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PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS          301

             The assumed form involves coefficients determined by  imposing minimizing conditions
             (Ritz-Galerkin  method)  or  orthogonality  conditions  (Galerkin  and  harmonic  balance
             methods),  which  effectively  converts  the  nonlinear  differential  equation  into  a  set  of
             nonlinear algebraic equations, solved iteratively. The incremental harmonic balance (IHB)
             method  (Lau et al.  1982; Ferri  1986), which has been found useful for the analysis of
             the cantilevered pipe system when the assumption of smallness of inertial nonlinearities
             is not  made, is also of  this class.
               A few more words may be in order here regarding the difficulties encountered in solving
             equations with large inertial nonlinearities, e.g. equation (5.39). Several of the well known
             numerical and combined analytical-numerical  methods for solving nonlinear differential
             equations fail, even though they  work with  large stiffness nonlinearities.+ On the other
             hand, two finite difference methods (Houbolt’s 4th order and an 8th order scheme) yield
             accurate results, but  both  introduce a  phase  shift and  the 4th  order  scheme also some
             numerical damping. Only the IHB  method  (Lau et al. 1982, 1983; Lau  & Yuen  1993;
             Semler et al. 1996) has proved to be totally satisfactory.
               All  of  these methods of  solution, despite some of  them having been developed only
             recently, may be regarded as ‘classical’, at least in their outlook. Also considered classical
             is the use of the Lyapunov second method (Hagedorn 1981; Hahn 1963) to establish local,
             global or a symptotic stabilityS  - see Appendix F.l. Finally, also classical is the use of
             Floquet theory for assessing stability of limit cycles or the type of bifurcation emanating
             therefrom (Appendix F. 1.2).
               Another  set  of  methods  have  come  into  prominence  over  the  past  20  years  or  so,
             collectively referred to as the modern methods of  nonlinear dynamics, which are at once
             more limited in  scope and more powerful than the classical methods (Guckenheimer &
             Holmes  1983). Typically one starts from knowledge of  the eigenvalues of  the linearized
             system - which  specify the  linear  behaviour - as  well  as  their  evolution  as  a  given
             parameter  (say,  the  dimensionless  flow  velocity,  u)  is  varied,  and  one  concentrates
             the  investigation to  the  case  where  one  of  the  eigenvalues has  a  zero  real  part.  The
             centre manifold  method  then  drastically  reduces  a  nonlinear  multidimensional system
             into a  simpler low-dimensional subsystem, which  nevertheless retains  all  the  pertinent
             information on the bifurcating mode, and  hence on the dynamics of  the system, in  the
             vicinity  of  u = u,,  (Appendices F.2  and  H.l).  However,  the  equations  on  the  centre
             manifold may  still be too complex, and further simplification may be desirable. To  this
             end, the method of  normal forms  (Appendices F.3 and  H.2) provides a  systematic way
             of  simplifying a  complex  nonlinear  system, by  retaining  only  the  essential  nonlinear
             terms which decide its dynamical behaviour. Therefore, these two methods together (or
             alternatively the combination of  the centre manifold and  averaging methods) constitute
             a powerful tool for obtaining the simplest possible subsystem, capable of  predicting the
             nonlinear dynamical behaviour for u  not too far away from a particular u,-. The use of
             symbolic  manipulation  computational  software  (e.g.  MACSYMA, MAPLE,  REDUCE

               +For example, the Picard iteration scheme with Chebyshev polynomials fails for large inertial nonlinearities,
             not only for the pipe problem, but for the van der Pol type of equation x + cx + x = -x2(i  + X) when c = -0.3
             but not when c = -0.1  (Semler et al. 1996).
               *While local  stability applies to a solution for motions in some prescribed domain, global  stability means
             that the solution is stable for all amplitudes. Similarly, asymptotic stability implies that the solution returns the
             system to its unperturbed state as t + 00,  while mere stability means that  it is returned to the neighbourhood
             of  that  state [see, e.g.  Hagedorn (1981) and Appendix F.l  for more precise definitions].
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