Perpmom Mech. Mack. Theory Vol. 29, No. 5, pp. 777-784, 1994

Copyrilght ~ 1994 Elsevier Science Lu:I Printed in Great Bntmn. All fillhts

0094-114X/94 $7.00 + 0.00

MAXIMUM MECHANICAL EFFICIENCY OF INFINITELY VARIABLE TRANSMISSIONS

HONG-SEN YAN Department of Mechanical Engineering. National Cheng Kung University. Tainan. Taiwan 70101.

R.O.C.

LONG-CHANG HSIEH Department of Power Mechanical Engineering. National Yunlin Polytechnic Institute, Hu.wei,

Taiwan 63208, R.O.C.

(Received 30 January 1991; in revised form 19 November 1991: received for publication 9 July 1993)

Almract--Equations for the mechanical efficiency of infinitely variable transmissions (Iv'r) are derived. As a result, we conclude that: (I) an IVT is of input-enupled type, (2) the mechanical efficiency of an IVT depends only on its allowable maximum speed ratio (R,)~,. (3) the mechanical efficiency of an IVT can only be improved by increasing (R,),_. the efficiency of planetary gear train ()In) and the eff~ency of continuonsly variable unit (~,). and (4) an IVT with five-member planetary war train of type ! power flow has maximum mechanical e~'kncy. A desiSn example is also presented.

N O M E N C L A T U R E

CVU--Continuously variable unit DT--Differential transmission

IV'r--infinitely variable transmission

;~ . P~. P,--Powers without frictional loss of the link of PGT adjacent to input axis. CVU and output axis . ~,. ~,--Powers with frictional loss of the link of PGT adjacent to input axis. CVU and output axis P(~cP~--Frictional power of PGT and CVU

--Planetary gear train R,--Velocity ratio of CVU

(R,).,~AIIowable maximum velocity ratio of CVU 7".. r . . T,--Torque of the link of PGT adjacent to input axis. CVU and output axis

Z,--Number of teeth of gear i w,. w,. m,--Angular velocity of the link adjacent to input axis. CVU and output axis

m,---Angular velocity of link i y,--Gear ratio of gear j to gear i

~,,,. )i,,. ~, --Elficiencies of IVT, POT and CVU

INTRODUCTION A differential transmission (DT) is a transmission with one continuously variable unit (CVU) and one differential unit. Planetary gear trains (PGT) with two degrees of freedom are usually used as differential units. A differential transmission is called an infinitely variable transmission (IVT) if it has the character of zero output velocity.

In the past, a number of studies have been done on the problems of power recirculation [I-7] and mechanical efficiency [8-1 I] ofdifl'erential transmissions. Chadda [12] presented the mechanical efficiency of IVT with certain type of power flow. Our purpose here is to focus on designing IVT with maximum mechanical efficiency by considering all types of power flow.

KINEMATIC ANALYSIS

We apply the fundamental circuit method [13] for the kinematic analysis of PGTs. For the sake of simplicity, notation if, j; k) is adopted to denote a fundamental circuit in which i and j are the two gears incident to the gear pair, and k is the transfer link. Let w,, m/and m, be the angular velocities of links i, j and k, respectively. Also, let 7j, be the gear ratio of gear j to gear i, i.e., 3'j, -- -1- Zj/Z,, in which Z/and Z, are the numbers of teeth of gearsj and i, respectively. The positive

MMT ~,S--J 777

778

~ G T Pm ~ [ (F'2)

H.-S. YAN and L.-C. HSW..I.i

Pout (F-2) I v

Pout

(a) Input-coupled (b) Output-coupled

Fig. I. Types of differential transmissions.

(or negative) sign means internal (or external) mesh of the gear pair. For fundamental circuit (i,j; k), its fundamental circuit equation is:

~,-~,,,coj+(~,,- i ~ : 0. (I)

Since each fundamental circuit can derive one equation of motion, a PGT with L I fundamental circuits can construct L~ equations of motion. And, these equations can be expressed as:

[,¢ ] [co] = 0 , ( 2 )

in which [A] is an L r × Ncoefficient matrix and [co] is an N × I angular velocity matrix. Substituting the known angular velocities into equation (2), all unknown angular velocities can be obtained.

According to the location of the CVU, differential transmissions can be divided into input- coupled and output-coupled types, Fig. I. If an output-coupled DT is used as an IVT and the output of the DT has zero velocity, we have a case that the member of the PGT adjacent to the CVU also has zero velocity. This is physically impossible when the input member of the DT is in motion. Therefore, only input-coupled DT can be used as an IVT.

TORQUE ANALYSIS

Let the subscripts a, b and v denote the three members of PGT adjacent to the input axis, output axis, and CVU, respectively. Then, P,,(Ps, P,,), T,,(Tb, To) and (o,(oJs, o~) are the power, torque and angular velocity of the member adjacent to the input axis (output axis, CVU). Based on static equilibrium and energy conservation theorem [I I], we have:

/fO a - - (Ob~

rs -- - ~,cu b ~ - co,,] "" (4)

Therefore, only the following three types of power flow are theoretically possible for input- coupled differential transmissions:

I. Type I, P, > 0, P, < 0 and Ps ~< 0, Fig. 2(a).

(a) Type I (b) Type t1 (c) Type m Fig. 2. Types of power flow for input-coupled IVT.

Maximum mechanical ¢t~ciency of IVT 779

2. Type II. P, < 0, P~ > 0 and Pb <~ O, Fig. 2(b), 3. Type III, Po > 0, P~ > 0 and Ps g 0, Fig. 2(c).

However. since it is physically impossible to have the condition such that P. > 0 and P~ > 0 while P#--0, type III power flow does not exist.

MECHANICAL EFFICIENCY

There are two kinds of frictional losses in differential transmissions, one is in PGT and the other is in CVU. Let r/~ be the mechanical efficiency of PGT, and ~, , P~ and ~b be the powers of PGT with frictional losses. Then 7s~, ]~ and ~b. by definition, can be expressed as:

P. = P., for P . > 0 .

= vln x Po, for P, < 0, (5)

P~ = e~, for e , > 0

= ~n x P~, for P~ < 0, (6)

P~ =/ '~, for e , > 0 ,

=JTpxXPb, for P#<0. (7)

Let P/be the frictional power of PGT. Since P / < 0, the frictional loss of PGT, [P/I, can be expressed as"

le/I = P. + ~ + P~. (s)

Let t/,. be the mechanical effffciency of CVU. Then, the frictional loss of CVU, ]P/~I, can be written as"

IP~,I = ~,, x (I - ~,,) for ~,, < 0,

- ~,, x (I - ~.)/~,,, for P. > 0. (9)

For input-coupled DT. the output power (P,.t) and input power (Pin) are:

eo,, = ~ (J0)

Pm= - Po,, + lefl ÷ lP~.l. 01)

Therefore, the mechanical efficiency of an IVT (~,~,) is:

~h~, -~ -Po.,/Pm. (12)

EQUATIONS OF MECHANICAL EFFICIENCY

In the following, we derive equations of mechanical efficiency for IVTs. Let (ca~),.~ be the value of ca,, for 0 ~< cas ~< cao and (CO~),~.o be the value of ca,. for cab ---- 0. where cao is the maximum output angular velocity. Then, the velocity ratio of the member adjacent to CVU, R,,. is:

R, -- (CO~)'~ , for J(ca,,)o,, J I> I(ca,,)o,,-ol, (ca~)o,~. o

_ (co..)o,-o for I(ca~)o, I ~ ](ca~L,,-o]. (13) (ca,.)o,~ '

For PGTs with two degrees of freedom and five (six) members, there are three (six) types as shown in Fig. 3 (Fig. 4) [14]. Since each PGT has three members adjacent to input axis, CVU and output axis. respectively, there are six (six, 12) permutations for the PGTs as shown in Fig. 3 [Figs 4(a)-(d) and Fig. 4(e)-(h)]. If member i is adjacent to the input axis, memberj to CVU, and member k to the output axis. i.e. ca,--co,, car--coj, and COb----ca,, we use i -*( j )" .k to denote this case.

Here, we use the IVT with five-member PGTs and case 4--*(2)--*5 as an example (Fig. 5) to illustrate the procedures of deriving the equations of mechanical effciency for |VTs. The PGTs with

780 H . - S . YAN a n d L . - C . H s ~

3

!'_F-W

(a)

-IC. (I~)

5 2

[,-J m

i

m I 1

Fig. 3. Planetary gear trains with five members.

(c)

five members have fundamental circuits (2, 3; 4) and (5, 3; 4). Their equations of motion for PGT can be expressed as:

{°'} [~-73:(7;2 - I ) 0 ] ~3 ={0}. 1 ( 7 . ! ) -753 - - (O e

CO b

Solving equat ion (14) for w,,. we have:

(14)

for = (1 - 7,2y.)~, . + 7.7~3fo~. (15)

Accord ing to the def in i t ion o f R,,. under condi t ions 7327s.~ < 0. foe > 0 and fob > 0. ~,, can be expressed as:

(fo,.).~-o _ (I - 7~,7,,) ~" = a,. a,. (~.- (I 6)

3 ~ 6 ~ 2

i '1"

(a) (b]

3-

(c) (d)

(e) (f) (g) (h)

Fig. 4. Planetary gear trains with six members.

Maximum mechanical ~'icicncy of Iv-r

t

HOTOR Fig. 5. An infinitely variable transmission.

K%%M T K~q !

781

Substituting equation (16) into equation (14), we have:

(17)

Then, based on equations (3)-(12) and equations (16) and (17), the mechanical efficiency of Iv'r , q,~,, is expressed as [10]:

-Po , , ~ , ( R , , - I) ~,v, = = (I 8)

Pm R,. -- r/~ ~,~

Also, under conditions 0 < 3'~:3'53 < I, co, > 0 and cob > 0, the mechanical efficiency of IVT is:

~l,,t ffi q n q " ( R ~ - I) (19) R,. - ~,~ ~,.

As a result, Table I lists equations of mechanical efficiency for all possible IVT with five-member PGT as shown in Fig. 3 and Table 2 lists equations of mechanical efficiency for all possible IVT with six-member PGT as shown in Figs 6(a)--(d). For IVT with six-member PGT shown in Figs 6(e)-(h), their mechanical efficiencies are the same as equations (18) and (19).

Therefore, we conclude that an IVT with type I power flow has better mechanical efficiency. And, an IVT with five-member PGT of type I power flow has maximum mechanical efficiency.

DESIGN EXAMPLE

Since an IVT has the kinematic property of infinite velocity ratio, it can be used as the power train for an exercising machine. The design constraints of this power train are:

(I) Since there is one gear pair between PGT and the input axis, the constraint of the angular velocity (co,) of the member of PTG adjacent to the input axis is 350 rpm ~< co, ~< 1800 rpm.

(2) The output o f this machine rotates from 0 to 540 rpm continuously. Since there is one gear pair between PGT and the output axis, the range of the angular velocity (¢o b) of the member of PGT adjacent to the output axis is 0 to coo. The constraint o f COo is 108 rpm ~< coo ~< 1800 rpm.

(3) For noise concern, all angular velocities of member of PGT are Ico, I ~< 1800 rpm. (4) The allowable maximum speed ratio (R~),o,. = 3, i.e. I ~< R~ ~< 3.

According to the conclusion in the previous section, we choose the IVT with five-member PGT and type I power flow. Based on Table !, we synthesize and select the IVT, shown in Fig. 5, as one of the possible designs. Here, carrier 4 is adjacent to the input axis, gear 2 adjacent to CVU, gear 5 adjacent to the output axis, and teeth numbers are Z, = 40, Z3 = 20 and Z5 = 80. When

782 H.-S. YAN and L.-C+ HSIEH

Table 1. Equations o f mechanical efficiency for 1VT with five-member PGT

n ' n 2 n ' n

I - . . J - I " " I

I I

Type ! power flow

P >O. Pv<0 and Pb<O

qpig(Rv-I )

qivt Rv_q pl(q v

Type li power flow

P < 0 , Pv>0 and Pb<0

Mechanica l e f f ic iency

Y32T~3 < 0 0 < Y32T53 < 1 co b > 0 1 < Y32Y53

5-4,-(2)-.4,-4 ~32¥53 < 0

(O b < 0 0 < y32Y53 < I 1 < Y32Y53

co b • 0 ~32~53 < 0 0 < Y32Y53

2"*" ( 5 ) " * ' 4

(l) b < 0 0 < ¥32~53 ~32~t53 < 0

O) b • 0 0 < ¥32~53 ¥32¥53 <: 0

5 - - " ( 4 ) " * " 2

O) b < 0 ¥32¥53 < 0 0 < ¥32"~53

(I) b • 0 0 < V32¥53 ~32~53 < 0

2 " - ~ ( 4 ) ' - ~ 5

(I) b < 0 ¥32~53 < 0 0 < ¥32~53

Y~2Y53 < 0 ¢0 b • 0 1 < Y32Y53 0 • Y32Ys3 < I

4 - . ~ ( 2 ) - * - 5 ~32¥53 < 0

(i) b < 0 0 • ¥32¥53< 1 1 < ¥32¥53

¥32Y53 < 0 0) b • 0 0 < Y32Y53 < 1 1 • Y32Y53

4 - ' * ' ( 5 ) " ~ 2 ¥32~$3 < 0

(o b < 0 1 • ~32~53 0 • ~32"1'$3 < I

qp lqv(Rv- I )

qWt Rv_l]pgqv

oJ, = oJ, = 540 rpm and 0 ~ oJ5 = ~ ~< 540 rpm, the angular velocities o f the other members o f PGT and R~ are:

( ! ) ¢o, = 0 r p m (2) 540 rpm ~< oJ 2 = co,. ~< 1620 rpm. (3) - 1629 rpm ~< co s ~ 540 rpm. (4) I

For fin = 0.98 and q~--0.8, the mechanical efficiency o f this tVT is between 0% to 88.45% (Fig. 6).

Maximum mechanical efftckncy of IVY

Table 2. Equations of mechanical effgiency for IVT with ~x-member PGT

783

' t

I ' 2 '

T. T.

Type ! power flow

P.>O, Pv<O and Pt,<O

qpg(Rv-I) qivt Rv_qplqv

Type ii power f low

P.<O. P,>Oand Pb<O

qpgqv(Rv-! ) qivt gv_qplqv Mechanical efficiency

¥3z¥53¥es < 0 % • 0 1 .c ¥3z'Ys3¥~s 0 .c "Z32¥s3¥~s < 1

6.-.4.-(2)--..-4 "s'3zYs3"f6s < 0

% < 0 0 .c 732Ys3Y6s < i 1 .c ~t3zYs3Y65

% > 0 ~3zTs3"l'es < 0 0 .c ¥3z¥,¥es

2--*.-(6)-..-4

% • 0 0 < ¥3z'YslZ6s ¥3z'Ys3¥es < 0

% • 0 0 ,c ¥~z"ts3¥6s 'Z~zYsj~6s < 0

6--~(4). -~2

CO b • 0 Y32¥S3Y6S < 0 0 < ¥32¥s3¥6s

% > 0 0 • Y3zYs3Y6s Y~zYs3Y6s < 0 2.-~.(4)-.o..6

COt. < 0 Y3frs3"Yes < 0 0 < ¥3zYs3¥6s

T3z'Ys~'Yes < 0 m b • 0 1 < T32YS3'/6S 0 < T32YS3¥6 s < I

4 ~ ( 2 ) ~ 6 ¥3z'Z,¥6s < 0

mb< 0 0 < Y32Ys3Y6S < l 1 < Y32Ys3Y6S

'Y32Ys3¥6s < 0 m b > 0 0 • ¥32¥s3¥os < ! 1 q: T32¥53¥6 $

4--~-(6)~2 Y3zYs3Yss < 0

co b < 0 1 < T32YS3T6S 0 < T32T$3¥ss<|

C O N C L U S I O N

Based on the concept o f fundamenta l circuit equat ion, static equil ibrium, and frictional loss, equat ions for the mechanical efficiency o f I V T are derived and listed in Tables i and 2. As the result, we conclude:

( I ) An I V T is o f input-coupled type. (2) IVT has only two types o f power flow which are type l (Po > 0, Fr < 0, Fb ~< 0)

784 H.-S. YAN and L.-C.

I.U.T. ffFl¢l[l~

!ll~t :Link 4 ~l,:Wiw 540vls 0.t~d:Li.~ ~ 0N :Link 2

lU'z

! GrJd Z TKTN 4 8 1 ~ ) Ylm 28

I GIAt3 TEfflt ~! Ullt 5 ~ U

4t~.

IPz g.8

I

188.8 216.8 324.8 432.8 548.8 (1151)

Fig. 6. Mechanical efficiency of IVT with type ! power flow (q~ = 0.98, q, - 0.8, I ~ R, ~; 3).

and type I! (P, < 0, P~ > 0, Pb ~< 0). The mechanical efficiency o f an IVY with type I power flow is bet ter than with type II power flow.

(3) The mechanical efficiency o f a n IVY is complete ly determined by its type o f power flow and al lowable m a x i m u m speed rat io (R , ) , , . ,

(4) The mechanical efficiency o f an IVY can only be improved by increasing (R~),~,, the efficiency o f P G T (qn) , and the efficiency o f C V U (ff~).

(5) An IVT with f ive-member P G T o f two degrees o f f reedom and type I power flow has m a x i m u m mechanical efficiency.

The result o f this work is beneficial to engineers for designing infinitely var iable t ransmissions with m a x i m u m mechanical efficiency.

Acknowledgemtnts--The authors arc grateful to the Mechanical Industry Research Laboratories of the Industrial Technology Research Institute (Chu-Tung, Taiwan) for the financial support of this project.

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(1989).