Page 277 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

258 SLENDER STRUCTURES AND AXIAL FLOW

parametric resonance was trying to establish itself, but was confused by the presence of
components of combination resonance.

4.5.4 Parametric resonances by analytical methods

The linear parametric resonance regions are frequently determined as special cases in
nonlinear analyses of the system (Section 5.9). This is done for two reasons: to validate
the more general nonlinear analysis in the linear limit, and to compare directly the linear
and nonlinear dynamics of the system.
The first such analysis, unusually published as a paper wholly devoted to linear
dynamics, is due to Ariaratnam & Namachchivaya (1986a) who study the principal
(subharmonic) resonances associated with the first and second modes of pipes with
supported ends (w 2: 2w,, r = 1,2) and the corresponding combination resonances (w 2
02 f wl) by means of nonlinear analytical methods. These methods are in fact the same as
those utilized to analyse the nonlinear dynamics of the system by Namachchivaya (1989)
and Namachchivaya & Tien (1989a,b) and are described briefly in Section 5.9. Basically,
the system is discretized into a two-degree-of-freedom one and transformed into first-order
form while using an elegant Hamiltonian formulation; then the method of averaging is
applied, via which the boundaries of the resonance regions are determined. The procedure
is mathematically complex but very powerful: it yields analytical expressions for the
resonance bounds, the minimum value of p below which resonance does not occur, and
so on. Also, by considering the stability of the solutions, it is shown which exist and which
do not; specifically, it is shown that for pipes with both ends supported the difference
combination resonance (w 2: 02 - wl) does not exist (cf. Figure 4.28).
The analytical results are compared to numerical ones obtained by the authors and
Paldoussis & Sundararajan (1975) - see the middle three regions of Figure 4.28. Agree-
ment is quite good, despite the fact that the analytical method is meant to be valid only in
the neighbourhood of the resonances, e.g. for w = 2wl + O(E); the discrepancy between
analytical and numerical results becomes important for p 3 0.4.

4.5.5 Articulated and modified systems
A two-segment articulated system hanging as a cantilever and conveying harmonically
perturbed flow as in (4.69) has been examined thoroughly for parametric resonances by
Bohn & Herrmann (1974a). The two pipe segments are of equal length, I, and no intercon-
necting springs are present, so that gravity is the only restoring force. Hence the following
dimensionless parameters are used: p = 38, U = U/ (igl)’” and 0 = [I/ ($g)] 52. The
resonance regions are determined by Bolotin’s method and Floquet multipliers.
Basically, the dynamical behaviour of this system is similar to that described in
Sections 4.5.1 and 4.5.2, for both simple and combination resonances. Of particular
importance is the dynamical behaviour just above the critical flow velocity for instability
in steadyjow, as shown in Figure 4.34: (a) for = 0.25, when stability is lost by flutter, at
-
uc. = 2.632; and (b) for p = 1.0, when stability is lost by divergence, at ccd = 1.732. The
dynamical behaviour in Figure 4.34(a) is similar to that in Figure 4.29(b), showing that
what is a region of stability for U < ti,. is essentially transformed into one of combination
resonances (quasiperiodic oscillation) for U > si,.; however, in this case also, there exists