran
smi
ssi
on
schem
e, it
is
set that th
e sch
e
me
co
de i
s
rep
r
e
s
e
n
ted by
a
lett
er with
t
w
o subscri
p
ts
. The firs
t subs
cript
rep
r
e
s
ent
s b
a
si
c driving
membe
r
s
an
d the se
co
nd
sub
s
cript re
pre
s
ent
s ba
si
c drive
n
me
mbers.
The first pa
rt
of the transmissi
on
code
is t
he high
speed g
ear tra
n
smi
ssi
on ba
sic
com
pone
nts.
The corn
er m
a
rk i
s
re
presented with 1;
the bas
i
c
co
mpone
nts for low sp
eed g
ear tra
n
smission
are
rep
r
e
s
ent
ed by 2. For t
he convenie
n
c
e of the fo
llo
w up the
o
retical
calculatio
n, input sh
aft is
r
e
pr
es
e
n
t
ed
b
y
, output shaft by
and the excentri
c shaft by
1
H
and
2
H
.
2. Deduc
tion
in Transmission Ra
tio
The tra
n
smi
s
sion
ratio
s
of
the first leve
l bevel gea
r
pair a
nd the
se
con
d
level
beveloid
gear a
r
e a
s
follows:
11
1
/
ba
iz
z
22
2
2
/(
)
ab
a
iz
z
z
(1)
The total trans
m
iss
i
on ratio
12
ii
i
.
Torq
ue a
c
ting
on the bevel pinion
1
/
a
TT
i
.
3. Force An
a
l
y
s
is and Calculation
With the
ad
vancem
ent i
n
networkin
g and
multi
m
edia te
ch
n
o
logie
s
e
n
a
b
les th
e
distrib
u
tion. A
l
though
en
cry
p
tion
can
p
r
o
v
ide multim
e
d
ia
conte
n
t o
n
ce
a
pie
c
e
o
f
digital
conte
n
t
is de
crypted,
the dish
one
st
custo
m
er
ca
n redi
strib
u
te it arbitrarily
s.
Tange
ntial force a
c
ting on t
he refe
ren
c
e
circle
11
1
200
0
/
ta
a
m
a
F
Td
(2)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Analysis o
n
the Key Para
m
e
ters of Aerosp
ace Mi
crom
iniaturization Decele
rato
r (Xiang
yan
g
Jin)
5727
In the formula,
1
ma
d
——Diam
e
te
r of the refere
nce
circle on
the bevel pini
on (mm
)
.
Torq
ue a
c
ting
on the bevel gear
whe
e
l
12
/
b
TT
i
.
Tange
ntial force a
c
ting on t
he refe
ren
c
e
circle
11
1
200
0
/
tb
b
m
b
F
Td
(3)
In the formula,
1
mb
d
——
Diamet
er of the refe
rence circle
o
n
the bevel g
ear wheel
(m
m).
Force on the
tooth of the gear with
small
difference [9, 10]
22
2
2
1.2
/
(
)
ta
a
b
F
Tz
d
z
(4)
The bea
rin
g
of internal be
veloid toothin
g
exc
entri
c
shaft with smal
l difference is installed
between the
external gear and t
he excentric
shaft. The layout
of
the output
mechani
sm shall
also
be con
s
idere
d
for th
e external
g
ear. Th
er
efo
r
e the dime
n
s
ion
of the e
x
centri
c shaft is
rest
ricte
d
to a ce
rtain de
gree. As exp
e
rien
ce
p
r
ov
es, the servi
c
e life of the
bearin
g of the
excentri
c shaft is the key factor to influence t
he
carryi
ng capability of this kind of
transmi
ssion.
The force dia
g
ram of the e
x
ternal gea
r is sh
own in Figure 2. Th
e normal force a
c
ting o
n
the external fear by the internal g
e
a
r
is
F. As it is double ecce
ntric,
theoretically,
2
/(
2
)
a
F
Tr
.
Figure 2. Forced e
s
tate of external ge
ar
Con
s
id
erin
g the uneve
n
force, it shall be
cal
c
ulate
d
by the followin
g
formul
a
2
0.6
/
a
FT
r
(5)
In the formula,
2
a
r
——Ra
d
iu
s of the refere
n
c
e ci
rcl
e
on the external b
e
veloid gea
r.
F
can be
resol
v
ed into
x
F
and
y
F
.
2
0.6
c
os
/
x
FT
r
ta
n
yx
FF
(6)
The m
a
ximu
m re
sulta
n
t fo
rce
a
c
ting
on
external
gea
r from va
riou
s
excentri
c
sh
a
fts is
as
below [9]:
ma
x
()
2
.
4
/
(
s
i
n
)
iW
W
W
QT
R
Z
Z
(7)
In the formula,
W
R
——Ra
d
iu
s of pin hole di
stributio
n ci
rcl
e
(mm);
W
Z
——Numbe
r
of the pin hol
es.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 10, Octobe
r 2013 : 572
5 –
5733
5728
By force poly
gon, the force
acting on the
external gea
r
1
R
is as belo
w
:
2
2
max
()
Rx
y
FF
Q
F
(8)
4. Anal
y
s
is o
n
Strength of Shaft and
Calculation
Safety coefficient che
cki
n
g
cal
c
ulatio
n
for
shaft in
clud
es the f
o
llowin
g
two
aspe
cts:
Fatigue
stren
g
th safety co
efficient
che
c
k
and
the
st
atic stre
ngth safety coefficient che
ck. T
he
fatigue stren
g
t
h
safety coef
ficient
i
s
che
c
ked
afte
r prel
iminary
cal
c
ul
ation a
nd
structural d
e
si
g
n
,
in a
c
cord
an
ce with
the f
a
ctors such
as
actu
al di
mensi
on, b
e
nding
mom
e
nt borne, tu
rning
moment dia
g
r
am an
d co
n
s
ide
r
ing fa
ctors
su
ch
a
s
st
re
ss
con
c
ent
rat
i
on,
surface state, si
ze
influen
ce a
n
d
the fatigue li
mit of the sh
aft materi
al
s f
o
r
cal
c
ulating
the fatigue
safety coeffici
ent
of dan
gerou
s se
ction
s
o
n
the shaft to see
whet
he
r t
he requi
rem
e
nts a
r
e m
e
t o
r
not.
Che
c
ki
ng
the safety co
efficient of th
e stati
c
stren
g
th is
to
cal
c
ulate
the
sa
fety coefficie
n
t of the
stat
ic
stren
g
th o
n
the da
nge
rou
s
se
ctions of
the shaft
in accordan
ce
with
yield strength
of
th
e shaft
materials
and the maximum trans
i
ent load ac
ting on the s
h
aft.
The formula
for che
cki
ng
and
cal
c
ulati
ng the fa
tigu
e strength
sa
fety coefficie
n
t of the
shaft is a
s
bel
ow:
22
/[
]
SS
S
S
S
S
(9)
In the formula,
S
——safety coefficient un
d
e
r actio
n
of bendin
g
mome
nt is con
s
id
ered;
S
——only the
safety co
e
fficient und
e
r
the a
c
tion
of torque
is
c
o
ns
ide
r
ed
;
2
S
——allo
wa
ble safety co
efficient cal
c
ulated in accordan
ce
with
fatigue stre
ng
th.
1
/(
)
m
K
S
1
/(
)
m
K
S
(10
)
The form
ula
for ch
eckin
g
and calculati
ng the st
ati
c
stren
g
th saf
e
ty coefficien
t of the
shaft is a
s
bel
ow
22
ss
s
s
ss
SS
SS
SS
(11
)
In the formula,
S
——only the s
a
fety c
oeffic
i
ent when bent is
c
o
ns
idered;
S
——only the
safety coeffici
ent whe
n
twisted is co
nsi
d
e
r
ed;
S
——allo
wa
ble safety co
ef
ficient calcul
ated in a
c
co
rdan
ce
with
static
yield stren
g
th
.
ma
x
m
a
x
//
ss
s
p
s
SW
M
S
W
T
(12
)
5. Impro
v
em
ent of Particl
e
S
w
arm Optimiz
a
tion
The pa
rticle
swarm
opti
m
ization
(PS
O
)
star
t
s
fro
m
a
set of random
soluti
ons
and
sea
r
che
s
for
the optimal solution
s thro
u
gh the it
erati
on.The corre
s
po
ndin
g
pa
rticle
is call
ed the
individual
pa
rticle
,
p
d
p
. The p
a
rticles will
up
d
a
te thei
r o
w
n
sp
eed
an
d
p
o
sition
s
acco
rding
to th
e
following two
formulas
[11].
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TELKOM
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Analysis o
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iniaturization Decele
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yan
g
Jin)
5729
12
*
*
*(
P
r
)
*
*(
P
r
)
V
V
c
r
an
d
p
Bes
t
es
ent
c
r
a
n
d
g
B
es
t
e
s
e
nt
(13
)
Pr
P
r
es
en
t
e
s
e
nt
V
(14
)
In this
formula,
V
—— the sp
eed of parti
cl
e;
Pr
es
e
n
t
—— the cu
rrent positio
n o
f
particle;
ran
d
—— the ra
nd
om numb
e
r b
e
twee
n [0, 1];
12
,
cc
—— the learning facto
r
, generally spe
a
k
ing;
—— the wei
ghted facto
r
, the value of
which i
s
usuall
y
from 0.1 to
0.9.
According t
o
formula (13
)
and (1
4), th
e position of
particl
e in ne
xt time is determin
e
d
by the
curre
n
t po
sition a
nd the
curre
n
t sp
eed.
T
h
e spee
d det
ermin
e
s the
distan
ce
an
d
the
dire
ction of speed d
e
termi
nes the
hea
d
i
ng directio
n of particl
es.
Acco
rdi
ng to
formula
(13
)
, the
curre
n
t sp
ee
d of pa
rticle
s is d
e
termi
n
ed by th
r
ee factors:
the origin
al spee
d,
the
individ
ual
extreme
pB
e
s
t
an
d
the
global
ex
treme
g
Bes
t
.
The gl
obal extreme
g
Bes
t
i
s
the
optim
al solution
at present. If
the algorith
m
appea
rs the
pr
em
ature co
nverge
nce,
the global extreme
g
Bes
t
must
be the l
o
cal o
p
timal solutio
n
. Com
b
ine
d
with formula
(13), if the
glo
bal extre
m
e i
s
cha
nged
(t
he
mutation o
peration will
be i
n
trodu
ce
d to
the gen
etic
al
gorithm
), the
headi
ng di
re
ction of pa
rticl
e
s
will be
cha
n
g
ed. Then th
e
particl
es
will
enter oth
e
r a
r
ea
s for
sea
r
ch. In the
su
bse
que
nt sea
r
ch
pro
c
e
ss, the
algorith
m
ma
y find the new individu
al e
x
treme
pB
e
s
t
and the global extre
m
e
g
Bes
t
.
Thro
ugh the
cyclin
g, the algorithm
will find the glob
al optimal solut
i
on. This is th
e basi
c
idea t
o
improve the p
a
rticle
swarm
optimizati
on
method
s prop
ose
d
in this p
aper.
Con
s
id
erin
g t
hat the
pa
rticle may fin
d
a
better po
siti
on u
nde
r th
e
influen
ce
of
curre
n
t
g
Bes
t
, so the muta
tion ope
ration
will be d
e
si
g
ned a
s
a
ran
dom op
erato
r
by the new
algorith
m
,
namely,
g
Bes
t
whi
c
h meets the
mutation conditions
w
ill be mutated accordi
ng to the certain
probability
. The cal
c
ul
ation
formula of
can be sho
w
n
as:
22
,(
)
0,
dd
k
andF
gB
est
F
othe
rs
(15
)
In this
formula,
k
sele
cts th
e any val
ue
b
e
twee
n [0.1,
0.3]. The val
u
e of
2
d
is
related to
the a
c
tual
problem
s, which is u
s
ually l
e
ss tha
n
the
maximum
of
2
.
d
F
can
be
set as th
e
theoreti
c
al o
p
timal value. The mini
mu
m situation i
s
co
nsi
dered
here.
2
is the group fitne
ss
varian
ce of p
a
rticle
swarm
s
, the cal
c
ul
at
ion formul
a of which ca
n be
sho
w
n a
s
:
22
1
()
n
ia
v
g
i
FF
F
(16
)
In this
formula,
n
—— the nu
mber of pa
rticles in pa
rticle
swarm
s
;
i
F
—— the fitness of the
i
parti
cle;
av
g
F
—— the cu
rrent averag
e fitness of parti
cle swa
r
m
s
.
The sp
ecifi
c
step
s
of
the mutation ope
ration
of
g
Bes
t
are: assu
ming
that we
have
obtaine
d a se
t of local opti
m
al point
1
X
through e
m
ployi
ng the pa
rticl
e
swarm opti
m
ization.
1
X
sho
u
ld
be transl
a
ted i
n
to
the bi
nary
cod
e
(i.e.
0,
1
strin
g
).
11
2
(,
,
.
.
.
,
)
D
X
xx
x
, in which
i
x
(
1
,
2
,
...
,
iD
) is the
i
position in the
bi
nary exp
r
e
ssi
on. A
fterward
s
, the m
u
tation op
eration
in
geneti
c
alg
o
ri
thm ha
s be
en
employe
d
to
cha
nge
1
X
into
2
X
, namely a probability is
adopted to
rand
omly cha
nge the
bits i
n
1
X
binary
exp
r
essio
n
[12]. Then
th
e corresp
ondi
ng co
ding will
b
e
also
cha
nge
d (from
0 to 1 or from
1
to 0) so
as to obtain a
grou
p of ne
w sol
u
tion
s. After
comp
ari
ng th
e fitness val
u
es
of
1
X
and
2
X
, it is
required t
o
elimin
ate th
e solution
s
which
are
poorer th
an
1
X
and
save the
solutio
n
s
whi
c
h a
r
e m
o
re
excelle
nt tha
n
1
X
. After finis
h
ing the
s
e
t
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number
of genetic
mutation, we
will get the
sol
u
tion
2
X
whi
c
h i
s
more optim
al tha
n
1
X
. Then
it
is necessa
ry
to adopt the parti
cle swarm alg
o
ri
thm to cond
uct the optimization in
the
neigh
borhoo
d
.
6. Applicatio
ns
6.1. The De
termination o
f
Desig
n
Variables
Acco
rdi
ng to the desig
n re
quire
ment
s, the va
riabl
es
can b
e
sho
w
n in Table 1 and there
are a total of eight variabl
e
s
.
Table 1. Para
meters of Micromini
a
turi
zat
i
on De
cel
e
rat
o
r (mm
)
Level Modulus
Teeth numb
e
r
of small gear
Teeth numb
e
r
of large gea
r
Teeth w
i
dth
Fi
rs
t l
e
v
e
l
m
1
Z
a1
Z
b1
b
1
Second level
m
2
Z
a2
Z
b2
b
2
6.2. The De
termination o
f
Desig
n
Co
nstrain
t
s
Acco
rdi
ng to the req
u
ireme
n
ts of
tran
smi
ssi
on ratio of
decelerator
12
1
12
2
(
)
120
0
ba
ab
a
zz
hx
zz
z
(17
)
Due to the li
mitations of t
he si
ze of d
e
c
ele
r
ato
r
, the
size of moto
r sh
ould
be l
e
ss than
82mm
×
12
0m
m×60
mm.
11
1
2
2
()
8
2
s
i
n
0
bb
gx
b
m
z
21
1
2
()
6
0
c
o
s
0
b
gx
b
b
(18
)
In this
formula,
1
b
—— the re
feren
c
e cone
angle of large
bevel gear.
The co
ntact f
a
tigue strengt
h of bevel ge
ar
2
3
11
1
1
()
AV
H
H
t
M
BH
E
L
S
K
H
P
a
KKK
K
F
u
gx
Z
Z
Z
Z
Z
Z
mz
b
u
(19
)
In this
formula,
t
F
—— the tangential force
of meshing g
ear;
u
—— the tran
smissio
n
ratio
of meshing g
ear.
The ben
ding
stren
g
th of the root of beve
l
gear
4
11
()
AV
F
F
t
F
SE
K
J
S
F
P
KK
K
K
F
gx
Y
Y
Y
Y
bm
(20
)
The ben
ding
stren
g
th of the root of thickened g
ear
5
22
()
t
A
V
F
F
FS
FP
F
g
x
KKK
K
Y
Y
bm
(21
)
Con
s
trai
nts o
f
coinci
den
ce
degree of bev
el gear p
a
ir transmi
ssion in
the first level
61
()
1
.
2
0
gx
(22
)
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TELKOM
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ISSN:
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Analysis o
n
the Key Para
m
e
ters of Aerosp
ace Mi
crom
iniaturization Decele
rato
r (Xiang
yan
g
Jin)
5731
In this
formula,
1
—— the coi
n
cid
e
n
c
e de
g
r
ee of bevel g
ear pai
r in the
first level.
The co
nst
r
ain
t
s of the coin
cide
nce deg
ree of
thicken
ed gea
r pai
r in the se
con
d
level
72
2
2
2
2
0
2
2
1
(
)
1
.
1
(
(
)
ta
n
t
a
n
ta
n
s
i
n
/
(
)
0
2
ba
a
a
a
b
a
b
gx
z
z
z
z
b
m
(23
)
In this
formula,
—— the helix angle of thicken
ed ge
a
r
(°
);
—— the eng
aging a
ngle o
f
thicken
ed g
ear pai
r (°
);
20
,
aa
a
b
——the
pressure a
ngle
of
the a
dden
du
m in the
cro
s
s
se
ction i
n
big
end
of thi
cke
ned
extern
al gea
r
and
th
e p
r
e
s
sure
a
ngle
of the
adde
ndum in
the small sect
ion of thicken
ed intern
al ge
ar.
In this
formula, the c
o
rner mark
a
rep
r
ese
n
ts the thicken
ed external g
ear a
nd
b
rep
r
e
s
ent
s th
e thicken
ed i
n
ternal
ge
ar.
2 repres
ent
the cro
s
s se
ction in
big
e
nd of thi
c
ken
e
d
gear an
d 0
re
pre
s
ent
s the
cro
s
s
se
ction
in
small
end.
The val
ue
ca
n be
calculat
ed a
c
cording
to
the followin
g
formul
as:
22
2
2
cos
ar
cc
o
s
a
a
aa
mz
r
22
0
0
co
s
arcc
os
b
ab
ab
mz
r
(24
)
The overl
ap
and interfe
r
e
n
ce
con
s
trai
n
t
s of t
ooth profile of intern
al meshi
ng thicken
ed
gear.T
he inte
rnal me
shi
ng
thickene
d ge
ar pai
r in
de
celerato
r is th
e one
-tooth transmi
ssion
a
nd
the tooth p
r
of
iles a
r
e
ea
sil
y
overlap
ped
and i
n
terf
e
r
e
d
. As the m
o
dification
co
e
fficients of
ea
ch
cro
s
s
se
ction
of thi
c
kene
d
gea
r
present
the li
nea
r ch
ange
s alon
g the
axial
dire
ction, whe
n
t
h
e
overlap
an
d i
n
terferen
ce
constrai
nts
of tooth
profile are esta
blis
h
ed, othe
r cro
s
s section
s
will
certai
nly mee
t
the req
u
ire
m
ents
as l
o
n
g
as the t
ooth
profile
s in th
e cro
ss
se
ctio
ns of
big e
nd
and
small en
d are not overlap
ped. The con
d
itions t
hat constrain the
cro
s
s se
ction
s
in big end
and
small en
d of internal thi
c
ke
ned ge
ar pai
r to
be overlap
ped an
d interf
ered a
r
e resp
ectively:
82
0
0
2
0
0
2
2
()
(
)
(
)
(
)
0
a
a
aa
b
b
ab
b
a
g
x
z
i
nv
z
i
nv
z
z
i
n
v
(25
)
In this
formula,
22
2
00
0
0
4
ar
cc
o
s
(
)
4
ab
aa
a
aa
dd
a
ad
22
2
00
0
0
4
ar
c
c
o
s
(
)
4
ab
aa
b
ab
dd
a
ad
a
—— the
cent
er
distan
ce
of
thicken
ed
ge
ar
pair,
whi
c
h
ca
n b
e
cal
c
u
l
ated by foll
o
w
ing
formula
22
2
()
co
s
2c
o
s
ba
mz
z
a
(26
)
92
2
2
2
2
2
2
2
()
(
)
(
)
(
)
0
aa
a
a
b
b
a
b
b
a
g
x
z
i
nv
z
i
nv
z
z
i
n
v
(27
)
In this
formula,
2
a
and
2
b
can take
cal
c
ulatio
n formula of
0
a
and
0
b
as
referenc
e.
The Matlab l
angu
age p
r
o
g
rammi
ng ha
s been e
m
pl
oyed and th
e runni
ng re
sults in
comp
uters ca
n be sh
own in Table 2.
Table 2. Re
sults of Optimi
zation in Mi
crominiatu
rization De
cel
e
rat
o
r (mm
)
Level Modulus
Teeth numb
e
r
of small gear
Teeth numb
e
r
of large gea
r
Teeth w
i
dth
Fi
rs
t l
e
v
e
l
m
1
=1 Z
a1
=18
Z
b1
=62
b
1
=9
Second level
m
2
=1.5
Z
a2
=35
Z
b2
=36
b
2
=12
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r 2013 : 572
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5733
5732
7. Finite Element An
aly
s
is on Bev
e
lo
id Gear
Duri
ng th
e g
ear'
s
me
shin
g p
r
ocess, t
he
stre
ss di
stribution va
ri
es
alon
g the
co
ntact
points.
Classi
cal ge
ar d
e
si
gn t
heo
ry ap
proximate
s
t
he dist
ribut
e
d
load a
s
co
nce
n
trated fo
rce
applie
d to the refere
nce circle, with som
e
errors in
th
e cal
c
ulatio
n. Calculation
by finite element
method
ca
n e
ffectively sim
u
late the
real
meshi
ng
process of th
e g
e
a
r
and
ob
se
rve the
cha
ngi
ng
con
d
ition
s
of
variou
s
stre
sse
s
. Cla
s
sica
l gea
r an
alysi
s
meth
od
su
ppo
se
s the g
ear to
be a
ri
gid
body, without
con
s
ide
r
ing
the influence
of the
gear deform
a
tion on the conta
c
t force o
n
gear
surfa
c
e
an
d t
he ove
r
lap
ra
tio of the g
e
a
r
pai
r [13
]. Fi
nite elem
ent,
however, trea
ts the g
e
a
r
a
s
a
flexible pie
c
e
.
In this th
esis, LS-DYNA
finite
elem
e
n
t analy
s
is software
is ad
opted fo
r fini
te
element
sim
u
lation an
alysis
on the m
e
shi
ng p
r
o
c
e
ss
of the ge
ar, with the
aim of analy
z
ing
distrib
u
tion of
its be
nding
stre
ss an
d contact
st
re
ss. The m
a
ximu
m co
ntact
stress of the
ge
ar
shall
be, at a
n
y time, on t
he me
shi
ng
positio
n of th
e gea
r tooth
and the
corre
s
po
ndin
g
st
re
ss
con
c
e
n
tration
is produ
ce
d
on the ro
ot of the gea
r tooth. The di
stributi
on of transi
ent co
nta
c
t
stre
ss at the root of a si
ngle
tooth is sho
w
n in Fi
gure
3.Ju
dgi
ng
by the eq
uivalent stress in
Figure 3, a
con
c
lu
sio
n
can be made
that the
contact stre
ss on tooth surf
ace is of lin
ear
distrib
u
tion
a
nd the
stre
ss on
both
si
de
s i
s
a little
g
r
eater. In
ge
n
e
ral, th
e
stre
ss di
strib
u
tio
n
is
even. The
a
nalyze
d
stre
ss
re
su
lt
sh
ows that the
gear de
sign
ed in thi
s
th
esi
s
meet
s t
h
e
requi
rem
ents for conta
c
t strength. Besi
des, it can
al
so be o
b
serv
ed that the overlap ratio of the
gear p
a
ir is greater tha
n
1.2
thro
ug
h the
me
shi
ng situation
of
tw
o
adj
acent teeth.
T
he
distrib
u
tion of
transie
nt con
t
act stre
ss at the
root of a single tooth is
sho
w
n in Fig
u
re 4.
Figure 3. Con
t
act stre
ss of external beve
l
oid
gear
Figure 4. Con
t
act stre
ss of tooth root in
external beve
l
oid gea
r
8. Conclusio
n
After analyzi
ng and re
se
arching the
decelerat
ors,
transmi
ssion
ratio dedu
cti
on, force
cal
c
ulatio
n and stre
ngth
analysi
s
for
decelerat
ors are do
ne to lay the fou
ndation for t
he
stru
ctural de
sign a
nd pro
v
ide t
heoreti
c
al ba
si
s for virtual proto
t
ype model
establi
s
hi
ng
and
comp
uter sim
u
lation.
The p
a
rticl
e
swarm optimi
z
ation an
d the
genet
i
c
alg
o
ri
thm have b
e
e
n
co
mbin
ed a
nd the
mutation
ope
rator optimi
z
a
t
ion mo
del
ha
s b
een
p
r
op
o
s
ed.
Thi
s
mo
del h
a
s solve
d
the
proble
m
s
of particle
swarm optimi
z
at
ion in mecha
n
ical de
sig
n
. For exampl
e, there a
r
e more variable
s
, it is
con
s
trai
ned a
nd it is easily pre
c
o
c
iou
s
. The impr
ove
d
particl
e swarm optimizatio
n has be
en al
so
su
ccessfully applie
d in the
optimum de
si
gn of microm
iniaturi
zation
decelerator.
Acco
rdi
ng to
the optimization re
sult
s,
t
he t
r
an
smi
ssi
o
n
rat
i
o of
d
e
c
e
lerat
i
o
n
sy
st
em,
t
h
e
torque a
nd fo
rce
suffered b
y
the compon
ents, the
tran
smissio
n
ratio
of whole sy
stem have bee
n
obtaine
d, whi
c
h ha
s provid
ed t
he ba
sis f
o
r the virtual
prototype mo
deling of de
celerato
r.
Ackn
o
w
l
e
dg
ements
This wo
rk
w
a
s sup
p
o
r
t
e
d
by
Scie
ntific Re
se
arch
Fund
of Heilongjia
ng P
r
ovinci
al
Educatio
n De
partme
n
t (No
.
1252
CGZ
H
18), T
w
elfth
Fi
ve-year Pla
n
Issue
s
for
Heilo
ngjian
g
High
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Analysis o
n
the Key Para
m
e
ters of Aerosp
ace Mi
crom
iniaturization Decele
rato
r (Xiang
yan
g
Jin)
5733
Educatio
n S
c
ientific
Re
sea
r
ch
(No.
HG
JXHB
211
079
2
)
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the
Yo
uth Scie
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