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Chapter 12
Toothed
Gearing
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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
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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12.4. Classification of Toothed Wheels
According to position of axes of shafts
Parallel Shafts
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SpurHelical
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12.4. Classification of Toothed WheelsAccording 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
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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
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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
Construction of an involute curve
../../Videos/Chapter 12/Kinematics with MicroStation Ch06 F Involute Gear Generation.mp4
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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
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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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12.5. Terms Used in Gears
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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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12.5. Terms Used in Gears
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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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12.5. Terms Used in 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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12.5. Terms Used in Gears
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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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12.5. Terms Used in Gears
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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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12.5. Terms Used in Gears
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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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12.5. Terms Used in Gears
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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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12.7 Condition for Constant Velocity Ratio of
Toothed Wheels–Law of Gearing
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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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,
components of these velocities along
the common normal MN must be equal.
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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
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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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12.7 Condition for Constant Velocity Ratio of
Toothed Wheels–Law of Gearing
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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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12.5. Terms Used in Gears
../../Videos/Chapter 12/Kinematics with MicroStation - Ch06 D Involute Engagement.flv
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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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12.5. Terms Used in Gears
../../Videos/Chapter 12/Kinematics with MicroStation Ch06 B Involute Gears.mp4
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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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12.5. Terms Used in Gears
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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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12.5. Terms Used in Gears
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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
12.10 Cycloidal Teeth
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../../Videos/Chapter 12/Epicycloid animation HINDI.mp4
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12.10 Cycloidal Teeth
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
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12.10 Cycloidal Teeth
Cycloidal
Conjugate pairs
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../../Videos/Chapter 12/Kinematics with MicroStation - Ch06 C Cycloidal Gears.mp4
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12.10 Cycloidal Teeth
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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12.11 Involute Teeth
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12.11 Involute Teeth
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From similar triangles
O2NP and O1MP
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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
Torque = FT × r
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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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Involute Gears Cycloidal Gears
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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Comparison Between Involute & Cycloidal Gears
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Involute Gears Cycloidal Gears
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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Comparison Between Involute & 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 - 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
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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 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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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)
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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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12.19 Interference in Involute Gears
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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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12.19 Interference in Involute Gears
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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.
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12.19 Interference in Involute Gears
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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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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.
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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.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
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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
12/25/2015
Mohammad Suliman Abuhiba, Ph.D., PE
85
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12.22 Min Number of Teeth on a Pinion for
Involute Rack in Order to Avoid Interference
12/25/2015
Mohammad Suliman Abuhiba, Ph.D., PE
86
-
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.
12/25/2015
Mohammad Suliman Abuhiba, Ph.D., PE
87
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