Page 303 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

284 SLENDER STRUCTURES AND AXIAL FLOW

The two components of the potential energy are derived next. Considering first the
strain energy expression (5.15) with E = 0, one can write
11
6 Lt2 “Irdt = ;El 6(K2)dsdt.

Utilization of the curvature expression (5.8) leads to

= E1 6’ LL[z”” + 42’ z”z”’ + z’’~ + z”” z’~] 6z ds dt + S(c5). (5.21)

The gravitational potential (5.10) may be dealt with in a similar manner. However,
since it will involve ax, a relationship between 6x and Sz needs to be found. This is done
by taking variations of the first of (5.1), the inextensibility condition, yielding

ax’ = - z’ 62‘ = -z’ (1 + iZ’2) 6z’ + S(c4);
d m
hence,
I’ [z’ 6z’ + iz’3 SZ’] ds. (5.22)
6x = -

After integrating the right-hand side of (5.22) by parts and noting that 6z = 0 at s = 0,
one obtains
Sx = - (z’ + iz’3) Sz + I’ (z” + pz”) 6z ds + S(c4). (5.23)

One can also prove quite easily (Semler 1991) that

(5.24)

Equation (5.24) is important, since terms of that form will arise from (5.23) in the process
of relating Sx to Sz.
Now, using (5.23) and (5.24), the variation of the gravitational energy (5.14) is obtained:

6Lt2%dt = -(rn+M)g [ I’ [- (z’ + ;2’3) 6z

+ (L - S) (z” + ;z” z’~) 6z] ds dt + O(2). (5.25)
Applying next the variational procedure to the right-hand side (rhs) of Hamilton’s
principle, equation (5.10), leads to