Page 475 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

462 Nonlinear Vibrations Chap. 14

14.1 PHASE PLANE
In an autonomous system, time t does not appear explicitly in the differential
equation of motion. Thus, only the differential of time, dt, appears in such an
equation.
We first study an automonous system with the differential equation
x + f { x , x ) = 0 (14.1-1)
where fix, x) can be a nonlinear function of x and i. In the method of state
space, we express the last equation in terms of two first-order equations as follows:
. (14.1-2)
y = - f ( x , y )
If X and y are Cartesian coordinates, the xy-plane is called the phase plane. The
state of the system is defined by the coordinate jc and y = x, which represents a
point on the phase plane. As the state of the system changes, the point on the
phase plane moves, thereby generating a curve that is called the trajectory.
Another useful concept is the state speed V defined by the equation

V = { x (14.1-3)
When the state speed is zero, an equilibrium state is reached in that both the
velocity of x and the acceleration y = x are zero.
Dividing the second of Eq. (14.1-2) by the first, we obtain the relation
- f i x , y )
dx = (f){x, y) (14.1-4)
Thus, for every point x, y in the phase plane for which 4>ixy) is not indeterminate,
there is a unique slope of the trajectory.
If y = 0 (i.e., points along the jc-axis) and fix, y) # 0, the slope of the
trajectory is infinite. Thus, all trajectories corresponding to such points must cross
the Jc-axis at right angles.
If y = 0 and fix, y) = 0, the slope is indeterminate. We define such points
as singular points. Singular points correspond to a state of equilibrium in that both
the velocity y = x and the force x = y = -f i x, y) are zero. Further discussion is
required to establish whether the equilibrium represented by the singular point is
stable or unstable.
Example 14.1-1
Determine the phase plane of a single-DOF oscillator:
X -h (o^x = 0
Solution: With y = i, this equation is written in terms of two first-order equations:
y = —ù)^x
X = y