Vol. 3, No. 8, 2008

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Nonlinear vibration of an edge-cracked beam with a cohesive zone, II: Perturbation analysis of Euler-Bernoulli beam vibration using a nonlinear spring for damage representation

Daniel A. Mendelsohn, Sridhar Vedachalam, Claudio Pecorari and Prasad S. Mokashi

Vol. 3 (2008), No. 8, 1589–1604

A nonlinear free-vibration analysis of an Euler–Bernoulli beam with an edge crack and a cohesive zone at the crack tip, represented by bending and shear springs, is presented. Restricting attention to bending nonlinearities, we suppose the beam is loaded statically in bending into the nonlinear region and small amplitude vibrations are then superposed. A two term perturbation expansion is used where the small parameter depends on the ratio of the first and second derivatives of the nonlinear moment-slope relations computed in Part I. The zeroth order term is the linear free-vibration solution (constant spring stiffness equal to the first derivative of the moment-slope relation). Each mode generates a second harmonic (first-order term) whose magnitude depends on the linear spring stiffness and on the small perturbation parameter. Key features of the zeroth and first-order solutions are studied as functions of the moment-slope relations computed in Part I, and the possibility of cohesive property characterization is discussed.

nonlinear beam vibrations, cracked beam, cohesive zone, material characterization
Received: 30 October 2007
Revised: 10 June 2008
Accepted: 15 August 2008
Published: 1 October 2008
Daniel A. Mendelsohn
Department of Mechanical Engineering
Scott Laboratory
The Ohio State University
201 W. 19th Avenue
Columbus, OH 43210
United States
Sridhar Vedachalam
Environmental Science Graduate Program
The Ohio State University
260 Ag. Engineering Building
590 Woody Hayes Drive
Columbus, OH 43210 United States
Claudio Pecorari
NDE System Development
Linkoping, Sweden, SE-582 22
Prasad S. Mokashi
Department of Mechanical Engineering
Scott Laboratory
The Ohio State University
201 W. 19th Avenue
Columbus, Ohio 43210
United States