Page 309 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

290 SLENDER STRUCTURES AND AXIAL FLOW

together with the dimensionless displacements

(5.45)

where W = q, but the barred quantity is used here for ‘symmetry’ with E. Hence, equa-
tions (5.36a,b) may be written as follows:

+
E”’)
- (,”” ,’ + ,” (r - &IJ - n)w’ w” - y = 0, (5.46a)
ct +% JBW‘ + 2% fig + Q2 w” - (r - n),” + w””
+ E”/’ + 2 ,I2 w”” + w’ ,u ,,”
- [ 3 El” wl’ + 4 E” w”’ + 2 E’ ,’’I’ ,r +
3 3 ]
+ (r - s& - n) (a”w’ +a”w” + ; w”2w”) = 0. (5.46b)

Note that the dissipative terms have been omitted for clarity, and that the nonlinear
inertial terms are not present in the current form of the equations. In fact, the only real
penalty incurred for the absence of nonlinear inertial terms is that one has to deal with
two equations, instead of just one.

5.2.8 Comparison with other equations for cantilevers
The nonlinear equations of motion obtained by different researchers are described and
compared in some detail, here for cantilevered pipes and in Section 5.2.9 for pipes with
fixed ends. In order to get a more ‘comparable’ set of equations, a standardization of the
notation has been imposed.

la) Bourrieres‘ work
This work is very original, all the more so since it was written in 1939. Bourribres (1939)
studied the case of planar motion of two interacting strings, one of them moving with
respect to the other. The pipe and the fluid represented by the strings are assumed to be
inextensible, and the string representing the fluid is supposed to be infinitely flexible. The
equations of motion of the pipe and the fluid are derived via the force-balance method.
The relationship between the shearing force Q and the bending moment A, together with
the condition of inextensibility, provides the nonlinear terms. Seven equations with nine
parameters are obtained, two of which are independent, with coordinate s and time t as
the two independent variables. After some algebraic manipulations, the fluid friction force
is eliminated, yielding the following five equations:

[(T + O)X’]’ - (Qz’)’ - (m + M)X - 2M UX’ - M U2x“ = 0,

[(T+@)z’]’- (ex’)’- (m+M)i”2MUZ’-MU2~”=0, (5.47)
d2 + z’~ = 1, Q = -At’, At = EZ(x’z’’ - z’x’’),
where T and 0 represent the tension in the pipe and the negative of the pressure force
in the fluid, respectively, and ( )’ = a( )/as.