Page 302 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 283

5.2.3 The equation of motion of a cantilevered pipe
Consider a small segment of the pipe and the fluid. By definition, the velocity of the pipe
element is
ar
V -- =xi+Zk, (5.16)
p- at
and the velocity of the fluid element is Vf = Vp + Ut, where Ut is the relative velocity
of the fluid element with respect to the pipe element, t being the unit vector along s. For
the cantilevered pipe, where the inextensibility condition is assumed to hold true, t has
the form t = (ax/as) i + (az/as)k. Consequently,

(5.17)

where D/Dt is the material derivative of the fluid element (Section 3.3.3). By analogy,
the accelerations of the pipe and of the fluid (used in Appendix G) are, respectively,
a2 r D2 r
aP = - af = 2. (5.18)
at2 ’ Dt
Hence, the total kinetic energy, 9, may be written as
I” + I”
T = irn (i2 z2) ds + M [(x + Ux’I2 + (Z + UZ’)~] ds, (5.19)

where the dots and primes denote a( )/at and a( )/as, respectively.
One important remark that ought to be made is that no variable term proportional
to U’ arises from expression (5.19) since, by expanding the integrand and by virtue of
the inextensibility condition, one obtains only a constant term, U2xf2 + U2zl2 = U2. This
illustrates the importance of the right-hand side of statement (5.10), which will provide
both linear and nonlinear components of the centrifugal force proportional to MU2.
The variational operations on 9 lead to

+ (z + Uz’)(SZ + U Sz’)] ds dt.
Integrating by parts and noting that x’Sx’ + 2’62’ = 0, one obtains

(5.20)

where XL = x(L) and ZL = z(L) are the longitudinal and lateral displacements of the free
end of the pipe.