Page 221 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I1 203

(4.21)

in which [dA;/dx](a/ax){ (dUi/&)(aw/ax)) in the second, fluid-moment-related term has
been neglected, as it is of second order for small taper angles; T(L) is given either
by equation (4.17) or by (4.18)-(4.20). It is obvious that the second and third terms
in equation (4.21), which are related to the fluid-related moment [equation (4.12)] are
quite small as compared to, say, the fourth term; indeed, for sufficiently small &I;/& and
dA,/dx, they may be neglected, and this is one of the reasons for not giving the derivation
of aAf/ax here in detail.
The boundary conditions are the same as for uniform tubular beams, e.g.
equations (3.77) or (3.78).
The equations of motion and boundary conditions may be rendered dimensionless by
the following set of nondimensional parameters:

4 = x/L, q = w/L, t = [EZ/(m + PeAe + ~iAi)]iL;f/L’,

6‘ = [Ai/Ae1{=09 0, = Ae(t)/Ae(O)> 0; = Ai(t>/Ai(O)- E = L/De(O),
vd = [Z/{E(m + PeAe + pjA;)]i!$Y*/L2, O = T(L)L2/EZ(0), IZ = FL2/EZ(0),

Ui = [piAi/EZI~$Ui(O)L, cu = [pe&/ErI~f~UvL = [peAe/ErI,,,(~.,c~/p,L>,)L, (4.22)
1
/2
Y = [PA~/EII{=ML~, ye 1 + P~/P, yi = (Pi/P - 1)s2,
where p = rn/(A, - A;). The equation of motion in dimensionless terms is then given by

(4.23)

4.2.2 Analysis and results

Some calculations have been conducted for conically tapered cantilevered tubular beams,
i.e. either conical in outer form or with a conical flow passage. The notation‘cylindrical-
conical’ or ‘conical-conical’ is used here, the first denoting a cylindrical outer shape and
a conical flow passage, while the second denotes conical outer and inner forms, as shown