Page 557 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

ANALYSIS FOR THE DERIVATION OF THE EQUATIONS OF MOTION 527

We now consider the force per unit length of the centreline due to gravity and the
pressure due to the surrounding fluid. For convenience, the pressure distribution of the
surrounding fluid acting on the external lateral surface per unit length of the pipe may
be replaced by the buoyancy force B (i.e. B = A,p,g) and the tensions A,p, and A,pL
applied on the top and bottom faces, where pe and p: are the pressures at levels PI and
Pi. The buoyancy R and gravity forces can be combined into a single force, called the
effective gravity force G, and the pressure force A,p, and the tension Q, can also be
combined into a single term QT. Let (Gxo, G,, GZo) denote components, referred to the
system (KO, yo, zo), of the effeciive gravity force G; then, we can write

Gq = (m - &pe )Sax,, Gy, = h - A,p,)ga,,,
(J.16)
G:o = On - Aupe)gazo, Qf = Q: +Aop,t

where m is the mass per unit length of the pipe, A, is the external cross-sectional area of
the pipe, pe is the density of the surrounding fluid, g is the acceleration due to gravity and
avo, a;,, are the direction cosines, referred to the system (XO, yo, ZO) of the gravitational
acceleration.
For the pipe vibrating in a quiescent fluid, fluid damping arises due to viscous effects
and due to the energy carried away by acoustic waves. The damping force arising from
these effects may be considered to be proportional to the pipe velocity. The components
of this force. referred to the system (XO, yo, ZO), may be written as

(J.17)

where c and c' are the coefficients of viscous damping due to the surrounding fluid,
associated with the lateral and axial motion of the pipe, respectively, and u, v, w are the
displacements of the pipe along the XO-, yo-, zo-axes.
Finally, components of the force resultant per unit length of the pipe centreline can be
written as follows:

(J. 18)

where upio, upyo, upzo are components of the pipe acceleration, M, and ML represent the
added mass per unit length, and R,,, R,, Ri, are components of the reaction force arising
from the internal flow.
Subject to the limitation that the cross-sectional dimensions of the pipe are small as
compared with the overall length of the pipe, the rotatory inertia about axes x and y can