Page 477 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

464 Nonlinear Vibrations Chap. 14

Integrating, we have

^ - I f ( x ) d x = E (14 .2-5)
L Jq
and by com parison with Eq. (14 .2-1) we find

U { x ) = - f f ( x ) cbc
•'n
(14.2-6)
dx
Thus, for a conservative system, the force is equal to the negative gradient of the
potential energy.
W ith y = X, Eq. (14.2-4) in the state space becomes

(14 .2-7)
dx y
W e note from this equation that singular points correspond to f(x) = 0 and
y = i = 0, and hence are equilibrium points. Equation (14.2-6) then indicates that
at the equilibrium points, the slope of the potential energy curve U(x) must be
zero. It can be shown that the m inim a of U(x) are stable equilibrium positions,
whereas the saddle points corresponding to the m axim a of U(x) are positions of
unstable equilibrium .

Stability of equilibrium. By exam ining Eq. (14.2-2), the value of E is
determ ined by the initial conditions of jc(0) and y(0) = i(0 ). If the initial condi­
tions are large, E w ill also be large. For every position jc, there is a potential U(x);
for motion to take place, E must be greater than U(x). Otherwise, Eq. (14.2-2)
shows that the velocity y = i is im aginary.
Figure 14.2-1 shows a general plot of U(x) and the trajectory y vs. x for
various values of E computed in Table 14.2-1 from Eq. (14.2-2).
For E = 1, U{x) lies below E = 1 only between jc = 0 to 1.2, x = 3.8 to 5.9,
and JC = 7 to 8.7. The trajectories corresponding to E = 7 are closed curves and
the period associated with them can be found from Eq. (14.2-2) by integration:
.X dx
= 2/
y j 2 [ E - U { x ) ]
where jCj and JC2 are extreme points of the trajectory on the x-axis.

For sm aller initial conditions, these closed trajectories become sm aller. For
E = 6, the trajectory about the equilibrium point x = 7.5 contracts to a point,
whereas the trajectory about the equilibrium point x = 5 is a closed curve between
X = 4.2 to 5.7.