Page 511 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)
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THE TIMOSHENKO EQUATIONS OF MOTION AND ASSOCIATED ANALYSIS 48 1
(iv) shear zero:
(E.8d)
where ( )’ = d/d(. It is evident that the boundary conditions in P are simpler and,
hence, for convenience, we proceed to determine the eigenfunctions associated with P
first.
For a clamped-clamped beam, after application of boundary conditions (E&) and
(E.8b) one obtains the characteristic function
+ 2 cash q COS p = 0, (E.9)
from which the eigenfrequencies w;, j = 1,2, . . ., may be obtained. The corresponding
eigenfunctions are
qj([) = -q;(cosh q; - cos p;) cosh (q;t) + (q; sinh q; - p: sin p;) sinh (q;{)
where p; and q; are as in (E.7), but with w; replacing 0; these eigenfunctions are not
normalized. P; and 5 are related via
Similarly, for a cantilevered beam one obtains
w2 sinh q sin p
-
-2 + - - - ($ + 2) cosh q cos p = 0, (E. 12)
A 4 P
E.3 THE INTEGRALS S,
These integrals, appearing in equation (4.43, have been evaluated analytically by Luu
(1983). A sample is given here
