Bevel gears are used to transmit power between intersecting shafts, typically at 90 degrees. This page explains the different types of bevel gears, their geometry, force calculations, and design equations used in engineering applications.
There are four basic types of bevel gears.
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 90o 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 90o 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)
Mp = torque on pinion shaft (Nm)
Fs = Separating Force (N)
Fp = Pinion Thrust (N)
Fg = 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.
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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 22o. 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 )
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 xand 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 b2 / (3.L2) = 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
Bevel gears are used where power needs to be transmitted between intersecting shafts, most commonly at 90°. Their geometry makes them suitable for compact, efficient direction changes in mechanical systems.
Different types of bevel gears are selected depending on load, speed, and smoothness requirements:
In engineering design, bevel gears are particularly valuable where space constraints require a change in shaft direction without using additional components such as belts or chains. Their ability to handle high loads while maintaining efficiency makes them essential in many mechanical power transmission systems.
Bevel gears are used to transmit power between intersecting shafts, usually at 90 degrees. They are commonly found in automotive differentials, gearboxes, machine tools, and industrial equipment where a change in direction of rotation is required.
The main types of bevel gears are straight bevel gears, zerol bevel gears, spiral bevel gears, and hypoid bevel gears. Each type is designed for different load, speed, and smoothness requirements.
Straight bevel gears have straight teeth and are typically used in low-speed applications, while spiral bevel gears have curved teeth that provide smoother and quieter operation, making them suitable for high-speed and high-load applications.
Hypoid bevel gears are mainly used in automotive differentials. Their offset shaft arrangement allows for larger pinion sizes, smoother operation, higher torque capacity, and quieter performance.
Zerol bevel gears are similar to spiral bevel gears but have a spiral angle of zero. They provide smoother operation than straight bevel gears without generating additional thrust forces, making them suitable as direct replacements.
Bevel gears generate thrust forces due to the conical shape of their pitch surfaces and the angle of the teeth. These forces act along the shaft and must be supported by appropriate bearing arrangements.
Bevel gears typically operate at a shaft angle of 90 degrees, but they can be designed for other angles depending on the application requirements.
Bevel gears are designed based on parameters such as number of teeth, pitch angles, module, and load capacity. Design calculations often follow standards such as BS ISO 10300 for strength and durability.
Bevel gears allow efficient transmission of motion between intersecting shafts, provide compact design solutions, and can handle a wide range of loads depending on the gear type.
Bevel gears require precise alignment and can generate significant thrust forces. Straight bevel gears in particular can be noisy and are not suitable for high-speed applications compared to spiral or hypoid gears.
