Chapter 12 - site.iugaza.edu.pssite.iugaza.edu.ps/mhaiba/files/2013/09/CH12-Toothed-Gearing-2015… · Angular velocity ratio is inversely proportional to ratio of distances of point

Chapter 12

Toothed

Gearing

12/25/2015

Mohammad Suliman Abuhiba, Ph.D., PE1

12.2. Friction Wheels

Motion & power

transmitted by gears

is kinematically

equivalent to that

transmitted by

friction wheels.

12/25/2015

Mohammad Suliman Abuhiba, Ph.D., PE

2

12.2. Friction Wheels

Wheel B will be rotated by wheel A as

long as the tangential force exerted by

wheel A does not exceed the maximum

frictional resistance between the two

wheels.

When the tangential force (P) exceeds the

frictional resistance (F), slipping will take

place between the two wheels.

Thus friction drive is not a positive drive.

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12.2. Friction

Wheels

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12.3. Advantages of Gear Drive

1. Transmit exact velocity ratio

2. Transmit large power

3. High efficiency

4. Reliable service

5. Compact layout

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12.3. Disadvantages of Gear Drive

1. Manufacture of gears require

special tools and equipment

2. Error in cutting teeth may

cause vibrations and noise

during operation

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12.4. Classification of Toothed Wheels

1. According to position of axes of shafts

2. According to peripheral velocity of gears

3. According to type of gearing

4. According to position of teeth on gear surface

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According to position of axes of shafts

Parallel Shafts

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SpurHelical

12.4. Classification of Toothed WheelsAccording to position of axes of shafts

Intersecting

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According to position of axes of shafts

Non-intersecting and non-parallel

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skew bevel gears worm gears


According to peripheral velocity of gears

a. Low velocity: less than 3 m/s

b. Medium velocity: between 3 & 15 m/s

c. High velocity: more than 15 m/s

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According to type of gearing

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Internal gearing

External gearing

Rack and pinion


According to position of teeth

on gear surface

a. Straight: spur

b. Inclined: helical

c. Curved: spiral

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12.5. Terms Used in Gears

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Construction of an involute curve

../../Videos/Chapter 12/Kinematics with MicroStation Ch06 F Involute Gear Generation.mp4

Pitch circle: imaginary circle at which we have

pure rolling action between the mating gears

Pitch point: common point of contact between

two pitch circles

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Pressure angle, f: angle between commonnormal to two gear teeth at the point of contact

and the common tangent at the pitch point.

Standard pressure angles are 14.5 ° & 20°

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6. Addendum: radial distance of a tooth from

pitch circle to top of tooth

7. Addendum circle: circle drawn through top of

teeth and is concentric with pitch circle.

8. Dedendum: radial distance of a tooth from

pitch circle to bottom of tooth

9. Dedendum circle (root circle): circle drawn

through bottom of teeth

Root circle diameter = Pitch circle diameter × cosf

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Clearance: radial distance from top of tooth to

bottom of tooth, in a meshing gear

Clearance circle: A circle passing through top of

meshing gear

Total depth: radial distance between addendum

& dedendum circles of a gear

Total Depth = addendum + dedendum

Working depth: radial distance from addendum

circle to clearance circle

Working depth = sum of addendums of two meshing

gears

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Tooth thickness: width of tooth measured along

the pitch circle

Tooth space: width of space between two

adjacent teeth measured along the pitch circle

Backlash: difference between tooth space &

tooth thickness, as measured along the pitch

circle

Theoretically, backlash should be zero

In actual practice some backlash must be allowed to

prevent jamming of teeth due to tooth errors &

thermal expansion

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Face of tooth: surface of gear tooth above pitch

surface

Flank of tooth: surface of gear tooth below pitch

surface

Top land: surface of top of tooth

Face width: width of gear tooth measured parallel

to its axis

Profile: curve formed by face and flank of tooth

Fillet radius: radius that connects root circle to

profile of tooth

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Circular pitch (pc): distance measured on

circumference of pitch circle from a point of one

tooth to corresponding point on next tooth.

D = Diameter of pitch circle

T = Number of teeth

Two gears will mesh together correctly, if the two

wheels have the same circular pitch:

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Diametral pitch (pd): ratio of number of teeth to

pitch circle diameter in inch

Module (m): ratio of pitch circle diameter in mm to

number of teeth

Module, m = D / T

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12.7 Condition for Constant Velocity Ratio of

Toothed Wheels–Law of Gearing

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a

b

E

D

F

../../Videos/Chapter 12/Kinematics with MicroStation - Ch06 A Law of Gearing.mp4

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The two teeth come in contact at point C

MN = common normal to the curves at C

O1M & O2N perpendicular to MN

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vC1 & vC2 = velocities of point C on

wheels 1 & 2 respectively

If the teeth are to remain in contact,

components of these velocities along

the common normal MN must be equal.

vC1 cos a = vC2 cos b

w1.O1C (O1M/O1C) = w2. O2C (O2N/ O2C)

w1.O1M = w2.O2N

w1/w2 = O2N/O1M = O2P/O1P

Angular velocity ratio is inversely

proportional to ratio of distances of point

P from O1 & O2.

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Common normal at point of contact between

a pair of teeth must always pass through the

pitch point

Fundamental condition which must be

satisfied while designing profiles for teeth of

gear

To have a constant angular velocity ratio for all

positions of wheels, point P must be the fixed

point (pitch point) for the two wheels.

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12.8 Velocity of Sliding of Teeth

Sliding between a pair of teeth in contact

at C occurs along the common tangent

Velocity of sliding = velocity of one tooth

relative to its mating tooth along the

common tangent at the point of contact

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12.8 Velocity of Sliding of Teeth

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Vs = VC1 sina – VC2 sinb

= (w1.O1C)(MC/O1C) – (w2.O2C)(NC/O2C)

= w1.MC – w2.NC = w1.(PM+PC) – w2.(PN-PC)

= w1.PM – w2.PN + (w2+w1)PC

w1.PM = w2.PN

Vs = (w2+w1)PC

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../../Videos/Chapter 12/Kinematics with MicroStation - Ch06 D Involute Engagement.flv

Path of contact: path traced by point of contact

of two teeth from beginning to end of

engagement

Length of path of contact: length of common

normal cut-off by addendum circles of the gear &

pinion

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../../Videos/Chapter 12/Kinematics with MicroStation Ch06 B Involute Gears.mp4

Arc of contact: path traced by a point on pitch

circle from beginning to end of engagement of a

given pair of teeth

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The arc of contact consists of two parts:

1. Arc of approach: portion of path of contact from

beginning of engagement to pitch point

2. Arc of recess: portion of path of contact from

pitch point to end of engagement

Contact ratio: ratio of length of arc of contact to

circular pitch. Number of pairs of teeth in

contact.

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12.9 Forms of Teeth

1. Cycloidal teeth

2. Involute teeth

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Cycloid = curve traced by a point on

circumference of a circle which rolls without

slipping on a fixed straight line

12.10 Cycloidal Teeth

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Epi-cycloid: curve traced by a point on

circumference of a circle When a circle rolls

without slipping on outside of a fixed circle


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../../Videos/Chapter 12/Epicycloid animation HINDI.mp4


Hypo-cycloid: curve

traced by a point on

circumference of a

circle when the

circle rolls without

slipping on inside of

a fixed circle

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../../Videos/Chapter 12/Hypocycloid animation HINDI.mp4


Cycloidal

Conjugate pairs

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../../Videos/Chapter 12/Kinematics with MicroStation - Ch06 C Cycloidal Gears.mp4


Assignment: Generate a conjugate Cycloidal

pair according to the following:

Pitch diameter of pinion (driver) = 100 mm

Pitch diameter of gear (driven) = 200 mm

Rolling circle diameter = 40 mm

Due Monday 23/11/2015

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12.11 Involute Teeth

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An involute of a circle

= plane curve

generated by a point

on a tangent, which

rolls on circle without

slipping or by a point

on a taut string which

is unwrapped from a

reel as shown.


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From similar triangles

O2NP and O1MP


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When power is being transmitted, maximum

tooth pressure is exerted along common normal

through the pitch point

This force may be resolved into tangential and

radial components

Torque = FT × r

12.12. Effect of Altering Centre Distance on

Velocity Ratio for Involute Teeth Gears

Centre of rotation of gear1 is shifted from O1 to O1'

Contact point shifts from Q to Q‘

Tangent M'N' to base circles intersects O1’O2 at

pitch point P‘

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12.12. Effect of Altering Centre Distance on

Velocity Ratio for Involute Teeth Gears

If center distance is changed within limits, velocity

ratio remains unchanged

Pressure angle increases with increase in center

distance

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Example 12.1

A single reduction gear of 120 kW with a pinion

250 mm pitch circle diameter and speed 650 rpm

is supported in bearings on either side. Calculate

the total load due to the power transmitted, the

pressure angle being 20°.

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Involute Gears Cycloidal Gears

Center distance for a pair of

gears can be varied within limits

without changing velocity ratio

Require exact center distance to

be maintained

Pressure angle, from start to

end of engagement, remains

constant. It is necessary for

smooth running & less wear of

gears.

Pressure angle is max at

beginning of engagement,

reduces to zero at pitch point,

starts decreasing and again

becomes max at end of

engagement. This results in less

smooth running of gears.

Comparison Between Involute & Cycloidal Gears

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Face & flank are generated by a single

curve

Double curves (epi-cycloid

& hypo-cycloid)

Easy to manufacture. The basic rack

has straight teeth that can be cut with

simple tools.

Not easy to manufacture

Interference occurs with pinions having

smaller number of teeth. This may be

avoided by altering heights of

addendum & dedendum of mating

teeth or pressure angle of the teeth.

Interference does not

occur at all

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Greater simplicity and

flexibility of involute gears

Since cycloidal teeth have

wider flanks, therefore

cycloidal gears are stronger

than involute gears, for the

same pitch.

Convex surfaces are in

contact

Contact takes place between

a convex flank & concave

surface, This results in less

wear.

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12.14. Systems of Gear Teeth

Four systems of gear teeth are in

common practice:

1. 14.5° Composite system

2. 14.5° Full depth involute system

3. 20° Full depth involute system

4. 20° Stub involute system

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12.15 Standard Proportions of

Gear Systems - Involute14 °

composite or

full depth

20° full

depth

20° stub

Addendum 1 m 1 m 0.8 m

Dedendum 1.25 m 1.25 m 1 m

Working depth 2 m 2 m 1.6 m

Minimum total depth 2.25 m 2.25 m 1.8 m

Tooth thickness 1.5708 m 1.5708 m 1.5708 m

Minimum clearance 0.25 m 0.25 m 0.2 m

Fillet radius at root 0.4 m 0.4 m 0.4 m

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12.16 Length of

Path of Contact

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../../Videos/Chapter 12/Kinematics with MicroStation - Ch06 D Involute Engagement.flv

12.16 Length of Path of Contact

Length of path of contact = length of common

normal cutoff by addendum circles of gear &

pinion

Length of path of contact = KL = KP + PL

KP = path of approach

PL = path of recess

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rA = O1L = Radius of addendum circle of pinion

RA = O2K = Radius of addendum circle of gear

r = O1P = Radius of pitch circle of pinion

R = O2P = Radius of pitch circle of gear

12.16. Length of Path of Contact

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12.16. Length of Path of Contact

Length of path of approach

From right angled triangle O2KN

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12.16 Length of Path of ContactLength of path of recess,

From right angled triangle O1ML

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12.16 Length of Path of Contact

Length of path of contact,

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12.17 Length of Arc of Contact

Arc of contact = path traced by a point on pitch

circle from beginning to end of engagement

In Fig. 12.11, arc of contact = EPF or GPH

GPH = arc GP + arc PH

Arc GP = arc of approach

Arc PH = arc of recess

Angles subtended by these arcs at O1 are

called angle of approach & angle of recess

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length of arc of approach (arc GP)

length of arc of recess (arc PH)

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Length of arc of contact

12.17. Length of Arc of Contact

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Contact ratio = number of pairs of teeth in

contact = ratio of length of arc of contact to

circular pitch

CR

12.18 Contact Ratio (CR)

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Example 12.2

A pinion having 30 teeth drives a gear having 80

teeth. The profile of the gears is involute with 20°

pressure angle, 12 mm module and 10 mm

addendum. Find the length of path of contact, arc

of contact and the contact ratio.

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Example 12.3

The number of teeth on each of the two equal

spur gears in mesh are 40. The teeth have 20°

involute profile and the module is 6 mm. If the arc

of contact is 1.75 times the circular pitch, find the

addendum.

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Example 12.4

Two involute gears of 20° pressure angle are in

mesh. The number of teeth on pinion is 20 and the

gear ratio is 2. If the pitch expressed in module is 5

mm and the pitch line speed is 1.2 m/s, assuming

addendum as standard and equal to one module,

find :

1. Angle turned through by pinion when one pair of

teeth is in mesh

2. Maximum velocity of sliding

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Example 12.5

A pair of gears, having 40 and 20 teeth respectively,

are rotating in mesh, the speed of the smaller being

2000 rpm. Determine the velocity of sliding

between the gear teeth faces at the point of

engagement, at the pitch point, and at the point of

disengagement if the smaller gear is the driver.

Assume that the gear teeth are 20° involute form,

addendum length is 5 mm and the module is 5 mm.

Also find the angle through which the pinion turns

while any pairs of teeth are in contact.

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Example 12.6

The following data relate to a pair of 20° involute

gears in mesh: Module = 6 mm, Number of teeth on

pinion = 17, Number of teeth on gear = 49 ; Addenda

on pinion and gear wheel = 1 module. Find

1. Number of pairs of teeth in contact

2. Angle turned through by pinion and gear when

one pair of teeth is in contact

3. Ratio of sliding to rolling motion when the tip of a

tooth on the larger wheel

i. is just making contact

ii. is just leaving contact with its mating tooth

iii. is at the pitch point

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Example 12.7

A pinion having 18 teeth engages with an internal

gear having 72 teeth. If the gears have involute

profiled teeth with 20° pressure angle, module of 4

mm and the addenda on pinion and gear are 8.5

mm and 3.5 mm respectively, find the length of

path of contact.

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12.19 Interference in Involute Gears

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Contact begins when

tip of driven tooth

contacts flank of

driving tooth.

Flank of driving tooth

first makes contact

with driven tooth at A,

and this occurs

before involute

portion of driving

tooth comes within

range.


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Contact occurs

below base circle

of gear 2 on non-

involute portion

of flank.

Involute tip of

driven gear tends

to dig out the

non-involute

flank of driver.


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If radius of addendum circle of pinion is

increased to O1N, point of contact L will move

from L to N.

When this radius is further increased, point of

contact L will be on the inside of base circle of

wheel and not on the involute profile of tooth on

wheel.


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Tip of tooth on pinion will then undercut the tooth

on wheel at root and remove part of involute

profile of tooth on the wheel.

Interference: when tip of tooth undercuts the

root on its mating gear


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If radius of addendum circle of wheel increases

beyond O2M, then tip of tooth on wheel will cause

interference with the tooth on pinion.

Points M & N are called interference points.

Limiting value of the radius of addendum circle

of pinion is O1N and of wheel is O2M.

Max length of path of approach, MP = r sinf

Max length of path of recess, PN = R sinf

Max length of path of contact,

MN = MP + PN = r sinf + R sinf = (r + R) sinf

Max length of arc of contact


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Example 12.8

Two mating gears have 20 and 40 involute teeth of

module 10 mm and 20° pressure angle. The

addendum on each wheel is to be made of such a

length that the line of contact on each side of the

pitch point has half the maximum possible length.

Determine the addendum height for each gear

wheel, length of the path of contact, arc of contact

and contact ratio.

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12.20 Minimum Number of Teeth on

Pinion in Order to Avoid Interference

Np = # of teeth on pinion

NG = # of teeth on wheel

m = Module of teeth

r = Pitch circle radius of pinion = m.Np / 2

m = Gear ratio = NG / Np = R / r

f = Pressure angle

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12.20 Minimum Number of Teeth on

Pinion in Order to Avoid Interference

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System of gear teeth Minimum number of

teeth on pinion

14.5° Composite 12

14.5° Full depth involute 32

20° Full depth involute 18

20° Stub involute 14

12.21. Minimum Number of Teeth on Gear

in Order to Avoid Interference

NG = Min number of teeth required on gear in

order to avoid interference

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Example 12.9

Determine minimum number of teeth required on a

pinion, in order to avoid interference which is to

gear with,

1. a wheel to give a gear ratio of 3 to 1

2. an equal wheel

The pressure angle is 20° & a standard addendum

of 1 module for the wheel may be assumed.

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Example 12.10

A pair of spur gears with involute teeth is to give a

gear ratio of 4 : 1. The arc of approach is not to be

less than the circular pitch and smaller wheel is the

driver. The angle of pressure is 14.5°. Find:

1. Least number of teeth that can be used on each

wheel

2. Addendum of wheel in terms of the circular pitch

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Example 12.11

A pair of involute spur gears with 16° pressure

angle and pitch of module 6 mm is in mesh. The

number of teeth on pinion is 16 and its rotational

speed is 240 rpm. When the gear ratio is 1.75, find

in order that the interference is just avoided;

1. addenda on pinion and gear wheel

2. length of path of contact

3. maximum velocity of sliding of teeth on either

side of the pitch point

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Example 12.12

A pair of 20° full depth involute spur gears having

30 and 50 teeth respectively of module 4 mm are in

mesh. The smaller gear rotates at 1000 rpm.

Determine:

1. sliding velocities at engagement and at

disengagement of pair of a teeth

2. contact ratio

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Example 12.13

Two gear wheels mesh externally and are to give a

velocity ratio of 3 to 1. The teeth are of involute

form ; module = 6 mm, addendum = one module,

pressure angle = 20°. The pinion rotates at 90 rpm.

Determine:

1. Number of teeth on pinion to avoid interference

on it and corresponding number of teeth on

wheel

2. Length of path and arc of contact

3. Number of pairs of teeth in contact

4. Maximum velocity of sliding

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12.22 Min Number of Teeth on a Pinion for

Involute Rack in Order to Avoid Interference

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Example 12.14

A pinion of 20 involute teeth & 125 mm pitch circle

diameter drives a rack. The addendum of both

pinion & rack is 6.25 mm.

1. What is the least pressure angle which can be

used to avoid interference?

2. With this pressure angle, find the length of the

arc of contact and the minimum number of

teeth in contact at a time.

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Documents

Chapter 12 - site.iugaza.edu.pssite.iugaza.edu.ps/mhaiba/files/2013/09/CH12-Toothed-Gearing-2015… · Angular velocity ratio is inversely proportional to ratio of distances of point