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On determination of vibration characteristics of a beam with a closing crack in bending vibrations

  • Scientific and Technical Section
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Strength of Materials Aims and scope

Abstract

We consider free vibrations of an elastic cantilever beam with an edge crack, which is simulated as a beam portion with a decreased moment of inertia. The weight of this portion is assumed to be constant, while the dimensions of the portion are determined by the energy criterion of equivalence. We propose an analytical approach to determination of natural frequencies and vibration modes, of a beam with an open or closing crack and to investigation of nonlinear distortions of the displacement wave and acceleration and deformation of various sections of a beam with a closing crack. The solution allows for the possibility that more than one vibration mode of a beam can be generated at the moment of the crack opening and includes the effect of the crack on the strain distribution in the beam volume. It is demonstrated that the approach we propose gives reliable relationships between various vibration characteristics of a beam and the crack parameters and makes it possible to solve an inverse problem of damage diagnostics.

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Abbreviations

L :

beam length

L c :

coordinate of the cracked section

h :

height of the cross section

b :

width of the cross section

a :

crack depth

γ:

relative crack depth

d :

parameter to be determined

F :

cross-sectional area

I :

moment of inertia of the section

I o :

moment of inertia of the cracked section

I m :

moment of inertion of mass

m L :

weight of a mass at the beam end

m :

intensity of the beam weight

E :

elastic modulus

ρ:

density

w(x) :

natural mode of vibration of a beam

θ(x):

distribution of the slopes of the elastic curve of a beam along its length

M(x) :

bending moment distribution along the beam length

Q(x) :

lateral force distribution along the beam length

ω:

natural angular frequency of an intact beam

ω o and ω c :

natural angular frequencies of a beam with an open crack and a beam with a closing one, respectively

f a=0 andf a :

resonance frequencies of intact and cracked specimens, respectively

K I :

stress intensity factor

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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. 3, pp. 5–23, May–June, 2000.

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Matveev, V.V., Bovsunovskii, A.P. On determination of vibration characteristics of a beam with a closing crack in bending vibrations. Strength Mater 32, 211–224 (2000). https://doi.org/10.1007/BF02509848

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