Stability Analysis - Mechanics of Structures: Variational and Computational Methods - page 650

FIGURE 11.1
Stability of equilibrium positions.

FIGURE 11.2
A rigid rod with spring.

Note that P is a nonli ne ar function of φ . To observe the response, plot P vs. φ . As show n

for the case of non-zero H in Fig. 11.3a, typically, the displacement va ri es smoothly as P

increases, i.e., there is a unique a nd stable state of equilibrium for each P .
An interesting case occurs for H

φ

= 0. Then

φ
L sin

k

P = (3)

φ

which is always greater than k 0 axis, it w ould appear that the P cannot

be evaluated. However, note that in t he limit as

/ L . On the φ =

φ → 0 , P is equal to k / L . Thus, with the

rod in a vertical position (at φ = 0), P can be applied and increased until reaching the

value P L, whereupon the bar could follow the path defined by (3). Then a small

change in P can cause a major change in

= k /

. The ph en omenon where there is more than
one equilibrium path is called bifurcation . The value P

φ

= k / L is the bifurcation point (see

Fig. 11.3b).