forced axi symmetric response of polar orthotropic linearly tapered circular plates forced axi symmetric vibrations of polar orthotropic linearly tapered circular plates are discussed on the basis of classical plate theory ritz method has been employed to obtain the solutions the deflection function and the bending moments for forced vibrations of the plate are presented for various values of taper parameter and rigidity ratio a comparison of results with those available in literature shows an excellent agreement

nonlinear oscillations and chaotic dynamics of an antisymmetric cross ply laminated composite rectangular thin plate under parametric excitation in this paper nonlinear oscillations and chaotic dynamics of a simply supported antisymmetric cross ply laminated composite rectangular thin plate under parametric excitation is investigated for the first time the governing equations of motion for the antisymmetric cross ply laminated composite plate are derived by using von karman type plate equation the geometric nonlinearity and nonlinear damping are included in the governing equations of motion a two dof parametrically excited nonlinear system including the quadratic and cubic nonlinear terms is obtained by using the galerkin method based on the fourier expansion and the temporal rescaling an asymptotic perturbation method is utilized to obtain four dimensional nonlinear averaged equations on the amplitude and the phase of nonlinear oscillations of the antisymmetric cross ply laminated composite plate based on the averaged equations the steady state nonlinear responses and their stabilities are determined by using numerical approach the relations between the steady state nonlinear responses and the amplitude and frequency of parametric excitation are obtained under the certain conditions the antisymmetric cross ply laminated composite plate may have two steady state nonzero solutions in which the jumping phenomenon occurs numerical simulation is used to discover the periodic and chaotic motions in the antisymmetric cross ply laminated composite rectangular thin plate it is observed from the numerical results that the multipulse orbits exist in the antisymmetric cross ply laminated composite rectangular thin plate

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

vibroacoustics of the piano soundboard non linearity and modal properties in the low and mid frequency ranges abstract the piano soundboard transforms the string vibration into sound and therefore its vibrations are of primary importance for the sound characteristics of the instrument an original vibro acoustical method is presented to isolate the soundboard nonlinearity from that of the exciting device here a loudspeaker and to measure it the nonlinear part of the soundboard response to an external excitation is quantitatively estimated for the first time at 40 db below the linear part at the ff nuance given this essentially linear response a modal identification is performed up to 3 khz by means of a novel high resolution modal analysis technique k ege x boutillon b david high resolution modal analysis journal of sound and vibration 325 4 5 2009 852 869 modal dampings which so far were unknown for the piano in this frequency range are determined in the mid frequency domain where fft based methods fail to evaluate them with an acceptable precision they turn out to be close to those imposed by wood a finite element modelling of the soundboard is also presented the low order modal shapes and the comparison between the corresponding experimental and numerical modal frequencies suggest that the boundary conditions can be considered as blocked except at very low frequencies the frequency dependency of the estimated modal densities and the observation of modal shapes reveal two well separated regimes below 1 khz the soundboard vibrates more or less like a homogeneous plate above that limit the structural waves are confined by ribs as already noticed by several authors and localised in restricted areas one or a few inter rib spaces presumably due to a slightly irregular spacing of the ribs across the soundboard

non linear vibration and thermal buckling of an orthotropic annular plate with a centric rigid mass abstract a computational analysis of the non linear vibration and thermal post buckling of a heated orthotropic annular plate with a central rigid mass is examined for the cases of immovably hinged as well as clamped constraint conditions of the outer edge first based on von karman s plate theory and hamilton s principles the governing equations in terms of the displacements of the middle plane of the problem are derived then upon assuming that harmonic responses of the system exist the non linear partial differential equations are converted into the corresponding non linear ordinary differential equations through elimination of the time variable by using the kantorovich time averaging method finally by applying a shooting method the fundamental responses of the non linear vibration and thermal post buckling of the plate are numerically obtained for some prescribed values of the parameters such as the material rigidity ratio temperature rise and so on the curves of the fundamental frequency versus specified amplitude and the thermal post buckled equilibrium paths of the plate are numerically presented

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

non linear steady state vibration of frames by finite element method a finite element method based on the virtual work principle to determine the steady state response of frams in free or forced periodic vibration is introduced the axial and flexural deformations are coupled by mean of the induced axial force along the element the spatial discretization of the deformations is achieved by the usual finite element method and the time discretization by fourier coefficients of the nodal displacements no unconventional element matrices are needed after applying the harmonic balance method a set of non linear algebraic equations of the fourier coefficients is obtained these equations are solved by the newtonian iteration method in terms of the fourier coefficient increments nodal damping can easily be included by a diagonal damping matrix the direct numerical determination of the fourier coefficient increments is difficult owing to the presence of peaks loops and discontinuities of slope along the amplitude frequency response curves parametric construction of the response curves using the phase difference between the response and excitation is recommended to provide more points during the rapid change of the phase i e at resonance for undamped natural vibration the method of selective coefficients adopted numerical examples on the duffing equation a hinged hinged beam a clamped hinged beam a ring and a frame are given for reasonably accurate results it is shown that the number of finite elements must be sufficient to predict at least the linear mode at the frequency of interest and the number of harmones considered must satisfy the conditions of completeness and balanceability which are discussed in detail

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

global and chaotic dynamics for a parametrically excited thin plate the global bifurcations and chaotic dynamics of a parametrically excited simply supported rectangular thin plate are analyzed the formulas of the thin plate are derived by von karman type equation and galerkin s approach the method of multiple scales is used to obtain the averaged equations based on the averaged equations theory of normal form is used to give the explicit expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues by maple program on the basis of the normal form global bifurcation analysis of the parametrically excited rectangular thin plate is given by a global perturbation method developed by kovacic and wiggins the chaotic motion of thin plate is found by numerical simulation

non linear vibrations of free edge thin spherical shells modal interaction rules and 1 1 2 internal resonance this paper is devoted to the derivation and the analysis of vibrations of shallow spherical shell subjected to large amplitude transverse displacement the analog for thin shallow shells of von karman s theory for large deflection of plates is used the validity range of the approximations is assessed by comparing the analytical modal analysis with a numerical solution the specific case of a free edge is considered the governing partial differential equations are expanded onto the natural modes of vibration of the shell the problem is replaced by an infinite set of coupled second order differential equations with quadratic and cubic non linear terms analytical expressions of the non linear coefficients are derived and a number of them are found to vanish as a consequence of the symmetry of revolution of the structure then for all the possible internal resonances a number of rules are deduced thus predicting the activation of the energy exchanges between the involved modes finally a specific mode coupling due to a 1 1 2 internal resonance between two companion modes and an axisymmetric mode is studied 2004 elsevier ltd all rights reserved

comments on the method of harmonic balance the advantages and limitations of the harmonic balance or describing function approximation scheme for solving nonlinear ordinary differential equations of oscillatory motion are discussed advantages include appicability to equations of any order and with large degrees of nonlinearity ease of determining limit cycle behavior and its stability and overall speed and efficiency the limitation rules are essentially those described by mickens 1983 it is pointed out that perturbation procedures provide better results when the degree of nonlinearity is small

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

motion planning for kinematic systems in this paper we present a general theory of motion planning for kinematic systems in particular the theory deals with e approximations of non admissible paths by admissible ones in a certain optimal sense the need for such an approximation arises for instance in the case of highly congested configuration spaces this theory has been developed by one of the authors in a previous series of papers it is based upon concepts from subriemannian geometry here we summarize the results of the theory and we improve on by developing in details an intricate case the ball with a trailer which corresponds to a distribution with flag of type 2 3 5 6

evidences of global bifurcations of an imperfect circular plate abstract the global bifurcations in modal interactions of an imperfect circular plate with one to one internal resonance are investigated the case of the third order subharmonic resonance in which an excitation frequency is near triple natural frequencies is considered the equations governing nonlinear oscillations of an imperfect circular plate are reduced to a system of non autonomous ordinary differential equations via galerkin s procedure the method of multiple scales is used to obtain a system of autonomous ordinary differential equations and then kovacic and wiggins method is used to investigate the global dynamics of the plate having found a sufficient condition under which silnikov type homoclinic orbit can exist we failed to observe any numerical evidences of global bifurcation

dtn based mixed dimensional coupling using a boundary stress recovery technique abstract a new computational approach for the mixed dimensional modeling of time harmonic waves in elastic structures is proposed a two dimensional 2d structure is considered which includes a part that is assumed to behave in a one dimensional 1d way the 2d and 1d structural regions are discretized by using 2d and 1d finite element fe formulations the hybrid model if designed properly is much more efficient than the standard 2d model taken for the entire problem one important issue related to such hybrid 2d 1d models is the way the 2d 1d coupling is done and the coupling error generated here three closely related coupling methods are considered they are all based on the dirichlet to neumann dtn map associated with the 1d problem on the interface the first coupling method is the dtn method in its usual form where the dtn map is calculated numerically and where the 1d and 2d problems are solved separately the second coupling method is the one devised by carka mear and landis cml which is equivalent to the former one but in which the 2d and 1d interface solutions are solved for simultaneously in the third coupling method the dtn map is enforced on the interface iteratively direct application of these three methods results in low accuracy as a recent study shows therefore these methods are used here in conjunction with a boundary stress recovery bsr technique originally proposed by hughes which provides the same order of accuracy for the stress as for the primary variable the performance of the three methods is demonstrated and they are compared via numerical examples conclusions are drawn on their relative merit

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

multi modal geometrical non linear free vibration of fully clamped composite laminated plates abstract the geometrically non linear free vibration of thin composite laminated plates is investigated using the hierarchical finite element hfem and the harmonic balance methods hbm von karman s non linear strain displacement relationships are employed and the mid 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 the convergence properties of the hfem and of the hbm are analyzed internal resonances are discovered and the consequent multi modal and multi frequency vibration of the plates is shown the variation of the plates mode shape and the effect of the fibres orientation are investigated

normal modes of vibration for non linear continuous systems a definition and a constructive methodology for normal nodes of motion are developed for a class of vibratory systems the dynamics of which are governed by non linear partial differential equations the definition for normal modes is given in terms of the dynamics on two dimensional invariant manifolds in the system phase space these manifolds are a continuation of the planes which represent the well known normal modes of the linearized system a local asymptotic approximation of the geometric structure of these manifolds can be obtained by using an approach which follows that used for generating center manifolds the procedure also provides the non linear ordinary differential equations which govern individual modal dynamics and a physical description of the system configuration when it is undergoing a modal motion in this paper the general theory is described for the application of vibrations of continuous media but it can easily be extended to other situations in order to demonstrate the power of the approach and its unique procedural aspects three examples involving beam vibrations are worked out in detail the examples are conservative simply supported beams the first demonstrates the methodology for a linear beam model the second is a beam on a non linear elastic foundation and the third example is a beam with non linear torsional springs attached at each end in these examples the simplicity of the mode shapes of the linearized model yields relatively simple calculations that do not obscure the important features of the procedure

nonlinear vibrations of loudspeaker like structures during high amplitudes of vibration nonlinearities affect the electroacoustical behavior of electrodynamic transducers and are responsible for audible distortions we distinguish two types of nonlinearities electrical and mechanical in this study attention is paid to the mechanical and geometrical properties of loudspeaker like structures the loudspeaker is viewed as a combination of an annular plate with a circular plate nonlinear vibrations of such a structure are investigated using the dynamic analog of the von karman equations furthermore the influence of both material properties and geometrical parameters is studied it is shown that nonlinear effects can be substantially reduced by choosing appropriate material and geometrical parameters

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

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

numerical computation of periodic responses of nonlinear large scale systems by shooting method abstract geometrically nonlinear vibrations of three dimensional elastic structures due to harmonic external excitations are investigated in the frequency domain the material of the structure is assumed to be linearly elastic the equation of motion is derived by the conservation of linear momentum in lagrangian coordinate system and it is discretized into a system of ordinary differential equations by the finite element method the shooting method is used to obtain the periodic solutions a procedure which transforms the initial value problem into a two point boundary value problem for the periodicity condition and then it finds the initial conditions which lead to periodic response is developed and presented for systems of second order ordinary differential equations the elmer software is used for computing the local and global mass and stiffness matrices and the force vector as well for computing the correction of the initial conditions by the shooting method stability of the solutions is studied by the floquet theory sequential continuation method is used to define the prediction for the next point from the frequency response diagram the main goal of the current work is to investigate and present the potential of the proposed numerical methods for the efficient computation of the frequency response functions of large scale nonlinear systems which often result from space discretization of real life engineering applications

nonlinear forced vibration of orthotropic rectangular plates using the method of multiple scales the method of multiple scales mms in conjunction with the galerkin method is used to analyze the nonlinear forced and damped response of a rectangular orthotropic plate subjected to a uniformly distributed harmonic transverse loading the effects of damping and in plane loads are considered the analysis considers simply supported as well as clamped plates for each case both movable and immovable edge conditions are considered by using mms all possible resonances such as primary subharmonic and superharmonic reso nances are studied for the undamped response without in plane loading comparisons of the mms results with those obtained by the finite element method show excellent agreement evelopment of composite materials comprising lam inates of orthotropic or multilayered anisotropic materi als recently has been receiving substantially growing research efforts due to the increasing demands for energy efficient high strength minimum weight aircraft designs many re searchers believe that the use of composite materials offers promising alternatives for aircraft designs thin laminated composite plates subjected to transverse periodic loadings could encounter deflections of the order of plate thickness or even higher responses of this kind cannot be predicted by using the linear theory consequently the need to study large amplitude deflection vibrations of composite structures is of paramount importance the literature survey shows that the equations of motion for the large deflection analysis of heterogeneous anisotropic plates using the von karman geometrical nonlinearity were first considered by whitney and leissa 1 based on these equa tions different methods of analysis have been developed by several researchers an excellent number of collections on nonlinear free and forced vibrations of composite plates cov ering the work through 1979 can be found in the comprehen sive book by chia 2 bert3 has conducted a survey on the dynamics of composite plates for the period 1979 81 a re view of the literature on nonlinear vibrations of plates can be found in the review paper by sathyamoorthy4 and the book by nayfeh and mook 5 large deflection analysis of symmetrically laminated rec equations of motion are presented in terms of the lateral displacement and stress function the equations are nondi mensionalized following the transformation introduced by brunelle and oyibo 9 though multimode analysis can be treated the present study is focused on single mode analysis a deflection function representing the first mode and satisfy ing the boundary conditions is assumed and subsequently the stress function is found next the galerkin method is applied to obtain the modal equation which is solved analytically by using the method of multiple scales mms 10 the mms also provides solutions for subharmonic and superharmonic reso nances the effects of damping ratio plate aspect ratio and in plane loading then are studied

normal modes of a non linear clamped clamped beam abstract non linear modal analysis approach based on invariant manifold method proposed earlier by shaw and pierre journal of sound and vibration 164 85 124 is utilized here to obtain the non linear normal modes of a clamped clamped beam for large amplitude displacements the results obtained for the fundamental normal mode are compared with the corresponding reported experimental and theoretical studies the effects of modal coupling are examined in greater detail the limitation of the present method for analyzing non linear behavior is highlighted

non linear interactions in asymmetric vibrations of a circular plate abstract in order to investigate the non linear asymmetric vibrations of a clamped circular plate on an elastic foundation the primary resonances of a plate with an internal resonance in which the natural frequencies of two asymmetric modes are commensurable are considered the response is expressed as an expansion in terms of the linear free oscillation modes and its amplitude is considered to be small but finite the method of multiple scales is used to reduce the non linear governing equations to a system of autonomous ordinary differential equations for amplitude and phase variables for a numerical example the case of internal resonance a commensurable relationship between natural frequencies 32 3 11 where the first subscript refers to the number of nodal diameters and the second subscript the number of nodal circles including boundary is considered when the frequency of excitation is near 11 there exist at most five stable steady state responses four of them are superpositions of traveling wave components and one is a superposition of standing wave components the result shows the interaction between modes corresponding to 11 and 32 by showing non vanishing amplitudes of the mode not directly excited when the frequency of excitation is near 32 similarly the interaction between modes is shown to exist all of the responses with non vanishing amplitudes of modes excited indirectly however turn out to be unstable which is a peculiar phenomenon

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

complexity of control affine motion planning in this paper we study the complexity of the motion planning problem for control affine systems such complexities are already defined and rather well understood in the particular case of nonholonomic or sub riemannian systems our aim is to generalize these notions and results to systems with a drift accordingly we present various definitions of complexity as functions of the curve that is approximated and of the precision of the approximation due to the lack of time rescaling invariance of these systems we consider geometric and parametrized curves separately then we give some asymptotic estimates for these quantities as a byproduct we are able to treat the long time local controllability problem giving quantitative estimates on the cost of stabilizing the system near a nonequilibrium point of the drift

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

the effect of the number of nodal diameters on non linear interactions in two asymmetric vibration modes of a circular plate in order to investigate the effect of the number of nodal diameters on non linear interactions in asymmetric vibrations of a circular plate a primary resonance of the plate is considered the plate is assumed to have an internal resonance in which the ratio of the natural frequencies of two asymmetric modes is three to one the response of the plate is expressed as an expansion in terms of the linear free oscillation modes and its amplitude is considered to be small but finite and the method of multiple scales is used in view of the corrected solvability conditions for the responses it has been found that in order for the modes to interact the ratio of the numbers of nodal diameters of two modes must be either three to one or one to one in this study the one to one case in which the modes have the same number of nodal diameters is examined the non linear governing equations are reduced to a system of autonomous ordinary differential equations for amplitude and phase variables by means of the corrected solvability conditions the steady state responses and their stability are determined by using this system the result shows very complicated interactions between two modes by telling existence of non vanishing amplitudes of the mode not directly excited

sound synthesis of a nonlinear string using volterra series this paper proposes to solve and simulate various kirchhoff models of nonlinear strings using volterra series two nonlinearities are studied the string tension is supposed to depend either on the global elongation of the string first model or on the local strain located at x second and more precise model the boundary conditions are simple dirichlet homogeneous ones or general dynamic conditions allowing the string to be connected to any system typically a bridge for each model a volterra series is used to represent the displacement as a functional of excitation forces the volterra kernels are solved using a modal decomposition the first kernel of the series yields the standard modes of the linearized problem while the next kernels introduce the nonlinear dynamics as a last step systematic identification of the kernels lead to a structure composed of linear filters sums and products which are well suited to the sound synthesis using standard signal processing techniques the nonlinear dynamic introduced through this simulation is significant and perceptible in sound results for sufficiently large excitations

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

on the steady state response and stability of non linear oscillators using harmonic balance abstract the harmonic balance hb method is used to study the steady state periodic response as well as its stability of a softening duffing oscillator the objectives of the work are 1 to show how the qualitative nature of the hb solution changes if one uses two harmonics rather than one in the approximate solution and 2 to show that the analysis of the mathieu equation which governs local stability of the steady state response must be based on an approximate rather than an exact analysis of this mathieu equation thus floquet theory incorrectly describes the stability of the approximate solution

free vibration analysis of stepped circular mindlin plates this paper presents the first known investigation on exact vibration of circular mindlin plates with multiple step wise thickness variations a stepped circular plate is divided into multiple annular and one circular segments along the locations of the step variations the governing differential equations for harmonic vibration of annual and circular segments are derived and an analytical method based on the domain decomposition technique is developed to solve the plate vibration problem exact vibration solutions are presented for circular mindlin plates of different edge support conditions and various combinations of step wise thickness variations the influence of plate boundary conditions plate thickness ratios step location ratios and step thickness ratios on the vibration behaviour of stepped circular mindlin plates are examined the exact vibration results can serve as benchmark values for researchers to validate their numerical methods for such circular plate problems

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

the harmonic balance method for bifurcation analysis of large scale nonlinear mechanical systems abstract the harmonic balance hb method is widely used in the literature for analyzing the periodic solutions of nonlinear mechanical systems the objective of this paper is to exploit the method for bifurcation analysis i e for the detection and tracking of bifurcations of nonlinear systems to this end an algorithm that combines the computation of the floquet exponents with bordering techniques is developed a new procedure for the tracking of neimark sacker bifurcations that exploits the properties of eigenvalue derivatives is also proposed the hb method is demonstrated using numerical experiments of a spacecraft structure that possesses a nonlinear vibration isolation device

analysis of non linear mode shapes and natural frequencies of continuous damped systems in this paper the aim is to find the non linear mode shapes and natural frequencies for a class of one dimensional continuous damped systems with weak cubic inertia damping and stiffness non linearities this paper presents general formulations for natural frequencies and mode shapes with all non linearity effects initially the non linear system with general boundary conditions is discretized and using a two dimensional manifold the model of cubic non linearities is constructed and the general equation of motion which governs non linear system is derived the method of multiple scales is then used to extend the non linear mode shapes and natural frequencies during this analysis it is realized that when the natural frequencies of the linear system become equal to the natural frequencies of the non linear system a one to one internal resonance will appear also there is a three to one internal resonance which is not dependent on the damping of the system finally general formulations of amplitude for vibrations natural frequencies and mode shapes of the non linear system are obtained in parametric forms thus a non linear problem with some simple integration can be solved the formulations are capable of handling any non linearities in inertia damping stiffness or any combination of them under any arbitrary boundary conditions

nonlinear dynamics of axially moving plates abstract the nonlinear dynamics for forced motions of an axially moving plate is numerically investigated using von karman plate theory and retaining in plane displacements and inertia the equations of motion are obtained via an energy method based on lagrange equations this yields a set of second order nonlinear ordinary differential equations with coupled terms the equations are transformed into a set of first order nonlinear ordinary differential equations and are solved via the pseudo arclength continuation technique the near resonance nonlinear dynamics is examined via plotting the frequency response curves of the system results are shown through frequency response curves time histories and phase plane diagrams the effect of system parameters such as the axial speed and the pretension on the resonant responses is also highlighted

nonlinear normal modes of structural systems via asymptotic approach abstract an asymptotic approach based on the method of multiple scales is employed to construct the nonlinear normal modes nnm s of self adjoint structural systems with arbitrary linear inertia and elastic stiffness operators general cubic inertia and geometric nonlinearities the methodology employed for constructing the approximate invariant manifolds of individual nnm s away from internal resonances and of the resonant modes near three to one internal resonances attempts to generalize previous studies based on asymptotic techniques the theory is applied to a hinged hinged uniform elastic beam carrying a lumped mass and undergoing axis stretching depending on the lumped mass relative to the beam mass and on its position along the span different classes of nonlinear normal modes and their stability are investigated

modal interaction in circular plates abstract the dynamic analogue of the von karman equations in polar form is used to study the symmetric response of a circular plate to a harmonic external excitation the method of multiple scales is used to determine a set of ordinary differential equations governing the modulation of the phases and amplitudes of oscillation for the case of an internal or autoparametric resonance the results show that the internal resonance is responsible for coupling of the modes involved and that the excited mode is not necessarily the dominant one furthermore the results show that for a choice of parameters the multi mode response loses its stability through a hopf bifurcation resulting in periodically or chaotically modulated motions of the plate

discrete dirichlet to neumann maps for unbounded domains abstract it is shown how to construct a discrete counterpart of the dirichlet to neumann dtn map for use on an artificial boundary b introduced in exterior boundary value problems following the idea of deakin and dryden this discrete map provides an approximate non reflecting or absorbing boundary condition for use in formulating a problem in the finite computational domain bounded by b different discrete maps are constructed for use with finite difference and finite element methods in the solution of some simple problems shows that use of the discrete dirichlet to neumann ddtn map yields nearly the same accuracy as that of using the continuous dtn map

nonlinear oscillations bifurcations and chaos of functionally graded materials plate abstract an analysis on the nonlinear dynamics of a simply supported functionally graded materials fgms rectangular plate subjected to the transversal and in plane excitations is presented in a thermal environment for the first time material properties are assumed to be temperature dependent based on reddy s third order plate theory the nonlinear governing equations of motion for the fgm plates are derived using hamilton s principle galerkin s method is utilized to discretize the governing partial equations to a two degree of freedom nonlinear system including the quadratic and cubic nonlinear terms under combined parametric and external excitations the resonant case considered here is 1 1 internal resonance and principal parametric resonance the asymptotic perturbation method is utilized to obtain four dimensional nonlinear averaged equation the numerical method is used to find the nonlinear dynamic responses of the fgm rectangular plate it was found that periodic quasi periodic solutions and chaotic motions exist for the fgm rectangular plates under certain conditions it is believed that the forcing excitations f1 and f2 can change the form of motions for the fgm rectangular plate

geometrically nonlinear flexural vibrations of plates in plane boundary conditions and some symmetry properties abstract this study is devoted to the derivation of some properties of the von karman equations for geometrically nonlinear models of plates with a boundary of arbitrary shape for applications to nonlinear vibration and buckling an intrinsic formulation of the local partial differential equations in terms of the transverse displacement and an airy stress function as unknowns is provided classical homogeneous boundary conditions with vanishing prescribed forces and displacements are derived in terms of the airy stress function in the case of a boundary of arbitrary geometry a special property of this operator crucial for some energy conserving numerical schemes and called triple self adjointness is derived in the case of an edge of arbitrary shape it is shown that this property takes a simple form for some classical boundary conditions so that the calculations in some practical cases are also simplified the applications of this work are either semi analytical methods of solution using an expansion of the solution onto an eigenmode basis of the associated linear problem or special energy conserving numerical methods

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

a new finite element formulation for electromechanical boundary value problems in this paper a new finite element formulation for the solution of electromechanical boundary value problems is presented as opposed to the standard formulation that uses scalar electric potential as nodal variables this new formulation implements a vector potential from which components of electric displacement are derived for linear piezoelectric materials with positive definite material moduli the resulting finite element stiffness matrix from the vector potential formulation is also positive definite if the material is non linear in a fashion characteristic of ferroelectric materials it is demonstrated that a straightforward iterative solution procedure is unstable for the standard scalar potential formulation but stable for the new vector potential formulation finally the method is used to compute fields around a crack tip in an idealized non linear ferroelectric material and results are compared to an analytical solution copyright 2002 john wiley sons ltd

basin boundaries in asymmetric vibrations of a circular plate abstract in order to investigate further nonlinear asymmetric vibrations of a clamped circular plate under a harmonic excitation we reexamine a primary resonance studied by yeo and lee corrected solvability conditions for non linear asymmetric vibrations of a circular plate journal of sound and vibration 257 2002 653 665 in which at most three stable steady state responses one standing wave and two traveling waves are observed to exist further examination however tells that there exist at most five stable steady state responses one standing wave and four traveling waves two of the traveling waves lose their stability by hopf bifurcation and have a sequence of period doubling bifurcations leading to chaos when the system has five attractors three equilibrium solutions one standing wave and two traveling waves and two chaotic attractors two modulated traveling waves the basin boundaries of the attractors on the principal plane are obtained also examined is how basin boundaries of the modulated motions quasi periodic and chaotic motions evolve as a system parameter varies the basin boundaries of the modulated motions turn out to have the fractal nature

the dirichlet to neumann map for two dimensional crack problems a coupled analytic finite element method is presented for two dimensional crack problems two classes of problems are studied the first considers problems where non linear constitutive processes occur in a region near the crack tip and the remotely applied loading can be characterized by the linear elastic k field and perhaps the t stress in this case the finite element method is applied in a circular region around the crack tip where non linear constitutive response is occurring and stiffness contributions associated with a numerically implemented dirichlet to neumann map are imposed on the circular boundary to account for the large surrounding elastic domain and the remote applied loading the second class of problems considers entirely linear elastic domains with irregular external boundaries and or complex applied loadings here the discrete dirichlet to neumann map is used to represent a circular region surrounding the crack tip and finite elements are used for the external region in this case the mixed mode stress intensity factors and the t stress are retrieved from the map

hardening softening behaviour in non linear oscillations of structural systems using non linear normal modes the definition of a non linear normal mode nnm as an invariant manifold in phase space is used in conservative cases it is shown that normal form theory allows one to compute all nnms as well as the attendant dynamics onto the manifolds in a single operation then a single mode motion is studied the aim of the present work is to show that too severe truncature using a single linear mode can lead to erroneous results using single non linear mode motion predicts the correct behaviour hence the nonlinear change of co ordinates allowing one to pass from the linear modal variables to the normal ones linked to the nnms defines a framework to properly truncate non linear vibration pdes two examples are studied a discrete system a mass connected to two springs and a continuous one a linear euler bernoulli beam resting on a non linear elastic foundation for the latter a comparison is given between the developed method and previously published results

motion planning and fastly oscillating controls we consider the motion planning problem where the nonholonomic constraints are given by a strong bracket generating distribution approximating a nonadmissible trajectory by an admissible one in the sub riemannian setting we prove a theorem which provides an exact asymptotic estimate of the interpolation entropy in the case of a free nilpotent algebra of first brackets this theorem shows that we can approximate in an asymptotically optimal way using sinusoidal fastly oscillating controls in the general case we obtain similar results however the estimates are less explicit

numerical methods and parallel algorithms for computation of periodic responses of plates numerical methods that compute the nonlinear frequency response curves of plates are presented the equation of motion of the plate is derived by assuming first order shear deformation theory and considering geometrical type of nonlinearity it is discretized by the finite element method using bilinear quadrilateral elements the frequency response curve is computed by shooting and continuation methods parallel implementation of the shooting method is presented and its scalability is investigated

numerical methods for thermally stressed shallow shell equations abstract we develop efficient and accurate numerical methods to solve a class of shallow shell problems of the von karman type the governing equations form a fourth order coupled system of nonlinear biharmonic equations for the transverse deflection and airy s stress function a second order finite difference discretization with three iterative methods picard newton and trust region dogleg is proposed for the numerical solution of the nonlinear pde system three simple boundary conditions and two application motivated mixed boundary conditions are considered along with the nonlinearity of the system boundary singularities that appear when mixed boundary conditions are specified are the main numerical challenges two approaches that use either a transition function or local corrections are developed to deal with these boundary singularities all the proposed numerical methods are validated using carefully designed numerical tests where expected orders of accuracy and rates of convergence are observed a rough run time performance comparison is also conducted to illustrate the efficiency of our methods as an application of the methods a snap through thermal buckling problem is considered the critical thermal loads of shell buckling with various boundary conditions are numerically calculated and snap through bifurcation curves are also obtained using our numerical methods together with a pseudo arclength continuation method our results are consistent with previous studies

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

motion planning for control affine systems satisfying low order controllability conditions abstractthis paper is devoted to the motion planning problem for control affine systems by using trigonometric polynomials as control functions the class of systems under consideration satisfies the controllability rank condition with the lie brackets up to the second order the approach proposed here allows to reduce a point to point control problem to solving a system of algebraic equations the local solvability of that system is proved and formulas for the parameters of control functions are presented our local and global control design schemes are illustrated by several examples

buckling and vibration of polar orthotropic circular plate resting on winkler foundation buckling and vibrational behavior of polar orthotropic circular plates of linearly varying thickness is presented on the basis of classical plate theory the plate is resting on winkler type foundation an approximate solution has been obtained by ritz method which employs basis functions based upon static deflection of polar orthotropic plates the effect of elastic foundation and that of orthotropy on the natural frequency of plate has been illustrated for different values of taper parameter flexibility parameter in plane force and nodal diameter the critical buckling load for clamped and simply supported plates have been obtained a comparison of results with those available in literature has been presented

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

periodic and non periodic combination resonance in kinematically excited system of rods a theoretical investigation has been made of periodic and non periodic combination resonances of the system of rods under vertical and horizontal kinematic excitations the elements of the system are connected with articulated points the coupling of the elements of the system through internal longitudinal forces which are transverse forces at the ends of neighboring rods are taken into account the equations of motion are obtained from the lagrange equations the mathematical analysis of the equations of motion is accomplished by using tondl s technique the steady state solutions and their stability are determined resonance curves for the stationary states have been determined numerically

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

forced asymmetric response of linearly tapered circular plates the eigenfunction method is used to analyze the asymmetric response of linearly tapered circular plates subjected to transverse loads uniformly distributed over an annular sectorial area of the plate the analysis is based on the classical plate theory numerical results are presented graphically for the transverse deflection and stresses of the plate for various combinations of plate and loading parameters results obtained as a particular case for a plate of constant thickness subjected to an off center half sine pulse point load are compared with previously published results and found to match exactly

interpreting the forced responses of a two degree of freedom nonlinear oscillator using backbone curves in this paper the backbone curves of a two degree of freedom nonlinear oscillator are used to interpret its behaviour when subjected to external forcing the backbone curves describe the loci of dynamic responses of a system when unforced and undamped and are represented in the frequency amplitude projection in this study we provide an analytical method for relating the backbone curves found using the second order normal form technique to the forced responses this is achieved using an energy based analysis to predict the resonant crossing points between the forced responses and the backbone curves this approach is applied to an example system subjected to two different forcing cases one in which the forcing is applied directly to an underlying linear mode and the other subjected to forcing in both linear modes additionally a method for assessing the accuracy of the prediction of the resonant crossing points is then introduced and these predictions are then compared to responses found using numerical continuation

a finite element method for domains with corners a new finite element method is devised for the numerical solution of elliptic boundary value problems with geometrical singularities in it the singularity is eliminated form the computational domain in an exact fashion this is in contrast to other common methods such as those which use a refined mesh in the singularity region or those which use special singular finite elements in them the singularity is treated as a part of the numerical scheme the new method is illustrated on an elliptic differential equation in a domain containing a re entrant corner numerical experiments show that the new method yields result which are generally much more accurate than those obtained by using the standard finite element method with mesh refinement in the singularity region both methods require about the same computing time

corrected solvability conditions for non linear asymmetric vibrations of a circular plate an investigation into non linear asymmetric vibrations of a clamped circular plate under a harmonic excitation is made we re examined a primary resonance studied by sridhar mook and nayfeh in which the frequency of excitation is near the natural frequency of an asymmetric mode of the plate we corrected their solvability conditions and found that in the absence of internal resonance the steady state response can have not only the form of standing wave but also the form of travelling wave which is a remarkable contrast to their conclusion

asymmetric vibrations and elastic stability of polar orthotropic circular plates of linearly varying profile abstract asymmetric vibration of polar orthotropic circular plates of linearly varying thickness subjected to hydrostatic in plane force are discussed on the basis of classical plate theory an approximate solution of the problem has been obtained by the ritz method which employs functions based upon the static deflection of polar orthotropic plates this method has a faster rate of convergence as compared to the polynomial co ordinate functions frequency parameters of the plate with elastically restrained edge conditions are presented for the three modes of vibrations for various values of taper parameter rigidity ratio flexibility parameter and buckling load parameter the critical buckling loads for elastically restrained edge conditions have been obtained a comparison of results with those available in the literature shows an excellent agreement