
From Equation 2.18, the characteristic roots of the overdamped system are
s
1
, s
2
¼zv
n
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
z
2
1v
n
q
¼1:1381 rad=min, 0:1953 rad=min (2:58)
The time constants in Equation 2.19 are t
1
¼1=s
1
¼0.8787 min and t
2
¼1=s
2
¼ 5.1213 min,
and from the unit step response in Equation 2.20, the response to a step of magnitude
c is
c
2
(t) ¼ Kc 1 þ
t
2
e
t=t
2
t
1
e
t=t
1
t
1
t
2
(2:59)
¼ 0:25 1
5:1213e
t=5:1213
0:8787e
t=0:8787
4:2426
, t 0(2:60)
The second-order differential equation in Equation 2.51 is in standard form; however, the second-
order differential equation for c
1
(t) contains the first derivative dc=dt on the right-hand side of the
equation