Page 97 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

80 SLENDER STRUCTURES AND AXIAL FLOW

uL i

Figure 3.8 (a) Definition of the coordinates and unit vectors associated with movements of the
free end of a cantilevered pipe (top), and the relationship between i, k, t and x for any point along
the cantilever (bottom); (b) velocity components for an element of the fluid in a cantilevered pipe;
(c) the same for an element of the fluid in a pipe with clamped ends.

The equation of motion is derived next, for a vertical cantilevered pipe, taking into
account gravity effects. The pipe is assumed to be inextensible, and use is made of the
curvilinear coordinate s. The derivation involves the evaluation of the various terms in
the Hamiltonian statement (3.57), following along similar lines to Benjamin's (1961a)
and Paldoussis' (1973a), but making use of the notation and relationships developed in
Section 3.3.1.
Some useful relationships will be obtained first, as follows: (i) recalling from (3.12)
that u = x - xo with xo = s here, then ,i = U; (ii) from (3.14), ax/as = [1 - (a~/as)~]'/~
with z = w, and hence ax/& 2 1 - id2, where ( 1' = a( )/as; (iii) from (3.17b), uL =
- so zw ds. Also, one may write r~ = iLi + iLk = ULi + WLk; from (3.25), tL = xii +
L 1 /2
zik 2: [I - iwf]i + wik; 6rL = BuLi + 6wLk; and the second term of (3.57) may be re-
written as

6" [MU2 8uL + MU(WL + Uw;) 6wL] dt, (3.58)
correct to S(c2), having made use of the order considerations expressed by (3.17a,b).
Hence, by grouping the terms implicitly involving a double integral into the first term,
Hamilton's principle is rewritten as

correct to (!?(e2).