A statistical method to estimate the vertical transmission through horizontally non-homogeneous media

A method to estimate the vertical transmission of radiation through horizontally non-homogeneous media is presented. In the medium, the cloud-like clusters of absorbers are assumed to be randomly located. A statistical method is applied to the distribution of the absorbing ability of the clusters. Various models for cluster shape have been assumed and related transmission properties have been examined in the intensity (column) and flux (slab) transmissions. The angular dependent extinction of radiation by scattering has also been considered in the flux transmission. An approximate method to convert a horizontally non-homogeneous medium to an equivalent homogeneous one is proposed. The non-homogeneity in the actual atmosphere is also discussed.     

The vibration and stability of orthotropic conical shells with non-homogeneous material properties under a hydrostatic pressure

In this paper, the vibration and stability of orthotropic conical shells with non-homogeneous material properties under a hydrostatic pressure are studied. At first, the basic relations have been obtained for orthotropic truncated conical shells, Young's moduli and density of which vary continuously in the thickness direction. By applying the Galerkin method to the foregoing equations, the buckling pressure and frequency parameter of truncated conical shells are obtained from these equations. Finally, carrying out some computations, the effects of the variations of conical shell characteristics, the effects of the non-homogeneity and the orthotropy on the critical dimensionless hydrostatic pressure and lowest dimensionless frequency parameter have been studied, when Young's moduli and density vary together and separately. The results are presented in tables, figures and compared with other works.     

Forced axisymmetric motion of circular,viscoelastic plates

Williams' method for forced motion of elastic systems is applied to circular, viscoelastic plates where the effects of rotatory inertia, transverse shear and time-dependent boundary conditions are included. The viscoelastic material is assumed to have a constant Poisson's ratio. A particular problem is solved for a symmetrically loaded, completely free plate. The material used is vulcanized rubber where the viscoelastic behavior in shear is used in specifying the material parameters of a three-element solid.     

Kinetics of metal-oxygen reaction by solid oxide electrolyte cells

Chronopotentiometric curves, generated by galvanostatic single steps, applied to solid oxide electrolyte cells, have been analysed on the basis of a dimensionless equation derived on the assumption that a scale of oxide grows at one of the electrode-electrolyte interfaces. This process is rate limiting for developing the charge transfer-diffusion overvoltage, and Wagner's theory on tarnishing under retarding electric field conditions, has been assumed for treating the kinetics of the growing scale. At constant temperature, the oxidation rate and oxidation rate constant have been measured as a function of the oxygen partial pressure in the range of pressures near the equilibrium pressure of the metal-oxide system.     

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.     

Dynamic analysis of circular and sector thick,layered plates

The finite element method is extended to the free vibration analysis of laminated thick plates with curved boundaries. Two elements are developed on the basis of Mindlin's thick plate theory in which the effects of thickness-shear deformation and rotary inertia are included. Both elements are derived in polar co-ordinates and can be joined together to handle annular as well as circular laminated anisotropic plate problems. Since axisymmetry has not been assumed, variations in material properties in the tangential direction can be dealt with. Numerical results are presented to demonstrate the influence of geometrical shape as well as that of thickness-shear deformation on the free vibrations of both homogeneous and layered plates. Comparisons between the numerical results obtained and those presented by other investigators confirm the accuracy of the new elements. The elements also can be used in the analysis of rectangular plates by assuming very large radii and very small subtended angle values.     

FORCED AND SELF-EXCITED VIBRATIONS OF PIPES CONTAINING MOBILE BOILING FLUID CLOTS

Numerical modelling of the dynamic behaviour of a pipe containing inner non-homogeneous flows of a boiling fluid has been carried out. Inasmuch as the efforts to solve this problem analytically are confronted by considerable difficulties connected with varying system mass, geometry and discontinuity of equation coefficients, computational techniques for simulating pipe dynamics have been developed based on using of numerical time integration methods and transfer matrix methods together with orthogonalization procedures relating to the space variables. The system vibrations at different values of the parameters of the flow non-homogeneity and its velocity are observed. The possibility of forming stable and unstable flows depending on the character of the non-homogeneity and the velocity of fluid clots has been found.     

Structural synthesis of sandwich beams with outer layers of box-section

It is proved by model measurements that, for sandwich beams constructed from two rectangular tubes and a damping layer glued between them, the following calculation methods can be applied. Static bending and shear stresses as well as deflections of simply supported beams may be calculated by Allen's formulae for sandwich beams with flexurally stiff faces. The first eigenfrequency and the loss factor can be determined by using the diagrams given in reference [1]. For the loss factors Ungar's formula gives a suitable approximation. A minimum cost design procedure is presented for a sandwich beam with constant cross-section. The unknown dimensions of the cross-section are determined which satisfy the design constraints and minimize the material costs. In a numerical example, constraints relating to the maximal dynamic stresses and deflection as well as local buckling of plate elements are considered. In the optimization the backtrack combinatorial discrete programming method is applied. A numerical comparison shows that the material costs of a sandwich beam are lower than those of a homogeneous box one.     

Vibrations of an orthotropic elliptic plate of non-uniform thickness and temperature

An analysis is presented of the free vibrations of an orthotropic elliptic plate whose temperature and thickness spatial variations both are parabolic along a line through the plate centre. An approximate but quite convenient frequency equation is derived by using Galerkin's method with a two-term deflection function. The upper bounds of frequencies corresponding to the first two modes of vibrations of a clamped orthotropic elliptic plate for various values of aspect ratio, taper constant and temperature constant are obtained.     

Stability of cantilever plates subjected to biaxial subtangential loading

The stability of cantilever plates which, mathematically, comprises a non-self-adjoint problem is investigated. It is assumed that the plate is acted upon by a subtangential biaxial edge load embodying the dead loading and the follower type loading as its limiting states. The scheme of modal expansions, containing the constrained rigid modes, together with Galerkin's method is employed and the stability of the plate in terms of subtangency and load parameters is analysed. As an example the kinetic stability analysis of a square cantilever plate is carried out in detail.     

Anderson-grüneisen parameter δ, bulk modulus and the compression curves of rhenium

The third-order elastic (TOE) constants of rhenium obtained from a model based on Keating's approach have been used to calculate the Anderson-Grüneisen (AG) parameter δ for this metal following the procedure suggested by Ramji Rao. The temperature dependence of the bulk modulus of rhenium has been calculated using Anderson's theory. The agreement with the experimental results of Fisher and Dever is good. The AG parameter has also been used to calculate the second Grüneisen constant q for rhenium. The variation of the lattice parameters of rhenium with hydrostatic pressure upto 500 kbars has been calculated using the theoretical TOE constants and Thurston's extrapolation formula. There is very good agreement with the experimental results of Liu et al.     

Large amplitude flexural vibration of eccentrically stiffened plates

The large amplitude free flexural vibrations of thin, orthotropic, eccentrically and lightly stiffened elastic rectangular plates are investigated. Clamped boundary conditions with movable in-plane edge conditions are assumed. A simple modal form of one-term transverse displacement is used and in-plane displacements are made to satisfy the in-plane equilibrium equations. By using Lagrange's equation, the modal equations for the nonlinear free vibration of stiffened plates are obtained for the cases when the stiffeners are assumed to be smeared out over the entire surface of the plate, and when the stiffeners are located at finite intervals. Numerical results are obtained for various possibilities of stiffening and for different aspect ratios of the plate. By particularizing the problem to different known cases, the results obtained here are compared with available analytical and experimental results, and the agreement is good.     

Dynamic behaviour of reactor structures subjected to impulsive loads

A reactor is modeled as a thin cylinder with one end capped by a solid circular plate and the dynamic behaviour of this structure is investigated when it is subjected to an impulsive load uniformly distributed over the circular plate. To simplify calculations, the load is assumed to be a step function with respect to time. As the fundamental equation of the cylinder under an axisymmetrical load, Donnell's equation is used and it is solved by the Laplace transformation method. From the results of the theoretical analysis, it becomes evident that large hoop stresses are induced in the neighbourhood of the junction of the plate and the cylinder.     

On Transverse Electric (TE) wave propagation in a warm moving magnetoplasma with sinusoidal electron-density distribution

It has been shown, using the first three moment equations in conjunction with Maxwell's equations, that the wave equation for the TE mode in a warm moving stratified magnetoplasma transforms, for an assumed sinusoidal distribution of the electron density, to the standard form of the Hill's differential equation.     

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.     

Axisymmetric vibrations of polar orthotropic mindlin annular plates of variable thickness

Analysis and numerical results are presented for the axisymmetric vibrations of polar orthotropic annular plates with linear variation in thickness, according to Mindlin's shear theory of plates. A chebyshev collocation technique has been employed to obtain the frequency equations for the transverse motion of such plates, for three different boundary conditions. Frequencies, mode shapes and moments for the first three modes of vibration have been computed for different plate parameters. A comparison of frequencies with the corresponding values obtained by classical plate theory leads to some interesting conclusions.     

Energy dependence of electron inelastic scattering mean-free-paths using synchrotron radiation photoelectron spectroscopy

The energy dependence of electron inelastic scattering mean-free-paths (mfp's) has been measured for thin films of Au, Ag and Mg for energies up to 150 eV by the overlayer method using synchrotron radiation photoemission. After the initial sharp drop in mfp at low energies (not seen in our data for Mg) mfp's for all three materials remain almost constant over the remaining range of kinetic energy. Some of the essentials of this technique are discussed; in particular the choice of substrate material, overlayer growth mechanisms and geometry of electron spectrometer are considered in detail. Increases in the metal-related 2p emission on oxidation of films of Mg and Al indicate that the mfp in the oxides is typically at least twice as great as in the metal and this ratio is energy-dependent. The dangers of using the “universal curve” of mfp versus energy are highlighted by this work.     

The elastic behaviour of InBi single crystals

Ultrasonic wave velocities have been measured by the pulse echo technique at 15MHz in single crystals of InBi grown by zone-melting. The elastic stiffness constant set has been computed from the velocity data by a least-mean-squares procedure. The elastic behaviour of this metallic compound is quite different in kind from that of the semiconducting III-V crystals; the volume compressibility does not follow Keyes' generalisation for the other III-V compounds. The elastic properties of InBi have been found to show the characteristics of a layer-like crystal with weak interlayer binding; this finding is illustrated by the linear compressibilities and by a compilation of cross-sections of the wave velocity and Young's modulus surfaces. The Debye temperature is 115°K.     

On the theory of Wien dissociation for weak electrolytes

Onsager's integral expression for the distribution function of oppositely charged ion pairs in a weak electrolyte in the presence of a uniform external electric field is rewritten as a sum of ordinary Bessel functions ranging from order zero to order plus infinity. Starting from this result a derivation of Onsager's expression for the dependence of the electrolyte's dissociation constant on the field intensity is given. It is shown that this requires only the zero order term in the above summation. Further, it is proved rigorously that the remaining higher order terms make no contribution to the dissociation constant. A non-zero contribution from these higher order terms would have implied dependence on the choice of the surface over which integration is performed at an intermediate stage of the analysis.     

Transmission of structure-borne sound from a beam into an infinite plate

An analytical study of transmission of structure-borne sound from a semi-infinite beam into an infinite, isotropic plate is presented. The beam is assumed to carry a torsional, a quasi-longitudinal and bending wave and the transmission is obtained with the help of the admittances of the beam and the plate. The analysis is restricted to the case of low frequencies but is otherwise general; thus due regard is given not only to the bending wave of the plate but also to the other propagating waves and to the local reactions. An interesting result from the study is that a bending wave on the beam will transfer a substantial part of its power into quasi-longitudinal and transverse waves in the plate, especially if the plate is thin compared with the beam. This is thought to be a factor that is important and not so easily quantifiable in the analysis of a complex structure. The local reactions on the other hand are of small importance for the power transmission from a torsional and quasi-longitudinal wave on the beam but may be important for the transmission of a bending wave, especially if the Young's modulus of the beam is larger than that of the plate.