Page 56 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CONCEPTS. DEFINITIONS AND METHODS 39
[ d2Wn;
Po.ni E Pni = -pRi 1 2(Ro/Ri)n ] cos ne __ (2.1 18b)
IZ 1 - (Ro/Ri)2n
I RO dt2 '
The subscript notation i, ni indicates the pressure on cylinder i due to nth mode vibration
of cylinder i, whereas 0, ni indicates the pressure on cylinder o due to the same vibration
of cylinder i.
Next, the loading on the shell and on the outer cylinder may be obtained from the
principle of virtual work, i.e. via

r2n
SWi,ni = lo {(-P;,~;R; de)(6wfIi sin ne + 8wni cos ne)),
(2.1 19)
r 2n
6Wu,ni = J', ((po,niRod~)(6v,,i ne + 8wni COS ne)).
sin
As seen in equations (2.1 18a,b), pi,ni and po,ni are functions of cos ne; hence, in view of
the orthogonality of sin ne and cos ne, only the 8wni component of the virtual displace-
ment contributes to the virtual work. Therefore, the forces on the inner shell and the outer
cylinder due to motions of the inner shell in the nth mode, denoted by F;,fli and Fo.,,i,
respectively, are given by

(2.120~1)

(2.120b)

In effect, to obtain these forces, the pressure field was transformed into a surface-force
field and projected onto the modal deformation vector in the eigenspace of this system.
Further, it is noted that if the shell oscillates in more than one mode, Fi,,i and Fo,nj will
still be the same, because, when projected onto the nth mode eigenvector, the contribution
of the additional modes is.zero, as a result of orthogonality of the cos ne for different n,
as per relationships (2.1 18a,b) and (2.1 19).
Similarly, if it is the outer shell that is flexible and oscillating while the inner one is
rigid, proceeding in the same manner one finds

d2wno/dt2 (Ro/R;)2J' + 1 d2w,,o
=
Fi,no = Fo,ni Fovno -pnR: - (2.121)
d2Wn;/dt2 ' (Ru/R;)2n - 1 1- dt2 '
There are obvious symmetries in the coefficients of d2wn;/dt2 and d2wno/dt2 in (2.120a,b)
and (2.121), which will be discussed later in subsection (d).
In the foregoing, rigid-body transverse motions of the cylinder were considered as
a particular case of shell motions with n = 1 [Figure 2.7(c)]. For transverse rigid-body
motions, however, the eigenvector or eigenfunction of motion becomes trivial, simplifying
to motion along specified directions; thus, one can then think of motions in the Cartesian
directions y and z, and work out the loading associated with oscillatory displacements w,
and w, in these directions, as shown in Figure 2.8. In this case the boundary conditions