Page 34 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CONCEPTS, DEFINITIONS AND METHODS 17
a way of solving the problem correctly while utilizing boundary conditions (2.29), but
the meaning of the domain 9 has to be expanded (Friedman 1956; Meirovitch 1967); an
example is given in Chapter 4 (Section 4.6.2).

2.1.5 Self-adjoint and positive definite continuous systems
The eigenvalue problem of equation (2.36), and thereby the system, is said to be se@-
adjointt if for any two comparison functions, u and u,
I s,
s, u2?[u] d9 = vZ[u] d9, u&[u] d9 = v&[u] d9, (2.42)

are satisfied. A consequence of self-adjointness is that the eigenvalues are real. Another
consequence is that a generalized or weighted orthogonality of the eigenfunctions then
holds true for nonrepeated eigenvalues; thus,

Furthermore, if
r r
1% uZ[u]d9 > 0 and J’, uA[u] d9 > 0

for all nonzero u, the operators are positive definite, and hence so is the system. The
consequence of this is that the eigenvalues of such a system are positive - refer to
Section 2.3 for the significance of this and back to Section 2.1.3 for further clarification
of the different usage of the word ‘eigenvalue’. In cases where 2 is only positive, rather
than positive definite, i.e. when the first integral (2.44) can be zero for some nonzero u,
while A remains positive definite, the system is called positive seniide$wite, and admits
solutions with A = 0.
Clearly, for the system of equations (2.22) and (2.23), the problem is self-adjoint. To
illustrate the case of a non-self-adjoint system in as simple a manner as possible while
still keeping in the framework of the examples already discussed, consider the system

Z[W] = EI(d4/dr4) + P(d2/dr2), A[w] = m; (2.45)
~(0) 0, ~’(0) = 0, EItv”(L) = 0, EZw”’(L) = 0. (2.46)
=
This could represent a cantilevered beam, subjected to a compressive tangential ‘follower’
force P, such that the boundary conditions remain unaffected. A follower force is one
retaining the same orientation to the structure in the course of motions of the system, in
this case remaining tangential to the free end.$ By applying the integrals (2.43) it is found
that the integrated out parts do not vanish [since Pu(l)v‘( 1) and Pu’(l)u( 1) are not zero
for the boundary conditions given].

‘If Y and A are complex operators, the equivalent property is for the eigenvalue problem to be Hennirion.
*In fact, such a compressive follower force could be generated by a light rocket engine [M,/(nzL) z 01
mounted on the free end of the cantilever, so that the force of reaction is always tangential to the free end.