Machine Design Databook 2010 Part 9 doc

The weight of the active coil of a helical springFor free-length tolerances, coil diameter tolerances, and load tolerances of helical compression springs DESIGN OF HELICAL COMPRESSION SP

The general expression for size factor

Wire diameter

SELECTION OF MATERIALS AND

STRESSES FOR SPRINGS

For materials for springs7

The torsional yield strength

The maximum allowable torsional stress for static

applications according to Joerres8;9;11

The maximum allowable torsional stress according to

Shigley and Mischke9

The shear endurance limit according to Zimmerli10

The torsional modulus of rupture

0:45sut cold-drawn carbon steel

0 :50sut hardened and tempered

carbon and low-alloy steel

0 :35sut austenitic stainless steel

and nonferrous alloys

for unpeened springs

for peened springs

The weight of the active coil of a helical spring

For free-length tolerances, coil diameter tolerances,

and load tolerances of helical compression springs

DESIGN OF HELICAL COMPRESSION SPRINGS

Design stress

The size factor

The design stress

Spring design stress, d, MPa (kpsi)

TABLE 20-11

Free-length tolerances of squared and ground helical compression springsa

Tolerances:mm/mm (in/in) of free length

Spring index (D=d)Number of active

TABLE 20-13

Load tolerances of helical compression springs

Tolerance:% of load, start with tolerance from Table 20-11 multiplied by LF

Deflection from free length to load, mm (in)Length

First load test at not less than 15% of available deflection; final load test at not more than 85% of available deflection

Source: Associated Spring, Barnes Group Inc., Bristol, Connecticut

lo
3dp

Source: K Lingaiah and B R Narayana Iyengar, Machine Design Data Handbook, Vol I, Suma Publishers, Bangalore, India, 1986, and K.Lingaiah, Machine Design Data Handbook, Vol 11, Suma Publishers, Bangalore, India, 1986

SPRINGS

The actual factor of safety or reliability factor

The wire diameter for static loading

The wire diameter where there is no space limitation

ðD ¼ cdÞ

ds ¼ e

naksz¼ 1 :89e

nad0:25 Metric ð20-49cÞ where ein kgf/mm2and d in mm

where na ¼ actual factor of safety or reliability factor

na¼ FðcompressedÞ

na ¼ free length
fully compressed length

free length
working length

¼ y þ a

where y is deflection under working load, m (mm),

a is the clearance which is to be added when determining the free length of the spring and

is made equal to 25% of the working deflection

Generally nais chosen at 1.25.

d ¼ 1:445

6naF

d ¼

6naF

Final dimensions (Fig 20-7d)

The number of active coils

The minimum free length of the spring

Outside diameter of cod of helical spring

Solid length (or height) of helical spring

Pitch of spring

Free length of helical spring lf or lo

Maximum working length of helical spring

Minimum working length of helical spring

Springs with different types of ends1;2;3

STABILITY OF HELICAL SPRINGS

The critical axial load that can cause buckling

d ¼ 1:77

naF

where

a ¼ clearance, m (mm)

n ¼ 2 if ends are bent before grinding

¼ 1 if ends are either ground or bent

¼ 0 if ends are neither ground nor bent

SPRINGS

The equivalent stiffness of springs

The critical load on the spring

The critical deflection is explicitly given by

REPEATED LOADING (Fig 20-9)

The variable shear stress amplitude

The mean shear stress

Design equations for repeated loadings1;2;3

2

ycr

lf þ
22

1 þ v

2 þ v

D lf

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SPRINGS

The Goodman straight-line relation

The Soderberg straight-line relation

Method 2

The static equivalent of cyclic load Fm Fa

The relation between e and f for brittle material

The static equivalent of cyclic load for brittle material

The relation between Fm, Fmax0 and Fmin

The diameter of wire for static equivalent load

The wire diameter when there is no space limitation

e

0:4

D0:3 SI ð20-67aÞ where F in N, ein MPa, D in m, and d in m

d ¼

3nað3Fmax
FminÞ

e

0:4

D0:3 USCS ð20-67bÞ where F in lbf, e in psi, D in in, and d in in

d ¼ 0:724

3na ð3Fmax
FminÞ

e

0:4

D0:3Metric ð20-67cÞ where F in kgf, e in kgf/mm2, D in mm, and d in mm

d ¼ 1:67

3nað3Fmax
FminÞ

e

0:57

c0:43 SI ð20-68aÞ where F in N, ein MPa, and d in m

d ¼

3nað3Fmax
FminÞ

e

0:57

c0:43 USCS ð20-68bÞ where F in lbf, e in psi, and d in in

d ¼ 0:64

3nað3Fmax
FminÞ

e

0:57

c0:43Metric ð20-68cÞ where F in kgf, ein kgf/mm2, and d in mm

SPRINGS

CONCENTRIC SPRINGS (Fig 20-10)

The relation between the respective loads shared by

each spring, when both the springs are of the same

length

The relation between the respective loads shared by

each spring, when both are stressed to the same value

The approximate relation between the sizes of two

concentric springs wound from round wire of the

same material

FIGURE 20-10 Concentric spring

Total load on concentric springs

The total maximum load on the spring

The load on the inner spring

The load on the outer spring

VIBRATION OF HELICAL SPRINGS

The natural frequency of a spring when one end of the

spring is at rest

F1 F2 ¼

D3 D1

3

d1 d2

4i2 i1

G1

F1 F2 ¼ D2 D1

d1 d2

3k1

F1 F2 ¼

D2 D1

0:75
d1 d2

2:5

ð20-71Þ where suffixes 1 and 2 refer, respectively, to springs

r

¼ 0:705

ffiffiffiffiffiffi k0 W

r

SI ð20-75Þ where

The natural frequency of a spring when both ends are

fixed

The natural frequency for a helical compression

spring one end against a flat plate and free at the

other end according to Wolford and Smith7

Another form of equation for natural frequency of

compression helical spring with both ends fixed

with-out damping effect

fn ¼ 22:3

k0 W

1=2

SI ð20-75aÞ where k0 in N/mm, W in N, fnin Hz,

g ¼ 9086:6 mm=s2

fn¼ 4:42

k0 W

1=2

USCS ð20-75bÞ where k0 in lbf/in, W in lbf, fn in Hz, g ¼ 32:2 ft=s2

fn¼ 1:28

k0 W

1=2

USCS ð20-75cÞ where k0 in lbf/in, W in lbf, fn in Hz,

g ¼ 386:4 in=s2

fn ¼
1

ffiffiffiffiffiffiffiffiffiffi 2k0g W

r

¼ 1:41

ffiffiffiffiffiffi k0 W

r

SI ð20-76Þ where k0 in N/m, W in N, fn in Hz,

g ¼ 9:0866 mm=s2

fn ¼ 44:6

k0 W

1=2

SI ð20-76aÞ where k0 in N/mm, W in N, fnin Hz,

g ¼ 9086:6 mm=s2

fn¼ 2:56

k0 W

1=2

USCS ð20-76bÞ where k0 in lb/ft, W in lbf, fnin Hz, g ¼ 32:2 ft=s2

fn¼ 8:84

k0 W

1=2

USCS ð20-76cÞ where k0 in lbf/in, W in lbf, fnin Hz,

g ¼ 386:4 in=s2

fn¼ 0:25

k0g W



1=2

SI ð20-76eÞ where

G ¼ shear modulus, MPa

STRESS WAVE PROPAGATION IN

CYLINDRICAL SPRINGS UNDER IMPACT

LOAD

The velocity of torsional stress wave in helical

com-pression springs

The velocity of surge wave (Vs)

The impact velocity (Vimp)

The frequency of vibration of valve spring per minute

fn ¼ 0 :11d

D2i

Gg



1=2

USCS ð20-76gÞ where

G ¼ modulus of rigidity, psi



1=2

SI ð20-76iÞ where V in m/s, G in MPa, g ¼ 9:8066 m=s2,  in g/cm3

V ¼

Gg



1=2

USCS ð20-76jÞ where V in in/s, G in psi, g ¼ 386:4 in=s2,  in lbf =in3

(It varies from 50 to 500 m/s.)

Vimp¼ 10:1

g

r

SI ð20-77aÞ where k0 in N/m, W in N

fn¼ 2676:12

ffiffiffiffiffiffi k0 W

r

Metric ð20-77bÞ where k0 in kgf/mm, W in kgf

fn¼ 530

ffiffiffiffiffiffi k0 W

r

USCS ð20-77cÞ where k0 in lbf/in, W in lbf

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SPRINGS

HELICAL EXTENSION SPRINGS (Fig 20-11

to 20-13)

For typical ends of extension helical springs

The maximum stress in bending at point A (Fig

20-12)

The constant K1 in Eq (20-78a)

The constant C1 in Eq (20-78b)

Recommended length

Cross center loop

or hook

I.D

required by design

FIGURE 20-11 Common-end configuration for helical extension springs Recommended length is distance from last body coil

to inside of end ID is inside diameter of adjacent coil in spring body (Associated Spring, Barnes Group, Inc.)

FIGURE 20-12 Location of maximum bending and torsional stresses

in twist loops (Associated Spring, Barnes Group, Inc.)

SPRINGS

The maximum stress in torsion at point B (Fig 20-12)

The constant C2 in Eq (20-78d)

For extension helical spring dimensions

FIGURE 20-13 Typical extension-spring dimensions (Associated Spring, Barnes Group, Inc.)

For design equations of extension helical springs

The spring rate

The stress

CONICAL SPRINGS [Fig 20-14(a)]

The axial deflection y for i coils of round stock may be

computed by the relation [Fig 20-14(a)]

The axial deflection of a conical spring made of

rectangular stock with radial thickness b and an

axial dimension h [Fig 20-14(c)]

For R1 , refer to Fig 20-12.

B ¼ 8DF d3 4C2
1

C2 ¼ 2R2

For R2 , refer to Fig 10-12.

In practice C2 may be taken greater than 4.

NONMETALLIC SPRINGS

Rectangular rubber spring (Fig 20-15)

Approximate overall dimension of the shock absorber

can be obtained by (Fig 20-15)

Spring constant K of an absorber

Dimensions of sleeve and core are found by empirical

relations

FIGURE 20-15 Rectangular rubber spring

TORSION SPRINGS (Fig 20-16)7

The maximum stress in torsion spring

The stress in torsion spring taking into consideration

the correction factor k0

The deflection

The stress in round wire spring

FIGURE 20-14 Conical and volute springs

L

D2¼
E 2F2

U ðFmax=FÞ2
1

SPRINGS

The stress is also given by Eq (20-90) without taking

into consideration the direct stress (F/A)

The expressions for k for use in Eq (20-90)

Equation (20-90) for stress becomes

The angular deflection in radians

The spring rate of torsion spring

The spring rate can also be expressed by Eq (20-95),

which gives good results

The allowable tensile stress for torsion springs

The endurance limit for torsion springs

Torsion spring of rectangular cross section

The stress in rectangular wire spring

Axial dimension b after keystoning

Another expression for stress for rectangular

cross-sectional wire torsion spring without taking into

consideration the direct stress (  ¼ F=A)

The spring rate

FIGURE 20-17 Torsion bar spring

Torsion bar springs

For allowable working stresses for rubber

compres-sion springs

sy¼ a¼

0 :78sut cold-drawn carbon steel

0 :87sut hardened and tempered

carbon and low-alloy steels

0 :61sut stainless steel

and nonferrous alloys

16Mt

d3Hollow circular

k01bh3G

k022bh2 a

aValues of k01and k02can be obtained from Table 20-9

TABLE 20-17 Factors for computing rectangular bars in torsion

Suggested allowable working stresses for rubber compression springs

Limits of allowable stressOccasional loading Cont or freq loadingb

4 SAE Handbook, Springs, Vol I, 1981.

5 Maleev, V L., and J B Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954.

6 Wahl, A M., Mechanical Springs, McGraw-Hill Book Company, New York, 1963.

7 Associated Spring, Barnes Group Inc., Bristol, CT, USA.

8 Jorres, R E., Springs; Chap 24 in J E Shigley and C R Mischke, eds., Standard Handbook of Machine Design, McGraw-Hill Book Company, New York, 1986.

9 Shigley, J E., and C R Mischke, Mechanical Engineering Design, 5th ed McGraw-Hill Company, New York, 1989.

10 Zimmerli, F P., Human Failures in Springs Applications, The Mainspring, No 17, Associated Spring Corporation, Bristol, Connecticut, Aug.-Sept 1957.

11 Shigley, J E., and C R Mischke, Standard Handbook of Machine Design, McGraw-Hill Book Company, New York, 1986.

12 Phelan, R M., Fundamentals of Mechanical Design, Tata-McGraw-Hill Publishing Company Ltd, New Delhi, 1975.

13 Lingaiah, K., Machine Design Data Handbook of Machine Design, 2nd edition, McGraw-Hill Publishing Company, New York, 1996).

14 Shigley, J E., and C R Mischke, Standard Handbook of Machine Design, 2nd edition, McGraw-Hill Publishing Company, New York, 1996.

Chironis, N P., Spring Design and Application, McGraw-Hill Book Company, 1961.

Norman, C A., E S Ault, and I F Zarobsky, Fundamentals of Machine Design, The Macmillan Company, New York, 1951.

Shigley, J E., Machine Design, McGraw-Hill Book Company, 1962.

SPRINGS

21

FLEXIBLE MACHINE ELEMENTS

SYMBOLS11;12;13

pivot arm length in Rockwood drive, m (in)

A ¼ 0:4ð
d2=4Þ useful area of cross-section of the wire rope, m2(in2)

dimension in Rockwood drive (Fig 21-5), m (in)

c dimension in Rockwood drive (Fig 21-5), m (in)

C center distance between sprockets (also with suffixes), m (in)

center distance between pulleys, m (in) capacity of conveyor, m3(ft3)

constant depends on the rope diameter, sheave diameter, chain, the bearing, and coefficient of friction [Eqs (21-59) to (21-62) and (21-86) to (21-103)] (also with suffixes)

C1 tooth width in precision roller and bush chains, m (in)

diameter of shaft, m (in) diameter of idler bearing, m (in) diameter of smaller pulley, m (in) diameter of rope, m (in)

pitch diameter of sprocket, m (in) d1 diameter of small sprocket, m (in)

hub diameter of pulley, m (in) d2 diameter of large sprocket, m (in)

da tip diameter of sprocket, m (in)

da1 tip diameter of small sprocket, m (in)

da2 tip diameter of large sprocket, m (in)

dc¼ fpFb equivalent pitch diameter, m (in)

df root diameter of sprocket, m (in)

dp pitch diameter of the V-belt small pulley, m (in)

dr diameter of roller pin, m (in)

diameter of large pulley, m (in) wire rope drum diameter, m (in) (Fig 21-4)

Dr diameter of reel barrel, m (in) Eq (21-76)

Dd diameter of the drum in mm as measured over the outermost

layer filling the reel drum

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Source: MACHINE DESIGN DATABOOK

Do diameter of the sheave pin, m (in)

E0 corrected elasticity modulus of steel ropes

(78.5 GPa ¼ 11.4 Mpsi), GPa (psi)

tension in belt, kN (lbf ) minimum tooth side radius, m (in)

Fa correction factor for instructional belt service from Table 21-27

Fc correction factor for belt length from Table 21-26

Fct centrifugal tension, kN (lbf )

Fd correction factor for arc of contact of belt from Table 21-25

F tangential force in the belt, required chain pull, kN (lbf )

Fs tension due to sagging of chain, kN (lbf )

F1 tension in belt on tight side, kN (lbf )

F2 tension in belt on slack side, kN (lbf )

Fc centrifugal force, kN (lbf )

values of coefficient for manila rope, Table 21-32 FR1 the minimum value of tooth flank radius in roller and bush

chains, m (in) FR2 the maximum value of tooth flank radius in roller and bush

chains, m (in)

g acceleration due to gravity, 9.8066 m/s2(32.2 ft/s2)

G tooth side relief in bush and roller chain, m (in)

h the thickness of wall of rope drum, m (in)

crown height, m (in) h1 depth of groove in rope drum, m (in)

H ¼ ðDd
DrÞ=2 depth of rope layer in reel drum, m (in)

number of V-belts, number of strands in a chain, transmission ratio

k ¼ ðe
1Þ=e variable in Eqs (21-2d), (21-4a), (21-6), and (21-123), which

ksg coefficient for sag from Table 21-55

l width of chain or length of roller, m (in)

minimum length of boss of pulley, m (in) minimum length of bore of pulley, m (in) length of conveyor belt, m (in)

length of cast-iron wire rope drum, m (in) outside length of coil link chain, m (in) K1 tooth correction factor for use in Eq (21-116a)

K2 multistrand factor for use in Eq (21-116a)

pitch length of V-belt, m (in) rope capacity of wire rope reel, m (in)

n number of times a rope passes over a sheave,

number of turns on the drum for one rope member speed, rpm

factor of safety

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n1 speed of smaller pulley, rpm or rps

speed of smaller sprocket, rpm or rps n2 speed of larger pulley, rpm or rps

speed of larger sprocket, rpm or rps

n0¼ nkd stress factor

PT power required by tripper, kW (hp)

pitch of the grooves on the wire rope drum, m (in) p1 distance between the grooves of two-rope pulley, m (in)

s the amount of shift of the line of action of the load from the

center line on the raising load side of sheave, m (in)

s the average shift of the center line in the load on the effort side

of the sheave, m (in)

S the distance through which the load is raised, m (in)

SA1 the minimum value of roller or bush seating angle, deg

SA2 the maximum value of roller or bush seating angle, deg

SR1 the minimum value of roller or bush seating radius, m (in)

SR2 the maximum value of roller or bush seating radius, m (in)

thickness of rim, m (in)

T tension in ropes, chains, kN (lbf )

TDmin minimum limit of the tooth top diameter, m (in)

TDmax maximum limit of the tooth top diameter, m (in)

v velocity of belt chain, m/s (ft/min)

w specific weight of belt, kN/m3(lbf/in3)

W width between reel drum flanges, m (in)

WB weight of belt, kN/m (lbf/in)

wc weight of chain, kN/m (lbf/in)

WI weight of revolving idler, kN/m (lbf/in) belt

z1 number of teeth on the small sprocket

z2 number of teeth on the large sprocket

1 unit tension in belt on tight side, MPa (psi)

2 unit tension in belt on slack side, MPa (psi)

c centrifugal force coefficient for leather belt, MPa (psi)

br breaking stress for hemp rope, MPa (psi)

 angle between tangent to the sprocket pitch circle and the

center line, deg

 coefficient of friction between belt and pulley

coefficient of journal friction

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 efficiency

!1 angular speed of small sprocket, rad/s

!2 angular speed of large sprocket, rad/s

Other factors in performance or in special aspects of design of flexible machine

elements are included from time to time in this chapter and being applicable only

in their immediate context, are not given at this stage.

BELTS

Flat belts

The ratio of tight side to slack side of belt at low

velocities

The power transmitted by belt

Power transmitted per m2(in2) of belt at low velocities

TABLE 21-1

Service correction factors, cs

Peak loads Light, steady load, such as steam engines, steam turbines, diesel engines, and

multicylinder gasoline engines

1.0Jerky loads, reciprocating machines such as normal-starting-torque squirrel-

cage motors, shunt-wound, DC motors, and single-cylinder engines

0.8Shock and reversing loads, full-voltage start such as squirrel-cage and

synchronous motors

0.6

TABLE 21-3

Values of coefficients cfor leather belts for use in Eqs (21-3) and (21-4)

Belt velocity, m/s (ft/min) 7.5 (1500) 10.0 (1950) 12.70 (2500) 15.0 (2950) 17.5 (3500) 20.0 (3950) 22.5 (4450) 25.0 (4950)

TABLE 21-2

Values of ðe
1Þ=e¼ k for various coefficients of frictions and arcs of contact

Arc of contact between the belt and pulley (, deg)

FLEXIBLE MACHINE ELEMENTS

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The ratio of tight to slack side of belt at high velocities

Power transmitted per m2 (in2) of belt at high

velocities

Equation (21-3a) in terms of tension on tight side (F1 )

and slack side of belt (F2), and centrifugal force (Fc)

The relation between the initial tension in the belt (F0 )

and the tension in the belt on the tight side (F1;max ) to

obtain maximum tension in the belt

The power transmitted at maximum tension in belt,

i.e., when F1 ¼ 2F0 , from Eq (21-1)

The power transmitted in actual practice taking into

consideration pulley correction factor (Kp ), velocity

correction factor (Kv), and service factor (Cs ) at

maximum tension in belt.

Stresses in belt (Fig 21-1c)

Tensile stress due to tension on tight side of belt

where F1 ¼ 1A; F2 ¼ 2A; Fc ¼ cA;

A ¼ a1t ¼ area of cross section of belt, m2(in2)

where Fa ¼ allowable tension in belt, N (lbf)

v ¼ velocity of belt, m/s (ft/min)

The tensile stress due to belt tension on account of

centrifugal force

The bending stress

The maximum belt stress

Stress due to twist in belt

For distribution of various stresses in belt

Coefficient of friction ( )

c¼ Fc a1t ¼ v29810



ð21-4lÞ where  ¼ specific weight of belt material N/dm3(lbf/in3)

2

¼ 0 for open belt

¼

Ea1D 2a2

 for half-crossed belt where a ¼ distance from centre of bigger pulley diameter to the point of twist of half-crossed belt and crossed belt >2D

a ¼ allowable stress in belt, MPa (psi) Refer to Fig 21-1C.

Refer to Table 21-4B for most commonly used belt materials in practice.

The values of Kpand Csare Table from Tables 21-4C and 21-4D, and Kvfrom Fig 21-1B, and also Table 21-4E for minimum pulley sizes.

Fa¼ allowable tension in belt, N (lbf)

v ¼ velocity of belt, m/s (ft/min)

For leather belts and belts of similar materialcis of importance only if v > 15%

FLEXIBLE MACHINE ELEMENTS

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TABLE 21-4A

Coefficients of frictions of leather belts on iron pulleys depending on velocity of belt

Velocity of belt, Coefficient of Velocity of belt, Coefficient of Velocity of belt, Coefficient of

Properties of some flat and round materials

Material Specification Size, in diameter, in 600 ft/min, lb/in Weight, lb/in3 of friction

At 6% elongation; 12% is maximum allowable value

Notes: d ¼ diameter, t ¼ thickness, w ¼ width The values given in this table for the allowable tension are based on a belt speed of 600 ft/min.Take Kv¼ 1:0 for polyamide and urethane belts

Source: Eagle Belting Co., Des Plaines, Illinois; table reproduced from J E Shigley and C R Mischke, Mechanical Engineering Design, Hill Book Company, New York, 1989

FLEXIBLE MACHINE ELEMENTS

Minimum pulley sizes for flat and round urethane belts (pulley diameters in inches)

Ratio of pulley speed to belt length, rev/(ft-min)

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TABLE 21-5

Coefficient of friction for belts depending on materials of pulley and belt

Pulley materialCast iron/steel

Thickness and width of leather belts

Relative strength of belt joints

Relative strength of joint to an equalType of joint section of solid leather, efficiency, %

Cemented, endless

90–100Cemented at factory

Rawhide, small holes 60–70

Rawhide, large holes 50–60

The cross section of the belt is given

For cross section and properties of belts

v

d
wv2g

 k

SI ð21-6aÞ

where P in kW, v in m/s, g ¼ 9:8066 m/s2,

w in N/m3, and d in MPa a1t ¼ 33 ;000P

v

d
wv2

g 104

 k

USCS ð21-6bÞ

where P in hp, v in ft/min, g ¼ 386:4 in/s2¼ 32:2 ft/s2,

w in lbf/in3, and din psi Refer to Tables 21-6A to 21-14.

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TABLE 21-10

Properties of leather belting for various purposes

PurposePower transmission

Splices Round belting for small machineSingle Double single and

Stitch tear resistance N/m 83,356

crack

TABLE 21-11

Tensile strength of fabric in finished rubber transmission belting

Tensile strength, N/m (kgf/mm) of width

Thickness of friction surface—rubber transmission belting

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BELT LENGTHS AND CONTACT ANGLES

FOR OPEN AND CROSSED BELTS

(Fig 21-1A)

Length of belt for open drive (Fig 21-1(A)a)

Length of belt for crossed drive (Fig 21-1(A)b)

Length of belt for quarter turn drive

For two-pulley open drive the center distance between

the two pulleys when the length of the belt is known

The unit elongation of belt is given by the equation

The relation between initial belt tension and final belt

tension

L ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4C2
ðD
dÞ2

q

¼1ðDLþ dsÞ ð21-7Þ

L ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4C2
ðD þ dÞ2

 L

4
0:393ðD þ dÞ

2

ðD
dÞ28

ð21-10Þ where

l¼
þ 2 sin
1

D
d 2C



ð21-10aÞ

s¼

sin
1

D
d 2C



ð21-10bÞ

 ¼
þ 2 sin
1

D þ d 2C



ð21-10cÞ

e ¼ ffiffiffi p

where  in MPa

e ¼ ffiffiffi p

where  in psi

e ¼ ffiffiffi p

where  in kgf/mm2

2 ffiffiffiffiffi F0

p

¼ p ffiffiffiffiffi F1

þ p ffiffiffiffiffi F2

ð21-12Þ where F0 ¼ initial belt tension, kN (lbf)

FLEXIBLE MACHINE ELEMENTS

FIGURE 21-1(A) Open and crossed belts.

FIGURE 21-1(B) Velocity correction factor for Kvfor use in Eq (21-4g) for leather belts

Belt stresses in open drive:f¼ ccentrifugal stress;2slack side stress;1tight side stress¼ 2þ n;neffective stress¼ u;b1,

b2bending stresses on pulleys 1 and 2 respectively;Gcreep angle ( angle over which creep takes place between belt and pulley).Lectrum S2¼ slack side F2; treibend¼ driving; Arbeitstrum S1¼ tight side F1; getrieben¼ driven

FIGURE 21-1(C) Stress distribution in belt (G Niemann, Maschinenelemente, Springer International Edition, Allied PublishersPrivate Ltd., New Delhi, 1978.)

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PULLEYS (Fig 21-2 and Fig 21-3)

C G Barth’s formula for the width of the pulley face

C G Barth’s empirical formula for the crown height

for wide belts

For rubber belts on well-aligned shafts, the crown

height

For poorly aligned shafts, the crown height

The rim thickness at edge for light-duty pulley

The rim thickness at edge for heavy-duty pulley for a

triple belt

The hub diameter of the pulley (Fig 21-2)

Arms

The bending moment on each arm

The section modulus of the arm at the hub

a ¼ 1:19a1 þ 10 mm for single belt SI ð21-13aÞ

a ¼ 1:1a1 þ 5 mm for double belt SI ð21-13bÞ Refer to Table 21-15 for width of pulley.

FIGURE 21-2 Cast-iron pulley.

INDIAN STANDARD SPECIFICATION

Cast-iron pulley

Minimum length of bore (Fig 21-2)

Half of the difference in diameters d1 and d2 (Fig 21-2)

The radius r1 near rim (Fig 21-2)

The radius r2 near rim (Fig 21-2)

Properties of solid woven fire-resistance conveyor belting for use in coal mines

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Nominal diameter,D, mm Crown,h, mm

Crown of cast iron and mild steel flat pulleys of diameters 400 to 2000 mma

Crownh of pulleys of widthNominal diameter,

Crown should be rounded, not angled; maximum roughness is Ro¼ AA 63 min

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The number of arms

Cross section of arms

Thickness of arm near boss (Fig 21-2)

The diameter of pulleys and arms in pulleys

The thickness of arm near rim

The radius of the cross-section of arms

Use webs for pulleys up to 200 mm diameter

for pulleys above 200 mm diameter and up to 400 mm diameter

for pulleys above 450 mm diameter

Use elliptical section

b ¼ 0:2943

ffiffiffiffiffiffiffi aD 4i

r

SI ð21-25aÞ

b ¼ 1:63

ffiffiffiffiffiffiffi aD i

r for single belt USCS ð21-25bÞ

b ¼ 0:2943

ffiffiffiffiffiffiffi aD 2i

r

SI ð21-26aÞ

b ¼ 1:253

ffiffiffiffiffiffiffi aD i

r for double belt USCS ð21-26bÞ Refer to Tables 21-18 to 21-21.

b1 —give a taper of 4 mm per 100 mm

r ¼3

TABLE 21-18

Minimum pulley diameters for given belt speeds and pliesa

Maximum belt speeds, m/s

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TABLE 21-19

Diameters of flat pulley and tolerances

Minimum pulley diameters for conveyor belting