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Efficiency of the method of spectral vibrodiagnostics for fatigue damage of structural elements. Part 3. Analytical and numerical determination of natural frequencies of longitudinal and bending vibrations of beams with transverse cracks. Solution

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Strength of Materials Aims and scope

Abstract

Based on the concepts of linear fracture mechanics, we derive analytical expressions for the determinational of natural frequencies of longitudinal and bending vibrations of beams which are rectangular in cross section, are fixed in different ways, have variable ratios of the section height to the beam length, and have transverse cracks of various types. The results of the analytical solution are compared with those obtained by the finite-element method as well as with experimental data obtained by the authors and other scientists. The analytical solution under consideration is shown to be quite simple and provide a fairly good accuracy of the results obtained. Using a cantilever beam with one or two symmetrical edge cracks or a central through crack as an example, we consider the possible dependence of the relative change in natural frequencies of vibration on the relative crack length, crack location, and the vibration mode of a beam. We discuss the possible methods of evaluating the crack size and location from the results of experimental determination of the change in natural vibration frequencies of a cracked beam.

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Abbreviations

ω0 :

natural frequency of an intact beam

ω0′ and ω0″:

natural frequency of a beam with a closing and nonclosing (open) crack

β′ and β″:

relative change of the natural frequency of the beam with a closing and open crack

L :

length of the beam

h :

height of the cross section

α:

coefficient of relative changes in the stiffness of cracked beam

a :

crack length

γ:

relative crack length

P(x) :

internal force in the cross section of a beam

S 1 andS 2 :

deometrical characteristics of the cross section of a beam

p(x) :

internal force

x c :

coordinate of the cracked section

θ:

constant whose value depends on the number and type of cracks

E n :

Euler numbers

References

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Authors
  1. V. V. Matveev
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  2. A. P. Bovsunovskii
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Additional information

Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 19–31, July–August, 1999.

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Matveev, V.V., Bovsunovskii, A.P. Efficiency of the method of spectral vibrodiagnostics for fatigue damage of structural elements. Part 3. Analytical and numerical determination of natural frequencies of longitudinal and bending vibrations of beams with transverse cracks. Solution. Strength Mater 31, 341–350 (1999). https://doi.org/10.1007/BF02511132

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  • DOI: https://doi.org/10.1007/BF02511132

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