Page 546 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

Appendix I

The Fractal Dimension from the

Experimental Pipe-vibration Signal

The delay-embedding method and the computation of what are generically called fractal
dimensions are relatively recent developments in dynamical systems theory. They are
briefly reviewed here, following Pdidoussis, Cusumano & Copeland (1992). A full intro-
duction may be found in Moon (1992), while a more theoretical review is given by
Eckmann & Ruelle (1985).
The basic assumption is that the dynamical steady state being analysed is evolving on
a low-dimensional manifold in the full phase space (which itself can have many, possibly
infinite dimensions). Knowledge of the dimensions of attractors over the operating range
of a system yields a firm estimate of the number of degrees of freedom needed to model
observed dynamics.
Here we shall use the correlation dimension developed by Grassberger & Proccacia
(1983a,b), which is the most widely applied dimension measure - largely because of
the ease with which it can be computed - see, e.g. Malraison et al. (1983), Brandstater
et al. (1983) and Cusumano & Moon (1995a,b).
To define the correlation dimension, let x(t) denote the steady-state solution under
consideration. It is assumed that x(t) is a finite dimensional state vector. We sample the
data at a fixed time-step At and obtain a data record

where xi x(iAt) and the time origin is taken to be zero. To measure the dimension of
this set, Grassberger & Proccacia define the correzation integral C(r) as

where H is the Heaviside step function, r is a scalar length scale and N,,, = $(N2 - N).
Note that in the limit, as N + 00, the two expressions (1.1) become equal. C(r) is the
cumulative distribution of length scales on the attractor; this statistical interpretation is
important for efficient computation. Grassberger & Proccacia define the correlation dimen-
sion d, by
In C(r)
d,.
=
lim -
r+o In Y
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