You are on page 1of 40

Chapter 4: Spur Gears

machine elements that
transmit motion by means
of successively engaging
teeth. The gear teeth act
like small levers.

Belt/Rope Drives -

Large center distance of the shafts

Chain Drives

Medium center distance of the shafts

Gear Drives

Small center distance of the shafts


Watch gear wheels

Gear train

Gear Pump

Printing machinery parts

Diesel engine builders

Rotary die cutting machines

Hoists and Cranes

Blow molding machinery

Boat out drives

Agricultural equipment
Automotive prototype and reproduction

Newspaper Industry

Plastics machinery

Motorcycle Transmissions

Polymer pumps

Automotive applications
Commercial and Military operations
Special gear box builders

Heavy earth moving vehicles

Canning and bottling machinery builders
Special machine tool builders
Book binding machines

Marine applications

Injection molding machinery

Military off-road vehicles

Stamping presses

Gears may be classified according to the relative

position of the axes of revolution. The axes may be
parallel, intersecting and neither parallel nor intersecting.
1. Gears for
parallel shafts

Spur Gears:

External contact

Internal contact

Parallel Helical gears

Heringbone gears


2. Gears for connecting

intersecting shafts Bevel Gears


Spiral bevel gears

Straight bevel gears


Crossed-helical gears

Worm & Worm Wheel









Addendum: The radial distance between the Pitch

Circle and the top of the teeth.
Arc of Action: Is the arc of the Pitch Circle between
the beginning and the end of the engagement of a
given pair of teeth.
Arc of Approach: Is the arc of the Pitch Circle
between the first point of contact of the gear teeth
and the Pitch Point.
Arc of Recession: That arc of the Pitch Circle
between the Pitch Point and the last point of contact
of the gear teeth.
Backlash: Play between mating teeth.

Base Circle: The circle from which is generated the

involute curve upon which the tooth profile is based.
Center Distance: The distance between centers of two
Chordal Addendum: The distance between a chord,
passing through the points where the Pitch Circle
crosses the tooth profile, and the tooth top.
Chordal Thickness: The thickness of the tooth
measured along a chord passing through the points
where the Pitch Circle crosses the tooth profile.
Circular Pitch: Millimeter of Pitch Circle circumference
per tooth.

Circular Thickness: The thickness of the tooth

measured along an arc following the Pitch Circle
Clearance: The distance between the top of a tooth and
the bottom of the space into which it fits on the meshing
Contact Ratio: The ratio of the length of the Arc of
Action to the Circular Pitch.
Dedendum: The radial distance between the bottom of
the tooth to pitch circle.
Diametral Pitch: Teeth per mm of diameter.

Face: The working surface of a gear tooth, located

between the pitch diameter and the top of the tooth.
Face Width: The width of the tooth measured parallel
to the gear axis.
Flank: The working surface of a gear tooth, located
between the pitch diameter and the bottom of the teeth
Gear: The larger of two meshed gears. If both gears
are the same size, they are both called "gears".
Land: The top surface of the tooth.


Line of Action: That line along which the point of contact

between gear teeth travels, between the first point of
contact and the last.
Module: Millimeter of Pitch Diameter to Teeth.
Pinion: The smaller of two meshed gears.
Pitch Circle: The circle, the radius of which is equal to the
distance from the center of the gear to the pitch point.
Diametral pitch: Teeth per millimeter of pitch diameter.
Pitch Point: The point of tangency of the pitch circles of
two meshing gears, where the Line of Centers crosses the
pitch circles.

Pressure Angle: Angle between the Line of Action and

a line perpendicular to the Line of Centers.
Profile Shift: An increase in the Outer Diameter and
Root Diameter of a gear, introduced to lower the
practical tooth number or acheive a non-standard
Center Distance.
Ratio: Ratio of the numbers of teeth on mating gears.
Root Circle: The circle that passes through the bottom
of the tooth spaces.
Root Diameter: The diameter of the Root Circle.
Working Depth: The depth to which a tooth extends
into the space between teeth on the mating gear.

Circular pitch p c

Diametral pitch T

Diametral pitch p d

Circular pitch D p c
Pitch diameter
Diametral pitch
Teeth X Circular pitch


Teeth on pinion

Circular pitch
Center dis tan ce

Teeth on Gear


on pinion Teeth on Gear

2 Diametral pitch

Base Circle Diameter Pitch Diameter Cos



= 1 Diametral Pitch
= 0.3183 Circular Pitch


= 1.157 Diametral Pitch

= 0.3683 Circular Pitch

Working Depth

= 2 Diametral Pitch
= 0.6366 Circular Pitch

Whole Depth

= 2.157 Diametral Pitch

= 0.6866 Circular Pitch


= 0.157 Diametral Pitch

= 0.05 Circular Pitch

Outside Diameter = (Teeth + 2) Diametral Pitch

= (Teeth + 2) Circular Pitch
Diametral Pitch

= (Teeth + 2) Outside Diameter


Tooth profile 1 drives tooth profile 2

by acting at the instantaneous
contact point K.
N1 N2 is the common normal of
the two profiles.

N1 is the foot of the perpendicular

from O1 to N1N2
N2 is the foot of the perpendicular
from O2 to N1N2.

Although the two profiles have

different velocities V1 and V2 at
point K, their velocities along
N1N2 are equal in both
Otherwise the two tooth
profiles would separate from
each other. Therefore, we have

O1 N1 1 O2 N 2 2


1 O2 N 2

2 O1 N1


We notice that the intersection

of the tangency N1N2 and the
line of center O1O2 is point P,
and from the similar triangles

O1 N1 P O2 N 2 P



Therefore, velocity ratio

1 O2 P

2 O1 P


Point P is very important to the

velocity ratio, and it is called the
pitch point. Pitch point divides
the line between the line of
centers and its position decides
the velocity ratio of the two teeth.
The above expression is the

fundamental law of geartooth action


From the equations 4.2 and 4.4,

we can write,

1 O2 P

2 2
O1 P
O1 N1


-ratio of the radii of the two base

circles and also given by;

O1 N1 O1 P cos
O2 N 2 O2 P cos



-centre distance between the base

O1O2 O1 P O2 P

O1 N1 O2 N 2

O1 N1 O2 N 2


4 .7

= pressure angle or the angle of

obliquity. It is angle between the common

normal to the base circles and the
common tangent to the pitch circles.

A common normal (the line of action)

to the tooth profiles at their point of
contact must, in all positions of the
contacting teeth, pass through a
fixed point on the line-of-centers
called the pitch point
Any two curves or profiles engaging
each other and satisfying the law of
gearing are conjugate curves, and
the relative rotation speed of the
gears will be constant (constant
velocity ratio).


To obtain the expected velocity ratio

of two tooth profiles, the normal line
of their profiles must pass through
the corresponding pitch point, which
is decided by the velocity ratio. The
two profiles which satisfy this
requirement are called conjugate


It is essential for correctly

meshing gears, the size of the
teeth ( the module ) must be
the same for both the gears.
Another requirement the
shape of teeth necessary for
the speed ratio to remain
constant during an increment
of rotation; this behaviour of
the contacting surfaces (ie.
the teeth flanks) is known as
conjugate action.