Page 173 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 155

The system is completed by the compatibility conditions at 61 = (or 62 = 0), imposing
the continuity of slope and bending moment at the pinned support:

Solutions are obtainable via an obvious extension of the method of Section 3.3.6(a),
eventually leading to an 8 x 8 determinant, in place of (3.84), which now is a function
of ts also (Chen & Jendrzejczyk 1985).
The qualitative dynamics of the system is illustrated in Figure 3.58(b). For l/L <
l,/L, 1, being a critical value depending on j?, the system loses stability by flutter at a
progressively higher flow velocity as l/L is increased, as compared to l/L = 0 which
corresponds to the basic cantilevered system; theoretically at least, the system is also
subject to divergence at higher flow velocity. For 1/L > Zc/L, the system loses stability

(ii)
\,---

(iii)

-

0 0.5
Time (s)

Figure 3.59 (a) Time histones of oscillation of a cantilevered pipe (p = 0.48) with an additional
simple support at l/L = 0.25, at various flow velocities: (i) Om/s; (ii) 6.6m/s; (iii) 19.Ods;
(iv) 24.2 m/s; (v) 25.2 m/s; (vi) 26.5 m/s (Chen & Jendrzejczyk 1985). (b) The precipitously
decreasing modal damping, <, towards zero as Ucf is approached for a similarly supported
pipe (p = 0.45) and different values of 1/L : ql/L = 0; A, l/L = 0.120; ., l/L = 0.194; 0,
I/L = 0.266 (Jendrzejczyk & Chen 1985).