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Although this is the same set of eigenvalues that we found for the string problem, the relationship to the frequency is quite different. The frequencies are

Ela4

m so that

As observed in the cases of the string and beam torsion, there is associated with the z'th natural frequency a unique deformation shape called the mode shape (or eigenfunction). These mode shapes can be obtained from the spatially dependent portion of the solution by evaluating the function, X,(x), for any known value of or,-. To find X, one substitutes any root back into either of Eqs. (2.236), recognizing that the constants D\ and D\ should now be written as D\. and />;. Using the first of these equations along with the knowledge that sinh(a,1) / 0, one finds that />,, = 0 and

where D\ can be anything. For example, choosing D\ — 1, one finds the mode shape to be

which is the same mode shape as obtained earlier for the vibrating string.

Example 9: Solution for Clamped-Free Beam

Consider the clamped-free beam as shown in Fig. 2.33, the boundary conditions of which reduce to conditions on X given by

X(0) = X'(0) = X"(l) = X"'(£) = 0. (2.244)

As with the previous example, one can show that this problem exhibits no nontrivial solution for the case of a — 0. Thus, we use the form of the general solution in Eqs. (2.221) for which a ^ 0. Along with the first two boundary conditions, this yields

0 0

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