Page 297 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

278 SLENDER STRUCTURES AND AXIAL FLOW

Unfortunately, the methods required for the study of nonlinear dynamics (and for
answering questions such as those in the previous paragraph) are much more complex than
those for linear dynamics, and they are not in everyone’s repertoire. Furthermore, they
cannot easily be ‘covered’ in a book such as this, properly requiring one or more books
of their own to do the job properly. One may distinguish ‘classical’ methods of nonlinear
dynamics exemplified by the treatment in Minorsky (1962), Hayashi (1964) and Andronov
et al. (1966), and ‘modern’ methods as exemplified by Guckenheimer & Holmes (1983),
Sanders & Verhulst (1985) and Glendinning (1994). Other useful general references are
Hirsh & Smale (1974), Nayfeh & Mook (1979), Hagedorn (1981), Rand & Armbruster
(1987), Arnold (1988), Anosov et al. (1988), Arrowsmith & Place (1990) and Nayfeh
(1981, 1993). An abbreviated treatment of some of the methods utilized extensively in
this chapter and elsewhere in the book is given in Appendix F. This appendix, along with
the specialized references cited therein, should be sufficient to guide the serious reader.
The more casual reader may skip over the mathematics and concentrate on the physical
interpretation of the results obtained in each case.
Most of the methods in Appendix F are concerned with local analysis, i.e. nonlinear
behaviour in the vicinity of a fixed point or limit cycle; this is a requirement for the
methods to work. More difficult is consideration of global dynamics aspects, e.g. the
possibility of a large-amplitude limit cycle circumscribing two fixed points, which has
not emanated from either. Although some aspects of global behaviour may sometimes be
discerned from local analysis, the complete dynamical picture can only be provided by
global analysis. The methods required for the latter will be discussed in ad hoc fashion
and without too much detail.
The possibility of chaos in nonlinear systems has been known ever since Henri PoincarC
at the turn of century, but it is fair to say that, for applied scientists, its existence lay
dormant until the 1960s and the advent of Lorenz’s work on thermal convection in the
atmosphere. A good layman’s introduction is given by Gleick (1987), and an excellent
engineering treatment by Moon ( 1992). More sophisticated mathematical treatments are
given by, among others, Thompson & Stewart (1986) and Wiggins (1988, 1990). Other
useful references are Berg6 et al. (1984), Devaney (1989), Parker & Chua (1989), Hao
(1990) and Tsonis (1992), among others.
Basically, chaos arises when, over some ranges of parameters, the system ceases being
predictable, in the sense that small changes in initial conditions may generate dispropor-
tionately large differences in the state of the system at any given time sufficiently long
afterwards. The system is deterministic, but it behaves as if it were random - but with a
most significant difference: its states are within specific regions of state-space, rather than
all over, as would be the case for a truly random system. Thus, the trajectories of system
response visit certain parts of the phase space, apparently randomly, but never others. A
fractal nature in such plots is often revealed, whereby a small such region, when blown
up, displays a similar character at a more microscopic scale.
Inevitably, specialized methods have been developed for the study of chaotic dynamics.
These will be described in abbreviated form as necessary in the sections that follow.

5.2 THE NONLINEAR EQUATIONS OF MOTION
In many of the early papers on nonlinear dynamics of pipes conveying fluid (Holmes 1977;
Ch’ng & Dowel1 1979; Lundgren et al. 1979; Bajaj et al. 1980; Rousselet & Herrmann