Vol. 4, No. 7-8, 2009

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Exponential solutions for a longitudinally vibrating inhomogeneous rod

Ivo Caliò and Isaac Elishakoff

Vol. 4 (2009), No. 7-8, 1251–1256
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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.

closed form solutions, rod vibration, exponential solutions
Received: 11 November 2008
Revised: 18 January 2009
Accepted: 19 January 2009
Published: 26 December 2009
Ivo Caliò
Dipartimento di Ingegneria Civile e Ambientale
Università di Catania
Isaac Elishakoff
Department of Mechanical Engineering
Florida Atlantic University
P.O. Box 3091
Boca Raton, FL 33431-0991
United States