Page 306 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 287

Recalling that u - CY(€*), w - O’(E), and using (5.6),

is obtained. Moreover, E is given by equation (5.33).
The expression for gravitational energy is the same as in the case of the cantilevered
pipe, i.e.
I”
%= -(m+M)g (xo+u)dxo. (5.35)

The final equations of motions are obtained once again via application of variational
techniques, this time with two independent variants, 6u and 6w. After many integrations
by parts, one finally obtains
(m + M) u + MU + 2 MU U’ + MU2 u“ + MU u’ - EAu”

- EI(w’”’ w’ + w” w”’) + (To - P - EA)w’ w” - (m + M)g = 0, (5.36a)
(WZ + M ) W + M 0 W’ + 2 M U W’ + M U2 W” - (To - P) W” + EZ w””

- EZ(3 u/fl ,,,’I + 4 ur! w’!/ + 2 u/ ,,,,!/!( + u///! + 2 ,,,,!2 ,+,//// + 8 ,,,I! ,,,l/’ + 2 ,,,!/3)
+ (To - P - EA) (u” W’ + U’ W” + 5 d2 w”) = 0, (5.36b)

where one now has two independent equations, instead of the one for a cantilevered pipe.
A Newtonian derivation is outlined in Appendix G.2.

5.2.5 Boundary conditions

Using variational methods, it is straightforward to derive boundary conditions for the
different cases considered. For the cantilevered pipe, the boundary conditions are the same
as for the linear case: w(0) = w’(0) = 0 and w”(L) = w”’(L) = 0. For the pipe fixed at
both ends, it is obvious that u(0) = w(0) = u(L) = w(L) = 0; in addition, if the pipe is
simply-supported, one obtains w”(0) = w”(L) = 0, while for the clamped-clamped pipe,
w’(0) = w’(L) = 0. Only two boundary conditions are necessary for u.

5.2.6 Dissipative terms
Dissipative terms have to be added to complete the equations. This can be done by
assuming that the internal dissipation of the pipe material is viscoelastic and of the
Kelvin-Voigt type (Snowdon 1968), i.e. that it is represented by CJ = E E + E* F, where
CT is the stress and E the strain. Following then the approach used by Stoker (1968), the
strain energy is modified, providing additional terms that can be written as

(5.37)

where a is the coefficient of Kelvin-Voigt damping in the material. Therefore, in equa-
tions (5.28) and (5.36a,b), EZ may be replaced by EZ(1 +a a/&) and EA by EA(l +