534.08;620.175.5 Using the approach proposed in Part 1, an approximate calculation of vibration parameters is made for an elastic body with a closing crack, in the region of a strong 1/2-order subharmonic resonance with the lower-harmonic amplitude of free vibration spectrum larger than the main amplitude of forced vibrations.

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We present a finite-element model to be used in a study of vibration of a beam with a closing crack. Some special features of numerical solution and methods for its fast realization are discussed. The results of experimental verification of the model are provided. Introduction.A fatigue crack is the most commonly encountered type of damage in the structures subjected to dynamic loading. In the course of cyclic deformation of an elastic body this crack tends to open during the tension half-cycle and close during the compression one, thus changing the body's stiffness in the process of cyclic deformation. The change in stiffness is usually modeled by an unsymmetrical piecewise-linear characteristic of the restoring force [1,2] or by a representative change in the exciting force [3].The closing crack being responsible for an essential nonlinearity of a vibrating system presents some difficulties in the analytical solution of a forced vibration problem for this system. The complexity of analytical solution of a vibration problem for cracked bodies and the difficulties encountered in assessment and prediction of variations of vibration characteristics of such bodies necessitate the application of numerical methods for solving these problems. A finite-element method is widely used for the solution of this type of problems.We put forward here a finite-element (FE) model of a beam with a closing crack, which is used for solving a forced vibration problem for this beam. A method of solution of differential equations is discussed, and the model reliability is assessed based on a comparison between the calculated and experimental data.Finite-Element Model for a Beam with a Closing Crack. The present finite-element model for a beam with an edge transverse crack (Fig. 1) is based on the FE model as discussed in [4]. The crack is assumed to affect only the beam's stiffness and have no influence on its mass and damping capacity. The effect of the crack on mechanical properties of an elastic body is allowed for through an equivalent change in stiffness of one of the FE model elements, which is arbitrarily called the damaged element. The crack is modeled by a modification of a stiffness matrix for this element.Stiffness of an elastic body is assumed to change instantaneously as the crack opens and closes. Thus, in the process of vibration the beam can be either in a conditionally undamaged state (the crack is closed) or damaged state (the crack is open). For each state we construct a respective stiffness matrix. When the crack is closed, its effect on the beam stiffness is ignored; in the case, the stiffness matrix is computed in the same way as for an intact beam. When the crack is open, an element with a modified stiffness is entered in the stiffness matrix in order to model the crack.The equivalent stiffness of the element that models the crack is determined by the method based on the strain energy balance. A change of potential energy in a section with a crack is found in terms of stress intensity factors by the fracture mech...

The vibrations of elastic bodies with closing cracks are essentially nonlinear. As a specific feature of these vibrations, we can mention the manifestation of so-called nonlinear effects, in particular, sub-and superharmonic resonances and the nonlinearity of vibrations at these resonances. We propose a method for the diagnostics of cracks based on the variation of the asymmetry of driving forces. The static component of the concentrated harmonic driving force changes the state of a crack: it becomes partially or completely open or closed. Moreover, the degree of nonlinearity of vibrations for any nonlinear resonance varies from the maximum level (in the absence of static component) to its almost complete absence in the case where the crack becomes open or closed in the course of vibrations under the action of the static component. The proposed method enables one to detect the presence of cracks without any preliminary data on the analyzed object in the intact state.Keywords: closing crack, sub-and superharmonic vibrations, diagnostics of cracks, nonlinear effects, asymmetry of the driving force.Introduction. The problem of detection of the cracks of subcritical sizes with an aim of prevention of the failures of machines and structures of different kinds is one of the most urgent problems of engineering diagnostics. The last decades were marked by the rapid development of the vibration methods of diagnostics of defects based, as a rule, on the analysis of the changes in the natural frequencies of vibrations and distortions of their shapes caused by the cracks [1, 2]. However, the sensitivity of these methods proved to be insufficient for practical application. Higher sensitivities to the presence of cracks are exhibited by the so-called nonlinear effects [3], i.e., sub-and superharmonic resonances, and the well-pronounced nonlinearity of vibrations at these resonances.In the case of vibration of a body containing a crack, the events of crack opening and closure are accompanied by the variations of stiffness of the entire structure simulated, for the sake of simplicity, by an asymmetric piecewise linear characteristic of the restoring force. The analytic investigations of forced vibrations of the mechanical systems with one degree of freedom and the indicated characteristic of the restoring force reveal the possibility of appearance of sub-[4-13] and superharmonic [7,8,10,12,14,15] resonance vibrations of various orders. Similar results were also obtained in the numerical investigations of the forced vibrations of rods with closing cracks [10,[16][17][18].The idea of modeling of the process of vibrations in cracked structural elements subjected to the action of static forces was realized by analyzing free [19,20] and forced [21] vibrations of rods. These investigations were carried out due to the necessity of more exact analysis of the conditions of loading of structural elements either operating in the fields of significant centrifugal forces (e.g., blades of gas-turbine engines) or carrying large static loads ...

The vibrations of elastic bodies with closing cracks are essentially nonlinear. As a specific feature of these vibrations, one can mention the manifestation of so-called nonlinear effects, e.g., nonlinear (i.e., sub-and superharmonic) resonances and the nonlinearity of vibrations for these resonances. The proposed method for the evaluation of the parameters of cracks (their sizes and location) is based on the analysis of the nonlinearity of vibrations in the neighborhood of a superharmonic resonance of order 2/1 and/or a subharmonic resonance of order 1/2 in the case of variation of the site of application of the driving force because, as follows from the results of numerical and experimental investigations, the level of nonlinearity of the vibrations of rods with closing cracks for nonlinear resonances depends not only on the parameters of the crack but also on the site of application of the driving forces.Keywords: closing crack, sub-and superharmonic vibrations, diagnostics of the crack, nonlinear effects. Introduction.Fatigue cracks form one of the most widespread types of defects in machines and structures operating under dynamic loads. As follows from the results of numerous experimental and theoretical investigations, cracks lead to a decrease in the natural frequencies of these objects and distort the form of vibrations. The relationship between the parameters of the crack (sizes and location), on the one hand, and the changes in the natural frequencies and forms of vibrations, on the other hand, is studied in numerous works [1,2]. However, the sensitivity of the methods of diagnostics of the cracks based on the changes in the natural frequencies and forms of vibrations appeared to be to be quite low. Later, it was shown that the so-called nonlinear effects [3], i.e., the sub-and superharmonic resonances, a strong nonlinearity of vibrations at these resonances, and the characteristics of damping of vibrations, are characterized by a higher sensitivity to the presence of cracks [4].In order to simplify the analysis of nonlinear effects, it is customary to assume that the stiffness of structures suffers instantaneous changes at the times of crack closure and opening. As a rule, this phenomenon is modeled by asymmetric piecewise-linear characteristics of the restoring forces. The analytic investigations of the forced vibrations of a mechanical system with one degree of freedom and the indicated characteristic of restoring forces make it possible to reveal sub-[5-14] and superharmonic [8,9,11,13,15,16] resonances of different orders. In addition, as shown in [3,7,8], the nonlinear effects strongly depend on the dissipative properties of the investigated system: the higher the level of damping of vibrations in the system, the lower the amplitudes of nonlinear resonances and the level of nonlinearity of vibrations at these resonances.The data of experimental investigations [4,[17][18][19] demonstrate that the growth of fatigue cracks is accompanied by a significant increase in the characteristics of damping o...