free and forced vibration analysis of a nonlinear system with cyclic symmetry application to a simplified model abstract this work program is devoted to studying the nonlinear dynamics of a structure with cyclic symmetry under conditions of geometric nonlinearity through the use of the harmonic balance method hbm in order to study the influence of nonlinearity due to the large deflection of blades a simplified model has been developed this approach leads to a system of linearly coupled second order nonlinear differential equations in which nonlinearity appears via cubic terms periodic solutions in both the free and forced cases are sought by applying hbm coupled with an arc length continuation method solution stability has been investigated using floquet s theorem in addition to featuring similar and nonsimilar nonlinear modes the unforced system is known to contain localized nonlinear modes that arise from branching point bifurcation at certain vibration amplitudes in the forced case these nonlinear modes give rise to a complex dynamic behavior many bifurcations can take place thus leading to strong or weak localization that may or may not be stable in this study special attention has been paid to the influence of excitation on dynamic responses several cases of excitation have been analyzed herein localized excitation and low engine order excitation in the case of low engine order excitation sensitivity of the response to a perturbation of this excitation type has been investigated and it has been shown that for a localized or sufficiently detuned excitation several solutions can coexist some of which are represented by closed curves in the frequency amplitude domain these various solutions overlap when increasing the force amplitude leading to forced nonlinear localization because closed curves are not tied up with the basic nonlinear solution they can easily be overlooked in this study they have been calculated using a sequential continuation with the force amplitude as a parameter

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 karman 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 omega n of one asymmetric mode of the plate thus the vibration is governed by the two degenerated modes corresponding to omega n 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

nonlinear structural response of laminated composite plates subjected to blast loading we are concerned with the theoretical analysis of the laminated composite plates exposed to normal blast shock waves as well as presenting correlation between the theoretical analysis and the experimental results of the strain time histories the laminated composite plate is clamped at all edges on the theoretical side of the study dynamic equations of the plates are derived by the use of the virtual work principle within the framework of the love theory of plates the geometric nonlinearity effects are taken into account with von karman assumptions then the governing equations of the laminated plate are solved by the runge kutta verner method a new displacement function is considered for the theoretical solution of the blast loaded clamped plate furthermore finite element modeling and analysis for the blast loaded composite plates are presented on the experimental side of the study tests have been carried out on the laminated composite plates with clamped edges for two different blast loads the results of theoretical and finite element methods are compared with the experimental results theoretical and finite element analyses results are in a good agreement there is a qualitative agreement between the analyses and experimental results in the first load case the predicted peak strains and response frequency are in an agreement with the experimental results for first load case thus the theoretical solution may be used for providing material in the preliminary design stage there is a difference between the analysis and experimental results in the second load case because of the extremely large deflections in this study the effects of loading conditions geometrical properties and material properties are separately examined on the dynamic behavior as well

geometrical non linear steady state forced periodic vibration of plates part ii stability study and analysis of multi modal response the steady state geometrically non linear periodic vibration of rectangular thin plates under harmonic external excitations is analyzed using the hierarchical finite element and the harmonic balance methods modal coupling due to internal resonance is detected and the consequent multi modal and multi frequency response is demonstrated the stability of the obtained solutions is investigated by studying the evolution of perturbations to the solutions and using floquet s theory

regularity and large time behaviour of solutions of a conservation law without convexity a l aide de la methode des caracteristiques generalisees on etudie la regularite et le comportement a grand des solutions faibles admissibles d une seule loi de conservation a 1 variable d espace avec un point d inflexion

theory and experiments for nonlinear vibrations of imperfect rectangular plates with free edges abstract large amplitude vibrations of completely free rectangular plates are investigated for the first time nonlinear higher order shear deformation theory is used and the nonlinear response to transverse harmonic excitation in the frequency neighborhood of the fundamental mode is studied geometric imperfections are taken into account the numerical analysis is carried out in two steps first the plate displacements and rotations are expanded in terms of chebyshev polynomials and a linear analysis is conducted to obtain the natural frequencies and modes of vibration then the energy functional is discretized by using the natural modes of vibration and a system of nonlinear ordinary differential equations with cubic and quadratic nonlinear terms is obtained a pseudo arc length continuation and collocation scheme is used for bifurcation analysis of periodic solutions moreover to validate the numerical results experimental tests have been conducted for the fundamental vibration mode at several harmonic excitation amplitudes by using a laser doppler vibrometer to characterize the nonlinear response of the plate with imperfections and closed loop control of the excitation the effects of geometric imperfections on the trend of nonlinearity and on natural frequencies are fully discussed and the convergence of the numerical solutions has been verified

geometrical non linear steady state forced periodic vibration of plates part i model and convergence studies a model for the steady state geometrically non linear periodic vibration of thin rectangular plates under harmonic external excitation is presented the equations of motion in the time domain are derived by applying the principle of virtual work and the hierarchical finite element method hfem these equations are transformed into the frequency domain by the harmonic balance method hbm and are solved by a continuation method the convergence properties of the model are discussed by applying it to isotropic and to composite laminated plates

a symplectic galerkin method for non linear vibration of beams and plates abstract non linear free vibration of beams and plates is conservative in the sense that the total energy of the system remains constant symplectic numerical integration aims to preserve the energy momenta and area volume of the phase space and is ideal in the study of non linear free vibration in this paper the lagrangian of continuous systems is discretized by galerkin s method to obtain the discrete hamiltonian and hamilton s equations symplectic numerical integration schemes are applied to the resulting ordinary differential equations to construct the phase diagram it is shown that due to modal coupling bounded quasi periodic and non periodic free vibrations are very common for continuous systems comparison is made with the runge kutta method it is shown that the runge kutta integration reduces the total energy of an undamped system while symplectic integration almost preserves it the symplectic schemes are extended to ordinary differential equations by means of the lagrangian and hamiltonian and the non linear vibration of beams and plates is studied by the schemes for the first time non linear modal coupling is emphasized the extension to shell structures involves the complication of curvature effects research is in progress

an unconditionally stable method for the euler equations we discuss how to combine a front tracking method with dimensional splitting to solve systems of conservation laws numerically in two space dimensions in addition we present an adaptive grid refinement strategy the method is unconditionally stable and allows for moderately highcflnumbers typically 1 4 and thus it is highly efficient the method is applied to the euler equations of gas dynamics in particular it is tested on an expanding circular gas front a wind tunnel with a step a double mach reflection and a shock bubble interaction the method shows very sharp resolution of shocks

non linear vibrations of a clamped circular plate with initial deflection and initial edge displacement part i theory abstract theoretical analyses are presented for axisymmetric non linear vibrations of a clamped circular plate of isotropic materials under uniformly distributed lateral loading with the effect of both initial deflection and initial edge displacement taken into consideration the dynamic analog of the marguerre equations is used and the steady state solutions are obtained by first applying the galerkin method and then the harmonic balance method actual calculations are carried out for the three degree of freedom galerkin system and the frequency response characteristics and typical waveforms are determined under various initial edge displacements including initially buckled cases effects of both initial deflection and initial edge displacement on the static deflection as well as the lower natural frequencies are also clarified

weighted poincare and sobolev inequalities and estimates for weighted peano maximal functions 1 introduction in this paper we derive local poincare and sobolev inequalities involving two weight functions the estimates we obtain include those proved by fabes kenig and serapioni in 8 for the one weight function case we also consider the notion of weighted peano derivatives and show that the peano derivative formed with a given weight function can be estimated in terms of a maximal function involving a second weight function in the special case of ordinary lebesgue measure results of this kind lead to the theorems of a p calderon and a zygmund about local differentiability of functions in sobolev spaces and of functions which are of bounded variation in the sense of tonelli see 3 and 4 all our results are based on variants of the basic estimate in 5 but we do not presuppose any facts from 5 let v x be a weight function on r i e let v x be nonnegative and locally integrable with respect to lebesgue measure we shall use the notation v e se v x dx ordinary lebesgue measure will be denoted ie l if bh x is the open ball with center x and radius h we say v satisfies the doubling condition with respect to lebesgue measure if

asymptotic behavior of solutions to scalar conservation laws and optimal convergence orders to n waves abstract the goal of this article is to develop a new technique to obtain better asymptotic estimates for scalar conservation laws general convex flux f u 0 is considered with an assumption lim u 0 uf u f u 1 we show that under suitable conditions on the initial value its solution converges to an n wave in l1 norm with the optimal convergence order of o 1 t the technique we use in this article is to enclose the solution with two rarefaction waves we also show a uniform convergence order in the sense of graphs a numerical example of this phenomenon is included

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 abstract 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 response 2 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

thermal effects on geometrically nonlinear vibrations of rectangular plates with fixed edges nonlinear forced vibrations and postbuckling of isotropic rectangular plates subjected to thermal variations are studied geometric imperfections are taken into account since they play a fundamental role the plate is modelled by using the von karman hypothesis and the equations of motion are obtained by using an energy approach plates with fixed edges and additional rotational elastic constraint are considered a pseudo arclength continuation method is used in order to obtain numerical results laboratory experiments have been performed on two plates of different thickness

nonlinear dynamic behavior of simply supported laminated composite plates subjected to blast load this paper deals with the analysis and discussion of nonlinear dynamic response of a laminated composite plate subjected to blast load dynamic equations of the plate are derived by the use of the virtual work principle the geometric nonlinearity effects are taken into account with the von karman large deflection theory of thin plates approximate solutions for a simply supported plate are assumed for the space domain the single term approximation functions are selected by considering the nonlinear static deformations of plate which is obtained using finite element method the galerkin method is used to obtain the nonlinear differential equations in the time domain the finite difference method is applied to solve the system of coupled nonlinear equations the results of approximate numerical analysis are obtained and compared with the literature and finite element results good agreement is found for the character and frequencies of vibrations

a moving mesh numerical method for hyperbolic conservation laws we show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in l1 r to within o n 2 by a piecewise linear function with o fn nodes the nodes are moved according to the method of characteristics we also show that a previous method of dafermos which uses piecewise constant approxima tions is accurate to o n 1 these numerical methods for conservation laws are the first to have proven convergence rates of greater than o fn 1 2 1 introduction it is well known that the solution of the hyperbolic conservation law

removable singularities of solutions of degenerate nonlinear elliptic equations on the boundary of a domain the removability of singularities of solutions for the dirichlet problem for degenerate nonlinear elliptic equations on the boundary of a domain is studied a method based on a priori energetic estimates of solutions to elliptic boundary value problems is used the growth in the vicinity of a boundary point finite or at infinity for generalized solutions is studied

nonlinear vibrations of viscoelastic rectangular plates abstract nonlinear vibrations of viscoelastic thin rectangular plates subjected to normal harmonic excitation in the spectral neighborhood of the lowest resonances are investigated the von karman nonlinear strain displacement relationships are used and geometric imperfections are taken into account the material is modeled as a kelvin voigt viscoelastic solid by retaining all the nonlinear terms the discretized nonlinear equations of motion are studied by using the arclength continuation and collocation method numerical results are obtained for the fundamental mode of a simply supported square plate with immovable edges by using models with 16 and 22 degrees of freedom and investigating solution convergence comparison to viscous damping and the effect of neglecting nonlinear viscoelastic damping terms are shown the change of the frequency response with the retardation time parameter is also investigated as well as the effect of geometric imperfections

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

theory and experiments for large amplitude vibrations of rectangular plates with geometric imperfections abstract large amplitude vibrations of rectangular plates subjected to harmonic excitation are investigated the von karman nonlinear strain displacement relationships are used to describe the geometric nonlinearity a specific boundary condition with restrained normal displacement at the plate edges and fully free in plane displacements not previously considered has been introduced as a consequence that it is very close to the experimental boundary condition results for this boundary condition are compared to nonlinear results previously obtained for i simply supported plates with immovable edges ii simply supported plates with movable edges and iii fully clamped plates the nonlinear equations of motion are studied by using a code based on pseudo arclength continuation method a thin rectangular stainless steel plate has been inserted in a metal frame this constraint is approximated with good accuracy by the newly introduced boundary condition the plate inserted into the frame has been measured with a 3d laser system in order to reconstruct the actual geometry and identify geometric imperfections out of planarity the plate has been experimentally tested in laboratory for both the first and second vibration modes for several excitation magnitudes in order to characterize the nonlinearity of the plate with imperfections numerical results are able to follow experimental results with good accuracy for both vibration modes and for different excitation levels once the geometric imperfection is introduced in the model effects of geometric imperfections on the trend of nonlinearity and on natural frequencies are shown convergence of the solution with the number of generalized coordinates is numerically verified

large amplitude finite element flexural vibration of plates stiffened plates large amplitude free flexural vibration of stiffened and unstiffened plates has been studied by using the finite element method an isoparametric quadratic plate bending element has been used both for the plate and the stiffener the dynamic version of von karman s field equations has been adopted and the formulation has been done in the total lagrangian coordinate system the in plane deformation and inertia have been taken into account the resulting nonlinear equations have been solved by the direct iteration technique using a linear mode shape as the starting vector the stiffener has been elegantly modeled so that it can be placed anywhere within the plate element and it need not follow the nodal lines this has increased the flexibility of the mesh generation considerably the arbitrary orientation and eccentricity of the stiffener have been incorporated in the formulation the shear deformation has been incorporated according to mindlin s hypothesis stiffened and unstiffened plates that have various boundary conditions have been analyzed and the results have been compared with those available in the literature

a method of fractional steps for scalar conservation laws without the cfl condition we present a numerical method for the n dimensional initial value problem for the scalar conservation law u xl x t i z7 f1 u x 0 u x x 0 uo x xv our method is based on the use of dimensional splitting and dafermos s method to solve the one dimensional equations this method is unconditionally stable in the sense that the time step is not limited by the space discretization furthermore we show that this method produces a subsequence which converges to the weak entropy solution as both the time and space discretization go to zero finally two numerical examples are discussed 0 introduction scalar conservation laws given their wide range of applications have been studied extensively over the years both from a mathematical physical and numerical point of view fundamental problems are the emergence of discontinuous solutions of the partial differential equation with the subsequent call for weak solutions which again results in subtle uniqueness questions existence and uniqueness were first proved for the general cauchy problem by conway and smoller 2 and later on by kuznetsov 9 vol pert 10 kruzkov 8 who used a viscosity method we will here use kruzkov s formulation of the entropy condition which is a mechanism to identify the unique physical solution we here study the cauchy problem

theory and experiments for large amplitude vibrations of empty and fluid filled circular cylindrical shells with imperfections abstract the large amplitude response of perfect and imperfect simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of some of the lowest natural frequencies is investigated donnell s non linear shallow shell theory is used and the solution is obtained by the galerkin method several expansions involving 16 or more natural modes of the shell are used the boundary conditions on the radial displacement and the continuity of circumferential displacement are exactly satisfied the effect of internal quiescent incompressible and inviscid fluid is investigated the non linear equations of motion are studied by using a code based on the arclength continuation method a series of accurate experiments on forced vibrations of an empty and water filled stainless steel shell have been performed several modes have been intensively investigated for different vibration amplitudes a closed loop control of the force excitation has been used the actual geometry of the test shell has been measured and the geometric imperfections have been introduced in the theoretical model several interesting non linear phenomena have been experimentally observed and numerically reproduced such as softening type non linearity different types of travelling wave response in the proximity of resonances interaction among modes with different numbers of circumferential waves and amplitude modulated response for all the modes investigated the theoretical and experimental results are in strong agreement

nonlinear damped vibrations of a laminated composite plate subjected to blast load this research deals with the derivation and solution of nonlinear dynamic equations of a clamped laminated plate exposed to blast load including structural damping effects dynamic equations of the plate are derived by the use of the virtual work principle the geometric nonlinearity effects are taken into account with the von karman large deflection theory of thin plates an appropriate approximate solution is assumed for the space domain considering the finite element analysis results for the large deformation static analysis of the laminated plate the galerkin method is used to obtain the nonlinear differential equations in the time domain these nonlinear differential equations include structural damping effects the finite difference method is applied to solve the system of coupled nonlinear equations the results of the theoretical analysis are compared with the experimental results good agreement is found for the peak values and frequencies effects of aspect ratio and damping on the dynamic response of the plate are examined the damping coefficient greatly affects the nonlinear dynamic response

source solutions and asymptotic behavior in conservation laws abstract we study the uniqueness of the solutions to the scalar conservation law u t u x 0 when the initial datum is a finite measure the case of a dirac mass is particularly emphasized it is shown how it provides a description of the asymptotic behavior of the solutions initiated by an arbitrary integrable function this behavior is proved to depend on one parameter in the case when is odd while it depends on two when is convex

asymmetric solutions in large amplitude free periodic vibrations of plates when the in plane inertia is neglected the equations of motion that represent free large amplitude periodic vibrations of perfect plates only contain cubic nonlinear terms it is here shown that unlike often assumed these equations admit asymmetric solutions for that purpose the time domain equations of motion are mapped into the frequency domain by the harmonic balance method keeping the constant term and the second harmonic in the truncated fourier expansion of the transverse displacements a test plate is analysed and a bifurcation due to a 1 2 internal resonance is found this bifurcation results in a branch of solutions where the equilibrium position is not the flat one and where odd and even harmonics coexist

piecewise self similar solutions and a numerical scheme for scalar conservation laws the solution of the riemann problem was a building block for general cauchy problems in conservation laws a cauchy problem is approximated by a series of riemann problems in many numerical schemes but since the structure of the riemann solution holds locally in time only and furthermore a riemann solution is not piecewise constant in general there are several fundamental issues in this approach such as the stability and the complexity of computation in this article we introduce a new approach which is based on piecewise self similar solutions the scheme proposed in this article solves the problem without the time marching process the approximation error enters in the step for the initial discretization only which is given as a similarity summation of base functions the complexity of the scheme is linear convergence to the entropy solution and the error estimate are shown the mechanism of the scheme is introduced in detail together with several interesting properties of the scheme

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

large time step shock capturing techniques for scalar conservation laws for a scalar conservation law u t f u x with f of constant sign the first order upwind difference scheme is a special case of godonov s method the method is equivalent to solving a sequence of riemann problems at each step and averaging the resulting solution over each cell in order to obtain the numerical solution at the next time level the difference scheme is stable and the solutions to the associated sequence of riemann problems do not interact provided the courant number nu is less than 1 by allowing and explicitly handling such interactions it is possible to obtain a generalized method which is stable for nu much larger than 1 in many cases the resulting solution is considerably more accurate than solutions obtained by other numerical methods in particular shocks can be correctly computed with virtually no smearing the generalized method is rather unorthodox and still has some problems associated with it nonetheless preliminary results are quite encouraging

shear deformable versus classical theories for nonlinear vibrations of rectangular isotropic and laminated composite plates in the present study i the classical von karman theory ii the first order shear deformation theory and iii the higher order third order shear deformation theory are compared for studying the nonlinear forced vibrations of isotropic and laminate composite rectangular plates in particular the harmonic response in the frequency neighborhood of the fundamental mode of rectangular plates is investigated and the response curves computed by using the three different theories are compared the boundary conditions of the plates are simply supported with immovable edges geometric imperfections are taken into account calculations for isotropic and laminated composite plates are presented and results are discussed for isotropic plates the frequency response curves for large amplitude vibrations obtained by using the three theories are almost coincident for laminated composite plates differences arise for relatively thick plates ratio between the thickness and the edge equal to 0 1 while for thin plates ratio between the thickness and the edge equal to 0 01 no difference is obtained for all cases the first order shear deformation with shear correction factor 3 2 and the higher order shear deformation theories give practically coincident results and differences are observed with respect to the classical von karman theory