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TECHNICAL PAPERS

Vibration of a Three-Dimensional, Finite Linear, Elastic Solid Containing Cracks

[+] Author and Article Information
I. Y. Shen

Department of Mechanical Engineering, University of Washington, Seattle, WA 98195

J. Appl. Mech 62(2), 282-288 (Jun 01, 1995) (7 pages) doi:10.1115/1.2895929 History: Received February 02, 1993; Revised October 11, 1993; Online October 30, 2007

Abstract

This paper is to determine vibrational eigensolutions m2, vm(r)]m = 1 of a three-dimensional, finite, linear, elastic solid C containing cracks in terms of crack configuration σc and eigensolutions n2, un(r)n = 1 of a perfect elastic solid P without the cracks. Use of Betti reciprocal theorem and the Green’s function of P expands vm(r) in terms of an infinite series of un(r). Substitution of the vm(r) series representation into the Kamke quotient of C and stationarity of the quotient result in a Fredholm integral equation whose nontrivial solutions predict λm2, and vm(r) of C . Finally, natural frequencies and mode shapes of a circular shaft of finite length containing a circumferential crack under torsional vibration are predicted through a two-term Ritz approximation of the Fredholm integral equation. The results differ significantly from those predicted by the method of flexibility matrices, when the ratio of the shaft length to the shaft radius is small.

Copyright © 1995 by The American Society of Mechanical Engineers
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