Gears
Gears are among the most important power
transmission elements. A gear is a rotating
machine element having cut teeth that
mesh with another toothed part (having
teeth of similar size and shape) in order to
transmit power. Two or more gears working
together are called a “transmission” (or
gear set) and can produce “Mechanical
Advantage”, and thus may be considered a
simple machine. The mechanical advantage
is a measure of the force or torque
amplification that is obtained using
mechanical devices.
When a gear meshes with a bigger gear (the size of the teeth must match, thus the
bigger gear has more teeth), a mechanical advantage is obtained, and the rotational
speeds and the torques of the two gears will be different. Since the input and output
power must be equal (ignoring friction losses), there is an inverse relationship
between the speed and torque ratios (the small gear will have higher speed and lower
torque and the larger gear will have lower speed and higher torque).
A transmission (or gear set) can be used to change the speed, torque, direction of
rotation, direction of a power source, or the type of motion.
The most common configuration for a gear is to
mesh with another gear; however, a gear can
also mesh with a non-rotating toothed part,
called a “rack”, thereby producing translation
instead of rotation, as shown in the figure. Such
arrangement is refered to as “rack and pinion”,
and it is commonly used in the steering systems
of automobiles.
Types of Gears
There are four principal types of gears:
 Spur gears: The simplest type of gears. The teeth are parallel to
the axis of rotation, as seen in the figure. It transmits rotation
between parallel shafts.
Fundamentals of Mechanical Design
Gears
Lecture Notes by: Dr. Ala Hijazi
Page 1 of 13
 Helical gears: The teeth are inclined with respect to the axis of
rotation, as seen in the figure. Same as spur gears, it transmits
rotation between parallel shafts, but it is less noisy than spur
gears, because of the gradual engagement of the teeth during
meshing, and thus it is more suitable for transmitting motion at
higher speeds.
In some cases, helical gears can also be used to transmit
rotation between perpendicular shafts, as seen in the figure.
 Bevel gears: The teeth are somehow similar to those of a spur
gear, but they are formed on conical surfaces instead of
cylinders. Bevel gears transmit rotation between intersecting
shafts. The gear shown in the figure has straight teeth where
this is the simplest type.
There are other type of bevel gears were the teeth form
circular arcs, and it is called spiral bevel gears, as shown in the
figure. With spiral bevel gears, the teeth engagement will be
more gradual (similar to helical gears), and thus it is less noisy
and it is suitable for higher speeds.
 Worms and worm gears: Transmit rotation between perpendicular
shafts (non-intersecting - there is an offset between them). The
worm resembles a screw which can be right handed or left handed.
Worm gear sets are usually used when high reduction in speed is
desired (speed ratios of 3 or higher). The unique thing about worms
and worm gears is that the rotation can be transmitted from the
worm to the worm gear, but not the opposite.
Nomenclature
Since spur gears are the
simplest type, it will be used for
illustration and to define the
primary parameters of gears
and their relations. The figure
illustrates the terminology of
spur gears.
 Pitch circle: the theoretical
circle upon which all gear
calculations are based, and
its diameter is called the
“pitch diameter”.
Fundamentals of Mechanical Design
Gears
Lecture Notes by: Dr. Ala Hijazi
Page 2 of 13







o Pitch circles of mating gears are "tangent"
to each other.
o The “centers distance” between two
mating gears is the sum of the pitch
radiuses of the two gears.
o The smaller of two mating gears is called
the “pinion” and the larger is called the
gear.
Addendum and Dedendum circles: the
circles defining the top and bottom faces of
the teeth.
Addendum “𝑎”: the radial distance from the pitch circle to the top surface of the
teeth.
Dedendum “𝑏”: the radial distance from the pitch
circle to the bottom surface of the teeth.
Clearance circle (or working depth circle): the
circle tangent to the addendum circle of the
mating gear. The radial distance between the
addendum circle and the clearance circle is called
the “working depth”.
Clearance “𝑐”: the distance between the tooth top
surface and the bottom surface of a mating gear.
Circular pitch “𝑝”: the distance measured on the "pitch circle" from a point on
one tooth to the corresponding point on an adjacent tooth. The circular pitch is
equal to the sum of "tooth thickness" and "width of space". The width of space
is slightly larger than the tooth thickness such that mating teeth can engage
easily without obstruction.
Module “𝑚”: is the ratio of pitch diameter to the number of teeth of a gear.
Module
(mm per tooth)
Circular pitch
𝑚=
𝑝=
𝜋𝑑
𝑁
𝑑
Pitch diameter
𝑁
Number of teeth
=𝜋𝑚
 Diametral pitch “𝑃”: the ratio of the number of teeth of a gear to the pitch
diameter (it is the inverse of the module, and it is used with gears sized in inches;
its unit is “teeth per inch”).
𝑃=
Fundamentals of Mechanical Design
Gears
𝑁
1
=
𝑑
𝑚
→
𝑝=
𝜋
𝑃
Lecture Notes by: Dr. Ala Hijazi
Page 3 of 13
Important Notes:
 The module (or the diametral pitch) determines the size of gear teeth.
 In order for gears to be able to mesh (work together), they must have the
same module (or diametral pitch).
 The pitch diameter of a gear (i.e., its size) is determined based on its module
(or diametral pitch) and the number of teeth:
𝑑=𝑚𝑁
Spur Gears Teeth Profile
Gears need to have constant angular velocity ratio during meshing. This property is
important for smooth transmission of power with minimal speed or torque variations
as pairs of teeth go into or come out of contact. The smooth transmission is important
for high-speed gearing (it is not very critical for low-speed gearing).
In order to obtain a constant angular velocity ratio, gear teeth must have a special
type of curved profile (not circular) that is called an "Involute" curve. Also, when the
profile of gear teeth are made as involute curves, the mating teeth surfaces will
simply roll on each other and there will be no sliding contact.
 During the meshing of gears, the
contact between a pair of gear
teeth occurs at a single
instantaneous point. This point of
contact (point c) occurs where the
two surfaces are tangent to each
other, and the forces will be
directed along the common
normal (line ab) which is also
called the “Line of Action” or the
“Pressure Line”.
 Rotation of the gears causes the location
of the contact point (point c) to move
across the respective tooth surfaces. For
gear teeth having involute profiles, as
the point of contact moves, it will move
along the pressure line which remains at
a constant angle.
 The angle between the pressure line and
the pitch line is called the “Pressure
Angle (∅)”.
Fundamentals of Mechanical Design
Gears
Lecture Notes by: Dr. Ala Hijazi
Page 4 of 13
Drawing Gear Teeth Involute Profile
An involute profile can be simply generated using a
cord wrapped around a circle, as seen in the figure.
 One end of the cord is fixed at a point on the
circumference of the circle (point c).
 A pin is attached at the other end of the cord
(point d).
 By moving the pin while the cord is tensioned, an
involute profile is generated (profile ab).
The same idea of generating an involute profile can be applied graphically (without
using a cord) as illustrated by the figure:
 The circle from which the involute profile starts is called the Base circle.
 A series of radial lines with constant angular
increments between them are drawn to
points A0, A1, .. , A4 on the circumference of
the circle.
 Tangent lines are drawn at points A1, .. , A4.
 Points B1, .. , B4 are defined along the tangent
lines knowing that the length of the line
segments A1 - B1, .. , A4 - B4 are as follows:
 Length of line A1 - B1 = Length of arc
segment between A1 - A0
 Length of line A2 - B2 = Length of arc segment between A2 - A0
 Length of line A3 - B3 = Length of arc segment between A3 - A0
 Length of line A4 - B4 = Length of arc segment between A4 - A0
 The involute curve is drawn starting from point A0 and passing through points B1,
.. , B4.
The radius of a base circle “𝑟𝑏 ”, which
is used to generate teeth involute
profile, depends on the radius of the
pitch circle “𝑟” and the pressure
angle “∅” (the angle between the
pressure line and the common
tangent of the pitch circles of mating
gears), as illustrated in the figure.
𝑟𝑏 = 𝑟 cos ∅
Fundamentals of Mechanical Design
Gears
Lecture Notes by: Dr. Ala Hijazi
Page 5 of 13
 Standard values of ∅ are 14.5°, 20° and 25° (20° is the most commonly used)
Important Note: mating gears must have the same pressure angle to be able to
mesh correctly.
A tooth profile starts from the dedendum circle and ends at the addendum circle
where the standard values of dedendum and addendum are defined as a function of
the module “𝑚” (or the diametral pitch “𝑃”):
o Dedendum:
𝑏 = 1.25 𝑚 =
o Addendum:
𝑎=𝑚=
1.25
𝑃
1
𝑃
 If the dedendum circle is smaller than the base circle (this happens when a small
pressure angle is used), the remaining portion of a tooth profile, which is not part
of the involute profile, is drawn as a tangent arc.
Contact Ratio
The contact ratio “ 𝑚𝑐 ” defines the average number of teeth pairs in contact during
meshing.
 If 𝑚𝑐 = 1 it means that only one pair of teeth is in contact at a time.
 To reduce noise and possibility of impact, it is recommended that 𝑚𝑐 ≥ 1.2
 The contact ratio increases with the number of teeth of a gear, and for this
reason, it is NOT recommended to use gears having less than ten teeth.
Interference
The contact of portions of teeth profiles that are not involute is called interference.
 Interference happens in gear pairs because the dedendum circle is smaller than
the base circle, and thus the involute portion of teeth profile is small.
 To reduce interference, larger pressure angles 20° or 25° are commonly used
(i.e., making the base circle smaller).
Fundamentals of Mechanical Design
Gears
Lecture Notes by: Dr. Ala Hijazi
Page 6 of 13
Gears Standards
Gears are usually made based on standards established by ISO (for metric sizes) or
ANSI (for inch sizes).
 The standards specify the different parameters used to define the geometry of
gears (such as, module, addendum, dedendum, tooth thickness, pressure angle,
etc.) and the relationships between these parameters.
 The standards were developed to attain interchangeability of gears having the
same pressure angle and module (or diametral pitch) regardless of the number of
teeth.
 The Table gives the preferred values of Module and Diametral Pitch according to
the ISO and ANSI standards.
 Gear manufacturers make a large variety of standard sized gears, and such gears
can usually be available as off-the-shelf components.
 When a standard gear is being used in a
machine, the gear is usually not shown in full
detail in the technical drawings, as shown in the
figure. Instead, a schematic representation is
used, and the gear is defined using its
geometric parameters. Detailed drawings of a
gear are needed when a special gear is designed
or when gears in assembly must be shown.
 The geometry of a standard spur gear is defined by "four parameters"; module (or
diametral pitch), pressure angle, number of teeth, and face width.
 A standard ISO spur gear having a module of 4 mm, 25 teeth, 20° pressure
angle, and 30 mm face width is designated as:
ISO- Spur gear 4M 25T 20PA 30FW.
Fundamentals of Mechanical Design
Gears
Lecture Notes by: Dr. Ala Hijazi
Page 7 of 13
 When the diameter of a gear is large, usually it is not made as
a solid gear, instead it is made in the shape of a rim, as seen in
the figure, in order to save material and reduce its weight.
Also, when the face width of a gear is small compared to its
diameter, usually the gear is made with a hub in order to
improve its stability (as seen in the figure).
Gear Ratio and Gear Trains
The gear ratio “𝑅” (also known as speed ratio) of a gear set is defined as the ratio of
the angular velocity of the input gear (driver gear) relative to that of the output gear
(driven gear):
𝑅 = 𝑛𝑑𝑟𝑖𝑣𝑒𝑟 ⁄𝑛𝑑𝑟𝑖𝑣𝑒𝑛
 The gear ratio is simply equal to the ratio of the number of teeth (or pitch
diameters) of the driven gear relative to driver gear:
𝑅=
𝑛𝑑𝑟𝑖𝑣𝑒𝑟 𝑁𝑑𝑟𝑖𝑣𝑒𝑛 𝑑𝑑𝑟𝑖𝑣𝑒𝑛
=
=
𝑛𝑑𝑟𝑖𝑣𝑒𝑛 𝑁𝑑𝑟𝑖𝑣𝑒𝑟 𝑑𝑑𝑟𝑖𝑣𝑒𝑟
Consider a pinion “2” driving a gear “3”, the speed
of the gear is:
𝑛
𝑁2
𝑅
𝑁3
𝑛3 = | 2| = |
𝑛2 |
The absolute-value sign is to give freedom in
choosing positive or negative directions.
A Gear Train is a gear set that consists of more than two
gears. In general, there can be several driving gears (input)
and several driven gears (output).
 Any gear that is not giving any input torque nor
taking any output torque is called an “idler” gear.
 For a gear train, the gear ratio is found as:
𝑅=
𝑃𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑑𝑟𝑖𝑣𝑒𝑛 𝑔𝑒𝑎𝑟𝑠 𝑡𝑜𝑜𝑡ℎ 𝑛𝑢𝑚𝑏𝑒𝑟𝑠
𝑃𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑑𝑟𝑖𝑣𝑒𝑟 𝑔𝑒𝑎𝑟𝑠 𝑡𝑜𝑜𝑡ℎ 𝑛𝑢𝑚𝑏𝑒𝑟𝑠
Fundamentals of Mechanical Design
Gears
Lecture Notes by: Dr. Ala Hijazi
Page 8 of 13
For the gear train shown in the figure,
pinion “2” is the input gear and gear
“6” is the output gear (other gears are
idlers), the gear ratio is found to be:
𝑅=
𝑁3 𝑁4 𝑁6
𝑁2 𝑁3 𝑁5
 Note that gears 2, 3 & 5 are drivers while gears 3, 4 & 6 are driven
And the speed of gear “6” is:
The sign is decided manually.
𝑛6 = −
Planetary gear trains
𝑛2
𝑅
In this type of trains, the axes of some of the
gears rotate about other gears, as seen in the
figure.
 Planetary gear trains include three main
components: a sun gear, a planet carrier with
one or more planet gears on it, and a ring gear.
 A planetary train have two degrees of
freedom, and thus needs to have two inputs.
 Planetary gear trains are commonly used for obtaining different gear ratios by
fixing one of the elements (the sun, the ring, or the planet carrier) and using one
of the remaining two elements as an input and the other as an output.
Parallel Helical Gears
Helical gears are mostly used to transmit motion between parallel shafts.
 The teeth are inclined with respect to the axis of rotation and the angle of
inclination of teeth is called the “Helix Angle”. The most commonly used value of
the helix angle is 30°.
 The shape of the teeth is an involute helicoid, and it can be produced by wrapping
a piece of paper shaped as parallelogram on the base cylinder.
 Contact of the teeth starts as a point then extends into a diagonal line across the
face of the tooth as teeth come into more engagement.
 Because of the gradual engagement of the teeth, helical gears can transmit heavy
loads at high speeds.
Fundamentals of Mechanical Design
Gears
Lecture Notes by: Dr. Ala Hijazi
Page 9 of 13
 A helical gear can have a right-hand or a lefthanded helix, as shown in the figure.
 In order for two helical gears to mesh, both
gears must have the same helix angle, but
one gear must have a
right-hand helix and the
other a left-hand helix, as
seen in the figure.
 Unlike spur gears which produce only radial load, a helical gear produces both
radial and thrust loads on the shaft supporting it.
 It is possible to cancel the resulting thrust loading by using two opposite-hand
helical gears mounted side by side on each shaft.

Another way to eliminate thrust load is to use
“Herringbone gears”. A herringbone gear is a special
type of double helical gear that combines two helical
gears of opposite hands, as seen in the figure.
 Because of the inclination of the teeth, the module, the pressure angle, and the
circular pitch, can be defined in two different directions; the normal direction
(normal to the teeth direction) and the transverse direction (parallel to the gear
face), as seen in the figure.

The figure shows the nomenclature of helical gears:
 Helix angle: 𝜓
 Transverse circular pitch: 𝑝𝑡
 Normal circular pitch: 𝑝𝑛
𝑝𝑛 = 𝑝𝑡 cos 𝜓
o The same relation applies to the module in the
normal and transverse directions:
𝑚𝑛 = 𝑚𝑡 cos 𝜓
 Normal and transverse pressure angles
𝜙𝑛 & 𝜙𝑡 are related to the helix angle as:
cos 𝜓 =
tan 𝜙𝑛
tan 𝜙𝑡
 According to standards, the values of the module and pressure angle are specified
in the normal direction, while, of course, the module in the transverse direction is
used for determining the "diameter of the pitch circle": 𝑑 = 𝑚𝑡 𝑁
Fundamentals of Mechanical Design
Gears
Lecture Notes by: Dr. Ala Hijazi
Page 10 of 13
 A helical gear is designated by its hand, module (normal), number of teeth, helix
angle, pressure angle (normal), and face width.
Example: RH Helical gear 4M 25T 30HA 20PA 30FW
o If we have a spur gear and a helical gear with the same module and number of
teeth, would the diameters of the two gears be equal?
Straight Bevel Gears
Bevel gears are used to transmit motion
between intersecting shafts. The shafts
usually make 90° angle with each other
but also other angles are possible.
 Similar to helical gears, bevel gears
produce both radial and thrust loads
on the shaft.
 Because bevel gears are cut on
conical surfaces (not cylindrical
surfaces as spur and helical gears),
they have a large end and a small end.
 The figure illustrates the terminology
of bevel gears:
 The pitch diameter is measured at
the large end of the tooth.
 The module and circular pitch are
calculated the same as in spur gears.
 The face width is measured along the conical surface. For considerations
related to teeth deflection, the face width of a bevel gear is usually kept small
relative to its diameter.
 The pitch angles are defined as shown in the figure and they are related to the
number of teeth as follows:
Pinion:
tan 𝛾 =
𝑁𝑝
𝑁𝐺
Gear:
tan Γ =
𝑁𝐺
𝑁𝑃
Important Note: The pitch angle for the pinion or the gear depend on the number
of teeth of both the pinion and the gear. Thus, unlike spur and helical gears, in
order to determine the geometry for any bevel gear, the number of teeth of the
mating gear needs to be known.
Fundamentals of Mechanical Design
Gears
Lecture Notes by: Dr. Ala Hijazi
Page 11 of 13
 Since the face width of bevel gears is kept relatively small, bevel gears usually
come with a hub to give better stability. Also, the hub helps in controlling the
mounting distance between the gear (or pinion) and the mating element.
 The Mounting Distance is the distance from the back surface of the gear (or pinion)
to the centerline of the mating element, as seen in the figure.
 A straight bevel gear is designated by its, module, number of teeth, number of
teeth of the pinion (or mating gear), pressure angle, and face width.
Example: Straight Bevel gear 4M 25GT 15PT 20PA 18FW
 Because bevel gears are used on intersecting shafts, it is
extremely difficult to have both gears mounted between
two bearings (straddle mounted). In most cases, one of the
two gears have to be mounted outboard of the bearings
(outboard mounted), as shown in the figure.
Worm Gears
The work principle of a worm and a worm gear is somehow similar to that of a power
screw moving a nut. The worm is similar to a screw, and as the worm rotates it bushes
the teeth of the mating worm gear causing it to rotate.
 If the worm has a single tooth (similar to a single thread screw), the worm will
need to rotate a number of revolutions equal to the number of teeth of the worm
gear for the worm gear to rotate once.
 Thus, the gear ratio is usually large, and it is found according to the number
of teeth as:
𝑅=
𝑁𝑤𝑜𝑟𝑚 𝑔𝑒𝑎𝑟
𝑁𝑤𝑜𝑟𝑚
 Where the number of teeth of the worm is usually from 1 to 3.
 Unlike other types of gears, in a worm gear set, the input shaft must be connected
to the worm (not the worm gear) since the worm gears are self-locking (the worm
gear cannot rotate the worm).
 The friction losses in worm gears are much higher than other types of gears
(because the sliding motion between the mating teeth surfaces).
 The figure illustrates the terminology of worm gears:
 The worm and worm gear have the same hand of helix, but the helix angles are
different.
Fundamentals of Mechanical Design
Gears
Lecture Notes by: Dr. Ala Hijazi
Page 12 of 13
 The lead angle for the worm, 𝜆, is
equal to the gear helix angle, 𝜓𝐺 .
 The lead “ 𝑙 ”and the axial pitch “𝑃𝑥 ”
of the worm are related as;
𝑙 = 𝑃𝑥 𝑁𝑤𝑜𝑟𝑚
 The lead “ 𝑙 ” and the lead angle “𝜆”
of the worm are related as;
tan 𝜆 =
𝑙
𝜋𝑑𝑤
Fundamentals of Mechanical Design
Gears
Lecture Notes by: Dr. Ala Hijazi
Page 13 of 13