Lecture #24
SPUR GEAR TEETH
CHARACTERISTICS
Course Name : DESIGN OF MACHINE ELEMENTS
Course Number: MET 214
In order to identify the shape commonly used with the teeth of a spur gear, consider the
formation of a curve that is referred to as an involute as shown below.
The cross section of the cylinder from which the string is unwrapped is referred to as
the base circle. Usually the end of the string is shown as being unwrapped from the
base circle and as a consequence it is the end of the string that traces out the involute as
shown in the diagram contained on the next slide.
As identified in the figure below, the string is keep taut when being unwound, and as a
consequence, the string always forms a tangent with the base circle regardless of how
much string has been unwrapped.
The line formed by any point on the involute and the corresponding point of tangency
existing with the base circle is referred to as the generating line. As is evident from the
figure, the length of the generating line segment existing between any point on the
involute and the corresponding point of tangency increases as more of the string is
unwrapped from the base circle.
As identified in the figure below, the base circle of a gear can be used to create a gear
tooth with an involute shape.
Although the next slide is from a different source than the illustration provided above,
and uses a different set of symbols to characterize an involute, several equations
fundamental to describing an involute are provided in the figure and accordingly, the
figure should compared with the previous illustrations with the objective of being able
to relate all illustrations to one another in a comprehensible manner.
As noted in the equations provided on the previous slide, the length of the string unwrapped
from the base circle, defined as 𝜌, is related to the roll angle, defined as 𝜃, by the radius of the
base circle, defined as
𝑑𝑏
2
.
Consequently, the length of 𝜌 is proportional to the roll angle 𝜃. Accordingly, if an involute is
formed from a base circle and attached to the base circle, and the base circle rotated at a
constant rate, the involute can be made to sweep across a tangent to the base circle at a
constant velocity. This is illustrated in the figure provided below. Since the radius of the base
circle is a, for one revolution of the base circle, the rise would be 2 𝜋 𝑎.
Accordingly, the involute satisfies one of the criteria required for a gear in order for the gear to
have the same speed ratio as the friction wheel implementation that was developed in a previous
lecture. The point of contact existing between gear teeth point must travel with a constant
velocity on the tangent line existing between the base circles in order for a pair of gears to
produce the same speed ratio as the friction wheel implementation. The properties of an
involute can be used by the teeth on both gears to enable a point of contact existing between the
gear teeth to move along the line of action at a constant rate as shown in the diagram below. The
line of action is shown in a vertical position to facilitate understanding of how the rotation of the
base circles effects velocity along the line of action.
As illustrated by the drawing provided below, a tangent to the involute at any point will
be perpendicular to the generating line for that point. This relationship can be proven to
be true for every point on an involute. The generating line is always normal to the
involute curve.
Accordingly, returning to the figure showing the action of two involutes in contact, repeated
below for convenience, the following point should be noted. Since the tangent of each involute is
perpendicular to it’s generating line, the tangents of two involutes in contact coincide only when
the generating line of one involute is a continuation of the generating line of the second involute.
Accordingly, the locus of the points of contact between two involutes is along the common
tangent to the two base circles as shown in the figure.
With the following considerations in mind, the Fundamental Law of toothed Gearing will
now be presented. The derivation presented below is taken from the text titled Design
of Machine elements 5th edition by Spotts.
Key features of gear teeth are illustrated in the figures below.
Pitch Diameter:
Diameter of pitch circle. The pitch circles or the circles that exists in the friction wheel
implementation of the meshing gears.
Pitch:
3 types of pitch are in common use.
a) Circular pitch
b) Diametral pitch
c) Metric module
a)
Circular pitch:
The distance from a point on a tooth of a gear at the pitch circle to a corresponding point on the
next adjacent tooth, measured along the pitch circle. The following relationship is a consequence of
a gear having an integral number of teeth spaced around the periphery of the gear.
N  D
where   circular pitch
N  number of teeth on gear
D  diameter of pitch circle
The following observation is evident from the requirement to meshing gears to form a mechanical
system of interlocking.
“The pitch of two gears in mesh must be identical”
As shown in the table below, standards have been established for gears.
The following key features are illustrated in the figures above.
1) addendum (a): The radial distance from the pitch circle to the outside of a tooth (top land)
Figures 8-11
2) dedendum (b): The radial distance from the pitch circle to the bottom of the tooth space.
3) Clearance (c) : The radial distance from the top of a tooth to the bottom of the tooth space of
the mating gear when the tooth is fully engaged.
c=b-a
4) Outside diameter (D0): The diameter of the circle that encloses the outside of the gear teeth.
Note that
D0 = D+ 2a
where
D= diameter of pitch circle
Using the definition for Pd and standards concerning gear teeth the following relationships may be
established.
D0 
N
1 N 2
2 
Pd
Pd
Pd
5) Root diameter: (DR) The diameter of the circle that contains the bottom of the tooth space. This
circle is called the root circle.
DR = D – 2b
6) Whole depth (ht) : The radial distance from the top of a tooth to the bottom of the tooth space.
Note that
ht = a + b
7) Working depth (hk): The radial distance that a gear tooth projects into the tooth space of a
mating gear. Note that
hk = a + a = 2a
and
ht = hk + c
8) Tooth thickness (t): The arc length, measured on the pitch circle from one side of a tooth to the
other side. This is sometimes called the circular thickness and has the theoretical value of one-half
of the circular pitch.
t
p


2 2 Pd
recall pPd  
9) Tooth space: The arc length, measured on the pitch circle, from the right side of one tooth to the
left side of the next tooth. Theoretically, the tooth space equals the tooth thickness. But for practical
reasons, the tooth space is made large. The difference between tooth space and tooth thickness is
referred to as backlash. Backlash provides space for lubrication, and relieves the demand for precision
on gear teeth dimensions.
10) Face width (F): The width of the tooth measured parallel to the axis of the gear.
11) Fillet: The arc joining the involute tooth profile to the root of the tooth space.
12) Face: The surface of a gear tooth from the pitch circle to the outside circle of the gear.
13) Flank: The surface of a gear tooth from the pitch circle to the root of the tooth space, including the
fillet.
14) Center distance (c): The distance from the center of the prism to the center of the gear; the sum of
the pitch radii of two gears in mesh.
C
DG DP DG  DP 


2
2
2
The center distance can also be expressed in terms of the diametrical pitch.
C
1  N G N P  N G  N P 



2  Pd
Pd 
2 Pd
15 Pressure angle: The pressure angle is the angle between the tangent to the pitch circles and the
line drawn normal (perpendicular) to the surface of the gear tooth see figure below.
As established previously,
where Db  diameter of base circle
Db  D cos
  pressure angle
Standard values of the pressure angle are established by gear manufacturers and the pressure
angle of two gears in mesh must be the same. Current standard pressure angles are 200 and 250.
Pressure angles of 141/20 are presently used only as replacements. Involute tooth forms for
different pressure angles are shown below