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Section 5–3 / Second-Order Systems 167
Hence the inverse Laplace transform of Equation (5–11) is obtained as
fort 0 (5–12)
From Equation (5–12), it can be seen that the frequency of transient oscillation is the
damped natural frequency vdand thus varies with the damping ratio z. The error signal
for this system is the difference between the input and output and is
fort 0
This error signal exhibits a damped sinusoidal oscillation. At steady state, or at t=q,
no error exists between the input and output.
If the damping ratio zis equal to zero, the response becomes undamped and
oscillations continue indefinitely. The response c(t)for the zero damping case may be
obtained by substituting z=0in Equation (5–12), yielding
fort 0 (5–13)
Thus, from Equation (5–13), we see that vnrepresents the undamped natural frequen-
cy of the system. That is,vnis that frequency at which the system output would oscillate
if the damping were decreased to zero. If the linear system has any amount of damping,
the undamped natural frequency cannot be observed experimentally. The frequency
that may be observed is the damped natural frequency vd, which is equal to
This frequency is always lower than the undamped natural frequency. An increase in z
would reduce the damped natural frequency vd.If zis increased beyond unity, the
response becomes overdamped and will not oscillate.
(2)Critically damped case (z=1): If the two poles of C(s)/R(s)are equal, the system
is said to be a critically damped one.
For a unit-step input,R(s)=1/sandC(s)can be written
(5–14)
The inverse Laplace transform of Equation (5–14) may be found as
fort 0 (5–15)
This result can also be obtained by letting zapproach unity in Equation (5–12) and by
using the following limit:
limzS 1
sinvd t
21 - z^2
=limzS 1
sinvn 21 - z^2 t
21 - z^2
=vn t
c(t)= 1 - e-vn^ tA 1 +vn tB,
C(s)=
v^2 n
As+vnB^2 s
vn 21 - z^2.
c(t)= 1 - cosvn t,
=e-zvn^ tacosvd t+
z
21 - z^2
sinvd tb,
e(t)=r(t)-c(t)
= 1 -
e-zvn^ t
21 - z^2
sin avd t+tan-^1
21 - z^2
z
b,
= 1 - e-zvn^ tacosvd t+
z
21 - z^2
sinvd tb
l-^1 CC(s)D=c(t)
