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A special class of
closed form solutions for inhomogeneous rods is investigated, arising from the
following problem: for a given distribution of the material density, find the axial
rigidity of an inhomogeneous rod so that the exponential mode shape serves as
the vibration mode. Specifically, for a rod clamped at one end and free at
the other, the exponentially varying vibration mode is postulated and the
associated semi-inverse problem is solved. This yields distributions of axial
rigidity which, together with a specific law of material density, satisfy the
governing eigenvalue problem. The results obtained can be used in the context of
functionally graded materials for vibration tailoring, that is, for the design of
a rod with a given natural frequency according to a postulated vibration
mode.
Keywords
closed form solutions, rod vibration, exponential solutions
