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An analysis of Gear Trains with particular reference to Epicyclic Gears

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Introduction

One of the most common uses of Gear Trains is in the gear boxes of cars. In the simplest form, the crash gear box, a collection of spur gears were arranged to give different ratios of input to output speed and in the case of the reverse gear, change the direction of the output rotation. The modern synchromesh gearbox is a development working on very similar principles.

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Another, and in many ways more interesting, type of train uses epicyclic gears.Many modern cars use a small epicyclic in the starter motor but much larger and more complicated versions have been used as the main gearbox. Before the 1939 - 1945 war the Daimler Car Company used a "pre-selector" Wilson box.

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In more recent years these were fitted to London Transport Buses and they were also to be found in some British Army armoured wheeled vehicles. The gear box, which is driven through a fluid fly wheel, allows the driver to pre-select the next gear to be used and then to make a very rapid change at the right moment. This ability is useful in heavy urban traffic and when driving across uneven country.

Gear Trains on Fixed Axes

The velocity ratio transmitted between two gears is the same as for two pitch circles rolling together. As the circular pitch must be the same for both wheels, the speed ratio is determined by the number of teeth.

$i.e.\;\;\;\;\;\;\;\;\;\;\frac{n_A}{n_B}\;=\;\pm \;\frac{t_B}{t_A}$

(1)

The positive sign is taken for Internal gearing

The negative sign is taken for External gearing.

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A Simple train is one of three or wheels connected in series.

The gear ratio is independent of the size of the intermediate wheels which are called Idlers

For an odd number of externally geared wheels the direction of rotation of the last wheel is the same as the first.

For an even number of wheels the direction of rotation of the llast wheel is opposite to that of the first.

A Compound Chains

When an intermediate shaft carries two wheels connected in series, the train becomes compound. e.g. If A gears with B which is compounded with C which then drives D.

$i.e.\;\;\;\;\;\;\;\;\;\;\frac{n_A}{n_D}\;=\;\frac{t_B}{t_A}\times \frac{t_D}{t_C}$

(2)

. Such arrangements can greatly increase or decrease the gear ratio.

Where the axis of the last wheel is in line with that of the first, the arrangement is known as a reverted train.

Epicyclic Trains.

One or more of the wheels of the gear train are now carried on an arm which itself rotates about the axis of the main wheels.

The central main wheel is called the Sun wheel. The wheels carried by the arm are known as The Planet wheels and the outside internal wheel is called the Annulus. In some epicycilic gear trains the "arm" has a number of "limbs" each carrying a planet wheel. In these cases the "arm" may be referred to as the Spider

There are a number of ways of analysing epicyclic gear trains. Three are described here:-

Tabular Method

The first line of the table is obtained by fixing the arm and then giving one wheel ( e.g. the wheel which later is to be fixed) one revolution.

Write down the the corresponding revolutions of the other wheels from their number of teeth.

As all the wheels and the arm can be turned about the same axis, an equal number of revolutions may be added (or subtracted) from each wheel.

Any two lines may be multiplied (or divided) through by the same factor.

Any two lines may be combined. By these means the given conditions are satisfied and the resulting gear ratio can be determined.

Note. The above may seem confusing and should be read in conjunction with the worked examples and in particular Example (3)

The Relative Velocity Method.

If the arm is fixed, then the gear becomes a simple or compound train. Thus the ratio of the speeds of any two wheels can be determined by using the methods of paragraph (2).

$i.e.\;\;\;\;\;\;\;\frac{n_A\;-\;n_{arm}}{n_B\;-\;n_{arm}}\;=\;\frac{speed\;of\;A}{speed\;of\;B}\;\;\;\;\;If\;the\;arm\;were\;fixed$

(3)

Where there is an intermediate shaft between A and B which carries two wheels C and D which are fixed together and arranged so that A drives C and D drives B, then the above equation can be modified to:-

$\frac{(n_H\;-\;n_{arm})}{(n_B\;-\;n_{arm})}\;=\;\frac{t_C}{t_A}\times \frac{t_B}{t_D}$

(4)

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which can be calculated from the number of teeth etc.

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For simple epicyclic operating with a fixed annulus there is nothing to choose between the above two methods. However if all the wheels are moving or when dealing with compound epicyclic, the second method gives a quicker and more direct solution. A comparison of the methods can be made from Examples (3) and (4)

The Determination of Torques

There are normally three torques acting on an epicyclic gear. These are:-

$\displaystyle \tau _1,$ the input Torque (in the sense of rotation of the input shaft)

$\displaystyle \tau _0,$ the output torque (opposite to the sense of rotation of the output shaft)

$\displaystyle \tau _C\;,$ the Torque on the casing.

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Assuming the casing to be fixed

$\eta \;\tau _1\;n_1\;=\;\tau _0\;n_0$

(5)

From considerations of the power and for equilibrium ( Taking signs into account )

$\tau _1\;+\;\tau _0\;+\;\tau _1\;=\;0$

(6)

For input and output shafts to have the same direction of rotation

$\;\;\;\;\;\;\tau _C\;=\;\tau _0\;+\;\tau _1\;\;\;\;\;\;numerically$

(7)

For input and output to have opposite directions of rotation.

$\tau _C\;=\;\tau _0\;+\;\tau _1\;\;\;\;\;\;\;numerically$

(8)

When there is no fixed wheel, there will, in general be three torques to satisfy the following equations. Of the torques one can be found from the horse-power and speed.

$\tau _1\;+\;\tau _2\;+\;\tau _3\;=\;0$

(9)

$And\;\;\;\;\;\;\;\tau _1\;n_1\;+\;\tau _2\;n_2\;+\;\tau _3\;n_3\;=\;0$

(10)

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Alternatively. The forces acting on each wheel (or compound pair) may be analysed separately, since if the gear is running at constant speed, there is no resultant force and no resultant torque on any intermediate wheel.

Worked Examples

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Example 1

Two shafts A and B in the same line are geared together through an intermediate parallel shaft C. The wheels connecting A and C have a diametral pitch of 9 and those connecting C and B 4, the least number of teeth on any wheel being not less than 15. The speed of B is to be about but not greater than 1/12 the speed of A and the ratio of each reduction is the same. Find suitable wheels, the actual reduction and the distance of shaft C from A and B.

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Example 2

The lead screw of a lathe has a right-hand single thread with a pitch of 1/4 in. The smallest change wheel has 20 teeth, the largest 120 teeth and the number of teeth on intermediate sizes increases in steps of 5. Find a gear train suitable for connecting the spindle and the lead screw when:- (a) A right hand screw with 26 threads per inch is to be cut. (b) A left-hand screw with 35 threads per inch is required.

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Example 3

In the Epicyclic Gear shown the internal wheels A and F and the compound wheel C-D rotate independently about the axis O. The wheels Band E rotate on pins fixed to the arm L. The wheels all have the same pitch and the numbers of teeth are : B and E 18, C 28, and D 26.

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If L makes 150 r.p.m. clockwise, find the speed of F when :- (a) The Wheel A is fixed. (b) The Wheel A makes 15 r.p.m. counter-clockwise.

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Example 4

In the epicyclic gear train shown in the diagram, wheel A and wheel E (30 teeth) are fixed to a sleeve Y which is free to rotate on spindle X. B(24 teeth) and C (22 teeth) are keyed to a shaft which is free to rotate in a bearing on arm F. D has 70 teeth. H has 15 teeth and is mounted on a shaftV which rotates at 100 r.p.m. Spindle X makes 300 r.p.m. in the same direction as V. All the teeth are of the same pitch.

Find the speed and direction of rotation of Z (U.L.)

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Example 5

A gear train is shown in the diagram in which the shaft X rotates at 500 r.p.m. Solid with shaft Xis the arm A on which the compound bevel wheels B and D can revolve freely together; Wheel B meshes with wheel C and wheel D with wheel E which is also solid with the spur wheel F; the latter meshes with with G on shaft Y which rotates in the opposite direction to X. The bevel wheel C is fixed by keying to the support.

The number of teeth on wheels C, D, E, F, and G are 20, 27, 32, 24, and 30 respectively. Determine the number of teeth required on wheel B to give a speed reduction of 20 to 1 between shafts X and Y

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Example 6

In the epicyclic gear unit shown in the diagram the input shaft is attached to the planet carrier

Z and the output shaft to the Sun wheel A. Sun wheel A_1 is fixed to the casing and does not rotate whist the planet wheels C and C_1 are compounded and rotate together. If all teeth are the same pitch and A and C each have 20 teeth, find the teeth onA_1 andC_1 so that the output shaft runs in the reverse direction and at half the speed of the input shaft.

If the input shaft transmits 14 h.p. at 1500 r.p.m.and the efficiency of the unit is 96% calculate the input and output torques and the fixing torque of the casing.

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Example 7

The diagram shows a system of gearing. The casing N is fixed. Wheel H is keyed to shaft B. Wheels K and L are fixed to each other and are free to rotate on shaft B. Wheel E is also free to rotate on shaft B. Wheels F and G are fixed to each other and are free to rotate on a pin mounted on wheel E. The number of teeth on the various wheels is shown on the diagram and all the teeth are of the same diametral pitch.

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If A rotates at 200 r.p.m., whilst C is held stationary, find te speed of B..Find also the torque on shaft C to hold it stationary under these conditions if 2 h.p. is being transmitted. (U.L.)

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Example 8

The diagram shows a compound epicyclic gear in which the casing C contains an epicyclic train and this casing is inside the larger casing D

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Determine the velocity ratio of the output shaft B to the input shaft A when the casing D is help stationary. The numbers indicate the number of teeth on the various wheels.

Wheel on A 80; Annular wheel B 160; Annular wheel on C 100; Annular wheel on D 120; Small pinion on F 20; Large pinion on F 66. (U.L.)

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Example 9

An epicyclic gear train has a fixed outer annulus, a sun wheel with 50 teeth and three planet wheels (each having 25 teeth) carried on a spider. The radius of the path of the centre of each planet wheel is 4 in. The weights of the various members and their radii of gyration about their respective polar axis given in the following table:-

Determine the total kinetic energy of the gear train in ft.lb., if the sun wheel is driven at 600 r.p.m. (U.L.)

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