Page 57 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

40 SLENDER STRUCTURES AND AXIAL FLOW

Figure 2.8 Rigid-body motion of the inner cylinder, in the y and z directions within the fluid-filled
annulus (see Figure 2.7).

for the fluid become
91 =-sino+-coso, dw, 91 =o.
dv,
(2.122)
ar Ri dt dt ar Ro
Proceeding as before, the pressure on the surface of the inner and outer cylinders is
determined and it is a function of both d2v,/dt2 and d2w,/dt2; in fact, the coefficients of
these accelerations are identical to those in (2.1 18a,b), but with n = 1; e.g.

P;,I; E pi; = -PR; (2.123)
1Ri
Then, the forces on the cylinder may be determined (i) either as before, by consid-
ering the virtual work associated with virtual displacements fie sin 0 + E, cos 0, or (ii)
directly, by integrating the pressure on the rigid cylinder via

F?’ 1.11 . = I’” -pi,& sin OdO, -p;,liR; COS Ode, (2.124)

and similarly for FZ,,i and F:,,;; thus, if following (i), the projection of the force field onto
the ‘mode’ concerned, has an immediate physical meaning in this case! It is obvious that
the same results as in equations (2.120a,b) are obtained, as should be, but with d2w,;/dt2
replaced by either d2v,/dt2 or d2wc/dt2.
A number of important conclusions are reached and insights gained from these results
in the following.

(a) The added mass concept

As is clear from (2.120a,b), the fluid loading is associated entirely with accelerations of
the structures, and hence accelerations of the fluid. This is physically reasonable: infinitely
slowly generated displacement of the shell away from its equilibrium position cannot, in