Page 553 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

ANALYSIS FOR THE DERIVATION OF THE EQUATIONS OF MOTION 523

For small deformations, the relationship between the unit axes is found to be
(Love 1927)

The same transformation matrix relates (x, y, z) to (xo, yo, zo). In (J.4) it is noted that
-aulas, aulas + w/Ro and @ are simply the angles of rotation about the x, y and z axes,
respectively; hence (J.4) could have been obtained by a sequence of small rotations via
the corresponding transformation (rotation) matrices.
Equation (5.4) has been obtained assuming no centreline extension; but, for small defor-
mations, the two frames obey the same relationship, even for the extensible case.

5.2 THE EXPRESSIONS FOR CURVATURE AND WIST

In this case, three coordinate systems are utilized: the Frenet-Serret and flexure-torsion
reference frames used in Section J.l, as well as an inertial system coincident with the
former. Then, after introducing the direction cosines relating these reference frames and
considering the derivatives of (e,, ey, e,) with respect to s, which are related to curvature
and twist, after very lengthy but straightforward manipulation (Love 1927, Chapter XXI;
Van 1986, Appendix B) one finds the curvature and twist of the deformed pipe in terms
of the deformation:

The axes associated with K, K’ and r* are defined in Figure 6.1.

J.3 DERIVATION OF THE FLUID-ACCELERATION VECTOR
Recall equation (6.14) in the main text,

avf avf
+
af = - U -, (5.6)
at as
where Vf is given by equation (6.9). By differentiating Vf with respect to t and s yields

(J.8)