This article provides a comprehensive look at bevel gears, include the following information:
A bevel gear is a toothed rotating component that transfers mechanical energy or shaft power between shafts that intersect at an angle, including perpendicular. It changes the axis of rotation and can modify the torque, either increasing or decreasing it, while inversely affecting the angular speed.
A bevel gear resembles a truncated cone, with teeth cut into its lateral surface that mesh with the teeth of other gears. The gear that supplies the shaft power is known as the driver gear, while the gear that receives the power is called the driven gear. Typically, the driver and driven gears have different numbers of teeth to create a mechanical advantage. The gear ratio is defined as the ratio of the number of teeth on the driven gear to the number on the driver gear, while the mechanical advantage refers to the ratio of the output torque to the input torque. This relationship is expressed by the following equation:
The mechanical advantage (MA) is determined by the relationship between various factors including the torques (τb and τa), the radii (rb and ra), and the number of teeth (Nb and Na) of the driven and driver gears, respectively. From the equation, it is evident that increasing the number of teeth on the driven gear results in a larger output torque.
Conversely, a higher mechanical advantage reduces the output speed of the driven gear. This relationship is described by the following 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 output power to input power. This differs from mechanical advantage, which focuses on amplifying forces or torques at the expense of speed. In bevel gears, power loss during transmission is primarily due to friction between the teeth surfaces and the loads on the bearings or housing. The efficiency of various types of bevel gears, compared to other gear types, is summarized in 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 |
Bevel gears come in various types, classified by their tooth profile and orientation. More complex types, such as spiral and hypoid bevel gears, have emerged from advancements in manufacturing processes like CNC machining.
A straight bevel gear is the most basic type of bevel gear, featuring teeth arranged in a straight line that intersects at the gear's axis when extended. The teeth are tapered, with the outer part, or heel, being larger than the inner part, or toe. Straight bevel gears have instantaneous lines of contact, which allows for greater tolerance in mounting. However, they are prone to vibration and noise, limiting their use to low-speed and static loading applications. A common application of straight bevel gears is in the differential systems of automotive vehicles.
Straight bevel gears are also the easiest to manufacture. The earliest method involved using a planer with an indexing head. More efficient manufacturing techniques have been developed, including the Revacycle and Coniflex systems used by Gleason Works.
A spiral bevel gear is a more complex type of bevel gear, distinguished by its curved and oblique teeth. Unlike the straight teeth of bevel gears, the spiral tooth orientation provides more overlap between teeth, leading to gradual engagement and disengagement. This smoother operation results in reduced vibration and noise. Additionally, the increased contact area allows spiral bevel gears to handle higher loads, making them capable of achieving greater load capacities while being smaller in size compared to straight bevel gears with equivalent capacity.
One drawback of spiral bevel gears is the increased thrust load they generate, which necessitates the use of more costly bearings. Typically, a rolling element thrust bearing is required for spiral bevel gear assemblies. Additionally, spiral bevel gears are manufactured in matched sets, and gear sets with the same design are not interchangeable unless specifically made to be interchangeable. These gear sets can be either right-hand or left-hand.
Spiral bevel gear teeth are generally shaped using gear generating machines, a process that ensures high precision and finish. To achieve the desired tooth bearing, lapping is also performed to refine the teeth further.
Zerol bevel gears are a variation of straight bevel gears, developed by Gleason Works. Unlike straight bevel gears, Zerol bevel gears feature teeth that are curved along their length. They resemble spiral bevel gears in profile, but they differ in their spiral angle: Zerol bevel gears have a 0° spiral angle, whereas spiral bevel gears typically have a 35° spiral angle.
Similar to straight bevel gears, Zerol bevel gears do not generate significant thrust loads, allowing the use of plain contact bearings. They can be used as substitutes for straight bevel gears without requiring changes to the housing or bearings. Additionally, the curvature of Zerol bevel gear teeth provides a slight overlap, similar to that of spiral gears, resulting in smoother operation compared to straight bevel gears.
Zerol bevel gear teeth are produced using a rotary mill cutter, which imparts a lengthwise curvature to the teeth. These gears are manufactured with high precision and are often finished through lapping or grinding to achieve the desired surface quality.
A hypoid bevel gear is a specialized type of bevel gear where the shafts are neither intersecting nor parallel. The offset between the two gear axes is referred to as the "offset." The teeth of hypoid bevel gears are helical, similar to those in spiral bevel gears. When a hypoid bevel gear has no offset, it essentially functions as a spiral bevel gear. The manufacturing and shaping processes for hypoid bevel gears are comparable to those used for spiral bevel gears.
Due to the offset, the spiral angle of the smaller gear (pinion) in a hypoid bevel gear set can be greater than the spiral angle of the larger gear. This means that the number of teeth on the gears does not directly correlate with their pitch diameters or theoretical operating diameters. This allows for the use of larger pinions with specific sizes of driven gears, which strengthens the pinion and provides a higher contact ratio with the larger gear. As a result, hypoid gears can transmit more torque and operate at higher gear ratios. Additionally, the offset allows for bearings to be positioned on both sides of the gears since their shafts do not intersect. However, an increased offset can reduce the overall efficiency of the gear system.
Hypoid gears run more smoothly and produce less vibration compared to spiral gears. However, they come with a drawback: the significant amount of sliding that occurs across the teeth's face can increase friction and wear. This necessitates the use of specialized lubricating oils to ensure smooth operation and longevity.
Miter gears are a type of bevel gear with a gear ratio of 1:1, meaning both the driver and driven gears have the same number of teeth. As they do not provide a mechanical advantage, their primary function is to change the direction of rotation. Typically, miter gears have axes that intersect perpendicularly. However, in some assemblies, shafts may intersect at various angles, known as angular miter bevel gears. These angles can range from 45° to 120°. Miter bevel gears can have teeth that are straight, spiral, or Zerol.
To gain a clearer understanding of gears and gear systems, it's essential to familiarize oneself with key terminology. The terms listed below describe various aspects of gears and their tooth profiles and are applicable to all types of gears, not just bevel gears.
The smaller gear in a bevel gear set that drives the larger gear.
The larger gear in a bevel gear set that is driven by the smaller pinion gear.
Also known as circular pitch, this is the distance between corresponding points on adjacent teeth of the same gear.
The diameter of the pitch circle, which is a critical design parameter for determining tooth thickness, pressure angles, and helix angles of the gear.
The ratio of the number of teeth to the pitch diameter of a gear.
The angle between the face of the pitch surface and the axis of the shaft.
The imaginary truncated cone where the base diameter corresponds to the pitch circle.
The angle between the line of force of the meshing teeth and the tangent to the pitch circle at the contact point. For proper meshing, gears must have the same pressure angle. The recommended pressure angle for straight bevel gears is 20°.
The angle between the shafts of the driver and driven gears.
The upper outline of the gear teeth, extending from the pitch circle to the top of the teeth.
The lower outline of the gear teeth, extending from the pitch circle to the bottom of the teeth.
The radial distance between the addendum and dedendum circles. Due to the slight taper of bevel gear teeth, this depth is not constant along the tooth. Addendum and dedendum angles are used to describe the teeth more accurately than the circles.
The angle between the top surface of the teeth (top land) and the pitch surface.
The angle between the bottom surface of the teeth (bottom land) and the pitch surface.
The variation in tooth depth along the face, measured perpendicular to the pitch surface.
The variation in space width along the face, measured on the pitch surface.
The variation in tooth thickness measured on the pitch surface.
The total depth of the teeth plus the clearance value.
The difference between the addendum of one gear and the dedendum of the mating gear.
The space between the mating gear teeth that exceeds their thickness. Different types of backlash are defined based on movement orientation:
The arc along the pitch circle.
The space between the surfaces of mating teeth.
The angular movement described by the backlash.
The linear movement perpendicular to the axis.
The linear movement parallel to the axis.
Backlash is crucial for preventing gear jamming due to contact. It allows lubricants to enter and protect the mating teeth surfaces and accommodates thermal expansion during operation.
The relationship between these terms is illustrated in the table of equations below.
| 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 tanφ
min |
| Tooth angle of gear | Outer cone distance, tooth thickness, dedendum of gear, and pressure angle |
(3.438/Ao)(T/2)+bog tanφ
min |
| 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 - T |
There 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 categorized into four distinct methods, summarized as follows:
Due to the conical shape of bevel gears, which introduces both depth and width taper, not all cutting techniques are applicable. For bevel gear cutting, metal cutting methods are generally classified into two categories: face hobbing and face milling.
Face Hobbing: Face hobbing is a continuous indexing gear generation process. This involves groups of cutting blades that cut all teeth gradually until the desired depth is achieved. As one blade group cuts one tooth, the next blade group enters the next tooth space. The cutting tool and the workpiece rotate simultaneously.
Face Milling: Face milling is a single indexing method where the cutting wheel or tool is fed to cut one tooth space and is then indexed to the next tooth location. The cutting tool and the workpiece are synched together to perform the cut. Each tooth is milled until all teeth are cut to the required depth. Face milling can be done by a two-tool planer, double rotary blade, single row mill cutter, or five-axis CNC milling machines.
Bevel gears offer a straightforward and effective solution for altering the axis of rotation in drivetrains. The choice of bevel gear type, as well as the manufacturing and finishing techniques, depends on the specific application. Below are some common 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 utilized in heavy machinery for both propulsion, similar to an automotive differential system, and for driving auxiliary units.
In the aviation industry, bevel gears are employed in power transmission systems for helicopters and aircraft accessory gearboxes.
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.
In marine transmissions, bevel gears are frequently utilized as part of the stern drive system. Typically, two bevel gear sets are employed between the engine and the propeller.
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