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There have been many investigations on the elastic deformation of uniform beams with one-axis o two-axes symmetrical cross-section, but few for the more general cases of an arbitrary cross-section. As is well known, in these general cases, bending vibrations in two perpendicular directions are coupled with torsional vibrations. To deal with the coupled vibration problem, we have to find the fundamental solutions which are not available in the literature. In applying the Fourier transform and its inverse to derive the fundamental solutions, we have to evaluate a special integral which determines the detailed expressions of the fundamental solutions. To this end, Mathematica is successfully used, and the basic two-point function, which is defined as the fundamental solution for the determinate of the adjoint-operator's cofactor matrix, is obtained in a closed form. Since the beam is treated as a continuous body and no discretization is involved, the present formulation provides the exact values of all natural frequencies and corresponding mode shapes. To demonstrate the validity of the present approach, the fundamental solutions that are derived are applied to the solution of the coupled bending-torsional vibration via the integral equation method.
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