Effect of vertical vibrations on a two-layer system with a deformable interface

The effect of single- and two-frequency vibrations on the behavior of a system consisting of two homogeneous viscous fluids bounded by rigid walls is analyzed. It is assumed that the system as a whole is under vertical vibrations obeying a certain law. An eigenvalue problem is obtained in order to analyze the stability of the relative equilibrium. The case of finite frequencies and arbitrary modulation amplitudes is treated along with the case of high frequencies and small modulation amplitudes. In the former case, the parametric resonance domains are examined depending on the parameters of the system. In the latter case, the high-frequency vibration is shown to create effective surface tension, thus flattening the interface, and can suppress instability when the heavy fluid is over the light one.     

Delamination of a two-layer corroded ring under radial vibrations

A two-layer ring, with account of its initial state caused by internal pressure and corrosion, leading to wear, is considered. The ring is subjected to an additional pressure causing its radial vibrations. It is shown that, under certain conditions, there is a vibration frequency at which the delamination of the ring is possible. Expressions are found which determine this frequency as a function of problem parameters, in particular, of the time of corrosive wear.     

Free vibrations of a cantilevered plate with a polymer coating

The article considers some of the special characteristics of the free vibrations of a steel cantilevered plate, with the formation of a polymer coating on the plate. It compares the results of an experimental investigation of the effect of internal stresses in the coating on the vibrations and on the static bending of the plate. It demonstrates the possibility of determining the internal stresses in the coating from the change in the frequency of the natural vibrations of a plate with a coating.Institute for the Mechanics of Metal-Polymer Systems, Academy of Sciences of the Belorussian SSR, Gomel'. Translated from Mehanika Polimerov, No. 3, pp. 541–544, May–June, 1970.     

Steady-state vibrations of a compound isotropic cylinder under a plane strain

We solve the problem of two-dimensional strain of a compound isotropic cylinder deformed by distributed pulsing forces. We study the influence of the geometric and elastic characteristics of the layers on the magnitude of the stresses and the lowest natural frequency of radial-symmetric vibrations of a three-component cylinder. Two figures Bibliography: 3 titles Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 103–108.     

Equations for the vibrations of a plate with general anisotropy

The exact form of the displacement field for an elastic layer with general anisotropy and nonuniform thickness is constructed. The low-frequency part of the solution, which satisfies the equations for the vibrations of a plate, is found.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 165, pp. 122–135, 1987.     

Longitudinal vibrations of a composite isotropic plate

Approximate integration of the longitudinal vibrations equations for composite plates is considered. A numerical investigation is performed of the potential energy distribution in a two-layered ring when a pulsating pressure acts on the inner contour.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 63–67, 1989.     

The uniform heating and bending of a two-layer plate

The problem of the uniform heating of a two-layer plate is solved. The transversely isotropic layer considered (a soft plate) is in ideal contact with a rigid isotropic thin elastically deformed layer. The ends of the plate are load-free. A boundary layer of the soft plate (a thin contact layer) is introduced, which enables the boundary conditions on the ends of the plate to be formulated in such a way that the problem has a bounded smooth solution [1]. The two-layer plate, generally speaking, is bounded along the axis perpendicular to the axes directed along the length and thickness of the plate. The resultant force and the resultant moment, applied to the end transverse sections, are equal to zero. The exact solution of the temperature problem is sought using the equations of the theory of elasticity. The plane problem of the bending of a two-layer plate acted upon by a uniformly distributed pressure applied to the side surface of an anisotropic layer is solved by a similar method. The ends of the rigid isotropic layer are clamped.     

Steady-state vibrations of an unbounded linear piezoelectric medium

We consider fundamental (Dirichlet and Neumann-type) boundary value problems in a theory of generalized plane strain for the steady-state vibrations of an infinite piezoelectric medium with transversely isotropic symmetry (6 mm). Using integral equation methods with the appropriate Sommerfeld-type radiation conditions, we prove existence and uniqueness results for the corresponding exterior boundary value problems. Exact solutions are obtained in the form of integral potentials.     

Steady-state vibrations of an unbounded linear piezoelectric medium

We consider fundamental (Dirichlet and Neumann-type) boundary value problems in a theory of generalized plane strain for the steady-state vibrations of an infinite piezoelectric medium with transversely isotropic symmetry (6 mm). Using integral equation methods with the appropriate Sommerfeld-type radiation conditions, we prove existence and uniqueness results for the corresponding exterior boundary value problems. Exact solutions are obtained in the form of integral potentials. (Received: September 27, 2005)     

Reinforcement of a thin plate by a thin layer

We study the bending of a thin plate, stiffened with a thin elastic layer, of thickness δ. We describe the complete construction of an asymptotic expansion with respect to δ of the solution of the Kirchhoff–Love model and give optimal estimates for the remainder. We identify approximate boundary conditions, which take into account the effect of the stiffener at various orders. Thanks to the tools of multi‐scale analysis, we give optimal estimates for the error between the approximate problems and the original one. We deal with a layer of constant stiffness, as well as with a stiffness in δ^?1. Copyright © 2007 John Wiley & Sons, Ltd.     

The postbuckling analysis of laminated circular plate with elliptic delamination

Based on the Von Karman plate theory, considering the effect of transverse shear deformation, and using the method of the dissociated three regions, the postbuckling governing equations for the axisymmetric laminated circular plates with elliptical delamination are derived. By using the orthogonal point collocation method, the governing equations, boundary conditions and continuity conditions are transformed into a group of nonlinear algebraically equation and the equations are solved with the alternative method. In the numerical examples, the effects of various elliptical in shape, delamination depth and different material properties on buckling and postbuckling of the laminated circular plates are discussed and the numerical results are compared with available data.     

Forced vibrations of a multiply connected isotropic plate

On the basis of A. I. Aleksandrovich's investigations the present authors worked out an approximate method of solving problems of forced flexural vibrations of a multiply connected isotropic plate. For an elliptic plate with circular holes numerical results are presented.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 105–111, 1987.     

Electroelastic thickness vibrations of a quartz SC-cut plate

Analytic solutions are obtained of the boundary value problem of steady motions of a thick piezocrystalline SC-cut quartz plate describing thickness modes of harmonic elastic vibrations. The plate endfaces have no electrodes and adjoin a vacuum or a rarefied gas. The measure of the difference between the found values of the five lowest eigenfrequencies of the thickness vibrations of a plate without electrodes and the values of the corresponding frequencies of a plate whose endfaces are covered by massive short-circuited electrodes is determined.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 83–85, 1988.     

Unilateral contact of a plate with a thin elastic obstacle

We study the problem of contact of an elastic body with a beam. The most attention is paid to describing boundary conditions on the possible contact set. Moreover, we study asymptotic properties of solutions and the energy functional as the rigidity parameters tend to infinity or the length of the beam (or the zone of possible contact) changes.     

Stochastic nonstationary temperature fields in a solid circular-cylindrical two-layer plate

We determine the nonstationary temperature field in a solid two-layer plate subject to heat sources whose power is a random function of time and the radial variable. The probability characteristics constructed for the stochastic nonstationary temperature fields are sufficient for engineering computations. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 104–108.     

On bending vibrations of a periodically inhomogeneous orthotropic plate

The matrix technique which has been developed by the author in connection with solving the problem of elastic wave propagation in an unbounded medium with periodic inhomogeneities is applied. The formalism is extended to the case of bending vibrations of inhomogeneous and periodically inhomogeneous orthotropic plates.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 179–181, 1989.     

On the vibrations of a plate with a concentrated mass and very small thickness

We consider the vibrations of an elastic plate that contains a small region whose size depends on a small parameter ε. The density is of order O(ε^–m) in the small region, the concentrated mass, and it is of order O(1) outside; m is a positive parameter. The thickness plate h being fixed, we describe the asymptotic behaviour, as ε→O, of the eigenvalues and eigenfunctions of the corresponding spectral problem, depending on the value of m: Low‐ and high‐frequency vibrations are studied for m>2. We also consider the case where the thickness plate h depends on ε; then, different values of m are singled out. Copyright © 2003 John Wiley & Sons, Ltd.     

The vibrations of a rotating thin piezoceramic cylindrical shell

G. H. Bryan [8] discovered and studied the dynamic phenomena that arise during the vibrations of thinwalled shells in natural modes while rotating uniformly about an axis of symmetry. In the present article a more detailed mathematical analysis has been conducted for the solution of the problem of the vibrations of a ring rotating with constant angular velocity under feasible conditions of perturbation of the vibrations and realistic mechanical properties of ceramic materials. It is established that two slowly precessing waves arise during the vibration of the ring, one of which (the one discovered by Bryan) is an inverse precession wave having an angular velocity of precession less than the angular velocity of rotation of the ring, while the second is a direct precession wave having angular velocity larger than the angular velocity of rotation of the ring. Under the actual working conditions for piezoceramic resonators the amplitudes of the direct and inverse wave become comparable. Consequently the vibrations present as a modulated standing wave, that is, the phenomenon of beats in the mode of steady-state normal vibration occurs. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 7–18.