Page 349 - Mechanical Engineers' Handbook (Volume 2)

340 Mathematical Models of Dynamic Physical Systems

( ) versus log . Bode diagrams for normalized first- and second-order systems are given
in Fig. 23. Bode diagrams for higher order systems are obtained by adding these first- and
second-order terms, appropriately scaled. A Nichols diagram can be obtained by cross plot-
ting the Bode magnitude and phase diagrams, eliminating log . Polar plots and Bode and
Nichols diagrams for common transfer functions are given in Table 8.

Frequency Response Performance Measures
Frequency response plots show that dynamic systems tend to behave like filters, ‘‘passing’’
or even amplifying certain ranges of input frequencies while blocking or attenuating other
frequency ranges. The range of frequencies for which the amplitude ratio is no less than 3
dB of its maximum value is called the bandwidth of the system. The bandwidth is defined
by upper and lower cutoff frequencies ,or by 0 and an upper cutoff frequency if
c
M(0) is the maximum amplitude ratio. Although the choice of ‘‘down 3 dB’’ used to define
the cutoff frequencies is somewhat arbitrary, the bandwidth is usually taken to be a measure
of the range of frequencies for which a significant portion of the input is felt in the system
output. The bandwidth is also taken to be a measure of the system speed of response, since
attenuation of inputs in the higher frequency ranges generally results from the inability of
the system to ‘‘follow’’ rapid changes in amplitude. Thus, a narrow bandwidth generally
indicates a sluggish system response.
Response to General Periodic Inputs
The Fourier series provides a means for representing a general periodic input as the sum of
a constant and terms containing sine and cosine. For this reason the Fourier series, together
with the superposition principle for linear systems, extends the results of frequency response
analysis to the general case of arbitrary periodic inputs. The Fourier series representation of
a periodic function ƒ(t) with period 2T on the interval t* 2T t t*is

a 0 n t b sin
n t
ƒ(t)
n
a cos
2 n 1 T n T
where
a t* 2T n t
1
n ƒ(t) cos dt
T t* T
b t* 2T n t
1
n
T t* ƒ(t) sin T dt
If ƒ(t) is defined outside the specified interval by a periodic extension of period 2T and if
ƒ(t) and its first derivative are piecewise continuous, then the series converges to ƒ(t)if t is
1
a point of continuity or to ⁄2 [ƒ(t ) ƒ(t )] if t is a point of discontinuity. Note that while

the Fourier series in general is infinite, the notion of bandwidth can be used to reduce the
number of terms required for a reasonable approximation.
6 STATE-VARIABLE METHODS
State-variable methods use the vector state and output equations introduced in Section 4 for
analysis of dynamic systems directly in the time domain. These methods have several ad-
vantages over transform methods. First, state-variable methods are particularly advantageous
for the study of multivariable (multiple-input/multiple-output) systems. Second, state-
variable methods are more naturally extended for the study of linear time-varying and non-