6
ConeBrakesandClutches
Thesebrakeshavetheadvantageofgreatertorqueforasmalleraxialforce
thaneithertypeofdiskbrakediscussedinChapter5.Themagnitudeofthe
improvementislimited,however,bytheobservationthatforsmallcone
anglesadisengagementforcemayberequired,dependingonthefriction
coefficient,becausetheinnerandouterconesmaytendtowedgetogether.
Thisisbecauseonengagementtheinnerconeisradiallycompressedandthe
outerconeisradiallyenlargedasthebrakeisengaged.Forsmallconeangles
theinducedfrictionforcedominatesthenormalforce,whichtendstoexpel
theinnercone,sothatanexternalforceisrequiredforseparation.This
characteristic,however,maybeusefulinthoseapplicationswhereabrakeis
toremainengagedinthepresenceofdisengagementforces.
I.TORQUEANDACTIVATIONFORCE
ThepertinentgeometryoftheconebrakeisshowninFigure1.Iftheinnerand
outer cones are concentric and rigid, the amount worn from the lining during
engagement will be given by
y ¼ kpr ð1-1Þ
where p denotes the pressure and r is the radius to the point where p acts.
Proportionality constant k may be evaluated by observing that the form of
relation (1-1) demands that the maximum pressure occur at the minimum
radius. Hence
y ¼ kp
max
r
i
ð1-2Þ
Copyright © 2004 Marcel Dekker, Inc.
Upon equating equations (1-1) and (1-2), we find that
p ¼ p
max

F
IGURE
1 Cone brake and its geometry (partially worn lining).
Chapter 6108
Copyright © 2004 Marcel Dekker, Inc.
wheretheelementofareaontheoutsideoftheinnerconeisgivenby
da¼2krd‘¼2kr
dr
sina
ð1-5Þ
andwherewehaveusedd‘sina=drandthePappustheoremfortheareaofa
surfaceofrevolution.Uponintegrationtheexpressionforthetorquebecomes
T¼
Akp
max
sina
r
i
r
2
0
Àr
2
i

ð1-6Þ
Sincethisexpressionvanishesforr
i
=0andforr
i

o
ð1-8Þ
Tofindtheactivationforce,wereturntoFigure1todiscoverthatitis
given by
F
a
¼
Z
A
ðp sin aþ Ap cos aÞda
¼ðsin a þ A cos aÞp
max
r
i
Z
A
1
r
2kr
dr
sin a
ð1-9Þ
¼ 2kp
max
1 þ
A
tan a

r
i

F
r
¼ 2kp
max
r
i
ðr
o
À r
i
Þ
A
tan a
À 1

ð1-11Þ
Clearly, a retraction force is necessary only when (A/tan a À 1) is greater than
zero. F
r
vanishes if
A
tan a
¼ 1
that is; if
A ¼ tan a
ð1-12Þ
The ratio of torque to activation force for a cone clutch or brake may be
obtained by dividing equation (1-6) by equation (1-9) to get
T
F

2
f/da
2
is positive whenever equation (1-15) holds,
f(A,a) is minimum along the curve
A ¼
1
tan a
ð1-16Þ
Because points on this curve represent the minimum torque that can be had
from a cone brake or clutch, it is clear that a design for such a unit should not
lie along this curve if it can be avoided.
Upon comparison of equation (3-3) with equation (1-8) we find that
equation (1-8) reduced to equation (3-3) when a = k/2. Consequently, we
may find what configuration of a cone brake or clutch can equal or exceed the
T/R ratio of a plate clutch or brake by solving
fðA; aÞ¼A ð1-17Þ
Chapter 6110
Copyright © 2004 Marcel Dekker, Inc.
Fromequation(1-14)wefindthatequation(1-17)holdswhenever
sina+Acosa=1.Hence,designsforwhichAisgreaterthan
A¼
1Àsina
cosa
ð1-18Þ
usuallyshouldbeavoidedbecauseaplateclutchhavingthesameinnerand
outerradiiwillprovidethesametorque,butwithsmalleraxialdimensions.
Thelastrelationthatisofinterestinthedesignofaconebrakeorclutch
istheconditionforwhichtheretractionforceiszero.Fromequation(1-11)it
isclearthatF

+ r
i
)F
a
].
Chapter 6112
Copyright © 2004 Marcel Dekker, Inc.
II.FOLDEDCONEBRAKE
Prototypeconebrakeshavebeendesignedandtestedforarangeofvehicle
sizes,fromtractorsandtrailerstosubcompactautomobiles[1].Boththelarge
andsmallsizesusedafoldedconedesign,asillustratedinFigures5and6,
eachwitha=27j.Althoughtheconebrakehasfewerpartsthandrum
brakes,thisadvantagemustbebalancedagainstthedisadvantageofrequiring
anoutboardwheelbearing.
Analysisofthefoldedconebrakewithasectorshoe,showninFigure5,
toobtaindesignformulasforthetorquecapabilityandtherequiredactiva-
tionforceisquitesimilartothatusedforsimpleconebrakesandclutches.
SincethebrakesillustratedinFigures5and6useasectorpad,webeginthe
analysisbyobservingfromFigure7(a)thatanelementofareaontheconical
surfacemaybewrittenas
da¼rdu
dr
sina
ð2-1Þ
F
IGURE
4DesignregionsintheA,aplaneforconeclutches/brakes.
Cone Brakes and Clutches 113
Copyright © 2004 Marcel Dekker, Inc.