Gear & Gear Ratio

The gear ratio is the relationship between the number of teeth on two gears that are meshed or two sprockets connected with a common roller chain, or the circumferences of two pulleys connected with a drive belt.

Depending on their construction and arrangement, geared devices can transmit forces at different speeds, torques, or in a different direction, from the power source. Gears are a very useful simple machine. The most common situation is for a gear to mesh with another gear, but a gear can mesh with any device having compatible teeth, such as other rotational gears, or linear moving racks. A gear's most important feature is that gears of unequal sizes (diameters) can be combined to produce a mechanical advantage, so that the rotational speed and torque of the second gear are different from that of the first. In the context of a particular machine, the term "gear" also refers to one particular arrangement of gears among other arrangements (such as "first gear"). Such arrangements are often given as a ratio, using the number of teeth or gear diameter as units. The term "gear" is also used in non-geared devices which perform equivalent tasks:

General

The smaller gear in a pair is often called the pinion; the larger, either the gear, or the wheel.

Mechanical advantage

The interlocking of the teeth in a pair of meshing gears means that their circumferences necessarily move at the same rate of linear motion (eg., metres per second, or feet per minute). Since rotational speed (eg. measured in revolutions per second, revolutions per minute, or radians per second) is proportional to a wheel's circumferential speed divided by its radius, we see that the larger the radius of a gear, the slower will be its rotational speed, when meshed with a gear of given size and speed. The same conclusion can also be reached by a different analytical process: counting teeth. Since the teeth of two meshing gears are locked in a one to one correspondence, when all of the teeth of the smaller gear have passed the point where the gears meet -- ie., when the smaller gear has made one revolution -- not all of the teeth of the larger gear will have passed that point -- the larger gear will have made less than one revolution. The smaller gear makes more revolutions in a given period of time; it turns faster. The speed ratio is simply the reciprocal ratio of the numbers of teeth on the two gears.

The torque ratio can be determined by considering the force that a tooth of one gear exerts on a tooth of the other gear. Consider two teeth in contact at a point on the line joining the shaft axes of the two gears. In general, the force will have both a radial and a circumferential component. The radial component can be ignored: it merely causes a sideways push on the shaft and does not contribute to turning. The circumferential component causes turning. The torque is equal to the circumferential component of the force times radius. Thus we see that the larger gear experiences greater torque; the smaller gear less. The torque ratio is equal to the ratio of the radii. This is exactly the inverse of the case with the velocity ratio. Higher torque implies lower velocity and vice versa. The fact that the torque ratio is the inverse of the velocity ratio could also be inferred from the law of conservation of energy. Here we have been neglecting the effect of friction on the torque ratio. The velocity ratio is truly given by the tooth or size ratio, but friction will cause the torque ratio to be actually somewhat less than the inverse of the velocity ratio.

In the above discussion we have made mention of the gear "radius". Since a gear is not a proper circle but a roughened circle, it does not have a radius. However, in a pair of meshing gears, each may be considered to have an effective radius, called the pitch radius, the pitch radii being such that smooth wheels of those radii would produce the same velocity ratio that the gears actually produce. The pitch radius can be considered sort of an "average" radius of the gear, somewhere between the outside radius of the gear and the radius at the base of the teeth.

The issue of pitch radius brings up the fact that the point on a gear tooth where it makes contact with a tooth on the mating gear varies during the time the pair of teeth are engaged; also the direction of force may vary. As a result, the velocity ratio (and torque ratio) is not, actually, in general, constant, if one considers the situation in detail, over the course of the period of engagement of a single pair of teeth. The velocity and torque ratios given at the beginning of this section are valid only "in bulk" -- as long-term averages; the values at some particular position of the teeth may be different.

It is in fact possible to choose tooth shapes that will result in the velocity ratio also being absolutely constant -- in the short term as well as the long term. In good quality gears this is usually done, since velocity ratio fluctuatons cause undue vibration, and put additional stress on the teeth, which can cause tooth breakage under heavy loads at high speed. Constant velocity ratio may also be desirable for precision in instrumentation gearing, clocks and watches. The involute tooth shape is one that results in a constant velocity ratio, and is the most commonly used of such shapes today.

Comparison with other drive mechanisms

The definite velocity ratio which results from having teeth gives gears an advantage over other drives (such as traction drives and V-belts) in precision machines such as watches that depend upon an exact velocity ratio. In cases where driver and follower are in close proximity gears also have an advantage over other drives in the reduced number of parts required; the downside is that gears are more expensive to manufacture and their lubrication requirements may impose a higher operating cost.

The automobile transmission allows selection between gears to give various mechanical advantages.

Spur gears

Spur gears are the simplest, and probably most common, type of gear. Their general form is a cylinder or disk (a disk is just a short cylinder). The teeth project radially, and with these "straight-cut gears", the leading edges of the teeth are aligned parallel to the axis of rotation. These gears can only mesh correctly if they are fitted to parallel axles.^[2]

Helical gears

Helical gears offer a refinement over spur gears. The leading edges of the teeth are not parallel to the axis of rotation, but are set at an angle. Since the gear is curved, this angling causes the tooth shape to be a segment of a helix. The angled teeth engage more gradually than do spur gear teeth. This causes helical gears to run more smoothly and quietly than spur gears. Helical gears also offer the possibility of using non-parallel shafts. A pair of helical gears can be meshed in two ways: with shafts oriented at either the sum or the difference of the helix angles of the gears. These configurations are referred to as parallel or crossed, respectively. The parallel configuration is the more mechanically sound. In it, the helices of a pair of meshing teeth meet at a common tangent, and the contact between the tooth surfaces will, generally, be a curve extending some distance across their face widths. In the crossed configuration, the helices do not meet tangentially, and only point contact is achieved between tooth surfaces. Because of the small area of contact, crossed helical gears can only be used with light loads.

Quite commonly, helical gears come in pairs where the helix angle of one is the negative of the helix angle of the other; such a pair might also be referred to as having a right handed helix and a left handed helix of equal angles. If such a pair is meshed in the 'parallel' mode, the two equal but opposite angles add to zero: the angle between shafts is zero -- that is, the shafts are parallel. If the pair is meshed in the 'crossed' mode, the angle between shafts will be twice the absolute value of either helix angle.

Note that 'parallel' helical gears need not have parallel shafts -- this only occurs if their helix angles are equal but opposite. The 'parallel' in 'parallel helical gears' must refer, if anything, to the (quasi) parallelism of the teeth, not to the shaft orientation.

As mentioned at the start of this section, helical gears operate more smoothly than do spur gears. With parallel helical gears, each pair of teeth first make contact at a single point at one side of the gear wheel; a moving curve of contact then grows gradually across the tooth face. It may span the entire width of the tooth for a time. Finally, it recedes until the teeth break contact at a single point on the opposite side of the wheel. Thus force is taken up and released gradually. With spur gears, the situation is quite different. When a pair of teeth meet, they immediately make line contact across their entire width. This causes impact stress and noise. Spur gears make a characteristic whine at high speeds and can not take as much torque as helical gears because their teeth are receiving impact blows. Whereas spur gears are used for low speed applications and those situations where noise control is not a problem, the use of helical gears is indicated when the application involves high speeds, large power transmission, or where noise abatement is important. The speed is considered to be high when the pitch line velocity (that is, the circumferential velocity) exceeds 5000 ft/min.^[3] A disadvantage of helical gears is a resultant thrust along the axis of the gear, which needs to be accommodated by appropriate thrust bearings, and a greater degree of sliding friction between the meshing teeth, often addressed with specific additives in the lubricant.

Double helical gears

Double helical gears, invented by André Citroën and also known as herringbone gears, overcome the problem of axial thrust presented by 'single' helical gears by having teeth that set in a 'V' shape. Each gear in a double helical gear can be thought of as two standard, but mirror image, helical gears stacked. This cancels out the thrust since each half of the gear thrusts in the opposite direction. They can be directly interchanged with spur gears without any need for different bearings.

Where the oppositely angled teeth meet in the middle of a herringbone gear, the alignment may be such that tooth tip meets tooth tip, or the alignment may be staggered, so that tooth tip meets tooth trough. The latter type of alignment results in what is known as a Wuest type herringbone gear.

With the older method of fabrication, herringbone gears had a central channel separating the two oppositely-angled courses of teeth. This was necessary to permit the shaving tool to run out of the groove. The development of the Sykes gear shaper now makes it possible to have continuous teeth, with no central gap.

Bevel gears

Bevel gears are essentially conically shaped, although the actual gear does not extend all the way to the vertex (tip) of the cone that bounds it. With two bevel gears in mesh, the vertices of their two cones lie on a single point, and the shaft axes also intersect at that point. The angle between the shafts can be anything except zero or 180 degrees. Bevel gears with equal numbers of teeth and shaft axes at 90 degrees are called miter gears.

The teeth of a bevel gear may be straight-cut as with spur gears, or they may be cut in a variety of other shapes. 'Spiral bevel gears' have teeth that are both curved along their (the tooth's) length; and set at an angle, analogously to the way helical gear teeth are set at an angle compared to spur gear teeth. 'Zero bevel gears' have teeth which are curved along their length, but not angled. Spiral bevel gears have the same advantages and disadvantages relative to their straight-cut cousins as helical gears do to spur gears. Straight bevel gears are generally used only at speeds below 5 m/s (1000 ft/min), or, for small gears, 1000 r.p.m.^[4]

Crown gear

A crown gear or contrate gear is a particular form of bevel gear whose teeth project at right angles to the plane of the wheel; in their orientation the teeth resemble the points on a crown. A crown gear can only mesh accurately with another bevel gear, although crown gears are sometimes seen meshing with spur gears. A crown gear is also sometimes meshed with an escapement such as found in mechanical clocks.

Hypoid gears

Hypoid gears resemble spiral bevel gears, except that the shaft axes are offset, not intersecting. The pitch surfaces appear conical but, to compensate for the offset shaft, are in fact hyperboloids of revolution. Hypoid gears are almost always designed to operate with shafts at 90 degrees. Depending on which side the shaft is offset to, relative to the angling of the teeth, contact between hypoid gear teeth may be even smoother and more gradual than with spiral bevel gear teeth. Also, the pinion can be designed with fewer teeth than a spiral bevel pinion, with the result that gear ratios of 60:1 and higher are "entirely feasible" using a single set of hypoid gears.^[5]

Worm gear

A worm is a gear that resembles a screw. It is a species of helical gear, but its helix angle is usually somewhat large(ie., somewhat close to 90 degrees) and its body is usually fairly long in the axial direction; and it is these attributes which give it its screw like qualities. A worm is usually meshed with an ordinary looking, disk-shaped gear, which is called the "gear", the "wheel", the "worm gear", or the "worm wheel". The prime feature of a worm-and-gear set is that it allows the attainment of a high gear ratio with few parts, in a small space. Helical gears are, in practice, limited to gear ratios of 10:1 and under; worm gear sets commonly have gear ratios between 10:1 and 100:1, and occasionally 500:1.^[6] In worm-and-gear sets, because the worm's helix angle is large, the sliding action between teeth is considerable, and the resulting frictional loss causes the efficiency of the drive to be usually less than 90 percent, sometimes less than 50 percent.so comparing to other gears it gives very less efficiency.^[7]

The distinction between a worm and a helical gear is made when at least one tooth persists for a full 360 degree turn around the helix. If this occurs, it is a 'worm'; if not, it is a 'helical gear'. A worm may have as few as one tooth. If that tooth persists for several turns around the helix, the worm will appear, superficially, to have more than one tooth, but what one in fact sees is the same tooth reappearing at intervals along the length of the worm. The usual screw nomenclature applies: a one-toothed worm is called "single thread" or "single start"; a worm with more than one tooth is called "multiple thread" or "multiple start".

We should note that the helix angle of a worm is not usually specified. Instead, the lead angle, which is equal to 90 degrees minus the helix angle, is given.

In a worm-and-gear set, the worm can always drive the gear. However, if the gear attempts to drive the worm, it may or may not succeed. Particularly if the lead angle is small, the gear's teeth may simply lock against the worm's teeth, because the force component circumferential to the worm is not sufficient to overcome friction. Whether this will happen depends on a function of several parameters; however, an approximate rule is that if the tangent of the lead angle is greater than the coefficient of friction, the gear will not lock.^[8] Worm-and-gear sets that do lock in the above manner are called "self locking". The self locking feature can be an advantage, as for instance when it is desired to set the position of a mechanism by turning the worm and then have the mechanism hold that position. Tuning gears on stringed musical instruments work that way.

If the gear in a worm-and-gear set is an ordinary helical gear only point contact between teeth will be achieved.^[9] If medium to high power transmission is desired, the tooth shape of the gear is modified to achieve more intimate contact with the worm thread. A noticeable feature of most such gears is that the tooth tops are concave, so that the gear partly envelopes the worm. A further development is to make the worm concave (viewed from the side, perpendicular to its axis) so that it partly envelopes the gear as well; this is called a cone-drive or Hindley worm.^[10]

A right hand helical gear or right hand worm is one in which the teeth twist clockwise as they recede from an observer looking along the axis. The designations, right hand and left hand, are the same as in the long established practice for screw threads, both external and internal. Two external helical gears operating on parallel axes must be of opposite hand. An internal helical gear and its pinion must be of the same hand.

A left hand helical gear or left hand worm is one in which the teeth twist counterclockwise as they recede from an observer looking along the axis.^[11]

Rack and pinion

A rack is a toothed bar or rod that can be thought of as a sector gear with an infinitely large radius of curvature. Torque can be converted to linear force by meshing a rack with a pinion: the pinion turns; the rack moves in a straight line. Such a mechanism is used in automobiles to convert the rotation of the steering wheel into the left-to-right motion of the tie rod(s). Racks also feature in the theory of gear geometry, where, for instance, the tooth shape of an interchangeable set of gears may be specified for the rack (infinite radius), and the tooth shapes for gears of particular actual radii then derived from that.

External vs. internal gears

An external gear is one with the teeth formed on the outer surface of a cylinder or cone. Conversely, an internal gear is one with the teeth formed on the inner surface of a cylinder or cone. For bevel gears, an internal gear is one with the pitch angle exceeding 90 degrees.^[11]

Parallel axis gears

See also

Gear nomenclature

Tooth contact nomenclature

Point of contact

Line of contact

A line of contact is a line or curve along which two tooth surfaces are tangent to each other.

Path of action

The path of action is the locus of successive contact points between a pair of gear teeth, during the phase of engagement. For conjugate gear teeth, the path of action passes through the pitch point. It is the trace of the surface of action in the plane of rotation.

Line of action

The line of action is the path of action for involute gears. It is the straight line passing through the pitch point and tangent to both base circles.

Surface of action

The surface of action is the imaginary surface in which contact occurs between two engaging tooth surfaces. It is the summation of the paths of action in all sections of the engaging teeth.

Plane of action

The plane of action is the surface of action for involute, parallel axis gears with either spur or helical teeth. It is tangent to the base cylinders.

Zone of action (contact zone)

Zone of action (contact zone) for involute, parallel-axis gears with either spur or helical teeth, is the rectangular area in the plane of action bounded by the length of action and the effective face width.

Path of contact

The path of contact is the curve on either tooth surface along which theoretical single point contact occurs during the engagement of gears with crowned tooth surfaces or gears that normally engage with only single point contact.

Length of action

Length of action is the distance on the line of action through which the point of contact moves during the action of the tooth profile.

Arc of action, Q_t

Arc of action is the arc of the pitch circle through which a tooth profile moves from the beginning to the end of contact with a mating profile.

Arc of approach, Q_a

Arc of approach is the arc of the pitch circle through which a tooth profile moves from its beginning of contact until the point of contact arrives at the pitch point.

Arc of recess, Q_r

Arc of recess is the arc of the pitch circle through which a tooth profile moves from contact at the pitch point until contact ends.

Contact ratio, m_c, ε

Contact ratio in general is the number of angular pitches through which a tooth surface rotates from the beginning to the end of contact.

Transverse contact ratio, m_p, ε_α

Transverse contact ratio is the contact ratio in a transverse plane. It is the ratio of the angle of action to the angular pitch. For involute gears it is most directly obtained as the ratio of the length of action to the base pitch.

Face contact ratio, m_F, ε_β

Face contact ratio is the contact ratio in an axial plane, or the ratio of the face width to the axial pitch. For bevel and hypoid gears it is the ratio of face advance to circular pitch.

Total contact ratio, m_t, ε_γ

Total contact ratio is the sum of the transverse contact ratio and the face contact ratio.

Modified contact ratio, m_o

Modified contact ratio for bevel gears is the square root of the sum of the squares of the transverse and face contact ratios. $m_o = (m_p^2 + m_F^2)^0.5$

Limit diameter

Limit diameter is the diameter on a gear at which the line of action intersects the maximum (or minimum for internal pinion) addendum circle of the mating gear. This is also referred to as the start of active profile, the start of contact, the end of contact, or the end of active profile.

Start of active profile (SAP)

The start of active profile is the intersection of the limit diameter and the involute profile.

Face advance

Face advance is the distance on a pitch circle through which a helical or spiral tooth moves from the position at which contact begins at one end of the tooth trace on the pitch surface to the position where contact ceases at the other end.

Backlash

Backlash is the error in motion that occurs when gears change direction. It exists because there is always some gap between the tailing face of the driving tooth and the leading face of the tooth behind it on the driven gear, and that gap must be closed before force can be transferred in the new direction. The term "backlash" can also be used to refer to the size of the gap, not just the phenomenon it causes; thus, one could speak of a pair of gears as having, for example, "0.1 mm of backlash." A pair of gears could be designed to have zero backlash, but this would presuppose perfection in manufacturing, uniform thermal expansion characteristics throughout the system, and no lubricant. Therefore, gear pairs are designed to have some backlash. It is usually provided by reducing the tooth thickness of each gear by half the desired gap distance. In the case of a large gear and a small pinion, however, the backlash is usually taken entirely off the gear and the pinion is given full sized teeth. Backlash can also be provided by moving the gears farther apart. For situations, such as instrumentation and control, where precision is important, backlash can be minimised through one of several techniques. For instance, the gear can be split along a plane perpendicular to the axis, one half fixed to the shaft in the usual manner, the other half placed alongside it, free to rotate about the shaft, but with springs between the two halves providing relative torque between them, so that one achieves, in effect, a single gear with expanding teeth. Another method involves tapering the teeth in the axial direction and providing for the gear to be slid in the axial direction to take up slack.

Epicyclic gearing

In an ordinary gear train, the gears rotate but their axes are stationary. An epicyclic gear train is one in which one or more of the axes also moves. Examples are the sun and planet gear system invented by the company of James Watt, in which the axis of the planet gear revolves around the central sun gear; and the differential gear system used to drive the wheels of automobiles, in which the axis of the central bevel pinion is turned "end over end" by the ring gear, the drive to the wheels being taken off by bevel gears meshing with the central bevel pinion. With the differential gearing, the sum of the two wheel speeds is fixed, but how it is divided between the two wheels is undetermined, so the outer wheel can run faster and the inner wheel slower on corners.

Shifting of gears

In some machines (e.g., automobiles) it is necessary to alter the gear ratio to suit the task. There are several methods of accomplishing this. For example:

There are several outcomes of gear shifting in motor vehicles. In the case of air pollution emissions, there are higher pollutant emissions generated in the lower gears, when the engine is working harder than when higher gears have been attained. In the case of vehicle noise emissions, there are higher sound levels emitted when the vehicle is engaged in lower gears. This fact has been utilized in analyzing vehicle generated sound since the late 1960s, and has been incorporated into the simulation of urban roadway noise and corresponding design of urban noise barriers along roadways.^[13]

Tooth profile

A profile is one side of a tooth in a cross section between the outside circle and the root circle. Usually a profile is the curve of intersection of a tooth surface and a plane or surface normal to the pitch surface, such as the transverse, normal, or axial plane.

The fillet curve (root fillet) is the concave portion of the tooth profile where it joins the bottom of the tooth space.²

As mentioned near the beginning of the article, the attainment of a non fluctuating velocity ratio is dependent on the profile of the teeth. Friction and wear between two gears is also dependent on the tooth profile. There are a great many tooth profiles that will give a constant velocity ratio, and in many cases, given an arbitrary tooth shape, it is possible to develop a tooth profile for the mating gear that will give a constant velocity ratio. However, two constant velocity tooth profiles have been by far the most commonly used in modern times. They are the cycloid and the involute. The cycloid was more common until the late 1800s; since then the involute has largely superseded it, particularly in drive train applications. The cycloid is in some ways the more interesting and flexible shape; however the involute has two advantages: it is easier to manufacture, and it permits the center to center spacing of the gears to vary over some range without ruining the constancy of the velocity ratio. Cycloidal gears only work properly if the center spacing is exactly right. Cycloidal gears are still used in mechanical clocks.

Undercut

Undercut is a condition in generated gear teeth when any part of the fillet curve lies inside of a line drawn tangent to the working profile at its point of juncture with the fillet. Undercut may be deliberately introduced to facilitate finishing operations. With undercut the fillet curve intersects the working profile. Without undercut the fillet curve and the working profile have a common tangent.

Pitch

Pitch is the distance between a point on one tooth and the corresponding point on an adjacent tooth.^[11] It is a dimension measured along a line or curve in the transverse, normal, or axial directions. The use of the single word “pitch” without qualification may be ambiguous, and for this reason it is preferable to use specific designations such as transverse circular pitch, normal base pitch, axial pitch.

Circular pitch, p

Circular pitch is the arc distance along a specified pitch circle or pitch line between corresponding profiles of adjacent teeth.

Transverse circular pitch, p_t

Normal circular pitch, p_n, p_e

Normal circular pitch is the circular pitch in the normal plane, and also the length of the arc along the normal pitch helix between helical teeth or threads.

Axial pitch, p_x

Axial pitch is linear pitch in an axial plane and in a pitch surface. In helical gears and worms, axial pitch has the same value at all diameters. In gearing of other types, axial pitch may be confined to the pitch surface and may be a circular measurement.

The term axial pitch is preferred to the term linear pitch. The axial pitch of a helical worm and the circular pitch of its wormgear are the same.

Normal base pitch, p_N, p_bn

Normal base pitch in an involute helical gear is the base pitch in the normal plane. It is the normal distance between parallel helical involute surfaces on the plane of action in the normal plane, or is the length of arc on the normal base helix. It is a constant distance in any helical involute gear.

Transverse base pitch, p_b, p_bt

Base pitch in an involute gear is the pitch on the base circle or along the line of action. Corresponding sides of involute gear teeth are parallel curves, and the base pitch is the constant and fundamental distance between them along a common normal in a transverse plane.

Diametral pitch (transverse), P_d

Diametral pitch (transverse) is the ratio of the number of teeth to the standard pitch diameter in inches.

Normal diametral pitch, P_nd

Normal diametral pitch is the value of diametral pitch in a normal plane of a helical gear or worm.

Angular pitch, θ_N, τ

Angular pitch is the angle subtended by the circular pitch, usually expressed in radians.

Cage gear

The cage gear, also called lantern gear or lantern pinion, has been used for centuries. Its teeth are cylindrical rods, parallel to the axle and arranged in a circle around it, much as the bars on a round bird cage or lantern. The assembly is held together by disks at either end into which the tooth rods and axle are set.

Gear materials

Numerous nonferrous alloys, cast irons, powder-metallurgy and even plastics are used in the manufacture of gears. However steels are most commonly used because of their high strength to weight ratio and low cost. Plastic is commonly used where cost or weight is a concern. A properly designed plastic gear can replace steel in many cases; It often has desirable properties. They can tolerate dirt, low speed meshing, and "skipping" quite well. Manufacturers have employed plastic to make consumer items affordable. This includes copy machines, optical storage devices, VCRs, cheap dynamos, consumer audio equipment, servo motors, and printers.

Tooth thickness

Circular thickness

Circular thickness is the length of arc between the two sides of a gear tooth, on the specified datum circle.

Transverse circular thickness

Transverse circular thickness is the circular thickness in the transverse plane.

Normal circular thickness

Normal circular thickness is the circular thickness in the normal plane. In a helical gear it may be considered as the length of arc along a normal helix.

Axial thickness

Axial thickness in helical gears and worms is the tooth thickness in an axial cross section at the standard pitch diameter.

Base circular thickness

Base circular thickness in involute teeth is the length of arc on the base circle between the two involute curves forming the profile of a tooth.

Normal chordial thickness

Chordal thickness is the length of the chord that subtends a circular thickness arc in the plane normal to the pitch helix. Any convenient measuring diameter may be selected, not necessarily the standard pitch diameter.

Chordial addendum (chordial height)

Chordal addendum (chordal height) is the height from the top of the tooth to the chord subtending the circular thickness arc. Any convenient measuring diameter may be selected, not necessarily the standard pitch diameter.

Profile shift

The profile shift is the displacement of the basic rack datum line from the reference cylinder, made non-dimensional by dividing by the normal module. It is used to specify the tooth thickness, often for zero backlash.

Rack shift

The rack shift is the displacement of the tool datum line from the reference cylinder, made non-dimensional by dividing by the normal module. It is used to specify the tooth thickness.

Measurement over pins

Measurement over pins is the measurement of the distance taken over a pin positioned in a tooth space and a reference surface. The reference surface may be the reference axis of the gear, a datum surface or either one or two pins positioned in the tooth space or spaces opposite the first. This measurement is used to determine tooth thickness.

Span measurement

Span measurement is the measurement of the distance across several teeth in a normal plane. As long as the measuring device has parallel measuring surfaces that contact on an unmodified portion of the involute, the measurement will be along a line tangent to the base cylinder. It is used to determine tooth thickness.

Modified addendum teeth

Full-depth teeth

Full-depth teeth are those in which the working depth equals 2.000 divided by the normal diametral pitch.

Stub teeth

Stub teeth are those in which the working depth is less than 2.000 divided by the normal diametral pitch.

Equal addendum teeth

Equal addendum teeth are those in which two engaging gears have equal addendums.

Long and short-addendum teeth

Long and short addendum teeth are those in which the addendums of two engaging gears are unequal.

See also

Notes

^ http://auto.howstuffworks.com/cvt1.htm
^ Spur gears on non-parallel shafts can mesh, but only point contact will be achieved, not line contact across the full width of the tooth; also the length of the path of contact may be too short.
^ Doughty and Vallance give the following information on helical gear speeds: "Pitch-line speeds of 4,000 to 7,000 fpm [20 to 36 m/s] are common with automobile and turbine gears, and speeds of 12,000 fpm [61 m/s] have been successfully used." -- p.281.
^ McGraw Hill Encyclopedia of Science and Technology, "Gear", p. 742.
^ McGraw Hill Encyclopedia of Science and Technology, "Gear, p. 743.
^ Doughty and Vallance, p. 287.
^ Doughty and Vallance, pp. 280, 296.
^ Doughty and Vallance, p. 296.
^ This appears wrong to me but it's in two of my references: Doughty and Vallance, p. 290; McGraw Hill Encyclopedia of Science and Technology, "Gear", p. 743.
^ McGraw Hill Encyclopedia of Science and Technology, "Gear", p. 744.
^ ^a ^b ^c ^d ^e ^f ^g ANSI/AGMA 1012-G05, "Gear Nomenclature, Definition of Terms with Symbols".
^ Further note on tooth force: If two rigid objects make contact, they always do so at a point (or points) where the tangents to their surfaces coincide -- that is, where there is a common tangent. The perpendicular to the common tangent at the point of contact is called the common normal. Ignoring friction, the force exerted by the objects on each other is always directed along the common normal. Thus, for meshing gear teeth, the line of action is the common normal to the tooth surfaces.
^ [http://www.worldcatlibraries.org/wcpa/top3mset/2930880 C Michael Hogan and Gary L Latshaw,The Relationship Between Highway Planning and Urban Noise , Proceedings of the ASCE, Urban Transportation Division Specialty Conference by the American Society of Civil Engineers, Urban Transportation Division, May 21 to 23, 1973, Chicago, Illinois

General sources

Venton Levy Doughty and Alex Vallance, Design of Machine Members, 4th edition; 1964; McGraw Hill. First edition was 1936. This is an engineering text on machine parts. It has two chapters on gears.

External links

A gear is a component within a transmission device that transmits rotational force to another gear or device. A gear is different from a pulley in that a gear is a round wheel which has linkages ("teeth" or "cogs") that mesh with other gear teeth, allowing force to be fully transferred without slippage. Depending on their construction and arrangement, geared devices can transmit forces at different speeds, torques, or in a different direction, from the power source. Gears are a very useful simple machine. The most common situation is for a gear to mesh with another gear, but a gear can mesh with any device having compatible teeth, such as other rotational gears, or linear moving racks. A gear's most important feature is that gears of unequal sizes (diameters) can be combined to produce a mechanical advantage, so that the rotational speed and torque of the second gear are different from that of the first. In the context of a particular machine, the term "gear" also refers to one particular arrangement of gears among other arrangements (such as "first gear"). Such arrangements are often given as a ratio, using the number of teeth or gear diameter as units. The term "gear" is also used in non-geared devices which perform equivalent tasks:

General

The smaller gear in a pair is often called the pinion; the larger, either the gear, or the wheel.

Mechanical advantage

Comparison with other drive mechanisms

The automobile transmission allows selection between gears to give various mechanical advantages.

Spur gears

Helical gears

Double helical gears

Bevel gears

Crown gear

Hypoid gears

Worm gear

We should note that the helix angle of a worm is not usually specified. Instead, the lead angle, which is equal to 90 degrees minus the helix angle, is given.

A left hand helical gear or left hand worm is one in which the teeth twist counterclockwise as they recede from an observer looking along the axis.^[11]

Rack and pinion

External vs. internal gears

Parallel axis gears

See also

Gear nomenclature

Tooth contact nomenclature

Point of contact

Line of contact

A line of contact is a line or curve along which two tooth surfaces are tangent to each other.

Path of action

Line of action

The line of action is the path of action for involute gears. It is the straight line passing through the pitch point and tangent to both base circles.

Surface of action

The surface of action is the imaginary surface in which contact occurs between two engaging tooth surfaces. It is the summation of the paths of action in all sections of the engaging teeth.

Plane of action

The plane of action is the surface of action for involute, parallel axis gears with either spur or helical teeth. It is tangent to the base cylinders.

Zone of action (contact zone)

Path of contact

Length of action

Length of action is the distance on the line of action through which the point of contact moves during the action of the tooth profile.

Arc of action, Q_t

Arc of action is the arc of the pitch circle through which a tooth profile moves from the beginning to the end of contact with a mating profile.

Arc of approach, Q_a

Arc of approach is the arc of the pitch circle through which a tooth profile moves from its beginning of contact until the point of contact arrives at the pitch point.

Arc of recess, Q_r

Arc of recess is the arc of the pitch circle through which a tooth profile moves from contact at the pitch point until contact ends.

Contact ratio, m_c, ε

Contact ratio in general is the number of angular pitches through which a tooth surface rotates from the beginning to the end of contact.

Transverse contact ratio, m_p, ε_α

Face contact ratio, m_F, ε_β

Face contact ratio is the contact ratio in an axial plane, or the ratio of the face width to the axial pitch. For bevel and hypoid gears it is the ratio of face advance to circular pitch.

Total contact ratio, m_t, ε_γ

Total contact ratio is the sum of the transverse contact ratio and the face contact ratio.

Modified contact ratio, m_o

Modified contact ratio for bevel gears is the square root of the sum of the squares of the transverse and face contact ratios. $m_o = (m_p^2 + m_F^2)^0.5$

Limit diameter

Start of active profile (SAP)

The start of active profile is the intersection of the limit diameter and the involute profile.

Face advance

Backlash

Epicyclic gearing

Shifting of gears

In some machines (e.g., automobiles) it is necessary to alter the gear ratio to suit the task. There are several methods of accomplishing this. For example:

Tooth profile

The fillet curve (root fillet) is the concave portion of the tooth profile where it joins the bottom of the tooth space.²

Undercut

Pitch

Circular pitch, p

Circular pitch is the arc distance along a specified pitch circle or pitch line between corresponding profiles of adjacent teeth.

Transverse circular pitch, p_t

Normal circular pitch, p_n, p_e

Normal circular pitch is the circular pitch in the normal plane, and also the length of the arc along the normal pitch helix between helical teeth or threads.

Axial pitch, p_x

The term axial pitch is preferred to the term linear pitch. The axial pitch of a helical worm and the circular pitch of its wormgear are the same.

Normal base pitch, p_N, p_bn

Transverse base pitch, p_b, p_bt

Diametral pitch (transverse), P_d

Diametral pitch (transverse) is the ratio of the number of teeth to the standard pitch diameter in inches.

Normal diametral pitch, P_nd

Normal diametral pitch is the value of diametral pitch in a normal plane of a helical gear or worm.

Angular pitch, θ_N, τ

Angular pitch is the angle subtended by the circular pitch, usually expressed in radians.

Cage gear

Gear materials

Tooth thickness

Circular thickness

Circular thickness is the length of arc between the two sides of a gear tooth, on the specified datum circle.

Transverse circular thickness

Transverse circular thickness is the circular thickness in the transverse plane.

Normal circular thickness

Normal circular thickness is the circular thickness in the normal plane. In a helical gear it may be considered as the length of arc along a normal helix.

Axial thickness

Axial thickness in helical gears and worms is the tooth thickness in an axial cross section at the standard pitch diameter.

Base circular thickness

Base circular thickness in involute teeth is the length of arc on the base circle between the two involute curves forming the profile of a tooth.

Normal chordial thickness

Chordial addendum (chordial height)

Profile shift

Rack shift

The rack shift is the displacement of the tool datum line from the reference cylinder, made non-dimensional by dividing by the normal module. It is used to specify the tooth thickness.

Measurement over pins

Span measurement

Modified addendum teeth

Full-depth teeth

Full-depth teeth are those in which the working depth equals 2.000 divided by the normal diametral pitch.

Stub teeth

Stub teeth are those in which the working depth is less than 2.000 divided by the normal diametral pitch.

Equal addendum teeth

Equal addendum teeth are those in which two engaging gears have equal addendums.

Long and short-addendum teeth

Long and short addendum teeth are those in which the addendums of two engaging gears are unequal.

See also

Notes

^ http://auto.howstuffworks.com/cvt1.htm
^ Spur gears on non-parallel shafts can mesh, but only point contact will be achieved, not line contact across the full width of the tooth; also the length of the path of contact may be too short.
^ Doughty and Vallance give the following information on helical gear speeds: "Pitch-line speeds of 4,000 to 7,000 fpm [20 to 36 m/s] are common with automobile and turbine gears, and speeds of 12,000 fpm [61 m/s] have been successfully used." -- p.281.
^ McGraw Hill Encyclopedia of Science and Technology, "Gear", p. 742.
^ McGraw Hill Encyclopedia of Science and Technology, "Gear, p. 743.
^ Doughty and Vallance, p. 287.
^ Doughty and Vallance, pp. 280, 296.
^ Doughty and Vallance, p. 296.
^ This appears wrong to me but it's in two of my references: Doughty and Vallance, p. 290; McGraw Hill Encyclopedia of Science and Technology, "Gear", p. 743.
^ McGraw Hill Encyclopedia of Science and Technology, "Gear", p. 744.
^ ^a ^b ^c ^d ^e ^f ^g ANSI/AGMA 1012-G05, "Gear Nomenclature, Definition of Terms with Symbols".
^ Further note on tooth force: If two rigid objects make contact, they always do so at a point (or points) where the tangents to their surfaces coincide -- that is, where there is a common tangent. The perpendicular to the common tangent at the point of contact is called the common normal. Ignoring friction, the force exerted by the objects on each other is always directed along the common normal. Thus, for meshing gear teeth, the line of action is the common normal to the tooth surfaces.
^ [http://www.worldcatlibraries.org/wcpa/top3mset/2930880 C Michael Hogan and Gary L Latshaw,The Relationship Between Highway Planning and Urban Noise , Proceedings of the ASCE, Urban Transportation Division Specialty Conference by the American Society of Civil Engineers, Urban Transportation Division, May 21 to 23, 1973, Chicago, Illinois

General sources

Venton Levy Doughty and Alex Vallance, Design of Machine Members, 4th edition; 1964; McGraw Hill. First edition was 1936. This is an engineering text on machine parts. It has two chapters on gears.

External links

The gear ratio is the relationship between the number of teeth on two gears that are meshed or two sprockets connected with a common roller chain, or the circumferences of two pulleys connected with a drive belt.

General description

In the picture to the right, the smaller gear has 13 teeth, while the second, larger gear has 21 teeth. The gear ratio is therefore 13/21 or 1/1.62 (also written as 1:1.62).

The first number in the ratio is usually the gear to which power is applied. In an automobile the first number is the gear receiving power from the engine.

This means that for every one revolution of the smaller gear, the larger gear has made 1/1.62, or 0.62, revolutions. In practical terms, the larger gear turns more slowly.

Suppose the largest gear in the picture has 42 teeth, the gear ratio between the second and third gear then is; 21/42 = 1/2 and for every revolution of the smallest gear the largest gear has only turned 0.62/2 = 0.31 revolution, a total reduction of around 1:3.

Since the number of teeth is also proportional to the circumference of the gear wheel (the bigger the wheel the more teeth it has) the gear ratio can also be expressed as the relationship between the circumferences of both wheels (where d is the diameter of the smaller wheel and D is the diameter of the larger wheel):

Because the gear teeth prevent any slippage at the interface of the two gears, we can assume that their velocities are the same at the contact point, and thus we can arrive at

In other words, the gear ratio is proportional to ratio of the gear diameters and inversely proportional to the ratio of gear speeds.

Counting the teeth derives the exact gear ratio, regardless of any variations in the diameter measurement. In the picture, each time the 13 teeth of the smaller gear make a revolution, 13 teeth of the larger gear will have moved, i.e. made 13/21 of a revolution or 0.62 of a revolution. As long as the gears remain meshed, the accounting of teeth and revolutions will remain perfect. So, for instance, gears can be used to construct a clock in which the minute hand moves precisely 12 times faster than the hour hand, regardless of the overall accuracy of the clock. For example, in one hour the minute hand moves once around the clock (1 C) and the hour hand moves 1/12 of the way around the clock (1/12 C).

Diameter measurements are useful for determining approximate gear ratios for non-gear linkages such as pulleys and belts. Smooth belts can slip, so even if exact pulley diameters are known quite exactly, the gear ratio may vary in operation, and may even depend on the load.

Belts can have teeth in them also and be coupled to gear-like pulleys. Special gears called sprockets can be coupled together with chains, as on bicycles and some motorcycles. Again, exact accounting of teeth and revolutions can be applied with these machines.

A belt with teeth, called the timing belt, is used in some internal combustion engines to exactly synchronize the movement of the camshaft with that of the crankshaft, so that the valves open and close at the top of each cylinder at exactly the right time relative to the movement of each cylinder. From the time the car is driven off the lot, to the time the belt needs replacing thousands of kilometers later, it synchronizes the two shafts exactly. A chain, called a timing chain, is used on some automobiles for this purpose, while in others, the camshaft and crankshaft are coupled directly together through meshed gears. But whichever form of drive is employed, on Four-stroke engines the crankshaft/camshaft gear ratio is always 2:1, which means that for every two revolutions of the crankshaft the camshaft will rotate through one revolution.

Automobile drivetrains generally have two or more areas where gearing is used: one in the transmission, which contains a number of different sets of gearing that can be changed to allow a wide range of vehicle speeds, and another at the differential, which contains one additional set of gearing that provides further mechanical advantage at the wheels. These components might be separate and connected by a driveshaft, or they might be combined into one unit called a transaxle.

A 2004 Chevrolet Corvette C5 Z06 with a six-speed manual transmission has the following gear ratios in the transmission:

In 1st gear, the engine makes 2.97 revolutions for every revolution of the transmission’s output. In 4th gear, the gear ratio of 1:1 means that the engine and the transmission’s output are moving at the same speed. 5th and 6th gears are known as overdrive gears, in which the output of the transmission is revolving faster than the engine.

The above Corvette has a differential ratio of 3.42:1. This means that for every 3.42 revolutions of the transmission’s output, the wheels make one revolution. The differential ratio multiplies with the transmission ratio, so in 1st gear, the engine makes 10.16 revolutions for every revolution of the wheels.

The car’s tires can almost be thought of as a third type of gearing. The example Corvette Z06 is equipped with 295/35-18 tires, which have a circumference of 82.1 inches. This means that for every complete revolution of the wheel, the car travels 82.1 inches. If the Corvette had larger tires, it would travel farther with each revolution of the wheel, which would be like a higher gear. If the car had smaller tires, it would be like a lower gear.

With the gear ratios of the transmission and differential, and the size of the tires, it becomes possible to calculate the speed of the car for a particular gear at a particular engine RPM.

For example, it is possible to determine the distance the car will travel for one revolution of the engine by dividing the circumference of the tire by the combined gear ratio of the transmission and differential.

It is possible to determine a car’s speed from the engine speed by multiplying the circumference of the tire by the engine speed and dividing by the combined gear ratio.

Wide-ratio vs. Close-ratio Transmission

A close-ratio transmission is a transmission in which there is a relatively little difference between the gear ratios of the gears. For example, a transmission with an engine shaft to drive shaft ratio of 4:1 in first gear and 2:1 in second gear would be considered wide-ratio when compared to another transmission with a ratio of 4:1 in first and 3:1 in second. This is because, for the wide-ratio first gear = 4/1 = 4, second gear = 2/1 = 2, so the transmission gear ratio = 4/2 = 2 (or 200%). For the close-ratio first gear = 4/1 = 4, second gear = 3/1 = 3 so the transmission gear ratio = 4/3 = 1.33 (or 133%), because 133% is less than 200%, the transmission with the 133% ratio between gears is considered close-ratio. However, not all transmissions start out with the same ratio in 1st gear or end with the same ratio in 5th gear, which makes comparing wide vs. close transmission more difficult.

Close-ratio transmissions are generally offered in sports cars, in which the engine is tuned for maximum power in a narrow range of operating speeds and the driver can be expected to enjoy shifting often to keep the engine in its power band.

Idler Gears

Note that in a sequence of gears chained together, the ratio depends only on the number of teeth on the first and last gear. The intermediate gears, regardless of their size, do not alter the overall gear ratio of the chain. But, of course, the addition of each intermediate gear reverses the direction of rotation of the final gear.

An intermediate gear which doesn't drive a shaft to perform any work is called an idler gear. Sometimes, a single idler gear is used to reverse the direction, in which case it may be referred to as a reverse idler. For instance, the typical automobile manual transmission engages reverse gear by means of inserting a reverse idler between two gears.

Idler gears can also transmit rotation among distant shafts in situations where it would be impractical to simply make the distant gears larger to bring them together. Not only do larger gears occupy more space, but the mass and rotational inertia (moment of inertia) of a gear is quadratic in the length of its radius. Instead of idler gears, of course, a toothed belt or chain can be used to transmit torque over distance.

Contents

General

Mechanical advantage

Comparison with other drive mechanisms

Spur gears

Helical gears

Double helical gears

Bevel gears

Crown gear

Hypoid gears

Worm gear

Rack and pinion

External vs. internal gears

Parallel axis gears

See also

Gear nomenclature

Tooth contact nomenclature

Point of contact

Line of contact

Path of action

Line of action

Surface of action

Plane of action

Zone of action (contact zone)

Path of contact

Length of action

Arc of action, Qt

Arc of approach, Qa

Arc of recess, Qr

Contact ratio, mc, ε

Transverse contact ratio, mp, εα

Face contact ratio, mF, εβ

Total contact ratio, mt, εγ

Modified contact ratio, mo

Limit diameter

Start of active profile (SAP)

Face advance

Backlash

Epicyclic gearing

Shifting of gears

Tooth profile

Undercut

Pitch

Circular pitch, p

Transverse circular pitch, pt

Normal circular pitch, pn, pe

Axial pitch, px

Normal base pitch, pN, pbn

Transverse base pitch, pb, pbt

Diametral pitch (transverse), Pd

Normal diametral pitch, Pnd

Angular pitch, θN, τ

Cage gear

Gear materials

Tooth thickness

Circular thickness

Transverse circular thickness

Normal circular thickness

Axial thickness

Base circular thickness

Normal chordial thickness

Chordial addendum (chordial height)

Profile shift

Rack shift

Measurement over pins

Span measurement

Modified addendum teeth

Full-depth teeth

Stub teeth

Equal addendum teeth

Long and short-addendum teeth

See also

Notes

General sources

External links

Contents

General

Mechanical advantage

Comparison with other drive mechanisms

Spur gears

Arc of action, Q_t

Arc of approach, Q_a

Arc of recess, Q_r

Contact ratio, m_c, ε

Transverse contact ratio, m_p, ε_α

Face contact ratio, m_F, ε_β

Total contact ratio, m_t, ε_γ

Modified contact ratio, m_o

Transverse circular pitch, p_t

Normal circular pitch, p_n, p_e

Axial pitch, p_x

Normal base pitch, p_N, p_bn

Transverse base pitch, p_b, p_bt

Diametral pitch (transverse), P_d

Normal diametral pitch, P_nd

Angular pitch, θ_N, τ

Arc of action, Q_t

Arc of approach, Q_a

Arc of recess, Q_r

Contact ratio, m_c, ε

Transverse contact ratio, m_p, ε_α

Face contact ratio, m_F, ε_β

Total contact ratio, m_t, ε_γ

Modified contact ratio, m_o

Transverse circular pitch, p_t

Normal circular pitch, p_n, p_e

Axial pitch, p_x

Normal base pitch, p_N, p_bn

Transverse base pitch, p_b, p_bt

Diametral pitch (transverse), P_d

Normal diametral pitch, P_nd

Angular pitch, θ_N, τ