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Chapter 12
Toothed
Gearing
1/2/2015
Dr. Mohammad Abuhiba, PE 1
12.2. Friction Wheels
Motion & power
transmitted by gears
is kinematically
equivalent to that
transmitted by
friction wheels.
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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
a. Parallel
b. Intersecting
c. Non-intersecting and
non-parallel
2. According to
peripheral velocity of
gears
a. Low velocity
b. Medium velocity
c. High velocity
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3. According to type of
gearing
a. Internal gearing
b. External gearing
c. Rack and pinion
4. According to position of
teeth on gear surface
a. Straight
b. Inclined
c. Curved
12.4. Classification of Toothed Wheels
According to position of axes of shafts
Parallel Shafts
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Spur
Helical
12.4. Classification of Toothed Wheels
According to position of axes of shafts
Intersecting
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12.4. Classification of Toothed Wheels
According to position of axes of shafts
Non-intersecting and non-parallel
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skew bevel gears worm gears
12.4. Classification of Toothed Wheels
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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12.4. Classification of Toothed Wheels
According to type of gearing
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Internal gearing
External
gearing
Rack and pinion
12.4. Classification of Toothed Wheels
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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12.5. Terms Used in Gears
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12.5. Terms Used in Gears
Construction of an involute curve
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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12.5. Terms Used in Gears
Pressure angle, f: angle between common
normal to two gear teeth at the point of contact
and the common tangent at the pitch point. The
standard pressure angles are 14.5 ° & 20°
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12.5. Terms Used in Gears
6. Addendum: radial distance of a tooth from
pitch circle to top of tooth
7. Dedendum: radial distance of a tooth from
pitch circle to bottom of tooth
8. Addendum circle: circle drawn through top of
teeth and is concentric with pitch circle.
9. Dedendum circle (root circle): circle drawn
through bottom of teeth
Root circle diameter = Pitch circle diameter ×
cosf
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12.5. Terms Used in Gears
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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12.5. Terms Used in Gears
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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12.5. Terms Used in Gears
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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12.5. Terms Used in Gears
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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12.5. Terms Used in Gears
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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12.5. Terms Used in Gears
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 wheel
& pinion
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12.5. Terms Used in Gears
Arc of contact: path traced by a point on pitch
circle from beginning to end of engagement of a
given pair of teeth
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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12.5. Terms Used in Gears
12.7 Condition for Constant Velocity Ratio of
Toothed Wheels–Law of Gearing
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a
b
E
D
F
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12.7 Condition for Constant Velocity Ratio of
Toothed Wheels–Law of Gearing
Consider the portions of two teeth
Let the two teeth come in contact at
point C
MN = common normal to the curves at C
From O1 & O2, draw O1M and O2N
perpendicular to MN.
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12.7 Condition for Constant Velocity Ratio of
Toothed Wheels–Law of Gearing
vC1 & vC2 = velocities of point C on
wheels 1 & 2 respectively
If the teeth are to remain in contact,
the components of these velocities
along the common normal MN must
be equal.
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12.7 Condition for Constant Velocity Ratio of
Toothed Wheels–Law of Gearing
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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12.7 Condition for Constant Velocity Ratio of
Toothed Wheels–Law of Gearing
The common normal to the two surfaces at
point of contact C intersects line of centers at
P which divides the center distance inversely
as the ratio of angular velocities
In order 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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12.7 Condition for Constant Velocity Ratio of
Toothed Wheels–Law of Gearing
The common normal at point of contact
between a pair of teeth must always pass
through the pitch point.
This is the fundamental condition which must
be satisfied while designing profiles for teeth
of gear wheels.
12.8 Velocity of Sliding of Teeth
The sliding between a pair of teeth in
contact at C occurs along the common
tangent to the tooth curves.
The velocity of sliding is the 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
Triangles O1PM & O2PN are similar
O1P/O2P = PM/PN
w2/ w1 = PM/PN
w1.PM = w2.PN
Vs = (w2+w1)PC
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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12.10 Cycloidal Teeth
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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12.10 Cycloidal Teeth
Hypo-cycloid: curve traced
by a point on the
circumference of a circle
when the circle rolls
without slipping on the
inside of a fixed circle
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12.10 Cycloidal Teeth
Cycloidal Conjugate
pairs
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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 the circle
without slipping or by a
point on a taut string
which in unwrapped from
a reel as shown.
12.11 Involute Teeth
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12.11 Involute Teeth
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From similar triangles
O2NP and O1MP
12.11 Involute Teeth
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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.
These components act along and at right angles
to the common tangent to the pitch circles.
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‘
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.
12.13. Comparison Between
Involute and Cycloidal Gears
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Involute Gears Cycloidal Gears
Face & flank are generated by a single
curve
double curves (epi-
cycloid & hypo-cycloid)
are required
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
12.13. Comparison Between
Involute and Cycloidal Gears
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Involute Gears Cycloidal Gears
Greater simplicity and
flexibility of the 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 and concave
surface, This results in less
wear.
12.13. Comparison Between
Involute and Cycloidal Gears
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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 - Involute
14 ° 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
When pinion
rotates in cw
direction, contact
between a pair of
involute teeth
begins at K &
ends at L.
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12.16. Length of
Path of Contact
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12.16. Length of
Path of Contact Length of path of contact = length of common normal
cutoff by addendum circles of wheel &
pinion.
Length of path of contact = KL = KP + PL
KP = path of approach
PL = path of recess
rA = O1L = Radius of addendum circle of pinion
RA = O2K = Radius of addendum circle of wheel
r = O1P = Radius of pitch circle of pinion
R = O2P = Radius of pitch circle of wheel
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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 Contact Length of path of recess,
From right angled triangle O1ML
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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12.17 Length of Arc of Contact
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12.17. Length of Arc of Contact length of arc of approach (arc GP)
length of arc of recess (arc PH)
Length of arc of contact
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12.18. Contact Ratio (CR)
Contact ratio = number of pairs of teeth
in contact = ratio of length of arc of
contact to circular pitch
CR
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Example 12.2
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 12.3
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 12.4
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. The number of pairs of teeth in contact
2. The angle turned through by the pinion and the gear
wheel when one pair of teeth is in contact
3. The 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.
The flank of driving tooth
first makes contact with
driven tooth at A, and this
occurs before involute
portion of driving tooth
comes within range.
Contact is occurring 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.
12.19 Interference in Involute Gears
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Fig. 12.13 Interference in involute gears
12.19 Interference in Involute Gears
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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.
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
12.19 Interference in Involute Gears
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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
12.19 Interference in Involute Gears
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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.t / 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 the 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 Wheel
in Order to Avoid Interference
NG = Min number of teeth required on wheel 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. The least number of teeth that can be used on
each wheel
2. The addendum of the 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. the addenda on pinion and gear wheel
2. the length of path of contact
3. the 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. The number of teeth on pinion to avoid
interference on it and the corresponding
number of teeth on the wheel
2. The length of path and arc of contact
3. The number of pairs of teeth in contact
4. The 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 and 125 mm pitch
circle diameter drives a rack. The addendum of
both pinion and rack is 6.25 mm. What is the least
pressure angle which can be used to avoid
interference ? 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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Evaluation Quiz
Quiz will be held on
Wednesday 31/12/2014
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