Malhotra - Analysis of a Cycloid Speed Reducer

Mechanism and Machine Theory Vol. 18, No. 6, pp. 491-499, 1983

0094-114X/83 $3.00+ .00 © 1983 Pergamon Press Ltd.

Printed in Great Britain.

ANALYSIS

OF A CYCLOID

SPEED REDUCER

S. K. M A L H O T R A t and M. A. PARAMESWARAN~; Indian Institute of Technology, Madras-600036, India (Received for publication 22 February 1983) Abstract--The cycloid speed reducer has the advantages of compactness, large ratios and high efficiency. Very little published information is available on its analysis and design. In this paper, a procedure to calculate the forces on various elements of the speed reducer as well as the theoretical efficiency is presented. Also the effects of design parameters otI forces and contact stresses are studied which will aid optimal design of this type of speed reducer.

INTRODUCTION

The cycloid speed reducer[l, 2] has been in use for several years. Compactness, large ratios (up to 90/I in single stage) and high efficiency are the advantages of this type of reducer. For this reducer, the input member is an eccentric shaft (Fig. I) on which the disc with n epitrochoidal lobes is mounted. The lobes are in contact with (n + I) rollers mounted on fixed pins. For each rotation of the input shaft the disc rotates by one lobe pitch in opposite direction, in other words for n rotations of input shaft the disc makes one revolution. The output is taken concentric with the input shaft by a set of output rollers which transmit only the rotation of the disc by rolling internally on the circumference of holes on the disc. The geometry and principle of operation are quite well known[l]. ° The following geometrical relations exist among the various dimensions. Some of these are represented for a 4 lobe arrangement in Fig. 2. R

-=n

(1)

r, = ne

(2)

r b = r~ + e

(3)

r

ct, = tan

11-sin nO + Rblr b sin ~ ] Lc~-s~--0-~

~]

(6) (7)

li = ra sin (~ti -- nO)

(R + r) r ' + e 2 - 2er

costr ~iJ]

Ir3+e2(R-I-r)-er(R-f-

2r) cos ( R 7,)1

(8)

~=R+r

(4)

D h = Dq + 2e

(5)

i'Senior. Design Engineer, FRP Research Centre. :[:Professor.

where R is the base circle radius for the epicycloid, r is the generating circle radius for the epicycloid, r a is the base circle radius of the disc, e is the eccentricity of input shaft with respect to the disc, r~ is the base circle radius of the housing rollers, R h is the radius of housing rollers pitch circle, D h is the diameter of holes on the disc, and Dq is output rollers pin diameter. For the instantaneous points of contact between the disc and the housing rollers (Fig. 3) the following relations can be derived.

In eq. (8),

)'i=

360(2i -- 1) 2(n+l)

where eti is the angle force Pi makes with the vertical, (Pi is the force between housing roller-/and the disc), 0 is rotation of the disc from symmetry position, Yi is angular position of the housing rollers, li is lever arm

~

OUTPUT ROLLERS"

OUTPUT SHAFT

Fig. I. Cycloid speed reducer schematic arrangement (taken from SM cyclo catalogue). 491

492

S . K . MALHOTRA and M. A. PARAMESWARAN

Fig. 2. oa : I n p u t / o u t p u t shaft centre, ob : Disc centre. A t 4 : O u t p u t pins. B~ 5 : Fixed pins.

,

OG

Fig. 3. Disc rotated through 0 clockwise from symmetric position.

Analysis of a cycloid speed reducer of force P~, p~ is radius of curvature of the disc at the point of contact of roller-i, and i is an integer 1,2,3 . . . . . Kudryastev[3] has dealt with the analysis of forces on the components based on the simplfied assumption of an infinite number of lobes on the disc. The present work attempts to provide a more exact analytical basis for the design of the cycloid reducer.

493

P,I~- ~ Qjr,,sin(flj+nO)=O. i=1

In the above equations, m

q = ~ , if m is even,

-

FORCE ANALYSIS

With ideally accurate geometry all the m output rollers as well as the (n + 1) housing rollers (Fig. 1) make contact with the disc. However, at any one time only m/2 or ( m - 1)/2 output rollers and n/2 or (n + 1)/2 housing rollers take part in torque transmission since only compressive forces can be transmitted at the contact points. Figure 3 shows the forces acting on various components of the reducer with 4 lobes in which the disc is turned by an angle 0 from the symmetry position. For this reducer, assuming no losses, and equating the input and output powers

m-1 2

, if m is odd.

and n

p = ~, if n is even,

-

n+l , if n is odd. 2

Assuming that the forces Pg and Qj are proportional to their respective distances from the centre of rotation

Pi = constant t,

-col

M~co, = [Qlr~ sin (111+ 40) + Q2r~ sin (1/2 + 40)] ~- (9) where M, is input torque, cog is input speed, Qj is the force between output roller-j and the disc, rw is the radius of output rollers pitch circle, flj is the angular position of hole-j in the disc with respect to the axis of symmetry, and j is an integer 1, 2, 3. . . . . . Torque equilibrium of the input shaft gives,

M~ = Fe cos (40 + ¢p)

(10)

where Fis the bearing reaction, and ~ois the angle force F makes with horizontal. The 3 equations of equilibrium for the disc are P, cos ~1 + P5 cos ~5 - Q, cos 40 - Q2 cos 40 - F sin ~ = 0

(11)

F cos ~0 - PI sin ~1 - P5 sin ~5 + Q, sin 40 + Q2 sin 40 = 0

(12)

P, lt + PJ5 - Q,rw sin (1/~ + 40) - Qzrwsin (112+ 40) = 0.

(13)

Generalising for the gear reducer with n lobes, (n + 1) housing rollers and m output rollers the above equations can be modified to

Ma = rw ~ Qj sin (1/j+ nO) nj=l

(14)

M. = Fe cos (nO + q))

(15)

Q i c o s n O - F s i n q~ = 0

~ P~cos e / i=1

(18)

j=l

(19)

and

sin (1/j+ nO)

constant.

(20)

The actual forces and their distribution on the rollers are easily computed from eqns (14) to (20) for any 0 (between 0 and 2n/n)[4], Figures 4 and 5 give typical results obtained for a 24 lobe reducer for M a = 10N mm ( r ~ = 4 8 m m , e = 2 m m , Rh = 75 mm , rw = 50mm , m = 12). The actual distribution of forces on the members will be different from the ideal due to manufacturing errors in profile, pitch, diameter, etc. The determination of the effect of these errors is made difficult not only by their random nature but also because the pattern of contact will depend on the combined effects of all the errors. However, as a first approximation, the programme as used for the ideal force analysis can be modified for a simple case in which certain rollers are assumed to be out of contact, i.e. forces Qj and P~ are zero for these rollers. Table 1 gives an idea of how the forces and their distribution 'are affected if one or more rollers (housing and output) are out of contact in the above 24 lobe reducer. It is seen that Pm,x increases by about 50~ for the case in which every third roller is out of contact. The above analysis is based on only one cycloidal disc transmitting power. In practice, 2 cycloidal discs spaced 180° apart are used for improving dynamic balance at high input speeds. If 2 discs are used, theoretically half the torque will be taken by each disc.

(16)

j~l POWER P

F cos ~ -- E i=1

q

Pgsin e~ + y' Qj sin nO = 0 j=l

(17)

LOSS

AND

EFFICIENCY

The rolling contact between the cycloidal disc and the rollers is the main factor in improving the efficiency

494

S. K, MALHOTRAand M. A. PARAMESWARAN 3 1~4

QI Q2 p

1.6 5

~. 1.2

0 =0*lor 15") 116

~T 0.8

@= 3.0"

'3

1 .2.

Q12

I

'~ 0.8 o

0.4

a,,/

0.4 Q7 to (;112= 0

01 2 3

8 9 10 11 12 NO

7

LLER

Qll

1.6

@=6*

& 5 6 7 8 9 1011 12

2

ROLLER NO.

912

19

1.6

0

11.

~1.2

=9*

1.2

0.1

o0.8

'--," 0.B o

119

0.4

0.4

011

F--

-

Qlf°Q5 =0

0

2345

10 41 12

678

1 i

ROLLER

ROLLER NO

10 11 12

7

NO

06

1.6

@ =12"

,7

1.2 -,0.8

23456

~6

%

-I

118

o,

0.4

119

T Q101111~12 2345 ROLLE

@ 9 10 11 12 NO

Fig. 4.

of the cycloid speed reducer. The various sources of power loss in a cycloid reducer are: (i) Bearing friction in the mounting of the disc on the input shaft. This mounting is usually on rolling bearings. (ii) Rolling contact friction between output rollers and holes in the disc. (iii) Rolling contact friction between housing rollers and the disc. (iv) Bearing friction in the mounting of the output rollers. The output rollers are usually hollow and mounted directly on the journals (pins) fixed on the output disc. (v) Bearing friction in the mounting of the housing rollers. The housing rollers are usually hollow and mounted directly on the journals (pins) fixed to the housing. For the elemental rotation dO of the cycloidal disc the rotations of the input shaft, output rollers and housing rollers are n dO, n dO and (n + l)d0, respectively. The five components of frictional work are

Dm d W, =f,,F(O) ~ n

dO

(21)

dW2 =f,: ~ Qj(O)n dO

(22)

j=l p

dW3 =f,s ~ Pi(O)(n + 1)dO

(23)

i=1

~

d W4 =f~, J=, Qj(O)--j-n Dq dO

(24)

dWs = f ~ , = Pi(O)

(n + 1)dO

(25)

dW = dW, + dW2 + dW3 + dW, + dWs

(26)

and the total frictional work is

wherefr,,f,: andf,~ are lever arms of rolling friction,f~,, ~: are sliding friction coefficients, D,, is mean diameter of input shaft bearing, Dp is housing rollers pin diameter, and D, is input shaft bearing rollers diameter.

Analysis of a cycloid speed reducer

495

1.0 e =O"or 15"

P4

0.8

P7 .-" 0.6

l/iTS2,3,0,,,;0.0 ;

0.4

,o

0.2 0.0

8

12 13 1 4 1 5 16 17 1 8 1 9 2 0 21

91011

22 2324 2 5

ROLLER NO. 1.0

O = 6.0"

0.8

) 19 ,118 '

,P17

P2~p ~:lp

-0.6 I1.

0.4 0.2

P16 i-

T

0.0

P2

3 to P15 = 0.0

2 3 4 S 6 7 8 9 1011 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 ROLLER NO.

1.0 0.8

6 =12"

~1o ~11

h

,P13

d.0.~,

P6

T

0.2 0.0 2345

9 10 11 12 131L,15 1617 1819 20 2122232&. 25

6

ROLLER

1.0

m

e

0,8 ~o.6

NO.

= 3.0*

PZ3 Pzz

Ps

_

P25 !4

o.

0.#

i

p 2i

5

0.2 0,0

IP6~ P7

P8 to P20 = 0.0

_

5 6 7 8 9 10 11 1213141516 1718 1920 2 1 2 2 2 3 2 4 2 5

ROLLERNO. 1.0 ~0.8

--

~3,~ is~6p.

O =9* P12

~-- 0.6

017 )8

Pll.

0.4

0.2 0

P2o P 1p2 2

--

P1 to 2

3

4

P40=O.O 5

6

7

8

T

,

m/ t

9 101112"13141516171819202122232425

ROLLERNO. Fig. 5.

=°-°

496

S. K. MALHOTRA and M. A. PARAMESWARAN Table 1. 24 Lobes, 25 Housing rollers, 12 Output rollers. 0 = if, M~ = 10 N m m FORCES ON OUTPUT ROLLERS Q j , N OUTPUT ROLLER" NO.

ALL ROLLERS ROLLER-2/)OES ROLLER-2 MAKE IDEAL NOT MAKE AND t~ DO NOT CONTACT CONTACT 02=0.0 MAKE CONTACT 132 =O.t,. =0.0 0.000

0.000

0. 000

2

0.800

0.000

0.000

1

3

1 .386

1 .600

2.317

t,

1 .600

1 .8/,8

0.000

5

1 .386

1 .600

2 .317

6

0 .800

0 .92/~

1 .388

7 f0 12

0 .000

0 .000

0 .000

FORCE$ON HOUSING ROLLERS, Pi ,N HOUSING ROLLE R NO.

ALL ROLLERS ROLLF.A- 2 MAKE IDEAL DOES NOT MAKE CONTACT CONTACT P2=0

ROLLERS- 2 AND 5 DO NOT MAKE CONTACT P2 =P5 =0.0

0 . 287

1

EVERY 3rd ROLLER DOES NOT MAKE C0 NTACT P3 :P6 :P9 :P12:0.0

0.322

0 .386

0 ./~33 0 .981

2

0 . 651

0.000

0 .000

3

0 .777

0 .870

1 . 0t~ t,

0 .00"0

4

0 .799

0 .89t,

1 .073

1 205 1 .166

5

0 .773

0 .865

0

6

0 .719

0 .80t.

0.961

0 .000

7

0 .650

0 .720

0.870

0 .970

8

0 .558

0 .625

0 .750

0 .8t, 2

9

0 .t,60

0 .51t,

0 .617

0.000

10

0 .352

0 .394

0.473

0 .531

11

0 .238

0 .267

0 .320

0.359

12

0 .120

0 .135

0 .162

0.000

13 t o 2 5

0 .000

0 .000

0 .ooo

0.000

T h e i n s t a n t a n e o u s efficiency is t h e n ,

rl~=

T h e o v e r a l l efficiency is t h e n ,

Man dO - d W Man dO

(27)

By i n t e g r a t i n g d W o v e r t h e d u r a t i o n o f o n e r o t a t i o n o f t h e i n p u t s h a f t , i.e. 1In r o t a t i o n o f t h e disc, t h e frict i o n a l w o r k p e r r o t a t i o n o f t h e i n p u t s h a f t c a n be determined

W=

000

Ma2n - W qo =

Ma2~

(29)

T h e r/i a n d 70 a r e e v a l u a t e d f o r g i v e n c o n d i t i o n s u s i n g n u m e r i c a l i n t e g r a t i o n t e c h n i q u e s s u c h as S i m p s o n ' s rule. By v a r y i n g t h e lever a r m o f rolling f r i c t i o n as well as

fj~ '~/n fqDmnt2'q" d W - -~ 30 r(O)dO (28)

+n do

j=i

Analysis of a cycloid speed reducer

[-

First 6 parameters are related as given in eqns (1)-(4). From eqns (2) and (3)

24 Lobes - 25 ROLLERS frl

=fr2=fr3 = f r ,

fsl=fs2=f s

00 o-~.----~____..._.~

497

f s = 0.0

rb=(n + l)e l

90

--

% gSO

.

~

0.08

From eqns (1) and (4),

R(n + 1)

-

Rb - - --

~o 7 0

o

o

(31)

If the speed ratio and the approximate overall size of gear reducer are given, n and R can be selected. Using eqns (14)-(20), the following equation for P~ can be derived.

L

,° l

(30)

0 .06

~'~-~--..~"'------~EZZ~/

I i J 0.002 0.004 o.oo6

J J 0.008 o.ol

Pi =

t r. . . .

Ma

sin(~-n0)

e

P

(32)

~ sin2 (~g -- nO)

Fig. 6.

i=1

the coefficient of sliding friction in the bearings, a comparative study of the effect of the two on 70 is carried out [4]. Figure 6 gives a typical curve for variation of r/o w i t h f and f~.

From Pi, the radii of curvature at the points of contact and Hertzian formula equation for contact stress (Sc)~ between housing rollers and the disc can be derived.

/ M, Esin(ai-nO) [ 1 + 1

EFFECT OF VARIATION OF DESIGN PARAMETERS

The various design parameters of a cycloid reducer are R, r, n, r a, r b, e, R~. r r and B.

(Sc)i : 0.591 / 2 ~

s i ~ 2(~ ~ n 0 ) U ~ _ ~ ]

20xll

24 Lobes ~ 25 Rolters

18

Pmox. Vs e 16

14

12

z ×8 o E o.

I

0.4

~

I

I

I

1

I

o.e

1,2

1.6

2.o

2.4

e,m m

Fig. 7.

=

(33,

S.K. MALHOTRAand M. A. PARAMESWARAN

498

where E is modulus of elasticity of material of the disc and rollers, B is width of the disc, and r R is radius of the housing rollers. It is seen that P~ depends upon e alone while (Sc)~ depends upon e, r e and B. If a suitable B ~re ratio is assumed (B/rR = 2 for the presesnt study), (S~)i is then a function of e and r e. (Sc)m~ is the largest among various (Sc)~ values. This can be computed by varying either e or rR, while keeping the other quantity fixed. Variation of Pmax(Pmax is the largest among various P,) with e is shown in Fig. 7, while Fig. 8 gives the variation of (S~)m~x with e for a given value of rg(r R = 5 m m in this case). In Fig. 9 are plotted values of e that give m i n i m u m (Sc)m~x for various values of rR. Figure 10 shows variation of (Sc)ma~ with r e for a given value of e (e = 2 m m in this case). Figures 7-10 are drawn for the reducer with 24 lobes and M~ = 2,400 N ram.

Following observations which will be helpful for design are made from above study on the effect of variation of e and rR: (i) It is found that Pmax decreases with increase of e, first at a faster rate, and then at a slower rate. (ii) For a given housing roller radius rg, (Sc)ma x first decreases with increase of e becomes m i n i m u m for a particular value of e and then increases with further increase of e. (iii) The value of e at which minimum (So)max occurs decreases with increase of roller radius rR. (iv) The value of (So)max (for a given value of e) decreases with increase of r R, first at a faster rate, then at slower rate and finally for very large values of r e, (S,)max increases with rR. As the diameter of housing rollers increase, width of disc also increases and hence the weight and size of gear reducer increases. For B/rR = 2, the practical limits for roller size will be as shown in Fig. 10.

6 xlO3 ~-

--|

5 ~E .EE 4

f

24 Lobes - 2 5 R o t t e r s

~-

(SC)max Vs. e Min(Sc)mox Occurs at : e= 2"Omm ; f°r tR=5'Omm

I--

/ r R = 5.0ram

z 3

~

2

-

1

-o 0.2

o

1

I

0.6

1.0

I

I

t

1.4 18 e ~ mm

I

I

I

2.2

2.6

30

Fig. 8. 2.4 24 L o b e s - 2 5

l

Rollers M,n (Sc)'mclx

5x 10:

2.0

24 L o b e s e=

o

E 7

l 4

1.6

~E

U3

E 3

1.2

Vs, r R

25 R o l l e r s

2.O rnm

(SC)mcix - MOlt. c o n t o c t s t r e s s between rollem and d i s c

m

5E

~2

08

E

E

E

1

O0

m L ~

5

] 6

__1__. 7 r R j Prim

Fig. 9.

L 8

_! 9

10

4

6

8

10

12

Fig. 10.

14

16

18

20

22

Analysis of a cycloid speed reducer CONCLUSIONS (i) A complete picture of forces on the various elements of a given cycloid speed reducer can be obtained using analysis given in section on Force Analysis. The procedure to calculate the forces can be modified without difficulty to analyse the effect of nonideal contact on the forces. This gives an idea how forces are affected due to the presence of manufacturing errors. (ii) Based on above theoretical study, the computation of efficiency shows that this type of speed reducer is more efficient compared to the conventional ones used for high reduction ratios, e.g. wormgears, epicyclic gears.

499

(iii) The effect of design parameters on forces and contact stresses is also analysed. This will be of great help in optimal design of this type of gear reducer.

REFERENCES

1. D. W. Botsiber and L. Kingston, Machine Design 28, 65-69 (1956). 2. R. Neumann, Maschinenbautecknik 26, 297-301 (1977). 3. V. N. Kudryatsev, Planetary Transmission (In Russian), (pp. 251-271). Machine Construction Publishers, Moscow. 4. S. K. Malhotra, Analysis of Cycloid Speed Reducer, M. S. Thesis, April 1982, Indian Institute of Technology. Madras-600036, India.

U N T E R S U C ~ N G DES CYCLO-GETRIEBES S. i. ~ l h o t r a

uad ~!. A. Parameswaraa

Kurzfassun~ - Die irifteverteilung auf des Au~enrollen und den "itmehzerrollen sowie der Wir~ungsgrad des Cyclo-Getriebes werdem amalytisch untersucht. Ausgehend yon den geo~etrischen Zusammenh&ngen der Abmessun@em werden allgemeing~itige Beziehungsgleichungen fqr die ~rifte und fdr die Verluste im Getriebe bei fehlerfreien Ein~riffsverh~dltmissen abgeleitet. Bei Vorgabe der Getriebeabmessungem und des Amtriebsmomentes sind diese Gleichungen leicht auf dem Computer zu l~sea. Obwohl der genauere EinfluB der Herstellungsfehler auf die Irifteverteilung sehr schwierig zu ermitteln wire, ~ann der Sonderfall ohne weiteres ~el~st werden, bei dem bestimmte Elememte des Getriebes als nichttragemd angenom~en werden. Als Beispiel wird elm Cyclo-Getriebe mit dem ~bersetzuagsverhiltnis 24/I umtersucht; eiaige Hinweise f~r die ionstru~tion werden gegeben.