Load

~ Pergamon Mech. Math. Theo O' Vol. 30. No. 4. pp. 553 567. 1995 Copyright ¢ 1995 Elsevier Science Ltd

0094-114X(94)00066-2 Printed in Great Britain. All rights reserved 0094-114X/95 $9.50 + 0.00

L O A D D I S T R I B U T I O N O F T I M I N G B E L T D R I V E S

T R A N S M I T T I N G V A R I A B L E T O R Q U E S

NESTOR A. KAROLEV and PETER W. GOLD Institut ffir Maschinenelemente und Maschinengestaltung, RWTH Aachen, Temptergraben 55,

52056 Aachen, Germany

(Received 28 October 1993; in revised Jbrm 19 October 1994, received Jor publication 2 November 1994)

Abstract--This paper is concerned with timing belt drives transmitting fluctuating torques. A method is presented for calculating the loads on belt and pulley teeth given the belt strand forces as a function of time. Belt material properties, a generally existing pitch difference as well as friction forces between belt groove and pulley top land are taken into account. It is found that, due to the friction hysteresis, the load distribution at a particular time is affected by the preceding loading states. Experimentally determined loads on a pulley tooth in a drive transmitting a reversed and repeated torque are compared with calculated ones.

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

F,, F,.*--belt tension force acting in position Fs--pitch equaling force Fu--total circumferential load Fv--initial belt tension Fw--shaft load F0--belt tension at the beginning of meshing F.--belt tension at the end of meshing G--weight force Q,---circumferential force transmitted by the belt

tooth i Qr,--force acting at the flank of the belt tooth i R,--friction force over the top land of the pulley

tooth i S,---circumferential force transmitted by the pulley

tooth i bl--backlash cG--belt groove stiffness constant ca--stiffness constant of a belt pitch cz--stiffness constant of a belt tooth

/--tooth position along the arc of contact

i, k--step number pb--unstretched belt pitch pp--pulley pitch t,~isplacement of the action point of the friction

force R, u,~isplacement of the belt tooth i relative to the

pulley v--belt speed z--number of teeth in complete mesh

%--pitch angle fl,),--angles in Fig. 4

6--constant tp--angle of pulley rotation /z-----coefficient of friction v,--tangential deflection of belt groove 0--angle of wrap around the pulley tooth oJ--angular velocity of driving and driven pulley,

respectively ~b--phase angle

I, ll--pulley position

1. I N T R O D U C T I O N

A l t h o u g h in recent years t i m i n g belt dr ives have been en joy ing a g rowing p o p u l a r i t y in mechan i ca l design, their o p e r a t i o n a l b e h a v i o r is still cons ide red to a large ex ten t unpred ic t ab le . C u r r e n t l i t e r a t u r e - - t e c h n i c a l b o o k s a n d des ign m a n u a l s - - s t i l l does no t p rov ide all the necessary infor - m a t i o n on belt a n d t o o t h forces a n d belt ma te r i a l proper t ies , which wou ld enab le engineers to op t imize a dr ive for a special case o f app l i ca t ion . Never theless , research on t im ing belt dr ives has been d o n e for years a n d v a l u a b l e results have been pub l i shed in scientific j o u r n a l s a n d in o the r scientific p u b l i c a t i o n s [1-6]. These s tudies a n d some o ther p u b l i c a t i o n s no t m e n t i o n e d here, deal with load d i s t r i b u t i o n o f t i m i n g belts exclusively u n d e r s t e a d y t o rque cond i t ions . In a n u m b e r o f

app l i ca t ions , however , f l u c t u a t i n g loads are preva len t . The a u t o m o t i v e indus t ry , especially, exerts in c o n n e c t i o n wi th the c a m s h a f t dr ive a great pressure for m o r e clar i ty a b o u t the o p e r a t i o n a l b e h a v i o r o f t i m i n g bel ts t r a n s m i t t i n g f luc tua t ing torques . However , no repor t has been f o u n d that discloses m e t h o d s for ca lcu la t ing belt a n d too th loads in such cases.

553

554 Nestor A. Karolev and Peter W. Gold

Experimental determination of tooth loads alone, as carried out in [7], does not seem suitable for providing the general knowledge that would allow a prediction of the loads and optimization of the drive.

The present study deals with timing belts with trapezoidal tooth profiles transmitting variable torques. It is assumed that the time function of the external load, i.e., the tension force in both belt sides, is known. Based on a belt spring model, introduced in Ref. [6], a method for calculating the load distribution along the arc of contact is developed. However, the approach presented here is quite different from that in previous studies [4, 6], as the load distribution is no longer regarded as constant in time. In order to prove the effectiveness of the analysis, theoretical results are compared with experimental ones.

2. A N A L Y S I S

2.1. Tooth loading

A pulley rotating in one direction but transmitting a bidirectional torque cannot be clearly identified as driving or driven. The belt sides also change between slack and tight. Therefore, it is reasonable to number the positions of belt and pulley teeth starting at the side of the first tooth engagement (Fig. 1). Thus, with increasing revolution angle q~, every tooth moves into a higher position and is consequently indicated by a higher number. The tooth loads are indexed in accordance with the tooth position. The belt pitch Pi and the tension forces Fi and F* , acting in it, are located between teeth i and i + 1 (see Figs 2 and 3). The tension force at the beginning of meshing is F0, and the tension F_., at the end of meshing, has the index z of the last tooth (Fig. 1).

Geometrically, the first belt tooth is completely engaged when the center line of the corresponding pulley groove is perpendicular to the belt strand. This position of the pulley is denoted by I. The belt tooth position is 1I and the enclosing pulley teeth are in positions 0I and 1 I, respectively (see pulley 1 in Fig. I). After a rotation of half the pitch angle ~p/2 the pulley comes into a position indicated by II in which the axis of symmetry of the pulley tooth having been in position 0I becomes perpendicular to the belt strand (in Fig. 1 pulley 2 is shown in position II). Generally, a certain pulley or belt tooth moves from position iI to position ili and after another ~0/2 degree rotation it logically takes the position (i + i)I. Consequently, for the sake of analysis the pulley motion is divided into ~p/2 degree steps. The need for that is explained later. Subject to the pulley position

pulley positionII pulley positionI

~ . . . . _ _ pulle(Z-1)I~-I,'~'c~--!"--beltyt;;tohthPOS't'ons - - " ~ 2 I {z-1)II p o s i t i o n s ~ OI II

zII

Fig, I. Pulley and belt tooth positions around the arc of contact,

y +1)I ,(i*1)I

Timing belt drives transmitting variable torques 555

I

(

I

' \ \ 2, ,'

a) b) Fig. 2. Forces on belt tooth.

a second subscript I or II will be attached to a force, pitch, deflection and to other position-related values, if the derived relation is not valid for both positions.

Neglecting the mass of the belt, a static force balance for the belt at every rotational step can be set up. Considering the belt tooth i, the load Qri acting on the tooth flank is balanced by the tension forces Fi and F*~ [Fig. 2(a)]. Qri is the resultant of all normal and tangential forces on the tooth flank. A force balance in tangential direction yields

Q,= Fi- F*_L, (I)

where Q~ is the circumferential load transmitted by the belt tooth, further just referred to as belt tooth load, The friction force R~ acting on the belt groove between the belt teeth i and i + 1 is expressed by

R, = F* - F,. (2)

Fi+1

V

i+I

Fi

2cR

Cz

Qi÷l ~_.~# ___~ -

i+1 I u'+'_

Pi L

h 4-+-- Fi

- + / J i

C 6 ~ i v

Pp i .I Fig. 3. Belt model.

r-O

i

- - I

+~- Ui

Fi*_ 1 I I I

bL+ui

i-1 h


The circumferential force transmitted by the pulley tooth is indicated by S/and can be expressed by the following equations:

S,= F * - F * , , (3)

o r

Si = Qi + Ri. (4)

Equations (3) and (4) are only valid for positive Q's (F~> F*~), otherwise the belt tooth i contacts the following pulley tooth i - 1 [see Fig. 2(b)]. The load then transmitted by that tooth is given by

S / _ , = F ~ - F i ,, (5)

o r

Si_ I = Qi + R i _ I .

Value and direction of the friction force are considered in section 2.3.

(6)

Hence,

V/max "~-- Ri max/¢G ; (12)

Ri <~ Rimax; (13)

v i ~ Vimax. (14)

2.2. Belt model

The belt tension members are modeled by a number of identical extension springs (spring constant 2CR) connected in series (Fig. 3). A belt pitch is made up of two springs and its length is expressed by

F,+ F* Pi = Pb "1- - - , (7)

2OR

or substituting F* from equation (2) into equation (7) by

Ri F, + -~

Pi = Pb -I- - - , (8) ¢R

where Pb and cR are the unstretched belt pitch and its stiffness constant, respectively. Assuming the elastic behavior of a belt tooth linear, it is simulated by a compression spring with a stiffness constant Cz. The location of the belt tooth center line, when the tooth just touches the preceding pulley tooth, marks the starting point of the coordinate u~ indicating the displacement of the tooth relative to the pulley. If the displacement u~ is negative, and its absolute value greater than the backlash bl (bl >/0), the belt tooth i contacts the following pulley tooth i - 1. Therefore, the relationship between u~ and the load Q~ is obtained by considering the following three cases

U i > 0 =~" Qi = UiCz (Q/> 0) (9a)

0 >1 ui >>- - b l =~ Qi = 0; (9b)

- b l > ui ~ Qi = (ui + bl)cz (Q i < 0) (9c)

Peeken et al. [6] found that the value of the friction force R~ is related to the deflection of the belt groove in tangential direction and extended the existent belt model by another spring with the spring constant cG. The friction force R/, acting at the pitch mid-point, is proportional to the deflection v/of the spring:

Ri = vi" ¢G. (1 O)

However, R/ cannot be larger than the maximum friction force R/max derived from Eytelwein's (capstan) formula:

Rimax - Fi(C ~'° - 1). (11)

Timing belt drives transmitting variable torques

i t r R, i

Si3

~-z ~ Si 6 A#,/ DISPLACEMENT ti B E

2 F*= consf

Fig. 4. Deformation characteristic of a belt tooth model.

557

In equation (11) the coefficient of friction M is assumed constant. 6 = + 1, depending on the direction of sliding explained in the next section. Under initial tension the wrap angle 0 is equal to the pitch angle %. But in a drive transmitting power the actual wrap angle is somewhat smaller, since the belt tooth inclines and moves radially outwards, as shown in Fig. 2, in order to fulfill the conditions of equilibrium. In this study, however, the resulting reduction of the friction force is not considered in the analysis.

Figure 4 shows the relationship between the pulley tooth load Si and the displacement ti of the action point of R~, obtained in an imaginary experiment. At A, which is the starting point, the belt tooth i just touches the pulley tooth i; i.e., Qi = 0 and ti -- t~0. The deflection v i is equal to zero, hence Ri = 0. There is no difference between the value of the tension forces F* and F * ~ . Retaining F * constant, the tooth load is applied by gradually decreasing F*i_~. In this way, the maximum friction force depends on the initial belt tension F * , but not on the load value S~ = F * - F * j . At first, both springs deform simultaneously until at B the positive maximum value of the friction force _.R I+)~max is reached. Then from B to C the belt slides over the pulley tooth land and only the deformation of the belt tooth spring increases. The unloading of the belt tooth begins at C. The friction force is again equal to zero at D and it reaches its negative maximum value R ~-) at E. /max With a further decrease of the load S~ the belt slides over the pulley again but in the opposite direction. Consequently, above the line AD the friction force is positive and below this line it is negative. At point F the tension F * ~ reaches its initial value, i.e., F * ~ = F * and S~ = 0, however, Q~ = IR ~g,a~x I and ti = t ~0. Neglecting in this case the belt elongation, tan fl = cc + Cz and tan 7 = Cz. Note that without considering cc (i.e., cG = oo) fl would be 90 ° and the pulley tooth load Si would change abruptly by R I ÷1 -- imax + IR c-~,,,x~ I if the displacement tg changed direction.

For comparison a load-displacement relationship of a real belt tooth is presented in Fig. 5. The characteristic is determined experimentally in Ref. [8] using the method stated above. The initial belt tension is F * = F* ~ = 30 N, however, in this case the load is applied by increasing F * . It is found that unlike the characteristic of a model tooth, that of a real belt tooth depends on the initial belt tension--the tooth stiffness increases with increasing initial belt tension. Still, this phenomenon, as well as the radial shifting of the belt tooth, mentioned above, are not considered in the present study.


15

Z 12

-'1-

o 9 0 I,,-.-

> -

6 O_

Z 0

,.--, 3 , < C3

-3

kJ °,,L "/ ^,~,~ .~ / I

\ \ / /~j /L v" /

OD

_ o / 0 ' ' ' " DISPLACEMENT t i

Fig. 5. Deformation characteristic of a real belt tooth.

2.3. Value and directions of the friction force Due to friction hysteresis, belt and tooth loads are affected by the foregoing loading states.

Therefore, it is not possible to determine the load distribution at a certain time or angle of revolution without considering the previous load history. This can be explained using the characteristic in Fig. 4. At a certain initial state M~ the pulley tooth load is S, and the corresponding displacement of the friction force action point is h~. After increasing the load gradually up to S;2, the displacement becomes t,~. But, if the load first reaches Si3 and then falls down to S~2, the final displacement will be t~2. Consequently, the tooth load and the value and direction of the friction force in particular can be determined correctly if the gradient of the tooth load does not change its sign within a certain time interval or between an initial and a final loading state. To meet this requirement, the rotational movement of the pulley has to be properly divided into steps. Changes in the tooth loading conditions result from the kinematics of the belt drive (because of alternating number of teeth in mesh) and from torque fluctuations which have to be considered while determining the step size. In this study, the step size is fixed at half the pitch angle, as for illustration of the procedure a further reduction of the step size is not required.

Figure 6 shows a pair of teeth at three consecutive positions--H, il i and (i + 1)I. Considering the tooth movement from one position into the other, the displacement of the acting point of the friction force represented by At/. and At(i+ 0~, respectively, is obtained by the sum of the relative displacement of the belt tooth and the relative elongation (shortening) of the belt portion:

F i l l - - Fil A / i l I : ( n i l I - - Uil ) --~ - - ; (15)

2CR

At(t+ ol = (u(i+ I)1 - uin) + 2CR

A reduction of the index i in the last equation by one yields

E l l - F( i - l ) l l At~l = (Un - u~_ ,)n) -t 2cR (16)


V

, t i I ~ •

Fi'I ~ Fu-1)I

I iI I u,i

AtiII : =tiI~

I hH ~

Fi'ii ~ Fci-~m

= v R ~

] i n I °''' (i-1)II

att~.l)L, ,'. tin

- - _ ~ ' ( i * l l l

F(,.I~I ~ 0-, ~ ~ :F~'

R(i.1)[ ~ I 0+I)I

-, } (i÷llI u(i4n

(i-1)I

iI

I J I [ I I

( i+ l ) I I ( i+ l ) I iII iI ( i -1) I ] ( i -1 ) I pulley tooth position

Fig. 6. Relative displacement between belt and pulley.

559

The deflection of the belt groove vi at a certain position (I or II) is the sum of the relative displacement At~ at this position and the deflection the groove has seen at the preceding tooth position:

Va = Atn + v(i-I)n (17a)

and

vilt = Atill + vn. (17b)

However, the last two equations are only valid if [vii < IVim,x l, otherwise Rimax is transmitted, i.e., vi = Vim,x. According to these considerations, the friction force Ri is determined either by equation (10) or by equation (11). In equation (11) (5 = 1 in case v i obtained by equations (17a) or (17b) is positive, otherwise 6 = - 1 .

2.4. Compatibi l i ty criterion

Referring to Fig. 3, it is obvious that force equilibrium alone does not provide sufficient equations for determining the tooth load Qi+ ~ given F* ~, Q~ and Ri. Inspection of the meshing geometry


gives that the difference between the belt pitch pi and the pulley pitch pp is equal to the difference between the displacement of the belt teeth i + 1 and i:

P i - Pp = ui+l - ui (18)

Substituting Pi from equation (8) into equation (18) and solving for ui+~ yields

Fi + R~/2 Ui+ I = U i - (Pp - - P b ) -I (19)

CR

Then, the load Qi+~ can be obtained by equation (9). The pitch difference in case the belt is unstretched, represented by pp --Pb, is a constant parameter of the drive and so is the force which would cause the belt pitch to equal the pulley one. This pitch equaling force Fs, first introduced by K6ster [3], can be expressed by the following equation

Fs = (pp - pb)CR (20)

F s can be determined experimentally only if the belt pitch is shorter than the pulley pitch [8]. Substituting pp--Pb from the last equation into equation (19) gives

ri + R , /2 - Fs ui+ l = ui-+ (21)

CR

One can use either equation (21) or equation (19). However, Fs can be obtained experimentally more easily than Pb and the distance between the neutral bending axis of the belt and the pulley outside diameter needed for determining pp [8].

2.5. Computa t ion procedure

Substituting Q~ from equation (9) into equation (1) and then Fi and R~ from equations (1) and (10) into equation (21), the value of u~+ ] can be evaluated given u~ and F* ]. For determining R~, the values from the foregoing pulley position used in equations (15) and (17b) or (16) and (17a) are needed. As F0 is given and R0 = 0, belt tooth loads Q~, friction forces R~, and belt tensions F~ around the arc of contact can be obtained iteratively by varying the initial value u~ until the calculated tension at the last tooth coincides with the given F.. The pulley tooth loads S~ are obtained by equations (4) and (6), respectively. A decrease of friction force due to the tooth tendency to move radially outwards can be considered empirically by diminishing the contact angle 0 in equation (11). Analysing a number of experiments, the contact angle over the pulley tooth in position lI is determined ~p/2.5 and in position z I I - ~p/2.

The pulley rotation is pursued step by step and, at every single pulley position, the distribution of the respective total circumferential load on the mating teeth is determined. The load distribution at the starting pulley position can be obtained "approximately" by any of the methods presented in Refs [4-6]. The effects of such "inaccuracy" on the load distribution are fading out with every step and virtually disappear after the first half "revolution".

3. E X P E R I M E N T A L A P P A R A T U S

For verification of the theory, the circumferential load Si acting on a pulley tooth is measured by using a pair of semiconductor strain gauges installed on a tooth made "easy" to bend by cutting slits in both tooth grooves and the experimental results are compared with calculated ones. The stiffness of the gauge tooth remains very high (12250 N/ram) and virtually does not affect the measurement. The experimental apparatus is sketched in Fig. 7. The belt (I), wound around two identical pulleys (2) and (3), experiences in the course of one pulley revolution a sinusoidal reversed and repeated torque caused by the weight G of two equal masses (4) fixed eccentrically on the pulley shafts. Consequently, the torque varies with a period of one revolution of the pulley. A geared motor (10) provides the losses and ensures a constant speed of 16 rpm. One shaft is mounted on a rocker which leans against a cantilever beam (6) with applied strain gauges for measuring the shaft force Fw. The initial tension Fv is adjusted accurately by rotating the threaded bolt (7). The pulley (2) with the gauged tooth is mounted on the other shaft. A slip ring assembly (not shown in the figure) transmits the data from the rotating pulley. An optoelectronic assembly (8), (9) gives


7

[-

6 5 3 4

//, _ _ ) _ _ 2 . . . .

~G 4 9 2

~ ' / 10

Fig. 7. Experimental assembly: (1) timing belt, (2) pulley with a gauge tooth, (3) pulley, (4) mass, (5) rocker, (6) cantilever beam with strain gauges, (7) threaded bolt, (8) optoelectronic assembly, (9) slit disc,

(10) geared motor.

561

the position of the teeth. The data are processed by a personal computer with data acquisition capability.

4. D I S C U S S I O N O F T H E O R E T I C A L A N D E X P E R I M E N T A L R E S U L T S

The test belt is section-XL (nominal belt pitch 5.08 mm, width 5 ram, number of belt teeth 98) made of polychloroprene rubber and with glass fibre tension members. The pulleys have 32 teeth each. The spring constant of the belt pitch CR and the tooth stiffness c z are experimentally determined in [8]. The stiffness of the belt groove CG is an average value obtained from a group of characteristics including that in Fig. 5 (see Ref. [8]). The pitch equaling force Fs is determined by a method represented in Ref. [8]. The values of these and other parameters used in the calculations are given in Table 1. The belt is subjected to a total circumferential load expressed by

sin(( In the experiments and calculations presented below the amplitude A is 60 N. The angle of

revolution of the pulley rp is zero when the slit tooth is in position 0I. Choosing the phase angle 0 = k . n / 2 (k = 0, 1,2, 3), Fu reaches crest and zero values when the pulley is in position II. Inspection of the drive geometry gives that in this position there are 16 teeth completely in mesh (z, = 16), while in position I there is one tooth more (zt = 17).

The shaft load under initial tension is 75 N. In drives with a fixed center distance, however, the shaft force increases somewhat after applying a torque as a result of increased belt tension mainly caused by the outwards shifting of the teeth [9]. As both Pt and Pv increase, it is supposed that the pitch difference is not affected and, therefore, setting the shaft load Fw ~ 2Fv, the error made by calculating the load distribution is not substantial. The belt strand forces F0 and F. are determined by the following equations

F0 = (fw - fu )/2 (23)

and

F. = (Fw + Fu)/2 (24) Figure 8 shows the calculated loads Si on the pulley teeth for the time of one pulley revolution,

i.e., the load distribution at 64 consecutive pulley positions (steps). The loads Si!j are connected

Table 1. Timing belt drive parameter

F w (N) F s (N) c~ (N/mm) cz (N/mm) c G (N/mm) /~ bl (mm)

75 37 4500 90 350 0.35 0.15


1 z

Q

=,

I

I...-

> -

a .

Fig. 8. Pulley tooth loads in a drive transmitting reversed and repeated torque.

with solid lines and the loads S~.k with broken lines. The subscripts j (j = 0, 2, 4 . . . . . 64) and k (k = 1, 3, 5 . . . . . 63) stand for the step number.

The load distribution at a certain step is represented in Fig. 8 by a curve lying in a plane parallel to the y - z plane. Figure 9 shows the load distribution at the time the load reaches its maximum value of 60 N (step No. 17 marked in Fig. 8 by an arrow). The influence of the previous load history on the load distribution becomes clear by comparing the load distribution in Fig. 9 with the one in Fig. 10 showing a case when a constant load Fu = 60 N is transmitted.

Z - Z

. z

c~- 6=- "¢ c=}

.._1 0 I g . .~ r r "

• " r C ) I--- "1- u _ C ) I .-- 0 C:) Z I - - 0 C )

>-

--J ~

0 . -

!/ -2

_/+ i

OII

F u = 6 0 ( s i n = / 2 ) , N

ll~ ,e ~ " " I p

,.~°,.O" V" ° " e - ' e - ' ~ - e ' " r .m~ ,v~" \ ,

r _,l ~ ' ~

I , I i I L

211 411 611 T O O T H

o . . . . o S i i I

m ta QilI

v - - - v Rii I I J I ~ I ~ I ~ I

8II 10II 12II 14II 16II 1BII

NUMBER i l I

Fig. 9. Distribution of the maximum circumferential load in a drive transmitting reversed and repeated torque.


Z - Z

-

0 <

" ~ 0

- r

C D CD

I--- > - .

_..I

O_

I i I I I i I l I

8 ,~ s i

t o i

F U :60 N ~' 6 ,"

m 4 ek-

u_° ~uz 2 I "~'~"°"e-'°'an" I/ ¢ o / I-.- u 0

-2 ~ " ~ v ' v - w/¢' o . . . . o Sil I

"v "~--v" ~" [] a Oil I

v----v Ril I -4 I l I I I I i I , I , I ,

011 211 411 611 811 1011 1211 1411 1611 1811

TOOTH NUMBER i I l Fig. 10. Load distribution in a drive transmitting a constant torque.

An experimental proof o f the load distribution in Fig. 9 could be given by measuring the load on all teeth around the arc o f contact at the same time. The distribution o f a steady load, however, can be indirectly determined by measuring the load on a single pulley tooth during its m o v e m e n t from one posit ion into another, as in this case at a certain posit ion every tooth experiences the same load as the foregoing one did. Figure l l (a) shows the result o f such measurement . For

Z

~ 6

O , , <

0

ZIZ F-- CD CD b--

2 ....1

SLIT TOOTH POSITION i l l 0 I I

011 211 411 611 811 1011 1211 1411 1611 10 , , , , , , , , , 10 ,

-2 OI

F u = 60 N ~~

•

: t,e ,

1

i

. . . . . S i calculated -J S i experimental ]

2 I 41 6 I 8 I 10I 12I 141 16I 181

SLIT TOOTH POSITION i1

Z

~- 6

r m <

O

- r I - - O O

~- 2 m

SLIT TOOTH POSITION i l l

211 411 611 811 1011 1211 1411 1611

i I i i i i I i

Fz

~ Z ~ Fu = 60 N ,! x

. . . . . S i calculated Si experimental

Ol 21 41 61 81 101 121 141 161

SLIT TOOTH POSITION il

Fig. II. Load on a gauged pulley tooth under steady torque conditions: (a) beginning of belt tooth climbing, (b) belt tooth climbing prevented by a roller.

181


comparison the calculated loads on the slit tooth are also given. In this case the pulley can be identified as driven and F 0 and F. as slack and tight side tension, respectively. The figure presents a case in which the belt tooth in the first few positions is not completely seated in the pulley groove due to low tension forces and large tooth interference. Although at this stage still not visible this phenomenon, detected by the gauge tooth, is actually the so called "tooth climbing" which begins

Z

4

< 0

2 "r F-- (D (D F--

0 F--

b'%

Z

o

"r p--

0 0 I--"

-2

-4

2

0

-2

-4

-6

(a) SLIT TOOTH POSITION ilI

011 21I 411 6II 011 1011 1211 1411 16II , l , , , , l , l , , , l , , l l , 80

a a'Ir'"" "~x 40

D? I o I ~t/,, ' . . . . . S i caLcuLated J [ - ~" - - - S i experimental ]

[ i , i , , , , , i , , , i i 1- o 0I 21 41 61 Bl 101 121 141 161 181

SLIT TOOTH POSITION il

(b) SLIT TOOTH POSITION i l l

011 211 411 611 811 1011 1211 I/+11 1611 I I I I I I I i i

- i x zx zx A A IX Z$ Z~ , lp~ . ~ _

IX LX

a ,.4

. . . . . Si calculated a

S i experimental aix A

60

40

20

0

-20

-40

-60

I

0I L I , I i I L I , I , I ~ I , I

21 41 6I 81 101 121 11+1 16I

SLIT TOOTH POSITION il F i g . 1 2 ( a , b)--Caption opposite.

i

181

% U -

O

U t'Y"

F---

Z

t l_

O

C ) .....J

<

F---- Z I.i.J rv"

t a d I t -

F--- CD

(c)

z

.Z

c2l . < ¢Z) J

-r- F-- O o


SLIT TOOTH POSITION i l I 011 211 t~ll 611 811 1011 1211 It+ll 1611

' I [ I I I I ' I ' I ' I ' /

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at approx . Fu > 0 . 7 5 . 2 F v (see Ref. [9]). As ou tward tooth d isp lacement and too th c l imbing in pa r t i cu la r are not cons idered by the theory, there is a significant difference in the shape of measured and ca lcula ted load d i s t r ibu t ion at the first few posi t ions. However , if the belt tooth is forced into the pul ley too th groove by an external force, e.g. by a roller placed on the belt back at belt too th

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position 0I, it remains in the groove due to friction at the tooth flank. In this case there is a good agreement between calculation and experiment as shown in Fig. I I(b).

Now consider Fig. 8 again. For a phase angle ~ = 0 trace the position of the slit tooth around the arc of contact step by step and connect the points marking the load in each tooth position. The result is a curve lying in a plane which is perpendicular to the x - y plane and at an angle of 45 ° to the y - z plane, provided the x and y axis have the same plotting scale. Figure 12(a) shows this curve in a plane view in comparison with the experimentally determined tooth loads. Figures 12(b), 12(c) and 12(d) present the load on the slit tooth for ~ equal to g/2, n and 3g/2, respectively.

The curves in Fig. 8 lying parallel to the x - z plane represent the pulley tooth load at a fixed position on the arc of contact for the time of one pulley revolution. Figure 13 shows the pulley and belt tooth loads and the friction forces at position 8II. This position is indicated in Fig. 8 by an arrow. Note that a negative belt tooth load at steps 1-7 and 43-65 means that the belt tooth which is in position 8II does not contact the pulley tooth which is in position 8II, but that in position 7II.

5. C O N C L U S I O N S

Under fluctuating torque conditions the load distribution is related to a certain total circumferential load acting at a certain moment and represents the state of force balance around the arc of contact at this moment. Due to the friction hysteresis every load distribution is affected by previous ones. Strictly speaking, the approach presented in this study should be generally applied, since in drives transmitting steady torques tooth load fluctuations (caused by the alternating number of teeth in mesh) also take place.

After having chosen a belt for a certain application the designer can vary only two parameters-- the initial tension Fv and the pitch difference Pp-Pb (by altering the pulley outer diameter). However, they both exert a significant influence on the tooth loads around the arc of contact. The effect of these and other parameters considered by the theory presented in this paper will be the subject of another report.

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Acknowledgement--The authors wish to express their appreciation to the Alexander von Humboldt Foundation in Bonn, Germany for sponsoring an essential part of this study.

R E F E R E N C E S 1. G. Gerbert, H. J6nsson et al., A S M E J. Mech. Design 100, 208-215 (1978). 2. T. Koyama and M. Kagotani, Bull. J S M E 22, 982-987 (1979). 3. L. K6ster, Untersuchung der Krfifteverhfiltnisse in Zahnriemenantrieben. Diss., Hochschule der Bundeswehr Hamburg

(1981). 4. M. R. Naji and K. M. Marshek, Trans. A S M E J. Mech. Transmissions Automn Design 105, 339-347 (1983). 5. N. Karolev, Optimierung der Kr~fteverhfiltnisse in Zahnriemenantrieben. Diss., TU Dresden (1987). 6. H. Peeken, F. Fischer and E. Frenken, Konstruktion 41, S. 183-I90 (1989). 7. T. Uchida, Y. Yamaji et aL, J S M E Proc. o f the Int. Conf. on Motion and Power Transmissions, Hiroshima, pp. 382-387

(Nov. 1991). 8. N. Karolev, Antriebstechnik 29, 55-58 (1990). 9. N. Karolev, Antriebstechnik 29, 69-73 (1990).

Zusammenfassung--In der vorliegenden Arbeit werden die Kr/ifteverh/iltnisse in Zahnriemenantrieben mit trapezf6rmigem Zahnprofil bei der t,)bertragung wechselnder Drehmomente untersucht. Unter der Voraussetzung, dab die zeitlichen Verltiufe yon Leer- und Lasttrumkraft bekannt sind, wird ein Iterationsverfahren zur Berechnung der Zug- und Zahnkrfifte entlang des Umschlingungsbogens vorgestellt. Die elastischen Eigenschafien der Zugstr/inge und Riemenzfihne sowie ein i.allg, vorhandener fertigungsbedingter Unterschied zwischen Riemen- und Scheibenteilung werden dabei ebenso beriick- sichtigt, wie die Reibkrfifie zwischen Scheibenzahnkopf und Riemenlfickengrund. Es wird gezeigt, dab die Verteilung der in einem bestimmten Augenblick wirkenden Umfangskraft auf die einzelnen Z/ihne entlang des Umschlingungsbogens auch yon der Vorgeschichte der Belastung abh/ingig ist. Um die Gtiltigkeit der Theorie zu prfifen, werden experimentell und theoretisch ermittelte Scheibenzahnkrfifte bei der I~bertragung eines reinen Wechseldrehmoments miteinander verglichen.

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