Page 201 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 183

see, e.g. Pierre & Dowel1
rest. This phenomenon is also known as mode localization ~
(1987), Bendiksen (1987) and Vakakis (1994).
The response of the system to a convected pressure perturbation is considered next,
of the form q exp[i(Rt - ~t)], i.e. with @(c) = exp(-iKc) in (3.149). In this case, the
solution is ~(t, t) = E",=, [C, exp(ih,() + @(()I exp(iwt), where @(() is the particular
solution; for a uniform, infinitely long pipe,
@(() = qeplKC[K4 - (U2 - f + n)K2 + 281'2URK - R2]-'.

The solution follows the same pattern as before, but @(<) comes into the picture; i.e.
4
Y(4) = C{an[a,-l + @"(O)] + b,[a, + @'(1)] + d,@(O) + en@(l))el*flt + @(().
n=l
Hence, a more complex form of (3.153) results, involving @(O) and Q(1). However, the
form of the solution is the same, and taking a, = alp] exp(iK), one eventually obtains
-"$ = qF(U, 8, Q, K, r, n), (3.155)
4 4
x[dnhn(l - e-'*,) - U,L,(I - e-l*n)K2 - KI + C[f(hn, K, e,, bn)]e-lK
={ ~ ~ ~ ~ n e l ~ + bnh,elhn - anh, - b,h,elK] I{ n=l I'

~
*
n
-
K
n=l K4 - U2K2 + 2B1/2UfiK - Q2
where u2 = u2 - f + n, and f is the same as the other expression in the numerator but
involving e,* and 6, instead of d, and a,, and +K for the last term. The interesting part of
this result is that F becomes infinite when either of the two bracketed expressions in the
denominator vanishes. Comparing with (3.151), it is seen that the second bracketed quan-
tity vanishes, if R coincides with one of the eigenfrequencies of the unsupported system:
Q = w. This is the 'normal' resonance condition. Then, comparing the first bracketed
expression to (3.153) with (3.154) substituted in it, it is clear that this too can vanish for
K = p, i.e. when the convection velocity of the pressure perturbation coincides with the
phase velocity of free waves in the pipe, a 'new' type of resonance.
Similar work on wave propagation in periodically supported pipes (with an additional
rotational stiffness present at each support) has been done by Singh & Mallik (1977).
The interested reader should also refer to Mead (1970, 1973).

3.8 ARTICULATED PIPES

It is recalled that, essentially, the incredible saga of the dynamics of cantilevered pipes
conveying fluid, in all its manifestations and variants, began with Benjamin's (1961 a,b)
work on articulated cantilevered pipes. Benjamin derived the correct statement of
Hamilton's principle for an articulated system, equation (3.10), in much the same way as
in Section 3.3.3, and in the process he discussed the incorrectness of previous derivations
of the equations of motion of cantilevered pipes. He also examined the mechanisms of
energy transfer and stability (Section 3.3.2), and illustrated the qualitatively predicted
dynamical behaviour by sample calculations and model experiments. Further work on the