Modern Control Engineering

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232 Chapter 5 / Transient and Steady-State Response Analyses

Note that if the value of kbis large, the response u(t)approaches a pulse signal, as shown in

Figure 5–50.

A–5–2. Gear trains are often used in servo systems to reduce speed, to magnify torque, or to obtain the

most efficient power transfer by matching the driving member to the given load.

Consider the gear-train system shown in Figure 5–51. In this system, a load is driven by a

motor through the gear train. Assuming that the stiffness of the shafts of the gear train is infinite

(there is neither backlash nor elastic deformation) and that the number of teeth on each gear is

proportional to the radius of the gear, obtain the equivalent moment of inertia and equivalent

viscous-friction coefficient referred to the motor shaft and referred to the load shaft.

In Figure 5–51 the numbers of teeth on gears 1, 2, 3, and 4 are N 1 ,N 2 ,N 3 , and N 4 , respectively.

The angular displacements of shafts, 1, 2, and 3 are u 1 ,u 2 , and u 3 , respectively.Thus,

and The moment of inertia and viscous-friction coefficient of each gear-train

component are denoted by J 1 ,b 1 ;J 2 ,b 2 ; and J 3 ,b 3 ; respectively. (J 3 andb 3 include the moment of

inertia and friction of the load.)

u 3 u 2 =N 3 N 4.

u 2 u 1 =N 1 N 2

x(t)

t

t

1

0

0

u(t)

1

L

Figure 5–50

Unit-step input and

the response of the

mechanical system

shown in Figure

5–49.

Shaft 1

Gear 2

Gear 1

Gear 3

Gear 4

Shaft 2

Shaft 3

J 1 ,b 1

N 1

Input torque

from motor

Tm(t)

u 1

N 2

N 3

N 4

u 2

u (^3) Load
torque
TL(t)
J 2 ,b 2
J 3 ,b 3
Figure 5–51
Gear-train system.
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