Page 304 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 285

=A+B. (5.26)

The first term, A, cancels the last term in equation (5.20), while with the use of equa-
tions (5.1) and (5.23) B is found to be

B = MU^/" [ t” + z’~ z” - z” l‘ (z’ z”) ds] Sz ds dt, (5.27)

and hence contributes all the centrifugal-force terms.
Finally, after many transformations and manipulations, and recalling that z = w as per
equation (5.1 l), the general equation of motion is found to be

(rn + M)ib + 2MUW‘(l + w”) + (rn + M)gw’ (1 + ; wJ2)
+W”[MU2(1 +w’2)+(Mii-(rn+M)g)(L-s)(1 +TW ‘2 )]
3
+
+ ~z[~””(i w’2) + 4 w’ w” w’ii + w”3]
- W” [ lL .I’ (rn + + W’ W’) & ds

+ 1‘ (;Mow‘’ + 2MU w’w’ + MU2 w‘w‘‘) ds

+
+ W’ is(rn M)(w’* + di?) ds = 0. (5.28)

The Newtonian derivation of this equation is given in Appendix G. 1.

5.2.4 The equation of motion for a pipe fixed at both ends
Here, as the inextensibility condition can no longer be applied, two equations are necessary:
one in the x- and the other in the z-direction. Moreover, since both ends of the pipe are
fixed, the right-hand side of expression (5.10) is now zero. Consequently, it is obvious that
the contribution of the fluid forces is not the same as in the case of the cantilevered pipe.
When a bar is subjected to tension, the axial elongation is accompanied by a lateral
contraction. Within the elastic range, the Poisson ratio u is constant (Timoshenko &
Gere 1961) and, for rubber-like materials, v 0.5. In the case where only a uniaxial
load is applied to an elastic body, the change of unit volume is proportional to 1 - 2 u.
Consequently, for rubber-like materials, the volume change due to uniaxial stress can
be considered zero, i.e. they are incompressible. In the case of a pipe, for any initial
volume of length dxo, this conservation of volume leads to dxo So = dxo( 1 + E) SI, where
SI represents the cross-sectional area of the pipe after elongation. For the incompressible
fluid inside the pipe, one also has Uo So = U1 SI, with Uo and U1 being the flow velocities
before and after elongation. Thus,

Ul(X0) = uo (SO/Sl) = uo (1 + E). (5.29)