Gear drives

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Spur Gears

Spur Gears
o are toothed wheels whose tooth elements are straight and parallel to the shaft axis.
(Faires, 1969)

o gears which have cylindrical pitch surfaces and operate on parallel axes, and the teeth
are straight and parallel to the axis. (Black & Adams, 1968)

o have teeth parallel to the axis of rotation and are used to transmit motion from one
shaft to another parallel shaft.

Of all types, the spur gear is the simplest and, for this reason, will be used to develop the
primary kinematic relationships of the tooth form. They are used to transmit motion and power
between parallel shafts.

FIGURE 4 -1: Spur Gear

schematic drawing

Terms used in Spur Gears1:

1. Pitch circle - it is an imaginary circle which by pure rolling action, would give the same
motion as the actual gear.
2. Pitch circle diameter - it is the diameter of the pitch circle. The size of the gear is
usually specified by the pitch circle diameter. It is also called as pitch diameter.

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3. Pitch point - it is a common point of contact between two pitch circles.

4. Pitch surface - it is the surface of the rolling discs which the meshing gears have
replaced at the pitch circle.
5. Pressure angle or angle of obliquity - it is the angle between the common normal to two
gear teeth at the point of contact and the common tangent at the pitch point. It is usually
denoted by φ. The standard pressure angles are 14.50 and 200.
6. Addendum - it is the radial distance of a tooth from the pitch circle to the top of the
tooth.
7. Dedendum - it is the radial distance of a tooth from the pitch circle to the bottom of the
tooth.
8. Addendum circle - it is the circle drawn through the top of the teeth and is concentric
with the pitch circle.
9. Dedendum circle - it is the circle drawn through the bottom of the teeth. It is also called
root circle.
10. Circular pitch - it is the distance measured on the circumference of the pitch circle from
a point of one tooth to the corresponding point on the next tooth.

FIGURE 4 – 2: Dimensions
and Nomenclature, Gear teeth

_______________________________________________
1
RS Khurmi & JK Gupta, A Textbook of Machine Design, 2005

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11. Diametral pitch - it is the ratio of number of teeth to the pitch circle diameter.
12. Module - it is the ratio of the pitch circle diameter to the number of teeth.
13. Clearance - it is the radial distance from the top of the tooth to the bottom of the tooth, in
a meshing gear. A circle passing through the top of the meshing gear is known as
clearance circle.
14. Total depth - it is the radial distance between the addendum and the dedendum circle of
a gear. It is equal to the sum of the addendum and dedendum.
15. Working depth - it is radial distance from the addendum circle to the clearance circle. It
is equal to the sum of the addendum of the two meshing gears.
16. Tooth thickness - it is the width of the tooth measured along the pitch circle.
17. Tooth space - it is the width of space between the two adjacent teeth measured along the
pitch circle.
18. Backlash - it is the difference between the tooth space and the tooth thickness, as
measured on the pitch circle.
19. Face of the tooth - it is surface of the tooth above the pitch surface.
20. Top land - it is the surface of the top of the tooth.
21. Flank of the tooth - it is the surface of the tooth below the pitch surface.
22. Face width - it is the width of the gear tooth measured parallel to its axis.
23. Profile - it is the curve formed by the face and flank of the tooth.
24. Fillet radius - it is the radius that connects the root circle to the profile of the tooth.
25. Path of contact - it is the path traced by the point of contact of two teeth from the
beginning to the end of engagement.
26. Length of the path of contact - it is the length of the common normal cut-off by the
addendum circles of the wheel and pinion.
27. Arc of contact - it is the path traced by a point on the pitch circle from the beginning to
the end of engagement of a given pair of teeth.
 Arc of approach - it is the portion of the path of contact from the beginning of the
engagement to the pitch point.
 Arc of recess - it is the portion of the path of contact from the pitch point to the end
of the engagement of a pair of teeth.

Design Formulae:
Velocity ratio – is the angular velocity of the driver divided by the angular velocity of the driven
gear.

 mω = = = = eq. 4 – 1

Gear ratio – is the ratio of number of teeth in the gear to the number of teeth in the pinion.
When the pinion is the driver, then

 mg = mω eq. 4 – 2

where: ω1- angular velocity driver, rad/s

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ω2 – angular velocity driven gear, rad/s

n1 – rpm of driver
n2 – rpm of driven gear
D1 – pitch diameter driver
D2 – pitch diameter driven gear
T1 – no. of teeth driver
T2 – no. of teeth driven gear
mg – gear ratio

Circular pitch,

 Pc = eq. 4 – 3

Diametral pitch,

 Pd = eq. 4 – 4

 PcPd = π eq. 4 – 5

Base pitch for involute gear,

 Pb = = = Pc cos ϕ eq. 4 – 6

where: Db – diameter of the base circle

D – pitch diameter
ϕ – pressure angle

Sample Problem: Find the tooth thickness on the tooth circle of a 200 full depth involute tooth
having a diametral pitch of 3, circular pitch of 1.04 and whole depth of tooth of 0.60.

SOLUTION
φ = 200
Pd = 3
Pc = 1.04
ht = dw = 0.60

Formula
t=

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t=

t = 0.524

Sample Problem: A pair of gears has a gear ratio of 3 and 60 gear teeth of a 14.50 full depth
tooth. The diametral pitch is 10. Compute the tooth thickness on the pitch circle.

SOLUTION

GR = 3
T = 60
φ = 200
Pd = 10

Formula
t=

t=

t = 0.157

Forces on Gear teeth

A simple single – reduction gear pair is shown below, where power is received from the motor
by the input shaft rotating at motor speed. The torque in the shaft can be computed from the
following equation:

 T = P/n eq. 4 -7

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where: T – torque
P – power
n – rotational speed

The torque is the product of the force acting tangent to the pitch circle of the pinion times the
pitch radius of the pinion. Tangential force, Ft is the force exerted by the pinion teeth on the
gear teeth. The torque on a gear is the product of the transmitted load, Wt, and the pitch radius of
the gear.

 T = Wt(R) = Wt (D/2) eq. 4 – 8

Systems of Gear teeth

1. Composite system (14.50) – it is used for general purpose gear.

2. Full depth involute system (14.50) – this used for gears with hobs for spur and helical
gears.
3. Full depth involute system (200) – it has stronger tooth than 14.50.
4. Stub involute system (200) – for use in heavy load applications.

Gear Materials

The gears may be manufactured from metallic or non-metallic. Metallic gears are made
from cast iron, steel and bronze. To reduce noise during gear engagement, non-metallic materials
are used like wood, rawhide, compressed paper and synthetic resins such as nylons.

Cast iron – widely used for the manufacture of gears due to its good wearing properties,
excellent in machinability and ease of producing complicated shapes by casting process.

Steel – used for high strength gears and may be plain carbon steel or alloy steel. Steel is
important for toughness and tooth hardness.

Metric module system

In SI, the common unit is millimeter (mm). The pitch of the gears in the metric system is
designated as module (m).Module is the pitch of the gear in millimeters divided by the number of
teeth.

Metric module

 m = Dg/Tg = Dp/Tp eq. 4 – 9

 m = 1/Pd eq. 4 – 10

 m = 25.4/Pd eq. 4 – 11

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where: m – metric module, mm

Pd – diametral pitch, inches
Dg – pitch diameter of the gear

Dp – pitch diameter of the pinion

Tg – number of teeth of the gear
Tp – number of teeth of the pinion

Design Exercises

PROBLEM: A 36-tooth pinion with a turning speed of 300 rpm drives 120-tooth gear of 14.20
involute full depth pressure angles. What would be the speed of the driven gear?
Ans. N = 90 rpm

SOLUTION

PROBLEM: Compute the circular pitch of a pair of gears having a ratio of 4 and center distance
of 10.23. Each gear has 72 teeth and pinion has 18 teeth. Ans. Pc = 0.71

SOLUTION

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PROBLEM: A 200 - pressure angle, 27-tooth spur gear has a diametral pitch of 5. Find the pitch
diameter, addendum, dedendum, outside diameter, and circular pitch.

SOLUTION

PROBLEM: A 250 - pressure angle, 43-tooth spur gear has a diametral pitch of 8. Find the pitch
diameter, addendum, dedendum, outside diameter, and circular pitch.

SOLUTION

PROBLEM: A 57- tooth spur gear is in mesh with a 23 – tooth pinion. The diametral pitch is 6
and pressure angle of 250. Find the contact ratio.

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SOLUTION

PROBLEM: A 57- tooth spur gear is in mesh with a 23 – tooth pinion. The diametral pitch is 6
and pressure angle of 250.Find VR, TR and GR.

SOLUTION

Assignment # __
Find machinery or device/machines that have spur gear application. Have a picture of it and
identify the details of the gear and compute the power needed to operate such machine. Submit
this requirement(s) on ___________.

Notes

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DESIGN PROBLEM # 4

Name: Rating:
Course/Yr: Date:

SPUR GEAR DESIGN

PROBLEM

Determine the torques and transmitted loads on the gear teeth in a 3-gear train containing a
pinion (14 teeth, 250, and Pd = 6), an idler (17 teeth), and a gear. Find the gear diameters and the
mean and alternating components of transmitted load on each gear. The pinion shaft passes 30 hp
at 300 rpm. The gear train has 2:1 ratio.

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DESIGN PROBLEM # 4 spur gear continuation

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