Efficiency of the method of spectral vibrodiagnostics for fatigue damage of structural elements. Part 1. Longitudinal vibrations, analytical solution

Matveev, V. V.

doi:10.1007/bf02767605

Cited by 14 publications

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“…5. This explanation makes sense, as the values of the constant component À 0 and the second-harmonic amplitude A 2 in the free vibration spectrum were obtained in [3] by the asymptotic method of nonlinear mechanics, which assumes a low value of the α parameter.…”

Section: Introductionmentioning

confidence: 99%

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Approximate analytical determination of vibrodiagnostic parameters of a cracked elastic body under subharmonic resonance. Part 2. Strong resonance

Matveev

Bovsunovskii

2008

Strength Mater

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Section: Introductionmentioning

confidence: 99%

“…method of nonlinear mechanics [3], a solution to equation (1) in the superharnomic resonance region is sought for in the form [1]…”

Section: Introductionmentioning

confidence: 99%

Approximate analytical determination of vibrodiagnostic parameters of a cracked elastic body under subharmonic resonance. Part 2. Strong resonance

Matveev

Bovsunovskii

2008

Strength Mater

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“…The examples of how to determine the parameters k and a for rods of rectangular cross section under tension and bending as well as for rectangular plates in bending under the conditions of deformation by natural modes of vibrations and in the presence of various types of mode I cracks were discussed in the publications [3,4] and [5,6], respectively; 2) in a superharmonic resonance n w = 0 2, in addition to the fundamental -first -harmonic À t 1 1 sin( ) n g -corresponding to the exciting force frequency n, there arises vibration with a spectrum of harmonic components of the fundamental resonance (n w = 0 ), which is determined using an asymptotic method of the nonlinear mechanics [1,7]. Here, w 0 is the natural frequency of an elastic body when the crack in it is closing [2],…”

Section: Introductionmentioning

confidence: 99%

Approximate analytical determination of vibrodiagnostic parameters of the presence of a crack in an elastic body under superharmonic resonance

Matveev

Boginich

2010

Strength Mater

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We discuss an approximate analytical method for calculating the main parameter of nonlinearity of vibration of an elastic body with a closing crack, which is simulated by a single-degree-of-freedom system with an asymmetric bilinear characteristic of the restoring force, under a strong 2nd-order superharmonic resonance.Introduction. As mentioned in [1,2], one of the least studied lines in the field of assessment of possible changes of the vibration state of a structural member during its long-term operation and in the development of vibrodiagnostic methods for detecting damages such as fatigue cracks is to establish relations between a closing crack parameters and the vibration parameters in nonlinear resonances.Some approximate analytical solutions were derived earlier for the determination of vibrodiagnostic parameters of the above-mentioned type of damage in an elastic body under forced vibration in the region of a strong and weak 1/2-order resonances [2, 3] as well as a weak 2nd-order superharmonic resonance [1].To further elaborate on the publication [1], we will discuss here an approximate analytical solution for the case of a strong superharmonic resonance.Approximate Solution Procedure. The procedure is based on the fundamentals as stated in [1, 2]: 1) in the case of a fairly small mode I crack we can disregard some difference in the modes of vibration of an elastic body between its deformation half-cycles, where the crack is open and closed, respectively, and for the given natural mode of vibration we can represent the body by a model of a single-degree-of-freedom system with an asymmetric bilinear characteristic of the restoring force. The forced vibration of this system is described by a differential equation of the form

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“…It should be noted that in the case of the simulation of the elastic system by the system with one degree of freedom, an analytical solution was initially considered for the region of the main resonance [20,21] where the analytical method of the nonlinear Krylov-Bogolyubov mechanics, first applied by Pisarenko [22,23] and subsequently widely used for the calculation of vibration of weakly nonlinear systems of the hysteresis type, was found to be efficient. In so doing, it was shown [22,23] that for such systems in the case of the main resonance, a first approximation gives sufficient accuracy since for possible dissipative properties of structural materials, the values of the amplitude of higher harmonics, along with the constant component, were found to be negligibly small.…”

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Approximate analytical method for determining the vibration-diagnostic parameter indicating the presence of a crack in a distributed-parameter elastic system at super- and subharmonic resonances

2010

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620.178.5:620.179 and A. P. YakovlevAn approximate method for calculating the vibration-diagnostic parameter indicating the presence of a crack in an elastic distributed-parameter system at super-and subharmonic resonances is considered. The method involves the finding of the non-linearity characteristic of an elastic system based on the analysis of its forced vibrations in the undamaged state and the use of results of the approximate analytical determination of parameters of nonlinearity for vibrations of an elastic body with a closing crack, which is simulated by a single-degree-of-freedom system with an asymmetric bilinear characteristic of the restoring force at the above-mentioned resonances.Keywords: super-and subharmonic resonances, vibration-based diagnostics of fatigue damage, closing crack, nonlinear vibrations, distributed-parameter elastic system. Introduction.Interest in studying vibrations of elastic bodies in the presence of damages such as fatigue cracks is motivated by the necessity of evaluating a possible change in the vibrational state of structural elements during their operation on the one hand, and by the development of the efficient methods of vibration-based diagnostics of this kind of damages, on the other. Here, one of the most investigated lines of research is establishing the calculated relationships between the parameters of the closing Mode I crack and vibration process at super-and subharmonic resonances. This is due to the complexity of the analytical solution of the problem on forced vibrations of a substantially nonlinear system derived.In this field, a number of investigations are known. Thus, in [1], the problem of transverse oscillations of a prismatic beam with a closing crack at subharmonic 2-nd order resonance under the action of a harmonic transverse concentrated load was considered using the Ostrogradsky-Hamilton principle and Ritz method. The weakening of the bending rigidity of the girder with an open crack was interpreted as a local decrease in the effective cross section of the beam in the limited region approximated by a triangular prism having a right angle at the crack tip and a height equal to the crack depth. It was assumed that in the region of the resonance, the beam performs vibrations close to one of the natural modes, and the mathematical model of such vibrations was represented by one ordinary differential equation in which the sign of the term corresponding to the so-called reduced depth of the crack was governed by the sign of the quasi-normal coordinate. Next, using the averaging method, the set of the time-independent differential equations was set up for determining the amplitude and phase of the second harmonic of the vibration process.The problem on forced vibrations of a rectangular plate containing longitudinal defects of continuity, such as closing cracks, was considered in [2] for the case of superharmonic second-order resonance of the first mode of transverse vibration. To solve it, the Ostrogradsky-Hamilton principle and Ritz method were a...

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Efficiency of the method of spectral vibrodiagnostics for fatigue damage of structural elements. Part 1. Longitudinal vibrations, analytical solution

Cited by 14 publications

References 0 publications

Approximate analytical determination of vibrodiagnostic parameters of a cracked elastic body under subharmonic resonance. Part 2. Strong resonance

Approximate analytical determination of vibrodiagnostic parameters of a cracked elastic body under subharmonic resonance. Part 2. Strong resonance

Approximate analytical determination of vibrodiagnostic parameters of the presence of a crack in an elastic body under superharmonic resonance

Approximate analytical method for determining the vibration-diagnostic parameter indicating the presence of a crack in a distributed-parameter elastic system at super- and subharmonic resonances

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