ASYMMETRIC NON-LINEAR FORCED VIBRATIONS OF FREE-EDGE CIRCULAR PLATES. PART 1: THEORY

In this article, a detailed study of the forced asymmetric non-linear vibrations of circular plates with a free edge is presented. The dynamic analogue of the von Kàrmàn equations is used to establish the governing equations. The plate displacement at a given point is expanded on the linear natural modes. The forcing is harmonic, with a frequency close to the natural frequency ω_kn of one asymmetric mode of the plate. Thus, the vibration is governed by the two degenerated modes corresponding to ω_kn, which are one-to-one internally resonant. An approximate analytical solution, using the method of multiple scales, is presented. Attention is focused on the case where one configuration which is not directly excited by the load gets energy through non-linear coupling with the other configuration. Slight imperfections of the plate are taken into account. Experimental validations are presented in the second part of this paper.     

Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws

In this article, we propose a new class of finite volume schemes of arbitrary accuracy in space and time for systems of hyperbolic balance laws with stiff source terms. The new class of schemes is based on a three stage procedure. First a high-order WENO reconstruction procedure is applied to the cell averages at the current time level. Second, the temporal evolution of the reconstruction polynomials is computed locally inside each cell using the governing equations. In the original ENO scheme of Harten et al. and in the ADER schemes of Titarev and Toro, this time evolution is achieved via a Taylor series expansion where the time derivatives are computed by repeated differentiation of the governing PDE with respect to space and time, i.e. by applying the so-called Cauchy–Kovalewski procedure. However, this approach is not able to handle stiff source terms. Therefore, we present a new strategy that only replaces the Cauchy–Kovalewski procedure compared to the previously mentioned schemes. For the time-evolution part of the algorithm, we introduce a local space–time discontinuous Galerkin (DG) finite element scheme that is able to handle also stiff source terms. This step is the only part of the algorithm which is locally implicit. The third and last step of the proposed ADER finite volume schemes consists of the standard explicit space–time integration over each control volume, using the local space–time DG solutions at the Gaussian integration points for the intercell fluxes and for the space–time integral over the source term. We will show numerical convergence studies for nonlinear systems in one space dimension with both non-stiff and with very stiff source terms up to sixth order of accuracy in space and time. The application of the new method to a large set of different test cases is shown, in particular the stiff scalar model problem of LeVeque and Yee [R.J. LeVeque, H.C. Yee, A study of numerical methods for hyperbolic conservation laws with stiff source terms, Journal of Computational Physics 86 (1) (1990) 187–210], the relaxation system of Jin and Xin [S. Jin, Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Communications on Pure and Applied Mathematics 48 (1995) 235–277] and the full compressible Euler equations with stiff friction source terms.     

INSTABILITY OF VIBRATION OF A MOVING-TRAIN-AND-RAIL COUPLING SYSTEM

This paper presents the derivation of the governing equations for the stability of vibration of an integrated system comprising a moving train and the railway track. The train consists of a convoy of articulated two-axle wagons. The equations are applicable to any arbitrary number of axles at arbitrary spacing. Each axle is modelled as a mass-spring-damper vibration unit. The railway track is an infinitely long Euler beam subjected to an axial compressive force and rests on a visco-elastic foundation. The governing equations for the integrated system are coupled differential equations, which can be transformed to algebraic equations by Fourier and Laplace transforms. Subsequent inverse Fourier transform and contour integration yield the instability equation. Critical parameter is identified. It follows by parametric studies on the instability of vibration due to different train configurations. Illustrative examples for trains having up to 20 wagons or 40 axles are given.     

On the integration of some classes of (<Emphasis Type="Italic">n</Emphasis>+1)-dimensional nonlinear equations of mathematical physics

This paper presents a method for construction of the families of particular solutions to some new classes of (n+1)-dimensional nonlinear partial differential equations (PDE). The method is based on the specific link between algebraic matrix equations and PDEs. Admittable solutions depend on arbitrary functions of n variables. Examples of deformed Burgers-type equations are given.     

Open and Closed World Models in Kaluza-Klein-Theory with Variables G and Λ

The field equation of higher dimensions theory, have been applied in the area of cosmology. The resulting differential equations are solved for open and closed. We derive a relation between the Einstein constant G(t) and the cosmological constant Λ(t) from the conservation law T _μ ^ν _;ν =0. We give a specific form of Λ(t) to solve the non-linear differential equations. Some cosmological parameters are calculated and some relevant cosmological problems are discussed.     

Families of solutions to the generalized Ginzburg-Landau equation and structural transitions between them

Solutions to the generalized Ginzburg-Landau equations for superconductors are obtained for a Ginzburg-Landau parameter κ close to unity. The families of solutions with arbitrary number n of flux quanta in a unit cell are analyzed. It is shown that under certain conditions, a cascade of phase transitions between different structures in a magnetic field appears near T _c . Algebraic equations are derived for determining the boundaries of coexistence of different phases on the {T, H ₀} plane.     

IMPROVEMENT OF THE SEMI-ANALYTICAL METHOD, FOR DETERMINING THE GEOMETRICALLY NON-LINEAR RESPONSE OF THIN STRAIGHT STRUCTURES. PART I: APPLICATION TO CLAMPED-CLAMPED AND SIMPLY SUPPORTED-CLAMPED BEAMS

In a previous series of papers (Benamar 1990 Ph.D. Thesis, University of Southampton; Benamaret al. 1991 Journal of Sound and Vibration149, 179-195;164, 399-424 [1-3]) a general model based on Hamilton's principle and spectral analysis has been developed for non-linear free vibrations occurring at large displacement amplitudes of fully clamped beams and rectangular homogeneous and composite plates. The results obtained with this model corresponding to the first non-linear mode shape of a clamped-clamped (CC) beam and to the first non-linear mode shape of a CC plate are in good agreement with those obtained in previous experimental studies (Benamaret al. 1991 Journal of Sound and Vibration 149, 179-195;164, 399-424 [2, 3]). More recently, this model has been re-derived (Azar et al. 1999 Journal of Sound and Vibration224, 377-395; submitted [4, 5]) using spectral analysis, Lagrange's equations and the harmonic balance method, and applied to obtain the non-linear steady state forced periodic response of simply supported (SS), CC, and simply supported-clamped (SSC) beams. The practical application of this approach to engineering problems necessitates the use of appropriate software in each case or use of published tables of data, obtained from numerical solution of the non-linear algebraic system, corresponding to each problem. The present work was an attempt to develop a more practical simple “multi-mode theory” based on the linearization of the non-linear algebraic equations, written on the modal basis, in the neighbourhood of each resonance. The purpose was to derive simple formulae, which are easy to use, for engineering purposes. In this paper, two models are proposed. The first is concerned with displacement amplitudes of vibrationW_max /H, obtained at the beam centre, up to about 0·7 times the beam thickness and the second may be used for higher amplitudes W_max/H up to about 1·5 times the beam thickness. This new approach has been successfully used in the free vibration case to the first, second and third non-linear modes shapes of CC beams and to the first non-linear mode shape of a CSS beam. It has also been applied to obtain the non-linear steady state periodic forced response of CC and CSS beams, excited harmonically with concentrated and distributed forces.     

TRANSIENT RESPONSE ANALYSIS OF LARGE-SCALE ROTOR-BEARING SYSTEM WITH STRONG NON-LINEAR ELEMENTS BY A TRANSFER MATRIX-NEWMARK FORMULATION ITEGRATION METHOD

This paper extends the transfer matrix technique (TMT) to the transient response analysis of a large complex non-linear rotor-bearing system by a transfer matrix-Newmark formulation itegration method. Firstly, the transfer matrix is obtained via the Newmark formulation. Secondly, the deflections and velocities at the stations, containing non-linear element, are determined by iteration. Finally, the deflections, velocities and accelerations of all stations are computed by TMT and the Newmark formulation consistent with the boundary conditions. In order to eliminate the numerical instability of TMT, the transfer vector {f^T ? ë^T}^T is used, instead of the traditional one {f^T ? ë^T}^T. Owing to the advantages of TMT and the Newmark formulation, this method can be applied to calculate the transient response of a large-scale rotor-bearing system with strong non-linear elements, and to analyze its stability. Two illustration examples are given, and the results agree well with those by Runge-Kutta method, and by modal synthetic method.     

VORTEX-INDUCED VIBRATIONS OF TWO SIDE-BY-SIDE EULER-BERNOULLI BEAMS

Vortex-induced vibration of two side-by-side elastic beams in a cross flow is numerically studied. The two beams are identical and fixed at both ends. In the numerical approach, the Euler-Bernoulli beam theory is used to model the beam vibration, and the laminar Navier-Stokes equations are solved to give the flow field. The flow equations are resolved using a finite element method and the flow-induced forces are calculated at every time step in order to correctly reflect the fluid-beam interaction. The beam response is calculated using the modal analysis method. Free vibrations of the two beams with three pitch ratios, T/D=1·13, 1·7 and 3·0, where T is the gap between the centers of the two beams and D is the beam diameter, are simulated at Re=800. Results obtained are compared with experimental measurements and other numerical results obtained assuming a two-degree-of-freedom (2-d.o.f.) model. The agreement is good in general. Correlation analysis is carried out, showing that the phase relation is different for differentT /D. The short-time Fourier transform (STFT) method is used to carry out the spectral analysis, along with the conventional auto-regressive moving averaging (ARMA) method for comparison. The STFT analysis shows that the time evolution of fluid force and beam vibration for T/D=1·13 and 3·0 are stationary. For theseT /D ratios, the STFT results are consistent with the ARMA results, but give a clearer picture of the higher order harmonics. For T/D=1·7, the time evolution is non-stationary. The STFT analysis shows that there are three types of frequency spectrum for the fluid force, with one, two, and three dominant frequencies respectively. The spectra intermittently change in a random way during the evolution. The ARMA results, though consistent with previous experiments, can only reveal a particular feature of the three different types of spectrum. This suggests that the STFT method is more appropriate to analyze the spectra of non-stationary time series in the study of flow-induced vibrations.     

Non-linear vibration analysis of multilayer beams by incremental finite elements,Part I: Theory and numerical formulation

An incremental variational equation for non-linear motions of multilayer beams composed of n stiff layers and (n ? 1) soft cores is derived from the dynamic virtual work equation by an appropriate integration procedure. The kinematical hypotheses of Euler-Bernoulli and Timoshenko beam theories are used to describe the displacement fields of the stiff layers and cores respectively. An efficient solution procedure of incremental harmonic balance method type, with use of finite elements, is developed. To demonstrate its capability, some problems in free non-linear vibrations of multilayer beams are treated by using the procedure. Results are compared with those available in the literature. The effects of damping are also included in this investigation but are described in Part II [1] of this paper in which a number of undamped and damped forced non-linear vibration problems are studied. Results in the form of tables and plots are also presented and comparisons are made with those available in the literature.     

Exact solution of the asymmetric Mindlin's plate equations applied to a disk

An exact solution is presented for the static and dynamic asymmetric response of a disk governed by Mindlin's plate equations forced by a pressure that varies radially as r^m. The static solution agrees with a modal solution adopting the dynamic Mindlin's plate equations in the limit when excitation frequency vanishes. This solution is useful in sizing magnitude and shape of surface asymmetries on a disk from pressure loading with slight eccentricity and circumferential non-uniformity.     

Periodic solutions of some infinite-dimensional Hamiltonian systems associated with non-linear partial difference equations I

We establish existence of a dense set of non-linear eigenvalues,E, and exponentially localized eigenfunctions,u _E , for some non-linear Schrödinger equations of the form $$Eu_E (x) = [( - \Delta + V(x))u_E ](x) + \lambda u_E (x)^3 ,$$ bifurcating off solutions of the linear equation with λ=0. The pointsx range over a lattice, ? ^d ,d=1,2,3,..., Δ is the finite difference Laplacian, andV(x) is a random potential. Such equations arise in localization theory and plasma physics. Our analysis is complicated by the circumstance that the linear operator ?Δ+V(x) has dense point spectrum near the edges of its spectrum which leads to small divisor problems. We solve these problems by developing some novel bifurcation techniques. Our methods extend to non-linear wave equations with random coefficients.     

Inverse scattering method and vector higher order non-linear Schrödinger equation

A generalized inverse scattering method has been developed for arbitrary n-dimensional Lax equations. Subsequently, the method has been used to obtain N-soliton solutions of a vector higher order non-linear Schrödinger equation, proposed by us. It has been shown that under a suitable reduction, the vector higher order non-linear Schrödinger equation reduces to the higher order non-linear Schrödinger equation. An infinite number of conserved quantities have been obtained by solving a set of coupled Riccati equations. Gauge equivalence is shown between the vector higher order non-linear Schrödinger equation and the generalized Landau–Lifshitz equation and the Lax pair for the latter equation has also been constructed in terms of the spin field, establishing direct integrability of the spin system.     

The Faddeev equations for the Heisenberg ferromagnet

The Faddeev equations for the three-magnon T-matrix of the Heisenberg ferromagnet with nearest neighbour interactions are derived for the cubic lattice in arbitrary dimensions. The extreme case of spin $\text{1}\text{2}$ is considered and the kinematical restriction, that only one spin deviation per site is possible, has been taken into account rigorously. Hence the T-matrix is unitary and suited for the study of bound state as well as scattering state properties. The analytic solution of the homogeneous Faddeev equations in one dimension is given.     

Semi-Lagrangian multistep exponential integrators for index 2 differential–algebraic systems

Implicit-explicit (IMEX) multistep methods are very useful for the time discretization of convection diffusion PDE problems such as the Burgers equations and the incompressible Navier–Stokes equations. In the latter as well as in PDE models of plasma physics and of electromechanical systems, semi-discretization in space gives rise to differential–algebraic (DAE) system of equations often of index higher than 1. In this paper we propose a new class of exponential integrators for index 2 DAEs arising from the semi-discretization of PDEs with a dominating and typically nonlinear convection term. This class of problems includes the incompressible Navier–Stokes equations. The integration methods are based on the backward differentiation formulae (BDF) and they can be applied without modifications in the semi-Lagrangian integration of convection diffusion problems. The approach gives improved performance at low viscosity regimes.     

Analyses of double exponential decays using modulating functions

Use of modulating functions (t^α(T - t)^β), with numerical integration over only the range of the decay (√^T₀), yields estimates of the decay rates directly. The effects of the choice of α and β, of noise and of truncation on the bias found in the extracted parameters is examined by analyzing simulated decays. It is recommended that such analyses be simply preludes to iterative, non-linear least squares analyses, which require estimated decay times.     

SUPER AND COMBINATORIAL HARMONIC RESPONSE OF FLEXIBLE ELASTIC CABLES WITH SMALL SAG

The paper deals with the analysis of cables in stayed bridges and TV-towers, where the excitation is caused by harmonically varying in-plane motions of the upper support point with the amplitude U. Such cables are characterized by a sag-to-chord-length ratio below &0uml;02, which means that the lowest circular eigenfrequencies for in-plane and out-of-plane eigenvibrations, ω₁and ω₂, are closely separated. The dynamic analysis is performed by a two-degree-of-freedom modal decomposition in the lowest in-plane and out-of-plane eigenmodes. Modal parameters are evaluated based on the eigenmodes for the parabolic approximation to the equilibrium suspension. Superharmonic components of the ordern , supported by the parametric terms of the excitation and the non-linear coupling terms, are registered in the response for circular frequency ω?ω₁/n. At moderate U, the cable response takes place entirely in the static equilibrium plane. At larger amplitudes the in-plane response becomes unstable and a coupled whirling superharmonic component occurs. In the paper a first order perturbation solution to the superharmonic response is performed based on the averaging method. For ω?(m/n)ω₁, m<n, the geometrical non-linear restoring forces gives rise to a substantial combinatorial harmonic component with the circular frequency (n/m)ω. Both entirely in-plane and coupled in-plane and out-of-plane responses occur. Based on an initial frequency analysis of the response, an analytical model for these vibrations is formulated with emphasis on superharmonics of the order n=3 and combinatorial harmonics of the order (n, m)=(3,2). All analytical solutions have been verified by direct numerical integration of the modal equations of motion.     

The interior field of a magnetized Einstein-Maxwell object

Using the harmonic map ansatz, we reduce the axisymmetric, static Einstein-Maxwell equations coupled with a magnetized perfect fluid to a set of Poisson-like equations. We were able to integrate the Poisson equations in terms of an arbitrary function M = M(ϱ, ζ) and some integration constants. The thermodynamic equation restricts the solutions to only some state equations, but in some cases when the solution exists, the interior solution can be matched with the corresponding exterior one.     

Investigation of glassy behaviour of flux grown Pb(Zn1/3Nb2/3)0.91Ti0.09O3 crystal

The glassy behaviour of flux grown single crystal of relaxor based lead zinc niobium titanate near its morphotropic phase boundary has been investigated. The frequency dependence of temperature at dielectric maxima (T′_m), which is governed by the production of polar nano-region at higher temperature, was analysed using different glass models and Power law. Various parameters like activation energy for relaxation, freezing temperature, relaxation frequency, etc. were determined after non-linear curve fitting. Further, the temperature dependence of dielectric constant, at temperatures much higher and lower than T_m, was analysed using two exponential functions to explain the degree of relaxation, production and freezing of polar regions, etc. Various other associated parameters are calculated by non-linear curve fitting and their significance has been explained.     

Robust and accurate filtered spherical harmonics expansions for radiative transfer

We present a novel application of filters to the spherical harmonics (P_N) expansion for radiative transfer problems in the high-energy-density regime. The filter we use is based on non-oscillatory spherical splines and a filter strength chosen to (i) preserve the equilibrium diffusion limit and (ii) vanish as the expansion order tends to infinity. Our implementation is based on modified equations that are derived by applying the filter after every time step in a simple first-order time integration scheme. The method is readily applied to existing codes that solve the P_N equations. Numerical results demonstrate that the solution to the filtered P_N equations are (i) more robust and less oscillatory than standard P_N solutions and (ii) more accurate than discrete ordinates solutions of comparable order. In particular, the filtered P₇ solution demonstrates comparable accuracy to an implicit Monte Carlo solution for a benchmark hohlraum problem in 2D Cartesian geometry.