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394               SLENDER STRUCTURES AND AXIAL FLOW

                    homoclinic  orbit,  along with the corresponding  time  trace is shown  in  Figure 5.58. For
                    low enough y (e.g. for y = 1 .O),  chaos is observed between the stable mixed-mode orbits.


                       2.0                                  2.0

                        1.0                                  1.0 -


                     e,  0.0                             6,  0.0 -


                       -1.0                                 -1.0 -

                       -2.0                                 -2.0-   1   I   I   I   I   1   I   I   I
                         -2.0   -1.0    0.0    1 .o   2.0      0.0   0.2   0.4   0.6   0.8   1.0
                     (a)                4                (b)                 7
                    Figure 5.58  (a) A phase portrait of a period-(1, 8) orbit and (b) the corresponding time trace for
                         a vertical articulated system near the point of double degeneracy (Champneys 1993).

                    5.9  NONLINEAR PARAMETRIC RESONANCES

                    As  shown  in  Section 4.5  both  by  linear  theory  and  by  experiments  conducted  in  the
                    1970s, the dynamics of pipes conveying harmonically perturbed flow is quite interesting,
                    especially  in the  case of  cantilevered  pipes.  It  was reasonable  to expect,  therefore, that
                    nonlinear study of the same system would soon follow - especially in view of the work in
                    Sections 5.5-5.8  conducted in the late 1970s and 1980s, showing that nonlinear dynamics
                    analysis  can  (i) provide  results  closer  to  reality,  (ii) elucidate  the  dynamical  behaviour
                    beyond  the  onset  of  instability,  (iii) give  new  insight  into  the  dynamics  even  before
                    the instability threshold,  (iv) reveal more interesting  fine structure in  the dynamics, and
                    (v) yield  entirely  new  results  (e.g. the  amplitude  of  the  motion).  The nonlinear  studies
                    of  parametric resonances of pipes conveying fluid, which began appearing in the second
                    half  of  the  1980s, demonstrate to the full one of  the tenets justifying  the space allocated
                    to dynamics of pipes conveying fluid in this two-volume book: that this system serves as
                    a crucible  for  the development,  illustration  and testing  of  new  dynamical  theory. Thus,
                    more  or less  at the  same time  and by  the  same authors,  in parallel  to the work  on the
                    pipe  problem  to  be  discussed  in  what  follows,  a  number  of  papers  have  appeared  on
                    the general theory of parametrically perturbed generic nonlinear systems subject to Hopf
                    bifurcations, e.g. by Bajaj (1986,  1987a) and Namachchivaya  & Ariaratnam (1987).
                      Most  of  the  work  done  in  this  area  is  analytical,  and most  of  that  makes  use  of  the
                    modem methods of nonlinear dynamics theory, but some numerical calculations have also
                    been done, as well as some new experiments.

                    5.9.1  Pipes with supported ends
                    Parametric  resonances  in  this  inherently  conservative  system  have  been  studied  by
                    Yoshizawa et al. (1986), Namachchivaya  (1989) and Namachchivaya  & Tien (1989a,b),
                    Chang & Chen (1994), and Jayaraman  & Narayanan  (1996).
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