the stress analysis and optimal design of differential

KAERl/CR-99/2000RN1OI

The Stress Analysis and Optimal Design of

Differential Planetary Reducer

DISCLAIMER

Portions of this document may be illegiblein electronic image products. Images areproduced from the best available originaldocument.

KAERI/CR-99/2000

The Stress Analysis and Optimal Design of

Differential Planetary Reducer

1

2000. 11.20.

6--

-1-

- Ill -

SUMMARY

I . Project Title

The stress analysis and optimal design of differential planetary reducer.

II. Objective and Necessity

The objective of this project is to make a optimal design through the stress,

analysis. Newly designed differential planetary gear have many features comparing

with conventional reduction gears with multiple gear trains in order to get a high

reduction ratio. Developing gears

with small seze. This reducer of

in one stage. This light weight,

are able to get a high efficiency and manufactured

planetary type is able to transmit high load torque

high efficiency differential planetary reducer, as a

new attempt of planetary reducer type, can obtain a high reduction ratio with the

simple mechanism which is impossible with the traditional planetary reducer type.

It has many advantages comparing with harmonic, RV, epi-cycle reducer in

technically and cost effective for high reduction ratio, compactness, and hightorque.

Japenese company has made success on manufacturing the high precision gears like

harmonic, RV, epi-cycle, and planetary gears, which were originally invented in

United State or Europe countries and generally applied to precise control system.

And nowdadys they share the large portion of the market in the world.

The developing differential planetary type reducer is possible to get a gear ratio

over thousands. of reduction ratio, hence it can be applied to the various industry

fields such as nuclear robotics in hostile work environments, extreme precision

control for radar of military industry, and iron industry, mine industry.

-lv -

III. Scopes and Contents

The scopes of this project are to make a optimal design and to carry out stress

analysis. We tarred out state-of-the-art by sui-veying the related patent and

technical document, and performed stress analysys to minirnite the weight. The

efficiency and applicable torque range is computed through the kinematic and

dynamic simulation. The detailed manufacturing drawing is generated through the

3-D modeling with the aids of stress analysis on the conceptually designed reducer.

IV. Results of the Project

The reducer is made of a single input/output planetary gear train, and inner gear

corresponding to each planetary is located at the outside to get high reduction ratio.

This simple and compact mechanism can perform high torque reduction. Also it

utilizes the standard involute teeth style for the easy manufacturing. Reducers can

obtain the reduction ratio from 300:1 to super reduction ratio.

V. Applications and Future Plans

The design data and mechanism developed in this project will be used in main

project for manufacturing to commercial products. The developing differential

planetary reducer can be utilized not only in general industrial machinary but in the

welding robot, construction work automation and robot actuator for hostile

environment.

Also these can be

precision control. The

eventually replace the

used for the military industry which require high torque,

development of compact, high ratio, high torque reducer will

imported high ratio reducer with this reducer.

-v-

CONTENTS

Chapter 1. Introduction

Section 1.

Section 2.

Section 3.

Necessity of research ................................................................................. 1

Objectives and contents of research ..................................................... 2

Economical, social, technical importance of research ...................... 3

Chapter 2. Contents and results of research

Section

Section

1.

2.

Section

1.

2.

3.

4.

Section

1.

2.

State-of-the-arts and technical results for

differential planetary rnducer .................................................................. 5

The theory and finite element analysis of planetary reducer 9......

Major specifications of planetary reducer ................................................. 9

Tooth equation of differential planetary reducer .................................... 11

3. Theoretical equation of planetary reducer ......................................... 14

Pressure angle for involute tooth profile .............................................. 14

Ratio automatic conclusion program of reducer ................................. 16

Conclusion of bending stress ...................................................................... 18

Conclusion Of contact stress ....................................................................... 32

4; Finite element analysis ............................................................................ 38

-vl -

Section 5. The results and discussion .................................................................... 40

1. Test results ......................................................................................................- 40

2. Analysis results ............................................................................................... 43

3. Discussion .......................................................................................................... 51

Chapter 3. Conclusions ................................................................................................. 53

Chapter 4. Application plan of developed results .................................. 54

Chapter 5. References .................................................................................................... 55

- WI-

1

2

al 1 X4 q? 7] %L+ +@A~ .......................................................................................... 1

X12397+42+* ............................................................................................ 2

41 3 3 9=t711%~Ql %%] “ ~}q “ 71e4 5-Q’% ..................................................... 3

X*}

xl

7J

1

2

3

4

2 71+33= ............................... 5++4 ZR71

e i3-4j71q &l*7]q “1%xl ................................... ............................................... 9

1. %-%7] q %+71 Q1 7% 34 %s- g 7+ 71AM ............................................ 9

2.3+5- ++A~7] ~ + ~+ xl+}+ ...........................................................................

X3 -8471 q 7uRi71 44 “1%’4 ...................................................................

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32

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2. 24 =]4 ........................................................................................................... 51

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a13’$z3~ ...................................................................................................................... 54

a] 4 %} 9771193qq g’#71]q ........................................................................... 55

q 5 x$ ~~zs~ ................................................................................................................. 56

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of pitch

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N~ NA

,-—%-9---$?—-—--- 7

i -—-—&?-————”—82

SEtmn&#+

(2-1)

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k= Wt k, kB km6~= k

v Fm -rfi(3-1)

-14-

= 974x ~“$o = 0.1948Kgf~9

Poweyw~=l, ooo ~t =1, 000x 2°& = 203.749N.

(3-3)

(3-4)

W~= 203.749/3=67.916N (3-5)

Kv=( A )B=( 70.72 )o-~=o.~z (3-7)A + {~t 70.72+4200x2.4%

-15-

q71~l,

A=50+56(1.() -B)=50+56(1.0 -0.63) =70.72 (3-7a)

B– (12– QV) 0.&7—(12–8)0”&7 so 63

4 “= 4. (3-7b)

k. Wt k. kB km(J*=kv Fwz J*

= 1.5x67.916 1X1 115

0.842 22x1.25 ~

= 20.238A4Pa (3-8)

-16-

(3-9)

2500vt=9r D2 n@/60000=(zx27x 4.733 J160000= 0.748wJs (3-lo)

Power =974 x (4.~3~00 “5) =0. 922 Kgfm71=3)=974 ~* (3-11)

9

Powey =1, ()()0X o“~~ = 668.449Nw~=l, ooo Vt.

(3-12)

W~= 668.449/3 = 222.816N (3-13)

F=12m=12X1.5=18mm (3-14)

KV=(-A+ fioou?’=(

70.7270.72 +d200x0.748

)o-~=o&3 (3-15)

-17-

= 35.941MPa (3-16)

2$00 )/60, 000= O.165m/svt=z D3 YzJ60,000=(zx72x

z7E=)=974 ‘owey =974x*= 11.1 Kgfmnb 9

PoweyWt=l, ooo ~t ‘1,000x o“i~5 =3,030.3N

(3-17)

(3-18)

(3-19)

wt=3,030.303/3 =l,010.1N (3-20)

-18-

F= 12m= 12x I.5= 18mm (3-21)

&=(*) ’=(_ ) 0.~=o.95 (3-22)70.72 +d200x0.165

= 1.5 X1010.1O1 1X1 106

0.952 22x1.5 ~ I= 131.083MPa (3-23)

2i5~0)/60, 000= 2.454m/sVt=z D4 nO/60,000= (ZX26.25X (3-24).

271ZZ) =974 ‘“y =974X ‘1 f~$5) =().273 Kgfm (3-25)Y

PowerWt=l; ooo Vt =1,000X 0.5

2.454= 203.749N (3-26)

-19-

W ~= 203.749/3 = 67.916N

= 105x67.916 1X1 1130.842 22x1.25 =

=16 .037MPa

(3-27)

(3-28)

(3-29)

(3-30)

-20-

2500 )/60, 000= 0.0346dsvt=?rDs nO/60,000= (zx71.25x 269 8 (3-31).

~E=)= 974 ‘O? =974X ‘26~::”5) =52.557 Kgfm (3-32)9

PoweT 0.5w~=l, ooo =1, OOOxo.0346Vt =14,450 .867N (3-33)

Wt= 14,450.867/3 = 4,816 .956N (3-34)

F=12m=12xl.25 =15mm (3-351

K.=(*) ’=(70.72

70.72 +~200x0.0346)o.~=o,964 (3-36)

= 1,5x4816.956 1X1 1.050.964 22X1.25 ().395

= 724.513MPa (3-37)

-21-

.../..\.-..~..... i} ; I

-tiiati”--””:fi‘-””-”f”-H f’ i i I 1

,>w

2%3.3 % q 21+ (Dynamic factor) C,+ K,

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a% 3.4 %!%~1 21+(Rim thickness factors) KB

(3-38)

-23-

I

~% 3.5 %%21+( Geometry factor) J (20° spur gear : standard addendum)

-24-

x3.3 ~%% 21+( Suggested application factors) &

Driven Machine

Light Moderate HeavyPower Source uniform

Shock Shock Shock

Uniform 1.00 1.25 1.50 L75

Light shock 1.20 1.40 1.75 2.25

Moderate shock 1.30 1.70 2.00 2.75

X3.4 =71 21+( Suggested size factors) KS

Diametral I Metric I Size factor

Pitch, Pd Module, m KS

>5 55 1.00

4 6 1.05

3 I 8 I 1.15

2 I 12 I 1.25

1;25 I 20 I 1.40

-25-

w, c. c, cm Cf(s== CD

C. dF I(3-39)

‘= ( Yn; :l.o) (3-39a)

-27-

2C 2X22.5 –~8.~5mm‘= ( mG+l.O) = (1.4+1.0) –

(3-40)

(3-40a)

~= ( Nlm)+( N4m) = (15x1.25)+(21x1.252 2

) =22.5mm (3-40b)

= 191 67.916x 1.5 1 1.15X10.842 18.75x22 0.082

= 387.382MPa (3-41)

2C 2x22.5

‘= ( w~+l.o) = (2.714 +1. o) ‘~2.~16~m(3-42)

w~ Ca c, c. Cf(7C =.C*

CV dF I

= 191 4,816 .956x1.5 1.05X10.964 12.1:6x22 0.11

=3, 129.206MPa

-29-

(3-43)

S3.5 @~# Al+(Elastic coefficient) CP

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\’& ---+--sI —--l J—~——-4

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D-m Uls,,.29?-=..,

,,“*unm x..<.,.”

—<—....—

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~~@oJ X3 ~ +22= (“C)~

A] 7L}(~)(v) (A) Z Ej 7il0] k 71 q

o 2.52 31.2 30.4 32.2

5 2.40 33.4 32.0 33.7

10 2.33 37.2 34.0 35.5

15 2.26 40.0 36.7 38.3

20 2.20 43.7 38.7 40.7

25 2.16 45.6 40.6 42.6

30 80.3 2.17 48.0 41.8 43.7

35 2.10 50.9 42.6 45.0

40 2.01 53.4 45.8 47.1

45 2.00 54.5 46.3 49.8

50 1.94 55.7 47.0 50.8

55 1.92 55.8 48.7 51.3

60 2.01 56.7 47.9 51.3

-36-

‘a$l N1 N4 zv~ N2 N3

E 3(T) K’&f”m 0.1948 0.273 ~52.55? 0.922 11.1036

~JW%( Wt ) N 67.916 67.916 4816.956: 222.816 1010.101

%9%+( db ) MPa 20.238 16.037 724.513 35.941 131.083AGMA

F?”J%q( 0, ) MPa 387.382 - 9129.ZW’ - -

~g+iq( ~b ) MPa 16.812 7.363 ..,?106/’”, 42.741 123.938ANSYS ,., -.:’-,}, , :

~“$%q( a. ) MPa 347.762 - ;’:..2O3V:“;;, - -,,.. :,,$

-38-

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-39-

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endina Strss

NisYs 5.5.35P3CT 18 200021:00:04MODAL SOLUTIC?JSTEP=1SUB =1TIEE=lsmT (AVG)

PowerGraphicsEFACET=lIlvREs=EazDEX ‘.443E-03Sm =.114595sux =16.812

m ;;?95m 3.825m 5.rj8m 7.s36m 9.391D 110246

n 13.102m 14.957_ 16.812

II

3ending Strss -

ANSYS 5.5.3SPOCT 18 200021:00:43IJODILSOLUTIONSTEP-lSUB =1T123E=1SINT (AVG)

PowerGraphicsEFACET=lAVRES-23atDEX = .00132Sm =.144489SMX =42 .741

-40-

I

STEP-1SUB =1TIME=lSINT (AVG)PowerGraphicsEF&CET-1AVRSS=SatDMX=.002361SEN ‘.041656SSX =123.938

F

31FORRFOX

:ending Strss I

I

ending Strss

ANSYS 5.5.3SPOCT 18 200021:02:21NODAL SOLUTIONSTEP=lSUB =1T131E=1SINT (AVG)PowerGraphicsEFACET-1AVRES=3iaZDEX = .186E-03S31N=.003726SEX =7.363

-41-

TIME= 1

S2NT (AVG)

PoverGraphics

EFACET=l

MWEs=uat

DMX ‘.015893

SMN ‘.002457

SMX =1064.>.F

NFCX

RF*OR

—

Bending Strss I

-42-

:oncact stx

ANSYS 5.5.3SPOCT 25 200017:42:30NODAL SOLOTIONSTEP-2sUB =4STI?IE=.001SSEQV (AvG)

PowerGraphicsEFACET-lAVRES=HatDMX ‘.744E-08SMN -1.767SIIX=347.762

PRK5.-95.117

II

:ontact stcess

ANSYS S.S.3SPOCT 2S 200017:43:39NODAL SOLUTICNSTEP-2SUB =45TICIE=.0015SEW (AVG)PowerGraphicsEFkCET-1AVRES=lIatD13X‘.744E-08SMN =1.767SEX ‘347.762

-43-

I ANSYS 5.5.3SPOCT 29 200022:19:12

t+ 11. ?$ NODAL SOLUTIONSTEP-4Sus =1sTIHE=2 .002SINT (AVG)PowerGraphicsEFACET=lAVRES=XatDMX = .046454SMN =.022741SMX =2031

NFOR~~oK

PRES-1204

;:\Downloads\ N4_NS. igs

I

AIJSYS 5.5.3SPOCT 29 200022:18:40NODAL SOLUTIONSTEP-4SU8 =15TIEE=2 .002SINT (AVG)PowerGraphicsEFACET=lkvREs-nacDHX = .0464S4SMN =.022741SE2 =2031

;:\Downloads\ N4_N5. igs

I

-44-

ol~-q +-@Q&5~~SCM415 ~%% ~

(Ns) (ANSYS)

Bending stress 724.513 1,064 1,427

Contact stress 3,129.206 2,031 —

-45-

[11

[21

[31

[41

[51

[61

[71

[81

[91

M.E.Stegemiller and D.R.Heuser, “A Three-Dimensional Analysis of

Base Flexibility of Gear Teeth,” Journal of Mechanical Design, Vol.

March 1993.

E.Yau, H.R.Busby and D.R.Heuser, “A Rayleigh-Ritz Approach to

Modeling Bending and Shear Deflections of Gear Teeth,” Computers

Structures Vol. 50, No. 5, pp. 705-713, 1994.

the

115,

&

Hsiang Hsi Lin, Chinwai Lee, F.B.Oswald and D.P.Townsend,’’Computer-Aided

Design of High Contact-Ratio Gears for Minimum Dynamic Load and Stress,”

Journal of Mechanical Desing, Vol. l15,March 1993.

Aizoh KUBO, Takashi KUBOKI and Teysuya NONAKA, “Estimation of

Transmission Error of Cylindrical Involute Getis by Tooth Contact Pattern,” “

JSME International Journal, Series III, Vol. 34, No. 2, 1991.

Robert L.Mott, P.E. , Machine Elements in Mechanical Design Macmillan.

Publishing Compang, Second edition,1992

Eliahu Zahavi, The Finite Element Method in Machine Design A Simon &

Schuster Englewood Cliffs, 1992.

Yeon-Su Kim and Sang-Hoon Choi, “Interference and Efficiency Analysis of

2K-H I Type Differential Gear Unit,” International Journal of the Korean

Society of Precision Engineering, Vol. I, No. I, June 2000.

W. Mierzejewski, T. Szopa, “Loads of Planet Wheels in Planetary Gears,”

Journal of Mechanical Design, Vol. 115, December 1993.

ADEL K. A1-Sabeeh, “Irregular Gears for Cyclic Speed Variation,” Mech.

Mach. Theory Vol. 26, No. 2, pp. 171-183, 1991.

-48-

[10]

[111

[12]

[131

[14]

[151

[161

[171

[181

[191

S.VIGAY-GAN and N.GANESAN, “Stress Analysis of Composite Spur

Gear Using the Finite Element Approach,” Computer & Structures, Vol. 46,

No. 5, pp. 869-875, 1993.

Akira YOSHIDA, Komei FUflTA, Kiichi MIYANISHI, Kenji HIGASHI, Yuji

OHUE and Yen LIU, “Effect of Standard Pressure Angle on the Fatigue

Strength of Nitrided Gears,” JSME International Journal, Series III, Vol. 31,

-49-

Appendix A %ks P] al t} program.

void CGearLoadDlg::BendingStress(double m_Nl, double m_N2,double m_N3,double

m_N4, double m_N5,

double m_MOTORKW, double m_MOTORRPM, double m_MODULE,

double m_CP, double m_CS, double m_I)

{ //m_i5 ‘?l q q m_CS~M~}%q ~1, m_CP% @A24 *

double b=O, N=O, ReduceRate=O, np=O, preX=O, PO=O;

ReduceRate=(l+(m_N3/m_Nl ))/( l-(m_N4*m_N3)/(m_N5*m_N2));

N=m_MOTORRPM/ReduceRate;

T=974*m_MOTORKW/’N;

Dg=m_N5*m_MODULE;

Dp=m_N4*m.MODULE;

np=m_Nl/m_N2*m_MOTORRPM;

Vt=3.14159*Dp*np/60000;

// if(Qv==O){

/* }

else{

Qv=m_Qv;

}*/

b=12*m_MODULE;

Kv= sqrt(78/(78+sqrt(Vt) ));

Wt=(2*T)/(3*Dg*0.001 *Kv)*9.81;//Ftq a

preX=Wt+Wt*tan( 20);

po=(3.14*( l-m_CS*m_CS)/m_CP+3 .14*( l-m_CS*m_CS)/m_CP);

SigmaB=sqrt(preX/(b*Dp)* (1/po)*(2/sin(m_I)* (l+Dp/Dg)));

UpdateData(FALSE);

m_Bresult=SigmaB;

OnInitDialogo;

}

void CGearLoadDlg”: :OnBendingo

{

BendingStress( m_Nl, m_N2, m_N3, m_N4, m–N5,

m_MOTORKW, m_MOTORRPM, m_MODULE,

m_CP, m_CS, m_I);

}

void CGearLoadDlg::Contact( double m_Nl, double m_N2,double m_N3,double m_N4,

double m_N5,

double m_MOTORKW, double m_MOTORRPM,

{

double rl=O, r_d=O, r_a=O, h=O, t=O, TT=O;

double m_MODULE)

double N=O, ReduceRate=O, np=O, preL=O, preSigmaC=O;

ReduceRate=(l+( m_N3/m_Nl ))/(1-(m_N4*m_N3 )/(m_N5*m_N2 ));

N=m_MOTORRPM/ReduceRate;

T=974*m_MOTORKW/’N;

Dg=m_N5*m_MODULE;

Dp=m_N4*m_MODULE;

np=m_Nl/m_N2*m_MOTORRPM;

Vt=3.14159*Dp*np/60000;

rl=m_N4*m_MODULE/2;

r_a=rl+m_MODULE;

r_d=rl–l.25*m_MODULE;

b=12*m_MODULE;

preL=(r_a-r_d)/2 ;//7] ~ Q1 %?l =01 ~= ~~=~.

Kv= sqrt(78/(78+sqrt(Vt) ));

Wt=(2*T)/(3*Dg*0.001 *Kv)*9.81;/@t~~ 21t}

// preSigmaC=T/(Kv *Dg/2);

// “if (preSigmaC<O){

// preSigmaC=preSigmaC* (’1 );

// }

// else

1/ preSigmaC=preSigmaC*l;

TT=(2*0.854)+(360/(2*m_N4));

t=(2*r_d)+sin(TT/2);

SigmaC=l.6*Wt*preL/(b*t*t/6);

UpdateData(FALSE);

m_Cresult=SigmaC;

OnInitDialogo;

}

void CGearLoadDlg: :OnContact( )

{

Contact( m_Nl, m_N2, m_N3, m–N4, m_N5,

m_MOTORKW, m_MOTORRPM, m_MODULE);

}

/prep7

/title,Bending Strength

m=l.25

N1=15

rl=m*N1/2

Ratio=21/15

rpm=2500/Ratio

T=974*0.5/rpm

force= (T*looo)/(3*rl) *9.81

! the number of tooth

! the radius of pitch circle

! the ratio of gear

! Torque

! Force

et,l,plane42

KEYOPT,l,l,O

KEYOPT,1,2,0

KEYOPT,1,3,3

KEYOPT,1,5,0

KEYOPT,1,6,0

mp,ex,l,191e6

mp,nuxy, 1,.3

r,l,12*m

*afin,deg

alpha=20 ! pressure angle

ne=9

xx=cos(alpha)

ra=rl+m

rd=rl–1.25*m

rb=rl*xx

r=rd–2.25*m

thetal =tan(alpha) *360/6.28-alpha

thick= 2*thetal+360/(2*Nl )

th=360/Nl

Csys,l

local,ll,l,0,0,0,90

n,l,r

n,6,rd

fill

n,ll,rb

n,20,ra

fill

fill,6,11,4

ngen,2,80,1,20, l,O,thick

fill,l,81,3,21,20,20,1

ngen,2,60,81,86, l,O,th–thick

fill,81,141,2,101,20,6,1

ngen,2,158,1,6,1,0,thick-th

fill,l,159,2,147,20,6.1

the number of element

the radius of addendum

the radius of deddendum

the radius of basic circle

thickness of rim

polar angle

the tooth thicness angle

Xdo,i,l,g,l

yy=(ra-rb)/ne

zz=rb+i*yy

w=(rl/zz)*xx

phi=acos(w)

theta2=tan(phi) *360/6.28-phi

theta3=thick-theta2

n,i+ll,zz,theta2

fill,i+ll,i+51,1,i+31

n,i+91,zz,theta3

fdl,i+51,i+91,1,i+71

*enddo

/pnum,node,l

e,l,2,22,21

egen,19,1,1

egen,4,20,1,19

egen,4,20,58,62

e,l,2,148,147

e,2,3,149,148

e,3,4,150,149

e,4,5,151,150

e,5,6,152,151

e,147,148,168,167

e,148,149,169,168

e,149,150,170,169

e,150,151,171,170

e,151,152,172,171

e,167,168,160,159

e,168,169,161,160

e,169,170,162,161

e,170,171,163,162

e,171,172,164,163

d,146,all,all

d,145,all,all

d,144,all,all

d,143,all,all

d,142,all,all

d,141,a11,a11

d,121,all,all

d,lOl,all,all

d,81,all,all

d,61,all,all

d,41,all,all

d,21,all,all

d,l,all,all

d,147,all,all

d,167,all,all

d,159,all,all

d,160,all,all

d,161,all,all

I d,162,all,all

d,163,all,all

d,164,a11,all

nrotat,all

f,60,fy,force

/solu

solve

/postl

plnsol,s,int,2

I 1

!

I

BIBLIOGRAPHIC INFORMATION SHEET

Performing Org. sponsoring org.StarndardReport No. INIS Subject Ccxie

Report No. Report No.

KAERI/cR-99/2ooo

Title / Subtitle

The Developmentof Radiation-HardenedRobotfor NuclearFacility

ProjectManagerKim SeungHo (QuantumOpticsDevelopmentTeam)

and Department

Researcher and IDepartment I

Seung ho Jung(”),Chang Hoi Kim(”),Yong Chil See(”)

PublicationPublisher

Publication2000. 12

Place Date

Page

Note

Classified

57 p. 111.& Tab. Yes(O ), NO ( ) SizeCm.

Open( O ), Rcstictd( ),

_ Class DocumentI

Report Type Research ReportI

Contract No. I

The objective of this project is to make a optimal design of differentialplanetar

reducer through the stress analysis. The developed gears are able a high efficienqand manufacturedwith small size. This reducer of planetarytype k able to transm~high rode torque in one stage. This light weight, high efficiency differentialplanetar

reducer,as a new attemptof planetaryreducertype, can obtain a high reductionratiwith the simple mechanism which is impossible with the traditionalplanetaryreducetype.

Subject Keywords I(About 10 words)

Reducer, Planetary reducer, Differential reducer, High reducer

the stress analysis and optimal design of differential

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