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

Boginich

2004

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We describe a procedure of approximate analytical determination of fatigue damage parameters of rectangular plates. As an initial damage characteristic we apply the relative value of the plate potential-strain-energy variation due to availability of a Mode I crack, and, based on this value, we find relations for determination of the plate natural frequency variation, as well as parameters of distortion of monoharmonic oscillations in the modes of main resonance and superharmonic resonance of the 2nd order.Introduction. Among numerous studies of vibration of cracked bodies, as applied to the problems of fatigue-damage vibrodiagnostic of such bodies, the vibration of plates has received comparatively little attention.Investigators usually consider changes in natural modes and frequencies of vibration as vibrodiagnostic parameters of damage, with primary emphasis on the determination of plates' natural frequencies as a more sensitive damage indicator. Specifically, Cornwell et al. [1] noted that the strain energy distribution method [2] as a modification of the experimental modal method applied in vibration tests of a 430 450 9 × × -mm plate using 31 accelerometers revealed only the notches of more than 50 mm in length. The findings [3-6] suggested a poor damage sensitivity of changes of vibration modes. However, it should be mentioned that a change of natural frequency is not always an adequately sensitive damage indicator. In particular, the tests of aluminum plates (measuring 250 225 2 5 × × . mm) with through-the-thickness notches 12 mm in length showed that the maximum change of frequency in five vibration modes did not exceed 1% [7]. Analysis of other vibrodiagnostic parameters, such as distortion of monoharmonicity of a vibration process and occurrence of sub-and superharmonic resonances [8][9][10][11], was applied to plates in [12] only.Just some of the above-mentioned works consider rectangular plates mainly with through-the-thickness edge or central cracks/notches. The known design-theoretical studies of vibration of rectangular plates consider both the finite-element numerical solutions [13] and numerical-analytical ones, which are based on the Levi-Nadai solutions with setting up of first-and second-kind Fredholm integral equations including singularity of crack-tip stresses [14] and use the finite Fourier transformation of discontinuous functions. In order to solve this problem, some version of the discrete method is used [15]. It consists in splitting a plate into a few subregions according to the crack pattern, choosing a set of shape functions for each subregion, setting up a continuity matrix for the whole region of the plate and an equation for eigenvalues by minimizing the energy functional [16].

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confidence: 91%

Vibrodiagnostic parameters of fatigue damage in rectangular plates. Part 1. A procedure of determination of damage parameters

Boginich

2004

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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

Boginich

2010

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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

“…For the determination of the lower-harmonic relative amplitude À 1 2 / in subharmonic resonance for a specific structural element, it is necessary to find α which depends on the type, relative size and location of a normal-rupture crack as well as on the relative dimenions and vibration mode of the structural element. For example, for a beam of rectangular cross section with a single transverse edge crack the parameter α is given by [4] where P x ( ) is the axial force in longitudinal vibrations or the bending moment in bending vibrations of a beam, S 1 is the cross-sectional area in longitudinal vibrations or the axial moment of inertia in bending vibrations, S 2 is cross-sectional area in longitudinal vibrations or section modulus in bending vibrations, b and h are the width and height of the beam cross section, l is the beam length, x cr is the coordinate of the cracked section, γ is the crack relative depth, and F 1 ( ) γ is the dimensionless function of the crack relative depth, which is involved in the expression for the normal stress intensity factor. Using the data on F 1 ( ) γ function as provided in [5], we arrive at the following expressions: The value of D h l x cr ( , ) depends on the relative height of the beam cross section (h l), crack location ( ), õ cr beam vibration mode (i) and, for an example, for a cantilever beam in longitudinal vibration, is given by …”

Section: Introductionmentioning

confidence: 99%

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

Bovsunovskii

2008

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