2. Introduction
Rolling Cylinders
The Fundamental Law of Gearing
Gear Tooth Nomenclature
Gear Types
Simple Gear Trains
Compound Gear Trains
Reverted Gear Trains
Epicyclic or Planetary Gear Trains
Examples
2
3. Gear trains are widely used in all kinds of mechanisms and
machines, from can openers to aircraft carriers. Whenever a change
in the speed or torque of a rotating device is needed, a gear train or
one of its cousins, the belt or chain drive mechanism, will usually be
used.
3
4. The simplest means of transferring rotary motion from one
shaft to another is a pair of rolling cylinders. They may be an
external set of rolling cylinders or an internal set. Provided
that sufficient friction is available at the rolling interface,
this mechanism will work quite well. There will be no slip
between the cylinders until the maximum available frictional
force at the joint is exceeded by the demands of torque
transfer.
4
5. 5
A variation on this mechanism is what causes your car or bicycle to move
along the road. Your tire is one rolling cylinder and the road the other
(very large radius) one. Friction is all that prevents slip between the two,
and it works well unless friction coefficient is reduced by the presence of
ice or other slippery substances.
A variant on the rolling cylinder drive is the flat or V belt. This
mechanism also transfers power through friction and is capable of quite
large power levels, provided enough belt cross section is provided.
Friction belts are used in a wide variety of applications from small sewing
machines to the alternator drive on your car, to multi horsepower
generators and pumps. Whenever absolute phasing is not required and
power levels are moderate, a friction belt drive may be the best choice.
They are relatively quiet running, require no lubrication, and are
inexpensive compared to gears and chain drives.
6. 6
The principal drawbacks to the rolling cylinder drive (or smooth belt)
mechanism are its relatively low torque capability and the possibility of
slip.
“The angular velocity ratio between the gears of a gear-set
remains constant throughout the mesh.”
An external set reverses the direction of rotation between
the cylinders and requires the negative sign. An internal
Gear-set or a belt or chain drive will have the same direction
of rotation on input and output shafts and require the positive
sign .
7. 7
Circular Pitch:
The circular pitch is the arc length along the pitch circle circumference
measured from a point on one tooth to the same point on the next. The
circular pitch defines the tooth size.
where d = pitch diameter
and N = number of teeth.
8. 8
Base Pitch:
The tooth pitch can also be measured along the base circle circumference and then is
called the base pitch pb.
Diametral Pitch:
The diametral pitch pd is:
pd = N/d
Module:
It is the reciprocal of diametral pitch with pitch diameter measured in millimeters.
m = d/N
In terms of diameters and number of teeth, the Velocity and Torque ratios are:
Gear Ratio:
The gear ratio mG is always > 1 and can be expressed in terms of either the velocity ratio or torque
ratio depending on which is larger than 1. Thus mG expresses the gear train's overall ratio independent
of change in direction of rotation or of the direction of power flow through the train when operated
either as a speed reducer or a speed increaser.
9. 9
SPUR GEARS:
SPUR GEARS are ones in which the teeth are parallel to
the axis of the gear. This is the simplest and least
expensive form of gear to make. Spur gears can only be
meshed if their axes are parallel.
HELICAL GEARS:
HELICAL GEARS are ones in which the teeth are at
a helix angle ψ with respect to the axis of the gear.
10. 10
HERRINGBONE GEARS :
HERRINGBONE GEARS are formed by joining two helical
gears of identical pitch and diameter but of opposite hand on
the same shaft. These two sets of teeth are often cut on the
same gear blank.
WORMS AND WORM GEARS:
If the helix angle is increased sufficiently, the result will be a
worm, which has only one tooth wrapped continuously
around its circumference a number of times, analogous to a
screw thread. This worm can be meshed with a special worm gear
(or worm wheel), whose axis is perpendicular to that of the worm.
Because the driving worm typically has only one tooth, the ratio of
the gear-set is equal to one over the number of teeth on the worm
gear.
“Perhaps the major advantage of the worm-set is that it can
be designed to be impossible to back drive.”
11. 11
RACK AND PINION:
Sometimes, the gear of a shaft meshes externally and internally with the gears in
a straight line. Such type of gear is called rack and pinion. The straight line gear
is called rack and the circular wheel is called pinion. A little consideration will
show that with the help of a rack and pinion, we can convert linear motion into
rotary motion and vice-versa.
BEVEL GEARS:
For right-angle drives, crossed helical gears or a worm-set can be used. For any
angle between the shafts, including 90°, bevel gears may be the solution. Just as
spur gears are based on rolling cylinders, bevel gears are based on rolling cones .
The angle between the axes of the cones and the included angles of the cones can
be any compatible values as long as the apices of the cones intersect. If they did
not intersect, there would be a mismatch of velocity at the interface.
12. 12
SPIRAL BEVEL GEARS:
If the teeth are parallel to the axis of the gear, it will be a straight bevel
gear. If the teeth are angled with respect to the axis, it will be a spiral
bevel gear, analogous to a helical gear. The cone axes and apices must
intersect in both cases. The advantages and disadvantages of straight bevel
and spiral bevel gears are similar to those of the spur gear and helical gear,
respectively, regarding strength, quietness, and cost.
13. A gear train is any collection of two or more meshing gears. A simple gear
train is one in which each shaft carries only one gear. The velocity ratio
(sometimes called train ratio) of this gear-set is found by expanding
equation:
Consider a simple gear train with four gears in series. The expression for this
simple train's velocity ratio is:
mv = (-N1/N2) (-N2/N3) (-N3/N4) = -N1/N4
It may be noted that when the number of intermediate gears are odd, the
motion of both the gears is like. But if the number of intermediate gears are
even, the motion of the driven or follower will be in the opposite direction
of the driver. These intermediate gears are called idle gears.
“Simple Gear Trains are limited to a ratio of about
10: 1”.
13
14. 14
When there are more than one gear on a shaft, it is called a compound train
of gear. Since the idle gears, in a simple train of gears do not effect the
speed ratio of the system. But these gears are useful in bridging over the
space between the driver and the driven.
To get a train ratio of greater than about 10:1 with spur, helical, or bevel
gears (or any combination thereof) it is necessary to compound the train.
The speed ratio of compound gear train is obtained as:
mv = (-N1/N2)x(-N3/N4)x(-N5/N6)
“The advantage of a compound train over a simple
gear train is that a much larger speed reduction from
the first shaft to the last shaft can be obtained
with small gears.”
*Since gears 2 and 3 are mounted on one shaft B, therefore N2 = N3. Similarly
gears 4 and 5 are mounted on shaft C, therefore N4 = N5.”
15. 15
When the axes of the first gear (i.e. first driver) and the last gear (i.e. last
driven or follower) are co-axial, then the gear train is known as reverted
gear train
Let T1 = Number of teeth on gear 1,
r1 = Pitch circle radius of gear 1, and
N1 = Speed of gear 1 in r.p.m.
Similarly,
T2, T3, T4 = Number of teeth on respective gears,
r2, r3, r4 = Pitch circle radii of respective gears, and
N2, N3, N4 = Speed of respective gears in r.p.m.
Since the distance between the centers of the shafts of
gears 1 and 2 as well as gears 3 and 4 is same, therefore
r1 + r2 = r3 + r4
mv = N1/N4 = T2 xT4/T1x T3
16. 16
The conventional gear trains described in the previous sections are all one-
degree-of-freedom (DOF) devices. Another class of gear train has wide
application, the epicyclic or planetary train. This is a two-DOF device. Two
inputs are needed to obtain a predictable output.
The epicyclic gear trains are useful for transmitting high velocity ratios with
gears of moderate size in a comparatively lesser space. They are used in the
back gear of lathe, differential gears of the automobiles, hoists, pulley
blocks, wrist watches etc.
17. 17
Velocity Ratios of Epicyclic Gear Trains
1. Tabular method. 2. Algebraic method.
1. Tabular method
Consider an epicyclic gear train:
Let TA = Number of teeth on gear A, and
TB = Number of teeth on gear B.
First of all, let us suppose that the arm is fixed. Therefore the axes of both the gears
are also fixed relative to each other. When the gear A makes one revolution
anticlockwise, the gear B will make TA / TB revolutions, clockwise. Assuming the
anticlockwise rotation as positive and clockwise as negative, we may say that when gear A makes + 1
revolution, then the gear B will make (– TA / TB) revolutions.
Secondly, if the gear A makes + x revolutions, then the gear B will make – x × TA / TB revolutions.
Thirdly, each element of an epicyclic train is given + y revolutions. And Finally, the motion of each
element of the gear train is added up.
18. 18
Example 1:
Find the gear ratio, circular pitch, base pitch, pitch diameters, pitch radii,
center distance, addendum, dedendum, whole depth, clearance, and outside
diameters of a gear-set with the given parameters.
Given: Diametral Pitch Pd = 6, 19-tooth pinion is meshed with a 37-tooth gear.
Solution:
1. Gear Ratio
2. Circular Pitch
3. Base Pitch
4. Pitch Diameters and Pitch Radii
19. 19
6. Center Distance
7. Addendum and Dedendum
9. Whole Depth
10.Clearance
11. Outside Diameters
20. 20
Example 2:
The gearing of a machine tool is shown in Figure. The motor shaft is connected to gear A
and rotates at 975 r.p.m. The gear wheels B, C, D and E are fixed to parallel shafts
rotating together. The final gear F is fixed on the output shaft. What is the speed of
gear F ? The number of teeth on each gear are as given below :
No. of teeth on each Gear,
TA = 20, TB = 50, TC = 25, TD = 75, TE = 26, TF = 65
Solution:
NA = 975 r.p.m. ;
NF = ?
mv = NA /NB x NC /ND x NE /NF = NA /NF
Or
NA /NF = TB /TAx TD /TC x TF /TE
So,
NA /NF = 50 /20x 75 /25x 65 /26
NF = NA /(18.75)
NF = 975 /(18.75)
NF = 52 r.p.m. Ans
21. 21
Example 3:
The speed ratio of the reverted gear train, as shown in Figure, is to be 12. The module pitch of gears A and
B is 3.125 mm and of gears C and D is 2.5 mm. Calculate the suitable numbers of teeth for the gears. No
gear is to have less than 24 teeth.
Given : Speed ratio, NA/ND = 12 ;mA = mB = 3.125 mm ; mC = mD = 2.5 mm
Solution:
Let NA = Speed of gear A,
TA = Number of teeth on gear A,
rA = Pitch circle radius of gear A,
NB, NC , ND = Speed of respective gears,
TB, TC , TD = Number of teeth on respective gears, and
rB, rC , rD = Pitch circle radii of respective gears.
Since the speed ratio between the gears A and B and between the gears C and D are to be same, therefore
NA/NB = NC/ND = √12 = 3.464
Also the speed ratio of any pair of gears in mesh is the inverse of their number of teeth, therefore
TB/TA = TD/TC = √12 = 3.464
We know that the distance between the shafts
x = rA + rB = rC + rD = 200mm
or mA . TA /2 + mB . TB /2 = mC . TC /2 + mD . TD /2 = 200
After solving we have,
TA = 28 , TB = 100, TC = 36, TD = 124 Ans
and
Speed ratio, NA/ND = TB/TA x TD/TC = 100/28 x 124/36= 12.3
22. 22
Example 4:
In an epicyclic gear train, an arm carries two gears A and B having 36 and 45
teeth respectively. If the arm rotates at 150 r.p.m. in the anticlockwise
direction about the centre of the gear A which is fixed, determine the speed
of gear B.
Given : TA = 36; TB = 45; NARM = 150 r.p.m (CCW); NA = 0
Solution:
At first we consult the table for epicyclic gear train,
Since,
NARM = 150 r.p.m (CCW); and NA = 0
So,
x + y = 0
or,
x = – y = – 150 r.p.m.
=> Speed of gear B, NB = y – x . TA / TB
and
NB = 150 + 150 . 36/45
NB = 270 r.p.m (CCW)
