Page 672 - Mechanical Engineers' Handbook (Volume 2)

7 Effects of Nonlinearities 663

Figure 38 (Continued)(b) Ramp response.

netic hysteresis effects can cause some system performance anomalies. The width of the
hysteresis band is dependent upon the amplitude of the input signal (the output is a function
of the input’s prior history as well as its present value).

7.1 Simple Nonlinearities
When the output of a nonlinear element depends only upon the present value of its input,
the element can often be described by a simple relationship between input and output am-
plitude. If this function is single valued, it is often possible to assess its effect on the system
by using linear approximations. One useful technique is to examine the small-perturbation
behavior of the system at a series of operating points along the input–output curve by
performing a linear analysis using the local slope at each operating point.
Another technique is describing function analysis, which is useful in estimating the
response of nonlinear systems to sinusoidal inputs. In general, a describing function is an
amplitude-dependent, frequency-dependent transfer function of a nonlinear element which
allows the system to be analyzed by conventional frequency-domain techniques. It is derived
1
from a Fourier analysis of the output of the nonlinear element to a sinusoidal input. For
simple nonlinearities that can be described by a single-valued output amplitude versus input
amplitude, the describing function is a simple gain that varies with input amplitude. In
concept, this gain is the average slope of the input–output curve for the particular input
amplitude being considered.
Saturation and deadzone are two of the most common nonlinearities encountered in
control of mechanical systems. Referring to Fig. 41a, an operating-point analysis would