Page 402 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)
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378               SLENDER STRUCTURES AND AXIAL, FLOW













                          .e                                      '.
                                               5.--  -
                          1  0.10 ;       I'\                     !
                                              \
                          6                -...                   A
                                                                  I
                             0.05  -              -. ->> .\;?    I     -
                                                     ,v\--  .-.-.-.  -.-
                                        ..
                                        ..       .-/  \J
                                        ..
                                        ..           , , , ,
                                        . .
                             0.00   ""I   .,  ,.  ,  ,   ,  ,  ,  ,   ,  ,  ,  ,   ,  ,  , ,
                             0.4

                             0.3



                             0.2


                             0.1



                             0.0
                                4    5     6     7    8     9     10   11
                          (b)                    Velocity, u
            Figure 5.48  (a) Bifurcation diagram for the system of  Figure 5.43(a), for p = 0.2 and  N = 3,
            obtained while ignoring the nonlinear inertial terms, showing the maximum generalized coordinate,
            91, versus u obtained with AUTO: -,   stable periodic solution; - . -, main branch of the unstable
            solution; - - -, unstable branch emerging from the bifurcation point marked by  a;  . . ., unstable
            branch connecting the two Hopf bifurcation points, amplified 20 times; A, limit points on the main
            branch. (b) The main  branch of  (a), together with results computed by  FDM (+) (Pdidoussis &
                                            Semler 1998).

            branches). To  give  additional proof  that  the  presumed  chaotic  solutions computed by
            FDM  are really  chaotic, the numerical  scheme developed by  Hairer et al. (1993) was
            used, since it is known to be particularly accurate in chaotic regimes, in the sense that it
            does not induce artificial chaos numerically. The results obtained confirm that the motion
            is indeed chaotic.
              In conclusion, it may be said that the addition of  a small mass at the end of  the fluid-
            conveying pipe enriches the  dynamics considerably, in  fact revealing the existence of
            a completely new dynamical system. Not only are different types of  periodic solutions
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