Download - ME-311 Machine Design - Lecture 19
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Instructor: Dr. Samiur Rahman Shah
NUST School of Mechanical and Manufacturing Engineering
ME-311 Machine Designecture 1!
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Clutches, Brakes, Couplings & Flywheels: Internal E#$anding Rim %lutches & 'ra(esE#ternal %ontracting Rim %lutches & 'ra(es)rictional %ontact *#ial %lutches
%one %lutches & 'ra(esEnerg+ considerations & Tem$erature rise)riction MaterialsMiscellaneous %lutches & %ou$lings
,age
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,age 3
*n a#ial clutch isone in hich themating surfaces
are $er$endicularto the rotationala#is%lutches ma+ /eof single $late ormulti-$late t+$e
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)rictional-contact a#ial clutches
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,age 0
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* clutch ith dis( ofe#ternal diameter D andinternal diameter d issho n
e are interested ino/taining a thrust forceF for re2uired tor2ue T and $ressure p
T o methods of sol ing
the $ro/lem are in useUniform ear 4 dis(is rigidUniform $ressure 4dis( is su$$le ,age 5
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)rictional contact a#ial clutches -uniform ear
,age 6
In this anal+sis7 the dis( is considered to /e rigid*t 8rst use7 the greatest ear is at the outerradius here tangential elocit+ is greatest
*fter sometime7 the ear e2ualises and theclutch ears e2uall+ o er the hole surface
The a#ial ear can thus /e e#$ressed as
In hich onl+ P and V ar+ along the radius of theclutch dis('+ de8nition7 ear is constant7 hence
t KPV f f w 21=
( ) 1C const PV == 2C prw = 2max3d
pr pC pr ai ===
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e can ta(e the earlier e#$ression hich is thecondition for ha ing the same or( done atradius r and at radius d /2Referring to 8gure7 e ha e an element of radiusr and thic(ness dr *rea of this element is 2πrdr )orce on this element is thus dF 9 2πprdr
e can 8nd the total normal force /+ integratingo er r hich aries /et een d /2 and D /2
Thus ith pr constant
,age
( )d Dd pdr d p prdr F a D
d a
D
d −=== ∫ ∫ 22
2
2
2
2
π π π
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The tor2ue is found /+ integrating the $roduct ofthe frictional force and the radius
'+ su/stituting the alue of F from the forcee2uation7 e ma+ o/tain a more con eniente#$ression for the tor2ue. Thus;
In use7 the force e2uation gi es the actuatingforce for the selected ma#imum $ressure p a . Thise2uation holds for an+ num/er of friction $airs orsurfaces. The tor2ue e2uation a/o e gi es tor2ueout$ut for one surface ,age <
( )2222
2
2
2
82 d D
d fprdr d fpdr fpr T a
D
d a
D
d −=== ∫ ∫
π π π
( )d D Ff T +=4
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• )rictional contact a#ial clutches - uniform $ressure
,age !
In this t+$e of clutch arrangement7 s$rings areem$lo+ed to o/tain a uniform $ressure o er the
hole dis( surface
hen uniform $ressure can /e assumed o er thearea of the dis(7 the actuating force F is sim$l+the $roduct of the $ressure and area. This gi es;
Tor2ue is thus an integral of the $roduct of thefrictional force and the radius
( )224
d D p
F a −= π
( )3322
2
122 d D
fpdr r fpT
D
d −== ∫
π π
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Since $ 9 $a from earlier e2uation7 e can rite
The a/o e tor2ue alue should /e multi$lied /+the num/er of surfaces to get o erall tor2ue for amulti-$late clutch
Tor2ue for uniform ear
)or uniform $ressure
,lot of the a/o e t o e2s. on right ,age 1=
−−= 22
33
3 d Dd D Ff
T
4
1 Dd fFD
T +=
( )( )2
3
1
1
3
1
Dd Dd
fFDT
−−=
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Dis( /ra(es
,age 11
There is no f undamentaldi>erence /et een adis( /ra(e and a clutch
%an /e of a slidingcali$er ?one side $iston@or a 8#ed cali$er ?dualside $iston@ t+$e
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The go erning a#ial ear e2uation is
here 9 ear in length units
f 1 9 o$erational conditions for motion t+$e7load and s$eedf 9 conditions for tem$erature andcleanlinessA 9 material ear $ro$ert+, 9 $ressure on materialB 9 elocit+ of sliding
t 9 time of o$eration ,age 1
KPVt f f w 21=
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The coordinate ṝ locates the line of action of FCf interest is e>ecti e radius r e 7 hich is thee2ui alent of an in8nitesimal radial thic(ness
,age 13
( )∫ ∫ ∫ −== 21 12θ
θ θ θ θ
o
i
o
i
r
r
r
r prdr prdrd F
( )∫ ∫ ∫ −== 21 2122θ
θ θ θ θ
o
i
o
i
r
r
r
r dr pr f drd fpr T
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The e2ui alent radius can thus /e found from fFr e = T
The locating coordinate ṝ of the acti ating force isfound /+ ta(ing moments a/out the #-a#is
,age 10
∫ ∫
==o
i
o
i
r
r
r
r e
prdr
dr pr
fF T
r
2
( ) ( )∫ ∫ ∫ −=== 21 221 coscossinθ
θ θ θ θ θ
o
i
o
i
r
r
r
r x dr pr drd r pr r F M
( ) e x fr F M r 1221 coscos θ θ
θ θ
−−==
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Dis( /ra(es 4 uniform ear
,age 15
)or ear to /e uniform7 PV has to /e constant,ressure p e#$ressed in terms of largestallo a/le $ressure p a ?occurring at the inner
radius r i @ as p = p a r i /r
Tor2ue e2uation then /ecomes
( ) ( )ioia r r r p F −−= 12 θ θ
( ) ( ) ( )2212122
1ioia
r
r ia r r r fprdr r fpT
o
i
−−=−= ∫ θ θ θ θ
2
1
2
22
io
io
ior
r ia
r
r ia
er r
r r r r
dr r p
rdr r pr
o
i
o
i +=−−==
∫ ∫
2coscos
12
21 io r r r
+−−
= θ θ θ θ
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• %ircular ?'utton or ,uc(@ $ad cali$er /ra(e
,age 1
In this situation7 r e = δe The actuating force is gi en /+
*nd the tor2ue is gi en /+av p R F
2π =
e fFr T =
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Energ+ considerations
,age 1<
hen a rotating mem/er of a machine is causedto sto$ /+ a /ra(e7 the (inetic energ+ of rotationmust /e a/sor/ed /+ the /ra(e 4 heat
Similarl+7 hen a clutch engages dri e /et eent o mem/ers of hich one is mo ing and theother is initiall+ at rest7 (inetic energ+ is dissi$ateddue to the initial sli$$age
* clutch or /ra(e ma+ /e destro+ed /+ it s o ngenerated heat%a$a/ilit+ of a clutch de$ends on
T+$e of material of clutch
*/ilit+ of the clutch to dissi$ate heat
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%onsidering the sim$li8ed diagram of clutch
riting the e2uations for moments of inertia s. Tfor the t o mem/ers
Instantaneous elocities are
,age 1!
11θ I T =− 22θ I T =
11
1 ω θ +−= t I T
22
2 ω θ += t
I T
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here and at t =0 The di>erence in elocities ?relati e elocit+@ isgi en /+
The clutching o$eration is com$leted at theinstant hen the relati e elocit+ or w 1 - w 2 /ecomes ero7 hence
,age =
11 ω θ = 22 ω θ =
+−+−=−= 2
2
1
1
21 ω ω θ θ θ t
I T
t I T
t I I
I I T
+−−=
21
2121
ω ω θ
( )( )21
21211 I I T
I I t
+−=
ω ω
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If e assume the clutch tor2ue duringengagement constant7 hence7 e get the rate ofenerg+-dissi$ation ?$o er units@ during the$rocess to /e
This sho s that the energ+ dissi$ation rate isgreatest at the start hen t = 0 . The total energ+dissi$ated during the clutching o$eration of/ra(ing o$eration is o/tained /+ integrating thea/o e e2uation from t = 0 to t = t 1
,age 1
+−−== t I I
I I T T T u
21
2121
ω ω θ
dt t I I
I I T T udt E
t t
∫ ∫
+−−== 11
021
2121
0ω ω
( )
( )21
2
2121
2 I I
I I E
+
−= ω ω
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Tem$erature rise
,age
The tem$erature rise of a clutch F /ra(e assem/l+can /e a$$ro#imated /+
hereGT 9 Tem$erature rise%$ 9 S$eci8c heat for materialm 9 Mass of clutch or /ra(e $arts
mC
E T
p
=∆
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If an o/Hect is at initial tem$erature T 1 in anen ironment of tem$erature T inf 7 then Ne ton scooling model is e#$ressed as
here T # 9 Tem$eraturesh
cr 9 C erall heat transfer coe cient
* 9 ateral heat transfer area%ooling of dis( /ra(e F clutch is go erned /+another Ne tonian e2uation
,age 3
−=−
−∞
∞ t mC Ah
T T T T
p
cr exp1
( ) ( ) ( )∞∞ −+=−= T T Ah f hT T Ah H cr CRoss υ
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hereJ loss 9 Rate of energ+ lossh %R 9 C erall heat transfer coe cienth r 9 Radial com$onent of h %R
h 3 9 Bentilation factor
,age 0
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,age 5
Than( +ou