Page 54 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CONCEF'TS. DEFINITIONS AND METHODS 37

Figure 2.7 (a) Cross-sectional view of two coaxial thin shells at rest, with the annulus filled with
a dense fluid; vi and wi are the displacements of the inner shell in the circumferential and radial
direction, respectively. (b) Three-dimensional view of the shell instantaneously deformed in the
n = 2 circumferential mode, with little axial variation (either because the shell is long and the
mode of axial deformation is small, or because idealized 2-D deformation has been assumed).
(c) Definition of the n = 1-4 circumferential modes.

shells are free to vibrate in a low axial mode number (e.g. in the first, beam-like mode), so
that gradients of displacements in the longitudinal direction are negligible, as compared
to the transverse directions [Le. in the plane of Figure 2.7(a)]. Alternatively, one could
assume that the mode of oscillation is purely two-dimensional, as shown in the example
of Figure 2.7(b). Hence, the displacement of the mean surface of the shell, generally of
the form {u, v, wJT, with u, v and w being, respectively, the axial, circumferential and
radial components, in this case simplifies to {w, wIT.