The fundamental nature of the component mode method, based on Fourier series, is examined and applied to a wide variety of non-uniform beam vibration problems. It is shown that a single segment guided-guided beam, coupled with a sine series approach is sufficient to derive an exact master frequency determinant for a general multi-segmented beam with arbitrary material and geometric properties in each segment and general boundary conditions at the ends. Similarly, a single segment simply-supported beam, coupled with a cosine series approach is also sufficient to derive an exact master frequency determinant for the general multi-segment beam problem. The method requires special treatment (via Stokes' transformation) at the end points of each component, while continuity between components is enforced with Lagrange multipliers. Results are shown for non-uniform beams including the effects of intermediate masses and springs.
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