Mechanics of Axially Moving and Vibrating in Transverse Direction Orthotropic Thermoelastic Web

We consider deformations and stability of axially moving orthotropic thermoelastic web. The web is modelled by a thin continuous plate moving with a constant velocity with small transverse vibrations and supported by a system of rollers. It is supposed that the plate is subjected to a combined thermomechanical loading including pure mechanical in-plane tension and also centripetal forces. Thermal strains corresponding to thermal tension and bending of the moving plate are taken into account. We formulate and analytically investigate the problem of out-of-plane thermomechanical divergence of orthotropic plate.     

Symmetric and anti-symmetric vibrations in micropolar thermoelastic materials plate with voids

The problem of symmetric and anti-symmetric vibrations in micropolar thermoelastic plate with voids has been investigated. The dispersive frequency equations are obtained for different surface waves propagating in the plate. The velocity curves are depicted for the symmetric and anti-symmetric vibrations, plate, Rayleigh and flexural waves. It is found that there exist two modes in the solution of frequency equation for the surface waves in micropolar thermoelastic plate with voids. We have observed that the first modes of velocity ratios of corresponding surface waves are lesser than those of second mode of vibration.     

Active damping of forced resonance vibrations of an isotropic shallow viscoelastic cylindrical panel under the action of an unknown mechanical load

Using a new approach, we consider the problem of active damping of the forced resonance vibrations of a viscoelastic isotropic cylindrical panel with simply supported edge faces. The mechanical load is assumed to be unknown; it can be found from the experimental data of a sensor. The problem is solved by the Bubnov–Galerkin method. A relation was obtained for the potential difference that has to be supplied to an actuator for the damping of resonance vibrations of the panel. We also study the influence of sizes of sensors and actuators, the dissipative properties of materials, and temperature on the efficiency of active damping of the forced resonance vibrations of a cylindrical panel.     

Influence of the temperature of dissipative heating on the damping of forced resonance vibrations of a simply supported viscoelastic cylindrical panel with the help of piezoelectric actuators

We consider the problem of active damping of the forced resonance vibrations of a viscoelastic cylindrical panel with the help of piezoelectric actuators. The end faces of the panel are assumed to be simply supported and heat-insulated. An analytical solution of the problem of electromechanics is obtained. On this basis, the dissipative function is calculated, and the energy equation is solved. We also analyze the influence of the temperature of dissipative heating on the efficiency of active damping of the forced vibrations of a cylindrical panel.     

A thermoelastic system of memory type in noncylindrical domains

In this paper, we consider a hyperbolic thermoelastic system of memory type in domains with moving boundary. The problem models vibrations of an elastic bar under thermal effects according to the heat conduction law of Gurtin and Pipkin. Global existence is proved by using the penalty method of Lions.     

The uncoupled problem of thermoelastic vibrations of a cylinder

Using the theory of functions of a complex variable, we construct an approximate solution of the problem of steady thermoelastic vibrations of a cylinder with curvilinear cavities, not taking account of the interaction of the strain and temperature fields. The results of numerical computations are given in the case of radially symmetric vibrations of a cylinder with a cavity. Two figures. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 107–112.     

Transient piezothermoelastic analysis of a cross-ply laminated cylindrical panel bonded to a piezoelectric actuator

In this study, the theoretical treatment of transient piezothermoelastic problem is developed for a cross-ply laminated cylindrical panel bonded to a piezoelectric actuator due to nonuniform heat supply. By using the exact solutions for cross-ply laminate and piezoelectric layer of crystal class mm2, the theoretical analysis of a transient piezothermoelasticity is developed for a simple supported cylindrical composite panel under the state of plane strain. Analysis of a piezothermoelastic problem leads to an appropriate electric potential applied to the piezoelectric layer which suppresses the induced thermoelastic displacement in the radial direction at the midpoint on the free surface of the cross-ply laminate. Some numerical results for the temperature change, the displacement, the stress in a transient state when the transient thermoelastic displacement is controlled are shown in figures.     

Numerical modeling of the active damping of forced thermomechanical resonance vibrations of viscoelastic shells of revolution with the help of piezoelectric inclusions

We consider the problem of active damping of forced resonance vibrations of viscoelastic shells of revolution with the help of piezoelectric sensors and actuators. Here, the interaction of electromechanical and thermal fields is taken into account. For modeling of vibrations, we use the Kirchhoff–Love hypotheses as well as hypotheses adequate to them and describing the distribution of temperature and electric field quantities. The shell temperature increases as a result of dissipative heating. For the active damping of vibrations, piezoelectric sensors and actuators are used. It is supposed that the electromechanical characteristics of materials depend on the temperature. The solution of this complex nonlinear problem has been obtained by the iterative method and finite element method. We have investigated the influence of temperature of dissipative heating on the efficiency of active damping of vibrations of a viscoelastic cylindrical panel with rigid restraint of its edges.     

Effect of memory dependent derivative on forced transverse vibrations in transversely isotropic thermoelastic cantilever nano-Beam with two temperature

The present research deals with the study of forced vibrations in transversely isotropic thermoelastic (TIT) nanoscale beam with two temperature (2T). Memory dependent derivative theory of thermoelasticity for clamped-free/cantilever nano-beam has been considered. The mathematical model is prepared for the nanoscale beam in a closed form with the application of Euler Bernoulli beam theory. Laplace transform method is employed to solve the problem. Forced vibrations due to exponential decaying time varying load acting vertically downward along the thickness direction of the nano-beam, Uniform load, Time harmonic load have been considered. Dynamic analysis for these forced vibrations and Static analysis has been carried out in this research. The dimensionless expressions for lateral deflection, thermal moment, temperature change, and axial stress are solved for these three forced vibrations. Response ratio has also been calculated. The analytical results have been numerically analysed using programming in MATLAB. The effect of kernel function has been depicted graphically on the lateral deflection, thermal moment, temperature change, axial stress and response ratio for all the three types of forced vibrations. Some particular cases have also been discussed.     

Wave propagation in a micropolar generalized thermoelastic body with stretch

In the present investigation, we discuss two different problems namely

1. Rayleigh-Lamb problem in micropolar generalized thermoelastic layer with stretch and
2. Rayleigh wave in a micropolar generalized thermoelastic half-space with stretch. The frequency and wave velocity equations for symmetric and anti-symmetric vibrations are obtained for the first problem. The frequency equation has also been derived for the second problem. The special cases of the above said problems of micropolar generalized thermoelasticity with stretch for Green-Lindsay and Lord-Shulman theory have been discussed in detail. Results of these analysis reduce to those without thermal and stretch effects

     

Investigation of frequencies of natural vibrations of a transversally isotropic cylindrical panel with a circular hole

We consider the problem of natural vibrations of a hingedly supported transversally isotropic cylindrical panel with a circular hole. The deformation of the shell is described by modified equations of the Timoshenko theory of shells. The numerical solution of the problem is constructed by the indirect method of boundary integral equations based on the sequential representation of Green functions.     

Boundary value problems in the theory of binary mixtures of thermoelastic solids

In this paper the diffusion model of the linear theory of thermoelasticity of binary mixtures is considered. The basic properties of wave numbers of the longitudinal and transverse plane waves are established. The interior boundary value problem (BVP) of steady vibration of binary mixtures of thermoelastic solids is formulated. The existence theorems of eigenfrequencies of the BVP of steady vibrations are proved. The connection between plane waves and existence of eigenfrequencies is established. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The coupled problem of thermoelastic vibrations of a cylinder with cavities

Using a method based on the machinery of analytic functions of complex variables, we construct the solution of the problem of steady planar strain of an isotropic cylinder with curvilinear cavities caused by uneven heating of the surfaces of the cylinder and the cavities. It is an exact solution of the equations and an approximate solution of the boundary conditions. The results of numerical studies are given for a hollow cylinder undergoing thermoelastic radially symmetric vibrations. Two figures. Bibliography: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 82–86.     

Nonlinear analysis of thermoelastic damping in axisymmetric vibration of micro circular thin-plate resonators

Thermoelastic damping is a source of dissipation in micro scale circular plate resonators. In contrast to previous researches, in this study thermoelastic damping is derived considering nonlinear effects. The microplate is assumed as a clamped circular plate. The microplate is modeled using the von Karman hypothesis along with Hamilton principle. Finally for harmonic vibrations, by using Kantorovich time averaging technique and perturbation techniques, thermoelastic damping is derived. The behavior of thermoelastic damping versus material properties, environmental temperature, plate radius and plate thickness are plotted. In this study the difference between linear and nonlinear analysis is shown for calculation of thermoelastic damping. The results show that the nonlinear analysis has a significant influence on thermoelastic damping coefficient.     

Minimization of stress levels by distribution of elastic parameters in elastic and thermoelastic bodies

We study the problem of minimizing stress levels in an isotropic elastic body by distributing elasticity parameters. We introduce a large class of differentiable functionals that we use to estimate stress levels. We obtain the necessary and sufficient conditions for global optimum and prove the uniqueness of the optimal stress field. We also study the problem of optimization of a thermoelastic body and derive a relation between the optimization results for elastic and thermoelastic bodies.Translated from Matematicheskie Metody i Fiziko-mekhanicheskie Polya, No. 26, pp. 43–50, 1987.     

Control-theoretic properties of structural acoustic models with thermal effects,I. Singular estimates

We consider a class of structural acoustics models with thermoelastic flexible wall. More precisely, the PDE system consists of a wave equation (within an acoustic chamber) which is coupled to a system of thermoelastic plate equations with rotational inertia; the coupling is strong as it is accomplished via boundary terms. Moreover, the system is subject to boundary thermal control. We show that—under three different sets of coupled (mechanical/thermal) boundary conditions—the overall coupled system inherits some specific regularity properties of its thermoelastic component, as it satisfies the same singular estimates recently established for the thermoelastic system alone. These regularity estimates are of central importance for (i) well-posedness of Differential and Algebraic Riccati equations arising in the associated optimal control problems, and (ii) existence of solutions to the semilinear initial/boundary value problem under nonlinear boundary conditions. The proof given uses as a critical ingredient a sharp trace theorem pertaining to second-order hyperbolic equations with Neumann boundary data.     

Three-dimensional mixed problems of thermoelasticity for isotropic plates

We obtain the homogeneous thermal solutions due to a temperature field for the three-dimensional thermoelastic problem for isotropic plates on whose plane faces homogeneous thermal and mixed mechanical conditions of flat face and diaphragm type are prescribed. This makes it possible to reduce the thermoelastic boundary-value problem to the corresponding elasticity problem. Translated fromTeoreticheskaya i Prikladnaya Mekhanika. No. 25, 1995, pp. 3–8.     

Global existence and exponential stability for a nonlinear thermoelastic Kirchhoff–Love plate

We study an initial–boundary-value problem for a quasilinear thermoelastic plate of Kirchhoff & Love-type with parabolic heat conduction due to Fourier, mechanically simply supported and held at the reference temperature on the boundary. For this problem, we show the short-time existence and uniqueness of classical solutions under appropriate regularity and compatibility assumptions on the data. Further, we use barrier techniques to prove the global existence and exponential stability of solutions under a smallness condition on the initial data. It is the first result of this kind established for a quasilinear non-parabolic thermoelastic Kirchhoff & Love plate in multiple dimensions.     

Exact Controllability and Boundary Stabilization of Torsional Vibrations of an Internally Damped Flexible Space Structure

In this paper, we study the exact controllability and boundary stabilization of the torsional vibrations of a flexible space structure (such as a solar cell array) modeled by a rectangular panel, incorporating the material damping of the structure. The panel is hoisted at one end by a rigid hub and the other end is totally free. For the attachment of this hub on one side of the panel, the hub dynamics leads to a nonstandard boundary condition. To incorporate internal damping of the material, we assume Voigt-type viscoelasticity of the structure. Exact controllability theory is established using the Hilbert uniqueness method by means of a control torque applied only on the rigid hub of the panel. At the same time, uniform exponential energy decay rate is obtained directly for the solution of this problem.     

Vibrations of elastic systems with a large number of tiny heavy inclusionsVibrations de systèmes élastiques avec un grand nombre de petites inclusions à forte densité

We consider a spectral problem modeling natural vibrations of a complex medium that consists of an elastic medium and tiny rigid inclusions. We study the asymptotic behaviour of solutions of this problem when the total number of inclusions and their density tend to infinity. We obtain a limit problem being a spectral problem for a linear fractional operator pencil that describes the macroscopic behaviour of the system (global vibrations). To cite this article: V. Rybalko, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 245–250.