Page 378 - Mechanical Engineers' Handbook (Volume 2)

8 Model Classifications 369

Although these assumptions appear to be rather limiting, the technique gives reasonable
results for a large class of control systems. In particular, the second assumption is generally
satisfied by higher order control systems with symmetric nonlinearities, since (a) symmetric
nonlinearities do not generate dc terms, (b) the amplitudes of harmonics are generally small
when compared with the fundamental term and subharmonics are uncommon, and (c) feed-
back within a control system typically provides low-pass filtering to further attenuate har-
monics, especially for higher order systems. Because the method is relatively simple and
can be used for systems of any order, describing functions have enjoyed wide practical
application.
The describing function of a nonlinear block is defined as the ratio of the fundamental
component of the output to the amplitude of a sinusoidal input. In general, the response of
the nonlinearity to the input
m( t) M sin t

is the output
n( t) N sin( t ) N sin(2 t ) N sin(3 t )
2
2
1
3
1
3
and, hence, the describing function for the nonlinearity is defined as the complex quantity
N 1
N(M) e j 1
M
Derivation of the approximating function typically proceeds by representing the fundamental
frequency by the Fourier series coefficients
A (M) T /2
2
1 n( t) cos td( t)
T T /2
B (M) T /2
2
1
T T /2 n( t) sin td( t)
The describing function is then written in terms of these coefficients as
2
2
A (M)
A (M)
B (M)
B (M) A (M)

1 / 2 exp j tan
1
1
1
1
1
N(M) j 1
M M M M B (M)
1
Note that if n( t) n( t), then the describing function is odd, A (M) 0, and there is
1
no phase shift between the input and output. If n( t) n( t), then the function is even,
B (M) 0, and the phase shift is /2.
1
The describing functions for a number of typical nonlinearities are given in Fig. 30.
Reference 9 contains an extensive catalog. The following derivation for a dead-zone nonlin-
earity demonstrates the general procedure for deriving a describing function. For the satu-
ration element depicted in Fig. 30a, the relationship between the input m( t) and output
n( t) can be written as
0 for D m D
n( t) KM(sin t sin t) for m D
1
1
KM(sin t sin t) for m D
1
1
Since the function is odd, A 0. By the symmetry over the four quarters of the response
1
period,