Effects of Frictional Loss on Bicycle Chain Drive Efficiency

icyclergysure-

tions.catingsultsand

Downloaded F

James B. SpicerAssociate Professor

Mem. ASMEe-mail: [email protected]

Christopher J. K.Richardson

Michael J. Ehrlich

Johanna R. Bernstein

The Johns Hopkins University3400 N. Charles St.

Room 102 Maryland HallBaltimore, MD 21218

Masahiko Fukuda

Masao Terada

Shimano Inc.Product Engineering Division

Sakai Osaka 590-77 Japan

Effects of Frictional Loss onBicycle Chain Drive EfficiencyChain drive efficiency has been studied to understand energy loss mechanisms in bdrive trains, primarily for derailleur-type systems. An analytical study of frictional eneloss mechanisms for chain drives is given along with a series of experimental meaments of chain drive efficiency under a range of power, speed and lubrication condiMeasurements of mechanical efficiency are compared to infrared measurements indithat frictional losses cannot account for the observed variations in efficiency. The reof this study indicate that chain tension and sprocket size primarily affect efficiencythat non-thermal loss mechanisms dominate overall chain drive efficiency.@DOI: 10.1115/1.1412848#

i

s

,

a

eerh

t

li

r

ible-ethe

as-venracy

otesrk

tireent

h-ultsis

r-ure-ea-

see-rs

s-ita-delsralls ofthe

ivetion-eat-ion

t

IntroductionEven though chain drives have been used on bicycles for o

100 years, there has not been much published work related toquantitative analysis of the efficiency of these drive trains. Troller chain that is currently used in most bicycle chain drivesalso used widely to drive industrial equipment. However, industrial applications the operating conditions and environmentthe chain can differ significantly from that for a bicycle. Wherechains in industrial drives can be isolated in well-lubricated enronments, bicycle chains are exposed to ambient conditionsthat continuous replacement of the lubricant, as can be accplished in industrial drives, cannot be performed easily. Alsoderailleur-type chain drives, chain offset conditions are encotered that generally do not exist in industrial chain drives. Finaindustrial drives are often designed so that the chain linetension are fixed, but in bicycles, gear changes, frame flexurechain tension variations provide for a dynamically varying opating environment. The purpose of this work is to present expmental results for the efficiency of bicycle chain drives undevariety of operating conditions and to explore the factors tgovern chain drive efficiency.

In general, the design of chain drives is fairly well-understo@1#. Owing to the variety of applications and the range of opering conditions for chain drives, there are numerous factorsmust be considered in specifying a chain drive. These factorsclude chain length, load ratings, roller impact velocity, rotationforces, contact forces, chordal action and chain vibration@2#. Mostof the design considerations center on assessing the loadpower transfer characteristics of the chain or relate to chaintime. Chain drive efficiency is not a primary design consideratperhaps owing to the non-critical nature of efficiency in maapplications. However, Hollingworth and Hills@3# performed adetailed theoretical and experimental study of chain contact foduring link articulation in heavy-duty chain drives. The resultsthis study were used to calculate the theoretical efficiency of ch

Contributed by the Power Transmission and Gearing Committee for publicatiothe JOURNAL OF MECHANICAL DESIGN. Manuscript received June 1999. AssociaEditor: C. M. Gosselin.

598 Õ Vol. 123, DECEMBER 2001 Copyright

rom: http://mechanicaldesign.asmedigitalcollection.asme.org/ on 09/17/20

verthe

heis

n-forasvi-uchom-in

un-lly,ndandr-ri-aat

odat-hatin-al

andife-onny

cesofain

drives assuming that frictional losses were primarily responsfor reductions in efficiency@4#. The models developed by Hollingworth and Hills@4# indicated that the efficiency of the chain drivshould increase with the number of sprocket teeth on each ofdriver and driven sprockets. Unfortunately, the underlyingsumptions regarding their expression for efficiency were not giand no experimental results were presented to verify the accuof the models in their work.

Unlike industrial chain drives where drive efficiency might nbe important, efficiency is of prime consideration in bicycle drivowing to the nature of the power source, the cyclist. Early woon bicycle drive efficiency was performed by Keller@5# who mademeasurements on the chain drive including the rear wheel andassembly. Measurements of efficiency were made for differtransmission systems~derailleur, internally geared hub, fixed!, us-ing chains exhibiting various conditions of repair~new, used, non-lubricated!, at a variety power transfer conditions. However, witout detailed analysis of Keller’s results, it appears that the resof this work do not wholly support the idea that the efficiencygoverned by friction. Work by Kidd et al.@6# has attempted toquantify drive efficiency in relation to models of drive train peformance based on chain contact forces. While static measments of these forces have agreed with models, efficiency msurement results have not appeared@7#. Consequently, the onlypublished results relating to bicycle drive train efficiency are thogiven by Keller@5# and no reports have been made giving a dtailed analysis of the bicycle chain drive indicating the factoaffecting loss.

In this work, the efficiencies of bicycle chain drives are invetigated both experimentally and theoretically to provide quanttive measurements of chain drive efficiency and to present mofor power loss. Assuming that frictional forces degrade the oveefficiency of the system, simple analytical models for the lossechain drives have been developed to estimate and identifyprimary mechanisms of frictional loss. These models for drlosses have been used to interpret experimental results. Addially, since losses due to friction ultimately are manifested as hing of the drive components, measurement of infrared emiss

n ine

© 2001 by ASME Transactions of the ASME

13 Terms of Use: http://asme.org/terms

r

f

h

:

romfor

pinork. If

allpro-n.ly-

n ofen-long-

undce,

acts

ats

haintheonat

turewo

, 1,k bynghefor

err of

the

Downloaded F

has been performed using a digital infrared camera systemverify assumptions made regarding frictional loss.

Modeling of Chain LossesWhile a variety of mechanisms exist for degrading the trans

of power in chain drives, the ultimate effect of each is to extraenergy from the system and convert it to a form that cannotrecovered usefully. For example, energy stored as elastic defotions of components might be recovered, but frictional losses cnot be recovered during drive operation. The predominantsumption among studies assessing chain drive efficiency isfriction is the dominant loss mechanism even though other lossuch as roller impact losses are known to occur. Impact losdescribe those interactions between the chain and the sprockeresult in non-local conversion of predominately elastic deformtions into heat. While the impact intensity can be modeled@8#, itsrelationship to drive efficiency has not been explored. Conquently, models of frictional losses will be developed herecomparison to experimentally determined efficiencies. Assumthat the losses in the chain drive result from friction betwecontacting components that execute motion relative to oneother, see Fig. 1, there are three significant sources for losscan be identifiedab initio. These sources are as follows:

I. Inner link bushing and chain pinII. Chain line offsetIII. Sprocket tooth, link roller and inner link bushing.

In general, the magnitude of a frictional force is given by tproduct of a coefficient of friction and a normal force. If a normcontact pressure acts over an area then the normal force mafound by integrating the contact pressure over this area suchfriction takes the following form:

F f5mEA0

PdA. (1)

With relative motion between the contacting surfaces, this frtional force performs work. This work is expressed as follows

Wf5ElFW f•d lW. (2)

In general, since the force of friction is in the opposite directionsurface movement, the work performed is negative and therenergy loss associated with friction. The rate at which this work

Fig. 1 Schematic of chain link showing elements that are in-volved with frictional loss. Dimensions and angles used inmodeling of frictional power loss are also shown.

Journal of Mechanical Design


to

ferctbema-an-as-thatsessest thata-

se-oringenan-that

ealy bethat

ic-

ofe is

is

performed can be represented by the power loss resulting ffriction. To calculate this power, a characteristic time intervalthe frictional work must be specified.

Case I: Inner Link Bushing and Chain PinConsider the frictional force that acts between the chain link

and the inner link bushing. This force is assumed to perform wonly during link engagement and departure during articulationit is assumed that the pin and bushing have a neat fit, thensurfaces can be assumed to be in contact. The normal forceducing friction is related to the chain tension during articulatioThis tension is derived in the Appendix using a static force anasis similar to the work by Tordion@2# and by Kidd et al.@7#. Theresults of this analysis show that the tension varies as a functiolink articulation during engagement and departure. The link tsion can be related to the surface contact pressure directed athe x-axis~along the chain line! and the contact area. If this pressure is constant along the width of the pin and only varies arothe circumference of the pin at the pin-bushing contact surfathen the link tension may be written as follows:

T5rtEu i

u f

Px~u!du. (3)

If the additional assumption is made that the contact pressureradially and is a constant, thenPx5P cosu. Finally, if the contactangle between the pin and the bushing is a constant such thuc52u i5u f5p/2, then the chain tension can be written as followin terms of the contact pressure:

T>2rtP or P>T/~2rt !. (4)

The work done by this force can be written as follows:

Wf5m IrE2a

0 EA0

PdAdb

5m IrE2a

0 EA0

T0

2rt

sinw

sin~w1a1b!dAdb (5)

Wf5m IrS p

2T0D sin~w!lnU11tan~a/2!/tan~w/2!

12tan~a/2!/tan~w/2!U (6)

where the tension has been expressed in terms of the free ctension the pressure angle and the articulation angle usingresults in the Appendix. Since the chain tension is large onlyone side of the drive, this loss only has significant contributionstwo points-at engagement on the front chain ring and at deparon the rear sprocket. Adding the contributions from these tpoints yields the following:

Wf Total5m lrS p

2T0D Fsin~w1!lnU11tan~a1/2!/tan~w1/2!

12tan~a1/2!/tan~w1/2!U

1sin~w2!lnU11tan~a2/2!/tan~w2/2!

12tan~a2/2!/tan~w2/2!UG (7)

where the subscripts on the angles refer to the front chain ringand the rear sprocket, 2. The average power dissipated per linthis source is written using the period of chain revolution alowith Eq. ~7!. The resulting expression must be multiplied by tnumber of half links in the chain to obtain the total power lossthis mechanism. The following result is obtained:

Pf Total5N1v1Wf Total

/2p. (8)

Note that the functional dependence of the total frictional powloss on the articulation angle and, consequently, on the numbeteeth on the sprockets is not clear owing to the form of Eq.~7!.However, if the articulation angles are small, tan(a/2).a/2, thenthe expression for the total work can be simplified such thattotal power dissipated becomes:

DECEMBER 2001, Vol. 123 Õ 599


t

a

s

g

h

l

e

ndaretheof

thee

esthat

omes-thesenter-a.

athewill

cingh isgleglenotinkhatng

theof

thorbes:

dhe

q.isiodsult

blyles

sult

ases-oxi-ain

Downloaded F

Pf Total'N1v1m IrS p

2T0D F 1

N11

1

N2G . (9)

Case II: Chain Line OffsetOwing to the way in which derailleur-type chain drives adju

gear ratio, the chain path is planar for only one ratio and mbend to accommodate non-coplanar sprockets for all other raFor this reason, unlike other chain drives, bicycle chains aresigned to accommodate relatively large lateral offsets. The chline for an offset drive condition is illustrated in Fig. 2. Note ththe sprocket teeth engage the chain at an angle, the offset asuch that the tooth contacts the inner surface of the innerplates. The path of interaction roughly corresponds to the deptthe tooth and frictional losses occur during engagement. Owinthe bending of the chain, additional contacting forces betweeninner and outer side plates occur producing loss. To expressexcess loss produced by side plate friction in the offset conditithe normal force must be estimated. It is assumed that the cbends at a single point such that the offset angle is produthrough the action at a single pin. Although the work by Kidet al. @7# indicates that the chain bending occurs through thetion of several links, the extreme situation assumed here shoyield an upper bound estimate. Using the results in the Appenfor the link tension yields an approximate expression for the nmal force. This normal force only acts during link engagementdeparture and contributes to frictional work only during articution. Using Eqs.~1! and~2! with results from the Appendix yieldsthe following:

Wf5m II T0 singE2a

0 sinw

sin~w1a1b!r 0db (10)

where the path for the frictional work is along an arc describedthe radius of contact and the instantaneous articulation angle.small offsets, sin(g)>g and the loss is linearly related to the offsangle. For each pin, there are two points of contact that prodoffset friction and, once again, only two articulation points needbe considered on the high tension portion of the chain. Sinceexpressions for work given in Eqs.~10! and~6! are identical withrespect to functional dependence on pressure and articulaangles, approximations made for the previous case can be immented for the case currently under consideration. If the articution angles for the two sprockets are small such that small anapproximations can be made for the tangent functions, thenfollowing result for the total power loss is obtained:

Pf Total'N1v1m II ~T0r 0 sing!F 1

N11

1

N2G . (11)

Comparison of the results for Cases I and II shows that the losfrom these two sources are similar in functional dependence.ditionally, if the coefficients of friction are the same and the rspective dimensional factors are similar, then the primary diffence occurs as a result of the offset angle. Since this angle is sgenerally, it can be assumed that chain line offset does not hasignificant effect on overall efficiency.

Fig. 2 Schematic of chain links showing offset condition. Theoffset angle, g, is related to the relative lateral displacement ofthe drive sprockets. Chain offset introduces additional contact-ing forces that increase frictional losses.

600 Õ Vol. 123, DECEMBER 2001


stustios.de-aint

ngle,ide

h ofto

thethe

on,ain

ceddac-ulddixor-ora-

byFort

ucetothe

tionple-la-glethe

sesAd-e-er-mallve a

Case III: Sprocket Tooth, Link Roller and Inner LinkBushing

The interactions among the sprocket tooth, the link roller athe inner link bushing during link engagement and departurethe most complicated of the interactions considered here. Aslink engages the chain ring, the roller executes a combinationslides and rotations until it finally assumes its final position onsprocket@3#. If the roller slides, then frictional losses occur at thtooth-roller interface. If the roller rotates, then frictional lossbetween the roller and the bushing occur. It can be assumedthe load transmitted from the tooth to the roller is transferred frthe roller to the bushing. This load results in high contact prsures between the tooth and the roller and produces wear onsurfaces. The corresponding pressure at the roller-bushing iface is significantly lower as a result of the larger contact are

To specify the path over which this force performs work,mechanism for the frictional loss must be identified. Sincecontact pressures between the tooth and the roller are large, itbe assumed that the roller rolls along the tooth surface produfrictional losses between the roller and the bushing. The patsimply related to the inner radius of the roller and the anthrough which the roller rotates relative to the bushing. This andepends on the articulation of the inner link and whether orthe link is engaging or departing a given sprocket. If the inner lis departing, then this angle is the articulation angle plus tproduced by the no-slip condition of the roller on the tooth duriarticulation. This latter angle,c, is related to the outer roller di-ameter and the geometry of the tooth and is approximatelysame for all sprockets. The work performed by this sourcefriction may be written as follows:

W̄f5m III T0r Ri~c6a!•a21E

2a

0 sin~a1b!

sin~w1a1b!db (12)

whereW̄f is the average work performed during articulation withe appropriate sign~1/2! being taken to describe departureengagement of the inner link. As before, this integration canperformed by a suitable transformation. The result is as follow

W̄f5m III T0r Ri~c6a!•a21@a cosw2sinw ln~cosa

1sina cotw!#. (13)

This result differs in form from that given by Hollingworth anHills @4# since their analysis considered a cranked link chain. Ttotal work is comprised of four terms of the form given in E~13!. Using this total work, the total power dissipated by thmechanism can be written directly using the chain rotation peras the time over which an average power is calculated. The refor the total power loss is given as follows:

Pf Total5m III T0r Ri

N1v1

4p (i 51

2

@~N1c1 /p21!d1i

1~N2c2 /p11!d i #

3F2p

Nicosw i2sinw i lnS cosS 2p

NiD1sinS 2p

NiD cotw i D G .

(14)

Since the sprocket tooth number is fairly high in this study~11to 52!, the angular range produced by link articulation is probaless than the roller rotation angle. Obviously, larger roller angwill result in larger losses just as larger articulation angles rein larger losses.

The frictional loss mechanisms considered in Case I and CIII should dominate under most operating conditions. If it is asumed that the coefficients of friction in these cases are apprmately equal then a total power loss can be written for the chdrive. This power loss is given as follows:

Transactions of the ASME


r

ln

o

ot

a

t

t

ith

p

r

us

d

rivethe

gesic-fra-heat

anitiveain aop-ition

ofreeed.

earef-

a-te-

n

n ofheper-ered

ifi-

ys

Downloaded F

Pf Total5mT0

N1v1

2p H p

2r(

i 51

2

sin~w i !lnU11tan~a i /2!/tan~w i /2!

12tan~a i /2!/tan~w i /2!U

1r Ri

2 (j 51

2

@~2c1 /a121!d1 j1~2c2 /a211!d2 j #

3@a j cosw j2sinw j ln~cosa j1sina j cotw j !#J . (15)

If it is assumed that this frictional power loss governs the oveefficiency of the drive, then an expression for the efficiency canwritten directly. Simply,

h5~12Pf Total/Pin!5~12Pf Total

/~pT0v1 /a1!!. (16)

Combining Eqs.~16! and ~17! yields:

h512m

p$ f ~r,w,a,r Ri

,c!% (17)

where the functionf represents the expression in Eq.~15! that iscontained by braces. First, it is noted that this expression forficiency is independent of the chain tension. This result is toexpected since both the input power and the frictional powerare directly proportional to the chain tension. Using the nomigeometrical constants for the drive in this study (N1552;N2511, 15, or 21;r/p50.144;r Ri

/p50.204), assuming a value foc5p/2 and m50.09 for the coefficient of friction~consistentwith reported values for lubricated steel/steel interfaces! yieldsvalues for the efficiency that range between 0.97 and 0.99.

Description of the Experimental ApparatusA chain drive test stand was designed to measure the overa

average mechanical efficiency of the chain drive from the frchain ring to the rear transmission components. To assess thficiency of the drive under various conditions, the power inputthe front chain ring was measured and was compared to the pthat was output by the rear drive sprocket. The ratio of the oupower to the input power was used to quantify the overall eciency of the system. To determine the powers in the drivedriven shafts, the torques transmitted by the shafts were measalong with the rotation rate of the drive shaft. Knowing the rotion rate of the drive shaft and the gear ratio, the rotation ratethe driven shaft could be calculated. The efficiency of the syswas calculated using the following formula:

%Efficiency5t2v2

t1v13100%. (18)

The essential elements of the test stand are shown schematin Fig. 3. The drive shaft torque was measured using a rotransformer torque sensor. This unit was equipped with a sspeed sensor that was used to measure the drive shaft rotationThe torque sensor differential outputs were amplified using lonoise instrumentation amplifiers. Since the torque sensor outwere harmonically varying signals with amplitudes proportionto the torque, the signals were measured using lock-in techniqto improve the signal-to-noise of recorded signal amplitudes.relate these signals to actual torque values, each of the tosensors was calibrated using known static loads. The chain dwas configured to the geometry found on bicycles with the dtance between the front chain ring and rear cassette being adjby mounting the entire output shaft assembly on an x-y trantional platform. Mounting the output drive components on thplatform allowed for accurate adjustment of the cassette fortance and offset from the front chain ring. Additionally, the drailleur unit could be adjusted independently to satisfy recomended mounting conditions. The output shaft torque w



allbe

ef-be

ossal

r

ll ornt

e ef-towerputffi-ndureda-of

em

callyaryaftrate.w-utsaluesToqueriveis-stedla-isis-

e-m-as

measured using a second rotary torque sensor. The entire dsystem was loaded using an electromagnetic brake mounted tooutput shaft.

An infrared camera system was used to acquire thermal imaof drive components while the drive was in operation. Since frtional losses result in heating of the drive components, the inred camera was useful in identifying those components wherewas generated. The primary component of this system wasinfrared camera that operated with an InSb planar array sensto wavelengths in the 3–5mm range. The output of this camerwas recorded using a digital image acquisition board installedhost computer. Images were acquired synchronously to driveeration so that components in the drive occupied the same posin the infrared image from frame to frame.

Experimental ProcedureTo assess chain drive efficiency effectively, measurements

efficiency were made over a range of operating conditions. Thmajor areas for investigation were identified and were pursuThese areas are described briefly as follows:

1 Configuration. Efficiency was measured as a function of gratio to assess the effects of drive configuration and thefects of offset were investigated.

2 Power/Rate. Efficiency variations with input power and rottion rate were measured to determine if load or radependent effects were present.

3 Lubrication. The effects of lubrication and de-lubrication oefficiency were quantified.

The first area was pursued as a result of the predicted variatioefficiency with drive configuration shown in the models here. Tmeasurements related to input power and rotation rate wereformed to assess efficiency under conditions that are encountduring bicycle operation@9#. Finally, lubrication effects were in-vestigated to identify whether or not chain lubrication had signcant effects on drive efficiency.

Experimental Results and Analysis

Configuration. First, preliminary measurements of efficiencwere made at 60 RPM, 100W with the following configurationthat are shown with the corresponding measured efficiency:

52– 11 no-offset: 91.4%

52– 15 no-offset: 93.2%

52– 21 no-offset: 95.0%.

Fig. 3 Experimental schematic of the chain drive test standshowing elements of the drive assembly.



602 Õ Vol. 123

Downloaded From: http://mecha

Table 2 Chain Drive Efficiencies for Different Drive Rotation Rates and Sprocket Configura-tions „constant input power 100 W …

30 RPM 40 RPM 50 RPM 60 RPM 70 RPM 80 RPM 90 RPM

52–11 96.7 95.0 92.8 90.9 89.3 87.5 85.452–15 97.8 96.5 94.6 93.0 91.0 89.3 87.252–21 - 97.8 95.9 94.4 92.8 91.3 89.8

Table 3 Chain Drive Efficiencies for Different Drive Powers and Sprocket Configurations „con-stant drive rotation rate 60 RPM …

50 W 75 W 100 W 125 W 150 W 175 W 200 W

52–11 81.0 87.4 91.0 93.0 94.4 95.3 95.852–15 83.2 89.8 93.2 95.1 96.5 97.5 98.052–21 85.5 91.1 94.4 96.0 97.2 98.1 98.6

Table 1 Drive Efficiencies for Different Chain Configurations

50 RPM100 W

60 RPM100 W

70 RPM100 W

60 RPM150 W

60 RPM175 W

52–11 92.5 91.1 88.7 94.6 95.552–15 94.7 92.3 90.4 96.2 97.552–21 95.2 93.8 92.0 97.4 98.2

iri

o

e

ttA,

e

si

i

yw

ndbi-

gthe

ares inls.the

tantingre-

aintedtestionlessin

inas

udyd

en-thisince

ex-pede ofwith

aand

Note that these measurements were made with the sprocketsno-offset or coplanar configuration which is not the usual opeing configuration for the chain. The effects of chain configuraton efficiency under normal conditions were assessed in testsing 30 minutes each and were conducted with the 52–15 comnation in the no-offset condition and with the 52–11 and 52–combinations in offset conditions. The results of these testssummarized in Table 1.

Comparing the data for 60 RPM 100 W tests shows thatoffset lowers the efficiency by, at most, 0.5% when measuremprecision is considered. Additionally, if the efficiencies are nmalized by efficiencies measured in the 52–15 configuration~bothsets of data were obtained with no offset!, then it appears that theoffset has a negligible effect on efficiency.

More importantly, the efficiencies show a consistent depdence on rear sprocket size. The larger the sprocket, the highemeasured efficiency regardless of the selected power or rotarate. In particular, if the results of modeling are used, the variain the measured efficiency follows the results obtained forcombined friction mechanisms considered in Cases I and III.suming an efficiency of 95.2%~as is found for 50 RPM 100 W52–21!, the combination of these two friction mechanisms canmade to yield this efficiency result ifm50.25 which might holdfor a partially lubricated interface. Using this value for the coficient of friction in the model yields an efficiency of 92.6% fothe 52–11 combination and 94.0% for 52–15. From the datathe table, the measured efficiencies are 92.5% and 94.7% retively. These results clearly indicate that the primary mechanfor chain drive loss could be friction at the pin/bushing interfaand friction at the bushing/roller interface.

Power and Rotation Rate. The measurements of efficiencas a function of drive configuration showed significant variatof efficiency with respect to the load on the drive train. To invetigate the effects of power and rotation rate on drive train eciency, more extensive measurements of efficiency under varpowers and rotation rates were completed. Measurementsrecorded sequentially at 5–8 minute intervals. The results of thmeasurements are given in Tables 2 and 3. Table 2 shows

, DECEMBER 2001

nicaldesign.asmedigitalcollection.asme.org/ on 09/17/20

n theat-onlast-bi-

21are

theentr-

n-r thetionionhes-

be

f-rin

pec-smce

yons-ffi-ingere

esethe

variation in efficiency as a function of sprocket combination adrive rotation rate. Table 3 shows the effects of sprocket comnation and input power on efficiency.

All recorded values in these tables have a precision~due toshort-term noise! of 60.2%. Comparing efficiency results amonTables 1–3 for corresponding operating conditions shows thatlong-term precision of the efficiency measurements is60.3% un-der normal operating conditions.

These results for efficiency show that higher efficienciesexpected for those configurations that have larger sprocketcombination. This result agrees with the frictional loss modeAnother trend, supported by the data in Table 2, indicates thatefficiency decreases with increasing rotation rate for conspower input. Also, by Table 3, efficiency increases with increaspower for constant rotation rate. Both of these trends can belated to the torque applied to the drive and ultimately to the chtension. The applied torque on the drive shaft can be calculafor each result in Tables 2 and 3 and the chain tension for eachcan be obtained. Analyzing the efficiency as a function of tensshows that the efficiency increases with chain tension regardof input power or rotation rate. Additionally, for a given chatension, the efficiency is independent of drive rotation rate~be-tween 40 and 80 RPM!. The dependence of efficiency on chatension is shown in Fig. 4 where the drive efficiency is showna function of the reciprocal chain tension.

This graph shows that for the tensions investigated in this st~76.2 to 305 N! that the reciprocal of the tension is linearly relateto the drive efficiency. For each of the linear fits to the experimtal data, the correlation coefficient exceeds 0.996. However,linear relationship cannot extend to extremely large tensions sthe extrapolated efficiencies at 1/T501/N are 101.4%~52–11!,103.2%~52–15! and 103.2%~52–21!. This experimentally mea-sured dependence of efficiency on chain tension can only beplained in an indirect sense using the models for loss developreviously since these models showed no direct dependencefficiency on tension. However, if the pressure angle changestension, then the calculated losses will vary with tension inmanner not considered previously. As is discussed by Kimball



dt

h

i

t

on

vt

val-ingre-the

er-ts ofthetheofIn

on

inin

f theeslley

Downloaded F

Barr @10#, the loads carried by various links could differ consierably from that assumed in the previous analysis owing to intional manipulations of sprocket pitch to mitigate the effectschain wear. Unfortunately, without accurate knowledge conceing the operation of the drive, a definitive interpretation of tdata cannot be complete.

Lubrication. In this phase of the study, the chain was thooughly degreased/cleaned using commercially available degring agents and was relubricated using one of three commercavailable lubricants: Lubricant 1~Castrol Wrench Force DryLube!; Lubricant 2~Pedro’s Syn Lube! or Lubricant 3~Generation4 White Lightning!. After lubrication, the chain was mounted othe test sand and the efficiency of the drive was measured. SLubricant 3 had been used in other sections of this study,re-test was completed for a reduced set of drive conditions.

The results in both Tables 4 and 5 indicate that the previtrends for efficiency as a function of configuration and as a fution of chain tension are still followed. However, these results aindicate that the actual lubricant used has little effect on the oall performance of the drive under laboratory conditions givenprecision of the measurement.

Fig. 4 Variation of chain drive efficiency with the reciprocal ofthe chain tension. For this graph, chain tension has been cal-culated using the measured torque values and the radius of thefront chain ring.

Table 4 Efficiencies for Different Drive Rotation Rates andSprocket configurations „input power 100 W …

Sub-Table 4.1: Lubricant 140 RPM 60 RPM 80 RPM

52–11 92.8 89.4 84.252–15 94.0 90.9 86.552–21 95.2 92.0 88.3


52–11 93.6 89.9 85.652–15 95.6 92.6 88.852–21 95.3 92.6 89.0


52–11 - - -52–15 94.2 91.1 87.252–21 - - -



-en-ofrn-e

r-eas-ally

nincehis

usc-

lsoer-he

The most notable aspect of these efficiency data is that allues are approximately 1.5 to 2% lower than the correspondvalues in Tables 2 and 3. This drop in efficiency could havesulted from wear in the chain. The chain was removed fromtest stand and was measured to have lengthened by;3 mm overits 52 link length~104 half links!.

Infrared Measurements During Chain Drive Operation.Infrared images of the chain drive during operation aid the intpretation of the efficiency measurements and support aspecthe modeling of chain drive loss. The data acquired usinginfrared camera were recorded as a series of images for whichintensity variations related to temperature variations in the fieldview and to the infrared emissivities of the drive components.Fig. 5 the infrared pixel intensity for positions on a chain pin,a derailleur pulley tooth and on the body of the pulley~midwaybetween the bearing and the pulley teeth! are shown as a functionof time for different input powers. These data, taken at 60 RPMthe 52–21 configuration, clearly show that the chain pin risestemperature more rapidly than the other locations regardless oinput power indicating that heating results from frictional lossnear the pin. Using Fig. 5, the data for the chain and the pu

Table 5 Efficiencies for Different Drive Powers and SprocketConfigurations „input rotation rate 60 RPM …

Sub-Table 5.1: Lubricant 150 W 100 W 150 W

52–11 78.1 89.2 92.952–15 79.8 90.7 94.352–21 83.3 92.3 95.3


52–11 80.4 89.9 93.552–15 83.2 92.5 95.652–21 83.0 92.2 95.5


52–11 - - -52–15 81.2 91.1 94.452–21 - - -

Fig. 5 Variation of infrared emission with time during chainoperation for different drive components. At time equal to 0 s,the unloaded chain drive was placed under the full power load-ing indicated.



n

u

l

e

h

as

n

aan

etsu

e

c

ric-dur-ig-ns.inkson.ency

oftionsion

dert of

notderlow

llain.thedriventser

also

theatedrticu-

ero

.ero

n-

Downloaded F

tooth indicate that the component temperature rises with the inpower. Unfortunately, the power loss in each of these cases isproportional to the input power owing to the dependence of eciency on chain tension. Using measured values for efficieunder the conditions for the data in Fig. 5~97.2% for 150 W,94.4% for 100 W and 85.5% for 50 W! indicates that 4.2 W ofpower were lost at 150 W input; 5.6 W at 100 W input and 7.3at 50 W input. Obviously, for a lower power loss, the temperatrise should be lower if the lost power is converted entirely to heThis is not observed in these measurements.

The effects of link delubrication on link temperature were ainvestigated by coating the outer surfaces of the chain links wan emissivity enhancing paint that provided uniform emissivacross the surfaces of the links. Individual links in the chain wthoroughly delubricated while adjoining links were maintaineda lubricated state. Thermal measurements on these links dudrive operation indicated that the surface temperatures of thebricated and delubricated links were the same within experimeuncertainty.

Discussion and ConclusionsTests of efficiency for the derailleur-type chain drive indica

that the overall efficiencies for the transfer of power from tfront drive sprocket to the rear sprocket range from 80.9%98.6% depending on the conditions of drive operation. Primfactors affecting the efficiency include the sizes of the sprocketthe drive and the tension in the chain.

It was found that larger sprockets provide more efficient trafer of power while smaller sprockets proved to be less efficieSimple, frictional loss models were developed that could be mto have sprocket size loss variations that agreed with those vtions measured experimentally. Typically, a 2–5% loss differewas measured between the 52–11 and the 52–21 sprocket conations depending on the drive operating conditions. Experimeresults indicated that the efficiency of the chain drive varied afunction of chain tension. It was found that the efficiency varilinearly with the reciprocal of the average chain tension withhighest efficiencies occurring at high chain tensions and lowelow chain tensions. The highest efficiency measured in the st98.6%, was recorded at a chain tension of 305 N and the low80.9%, at 76.2 N.

It was found that chain line offset and chain lubrication havnegligible effect on efficiency under laboratory conditions. Calclations of frictional loss resulting from offset indicate that this loshould be small compared to those produced by other menisms. Lubrication effects on chain efficiency were tested usthree different chain lubricants under a variety of test configutions. No significant quantifiable effect of lubrication could binferred from these tests.

Fig. 6 Diagram showing chain contact forces during linkarticulation.

604 Õ Vol. 123, DECEMBER 2001


putnot

ffi-cy

Wreat.

soith

ityreinringlu-

ntal

teetoryin

s-nt.deria-cembi-ntals ad

het atdy,est,

au-ssha-

ingra-e

Infrared measurements of drive components indicate that ftional losses in the chain cause the chain temperature to riseing operation. This increase in temperature failed to provide snificant correlation with losses under various operating conditioInfrared measurements on lubricated and delubricated chain lshowed that the frictional heating did not depend on lubricatiThese infrared measurements agreed with measured efficiresults.

From the results of this study, it appears that the efficiencythe bicycle chain drive depends intimately on the chain operaas it engages and departs from the sprockets on the high-tenpart of the drive. Owing to the high efficiencies measured unhigh chain tensions, friction can only account for a few percenthe overall losses. Most probably, mechanical losses that areconverted to thermal energy in the drive account for the remainof the measured loss. These mechanical losses most likely arefrequency vibrations~,100 kHz! that are damped by the overastructure and do not produce localized heating in the drive trVariations in link engagement and departure produced bychain tension change these mechanical losses. For a givenrotation rate, larger sprockets will produce fewer of these eveper unit time and thereby incur less loss. It is likely that highchain tensions restrict the link motion during engagement andreduce the associated losses.

Appendix Chain Tension Analysis at Engagement andDeparture

Consider the diagram in Fig. 6 where the forces acting onchain link during engagement/departure are shown. The indicforces act at the angles shown. Note that the instantaneous alation angle ranges from2a ~prior to link articulation! to 0 ~whenthe link is fully engaged on the sprocket!.

Requiring that the sum of forces in the x-direction equals zyields the following:

~ i! T15T0

sinw

sin~w1a1b!.

Equation~i! gives the tension in the link during link articulationRequiring that the sum of forces in the y-direction equals zyields the following:

~ ii ! N1T1 cos~w1a1b!5T0 cosw

Solving for the normal force yields the following:

~ iii ! N5T0

sin~a1b!

sin~w1a1b!.

Note that for b52a, N50 and for b50, N5T0 sina/ sin(w1a). Both the link tension and the tooth normal force are cotinuous functions of the instantaneous articulation angle.

Nomenclature

A0 5 frictional force area of contactF f 5 frictional force magnitude

lW 5 path for frictional workN 5 sprocket tooth, normal contacting force

N1 ,N2 5 number of teeth on the drive, driven sprocketp 5 chain pitchP 5 normal contact pressure

Pf 5 frictional power lossPf Total 5 total frictional power loss

Pin 5 power delivered by drive sprocketPx 5 surface contact pressure directed along x-axisr 0 5 radius of contact during offset operation

r Ri 5 inner roller radius



t

f

of

J.

/

s,’’

Downloaded F

r R0 5 outer roller radiust 5 width of the inner link bushing

T 5 link tensionT0 5 free chain tensionT1 5 tension acting along partially engaged chain link

a,a i 5 articulation angle, articulation angle for sprocket wii teeth

b 5 instantaneous articulation angleg 5 chain offset angled 5 no-slip distance traveled by roller on sprocket tooth

surfaceh 5 efficiency of chain driveu 5 angle of integration for contact pressurem 5 coefficient of frictionr 5 link pin radiust 5 time interval for fictional work

t1 ,t2 5 drive and driven shaft torquesw 5 tooth pressure anglec 5 absolute roller rotation angle,c5d/r R0

v1 ,v2 5 drive and driven shaft angular frequencies



h

References@1# Vogwell, J., 1994, ‘‘Chain Drives,’’Rotary Power Transmission Design, K.

Hurst ed., McGraw-Hill Europe, pp. 79–95.@2# Tordion, G. V., 1996, ‘‘Chains,’’Mechanical Design Handbook, H. A. Rothbart

ed., McGraw Hill, pp. 25.1–25.12.@3# Hollingworth, N. E., and Hills, D. A., 1986, ‘‘Forces in Heavy-Duty Drive

Chain during Articulation,’’ Proc. Inst. Mech. Eng.200, No. C5, pp. 367–374.@4# Hollingworth, N. E., and Hills, D. A., 1986, ‘‘Theoretical Efficiency o

Cranked Link Chain Drive,’’ Proc. Inst. Mech. Eng.200, No. C5, pp. 375–377.@5# Keller, J., 1983, ‘‘Der Wirkungsgrad im Fahrradantrieb,’’Radmarkt, No. 12,

pp. 71–73.@6# Kidd, M., Loch, N., and Reuben, R., 1999, ‘‘Experimental Examination

Bicycle Chain Forces,’’ Exp. Mech.,39, No. 4, pp. 278–283.@7# Kidd, M., Loch, N., and Reuber, R., 1996, ‘‘Bicycle Chain Efficiency,’’The

Engineering of Sport, S. Haake, ed., Balkema, Rotterdam, pp. 217–220.@8# Chew, M., 1985, ‘‘Inertia Effects of a Roller Chain on Impact Intensity,’’

Mech. Trans. Auto. Des.107, pp. 123–130.@9# Martin, J., 1996, ‘‘Aerodynamics and Cycling,’’ http://www.krs.hia.no

;stephans/aero.htm.@10# Kimball, D. S., and Barr, J. H., 1936,Elements of Machine Design, John Wiley

and Sons, New York, p. 410.@11# Juden, C., 1997 ‘‘Measurements of Efficiency of Chain and Shaft Drive

http://www.soroos.net/hbs/chains/sn1I2I2.html.



Documents

Effects of Frictional Loss on Bicycle Chain Drive Efficiency