Improvement of the semi-analytical method, based on Hamilton's principle and spectral analysis, for determination of the geometrically non-linear response of thin straight structures. Part III: steady state periodic forced response of rectangular plates

In Parts I and II of this series of papers, a 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, has been developed for beams and fully clamped rectangular plates.¹ Simple explicit formulae have been derived, which allowed, via the so-called first formulation, direct calculation of the basic function contributions to the first three non-linear mode shapes of clamped-clamped and clamped-simply supported beams, and the two first non-linear mode shapes of FCRP. Also, in Part I of this series of papers, this approach has been successively extended, in order to determine the amplitude-dependent deflection shapes associated with the non-linear steady state periodic forced response² of clamped-clamped beams, excited by a concentrated or a distributed harmonic force in the neighbourhood of the first resonance.This new approach has been applied in the present work to obtain the NLSSPFR formulation for FCRP, SSRP, and CCCSSRP, leading in each case to a non-linear system of coupled differential equations, which may be considered as a multi-dimensional form of the well-known Duffing equation. The single-mode assumption, and the harmonic balance method, have been used for both harmonic concentrated and distributed excitation forces, leading to one-dimensional non-linear frequency response functions of the plates considered. Comparisons have been made between the curves based on these functions, and the results available in the literature, showing a reasonable agreement, for finite but relatively small vibration amplitudes. A more accurate estimation of the FCRP non-linear frequency response functions has been obtained by the extension of the improved version of the semi-analytical model developed in Part I for the NLSSPFR of beams, to the case of FCRP, leading to explicit analytical expressions for the “multi-dimensional non-linear frequency response function”, depending on the forcing level, and the amplitude of the response induced in the range considered for the excitation frequency.     

Non-linear vibrations of three-layer beams with viscoelastic cores,II: Experiment

Experimental results are presented for large amplitude, forced motion of damped, three-layer beams. The beams are constructed with a viscoelastic material constrained between stiff, elastic, outer layers. The sandwich beam is axially restrained; therefore large amplitude displacements cause non-linear response. When the beam is forced at one-half of the lateral vibration resonant frequency, superharmonic response occurs. The experiment is briefly described and frequency response characteristics, spatial shapes and a measure of superharmonic response are presented. The results are compared with predictions from a previously developed theory.     

A SEMI-ANALYTICAL APPROACH TO THE NON-LINEAR DYNAMIC RESPONSE PROBLEM OF BEAMS AT LARGE VIBRATION AMPLITUDES, PART II: MULTIMODE APPROACH TO THE STEADY STATE FORCED PERIODIC RESPONSE

The semi-analytical approach to the non-linear dynamic response of beams based on multimode analysis has been presented in Part I of this series of papers (Azrar et al., 1999 Journal of Sound and Vibration224, 183-207 [1]). The mathematical formulation of the problem and single mode analysis have been studied. The objective of this paper is to take advantage of applying this semi-analytical approach to the large amplitude forced vibrations of beams. Various types of excitation forces such as harmonic distributed and concentrated loads are considered. The governing equation of motion is obtained and can be considered as a multi-dimensional form of the Duffing equation. Using the harmonic balance method, the equation of motion is converted into non-linear algebraic form. Techniques of solution based on iterative-incremental procedures are presented. The non-linear frequency and the non-linear modes are determined at large amplitudes of vibration. The basic function contribution coefficients to the displacement response for various beam boundary conditions are calculated. The percentage of participation for each mode in the response is presented in order to appraise the relation to higher modes contributing to the solution. Also, the percentage contributions of the higher modes to the bending moment near to the clamps are given, in order to determine accurately the error introduced in the non-linear bending stress estimated by different approximations. Solutions obtained in the jump phenomena region have been determined by a careful selection of the initial iteration at each frequency. The non-linear deflection shapes in various regions of the solution, the corresponding axial force ratios and the bending moments are presented in order to follow the behaviour of the beam at large vibration amplitudes. The numerical results obtained here for the non-linear forced response are compared with those from the linear theory, with available non-linear results, based on various approaches, and with the single mode analysis.     

Bright and dark spatial solitons in non-Kerr media

An overview of the theory of self-guided optical beams, spatial optical solitons supported by non-Kerr non-linearities, is presented. This includes bright and dark solitons in optical media with intensity-dependent non-linear response as well as two-component solitary waves supported by parametric wave mixing in quadratic or cubic media. The properties of non-linear spatially localized waves are discussed for qualitatively different types of soliton bearing non-integrable non-linear models, including the scalar model described by a generalized non-linear Schrödinger equation and the models of the second- and third-harmonic generation. Special attention is paid to the recent advances of the theory of soliton stability and soliton internal modes.     

Effects of non-linear damping on random response of beams to acoustic loading

Effects of both non-linear damping and large deflection are included in a theoretical analysis in an attempt to explain experimental phenomena observed for aircraft panels excited at high sound pressure levels: that is, the broadening of the strain response peak and the increasing of the modal frequency. Two non-linear damping models are considered in the analysis, with a single-mode approach. The root-mean-square (RMS) maximum deflection, the RMS maximum strain, the spectral density function of maximum strain and the equivalent linear frequency for simply supported and clamped beams are obtained. It is demonstrated that non-linear damping contributes significantly to the broadening of the strain response peak and to the RMS maximum deflection and strain, and frequency.     

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.     

The role of median surface curvature in large amplitude flexural vibrations of thin shells

General conclusions regarding the non-linear vibration of structural components like curved beams, rings and thin shells are derived from the study of two specific examples, the circular ring and shallow spherical shell. It is shown that whereas the non-linear behaviour of flat plates and straight bars is generally of a hardening type, the behaviour of thin structural elements that have a finite curvature of the undeformed median surface in one or both principal axis directions may be of the hardening or softening type, depending on the structural parameters as well as on whether the shell is open or closed. It is seen that with careful judgment in the use of mode shapes of one or more terms, the resulting modal equations help one to appreciate much better the physics of the problem, whereas sophisticated mathematical models tend to obscure this.     

二次谐波转换对激光脉宽影响的实验研究

 实验研究了KDP晶体二次谐波转换非线性效应对脉宽测量精度的影响。结果表明：在小信号近似下,长度为1～3cm的KDPⅡ类晶体,对于近高斯脉冲,其基频脉宽与倍频脉宽存在21/2的关系;对于非近高斯脉冲,基频脉冲半宽与倍频脉冲波形的1/4高宽相等;在输入功率密度和能量转换效率较高时,基频与倍频脉宽无固定关系。     

Dynamic non-linear response of beams subjected to impact loads

A non-linear theory is presented for plane deformation of beams which allows for longitudinal stretching as well as for cross-sectional stretching and shearing. The exact strain measures for this theory are also deduced. The longitudinal and flexural motions are coupled in the theory. If the cross section is constrained from stretching, the resulting theory may be classified as a non-linear Timoshenko beam theory. The equations of the latter theory are used to study the motion of beams under impact loads.     

Lattice bending in x-ray interferometers

This paper investigates both theoretically and experimentally the effect of lattice bending on the output signals of a two-crystal x-ray interferometer of the Laue LLL type. The cross section intensity of the outgoing beams is modulated by the moiré effect produced by the overlapping of the analysing lattice on the x-ray standing field in front of it. Since the intensities of the transmitted and diffracted beams are integrated, the moiré pattern causes loss of visibility in the x-ray fringes and a non-linear phase shift, which depends on the pitch alignment of the analysing crystal with respect to the fixed crystal. The analysis of this phase shift allows the lattice curvature to be estimated.     

LARGEST LYAPUNOV EXPONENT AND ALMOST CERTAIN STABILITY ANALYSIS OF SLENDER BEAMS UNDER A LARGE LINEAR MOTION OF BASEMENT SUBJECT TO NARROWBAND PARAMETRIC EXCITATION

The first order approximate solutions of a set of non-liner differential equations, which is established by using Kane's method and governs the planar motion of beams under a large linear motion of basement, are systematically derived via the method of multiple scales. The non-linear dynamic behaviors of a simply supported beam subject to narrowband random parametric excitation, in which either the principal parametric resonance of its first mode or a combination parametric resonance of the additive type of its first two modes with or without 3:1 internal resonance between the first two modes is taken into consideration, are analyzed in detail. The largest Lyapunov exponent is numerically obtained to determine the almost certain stability or instability of the trivial response of the system and the validity of the stability is verified by direct numerical integration of the equation of motion of the system.     

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.     

Non-linear free vibrations of stepped thickness beams

The non-linear free vibrations of stepped thickness beams are analyzed by assuming sinusoidal responses and using the transfer matrix method. The numerical results for clamped and simply supported, one-stepped thickness beams with rectangular cross-section are presented and the effects of the beam geometry on the non-linear vibration characteristics are discussed. The results are also compared with those obtained by a Galerkin method in which the linear mode function of the beam is used. The use of a Galerkin method seems to considerably overestimate the non-linearity of the stepped thickness beam in certain cases.     

From topological charge information to sets of solitons in quadratic non-linear media

We describe the principle of operation of a new class of optical devices operating in quadratic non-linear media that mix wave front topological charge dislocations nested in focused light beams and produce certain patterns of bright spatial solitons. Central to the device behaviour is the orbital angular momentum of the light beams.     

Second-harmonic generation by intense beams containing phase dislocations:self-breaking into sets of solitons

Second-harmonic generation of light in bulk quadratic non-linear media with intense input beams containing phase dislocations is studied numerically under conditions for TypeI phase-matching. We investigate how, above a threshold light intensity, the input beams self-split along the azimuthal direction into a pattern of separate beams which then form a set of spatial solitons. The mechanism of such a behaviour is the azimuthal modulational instability of the ring-shaped, mutually trapped fundamental and second-harmonic beams containing the phase dislocations. This revised version was published online in November 2006 with corrections to the Cover Date.     

Non-linear free torsional vibrations of thin-walled beams with bisymmetric cross-section

A finite element method for studying non-linear free torsional vibrations of thin-walled beams with bisymmetric open cross-section is presented. The non-linearity of the problem arises from axial loads generated at moderately large amplitude torsional vibrations due to immovability of end supports. The derivation of the fundamental differential equation of the problem is based on the classical assumption of a thin-walled beam with a non-deformable cross-section. The non-linear eigenvalue problem is solved iteratively by series of linear eigenvalue problems until the required accuracy is obtained. Non-linear frequencies, fundamental mode shapes and axial loads computed for various amplitude of torsional vibrations of thin-walled I beams are included.     

椭圆响应强非局域非线性介质中的二维异步分数傅里叶变换及光束传输特性

研究了椭圆响应强非局域非线性介质中的光束传输问题. 结果表明:任意光束在这类介质中传输时均遵守二维异步分数傅里叶变换的传输规律. 基于二维异步分数傅里叶变换这一数学工具, 可很方便地对光束的传输进行解析求解并分析其性质. 利用二维异步分数傅里叶变换的性质, 讨论了一般光束的传输性质; 分析了孤子和二维异步呼吸子的形成条件; 得出了孤子/呼吸子的相互作用规律. 关键词： 椭圆响应 强非局域非线性 孤子 呼吸子     

IMPROVEMENT OF THE SEMI-ANALYTICAL METHOD, FOR DETERMINING THE GEOMETRICALLY NON-LINEAR RESPONSE OF THIN STRAIGHT STRUCTURES: PART II—FIRST AND SECOND NON-LINEAR MODE SHAPES OF FULLY CLAMPED RECTANGULAR PLATES

In a previous series of papers, a semi-analytical model based on Hamilton's principle and spectral analysis has been developed for geometrically non-linear free vibrations occurring at large displacement amplitudes of clamped-clamped beams and fully clamped rectangular homogeneous and composite plates. In Part I of this series of papers, concerned with geometrically non-linear free and forced vibrations of various beams, a practical simple “multi-mode theory”, based on the linearization of the non-linear algebraic equations, written in the modal basis, in the neighbourhood of each resonance has been developed. Simple explicit formulae, ready and easy to use for analytical or engineering purposes have been derived, which allows direct calculation of the basic function contributions to the first three non-linear mode shapes of the beams considered. Also, various possible truncations of the series expansion defining the first non-linear mode shape have been considered and compared with the complete solution, which showed that an increasing number of basic functions has to be used, corresponding to increasingly sized intervals of vibration amplitudes; starting from use of only one function, i.e., the first linear mode shape, corresponding to very small amplitudes, for which the linear theory is still valid, and ending by the complete series, involving six functions, corresponding to maximum vibration amplitudes at the beam middle point up to once the beam thickness. For higher amplitudes, a complementary second formulation has been developed, leading to reproduction of the known results via the solution of reduced linear systems of five equations and five unknowns. The purpose of this paper is to extend and adapt the approach described above to the geometrically non-linear free vibration of fully clamped rectangular plates in order to allow direct and easy calculation of the first, second and higher non-linear fully clamped rectangular plate mode shapes, with their associated non-linear frequencies and non-linear bending stress patterns. Also, numerical results corresponding to the first and second non-linear modes shapes of fully clamped rectangular plates with an aspect ratio α=0·6 are presented. Data concerning the higher non-linear modes, the aspect ratio effect, and the forced vibration case will be presented later.     

Optimal design of beams in torsion under periodic loading

The optimal design of beams in torsion under harmonically varying torques is discussed. The analysis covers the cases when the excitation frequency is either less than or greater than the fundamental frequency of the beam. The beams analyzed are in the main assumed to have rectangular cross-section but the theory is easily extended to other section shapes. In each case the problem is stated in variational form with the introduction of constraints through Lagrange multipliers. The mathematical analysis of the various problems presented results in a system of non-linear differential equations with associated boundary conditions. The solutions given for some of the cases provide expressions for the design variable and the response, along the length of the beam, in terms of the forcing frequency and some constants which can be determined for the particular problem. The computed results and data are given in tabular form and some optimum profiles are shown graphically.     

The effects of large vibration amplitudes on the axisymmetric mode shapes and natural frequencies of clamped thin isotropic circular plates. Part I: iterative and explicit analytical solution for non-linear transverse vibrations

The effects of large vibration amplitudes on the first two axisymmetric mode shapes of clamped thin isotropic circular plates are examined. The theoretical model based on Hamilton's principle and spectral analysis developed previously by Benamar et al. for clamped-clamped beams and fully clamped rectangular plates is adapted to the case of circular plates using a basis of Bessel's functions. The model effectively reduces the large-amplitude free vibration problem to the solution of a set of non-linear algebraic equations. Numerical results are given for the first and second axisymmetric non-linear mode shapes for a wide range of vibration amplitudes. For each value of the vibration amplitude considered, the corresponding contributions of the basic functions defining the non-linear transverse displacement function and the associated non-linear frequency are given. The non-linear frequencies associated to the fundamental non-linear mode shape predicted by the present model were compared with numerical results from the available published literature and a good agreement was found. The non-linear mode shapes exhibit higher bending stresses near to the clamped edge at large deflections, compared with those predicted by linear theory. In order to obtain explicit analytical solutions for the first two non-linear axisymmetric mode shapes of clamped circular plates, which are expected to be very useful in engineering applications and in further analytical developments, the improved version of the semi-analytical model developed by El Kadiri et al. for beams and rectangular plates, has been adapted to the case of clamped circular plates, leading to explicit expressions for the higher basic function contributions, which are shown to be in a good agreement with the iterative solutions, for maximum non-dimensional vibration amplitude values of 0.5 and 0.44 for the first and second axisymmetric non-linear mode shapes, respectively.