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Gear Nomenclature

Gear Nomenclature - For Mechanical Engineers

This page aims to present basic gear nomenclature.
When you click on the alphabetically listed gear related words, you can confirm the meaning of each terminology.

Index :

A
Abrasive wear
Accuracy of gears
Active flank
Addendum flank
Addendum
Adhesive wear

B
Backlash
Base circle
Base cylinder / Base diameter / Base circle
Bevel gears
Bottom land

C
Calculation of gear strength
Carburizing
Center distance
Chamfering gear
Clearance
CP rack and pinion
Crowning
Cutter module

D
Drive and driven gears
Dedendum
Dedendum flank

E
Effective facewidth
End relief
External gear

F
Face width
Fillet
Flank line

G
Gearbox
Gear drawing
Gear forging and casting
Gear honing
Gear lapping
Gear lubrication
Gear material
Gear rack
Gear ratio
Gear rolling
Gear shaving
Gear strength
Gear train
Gear tooth profile modifications
Grinding burn
Grinding crack
Ground gears

H
Heat treatment machine for gear
Heat treatment of gears
Helical gear
Helical rack
Hob

I
Idler gears
Induction heardening of gears
Internal gear
Intersecting shaft gears
Involute curve / Involute gear
Involute tooth form

L
Lead angle
Length of path of contact

M
Measurement over balls
Measurement over rollers
Miter gear
Module

N
Nominal pressure angle
Nonintersecting and nonparallel shaft gears
Normal chordal tooth thickness
Normal space width
Normal module
Normal pitch
Normal pressure angle
Normal profile
Normal space width
Normal tooth thickness
Number of teeth

O
Overlap angle / Overlap ratio
Overlap length

P
Pair of gears
Parallel shaft gears
Pinion
Pinion and wheel
Pinion cutter
Pitch
Pitch circle / Pitch cylinder / Pitch point / Pitch surface
Pitch circle diameter
Planetary gear drive system
Polishing
Pressure angle
Profile shift
Profile shift coefficient

R
Rack and pinion
Rack type cutter
Referene circle / Reference cylinder / Reference diameter / Reference surface
Root circle
Root cylinder
Root diameter
Root relief
Root surface

S
Scratching
Screw gear
Scuffing
Shaft angle
Span thickness of gear
Semi-topping
Speed reducing ratio / Speed increasing ratio
Spur gear
Spiral bevel gear
Stainless steel gears
Straight bevel gear

T
Tempered gears
Thrust
Tip circle
Tip cylinder
Tip diameter
Tip relief
Tip surface
Tool pitch
Tooth
Tooth depth
Tooth flank
Tooth profile
Tooth space
Tooth tip
Tooth tip interference / Meshing interference
Top land
Topping
Total angle of transmission / Total contact ratio
Transverse angle of transmission / Transverse contact ratio
Transverse module
Transverse pitch
Transverse pressure angle
Transverse profile
Transverse space width
Transverse tooth thickness
Twist angle (Helix angle)

U
Undercut
Usable flank

W
Wear
Worm
Worm gear

Contents :

Abrasive wear

Abrasive wear is called abrasive wear, which is caused by a cutting action caused by a hard object such as a foreign matter, or an action in which a hard gear shaves a soft tooth surface due to a difference in the height of the tooth surface.
In particular, the former is called three-way wear, and the latter is called two-way wear.
Abrasive wear is caused by the cutting action which causes the grooves on the line to run irregularly in the sliding direction.

Accuracy of gears

A rattle or a loud sound may cause trouble when a gear is used to transmit rotational or power. In addition, if the tooth feel is poor and wear of the tooth surface progresses, or if the tooth is fixed in the worst case, the function of the gear cannot be fully exerted. In order to suppress such problems, it is necessary to improve the accuracy of the gears.
Gear accuracy can be classified into three main types: (1) tooth profile accuracy, (2) tooth line accuracy, and (3) assembly accuracy.
(1) Tooth profile accuracy includes tooth profile accuracy for examining tooth profile accuracy, tooth groove deflection for checking deflection when machining teeth, and single/accumulated pitch error for examining tooth pitch accuracy. In order to suppress rotational backlash and noise generation, it is necessary to improve the accuracy of these components.
(2) Tooth line accuracy affects tooth feel. Poor tooth contact can lead to defects on the tooth surface (pitting, spolling, sticking, etc.), leading to serious defect events.
(3) Assembly accuracy includes the center distance accuracy of the gears and the degree of parallelism of the axes. The center distance accuracy greatly contributes to the backlash, and if the backlash is large, the meshing ratio of the teeth decreases and the rotational play increases. Conversely, if the backlash is small, the gear may not turn. Poor shaft parallelism can cause a change in the engagement ratio during one revolution, resulting in rotational play and noise.

Active flank

The active flank is the part of the tooth flank that is in contact when two gears mesh.

Addendum flank

The addendum flank is the portion of the tooth flank between the reference surface and the tooth tip surface.

Addendum

The addendum means the part which is from the reference surface to the tooth crest for the tooth of the gear. In addition, the addendum means the value which the addendum root circle radius minus the reference circle radius.

Adhesive wear

Wear in which the contact portion of the gear is in solid contact and the surface is roughened by repeated adhesion and peeling is called adhesion wear.
Adhesive wear is primarily seen in low speed operating environments.
Examples of suppression of adhesion wear include optimization of lubrication, addition of an extreme pressure material to lubricating oil, surface modification such as coating, change of gear material, tooth surface roughness, tooth shape modification, and the like.

Backlash

The space created between the tooth surfaces of corresponding mating gears is called the backlash. It is necessary for the rotation as without it, heat seizure due to the insufficient lubrication and tooth interference are likely to occur. However, increasing the backlash may cause more vibration and noise. Therefore, it is necessary to specify the backlash appropriate for the application.

Backlash

Base circle

Base circle of a gear is the base circle for the involute curve. An involute curve is the trace of a point at the end of a taut string that unwinds from a cylinder, and this cylinder is called the base circle. The tangent lines to the involute curve and the base circle have the characteristic of being always perpendicular to each other. As a result, the direction of the force acting at the contact point of the gears with involute curves is constantly along the common tangent to the two base circles.

The base circle diameter is given by the equation db = d cos α.

Furthermore, the examples of gear terminology related to pitch circle include pitch circle diameter, involute curve, pressure angle, etc.

Working Gear Nomenclature
Involute curve
Meshing of involute gear

Base cylinder / Base diameter / Base circle

When drawing an involute curve, the circle which acts as the foundation for the curve is called the base circle. For cylindrical gears, the curve connecting the base circles and perpendicular to the gear axis is called the base cylinder. The base diameter db is determined by the following formula using the pitch circle diameter d and the standard pressure angle α:

db = d cos α

Involute curve

Bevel gears

A gear pair having a crossing axis and a reference plane having a conical surface is called a bevel gear. The bevel gear is used when the power transmission direction is to be changed (typically 90 degrees).

Bottom land

In the tooth groove, the part between the two fillets is called the bottom land.

Bottom land / Fillet / Root Cylinder

Calculation of gear strength

Gear damage is divided into four main types: 1.tooth breakage, 2.tooth surface fatigue, 3.scoring and 4.wear.
In order to prevent these four damages at the design stage, it is necessary to calculate the strength of the gear.
1. Tooth fracture is caused by two main factors: one-shot fracture due to an impulsive overload on the tooth and the other fatigue fracture due to a variable load. These are evaluated by calculating the stress acting on the dangerous section as the bending load. Risk sections for assessing tooth breakage include the LEWIS method and the HOFER method.
2. Fatigue on the tooth flank is assessed by the surface pressure on the tooth flank. Since the gears are in contact with each other on curved surfaces, calculations are performed based on the contact stress of Hertz.
3. Scoring is more likely to occur during high-speed rotation due to welding of metal caused by breakage of the lubricant film in the tooth surface contact area. Equations for scoring include the ALMEN equation BLOK and the NIEMANN equation.
4. Wear is also affected by metal contact due to oil film breakage, which is particularly likely to occur under high load conditions. Wear is also evaluated based on the surface pressure generated on the tooth surface, but is difficult to predict.

Carburizing

Carburization quenching is a type of heat treatment used for gears.
Carburization is an operation in which carbon is impregnated into the surface of a steel material having a small amount of carbon component to bring the steel material into a high carbon state. A series of processes for hardening a surface layer by carburizing a steel material, followed by quenching and tempering is called carburizing and quenching. When the carburizing hardening is performed, the hardness of the surface portion of the steel material having a high carbon concentration increases, while the hardness of the inside remains low, and as a result, the steel material becomes tough and rich in high abrasion property. When the carburizing and quenching is performed, it is also possible to prevent curing of only a part of the target portion by using a carburizing preventive agent. In addition, since the gear after the carburizing hardening is deformed, if high accuracy is required for the gear, grinding finishing of the gear is separately required.

Center distance

When a pair of gears mesh, the distance between the centers of the meshing gears' shafts is called its Center Distance and it is calculated as the half of the sum of two gears' pitch diameters.

Formula for Center Distance : a = (d1 + d2) / 2

Working Gear Nomenclature

Chamfering gear *

Chamfering the edge face of a gear includes deburring, round chamfering and square chamfering as in pic 13.7. Deburring is to remove burr generated by gear cutting and to trim off any sharp edges, while round chamfering and square chamfering are used to make meshing of gears smoother when switching meshing by sliding gear.

The machine tool for such processes is a gear tooth chamfering machine, which usually uses a pencil-like or hollow mill cutter, but sometimes hob forming tool is used.

Pic 13.7 Chamfering gear (From left to right : Deburring / Round chamfer / Square chamfer)

Clearance

The distance between the tooth top land of one gear and the root of the mating gear measured on the centerline connecting the gears is called the clearance.

CP rack and pinion

CP stands for the abbreviation of Circular Pitch. Normally, the pitch of a gear is determined by πm. In CP rack and pinion, the value of the module is adjusted so that the pitch becomes a good number (for example, an integer).

Crowning

Similar to above tooth profile modifications, there is a crowning method to improve meshing when gears deflect. Crowning is a process whereby the straight tooth surface parallel to the shaft is modified to a convex arc surface as shown in Figure 3.10.

It is effective when, due to load on the gear, the gear shaft deflects causing edge contact (the phenomenon of contacting only one edge of the tooth surface).

Again, caution should be exercised to avoid crowning more than necessary as it will shorten the life of the gear.

Edge contact
Crowning and end relief

Cutter module

The value obtained by dividing the tool pitch by the circumferential ratio π is called the tool pitch.

Drive and driven gears

Of the pair of gears, the gear that rotates the other gear is called the drive gear while the gear being rotated by the other gear is called the driven gear. The drive gear is connected to the input shaft and the driven gear is attached to the output shaft.

Dedendum

The tooth root means the part which is from the reference surface to the bottom land for the tooth of the gear. In addition, the dedendum means the value which the reference circle radius minus the dedendum circle radius.

Dedendum flank

The dedendum flank is the portion of the tooth flank between the reference surface and the root surface.

Effective facewidth

The portion of the face width which is in mesh with the mating gear is called the effective face width of the gear. It is mainly used to calculate the tooth surface strength.

End relief

Somewhat similar to crowning, there is the end relief method. The objectives and the effects are the same as in crowning.
While crowning draws a gentle convex surface, end relief involves shaving of both ends of the tooth surface.
Because the removal produces a flat surface it is not as effective as crowning, but it has the benefit of technically easier production.

External gear

The external gear means the gear which the diameter of the tooth crest is bigger than the diameter of the bottom land, and engaging by 2 gears is circumscribed. The rack is treated as the external gear.

Face width

The length of gear tooth is called its face width.

Working Gear Nomenclature

Fillet

The surface between the tooth surface and the bottom land is called the fillet.

Bottom land / Fillet / Root Cylinder

Flank line

The flank line is the line along the face width of the tooth at which the reference surface intersects the tooth flank. The workmanship of the flank line greatly influences the tooth contact so that its accuracy becomes an important inspection item in gear designs. While the flank line of spur gears matches the gear’s direction of its rotational axis, the flank line of helical gears does not match the rotational axis due to its twisting angle.

Gearbox

A gearbox is a gear system that converts input such as rotational speed, rotational direction, torque, etc. into an appropriate one through several stages of gears and outputs the same. Gearboxes range from simple gears and housings to large gears (e.g., automobile transmissions) with gears, shafts, bearings to support shafts, and transmission mechanisms.

Gear drawing

When preparing a drawing of a gear, the shape, dimensions, tolerances, etc. of the gear are shown, and the gear specifications such as module, number of teeth, material, accuracy grade, displacement amount, finishing method, pressure angle, tooth shape, etc. are described. In addition, the drawings shall also include matters relating to processing, assembly, inspection, heat treatment, hardness, etc., as necessary. When the gear is illustrated, it is necessary to appropriately use a line such that the tip circle is represented by a thick solid line, the reference plane is represented by a thin dashed-dotted line, or the like.

Gear forging and casting *

Forging with a forging die (mainly for bevel gearing) and casting with die-cast (mainly for nylon gearing) are partly used for mass production.

Gear honing *

Gear honing is highly-efficient processing method, where gear-shaped grind stone is used instead of gear-shaped shaving cutter. This method is used for finishing quenched gears.
The gears used for cutting, shaving and honing are expected to have undercut in the preprocessing process.

Gear lapping *

Rotating the meshing part of a pair of gears greased with oil dissolved with polishing powder to finish the place where the contact pressure is high to improve the gear’s contact is called lapping, and the machine that performs lapping is called a gear lapping machine. However, pitch error is hardly corrected with this method.

Gear lubrication

The main purpose of lubricating the gears is as follows :
- Reducing wear of tooth flanks by improving slippage between tooth flanks
- Cooling the tooth flank to suppress temperature rise due to heat generation
- Prevention of noise and vibration between tooth flanks
As a result, lubrication of the gears can prevent damage to the gears and extend their life.
In order to lubricate the gear correctly, it is important to select the lubricating method and the lubricating oil. The main lubrication methods for gears are oil bath type (splash type), forced lubrication type, grease filling type, etc. In addition, viscosity, extreme pressure, thermal and oxidative stability, water separation, and antifoaming properties are important in selecting lubricating oils.

Gear material

There is no particular choice of material for the gear, and the choice will depend on the application. In general, iron-based materials such as steel and cast iron, non-ferrous metal materials such as bronze, and resin-based materials such as duracon and nylon are used. These are greatly contributed from the viewpoints of good slidability and ease of production. For this reason, materials with high seizure properties, such as aluminum and titanium, are not generally used because of the occurrence of tooth surface galling and the like.
Among iron-based materials, carbon steel and alloy steel are used for high hardness and strength, stainless steel is used for corrosion resistance, and cast iron is used for easy manufacture (mass production or large scale) rather than precision.
Resin-based materials are used under light load, and they are lightweight and do not rust. Nylon-based products can be produced in large quantities at very low cost.

Gear rack

A gear rack is categorized into a parallel shaft type gear with straight tooth trace and rod-shaped body.
By meshing with a pinon, it can transform a rotational motion into a linear motion. There also exists a helical rack which has its tooth trace at a diagonal.

Gear ratio

Gear ratio is determined by the following formula where the number of teeth of the large gear is Zb and the number of teeth of the small gear is Zs :

U = Zs / Zb (numerator: small gear, denominator: large gear)

When the number of rotations of the large gear is Nb, and the corresponding number for the small gear is Ns, the rotational ratio and the gear ratio have the following relationship :

Nb / Ns = Zs / Zb

Also, if the torque occurring in the large gear is Tb and the same in the small gear is Ts, the following relationship exists :

Ts / Tb = Zs / Zb

Values similar to the gear ratio are speed reducing ratio and speed increasing ratio.

Gear rolling *

Gear rolling is the new method to manufacture gears. In this method, tooth profile is formed by pressing a gear-shaped die against rotating gear material (plastic processing method). This method can be divided into free rolling whose rotation of die is free, and forced rolling where rotating ratio of die and gear material is fixed (equal to the ratio of number of teeth of gear and die). There are also cold rolling and hot rolling where material is heated by high frequency heat. This method is high in production efficiency, relatively high in pitch precision, and excellent in the wear resistance of the tooth flank, but the kind of material to be used for gear is limited.

Gear shaving *

Shaving is commonly used to finish gears precisely and productively. In this method, a quenched polishing gear with many narrow grooves on the tooth flank to use as a blade (also known as rack) and is used as a cutter (see the shaving cutter in pic 13.4), and it meshes with machined gear in relation to that of a screw gear as in pic 13.5 and cuts using a relative slipping motion and finishes the tooth flank in a short time generating threadlike chips. In many cases, crowning is also performed at the same time as shown in pic 13.6. However, this method is not used for the materials whose surface hardness is great. The maximum suitable hardness is a Rockwell hardness of 38.

Pic 13.4 Shaving cutter’s blade
Pic 13.5 Shaving of helical gear
Pic 13.6 Crown shaving

Gear strength

The calculation of the strength of the gear required for the design of the gear is generally performed by calculating the "bending strength of the tooth" and the "tooth surface strength". At the same time, the strength against burn-in (scoring strength) may be obtained. In this case, a calculation formula called a Lewis formula for the bending strength of the tooth and a Hertz formula for the tooth surface strength is widely used. The defects caused in the teeth of the gear due to the insufficient strength include breakage when the bending strength of the teeth is insufficient, wear of the tooth surface when the tooth surface strength is insufficient, and the like.

Gear train

Gear trains involve multiple pairs of gears in mesh.

Gear tooth profile modifications

The gears with the correct involute tooth forms can transmit power efficiently and correctly. However in reality, due to the fact gears and their surrounding components are not rigid bodies but elastic and are subject to deflection by load, the gear mesh may not always correspond to the theoretical positions.  When this happens, it may lead to a shortened life and increased noise.  However, by removing a part of the traditional involute tooth form by chamfering, it is possible to avoid these effects of changes in the mesh due to tooth deflection.

This type of modification is useful in reducing these problems, but too much adjustment may worsen the mesh and therefore caution is advised.

There are modifications of tooth tips and tooth roots, but generally tooth tip modification is more commonly used.

Tooth profile modification

Grinding burn

The discoloration phenomenon caused by the formation of an oxide film on the processed surface by the heat at the time of grinding and the interference of incident light is called grinding burning.
The color tone changes depending on the temperature generated during grinding.
Since the hardness is lowered in the portion where the grinding burn occurs, the portion is liable to become a damage starting point.
In order to suppress grinding burn, it is necessary to suppress heat generation by performing sufficient lubrication or using a grindstone having a good cutting quality.

Grinding crack

The phenomenon of surface cracking occurring at the time of grinding or after grinding is called grinding cracking.
It is characterized by drawing a fine pattern of cracks with regularity.
Grinding cracking is caused by an increase in temperature during grinding, and may cause grinding cracking if the grinding burn becomes severe.

Ground gears

A gear with a ground tooth surface is called ground gear.
Tooth grinding makes it possible to improve the accuracy of teeth and the mirror surface of tooth surfaces, resulting in low vibration, low sound, and improved fatigue strength. The tooth flank is also ground to remove distortion after heat treatment.

Heat treatment machine for gear *

Heat treatment machine which performs heat treatment evenly and productively while reducing distortion during quenching requires high technology. Some heat treatment machines specialize in the processes.

Heat treatment of gears

When steel is used for gears, heat treatment is generally performed to obtain a metallographic structure suitable for use.
The higher the carbon content of steel, the easier it is to burn, and the harder it is to harden and increase its strength.
However, in this state, stress is generated inside the metal, which may cause problems such as residual strain, brittleness, and tendency to rust.
Therefore, appropriate tempering is required.
Hardening includes nitriding, carburizing, high-frequency hardening, flame hardening aimed at local hardening, subjective hardening aimed at overall hardening, and tempering.

Helical gear

A helical gear belongs to a parallel shaft type gear whose tooth trace is twisted.
It has a low noise level due to its twisted tooth trace and is suitable for high speed rotation. However, it is necessary to take measures as it creates axial thrust force.

Helical rack

A rack whose teeth are cut obliquely to the direction of travel is called a helical rack. The helical rack is used in combination with a helical gear.

Hob

A hob is a cylindrical tool with cutting edges on its outer surface which is used to cut gear teeth. A hob is attached to a gear cutting machine called a hobbing machine and cuts teeth on a gear blank as they rotate. A hob is used in the gear form generating method and can cut spur and bevel gears and worm wheels.

When gear cutting, different hobs are in general needed depending for example on the desired gear’s pitch (module, etc.), pressure angle, and the size of the hobbing machine. However, for different number of teeth, it is theoretically possible to use the same hob.

Examples of the practical use of the term “hob” are :
“To cut gears to match module 2, pressure angle 20˚, we prepared the appropriate hob to cut the gears.”
“The cutting edges of the hob became worn and we re-sharpened the hob.”

Other terms related to “hob” include hobbing machine, pinion cutter, carbide hob, form generating gear cutting, etc.

Image of hob

Idler gears

A gear used to adjust rotational direction or the center distance between two gears without affecting the speed ratio is called an idler gear. If three gears are arranged as shown in the diagram below and if the right gear 1 is chosen as the driver and the left gear 3 is the driven gear, then the middle gear 2 becomes the idler gear.

It appears at first glance that Gear 2 would affect the speed reduction ratio. However it becomes clear that Gear 2 is an idler gear if the speed reduction ratio is actually calculated. If we specify the overall speed reduction of the gear mechanism as i, then i = (Z2 / Z1 ) x ( Z3 / Z2 ) = ( Z3 / Z1 ). Thus, the number of teeth in Gear 2 does not affect the overall speed ratio. The reasons for inserting an idler gear have been mentioned before, but in what kind of situations are idler gears used? One most frequently used example is the planetary gears in a planetary gear mechanism. Fix the planetary gear carrier D, and specify the input shaft as the sun gear A and the output shaft as the internal gear C. In that case, the planetary gears become the idler gears and can transmit force without affecting the speed reduction ratio.

Single Stage Gear Train with an Idler Gear
An example of a planetary gear system

Induction heardening of gears

A high-frequency induction heater is used to harden only the tooth surface of the gear. For hardening, carbon-containing materials such as S35 to 50C and SCM435 to 445 are used. Because carbon is diffused throughout, the hardened layer is thicker than carburized hardening. The hardness of the induction hardening is about 50 to 60 HRC on the Rockwell C scale.

Internal gear

An internal gear has the teeth cut on the inside of a cylinder and meshes with an external spur gear.
It is frequently used in planetary gear mechanisms, but sometimes it causes interference when meshing with an external gear.
In general use, the external gear usually drives the internal gear.

Intersecting shaft gears

Intersecting shaft gears have the pair’s shafts intersecting each other. They are used to change the direction of power transmission. They are applicable to bevel gears.

Involute curve / involute gear

When a string wound on a circle is unwrapped under tension, the trajectory traced by a fixed point on the string is called the involute curve. When the gear tooth is formed using the involute curves, it is called the involute gear.

Involute curve

Involute tooth form

The involute tooth form is the most general gear tooth form. The involute curve is the trace that the end of a taut string produces as it is unwound from a cylinder, and the gear tooth whose cross section is the involute curve is called the involute tooth form. When two gears with involute curves mesh together, the contact point moves along the common tangent of the two base circles. Also, because the contact surface is always perpendicular to the common tangent, it has the feature of having the direction of the acting force along the same common tangent. That is to say, with involute shaped gears, the force is transmitted in one direction securely from the beginning to the end of the mesh. The characteristics of the involute gears include that even with some errors in the center distance, they can still mesh and they are easy to generate. Especially regarding the center distance, there are times when the gear shaft deflects due to the transmission of power or when the cumulative tolerance of surrounding components builds up, the ability to mesh correctly with some center distance errors is an important  characteristic.

It should be noted that examples of the related terminology to involute gear forms can be listed as cycloid gear form, trochoid gear form,  etc.

Involute Curve

Lead angle

The spiral curve wrapped around a gear cylinder or cone and having a certain fixed incline angle relative to the gear axis is called a helix curve. This incline angle is called the lead angle β.

Length of path of contact

When two gears are meshing, the length of the line of action between the respective tooth tip circles is called the length of path of contact.

Measurement over balls

When the number of teeth is an even number, balls determined from the tooth profile are applied to two tooth grooves that can form a diameter, and when the number of teeth is an odd number, balls are applied to two tooth grooves that are closest to the diameter, and the distance measured through the two balls is called an overball dimension. It is used in gears that are not very large.

Measurement over rollers

The overpin size is measured by replacing the ball used for the overball size with a roller (pin).

More details: TOOTH THICKNESS

Miter gear

A miter gear is a bevel gear and when used with another bevel gear of the same number of teeth (speed ratio of 1), it is called a miter gear.
When its teeth are twisted, it is called a spiral miter gear.

Module

The unit to indicate the size of gear teeth is called the module. Module m is determined by the pitch diameter d and the number of teeth z.
m = d / z
Generally, the selection of the size of the module is made from among the standardized list.

Comparative size of various rack teeth

Nominal pressure angle

The angle representing the inclination of the rack tooth shape is called the tool pressure angle. It seems that the tool pressure angle is generally set to 20 deg.

Nonintersecting and nonparallel shaft gears

These are gear pairs where the shafts are crossed but offset from each other. Screw gears and worm gears belong to this group.

Normal chordal tooth thickness

Normal chordal tooth thickness is the shortest distance between intersection points of reference circle on normal plane and both sides of tooth surface.

Normal chordal tooth thickness

Normal module

Normal module is the module seen from normal plane, and normal pitch is divided by pi. Transverse module ms and normal module mn are linked by following expression :

mn / ms = tan β
(β = twisting angle on pitch circle)

Normal pitch

Normal pitch is the pitch on normal plane corresponding to basic rack which defines tooth profile of helical gear.

Normal pressure angle

Normal pressure angle is formed by normal tooth profile edge and reference circle radius on the point where normal tooth profile intersects reference circle. To define a criterion for rack, pressure angle on normal plane of basic rack that defines tooth profile of helical gear is called normal pressure angle.
Transverse pressure angle αs and normal pressure angle αn are linked by following expression :

tan αs = tan αn / cos β
(β = twisting angle on pitch circle)

Therefore, transverse pressure angle ≥ normal pressure angle.

Normal pressure angle

Normal profile

Normal profile is tooth profile curve appears when cutting at right angle to tooth trace. This term is mainly used for gear such as helical gear whose tooth trace is twisted. When cutting pitch cylinder of helical gear at right angle to tooth, its trajectory becomes oval, and this is the characteristic of normal profile.

Normal space width

Normal space width is arc length between intersection points of reference circle on normal plane and two tooth faces which face tooth space.

Normal tooth thickness

Normal tooth thickness is arc length between intersection points of reference circle on normal plane and both sides of tooth surface.

Number of teeth

The number of tooth forms placed around the circumference of a gear is called the number of teeth of the gear. For example, the number of teeth of the figure below is 10.

Number of teeth

Overlap angle / Overlap ratio

Considering one tooth of a screw gear, the angle formed by the surface containing both ends of the standard tooth trace (intersection of tooth face and standard surface) and the cross section normal to the gear shaft is called the overlap angle. The overlap angle is equal to the value of the total angle of transmission less the transverse angle of transmission.
Also, the value obtained by dividing the overlap angle with the matching angle is called the overlap ratio.

Overlap length

On screw gears, if the face width is wf and the helix angle is β, then wf tan β is called the overlap length. The overlap length for spur gears is 0.

Pair of gears

A pair of gears is a mechanism in which one gear pushes the other by sequentially engaging their teeth as shown in the figures below.

Pair of Gears (gears)
Pair of Gears (rack and pinion)

Parallel shaft gears

Pairs of parallel shaft gears have their shafts parallel to each other. The rotation of the input shaft is opposite to that of the output shaft. Spur and helical gears belong to this group.

Pinion and wheel

Of the pair of gears, the smaller gear is called a pinion and the larger one a wheel.

Pinion

Of the pair of gears, the gear with the smaller number of teeth is called the pinion (in this case, the gear with the larger number of teeth is called the gear). In a rack and pinion mechanism, a cylindrical gear that meshes with a rack is referred to as a pinion.

Pinion cutter

A pinion cutter is a cutter in the shape of spur gear. In other words, it’s considered to be a gear-shaped rack cutter. Whereas a rack just moves in parallel vertically, a pinion cutter rotates while moving up and down, and forms gears with tooth profile which meshes with cutter accurately. Another big feature is that the pinion cutter can form internal gears (ring gears) which have gear teeth on the inside of the cylinder.

Pitch

The space between the adjacent teeth is called the pitch. The pitch p is obtained by multiplying module m by π.
p = π m

Pitch

Pitch circle / Pitch cylinder / Pitch point / Pitch surface

The point of contact of a pair of gears in mesh where the angular velocity ratio is determined is called the pitch point. In parallel axis or intersecting axis gears, the pair makes the rolling contact at the pitch point. The trajectory gained by rotating the pitch point around the respective rotary axis of gears is called the pitch circle. From this definition, the contact point of the pitch circles is the pitch point. In addition, the curved surface that contains the pitch circle on which the gears have the rolling contact is called the pitch surface. In particular, the pitch surface of cylindrical gears is called the pitch cylinder.

Pitch circle diameter (PCD)

Standard pitch circle is called standard circle or pitch circle and its diameter is obtained as the product of the number of teeth and the module. The pitch circle diameter is d = zm (where z = number of teeth, m = module). The pitch circle diameters are important numbers in locating the shafts of two mating gears. The center distance of two meshing gears is expressed as the sum of the pitch circle radii of the two gears. Since the module of the meshing gears are the same, center distance a = (d₁ + d₂)/2 = (z₁ m + z₂ m)/ 2.

The pitch circle of a gear does not physically exist as a form. Consequently, it is not possible to measure it with an instrument. There are various opinions, but for spur gears it is generally likened to two disks equal to the pitch diameters which are side by side and transmitting power by friction.

The examples of gear terminology related to pitch circle can be listed as pitch circle diameter, meshing pitch circle diameter, outside diameter, root diameter, etc.

Pitch circle diameter (reference diameter)

Planetary gear drive system

A planetary gear drive system is used when a large speed reduction in a compact space is desired and is made up of four basic elements: a sun gear in the center, planet gears which are in mesh with the sun gear and an internal gear, the internal gear on the outside and a carrier on which the planetary gears are mounted.

The following are the characteristics of planetary gear drive systems :

  1. As a gear system, its volume can be made small, requiring little space for placement in a mechanism
  2. Large speed reduction ratio is obtained in a small number of stages
  3. It is capable of transmitting a large torque
  4. The input shaft and the output shaft are in-line
  5. Using the three elements, the sun gear rotation, the planetary gear revolution and internal gear rotation, alternately as the fixed, input and output elements, several variations of transmission ratios and rotational directions can be obtained with a single unit
  6. The structure tends to become complex, sometimes leading to internal gear interferences

Planetary gear drive systems can be classified into three types by the role each element plays.

  1. Planetary type
    The internal gear is fixed, and the sun gear is rotated as the input to drive the planet gears, thus rotating the carrier shaft.
  2. Solar type
    The sun gear is fixed, and the internal gear is rotated as the input to drive the planet gears, thus rotating the carrier shaft.
  3. Star type
    The carrier is fixed so that the planetary gears rotate on fixed axes, and the sun gear is rotated as the input, driving the internal gear in the opposite direction.

As an example of the applications of planetary gear drive systems, a harmonic drive can be cited. However, a more familiar example is a hand operated pencil sharpener.

An example of a planetary gear system
Planetary gear mechanism

Design points and application examples of planetary gear mechanisms

As an example of a speed reducer with in-line input and output shafts, a planetary gear mechanism can be named. Compared to spur gear based speed reducers with the same reduction ratio, this mechanism has a less number of components and it is able to create easy to assemble, compact speed reducers.
A planetary gear mechanism is made up gears (sun gear, planet gears and internal gear) and a planet gear carrier. In addition, because the planetary gears are meshed with both the sun gear and the internal gear, the design of the shaft bearings and the amount of the backlash become important points to consider.
If ball or rolling bearings can be incorporated into planet gears, it is relatively easy to manage the gear shaft spacing, making it possible to produce smooth running, high efficiency speed reducers. However, if there is a significant load on the bearings, the life expectancy may be reduced.
When compact designs are needed, such as for gearmotor speed reducers, sleeve bearings are used to obtain a longer life. On the other hand, since the sleeve bearings rely on the oil film formation on the sliding surface, the accuracy of the fit is less than for ball bearings, resulting in possible reduction in the efficiency of the unit. Therefore, special attention should be paid to the clearance of the bearings and the backlash of the gears.
The application examples are generally in gearmotors where very compact and high efficiency units have been developed recently. Especially notable uses include speed increasers for wind power generation and electric actuators for opening and closing valves.
For gearmotors and wind power generators, the internal gear is fixed. For electric valve actuators, the outside surface is machined as a worm wheel so that the valve can be opened/closed by hand-operating the worm. Normally, when the motor is stopped, the valve is opened/closed by turning the worm by hand. However, even if the worm is turned by mistake while the motor is running, the hand operation is possible without interference from the motor movement. This is an example of a mechanism design where the fail-safe function can be incorporated into the system using the planetary gear mechanism.
As a new application of planetary gear drive, the improvements in the product precision is allowing stacking of the units to make multi-stage planetary gear mechanisms. Now high reduction ratio, high efficiency and compact speed reducers are used in a wide range of fields from automatic transmission in automobiles to medical equipment, industrial machines, food machinery, etc.

Polishing

Abrasion in which the roughness protrusion of the tooth surface contact portion is scraped and the surface comes close to the mirror surface is called polishing.
The polishing itself is not regarded as a defect, but the smoothing of the surface may disadvantageously cause the formation of an oil film, and may cause damage such as scuffing and surface fatigue.

Pressure angle

In a word, pressure angle can be expressed as the slant of the teeth. However, there are two ways to look at pressure angle. One is the pressure angle of the rack used in tooth cutting and the other is the pressure angle when the gears mesh. Theoretically, both are the same, but differ in the way they are defined.
First, let’s explain the pressure angle of the rack tool.

Standard basic rack tooth profile

On the rack tool, the pressure angle is the tilt angle of the tooth surface relative to the normal to the pitch line.

Gear Terminology

About the other pressure angle involved in meshing of gears, it is the angle between the center line connecting the two gears and the common tangent to the two base circles. In standard spur gears, the normal line to the center line matches the normal line to the pitch line of the rack tool. Also, the common tangent to the base circles is perpendicular to the tooth and the pressure angle of the rack tool becomes identical to the pressure angle of the gear mesh.
At the point of gear mesh, the driving force is transmitted in the direction perpendicular to the tooth. Therefore, the force acting on the gear shaft is tilted by the amount of the pressure angle relative to the normal to the center line.
Generally pressure angles are 20°, but there are occasions when special pressure angles such as 14.5° and 17.5° are used.

Profile shift

When forming gear teeth, the movement of the teeth cutting tool in the radial direction is called the profile shift and the amount moved is called the profile shift amount. The teeth thickness can be made thicker or thinner by shifting the profile. The direction of the shift is + for thicker teeth and is – for thinner teeth. The main reasons for shifting profiles are to avoid tooth root undercut when cutting gears, optimizing the meshing conditions by adjusting the center distance between gears, etc.

Left: Meshing of standard spur gear with a rack (α = 20°, z = 12, x = 0) / Right: Meshing of profile shifted spur gear with a rack (α = 20°, z = 12, x = + 0.6) Comparison with positive shifted tooth profile

Profile shift coefficient

The value obtained by dividing the amount of profile shift by module m is called the profile shift coefficient.

Rack and pinion

A rack with teeth cut on a flat bar and a pinion meshing with it as a pair is called the rack and pinion. The pair converts a rotational motion into a linear motion, or a linear motion into a rotational motion. For each rotation of the pinion, the rack moves the number of teeth times the pitch of the gears. Rack and pinion can be made with helical gears with the slanted gear teeth as well, but caution must be exercised due to the generation of thrust forces.

Rack type cutter

A rack type tool that forms an involute tooth profile is called a rack cutter. By translating the rack cutter at the same speed as the peripheral speed of the workpiece (the material to be cut) on the pitch circle of the gear, an involute tooth shape can be formed on the workpiece.

Nominal pressure angle and Nominal pitch of the cutter

Referene circle / Reference cylinder / Reference diameter / Reference surface

As the basis for expressing the size of gears, the diameter determined as the number of teeth times the module (zm) is called the pitch or reference diameter.
In addition, the surface extracted by the plane perpendicular to the gear shaft and containing the pitch circle is called the reference surface.
This reference surface for cylindrical gears becomes the reference cylinder and for bevel gears the reference cone.

Root circle

The circle concentric with the reference circle and obtained by connecting the bottoms of the teeth is called the root circle. The tip of the rack shaped cutters and shaving cutters reaches as far as the root circle.

The formula for root circle diameter (df) is  df  = d – 2.5 m  (d : reference diameter m : module).

Also, the distance from the reference circle to the root circle is called the dedendum (hf) and hf  = 1.25 x m.

Working Gear Nomenclature

Root cylinder

The root surface of cylindrical gears is called the root cylinder.

Bottom land / Fillet / Root Cylinder

Root diameter

The diameter of the root circle is called the root diameter. The root diameter is generally determined by the tooth cutting tool.

Root relief

Root relief is modifying dedendum by thinning when root interference occurs.

Root surface

The rotating surface containing the bottom of the teeth grooves is called the root surface.

Scratching

Scratching is a scratch that is generated at the contact portion of the gear and has a linear groove in the sliding direction.
The location where scratching occurs is not specified, and the length of the scratch varies.
In order to suppress scratching, it is effective to prevent foreign matter from entering from the outside.

Screw gear

A screw gear is a type of helical gear used with skewed axes.
It is used in low efficient and light load applications and requires ample lubrication while in use.

Scuffing

Strong adhesive wear caused by metal contact between tooth surfaces due to oil film breakage and repeated fusion and peeling at the contact portion is called scuffing.
Oil film breakage includes a case in which the contact portion becomes high temperature under high-speed and high-load conditions and the oil film breaks (high-temperature scuffing), a case in which the wedge film of the oil is not formed under low-speed and high-load conditions (low-temperature scuffing), and a case in which the roughness of the tooth surface becomes large due to wear or deformation during operation (fatigue scuffing).

Span thickness of gear

The distance spanning multiple consecutive teeth of an external gear or multiple consecutive grooves of an internal gear by a parallel flat surfaces is called the span thickness of the gear. A tooth thickness micrometer is generally used to measure the tooth thickness.

Shaft angle

Shaft angle is the angle formed by the shafts of a pair of gears. It is necessary to specify the angle for intersecting shaft gears to change power transmission directions and also for nonintersecting and nonparallel shaft gears.

Shaft angle

The distance between two planes, which can be measured by sandwiching the outermost left and right tooth flanks of multiple consecutive teeth in the case of external gears, or multiple consecutive tooth grooves in the case of internal gears, is called the crossing thickness. A tooth thickness micrometer is generally used to measure the stray tooth thickness.

Semi-topping

When cutting gear teeth, the chamfering of the top of the gear is called semi-topping. To do this, a special rack shaped cutter which cuts the tooth and the chamfer at the same time is utilized. Use of this method is effective in preventing burrs and damages to the sharp corners at the top of gears.

Magnitude of semitopping

Speed reducing ratio / Speed increasing ratio

In a pair of gears, when the rotational speed of the output shaft, Nout, is less than that of the input shaft, Nin, Nout / Nin is called the speed reducing ratio. In this instance, the number of teeth of the output shaft is larger than that of the input shaft. The opposite of the above case is the speed increasing ratio.

Spiral bevel gear

The bevel gear with twisted tooth trace is called a "spiral bevel gear".
It is more difficult to make than a straight bevel gear, but is quieter, carries higher load and can be used in high speed applications.

Spur gear

A spur gear is categorized as a parallel shaft type gear, and is a cylindrical gear whose tooth trace is parallel to its axis.
In addition, its characteristics include no thrust generation while in use, ease of manufacturing, and ability to be made to a high degree of precision.

Stainless steel gears

Stainless gears are used when corrosion resistance is required for carbon steel. Austenite, which has excellent corrosion resistance, and martensite, which has moderate strength and wear resistance, are commonly used as materials for gears.

Straight bevel gear

A bevel gear is a conical gear belonging to the intersecting axis gears category.
A bevel gear with the straight tooth trace is called a "straight bevel gear".
The straight bevel gear is easier to manufacture than the spiral bevel gear and has the characteristic of not generating the thrust force in the negative direction.

Tempered gears

The steel is hardened and tempered at a high temperature of 400 to 650℃, and the metal structure is made into tall steet or sorbite to obtain toughness. This heat treatment is called tempering, and the gear using this tempering material is called tempering gear. The standard temper hardness is about 200 to 270 HB in terms of Brinell hardness. The tempered material can be machined even after heat treatment and can be formed with high precision.

Thrust

Thrust is a pushing force, and in gears, it is the pushing force acting parallel to the rotating shaft which is called a thrust force. It is the axial force that is created when transmitting power by gears. It does not occur with spur gears but it is definitely created with helical gears.

data-fontsize="16" data-lineheight="24">Regarding the thrust force that occurs with helical gears :

Because the teeth of helical gears are slanted relative to the gear shaft, the transmitted force is divided into the rotational direction and thrust direction. Since the force acting on the tooth surface is perpendicular to the contact surface, the transmitted force in helical gear is tilted by the amount of the twist or helix angle (β). This force is divided into orthogonal to the gear shaft component and parallel to the gear shaft component where the former becomes the transmitted power and the latter becomes the thrust force. In this case, the thrust force is Ft tan β.

Direction of forces acting on a gear
Forces acting upon a gear

Due to the thrust force, a transmitted force comes into play which is parallel to the helical gear shafts and in the direction to separate the gears. For that reason, when using helical gears, it is necessary to secure the gear in the axial direction of the shaft.  Also, this securing requires the design consideration to take this thrust force into account.

Direction of forces acting on a helical gear mesh

It should be noted, the examples of gear terminology related to thrust includes helical gear, tangent force, radial force, etc.

Tip circle (Outside circle)

The circle concentric with the reference circle and obtained by connecting the tooth tips is called the tooth tip circle or tip circle for short (also sometimes called the outside circle). In the case of standard spur gears, the tip circle diameter (da) is  da = d +2m (d : reference diameter m : module). Also, the tip circle diameter is an important dimension with regard to the design of its meshing gear and components surrounding it. Incidentally, the distance from the tip circle to the root circle is called the tooth depth and represents the height of the gear tooth. Within the tooth depth, the distance from the tip circle to the reference circle is called the addendum (ha) and when there is no profile shift, the addendum is ha = m.

Working Gear Nomenclature

Tip cylinder

The tip surface of the cylindrical gear is called the tip cylinder.

Tip diameter

The diameter of the tip circle is called the tip diameter. The tip diameter generally depends on the outer diameter of the piece.

Tip relief

Tip relief is modifying addendum by thinning when tip interference occurs.

Tip surface

The rotating surface that contains the gear tooth tip is called the tip surface.

Tip surface / Tooth tip / Top land

Tool pitch

The distance between adjacent teeth of a rack tooth profile is called the tool pitch.

Tooth

The protruded part of a gear that engages the mating gear and transmits power is called the tooth.

Tooth depth

The tooth depth means the value which the addendum circle radius minus the deddendum circle radius. In addition, the tooth depth means the height of the gear.

Tooth flank

The tooth flank is the surface between the tooth tip and the tooth root. A pair of gears transmits power through the tooth flank. Therefore, any inappropriate tooth contact may cause problems such as tooth wear, occurrence of flaking and pitting as well as deterioration of power transmission efficiency.

Tooth profile

Tooth profile is the curve that determines shape of gear and has two types :

Tooth profile is determined from contact configuration (rolling or sliding) of meshing part, performances such as tooth strength, and manufacturability.

Tooth space

The space between the adjoining teeth is called the tooth space which becomes the concave portion opposite to the tooth of the meshing gear. When the gears are meshed, the tooth of one gear enters the tooth space of the mating gear.

Tooth tip

The intersection of the tip surface and the tooth surface is called the tooth tip.

Tip surface / Tooth tip / Top land

Tooth tip interference / Meshing interference

A defect in which the tooth tip of one of the gears bites into the other tooth surface when the gears are engaged is called a tooth tip interference.
In order to avoid the tooth tip interference, it is necessary to take measures such as removing work (chamfering, etc.) in the vicinity of the tooth tip and using a dislocation gear.

Top land

The individual tip surface of the gear is called the top land.

Tip surface / Tooth tip / Top land

Topping

Topping is the method in which the tooth cutting tool’s roots shave the top land of the gear. For example, when a gear is cut using the normal tool, it leaves the cylindrical outer surface of the blank. When a topping gear cutter is used, the gear’s tooth form generation, outside diameter finishing and tooth tip deburring can all be done at the same time, resulting in gears with smaller OD run-out and effectively preventing the creation of burr at the tooth tip.

Total angle of transmission / Total contact ratio

When two gears mesh, the rotating angle from the time the teeth start to touch each other to when the teeth separate is called the total angle of transmission. Also, the value obtained by the total angle of transmission divided by the angular pitch is called the total contact ratio.

Transverse angle of transmission / Transverse contact ratio

Looking at the transverse cross section of two meshing gears, the rotating angle from the time the teeth start to touch each other to when the teeth separate is called the transverse angle of transmission. Also, the value obtained by the transverse angle of transmission divided by the angular pitch is called the transverse contact ratio. Furthermore, the transverse contact ratio is equal to the length of the path of contact divided by the transverse normal pitch. For spur gears, the total and the transverse angles of transmission are the same and it is unnecessary to use the distinguishing terms of “total” and “transverse”.

Transverse module

Transverse module is the module seen from normal section, and transverse pitch is divided by pi.

Transverse pitch

Transverse pitch is arc distance between points where normal tooth profile and reference circle of neighboring teeth intersect.

Transverse pressure angle

Transverse pressure angle is formed by transverse tooth profile edge and reference circle radius on the point where transverse tooth profile intersects reference circle. To define a criterion for rack, transverse pressure angle is tilt of tooth profile line when cutting rack at a right angle to shaft.

Transverse pressure angle

Transverse profile

Transverse profile (axially right angle profile) is tooth profile curve that appears when cutting at right angle to axis of reference surface. This term is mainly used for gear such as helical gear whose tooth trace is twisted in comparison with normal profile.

Transverse space width

Transverse space width is arc length between intersection points of reference circle and two tooth surfaces which face tooth space.

Transverse tooth thickness

Transverse tooth thickness is arc length between intersection points of reference circle and both sides of tooth surface.

Twist angle (Helix angle)

The amount of twist that the teeth of helical gears have relative to the shaft is called the helix angle or the pitch cylinder helix angle. In the general helical gears, the involute curve of the tooth surface is tilted relative to the gear shaft by the amount of the helix angle. In other words, if the tooth is cut perpendicular to this tooth direction, the same involute tooth shape as the standard spur gears can be obtained. This type of gear is called the normal module helical gear. In this case, the teeth are formed simply by tilting the hob or rack shaped cutting tool by the amount of helix angle relative to the gear shaft. There also exist axial module helical gears with different tooth form reference surfaces.

Fundamental relationship of a helical gear (Right-hand)

Undercut

The undercut of gears is also called deeper cutting and indicates the phenomenon of cutting the root of the gear deeper than the involute tooth curve. This can happen when there is interference between a tooth cutting tool and a gear or between the two meshing gears. When the undercutting is large, the root of the gear becomes narrower and the tooth form becomes weak in its bending strength.

The main reason for undercutting to occur is, when cutting a gear with a small number of teeth (for standard gear with 20° pressure angle, number of teeth z = 17 or less), the tip of the cutting tool goes beyond the interference point.

To prevent undercutting without changing the number of teeth, a method called profile shifting is used. Profile shifting means changing the depth of the cutting tool and in the case of undercut, a positive profile shift is used.  In this case, the root of the gear tooth becomes thicker.  There is a limit to the amount of positive profile shift, and caution should be exercised since when the shift is excessive (x = +0.5), the tip of the tooth becomes pointed.

In addition, the examples of gear terminology related to undercut are involute tooth form, profile shifted gear, positive profile shift, etc.

Comparison with positive shifted tooth profile
Generation of positive shifted spur gear (α = 20 degrees, z = 10, x = +0.5)
Generation of negative shifted spur gear (α = 20 degrees, z = 10, x = -0.5)

Usable flank

The portion of the tooth flank which can be used as the active flank is called the usable flank.

Wear

A phenomenon in which a tooth surface is worn by wear and teeth are lost is called wear and tear.
In such a state, the teeth cannot be engaged with each other, and the body of the gear is not formed.

Worm

A worm is a screw shaped gear used in a set with a worm gear in a non-parallel, non-intersecting axes.
A large speed reduction can be obtained with a single stage.

Worm gear

A worm gear is a cylindrical gear used with a worm as a set in non-parallel, non-intersecting axes applications.
While a large speed reduction ratio can be obtained with a single stage, caution must be exercised due to thrust generation and need for lubrication.

NOTE:
The articles with * marks are reproduced with the permission.
Masao Kubota, Haguruma Nyumon, Tokyo : Ohmsha, Ltd., 1963.

Related links :
Gear Rack and Pinion - A detailed description of Gear Rack and Pinion

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