Introduction
Gears are machine elements used to transmit rotary motion
between two shafts, normally with a constant ratio. The pinion
is the smallest gear and the larger gear is called the gear
wheel.. A rack is a rectangular prism with gear teeth machined
along one side- it is in effect a gear wheel with an infinite pitch
circle diameter. In practice the action of gears in transmitting
motion is a cam action each pair of mating teeth acting as
cams. Gear design has evolved to such a level that throughout
the motion of each contacting pair of teeth the velocity ratio of
the gears is maintained fixed and the velocity ratio is still fixed
as each subsequent pair of teeth come into contact. When the
teeth action is such that the driving tooth moving at constant
angular velocity produces a proportional constant velocity of the
driven tooth the action is termed a conjugate action. The teeth
shape universally selected for the gear teeth is the involute
profile.
Consider one end of a piece of string is fastened to the OD of
one cylinder and the other end of the string is fastened to the
OD of another cylinder parallel to the first and both cylinders are
rotated in the opposite directions to tension the string(see figure
below). The point on the string midway between the cylinder P
is marked. As the left hand cylinder rotates CCW the point
moves towards this cylinder as it wraps on . The point moves
away from the right hand cylinder as the string unwraps. The
point traces the involute form of the gear teeth.
1
The lines normal to the point of contact of the gears always
intersects the centre line joining the gear centres at one point
called the pitch point. For each gear the circle passing through the
pitch point is called the pitch circle. The gear ratio is proportional
to the diameters of the two pitch circles. For metric gears (as
adopted by most of the worlds nations) the gear proportions are
based on the module.
m = (Pitch Circle Diameter (mm)) / (Number of teeth on gear).
In the USA the module is not used and instead the Diametric Pitch
Pd is used
d p = (Number of Teeth) / Diametrical Pitch (inches)
2
Profile of a standard 1mm module gear teeth for a gear with Infinite radius, Rack .
Other module teeth profiles are directly proportion . e.g. 2mm
module teeth are 2 x this profile
Many gears trains are very low power applications with an object of
transmitting motion with minimum torque e.g. watch and clock
mechanisms, instruments, toys, music boxes etc. These
applications do not require detailed strength calculations.
Standards
 AGMA 2001-C95 or AGMA-2101-C95 Fundamental Rating
factors and Calculation Methods for involute Spur Gear and
Helical Gear Teeth
 BS 436-4:1996, ISO 1328-1:1995..Spur and helical gears.
Definitions and allowable values of deviations relevant to
corresponding flanks of gear teeth
 BS 436-5:1997, ISO 1328-2:1997..Spur and helical gears.
Definitions and allowable values of deviations relevant to radial
composite deviations and runout information
 BS ISO 6336-1:1996 ..Calculation of load capacity of spur and
helical gears. Basic principles, introduction and general
influence factors
 BS ISO 6336-2:1996..Calculation of load capacity of spur and
3


helical gears. Calculation of surface durability (pitting)
BS ISO 6336-3:1996..Calculation of load capacity of spur and
helical gears. Calculation of tooth bending strength
BS ISO 6336-5:2003..Calculation of load capacity of spur and
helical gears. Strength and quality of materials
If it is necessary to design a gearbox from scratch the design process
in selecting the gear size is not complicated - the various design
formulae have all been developed over time and are available in the
relevant standards. However significant effort, judgment and expertise
are required in designing the whole system including the gears, shafts,
bearings, gearbox, and lubrication. For the same duty many different
gear options are available for the type of gear, the materials and the
quality. It is always preferable to procure gearboxes from specialized
gearbox manufacturers
4
Terminology - spur gears
 Diametral pitch (d p )...... The number of teeth per one inch of
pitch circle diameter.
 Module. (m) ...... The length, in mm, of the pitch circle diameter
per tooth.
 Circular pitch (p)...... The distance between adjacent teeth
measured along the are at the pitch circle diameter
 Addendum ( h a )...... The height of the tooth above the pitch
circle diameter.
 Centre distance (a)...... The distance between the axes of two
gears in mesh.
 Circular tooth thickness (ctt)...... The width of a tooth measured
along the are at the pitch circle diameter.
 Dedendum ( h f )...... The depth of the tooth below the pitch circle
diameter.
 Outside diameter ( D o )...... The outside diameter of the gear.
 Base Circle diameter ( D b ) ...... The diameter on which the
involute teeth profile is based.
 Pitch circle dia ( p ) ...... The diameter of the pitch circle.
 Pitch point...... The point at which the pitch circle diameters of
two gears in mesh coincide.
 Pitch to back...... The distance on a rack between the pitch circle
diameter line and the rear face of the rack.
 Pressure angle ...... The angle between the tooth profile at the
pitch circle diameter and a radial line passing through the same
point.
 Whole depth...... The total depth of the space between adjacent
teeth.
5
6
Spur Gear Design
The spur gear is is simplest type of gear manufactured and is
generally used for transmission of rotary motion between parallel
shafts. The spur gear is the first choice option for gears except
when high speeds, loads, and ratios direct towards other
options. Other gear types may also be preferred to provide more
silent low-vibration operation. A single spur gear is generally
selected to have a ratio range of between 1:1 and 1:6 with a pitch
line velocity up to 25 m/s. The spur gear has an operating efficiency
of 98-99%. The pinion is made from a harder material than the
wheel. A gear pair should be selected to have the highest number
of teeth consistent with a suitable safety margin in strength and
wear. The minimum number of teeth on a gear with a normal
pressure angle of 20 degrees is 18.
The preferred numbers of teeth (N) are as follows
12 13 14 15 16 18 20 22 24 25 28 30 32 34 38
40 45 50 54 60 64 70 72 75 80 84 90 96 100
120 140 150 180 200 220 250
Materials used for gears
Mild steel is a poor material for gears as it has poor resistance to
surface loading. The carbon content for unhardened gears is
generally 0.4 % (min) with 0.55 % (min) carbon for the
pinions. Dissimilar materials should be used for the meshing gears
- this particularly applies to alloy steels. Alloy steels have superior
fatigue properties compared to carbon steels for comparable
strengths. For extremely high gear loading case hardened steels
are used the surface hardening method employed should be such to
provide sufficient case depth for the final grinding process used.
7
Material
Notes
Ferrous metals
Low Cost easy to machine with
high damping
applications
Large moderate power, commercial
gears
Power gears with medium rating to
Cast Steels
Low cost, reasonable strength
commercial quality
Good machining, can be heat
Power gears with medium rating to
Plain-Carbon Steels
treated
commercial/medium quality
Heat Treatable to provide
Highest power requirement. For
Alloy Steels
highest strength and durability
precision and high precision
Good corrosion resistance. Non- Corrosion resistance with low power
Stainless Steels (Aust)
magnetic
ratings. Up to precision quality
Hardenable, Reasonable
Low to medium power ratings Up to
Stainless Steels (Mart)
corrosion resistance, magnetic high precision levels of quality
Non-Ferrous metals
Light weight, non-corrosive and Light duty instrument gears up to
Aluminum alloys
good machineability
high precision quality
Low cost, non-corrosive,
low cost commercial quality gears.
Brass alloys
excellent machinability
Quality up to medium precision
Excellent machinability, low
For use with steel power gears.
Bronze alloys
friction and good compatibility
Quality up to high precision
with steel
Light weight with poor corrosion Light weight low load gears. Quality
Magnesium alloys
resistance
up to medium precision
Low coefficient of thermal
Special gears for thermal
Nickel alloys
expansion. Poor machinability
applications to commercial quality
High strength, for low weight,
Special light weight high strength
Titanium alloys
good corrosion resistance
gears to medium precision
Low cost with low precision and High production, low quality gears
Di-cast alloys
strength
to commercial quality
Low cost, low quality, moderate High production, low quality to
Sintered powder alloys
strength
moderate commercial quality
Non metals
Cast Iron
Acetal (Delrin
Wear resistant, low water
absorption
Long life , low load bearings to
commercial quality
Phenolic laminates
Low cost, low quality, moderate
strength
High production, low quality to
moderate commercial quality
8
Nylons
No lubrication, no lubricant,
absorbs water
Long life at low loads to commercial
quality
PTFE
Low friction and no lubrication
Special low friction gears to
commercial quality
Equations for basic gear relationships
It is acceptable to marginally modify these relationships, e.g., to modify
the addendum / dedendum to allow Centre Distance adjustments. Any
changes modifications will affect the gear performance in good and bad
ways...
Addendum
Base Circle diameter
Centre distance
Circular pitch
Circular tooth thickness
Dedendum
Module
Number of teeth
Outside diameter
h a = m = 0.3183 p
Db = d.cos α
a = ( d g + d p) / 2
p = m.π
ctt = p/2
h f = h - a = 1,25m = 0,3979 p
m = d /n
z=d/m
D o = (z + 2) x m
Pitch circle diameter
d = n . m ... (d g = gear & d p = pinion )
Whole depth(min)
Top land width(min)
h = 2.25 . m
t o = 0,25 . m
9
10
Module (m)
The module is the ratio of the pitch diameter to the number of teeth.
The unit of the module is millimeters. Below is a diagram showing the
relative size of teeth machined in a rack with module ranging from
module values of 0,5 mm to 6 mm
The preferred module values (in mm) are
0,5 0,8 1.0 1,25
6
8
10
12
1,5 2,5
16
20
3
4
5
25 32
40
50
11
Normal Pressure angle (φn)
An important variable affecting the geometry of the gear teeth is the
normal pressure angle. This is generally standardized at 20o. Other
pressure angles should be used only for special reasons and using
considered judgment. The following changes result from increasing
the pressure angle





Reduction in the danger of undercutting and interference
Reduction of slipping speeds
Increased loading capacity in contact, seizure and wear
Increased rigidity of the toothing
Increased noise and radial forces
Gears required to have low noise levels have pressure angles 15o
to17.5o
Contact Ratio
The gear design is such that when in mesh the rotating gears have
more than one gear in contact and transferring the torque for some
of the time. This property is called the contact ratio. This is a ratio
of the length of the line-of-action to the base pitch. The higher the
contact ratio the more the load is shared between teeth. It is good
practice to maintain a contact ratio of 1.2 or greater. Under no
circumstances should the ratio drop below 1.1.
A contact ratio between 1 and 2 means that part of the time two
pairs of teeth are in contact and during the remaining time one pair
is in contact. A ratio between 2 and 3 means 2 or 3 pairs of teeth
are always in contact. Such as high contact ratio generally is not
obtained with external spur gears, but can be developed in the
12
meshing of an internal and external spur gear pair or specially
designed non-standard external spur gears.
(Rgo2 - Rgb2 )1/2 + (Rpo2 - Rpb2 )1/2 - a sin φ
contact ratio m = p cos φ
R go = D go / 2..Radius of Outside Dia of Gear
R gb = D gb / 2..Radius of Base Dia of Gear
R po = D po / 2..Radius of Outside Dia of Pinion
R pb = D pb / 2..Radius of Base Dia of Pinion
p = circular pitch.
a = ( d g+ d p )/2 = center distance.
Spur gear Forces, torques, velocities & Powers













F = tooth force between contacting teeth (at angle pressure
angle φ to pitch line tangent. (N)
F t = tangential component of tooth force (N)
F s = Separating component of tooth force
α= Pressure angle
d 1 = Pitch Circle Dia -driving gear (m)
d 2 = Pitch Circle Dia -driven gear (m)
ω 1 = Angular velocity of driver gear (Rads/s)
ω 2 = Angular velocity of driven gear (Rads/s)
z 1 = Number of teeth on driver gear
z 2 = Number of teeth on driven gear
P = power transmitted (Watts)
M = torque (Nm)
η = efficiency
Tangential force on gears F t = F cos φ
13
Separating force on gears F s = F t tan φ
Torque on driver gear T 1 = F t d 1 / 2
Torque on driver gear T 2 = F t d 2 / 2
Speed Ratio = ω 1 / ω 2 = d 2 / d 1 = z 2 /z 1
Input Power P 1 = T1 .ω 1
Output Power P 2 = η.T 1 .ω 2
Spur gear Strength and durability calculations
Designing spur gears is normally done in accordance with
standards the two most popular series are listed under
standards
above:
The notes below relate to approximate methods for estimating
gear strengths. The methods are really only useful for first
approximations and/or selection of stock gears (ref links
below). — Detailed design of spur and helical gears is best
completed using the standards. Books are available providing
the necessary guidance. Software is also available making the
process very easy. A very reasonably priced and easy to use
package is included in the links below (Mitcalc.com)
The determination of the capacity of gears to transfer the
required torque for the desired operating life is completed by
determining the strength of the gear teeth in bending and also
14
the surface durability, i.e., of the teeth ( resistance to
wearing/bearing/scuffing loads ) ..
The equations below are based on methods used by
Buckingham..
Bending
The basic bending stress for gear teeth is obtained by using the Lewis
formula
σ = Ft / ( ba. m. Y )





F t = Tangential force on tooth = Wt (alşo used) in [N]
σ = Tooth Bending stress (MPa)
b a = Face width (mm) = F (in some books)
Y = Lewis Form Factor
m = Module (mm)
Note: The Lewis formula is often expressed as
σ = Ft / ( ba. p. y )
Where y = Y/π and p = circular pitch
When a gear wheel is rotating the gear teeth come into contact
with some degree of impact. To allow for this a velocity factor is
introduced into the equation. This is given by the Barth
equation for milled profile gears.
K v = 6,1 / (6,1 +V )
V = the pitch line velocity in m/s and when d is in m/s,
15
V = d.ω/2
(m/s)
Or when d is in mm,
V = πdn/60,000
(m/s)
Note: This factor is different for different gear conditions
i.e., K v = ( 3.05 + V )/3.05 for cast iron, cast profile gears.
The Lewis formula is thus modified as follows
σ = K v.Ft / ( ba. m. Y )
In general, for AGMA STRESS EQUATION
KV = [A + (200 V)1/2]B / A
Where
B = 0,25 (12 – QV)2/3 and
A = 50 + 56 (1-B)
σ = [Wt/(bmtJ)] (KO KV KS KH KB)
(14-16)
KO = The Overload Factor, (Figures 14-17 and 14-18)
KV = The Dynamic Factor, (Eq.14-27)
KS = The Size Factor, (KS = 0,8433 [bmt (Y)1/2]0.0535 )
14-10.
KH = Km = The Load Distributıon Factor, and
KB = The Rim-Thickness Factor.
16
Surface Durability
This calculation involves determining the contact stress between the
gear teeth and uses the Herz Formula
σ w = 2.F / ( π .b .l )
σ w = largest surface pressure
F = force pressing the two cylinders (gears) together
l = length of the cylinders (gear)
b = halfwidth =
d 1 ,d 2 Are the diameters for the two contacting cylinders.
ν 1, ν 2 Poisson ratio for the two gear materials
E 1 ,E 2 Are the Young's Modulus Values for the two gears
To arrive at the formula used for gear calculations the following
changes are made
F is replaced by F t/ cos α
d is replaced by 2.r
l is replaced by W
The velocity factor K v as described above is introduced.
Also an elastic constant Z E is created
17
When the value of E used is in MPa then the units of Cp are √ MPa =
KPa The resulting formula for the compressive stress developed is as
shown below
The dynamic contact stress c developed by the transmitted torque
must be less than the allowable contact stress Se. (α = φ.).
Note: Values for Allowable stress values Se and ZE for some materials
are provided at Gear Table
r1 = d1 sin φ /2
r2 = d2 sin φ /2
Important Note: The above equations do not take into account the
various factors which are integral to calculations completed using the
relevant standards. These equations therefore yield results suitable
for first estimate design purposes only...
18
Design Process
To select gears from a stock gear catalogue or do a first
approximation for a gear design select the gear material and
obtain a safe working stress e.g Yield stress / Factor of Safety.
/Safe fatigue stress








Determine the input speed, output speed, ratio, torque to
be transmitted
Select materials for the gears (pinion is more highly loaded
than gear)
Determine safe working stresses (uts /factor of safety or
yield stress/factor of safety or Fatigue strength / Factor of
safety )
Determine Allowable endurance Stress Se
Select a module value and determine the resulting
geometry of the gear
Use the Lewis formula and the endurance formula to
establish the resulting face width
If the gear proportions are reasonable then - proceed to
more detailed evaluations
If the resulting face width is excessive - change the module
or material or both and start again
The gear face width should be selected in the range 9-15 x
module or for straight spur gears-up to 60% of the pinion
diameter.
19
Internal Gears
Advantages:
1. Geometry ideal for epicyclic gear design
2. Allows compact design since the center distance is less
than for external gears.
3. A high contact ratio is possible.
4. Good surface endurance due to a convex profile surface
working against a concave surface.
Disadvantages:
1. Housing and bearing supports are more complicated,
because the external gear nests within the internal gear.
2. Low ratios are unsuitable and in many cases impossible
because of interferences.
3. Fabrication is limited to the shaper generating process, and
usually special tooling is required.
20
21
Lewis form factor,Y.
Table of Lewis Form Factors for different tooth forms and pressure angles
No
Teeth
Load Near Tip of Teeth
14 1/2 deg
Y
y
20 deg FD
Y
y
Load at Near Middle of Teeth
20 deg Stub 25 deg
Y
y
Y
14 1/2 deg
y
Y
y
20 deg FD
Y
y
10
0,176 0,056 0,201 0,064 0,261 0,083 0,238 0,076
11
0,192 0,061 0,226 0,072 0,289 0,092 0,259 0,082
12
0,21
13
0,223 0,071 0,264 0,084 0,324 0,103 0,293 0,093 0,377 0,12
14
0,236 0,075 0,276 0,088 0,339 0,108 0,307 0,098 0,399 0,127 0,468 0,149
15
0,245 0,078 0,289 0,092 0,349 0,111 0,32
16
0,255 0,081 0,295 0,094 0,36
17
0,264 0,084 0,302 0,096 0,368 0,117 0,342 0,109 0,446 0,142 0,512 0,163
18
0,27
19
0,277 0,088 0,314 0,1
20
0,283 0,09
21
0,289 0,092 0,326 0,104 0,399 0,127 0,377 0,12
22
0,292 0,093 0,33
23
0,296 0,094 0,333 0,106 0,408 0,13
24
0,302 0,096 0,337 0,107 0,411 0,131 0,396 0,126 0,509 0,162 0,572 0,182
25
0,305 0,097 0,34
26
0,308 0,098 0,344 0,109 0,421 0,134 0,407 0,13
27
0,311 0,099 0,348 0,111 0,426 0,136 0,412 0,131 0,528 0,168 0,588 0,187
28
0,314 0,1
29
0,316 0,101 0,355 0,113 0,434 0,138 0,421 0,134 0,537 0,171 0,599 0,191
30
0,318 0,101 0,358 0,114 0,437 0,139 0,425 0,135 0,54
31
0,32
0,067 0,245 0,078 0,311 0,099 0,277 0,088 0,355 0,113 0,415 0,132
0,32
0,102 0,415 0,132 0,49
0,115 0,332 0,106 0,43
0,086 0,308 0,098 0,377 0,12
0,443 0,141
0,156
0,137 0,503 0,16
0,352 0,112 0,459 0,146 0,522 0,166
0,386 0,123 0,361 0,115 0,471 0,15
0,534 0,17
0,102 0,393 0,125 0,369 0,117 0,481 0,153 0,544 0,173
0,49
0,156 0,553 0,176
0,105 0,404 0,129 0,384 0,122 0,496 0,158 0,559 0,178
0,390 0,124 0,502 0,16
0,565 0,18
0,108 0,416 0,132 0,402 0,128 0,515 0,164 0,58
0,352 0,112 0,43
0,101 0,361 0,115 0,44
0,522 0,166 0,584 0,186
0,137 0,417 0,133 0,534 0,17
0,14
0,185
0,592 0,188
0,172 0,606 0,193
0,429 0,137 0,554 0,176 0,611 0,194
22
32
0,322 0,101 0,364 0,116 0,443 0,141 0,433 0,138 0,547 0,174 0,617 0,196
33
0,324 0,103 0,367 0,117 0,445 0,142 0,436 0,139 0,55
34
0,326 0,104 0,371 0,118 0,447 0,142 0,44
35
0,327 0,104 0,373 0,119 0,449 0,143 0,443 0,141 0,556 0,177 0,633 0,201
36
0,329 0,105 0,377 0,12
37
0,33
38
0,333 0,106 0,384 0,122 0,455 0,145 0,452 0,144 0,565 0,18
39
0,335 0,107 0,386 0,123 0,457 0,145 0,454 0,145 0,568 0,181 0,655 0,208
40
0,336 0,107 0,389 0,124 0,459 0,146 0,457 0,145 0,57
43
0,339 0,108 0,397 0,126 0,467 0,149 0,464 0,148 0,574 0,183 0,668 0,213
45
0,34
50
0,346 0,11
55
0,352 0,112 0,415 0,132 0,48
60
0,355 0,113 0,421 0,134 0,484 0,154 0,491 0,156 0,603 0,192 0,713 0,227
65
0,358 0,114 0,425 0,135 0,488 0,155 0,496 0,158 0,607 0,193 0,721 0,23
70
0,36
75
0,361 0,115 0,433 0,138 0,496 0,158 0,506 0,161 0,613 0,195 0,735 0,234
80
0,363 0,116 0,436 0,139 0,499 0,159 0,509 0,162 0,615 0,196 0,739 0,235
90
0,366 0,117 0,442 0,141 0,503 0,16
100
0,368 0,117 0,446 0,142 0,506 0,161 0,521 0,166 0,622 0,198 0,755 0,24
150
0,375 0,119 0,458 0,146 0,518 0,165 0,537 0,171 0,635 0,202 0,778 0,248
0,105 0,38
0,14
0,175 0,623 0,198
0,553 0,176 0,628 0,2
0,451 0,144 0,446 0,142 0,559 0,178 0,639 0,203
0,121 0,454 0,145 0,449 0,143 0,563 0,179 0,645 0,205
0,65
0,207
0,181 0,659 0,21
0,108 0,399 0,127 0,468 0,149 0,468 0,149 0,579 0,184 0,678 0,216
0,408 0,13
0,474 0,151 0,477 0,152 0,588 0,187 0,694 0,221
0,153 0,484 0,154 0,596 0,19
0,115 0,429 0,137 0,493 0,157 0,501 0,159 0,61
0,704 0,224
0,194 0,728 0,232
0,516 0,164 0,619 0,197 0,747 0,238
23
200
0,378 0,12
0,463 0,147 0,524 0,167 0,545 0,173 0,64
300
0,38
0,122 0,471 0,15
Rack
0,39
0,124 0,484 0,154 0,55
0,534 0,17
0,554 0,176 0,65
0,175 0,566 0,18
0,66
0,204 0,787 0,251
0,207 0,801 0,255
0,21
0,823 0,262
24