Geometrically nonlinear vibrations of rectangular plates carrying a concentrated mass

Nonlinear forced vibrations of rectangular plates carrying a central concentrated mass are studied. The plate is assumed to have immovable edges and rotational springs; numerical results are presented for clamped plates. The Von Kármán nonlinear plate theory is used, but in-plane inertia in both the plate and the mass is retrained. The problem is discretized into a multi-degree-of-freedom (dof) system by using an energy approach and Lagrange equations taking damping into account. A pseudo-arclength continuation method is used in order to obtain numerical solutions. Results are presented as both (i) frequency-amplitude curves and (ii) time domain responses. The effect of gravity and the effect of the consequent initial plate deflection are also investigated.     

Multi-frequency excitation of stiffened triangular plates for large amplitude oscillations

Free and forced vibrations of triangular plate are investigated. Diverse types of stiffeners were attached onto the plate to suppress the undesirable large-amplitude oscillations. The governing equation of motion for a triangular plate, based on the von Kármán theory, is developed and the nonlinear ordinary differential equation of the system using Galerkin approach is obtained. Closed-form expressions for the free undamped and large-amplitude vibration of an orthotropic triangular elastic plate are presented using the two well-known analytical methods, namely, the energy balance method and the variational approach. The frequency responses in the closed-form are presented and their sensitivities with respect to the initial amplitudes are studied. An error analysis is performed and the vibration behavior, as well as the accuracy of the solution methods, is evaluated. Different types of the stiffened triangular plates are considered in order to cover a wide range of practical applications. Numerical simulations are carried out and the validity of the solution procedure is explored. It is demonstrated that the two methods of energy balance and variational approach have been quite straightforward and reliable techniques to solve those nonlinear differential equations. Subsequently, due to the importance of multiple resonant responses in engineering design, multi-frequency excitations are considered. It is assumed that three periodic forces are applied to the plate in three specific positions. The multiple time scaling method is utilized to obtain approximate solutions for the frequency resonance cases. Influences of different parameters, namely, the position of applied forces, geometry and the number of stiffeners on the frequency response of the triangular plates are examined.     

Transitional periodic domain structure in thin films of magnetically uniaxial materials

The paper gives the theory of transitional domain structure in thin films of uniaxial ferromagnets with easy axis perpendicular to the film. It is proved that in a certain range of thicknesses this structure is energetically more advantageous than the plate structure originally proposed by Kittel. The proposed model explains the transition from the Kittel structure to the homogeneously magnetized film.The authors thank Z. Málek C. Sc. and V. Janovec C. Sc. for valuable remarks and V. Kamberský for help in the numerical calculations.     

Periodic geometrically nonlinear free vibrations of circular plates

The geometrically nonlinear free vibrations of thin isotropic circular plates are investigated using a multi-degree-of-freedom model, which is based on thin plate theory and on Von Kármán's nonlinear strain-displacement relations. The middle plane in-plane displacements are included in the formulation and the common axisymmetry restriction is not imposed. The equations of motion are derived by the principle of the virtual work and an approximated model is achieved by assuming that the in-plane and transverse displacement fields are given by weighted series of spatial functions. These spatial functions are based on hierarchical sets of polynomials, which have been successfully used in p-version finite elements for beams and rectangular plates, and on trigonometric functions. Employing the harmonic balance method, the differential equations of motion are converted into a nonlinear algebraic form and then solved by a continuation method. Convergence with the number of shape functions and of harmonics is analysed. The numerical results obtained are presented and compared with available published results; it is shown that the hierarchical sets of functions provide good results with a small number of degrees of freedom. Internal resonances are found and the ensuing multimodal oscillations are described.     

Geometrically nonlinear free vibration of shear deformable piezoelectric carbon nanotube/fiber/polymer multiscale laminated composite plates

The nonlinear free vibration of carbon nanotubes/fiber/polymer composite (CNTFPC) multi-scale plates with surface-bonded piezoelectric actuators is studied in this paper. The governing equations of the piezoelectric nanotubes/fiber/polymer multiscale laminated composite plates are derived based on first-order shear deformation plate theory (FSDT) and von Kármán geometrical nonlinearity. Halpin–Tsai equations and fiber micromechanics are used in hierarchy to predict the bulk material properties of the multiscale composite. The carbon nanotubes are assumed to be uniformly distributed and randomly oriented through the epoxy resin matrix. A perturbation scheme of multiple time scales is employed to determine the nonlinear vibration response and the nonlinear natural frequencies of the plates with immovable simply supported boundary conditions. The effects of the applied constant voltage, plate geometry, volume fraction of fibers and weight percentage of single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs) on the linear and nonlinear natural frequencies of the piezoelectric nanotubes/fiber/polymer multiscale composite plate are investigated through a detailed parametric study.     

A Tentative Expression of the Károlyházy Uncertainty of the Space-Time Structure Through Vacuum Spreads in Quantum Gravity

In the existing expositions of the Károlyházy model, quantum mechanical uncertainties are mimicked by classical spreads. It is shown how to express those uncertainties through entities of the future unified theory of general relativity and quantum theory.     

Discontinuity of the dynamic critical exponents ofO(n)-symmetric systems

Then-component model of Sasvári, Schwabl, and Szépfalusy is analyzed by means of renormalized field theory to two-loop order. Singular two-loop contributions have a significant effect on the nature of the stability boundary of the dynamic scaling region in then—d plane. Along this boundary the dynamic critical exponents exhibit unexpected discontinuities.     

Power flow model of flexural waves in finite orthotropic plates

In this paper, an approximate power flow model is developed for the analysis of the flexural waves in finite orthotropic plate transversely vibrating in the medium-to-high frequency ranges. The derived energy equation for the model is expressed with time- and locally space-averaged farfield wave energy density. It could be the more general form than that of the conventional power flow analysis model seen in the isotropic plate. With the derived model, dynamic characteristics varying with the direction can be expressed. To verify the validity and accuracy of the model, numerical analyses are performed for the case where a finite rectangular plate is excited by a harmonic point force, and the calculated results expressed with the energy levels are compared with classical modal solutions by changing the frequency and the damping loss factor of the plate. The dominant power transmission paths in the plate are also predicted from the distribution of the approximate intensity fields.     

Nonlocal vibration of coupled DLGS systems embedded on Visco-Pasternak foundation

In this paper, vibration analysis of the coupled system of double-layered graphene sheets (CS-DLGSs) embedded in a Visco-Pasternak foundation is carried out using the nonlocal elasticity theory of orthotropic plate. The two DLGSs are coupled by an enclosing viscoelastic medium which is simulated as a Visco-Pasternak foundation. Considering the Von Kármán nonlinear strain-displacement-relations, the motion equations are derived using the Hamilton's principle. Differential quadrature method (DQM) is applied to obtain the frequency ratio for various boundary conditions. The detailed parametric study is conducted, focusing on the combined effects of the nonlocal parameter, aspect ratio, graphene sheet's size, boundary conditions and the elastic and viscoelastic medium coefficients on the frequency ratio of CS-DLGSs. In this coupled system, two case of DLGSs vibration are investigated and compared with each other: (1) In-phase vibration (2) Out-of-phase vibration. The results indicate that the frequency ratio of the CS-DLGSs is more than the single-layered graphene sheet (SLGS). The results are in good agreement with the previous researches.     

Dynamic stress concentration in a ribbed plate using the acoustical wave propagator technique

This paper presents an explicit acoustical wave propagator technique to investigate the flexural wave scattering and dynamic stress concentration in a ribbed plate based on Mindlin plate theory. A scheme combining Chebyshev polynomial expansion and fast Fourier transformation is introduced to implement the operation of the acoustical wave propagator. The exact analytical solutions are also presented to demonstrate the validity of the present technique. The wave propagation patterns are used to investigate the effect of rib on the flexural wave scattering and distribution of dynamic stress concentration. Dynamic stress concentration factors around the discontinuities are examined in detail.     

Stability and vibration of isotropic,orthotropic and laminated plates according to a higher-order shear deformation theory

A higher-order shear deformation theory is used to determine the natural frequencies and buckling loads of elastic plates. The theory accounts for parabolic distribution of the transverse shear strains through the thickness of the plate and rotary inertia. Exact solutions of simply supported plates are obtained and the results are compared with the exact solutions of three-dimensional elasticity theory, the first-order shear deformation theory, and the classical plate theory. The present theory predicts the frequencies and buckling loads more accurately when compared to the first-order and classical plate theories.     

Uniform product formulae with application to the Feynman—Nelson integral for open systems

The product formula for perturbations of propagators by Faris is generalized: we show that one can use arbitrary partitions of a given interval and perform the limit uniformly with respect to the partition norm. The results are applied to express solutions of the Schrödinger equation, in particular for some complex and time-dependent potentials, by means of Nelson-type path integrals.Nuclear Centre, Charles University, Areál Troja, Povltavská ul. 18000, Prague Czechoslovakia     

A phenomenological theory of the thermal conductivity of thin films

A phenomenological theory is presented for the lattice thermal conductivity of thin plates (or films) in a contact with another material (e.g. a thin plate covered by an oxide film, a thin film evaporated on a substrate, etc.). In such heterogeneous systems, it is necessary to consider — besides reflections — the exchange of phonons between the phases of which the system is composed. Boundary conditions are formulated for the phonon distribution functions. It is shown that the surrounding material (or the substrate) can exert considerable influence upon the effective thermal conductivity of the plate (film).In conclusion, the authors express their sincere gratitude to . Bárta for valuable discussions on the subject of this paper.     

Decagonal quasicrystal plate with elliptic holes subjected to out-of-plane bending moments

In the present paper, we consider only the ideal elastic behavior, neglecting the dissipation associated with the atomic rearrangements. Under these conditions, the decagonal quasicrystal plate bending problems have been discussed. The Stroh-like formalism for the bending theory of decagonal quasicrystal plate is developed. The analytical solutions for problems of decagonal quasicrystal plate with elliptic hole subjected to out-of-plane bending moments are obtained directly by using the forms. The resultant bending moments around the hole boundaries are also given explicitly. When the phonon–phason coupling is absent, the results reduce to the corresponding solutions for the isotropic elastic plates.     

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.     

Use of lagrange multipliers with polynomial series for dynamic analysis of constrained plates part I: Polynomial series

A theory for prediction of the dynamic response of a constrained plate is presented here. The boundaries of the plate may be partially fixed, its dynamic loading is due to elastically mounted vibrating machines and its constraints include beam-like stiffeners. The theory yields the eigenvalues and modal shapes of the plate and stiffeners which comprise the system. The solution, given in Part I, is based on Galerkin's method combined with use of special polynomial series presented by Kantorovich and Krylov. These eigenvalues are used in Part II [1] for response analysis of the complete system and the eigenvalues of the complete system will be obtained by the application of Lagrange equations and multipliers. The various coefficients used in the process are presented in the Appendices to the work. Comparisons with published results show good agreement.     

Nonclassical states generated via four-wave mixing

The nonclassical properties of quantum fields generated by four- or three-wave mixing are investigated on the basis of assumptions of two-photon coherent (squeezed) input states and the approximation of strong pumping. The influence of losses is shown supposing the model of mutually independent phonon reservoirs. The possibility of producing squeezed and sub-Poisson light outputs is demonstrated in single or superposed modes using computer solutions.The author wishes to acknowledge helpful discussions with Dr. J. Peina and Dr. M. Kárská.     

Size-dependent resonance and buckling behavior of nanoplates with high-order surface stress effects

This work presents a theoretical study of the resonance frequency and buckling load of nanoplates with high-order surface stress model. A classical thin plate theory based on Kirchhoff–Love assumption is implemented with surface effects. Circular and rectangular nanoplates with simply supported end conditions are exemplified. The size-dependent solutions are compared with the simplified solutions based on simple surface stress model, and also on the classical theory of elasticity. We aim to explore the scope of applicability so that the modified continuum mechanics model could serve as a refined approach in the prediction of mechanical behavior of nanoplates.