GEAR DRIVES

Dr. S. Solomon Raj

Associate Professor, Department of Mechanical Engineering
 KINEMATICS OF GEARS

Gears are toothed, cylindrical wheels used for transmitting motion and power from one
rotating shaft to another.

The teeth of a driving gear mesh accurately in the spaces between teeth on the driven
gear.

The driving teeth push on the driven teeth, exerting a force perpendicular to the radius
of the gear.
Thus, a torque is transmitted, and because the gear is rotating, power is also
transmitted.

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

3/8/2023 S. Solomon Raj, PhD, MED, CBIT
 Speed Reduction Ratio
Often gears are employed to produce a change in the speed of rotation of the
driven gear relative to the driving gear.

The amount of speed reduction is dependent

on the ratio of the number of teeth in the
pinion to the number of teeth in the gear
according to this relationship:

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

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 Spur gears

have teeth that are straight and arranged

parallel to the axis of the shaft that carries the
gear. The curved shape of the faces of the spur
gear teeth have a special geometry called an
involute curve

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

 SPUR GEAR GEOMETRY INVOLUTE-TOOTH FORM

The most widely used spur gear tooth form is the full depth involute form.

The involute is one of a class of geometric curves called conjugate curves.

When two such gear teeth are in mesh and rotating, there is a constant angular
velocity ratio between them:
From the moment of initial contact to the moment of disengagement, the speed
of the driving gear is in a constant proportion to the speed of the driven gear. The
resulting action of the two gears is very smooth.
If this were not the case, there would be some speeding up and slowing down
during the engagement, with the resulting accelerations causing vibration, noise,
and dangerous torsional oscillations in the system.

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

 Law of gearing states that the common normal at the point of contact
between a pair of teeth must always pass through the pitch point for all
positions of mating gear. This law forms the basis for the gear profile design.
This is a must condition for the two gears to perform properly.

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

 It has been found that only involute and cycloidal
curves satisfy the fundamental law of gearing.

Cycloidal tooth offers the following advantages

compared with involute tooth:

(i) In case of cycloidal gears, a convex flank on one

tooth comes in contact with the concave flank
of the mating tooth. This increases the contact
area and also the wear strength. In involute
gears, the contact is between two convex
surfaces on mating teeth, resulting in smaller
contact area and lower wear strength.

(ii) The phenomenon of interference does not occur

at all in cycloidal gears.

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

✓However, cycloidal teeth are rarely used in practice due to the following
disadvantages:

(i) Cycloidal tooth is made of two curves— hypocycloid curve below the pitch circle and
epicycloid curve above the pitch circle. It is very diffi cult to manufacture an accurate
profi le consisting of two curves. The profi le of an involute tooth is made of a single
curve and only one cutter is necessary to manufacture one complete set of pinion and
gear. This results in reduction in manufacturing cost.

(ii) In case of an involute profi le, the common normal at the point of contact always
passes through the pitch point P and maintains a constant inclination a with the
common tangent to the two pitch circles. The angle a is called the pressure angle.
Therefore, the pressure angle remains constant in involute tooth. In case of cycloidal
tooth, the pressure angle varies. The pressure angle has maximum value at the
beginning of engagement and reduces to zero when the point of contact coincides
with the pitch point. It again increases to maximum value in the reverse direction.

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

 SPUR GEAR NOMENCLATURE AND GEAR-TOOTH FEATURES

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

Contact Ratio

When two gears mesh, it is essential for smooth operation that a second tooth
begin to make contact before a given tooth disengages. The term contact ratio
is used to indicate the average number of teeth in contact during the
transmission of power. A recommended minimum contact ratio is 1.2 and
typical spur gear combinations often have values of 1.5 or higher.

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

(i) Pinion A pinion is the smaller of the two mating gears.

(ii) Gear A gear is the larger of the two mating gears.

(iii) Velocity Ratio (i) Velocity ratio is the ratio of angular velocity of the driving gear
to the angular velocity of the driven gear. It is also called the speed ratio.

(iv) Transmission Ratio (i’) The transmission ratio (i’) is the ratio of the angular speed
of the fi rst driving gear to the angular speed of the last driven gear in a gear train.

(v) Pitch Surface The pitch surfaces of the gears are imaginary planes, cylinders or
cones that roll together without slipping.
(vi) Pitch Circle The pitch circle is the curve of intersection of the pitch surface of
revolution and the plane of rotation. It is an imaginary circle that rolls without
slipping with the pitch circle of a mating gear. The pitch circles of a pair of mating
gears are tangent to each other.

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

Circular Pitch The circular pitch (p) is the distance measured along the pitch circle
between two similar points on adjacent teeth. Therefore,

Diametral Pitch The diametral pitch (P) is the ratio of the number of teeth to the
pitch circle diameter. Therefore,

Module The module (m) is defined as the inverse of the diametral pitch.
Therefore,

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

 The centre to centre distance between two gears having zp and zg teeth is given
by

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

 STANDARD SYSTEMS OF GEAR TOOTH
All standard systems prescribe the involute profi le for gear tooth. The reasons are as
follows:
(i) The involute profile satisfies the fundamental law of gearing at any centre
distance.

(ii) All involute gears of a given module and pressure angle are completely
interchangeable.

(iii) All involute gears of a given module and pressure angle can be machined from
one single tool.

(iv) The basic rack of an involute profi le has straight sides. It is comparatively easy to
machine straight sides. Further, straight sides can be more accurately machined
compared with a curved surface.
(v) A slight change in the centre distance, which might be caused by incorrect
mounting, has no effect upon the shape of the involute. In addition, the pitch point is
still fi xed and the law of gearing is satisfi ed. Therefore, the velocity ratio remains
constant.

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

There are three standard systems for the shape of gear teeth. They are as follows:
(i) 14.5° full depth involute system
(ii) 20° full depth involute systems
(iii) 20° stub involute system

As the number of teeth on the gear is increased, the involute outline becomes
straighter and straighter. When the number of teeth is infi nity or when the pitch
circle radius approaches infinity, the gear becomes a rack with straight-sided teeth.
This rack is called the ‘basic’ rack, which is standardized in each system of gearing.

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

3/8/2023 S. Solomon Raj, PhD, MED, CBIT
The standard proportions of the gear tooth in
terms of module m, for 20° full depth system
are rewritten here. One of the methods of strengthening the gear tooth
is ‘crowning’. During operation, there is uneven
distribution of pressure along the face width of the
addendum (ha) = (m)
tooth due to the following reasons:
dedendum (hf ) = (1.25 m) (i) Inaccuracies of tooth profile caused by
machining errors and distortion during heat
clearance (c) = (0.25 m)
treatment
working depth (hk) = (2 m) (ii) (ii) Errors in assembly
(iii) Elastic deflection of shaft due to gear tooth
whole depth (h) = (2.25 m)
forces and bearing reactions
tooth thickness (s) = (1.5708 m)
tooth space = (1.5708 m)
fi llet radius = (0.4 m)
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 FORCE ANALYSIS

Pr is a separating force, which is always

directed towards the centre of the gear.

The tangential component Pt acts at the pitch

circle radius. Therefore,

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

 The above analysis of the gear tooth force is based on the following
assumptions:
(i) As the point of contact moves, the magnitude of the resultant
force PN changes. This effect is neglected in the above analysis.

(ii) It is assumed that only one pair of teeth takes the entire load. At
times there are two pairs, which are simultaneously in contact and
share the load. This aspect is neglected in the analysis.

(iii) The analysis is valid under static conditions, i.e., when the gears
are running at very low velocities. In practice, there is dynamic force
in addition to force due to power transmission. The effect of this
dynamic force is neglected in the analysis.
3/8/2023 S. Solomon Raj, PhD, MED, CBIT
 GEAR TOOTH FAILURES
There are two basic modes of gear tooth failure— breakage of the tooth due to static
and dynamic loads and the surface destruction. The complete breakage of the tooth
can be avoided by adjusting the parameters in the gear design, such as the module and
the face width, so that the beam strength of the gear tooth is more than the sum of
static and dynamic loads. The surface destruction or tooth wear is classified according
to the basis of their primary causes. The principal types of gear tooth wear are as
follows:
(i) Abrasive Wear Foreign particles in the lubricant, such as dirt, rust, weld spatter
or metallic debris can scratch or brinell the tooth surface. Remedies against this
type of wear are provision of oil filters, increasing surface hardness and use of high
viscosity oils. A thick lubricating film developed by these oils allows fi ne particles
to pass without scratching.
(ii) Corrosive Wear The corrosion of the tooth surface is caused by corrosive elements,
such as extreme pressure additives present in lubricating oils and foreign materials due
to external contamination. These elements attack the tooth surface, resulting in fi ne
wear uniformly distributed over the entire surface. Remedies against this type of wear
are, providing complete enclosure for the gears free from external contamination,
selecting proper additives and replacing the lubricating oil at regular intervals.

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

(iii) Initial Pitting The initial or corrective pitting is a localized phenomenon,
characterized by small pits at high spots. Such high spots are progressively worn out
and the load is redistributed. Initial pitting is caused by the errors in tooth profi le,
surface irregularities and misalignment. The remedies against initial pitting are
precise machining of gears, adjusting the correct alignment of gears so that the load
is uniformly distributed across the full face width, and reducing the dynamic loads.

Scoring Excessive surface pressure, high surface speed and inadequate supply of
lubricant result in the breakdown of the oil film. This results in excessive frictional heat
and overheating of the meshing teeth. Scoring is a stick-slip phenomenon, in which
alternate welding and shearing takes place rapidly at the high spots. Here, the rate of
wear is faster. Scoring can be avoided by selecting the parameters, such as surface
speed, surface pressure and the flow of lubricant in such a way that the resulting
temperature at the contacting surfaces is within permissible limits. The bulk
temperature of the lubricant can be reduced by providing fins on the outside surface
of the gear box and a fan for forced circu1ation of air over the fins.

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

3/8/2023 S. Solomon Raj, PhD, MED, CBIT
 NUMBER OF TEETH

There is a concept of ‘hunting’ tooth for uniform distribution of tooth wear. Suppose
(zp = 20) and (zg= 40), then after every two revolutions of the pinion, the same pair of
teeth will engage. If however, we take (zp = 20) and (zg = 41), the pinion will rotate 41
times before the same pair of teeth will engage again. This extra tooth is called the
hunting tooth. It results in more even distribution of wear. For the provision of hunting
tooth, it should be permissible to alter the velocity ratio slightly.

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

 BEAM STRENGTH OF GEAR TOOTH

The analysis of bending stresses in gear tooth was done by Wilfred Lewis in his paper,
‘The investigation of the strength of gear tooth’ submitted at the Engineer’s Club of
Philadelphia in 1892. Even today, the Lewis equation is considered as the basic equation
in the design of gears.

In the Lewis analysis, the gear tooth is treated as a cantilever beam as shown in . The
tangential component (Pt) causes the bending moment about the base of the
tooth. The Lewis equation is based on the following assumptions:
(i) The effect of the radial component (Pr), which induces compressive stresses, is
neglected.
(ii) It is assumed that the tangential component (Pt) is uniformly distributed over the
face width of the gear. This is possible when the gears are rigid and accurately machined.
(iii) The effect of stress concentration is neglected.
(iv) It is assumed that at any time, only one pair of teeth is in contact and takes the total
load.

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

3/8/2023 S. Solomon Raj, PhD, MED, CBIT
In the above equation, Y is called the Lewis form
factor. Equation gives the relationship between the
tangential force (Pt) and the corresponding stress
σ b.

When the stress reaches the permissible

magnitude of bending stresses, the
corresponding force (Pt) is called the beam
strength.
Therefore, the beam strength (Sb) is the
maximum value of the tangential force that
the tooth can transmit without bending failure.
Replacing (Pt) by (Sb), Eq. is modified in the
following way:

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

 Y=yπ

Small y values are given in Pg:: 204 & 232/Kmahadevan

Earle Buckingham has suggested that the

endurance limit stress of gear tooth is
approximately one-third of the ultimate tensile
strength of the material

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

 EFFECTIVE LOAD ON GEAR TOOTH

In gear design, the maximum force (due to maximum torque) is the criterion. This
is accounted by means of a service factor. The service factor Cs is defi ned as

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

When gears rotate at very low speed, almost at zero velocity, the transmitted load (Pt)
can be considered to be the actual force present between two meshing teeth. However,
in most of the cases, the gears rotate at an appreciable speed and it becomes necessary
to consider the dynamic force resulting from the impact between mating teeth. The
dynamic force is induced due to the following factors:
(i) inaccuracies of the tooth profile;
(ii) errors in tooth spacing;
(iii) misalignment between bearings;
(iv) elasticity of parts; and
(v) inertia of rotating disks.

There are two methods to account for the dynamic load—approximate estimation
by the velocity factor in the preliminary stages of gear design and precise
calculation by Buckingham’s equation in the final stages of gear design.

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

 Dynamic load

Barth- velocity factors-initial Buckingham-final stages

For precision gears with

For ordinary cut gears, with For accurately hobbed and
shaving, grinding
velocity less than 10M/sec generated gears
and lapping operations and
with v < 20 m/s,
with v > 20 m/s,
Cv=3/(3+v) Cv=6/(6+v)
Cv=5.6/(5.6+sqrt of v)

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

 The above equations are found in pg no: 207/KMD

ESTIMATION OF MODULE BASED ON BEAM STRENGTH

In order to avoid failure of gear tooth due to bending,

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 WEAR STRENGTH OF GEAR TOOTH

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 WEAR STRENGTH OF GEAR TOOTH

Therefore, the wear strength is the maximum value of the tangential force that the
tooth can transmit without pitting failure.

Internal gears
3/8/2023 S. Solomon Raj, PhD, MED, CBIT
 In order to avoid failure of gear tooth due to pitting,

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

 INTERFERENCE BETWEEN MATING SPUR GEAR TEETH
For certain combinations of numbers of teeth
in a gear pair, there is interference between 1. If a designer wants to be sure that
the tip of the teeth on the pinion and the fillet there will not be
or root of the teeth on the gear. interference between any two gears
when using the
Obviously this cannot be tolerated because the 14 12, full-depth, involute system, the
gears simply will not mesh. The probability pinion of the
that interference will occur is greatest when a gear pair must have no fewer than 32
small pinion drives a large gear, with the worst teeth.
case being a small pinion driving a rack. 2. For the 20°, full-depth, involute
system, using no
It is the designer’s responsibility to ensure that fewer than 18 teeth will ensure that no
interference does not occur in a given interference
application. The surest way to do this is to occurs.
control the minimum number of teeth in the 3. For the 25°, full-depth, involute
pinion to the limiting values system, using no
fewer than 12 teeth will ensure that no
interference
occurs.

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

 Overcoming Interference

If a proposed design encounters interference, there are ways to make it work. But
caution should be exercised because the tooth form or the alignment of the mating
gears is changed, causing the stress and wear analysis to be inaccurate. With this in
mind, the designer can provide for undercutting, modification of the addendum on
the pinion or the gear, or modification of the center distance:

Undercutting is the process of cutting away the material at the fillet or root of
the gear teeth, thus relieving the interference.

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

 ESTIMATION OF MODULE BASED ON BEAM STRENGTH
In order to avoid failure of gear tooth due to bending, Sb > Peff

Introducing a factor of safety, Sb = Peff (fs)

3/8/2023 S. Solomon Raj, PhD, MED, CBIT

 ESTIMATION OF MODULE BASED ON WEAR STRENGTH

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3/8/2023 S. Solomon Raj, PhD, MED, CBIT