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].

The paper discusses the results of analytical determination of vibrodiagnostic parameters of fatigue damage in the form of Mode I straight cracks of constant depth in a rectangular homogeneous uniform-thickness plate with various plate fastening conditions and vibration modes. It is demonstrated that, in the range of the relative crack depth up to 0.3, the parameter of distortion of vibration monoharmonicity in the 2nd-order superharmonic resonant mode is one-two orders of magnitude higher than the value of a relative change of the plate's natural frequency.

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The paper addresses the analytical determination of vibrodiagnostic parameters that describe the presence of normal-rupture flat through-the-thickness and surface semi-elliptical central cracks in a rectangular homogeneous plate of constant thickness for various plate fixing conditions and vibration modes. It is shown that the most sensitive vibrodiagnostic damage parameter of a plate is the variation of the logarithmic decrement in the case of a through-the thickness crack and the second-harmonic ratio of the vibration process in superharmonic resonance in the case of a surface crack.Introduction. In Part 2 [1] using the general input relationships for a rectangular plate [2] we derived analytical expressions for the determination of a relative energy characteristic and some vibrodiagnostic parameters of fatigue damage in the form of flat surface cracks of constant depth.Here, we will apply similar approaches [2] to discuss the results of calculation of vibrodiagnostic damage parameters for normal-rupture central through-the-thickness and surface semi-elliptical cracks using the data on pre-determined relative energy characteristic of damage in a plate. Determination of Relative Energy Characteristic of Damage in a Plate.For the fatigue damage type under consideration in a rectangular plate (Fig. 1), we use the general input expression for the relative energy characteristic of damage [1]