Page 172 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

154 SLENDER STRUCTURES AND AXIAL FLOW

‘cf

I I
I I
I
I
I I
I I
I I
I
I
I
I I
I
I I
I
I Stable
I I
I I
1,IL -
I
0 7 1
(b) 1 lL
Figure 3.58 (a) Schematic of a pipe fixed at the upstream end and with a simple support at
& = 1/L; (b) qualitative stability diagram, showing the dimensionless flow velocities u versus 1/L
and the definition of lc/L; &d and urf are the dimensionless critical flow velocities for divergence
and flutter, respectively (Chen & Jendrzejczyk 1985).
The domain of the problem 9 = [0, 13 is broken into two, e1 = [0, &] and $2 = [0,
1 - with Cs = 1/L, wherein the dimensionless displacements of the pipe are ~1 and
~72, respectively. The equations of motion are then given by

(3.1 15)

cf. equation (3.1); the dimensionless quantities are the same as before, based on the overall
length L. The boundary conditions are