Modern Control Engineering

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234 Chapter 5 / Transient and Steady-State Response Analyses

The relationship between J1eqandJ3eqis thus

and that between b1eqandb3eqis

The effect of J 2 andJ 3 on an equivalent moment of inertia is determined by the gear ratios

and For speed-reducing gear trains, the ratios, and are usually less than unity.

If and then the effect of J 2 andJ 3 on the equivalent moment of inertia J1eq

is negligible. Similar comments apply to the equivalent viscous-friction coefficient b1eqof the gear

train. In terms of the equivalent moment of inertia J1eqand equivalent viscous-friction coefficient

b1eq, Equation (5–66) can be simplified to give

where

A–5–3. When the system shown in Figure 5–52(a) is subjected to a unit-step input, the system output

responds as shown in Figure 5–52(b). Determine the values of KandTfrom the response curve.

Solution.The maximum overshoot of 25.4%corresponds to z=0.4. From the response curve

we have

Consequently,

tp=

p

vd

=

p

vn 21 - z^2

=

p

vn 21 - 0.4^2

= 3

tp= 3

n=

N 1

N 2

N 3

N 4

J 1 eq u

$

1 +b 1 eq^ u

1 +nTL=Tm

N 1 N 2  1 N 3 N 4 1,

N 3 N 4. N 1 N 2 N 3 N 4

N 1 N 2

b 1 eq= a

N 1

N 2

b

2

a

N 3

N 4

b

2

b 3 eq

J 1 eq= a

N 1

N 2

b

2

a

N 3

N 4

b

2

J 3 eq

+

• 
  

R(s) C(s)

(a)

(b)

c(t)

1

03 t

0.254

K

s(Ts+ 1)

Figure 5–52

(a) Closed-loop

system; (b) unit-step

response curve.

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