On the analysis of large amplitude vibrations of non-uniform rectangular plates

An analysis of the large amplitude vibrations of non-uniform rectangular plates is presented. The method is based on Berger's assumption of neglecting the second invariant of the middle surface strain in the expression corresponding to the total potential energy of the system, in conjunction with a Galerkin procedure.     

Large amplitude vibrations of rectangular plates with non-uniform elastic edge supports

Large amplitude vibrations of a rectangular plate with parabolically varying rotational constraints on two opposite edges are studied. The problem has been solved by both Berger's approach and the more rigorous von Kármán approach. Galerkin's method is used to obtain an ordinary differential equation in the modal function, the solution to which is given in terms of elliptic functions.     

On the non-linear vibration of circular plates of variable thickness elastically restrained along the edges

Berger's technique of neglecting the second strain invariant in the middle plane of the plate is used in an investigation of the effect of non-linearity on the frequency of a vibrating circular plate of variable thickness which is elastically restrained along its circumference. A simple polynomial expression for the deflection function and a Galerkin method are used in the analysis.     

Non-linear dynamic behaviour of plates on elastic foundations

The large amplitude transverse vibration characteristics of a simply-supported isotropic equilateral triangular plate on elastic foundation have been investigated by following Berger's well-known approximate method in conjunction with a special type of co-ordinates known as tri-linear co-ordinates. A second order non-linear differential equation for the unknown time function has been obtained and solved, as usual, in terms of Jacobian elliptic functions. Results obtained from numerical calculations are presented graphically.     

Vibration of cylindrical shells of bimodulus composite materials

A theory is formulated for the small amplitude free vibration of thick, circular cylindrical shells laminated of bimodulus composite materials, which have different elastic properties depending upon whether the fiber-direction strain is tensile or compressive. The theory used is the dynamic, shear deformable (moderately thick shell) analog of the Sanders best first approximation thin shell theory. By means of tracers, the analysis can be reduced to that of various simpler shell theories, namely Love's first approximation, and Donnell's shallow shell theory. As an example of the application of the theory, a closed form solution is presented for a freely supported panel or complete shell. To validate the analysis, numerical results are compared with existing results for various special cases. Also, the effects of the various shell theories, thickness shear flexibility, and bimodulus action are investigated.     

Some problems of analysis and optimization of plates and shells

The classical optimization problems of plates and shells to satisfy a priori given geometry and dynamical characteristics are considered. Orthotropic plates and shells with variable thickness and low transverse stiffness are analyzed. First, some useful theorems and their proofs are given. Then the finite approximation of the problem related to optimization of free vibrations of shells with transverse deformation and rotary inertia is discussed. The varational iteration (MVI) and Bubnov-Galerkin (MB) methods are applied, and their convergence and suitability for application to plates and shells analysis are discussed and numerically evaluated.     

Unsteady fluid dynamic forces on a simply-supported circular cylinder of finite length conveying a flow,with applications to stability analysis

An exact analytic expression for the unsteady fluid pressure acting on the internal walls of a simply-supported circular cylindrical tube of finite length, carrying flow, is presented. The generalized force coefficients corresponding to specific modes of deformation are given explicitly. The results are applied to two problems: (1) the interaction of flow and buckling of thin-walled cylindrical shells subjected to lateral pressure and/or end thrust; (2) the aeroelastic stability of the shells. The second problem is aimed at resolving some controversy about post-divergence flutter oscillation of cylindrical shells or plates exposed to a subsonic flow. The shell equation, of the Morley type, is solved by Galerkin's method and an analytic approach is used to examine the stability of the system. It is important that damping be taken into account in the analysis. The undeformed configuration is always unstable when the flow speed exceeds the minimum divergence boundary.     

Fundamental frequency of vibration of clamped plates of arbitrary shape subjected to a hydrostatic state of in-plane stress

The present paper makes use of conformal mapping and Galerkin's method to determine the fundamental frequency of vibration of clamped plates of arbitrary shape subjected to uniform in-plane stress. Numerical results are presented for several plates of regular polygonal shape. Frequency curves obtained by means of Lurie's expression are also given for simply supported plates.     

Effects of large amplitude and transverse shear on vibrations of triangular plates

In this paper an analytical investigation of large amplitude free flexural vibrations of isotropic and orthotropic moderately thick triangular plates is carried out. The governing equations are expressed in terms of the lateral displacement, w, and the stress function, F, and are based on an improved non-linear vibration theory which accounts for the effects of transverse shear deformation and rotatory inertia. Solutions to the governing equations are obtained by using a single-mode approximation for w, Galerkin's method and a numerical integration procedure. Numerical results are presented in terms of variations of non-linear frequency ratios with amplitudes of vibrations. The effects of transverse shear, rotatory inertia, material properties, aspect ratios, and thickness parameters are studied and compared with available solutions wherever possible. Present results are in close agreement with those reported for thin plates. It is believed that all of the results reported here that are applicable for moderately thick plates are new and therefore, no comparison is possible.     

Shear and rotatory inertia effects on large amplitude vibration of skew plates

The large amplitude free flexural vibration of elastic, isotropic skew plates is investigated, the effects of transverse shear and rotatory inertia being included. By use of Galerkin's method and the extended Berger approximation, solutions are obtained on the basis of an assumed vibration mode. The non-linear period vs. amplitude behavior is of the hardening type and the non-linear period is found to increase when the effects of transverse shear and rotatory inertia are considered in the analysis. The influence of these effects on aspect ratios and skew angles of thin and moderately thick skew plates is investigated both at small and large amplitudes.     

The thermalization length by the green function method

We study the concept of the thermalization lenght by the Green function method, using Sobolev's escape and destruction probabilities. A unique analytical procedure yields the usual numerical estimates for the noncoherent scattering case (with complete redistribution in frequency) as well as for the case of coherent scattering. The on-the-spot approximation is used to remove the assumption of an infinite geometry. Rybicki and Hummer's longest flight approximation is shown to yield the on-the-spot approximation. A generalization to the multilevel case is sketched.     

The neutron scattering and exciton spectrum due to impurities in the molecular crystals with the resonance interaction

The differential cross-section of neutron scattering by excitons and the frequency equation for exciton chain with point impurity are obtained by using the exact solution, received in our previous papers [1, 2] for the case of ideal exciton chain problem. All results are compared with those given by the known approximate solution of the same problem [4] based on the approximation of Pauli's operators of creation and annihilation of excitons by Bose's ones.     

Potential-moving,Migdal's recursion formula,differential renormalization and duality

Migdal's original recursion formula is rederived as a low-temperature approximation by an isotropic type of potential-moving. For self-dual spin or gauge systems this transformation is shown to be differentiably conjugate to another one, which is obtained as a high-temperature approximation. The conjugation relation is established through the duality mapping.This explains the mechanism leading to some exact results obtained with Migdal's differential renormalization equation. The last equation is also explicitly rederived as the result of potential-moving approximations inspired by the methods of differential renormalization in real space.Some applications and extensions of the above results are finally considered in connection with an approach, which was recently proposed for systematically improving Migdal's approximation.     

Ultrasonic waves in porous ceramics with non-spherical holes

Formulae describing the effective elastic moduli of a porous ceramic medium are derived using Eshelby's solution and Maxwell's approximation. The ceramic medium is considered as an infinite matrix, which has uniform elastic properties and encloses non-spherical pores. A numerical evaluation of the velocity of an ultrasonic wave in the ceramic, as function of the porosity and pore shape, is presented. The theoretical results were combined with those obtained experimentally for different firing temperatures of the ceramic.     

Free vibration of antisymmetric,angle-ply laminated plates including transverse shear deformation by the finite element method

A finite element formulation of the equations governing the laminated anisotropic plate theory of Yang, Norris and Stavsky, is presented. The theory is a generalization of Mindlin's theory for isotropic plates to laminated anisotropic plates and includes shear deformation and rotary inertia effects. Finite element solutions are presented for rectangular plates of antisymmetric angle-ply laminates whose material properties are typical of a highly anisotropic composite material. Two sets of material properties that are typical of high modulus fiber-reinforced composites are used to show the parametric effects of plate aspect ratio, length-to-thickness ratio, number of layers and lamination angle. The numerical results are compared with the closed form results of Bert and Chen. As a special case, numerical results are presented for thick isotropic plates, and are compared with those for 3-D linear elasticity theory and Mindlin's thick plate theory.     

An improved separability approximation for line radiative transport in nonhomogeneous media

A simple modification to the constant half-width approximation of Wilson and Greif, which is an extension of the well-known Curtis-Godson method to the treatment of temperature variations along the integration path, is introduced which permits a more accurate evaluation of line radiative transport in non- homogeneous gases. To demonstrate the method's accuracy, comparisons are made with Wilson and Greif and numerical frequency integrated results for the line equivalent width and radiative flux in a planar slab with prescribed Planck function and line half-width spatial variations are chosen to represent typical shock layer conditions. It is found that the modified procedure reduces the inaccuracies inherent in Wilson and Greif's approximation by factors ranging from 5 to 10, while retaining the latter method's ease of application.     

The t2 ← e transition at the Cu impurity in ZnS crystal

The excitation energy for the localized t₂ ← e transition is calculated by the Green's function method in CNDO approximation using a parent system model.     

Evaluation of force constant ratio in the AlCu system

Band model calculations, using Mannheim's theory of nearest neighbour central force approximation in a cubic crystal, have been carried out to estimate the impurity-host to host-host coupling ratio λ'/λ in the AlCu system. The obtained value of 1.22 for λ′/λ suggests about 20 per cent stiffening of the impurity to host binding.     

The exact nonlinear driven response of a one-dimensional gas

The exact nonlinear driven response of a one-dimensional gas of hard rods to a time dependent uniform external field is obtained in terms of the single particle Green's function. Limiting cases of very large and very small field are examined and, for the latter, comparison is made between the linear approximation to the exact response and the response obtained from the linear response theory perturbation approximation. Strong disagreement between these results is observed. This supports van Kampen's contention that a macroscopic linear response to an external perturbation is not a manifestation of microscopic linearity.     

The forced motion of viscoelastic thick cylindrical shells

This paper presents a theoretical analysis of a dynamic boundary value problem of the axially-symmetric motion of isotropic, homogeneous, linearly-viscoelastic, thick, cylindrical shells subjected to time-dependent surface tractions and/or time-dependent boundary conditions. Williams' modal-acceleration method has been used to treat the time-dependent boundary conditions. Two forms of the correspondence principle are used to obtain the governing differential equations and the quasi-static solutions. A numerical example is given to study the transient response of a cylindrically hollow rod subject to longitudinal impacts.