Study of the Resonant Vibrations and Dissipative Heating of Thin-Walled Elements Made of a Physically Nonlinear Material

An approximate formulation is given to a dynamic coupled thermomechanical problem for physically nonlinear inelastic thin-walled structural elements within the framework of a geometrically linear theory and the Kirchhoff–Love hypotheses. A simplified model is used to describe the vibrations and dissipative heating of inhomogeneous physically nonlinear bodies under harmonic loading. Nonstationary vibroheating problem is solved. The dissipative function obtained from the solution for steady-state vibrations is used to simulate internal heat sources. For the partial case of forced vibrations of a beam, the amplitude–frequency characteristics of the field quantities are studied within a wide frequency range. The temperature characteristics for the first and second resonance modes are compared.     

Resonance Vibrations and Dissipative Heating of Thin-Walled Laminated Elements Made of Physically Nonlinear Materials

The dynamic thermomechanical problem for thin-walled laminated elements is formulated based on the geometrically linear theory and Kirchhoff–Love hypotheses. A simplified model of vibrations and dissipative heating of structurally inhomogeneous inelastic bodies under harmonic loading is used. The mechanical properties of materials are described using strain-dependent complex moduli. A nonstationary vibration-heating problem is solved. The dissipative function, derived from the stationary solution, is used to specify internal heat sources. The amplitude–frequency characteristics and spatial distributions of the main field variables are studied for a sandwich beam subjected to forced vibrations     

Model of a fracture as an emitter of elastic vibrations

During the past two decades a new method has begun to be intensively developed for the investigation of fracture processes which is based on recording the mechanical vibrations generated by the defects of a medium [1]. The new method's problems include: extraction of a useful signal from the extraneous noises, identification of the type of defect, determination of its characteristic dimensions, and an estimate of the danger of the situation which has developed. The solution of the problems indicated has great meaning in such practical applications as nondestructive quality control and the engineering diagnostics of materials and manufactured goods. Therefore, the investigation of the spectrum of the signals produced by the formation of macroscopic fractures, such as the terminal and, consequently, most dangerous phase of fracture, is of great interest. The kinematical characteristics of a fracture as an emitter of elastic vibrations are formulated in this paper. The spatial and time spectra of the dynamical motions caused by the appearance of a growing fracture in a thin plate are discussed. Relationships are derived between the spectral characteristics of propagating disturbances and the parameters of the fracture.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 160–166, March–April, 1976.The author thanks L. I. Slepyan for valuable discussion and attention to this research.     

Vibrations and Dissipative Heating of a Viscoelastic Beam under a Moving Load

The vibrations and dissipative heating of an infinite viscoelastic beam under a moving load are studied on the basis of Timoshenko beam theory. The influence of transverse-shear strain and rotary inertia on the critical velocities of the moving load, the amplitude of bending vibration, and the temperature of dissipative heating is analyzed__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 1, pp. 69–76, January 2005.     

Damping of Free Elastic Vibrations in Linear Systems

The possibility of using a Mises truss as an absorber of free elastic vibrations in a linear elastic system is examined. The nonlinear normal mode method is used to analyze nonlinear vibrations. A local nonlinear normal mode is shown to be favorable for vibration damping__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 2, pp. 110–117, February 2005.     

Control of nonlinear vibrations of vibrating ring microgyroscope

The influence of the basement rotation on the variations in the spectrum of vibration frequencies of thin elastic shells and rings was known already at the end of the 19th century [1]. The physical phenomenon of inertness of elastic waves occurring free vibrations of an axisymmetric body, first explained in [2], were practically used in developing new types of gyros [2–6]. The foundations of the theory of wave gyros were laid in [2, 4], and the errors of such gyroscopes for various shapes of the vibrating resonator were studied in [2, 4, 7, 8]. It was shown that the error of the resonator manufacturing (the variable density, thickness, anisotropy of the material elastic properties, etc.) [2, 8] and the geometric nonlinearity of the resonator flexural vibrations studied in [2, 7] lead to splitting of the natural frequency of flexural vibrations [2], which is reflected in the wave picture of the resonator vibrations and characterizes the gyroscope precision.In the present paper, we study the errors of the vibrating microgyroscope which arise because of nonlinear elastic properties of the ring resonator material. We construct a control of the potential on the electrodes which allows one to maintain the prescribed amplitude of the normal resonator deflection and compensate for the gyroscope errors arising because of the nonlinear elastic properties of the material.     

Behavior of a floating elastic plate during vibrations of a bottom segment

The Wiener-Hopf technique is used to obtain an analytical solution for the problem of vibrations of a floating semi-infinite elastic plate due to earthquake-induced vibrations of a bottom segment. An explicit solution is obtained ignoring the inertial term. The surface-wave amplitudes and ice-plate deflection are studied numerically as functions of the frequency and position of the vibrating bottom segment, ice thickness, and fluid depth.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 98–108, March–April, 2005.     

Vibrations and Dissipative Heating of a Cylindrical Panel under a Periodic Moving Load

An analytic solution is obtained to describe the vibrations and dissipative heating of a simply supported infinite cylindrical panel under periodic normal loads moving over its surface with a constant velocity. Special attention is focused on resonant vibrations, which result in the most intensive dissipative heating. It is additionally assumed that the material of the panel is viscoelastic, its properties are independent of temperature, and Poisson’s ratio is real. The influence of thickness, radius of curvature, load velocity, and viscoelastic properties on the thermal state of the panel is analysed against the thermal state of the plate__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 4, pp. 100–109, April 2005.     

Resonant Vibrations and Dissipative Heating of an Infinite Piezoceramic Cylinder

The monoharmonic radial vibrations and dissipative heating of an infinite hollow piezoceramic cylinder are studied in dynamic formulation, taking into account the temperature dependence of the complex electromechanical characteristics over a wide range of temperatures, including depolarization temperatures. The influence of the heat exchange conditions, the level of electric load, and geometry on the thermoelectromechanical characteristics is studied in the case of forced vibrations at the first resonance__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 3, pp. 101–107, March 2005.     

Axisymmetric Vibrations and Dissipative Heating of a Laminated Inelastic Disk

The coupled thermomechanical dynamic behavior of an inhomogeneous body is investigated for a partial case where a laminated inelastic disc is subject to forced axisymmetric vibrations and dissipative heating. The problem is solved in complete and approximate formulations. In the former case, the behavior of the material is described using generalized flow theory. In the latter case, the behavior of the material is characterized by complex moduli. The spatial distributions of the field quantities and the temperature– and amplitude–frequency characteristics of the disc are analyzed. The results are compared.     

Convective flows in a cylindrical fluid zone in a high-frequency vibration field

Convective flows of a nonuniformly heated fluid in a cylindrical fluid zone in a high-frequency longitudinal vibration field are studied. Vibration frequencies which are high as compared with dissipative decrements and capillary frequencies, but small as compared with acoustic frequencies are considered. The general method formulated earlier for describing the behavior of inhomogeneous fluids under the influence of high-frequency vibrations is used. The interaction between the vibrational flow mechanisms and thermocapillary effects on a free surface is analyzed.Perm'. Marseilles (France). Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 53–61, September–October, 1994.     

Asymptotic analysis of the behavior of an elastic bar under aperiodic intensive loading

It is established that when aperiodic loads of large intensity act on an elastic bar, the higher modes of stability loss have the highest rates of growth of deflections. A method is indicated for determining the numbers of these modes, when the effect of the longitudinal vibrations on the transverse vibrations is taken into account and when it is not taken into account. A comparison of the results obtained with results of other authors [1–7] is presented.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 160–170, May–June, 1972.     

Natural vibrations of a hummock ridge in an elastic ice sheet floating on the surface of an infinitely deep fluid

The properties of the natural vibrations of a hummock ridge in an elastic ice sheet are investigated. Typical shapes of the dispersion curves for symmetric and antisymmetric boundary waves which propagate along the hummock and damp exponentially with distance from the latter are obtained. It is shown that natural vibrations can initiate failure of the ice sheet at a certain distance from the hummock. Under compression this process leads to the formation of a parallel hummock ridge.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 99–105, November–December, 1995.     

Vibrations of a system of fluid-filled cylindrical shells on an elastic base

The resonance synchronization of the vibrations of a system of shells is studied. It is established that the vibrations of isolated components excite the vibrations of the whole system, consisting of three cylindrical shells on a common elastic base.__________Translated from Prikladnaya Mekhanika, Vol. 40, No. 12, pp. 134–137, December 2004.     

Rolling friction as a viscoelastic dissipative process

The viscous additions to the stress tensor on the half–space surface over which a ball moves are calculated with the use of a viscosity (dissipative) tensor. The rolling friction force which corresponds to the Coulomb law and which is proportional to the velocity and which is simultaneously the lower estimate for the sliding friction force is found. Expressions for the radial and vertical displacements on the surface of an elastic half–space are given.     

Investigation of wave fields in a fluid layer and elastic half-space excited by a near-bottom source of vibrations

Questions of a study of acoustic and seismic wave propagation in the ocean and the underlying medium, associated with problems of marine hydrolocation (sonar) and the investigation of tsunami wave predecessors caused by underwater volcanic eruptions are of great interest at this time. The simplest model, a point source of vibrations located in a fluid layer at a certain range from the bottom, is used to describe the mentioned wave processes. Wave fields in a fluid and elastic base are investigated in this paper, analytical formulas are obtained, and results of a numerical analysis are presented.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 49–51, July–August, 1984.     

Radial Vibrations and Heating of a Ring Piezoplate under Electric Excitation Applied to Nonuniformly Electroded Surfaces

Radial vibrations and dissipative heating of a polarized piezoceramic ring plate are studied. The plate is excited by a harmonic electric field applied to nonuniformly electroded surfaces of the plate. The viscoelastic behavior of piezoceramics is described in terms of complex quantities. An analytical solution is found in the case of quasistatic harmonic loading. The dynamic nonlinear problem of coupled thermoviscoelasticity is solved with regard for the temperature dependence of the properties of piezoceramics by step-by-step integration in time, using the numerical methods of discrete orthogonalization and finite differences. A numerical analysis is conducted for TsTStBS-2 piezoceramics to study the influence of partial electroding on the stress–strain distribution, natural frequency, and amplitude–frequency and temperature–frequency characteristics     

Dissipative Heating of a Viscoelastic Cylinder under a Load Steadily Moving over Its Surface

The Fourier method is used to find the analytical solutions to two-dimensional quasistatic problems of stationary polyharmonic vibrations and dissipative heating of a linearly viscoelastic cylinder. The influence of the cylinder thickness and the width of the loading area on the thermomechanical state of the cylinder is studied based on numerical data__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 2, pp. 23–32, February 2005.     

Hydroelastic interaction of a wedge-shaped construction entering a liquid

The aim of the paper is to create a mathematical model of the hydroelastic interaction of a wedge-shaped construction with a liquid that it enters. The various relevant factors — velocity and acceleration of elastic vibrations, influence of catinary forces, initial bending, and mobility of brackets — are to be taken into account as fully as possible.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 118–125, May–June, 1993.     

Role of two-dimensional acoustic wave cutoff in the instability of subsonic MHD flows

One of the important factors affecting the structure of the natural vibrations and the conditions under which they build up in an inhomogeneous subsonic flow may be the cutoff of non-one-dimensional sound waves expressed in the strong reflection of such waves from the critical sections (caustics). In this study the case of natural two-dimensional acoustic perturbations in an inhomogeneous subsonic conducting gas flow in the presence of critical sections is subjected to an asymptotic analysis. Special attention is paid to the conditions of growth of the two-dimensional acoustic perturbations in the internal resonator formed by two critical sections and the walls of an MHD channel.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 26–36, March–April, 1988.The authors are grateful to seminar participants L. M. Biberman and G. A. Lyubimov for useful discussions.