Page 242 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I1 223

there is pressurization p vis-&vis the outer ambient fluid, then T(L) = T - pA( 1 - 2v)
approximately - see Section 3.3.2.
Using equations (4.30)-(4.31b), (4.33) and (4.34) and retaining p9 and w as variables,
the following system of two differential equations may be obtaincd:

+ [(M + m)(L - x)g + ST(L)]- + k‘GA,
ax
a2 + (4.35)
as?
EI,, ~ + [k’GA, - (M + m)g(L - x) - ST(L)I
- - a2p9
- (If +I,)- at2 = 0.
It should be noted that equations (4.35) are not identical to those derived via Hamilton’s
principle. This is discussed in Appendix E.1. Here suffice it to say that the dynamical
behaviour as obtained by the two sets of equations is sensibly the same for physically
realistic conditions.
The system may be expressed in dimensionless terms by defining the following quan-
tities:
= x/L, q = w/L, r = [El,/(M + m)]’/2t/L3,

u = (M/EI,)‘/’UL, p = M/(M + m), y = (M + m)L3g/EI,, (4.36)
A = k’GA,L’/EI,, CJ = (7, + I,)/[(M + m)L’], TL = T(L)L2/El,,

fA = FAL3/EIp, E = L/2a,
where a is the internal radius of the pipe. It is noted that for a given pipe material (i.e. for
a given Poisson ratio, u). A and E are interrelated:
8k’~’a’
A= (4.37)
(1 +a2)(1 + v)’
where a, defined earlier, is equal to a/(a + h), h being the wall thickness of the pipe.
Substituting these terms into equations (4.35) gives the dimensionless equations of motion:

It is noted that the equations of motion are not in their final form, as the fluid-dynamic
force f~ is yet to be derived, in Sections 4.4.3 and 4.4.4. The parameter E does not
appear explicitly in equations (4.38), but it does in the expression for f~ in Section 4.4.4.
It should also be noted that in equations (4.35) and (4.38), internal damping within the
material of the pipe is neglected; if it is not, it may be modelled by a hysteretic damping