BY : Prof Dr. Nabil Gadallaheng.modern-academy.edu.eg/E-learning/Mech/Mach. Design II - MNF...

MILITARY TECHNICAL COLLEGE

MECHNICAL DESIGN AND PRODUCTION DEPARTMENT

LECTURES ON MACHINE Design II

BY : Prof Dr. Nabil Gadallah

• Time Schedule

– Lecture 3 hrs/week

– Exercise 2 hrs/week

• References:

– Mechanical Engineering Design, J. E. Shigley

• Required tools for lectures and exercises:

– Notebook size A3 (5 mm)

– Triangles set.

– Eraser, Pencils

INDEX

CHAPTER 13 : SPUR GEARS

CHAPTER 14: HELICAL, WORM AND BEVEL GEARS

CHAPTER 12: LUBRICATION AND JOURNAL BEARINGS

CHAPTER 11: ROLLING CONTACT BEARINGS

CHAPTER 10: MECHANICAL SPRINGS

CHAPTER 17: FLEXIBLE MECHANICAL ELEMENTS

Helical gears Spur gears

Crossed helical

(spiral) gears Bevel gears

Worm gear Spiral bevel gears

Rack and pinion gears

Internal gears

Mercedes-Benz Actros,

manual transmission

http://www.fox-prints.com/images/chain_drive.jpg

CHAPTER 13 Spur Gears

13.1- Introduction:

- The simplest means of transferring rotary motion of one shaft to

another is a pair of roller cylinders. Provided that sufficient friction is

available at the rolling interference.

- Preventing slipping requires some meshing teeth to the rolling

cylinders. They, then becomes gears and they are together called a

gear set. The smaller one is called the pinion and the other is the gear.

13.2- Theory of gearing:

Fundamental law of gearing:

- The common normal of the tooth

profiles at all contact points

within the mesh, must always pass

through a fixed point on the line

of centers called the pitch point.

- The angular velocity ratio remains

constant through the mesh.

- When the tooth profiles are designed so as to produce a constant angular velocity ratio during meshing, they are said to have conjugate action.

- One of the solution is the involute profile, which is in universal use for gear teeth.

- To transmit motion at a constant angular velocity ratio, the pitch point must remain fixed, that is all the lines of action for every instantaneous point of contact must pass through the same point.

- In the case of the involute profile, all points of contact occur on the same straight line, that is all normal to the tooth profiles at the point of contact coincide with the line of action, and thus these profiles transmit uniform rotary motion.

13.3- Basic Definition:

As shown in figure before, the following definitions will be defined:

1- The pitch circle: •Is the surface of the theoretical rolling cylinder (Tangent to each other).

•The pitch circle is a theoretical surface upon which all calculation are

usually based.

2- The module (m):

Is the ratio of the pitch circle diameter to the number of teeth

(m = d / N or d = m N)

3- The circular pitch (P):

Is the distance measured on the pitch circle from a point on one tooth to

a corresponding point on an adjacent tooth.

(P = tooth thickness + space width).

4- The addendum (a): (Added on):

Is the radial distance between the top land and the pitch circle.

5- The dedendum (b): (Deduced from):

Is the radial distance from the bottom land to the pitch circle.

6- The clearance (C):

Is the amount by which the dedendum in a given gear exceeded the addendum of its mating gear.

7- Addendum circle:

da = d + 2a

= d + 2m

= m N+2m

8- Dedendum circle:

dd = d – 2b

= d – 2 (1.25 m)

= m N - 2.5 m

SPUR GEAR

m=4.5 z=22

13.4- Mesh geometry:

• Pressure line:

Is the tangent to both base circles of the pinion and gear.

It represents the direction in which the resultant force

acts between the gears.

• The pressure angle ():

Is the angle between the pressure line and the pitch line.

(= 14.5o, 20o, 25o)

• Base circle diameter:

db = d cos

13.5- Contact ratio:

Z

- The points of beginning and leaving contact define the mesh of the

pinion and gear. The distance between these points along the line of

action (pressure line) is called the length of action, Z, where:

sin)cos()()cos()( 2222 CrarrarZ gggppp

Where:

• rp, rg : are the pitch circle radii of pinion and gear.

• ap, ag : are the addenda of pinion and gear

• C : is the center distance

- The parameter, Z, my be defined by the intersection of the

respective addendum circles with the line of action (pressure line).

- The contact ratio defines the average number of teeth in contact at

any one time. It is calculated from:

bP

Zcm cosPPb

2.1cm preferred (1.4-1.6)

P

qm t

c Or , where:

tq : Is the arc of action

P : Is the circular pitch

13.6- Interference:

- The involute tooth form is only defined outside of the base circle.

- If the base circle is larger than the dedendum circle, then the portion

of the tooth below the base circle will not be an involute.

- This part (portion) will interfere with the tip of the tooth on the

mating gear, which is an involute. The actual effect is that the

involute tip of the tooth tends to dig out the non-involute part of the

mating gear.

- To avoid interference, the minimum number of teeth:

2minsin

2N

minN

32 14.5

17 20

12 25

minN

13.7- Speed ratio:

D2

V

D1

n1

n2

C=centre distance

Transmission ratio:

2

1

i

: Is the angular velocity

= 60

2 n

n : Is the r.p.m

2

1

n

ni

2211 rrV

p

g

p

g

p

g

N

N

mN

mN

d

d

d

d

r

r

1

2

1

2

2

1

1

2

1

2

2

1

2

1

N

N

d

d

n

ni

13.8- Force analysis of spur gears:

- Designate number 1 for the frame of the machine.

- Designate input gear (pinion) as gear 2.

- Designate successive gears as 3, 4,….. etc

- Designate shafts using lowercase letters a, b, c, ….etc

Pinion2

Gear

n2a

n3 b

3

Wa2

W23

Ta2

a

2

W32

Wb3

b

3

Tb3

Wa2t Ta2

Wa2 Wa2r

a

2

W32

W32W32r

t

- W : Total force between teeth

- Wt : Tangential force (transmitted load)

- Wr : Radial force

- W23 : means force exerted by gear 2 against gear 3.

- W2a : means force exerted by gear 2 against shaft a.

22 rt WWW

cosWW t

d

T

r

TW t 2

HT .TH

VWH t .

sinWW r

60

dnV

,

,

13.9- Tooth stresses analysis:

Limiting design factors in specifying the capacity of any gear drive:

1- The heat generated during operation.

2- Failure of the teeth by breakage.

3- Fatigue failure of the tooth surfaces.

4- Abrasive wear of the tooth surface.

5- Noise as a result of high speeds, heavy loads, or mounting

inaccurcies.

F

RootWt

WWt

Wr

13.9.1- Fatigue failure due to bending stress:

FmJ

Wt

J

(1)

: AGMA (American Gears Manufacturer Association) form factor

(from tables 13-4 (P.497), 13-5 (P.498)).

This factor includes the fatigue stress concentration factor.

F : Face width

m : Module

When a pair of gears is driven at moderate or high speeds and noise is

generated, it is certain that dynamic effects are present. Then a velocity

factor (Kv) must be introduced in the above equation (1), therefore:

FmJK

W

v

t (2)

Where:

VKv

20050

50

(for gears finished by hobbing of shaping)

VKv

20078

78

(for precise gears (ground))

V is the pitch line velocity in m/sec.

Table (13-4): AGMA geometry factor J for teeth having Φ=20o, a=1m, b=1.25m

Table (13-5): AGMA geometry factor J for teeth having Φ=25o, a=1m, b=1.25m

- Fatigue strength:

Where:

medcbaee kkkkkkSS '

eS :Endurance limit of the gear tooth.

'

eS :Endurance limit of rotating beam specimen

ak :Surface factor

bk :Size factor

ck :Reliability factor

dk :Temperature factor

ek :Modifying factor for stress concentration

mk :Miscellaneous effects factor

Surface finish:

The surface factor should always correspond to a machined finish, even

when the flank of the tooth is ground or shaved. The reason for this is

that the bottom land is usually not ground, but left as the original

machined finish. Figure (13-25) shows a chart corresponding to a

surface factor.

Figure (13-25): Surface finish factor for cut, shaved and ground gear

Size factor: The size factor is related to the module as shown in table (13-7)

Factor Kb Module m Factor Kb Module m

0.843 11 1.000 1-2

0.836 12 0.984 2.25

0.824 14 0.974 2.5

0.813 16 0.965 2.75

0.804 18 0.956 3

0.796 20 0.942 3.5

0.788 22 0.930 4

0.779 25 0.920 4.5

0.770 28 0.910 5

0.760 32 0.902 5.5

0.752 36 0.894 6

0.744 40 0.881 7

0.736 45 0.870 8

0.728 50 0.860 9

0.851 10

Table (13-7): Size factor for spur gear teeth

Reliability factor:

The reliability factor is given in table (13-8)

0.9999 0.999 0.99 0.95 0.9 0.5 Reliability R

0.702 0.753 0.814 0.868 0.897 1.0 Factor Kc

Temperature factor:

The temperature factor is given by equation (13-30):

500350 5.0

350 1

T

Tkd

Stress concentration factor:

The fatigue stress concentration factor, Kf has been incorporated into

the geometry factor, J. Since it is fully accounted for, Ke=1 for gears.

Table (13-8): Reliability factor

Miscellaneous effects:

Gear that always rotate in the same direction and are not idlers are

subjected to a tooth force that always acts on the same side of the tooth.

Thus the fatigue load is repeated but not reversed and the tooth is said to

be subjected to one-way bending. This means that the endurance limit

should be increased, therefore in this case:

1400 26)-(13 figure from

1400 33.1

MpaS

MpaSk

ut

ut

m

1

2Time

Load

2

1

3

Driving

Load

Time

One-way bending Figure (13-26): Miscellaneous effects factors

for one way bending

For two-way bending

Km=1

1

2Time

Load

2

1

3

Driving

Load

Time

- Factor of safety:

Generally:

:is the over load factor.

Values recommended by AGMA are listed in table (13-9)

e

G

Sn

mo

G

KK

nn ,

Where:

oK

Driven machinery Source of power

Heavy shock Moderate shock Uniform

1.75 1.25 1 Uniform

2.0 1.5 1.25 Light shock

2.25 1.75 1.5 Medium shock

Table (13-9): Overload correction factor, Ko

:is an GMA load distribution factor which accounts for the

possibility that the tooth force may not be uniformly distributed

across the full face width. Its value is given in table (13-9).

mK

Face Width, mm Characteristic of support

400 up 225 150 0-50

1.8 1.5 1.4 1.3

Accurate mounting, small bearing

clearance, minimum deflection,

precision gears

2.2 1.8 1.7 1.6 Less rigid mountings, less accurate

gears, contact across full face

Over 2.2 Accuracy and mounting such that less

than full face contact exists

Table (13-10): Load distribution factor, Km

Generally, AGMA recommended that, n>2 to guard against

fatigue failure.

13.9.2- Fatigue contact stress (Surface durability):

To assure a satisfactory life, the gears must be designed so that the

dynamic surface stresses are within the surface endurance limit of

the material. From Hertz theory, the surface contact stress can be

obtained as follows:

: is the Hertzian contact stress

IFdC

WC

pv

tpH

Where:

H

pC : Elastic coefficient (factor), from table (13-11)

=

G

G

p

p

EE

2211

1

: Poisson ratio

3

: is the Hertzian contact stress E

vv KC : Dynamic velocity factor

F : Face width

I : AGMA geometry factor = 12

sincos

G

G

m

m

p

g

GN

Nm ,

(The plus sign for external spur gear and the minus sign for internal spur gear)

Gear Modulus of

elasticity E,

GPa

Pinion Tin

bronze

Aluminum

bronze

Cast

iron

Nodular

iron

Malleabl

e iron Steel

158 162 174 179 181 191 200 Steel

154 158 168 172 174 181 170 Malleable

Iron

152 156 166 170 172 179 170 Nodular iron

149 154 163 166 168 174 150 Cast iron

141 145 154 156 158 162 120 Aluminum

bronze

137 141 149 152 154 158 110 Tin bronze

Table(13-11): Values of the elastic coefficient, Cp

- Surface fatigue strength:

Where:

: is the theoretical contact strength (correspond to).

: is the Brinell hardness of the softer of the two contacting surfaces

- The AGMA recommended that this contact fatigue strength be

modified in a manner quite similar to that used for the bending

endurance limit. The equation is:

- The surface fatigue strength for steels is given as:

MpaHBSC 7076.2

The above equation corresponds to a life of stress application.

cS

HB

C

RT

HLH S

CC

CCS

Where:

: is the corrected fatigue strength or Hertzian strength HS

LC : is the life factor

HC : is the hardness ratio factor, for spur gear =1

TC : is the temperature factor, for 1 ,120 T

o CCT

RC : is the reliability factor

Life modification factor:

This factor is used to increase the strength when the gear is to be used

for short periods of time. Its value is given in table (13-12).

Reliability modification factor:

This factor as represented by AGMA is given in table (13-12)

Hardness ratio factor:

This factor was included by AGMA to account for the difference in

strength due to the fact that one of the mating gears might be softer

than the other. However for spur gear =1 HC

HC

Temperature factor:

AGMA makes no recommendation on values to use for temperature

factor when the temperature exceeds . Co120

Reliability

factor, CR

Reliability, R Life factor, CL Cycle of life

0.80 Up to 0.99 1.5 104

1.0 0.99-0.999 1.3 105

1.25 up 0.999 up 1.1 106

1 108 up

Table (13-12): Life & Reliability modification factor

Factors of safety to guard against surface failures should be selected

using the outlines in the proceeding section. Therefore,

: is the over load factor, given in table (13-9)

- Factor of safety:

Where the value of the permissible transmitted load is given when

mo

G

CC

nn

Where:

mm KC

oo KC

: is an AGMA load distribution factor given in table (13-10).

t

Pt

GW

Wn

,

HH S

IFdC

WCS

pv

Pt

pH

,

This is necessary because H and tW in equation (3) are not

linearly related.

2

H

HG

Sn

Or in terms of stress, the safety factor can be obtained as:

CHAPTER 14 Helical, Worm and Bevel Gears

14.1- Helical gears:

- Helical gears are used to transmit motion between parallel

shafts, as shown in Figure.

Crossed helical gear Helical gear

14.1.1- Helical gears kinematics:

1- Helical gears used to transmit motion between parallel shafts.

2- The initial contact of helical gear teeth is a point which changes into

a line as the teeth come into more engagement. The line is diagonal

across the face of the tooth. This gradual engagement of the teeth

gives helical gears the ability to transmit heavy loads at high speeds.

3- Helical gears subject the shaft bearings to both radial and thrust

loads.

4- The helix angle ( ) is the same on each gear, but one gear must

have a right hand (R.H) helix and the other a left hand (L.H) helix,

see Figure below.

Left hand helix Right hand helix

5- Figure (14-3) represents a portion of the top view of a helical rack.

Lines ab and cd are the centerlines of two adjacent helical taken on

the pitch plane.

Figure (14-3): Nomenclature of helical gears

6- The distance ac is the traverse circular pitch, (or ) in the plane

of rotation (usually called the circular pitch).

7- The distance ae is the normal circular pitch, and is related to the

traverse circular pitch by the relation:

(14-1)

8- The distance ad is the axial pitch, and is related to the traverse

circular pitch by the relation:

(14-2)

9- The module in the normal direction, is related to the module in

the transversal direction, (or ) by the relation:

(14-3)

tp p

np

costn pp

xp

tan

t

x

pp

nm

tm m

cosmmn

10- The pitch circle diameter of the helical gear is given by:

(14-4)

11- The pressure angle in the normal direction, is related to the

pressure angle in the direction of rotation (transversal), by the

relation:

(14-5)

12- Addendum circle: (14-6)

Dedendum circle: (14-7)

Nm

mNd

n

cos

n

t

t

n

tan

tancos

na mmdd 2

nd mmdd 5.2

13- Virtual number of teeth of a helical gear, is related to the actual

number, N by the relation:

(14-8)

The virtual number of teeth of a helical gear is shown in Fig. (14-4).

This figure shows a cylinder cut by an oblique plane at an angle to a

right section. The oblique plane cuts out an arc having a radius of

curvature of R. This radius represents the apparent radius of a helical

gear tooth when viewed in the direction of the tooth elements.

Therefore, a gear of the same pitch and with the radius R will have a

greater number of teeth because of the increased radius. In the helical

gear design this is called the virtual number of teeth.

\N

3

\

cos

NN

D

R

a

b

Figure (14-4): A cylinder cut by an oblique plane

14.1.2- Force analysis of helical gears:

Figure (14-5) is a three dimensional view of the forces acting against

a helical gear tooth. The point of application of the forces is in the

pitch plane and in the center of the gear face. From the geometry of

the figure, it is clear that:

pitch cylinder

x

y

z

tooth element

Wa

Wr

Wt

W

t

n

Figure (14-5): Tooth forces acting on a helical gear

a- The force W is first resolved into two components:

(14-9)

b- The force is resolved into two components:

(14-10)

Usually is given, and the other forces are required, therefore:

(14-11)

(14-12)

(14-13)

n

nr

WW

WW

cos

sin

1

sincos

coscos

na

nt

WW

WW

1W

coscos n

tWW

tW

cos

tansin

coscos

ntn

n

tr W

WW

t

n

tan

tancos

(14-14)

and (14-15)

(14-16)

ttr WW tan

sincoscoscos

n

n

ta

WW

tanta WW

Stress analysis of helical gears: -14.1.3

a- Fatigue failure due to bending stress:

The equation for fatigue bending stress is the same as in the spur gear:

(14-17)

(14-18)

(14-19)

Where:

FmJK

W

v

t

e

G

Sn

mo

G

KK

nn

: Bending stress

tW : Transmitted load m

: Face width

vK : Dynamic or velocity factor

F

: Transversal module

For helical gears, the velocity factor (Kv) is given as:

(14-20) V

Kv20078

78

(Since all helical gears have high precision ground teeth)

V is the pitch line velocity in (m/sec).

medcbaee kkkkkkSS '

(Where, the factors is the same as given in the case of spur gears)

J : Geometry factor (from Fig. (14-8 a,b))

Figure (14-8): Geometry factor for helical gears

(a)- Geometry factor for gears mating with a 75 teeth

(b)- J-factor multipliers when tooth numbers other than 75 are used

in the mating gear.

oK :is the over load factor.




1.75 1.25 1 Uniform


Driven machinery

Source of power


mK :is an GMA load distribution factor. Its value is given in table (14-1).

Face Width, mmCharacteristics of support

0-50 150 225 400 up

Accurate mounting, small bearing clearance,

minimum deflection, precision gears1.2 1.3 1.4 1.7

Less rigid mountings, less accurate gears, contact

across full face1.5 1.6 1.7 2.0

Accuracy and mounting such that less than full face

contact existsOver 2.0


The surface contact stress can be obtained as follows:


IFdC

WC

pv

tpH

Where:

H


=

G

G

p

p

EE

2211

1

: Poisson ratio

:Fatigue contact stress (Surface durability) -b

(14-21)

t

Pt

GW

Wn

,

(14-22) mo

G

CC

nn ,

: is the modulus of elasticity E


F : Face width

pd : Pitch diameter of pinion


137 141 149 152 154 158 110 Tin bronze

141 145 154 156 158 162 120 Aluminum

bronze

149 154 163 166 168 174 150 Cast iron

152 156 166 170 172 179 170 Nodular iron

154 158 168 172 174 181 170 Malleable

Iron

158 162 174 179 181 191 200 Steel

Tin

bronze

Aluminum

bronze

Cast

iron

Nodular

iron

Malleabl

e iron Steel

Gear Modulus of

elasticity E,

GPa

Pinion


I : AGMA geometry factor = 12

sincos

G

G

N

tt

m

m

m

p

g

GN

Nm ,


Where:

Nm : Load sharing ratio and is found from the equation:

Z

pm N

N95.0

(14-23)

(14-24)

Where:

Np : Normal base pitch and it is related to the normal circular pitch

by the relation:

nnN pp cos

np

Z : Length of the line of action in the transverse plane and is given by

the equation:

tgpbGggbppp rrrarrarZ sin)(()()( 2222

(14-25)

(14-26)

gp rr ,

Where:

: The pitch circle radii of pinion and gear.

bgbp rr , : The base circle radii of pinion and gear tr cos

gp aa , : The addenda of pinion and gear = nm

Note: ):26-14Certain precaution must be taken in using equation (

1- The tooth profiles are not conjugate below the base circle, and

consequently, if either or is larger than,

,that term should be replaced by .

2- The effective outside radius is sometimes less than owing to

rounding of the tips of the teeth. When this is the case, always use

the effective outside radius instead of .

22)( bppp rar 22 ()( bGgg rar

tgp rr sin)( tgp rr sin)(

ar

ar


Where:

: is the theoretical contact strength (correspond to). : is the Brinell hardness of the softer of the two contacting

surfaces





MpaHBSC 7076.2

cS

HB

C

RT

HLH S

CC

CCS

Where:

HS : is the corrected fatigue strength or Hertzian strength.

LC : is the life factor, and is given in table (13-12)

HC : is the hardness ratio factor. Its value is given in Fig. (14-9).


o CCT

RC : is the reliability factor, and is given in table (13-12).

Figure (14-9): Hardness ratio factor CH for helical gears.

The factor K is the Brinell hardness of the pinion divided by the

Brinell hardness of the gear. Use CH = 1 when K < 1.2.

14.1.4- Direction of axial force:

Direction of axial force depends on:

-Helix angle (L.H or R.H)

-Gear (Driving or Driven)

-Direction of rotation (C.W or C.C.W)

Figure (14-9): Direction of axial force

wtwa

wr

wt

wr

wtwa

wrwr

wa wa wt

Figure (14-10): Force analysis of helical gear

RBY

RBZ

Y

RAZ

RAYMa

wr

x

wt

x

Z

B

ZY

Ma = Wa. r = Wa. d/2

wa x

A

wr

x

wa

r


14.2- Bevel gears: Bevel gears are usually used to transmit motion between two

intersecting shafts (at any angle but usually ). A bevel gear set is

shown in Fig. below.

o90

Spiral bevel

gears Straight bevel

gears

Spiral bevel


gears

Hypoid bevel gears

14.2.1- Bevel gears kinematics:

The terminology of bevel gears is illustrated in Fig. (14-21).

Figure (14-21): Terminology of bevel gears

g

p

L

F - face

pinion pitch dia dp

pinion pitch cone

pitch cone angles

gear pitch dia dg

gear pitch cone

Back cone

back cone radius rb

Spiral bevel


gears


1- The pitch circle of bevel gears is measured at the large end of the

tooth.

2- The pitch diameter and the circular pitch are calculated in the same

manner as for spur gears as:

(14-27)

3- The pitch angles are defined by the pitch cones meeting at the

apex, as shown in Fig. (14-21). They are related to the tooth

numbers as follows:

(14-28)

Where are respectively the pitch angles of the pinion and gear.

mp

mNd

c

,

iN

N

N

N

p

g

g

p

tan

tan

,

4- The Face width is given as;

(14-29)

Whichever is smaller.

mFA

Fo

10or 3

14.2.2- Force analysis of bevel gears:

In determining shaft and bearing loads for bevel gear application, the

usual practice is to use the tangential (transmitted) load which would

occur if all the forces were concentrated at the midpoint of the tooth.

While the actual resultant forces occur somewhere between the

midpoint and the large end of the tooth.

(Calculated at the midpoint of the tooth) (14-30)

Where is the pitch radius of the gear under consideration at the

midpoint of the tooth and is determined as (see Fig. below):

(14-31)

The average velocity is determined from: (14-32)

avav

tV

H

r

TW *

avr

sin2

sin2

Frr

Frr

gav

pav

g

p

avav rV


pit

ch d

iam

eter

dp

Cone distance Ao

'Face width, F

p

Determination of average diameter p

d dav

p

F/2

54

32

1

Figure (14-22): Bevel gear tooth forces

Figure (14-22) is a three dimensional view of the forces acting at the

center of the tooth. From the geometry of the figure, it is clear that:

W

Wa

Wr

Wt

axis ofrevolution

perpendicular to cone

Figure (14-22): Bevel gear tooth forces

a- The force W is first resolved into two components:

(14-33)

b- The force is resolved into two components:

(14-34)

Usually is given, and the other forces are required, therefore:

(14-35)

(14-36)

Where: :is the pressure angle

Note: are used for shaft and bearing design (reactions

and bending moment diagrams)

sin

cos

*

1

*

WW

WWt

*

1W

sinsin

cossin

*

*

WW

WW

a

r

*

tW

cos

*

tWW

sintan

costan

**

**

ta

tr

WW

WW

*** ,, art WWW

Wrp

Wrg

WtgWag

Gear

Wap

Wtp

Pinion

Stress analysis of bevel gears: -14.2.3

are used for check analysis of bevel gears, where the

resultant force in this case occurs at the back cone of the teeth.

a- Fatigue failure due to bending stress: The equation for fatigue bending stress is the same as in the spur gear:

(14-37)

(14-38)

(14-39)

FmJK

W

v

t

e

G

Sn

mo

G

KK

nn

: Bending stress

tW

m

: Face width

vK : Dynamic or velocity factor

F

: Module

art WWW ,,

: Transmitted load, , , V

H

r

TWt

2

22

drV

mNdr

2

22

drV

mNdr

For bevel gears, the velocity factor (Kv) is given as:

(14-40)

V is the pitch line velocity in (m/sec).

medcbaee kkkkkkSS '

(Where, the factors is the same as given in the case of spur gears)

(for gears finished by hobbing of shaping)

VKv

20078

78

(for precise gears (ground)) (14-41)

VKv

20050

50

J : Geometry factor (from Fig. (14-23))

Figure (14-23): Geometry factor J straight for bevel gears

oK :is the over load factor.




1.75 1.25 1 Uniform


Driven machinery

Source of power


mK :is an GMA load distribution factor. Its value is given in table (14-2).


Both gears outboard One gear outboard Both gears

inboard

Application

1.25-1.4 1.1-1.25 1-1.1 General industrial

1.1-1.25 1-1.1 Automotive

1.25-1.5 1.1-1.40 1-1.25 Aircraft

Both gears inboard One gear outboard Both gears outboard

Figure (14-24): Mounting of bevel gears

The surface contact stress can be obtained as follows:


IFdC

WC

pv

tpH

Where:

H


=

G

G

p

p

EE

2211

1

: Poisson ratio

:Fatigue contact stress (Surface durability) -b

(14-42)

t

Pt

GW

Wn

,

(14-43) mo

G

CC

nn ,

: is the modulus of elasticity E


F : Face width,

pd : Pitch diameter of pinion


c

o

p

mF

AF

10or

3

Gear Modulus of

elasticity E,

GPa

Pinion Tin

bronze

Aluminum

bronze

Cast

iron Steel

195 199 203 232 207 Steel

178 182 186 203 130 Cast iron

174 178 182 199 120 Aluminum

bronze

170 174 178 195 110 Tin bronze

I : AGMA geometry factor (from Fig. (14-25))

Figure (14-25): Geometry factor I straight for bevel gears


Where:

: is the theoretical contact strength (correspond to).

: is the Brinell hardness of the softer of the two contacting surfaces





MpaHBSC 7076.2

cS

HB

C

RT

HLH S

CC

CCS

Where:

HS : is the corrected fatigue strength or Hertzian strength.

LC : is the life factor, and is given in table (13-12)

HC : is the hardness ratio factor =1


o CCT

RC : is the reliability factor, and is given in table (13-12).


14.3- Worm gears: A worm gear is used when a large speed

reduction ratio is required between crossed

axis shafts which do not intersect. A basic

helical gear can be used but the power which

can be transmitted is low. A worm drive

consists of a large diameter worm wheel

with a worm screw meshing with teeth on

the periphery of the worm wheel. The worm

is similar to a screw and the worm wheel is

similar to a section of a nut. As the worm is

Spiral bevel

gears

rotated the worm wheel is caused to rotate due to the screw like action

of the worm. The size of the worm gearset is generally based on the

centre distance between the worm and the worm wheel.

Spiral bevel


gears

- Worm gearset are either single or double enveloping.

- In single enveloping set, the worm wheel has its width cut into a

concave surface, thus partially enclosing the worm when in mesh.

- In double enveloping set, in addition to having the worm wheel width

cut concavely, this type has the worm length cut concavely. The result

is that both the worm and gear partially enclose each. A double

enveloping set will have more teeth in contact and will have area

rather than line contact, thus permitting greater load transmission.

Single enveloping gearset Double enveloping gearset

Spiral bevel


gears

Spiral bevel


gears

Figure (14-13): Terminology of worm gears

14.3.1- Worm gearing kinematics:

The terminology of bevel gears is illustrated in Fig. (14-13).

Spiral bevel


gears

Pitch cylinder

Helix

Lead LLead angle

pitch diameter dw

Worm Axial pitch px


Spiral bevel


gears


Spiral bevel


gears

• Worm and gear have same hand of helix.

• Helix angles are quite different.

• Worm has large helix angle, gear has small helix angle.

• Specify lead angle λ on worm and helix angle ψg on the gear. These

are the same for a 90° shaft angle.

(λ = ψg for 90o shafts)

• The axial pitch (px) of the worm and the transverse circular pitch (pt)

of the gear are equal for 90o shafts.

(px = pt for 90o shafts)

• The pitch diameter of the worm is not related to the number of teeth.

It is chosen such that:

Where:

C: is the center distance.

0.875 0.875

3 1.7W

C Cd

Spiral bevel


gears

• The pitch diameter of the gear is the diameter measured on a plane

containing the worm axis.

• The lead (L):

Where:

Nw : Number of teeth of worm (number of starts)

• The face width Fg of the worm gear should be made equal to the

length of a tangent to the worm pitch circle between its points of

intersection with the addendum circle.

G tG

N pd

x WL p N

tanW

L

d

14.3.2- Force analysis of worm gearing:

Figure (14-15) is a three dimensional view of the forces acting on the

worm. From the geometry of the figure, it is clear that:

n

Wy

Wx

Wz

t

W

pitch helixpitch cylinder

x

y

z

W Wf =

Wcos

Wsin

Figure (14-15): Drawing of the pitch cylinder of a worm,

showing the forces exerted upon it by the worm gear.

Spiral bevel


gears

Figure (14-15): Drawing of the pitch cylinder of a worm,

showing the forces exerted upon it by the worm gear.

1- Neglecting friction:

Force exerted by gear onto the worm is W.

Subscript W and G represent forces acting on Worm and Gear

respectively.

Gear axis is parallel to x, worm axis is parallel to z, right handed

coordinate system

cos sin

sin

cos cos

x y z

x n

y n

z n

W W W

W W

W W

W W

W i j k

Wt Ga x

Wr Gr y

Wa Gt z

W W W

W W W

W W W

2- Including friction:

• Relative motion between worm and gear is sliding

- Friction is important.

- Need to introduce coefficient of friction .

• Given a load W acting normal to the tooth profile, with a

component in negative x direction, in positive z

direction

Where: : is the coefficient of friction between teeth.

fW W

cosW sinW

cos sin cos

sin

cos cos sin

x n

y n

z n

W W

W W

W W

• Experimentally, coefficient of friction is dependent on sliding

velocity

Where:

VG = pitch line velocity of gear

Vw = pitch line velocity of worm

cos

W G S

WS

VV

V V V

Figure (14-17): representative

values of the coefficient of friction

for worm gears.

Curve A when more friction is

expected (C.I).

Curve B for high quality materials

(case hardened worm mating with

a phosphor-bronze gear)

Efficiency:

cotcos

tancos

friction)with ( .

friction)without ( .

friction power with required

frictionout power with required

n

n

wt

wt

W

W

14.3.3- Power rating of worm gearing:

lo HHH

Where:

- Hl: is the losses power, and is given as:

Hl = Wf x Vs

Where:

Wf : is the friction force (= W)

Vs : is the sliding speed

Hl = W x Vs

- Ho: is the output power, and is given as:

Ho= Wtg x Vtg

Where:

260

2 gg

ggtg

dnrV

- The maximum (permissible) tangential force on the gear is given as:

vmegstg KKFdKW 8.0

Where:

: Conversion factor = 0.0131

: Material factor (from table 1) sK

gd : Pitch diameter of the gear

eF : Effective face width

= smaller of ( or Fg (where Fg is shown in figure)) wd

3

2

FgmK : Ratio correction factor (from table 2)

: Velocity factor (from table 3) vK

Table 1: Material factor (Ks) for cylindrical worm gears

Centrifugal

Cast Bronze

Static-Chill

Cast Bronze

Sand-Cast

Bronze

Face width,

mm

1000 800 700 Up to 75

975 780 665 100

940 760 640 125

900 720 600 150

850 680 570 175

800 640 530 200

750 600 500 225

Table 2: Ratio correction factor (Km)

Km Tr. Ratio Km Tr. Ratio Km Tr. Ratio

0.825 30 0.724 8.0 0.500 3.0

0.815 40 0.744 9.0 0.554 3.5

0.785 50 0.760 10 0.593 4.0

0.745 60 0.783 12 0.620 4.5

0.687 70 0.799 14 0.645 5.0

0.622 80 0.809 16 0.679 6.0

0.490 100 0.820 20 0.706 7.0

Kv Vs)m/s) Kv Vs)m/s) Kv Vs)m/s)

0.216 7.2 0.47 1.50 0.649 0.005

0.200 8 0.45 1.80 0.646 0.008

0.187 9 0.42 2.00 0.644 0.050

0.175 10 0.395 2.25 0.638 0.100

0.168 11 0.375 2.50 0.631 0.150

0.156 12 0.360 2.80 0.625 0.200

0.148 13 0.340 3.00 0.615 0.300

0.140 14 0.310 3.60 0.600 0.400

0.134 16 0.285 4.00 0.590 0.500

0.106 20 0.265 4.50 0.560 0.750

0.089 25 0.258 5.00 0.530 1.000

0.079 30 0.235 6.00 0.500 1.250

Table 3:

Velocity

factor

(Kv)

CHAPTER 12

Sliding (Journal) Bearings 12.1- Introduction:

- Bearings are supports for rotating shafts. Generally a shaft can be

effectively supported by two bearings, one at each end.

- In case of long shafts and when a shaft carry a number of mountings

(such as gears, pulley, ….etc), intermediate supports or bearings are

usually provided, to reduce the unwanted deflection of the shaft.

12.2- Classification of bearings:

According to type of contact of shaft with bearing or according to the

kind of friction generated in active surfaces, bearings can be classified

into:

A- Sliding bearings:

- Sliding bearings are those bearings where shaft is in direct contact

with bearing and is sliding on its cylindrical surfaces.

- Because of the nature of contact, the friction between the mating

parts is usually high, so these bearings require more lubrication.

- The lubrication does not totally eliminate contact between the surfaces.

B- Anti-friction bearings:

- These bearings are known as rolling bearings in which a pure rolling

motion is achieved in place of the sliding motion which occurs in

sliding bearings.

- As the rolling friction is much less than the sliding friction, rolling

friction are called anti-friction bearings.

- Generally, both types of bearings (sliding bearings & anti-friction

bearings) are divided according to the direction of the applied load.

- Advantages of plain bearings:

1- They have a very low coefficient of friction if properly

designed and lubricated.

2- They have very high load-carrying capabilities.

3- Their resistance to shock and vibration is greater than

rolling-contact bearings.

4- The hydrodynamic oil film produced by plain bearings

damps vibration, so less noise is transmitted.

5- They are less sensitive to lubricant contamination than

rolling-contact bearings.

- Advantages of rolling-contact bearings

1- At low speeds, ball and roller bearings produce much less

friction than plain bearings.

2- Certain types of rolling-contact bearings can support both

radial and thrust loading simultaneously.

3- Rolling bearings can operate with small amounts of

lubricant.

4- Rolling-contact bearings are relatively insensitive to

lubricant viscosity.

5- Rolling-contact bearings have low wear rates and require

little maintenance.

Types of load

Thrust load-1 Radial load-2

Combined load-3

12.3- Sliding (Journal) bearings:

- According to the direction of the applied load, sliding bearings are

dived into:

A- Radial bearings(Journal bearings):

- The journal bearings are used to support only the normal or radial

loads (loads acting perpendicular to the shaft axis).

- The journal bearings rotates inside a stationary bush or sleeve. The

journal is that part of the shaft which is in contact with the bearing.

B- Thrust bearings:

- Thrust bearings are used where loads acting along shaft axis are to

be supported.

Radial bearings (Journal bearings): -12.3.1

1- Solid bearing:

Hole for bolting the bearing

Hole to introduce lubricant

- Is the simplest form of the journal bearings.

- This is usually made of C.I.

- As the name implies, this is consists of one block in which a hole is

bored to receive the journal.

- The rectangular base of the bearings has two holes which are used for

bolting down the bearing.

- A hole provided at the top is used to introduce lubricant into the

bearing.

- This type of bearing is used for light duty service only. (low speeds,

low loads).

- The drawback of this bearing is that it has to be discarded once the

inner surface of the bearing gets worn-out as there is no provision for

adjustment for wear.

2- Bushed bearing:

- Bushed bearings consist mainly of two parts, the body and the bush.

- The body or the main block is made of cast iron.

- The bush being usually made of soft materials like brass, bronze,

undergoes wear and can be periodically replaced.

- The bush can be fixed either by:

a- Pressing fit into place (H7,H8/n6.p6,r6,s6).

b- Set screw, which is fitted half in the main block and half in the bush

and may help to prevent relative motion or axial movement of the

bush in the block.

- Bushes re standardized are defined by d X DX L

(Bush 25H7 X 32r6 X 50).

- The oil hole provided at the top is used

to introduce the required lubricant.

3- Pedestal (Split) Bearing

- Pedestal bearings consist mainly of a

pedestal, a cap and a bush split into two

halves called 'brasses.'

- Easy assembly of the unit and the

periodical replacement of the brasses is

made by the split parts.

- Flanges are provided to prevent the axial

movement.

- For long shafts requiring intermediate

supports, pedestal bearings (Plummer

blocks) are preferred in place of ordinary

bushed bearing.

Examples of pedestal bearings:

12.4- Types of lubrication:

1- Hydrodynamic lubrication:

- The most effective technique in journal bearings.

- The surface of the mating parts are separated by a

relatively thick

film of lubricant.

- The film pressure is created by the moving surfaces itself.

- Surface wear does not occur.

- Film thicknesses 0.008-0.020 mm.

- f=0.002-0.010.

2- Hydrostatic lubrication:

- The lubricant is introduced at a pressure high enough to

separate the surfaces with a thick film of lubricant.

- Continuous flow of lubricant to the sliding interface.

- e.g air hockey, hovercraft.

- f=0.002-0.010

Air hockey Hovercraft

3- Elastohydrodynamic lubrication:

- When a lubricant is introduced between surfaces which are in

rolling contacts, such as mating gears, cams or rolling bearing.

- Elastohydrodynamic: occurs if the contacting surfaces

nonconforming as with the gear teeth or cam and follower.

Small contact patch allows a full hydrodynamic film to form.

- Depends on elastic deformation of parts.

•

ω

Lubricant film Elastic deformation

4- Boundary lubrication:

- Insufficient surface area, drop in the velocity of the

moving surface, increase load or increase in the lubricant

temperature, may prevent the built up of enough film

thickness.

- f=0.05-0.20.

5- Solid film:

- When bearings must be operated at extreme temperature, a

solid film lubricant such as graphite may be used because

the ordinary mineral oils are not satisfactory.

- Low coefficient of friction.

12.5- Viscosity:

- Viscosity is a measure of fluid’s resistance to shear.

- Viscosity, , for fluids is analagous to shear modulus, G, for solids.

- Temperature increases, viscosity decreases.

- Pressure increases, viscosity increases.

- To derive the absolute viscosity we consider two parallel surfaces,

one moving relative two the other with a fluid trapped between the

two surfaces.

Surface

Platey

h

u

F

dy

du

A

F Newton’s law of viscous flow

Where: Is the absolute (dynamic) viscosity.

- The shear stress is proportional to the rate of change of velocity

w.r.t the y.

- Assume = constant

- Units of viscosity:

a- (N/m2)/s-1 = pa.s

b- The Poise, P = dyn.s/cm2, dyn = gm.cm/s2

Cp = 1/100 P

h

u

dy

du

h

u

12.6- Petroff’s Law:

- r : Shaft radius

- c : Clearance (filled with oil)

- l : length of bearing

- N : rev/s

- If the shaft rotates at N (rev/s), then its surface velocity,

u = ω.r = 2лN.r

c

Nr

h

u

2

- The torque:

- If W is the radial force acting on the bearing, then the pressure:

- The frictional force is fW, where f is the coefficient of friction.

- The frictional torque T:

C

Nlr

rrlc

rrl

r

rFT

324

2Nr2

2

A

rl

WP

2

flPr

rrlPf

rWfT

22

2

(2)

(1)

- From equations (1) and (2), the coefficient of friction:

c

r

P

Nrf

22 Petroff’s Law (1383)

12.7- Sliding bearings nomenclature:

- O : Center of the journal

- O’ : Center of the bearing

- c : The radial clearance (difference in the radii of the bearing and

journal)

- e : Eccentricity

- h : Oil film thickness at any point

- ho : Minimum oil film thickness and it occurs at the line of centers.

- : e/c ……. Eccentricity ratio

12.8- Sliding bearings design:

Two groups of variables in the design of journal bearings:

1- Selected (chosen) parameters:

A- The viscosity, .

B- The load per unit of projected bearing area, P.

C- The speed N (rev/s)

D- The bearing dimensions (r, c, β, l)

2- Dependent variables:

A- The coefficient of friction, f.

B- The temperature rise, ΔT.

C- The oil flow, Q.

D- The minimum oil film thickness, ho.

E- Angle for maximum oil film pressure.

12.9- Bearing characteristic number (Sommerfeld number):

This quantity is defined by the equation:

Where:

S: bearing characteristic number

r: journal radius

c: radial clearance

: absolute viscosity

N: speed (rev/s)

P: load per unit of projected bearing area.

P

N

c

rS

2

12.10- Design steps:

1- Temperature rise:

- The average working temperature of the oil:

Where:

Ti : is the inlet temperature

ΔT : is the temperature rise

- The dimensionless temperature rise variable is:

Where:

γ : density of oil (861 kg/m3)

CH : specific heat of the lubricant (1760 J/kg Co)

2

TTT iav

P

TCT H

var

Procedure for determining the temperature rise:

a- Estimate the average temperature of the oil.

b- Find for the chosen oil at Tav. (from figure 12-10& 12-11)

c- Calculate the bearing characteristic number.

d- Find the temperature rise variable Tvar. (from figure 12-12)

e- Determine the temperature rise ΔT from the relation:

2

2015

TTT

CCT

iav

OO

P

N

c

rS

2

P

TCT H

var

Figures (12-10, 12-11): Viscosity temperature chart

Figure (12-12): Chart for temperature variable

f- Find Tav from the relation:

g- Repeat again from step (b) until to get two successive Tav very close.

h- According to the last Tav, we have to get and S.

2

TTT iav

oCold

av

new

av TT 2

2- From figure (12-13) and for certain l/d and S→ Find ho/c and .

(The minimum oil film thickness variable and eccentricity ratio)

Figure (12-13): Chart for temperature variable

3- From figure (12-14) and for certain l/d and S→ Find position of

minimum of minimum film thickness Φo.

Figure (12-14): Chart for determining the position of the

minimum oil film hickness

4- From figure (12-16) and for certain l/d and S→ Find the coefficient

of friction variable (r/c)f.

Figure (12-16): Chart for coefficient of friction variable

5- The torque required to overcome friction:

Where:

f : Coefficient of friction

W : Radial load on bearing

r : Radius of bearing

6- The power loss (due to friction):

rWfT

NT

TH

2

7- From figure (12-17) and for certain l/d and S→ Find the flow

variable Q/rcNl

Figure (12-17): Chart for flow variable

- NOTE: The amount of oil supplied

to the bearing must be > Q

8- From figure (12-18) and for certain l/d and S→ Find the side

leakage variable (Qs/Q).

Figure (12-18): Chart for side leakage variable

9- From figures (12-19) & (12-20) and for certain l/d and S→ Find the

maximum film pressure variable (P/Pmax) → figure (12-19) & its

angular location (θPmax) and the terminating position of the oil film

(θPo) → figure (12-20).

Figure (12-19): Chart for maximum film pressure variable

Figure (12-20): Chart for finding the terminating position of the lubricant

film and the position of maximum film pressure

11.1- kinds of rolling contact bearings:

CHAPTER 11

Rolling Contact Bearings

- As the rolling friction is much less than the sliding friction, rolling

bearings are called 'antifriction bearings‘.

Ball bearings Roller bearings

- Each of these may be subdivided into:

a- Radial bearings

b-Thrust bearings

c-Radial-Thrust bearings

11.1.1- Ball Bearings:

a- Radial Ball Bearings:

Face outer ring

inner ring

inner ring ball rqace

outer ring ball race

corner radius

corner radius

separatorBalls

outsidediameter Bore

shoulders

The rolling contact bearings consists as a rule of:

- Inner race

- Outer race (the inner race being pushed or pressed onto the shaft and

the outer race being secured in the housing).

- The rolling elements (balls) at a certain distance from one another.

- Cage or separator (The rolling elements roll within a separator or a

cage, to prevent the rolling elements from rubbing each other and

to hold the rolling elements at a constant distance from each other).

- Running tracks of both races of these bearings are sufficiently deep

to carry axial loads in addition to radial loads at high speeds.

- Balls are inserted in bearings in such a way that the inner race is placed eccentrically

against the outer race, after which the inner race is returned to a concentric position.

Then the balls are uniformly spaced along periphery and the two parts separator (cage)

are installed from either side to be riveted.

filling slot

Conrad method

b- Thrust Ball Bearings:

- Exist in two types:

- They consist of 4 elements:

a- Shaft race.

b- Housing race. c- Steel balls (Rolling elements). d- Cage or separator.

Single row thrust

Ball bearings

Double row thrust

Ball bearings

c- Radial-Thrust Ball Bearings:

Deep groove Ball Angular contact ball

bearings bearings

There are many kind of Radial-Thrust Ball Bearings, the most common

in industrial use are:

11.1.2- Roller Bearings:

a- Radial roller Bearings:

For radial roller bearing, the contact between the rolling elements

and the run races is a line contact and not a point contact as in ball

bearing. So, the load carrying capacity of roller bearing is higher

than this is of the ball bearings with the same dimension. They are

proper to be used for great impact loading.

Straight roller Needle

b- Thrust Roller Bearings:

Cylindrical Roller Spherical roller Taper Roller

c- Radial-Thrust roller Bearings:

Taper roller Spherical roller

- Load is transferred through elements in rolling contact rather than

sliding contact.

- It consists of:

1- Inner ring (race).

2- Outer ring (race).

3- Rolling elements.

- Other elements:

4- Cage.

5- Sealing.

6- Dust protection.

Ball

Bearing

Straight

roller

Needle

Taper

roller

Spherical

roller

Roller

Bearing

Rolling Elements:

- The life of an individual bearing is defined as the total number of

revolutions, or the number of hours at a given constant speed, of

bearing operation required for the failure criteria to develop.

- Rating Life:

Is the number of revolutions or working hours of operation at a given

constant speed, that 90% of a group of bearings will complete or

exceed before failure. Rating life is termed as L10.

11.3- Bearing load:

- If two groups of identical bearings tested under different loads F1, F2,

they will have different respective lives L1, L2, then:

11.2- Bearing Life:

a

F

F

L

L

1

2

2

1

Where: a = 3 for ball bearings

= 10/3 for roller bearings

(1)

- Basic load rating (C):

- Is the constant radial load which a group of identical bearings can

endure for a rating life of 1 million revolutions of the inner ring.

-If L1= 106 of revolutions, F1= C, then:

a

C

F

L

2

2

1

a

F

F

L

L

1

2

2

1

a

F

CL

Where: L in millions of revolutions.

(2)

Equation (2) can be written as:

aFLC /1

- In general:

a

R

D

R

DeqR

n

n

L

LFC

/1

Where, the subscripts:

- D → for designed or required values

- R → for rated values (from catalog)

- LR (standard) 500 hrs for ISO system

3000 hrs for Timken

- nR (standard)

100/3 r.p.m for ISO system

500 r.p.m for Timken

- ISO system → (SKF, ZKL, FAG…..)

- LD : Designed for L10 life (i.e, the reliability is 90%)

- For a reliability other than 90%:

a

a

R

D

R

DeqR

Rn

n

L

LFC

17.1/1

/1

/1ln

1

84.6

1

- Generally:

a

a

R

D

R

DeqaR

Rn

n

L

LFKC

17.1/1

/1

/1ln

1

84.6

1

Where:

Ka :Application factor

11.4- Selection of Ball Bearings:

areq YFVXFPF

Where:

Feq = equivalent radial load.

Fr = applied radial load.

Fa = applied thrust load.

X = radial load coefficient.

Y = axial load coefficient.

V = a rotation factor:

1 for rotating inner ring

1.2 for rotating outer ring

eFPF req r

a

F

F when -

eYFVXFPF areq r

a

F

Fen wh-

e → is given in the bearing tables according to the value of , Where

Co is the static basic load rating. o

a

C

F

Fa/Co e X Y

0.025 0.22 0.56 2

0.04 0.24 0.56 1.8

0.07 0.27 0.56 1.6

0.13 0.31 0.56 1.4

0.25 0.37 0.56 1.2

0.5 0.44 0.56 1

Table for X, Y for deep groove ball bearings:

11-5- Selection of Taper Roller Bearings

The nomenclature for a taper roller bearings is shown in the figure

Below.

- The inner ring is called the cone, and the outer ring is called

the cup.

- It can be seen that, a tapered roller bearing is separable in

that the cup can be removed from the cone and roller

assembly.

- This type of bearing can carry both radial and axial loads or

any combinations of the two.

- However, even when an external axial load is not present,

the radial load will induce a thrust (axial) reaction within

the bearing because of the taper.

- The mounting of bearings can be as follows:

a- O-configuration (back-to-back).

b- X-configuration (face-to-face).

eFPF req r

a

F

F when -

eYFFPF areq r

a

F

Fen wh4.0 -

The values of factors e and Y will be found in the bearing tables.

- All the requisite equations for the various bearings arrangements

and load cases are given as follows:

Designation of Bearing (bearing codes): -6-11 - Rolling element bearings re categorized by a code made up of two

sections:

A- section 1:

The code for the bearing series which is further divided into:

- A type code,

- A diameter series and

- in many cases a width series.

B- section 2:

The code for the bore diameter.

- Type code:

The first digit, letter of the bearing code define the bearing type.

•1 Self aligning ball

•2 Type 1 but wider

•3 Double row angular contact

•4 Double row ball

•6 Single row ball (deep groove)

•7 Single row angular contact

•16 Type 6 but narrower

•22 Self aligning roller

•23 Type 22 but wider

•51 Thrust ball

•M Radial ball with filling slots

•N Cylindrical roller

•HJ - Separate thrust collar)

•QJ Single row duplex ball

- Diameter and width series (dimension series):

- The second pair of digits define the dimension series.

- The first number is from the width series (0, 1, 2, 3, 4, 5 and 6)

- The second number is from the diameter series (outside diameter).

(8, 9, 0, 1, 2 ,3 and 4).

- The most common sizes being defined as follows:

•0 Extra light

•1 Extra light thrust

•2 Light

•3 Medium

•4 Heavy

- Note:

For 02, 03, 04 the zero is ignored.

- Example: 0 2 (0 is width series, 2 is diameter series).

- Bore code:

- Bores from 10-17 mm:

Bore diameter code

10 00

12 01

15 02

17 03

20 04

- Bores from 20-480 mm:

Code no.= Bore diameter / five

- Designation of Bearing:

15

6202

25

6205

20 17 12 10 Bore diameter

6204 6203 6201 6200 Designation

Number

10 0 200 300 400

EL L M H

•EL Extra Light series

•L Light series

•M Medium series

•H Heavy series

Rolling element type

10.1- Introduction:

CHAPTER 10

Mechanical Springs

- A spring is a flexible elastic object used to store mechanical energy.

- Springs are usually made out of hardened steel. Small springs can be

wound from pre-hardened stock, while larger ones are made from

annealed steel and hardened after fabrication.

- Mechanical springs are used to:

1- Provide flexibility,

2- store or absorb energy.

- Mechanical springs are classified as:

Wire springs Flat springs Special shaped springs

• Helical spring

(Round or Square)

• Leaf spring

• Belleville spring

Tensile Compressive Torsional

Flat springs

Leaf spring Belleville spring

(a) Five in Series

(b) Six in Parallel

(c) Combination of

Parallel and Series.

http://www.mech.uwa.edu.au/DANotes/springs/intro/discSpringsBIG.gif

10.2- Stresses in Compression Coil springs:

- Subjected to shear stresses only.

- The maximum shear stress in the wire cross section:

Where:

.

.

.

.

A

F

J

rT

max

2.D

FT

2

dr

32

4dJ

4

2dA

D: mean diameter of the spring

d: diameter of the wire

23

24max

48

432

22

d

F

d

FD

d

F

d

dDF

Define as spring index

d

DC

Cd

FD 5.01

83max

Let

CKs

5.01

3max

8

d

FDKs

Where:

sK : is the shear stress multiplication factor

- To include the effect of curvature, Wahl suggested a modification on

the formula to be:

3max

8

d

FDK

Where:

CC

CK

165.0

44

14

(Wahl correction factor)

- This factor includes the effect of direct shear together with another

effect due to curvature. Therefore, by defining:

scKKK Where:

cK : is the effect of curvature alone.

- Investigations reveal that, for static loads, the curvature stress can be

neglected. Therefore:

- For fatigue loads is used as a fatigue strength reduction factor.

sKK

cK

10.3- Deflection of Compression Coil springs:

- The static energy in the spring:

GJ

lTU

2

2

Where:

l : is the total length of spring wire.

DN

N : is the number of active coils.

DT NNN

TN : total number of coils.

DN : dead (inactive) number of coils.

2

DFT

-

-

-

32

4dJ

-

- G : shear modulus.

Gd

NDF

dG

DNDFU

4

32

4

22

4

3224

)()(

- The deflection: F

Uy

Gd

NFDy

4

38

- The spring constant (rate or stiffness): ykF

ND

Gd

y

Fk

3

4

8

Types of ends for compression springs:

a- Both ends plain, ND=1/2

b- Both ends squared, ND=1

c- Both ends squared and

ground, ND=2

d- Both ends plain and ground, ND=1

10.4- Springs materials:

- Types of Materials:

• Hard drawn high carbon steel

• Oil tempered high carbon steel

• Stain less steel

- Light-duty springs

• Copper or nickel based alloys.

- Experiments show that the tensile strength of the spring materials are

related to the wire size by the equation:

mutd

AS

A

Where:

: is a constant related to the strength.

-

-

- m : is the slope of the line on the log-log plot.

uty SS 75.0

- An approximate relation-ship between yield strength and ultimate

strength in tension:

- By applying the distortion-energy theory:

ysy SS 577.0

- Table (10-2) constants used to estimate the tensile strength of spring steels

10.5- Fatigue loading:

- Springs are subjected to a fluctuating stress.

3

minmax 8

2 d

DFK

FFF m

smm

-

- 3

minmax 8

2 d

DFK

FFF a

saa

Fa

Fa

Fm

in

Fm

ax

Fm

Time

Force

- Static safety factor:

max

sy

s

Sn -

ysy SS 577.0-

uty SS 75.0-

mutd

AS - (A & m constants given from table (10-2))

- am max

- Fatigue safety factor:

a

seSn

-

springs peenedfor MPa 465

springs unpeenedfor MPa 310'

seS-

- These values are corrected for surface finish & size but not for reliability,

temperature or stress concentration.

cesese kkSS '

Where:

ck : is the reliability factor.

ek : is the stress concentration factor , cK

1

s

cK

KK

c

csese

K

kSS '

10.6- Extension springs:

A

F

tb

I

cM

23

m

d

4

d

r32F

FK

tb

i

m

r

rK

http://www.mech.uwa.edu.au/DANotes/springs/intro/torsSpringsBIG.jpeg

10.7- Torsion springs:

3d

r32F

K

14

14 2

CC

CCK

i

- Normal stress:

For inner race

14

14 2

CC

CCK

oFor outer race

E4d

64FrDN

- The angular deflection (in radian):

Torque required to wind up the spring one

turn.

- The spring rate:

DN

E

k

64

d

N.m/rad Fr

4

DN

Ek

2.10

d4'

This value of corrected to: DN

Ek

8.10

d4' 'k

CHAPTER 17 Flexible Mechanical Elements

Belt Drives 17.1- Introduction:

- Flexible mechanical elements such as belts, ropes or chains are used

for the transmission of power over comparatively long distances.

- When these these elements are employed, they usually replace a

group of gears, shafts, and bearings.

- They, thus greatly simplify a machine and consequently are a major

cost-reducing elements.

17.2- Belt drives:

- A belt drive is a method of transferring rotary motion between two

parallel shafts. A belt drive includes one pulley on each shaft and one or

more continuous belts over the two parallel pulleys. The motion of the

driving pulley is transferred to the driven pulley via the friction between

the belt and the pulley.

- Belts have the following characteristics:

1- Easy, flexible equipment design, as tolerances are not important.

2- Isolation from shock and vibration between driver and driven system.

3- Driven shaft speed conveniently changed by changing pulley sizes.

4- Belt drives require no lubrication.

5- Maintenance is relatively convenient.

6- Very quiet compared to chain drives, and direct spur gear drives.

- Kind of Belts:

1- Flat belt drive:

2- V- belt drive:

3- Timing belt drive:

- Chin drive:

http://www.fox-prints.com/images/chain_drive.jpg

17.3- Flat Belt drives:

17.3.1- Geometrical relations:

: wrap angle (contact angle), s for small & l for large pulleys. -

d : diameter of small pulley. -

D : diameter of large pulley. -

C : center distance. -

D

d

s

l

sl dDdDCL 2

14

22

- From geometry, the length of the belt:.

C

dDdDCL

422

2

- 0r from shiegly:

( in radians)

dDdDCLc 2

422

- For crossed belt:.

C

dDdDCLc

422

2

- 0r from shiegly:

( in radians)

17.3.2- Belt tensions:

1P : Tension of tight side. -

2P : Tension of loose side. -

m : Mass of unit length (of one meter) of the belt (kg). -

f : Coefficient of friction between belt & pulley.. -

P1P2

d

dd

P+dp

mv2d

fdN

v

P

x

y

P

- An element of the belt is subjected to:

(1)

dN

i) Tension &

fdN

dmvr

vmrd 2

2

dPP

02

cos2

cos fdNd

Pd

dPP

02

sin2

sin 2 dNdmvd

Pd

dPP

ii) Normal & frictional forces &

iii) Centrifugal forces =

- An equilibrium in x & y directions leads to:

(2)

- For small value of : d

22

sin ,12

cos ddd

- Substitute in equation (1):

fd

mvP

dP

2

f

dPdNfdNdP 0

1

2 02

P

P

dfmvP

dP

femvP

mvP

2

2

21

feP

P

2

1

(3)

- Substitute from equation (3) in equation (2) and neglecting second

order terms:

(4)

- Integrating equation (4):

- Neglecting the centrifugal forces:

Euler equation

- For V-belts:

2

2

211

212

dPPT

DPPT

vPPH 21

2

sin

22

21

f

emvP

mvP

d

Di

- Transmitted torque:

- Transmission ratio:

- Power:

- Condition of maximum power transmission:

PPP

PPP

i

i

2

1

iP

2

21 PPPi

: initial tightening force. -

BY : Prof Dr. Nabil Gadallaheng.modern-academy.edu.eg/E-learning/Mech/Mach. Design II - MNF...

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Transcript of BY : Prof Dr. Nabil Gadallaheng.modern-academy.edu.eg/E-learning/Mech/Mach. Design II - MNF...