Page 285 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

266 SLENDER STRUCTURES AND AXIAL FLOW

is very similar to the case of internal flow; as this is discussed in Chapter 8 (Volume 2),
it will not be duplicated here.
An alternative method is to obtain the eigenvalues and eigenfunctions {A;, x;(()} of the
conservative part of (4.80), q”” + u2q” + ij = 0, as well as those of its adjoint, [A;, +;($)};
these are in fact the same as for problem (2.52), and are given by equations (2.59). Then,
introducing xj(()q; (t) into equation (4.80) and using the biorthogonality relation
(2.57), another form of equation (4.81) is obtained, in which [MI and [K] are diagonal.
The presence of the Coriolis term in (4.80), however, means that [C] is not diagonal.
Hence, even more than for the problem in Section 2.1.6, this method offers no special
advantage, since it cannot diagonalize the nonhomogeneous problem ‘in one step’ as it
would if this were an ordinary mechanical system.
Let us now turn our attention to the forced response of a cantilevered pipe with a tip
point muss, A, subjected to an arbitrary force field, f(e, t). The dimensionless equations
of motion and boundary conditions in this case are

q”” + u2q” + 2purj’ + a?j + ij = f((, t); (4.85a)

q(0) = q’(0) = 0, q”(1) = q”’(1) - pij(1) = 0, (4.85b)
where p = A/[(M + m)L]. An alternative way of formulating the problem leads to

q”” + u2q” + 2/9’/*U?j’ + a?j + [l + WUs(6 - 1)lij = f((, t). (4.86a)
q(0) = q’(0) = 0, q”(1) = q”’(1) = 0, (4.86b)
in which Us(( - 1) is the Dirac delta function. As hinted in Section 2.1.4, the decoupling
of the equations in this case poses some interesting problems, because the boundary
conditions in (4.85a,b) are time-dependent. Three possible procedures immediately spring
to mind, as follows:
Method (a): to utilize the eigenfunctions +;(e) of the problem q’”’ + ij = 0 subject to
boundary conditions (4.85b) to discretize the system;
Method (b): to utilize these same eigenfunctions +;(() but apply them to an ‘expanded
domain’ of the problem (Friedman 1956), which effectively means that the time-dependent
boundary conditions are added to the equation of motion, so that one obtains

Method (c): to utilize the cantilever beam eigenfunctions, 4; (e), directly to decouple
equation (4.86a).
In principle, one can show directly which of these methods are correct or otherwise,
but here we shall do so by means of sample computations. To simplify matters and
since the main point of interest is the decoupling procedure, we consider the homoge-
neous undamped version of this system: f((, t) = 0, a = 0. The results are presented in
Table 4.6, for two-mode discretization in all cases. For the same value of p, the values
of WI and w2 for u = 0 are the same whether B = 0 or = 0.1, and hence they are not