Page 510 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

480 SLENDER STRUCTURES AND AXIAL FLOW

Therefore, the overall conclusion is that, except for extreme conditions, either set
of equations may be used - leaving aside the thorny question as to which set is the
correct one.

E.2 THE EIGENFUNCTIONS OF A TIMOSHENKO BEAM
Neglecting rotatory inertia and flow effects, equations (4.38) reduce to

Eliminating q or + from one of these equations, one obtains

a2q
a4q 1 a4q +--0, a4+ 1 a4+ a2+
__-____
ap A apat2 at* at4 A apat2 +--0, (E.4)
at2
and hence the eigenvalue problem associated with just one of them needs to be considered.
Letting q = Y (6) exp(iwt), 1c. = W(4) exp(iwt), this is associated with
d4Y w2 d2Y
+
- - - w2Y = 0,
-
dt4 A dC2
and the same for W. Proceeding as for an Euler-Bernoulli beam (cf. Section 2.1.3), the
solution of (E3 is of the form

Y = cosh q$ + B sinh q5 + C cos p4: + D sin pt, (E.6)
where

and similarly for ly. After some manipulation, malung use of (E.3), it is easy to obtaint
‘separated forms’ of boundary conditions (4.39a,b) as follows:
(i) displacement zero:
Y = 0 and @’” = 0; (E.8a)

(ii) slope zero:

W=0 and (~)Y”’+[I+(~)2]y’=o; (E.8b)

(iii) bending moment zero:

W’= 0 and Y”+ (E&)

+For example, if P = 0, differentiating the first of (E.3) with respect to 6 and then substituting the second
one (with P = 0) into it leads to the second of (E.8a).