Page 494 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

464 SLENDER STRUCTURES AND AXIAL FLOW

Figure A.1 (a) Diagram of the pipe under consideration oscillating in a horizontal plane;
(b) definition of coordinate systems and displacements; (c,d) an element of the fluid and of the
pipe, respectively, with the forces and moments acting thereon.

Force and moment balances in the z- and x-directions, as in Section 3.3.2, after simpli-
fication and substitution of s 2: x, give

+M
a dU
-((PA - T) +M - = 0.
ax dt
Integration of (AS) from x to L and substitution in (A.4) gives
a2 a2w
+ 2MU __ + (M +m) - 0,
=
axat at2
which is the same as equation (3.38) once terms involving T, 7, g, c and E* have been
deleted.
The same equation was obtained by Ginsberg (1973). His derivation, however, is flawed
in two ways. (i) Having approximated i 2 I, irrespective of order-of-magnitude consid-
erations, the second terms of v,~ and arel in (A.2) are absent, and hence so is the last