Bevel gears
Bevel gear on roller shutter door.
Bevel gear lifts floodgate by means of central screw.
Bevel gears are gears where the axes of the two shafts intersect and the tooth-bearing
faces of the gears themselves are conically shaped. Bevel gears are most often
mounted on shafts that are 90 degrees apart, but can be designed to work at other
angles as well. The pitch surface of bevel gears is a cone.
Introduction
An important concept in gearing is the pitch surface. In every pair of meshing gears,
each gear has a pitch surface. The pitch surfaces are the surfaces of imaginary smooth
(toothless) bodies that would produce the same gearing relationship by frictional
contact between their faces as the actual gears do by their tooth-to-tooth contact. They
are a sort of "average" surface that one would get by evening out the peaks and
valleys of the individual teeth. For an ordinary gear the pitch surface is a cylinder. For
a bevel gear the pitch surface is a cone. The pitch cones of meshed bevel gears are
coaxial with the gear shafts; and the apexes of the two cones are at the point of
intersection of the shaft axes. The pitch angle is the angle between the face of the
cone and the axis. The most familiar kinds of bevel gears, such as those in the picture
at the beginning of this article, have pitch angles of less than 90 degrees. They are
"pointy". This type of bevel gear is called an external bevel gear because the teeth are
facing outwards. It is possible to have a pitch angle greater than ninety degrees, in
which case the cone, rather than forming a point, forms a sort of conical cup. The
teeth are then facing inwards, and this type of gear is called an internal bevel gear. In
the border line case, a pitch angle of exactly 90 degrees, the teeth point straight
forward. In this orientation, they resemble the points on a crown, and this type of gear
is called a crown bevel gear or crown gear.
Teeth
Independently from the operating angle, the gear axes must intersect (at the point O)
There are two issues regarding tooth shape. One is the cross-sectional profile of the
individual tooth. The other is the line or curve on which the tooth is set on the face of
the gear: in other words the line or curve along which the cross-sectional profile is
projected to form the actual three-dimensional shape of the tooth. The primary effect
of both the cross-sectional profile and the tooth line or curve is on the smoothness of
operation of the gears. Some result in a smoother gear action than others.
The teeth on bevel gears can be straight, spiral or "zero".
In straight bevel gears the teeth are straight and parallel to the generators of
the cone. This is the simplest form of bevel gear. It resembles a spur gear, only
conical rather than cylindrical. The gears in the floodgate picture are straight
bevel gears. In straight, when each tooth engages it impacts the corresponding
tooth and simply curving the gear teeth can solve the problem.
Spiral bevel gears have their teeth formed along spiral lines. They are
somewhat analogous to helical gears, a cylindrical type, in that the teeth are
angled; however with spiral gears the teeth are also curved. The advantage of
the spiral tooth over the straight tooth is that they engage more gradually. The
contact between the teeth starts at one end of the gear and then spreads across
the whole tooth. This results in a less abrupt transfer of force when a new pair
of teeth come in to play. With straight bevel gears, the abrupt tooth
engagement causes noise, especially at high speeds, and impact stress on the
teeth which makes them unable to take heavy loads at high speeds without
breaking. For these reasons straight bevel gears are generally limited to use at
linear speeds less than 1000 feet/min; or, for small gears, under 1000 r.p.m.1
Zero bevel gears are an intermediate type between straight and spiral bevel
gears. Their teeth are curved, but not angled.
The bevel gear planer was invented by William Gleason at Gleason Works in 1874.
These gears permit minor adjustment during assembly and allow for some
displacement due to deflection under operating loads without concentrating the load
on the end of the tooth. For reliable performance, Gears must be pinned to shaft with a
dowel or taper pin.
Hypoid bevel gears can engage with the axes in different planes. This is used in
many car differentials. The ring gear of the differential and the input pinion gear are
both hypoid. This allows input pinion to be mounted lower than the axis of the ring
gear. Hypoid gears are stronger, operate more quietly and can be used for higher
reduction ratios. They also have sliding action along the teeth, potentially reducing
efficiency.
Miter gears are mating bevel gears with equal numbers of teeth and with axes at right
angles.
Skew bevel gears are those for which the corresponding crown gear has teeth that are
straight and oblique.
Applications
Bevel gears on grain mill.Note wooden teeth inserts on one of the gears.
A good example of bevel gears is seen as the main mechanism for a hand drill. As the
handle of the drill is turned in a vertical direction, the bevel gears change the rotation
of the chuck to a horizontal rotation. The bevel gears in a hand drill have the added
advantage of increasing the speed of rotation of the chuck and this makes it possible
to drill a range of materials.
The bevel gear find its application in locomotives, marine applications, automobiles,
printing presses, cooling towers, power plants, steel plants, defence and also in
railway track inspection machine.
Bevel gears are used in differential drives, which can transmit power to two axles
spinning at different speeds, such as those on a cornering automobile.
Spiral bevel gears are important components on all current rotorcraft drive systems.
These components are required to operate at high speeds, high loads, and for an
extremely large number of load cycles. In this application, spiral bevel gears are used
to redirect the shaft from the horizontal gas turbine engine to the vertical rotor
Advantages
This gear makes it possible to change the operating angle
Disadvantages
One wheel of such gear is designed to work with its complementary wheel and
no other.
Must be precisely mounted.
The axes must be capable of supporting significant forces.
straight bevel gear terminology
Straight Bevel Gear Force equations
Nomenclature
z p = Number of teeth on pinion
z g = Number of teeth on gear
= Pressure Angle of Teeth.
p = Pitch Angle (pinion)....= tan-1 (z p / z g )
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)
Spiral Bevel Gear
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.
n = Normal Pressure angle..
= pitch cone angle
= Helix angle
Pinion Thrust F p = F t [ (tan n sin / cos ) tan cos ]
Note: ( + ) if helix angle is as shown and ( -) if helix angle is opposite to that shown
Hypoid Bevel Gear
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
rations 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..
Straight Bevel Gear Strength and Durability Equations
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 g
Equivalent Number of teeth on pinion = z ep = z p / cos p
Bending Strength Equations
The basic lewis formula for spur gear teeth is shown as follows
= F t / ( W. m. Y )
F t = Tangential force on tooth (N)
= 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 be considering and
element dx..
The Lewis formula applied to the element is as follows
To obtain the allowable torque T transmitted by the multiply both sides by r xand integrating results in
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 then 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 a 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
Fb Ft/Kv
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)
Surface Durability Equations
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 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 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
Fw Ft/Kv
