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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.
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