Efficiency of the method of spectral vibrodiagnostics for fatigue damage of structural elements. Part 4. Analysis of distortion of harmonicity of vibration cycle of beams with closing transverse cracks

Using an asymptotic method of nonlinear mechanics and concepts of linear fracture mechanics, we derive analytical expressions that relate the constant component and amplitudes of even harmonics, and the ratio of maximum deviations and half-periods of displacements of opposite signs in a vibration cycle of prismatic beams to the length of a closing transverse crack. The analytical solution is compared to the data of experimental determination of these parameters in bending vibrations of rectangular cantilever beams with an edge transverse crack. Analysis of the results of analytic and experimental investigations has shown that harmonic analysis of the strain waveform of beam-type elements in resonance vibrations can be used as an efficient diagnostic method for detecting fatigue damage such as closin cracks. Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 1, pp. 5–12, January–February, 2000.     

On determination of the natural frequency of transverse and longitudinal vibrations of a cracked beam. Part 1. analytical approach

We consider a perfectly elastic cantilever beam with a crack. The influence of the crack depth and location on the vibration mode of the beam is simulated by a local change in the cross section. This problem is solved by the Rayleigh method. Fairly simple formulas for the determination of the natural frequency of the first mode of bending and longitudinal vibrations of a cantilever beam with an open and closing crack are derived. Institute for Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 2, pp. 35–44, March–April, 1999.     

The influence of elastic-dissipative coupling parameters on resonant vibrations of systems with violated cyclic symmetry

This paper deals with the regularities of the influence of the variation of the elastic-dissipative coupling parameters of subsystems on the formation of resonant vibrations of systems with a violated cyclic symmetry, like shrouded blade assemblies, and gives the results of numerical calculations. The formation of the amplitude-frequency characteristics and the distribution of resonant vibration amplitudes of the systems under study is shown to depend on the relationship between elastic and dissipative characteristics of coupling of the subsystems. When the elastic coupling is stiff, which provides localization of vibrations of the elastically-coupled system, the introduction of dissipative coupling does not cause any changes in the behavior of the amplitude-frequency characteristics, and the degree of localization is almost independent of the dissipative properties of the coupling. As the stiffness of the elastic-dissipative coupling grows, the localization of vibrations shows up only at certain values of the dissipative characteristics of the coupling. Institute for Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 1, pp. 102–115, January–February, 1999.     

Longitudinal vibrations of a cantilever beam with a transverse crack. Part 2. Piecewise-linear model

A model of vibrations of a cracked body that is based on the difference in the stiffness characteristics in tension and compression is represented as a linear system with piecewise-constant parameters. We determine the parameters of vibrations of this system. We plot the relative crack size as a function of the difference in amplitudes in tension and compression, which is related to the vibration amplitude of a crack-free body with account for the resistance. The approaches described are applicable to evaluation of the “survival” of a structure (i.e., evaluation of the residual durability of a cracked structure). Zaporozh'e State University, Zaporozh'e, Ukraine. Translated from Problemy Prochnosti, No. 5, pp. 78–85, September–October, 1999.     

Efficiency of the method of spectral vibrodiagnostics for fatigue damage of structural elements. Part 2. Bending vibrations, analytical solution

We discuss the possibility of efficient application of the harmonic analysis of strain cycles formed in an elastic rod in the process of natural or resonance vibrations to the evaluation of the degree of fatigue damage. Fatigue damage is simulated by discontinuities of the material resulting in different values of the modulus of elasticity in the processes of tension and compression. In determining the potential strain energy of a rod in the presence of a crack on the basis of linear fracture mechanics, this model enables us to deduce expressions for relative changes in the stiffness of the rod in the presence of edge and surface cracks. By using the asymptotic method of nonlinear mechanics, we find the second approximation to the solution of the differential equation of vibrations of the rod whose nonlinearity is explained by the effect of “breathing” of the discontinuity formed in the material. It is shown that the discontinuity of the material caused by fatigue damage is responsible both for changes in the natural frequency of vibrations of the rod and for significant contributions of the constant component and even harmonics to the spectrum of strain cycles in the rod. The values of the constant component and the amplitudes of even harmonics (mainly of the second one) may serve as an efficient parameter for the detection of fatigue damage to materials. Institute for Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 6, pp. 9–22, November–December, 1998.     

On determination of vibration characteristics of a beam with a closing crack in bending vibrations

We consider free vibrations of an elastic cantilever beam with an edge crack, which is simulated as a beam portion with a decreased moment of inertia. The weight of this portion is assumed to be constant, while the dimensions of the portion are determined by the energy criterion of equivalence. We propose an analytical approach to determination of natural frequencies and vibration modes, of a beam with an open or closing crack and to investigation of nonlinear distortions of the displacement wave and acceleration and deformation of various sections of a beam with a closing crack. The solution allows for the possibility that more than one vibration mode of a beam can be generated at the moment of the crack opening and includes the effect of the crack on the strain distribution in the beam volume. It is demonstrated that the approach we propose gives reliable relationships between various vibration characteristics of a beam and the crack parameters and makes it possible to solve an inverse problem of damage diagnostics. Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 3, pp. 5–23, May–June, 2000.     

The effect of damping and force application point on the non-linear dynamic behavior of a cracked beam at sub-and superresonance vibrations

The finite element model of an elastic body with a closing crack is presented. Using this model, we show that the non-linear distortions of vibrations at superharmonic and subharmonic resonances are the high sensitive indicators of crack presence. The intensity of these indicators manifestation substantially depends on the level of damping in the system and on the load application point. __________ Translated from Problemy Prochnosti, No. 5, pp. 61–67, September–October, 2006. Report on International Conference “Dynamics, Strength, and Life of Machines and Structures” (1–4 November 2006, Kiev, Ukraine).     

Numerical study of vibrations of a nonlinear mechanical system simulating a cracked body

The paper presents the results of a numerical study of forced and decaying vibrations of a system simulating a body with a closing crack under the action of various modes of a nonlinear restoring force and nonlinear viscous friction or the hysteresis-type energy dissipation. We obtained the general patterns of appearance of higher harmonics of the Fourier expansion of time dependence of vibration of a cracked body model. The sensitivity of the higher-harmonic method to the presence of a crack is compared with that of some other vibration damage indicators. Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 6, pp. 65–80, November–December, 1999.     

Fracture analysis in 2D magneto–electro–elastic media by the boundary element method

An integral formulation for 2D cracked infinite anisotropic magneto–electro–elastic media is presented. Based on the method proposed by Garcia-Sanchez et al. (Comput Struct 83: 804–820, 2005), the hypersingular kernels are analytically transformed into weakly singular and regular integrals with known analytical solution. Special quadratic discontinuous crack tip elements are employed to model the singular characteristics of the stresses, electric displacements and magnetic inductions. The extended stress intensity factors at the crack tips are calculated using the extended discontinuous displacements at crack tip elements based on one point extended displacement formulation. Some results for curved cracks in magneto–electro–elastic media are also presented.     

Procedure for numerical investigation of forced vibrations of three-dimensional structures with nonlinear elastic elements based on the finite-element method

We describe a numerical procedure which allows one to investigate forced vibrations of complex three-dimensional structures with nonlinear elastic elements, develop a computational dynamic model on the basis of the finite-element method and the method of generalized coordinates, and investigate vibrations of an actual structure of a load-bearing mounting of an aircraft engine. We also analyze the effect of nonlinear elastic elements on the dynamic behavior of the structure and evaluate the efficiency of their application. Kiev State Technical University of Civil Engineering and Architecture, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 107–116, July–August, 1997.     

Analytic evaluation of the stress-strain state of a pipeline elastically bent in the ground

We construct a mathematical model of three-dimensional bending with tension and torsion of a prestressed curvilinear pipeline in an elastic compliant soil with longitudinal and torsional frictiontype reactions. The model has the form of a system of 12 ordinary linear differential equations. It takes into account large initial displacements of the pipeline and the action of temperature and pressure linearly distributed along the axis of the pipe. The analytic solution of the problem obtained in elementary functions is of significant practical interest for the inspection and design-basis strength analyses of underground, surface, and above-ground pipelines. Scientific and Engineering Center of Materials-Science Support of the Production and Certification of Equipment of Nuclear Power Plants at the Institute for Problems of Strength of the National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 2, pp. 151 – 160, March – April, 1998.     

Measurement of viscoplastic properties of a liquid in experiments with a torsion viscosimeter

The results of a numerical solution of the problem of vibrations of a torsion viscosimeter filled with an incompressible viscoplastic liquid are presented. It has been shown that in the vicinity of the rotation axis there appears a dead zone, whose boundary changes in the process of vibrations. The influence of plastic properties of the liquid on the characteristics of viscosimeter vibrations has been determined. A method for identifying viscoplastic properties by the observed parameters of vibrations is proposed. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 80, No. 1, pp. 124–127, January–February, 2007.     

On determination of the natural frequency of transverse and longitudinal vibrations of a cracked beam. Part 2. Experimental and calculation results

The results of experimental investigation of the relationship between the natural frequency of the first mode bending and longitudinal vibrations of titanium alloy and alloyed steel cantilever beams and the crack parameters (crack depth and location) are presented together with the data on calculation of the relative change of the vibration frequency of cracked beams, which are obtained by the proposed analytical approaches. Based on a comparative analysis, we choose fairly simple and sufficiently accurate formulas for calculating the frequencies of bending and longitudinal vibrations of a cantilever beam with an open or closing crack. Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 3, pp. 45–53, May–June, 1999.     

Wedging of an edge crack in the elastic wedge

We study the problem of wedging of an elastic wedge realized along the edge crack with the help of a hard wedge. The analytic solution of the problem is obtained by the Wiener–Hopf method. We determine the stress intensity factors, the distributions of stresses on the continuation of the crack and in the contact zone, and the circular displacements of the crack lips.     

Analysis of the field of a plane SH-wave scattered by a finite crack on the interface of materials

We solve the problem of diffraction of plane elastic SH-waves on a crack of finite width located in the plane interface of two different perfectly joined isotropic elastic materials. The problem is solved by the Wiener-Hopf method. An approximate solution of the Wiener-Hopf equation is obtained. The numerical results in the far-field zone are presented for a broad frequency band. The specific features of distribution of the radiation field in the zone of critical angles are analyzed. The probability of excitation of lateral waves is investigated and the possibility of detection of defects is discussed. A simple experimental procedure is proposed for the NDT evaluation of crack width in a test bench based on measuring the modulus of the field of displacements in the zone of critical angles. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 42, No. 6, pp. 5–16, November–December, 2006.     

Forced stationary vibrations of flexible plates induced by random narrow-band excitations

We consider some problems related to the analysis of the modes of forced vibrations of elastic plates induced by random narrow-band excitations. The investigation of nonlinear vibrations of a plate is performed within the framework of a reduced discrete model constructed by using the Hamilton variational principle, finite-element methods, and generalized coordinates. To study the dynamical state of the analyzed structure, we perform numerical simulation of the vibrations induced by a particular realization of narrow-band excitations. Each realization of random narrow-band excitations is obtained with the help of a shaping filter of the second order and describes harmonic oscillations with slowly varying amplitude and phase. We also consider the scenarios of transitions between different modes of vibrations of the plate and estimate the time of its residence in different dynamical states. Kiev National University of Civil Building and Architecture, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 2, pp. 105–117, March–April, 2000.     

Experimental-numerical investigation of the dynamic stability of flexural-torsional vibrations of compressor blades under conditions of attached and separated flow. Part 2. Cross aerodynamic characteristics

We consider the results of investigation of cross aerodynamic characteristics of compressor blades and their dynamic stability under flexural-torsional vibrations for the cases of attached and separated flows in broad ranges of variations of the phase shift, the ratio of amplitudes of the translational and angular components of displacements, the angle of attack, the reduced frequency of vibrations, and the geometry of the blade cascade. Translated from Problemy Prochnosti, No. 4, pp. 89–100, July–August, 2009.     

Free vibrations of spherical belts with circular and longitudinal ribs

We propose a procedure for the numerical evaluation of the frequencies of free vibrations of spherical belts with circular and longitudinal ribs, determine those frequencies, and study the influence of the parameters of the ribs on the natural frequencies of free vibrations of shell structures reinforced by ribs. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 3, pp. 124–127, May–June, 1998.     

Stress intensity near a crack in the composition of a half space and a layer under harmonic loading

We consider a three-dimensional problem for an elastic bimaterial body formed by a half space and a layer and containing a circular crack. The surface of the crack is subjected to the action of stationary torsional forces. The problem is reduced to the solution of a system of boundary integral equations of Helmholtz-potential type for the unknown jumps of displacements on the crack surfaces. The dynamic stress concentration in the vicinity of the crack contour is investigated. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 2, pp. 27–32, March–April, 2008.     

Bilateral bending of a plate with nonsymmetric through-thickness arc crack with allowance for the contact of its edges

The problem of bending of plates with nonsymmetric through-thickness arc crack with distributed moments at infinity with allowance for the contact of the crack edges has been examined on the basis of the classical theory of plate bending. The conditions for the existence of the solution of the problem in this formulation have been established. A numerical analysis of intensity factors of forces and moments, contact force between the crack edges has been performed, the results of which have been graphically illustrated. __________ Translated from Problemy Prochnosti, No. 5, pp. 135–141, September–October, 2006. Report on International Conference “Dynamics, Strength, and Life of Machines and Structures” (1–4 November 2006, Kiev, Ukraine).