VORTEX-INDUCED OSCILLATIONS 613
convenient measure of the damping is the logarithmic decrement, , which
is defined as the natural logarithm of the ratio of any two successive ampli-
tudes of oscillations. It may be expressed as
2 π,/(1 ,^2 )1/2 (14.74b)where,C/Cc. When the frequency of the exciting force F(t) coincides
with the natural frequency fn, resonance takes place.
In a real structure, the mass distribution, ms, along the pile may be
non-uniform; the mass of entrained water, mw, in a hollow pile and the
added mass, ma, resulting from the motion of the pile in water should also
be accounted for. The mass per unit length is then
mmsmwma. (14.75)The added mass may be expressed as maCa πD^2 /4. For an isolated cylin-
der,Ca1.Cais related to CM, the inertia coefficient, as CaCM 1. The
pile may not be completely immersed; in this case, the mass of entrained
water and the added mass are to be considered only for the immersed part
of the pile.
In the analysis of dynamic response, the real structure is replaced by
an equivalent cylinder of the same cross-section but of a length equal to
the depth of water (Fig. 14.21). Both the real structure and the equivalent
cylinder have the same mode shape, natural frequency, and inertial prop-
erties. By this artifice, the experimental results pertaining to a cantilever
with the fixed end at the sea bed can be used to analyse the real structure.
In this section only the simple case of the vertical pile with a uniform dis-
tribution of mass and without any end mass or constraints is considered.
Piles with lengths greater than the water depth and with end constraints
and end masses are treated fully in Hallam, Heaf and Wootton (1978),
with worked examples.
Fig. 14.21 Equating a real structure to an equivalent structure