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This article provides a comprehensive look at bevel gears, include the following information:
A bevel gear is a toothed rotating machine element used to transfer mechanical energy or shaft power between shafts that are intersecting, either perpendicular or at an angle. This results in a change in the axis of rotation of the shaft power. Aside from this function, bevel gears can also increase or decrease torque while producing the opposite effect on the angular speed.
A bevel gear can be imagined as a truncated cone. On its lateral side, teeth that have been milled interlock with other gears with their own sets of teeth. The gear transmitting the shaft power is called the driver gear, while the gear where power is being transmitted is called the driven gear. The number of teeth of the driver and driven gear is usually different to produce a mechanical advantage. The ratio between the number of teeth of the driven to the driver gear is known as the gear ratio, while the mechanical advantage is the ratio of the output torque to the input torque. This relationship is shown by the following equation:
\begin{equation} \ MA = \frac{T_b}{T_a} = \frac{r_b}{r_a} = \frac{N_b}{N_a} \end{equation}
MA is the mechanical advantage, τb and τa are the torques, rb and ra are the radii, and Nb and Na are the numbers of teeth of the driven and driver gears, respectively. From the equation, it can be seen that increasing the number of teeth of the driven gear produces a larger output torque.
On the other hand, producing a larger mechanical advantage decreases the driven gear's output speed. This is expressed by the equation:
\begin{equation} \ MA = \frac{W_a}{W_b} \end{equation}
ωª and ωb are the driver and driven gears‘ angular speeds, respectively. In general, a gear ratio of 10:1 is recommended for a bevel gear set. For increasing the speed of the driven gears, a gear ratio of 1:5 is suggested.
Note that bevel gears are usually a paired set and should not be used interchangeably. Bevel gears are assembled in a specific way due to their inherent transmission of both thrust and radial loads, in contrast with spur gears which mostly transmit radial loads only. All bevel gears are assembled at the optimum position for best performance.
Efficiency is defined as the ratio of the output power to the input power. Note that this is different from mechanical advantage, which is concerned with amplifying of forces or torques by sacrificing speed. When it comes to bevel gears, loss of power during transmission is attributed to friction due to sliding between teeth surfaces and loads applied to the bearings or housing. The efficiency of different types of bevel gears compared with other types is summarized by the table below.
Type of Gear Approximate Range of Efficiency Type of Load Imposed in Bearings Straight Bevel Gear 97 – 99.5% Radial and thrust Spiral Bevel Gear 97 – 99.5% Radial and thrust Zerol Bevel Gear 97 – 99.5% Radial and thrust Hypoid Bevel Gear 90 – 98% Radial and thrust External Spur Gears 97 – 99.5% Radial Internal Gears 97 – 99.5% Radial Worm Gear 50 – 90% Radial and thrust Common Types of GearsThere are different types of bevel gears as categorized by their tooth profile and orientation. The more complicated types, such as the spiral and hypoid bevel gears, resulted from the further development of manufacturing processes like CNC machining.
A straight bevel gear is the simplest form of a bevel gear. The teeth are in a straight line that intersects at the axis of the gear when extended. The teeth are tapered in thickness, making the outer, or heel part of the tooth larger than the inner part, or toe. Straight bevel gears have instantaneous lines of contact, permitting more tolerance in mounting. A downside to using this type is the vibration and noise. This limits straight bevel gears to low-speed and static loading applications. CA common application of straight bevel gears is differential systems in automotive vehicles.
Straight bevel gears are the easiest to manufacture. The earliest manufacturing method for producing a straight bevel gear is by using a planer with an indexing head. More efficient manufacturing methods have been made following the introduction of Revacycle and Coniflex systems, employed by Gleason Works.
A spiral bevel gear is the most complex form of bevel gear. The teeth of spiral gears are curved and oblique, in contrast to the teeth orientation of straight bevel gears. This results in more overlap between teeth, which promotes gradual engagement and disengagement upon tooth contact. This improved smoothness results in minimal vibration and noise produced during operation. Also, because of higher load sharing from more teeth in contact, spiral bevel gears have better load capacities. This allows them to be smaller compared to straight bevel gears with the same capacity.
A disadvantage of spiral bevel gears is the larger thrust load exerted, which requires more expensive bearings. A rolling element thrust bearing is usually required for spiral bevel gear assemblies. Also, spiral bevel gears are made in matched sets. Different gear sets with the same design are not interchangeable unless built to be interchangeable on purpose. Spiral bevel gear sets are made either right-hand or left-hand.
Spiral bevel gear teeth are typically shaped by gear generating machines, which will be discussed in depth later. This process creates high accuracy and finish. Also, lapping is done to finish the teeth and further obtain the desired tooth bearing.
This type is a modification of a straight bevel gear trademarked by Gleason Works. Zerol bevel gears have teeth curved in a lengthwise direction. These gears are also somewhat similar to spiral bevel gears in their profile. Their difference is the spiral angle; Zerol types have 0° spiral angles, while spiral types have 35°.
As with straight bevel gears, Zerol types do not produce excessive thrust loads. Thus, plain contact bearings can be used. Zerol types can be substituted with straight bevel gears without changing the housing or bearings. Moreover, due to their curvature, Zerol bevel gear teeth have a slight overlapping action similar to spiral gears. This makes the gears run smoother than straight bevel gears.
Zerol bevel gear teeth are generated by a rotary mill cutter. The curvature of this cutter creates the lengthwise curvature of the tooth. Zerol bevel gears are with high precision, often finished by lapping or grinding.
A hypoid bevel gear is a special type of bevel gears in which the axes of the shafts are not intersecting nor parallel. The distance between the two gear axes is called the offset. The teeth of hypoid bevel gears are helical, similar to spiral bevel gears. A hypoid bevel gear designed with no offset is simply a spiral bevel gear. The manufacture and shaping of hypoid types are similar to spiral bevel gears.
Because of the offset, the spiral angle of the smaller gear (pinion) of a hypoid bevel gear set can be made larger than the spiral diameter of the larger gear. The ratio of the number of teeth of the gears is not directly proportional to the ratio of their pitch diameter or the theoretical operating diameter of the gear. This makes it possible to match larger pinions to a particular size of a driven gear, making the pinion stronger and allowing it a higher contact ratio to the larger gear. In turn, the hypoid gear is allowed to transmit more torque and operate at higher gear ratios. Also, with enough offset, bearings on both sides of the gears can be placed since their shafts are not intersecting. The trade-off, however, is the decrease in efficiency as the offset is increased.
Hypoid gears operate more smoothly with minimal vibration compared to spiral gears. However, the downside of using hypoid gears, aside from the aforemented efficiency issue, is the high amount of sliding that takes place across the face of the teeth. This means special lubricating oils must be used.
This is a type of bevel gear with a gear ratio of 1:1, meaning the driver and driven gears have the same number of teeth. Because a miter gear does not produce any mechanical advantage, the function of this type is limited to changing the axis or rotation. Usually, miter gears have axes that intersect perpendicularly. In some assemblies, The shafts are aligned to intersect at any angle. These are known as angular miter bevel gears. Shaft angles of angular miter bevel gears can range from 45° to 120°. Miter bevel gear teeth cuts can be straight, spiral, or Zerol.
To better understand gears and gear systems, one must first look at the terminology. Below are some of the terms used to describe gears and their tooth profile. These are applicable for all types of gears, not only bevel gears.
The smaller bevel gear in a bevel gear set.
The larger bevel gear in a bevel gear set.
Also known as circular pitch, pitch is the distance from one point on a tooth to the corresponding point of the adjacent tooth on the same gear.
The diameter of the pitch circle. This is a predefined design dimension where other gear characteristics such as tooth thickness, pressure angles, and helix angles are determined.
The ratio of the number of teeth and the pitch diameter.
The angle between the face of the pitch surface and the shaft axis.
The imaginary truncated cone wherein the base diameter is the pitch circle.
A predefined value that is described by the angle between the line of force of the meshing teeth and the line tangent to the pitch circle at the contact point. Gears must have the same pressure angle in order to mesh. The recommended pressure angle for straight bevel gears is 20°.
A predetermined value that defines the angle between the driven and driver shafts.
The upper outline of the gear teeth.
The bottom outline of the gear teeth.
The radial distance between the addendum and dedendum circles of a gear. Note that the teeth of a bevel gear are slightly tapered, thus, the total depth is not constant along the tooth. Because of this, the addendum and dedendum angles are used to describe the teeth instead of the addendum and dedendum circles.
The angle between the face of the upper surface of the teeth, or top land, and the pitch surface.
The angle between the bottom surface of the teeth, or bottom land, and the pitch surface.
The change in tooth depth along the face, measured perpendicular to the pitch surface.
The change of the space width along the face, measured on the pitch surface.
The change of tooth thickness measured on the pitch surface.
The total depth of the teeth plus the value of the clearance.
The difference between the addendum of a gear to the dedendum of the mating gear.
The amount of space that exceeds the thickness of a mating gear tooth. For bevel gears, there are different types of backlash depending on the orientation of the movement. These are:
The arc along the pitch circle
The space between the surface of the mating teeth
The described as the angular movement
The linear movement perpendicular to the axis
The linear movement parallel to the axis
Backlash is necessary to prevent the gears from jamming due to contact. This space allows for lubricants to enter and protect the surfaces of the mating teeth. Also, the backlash allows for thermal expansion during operation.
The relationship between these terms is shown in the table of equations below.
Straight Bevel Gear Formulas (20° Pressure Angle, 90° Shaft Angle) To Find Having Formula Pitch diameter of pinion Number of pinion teeth and diametral pitch d = Np / Pd Pitch diameter of gear Number of gear teeth and diametral pitch D = Ng / Pd Pitch angle of pinion Number of pinion teeth and number of gear teeth γ = tan^-1(Np / Ng) Pitch angle of gear Pitch angle of pinion Γ= 90°-γ Outer cone distance of pinion and gear Gear pitch diameter and pitch angle of gear Ao = D / (2sinΓ) Circular pitch of pinion and gear Diametral pitch p = 3.1416 / Pd Dedendum angle of pinion Dedendum of pinion and outer cone distance δp = tan-1(bop / Ao) Dedendum angle of gear Dedendum of gear and outer cone distance δg = tan-1(bog / Ao) Face angle of pinion blank Pinion pitch angle and dedendum angle of gear γo = γ + δg Face angle of gear blank Gear pitch angle and dedendum angle of pinion Γo = Γ + δp Root angle of pinion Pitch angle of pinion and dedendum angle of pinion γr = γ - δp Root angle of gear Pitch angle of gear and dedendum angle of gear Γr = Γ - δg Outside diameter of pinion Pinion pitch diameter of gear, pinion addendum, and pitch angle of pinion do = d +2aop cosγ Outside diameter of gear Pitch diameter of gear, gear addendum, and pitch angle of gear Do = D + 2aog cosΓ Pitch apex to crown of pinion Pitch diameter of gear, addendum, and pitch angle of pinion xo = (D/2) - aop sinγ Pitch apex to crown of gear Pitch diameter of pinion, addendum, and pitch angle of gear Xo = (d/2) - aog sinΓ Circular tooth thickness of pinion Circular pitch and gear circular tooth thickness t = p - T Chordal thickness of pinion Circular tooth thickness, pitch diameter of pinion and backlash tc = t - (t3/6d2) - (B/2) Chordal thickness of gear Circular tooth thickness, pitch diameter of gear and backlash Tc = T - (T3/6D2) - (B/2) Chordal addendum of pinion Addendum angle, circular tooth thickness, pitch diameter, and pitch angle of pinion acp=aop + (t2 cosγ / 4d) Chordal addendum of gear Addendum angle, circular tooth thickness, pitch diameter, and pitch angle of gear acg=aog + (T2 cosΓ / 4D) Tooth angle of pinion Outer cone distance, tooth thickness, dedendum of pinion, and pressure angle (3.438/Ao)(t/2)+bop tanφmin
Tooth angle of gear Outer cone distance, tooth thickness, dedendum of gear, and pressure angle (3.438/Ao)(T/2)+bog tanφmin
Spiral Bevel Gear Formulas (20° Pressure Angle, 90° Shaft Angle) To Find Having Formula Pitch diameter of pinion Number of pinion teeth and diametral pitch d = Np / Pd Pitch diameter of gear Number of gear teeth and diametral pitch D = Ng / Pd Pitch angle of pinion Number of pinion teeth and number of gear teeth γ = tan-1(Np / Ng) Pitch angle of gear Pitch angle of pinion Γ= 90°-γ Outer cone distance of pinion and gear Pitch diameter of gear and pitch angle of gear Ao = D / (2sinΓ) Circular pitch of pinion and gear Diametral pitch p = 3.1416 / Pd Dedendum angle of pinion Dedendum of pinion and outer cone distance δp = tan-1(bop / Ao) Dedendum angle of gear Dedendum of gear and outer cone distance δg = tan-1(bog / Ao) Face angle of pinion blank Pitch angle of pinion dedendum angle of gear γo = γ + δg Face angle of gear blank Pitch angle of gear and dedendum angle of pinion Γo = Γ + δp Root angle of pinion Pitch angle of pinion and dedendum angle pinion γr = γ - δp Root angle of gear Pitch angle of gear and dedendum angle of gear Γr = Γ - δg Outside diameter of pinion Pitch diameter, addendum, and pitch angle of pinion do = d +2aop cosγ Outside diameter of gear Pitch diameter, addendum, and pitch angle of gear Do = D + 2aog cosΓ Pitch apex to crown of pinion Pitch diameter of gear, pitch angle, and addendum of pinion xo = (D/2) - aop sinγ Pitch apex to crown of gear Pitch diameter of gear, pitch angle, and addendum of gear Xo = (d/2) - aog sinΓ Circular tooth thickness of pinion Circular pitch of pinion and circular pitch of gear t = p - TThere are four main methods of manufacturing gears. These are metal cutting, casting, forming, and powder metallurgy. Metal cutting is the most widely used process because of its dimensional accuracy. The second two, casting and forming, are used in special circumstances- for example, producing a large gear through casting, which reduces machining expenses by casting closer to the final shape. Another form of casting, known as injection molding, is used to manufacture plastic gears. Forming, on the other hand, can take the form of cold drawing or forging. Cold drawing involves a stock pulled or extruded into a series of dies to form the shape of the gear. Forging presses the stock against dies with the desired tooth configuration. Because of work hardening through continuous deformation, the resulting gear is harder, with a more contoured grain flow.
Gear cutting can be divided into four more classifications summarized below.
Because of the conical shape resulting in a depth and width taper, not all techniques can be applied for bevel gears. For bevel gear cutting, metal cutting techniques can be categorized in to two: face hobbing and face milling.
The use of bevel gears is one of the simplest and most efficient methods of changing a drivetrains‘ axis of rotation. The type of bevel gear and the manufacturing and finishing processes used depend on the type of application. Below are some of the applications of bevel gear systems.
The most popular application of bevel gears is in the differential of an automotive vehicle. The differential is the part of the front or rear axle assembly that allows the wheels to rotate at different speeds. This allows the vehicle to turn corners while maintaining handling and traction. The driveshaft is connected to the hypoid gear assembly, which consists of a pinion and a ring gear. The ring gear is mounted to the carrier with other bevel gears in a planetary gear train.
Bevel gears are used by heavy equipment either for propulsion, the same as an automotive differential system, or for auxiliary units.
Bevel gears are used in the aviation industry for power transmission systems of helicopters and aircraft accessory gearbox drivers.
An example of industrial plant equipment that uses bevel gears is cooling tower fans. The motor is usually mounted at the deck of the cooling tower with the shaft axis oriented horizontally. A gearbox assembly reduces the speed and increases the torque while also reorienting the axis of rotation vertically.
Bevel gears are commonly used in marine transmission as part of the stern drive. There are two bevel gear sets used between the engine and the propeller.
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