reduced basis technique for nonlinear vibration analysis of composite panels abstract a reduced basis technique and a computational procedure are presented for the nonlinear free vibrations of composite panels the computational procedure can be conveniently divided into two distinct steps the first step involves the generation of various order perturbation vectors using linstedt poincare perturbation technique the second step consists of using the perturbation vectors as basis vectors computing the amplitudes of these vectors and the nonlinear frequency of vibration via a direct variational procedure the analytical formulation is based on a form of the geometrically nonlinear shallow shell theory with the effects of transverse shear deformation rotatory inertia and anisotropic material behavior included the panel is discretized by using mixed finite element models with the fundamental unknowns consisting of both the nodal displacements and the stress resultant parameters of the panel the potential of the proposed technique is discussed and its effectiveness is demonstrated by means of numerical examples

the effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures part i simply supported and clamped clamped beams abstract a method for calculating the non linear mode shapes and natural frequencies of fully clamped beams at large vibration amplitudes is presented and results are compared with those of previous studies and of experimental measurements first the transverse displacement is assumed to be harmonic and is expanded in the form of a finite series of functions then the non linear deformation energy is expressed by taking into account the non linear terms due to the axial strain induced by large deflections a set of non linear algebraic equations which reduces to the classical linear eigenvalue problem when non linear terms are neglected is determined through hamilton s principle it is also shown that unless a condition is imposed on the contribution of one mode the solution of this set leads to the linear case consequently in order to obtain a numerical solution for the non linear problem in the neighbourhood of a given mode the contribution of this mode is chosen and those of other modes are calculated in this paper part i of a series of three papers this method is applied to obtain the first three non linear mode shapes of clamped clamped and simply supported beams the results obtained corresponding to the fundamental non linear mode shape are in good agreement with those of a previous theoretical and experimental study in particular high values of increase of beam curvatures are noticed near the clamps causing a highly non linear increase in bending strain with increasing deflection

computational formulation for periodic vibration of geometrically nonlinear structures part 2 numerical strategy and examples abstract in this paper the numerical strategy for solving the matrix amplitude equation with parameter is discussed in detail the matrix amplitude equation is the result of application of the galerkin method for analysis of periodic solutions of the geometrically nonlinear structures a theoretical background of the proposed method of analysis is given in a companion paper by lewandowski lewandowski r 1996 computational formulation for periodic vibration of geometrically nonlinear structures part 1 theoretical background international journal of solids and structures34 15 1925 1947 an example application of general theory is also presented numerical results are given and discussed to show some interesting behaviors of nonlinear structures and to illustrate the numerical efficiency and validity of the suggested method

finite element method for nonlinear free vibrations of composite plates a multimode time domain modal formulation based on the finite element method for large amplitude free vibration of thin composite plates is presented accurate frequency maximum deflection relations can be predicted for the fundamental and the higher nonlinear modes a modal participation is defined and accurate and convergent frequencies can be determined with minimum number of linear modes a procedure for the selection of initial conditions for periodic plate response is presented convergence of frequency with gridwork refinement and number of linear modes is studied the classical single mode elliptic function frequency solutions for simply supported beams and square plates are assessed examples of orthotropic and composite plates are given and the characteristics of nonlinear response are studied

a finite element method for non linear forced vibrations of beams geometric non linearities for large amplitude free and forced vibrations of beam are investigated longitudinal displacement and inertia are included in the formulation the finite element method is used the harmonic force matrix is introduced and derived various out of plane and inplane boundary conditions are considered results showing the dependence of the amplitude on the frequency ratio and on the strain are presented for different boundary conditions and loads it is concluded that the effects of longitudinal deformation and inertia are to reduce the non linearity

some computational aspects of the stability analysis of nonlinear structures abstract the perturbation method and the continuation method are the two most popular techniques for the solution of finite element equations that describe instability phenomena this paper presents a summary of the principles involved and describes some trends in the development of these techniques

lagrange type formulation for finite element analysis of non linear beam vibrations abstract a lagrange type formulation for finite element analysis of non linear vibrations of immovably supported beams is presented two equations of motion coupled in axial and transverse displacements are derived by using lagrange s equations by neglecting the in plane inertial effects these equations are written in terms of the transverse displacement alone upon defining certain properties for the non linear oscillatory behaviour of the transverse displacement the governing equation is reduced to an equation in space alone from which the eigenvalue like quantity is computed the governing equation is solved in two ways a direct iteration technique is used in the first method to compute a numerically exact mode shape and the corresponding frequency a rayleigh quotient type of formulation similar to linear vibration analysis is used in the second approach to evaluate the frequency of vibration for a fundamental mode which is determined from a linear fem model and is maintained constant at all amplitudes numerical results are compared with available results and they corroborate the observations of earlier research workers

a quartic c 3 spline collocation method for solving second order initial value problems abstract this paper introduces and analyzes a new formula for integrating special second order periodic and nonperiodic initial value problems in ordinary differential equations the presented formula is based on quartic c3 splines as an approximation to the exact solution of the initial value problem y x f x y y 0 y0 y 0 y 0 it turns out that the proposed method is a continuous extension of the well known fourth order numerov s method and hence possesses nonvanishing intervals of periodicity and absolute stability

computational formulation for periodic vibration of geometrically nonlinear structures part 1 theoretical background abstract a general computational formulation for geometrically nonlinear structures excited by harmonic forces and executing periodic motion in a steady state is presented the equations of both continuous and discretized models are reformulated to obtain the motion equation in a more suitable form to a further analysis the multi harmonic solution of motion equation is written in a form of truncated fourier series next the galerkin ritz and the harmonic balance method are discussed in a context of their equivalency in derivation of the matrix amplitude equation the matrix amplitude equation as well as the associated tangent matrix are given in an explicit form the stability of steady state solution is discussed by using the floquet theory the numerical algorithm and an example application are described in a companion paper by lewandowski lewandowski r computational formulation for periodic vibration of geometrically nonlinear structures part 2 numerical strategy and numerical examples international journal of solids and structures in preparation

re investigation of large amplitude free vibrations of beams using finite elements the dynamic finite element matrix equations are reduced to a scalar second order non linear differential equation solved by using the direct integration method

application of the ritz method to the analysis of non linear free vibrations of beams this paper presents an analytical solution for geometrically non linear free vibrations of beams with elastically supported ends in the horizontal direction the equation of motion is obtained by employing hamilton s principle and assuming that horizontal inertia forces can be neglected the ritz method with a continuum solution and an iterative procedure are used for determining the frequencies and non linear modes of vibrations the orthogonality conditions for these modes are also discussed numerical results for various beam boundary conditions are presented and compared with available results

non linear free vibrations of beams by the finite element and continuation methods abstract a computational method is presented for determining backbone curves of arbitrary geometry as an example beam vibration is considered the beam response is expanded into a truncated fourier series with respect to time after starting from the integral of action the variational approach and the finite element method are used to formulate the non linear eigenvalue problem the continuation method is adopted to solve the resulting non linear eigenvalue problem and to obtain the non linear frequencies and modes of vibration numerical results for various beams are presented and compared with available results to demonstrate the accuracy and applicability of the method moreover the bifurcation points on some beams backbone curves are found and reported for the first time

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 abstract 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 vibration 224 183 207 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

exploring space frequency co occurrences via local quantized patterns for texture representation local binary pattern lbp has shown its power in texture classification and face recognition however the lbp operator is performed in the original image space and it lacks deeper pixel interactions to capture a richer description in this paper we propose to explore space frequency co occurrences via local quantized patterns for texture representation the proposed method proceeds in two channels in each channel the multi resolution spatial maps are first obtained by specific spatial filtering and local frequency features spectral maps are subsequently extracted by applying the local fourier transform to the spatial map two types of quantization via global thresholding are employed to quantize the spatial and spectral maps into three and two levels respectively the quantized patterns are then jointly encoded to construct a space frequency co occurrence histogram finally the two channel histograms are combined to characterize the texture without resort to the texton based representation our method directly encodes the joint information in the space and frequency domains while preserving the robustness to image rotation illumination scale and viewpoint changes extensive experiments have been conducted on three well known texture databases and our method achieves the best classification results compared with state of the art approaches investigated highlightstwo channel space frequency feature spaces are designed to obtain invariant features two types of quantization via global thresholding are developed the joint space frequency coding is explored to capture a richer description our method outperforms the state of the art on three well known texture datasets

the effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures part iii fully clamped rectangular isotropic plates measurements of the mode shape amplitude dependence and the spatial distribution of harmonic distortion abstract in this work the amplitude dependence of the resonance frequency and the spatial distribution of the first and second harmonic components of the dynamic response of a fully clamped rectangular homogeneous plate have been investigated experimentally by using electrodynamic point excitation at the centre of the plate and a non contacting optical vibration transducer for response measurement the spatial distribution of the complex harmonic distortion has been investigated in order to reach high amplitudes of vibration under distributed loading a set of tests were also carried out with a progressive wave tunnel in which the plates tested were subjected to high level acoustic excitation at overall sound pressure levels up to 155 db the results of such experimental measurements are given and compared with theoretical predictions

some erroneous finite element formulations of non linear vibrations of beams and plates the non linear vibrations of straight beams and flat plates are governed by differential equations which effectively involve a cubic non linearity this paper brings out the source of errors in some finite element formulations of these non linear problems which are based on the introduction of a linearizing function in the expression for the strain energy the magnitude of this error is derived it is explicitly shown that the discrepancy between the results of approximate analytical solutions and those of these finite element solutions is attributable to these errors in the finite element formulations

a finite element time domain modal formulation for large amplitude free vibrations of beams and plates abstract a finite element time domain modal fomulation is presented for the large amplitude free vibration of plates the procedure of deriving the non linear modal equations of motion is simple and general accurate frequency maximum deflection relations can be obtained for the fundamental and higher non linear modes the percentage of participation from each linear mode determines the number of modes to be included for accurate frequency results numerical examples on beams and square plates are given and compared with classical continuum analytical solutions a plate of complex planform and boundary conditions is also given to show the versatility of the presented modal formulation

non linear vibration of beams with internal resonance by the hierarchical finite element method the hierarchical nite element hfem and the harmonic balance methods hbm are used to investigate the geometrically non linear free and steady state forced vibrations of uniform slender beams the beam analogue of von kah rmah n s non linear strain displacement relationships are employed and the middle plane in plane displacements are included in the model the equations of motion are developed by applying the principle of virtual work and are solved by a continuation method 1 3 and 1 5 internal resonances are discovered and their consequences are discussed the convergence properties of the hfem are analyzed and it is demonstrated that the hfem model requires far fewer degrees of freedom than the h version of the fem models presented in the literature 1999 academic press

an asymptotic numerical method for large amplitude free vibrations of thin elastic plates abstract an asymptotic numerical method has been developed for large amplitude free vibrations of thin elastic plates it is based on the perturbation method and the finite element method this method eliminates the major difficulties of the classical perturbation methods namely the complexity of the right hand sides and the limitation of the validity of the solution obtained the applicability of this method to non linear vibrations of plates is clearly presented based on the von karman theory and the harmonic balance method a cubic non linear operational formulation has been obtained by using the mixed stress displacement hellinger reissner principle a quadratic formulation is given the displacement and frequency are expanded into power series with respect to a control parameter the non linear governing equation is then transformed into a sequence of linear problems having the same stiffness matrix which can be solved by a classical fem needing one matrix inversion a large number of terms of the series can be easily computed with a small computation time the non linear mode and frequency are then obtained up to the radius of convergence taking the starting point in the zone of validity the method is reapplied in order to determine a further part of the non linear solution iteration of this method leads to a powerful incremental method in order to increase the validity of the perturbed solution another technique called pade approximants is shrewdly incorporated the solutions obtained by these two concepts coincide perfectly in a very large part of the backbone curve comprehensive numerical tests for non linear free vibrations of circular square rectangular and annular plates with various boundary conditions are reported and discussed

semi analytical approach to the non linear dynamic response problem of s s and c c beams at large vibration amplitudes part i general theory and application to the single mode approach to free and forced vibration analysis abstract in a previous series of papers 1 3 a general model based on hamilton s principle and spectral analysis was developed for non linear free vibrations occurring at large displacement amplitudes of fully clamped beams and rectangular homogeneous and composite plates as an introduction to the present work concerned with the forced non linear response of c c and s s beams the above model has been derived using spectral analysis lagrange s equations and the harmonic balance method then the forced case has been examined and the analysis led to a set of non linear partial differential equations which reduces to the classical modal analysis forced response matrix equation when the non linear terms are neglected on the other hand if only one mode is assumed this set reduces to the duffing equation very well known in one mode analyses of non linear systems having cubic non linearities so it appeared sensible to consider such a formulation as the multidimensional duffing equation in order to solve the multidimensional duffing equation in the case of harmonic excitation of beam like structures a method is proposed based on the harmonic balance method and a set of non linear algebraic equations is obtained whose numerical solution leads in each case to the basic function contribution coefficients to the displacement response function these coefficients depend on the excitation frequency and the distribution of the applied forces the frequency response curve obtained here exhibits qualitatively a classical non linear behaviour with multivalued regions in which the jump phenomenon could occur quantitatively the analytical results obtained here without assuming any limitation to the scale of the excitation is identical to that obtained by the multiple scale method which assumes small values of the scaling parameter attention was focused on the assumed one mode in order to improve the results obtained in the case of free vibrations the analytical solution obtained by elliptic functions has been expanded into power series of higher orders using the symbolic manipulation program maple it has been shown that extreme care must be taken in the choice of the polynomial approximation which is valid only in a zone limited by a radius of convergence the use of pade approximants permitted considerable increase in the zone of validity of the solution obtained for very large vibration amplitudes

large amplitude free vibrations of beams a discussion on various formulations and assumptions various analytical formulations of the problem of large amplitude free vibrations of simply supported beams with immovable ends based on the rayleigh ritz method with one term approximations for axial and transverse displacements are presented in this paper many controversial points raised by different investigators at different times such as whether axial displacements should be included in the strain energy or not and whether or not the strain displacement relationship can be linearized are discussed herein it is observed that the formulation wherein the axial displacement is neglected and further the quadratic term in the strain displacement relation is linearized leads to an equation of motion which when solved based on the simple harmonic oscillations assumption yields exactly the same non linear frequency as do the perturbation method the ritz galerkin method and the elliptical integral solution with axial displacement included no linearization of non linear terms and without the harmonic oscillations assumption

a finite element method for nonlinear forced vibrations of rectangular plates the finite element method has been extended to determine the response of large amplitu de forced vibrations of thin plates a harmonic force matrix of a rectangular element under uniform harmonic excitation is developed for nonlinear forced vibration analysis in plane deformation and inertia are both considered in the formulation results obtained are compared with simple elliptic response perturbation and other approximation solutions nomenclature a nondimensional amplitude of vibration wmax z b nondimensional forcing amplitude factor a b plate length and width c d extensional and bending stiffness matrices c ratio of volumes under plate mode shape and square of mode shape e young s modulus e midsurface strains f0 amplitude of uniform harmonic pressure f ff2 linearizing functions h plate thickness k element linear stiffness matrix k element nonlinear stiffness matrix length of loaded plate element m element mass matrix

an arc length method including line searches and accelerations this paper describes a method for introducing line searches into the arc length solution procedure such line searches may be used at each iteration to calculate an optimum scalar step length which scales the normal iterative vector in practice a loose tolerance is provided so that on many iterations the line searches are avoided however on difficult iterations the line searches are shown to lead to a substantial improvement in the convergence characteristics a simple single parameter acceleration is also developed using line search concepts the new arc length method is applied to both the geometrically nonlinear analysis of shallow shells and the materially nonlinear analysis of reinforced concrete beams and slabs significant improvements are demonstrated in relation to the standard arc length method

on nonlinear free vibrations of simply supported uniform beams abstract the problem of large amplitude free vibrations of simply supported uniform beams with immovable ends is examined the equation of motion for this problem is found to be of duffing type the solution of the equation of motion that is the frequency vs amplitude relation is examined here by several methods including the perturbation method the elliptic function method the harmonic balance method and the method in which one assumes simple harmonic oscillations it is found that the second approximation of the harmonic balance method yields good results which are in excellent agreement with the exact solution