Page 479 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

466 Nonlinear Vibrations Chap. 14

For E = S one of the maxima of U(x) at x = 6.5 is tangent to E = S and the
trajectory at this point has four branches. The point x = 6.5 is a saddle point for
E = S and the motion is unstable. The saddle point trajectories are called sépara­
trices.
For E > 8, the trajectories may or may not be closed. E = 9 shows a closed
trajectory between x = 3.3 to 10.2. Note that at x = 6.5, dU/dx = - f i x) = 0 and

y = X ^ 0 for £ = 9, and hence equilibrium does not exist.

14.3 STABILITY OF EQUILIBRIUM

Expressed in the general form
P( x, y)
dx Q{x,y) (14.3-1)
the singular points (x^, y^) of the equation are identified by
P(X,, y,) = Q{x„ y j = 0 (14.3-2)
Equation (14.3-1), of course, is equivalent to the two equations

^ - Q ( . x . y )
(14.3-3)
% - n x . y )
from which the time dt has been eliminated. A study of these equations in the
neighborhood of the singular point provides us with answers as to the stability of
equilibrium.
Recognizing that the slope dy/dx of the trajectories does not vary with
translation of the coordinate axes, we translate the u, v axis to one of the singular
points to be studied, as shown in Fig. 14.3-1. We then have
X = x^ + u
y =y, + V
(14.3-4)
dy _ dv
dx du
If Fix, y) and Qix, y) are now expanded in terms of the Taylor series about the

y

i X s . Y s )
0^ X

Figure 14.3-1.