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302 SLENDER STRUCTURES AND AXIAL FLOW
and MATHEMATICA) renders these tools even more potent (Rand 1984; Rand &
Armbruster 1987).
Another powerful set of numerical tools, developed relatively recently, after the
concepts of bifurcation theory were established, are continuation or homotopy methods,
nowadays available in computer packages. These methods, exemplified by AUTO (Doedel
1981; Doedel & KernCves 1986), ‘follow’ a particular type of solution as it evolves in
phase space as a result of varying a particular set of system parameters, and can detect
the birth of a new type of solution via stability considerations. They are an invaluable
tool in constructing bifurcation diagrams, which at a glance summarize the changes in
dynamical behaviour occurring as the parameter in question is varied (Appendix F.5).
As noted in the foregoing, some of these methods are outlined in Appendix F. In what
follows, similarly to the approach in Chapters 3 and 4, the methods used in each case
are mentioned without much detail, and then the results are presented and discussed. In a
few cases, however, e.g. in Sections 5.5.2 and 5.7.3(a,b), the analysis is outlined in fair
detail, to give an appreciation of the power of these methods.
5.5 PIPES WITH SUPPORTED ENDS
5.5.1 The effect of amplitude on frequency
Perhaps the earliest study on nonlinear aspects of the dynamics of pipes with supported
ends conveying fluid was conducted by Thurman & Mote (1969b), paralleling closely
another on the nonlinear oscillation of axially moving strips (Thurman & Mote 1969a).
In this study, motions in both the lateral and axial directions are taken into account, via
equations (5.62). When these equations are rendered dimensionless, the additional nondi-
mensional quantity d = AL2/Z emerges, which plays an important role in the dynamics
of the system: all nonlinear terms are multiplied by (d - I‘ >.
The equations of motion are analysed by means of a hybrid method incorporating
elements of Lindstedt’s perturbation method and the Krylov-Bogoliubov method. The
main finding is that the nonlinear natural frequencies prior to divergence are higher than
the linear ones (i.e. the period of oscillation is lower). The discrepancy becomes progres-
sively larger with increasing flow velocity, u,+ as shown in Figure 5.3. To understand
why, it is recalled that linear theory shows a very precipitous reduction in frequency with
u close to the point of divergence (Figure 3.10), meaning that the effective stiffness of
the pipe is diminished very rapidly; on the other hand, the nonlinear tension effects are
not diminished. Hence, the relative discrepancy between linear and nonlinear analysis for
a given tension increase is dramatically magnified with u. Finally, the flattening of the
curves for each value of u/n in Figure 5.3 corresponds to a ‘saturation’ of the method of
solution - carried to the second perturbation; beyond each local minimum, the accuracy
of the result becomes doubtful. The important question of how these nonlinearities affect
the transition to divergence was not addressed.
It should be remarked that in Thurman & Mote’s work both axial motion and vari-
able incremental tension associated with large deformations are taken into account - see
equations (5.62). In contrast, the nonlinearity considered in Holmes’ work (1977, 1978),
+It is recalled that the dimensionless flow velocity, % in Section 5.2, is now denoted by u throughout the
rest of this chapter, for conformity with Chapters 3 and 4.