Page 380 - Mechanical Engineers' Handbook (Volume 2)

8 Model Classifications 371

Figure 30b Saturation nonlinearity: (i) nonlinear characteristic; (ii) sinusoidal input wave shape; (iii)
output wave shape; (iv) describing-function coefficients; (v) normalized describing function.

Phase-Plane Method
The phase-plane method is a graphical application of the state-space approach used to char-
acterize the free response of second-order nonlinear systems. While any convenient pair of
state variables can be used, the phase variables originally were taken to be the displacement
and velocity of the mass of a second-order mechanical system. Using the two state variables
as the coordinate axis, the transient response of a system is captured on the phase plane as
the plot of one variable against the other, with time implicit on the resulting curve. The
curve for a specific initial condition is called a trajectory in the phase plane; a representative
sample of trajectories is called the phase portrait of the system. The phase portrait is a