Class 28 - Modeling Of A Gear Train

System Modeling Coursework

Class 28: Modeling of Gear Train

P.R. VENKATESWARAN Faculty, Instrumentation and Control Engineering, Manipal Institute of Technology, Manipal Karnataka 576 104 INDIA Ph: 0820 2925154, 2925152 Fax: 0820 2571071 Email: [email protected], [email protected] Web address: http://www.esnips.com/web/SystemModelingClassNotes

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. July – December 2008

prv/System Modeling Coursework/MIT-Manipal

2

Contents • • • •

Uses of Gear Trains Description Derivation of Gear Ratio Numerical

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

3

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. July – December 2008

prv/System Modeling Coursework/MIT-Manipal

4

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. July – December 2008

prv/System Modeling Coursework/MIT-Manipal

5

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 Torque Similarly, N2 , r2 , θ2 ,J2 , B2 ,T2 can be respectively defined for Gear 2 July – December 2008

prv/System Modeling Coursework/MIT-Manipal

6

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,

d 2θ1 dθ1 J1 2 + B1 + Tl = Tm dt dt

• 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

dθ 2 d 2θ 2 J 2 2 + B2 + Tl = T2 dt dt July – December 2008

prv/System Modeling Coursework/MIT-Manipal

7

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 =θ1r1=θ2r2. Therefore

θ 2 r1 = θ1 r2 • The number of gears in each gear is proportional to its radius. i.e. N1αr1 and N2αr2. Hence, N1 r1 θ 2 = = N 2 r2 θ1 July – December 2008

prv/System Modeling Coursework/MIT-Manipal

8

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 = T1θ1=T2θ2. Therefore

T1 θ 2 = T2 θ1

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

9

Work done by the gears • Differentiating the equation of work done, we get

dθ1 dθ 2 T1 = T2 dt dt T1ω1 = T2ω2 • Differentiating one more time, the relation will become

T1 ω2 a2 = = T2 ω1 a1 July – December 2008

prv/System Modeling Coursework/MIT-Manipal

10

Summing up… N1 r1 θ 2 T1 ω2 a2 = = = = = N 2 r2 θ1 T2 ω1 a1 • 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
prv/System Modeling Coursework/MIT-Manipal

11

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.

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

12

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, N1 r1 = N 2 r2

where r1, r2 = radius of Gear 1 and Gear 2 N1, N2=Number of teeth in Gear 1 and Gear 2. Therefore, the ratio of diameters = N1 = r1 = 20 = 2 N2

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

r2

20

13

Answers b) If Gear 1 is rotated by an angle of 400, then what will be the angular displacement of Gear 2 N1 θ 2 = N 2 θ1

Displacement for Gear 2 is ,

N1 20 0 0 θ2 = × θ1 = × 40 = 80 N2 10 July – December 2008

prv/System Modeling Coursework/MIT-Manipal

14

Answers c)

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

N1 ω 2 = N 2 ω1 Angular speed of gear 2,

N1 20 ω2 = ω1 = × 30 = 60rad / sec N2 10

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

15

Answers d) If the angular acceleration of Gear 2 is 4 rad/sec2 then find the angular acceleration of Gear 1 N1 a 2 = N 2 a1 Angular acceleration of Gear 1 is

N2 10 a1 = a2 = × 4 = 2rad / sec 2 N1 20

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

16

Answers e)

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

N1 T1 = N 2 T2 Torque acting on Gear 2,

N2 10 T2 = T1 = × 5 = 2.5 N − m N1 20

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

17

References • Control Systems – A. Nagoor Kani – RBA Publications, Chennai

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

18

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… July – December 2008

prv/System Modeling Coursework/MIT-Manipal

19