Page 344 - Mechanical Engineers' Handbook (Volume 2)

5 Approaches to Linear Systems Analysis 335

1. Overdamped Response (
1). The system poles are real and distinct. The response
of the second-order system can be decomposed into the response of two cascaded
first-order systems, as shown in Fig. 18.
2. Critically Damped Response (
1). The system poles are real and repeated. This
is the limiting case of overdamped response, where the response is as fast as possible
without overshoot.
3. Underdamped Response (1
0). The system poles are complex conjugates. The
response oscillates at the damped frequency . The magnitude of the oscillations
d
and the speed with which the oscillations decay depend on the damping ratio
.
4. Harmonic Oscillation (
0). The system poles are pure imaginary numbers. The
response oscillates at the natural frequency and the oscillations are undamped
n
(i.e., the oscillations are sustained and do not decay).
The Complex s-Plane
The location of the system poles (roots of the characteristic equation) in the complex s-plane
reveals the nature of the system response to test inputs. Figure 19 shows the relationship
between the location of the poles in the complex plane and the parameters of the standard
second-order system. Figure 20 shows the unit impulse response of a second-order system
corresponding to various pole locations in the complex plane.

Transient Response of Higher Order Systems
The response of third- and higher order systems to test inputs is simply the sum of terms
representing component first- and second-order responses. This is because the system poles
must either be real, resulting in first-order terms, or complex, resulting in second-order
underdamped terms. Furthermore, because the transients associated with those system poles
having the largest real part decay the most slowly, these transients tend to dominate the
output. The response of higher order systems therefore tends to have the same form as the
response to the dominant poles, with the response to the subdominant poles superimposed
over it. Note that the larger the relative difference between the real parts of the dominant
and subdominant poles, the more the output tends to resemble the dominant mode of re-
sponse.
For example, consider a fixed linear third-order system. The system has three poles.
Either the poles may all be real or one may be real while the other pair is complex conjugate.
This leads to the three forms of step response shown in Fig. 21, depending on the relative
locations of the poles in the complex plane.

Figure 18 Overdamped response of a second-order system decomposed into the responses of two first-
order systems.