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PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 34 1

Static restabilization

Divrrgeriw. 1st mode

Gravity parameter. y

- 0

-0.2 0 0.2 0.4
Displacement, H’( 1. 71 (C)

Figure 5.25 (a) Stability boundaries for the system of Figure 5.23 obtained by direct eigenvalue
analysis, except for the ‘global oscillations’ region obtained numerically from the equations of the
nonlinear system: (b) phase portrait of the system showing the saddle node at (0) and the two stable
equilibria (fl) for y = -60, and u = 7.5; (c) three saddles (0) and (f2}, two stable equlibria (fl]
and global oscillations for y = -60 and u = 13.1. In all cases B = 0.18, LY = 5 x K = 100.
& = 0.8 (Pai’doussis & Semler 199313).

where

(5.121)

[MI is the matrix of the eigenvalues with negative real parts; for this 4-D system the last
row of [A] involves a scalar.