0195136047.pdf

804 BASIC CONTROL SYSTEMS

are

s 1 , 2 =−

F

2 J

±

√(

F

2 J

) 2

−

K

J

The system is:

(i) Overdamped if

(

F

2 J

) 2

>

K

J

(ii) Underdamped if

(

F

2 J

) 2

<

K

J

(iii) Critically damped if

(

F

2 J

) 2

=

K

J

To make the servomechanism fast acting, the underdamped case is desirable. For the

underdamped situation, the roots are given by

s 1 , 2 =−

F

2 J

±j

√

K

J

−

(

F

2 J

) 2

which can be expressed as

s 1 , 2 =−ξωn±jωd

whereωd =ωn

√

1 −ξ^2 is known asdamped frequency of oscillation. The damped

oscillations in our example occur at a frequency of

ωd= 30

√

1 − 0. 2222 = 29 .25 rad/s

(b) Starting from Equation (16.2.22), with a step command of magnituder 0 and a step change

in load of magnitudeTL, assuming the system to be initially at rest, one obtains

C(s)·(s^2 J+sF+K)=

Kr 0

s

−

TL

s

or

C(s)=

(K/J )r 0

s

(

s^2 +s

F

J

+

K

J

)−

TL/J

s

(

s^2 +s

F

J

+

K

J

)

IfTL=0, at steady state,cSS=r 0. In the presence of a fixed load torque,cSS=r 0 −TL/K.

The position lag error for our example is given by

TL

K

=

1. 5 × 10 −^3

0. 0135

= 0 .111 rad

(c) The loop gain must be increased by a factor of 0. 111 / 0. 025 = 4 .44. Hence the new value

of the amplifier gain is

Ka′= 4. 44 Ka= 4. 44 × 100 =444 V/V