Toothed gearing.pdf

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CHAPTER 2
TOOTHED GEARING
KINEMATICS OF MACHINERY

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KINEMATICS OF MACHINERY
1.1 Introduction
Like belts, ropes and chains, toothed gears
also transmit power from one rotating shaft to
another, which may be parallel, intersecting or
skewed (i.e., which are neither parallel nor
intersecting)..

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KINEMATICS OF MACHINERY
Toothed gears are used:
Where the distance between the axes of connecting
shafts is short.
The speed of shafts is low and the belt drive is not
recommended.
The speed or the velocity ratio of the connecting
shafts is to be maintained constant.
The torque transmitted is high.
Whenever step up or step down of the speed
required

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Types of Toothed gears:
Axes parallel - spur or helical gears.
Axes intersecting - Bevel gears.
Axes neither parallel nor intersecting - spiral,
worm gears.

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Spur gear Helical gears Bevel gears
Worm gear Rack and pinion
Fig.1 Types of gears
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Spur gears:
Spur gears used to connect two parallel shafts. They are
the most common type of gears.
They have straight teeth parallel to the axes of the shaft.
Each time a gear tooth engages a tooth on the other gear,
the teeth collide, and this impact makes a noise.
It also increases the stress on the gear teeth.
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Helical gears:
The gears are used to connect two parallel shafts and
having teeth inclined (or curved) to the axes of the shafts
are known as helical gears.
Two mating gears have the same helix angle, but have
teeth of opposite hands.
When two teeth on a helical gear system engage, the
contact starts at one end of the tooth and gradually
spreads as the gears rotate, until the two teeth are in full
engagement.
This gradual engagement makes helical gears operate
much more smoothly and quietly than spur gears.
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Rack and pinion
The rack is like a gear whose axis is at infinity.
Racks are straight gears that are used to
convert rotational motion to translational motion
by means of a gear mesh.
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Bevel gears
Useful when the direction of a shaft's rotation
needs to be changed.
They are used to connect two intersecting shafts.
Bevel gears are usually mounted on shafts that
are 90 degrees apart, but can be designed to
work at other angles as well.
The teeth on bevel gears can be straight, spiral or
hypoid.
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Worm and worm gear:
The gears are used to connect two non-parallel
and non-intersecting shafts.
Worm gears are used when large gear
reductions are needed.
It is common for worm gears to have reductions
of 20:1, and even up to 300:1 or greater.
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Terminologies used in Gears:
Gear Ratio (G): Gear ratio is defined as the number
of teeth on the follower gear (driven gear) to the
number of teeth on the driver gear.
G = T2
/T1
Where T is the number of teeth on gear.
Addendum Circle: It is a circle passing through the
tips of teeth.
Addendum: It is the radial height of a tooth above the
pitch circle. Its standard value is one module.
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• Dedendum circle or Root Circle: It is a circle
passing through the roots of the teeth.
• Dedendum: It is the radial depth of a tooth below
the pitch circle
• Space Width: It is the width of the tooth space
along the pitch circle.
• Tooth Thickness: It is the thickness of the tooth
measured along the pitch circle.
• Backlash: It is the difference between the space
width and the tooth thickness along the pitch circle.
Backlash = Space width — Tooth thickness
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Face Width: The length of the tooth parallel to the
gear axis is the face width.
Top Land: It is the surface of the top of the tooth.
Bottom Land: The surface of the bottom of the tooth
between the adjacent fillets.
Face: Tooth surface between the pitch circle and the
top land.
Flank: Tooth surface between the pitch circle and
the bottom land including fillet.
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Velocity Ratio (VR): Velocity Ratio is defined as the
angular velocity of the follower gear (driven gear) to
the angular velocity of the driver gear.
VR= = N2
/N1
T1
/T2
or G = 1/VR
Pitch circle diameter or Pitch Diameter: (D)
It is the diameter of a circular disc which by pure
rolling action would transmit the same motion as of the
gear wheels. In case of spur gear, this represents the
diameter of the cylindrical disc which the gear has
replaced.
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Pitch Surface: It is the surface of the disc which the
toothed gear has replaced at the pitch circle.
Pitch point: It is the point of contact of the two pitch
circles of the mating gears.
Pitch line: It is the line of contact of two pitch
surfaces.
Pitch: As applied to toothed gears, is expressed as
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Circular pitch (Pc
) - It is the distance measured along
the circumference of the pitch circle from a point on
one tooth to the corresponding point on the adjacent
tooth.
Pc
= D / T where ‘D’ is pith circle diameter and ‘T’ is
number of teeth.
Diametral pitch: (PD
) or DP - It is defined as the
number of teeth per unit length of the pitch circle
diameter. This is commonly adopted in F.P.S. system.
PD
= T/D
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Module (m): - It is expressed as the length of the
pitch circle diameter per tooth and it is the reciprocal
of diametrical pitch. It is commonly adopted in metric
system.
m =D/T or D = mT
Pc
= m
For two meshing gears, their module pitch or their
circular pitch should be same.
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Cycloidal Teeth
Cycloid: It is the locus of a point on the circumference
of a circle that rolls without slipping on a fixed straight
line.
Epicycloid: It is the locus of a point on the
circumference of a circle that rolls without slipping on
the circumference of another circle.
Hypocycloid: It is the locus of a point on the
circumference of a circle that rolls without slipping inside
the circumference of another circle.
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Involute Teeth
Involute is the locus of a point on a straight line which
rolls on the circumference of a circle without slipping.
Also it is the path traced out by the free end of a taut
string being unwound from the circumference of a
circle. The circle on which the straight line rolls is
called ‘Base Circle’.
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Properties or characteristics of Involute teeth
The shape of the involute profile is dependent
only on the dimensions of the base circle.
For a pair of involute gears in mesh the angular
velocity ratio is inversely proportional to the size
of base circles.
Pitch diameters of two mating involutes are
directly proportional to their base circle
diameters.
The normal to the involute at a given point is the
tangent drawn from that point to the base circle.
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The common normal to the two involute profiles
of the meshing teeth always passes through the
pitch point.
Involute is the only tooth form that is not sensitive
to the centre distance of their base circles.
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INVOLUTOMETRY
Involutometry is a methodology by which the
dimensions of a gear tooth are determined. In
considering involute for a tooth form, it is necessary to
be able to calculate certain properties of the involute.
Derivation ……
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Determination of Length
of Path of Contact of two
meshing Involute gears
Derivation…
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ARC OF CONTACT :
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INTERFERENCE
IN
INVOLUTE GEARS
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Interference in Involute Rack & pinion:
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MINIMUM NUMBER
OF TEETH ON PINION
TO AVOID
INTERFERENCE
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MINIMUM NUMBER OF TEETH ON PINION MESHING
WITH A RACK
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Numericals from
Chapter : Toothed Gearing
End of Chapter