Wednesday, February 27, 2008

Introduction to Gears

Gears are used in all types of machinery. They can be as small as 3mm diameter to 11m diameter. They are used to transmit rotary motion from one shaft to another by means of toothed wheels, which are in direct mesh with each other. In contrast, chain wheels transmit motion from one wheel to another by means of chain connection. They are called sprockets. Gears are used for power transmission as in automobile gearbox and in control elements.

Spur gears:
Spur gears are the most commonly used gears, are used to transmit motion between parallel shafts. These impose load on bearings. Its teeth are straight and parallel to the axis. These can be either internal type or external type. An external spur gear of infinite radius, called as rack. Rack has straight-sided tooth (for involute tooth profile gears).

Helical gears:
In helical gears, the tooth profile in transverse plane gets gradually rotated along the helix angle as we move along the axis. The hand of the helix determines whether it is a right handed helical gear or a left handed helical gear. Helical gears can be external or internal type. However, internal helical gears are not very common in external helical gears, the hands of the mating gears are of opposite hands, whereas for internal helical gear meshing with external gear, they are of the same hand. The mating gear should always have the same helix angle.
Sometimes double helical gears are also used where they have both right handed and left handed helical teeth on each gear. Normally there is a gap between the two helices. However, there are gears which have no gap between the helices double helical gear are also called chevron gears. Single helical gear impose both radial and thrust load on bearings. Double helical gear normally impose radial loads as the thrust loads due to opposite helices are in the opposite directions and, therefore, they cancel out. The spur gears can be considered as a special case of gears with helix angle equal to zero.

Terms Used in Gears

The following terms, which will be mostly used in this chapter, should be clearly understood at this stage. These terms are illustrated in fig.

1) Pitch circle : It is an imaginary circle which by pure rolling action, would give the same motion as the actual gear.

2) Pitch circle diameter : It is the diameter of the pitch circle. The size of the gear is usually specified by the pitch circle diameter. It is also called as pitch diameter.

3) Pitch point : It is a common point of contact between two pitch circles.

4) Pitch surface : It is the surface of the rolling discs which the meshing gears have replaced at the pitch circle.

5) Pressure angle or angle of obliquity : It is the angle between the common normal to two gear teeth at the point of contact and the common tangent at the pitch point. It is usually denoted by The standard pressure angles are 14½o and 20o.

6) Addendum : It is the radial distance of a tooth from the pitch circle to the top of the tooth.

7) Dedendum : It is the radial distance of a tooth from the pitch circle to the bottom of the tooth.

8) Addendum circle: It is the circle drawn through the top of the teeth and is concentric with the pitch circle.

9) Dedendum circle: It is the circle drawn through the bottom of the teeth. It is also called root circle.
Note : Root circle diameter = Pitch circle diameter X cos where  is the pressure angle.

10) Circular pitch: It is the distance measured on the circumference of the pitch circle from a point of one tooth to the corresponding point on the next tooth. It is usually denoted by Pc , Mathematically.

Circular pitch, Pc = D/T
where D = Diameter of the pitch circle, and
T = Number of teeth on the wheel.

A little consideration will show that the two gears will mesh together correctly, if the two wheels have the same circular pitch.

Note : If D1 and D2 are the diameters of the two meshing gears having the teeth T1 and T2 respectively, then for them to mesh correctly.
Pc = D1/T1 = D2/T2 or D1/D2 = T1/T2

11) Diametral pitch : It is the ratio of number of teeth to the pitch circle diameter in millimeters. It is denoted by Pd Mathematically,

Diametral pitch, Pd = T/D = /Pc
Where T = Number of teeth, and
D = Pitch circle diameter.

12) Module: It is the ratio of the pitch circle diameter in millimeters to the number of teeth. It is usually denoted by m. Mathematically,

Module, m = D/T

Note : The recommended series of modules in Indian Standard are 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10, 12, 16, and 20. The modules 1.125, 1.375, 1.75, 2.25, 2.75, 3.5, 4.5 5.5, 7, 9, 11, 14 and 18 are of second choice.

13) Clearance: It is the radial distance from the top of the tooth to the bottom of the tooth, in a meshing gear. A circle passing through the top of the meshing gear is known as clearance circle.

14) Total depth: It is the radial distance between the addendum and the dedendum of a gear. It is equal to the sum of the addendum and dedendum.

15) Working depth: It is the radial distance from the addendum circle to the clearance circle. It is equal to the sum of the addendum of the two meshing gear.

16) Tooth thickness: It is the width of the tooth measured along the pitch circle.

17) Tooth space: It is the width of space between the two adjacent teeth measured along the pitch circle.

18) Backlash: It is the difference between the tooth space and the tooth thickness, as measured along the pitch circle. Theoretically, the backlash should be zero, but in actual practice some backlash must be allowed to prevent jamming of the teeth due to tooth errors and thermal expansion.

19) Face of tooth: It is the surface of the gear tooth above the pitch surface.

20) Flank of tooth: It the surface of the gear tooth below the pitch surface.

21) Top land: It is the surface of the top of the tooth.

22) Face width: It is the width of the gear tooth measured parallel to its axis.

23) Profile: It is the curve formed by the face and flank of the tooth.

24) Fillet radius: It is the radius that connects the root circle to the profile of the tooth.

25) Path of contact: it is the path traced by the point of contact of two teeth from the beginning to the end of engagement.

26) Length of the path of contact : It is the length of the common normal cut-off by the addendum circles of the wheel and pinion.

27) Arc of contact: It is the path traced by a point on the pitch circle from the beginning to the end of engagement of a given pair of teeth. The arc of contact consists of two parts, …i.e.,
(a) Arc of approach : It is the portion of the path of contact from the beginning of the engagement to the pitch point.
(b) Arc of recess : it is the portion of the path of contact from the pitch point to the end of the engagement of a pair of teeth.
Note : The ratio of the length of arc of contact to the circular pitch is known as contact ratio i.e. number of pairs of teeth in contact.

Gear Materials

The material used for the manufacture of gears depends upon the strength and service conditions like wear, noise etc. The gears may be manufactured from metallic or non-metallic materials. The metallic gears with cut teeth are commercially obtainable in cast iron, steel and bronze. The non-metallic materials like wood, raw hide, compressed paper and synthetic resins like nylon are used for gears, especially for reducing noise.
The cast iron is widely used for the manufacture of gears due to its good wearing properties, excellent machinability and ease of producing complicated shapes by casting method. The cast iron gears with cut teeth may be employed, where smooth action is not important.
The steel is used for high strength gears and steel may be plain carbon steel or alloy steel. The steel gears are usually heat treated in order to combine properly the toughness and tooth hardness.
The phosphor bronze is widely used for worm gears in order to reduce wear of the worms which will be excessive with cast iron or steel.

Comparison Between Involute and Cycloidal Gears

In actual practice, the involute gears are more commonly used as compared to cycloidal gears, due to the following advantages:

Advantages of involute gears:
Following are the advantages of involute gears:
1) The most important advantage of the involute gears is that the center distance for a pair of involute gears can be varied within limits without changing the velocity ratio. This is not true for cycloidal gears which requires exact center distance to be maintained.

2) In involute gears, the pressure angle, from the start of the engagement of teeth to the end of the engagement, remains constant. It is necessary for smooth running and less wear of gears. But in cycloidal gears, the pressure angle is maximum at the beginning of engagement, reduces to zero at pitch point, starts increasing and again become maximum at the end of engagement. This results in less smooth running of gears.

3) The face and flank of involute teeth are generated by a single curve where as in cycloidal gears, double curves (i.e. epi-cycloid and hypo-cycloid) are required for the face and flank respectively. Thus the involute teeth are easy to manufacture than cycloidal teeth. In involute system, the basic rack has straight teeth and the same can be cut with simple tools.

Note : The only disadvantage of the involute teeth is that the interference occurs with pinions having smaller number of teeth. This may be avoided by altering the heights of addendum and dedendum of the mating teeth or the angle of obliquity of the teeth.

Advantages of cycloidal gears :
1) Since the cycloidal teeth have wider flanks, therefore the cycloidal gears are stronger than the involute gears, for the same pitch. Due to this reason, the cycloidal teeth are preferred specially for cast teeth.

2) In cycloidal gears, the contact takes place between a convex flank and concave surface, where as in involute gears, the convex surface are in contact. This condition results in less wear in cycloidal gears as compared to involute gears. However the difference in wear is negligible.

3) The cycloidal gears, the interference does not occur at all. Though there are advantages of cycloidal gears but they are outweighted by the greater simplicity and flexibility of the involute gears.

Systems of Gear Teeth

The following four systems of gear teeth are commonly used in practice :
1. 14½o Composite system,
2. 14 ½o Full depth involute system,
3. 20o Full depth involute system, and
4. 20o Stub involute system.

The 14½o composite system is used for general purpose gears. It is stronger but has no interchangeability. The tooth profile of this system has cycloidal curves at the top and bottom and involute curve at the middle portion. The teeth are produced by formed milling cutters or hobs. The tooth profile of the 14½o full depth involute system was developed for use with gear hobs for spur and helical gears.
The tooth profile of the 20o full depth involute system may be cut by hobs. The increase of the pressure angle from 14½o to 20o results in a stronger tooth, because the tooth acting as a beam is wider at the base. The 20o stub involute system has a strong tooth to take heavy loads.

Standard Proportions of Gear Systems

S. No.


14½o composite or full depth involute system

20o full depth involute system

20o slub involute system



1 m

1 m

0.8 m



1.25 m

1.25 m

1 m


Working depth

2 m

2 m

1.60 m


Minimum total depth

2.25 m

2.25 m

1.80 m


Tooth thickness

1.5708 m

1.5708 m

1.5708 m


Minimum clearance

0.25 m

0.25 m

0.2 m


Fillet radius at root

0.4 m

0.4 m

0.4 m

Law of Gearing

The law of gearing states the conditions which must be fulfilled by the gear tooth profiles to maintain a constant angular velocity ratio between two gears. It states that the common normal at the point of contact of the two teeth should always pass through a common pitch point.