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# A Note on the Use of Approximate Solutions for the Bending Vibrations of Simply Supported Cracked Beams

[+] Author and Article Information
L. Rubio

Department of Mechanical Engineering, University Carlos III of Madrid, Avenida de la Universidad 30, 28911 Leganés, Madrid, Spain

J. Fernández-Sáez1

Department of Continuum Mechanics and Structural Analysis, University Carlos III of Madrid, Avenida de la Universidad 30, 28911 Leganés, Madrid, Spainppfer@ing.uc3m.es

1

Corresponding author.

J. Vib. Acoust 132(2), 024504 (Mar 17, 2010) (6 pages) doi:10.1115/1.4000779 History: Received February 14, 2009; Revised August 07, 2009; Published March 17, 2010; Online March 17, 2010

## Abstract

The main goal of this note is to discuss the applicability of approximate closed-form solutions to evaluate the natural frequencies for bending vibrations of simply supported Euler–Bernoulli cracked beams. From the well-known model, which considers the cracked beam as two beams connected by a rotational spring, different approximate solutions are revisited and compared with those found by a direct method, which has been chosen as reference.

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## Figures

Figure 2

Variation in natural frequencies with crack severity for a crack located at β=0.40; (—) direct method, (−−) perturbation method (Eq. 13), (⋅×⋅) Zhong and Oyadiji method (Eq. 27), and (−◇−) Ritz method: (a) first, (b) second, (c) third, and (d) fourth modes

Figure 3

Crack severity values for which the solution differs from the direct (reference) solution with less than 5%; (−−) ωi calculated from Eq. 13 and (−×−) ωi calculated from Eq. 27: (a) first, (b) second, (c) third, and (d) fourth modes

Figure 1

Variation in natural frequencies with crack severity for a crack located at β=0.20; (—) direct method, (−−) perturbation method (Eq. 13), (⋅×⋅) Zhong and Oyadiji method (Eq. 27), and (−◇−) Ritz method: (a) first, (b) second, (c) third, and (d) fourth modes

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