Solution of some dynamic problems of viscoelasticity theory by the averaging method

Damping of longitudinal vibrations of a viscoelastic bar and transverse vibrations of a freely supported orthotropic plate is investigated by the averaging method. The problem of the forced movement of a viscoelastic cylinder enclosed in a thin elastic shell under the effect of an arbitrarily time-varying internal pressure is studied.Institute of Cybernetics at the Computer Center Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, No. 2, pp. 246–252, March–April, 1970.     

Steady-state vibrations of a thin two-layer plate with delamination

We consider the problem of harmonic vibrations of a thin two-layer plate with horizontal crack. The problem is solved with the help of the null-field approach. The influence of the shape of the crack contour on the amplitude-frequency characteristics of plate vibrations is investigated. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 83–89, April–June, 1998.     

Investigation of the natural vibrations of plates made of composite materials with the help of the finite-element method

The modes and frequencies of small natural vibrations in a cavity of thin plates are investigated on the basis of the method of finite elements in displacements. The effect of the rigidity characteristics of a material on the natural frequencies and modes of vibrations of flat cantilever vanes whose tapered edge is restrained is studied. An example is given of the use of the finite-element method to determine the natural vibrations of an orthotropic plate in the shape of an airplane wing. Quadrangular and modified triangular bending finite elements are used to simulate a continuous system. A mass-element matrix is constructed on the basis of a four-term ("contracted") polynomial determining the deflection that permits a significant reduction of the order of the solving system.Translated from Mekhanika Polimerov, No. 2, pp. 284–288, March–April, 1976.     

Application of a refined theory of torsional vibrations to the calculation of natural frequencies and damping coefficients of orthotropic composite cantilever rods

For calculating the natural complex frequencies of torsional vibrations of rectangular orthotropic composite cantilever rods, a theory taking into account the normal stresses and inertial forces acting in the axial direction is employed. The results obtained are compared with those found by using the classical theory of torsional vibrations of rods, the theory of vibrations of thin orthotropic plates, and the FEM. It is found that the difference between the natural frequencies given by the classical and refined theories depends on relations between geometrical sizes of a rod and between its axial elastic modulus and shear moduli, and on the number of the mode of torsional vibrations.     

Landau-Hopf scenario of passage to turbulence in some problems of elastic stability theory

We consider a boundary value problem for a nonlinear integro-differential equation modeling the vibrations of a pipe transporting a fluid and the vibrations of a thin cylinder in an axisymmetric flow. For the boundary conditions we take the hinge support conditions. We show that the Landau-Hopf scenario of passage to turbulence can be realized in such a problem, whereby stable invariant tori of increasing dimensions are generated as the main bifurcation parameter increases.     

Determination of the stress state in a half-space in the vicinity of cylindrical defects appearing on the surface under torsional vibrations

We solve the problem of the determination of the stress state under torsional vibrations of a half-space with a cylindrical defect (a crack or a thin rigid inclusion) that crosses its surface. The method of solution is based on the use of discontinuous solutions of the equations of torsional vibrations and consists in the reduction of the initial boundary-value problems to integral equations for the unknown jumps of an angular displacement or a tangential stress.     

Free Vibrations of a Thin Elastic Orthotropic Cylindrical Panel with Free Еdges

Mechanics of Composite Materials - Using a system of equations corresponding to the classical theory of orthotropic cylindrical shells, the free vibrations of a thin elastic orthotropic cylindrical...     

High-frequency vibrations of isotropic plates

Using the method of A. L. Gol'denveizer we reduce the three-dimensional problem of high-frequency steady-state vibrations of a thin isotropic plate to a series of two-dimensional problems. We construct two recursive processes that describe the basic stress-strain state of the plate. It is established that the first coefficient of the asymptotic expansion of the natural frequency of vibrations is determined by the boundary conditions on the plane faces of the plate and is independent of the conditions on its lateral surface. Therefore to solve the problem it is necessary to take account of at least two approximations of the recursive processes. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 82–85.     

Dynamic method of determining the mechanical characteristics of the surface layer of polymers

A thin polymer film is treated as a three-layered plate, the inner layer being characterized by the bulk and the outer layers by the surface properties of the polymer. Using the equations describing the vibrations of a cantilever element of rectangular section, the authors propose formulas for the thickness, modulus of elasticity, and mechanical loss factor of the surface layer. The mechanical characteristics of the surface layers of PMM and nitrocellulose have been determined by electromagnetically exciting and photometrically recording vibrations in film specimens of varying thickness.Institute of the Chemistry of High-Molecular Compounds, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 3, pp. 548–550, May–June, 1969.     

Various methods of determining the natural frequencies and damping of composite cantilever plates. 1. Exact solution for the binomial model of deformation

On the basis of the classical theory of thin anisotropic laminated plates the article analyzes the free vibrations of rectangular cantilever plates made of fibrous composites. The application of Kantorovich's method for the binomial representation of the shape of the elastic surface of a plate yielded for two unknown functions a system of two connected differential equations and the corresponding boundary conditions at the place of constraint and at the free edge. The exact solution for the frequencies and forms of the free vibrations was found with the use of Laplace transformation with respect to the space variable. The magnitudes of several first dimensionless frequencies of the bending and torsional vibrations of the plate were calculated for a wide range of change of two dimensionless complexes, with the dimensions of the plate and the anisotropy of the elastic properties of the material taken into account. The article shows that with torsional vibrations the warping constraint at the fixed end explains the apparent dependence of the shear modulus of the composite on the length of the specimen that had been discovered earlier on in experiments with a torsional pendulum. It examines the interaction and transformation of the second bending mode and of the first torsional mode of the vibrations. It analyzes the asymptotics of the dimensionless frequencies when the length of the plate is increased, and it shows that taking into account the bending-torsion interaction in strongly anisotropic materials type unidirectional carbon reinforced plastic can reduce substantially the frequencies of the bending vibrations but has no effect (within the framework of the binomial model) on the frequencies of the torsional vibrations.Institute of Engineering Science Russian Academy of Sciences, St. Petersburg, Russia. St. Petersburg State University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 6, pp. 759–769, November–December, 1996.     

Asymptotic analysis of three-dimensional dynamic equations of a thin plate : PMM vol. 38, n 6, 1974, pp. 1072–1078

Two-dimensional dynamic equations of thin plate vibrations are obtained from the three-dimensional dynamic equations of elasticity theory on the basis of an asymptotic method [1 – 3], Such an approach permits establishing the limits of applicability of the two-dimensional dynamic equations and the corresponding boundary and initial conditions, and indicating the means of obtaining refined results.The question of the construction of an inner state of stress of a thin plate under dynamic conditions is examined herein. The possibility of considering states of stress with distinct variability in time and in the coordinates and with a distinct relationship between the displacement intensities, is taken into account.     

Free bending vibrations of an isotropic plate with free holes

A solution of the problem of the free bending vibrations of a thin finite isotropic plate with stiffly clamped outer edge is obtained by using functions of a complex variable. Results of numerical investigations are presented for an elliptic plate with acircular hole.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 67–70, 1989.     

Free vibrations of thin annular plates made from functionally graded material

Subject of the consideration is thin annular plate made of a two-phase functionally graded composte. The plate has periodically inhomogeneous microstructure slowly varying in space: the λ-periodic structure along circular coordinate, but smoothly graded apparent (averaged) properties in the perpendicular, radial direction. The aim of the contribution is to derive and apply a deterministic macroscopic model describing the free vibrations of this plate. Modeling procedure is based on tolerance averaging technique. We received, equations system with smooth coefficients. We made numerical solution of this problem, using finite difference method, and analyze influence of material proportion and microstructure size on first frequency of free vibrations. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical analysis of free vibrations of laminated composite conical and cylindrical shells: Discrete singular convolution (DSC) approach

A numerical study on the free vibration analysis for laminated conical and cylindrical shell is presented. The analysis is carried out using Love's first approximation thin shell theory and solved using discrete singular convolution (DSC) method. Numerical results in free vibrations of laminated conical and cylindrical shells are presented graphically for different geometric and material parameters. Free vibrations of isotropic cylindrical shells and annular plates are treated as special cases. The effects of circumferential wave number, number of layers on frequencies characteristics are also discussed. The numerical results show that the present method is quite easy to implement, accurate and efficient for the problems considered.     

Parametric resonances in the problem of longitudinal impact on a thin rod

The longitudinal impact on a thin elastic rod, which generates a periodic system of longitudinal waves in it, is considered. At definite values of the parameters of the problem in the linear approximation, these waves induce parametric resonances, which are accompanied by an unlimited increase in the amplitude of the transverse vibrations. To obtain finite values of the amplitudes, a quasilinear system is considered in which the effect of the transverse vibrations on the longitudinal vibrations is taken into account. This system was previously solved using the Bubnov–Galerkin method and beats accompanied by energy transfer between the transverse and longitudinal vibrations were obtained. In this work, an approximate analytical solution of the system has been derived that is based on double-scale expansions. A qualitative analysis of this solution has been carried out. An estimate of the maximum transverse bending has been obtained for various methods of loading. Both shortand long-term pulses have been considered. It has been shown that, in the case of a spontaneously applied long-term pulse that is lower than the Euler critical load, intensive transverse vibrations can occur.     

Integration of the equations of planar and flexural vibrations of anisotropic plates in special functions of generalized complex variables

On the basis of the method of successive approximations in formal series we construct systems of particular solutions of the equations of planar and flexural harmonic vibrations of thin rectilinear anisotropic plates with a plane of symmetry of elastic properties parallel to the faces. The solutions constructed are classified as special sk-functions of generalized complex variables. Translated fromTeoreticheskaya i Prikladnaya Mekhanika No. 24, 1993, pp. 54–61.     

Interaction of a harmonic torsional wave with ring-shaped defects in an elastic body

We have solved the problem of determination of the stressed state in an isotropic elastic body near ring-shaped defects (a crack or a thin rigid inclusion) as a result of the action of a harmonic torsional wave. The method of solution is based on the use of discontinuous solutions of the equation of torsional vibrations and lies in the reduction of the initial boundary-value problems to integral equations for the unknown jumps of angular displacement or tangential stress.     

Basic properties of special functions of generalized complex variables

For special functions of generalized complex variables that are regular particular solutions of the equations of plane and flexural vibrations of thin rectilinear anisotropic plates in domains of interior and annulus type we prove that the formal series representations converge absolutely and uniformly, establish a differentiation formula, and prove that the system of solutions is a basis. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 62–68.     

Exact Controllability and Asymptotic Analysis for Shallow Shells

The authors consider the exact controllability of the vibrations of a thin shallow shell, of thickness 2εwith controls imposed on the lateral surface and at the top and bottom of the shell. Apart from proving the existence of exact controls, it is shown that the solutions of the three dimensional exact controllability problems converge, as the thickness of the shell goes to zero, to the solution of an exact controllability problem in two dimensions.