8. gear trains

7/29/2019 8. Gear Trains

1/16

1

8. GEAR TRAINS

A gear train is a combination of several gear arranged in a chainfashion such that the follower of one combination is the driver ofthe next, and so on.

Gear trains are widely used in many kinds of mechanismswhenever a change in speed or torque of rotating members isrequired.

8.1. ANGULAR VELOCITY RATIO

Angular velocity ratio is the ratio of the output angular velocity tothe input angular velocity and it is designated by e.

In gear trains it is necessary to identify sense of rotation for eachgear.

In a simple gear train, in many drive applications, an idler gearisused to bridge over the space between the driver and follower.

)1(input

output

driver

followere


2/16

2

This idler gear does not affect the velocity ratio of the system.

The velocity ratio ein a gear train is determined in terms of thenumber of teeth of the drivers and followers in the chain.

For the external gear train shown in fig (a),ignoring the sense ofrotation, the angular velocity ratio is

The diametral pitch of a pair of gears in mesh is defined as

Noting that the peripheral velocity at the point of contact to be

)2(input

outpute

)3(d

N

diameterpitch

geartheonteethofnumberP

)4(3322 ddV


3/16

3

The velocity can be determined in terms of the diametral pitch.

the velocity ratio is given by

where, N2= number of teeth on the driver, andN3= number of teeth on the follower.

)6(

)5(

3

2

2

3

3

3

2

2

N

N

or

P

N

P

NV

)7(3

2

2

3

NNe

input

output


4/16

4

For the internal gear train shown in fig (b), the velocity ratio isgiven by

It can be written also

where the sign is to be placed by inspection.

In general, for a gear train where some gears are drivers andothers are followers, the velocity ratio is given by

)9(3

2

N

N

followertheonteethofnumber

drivertheonteethofnumbere

input

output

)8(3

2

2

3

N

N

einput

output

)10(followertheonteethofnumbersteethofproduct

drivertheonteethofnumbersteethofproduct

einput

output


5/16

5

For the gear train shown in the figure below, the velocity ratio isfound as follows:

For the gear train the velocity ratio is

which, in terms of the teeth numbers, is obtained to be

6

5

5

6

3

4

3

3

4

2

3

2

2

31

N

Ne

N

Ne

N

Ne

)11(321 eeee

)12(6

5

4

2

N

N

N

Ne


6/16

6

From the above velocity ratio equation, it can be noted that:

i. the idler gear 3 does not affect the velocity ratio, and

ii. the velocity ratio is given by

In analyzing gear trains, it is convenient to express angular

speeds in revolutions per minute, rpm.

8.2. TYPES OF GEAR TRAINS

Gear trains are of three types.

i. A simple gear train is one in which only one gear is mounted on

each shaft (see fig 1) the gears are in pure series connection

the gear ratio is usually limited to the ratio 1:10

otherwise, the gear set will become very large and expensive.

)13(followertheonteethofnumbersteethofproduct

drivertheonteethofnumbersteethofproducte


7/16

7

ii. Acompound gear trainis one in whichmore than one gears are mounted onthe gear shaft. (see fig 2)

arranged in parallel or parallel-series

iii. A planetary or epicyclic gear trainisone in which the mounting shaft of oneor more gear is not stationary relativeto mounting shafts of other gears.

Planetary gear shafts rotate about sun

and/or ring gear shafts.

8.3. REVERTED GEAR TRAIN

When the axes of the first and lastgears are co-axial, the gear train is

known as a reverted gear train.


8/16

8

The circular pitch p is defined as

For the same diametral pitch P,

From this we obtain the relationship

and the velocity of the output shaft is given by

)14(

N

dp

)15(PN

dp

)16()()(

)(1

)(1

)(21)(

21

4321

4321

4321

NNNN

NNP

NNP

dddd

)17(1

4

3

2

1

4

N

N

N

N


9/16

9

8.4. PLANETARY GEAR TRAINS

A planetary gear trainis one in which the axes of some of thegears may have planetary motion.

A planetary gear train includes a sun gear, about which one ormore planet gears rotate, and a planet gear carrieror arm.

Have two degree of freedom.


10/16

10

8.5. METHODS OF ANALYSIS OF PLANETARY GEAR TRAINS

There are three methods of the analysis of planetary gear trains

i. the formula method

ii. the tabulation method, and

iii. the instantaneous center method

8.5.1. Solution of Planetary Gear Trains by Formula Method

Consider the planetary gear train shown

Gear 2 is the sun gear and arm 3 is the planet carrier.

Gear 4 and 5, the planet gears fixed to the arm, revolve aboutthe sun gear while also rotating about their own axes.


11/16

11

The velocity of gear 2 relative to the arm 3 is

Also, the velocity of gear 5 relative to the arm is

Dividing equation (19) by equation (18), we obtain

Equation (20) expresses the relative velocity of gear 5 relative togear 2 where both velocities are taken relative to the arm 3.

The ratio (5-3)/(2-3) is the same whether the arm is rotatingor not, and is proportional to the tooth numbers.

i.e. the ratio is the angular velocity ratio of the gear train.

)18(3223

)19(3553

)20(32

35

23

53

)21(32

35 e


12/16

12

Equation (21) is expressed in a more convenient form as

where ,

F is the angular velocity of the first gear in the train

L is the angular velocity of the last gear in the train

A is the angular velocity of the arm or planet gear carrier,

is velocity of last and first gears relative to the arm.

This ratio is equal to the product of teeth numbers of drivergears over product of teeth numbers on driven gears.

Note: In applying equation (22) the following conditions mustbe satisfied.

i. The last and first gears must be gears that mesh with the gear orgears that have planetary motion

ii. The first and last gears must be on parallel shafts because angularvelocities must be parallel in order to treat them algebraically.

)22(

FA

LA

AF

ALe

FA

LA


13/16

13

8.5.2 . Solution of Planetary Gear Trains by Tabulation Method

Consider the planetary gear train composed of a sun gear 2, aplanet gear carrier 3, a planet gear 4 and an internal gear 5

which is in mesh with the planet gear as shown below.

Given certain values for the speed of the sun gear and theplanet gear carrier, we want to determine the speed of internalgear.


14/16

14

In applying the tabulation method, we carry out the followingthree steps.

i. First, lock the gears to the arm and rotate the arm one

positive turn. This causes the gears to rotate one positiveturn with the arm.

ii. Next, fixing the arm, rotate one or more of the sun gears andapply the relation for the angular velocity ratio.

iii. Tabulate the resulting motion and add the turns of each gearin steps (i) and (ii) to satisfy the given conditions of motion.


15/16

15

8.6. PLANETARY GEAR TRAINS WITH TWO INPUTS

A planetary gear train with two inputs is shown below.

The output speed is obtained by superposing as follows:

)23(12

2211 fixedheldfixedheldoutee


16/16

16

8. gear trains

Documents