2 Gear Basics

Gear Basics and geometry
Pitch Circle, Pitch Diameter and pitch point

Basically, gears are modifications of wheels, with gear teeth added to prevent slipping
and to assure that their relative motions are constant.
However, it should be noted that the relative surface velocities of the wheels are
determined by the diameters of the wheels.
It should also be noted that for a certain center distance and velocity ratio there is only
one pair of wheel that satisfy this conditions.
For Example for a center distance of 300 mm and velocity ratio of 2/3 only the two
tangent wheels of diameter 240 and 360 mm will satisfy the required center distance
and velocity ratio. These diameters are the diameters of the pitch circles which are
located at the same canter distance to produce a gearing ratio of 2/3.
The pitch circle, (Figure 1) corresponds to the diameter of the wheel. Thus the angular
velocity of a gear is determined by the diameter of this imaginary pitch circle. All
design calculations relating to gear performance are based on the pitch-circle diameter
or, more simply, the pitch diameter (PD).
For two gears to operate properly, their pitch circles must be tangential to each other.
The point at which the two pitch circles are tangent, at which they intersect the
centerline connecting their centers of rotation, is called the pitch point.
Figure 1 Tangent pitch circles between two gears produce a pitch point Gear t
Gearing Law
For correct gearing, the velocity ratio between any pairs of gears must be constant at
every instant. Also, to provide uniform pressure and motion and to minimize friction
and wear, gears are designed to have rolling motion between mating teeth rather than
sliding motion To achieve these conditions, the common normal at the point of
contact of mating teeth must always pass through the pitch point at any instant. This
condition is known as "Gearing Law " and is illustrated in Figure 1 and 2.
Figure 2 Line of action and pressure angle
Gear Pressure Angle

It is the acute angle at the pitch point between the line of action and the common
tangent to the pitch circles, figure 2. This angle determines the direction of the
interaction force between the mating teeth. Consequently the pressure angle
determines the values of the thrust and tangential components of the transmitting
forces as well as the transmitting torque between the mating gears. Common values of
pressure angles are 14½ ° and 20° .
Gear Profiles
There are a few standard profiles of teeth which satisfy the gearing law. Some of these
profiles are
a) Involute profile.
b) Cycloidal profile.
c) Novikov profile.
Practically most of industrial gears are made such that its tooth profile is involute.
This is because the involute system has the following advantages:-
a) The pressure angle is constant throughout from the commencement of the

engagement to the end of the engagement,
b) Variation of center distance within limits does not affect the velocity ratio of
the mating gears.
c) The profile of tooth flank and tooth face is single curve.
d) The profile of the basic rack in involute system is a straight line which
simplified the manufacturing process.
Involute Curve
This is the curve that is generated by a point on a straight line when the line rolls
around a base circle. A somewhat simpler method
of developing an involute curve is that shown in
Figure 2, by unwinding a tautly held string from a
base circle; point A generates involute curve.
The involute shape depends on the diameter of its

base circle.
The diameter of the base circle = DP cos ψ
= M×N× cos ψ
Where M is the gear module, N is the no. of gear

teeth and ψ is the pressure angle
Accordingly the shape of the involute curve is

affected by the the gear module, the No. of
gear teeth and the pressure angle
Gear Module
The gear module defined as the pitch diameter divided by the number of teeth, or
M = PD/N.
Accordingly PD= M×N
Any two gears having the same module will mesh properly if they are mount so as
have the correct distances and relationship.
+Preferred value Values of

Module M
0.6 1.5 4 10
0.8 2 5 12
1 2.5 6 16
1.25 3 8 20
Gear Tooth Dimensions
The important tooth elements can be specified in terms of the module and are as
follows:
Figure 2 Gear tooth nomenclature

1. Addendum: the radial distance from the pitch circle to the outside diameter.
Addendum = M mm
2. Dedendum: the radial distance from the pitch circle to the root circle. It is
equal to the addendum plus the clearance, which is provided to prevent the
outer corner of a tooth from touching against the bottom of the tooth space.
i. Dedendum = 1,25 M mm
3. Tooth depth = 2.25 M mm

4. Tooth clearance = 0.25 M mm
5. Pitch diameter DP= M×N ,,
6. Addendum diameter ( Blank diameter) = M×(N+2) mm
7. Dedendum diameter (root diameter) = M×(N-2.5) mm
8. Circular pitch: the distance between corresponding points of adjacent teeth,
measured along the pitch circle. It is numerically equal to πM.
9. 4. Tooth thickness: the thickness of a tooth, measured along the pitch circle.
When tooth thickness and the corresponding tooth space are equal, no
backlash exists in a pair of mating gears.
10. 5. Face width: the length of the gear teeth in an axial plane.
11. 6. Tooth face: the mating surface between the pitch circle and the addendum
circle.
12. 7. Tooth flank: the mating surface between the pitch circle and the root circle.
Physical Requirements of Gears
A consideration of gear theory leads to five requirements that must be met in order for
gears to operate satisfactorily:
1. The actual tooth profile must be the same as the theoretical profile.
2. Tooth spacing must be uniform and correct.
3. The actual and theoretical pitch circles must be coincident and be concentric with
the axis of rotation of the gear.
4. The face and flank surfaces must be smooth and sufficiently hard to resist wear and
prevent noisy operation.
5. Adequate shafts and bearings must be provided so that desired center-to-center
distances are retained under operational loads.
The first four of these requirements are determined by the material selection and
manufacturing process. The various methods of manufacture that are used represent
attempts to meet these requirements to varying degrees with minimum cost, and their
effectiveness must be measured in terms of the extent to which the resulting gears
embody these requirements.
Gear Manufacturing
Whether produced in large or small quantities, in cells, or job shop batches, the
sequence of processes for gear manufacturing requires four sets of operations:
1. Blanking
2. Gear cutting
3. Heat treatment
4. Grinding
Blanking refers to the initial forming or machining operations that produce a semi
finished part ready for gear cutting, starting from a piece of raw material. Good-
quality blanks are essential in precision gear manufacturing.
Gear Cutting Methods

The most common and accurate method of production of gears is by machining. The
different methods of production of gears by machining operations are:-
1. Formed cutter method:

a. By a formed disc cutter in a milling machine.
b. By a formed end mill in a milling machine.
c. By a formed single point tool in a shaping or planing machine.
e. By a formed cutter in a broaching machine.
3. Generating method:
a. By a rack tooth cutter in a gear cutting machine.
b. By a pinion cuter in a gear cutting machine.
c. By a hob cutter in a gear cutting machine.
d. By a bevel gear generator.
Solved Example
A spur gear of 40 teeth, 4 mm module and 20 degree pressure angle. It required to
determine the followings:-
a) Tooth addendum
b) Tooth dedendum
c) Tooth depth
d) Diameter of pitch circle
e) Diameter of addendum circle ( blank diameter)
f) Diameter of the root circle
g) Diameter of base circle
Answers
a) Tooth addendum = 4 mm
b) Tooth dedendum = 1.25× 4 =5 mm
c) Tooth depth =2.25 × 4 = 9 mm
d) Diameter of pitch circle = 4 × 40 = 160 mm
e) Diameter of addendum circle ( blank diameter) = 4 (40+2) = 168 mm
f) Diameter of the root circle = 4 × (40-2.5) = 150 mm
h) Diameter of base circle = 160 × cos20= 150.34
Important Note
The addendum circle (external circle of the gear blank) and the root circle could be
defined physically. However the pitch circle and the base circle are imaginary circles.

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Gear Basics and geometry

Pitch Circle, Pitch Diameter and pitch point

Figure 2 Line of action and pressure angle

Gear Pressure Angle

a) The pressure angle is constant throughout from the commencement of the

The involute shape depends on the diameter of its

Where M is the gear module, N is the no. of gear

Accordingly the shape of the involute curve is

+Preferred value Values of

Figure 2 Gear tooth nomenclature

3. Tooth depth = 2.25 M mm

Gear Cutting Methods

1. Formed cutter method:

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