Page 61 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

44 SLENDER STRUCTURES AND AXIAL FLOW

It is easy, therefore, to appreciate that in this case there exists an added mass matrix, of
[:I, mio ] (2.135)
the form

nz0,
which couples hydrodynamically the motions of the two shells; here the subscript n has
been suppressed. The corresponding vector is (d2w,;/dr2, d2w,,/dt2)T; nz;; and m,; are
the negatives of the coefficients of d2w,;/dt2 in (2.120a,b), while m;, and moo are the
corresponding quantities from (2.121). It is obvious that the matrix must be symmetric,
as a consequence of the reciprocity principle in mechanics.
Consider next the situation of rigid-body motion (n = 1) of both the inner and outer
cylinders. In this case
[MIX + [CIX + [K]x = -[M’]X, (2.136)

where x = {y;, yo, z;, z,)~ and

(2.137)

in which m!;.?’ = mf = ,OITR?L[(R,/R~)~ + l]/[(Ro/Ri)2 - 11 and so on, as given by
expressions (2.120a,b) and (2.121) for n = 1. Thus, coupling ofthe motions of the two
cylinders arises. This means that if, for example, the inner cylinder is given some initial
displacement or velocity at r = 0, the outer cylinder would also vibrate for r > 0.
It is noted in (2.137) that, because of symmetry, there is no fluid coupling between y-
and z-motions; i.e. acceleration of one cylinder in one direction generates a symmetric flow
field, with no force resultant in the other direction. Generally, however, for asymmetric
systems, such cross-coupling does exist, and matrix (2.137) would be fully populated, ;.e.
rn;: and similar terms would no longer be null; furthermore, m? # in;, and SO on.

(e) Effects of various parameters on added mass

Tables, figures and lists of results for added mass coefficients in a variety of systems are
given by Blevins (1979), Chen (1987), Gibert (1988) and Naudascher & Rockwell (1994).
Hence, we shall confine ourselves here to making some general comments on parameters
affecting the added mass, of which the reader should be aware.

(i) General effects of geometry. In general, proximity to other structures affects
the added mass of the vibrating one; e.g. proximity to a rigid wall signifies increased
accelerations (for inviscid fluid) and hence larger added mass, as already remarked in
the foregoing, especially in connection with the system of two coaxial cylinders or shells
(Figure 2.7). Of equal interest is the case of eccentrically located cylinders (see also
Chapter 11). A useful result (Gibert 1988) is that the added mass coefficient, C;:, is
given by

2(r - 1) [r - 1 - Je(2r - 2 -e)]
-- - , for r< 1.1, (2.138)
cnl (r - 1 - e)2