Page 284 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I1 265

to bracket the flutter conditions sufficiently closely, if this type of testing were used for
the system motivating this study: the determination of the flutter boundary of aircraft via
forced vibration testing.
Another set of forced vibration experiments was conducted by Shilling & Lou (1980)
with a vertical cantilever (made of PVC and of Di = 17.6mm) with several masses
attached along the length [see Figure 3.67(a)]. The upstream support together with a
suction pipe and a motor-pump unit were mounted on a horizontal track and oscillated by
an exciter; the test pipe was partially or totally immersed in water, or totally surrounded
by air. It was found that, with internal flow, the response is richer in higher harmonics.
Also, immersion appears to greatly enhance the modal content of the vibration, but this
clearly depends on the forcing frequency; see also Sections 4.2.4, 4.3.2 and 4.4.10.

4.6.2 Analytical methods for forced vibration

Consider a simplified form of equation (3.70) for a cantilever conveying fluid, subject to
a forcing function,

qrn’ + uy + zp”’u?j‘ +a$ + ij = f(& t), (4.80)

with boundary conditions (3.78). By means of the Galerkin method (Section 2.1.6), this
equation may be discretized into

[MI($) + [C1(4}+ [KI(q) = {el, Qj<t> = 1 4j(t>f(t* (4.81)

@j being the jth eigenfunction of a cantilever beam, and Qj the corresponding element of
[e}. Here both [C] and [K] are nondiagonal, nonsymmetric matrices, functions of u. To
decouple the system, the methods described in equations (2.16)-(2.19) may be utilized,
in which the system is first transformed into one of first order,
[Bl[il+ [EIIzJ = {FI. (4.82)

The asymmetry of [C] and [K] means that [B] and [E] are also nonsymmetric. Hence, to
decouple this system, one proceeds (PaTdoussis 1973b; Section 4) to solve the eigenvalue
problem (p[B] + [EJ)(u} = (0) and its adjoint @[BIT + [E]T)[~} = (O}, from which the
same set of eigenvalues pj are obtained, but two different sets of eigenvectors, x, and
$,, leading to modal matrices [A] and [N]. Because of the weighted biorthogonality of
the x, and $,, [NIT[B][A] and [NIT[E][A] are diagonal, an easily proved result. Hence,
introducing the transformation
(7.1 = [AI{O (4.83)

into (4.82) and pre-multiplying by [NIT, one obtains an equation of the form
[Jlt41+ [LIIO = [NIT{F} = {@I? (4.84)

in which [J] and [L] are diagonal; thus the system has been decoupled and hence is easily
solvable. A particular example of excitation f((, t) due to a random pressure field (e.g.
turbulence-induced excitation) is given in Paidoussis (1973b) for external axial flow which