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Strength of Materials

, Volume 30, Issue 6, pp 564–574 | Cite as

Efficiency of the method of spectral vibrodiagnostics for fatigue damage of structural elements. Part 2. Bending vibrations, analytical solution

  • V. V. Matveev
  • A. P. Bovsunovskii
Scientific and Technical Section

Abstract

We discuss the possibility of efficient application of the harmonic analysis of strain cycles formed in an elastic rod in the process of natural or resonance vibrations to the evaluation of the degree of fatigue damage. Fatigue damage is simulated by discontinuities of the material resulting in different values of the modulus of elasticity in the processes of tension and compression. In determining the potential strain energy of a rod in the presence of a crack on the basis of linear fracture mechanics, this model enables us to deduce expressions for relative changes in the stiffness of the rod in the presence of edge and surface cracks. By using the asymptotic method of nonlinear mechanics, we find the second approximation to the solution of the differential equation of vibrations of the rod whose nonlinearity is explained by the effect of “breathing” of the discontinuity formed in the material. It is shown that the discontinuity of the material caused by fatigue damage is responsible both for changes in the natural frequency of vibrations of the rod and for significant contributions of the constant component and even harmonics to the spectrum of strain cycles in the rod. The values of the constant component and the amplitudes of even harmonics (mainly of the second one) may serve as an efficient parameter for the detection of fatigue damage to materials.

Keywords

Stress Intensity Factor Fatigue Damage Edge Crack Transverse Crack Constant Component 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notation

a

amplitude of the first harmonic of vibrations

b

width of the cross section

dc

depth of a surface crack

E0

the longitudinal modulus of elasticity of an intact material

E′

longitudinal tensile modulus of elasticity of a damaged material

Er

reduced modulus of elasticity in bending

Fc

area of the crack

HI(γc)

dimensionless function of the relative actual length of edge cracks

h

height of the cross section

I

moment of intertia of the cross-sectional area

j

numbers of even harmonics (j=2, 4,…)

K

generalized stiffness of an intact beam

K′

generalized stiffness of a damaged beam

KI

mode I stress intensity factor

l andlc

current and actual values of the length of an edge crack and the half-lenghh of a surface crack, respectively

L

length of the beam

M

generalized mass of the beam

M(x)

bending moment in an arbitrary cross section of the beam

\(\overline M \)(x)

normalized function of the distribution of the bending moment along the length of the beam

M0

maximum bending moment

Mc=M(xc)

bending moment in a section with a crack

Π

potential strain energy

r

relative thickness of the damaged surface layer

t

thickness of the damaged surface layer

u

displacement (deflection) of the beam

W

section modulus

X(x)

natural mode of vibrations

x

coordinate of an arbitrary section

x0

coordinate of the section with the maximum value of the bending moment

xc

coordinate of the cracked section

α=(E0Er)/E0

(K−K′)/K,\(\bar \alpha \)=(E 0E′)/E 0

β′

0−ω0′)/ω0,

β″

0−ω0″)/ω0

γ

relative length of edge cracks (γ=l/h for a single crack and γ=2l/h for two symmetric cracks)

Δ

displacement of the neutral layer forE′εE 0

longitudinal strain

0

amplitude of the first harmonic of strain cycles in the surface layer

λI(γ)

dimensionless function of the relative length of a mode I edge crack

μ

Poisson’s ratio

σ

normal stress

σc

maximum normal stress in a crack-free section with coordinatex c

ϕ

phase of the first harmonic

ω0

angular natural (resonance) frequency of the intact beam

ω0′ and ω0

angular natural (resonance) frequency of the beam for the cases of asymmetric and symmetric (about the neutral layer) discontinuities in the material, respectively

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References

  1. 1.
    V. V. Matveev, “Efficiency of the method of spectral vibrodiagnostics for fatigue damage of structural elements. Part 1. Longitudinal vibrations. Analytic solution”,Probl. Prochn., No.6, 5–20 (1997).Google Scholar
  2. 2.
    G. S. Pisarenko, A. P. Yakovlev, and V. V. Matveev,Strength of Materials. Handbook [in Russian], Naukova Dumka, Kiev (1988).Google Scholar
  3. 3.
    Fracture. Vol. 2:Mathematical Fundamentals of the Theory of Fracture [Russian translation], Mir, Moscow (1975).Google Scholar
  4. 4.
    Yu. Murakami (ed.),Stress Intensity Factors. Handbook [Russian translation], Vol. 1, Mir, Moscow (1990).Google Scholar

Copyright information

© Kluwer Academic/Plenum Publshers 1999

Authors and Affiliations

  • V. V. Matveev
  • A. P. Bovsunovskii

There are no affiliations available

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