67071_19.pdf

CHAPTER

19 COUPLINGS, CLUTCHES, AND BRAKES

S Y M B O L S 8 , 9 , 1 °

Ar Ac b

C

Cl C2 d

al

a2 d ~ D

D1

D2

Oi Do O m el, e2, e3 E

distance between center lines of shafts in Oldham's coupling, m (in)

area, m 2 (in 2) external area, m 2 (in 2) radiating surface required, m 2 (in2) contact area of friction surface, m ~ (in 2) width of key, m (in) width of shoe, m (in) width of inclined face in grooved rim clutch, m (in) width of spring in centrifugal clutch, m (in) width of wheel, m (in) width of operating lever (Fig. 19-16), m (in)~ heat transfer coefficient, kJ/m ~ K h (kcal/m~/°C/h) specific heat of material, kJ/kg K (kcal/kg/°C). radiating factor for brakes, kJ/m 2 K s (kcal/mL/min/°C) diameter of shaft, m (in) diameter of pin, roller pin, m (in) diameter of bolt, m (in) diameter of pin at neck in the flexible coupling, m (in) diameter of hole for bolt, m (in) outside diameter of bush, m (in) diameter of wheel, m (in) diameter of sheave, m (in) outside diameter of flange coupling, m (in) inside diameter of disk of friction material in disk clutches and

brakes, m (in) outside diameter of disk of friction material in disk clutches and

brakes, m (in) inside diameter of hollow rigid type of coupling, m (in) outside diameter of hollow rigid type of coupling, m (in) mean diameter, m (in) dimensions shown in Fig. 19-16, m (in) energy (also with suffixes), N m (lbf in) Young's modulus of elasticity, GPa (Mpsi)

19.1

19.2 CHAPTER NINETEEN

F~

F2 F'a rb Fc Fn rx, F~

Fo

H

i-ia i

i l

i2 {

I kl

k s l

L Mt ;Vl,a n

?/1, n2

n

P N #N P

operating force on block brakes, kN (lbf); force at each pin in the flexible bush coupling, kN (lbf)

total pressure, kN (lbf) force (also with suffixes), kN (lbf) actuating force, kN (lbf) tension on tight side of band, kN (lbf) the force acting on disks of one operating lever of the clutch

(Fig. 19-16), kN (lbf) tension on slack side of band, kN (lbf) total axial force on i number of clutch disks, kN (lbf) tension load in each bolt, kN (lbf) centrifugal force, kN (lbf) total normal force, kN (lbf) components of actuating force F acting at a distance c from the

hinge pin (Figs. 19-25 and 19-26), kN (lbf) tangential force at rim of brake wheel, kN (lbf) tangential friction force, kN (lbf) acceleration due to gravity, 9.8066 m/s 2 (9806.6 mm/s 2)

(32.2 ft/s 2) thickness of key, m (in) thickness of central disk in Oldham's coupling, m (in) thickness of operating lever (Fig. 19-16), m (in) depth of spring in centrifugal clutch, m (mm) rate of heat to be radiated, J (kcal) heat generated, J (kcal) the rate of dissipation, J (kcal) number of pins, number of bolts, number of rollers, pairs of friction surfaces number of shoes in centrifugal clutch number of times the fluid circulates through the torus in one

second number of driving disks number of driven disks number of operating lever of clutch moment of inertia, area, m 4, cm 4 (in 4) load factor or the ratio of the actual brake operating time to the

total cycle of operation speed factor length (also with suffixes), m (in) length of spring in centrifugal clutch measured along arc,

m (in) length of bush, m (in) dimension of operating lever as shown in Fig. 19-16 torque to be transmitted, N m (lbf in) allowable torque, N m (lbf in) speed, rpm speed of the live load before and after the brake is applied,

respectively, rpm number of clutching or braking cycles per hour power, kW (hp) normal force (Figs. 19-25 and 19-26), kN (lbf) frictional force (Figs. 19-25 and 19-26), kN (lbf) unit pressure, MPa (psi)

COUPLINGS, CLUTCHES, AND BRAKES 19.3

Pa

Pb P

p,

rm rmi

rmo R R c

Rd

Rr Rx, Ry t

Ta Tar A T tc v

721, V2

W

Y o"

ob ! orb O'c(max)

Crdb T

% Tdl Td2

rf Ts oz

unit pressure acting upon an element of area of the frictional material located at an angle 0 from the hinge pin (Figs. 19-25 and 19-26), MPa (psi)

maximum pressure between the fabric and the inside of the rim, MPa (psi)

allowable pressure, MPa (psi) maximum pressure located at an angle Oa from the hinge pin

(Figs. 19-25 and 19-26), MPa (psi) bearing pressure, MPa (psi) total force acting from the side of the bush on operating lever

(Fig. 19-16), kN (lbf) the force acting from the side of the bush on one operating

lever, kN (lbf) radius, m (in) distance from the center of gravity of the shoe from the axis of

rotation, m (in) mean radius, m (in) mean radius of inner passage of hydraulic coupling, m (in) mean radius of outer passage in hydraulic coupling, m (in) reaction (also with suffixes), kN (lbf) radius of curvature of the ramp at the point of contact

(Fig. 19-21), m (in) radius of the contact surface on the driven member (Fig. 19-21),

m (in) radius of the roller (Fig. 19-21), m (in) hinge pin reactions (Figs. 19-25 and 19-26), kN (lbf) time of sing!e clutching or braking operation (Eq. 19-198), s ambient or initial temperature, °C (°F) average equilibrium temperature, °C (°F) rise in temperature of the brake drum, °C (°F) cooling time, s (min) velocity, m/s speed of the live load before and after the brake is applied,

respectively, m/s axial width in cone brake, m (in) width of band, m (in) work done, N m (lbf in) weight of the fluid flowing in the torus, kN (lbf) weight lowered, kN (lbf) weight of parts in Eq. (19-136), kN (lbf) weight of shoe, kN (lbf) deflection, m (in) stress (also with suffixes), MPa (psi) allowable or design stress in bolts, MPa (psi) design bearing stress for keys, MPa (psi) maximum compressive stress in Hertz's formula, MPa (psi) design bending stress, MPa (psi) shear stress, MPa (psi) allowable or design stress in bolts, MPa (psi) design shear stress in sleeve, MPa (psi) design shear stress in key, MPa (psi) design shear stress in flange at the outside hub diameter, MPa (psi) design shear stress in shaft, MPa (psi) one-half the cone angle, deg pressure angle, deg


032

friction angle, deg one-half angle of the contact surface of block, deg coefficient of friction factor which takes care of the reduced strength of shaft due to

keyway running speed of centrifugal clutch, rad/s speed at which the engagement between the shoe of centrifugal

clutch and pulley commences, rad/s

S U F F I X E S

a axial d dissipated, design g generated 1, i inner 2, o outer n normal x x direction y y direction 0 tangential # friction

Other factors in performance or in special aspects are included from time to time in this chapter and, being applicable only in their immediate context, are not included at this stage.

Particular Formula

19.1 COUPLINGS

C O M M O N F L A N G E C O U P L I N G (Fig. 19-1) i = 20d + 3

The commonly used formula for approximate where d i n m number of bolts i = 0.5d + 3

where d in in

7rd 3 Mt = -i-g- r/Z~

The torque transmitted by the shaft

The torque transmitted by the coupling

SI (19-1a)

u s e s (19-1b)

(19-2)

1000P M t = ~ SI (19-3a)

where Mt in N m; P in kW; ~ in rad/s

63,000P Mt = ~ u s e s (19-3b)

n

where Mt in lbf in; P in hp, n in rpm

9550P M t - ~ SI (19-3c)

n


Particular Formula

~ t ~ h p~-

• "--t ~ ~ ~ ; . ~ ~T-~- - -

.tJ. "- I-,

............. --t i

FIGURE 19-1 Flange coupling.

The torque transmitted through bolts

The torque capacity which is based on bearing of bolts

The torque capacity which is based on shear of flange at the outside hub diameter

The friction-torque capacity of the flanged coupling which is based on the concept of the friction force acting at the mean radius of the surface

The preliminary bolt diameter may be determined by the empirical formula

The bolt diameter from Eqs. (19-2) and (19-4)

The bolt diameter from Eqs. (19-3) and (19-4)

where Mt in N m; P in kW; n in rpm

159P m t = n'

where Mt in N m; P in kW; n' in rps

M t = i - - ~ % 2

Mt = i(d] 11)O'b D1 2

M t = t(rrDz)rf D2 2

M t = i# Fbr m

D + d where rm = ~ = mean radius

2

Fb = tension load in each bolt, kN (kgf)

0.5d dl = ~

vq

d2%rl

< : V 2/-q-~;

8000P dl -- Y ~ ' ~ 1

SI (19-3d)

(19-4)

(19-5)

(19-6)

(19-7)

(19-8)

(19-9)

SI (19-10a)


Particular Formula

The diameter of shaft from Eqs. (19-2) and (19-3)

The average value of diameter of the bolt circle

The hub diameter

The outside diameter of flange

where d 1 , D1 in m; P in kW; ~'b in Pa; co in rad/s

1273P d 1 - - V ~ n - ~ _ b SI (19-10b)

where dl, D1 in m; P in kW; Tb in Pa; n' in rps

76,400P dl = V ~ SI (19-10c)

where dl, D1 in m; P in kW; Tb in Pa; n in rpm

50,400P dl = V ~ u s e s (19-10d)

where dl, Dl in in; P in hp; "76 in psi; ~o in rpm where i = effective number of bolts doing work should be taken as all bolts if they are fitted in reamed holes and only half the total number of bolts if they are not fitted into reamed holes

d = ~16,000PTrrl~Zs

where P in kW; d in m

/100,800P

where P in hp; d in in

d = 3/152,800P

V 7rrlnTs

where P in kW; d in m

3/2546P

where P in kW; d in m; n' in rps

D~ = 2d + 0.05

where O l in m

Dl = 2 d + 2

D2 = 1.5d + 0.025

where D2 in m

D2 = 1.5d + 1

D = 2.5d + 0.075

where D in m

D = 2.5d + 3

SI (19-11a)

u s e s (19-11b)

SI (19-1 lc)

SI (19-1 ld)

SI (19-12a)

u s e s (19-12b)

SI (19-13a)

u s e s (19-13b)

SI (19-14a)

u s e s (19-14b)


Particular Formula

The hub length

MARINE TYPE OF FLANGE COUPLING

Solid rigid type [Fig. 19-2(a), Table 19-11

The number of bolts

The diameter of bolt

T y O " - ) i ' - - ! ~ .{.-'~ t ~-/--Taper ,[.......~2L.~.~ -

D1

~t4 "-÷--

(a) Solid rigid type (b) Hollow rigid type

FIGURE 19-2 Rigid marine coupling.

The thickness of flange

The diameter of the bolt circle


1 = 1.25d + 0.01875

where l in m and d in m

Sl (19-14c)

! = 1.25d + 0.75 USCS (19-14d)

where I and d in in

i = 3 3 d + 5 SI (19-15a)

where d in in

i = 0.85d+ 5 USCS (19-15b)

where d in in

fld "r s 4 : V.2 - 4

based on torque capacity of the shaft

/ tD2~f

" -- V 4--F5i~ based on torque capacity of flange

t = 0.25 to 0.28d

D1 = 1.4d to 1.6d

D = D1 + 2d to 3d

in 100

(19-16a)

(19-16b)

(19-17)

(19-18)

(19-19)

Taper of bolt 1 in 100

19.8 C H A P T E R N I N E T E E N

TABLE 19-1 Forged end type rigid couplings (all dimensions in mm)

Number coupling

Recessed Spigot flange flange

Shaft Flange Pitch diameter outside Locating circle Bolt Bolt hole

diameter, Flange diameter, Recess Spigot diameter, size, diameter, Number Max Min D width, t De depth, cl depth, Cz D1 dl de H8 of bolts

R1 S1 R2 $2 R3 $3 R4 $4 R5 $5 R6 $6 R7 $7 R8 $8 R9 $9 R10 S10 R l l S l l R12 S12 R13 S13 R14 S14 R15 S15 R16 S16 R17 S17 R18 S18 R19 S19 R20 $20 R21 $21 R22 $22 R23 $23 R24 $24

53 100 17 50 6 4 70 M10 11 4 45 36 120 22 60 6 4 85 M12 13 4 55 46 140 22 75 7 5 100 M14 15 4 70 55 175 27 95 7 5 125 M16 17 6 80 71 195 32 95 7 5 140 M18 19 6 90 81 225 32 125 7 5 160 20 21 6

110 91 265 36 150 9 7 190 24 25 6 130 111 300 46 150 9 7 215 30 32 6 150 131 335 50 195 9 7 240 33 34 8 170 151 375 55 195 10 8 265 36 38 6 190 171 400 55 240 10 8 290 36 38 8 210 191 445 65 240 10 8 315 42 44 8 230 211 475 70 280 10 8 340 45 46 8 250 231 500 70 280 10 8 370 45 46 10 270 251 560 80 330 10 8 400 52 55 10 300 271 600 85 330 10 8 410 56 60 10 330 301 650 90 400 10 8 480 60 65 10 360 331 730 100 400 10 8 520 68 72 10 390 361 775 105 480 11 9 570 72 76 10 430 391 875 110 480 11 9 620 76 80 12 470 431 900 115 560 11 9 670 80 85 12 520 471 925 120 560 12 10 730 90 95 12 571 521 1000 125 640 12 10 790 110 105 12 620 571 1090 130 720 12 10 850 110 115 12

F a , , - - t " c~c~

¢


Particular Formula

Hollow rigid type [Fig. 19-2(b)]

The minimum number of bolts

The mean diameter of bolt


The empirical formula for thickness of flange

The diameter of bolt circles

For design calculations of other dimensions of marine hollow rigid type of flange coupling

For dimensions of fitted half couplings for power transmission

i = 50Do SI (19-20a)

where Do in m

i=1.25Do USCS (19-20b)

where Do in in

dl = ~ ( 1 -- K4)D37s 2iD1Tb (19-21)

where K = m Oi Do

(1 - K4)D3Ts t = 8DZT f (19-22)

t = 0.25 to 0.28Do (19-23)

D1 = 1.4Do (19-24)

The method of analyzing the stresses and arriving at the dimensions of the various parts of a marine hollow flange coupling is similar to that given for the marine solid rigid type and common flange coupling.

Refer to Table 19-2.

PULLEY FLANGE COUPLING (Fig. 19-3)

The number of bolts

The preliminary bolt diameter

i - - 20d + 3 SI (19-25a)

where d in m

i = 0 .5d+ 3 USCS (19-25b)

where d in in

0.5d dt = ~ (19-26)

Vl

il' [

~Taper 1 in 20 FIGURE 19-3 Pulley flange coupling.

o~

o.,,~ r~

J r~

i I

I..-p I-~ (

G

)

} • ~G-~I I

1 ~11= 1'(3

~C! ~'

11 iJ i

I •

~---p

o

°~

.~~

..= o z~

,=.~

,=.~

,=

.~

¢'q ¢'-,I

¢'q ¢.~

ee'~

.~=

t~

t~

0',

¢",1 t~

O

X

~"

O0

e~

~

O0

O

0

t"q

O0

¢",1 ¢",1

t~

t~

,"- ~

,-'-~ ,=='~

¢'4 ¢

~

~"

19.10


Particular Formula

The width of flange l 1 (Fig. 19-3)

The hub length l

The thickness of the flange

The hub diameter

The average value of the diameter of the bolt circle


PIN OR BUSH TYPE FLEXIBLE COUPLING (Fig. 19-4, Table 19-3)

Torque to be transmitted

11 = ½ d + 0.025

where 11 and d in m

ll = ½ d + 1.0

where d in in

l = 1.4d + 0.0175

where l and d in m

l = 1.4d + 0.7

where l and d in in

t = 0.25d + 0.007

where t and d in m

t = 0.25d + 0.25

where t and d in in

D 2 = 1.8d + 0.01

where D 2 and d in m

D 2 = 1 . 8 d + 0 . 4

where D 2 and d in in

D1 = 2d + 0.025

where D 1 and d in m

D 1 -- 2d + 1.0

where D1 and d in in

D = 2.5d + 0.075

where D and d in m

D = 2 d + 3 . 0

where D and d in in

SI (19-27a)

USCS (19-27b)

SI (19-28a)

USCS (19-28b)

SI (19-29a)

USCS (19-29b)

SI (19-30a)

USCS (19-30b)

SI (19-31a)

USCS (19-31b)

SI (19-32a)

USCS (19-32b)

M t = iF D1 2

M t = ipbld' ( D1

where

Pb = bearing pressure, MPa (psi) F = force at each pin, kN ( l b f ) = pbld' d t = outside diameter of the bush, m (in)

(19-33a)

(19-33b)


I II

121

l

~ e ~ M .

=f

0.1d

Shear stress in pin

Bending stress in pin

F "rp = 0.785d 2

where

Tp = allowable shearing stress, MPa (psi) dp = dl = diameter of pin at the neck, m (in)

F ( 5 + O- b - -

71" 3

35 dp

(19-34)

(19-35)

OLDHAM COUPLING (Fig. 19-5)

The total pressure on each side of the coupling

The torque transmitted on each side of the coupling

Power transmitted

The diameter of the disk

The diameter of the boss

F = l p D h (19-36)

where h = axial dimension of the contact area, m (in)

pD2h M, = 2Fl = 6 (19-37)

where

D = the distance to the pressure area centroid / = ] from the center line, m (in)

p = allowable pressure ~>8.3 MPa (1.2 kpsi)

p = pD2hn SI (19-38a) 57,277

where P in kW

p = pD2hn USCS (19-38b) 378,180

where P in hp; D, h in in; p in psi

D = 3d + a (19-39)

D2 = 2d (19-40)

o~

=

o.,,I

m =

oil

r~

|~

~

= ~

"~

=~

0{0

v') ~.0

I ~ 0"~

~ ~

O~

tt~

,~

~

tt-) ~

t'~

~ ~

tt~

0"~ q

~

~

I I

O0

~ ,==<

('~

t ~ ~

~ ~

~ ~

O0

~ ~

('¢') ~

0

~ t~

,~

~

~ ~

~T ~

~T ~T

~T ~T

~T ~

q q

q q

~ q

~ q

q q

(3 ,,L

r (3')

v-

I ', -i~

L Ip .! ~-~Ci.--)t .~

~ '-

'o

r ,.

~ ~

o~

0";

(-

o..

8 >,,

o

a

19.13


Particular Formula

T o n g u ~

t;..N

_ _ u ~ D II __

Flange ~ ii I .[

FIGURE 19-5 Oldham's coupling.

Length of the boss

Breadth of groove

The thickness of the groove

The thickness of central disk


-~3,-b "~ k~\\\\\\\\\\\\\\\\\\\\\\\\\\\\~ --------

. . . . . . . . . . r - - - - - : . __ l ) {_ h

• !-- I :4 FIGURE 19-6 Muff or sleeve coupling.

l = 1.75d (19-41)

D w = ~ (19-42)

W h~ = ~ (19-43a)

W h = ~ (19-43b)

t = 3 d (19-44)

MUFF OR SLEEVE COUPLING (Fig. 19-6)

The outside diameter of sleeve

The outside diameter of sleeve is also obtained from equation

The length of the sleeve (Fig. 19-6)

Length of the key (Fig. 19-6)

The diameter of shaft

D = 2d + 0.013 SI (19-45a)

where D, d in m

O = 2d + 0.52 USCS (19-45b)

where D, d in in

/ 16Mt D = V ~ T d 1~] ~ K4 ) (19-46)

d where K = -

D

l = 3.5d (19-47)

l = 3.5d (19-48)

/16Mr d = V~7~.--~a (19-49)

where Mt is torque obtained from Eq. (19-2)


Particular Formula

The width of the keyway

The thickness of the key

b ..._ 2Mt

2Mr

(19-50)

(19-51)

F A I R B A I R N ' S L A P - B O X C O U P L I N G (Fig. 19-7)

The outside diameter of sleeve

The length of the lap

Use Eqs. (19-45) or (19-46)

l = 0.9d + 0.003

where l, d in m

SI (19-52a)

1 = 0.9d + 0.12

where 1, d in in

USCS (19-52b)

The length of the sleeve L = 2.25d + 0.02

where L, d in m

SI (19-53a)

L -- 2.25d + 0.8

where L, d in in

, ~ ~ ~ ~ L ~ I ~ per °f lap , 1 i n 1 2 -

I " ~

FIGURE 19-'/ Fairbairn's lap-box coupling.

USCS (19-53b)

,~ '--rr;-'l '--rrr-'___l 4 ( ! i f I ~1~ !o__[ '

I I ~IOl I q, WI I l I ~-c-~ J

, - . . . . .

FIGURE 19-8 Split muff coupling.

SPLIT M U F F C O U P L I N G (Fig. 19-8)

The outside diameter of the sleeve

The length of the sleeve (Fig. 19-8)

D = 2d + 0.013

where D, d in m

D = 2d + 0.52

where D, d in in

1 = 3.5d or 2.5d + 0.05

where 1, d in m

SI (19-54a)

USCS (19-54b)

SI (19-55a)


Particular Formula

The torque to be transmitted by the coupling

l = 3.5d or 2.5d + 2.0

where l, d in in

7rdZ o-t #id Mt = 16

where

dc - core diameter of the clamping bolts, m (in) i -- number of bolts

USCS (19-55b)

(19-56)

SLIP C O U P L I N G (Fig. 19-9)

The axial force exerted by the springs

With two pairs of friction surfaces, the tangential force

The radius of applications of Fo with sufficient accuracy

The torque

The relation between D l and D 2

It ili j .... 1

FIGURE 19-9 Slip coupling.

7r Fa = -~ (D 2 - DZ)P (19-57)

Fo -- 2#Fa (19-58)

r m = ~

D m _. D2 -+ D1 (19-59) 2 4

Mt = 0.000385(D 2 - D~)(D2 + DI)#p SI (19-60a)

Mt = 0.3927(D 2 - D2)(D2 + D, )#p

USCS (19-60b)

where the values of # and p may be taken from Table 19-4

D 2 = 1.6 (19-61) Ol

where D1 and D 2 are the inner and outer diameters of disk of friction lining


Particular Formula

S E L L E R S ' C O N E C O U P L I N G (Fig. 19-10)

The length of the box

The outside diameter of the conical sleeve

Outside diameter of the box

The length of the conical sleeve

L = 3.65d to 4d

D1 = 1.875d to 2d + 0.0125 SI

where D, d in m

D~ = 1.875d to 2d + 0.5 USCS

where D, d in in

D 2 -- 3d

1-- 1.5d

I., L = 3 65d ~_1 ,~ r4

_( , - o~5~ "" ~

. . . . . . . . . _ 1 . I

~ ' " " " I[ ---~ 1< 040d r3 " '

FIGURE 19-10 Sellers, cone coupling. FIGURE 19-11 Hydraulic coupling.

(19-62)

(19-63a)

(19-63b)

(19-64)

(19-65)

H Y D R A U L I C C O U P L I N G S (Fig. 19-11)

Torque transmitted

Percent slip between primary and secondary speeds

The mean radius of inner passage (Fig. 19-1 i)

The mean radius of outer passage (Fig. 19-11)

The number of times the fluid circulates through the torus in one second is given by

Mt Ksn 2 W(r2mo 2 --- ~ r m i ) (19-66)

1.42 (approx.) where K = coefficient = TOT

S - - f / p m MS

np x 100 (19-67)

where np and ns are the primary and secondary speeds of impeller, respectively, rpm

2 ( r32 - r~ ) (19-68a) ?'me ~-- "~ r~ r 2

2(r3 rmo=~ r] r2 (19-68b)

13,000Mt i - - (19-69) nW(r2o 2 w r m i )

r~

J r~

o~

0

~ ~

,.~

O

O

O.,

~ (3

~o

o

o

o ~

"~,

. o

o o

~ ;>

.1

oo

oo

o

o o

o

oo

oo

o

O

O

0o

O

,-~

t'~

l O

o

o o

oo

c

5o

0o

0

o

~"

O

O~

~

O

O

O

O

o0

0

o

O~

OO

0o

o

,-" O

/

0o

t'xl ¢-,I

t'~

,--~

X

¢',1 ¢

~

t'q

t"q

O

O

O

O

O

O

O

O

'~

"~

"" -"

o o

o o

o o

o o

~5 o

¢5

o

...N

~ ~

O

~ O

N

~~

~~

N

0

~N

N

o o

N

~ ~

~ .~

.~

~

~

,.a ~.a

• ~ O

O

~

O

O

O

O

~ O

O

• ~ =

= o ~

= =

= =

~.~

=

= O

O

O

O

O

O

~

O

O

o.~

_.~

.~.~

.~

.~

.~

~ o

._~.~

_

o ~

~ ~

~ ~

o ~

0 0

0 ..~

,.~

~ ~

~ ,-~-, 0

.~ ~

.,,N

~ ¢~

¢~ ¢~

,-- O

O

O

O

.,..~

~~

tr~

~.~

~.~ ~.

o

0

~~

×

o~

~

~_~.= -~ ~.~ ~

19.18


Particular Formula

Power t ransmit ted by torque converter Mt - Kn2D 5 (19-70)

where

K = coeff ic ient- -var ies with the design n = speed of driven shaft, rpm

D = outside diameter o f vanes, m (in)

19.2 C L U T C H E S POSITIVE CLUTCHES (Fig. 19-12)

Jaw clutch coupling

The area in shear

The shear stress assuming that only one-ha l f the total number o f jaws i is in actual contac t

a = 2.2d + 0.025 m c = 1.2d ÷ 0.03 m f = 1.4d + 0.0055 m g = d ÷ 0.005 m h = 0.3d + 0.0125 m i -- 0.4d ÷ 0.005 m j = 0.2d + 0.0375 m k = 1.2d + 0.02 m 1 = 1 . 7 d + 0.0584 m

A _._

0.5(a- b)h sin o~

a = 2.2d + 1.0 in c = 1 . 2 d + l . 2 i n f -- 1.4d + 0.3 in g = d + 0.2 in h = 0.3d + 0.5 in i = 0.4d ÷ 0.2 in j = 0 . 2 d + 0 . 1 5 i n k -- 1.2d + 0.8 in 1 = 1 . 7 d ÷ 2.3 in

(19-71)

(19-72)

7- =

4Fo sin c~

i(a - b)h cos oz (19-73)

7 - - -

2.8Fo

i ( a - b ) h for tan c~ = 0.7 (19-74)

where c~ = angle made by the shearing plane with the direction of pressure

, r - l - - - I - r l I IL.~., a / ]

k_; . . . . . . . t

(a)

FIGURE 19-12 Square-jaw clutch.

(b)

h

(c)


Particular Formula

FRICTION CLUTCHES

Cone clutch (Fig. 19-13)

The axial force in terms of the clutch dimensions

Axial force in terms of normal force (Fig. 19-13)

The tangential force due to friction

Torque transmitted through friction

Power transmitted

Fa = 7rDmpb sin o~ (19-75)

where

D m = ½ (D1 + D2) (approx.) c~ = one-half the cone angle, deg

= ranges from 15 ° to 25 ° for industrial clutches faced with wood

= 12.5 ° for clutches faced with asbestos or leather or cork insert

F a = F,, sin c~ (19-76)

F o = #F , sin c~ (19-77)

Mt = #FaDm (19-78) 2 sin c~

p = #FuDmn SI (19-79a) 19,100 sin c~k/

p = #FuDmn USCS (19-79b) 126,000 sin c~kl

P = 7r#pDz bn SI (19-79c) 19,100kt

a c

L " - _ - - - -

FIGURE 19-13 Cone clutch.

D

Fa


Particular Formula

The force necessary to engage the clutch when one member is rotating

The ratio (Dm/b)

The value of D m in commercial clutches

P = 7r#pD2mbn USCS 126,000kt

where kt = load factor from Table 14-7

Refer to Table 19-4 for p.

F'~ = Fn (sin a + # cos a)

Dm -- 4.5 to 8 q = b

/Pklq D i n - - 18.2 V ~ - ~

/Pktq O m : 34.2 V~p-- ~

O m -- 5d to 10d

(19-79d)

(19-80)

(19-81)

SI (19-82a)

USCS (19-82b)

(19-82c)

DISK C L U T C H E S (Fig. 19-14)

The axial force

The torque transmitted

. . . . .

Splined sleeve [

FIGURE 19-14 Multidisk clutch.

Fa = 17rpD 1 (D 2 _ D1 )

Refer to Table 19-4 for p.

M t = 1 p F a D m

where

2 (032 - Om --- -~ ( D 2 - 0 2)

for uniform pressure distribution and

Om = 1 ( 0 2 _jr_ O l )

for uniform wear

(19-83)

(19-84)

(19-85a)

(19-85b)

Power transmitted ilzFan ( D3 - D~ ) P = 28,650k--------~ D22 - d 2 SI

P : 189,000k---"---~ D 2 - d 2 USCS

for uniform pressure

(19-86a)

(19-86b)


Particular

The clutch capacity at speed nl

Formula

The speed factor

where Fa = 7rp -

p = 7ri#pnD l (D~ - D~)

76,400kt

p = 7ri#pnD 1 (D~ - D 2)

504,000kt

for uniform wear

SI

SI

(19-87a)

(19-87a)

where n - speed at which the capacity of clutch to be determined, rpm

D I M E N S I O N S OF D I S K S (Fig. 19-15)

The maximum diameter of disk

The minimum diameter of disk

The thickness of disk

The number of friction surfaces

The number of driving disks

The number of driven disks

Dffl t

2

FIGURE 19-15 Dimensions of disks.

D 2 = 2.5 to 3.6D1 (19-90)

D I - 4 d (19-91)

h = 1 to 3 mm (19-92)

i = i l + i 2 - 1 (19-93)

i il = ~ (19-94)

i i2 = ~ + 1 (19-95)

ks = 0.1 + O.O01n (19-89)

P = design power at speed, n ks = speed factor obtained from Eq. (19-89)

where

P n 1 P1 = nks (19-88)


Particular Formula

DESIGN OF A TYPICAL CLUTCH OPERATING LEVER (Fig. 19-16)

The total axial force on i number of clutch disk or plates

FIGURE 19-16 A typical clutch operating lever.

F~a = iTrp'D1 (D 1 - D2)

where p' = actual pressure between disks

4Mta F" = iTr#(D1 - D)D2m ' MPa (psi)

Mta -- allowable torque, N m (lbf in)

(19-96)

The force acting on disks of one operating lever of the clutch (Fig. 19-16)

The total force acting from the side of the bushing (Fig. 19-16)

The force acting from the side of the bushing on one operating lever (Fig. 19-16)

The thickness of the !ever very close to the pin (Fig. 19-16)

The diameter of the pin (Fig. 19-16)

F" F1 = -~- (19-97)

where i t - number of operating levers

P = {Pl (19-98)

L cot(a + 0) - el - # P1 = F1 (19-99)

e2 + # e3 + ~

[ 6_F_ale3 ]1/3 b . , (19-100)

h - - ( _ ~ ) t C r d b

where ~rdb -- design bending stress for the material of the levers, MPa (psi)

Ratio of b/h - 0.75 to 1

d = 2~r V ~T-~T d (19-101)

where

Fr = resultant force due to F1 and P1 cot(oz -+- q~) on the pin, kN (lbf)

~-d = design shear stress of the material of the pin, MPa (psi)


Particular Formula

EXPANDING-RING CLUTCHES (Fig. 19-17)

Torque transmitted [Fig. 19-17(a)]

F s --aim F" ' ' ^ ~ F..

,,,.N1 ~_

(a) (b)

FIGURE 19-17 Expanding-ring clutch.

The moment of the normal force for each half of the band [Fig. 19-17(a)]

The force applied to the ends of the split ring to expand the ring [Fig. 19-17(a)]

If the ring is made in one piece (Fig. 19-7(b)] an additional force required to expand the inner ring before contact is made with inner surface of the shell

The total force required to expand the ring and to produce the necessary pressure between the contact surfaces

Mt = 2#pwr 20

where

0 = one half the total arc of contact, rad w = width of ring, m (in)

Mo = pwrL

when 0 ~ 7r rad

Fs = pwr

w,3 (1 ,) F e = 6L d l - d

where

d l = original diameter of ring, m (in) d = inner diameter of drum, m (in) w = width of ring, m (in) t = thickness of ring, m (in)

F=F~.+Fe

E w t 3 ( 1 1) F =pwr + - - - ~ -dll--d

(19-102)

(19-103)

(19-104)

(19-105)

(19-106)

(19-107)

RIM CLUTCHES (Fig. 19-18)

When the grooved rim clutch being engaged, the equation of equilibrium of forces along the vertical axis

After the block is pressed on firmly the equation of equilibrium of forces along the vertical axis

Torque transmitted

G = G (sin a + # cos a)

F~ = F~ sin a

M t -- ½ili2FoD = ili2#flD2bp

where

il = number of grooves in the rim i 2 =-- number of shoes b = inclined face, m (in) 2/3 = angle of contact, rad

(19-108)

(19-109)

(19-110)


Particular Formula

I

FIGURE 19-18 Grooved rim clutch.

~o

~ _ 2

The width of the inclined face

Frictional force

Torque transmitted in case of a flat rim clutch when il = 1 and the number of sides b is only one-half that of a grooved rim

D = pitch diameter, m (in) 2c~ = V-groove angle, deg

b = 0.01D + 0.006 m

b - 0.0 1D + 0.25 in

Fo = #F~

where Fn' = 2/3Obp

i M t = -~ #/3DZbp

SI

USCS

(19-I l ia)

(19-1lib)

(19-112a)

(19-112b)

(19-113)

C E N T R I F U G A L C L U T C H (Fig . 19 -19)

Design of shoe

Centrifugal force for speed 031 (rad/s) at which engagement between shoe and pulley commences

Centrifugal force for running speed 03 2 (rad/s)

The outward radial force on inside rim of the pulley at speed 032

The centrifugal force for 031 - - 0.75032

w COl2 r Fcl-g Fc2 - w 0322 r

g

Fc= Fc2-F~I

eC-g

, 7w 03~r

(19-114)

(19-115)

(19-116a)

(19-116b)

(19-117)


Particular Formula

-- 4d -q Control spring ~Z,~///////,,////_ShOeb = 1.70di~~ lining • / L = 4.30d ~ . ] Oil-thrower ring-l\

, 'o. OOo. oo ~ / ' ~ ~ --1.70d~i- ~ J ~ ] ~ ~ ~ ~ " Cover

1:I --II I plate ' ~ ) ~ ' " ~ ' ~ ' - - " " ~ " \\\]- ~ ~ ~ Z ~ . ~ x ~ ' ~ ~ ~ Spider

=ttt f ~' ~--Carrierrlm 0.60d

bush

5.70d ,

FIGURE 19-19 Centrifugal clutch.

Torque required for the maximum power to be transmitted

The equation to calculate the length of the shoe (Fig. 19-19)

, w ( 2 _ ov2)rr, Mt = 4#r Fc - 4# g

where r' = inner radius of the rim

l : r c = w (~vz_~v~)r bp gbp

(19-118)

(19-119)

Spring The central deflection of flat spring (Fig. 19-19) which is treated as a beam freely supported at the points where it bears against the shoe and loaded centrally by the adjusting screw

The maximum load exerted on the spring at speed OVl

The cross section of spring can be calculated by the equation

For other proportionate dimensions of centrifugal clutch

1 Wl 3

Y = 48E1

w W = Fcl = - w21r

g

I . _ _ _ _

bh 3 Wl 3

12 48Ey

Refer to Fig. 19-19.

(19-120)

(19-121)

(19-122)

COUPLINGS, CLUTCHES, AND BRAKES 1 9 . 2 7

Particular Formula

OVERRUNNING CLUTCHES

Roller clutch (Fig. 19-20)

The condition for the operation of the clutch

The force crushing the roller

The torque transmitted

The allowable load on roller

The roller diameter

The number of roller

L O G A R I T H M I C SPIRAL ROLLER CLUTCH (Fig. 19-21) The radius of curvature of the ramp at the point of contact (Fig. 19-21)

The radius vector of point C (Fig. 19-21)

The radius of the contact surface on the driven member in terms of the roller radius and functions angles ~ and ~b (Fig. 19-21)

The tangential force

clutch roller

(a)

F ' F

Driver

Driven

FIGURE 19-20 Roller clutch.

a < 2~b (19-123)

where ~b = angle of friction, # varies from 0.03 to 0.005

For ~b = angle 1 °43', the angle a < 3026 '

Fo F = ~ (19-124)

tan a

where Fo = tangential force necessary to transmit the torque at pitch diameter D

Mc = ½FoD (19-125)

F a < iock'ld

where

k' = coefficient of the flattening of the roller

4.64ac = ~ (19-126)

for crc = allowable crushing stress

= 1035.0 MPa (150 kpsi)

d = 0.1D to 0.15D

i = ~'(D + d) 2d (19-126a)

Rc = 2(Rd -- Rr) sin 2~b s-~n2~

(19-127)

cosO ) Rd = Rc 1 + cos(2~b + ~) (19-129)

Fo = F s i n ~ (19-130)

sin 2q5 Rv = cos(2~ + ~) Rr (19-128)


Part i cu lar F o r m u l a

Fo H ,,t~,J.~¢.I,~OujIHH//

, * ~ : . 'QA_

2 ,

i

. . . .

FIGURE 19-21 Logarithmic spiral roller-clutch.

The normal force

The torque transmit ted

The max imum compressive stress at the surface area of contact between the roller and the cage made of different materials

The max imum compressive stress at the surface area of contact between the roller and the cage for V c - - 'O r - - 0.3

The max imum compressive stress at the surface area of contact between the roller and the cage made of same material (E,: = E r - - E ) and vc = Vr = 0.3

Fo Fn = ~ = F cos 4) (19-130a)

tan 4~

i F n R d M, = ~ (19-130b)

cot 4~

where

24~ = angle of wedge, deg (usually q5 varies from 3 ° to 12 °)

i = number of rollers in the clutch

Oc(max) --" 0 . 7 9 8

O'c(max) =

O'c(max) - - 0.418

(1 1) F R-~ + R-~.

2 1 - vc 1 - V r +

Er Ec

1 1

1 1

1/2

[ 1 11 l

1/2

F/F/ O'c(max) - - 0.418 V/-~r if R,. >> R r

O'c(max) -~- 0.418 ~ /2FE V l d

where

d = 2Rr = diameter of roller, m (ram) l - length of the roller, m (mm)

1/2

(19-131)

(19-132)

(19-133a)

(19-133b)

(19-133c)


Particular Formula

The design torque transmitted by the clutch

For further design data for clutches

Mtd = ildRdcrc(max) tan 0.35E

where 2~b varies from 3 to 6 deg.

Refer to Tables 19-5, 19-6, 19-7.

(19-134)

TABLE 19-5 Preferred dimensions and deviations for clutch facings (all dimensions in mm)

Outside diameter Deviation Inside diameter Deviation Thickness Deviation

120,125, 130 0 135,140, 145 -0.5 150, 155, 150 170,180, 190 200,210, 220 0 230,240, 250 -0.8 260,270,280 290,300

325, 350 0

-1.0

80, 85, 90 +0.5 95, 100, 105 0 110

120, 130, 140 +0.8 150 0 175, 203 + 1.0

0

3,3.5,4 :t:0. 1

19.3 BRAKES

ENERGY EQUATIONS

C a s e o f a ho i s t ing drum lower ing a load:

The decrease of kinetic energy for a change of speed of the live load from Vl to v2

The change of potential energy absorbed by the brake during the time t

The change of kinetic energy of all rotating parts such as the hoist drum and various gears and sheaves which must be absorbed by the brake

F(v21 - v~) Ek = 2g (19-135a)

where vl, ~2 --- speed of the live load before and after the brake is applied respectively, m/s

F = load, kN (lbf)

F Ep = 2(Vl + Vz)t (19-135b)

Wk2o(03 2 CO 2 ) Er = Z - (19-136)

2g

where

ko -- radius of gyration of rotating parts, m (mm) 031,032 = angular velocity of the rotating parts, rad/s


TABLE 19-6 Service factors for clutches

Type of service

Service factor not including starting factor

Driving machine Electric motor steady load 1.0 Fluctuating load 1.5 Gas engine, single cylinder 1.5 Gas engine, multiple cylinder 1.0 Diesel engine, high-speed 1.5 Large, low-speed 2.0

Driven machine Generator, steady load 1.0 Fluctuating load 1.0 Blower 1.0 Compressor depending on number 2.0-2.5

of cylinders Pumps, centrifugal 1.0 Pumps, single-acting 2.0 Pumps, double-acting 1.5 Line shaft 1.5 Wood working machinery 1.75 Hoists, elevators, cranes, shovels 2.0 Hammer mills, ball mills, crushers 2.0 Brick machinery 3.0 Rock crushers 3.0

TABLE 19-7 Shear strength for clutch facings

Type Facing material

Shear strength

MPa kgf/mm 2

A Solid woven or plied fabric with 7.4 0.75 or without metallic reinforcement

B Molded and semimolded 4.9 0.50 compound

m , , ,== m

m

9 1 - -

(a) Block and Wheel (b) Wear of Block

FIGURE 19-22 Single-block brake.

Particular Formula

The work to be done by the tangential force Fo at the brake sheave surface in t seconds

The tangential force at the brake sheave surface

Torque transmitted when the blocks are pressed against flat or conical surface

The operating force on block in radial direction (Fig. 19-22)

Torque applied at the braking surface, when the blocks are pressed radially against the outer or inner surface of a cylindrical drum (Fig. 19-22)

Wk FoTrD(nl + n2)t

2 x 6 0

F 0 - - 38.2(&, + Ep + Er)

D(nl + n2)

M, = #F, Dm 2

where F, = total normal force, kN (lbf)

F=Fo(20+s in20) # 4 sin 0

D ( 4sin0 ) M t = # F ~ - 2 0 + s i n 2 0

(19-137)

(19-138)

(19-139)

(19-140)

(19-141)


Particular Formula

1.30

~- 1.25 ¢N ¢-

• g 1.20 +

~D e~ 1.15 ~D

,- 1.10 . n

1.05 J

1.00 - ~ " 0 10

i , /

/

, / /

11 f

20 30 40 50 60 70 80 90 O, degree

FIGURE 19-23 (4 sin 0)/(20 + sin 20) plotted against the semiblock angle 0.

The tangential frictional force on the block (Fig. 19-22)

Torque applied when 0 is less than 60 °

4 sin 0 ) Fo = # F 20 + sin20 (19-142)

Refer to Fig. 19-23 for values of

D (approx.) M, = # F ~-

where F = #pa(brO)

4 sin 0

20 + sin 20"

(19-143)

BRAKE F O R M U L A S

Block brake formulas

For block brake formulas

Band brake formulas

For band brake formulas

The magnitude of pressure between the band and the brake sheave

The practical rule for the band thickness

Width of band

Refer to Table 19-8 for formulas from Eqs. (19-144) to (19-148)

Refer to Table 19-8 for formulas from Eqs. (19-149) to (19-157)

F1 + F 2 P = D----~ (19-158)

h = 0.005D (19-159)

F1 w -- ~ (19-160)

hod


TABLE 19-8 Formulas for block, simple, and differential band brakes

Type of brake and rotation Force at the end of brake handle, kN (kgf)

Block brake

;z :!

Rotation in either direction

Block brake Clockwise

IF

b ~~ Counterclockwise

Block brake Clockwise

Counterclockwise p---a =p b =',

(c)

Simple band brake

F _ ~ o

FieF2 (a)

I F

Clockwise

Counterclockwise

Simple band brake

F ~ o Clockwise

F I ~ F£ ~ Counterclockwise /'M -+F

(b)

* For counterclockwise direction (c/a) must be less than (1/#) or brake will be self-locking.

F = Fo a (19-144) #(a+b)

Foa (1 c) (19-145) e=77g -a

Foa (1 c) (19-146)

Foa(1 c) (19-147) F=~---~ ;+-a

~oa(1 c)" ~19148, F = a-7-~ ;-a

~ ~0__~(a e~0_e~°)l ~19 149,

~ ~0__~(a e~0_l 1) ~19 1,0,

r=~°-~( 1 ) a e~0 1 ~19151,

~ ~o__~(a e~0e~°)l ~19,,2,

COUPLINGS, CLUTCHES, AND BRAKES 19 .33

TABLE 19-8 Formulas for block, simple, and differential band brakes (Cont.)

Type of brake and rotation Force at the end of brake handle, kN (kgf)

Differential band brake

Differential band brake

1=0

I F0

~ F

v

Clockwise

Counterclockwise

If b 2 -- bl F is the same for rotation in either direction

Clockwise

Counterclockwise

F=F°( b2eu°+bl ) a e u ° - 1 (19-153)

F=F°( ble"°+b2)a e ~0- 1 (19-154)

F=FO__)b(ble"°+l)* (19-155) a e u° + 1

F -- ~F°( b2eu°-bl ~° - 1 (19-156)

F = --~F°(b2-ble"°) "° - 1 (19-157)

* F o r the a b o v e two cases, i f b2 = bl -- b.

** In this case i f b 2 < bl e"°, F will be negat ive or zero a n d the b r a k e works a u t o m a t i c a l l y or the b r ake is " se l f - lock ing ."

Particular Formula

Suitable d rum diameter according to Hagenbook

Suitable drum diameter in terms of frictional horsepower

( t)lj3 ( /)lj3 -~- < 10D < -~- SI (19-161)

where Mt in N m and D in m

--~ < D < ~ U S C S (19-162)

where Mt in lbf and D in in

(79.3#P) 1/3 < 100D < (105.8#P) 1/3 SI (19-163a)

where P in kW and D in m

(60#P) 1/3 < D < (80#P) 1/3 U S C S (19-163b)

where P in hp and D in in

# P is taken as the max imum horsepower to be dissipated in any 15-min period


Particular Formula

CONE BRAKES (Fig. 19-24)

The normal force

The radial force

The tangential force or braking force

The braking torque

sin a

Fa Fr=

tan a

Fo = #Fn = ~Fa sin o~

UFoD Mt = 2 sin

where D = mean diameter, m (mm)

CONSIDERING THE LEVER (Fig. 19-24)

The axial force aF F.= h

The relation between the operating force F and the braking force Fo

The area of the contact surface using the designation given in Fig. 19-24

The average pressure between the contact surfaces

F = hFo sin a #a

~Dw A = ~

C O S O l

where

w = axial width, m (mm) = half the cone angle, deg

F. F~ - - - m - -

F a y A 7rDw tan a

Take

a = from 10 ° to 18 ° w = 0.12D to 0.22D

fiA,

,2.'

Fa l

i (a) (b) FIGURE 19-24 Cone brake.

(19-164)

(19-165)

(19-166)

(19-167)

(19-168)

(19-169)

(19-170)

(19-171)


Particular Formula

DISK BRAKES

The torque t ransmit ted for i pairs of friction surfaces

The axial force t ransmit ted

For design values of brake facings

TABLE 19-9 Design values for brake facings

7ri#plD, (D 2 - D 2) Mt = 8 (19-172)

Fa = ½7rplDl(D2 - D1) (19-173)

where Pl = intensity of pressure at the inner radius, M P a (psi)

Refer to Table 19-9.

Facing material

Design coefficient of friction/~ MPa

Permissible unit pressure

1 m/s 10 mls

kgf]mm 2 MPa kgf]mm 2

Cast iron on cast iron Dry 0.20 Oily 0.07

Wood on cast iron 0.25-0.30 Leather on cast iron

Dry 0.40-0.50 Oily 0.15

Asbestos fabric on metal Dry 0.35-0.40 Oily 0.25

Molded asbestos on metal 0.30-0.35

0.5521-0.6824 0.0563-0.0703 0.1383-0.1726 0.0141-0.0176

0.0549-0.1039 0.0056-0.0106

0.6209-0.6894 0.0633-0.0703 0.1726-0.2069 0.0176-0.0211

1.0395-1.2062 0.106-0.123 0.2069-0.2756 0.0211-0.0281

Note: 1 kpsi = 6.894757 MPa or 1 MPa = 145 psi.

INTERNAL EXPANDING-RIM BRAKE

Forces on Shoe (Fig. 19-25)

F O R C L O C K W I S E R O T A T I O N The max imum pressure

The momen t Mtl z of the frictional forces

The momen t of the normal forces

sin 0 a

Pa- -P sin0 (19-174)

#pabr I i 2 Mttz -- sin 0 a sin O(r - a cos 0) dO

1

(19-174a)

pabra I i 2 Mtn = sin 0----~ sin 20dO (19-175)

1


Particular Formula

ly / / ~ ° ~/° 0 ;: N IdN 7 Fx ~ ~ dNcosO

Rxl ,, x

r "-I ay

FIGURE 19-25 Forces on the shoe. (J. E. Shigley, Mechanical Engineering Design, 1962, courtesy of McGraw-Hill.)

The actuating force

The torque Mt applied to the drum by the brake shoe

The hinge-pin horizontal reaction

The hinge-pin vertical reaction

F Mtn - Mtl L

m t - - #pabr 2 (cos 0! - COS 02)

sin Oa

R x - - ~ sin Oa sin 0 cos 0 dO I

Ry - sin Oa sin 2 0 dO 1

+ # sin 0 cos 0 - Fy 1

(19-176)

(19-177)

(19-178)

(19-179)


Particular Formula

F O R C O U N T E R C L O C K W I S E R O T A T I O N (Fig. 19-25)

EXTERNAL CONTRACTING-RIM BRAKE

Forces on shoe (Fig. 19-26)

F O R C L O C K W I S E R O T A T I O N The m o m e n t Mt. of the friction forces Fig. 19-26

The m o m e n t of the normal force

F = Mtn + Mtl~ ( 19-180) C

R x = ~ sin Oa sin 0 cos 0 dO 1

I; ) + # sin 20dO - F x 1

Ry - sin O~ sin 2 0 dO 1

- # sin 0 cos 0 dO - Fy 1

# p a b r Ii 2 Mt~ --- sin Oa sin O(r -- a cos O) dO

1

Pabr l °2 = sin 2 0 dO Mtn sin Oa 01

(19-181)

(19-182)

(19-183)

(19-184)

\

Fy

1

dN sine -c

IJ dN cose

dN cose

% R x

Ry

FIGURE 19-26 Forces and notation for external-contracting shoe. (J. E. Shigley, Mechanical Engineering Design, 1962, courtesy of McGraw-Hill.)


Part icu lar F o r m u l a

The actuating force

The horizontal reaction at the hinge-pin

The vertical reaction at the hinge-pin

FOR COUNTERCLOCKWISE ROTATION

F = Mtn + Mt# (19-185) ¢

= ~ sin 0 cos 0 dO Rx sin Oa o~

) + # sin 2 0 dO - Fx (19-186) 01

Ry - sin Oa # sin 0 cos 0 dO i

I ) - sin 2 0 dO + Fy (19-187) 01

F = Mtn - Mtu ( 19-188) ¢

Rx - sin Oa sin 0 cos 0 dO !

- # s i n 2 0 dO - Fx ( 1 9 - 1 8 9 ) 1

Ry - sin Oa - # sin 0 cos 0 dO !

- sin 2 0d0 + Fy (19-190) i

H E A T I N G O F B R A K E S

Heat generated from work of friction

Heat to be radiated for a brake lowering the load

The heat generated is also given by the equation

Hg = #pAcv J (joules) SI (19-191a)

#pAcv USCS (19-191b) He,= 778

H = Wh J (joules) SI (19-192a)

Wh H = - - USCS (19-192b)

778

where h = total height or distance, m (ft)

He = 754ktP SI (19-193a)

where P in kW and Hg in J/s

He, = 42.4ktP USCS (19-193b)

where P in hp


Particular Formula

The rise in temperature in °C of the brake drum or clutch plates

The rate of heat dissipation

The required area of radiating surface

Approximate time required for the brake to cool

Gagne's formula for heat generated during a single operation

For additional design data for brakes

A T = H m C (19-194)

where

m = mass of brake drum or clutch plates, kg C = specific heat capacity

= 500 J/kg °C for cast iron or steel = 0.13 Btu/lbm °F for cast iron = 0.116 Btu/lbm °F for steel

Ha = C2 ATAr SI (19-195a)

where Hd in J.

H d -- 0.25C2 A T A r Metric (19-195b)

where C2 -- radiating factor from Table 19-13 Hd in kcal.

754kiN A r - - ~ SI (19-196a)

C2 A T

where A r in m 2

0.18ktN A r = ~ SI (19-196b)

C2 A T

where A r in mm 2

WrC2 In A T tc = (19-197)

KAr

where K = a constant varying from 0.4 to 0.8

n c [3600 + -

(19-198)

where (Tav - Ta) = temperature difference between the brake surface and the atmosphere, °C

Refer to Table 19-15 for values of C.

Refer to Tables 19-11 to 19-17.

19.40 C H A P T E R N I N E T E E N

T A B L E 19-10 Working pressure for brake blocks

Pressure

Rubbing Wood blocks velocity, m/s MPa kgf/mm 2

Asbestos fabric

MPa kgf/mm 2 MPa

Asbestos blocks

kgf/mm 2

1 0.5521 0.0563 0.6894 0.0703 1.1032 0.1125 2 0.4482 0.0457 0.5521 0.0563 1 .0346 0.1055 3 0.3452 0.0352 0.4138 0.0422 0.8963 0.0914 4 0.2412 0.0246 0.2756 0.0281 0.6894 0.0703 5 0.1726 0.0176 0.2069 0.0211 0.4825 0.0492

10 0.1726 0.0176 0.2069 0.0211 0.2756 0.0281

Note: 1 kpsi = 6.894754 MPa or 1 MPa = 145 psi.

T A B L E 19-11 Comparison of hoist brakes

Block brakes

Brake Double characteristics block

Band brakes

V-grooved Both directions sheave Simple of rotation

Axial brakes

Cone Multidisk

F Force ratio

ro

Average numerical value

Relative value

Travel at lever end

Average travel, mm (in)

b b sin a b b(e u° + 1)

#a #a a(e "° - 1 ) a(e ~° - 1 )

0.667 0.282 0.0323 0.165

20.6 8.7 1 5.1

hi aa hla hlaO hlaO b b sin a 27rb 47rb

8.0 (0.313) 18.8 (0.74) 74.5 (2.943) 37.36 (1.471)

Maximum capacity P, 1512.7 18.9 227.0 75.6 kW (hp) (2000) (25) (300) (100)

b sin a #a

0.161

5.0

b hla b sin c~

32.8 (1.292)

37.8 (50)

b

n#a

0.097

3.00

ih'l a b

5.56 (0.219)

90.8 (120)

a hi = the normal distance between the sheave and the stationary braking surface to prevent dragging. b h -- b in Fig. 19-21.

T A B L E 19-12 Service factors for typical machines

Type of driven machine

Electric motor steam or water turbine

High-speed steam or gas engine

Service factors for prime movers

Petrol engine Oil engine

___4 Cyl a ___4 Cyl _>6 Cyl ___4 Cyl

Alternators and generators (excluding welding generators), induced-draft fans, printing machinery, rotary pumps, compressors, and exhausters, conveyors

Woodworking machinery, machine tools (cutting) excluding planing machines, calenders, mixers, and elevators

Forced-draft fans, high-speed reciprocating compressors, high speed crushers and pulverizers, machine tools (forming)

Rotary screens, rod mills, tube, cable and wire machinery, vacuum pumps

Low-speed reciprocating compressors, haulage gears, metal planing machines, brick and tile machinery, rubber machinery, tube mills, generators(welding)

1.5 2.0 2.5 3.0 3.5 5.0

2.0 2.5 3.0 3.5 3.0 5.5

2.5 3.0 3.5 4.0 4.5 6.0

3.0 3.5 4.0 4.5 5.0 6.5

3.5 4.0 4.5 5.0 5.5 7.0

T A B L E 19-13 Radiating factors for brakes

Temperature difference, AT

Radiating factor, Cz CzA T

W]m z K cal/m z s °C W]m z cal/m 2 s

55.5 12.26 2.93 681.36 162.73 111.5 15.33 3.66 1703.41 406.83 166.5 16.97 4.05 2827.66 675.34 222.6 18.40 4.39 4088.19 976.40

T A B L E 19-15 Values of beat transfer coefficient C for rough block surfaces

Heat-transfer coefficient, C Velocity, v, m/s W/m e K kcal/m 2 h °C

0.0 8.5 7.31 6.1 14.1 12.13

12.2 18.8 16.20 18.3 22.5 19.30 24.4 25.6 22.00 30.5 29.0 24.90

T A B L E 19-14 p v values as recommended by Hutte for brakes

Service SI

p v

Metric

Intermittent operations with long rest periods and poor heat radiation, as with wood blocks

Continuous service with short rest periods and with poor radiation

Continuous operation with good radiation as with an oil bath

26.97

13.73

40.70

2.75

1.40

4.15

1 9 . 4 2 CHAPTER NINETEEN

T A B L E 19-16 V a l u e s o f e u°

Leather belt on

Wood Proportion of contact to whole Steel band on Slightly greasy Very greasy circumference cast iron/~ = 0.18 /z = 0.47 /z = 0.12

Cast iron

Slightly greasy = 0.28

Damp /z = 0.38

0.1 1.12 1.34 1.08 1.19 1.27 0.2 1.25 1.81 1.16 1.42 1.61 0.3 1.40 2.43 1.25 1.69 2.05 0.4 1.57 3.26 1.35 2.02 2.60 0.425 1.62 3.51 1.38 2.11 2.76 0.45 1.66 3.78 1.40 2.21 2.93 0.475 1.71 4.07 1.43 2.31 3.11 0.500 1.76 4.38 1.46 2.41 3.30 0.525 1.81 4.71 1.49 2.52 3.50 0.6 1.97 5.88 1.57 2.81 4.19 0.7 2.21 7.90 1.66 3.43 5.32 0.8 2.47 10.60 1.83 4.09 6.75 0.9 2.77 14.30 1.97 4.87 8.57 1.0 3.10 19.20 2.12 5.81 10.90

T A B L E 19-17 Coef f ic ient o f fr ict ion and permiss ib le var iat ions on d i m e n s i o n s for a u t o m o t i v e b r a k e s l ining

Type and class of brake lining

Range of coefficient Permissible of friction, variation in /~ /z, %

Tolerance on width Tolerance on thickness for sizes, mm for sizes, mm

<5 mm >5 mm <5 mm >5 mm thickness thickness thickness thickness

Type I--rigid molded sets or flexible molded rolls or sets Class A--medium friction Class B--high friction

Type II--rigid woven sets or flexible woven rolls or sets Class A--medium friction Class B--high friction

0.28-0.40 +30, -20 +0 + 0 +0 +0 0.36-0.45 +30, -20 -0.2 -0.3 -0.8 -0.8

0.33-0.43 +20, -30 0.43-0.53 +20, -30

REFERENCES

1. Shigley, J. E., Machine Design, McGraw-Hi l l Book C o m p a n y , New York , 1962. 2. Maleev, V. L. and J. B. H a r t m a n , Machine Design, In te rna t iona l Tex tbook C o m p a n y , Scranton,

Pennsylvania , 1954. 3. Black, P. H., and O. E. Adams , Jr., Machine Design, McGraw-Hi l l Book C o m p a n y , New York , 1968. 4. N o r m a n , C. A., E. S. Ault , and I. F. Zarobsky , Fundamentals off Machine Design, The Macmi l lan Company ,

New York , 1951.


5. Spotts, M. F., Machine Design Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1964. 6. Spotts, M. F., Design of Machine Elements, Prentice-Hall of India Ltd., New Delhi, 1969. 7. Vallance, A., and V. L. Doughtie, Design of Machine Members, McGraw-Hill Book Company, New York,

i951. 8. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative

Society, Bangalore, India, 1962. 9. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Metric

Units), Suma Publishers, Bangalore, India, 1986. 10. Lingaiah, K., Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers,

Bangalore, India, 1986. 11. Lingaiah, K., Machine Design Data Handbook, McGraw-Hill Publishing Company, New York, 1994. 12. Bureau of Indian Standards, New Delhi, India,

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