Page 192 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

174 SLENDER STRUCTURES AND AXIAL FLOW

are considered, which satisfy (3.13 1) provided that

(3.135)

Applying the boundary conditions, the frequency equation is obtained, sin(K1 - ~2) = 0,
and the dimensionless frequencies are found to be
nn(c2 -E*)
on = n = 1,2,3.. (3.136)
[c2 - E2(1 - 8)]1/2’

The corresponding mode shapes are given by

where the wavenumber and phase velocity are

and up = w,/K,; up is related to its dimensional counterpart, V,, via wp = [(M +
m)/T]1/2V,. A number of useful observations can now be made. Wave propagation in
this system is not frequency-dispersive, since the phase velocity is not a function of
wavenumber (wavelength). Another manifestation of this is that the ratio of the frequency
with flow to that without is independent of n. Finally, when U = 0, up is infinite, and
the system vibrates with the same phase, whereas for E # 0, a finite up is obtained. This
means that for U # 0, no classical normal modes exist [cf. Section 3.4.1 and Figure 3.131:
various parts of the system pass through their equilibrium position at different times; i.e.
the modal form contains a travelling wave component.
We next consider wave propagation in a beam (EZ # 0), but taking u = 0 in equa-
tion (3.75), as discussed by Meirovitch (1967). In this case, the phase velocity, up, is
a function of the wavenumber (wavelength): up = K; hence, the beam is a frequency-
dispersive medium. A general nonharmonic waveform may be thought of as a super-
position of harmonic waves, q(& t) = E, A, COS[K, ([ - v,,t)]; since each component
travels with different phase velocity, the wave form will change as the wave propagates
along the beam, as a result of dispersion. If u # 0, the situation is further complicated.
This is discussed next, for a pipe on an elastic foundation.

3.7.2 Infinitely long pipe on elastic foundation
This problem has been considered by Roth (1964) and Stein & Tobriner (1970), and what
is presented here is a summary of some of their work.
A form of equation (3.70) is used for the pipe on a generally dissipative elastic foun-
dation, but dissipation may also come from other sources, i.e.
+
+
q”” + (u2 - r + n)~” 2pZulj’ + k~ + ~li ij = 0; (3.138)
no Poisson-ratio effects are considered, however, since no pressurization-induced tension
can arise in the absence of end constraints. Since L could be infinite. in the dimensionless