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Figure 2.36 Mode shapes for first three modes of a spring-restrained, hinged-free beam in bending, k = 1; wt = (1.24792 )2 J El/(mi4), co2 = (4.03114 )2 J EI/(mi4), and co} = (7.13413 fjEl/dnt4).

Figure 2.37 Variation of lowest eigenvalues a, t versus dimensionless spring constant k.

Figure 2.38 Mode shape for fundamental mode of the spring-restrained, hinged-free beam in bending; k = 50 and w¡ = (1.83929)2^EI/(me4).

Example 11: Solution for Free-Free Beam

The case of a uniform beam that is unconstrained at both ends, Fig. 2.39, may be considered as a crude first approximation to a freely flying vehicle. Their elastic and rigid dynamic properties are quite similar. In both instances these properties can be described in terms of a modal representation. The boundary conditions for this case require that

X"(0) = X"'(0) = X"(£) = X"\C) = 0. (2.265)

The spatially dependent portion of the general solution to be used here will again involve the sums and differences of the trigonometric and hyperbolic functions. Two of the E¡ s can be eliminated by applying the boundary conditions at x = 0 so that

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