828 BASIC CONTROL SYSTEMSG(s)=
10
s^2 +s− 2
(i) Is the system stable?
(ii) If it is unstable, can it be made stable
by employing a feedback path with a
transfer functionHaroundG(s)? If so,
findH.
(b) Redo part (a) for the case whereG(s)=10
s(s^2 +s− 2 )16.2.40The output response of a second-order servomech-
anism is given as
c(t)
r 0
= 1 − 1. 66 e−^8 tsin( 6 t+37°)
when the input is a step function of magnituder 0.
(a) Determine the damped frequency of oscilla-
tion.
(b) Obtain the value of the damping ratio.
(c) Find the natural frequency of the system.
(d) Evaluate the loop gain if the inertia of the
output member is 0.01 kg·m^2 and the viscous
coefficient is 0.2 kg·m^2 /s.
(e) To what value should the loop gain be in-
creased if the damping ratio is not to be less
than 0.4?
(f) Obtain the closed-loop transfer function.
(g) Find the corresponding open-loop transfer
function.
(h) When the system is operated in open-loop
configuration, determine the complete output
response for a unit-step input.16.2.41Figure P16.2.41 shows the block diagram of a
control system. Determine:
(a) The closed-loop transfer function.
(b) The frequency of oscillation of the output vari-
able in responding to a step command before
reaching steady state.
(c) The percent maximum overshoot in part (b).
(d) The time required for the output to reach up
to 99% of steady state in part (b).
*16.2.42The feedback control system is characterized by
d^2 c
dt^2
+ 6. 4dc
dt
= 160 e
wherecis the output variable ande=r− 0. 4 c. De-
termine the damping ratioξ, the natural frequency
ωn, and percent maximum overshoot.
16.2.43A second-order servomechanism with the config-
uration of Figure 16.2.7 has the following param-
eters:
Open-loop gainK= 24 × 10 −^4 N·m/rad
System inertiaJ= 1. 4 × 10 −^5 kg·m^2
System viscous-friction coefficientF = 220 ×
10 −^6 kg·m^2 /s
(a) Find the damping ratio.
(b) If the loop gain has to be increased to 250×
10 −^4 N·m/rad in order to meet the accuracy
requirements during steady-state operation,
determine the error-rate damping coefficient
needed, while keeping the damping ratio un-
changed.
16.2.44The system of Problem 16.2.41 is modified, as
shown in Figure P16.2.44, to include error-rate
damping. Find the value of the error-rate factor
Keso that the damping ratio of the modified char-
acteristic equation is 0.6.
16.2.45For the system shown in Figure P16.2.45, deter-
mine the value of the output-rate factor that yields
a response (to a step command) with a maximum
overshoot of 10%.
*16.2.46Redo Problem 16.2.45 for the system whose block
diagram is depicted in Figure P16.2.46.+
− s +^2 sRC 1 10 (s)
Figure P16.2.41+−s + 2 sRE 1 + sKe 10 CFigure P16.2.44