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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].