Page 388 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

364 SLENDER STRUCTURES AND AXIAL FLOW

-
9.0
r
x
0-0-0-0
8
5
.- - 8.0
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c
e,
- :/
-
C 7.0
CJ
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‘C -
6.0
5.0

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m
.“ o A 0 Theoretical
Y
7.0
6.0 ’ I

2 3 4 5 6 I 8
fb) Number of modes, N

Figure 5.39 Convergence of the critical flow velocities for (a) the Hopf and (b) the first
period-doubling bifurcations with the number of modes, N, in the Galerkin discretization of the
system of Figure 5.30. In (b): 0, ‘quadratic’ (Le. quadratically smoothed; n = 2) model; A, ‘cubic’
(n = 3); 0, quintic (n = 5) (Paidoussis, Li & Rand 1991a).

Table 5.4 Theoretical values from (a) Paidoussis et al. (1991a) for N = 5 and
various n (hence the range) and (b) from Paidoussis & Semler (1993a) for N = 4
and n = 3 compared with experiment, in terms of the main bifurcations of the
fluttering cantilevered pipe impacting on motion-limiting restraints.

Bifurcation U Theory (a) Experiment Theory (b)
Hopf UH 8.40 8.04 8.40
1st period doubling upd 8.63-8.94 8.43 9.05
Chaos uch 8.68-8.97 8.72 9.20
‘Restabilization’ u rs - -9.0 9.65

is better than expected for a system such as this. However, the amplitude of motion is
still much larger than in the experiments.
This final weakness of the model was overcome using the full nonlinear equation of
motion, equation (5.42), by PaYdoussis & Semler (1993a). Typical results are shown in