Page 86 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 69

pipes with supported ends. An example of a system that loses stability by a Hamiltonian
Hopf bifurcation is the column subjected to a tangential follower load, a nonconservative
circulatory system, for which u,’ = gC = 20.05.
Finally, Figure 3.4(d) shows another form of coupled-mode flutter, for which Done &
Simpson’s (1977) nomenclature of Pai;doussis ’ (coupled-mode)Jlutter will be retained, to
distinguish it from the Hamiltonian Hopf bifurcation of Figure 3.4(c). The distinguishing
feature is that in this case the bifurcation originates directly form a divergent state; hence,
at the onset of flutter (u = u,), the frequency of oscillation is zero [Rr(w) = 01, and
then %e(w) # 0 for u > u,. This kind of bifurcation will be found to arise for pipes with
supported ends (Section 3.4), as well as for other systems (e.g.in Chapter 8).

3.3 THE EQUATIONS OF MOTION

3.3.1 Preamble
The linear equation of motion for a pipe conveying fluid will be derived in the next two
sections by the Newtonian and the Hamiltonian approaches. Before embarking on these
derivations, however, it is useful to introduce some basic concepts.
The first is related to the description of the system via either Eulerian or Lagrangian
coordinates, differentiated by the concepts of spatial position and particle individuality,
respectively. In the Eulerian description the coordinates are fixed in space and may not
be populated by the same material particles as time varies; these are the coordinates
commonly used in fluid mechanics (e.g. in Section 2.2). In the Lagrangian description,
coordinates are identified with individual particles (or elemental volumes surrounding
marked points in the continuum).
To fix ideas, let us consider the longitudinal vibration of a bar, i.e. a one-dimensional
continuum. In the Eulerian description, the position x, fixed in space, may be used as
the independent space variable, and the deflection field described as u(x, t); as the bar
vibrates, different particles or material points at different times will be located at x. In the
Lagrangian description, a given particle may be identified by its position at a given time
(say, r = 0) or, more usefully, by its position when the bar is undeformed, x = XO. This
particle will be at a different x as time varies, but will be identified with xo always (Hodge
1970). Clearly, the deflection field may equally be described in terms of u(x0, t). This is
the more ‘mechanical’ description and it is the foundation of Lagrangian dynamics, for
instance.
Similarly, in the case of flexural oscillations of the pipe, treated as a beam, two coor-
dinate systems may be utilized: the Eulerian (x, z) or the Lagrangian (xg, ZO) - see
Figure 3.5(a). The equilibrium configuration is along the x-axis, and hence (XO, a)
(XO, 0) in this case. The lateral deflection of the pipe may be described as w(x, t) in Eule-
rian coordinates or W(Q, r) in the Lagrangian ones; however, as we can see, there is also
change in the axial or x-position of each point, i.e. u(x, t) or u(x0, t). If we consider a
point P, which in the undeformed state is at PO, then its deflection is

U=X-XO and W=Z-ZO=Z. (3.12)
In what follows we shall use both sets of coordinates, but the usefulness of this discus-
sion will become most evident when the nonlinear equations of motion are derived in
Chapter 5.