Page 262 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I1 243

Theoretical studies with the correct equations of motion were conducted by Gins-
berg (1973) for pinned-pinned pipes and by Paldoussis & Issid (1974) and Pafdoussis
& Sundararajan (1975) for cantilevered and clamped-clamped pipes. Experiments were
conducted by Paldoussis & Issid (1976). The work to be presented here is based mainly
on these studies.
Further work was done by Ariaratnam & Namachchivaya (1986a) on pipes with
supported ends, Bohn & Henmann (1974a) on the articulated system, and Noah &
Hopkins (1980) on an elastically supported cantilever [Figure 3.61(c)], to be discussed
in Sections 4.5.4 and 4.5.5. Also, a great deal of work on the nonlinear dynamics of the
system has been done in recent years, to be discussed in Section 5.9.
In what follows, we shall distinguish between simple parametric and combination
resonances, which will be defined in due course.

4.5.1 Simple parametric resonances
For conservative systems, simple parametric resonances occur over specific ranges of o in
the vicinity of 2w,/k. k = 1, 2, 3, . . _, where w, is one of the real eigenfrequencies of the
system. (‘Simple’ is used in this book to differentiate parametric resonances associated
with one eigenfrequency from combination resonances, defined in Section 4.5.2, involving
two; however, they are often just referred to as ‘parametric resonances’ for simplicity.)
As p + 0, the resonances occur at w/w, = 2/k, and for larger p over a range of w in the
neighbourhood of these values. For nonconservative systems, there is a minimum value
of p below which parametric resonances are impossible, and for higher p they occur in
the vicinity of w/%e(w,) = 2/k, %e(w,) being the frequency of oscillation associated
with the n th complex eigenfrequency, w,.
One may distinguish primary resonances, corresponding to odd values of k, of which
the principal one (k = 1 so that w/w,, = 2), a subharmonic resonance, is of particular
importance, and secondary resonances, corresponding to even values of k. For the pipe
problem, as the w,, vary with u, it is expected that the ranges of w necessary to induce
parametric resonances will vary accordingly.
The easiest way of determining the regions of existence of parametric resonance is via
a Fourier series solution approach, usually known as Bolotin’s method (Bolotin 1964).
To this end, the equation of motion, equation (3.70), is discretized by Galerkin’s method,
~(6, t) = N 4r (6) qr(t). where the 4, are the beam eigenfunctions with the appropriate
boundary conditions, leading to an equation similar to (3.86) but with the U terms retained;
with (4.69) substituted therein, one obtains

ij + {F + 2j3”2~o(l + p COS wt)B]q
+ {A + [u~(I + p COS ~t)~ j31/2uop~ wt - r
sin
y
-
-
+ n(l - 2vS)lC + [y + /3%0pw sin wt]D + yB)q = 0, (4.70)
in which q = (41, q2, . . . , q~]~, A is the diagonal matrix with elements A:, h, being the
rth dimensionless beam eigenvalue associated with @,, F is a diagonal matrix with
elements ah: + cr, and B, C and D are square matrices with elements b,,, c,, and d,,
defined in equation (3.87).