MOTION EQUATIONS OF MULTILAYERED ELASTIC ELECTROCONDUCTIVE PLATES IN A MAGNETIC FIELD

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A meshfree method for transverse vibrations of anisotropic plates

A meshfree approach, called displacement boundary method, for anisotropic Kirchhoff plate dynamic analysis is presented. This method is deduced from a variational principle, which uses a modified hybrid functional involving the generalized displacements and generalized tractions on the boundary and the lateral deflection in the domain as independent variables. The discretization process is based on the employment of the fundamental solutions of the static problem operator for the expression of the variables involved in the functional. The stiffness and mass matrices obtained for the dynamic model are frequency-independent, symmetric and positive definite and their computation involves boundary integrals of regular kernels only. Due to its features, the final resolving system can be solved with the classical approaches by using standard numerical procedures. To assess the formulation, the free vibrations of some anisotropic plates were calculated and the results compared with those obtained using other solution techniques. The present results are in good agreement with those found in the literature showing the accuracy and effectiveness of the proposed approach.     

Variational-difference method in the problem of the vibrations of laminated plates

Highway Institute, Kiev. Translated from Prikladnaya Mekhanika, Vol. 28, No. 8, pp. 37–42, August, 1992.     

On the flexural vibrations of sandwich plates

A simplified approach to sandwich plate free vibrations, based on the superposition method, can provide approximations to the lowest modes for various boundary conditions, showing a characteristic influence of the bending-shear stiffness ratio.

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Sommario Un'analisi semplificata delle vibrazioni naturali di pannelli sandwich in diverse condizioni di vincolo, con il metodo di sovrapposizione degli effetti, consente l'individuazione approssimata dei modi più bassi, mostrando un comportamento vibratorio caratteristico al variare del rapporto tra le rigidezze di flessione e taglio.

     

Non-linear longitudinal vibrations of piezoceramics excited by weak electric fields

Typical non-linear effects, e.g. dependence of the resonance frequency on the amplitude, superharmonics in spectra and a non-linear relationship between excitation voltage and vibration amplitude as well as jump phenomena are observed in experiments with piezoceramics excited at resonance by weak electric fields. These non-linear effects can be observed for both the piezoelectric 31- and the 33-effect. In contrast to the well-known non-linear effects exhibited by piezoceramics in the presence of strong electric fields, these effects are not described in detail in the literature.In this paper, we attempt to model these phenomena using an electric enthalpy density to capture the cubic-like effects observed in the experiments. The equations of motion for the system under consideration are derived via the Ritz method using Hamilton's principle. The ‘non-linear’ parameters are identified and the numerical results are compared to those obtained experimentally. The effects described herein may have a significant influence in structures excited close to resonance frequencies via piezoelectric elements.     

On the theory of transverse bending of elastic plates

Departing from a self-contained two-dimensional formulation of the linear-theory problem of transverse bending of plates, three distinct topics are considered.The first of these concerns the integration problem for the case of orthotropy, specifically in regard to the factorization of a certain sixth-order master-equation.The second topic concerns the boundary layer aspects of contracted or reduced boundary conditions for the interior solution contribution for the case of isotropic plates. The analysis of this is based on a new form of the well-known general solution in terms of a deflection and a stress function variable, with this new form making it possible to distinguish between first- and second-order transverse shear deformation effects; the former being associated with the edge zone and the latter with the interior domain of the plate, with the shear correction terms for the interior being generalizations of the Timoshenko shear correction terms for beams.The third topic is a new system of contracted boundary conditions, both for the stress and for the displacement boundary value problem, in such a way that first-order transverse shear deformation effects are explicitly incorporated in the interior-domain solution contribution.     

Magnetoelastic buckling of thin plates in a uniform transverse magnetic field

The buckling of thin soft ferromagnetic elastic plates in a uniform transverse magnetic field is discussed. Based on a linearized version of the general magnetoelastic equations, a plate equation is derived together with the pertinent boundary conditions. By searching a nontrivial solution of this homogeneous system, a buckling value can be found. As an example, the circular plate is considered. In this paper, Maxwell stresses are used, but it is shown that identical results can be obtained by using other models (especially Amperian current) for the magnetoelastic stresses.

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Zusammenfassung In dieser Arbeit wird das Knicken dünner ferromagnetischer, elastischer Platten in uniformen magnetischen Feldern behandelt. Die Ableitung einer Plattengleichung zusammen mit den bezüglichen Randesbedingungen wird auf die linearisierten Gleichungen für weiche ferromagnetische Medien begründet. Durch Lösung dieses Systems wird eine Knicklast ermittelt. Die Methode wird anhand des Beispieles der kreisförmigen Platte illustriert. In dieser Abhandlung werden Maxwell Spannungen verwendet, aber es wird gezeigt dasz andere Modelle (insonderheit Amperesche Stromstärke) für die magnetoelastischen Spannungen das gleiche Resultat bilden.

     

Rayleigh problem in slip flow with transverse magnetic field

An analytical continuum solution of the Rayleigh problem in slip flow with applied magnetic field is obtained using a modified initial condition and slip boundary conditions. The results are uniformly valid for all times and show that the velocity slip and the local skin friction coefficient remain almost unaffected by the imposition of the magnetic field for small times. They increase however with the magnetic field for large times. The present results reduce to the corresponding results of the hydrodynamic case when there is no magnetic field.Nomenclature A constant - b characteristic length - B magnetic field vector - B ₀ magntidue of the applied magnetic field normale to the plate - B _x magnitude of the induced magnetic field parallel to the plate - C slip coefficient, (2–f)/f - C _f skin friction coefficient, - C _D average drag coefficient - erfc(_x) complementary error function, - E electric field vector - f Maxwell's reflection coefficient - H _a Hartmann number, (B ₀ ² b ²/)^1/2 - nondimensional magnetic parameter - J current vector - Kn=L/b Knudsen number - L mean free path - M Mach number - p constant parameter - P _m magnetic Prandtl number, Re _m/Re= ₀ - q velocity vector - Re Reynolds number, Ub/ - Re _m magnetic Reynolds number, ₀ Ub - t time - nondimensional time, tU/b - u velocity of the fluid parallel to the plate - nondimensional velocity, u/U - U velocity of the plate - Laplace transform of - x, y coordinates along and normal to the plate respectively - y nondimensional distance, y/b - Z nondimensional parameter, 1/Re ^1/2 Kn - ratio of specific heats - boundary layer thickness - velocity slip - viscosity - ₀ magnetic permeability - kinematic viscosity - nondimensional time parameter, ( /Re)^1/2/Kn - density - electrical conductivity     

Characteristic orthogonal polynomials in the study of transverse vibrations of nonhomogeneous rectangular orthotropic plates of bilinearly varying thickness

Effect of nonhomogeneity on the vibrational characteristics of thin orthotropic rectangular plates of bilinearly varying thickness has been studied using boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method. The thickness variation is taken as the Cartesian product of linear variations along two concurrent edges of the plate. The orthogonal polynomials in two variables are generated using the Gram-Schmidt process. The nonhomogeneity of the plate material is assumed to arise due to linear variations in Young’s moduli, shear modulus and density of the plate with the in-plane coordinates. Numerical results have been computed for four different combinations of clamped, simply supported and free edges. Effect of thickness variation together with varying values of aspect ratio and nonhomogeneity on the natural frequencies is illustrated for the first three modes of vibration. Three dimensional mode shapes have been presented. Comparison has been made with the known results.     

Experimental studies of turbulence in mercury flows with transverse magnetic fields

The differential pressure reading from a static hole pair is utilized for determination of the local wall shear stress. Both the hole diameter and forward-facing angle are varied to test the sensitivity of the device. The static hole pair in tested in a two-dimensional zero pressure gradient turbulent boundary layer on a smooth surface. The calibrating values for the local wall shear is determined from the universal scaling laws for the mean velocity profile in the inner part of the turbulent boundary layer. The static hole pair is found to be sensitive to imperfections in the manufacturing process, and needs an individual calibration in order to make accurate measurements of the local skin friction possible.