Page 93 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

76 SLENDER STRUCTURES AND AXIAL FLOW

instance,+ for metals and certain types of rubber-like materials, and over frequency ranges
of practical interest, energy dissipation can adequately be accounted for by hysteresis;
then, when a specimen of such a material is subjected to harmonic loading with a (real)
circular frequency 52, the energy dissipation per cycle can be calculated by taking the
Young’s modulus to be complex, in the form E(l + pi), where E and p are constants
independent of 52, and p << 1. This implies that the small stresses related to hysteresis
are in quadrature with the principal, linear-elastic stresses. This representation remains a
reasonable approximation for lightly damped oscillation - i.e. provided that sm(52) <<
%e (52) when 52 = %e(52) + i9m(52); however, if there is another source of damping
(e.g. flow-induced damping in cantilevered pipes conveying fluid) such that the overall
damping is large, misleading results may be obtained. Nevertheless, within the limits of its
applicability [e.g. close to a flutter boundary or for lightly damped conservative systems
where sm(f.2) << %e(52)], the hysteretic model is very convenient. In that case, the first
term of equation (3.38) may be replaced by

(3.39)

Finally, a variant of the equation of motion, first introduced by Gregory & Paidoussis
(1966a) for experimental convenience (Section 3.5.6) will be discussed. For simplicity,
consider the horizontal system with dU/dt = 0 and neglect dissipation. Then suppose
that the downstream end of the pipe is fitted with a convergent nozzle, assumed to be
weightless and very short compared to the total length of the pipe. The discharge velocity
Uj is given by Uj = U(A/Aj), where A, is the terminal cross-sectional area of the nozzle
flow passage. Equation (3.36) in this case simplifies to

(T - pA)l - (T - pA) = 0; (3.40)
x=L
consideration of momentum at x = L - cf. the second and third terms of equation
(2.63) - gives
I
(PA - T)/ = MU(Uj - U), (3.41)
x=L
which, in view of (3.40), applies for all x. Hence, substituting into (3.34), simplified
according to the assumptions made here, yields the modified equation of motion
a4 a2 a2w a2
+
+
EI - +-MUU. - 2MU - (M +m)- = 0. (3.42)
ax4 ax2 axat at2
3.3.3 Hamiltonian derivation

The difficulty in deriving an expression of Hamilton’s principle for this problem lies in the
fact that the system is open, with in-flow and out-flow of mass and momentum. Housner’s
(1952) derivation of the equation of motion for pipes with supported ends by means of

+See also Payne & Scott (1960), Snowdon (1968) and the workshop proceedings edited by Snowdon (1975)
and Rogers (1984).