Page 343 - Mechanical Engineers' Handbook (Volume 2)

334 Mathematical Models of Dynamic Physical Systems

Second-Order Transient Response
The standard form of the I/O equation for a second-order system is

2
dy dy
2
y(t) u(t)
2
2
dt 2 n dt n n
with transfer function
Y(s) 2 n
H(s)
U(s) s 2
s 2 n
2
n
The system poles are obtained by applying the quadratic formula to the characteristic equa-
tion as
p 1.2
j 1
2
n
n
where the following parameters are defined:
is the damping ratio, is the natural fre-
n
quency, and 1
2 is the damped natural frequency.
d n
The nature of the response of the standard second-order system to a step input depends
on the value of the damping ratio, as depicted in Fig. 17. For a stable system, four classes
of response are defined.

Figure 17 Response of a second-order system to a unit step input for selected values of the damping
ratio.