Bevel Gears

Chapter 1: What is a Bevel Gear?

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. At its lateral side, teeth are milled which interlock to other gears with its own set 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 are 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 mechanical advantage is the ratio of the output torque to the input torque. This relationship is shown by the following equation:

MA is the mechanical advantage, τb and τa are the torques, rb and ra are the radii, and Nb and Na are the number 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 gears output speed. This is expressed by the equation:

ωª and ωb are the driver and driven gears’ angular speed, respectively. In general, a gear ratio of 10:1 is recommended for a bevel gear set. For increasing the speed of the driven gear, 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 its 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 its optimum position for best performance.

Chapter 2: Efficiency of a Bevel Gear

Efficiency is defined as the ratio of the output power to the input power. Note that this is different with mechanical advantage that is concerned with the amplification 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. Efficiency of different types of bevel gears compared with other types are 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

Chapter 3: Types of Bevel Gears

There are different types of bevel gears according to their tooth profile and orientation. The more complicated types such as the spiral and hypoid bevel gears resulted from further development of manufacturing processes such as CNC machining.

• Straight Bevel Gears

  This is the simplest form of a bevel gear. The teeth are in a straight line which 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 in using this type is the vibration and noise. This limits straight bevel gears to low-speed and static loading applications. Common application of straight bevel gears are 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.

  

  
  

• Spiral Bevel Gears

  This is the most complex form of bevel gears. 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 in size 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 purposely built to. Spiral bevel gear sets are made either right-hand or left-hand.

  Spiral bevel gear teeth are typically shaped by gear generating types of 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.

• Zerol Bevel Gears

  This type is a modification of a straight bevel gear trademarked by Gleason Works. Zerol bevel gears have teeth curved in the lengthwise direction. These gears are also somewhat similar to spiral bevel gears in terms of its profile. Their difference is the spiral angle; Zerol types have 0° spiral angles while spiral types have 35°.

  

  
  

  Like the 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 its 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 makes the lengthwise curvature of the tooth. Zerol bevel gears are cut at a high precision, often finished by lapping or grinding.

• Hypoid Bevel Gears

  This is a special type of bevel gears where the axes of the shafts are not interesting 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. 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 are 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 have a higher contact ratio to the larger gear. In turn, it allows hypoid gears 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 being increased.

  

  
  

  Hypoid gears operate smoother with minimal vibration than spiral gears. The downside of using spiral gears, aside from the efficiency issue mentioned earlier, is the high sliding that takes place across the face of the teeth. This means special lubricating oils must be used.

• Miter Bevel Gears

  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. The purpose of this type is limited to changing the axis or rotation. It does not produce any mechanical advantage. 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.

  

  
  

Chapter 4: Geometry and Terminologies

To better understand gears and gear systems, one must first look at its terminologies. 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.

Pinion

The smaller bevel gear in a bevel gear set.

Gear

The larger bevel gear in a bevel gear set.

Pitch

Also known as circular pitch, is the distance from one point on a tooth to the corresponding point of the adjacent tooth on the same gear.

Pitch diameter

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.

Diametral pitch

The ratio of the number of teeth and the pitch diameter.

Pitch angle

The angle between the face of the pitch surface and the shaft axis.

Pitch surface

The imaginary truncated cone wherein the base diameter is the pitch circle.

Pressure angle

A predefined value which 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 angles for straight bevel gears is 20°.

Shaft angle

A predetermined value that defines the angle between the driven and driver shafts.

Addendum

The upper outline of the gear teeth.

Dedendum

The bottom outline of the gear teeth.

Total depth

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.

Addendum angle

The angle between the face of the upper surface of the teeth or top land and the pitch surface.

Dedendum angle

The angle between the bottom surface of the teeth or bottom land and the pitch surface.

Depth of taper

The change in tooth depth along the face measured perpendicular to the pitch surface.

Space width taper

The change of the space width along the face measured on the pitch surface.

Thickness taper

The change of tooth thickness measured on the pitch surface.

Working depth

The total depth of the teeth plus the value of the clearance.

Clearance

The difference between the addendum of a gear to the dedendum of the mating gear.

Backlash

The amount of space that exceeds the thickness of a mating gear tooth. For bevel gears, there are different types of backlash depending on orientation of the movement. These are:

• Circular

  The arc along the pitch circle

• Normal

  The space between the surface of the mating teeth

• Angular

  The described as the angular movement

• Radial

  The linear movement perpendicular to the axis

• Axial

  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 thermal expansion during operation.

The relationship between these terms are shown by 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 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

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 - T

Chapter 5: Manufacturing Processes

There are three main methods of manufacturing gears. These are metal cutting, casting, and forming. Metal cutting is the most widely used process because of its dimensional accuracy. The other two, casting and forming, are used in special circumstances such as 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 be cold drawing or forging. Cold drawing involves a stock to be 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.

• Rotating threaded tool: hobbing, generating
• Rotating and reciprocating tool: shaping, shaving, generating
• Rotating disc wheel: milling, form grinding, thread grinding
• Linear motion tool: broaching, punching

Because of its 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 into two: 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 two-tool planer, double rotary blade, single row mill cutter, or five-axis CNC milling machines.
  
  

  

  
  

Chapter 6: Bevel Gear Applications

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 manufacturing and finishing processes used depends on the type of application. Below are some of the applications of bevel gear systems.

Automotive

The most popular application of bevel gears is 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 consisting of a pinion and a ring gear. The ring gear is mounted to the carrier with other bevel gears in a planetary gear train.

Heavy Equipment

Bevel gears are used by heavy equipment either for propulsion, the same as an automotive differential system, or for auxiliary units.

Aviation

Bevel gears are used in the aviation industry for power transmission systems of helicopters and aircraft accessory gearbox drivers.

Industrial Plant Equipment

An example of an industrial plant equipment that uses bevel gears are 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.

Marine

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.

Conclusion:

• Bevel gears are rotating machine elements used to transmit mechanical power between two intersecting shafts, either perpendicular or at an angle. Aside from changing the axis of rotation, bevel gears can also produce a mechanical advantage by increasing the output torque.
• Producing a mechanical advantage, however, decreases the angular speed of the driven shaft. Thus, bevel gears can also be used as speed reduction mechanisms.
• Efficiency is the ratio between output power and input power. Power loss from bevel gears are mostly due to friction from sliding contact. This is then dissipated as heat which is usually removed by lubricating oils.
• Bevel gears are classified according to the tooth profile and orientation. The types of bevel gears are straight, spiral, Zeroil, and hypoid.
• Efficiencies of bevel gears range from 97-99.5%, except for the hypoid bevel gears with an efficiency of 90-98%. A larger offset of a hypoid gear causes further decrease in efficiency.
• There are many terms used to describe gears. The most important for bevel gears are the pitch diameter, pressure angle, shaft angle and number of teeth. These are the key values which will define the geometry of the gear.
• There are three main methods of manufacturing gears: cutting, casting, and forming. Among the three, cutting is the most widely used.
• Gear cutting is further broken down into several methods. One is by using a rotating threaded tool such as a hob. Next is by using a rotating or reciprocating cutting tool that mates together with the gear blank. Third is cutting using a rotating disc wheel as seen from milling processes. And lastly, gear cutting using a linear shaper or broaching tool.
• The most popular application of bevel gears is the automotive differential. This is seen not only on automotive vehicles, but also in light and heavy equipment. Other main uses are in the aviation and marine industry.