Page 269 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

250 SLENDER STRUCTURES AND AXIAL FLOW

recalled, however, that both damping due to Coriolis forces and parametric excitation are
proportional to uo for a constant p; hence, for a given uo the magnitudes of the parametric
excitation and of damping are predetermined, and as uo is decreased they both decrease
proportionally.+ Thus one cannot say a priori for what range of uo parametric resonances
may occur, if at all, for a given range of p.
Hence, the dynamics of cantilevered pipes with pulsating flow is quite different from
that of columns with harmonically perturbed end-load. The combined effects of the
nonconservative follower and Coriolis forces fundamentally affect both the existence and
the extent of the parametric resonance regions.

4.5.2 Combination resonances

Another type of parametric oscillation is due to combination resonances, which occur in
the neighbourhood of w = (0, f w,)/k, k = 1,2, . . ., where in this case n # m. They are
not obtainable by Bolotin’s method, because the oscillation is not periodic but quasiperi-
odic, but they may be obtained semi-analytically by Floquet analysis - see Meirovitch
(1970) and Appendix F. 1.2.
For the analysis, equations (4.70) are rewritten into first-order form, u = {q, q}T, and then
integrated numerically with 2N different initial conditions: { 1, 0, 0, . . .)T, {0, 1, 0, . . .}T, etc.
The solutions thus obtained after one period, u1 (T), u*(T), . . . , u*~(T) are used to construct
the so-called fundamental or ‘monodromy’ matrix of the system,

[Y] = [u’(T), U*(T), . . . , U2N(T)]. (4.75)
The characteristic or Floquet multipliers of the system are given by the eigenvalues of [ Y].
If at least one of them has an absolute value greater than unity, the system is unstable (in
the linear sense), giving rise to (i) a parametric resonance if this Floquet multiplier is real,
and (ii) a combination resonance if it is complex. In this sense, combination resonances
correspond to quasiperiodic motions - see Section 5.9.
A typical set of results for a clamped-clamped pipe are shown in Figure 4.28. A small
amount of damping has been added (a = lop3), so as to eliminate the profusion of very
narrow resonance regions that would otherwise clutter up the figure. Regions of both
simple and combination parametric resonance are seen to exist. The various resonances
are well ordered and easily identifiable, as would for instance be the case for a column
subjected to a harmonically perturbed end-load. Indeed, the analogy applies to the extent
that combination resonances of the difference types do not materialize for these boundary
conditions (Iwatsubo et al. 1974; Ariaratnam & Namachchivaya 1986a). In this figure,
001, the zero-flow first-mode undamped eigenfrequency, is used simply as a normalization
factor throughout. As a result of this, however, the upper resonance regions in the figure
appear to be more extensive than the lower ones; had w/q2 been used instead of w/wol
for the second mode, the same width as for the principal resonance of the first mode
would have been obtained. The minimum value of p below which resonance does not

+A different behaviour is obtained if the excitation force can be varied independently of damping. Calcu-
lations in which an external axial load Fo cos OT is imposed and u is kept constant show that parametric
resonances may then occur even for small u.