Class 28 - Modeling of a Gear Train

System Modeling Coursework

P.R. VENKATESWARANFaculty, Instrumentation and Control Engineering,

Manipal Institute of Technology, ManipalKarnataka 576 104 INDIAPh: 0820 2925154, 2925152

Fax: 0820 2571071Email: [email protected], [email protected]

Web address: http://www.esnips.com/web/SystemModelingClassNotes

Class 28: Modeling of Gear Train

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http://www.esnips.com/web/SystemModelingClassNotes

July – December 2008 prv/System Modeling Coursework/MIT-Manipal 2

WARNING!

•

I claim no originality in all these notes. These are the compilation from various sources for the purpose of delivering lectures. I humbly acknowledge the wonderful help provided by the original sources in this compilation.

•

For best results, it is always suggested you read the source material.


Contents

•

Uses of Gear Trains•

Description

•

Derivation of Gear Ratio•

Numerical


Use of Gear Trains

•

Gear Trains are used in control systems to alter the speed to torque ratio of the rotational power transmitted from motor to load. This is necessary to match the torque requirement of the load to that of the motor.

•

Usually a servomotor operates at high speed but has low torque. To drive a load with high torque and low speed using a servomotor, the torque magnification and speed reduction are achieved by gear trains.

•

The gear train in mechanical rotational system is analogous to transformer in electrical system.


Description

•

Consider the motor driving a load through a gear train as shown in figure.

•

Let the gear train consists of two gears with teeth N1

and N2

.•

The gear connected to motor shaft is called primary gear and the gear connected to the load shaft is called the secondary gear.


Notations

•

N1

-

Number of teeth in Gear 1•

r1

–

Radius of Gear 1•

θ1

–

Angular displacement of the shaft•

J1 –

Moment of inertia of motor and Gear 1

•

B1

–

Viscous friction coefficient of motor and Gear 1•

T1

–

Load Torque on Gear 1•

Tm

–

Torque developed by motor•

Tl

–

Load TorqueSimilarly, N2 , r2 ,

θ2

,J2 , B2 ,T2 can be respectively defined for Gear 2


Torque equations

•

The torque developed by motor is balanced by the sum of load torque requirement on Gear 1 and opposing torques due to J1

and B1

. Hence, the torque balance equation for motor shaft is given by,

•

The torque transmitted to Gear 2 is balanced by the sum of load torque and opposing torques due to J2

and B2

. Hence the torque balance equation for load shaft is given by

ml TTdtdB

dtdJ =++ 1

121

2

1θθ

22

222

2

2 TTdt

dBdt

dJ l =++θθ


Relation for displacement

•

When the motor drives the load, the linear distance traveled by each gear is the same. The linear distance traveled by a gear is

given by the product of the radius and angular displacement.•

Linear distance traveled by the gear =θ1

r1

=θ2

r2

. Therefore

•

The number of gears in each gear is proportional to its radius. i.e. N1

αr1

and N2

αr2

. Hence,

2

1

1

2

rr

=θθ

1

2

2

1

2

1

θθ

==rr

NN


Work done by the gears

•

In ideal gear train system, there is no power loss in transmission. Hence work done by both the gears is equal. The work done by a gear is given by the product of torque action on it and its angular displacement.

•

Work done by the gear = T1

θ1

=T2

θ2

. Therefore

1

2

2

1

θθ

=TT


Work done by the gears

•

Differentiating the equation of work done, we get

•

Differentiating one more time, the relation will become2211

22

11

ωω

θθ

TTdt

dTdtdT

=

=

1

2

1

2

2

1

aa

TT

==ωω


Summing up…

•

When N1

>N2

, the gear train increase the speed and reduce the torque

•

When N1

=N2

, there is no change in speed and torque.•

When N1

<N2

, the gear train decrease the speed and increase the torque

1

2

1

2

2

1

1

2

2

1

2

1

aa

TT

rr

NN

=====ωω

θθ


Numerical

• A gear train consisting of two gears is used to drive a load. One gear consists of 20 teeth and the other has 10 teeth. a)

What is the ratio of the diameters of the gears?

b)

If Gear 1 is rotated by an angle of 400, then what will be the angular displacement of Gear 2

c)

If the angular speed of Gear 1 is 30 rad/sec then what is the value of angular speed of Gear 2?

d)

If the angular acceleration of Gear 2 is 4 rad/sec2

then find the angular acceleration of Gear 1

e)

If the torque acting on Gear 1 is 5N-m, then find the torque on Gear 2.


Answers

a)

What is the ratio of the diameters of the gears?The diameter of the gear is proportional to radius. Hence the ratio of the radius is equal to the ratio of the diameters.From the equation of the gear ratio,

where r1

, r2

= radius of Gear 1 and Gear 2N1

, N2

=Number of teeth in Gear 1 and Gear 2.Therefore, the ratio of diameters =

2

1

2

1

rr

NN

=

22020

2

1

2

1 ===rr

NN


Answers

b)

If Gear 1 is rotated by an angle of 400, then what will be the angular displacement of Gear 2

Displacement for Gear 2 is ,

1

2

2

1

θθ

=NN

001

2

12 8040

1020

=×=×= θθNN


Answers

c)

If the angular speed of Gear 1 is 30 rad/sec then what is the value of angular speed of Gear 2?

Angular speed of gear 2,

sec/60301020

12

12 rad

NN

=×== ωω

1

2

2

1

ωω

=NN


Answers

d)

If the angular acceleration of Gear 2 is 4 rad/sec2

then

find the angular acceleration of Gear 1

Angular acceleration of Gear 1 is 1

2

2

1

aa

NN

=

22

1

21 sec/24

2010 rada

NNa =×==


Answers

e)

If the torque acting on Gear 1 is 5N-m, then find the torque on Gear 2.

Torque acting on Gear 2, 2

1

2

1

TT

NN

=

mNTNNT −=×== 5.25

2010

11

22


References

•

Control Systems –

A. Nagoor

Kani

–

RBA Publications, Chennai


And, before we break…

•

Oh! The worst of all tragedies is not to die young, but to live until I am seventy five and yet not ever truly to have lived.–

Martin Luther King Jr.

Thanks for listening…

Class 28 - Modeling of a Gear Train

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