mechanical design data book - marcus...

Mechanical Design Data Book

Shinto

Typewritten Text

Shinto

Typewritten Text

Shinto

Typewritten Text

http://www.meadinfo.org/

http://www.meadinfo.org/

1

Design Data Hand Book Contents:-

1 Friction Clutches

• Single plate clutches…………………………………………………………………05 • Multi plate clutches……………………………………………………………………05 • Cone clutches………………………………………………………………………………………06 • Centrifugal clutches……………………………………………………………………06

2 Brakes

• External Contracting Brakes…………………………………………………08 • Internal Expanding Brake…………………………………………………………09 • Band Brakes……………………………………………………………………………………………10 • Thermal Considerations………………………………………………………………11

3 Belt Drives

• Geometrical Relationships………………………………………………………12 • Analysis of Belt Tensions………………………………………………………13 • Condition for Maximum Power…………………………………………………13 • Selection of Flat Belts from the Manufacture’s

Catalogue…………………………………………………………………………………………………13 • Selection of V-Belts……………………………………………………………………15

4 Chain Drives

• Roller Chains………………………………………………………………………………………20 • Geometrical Relationships………………………………………………………20 • Power Rating of Roller Chains……………………………………………21 • Sprocket Wheels…………………………………………………………………………………24

5 Rolling Contact Bearings

• Stribeck’s Equation………………………………………………………………………25 • Equivalent Bearing Load……………………………………………………………26

A MEADinfo Publication Shinto Mathew

2

• Load Life Relationship………………………………………………………………26 • Selection of Bearing from the Manufacture’s

Catalogue…………………………………………………………………………………………………27 • Selection of Taper Roller Bearings………………………………32 • Design for Cyclic Load and Speed……………………………………38 • Bearing With a Probability of Survival Other Than

90 Percent………………………………………………………………………………………………38

6 Sliding Contact Bearings

• Effect of Temperature on Viscosity………………………………39 • Hydrostatic Step Bearing…………………………………………………………40 • Energy Losses in Hydrostatic Bearing…………………………40 • Reynold’s Equation…………………………………………………………………………41 • Raimondi and Boyd Method…………………………………………………………41 • Temperature Rise………………………………………………………………………………43 • Bearing Design –Selection of Parameters…………………44

7 Spur Gears

• Standard System of Gear Tooth……………………………………………45 • Force Analysis……………………………………………………………………………………50 • Beam Strength of Gear Tooth…………………………………………………47 • Effective Load on Gear Tooth………………………………………………48 • Estimation of Module Based on Beam Strength………50 • Wear Strength of Gear Tooth…………………………………………………50 • Estimation of Module Based on Wear Strength………51 • Gear Design for Maximum Power Transmitting

Capacity……………………………………………………………………………………………………51

8 Helical Gears

• Virtual Number of Tooth……………………………………………………………52 • Tooth Proportions……………………………………………………………………………53 • Beam Strength of Helical Gears…………………………………………54 • Effective Load on Gear Tooth………………………………………………54 • Wear Strength of Helical Gears…………………………………………55

9 Bevel Gears

• Force Analysis……………………………………………………………………………………57


3

• Beam Strength of Bevel Gears………………………………………………58 • Wear Strength of Bevel Gears………………………………………………59 • Effective Load on Gear Tooth………………………………………………60

10 Worm Gears • Proportions of Worm Gears………………………………………………………62 • Force Analysis……………………………………………………………………………………64 • Friction in Worm Gears………………………………………………………………64 • Strength Rating of Worm Gears……………………………………………65 • Wear rating of worm gears………………………………………………………67


4

FRICTION CLUTCHES Notations:- D = outer diameter of friction disk. d = inner diameter of friction disk. p = intensity of pressure. P = total operating force. ( ) ftM = torque transmitted by friction. z = number of pairs of contacting surfaces, for single plate clutch z=one. (z = number of plates – 1). µ = coefficient of friction.

ap = intensity of pressure at the inner edge. α = semi cone angle.

dr = radius of the drum.

gr = radius of the centre of gravity of the shoe in engaged position. m = mass of each shoe.

cfP = centrifugal force. =sP Spring force

2ω = running speed. (Rad/sec)

1ω = speed at which engagement starts. (Rad/sec)


5

Single Plate & Multi Plate Clutches Uniform pressure theory

)(4

22 dDP −=π

( ))()(

3 22

33

dDdDPz

M ft −−

=μ

Uniform wear theory

)(2

dDdp

P a −=π

( ) )(4

dDPzM ft +=μ


6

Cone Clutches Uniform pressure theory

)(4

22 dDP −=π

( ))()(

sin3 22

33

dDdDPzM ft −

−=

αμ

Uniform wear theory

)(2

dDdp

P a −=π

( ) )(sin4

dDPzM ft +=α

μ

Centrifugal Clutches

1000

21 g

s

rmP

ω=

( )1000

)( 21

22 ωωμ −

=zrmr

M dgft

Note: - here z = number of shoes.


7

Brakes Notations:- E = total energy absorbed by the brake. K.E = kinetic energy absorbed by the brake. P.E = potential energy absorbed by the brake. m = mass of the system. I = mass moment of inertia of the rotating body. k = radius of gyration.

21,vv = Initial and final velocities of the system 21,ωω = Initial and final angular velocities of the body

tM = braking torque. θ = angle through which the brake drum rotates during the braking period.

mghEP

mkEK

IEK

vvmEK

=

−=

−=

−=

.

)(21.

)(21.

)(21.

22

21

2

22

21

22

21

ωω

ωω

θtME =


8

External Contracting Brakes Block brake with short shoe

NRM t μ= Where tM = Braking Torque R = Radius of the Brake Drum μ = Coefficient of Friction

N = Normal reaction

plwN = Where p = Permissible pressure between the block and the brake drum l = length of the block w = width of the block

)( PNRNR

Y

X

−== μ

Nb

caP ×−

=)( μ

Pivoted block brake with long shoe

φcosmaxPP =

θθθ2sin2

sin4+

=Rh


9

θμ sin2 max2wpRM t =

)2sin2(21

max θθμ += RwpRY Internal expanding brake

( ) ( )[ ]max

2121max

sin42cos2coscoscos4

φθθθθμ −−−

=hRRwp

M f

( ) ( )[ ]

max

1212max

sin42sin2sin2

φθθθθ −−−

=RwhpM n

max

21max2

sin)cos(cos

φθθμ −

=wpR

M t

CMM

P fn −= (Clock wise rotation of the brake drum)

CMM

P fn += (Anti clock wise rotation of the brake drum) 0

20

max 9090 >= θφ when 0

22max 90<= θθφ when Where

maxp = maximum intensity of pressure. μ = coefficient of friction.

)2sin2(21

max θθ += RwpRX


10

fM = moment due to friction. nM = moment due to normal force. tM = elemental torque due to frictional force.

R = radius of the brake lining. w = face width of frictional lining. Band Brakes

1P = tension on the tight side of the band. 2P = tension on the loose side of the band. θ = angle of wrap (rad).

tM = torque capacity of the brake. R = radius of the brake drum.

RPPM t )( 21 −=

RwPp =

RwPp 1

max =

p = intensity of pressure. w = width of the frictional lining. Differential band brake.

lebaP

p)(2

μθ×−=


11

Thermal Considerations

mcE

t =Δ

Where tΔ = temperature rise of the brake drum assembly ( C0 ) E = total energy absorbed by the brake m = mass of the brake drum assembly c = specific heat of the brake drum material


12

Belt Drives GEOMETRICAL RELATIONSHIPS Open belt drive

)2

(sin2180 1

CdD

s−

−= −α

)2

(sin2180 1

CdD

b−

+= −α

CdDdDCL

4)(

2)(2

2−+

++=π

Cross belt drive

)2

(sin2180 1

CdD

bs+

+== −αα

CdDdDCL

4)(

2)(2

2++

++=π


13

Analysis of belt tension

αfemvPmvP

=−−

22

21

(For flat belts)

)21sin(

22

21

θαfe

mvPmvP

=−−

(For V-belts)

Power transmitted= vPP )( 21 −

Condition for maximum power transmission

mP

v i

3=

SELECTION OF FLAT BELT FROM THE

MANUFACTURES CATALOGUE

)()( max kWFkW a= Where max)(kW = power transmitted by the belt for the design purpose


14

)(kW = actual power transmitted by the belt

aF = load correction factor

Type of load

aF

(i) Normal load 1.0

(ii) Steady load, e.g. centrifugal pumps-fans-light machine tools-conveyors

1.2

(iii) Intermittent load, e.g. heavy duty fans-blowers-compressors- reciprocating pumps-line shafts-heavy duty machines

1.3

(iv) Shock load, e.g. vacuum pumps-rolling mills-hammers-grinders

1.5

Arc of contact factor

sα (degrees) 120 130 140 150 160 170 180 190 200

dF 1.33 1.26 1.19 1.13 1.08 1.04 1.00 0.97 0.94

HI-SPEED 0.0118 kW per mm width per ply FORT 0.0147 kW per mm width per ply


15

Standard widths of the belt are as follows

3-Ply 25 40 50 63 76 4-Ply 40 44 50 63 76 90 100 112 125 1525-Ply 76 100 112 125 152 6-Ply 112 125 152 180 200

dcorrected FkWkW ×= max)()( For HI-SPEED belt,

Corrected kW rating= (5.08)0.0118v

For FORT belt,

Corrected kW rating= (5.08)0.0147v

SELECTION OF V-BELTS

Dimensions of standard cross-sections Belt Section Width

W(mm) Thickness

T(mm) Minimum pitch

diameter of pulley(mm)A 13 8 125 B 17 11 200 C 22 14 300 D 32 19 500 E 38 23 630


16

Conversion of inside length to pitch length of the belt Belt section A B C D E

Difference between pitch length and inside length (mm)

36

43

56

79

92

Preferred values for pitch diameters (mm)

125 132 140 150 160 170 180 190 200 212 224236 250 265 280 300 315 355 375 400 425 450475 500 530 560 600 630 670 710 750 800 9001000


17

ld

a

FFbeltofratingkWFkWinpowerdtransmittebeltsofNumber

×××

=___

)___(__

Where aF = correction factor for industrial service

dF = correction factor for arc of contact

lF = correction factor for belt length


18


19

iL Belt section

A B C D E 3658 - 1.11 1.00 0.90 - 4013 - 1.13 1.02 0.92 - 4115 - 1.14 1.03 0.92 - 4394 - 1.15 1.04 0.93 - 4572 - 1.16 1.05 0.94 - 4953 - 1.18 1.07 0.96 - 5334 - 1.19 1.08 0.96 0.94 6045 - - 1.11 1.00 0.96 6807 - - 1.14 1.03 0.99 7569 - - 1.16 1.05 1.01

dF

sα (Degrees)

0.9

0.8

0.7

0.6

0.5

120 150 180


20

Chain Drives

Roller Chains Dimensions and breaking loads of roller chains

ISO chain number

Pitch p (mm)

Roller diameter

1d (mm)

Width 1b

(mm)

Transverse pitch tp (mm)

Breaking load for single strand

chain (kN)

06 B 9.525 6.35 5.72 10.24 10.7 08 B 12.70 8.51 7.75 13.92 18.2 10 B 15.875 10.16 9.65 16.59 22.7 12 B 19.05 12.07 11.68 19.46 29.5 16 B 25.40 15.88 17.02 31.88 65.0 20 B 31.75 19.05 19.56 36.45 98.1 24 B 38.10 25.40 25.40 48.36 108.9 28 B 44.45 27.94 30.99 59.56 131.5 32 B 50.80 29.21 30.99 58.55 172.4 40 B 63.50 39.37 38.10 72.29 272.2

Geometric Relationships

Velocity ratio,1

2

2

1

zz

nni ==

Average velocity, 31060 ×=

zpnv

Length of the chain, pLL n ×= Number of links in the

chain, ⎟⎠⎞

⎜⎝⎛×⎟

⎠⎞

⎜⎝⎛ −

+⎟⎠⎞

⎜⎝⎛ +

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

apzzzz

paLn

21221

222

π

Where a = centre distance between the axis of the driving and driven sprockets.


21

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦⎤

⎢⎣⎡ −

−⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +

−+⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +

−=2

122

2121

28

224 πzzzzLzzLpa nn

POWER RATING OF ROLLER CHAINS

10001vPkW =

Where 1P = allowable tension in the chain (N) v = average velocity of chain


22

kW rating of chain = ( )

21

___KK

KdtransmittebetokW s

××

Where sK = service factor

Multiple strand factors )( 1K Number of strands

1K1 1.0 2 1.7 3 2.5 4 3.3 5 3.9 6 4.6


23

Tooth correction factor )( 2K Number of teeth on the

driving sprocket 2K15 0.8516 0.9217 1.0018 1.0519 1.1120 1.1821 1.2622 1.2923 1.3524 1.4125 1.4630 1.73


24

SPROCKET WHEELS


25

Rolling Contact Bearing Stribeck’s Equation

( ) ...............2cos2cos2 3210 +++= ββ PPPC

βδδ cos

1

2 =

32

1

2

1

2⎟⎟⎠

⎞⎜⎜⎝

⎛=

PP

δδ

MPC 10 = Where, ( ) ( )[ ]2525 2cos2cos21 ββ ++=M 0C = Static load ..., 21 δδ = radial deflections at the respective balls.

z360

=β

Where z is number of balls

⎟⎠⎞

⎜⎝⎛

Mz is practically constant and Stribeck suggested a value of

5 for ⎟⎠⎞

⎜⎝⎛

Mz

10 51 zPC ⎟⎠⎞

⎜⎝⎛=


26

21 kdP = Where d is, the ball diameter and factor k

depends upon radii of curvature at the point of contact and on the modulii of elasticity of the materials. Stribeck’s Equation

5

2

0zkdC =

Equivalent Bearing Load

ar YFXFP += Where, P= equivalent dynamic load rF = radial load aF = axial or thrust load X and Y are radial and thrust factors respectively and there values are given in the manufactures catalogue. Load Life Relationship

p

PCL ⎟⎠⎞

⎜⎝⎛=

Where L = bearing life (in million revolutions) C = dynamic load capacity (N) p = 3 (for ball bearing) p = 10/3 (for roller bearing)


27

Relationship between life in million revolutions and and life in working hours is given by

61060 hnL

L =

Where hL =bearing life (hours) n = speed of rotation (rpm) Selection of bearing from manufacture’s

catalogue

X and Y factors for single-row deep groove ball bearings

⎟⎟⎠

⎞⎜⎜⎝

⎛

0CFa

eFF

r

a ≤⎟⎟⎠

⎞⎜⎜⎝

⎛

eFF

r

a >⎟⎟⎠

⎞⎜⎜⎝

⎛

e

X

Y

X

Y

0.025 0.040 0.070 0.130 0.250 0.500

1 1 1 1 1 1

0 0 0 0 0 0

0.56 0.56 0.56 0.56 0.56 0.56

2.0 1.8 1.6 1.4 1.2 1.0

0.22 0.24 0.27 0.31 0.37 0.44

ar YFXFP +=


28

Dimensions and static and dynamic load capabilities of single–row deep groove ball

bearings.

Principal dimensions

(mm)

Basic load ratings(N)

Designation

d D B C 0C

10 19 5 1480 630 61800 26 8 4620 1960 6000 30 9 5070 2240 6200 35 11 8060 3750 6300

12 21 5 1430 695 61801 28 8 5070 2240 6001 32 10 6890 3100 6201 37 12 9750 4650 6301

15 24 5 1560 815 61802 32 9 5590 2500 6002 35 11 7800 3550 6202 42 13 11400 5400 6302

17 26 5 1680 930 61803 35 10 6050 2800 6003 40 12 9560 4500 6202 47 14 13500 6550 6303 62 17 22900 11800 6403

20 32 7 2700 1500 61804 42 8 7020 3400 16400 42 12 9360 4500 6004 47 14 12700 6200 6204 52 15 15900 7800 6304 72 19 30700 16600 6404

25

37 7 3120 1960 61805 47 8 7610 4000 16005 47 12 11200 5600 6005


29

52 15 14000 6950 6205 62 17 22500 11400 6305

80 21 35800 19600 6405 30 42 7 3120 2080 61806

55 9 11200 5850 16006 55 13 13300 6800 6006 62 16 19500 10000 6206 72 19 28100 14600 6306 90 23 43600 24000 6406

35 47 7 4030 3000 61800 62 9 12400 6950 16007 62 14 15900 8500 6007 72 17 25500 13700 6207 80 21 33200 18000 6307 100 25 55300 31000 6407

40 52 7 4160 3350 61808 68 9 13300 7800 16008 68 15 16800 9300 6008 80 18 30700 16600 6208 90 23 41000 22400 6308 110 27 63700 36500 6408

45 58 7 6050 3800 61809 75 10 15600 9300 16009 75 16 21200 12200 6009 85 19 33200 18600 6209 100 25 52700 30000 6309 120 29 76100 45500 6409

50 65 7 6240 4250 61810 80 10 16300 10000 16010 80 16 21600 12300 6010 90 20 35100 19600 6210 110 27 61800 36000 6310 130 31 87100 52000 6410

55 72 9 8320 5600 61811


30

90 11 19500 12200 16011 90 18 28100 17000 6011 100 21 43600 25000 6211 120 29 71500 41500 6311 140 33 99500 63000 6411

60 78 10 8710 6100 61812 95 11 19900 13200 16012 95 18 29600 18300 6012 110 22 47500 28000 6212 130 31 81900 48000 6312 150 35 108000 69500 6412

65 85 10 11700 8300 61813 100 11 21200 14600 16013 100 18 30700 19600 6013 120 23 55900 34000 6213 140 33 92300 56000 6313 160 37 119000 78000 6413

70 90 10 12100 9150 61814 110 13 28100 19000 16014 110 20 37700 24500 6014 125 24 61800 37500 6214 150 35 104000 63000 6314 180 42 143000 104000 6414

75 95 10 12500 9800 61815 115 13 28600 20000 10615 115 20 39700 26000 6015 130 25 66300 40500 6215 160 37 112000 72000 6315 190 45 153000 114000 6415


31

Dynamic load capacity p

PCL ⎟

⎠⎞

⎜⎝⎛=


32

Selection of Taper Roller Bearings

YFF r

a5.0

=

Where Y is the thrust factor

Equivalent dynamic load for single row taper roller bearings is given by

( )( ) eFFwhenYFFPeFFwhenFP

raar

rar

>+=≤=

4.0

Dimensions, Dynamic capabilities and calculation factors for

single row taper roller bearing d D B C Designation e Y

20 42 15 22900 32004X 0.37 1.6 47 15.25 26000 30204 0.35 1.7 52 16.25 31900 30304 0.30 2.0 52 72.25 41300 32304 0.30 2.0

25 47 15 25500 32005X 0.43 1.4 52 16.25 29200 30205 0.37 1.6 52 19.25 34100 32205B 0.57 1.05 52 22 44000 33205 0.35 1.7 62 18.25 41800 30305 0.30 2 62 18.25 35800 31305 0.83 0.72 62 25.25 56100 32305 0.30 2

30 55 17 33600 32006X 0.43 1.4 62 17.25 38000 30206 0.37 1.6 62 21.25 47300 32206 0.37 1.6 62 21.25 45700 32206B 0.57 1.05


33

30 62 25 60500 33206 0.35 1.7 72 20.75 52800 30306 0.31 1.9 72 20.75 44600 31306 0.83 0.72 72 28.75 72100 32306 0.31 1.9

35 62 18 40200 32007X 0.46 1.3 72 18.25 48400 30207 0.37 1.6 72 24.25 61600 32207 0.37 1.6 72 24.25 57200 32207B 0.57 1.05 72 28 79200 33207 0.35 1.7 80 22.75 68200 30307 0.31 1.9 80 22.75 57200 31307 0.83 0.72 80 32.75 89700 32307 0.31 1.9 80 32.75 88000 32307B 0.54 1.1

40 68 19 49500 32008X 0.37 1.6 75 26 74800 33108 0.35 1.7 80 19.75 58300 30208 0.37 1.6 80 24.75 70400 32208 0.37 1.6 80 32 96800 33208 0.35 1.7 85 33 114000 T2EE040 0.35 1.7 90 25.25 80900 30308 0.35 1.7 90 25.25 69300 31308 0.83 1.72 90 35.25 110000 32308 0.35 1.7

45 75 20 55000 32009X 0.40 1.5 80 26 79200 33109 0.37 1.6 85 20.75 62700 30209 0.40 1.5 85 24.75 74800 32209 0.40 1.5 85 32 101000 33209 0.40 1.5 95 29 84200 T7FC045 0.88 0.68 95 36 140000 T2ED045 0.33 1.8 100 27.25 101000 30309 0.35 1.7 100 27.25 85800 31309 0.83 0.72 100 38.25 132000 32309 0.35 1.7 100 38.25 128000 32309B 0.54 1.1

50 80 20 57200 32010X 0.43 1.4


34

50 80 24 64400 33010 0.31 1.9 85 26 80900 33110 0.40 1.5 90 21.75 70400 30210 0.43 1.4 90 24.75 76500 32210 0.43 1.4 90 32 108000 33210 0.40 1.5 100 36 145000 T2ED050 0.35 1.7 105 32 102000 T7FC050 0.88 0.68 110 29.25 117000 30310 0.35 1.7 110 29.25 99000 31310 0.83 0.72 110 42.25 161000 32310 0.35 1.7 110 42.25 151000 32310B 0.54 1.1

60 95 23 76500 32012X 0.43 1.4 95 27 85800 33012 0.33 1.8 100 30 110000 33112 0.40 1.5 110 23.75 91300 30212 0.40 1.5 110 29.75 119000 32212 0.40 1.5 110 38 157000 33212 0.40 1.5 115 39 157000 T5ED060 0.54 1.1 115 40 183000 T2EE060 0.33 1.8 125 37 145000 T7FC060 0.83 0.72 130 33.5 161000 30312 0.35 1.7 130 33.5 134000 31312 0.83 0.72 130 48.5 216000 32312 0.35 1.7 130 48.5 205000 32312B 0.54 1.1

70 110 25 95200 32014X 0.43 1.4 110 31 121000 33014 0.28 2.1 120 37 161000 33114 0.37 1.6 125 26.25 119000 30214 0.43 1.4 125 33.25 147000 32214 0.43 1.4 125 41 190000 33214 0.40 1.5 130 43 220000 T2ED070 0.33 1.8 140 39 168000 T7FC070 0.88 0.68 140 32 264000 T4FE070 0.44 1.35 150 38 209000 3014 0.35 1.7


35

70 150 38 176000 31314 0.83 0.72 150 54 275000 32314 0.35 1.7 150 54 264000 32314B 0.54 1.1

80 125 29 128000 32016X 0.43 1.4 125 36 157000 33016 0.28 2.1 130 37 168000 33116 0.43 1.4 140 28.25 140000 30216 0.43 1.4 140 35.25 176000 32216 0.43 1.4 140 46 233000 33216 0.43 1.4 145 46 264000 T2ED080 0.31 1.9 170 42.5 255000 30316 0.35 1.7 170 42.5 212000 31316 0.83 0.72 170 61.5 358000 32316 0.35 1.7 170 61.5 336000 32316B 0.54 1.1

90 140 32 157000 32018X 0.43 1.4 140 39 205000 33018 0.27 2.2 150 45 238000 33118 0.40 1.5 155 46 270000 T2ED090 0.33 1.8 160 32.5 183000 30218 0.43 1.4 160 42.5 238000 32218 0.43 1.4 190 46.5 308000 30318 0.35 1.7 190 46.5 251000 31318 0.83 0.72 190 67.5 429000 32318 0.35 1.7

100 145 24 119000 T4CB100 0.48 1.25 150 32 161000 32020X 0.46 1.3 150 39 212000 33020 0.28 2.1 165 47 292000 T2EE100 0.31 1.9 180 37 233000 30220 0.43 1.4 180 49 297000 32220 0.43 1.4 180 63 402000 33220 0.40 1.5 215 51.5 380000 30320 0.35 1.7 215 56.5 352000 31320X 0.83 0.72 215 77.5 539000 32320 0.35 1.7

150 225 48 347000 32030X 0.46 1.3


36

150 270 49 402000 30230 0.43 1.4 270 77 682000 32230 0.43 1.4 320 72 765000 30330 0.35 1.7 320 82 837000 31330X 0.83 0.72

200 280 51 446000 32940 0.40 1.5 310 70 704000 32040X 0.43 1.4 360 64 737000 30240 0.43 1.4 360 104 1140000 32240 0.40 1.5

300 420 76 990000 32960 0.40 1.5


37


38

Design for Cyclic Load and Speeds

3

3

⎥⎦

⎤⎢⎣

⎡ΣΣ

=N

BPPe

Bearing With a Probability of Survival Other

Than 90 Percent

b

e

e

R

RLL

1

90

90 1log

1log

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎠⎞

⎜⎝⎛

=⎟⎟⎠

⎞⎜⎜⎝

⎛

Where b = 1.17


39

Sliding Contact Bearing

Effect of Temperature on Viscosity


40

Hydrostatic Step Bearing The following notations are used in the analysis, W = Trust load

0R = outer radius of the shaft

iR = inner radius of the shaft

iP = supply of inlet pressure

oP = outlet or atmospheric pressure

0h = fluid film thickness Q = flow of the lubricant μ = viscosity of the lubricant

⎟⎟⎠

⎞⎜⎜⎝

⎛=

ie

i

RR

hPQ

0

30

log6μ

π

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

ie

ii

RRRRP

W0

220

log2

π

Energy Losses in Hydrostatic Thrust

Bearing

)10)(()( 60

−−= PPQkW ip pkW )( = power loss in pumping

0

440

2

6

)(1005.58

1)(h

RRnkW if

−⎥⎦⎤

⎢⎣⎡

×=

μ

fkW )( = power loss due to friction


41

fpt kWkWkW )()()( +=

tkW )( = total power loss Reynold’s Equation

⎟⎠⎞

⎜⎝⎛∂∂

=⎥⎦⎤

⎢⎣⎡

∂∂

∂∂

+⎥⎦⎤

⎢⎣⎡

∂∂

∂∂

xhU

zph

zxph

xμ633

Raimondi and Boyd Method

Dimensionless performance parameters for full journal bearings with side flow

⎟⎠⎞

⎜⎝⎛

dl

ε ⎟

⎠⎞

⎜⎝⎛

ch0

S φ fcr⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛lrcn

Q

s ⎟⎟

⎠

⎞⎜⎜⎝

⎛QQs ⎟⎟

⎠

⎞⎜⎜⎝

⎛

maxpp

∞

0 1.0 ∞ 70.92 ∞ π 0 _ 0.1 0.9 0.240 69.10 4.80 3.03 0 0.826 0.2 0.8 0.123 67.26 2.57 2.83 0 0.814 0.4 0.6 0.0626 61.94 1.52 2.26 0 0.764 0.6 0.4 0.0389 54.31 1.20 1.56 0 0.667 0.8 0.2 0.021 42.22 0.961 0.760 0 0.495 0.9 0.1 0.0115 31.62 0.756 0.411 0 0.358 0.97 0.03 _ _ _ _ 0 _ 1.0 0 0 0 0 0 0 0

1

0 1.0 ∞ 85 ∞ π 0 _ 0.1 0.9 1.33 79.5 26.4 3.37 0.150 0.540 0.2 0.8 0.631 74.02 12.8 3.59 0.280 0.529 0.4 0.6 0.264 63.10 5.79 3.99 0.497 0.484 0.6 0.4 0.121 50.58 3.22 4.33 0.680 0.415 0.8 0.2 0.0446 36.24 1.70 4.62 0.842 0.313 0.9 0.1 0.0188 26.45 1.05 4.74 0.919 0.247


42

0.97 0.03 0.00474 15.47 0.514 4.82 0.973 0.152 1.0 0 0 0 0 0 1.0 _

½ 0 1.0 ∞ 88.5 ∞ π 0 _ 0.1 0.9 4.31 81.62 85.6 3.43 0.173 0.523 0.2 0.8 2.03 74.94 40.9 3.72 0.318 0.506 0.4 0.6 0.779 61.45 17.0 4.29 0.552 0.441 0.6 0.4 0.319 48.14 8.10 4.85 0.730 0.365 0.8 0.2 0.0923 33.31 3.26 5.41 0.874 0.267 0.9 0.1 0.0313 23.66 1.60 5.69 0.939 0.206 0.97 0.03 0.00609 13.75 0.610 5.88 0.980 0.126 1.0 0 0 0 0 _ 1.0 0

¼ 0 1.0 ∞ 89.5 ∞ π 0 _ 0.1 0.9 16.2 82.31 322.0 3.45 0.180 0.515 0.2 0.8 7.57 75.18 153.0 3.76 0.330 0.489 0.4 0.6 2.83 60.86 61.1 4.37 0.567 0.415 0.6 0.4 1.07 46.72 26.7 4.99 0.746 0.334 0.8 0.2 0.261 31.04 8.8 5.60 0.884 0.240 0.9 0.1 0.0736 21.85 3.50 5.91 0.945 0.180 0.97 0.03 0.0101 12.22 0.922 6.12 0.984 0.108 1.0 0 0 0 0 _ 1.0 0

c = R-r Where c = radial clearance (mm) R = radius of bearing r = radius of journal

ce

=ε

Where e =eccentricity ratio, ε = eccentricity ratio

⎟⎠⎞

⎜⎝⎛−=

ch01ε

Where 0h =film thickness


43

⎟⎠⎞

⎜⎝⎛

ch0 is called the minimum film thickness variable

The Sommerfed number is given by

pn

crS sμ2

⎟⎠⎞

⎜⎝⎛=

Where sn =journal speed p = unit bearing pressure The Coefficient of Friction Variable (CFV) is given by

fcrCFV ⎟⎠⎞

⎜⎝⎛=)(

Where f is the coefficient of friction

Frictional power 6102

)(fWrn

kW sf

π=

The Flow Variable (FV) is given by

lrcnQFV

s

=)(

Where l = length of the bearing Q= flow of the lubricant Temperature Rise

)()(3.8

FVCFVpt =Δ

2tTT iav

Δ+=


44

Bearing Design – Selection of Parameters


45

Spur Gears The pitch circle diameter is given by

mzd =1

Centre to centre distance,

2)( gpn zzm

a+

=

Here transmission ratio g

p

p

g

nn

zz

i ==

Standard System of Gear Tooth

Choice 1 (preferred)

1.00 5.00

1.25 6.0

1.508.00

2.00 10.00

2.5 12.00

3.00 16.00

4.0 20.00

Choice2 1.12 5.5

1.3757.00

1.759.00

2.25 11.00

2.75 14.0

3.50 18.00

4.5

Addendum ( )ah =(m) Dedendum ( )fh =1.25m Clearance(c) =0.25m Tooth thickness = 1.5708m Fillet radius = 0.4m


46

Force Analysis

nkWM t π2

)(1060 6×=

1

2dmp t

t =

αtantr PP =

αcost

NP

P =

Number of Teeth

α2min sin2

=z

Pressure angle ( )α 05.14 020 025

minz (Theoretical) 32 17 11

minz (Practical) 27 14 9

Face Width (3m)<b< (12m) In preliminary stages of gear design, the face width assumed as ten times of module.


47

Beam Strength of Gear Tooth YmbS bb σ=

Values of the Lewis form factor Y for 20 0 full depth involute

system z Y z Y z Y

15 0.289 27 0.348 55 0.415 16 0.295 28 0.352 60 0.421 17 0.302 29 0.355 65 0.425 18 0.308 30 0.358 70 0.429 19 0.314 32 0.364 75 0.433 20 0.320 33 0.367 80 0.436 21 0.326 35 0.373 90 0.442 22 0.330 37 0.380 100 0.446 23 0.333 39 0.386 150 0.458 24 0.337 40 0.389 200 0.463 25 0.340 45 0.399 300 0.471 26 0.344 50 0.408 Rack 0.484


48

Effective Load on Gear Tooth

(1)For ordinary and commercially cut gears made with form cutters with v<10m/s

vCv +

=3

3

(2) For actually hobbled and generated gears with v<20m/s,

vCv +

=6

6

(3) For precision gears with shaving, grinding and lapping operations and with v>20m/s,

v

Cv+

=6.5

6.5

The pitch line velocity is given by

31060'

×=

ndv π

The effective load between two meshing teeth is given by

v

tseff C

PCP =

n the final stages of gear design, when the gear dimensions are known, the errors specified and the quality of gears determined, the dynamic load is calculated by the equations derived by Prof. Spotts. The effective load is given by ( )dtseff PPCP +=

where dP is the dynamic load Depending upon the materials of the pinion and the gear, there are three equations for the dynamic load. (1) Steel Pinion with steel gear:

( )22

21

21

2530 rr

rbrzenP pp

d+

=


49

(2) C.I Pinion with C.I gear:

( )22

21

21

3785 rr

rbrzenP pp

d+

=

(3) Steel Pinion with C.I Gear

( )22

21

21

92.03260 rr

rbrzenP pp

d+

=

e = sum of errors between two meshing teeth (mm) gp eee += where pe =error for pinion ge =error for gear

Type of driven

machines

Source of power

Electric motor

Turbine/Multi

cylinder engine

Single-cylinder

engine

Generators-feeding

mechanisms-belt conveyors-blowers-compressors-agitators

and mixers

1.10

1.25

1.50

Machine tools-hoist and

cranes-rotary drives-piston pumps-distribution pumps

1.25

1.50

1.75

Blanking and shearing presses -rolling mills-centrifuges-steel

work machinery

1.75

2.00

2.25


50

Estimation of Module Based on Beam

Strength

( ) ( )

31

6

3

1060

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛

×=

YSmbznC

fsCkWmut

v

s

π

Wear Strength of Gear Tooth

( )4.1

11cossin 212 EE

K c +=

αασ


51

KbQdS pw1=

pg

g

zzz

Q−

=2

Expression for the load stress factor K can be simplified when all the gears are made of steel with a 20 0 pressure angle . in this special case,

221 207000 mmNEE ==

020=α 2))(81.9(27.0 mmNBHNc =σ

where BHN=Brinell Hardness Number. Therefore,

2

10016.0 ⎟

⎠⎞

⎜⎝⎛=

BHNK

Estimation of Module Based on

Wear Strength

( ) ( )

31

2

61060

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎠⎞

⎜⎝⎛

×=

QKmbCnz

fsCkWmvpp

s

π

Gear Design for Maximum Power Transmitting Capacity

dw PS 2=

2w

dtS

PP ==


52

Helical gears

ψcosPPn =

ψcosmmn =

nm = normal module m = transverse module

ψtanppa =

αα

ψtantan

cos n=

ψcosnzm

d =

ψcos2)( 21 zzm

a n +=

p

g

g

p

zz

i ==ωω

Where i=speed ratio for helical gear Suffixes p and g refer to the pinion and gear respectively a is the centre to centre distance between two helical gears having 1z and 2z as the number of teeth. The normal pressure angle is usually 020 .

Virtual number of teeth

ψ31

coszz =


53

Tooth proportions In helical gears, the normal module nm should be selected from standards. The first preference values of the normal

module are nm (mm) 1, 1.25, 1.5, 2, 2.5,3,4,5,6,8 and10. The standard proportions of the addendum and dedendum are, Addendum na mh =)( Dedendum nf mh 25.1)( = Clearance nmc 25.0)( = Addendum circle diameter ad is given by

⎥⎦

⎤⎢⎣

⎡+= 2

cosψzmd na

Dedendum circle diameter fd is given by

⎥⎦

⎤⎢⎣

⎡−= 5.2

cosψzmd nf

ψπsin

nmb ≥

This is the minimum face width. Force Analysis

=tp Tangential component

=rp Radial component

=ap Axial or thrust component

ψα coscos nt pp = ψtanta pp =


54

⎥⎦

⎤⎢⎣

⎡=

ψα

costan n

tr pp

dmp t

t2

=

Beam strength of helical gears

YmS bnb σ=

Effective load on gear tooth

nkWM t π2

)(1060 6×=

dMP t

t2

=

v

tseff C

PCP =

sC = service factor (from table)

vC = velocity factor The velocity factor ,

vCv

+=

6.56.5

Dynamic load is given by


55

22

21

21

2530 rr

rbrzenP pp

d+

=

)coscos( ψαndtseff PPCP +=

)( fsPS effb = Wear strength of helical gears

ψ2cosKbQd

S pw =

11

12

pg

g

zzz

Q+

=

pg

g

zzz

Q+

=2

for internal helical gear

pg

g

zzz

Q−

=2

4.1

11cossin21

2⎥⎦

⎤⎢⎣

⎡+

=EE

Knnc αασ


56

=nα Normal pressure angle )20( 0

2

10016.0 ⎟

⎠⎞

⎜⎝⎛=

BHNK

)( fsPS effw =


57

Bevel Gears

γcos2Drb =

γcos1 zz =

g

p

zz

=γtan

p

g

zz

=Γtan

2πγ =Γ+

The cone distance 0A is given by 22

0 22 ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛= gp DD

A

Force Analysis

⎥⎦

⎤⎢⎣

⎡−=

2sin

2γbD

r pm

Where mr = radius of the pinion at the mid point along the face width b = face width of the tooth


58

m

tt r

MP =

αtants PP =

Where tP = tangential or useful component which is perpendicular to the plane of the paper.

sP = the separating force between the two meshing teeth

γαγα

sintancostan

ta

tr

PPPP

==

Beam Strength of Bevel Gears

⎥⎦

⎤⎢⎣

⎡−=

0

1AbYmbS bb σ

Where bS beam strength of the tooth m = module at the large end of the tooth b = face width bσ = permissible bending stress ( 3utS ) Y = Lewis form factor based on formative number of teeth

0A = cone distance

DM

P tt

2=

face width of the bevel gear is generally taken as (10 m) or ( 30A ) whichever is smaller


59

∴ b = (10 m) or ( )30A (Whichever is smaller)

WEAR STRENGTH OF BEVEL GEARS Buckingham’s equation

KbQdS pw1=

Where wS = wear strength b = face width of gears Q = ratio factors 1

pd = pitch circle diameter of the formative pinion K = material constant

bp rd 21 =

γcos75.0 KbQD

S pw =

(Buckingham’s equation)

γtan2

pg

g

zzz

Q+

=

4.1

11cossin2

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

=gp

c EEK

αασ

When pinion as well as the gear is made of steel with 020 pressure angle, the value of K is given by

2

10016.0 ⎟

⎠⎞

⎜⎝⎛=

BHNK


60

EFFECTIVE LOAD ON GEAR TOOTH

nkWM t π2

)(1060 6×=

DM

P tt

2=

v

tseff C

PCP =

sC = service factor (from table)

Type of driven machines

Source of power

Electric motor

Turbine/Multi

cylinder engine

Single-cylinder

engine

Generators-feeding

mechanisms-belt conveyors-blowers-compressors-agitators

and mixers

1.10

1.25

1.50

Machine tools-hoist and

cranes-rotary drives-piston pumps-distribution pumps

1.25

1.50

1.75

Blanking and shearing presses -rolling mills-centrifuges-steel

work machinery

1.75

2.00

2.25

vC = velocity factor

The velocity factor for cut teeth is given by


61

vCv +

=6

6

For general teeth,

vCv

+=

6.56.5

Dynamic load is given by

22

21

211

2530 rr

rrbzenP pp

d+

21,rr Radii of the pinion and gear respectively 1b Axial width of the gear blank

⎥⎦

⎤⎢⎣

⎡−=

2sin

21γbD

r p

⎥⎦

⎤⎢⎣

⎡−=

2cos

22γbD

r g

)( dtseff PPCP += Stress in gear tooth due to bending

)( fsPS effb = Stress in gear tooth due to pitting

)( fsPS effw =


62

Worm Gears Notations:-

1z = number of starts on the worm 2z = number of teeth on the worm wheel

q = diametral quotient m = module

1d = pitch circle diameter of the worm 1ad = outer diameter of the worm

2ad = outer diameter of the worm wheel

2d = pitch circle diameter of the worm wheel l = lead of the worm

xp = axial pitch of the worm a = the centre distance i = the speed ratio. F = the effective face width

rl = the length of the root of the worm gear teeth.

Proportions of Worm Gears


63

md

q 1=

1zpl x= 22 mzd = mp x π=

1mzl π=

)(21

2zqma +=


64

1

2

zzi =

)1(2 += qmF

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+= −

cdFcdl

aar 2

sin)2(1

11

Force Analysis tP )( 1 = tangential component on the worm aP )( 1 = axial component on the worm rP )( 1 = radial component on the worm

11

2)(

dM

P tt =

( )( )γμγα

γμγαcossincossincoscos)()( 11 +

−×= ta PP

)cossin(cossin)()( 11 γμγα

α+

×= tr PP

Friction in worm gears sv = rubbing velocity 1v = pitch line velocity of the worm

2v = pitch line velocity of the worm wheel

)1000)(60(11

1nd

vπ

=

γπ

cos)60000(11ndvs =

( ))cot(

tancosγμαγμαη

+−

=cas


65

Strength Rating Of Worm Gears

γγ

cos65.17)(cos65.17)(

2222

2111

dmlSXMdmlSXM

rbbt

rbbt

==

1)( tM , 2)( tM = permissible torque on the worm wheel

1bX , 2bX = speed factors for the strength of worm and worm wheel

1bS , 2bS = bending stress factors for worm and worm wheel


66

m = module rl = the length of the root of the worm gear teeth. 2d = pitch circle diameter of the worm wheel γ = lead angle of the worm Power transmitting capacity of the worm gear based on the beam strength is given by

610602×

= tnMkW

π

Where )( tM is the lower value between 1)( tM and 2)( tM .


67

Wear Rating of Worm Gears

mdYSXM

mdYSXM

Zcct

Zcct8.1

2224

8.12113

)(64.18)(

)(64.18)(

=

=

3)( tM , 4)( tM = permissible torque on the worm wheel

1cX , 2cX = speed factors for the strength of worm and worm wheel

1cS , 2cS = surface stress factors of the worm and worm wheel

zY = zone factor Thermal Considerations

kWH g )1(1000 η−= Where gH = rate heat generation


68

η = efficiency of the of the worm gear (fraction) kW = power transmitted by the gears

AttkH d )( 0−=

Where dH = rate of heat dissipation k = overall heat transfer coefficient of housing walls ( )CmW 02 t = temperature of the lubrication oil. ( C0 )

0t = temperature of the surrounding air ( C0 ) A = effective surface area of housing

kAkWtt

AttkkW

)1(1000)1(1000

)(

0

0

ηη−

+=

−−

=


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