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

Abstract

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].

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

“…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],…”

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

“…For this purpose, the steady-state forced vibrations of a solid elastic body should be considered at a given mode of the harmonic loading with the frequency n w = 1 n j at a superharmonic resonance of the nth order and n w = n j at a subharmonic resonance of the order 1 n (w j is the frequency of the resonant jth natural mode vibrations of the elastic body). The nonlinearity parameter a of this elastic system under its deformation under conditions of forced vibrations is found in terms of the energy characteristic of the damage k [25]:…”

Approximate analytical determination of vibrodiagnostic parameters of a cracked elastic body under subharmonic resonance. Part 1. Weak resonance

“…Examples of how to determine the α parameter for beam-type elements with normal-rupture cracks under longitudinal and bending vibrations are discussed in [20] and for rectangular plates in [21].…”

“…where A q 1 0 is given by (21). Since we consider the excitation frequency range ν ω > 0 , the value π γ π 2 1 < < should be taken in the calculation.…”