Page 347 - Mechanical Engineers' Handbook (Volume 2)

338 Mathematical Models of Dynamic Physical Systems

Table 7 Transient Performance Measures Based on Step Response
Formula for a
Performance Measure Definition Second-Order System
Time required for the response
Delay time, t d
to reach half the final value
for the first time
Time required for the response
10–90% rise time, t r
to rise from 10 to 90% of
the final response (used for
overdamped responses)
Time required for the response
0–100% rise time, t r t
to rise from 0 to 100% of r
d
the final response (used for where cos 1
underdamped responses)
Time required for the response
Peak time, t p
t
to reach the first peak of the p d
overshoot
The difference in the response M e
/ 1 2
p
Maximum overshoot, M p
between the first peak of
the overshoot and the final
response
Percent overshoot, PO The ratio of maximum PO 100e
/ 1 2
overshoot to the final
response expressed as a
percentage
The time required for the 4
Setting time, t s t (2% band)
response to reach and stay s
n
within a specified band 3
centered on the final t (5% band)
s
response (usually 2% or 5%
n
of final response band)
called the period. Periodic inputs are important because these are ubiquitous: rotating un-
balanced machinery, reciprocating pumps and engines, ac electrical power, and a legion of
noise and disturbance inputs can be approximated by periodic inputs. Sinusoids are the most
important category of periodic inputs, because these are frequently occurring and easily
analyzed and form the basis for analysis of general periodic inputs.

Frequency Response
The frequency response of a system is the steady-state response of the system to a sinusoidal
input. For a linear system, the frequency response has the unique property that the response
is a sinusoid of the same frequency as the input sinusoid, differing only in amplitude and
phase. In addition, it is easy to show that the amplitude and phase of the response are
functions of the input frequency, which are readily obtained from the system transfer func-
tion.
Consider a system defined by the transfer function H(s). For an input
u(t) A sin t
the corresponding steady-state output is