Page 446 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

418 SLENDER STRUCTURES AND AXIAL FLOW

The initial (unstrained) centreline

centreline

Figure 6.1 Kinematics of the curved pipe system; definition of coordinate systems and of coor-
dinates used.

of the pipe, a particle on the pipe that was initially at PO, occupies a new position P.
The second, moving reference frame (x, y, z) has its origin at P and is attached to the
deformed centreline in a manner analogous to the (XO, yo, zo) system. The orientation of
the (x, y, z) system is completely determined by the three Eulerian angles $, 8 and $.
The orientation of the (x, y, z) system is completely defined by the sequence of small
Eulerian rotations: $ about the zo-axis, 8 about the new x-axis, and finally $ about the
new z-axis. The components of Pg, referred to the (XO, yo, zo) system, are denoted by
u, w and w. Thus u and w are the transverse displacements, while w is the longitudinal
displacement. Furthermore, during the motion of the pipe, a plane section at PO rotates