There are four basic types of bevel gears.

- Straight bevel gears.. These gears have a conical pitch surface and straight teeth tapering towards an apex
- Zerol bevel gears. These are similar to a bevel gears except the teeth are curved. In essence, Zerol Bevel Gears are Spiral Bevel Gears with a spiral angle of zero.
- Spiral bevel gears. The teeth are curved teeth at an angle allowing tooth contact to be gradual and smooth
- Hypoid bevel gears..These gears are similar to spiral bevel except that the pitch surfaces are hyperboloids rather than cones. Pinion can be offset above, or below,the gear centre, thus allowing larger pinion diameter, and longer life and smoother mesh, with additional ratios e.g 6:1, 8:1, 10:1

BS 545:1982..Specification for bevel gears (machine
cut)..(Obsolescent)

BS ISO 10300-1:2001..Calculation of load capacity of bevel gears.
Calculation of load capacity of bevel gears. Introduction and
general influence factors

BS ISO 10300-2:2001..Calculation of load capacity of bevel gears.
Calculation of surface durability (pitting)

BS ISO 10300-3:2001..Calculation of load capacity of bevel gears.
Calculation of tooth root strength

These are gears cut from conical blanks and connect
intersecting shaft axes. The connecting shafts are
generally at 90^{o} and sometimes one shaft drives a
bevel gear which is mounted on a through shaft resulting in two
output shafts. The point of intersection of the
shafts is called the apex and the teeth of the two gears converge
at the apex. The design of bevel gears results in
thrust forces away from the apex. With the bearing
limitations the gears have to be carefully designed to ensure
that they are not thrown out of alignment as they are
loaded.

Straight bevel gears are used widely in machine drive systems to
effect 90^{o} direction changes. and in differential
drives They have the same limitations as spur gears
and they are therefore not used on high duty high speed
applications. Straight bevel gears are low cost units
supplied with ratios from 1:2 to 4:1.

z _{p} = Number of teeth on pinion

z _{g} = Number of teeth on pinion

α = Pressure Angle of Teeth.

ε _{p} = Pitch Angle (pinion)....=
tan^{-1} (z _{p} / z _{g} )

ε _{g} = Pitch Angle (gear)....=
tan^{-1} (z _{g} / z _{p} )

P _{p} = Power at Pinion shaft (kW)

n _{p} = Rotational speed of pinion shaft
(revs/min)

d _{p} = Pinion Pitch Circle diameter (mm)

M_{p} = torque on pinion shaft (Nm)

F_{s} = Separating Force (N)

F_{p} = Pinion Thrust (N)

F_{g} = Gear Thrust (N)

The advantage of Zerol bevel gears compared to
straight bevel gears is that they operate with a smooth localised
point contact as opposed to a line contact enabling smoother
operation with low vibration levels and higher
speeds. Because there is no spiral angle and no
additional developed thrust these gears can be used as direct
replacements for straight bevel gears. These gears
normally have a pressure angle of 20 ^{o}. The
minimum number of teeth on the pinion is 14. The
design of Zerol gears is relatively specialised and they are
manufactured using special "Gleason" machine
tools..

These are produced using a spiral gear form which results in a smoother drive suitable for higher speed higher loaded applications. Again satisfactory performance of this type of gear is largely dependent upon the rigidity of the bearings and mountings.

α _{n} = Normal Pressure
angle..

ε = pitch cone angle

γ = Helix angle

Pinion Thrust F _{p} = F
_{t} [ (tan α
_{n} sin ε / cos
γ ) ± tan g cos ε ]

Note: ( + ) if helix angle is as shown and (
-) if helix angle is opposite to that shown

Hypoid gears are best for the applications requiring large speed
reductions with non intersecting shafts and those applications
requiring smooth and quiet operation. Hypoid gears are
generally used for automotive applications. The
minimum number of teeth for speed ratios greater than 6 :1 is
eight although 6 teeth pinions can be used for ratios below
6:1. Hypoid gears have pressure angles between 19 and
22^{o}. The design of hypoid gears is
relatively specialised and they are manufactured using special
"Gleason" machine tools..

Designing bevel gears is normally done in accordance with
standards as listed under specifications 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 bevel gears should only
be completed using the relevant standards. Books are
available providing the necessary guidance. Software
is also available making the process very easy.

The equations are basically modified spur gears
equations using a spur gear equivalent number of teeth z
_{e}

Equivalent Number of teeth on gear = z _{eg} = z
_{g} / cos ε _{g}

Equivalent Number of teeth on pinion = z _{ep} = z
_{p} / cos ε _{p}

The basic lewis formula for spur gear teeth is shown as follows

s = F
_{t} / ( W. m. Y )

- F
_{t}= Tangential force on tooth (N) - s = Tooth Bending stress (MPa)
- W = Face width (mm)
- Y = Lewis Form Factor
- m = Module (mm)

The Lewis formula is modified to provide the allowable tangential
force F _{b} based on the allowable bending Stress S
_{b}

F _{a} = S _{b}.W. m.
Y

It is clear that a bevel gear does not have a uniform section or a uniform module and therefore it is necessary to start an analysis by considering an element dx..

The Lewis formula applied to the element is as follows

To obtain the allowable torque (M) transmitted by the gears
multiply both sides by r _{x}and integrate the resulting
equations as shown below

The module varies along the gear teeth in proportion to the
radius from the apex along the pitch cone.

Thus ..m / m _{x} = L / x where m = module at x = L

A similar relationship holds for for r _{x}. i.e for r
_{x} /r = x /L

Substituting these relationships into the integration equation
results..

d _{x} varies from x = (L -b) to x = L the integration
can be solved as follows:

The face width is considered to be limited to 1/3 of the cone
distance when the factor b^{2} / (3.L^{2}) = 1/27
is so small compared to the other factors that it can be
reasonably ignored . Then dividing by r to arrive at
the Lewis equation for the allowable bending load

The allowable bending load F _{b} must be greater than
the dynamic load which is the actual bending load calculated from
the transmitted torque modified by the Barth formula as
identified in the notes on spur gears i.e

**F _{b} ³ F _{t} / K
_{v}**

K _{v} is given by the Barth equation for milled profile
gears.

**K _{v} = 6,1 / (6,1 +V
)**

Note: This factor is different for different gear conditions i.e
K _{v} = ( 3.05 + V )/3.05 for cast iron, cast profile
gears.

V =Average velocity of gear face = 0.0000524.n.d
_{mean}

d _{mean} is the mean pitch circle diameter (mm)..

n = Rotational speed of gear (rpm)

The gear durability equation is based on the Hertz contact stress
equation and its application to gears.

The allowable tangential wear load F _{w} is calculated
as follows

**F _{w} = d _{p}. K.
Q^{'} / cos ε
_{p}**

d _{p} = Pitch diameter measured at the back of
tooth

Q^{'} = 2 z _{eg} /( z _{ep} + z
_{ep} )

z _{eg} & z _{ep} are the equivalent number
of teeth on the gear and pinion as defined above

K = Wear Load Factor see table Gear
table

The allowable load F _{w} must be greater than the
dynamic bending load which is the actual load calculated from the
transmitted torque modified by the Barth formula as identified in
the notes on spur gears i.e

**F _{w} ³ F _{t} / K
_{v}**

- Boston Gears - Miter Gears ...A supplier of bevel gears
- NASA TEsts on Spiral Bevel Gears ...Interesting short note.
- Neeterdrive.co.uk ... UK supplier of Bevel gears and associated components
- SEW Eurodrive...All the information on Gearboxes you will need
- Quality Transmission Components...Supplier with downloadable Gear Design Handbook
- Stock Drive Products= Sterling Instruments...Supplier with large quantity of downloadable drive information
- hewitt-topham...UK gear supplier
- Lenze...Drive system supplier with geared motor section