Page 31 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

14 SLENDER STRUCTURES AND AXIAL FLOW

where 6(x - L) is the Dirac delta function; boundary conditions (2.29) then reduce to
(2.23). According to Galerkin's method, the solution of (2.30) may be expressed as

N
(2.31)

where the +, (x) are appropriate comparison fiinctions, i.e. functions in the same domain,
9 = [0, L], satisfying all the boundary conditions (both geometrical and naturalt), and
qj(t) are the generalized coordinates of the discretized system which will eventually
emerge by application of this method (Meirovitch 1967). It is now clear why it is advan-
tageous to recast this problem into the form of equations (2.30) and (2.23), for it is then
possible to use $j(x) 4,(x), i.e. to use the eigenfunctions given by (2.27) as suitable
comparison functions: suitable, since they satisfy the boundary conditions associated with
(2.30), and also convenient, since they are already known.
When approximation (2.31) is substituted into the left-hand side of (2.30), the result
will generally not be zero, but equal to an error function, which may be denoted by %[WN].
Galerkin's method requires that

(2.32)

i.e. over the domain, the integrated error, weighted by +r(x),t should be zero (Finlayson
8t Scriven 1966).
Thus, in this example, substituting approximation (2.31) with +,(x) = 4,(x) into equa-
tion (2.30), multiplying by &(x) and integrating over 9 = [0, L], leads to

N
[EZA')#2irj + [mLGrj + Me&(L)4j(L)]qj] = 0, r = 1,2, . . . , N, (2.33)
j=l

in view of the orthogonality of eigenfunctions (2.27), i.e.
rL

where 6,j is the Kronecker delta (0 for r # j and 1 for r = j). Clearly the system is now
discretized. Thus, if a two-mode approximation (N = 2) is utilized, equation (2.33) may
be written in the following matrix form:

The eigenvalues and eigenfrequencies of this matrix system are approximations
of the lowest two of the continuous system; thus, if Me = imL, then 521 =
2.018(EZ/n~L~)'/~, = 17. 165(EI/mL4)1/2. The corresponding eigenvectors give, in a
522
+Geometrical boundary conditions are of the type w = 0, while riarural ones involve forces or moments,
Ir=O
e.g. ~l(a3~/a~3)( = 0.
.r=L
tThe weighting function comes in 'naturally' if Galerkin's method is derived via variational techniques.