Longitudinal vibrations of a cross-cracked cantilever beam. Part 1. Small vibrations

Based on the analysis of solutions to differential equations of longitudinal vibrations of a macroelastic body whose nonlinearity is due to “breathing” of the damaged material under tension-compression cycling, we estimate the suitability of the deformation-cycle characteristic of a body under free and forced vibrations for the diagnostics of fatigue damage in structural elements. We consider the following problems: the construction of a simplified model of longitudinal vibrations of mechanical elements with cross cracks, the investigation of free vibrations of a cracked body including resistance forces, the forced vibrations of a cracked body taking into account a perturbation force including and excluding resistance forces, and the study of vibrations of a beam with resistance different from the linear one. We show that the presence of a macrocrack is responsible for the occurrence of representative higher harmonics in the deformation-cycle spectrum. They can serve as a sensitive and efficient vibration-diagnostic parameter to detect cracks in structural elements. Zaporozh'e State University, Zaporozh'e, Ukraine. Translated from Problemy Prochnosti, No. 2, pp. 23–34, March–April, 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.     

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

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. __________ Translated from Problemy Prochnosti, No. 3, pp. 5–16, May–June, 2008.     

Generalization of the continuous and discrete structural approaches to the solution of problems of the dynamics of laminated composite plates and structures with stratification. Part 2. Analytic decomposition of unknown functions. Results and comparisons

A three-dimensional solution of the problem of determination of the frequencies and modes of natural vibrations of multilayer plates is obtained on the basis of the method of analytic separation of variables. We compare the models with analytic separation of variables and polynomial approximation of unknown functions over the thickness of the structure. Within the framework of the continuous structural approach, the applied model enables one to determine the frequencies and modes of natural vibrations for a broad range of problems with sufficiently high accuracy. If it is necessary to improve the accuracy of results, then one can pass to the discrete structural version of the model. Ukrainian Transport University, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 6, pp. 93–99, November–December, 1999.     

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.     

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.     

Analysis of flexural-flexural-torsional nonlinear vibrations of twisted rotating beams with cross-sectional deplanation

We present results of the investigations on flexural-flexural-torsional nonlinear vibrations of twisted rotating beams described by the system of three nonlinear integro-differential equations expressed in partial derivatives. Cross-sectional deplanation of beams is adjusted to the equations and it is assumed that centre of gravity and shear centre are located at different points. Vibration systems are expressed as a row by free forms of the linear problem. Free vibrations are studied with the help of the Show–Pierre nonlinear normal mode. Translated from Problemy Prochnosti, No. 2, pp. 112–124, March–April, 2009.     

Efficiency of the method of spectral vibrodiagnostics for fatigue damage of structural elements. Part 1. Longitudinal vibrations, analytical solution

On the basis of an analysis of the asymptotics of higher approximations to solutions of a differential equation of longitudinal vibrations of a macroelastic body whose nonlinearity is explained by the hysteresis and “breathing” of the damaged material in the process of its cyclic deformation under conditions of tension-compression, we estimate the applicability of the spectral characteristics of strain cycles of the body under conditions of its natural or resonance vibrations to the diagnostics of fatigue damage of structural elements. It is shown that if nonlinearity is explained solely by the hysteresis of the material, then even the most representative first higher odd harmonics revealed in the vibration spectrum have very small amplitudes and the analysis of their variations is, in fact, similar to the analysis of the variations of the logarithmic decrement of vibrations (regarded as an integral characteristic of the dissipative properties of the material of the investigated element) but has lower resolution. At the same time, the discontinuities of the material appearing in the process of strain cycling are characterized by formation of numerous microcracks or a macrocrack and result in more clear detection of the representative constant component and even harmonics (mainly the second and fourth ones) in the spectrum of strain cycles of a vibration-insulated structural element in the process of its natural or resonance vibrations. These characteristics may serve as sensitive and efficient parameters for detection of fatigue damage to the material. Translated from Problemy Prochnosti, No. 6, pp. 5–20, November–December, 1997.     

A study of the dissipative properties of homogeneous materials deposited as coatings. Part 2. Copper condensates with different microstructural characteristics

The dependence of dissipative properties of copper condensates on the grain size has been studied. Condensates with a grain size of 0.9–3.7 μm were obtained by electron-beam vapor deposition of copper as coatings on a titanium substrate. The oscillograms of damped bending vibrations of the substrate-coating system were recorded at a frequency of 140–150 Hz in the temperature range 20–350°C. The values of true logarithmic vibration decrement for copper condensates were determined by the self-consistent calculation of energy loss factors of the substrate and coating by the method proposed in the part 1. It has been shown that the amplitude dependences of the true logarithmic vibration decrement of copper condensates at room and elevated temperature are largely determined by their structure. __________ Translated from Problemy Prochnosti, No. 2, pp. 149–158, March–April, 2008.     

On the procedure of evaluating the damping ability of the materials of specimens subjected to torsional and bending vibrations

We describe a method for the design and numerical analysis of compact mechanical vibrating systems with generators of vibrations on their inertial weights and establish the conditions of damping of unwanted vibrations in these systems aimed at guaranteeing the absence of favorable conditions for the introduction of systematic errors to experimental results. The designed and manufactured vibrating systems are dynamically balanced, and predicted values of the frequencies of their natural vibrations satisfactorily agree with the experimental data. Institute for Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 145–152, July–August, 1997.     

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.     

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.     

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.     

Flexural vibrations of semiconductor wafers in the presence of a pulsed thermal excitation source

An analysis is made of the action of current pulses on a metal film lying on a semiconductor wafer. It is shown that the passage of current pulses of amplitude (1–7)×10¹⁰ A/m² and duration 50–200 μs across aluminum tracks excites sound. Experimental dependences of the energy of the flexural vibrations as a function of the duration and amplitude of the pulsed action were obtained. It was observed for the first time that the melting of a metal-semiconductor contact is accompanied by an abrupt increase in the energy of the flexural vibrations of the wafer. Pis’ma Zh. Tekh. Fiz. 25, 57–63 (March 12, 1999)     

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.     

Investigation into nonlinear vibrations of composite plates using the <Emphasis Type="Italic">R</Emphasis>-function theory

We have developed an effective approach to the solution of problems on geometrically nonlinear vibrations of orthotropic multilayer plates of irregular shapes in a classical statement based on the use of the R-function theory, Ritz variational method and Bubnov-Galerkin method. Using the proposed method, problems of vibrations of both multilayer rectangular plates and plates of complex geometries have been solved. The effect of the domain geometry and boundary conditions on the amplitude-frequency characteristics has been investigated. __________ Translated from Problemy Prochnosti, No. 5, pp. 101–113, September–October, 2007.     

Unsteady wave flows of a gas suspension with account for phase changes

A system of equations of unsteady motion of a gas suspension with account for heat and mass transfer is presented. As a result of the computational experiment, the effect of the gas vibrations at the channel inlet on the motion of the gas suspension and the rate of heat and mass transfer is studied. Specifically, it is shown that, in the considered parametric region, the gas vibrations enhance the interphase heat and mass transfer. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 73, No. 3, pp. 487–492, May–June, 2000.     

Experimental-numerical investigation of the dynamic stability of flexural-torsional vibrations of compressor blades under conditions of attached and separated flow. Part 1. Procedure and equipment for the determination of nonstationary aerodynamic characteristics

We present the results of investigations of the aerodynamic stability of flexural-torsional vibrations of compressor blades under conditions of attached and separated flow with regard for the cross and mutual aerodynamic links of the blades in a broad range of variations of the phase shift, the ratio of the amplitudes of translational and angular components of their displacements, and the angle of attack. A procedure and equipment used for the experimental determination of the nonstationary aerodynamic characteristics of the blades are described on the basis of the model of flat-plate airfoil cascades. __________ Translated from Problemy Prochnosti, No. 2, pp. 129–142, March–April, 2007.     

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.