geometrically nonlinear free vibrations of simply supported isotropic thin circular plates nonlinear free axisymmetric vibration of simply supported isotropic circular plates is investigated by using the energy method and a multimode approach in plane deformation is included in the formulation lagrange s equations are used to derive the governing equation of motion using the harmonic balance method the equation of motion is converted into a nonlinear algebraic form the numerical iterative method of solution adopted here is the so called linearized updated mode method which permits the authors to obtain accurate results for vibration amplitudes up to three times the plate thickness the percentage of participation of each out of plane basic function to the deflection shape and to the bending stress at the plate centre and of each in plane basic function to the membrane stress at the centre are calculated in order to determine the minimum number of in and out of plane basic functions to be used in order to achieve a good accuracy of the model the nonlinear frequency the nonlinear fundamental mode shape and their associated nonlinear bending and membrane stresses are determined at large amplitudes of vibration the numerical results obtained here are presented and compared with available published results based on various approaches and with the single mode solution the limit of validity of the single mode approach is also investigated

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

nonlinear vibration and dynamic stability analysis of composite plates large amplitude flexural vibration characteristics of composite plates under transverse harmonic pressure or periodic in plane load are investigated here using the shear deformable finite element method the nonlinear stiffness matrix is formulated based on von karman s assumptions to obtain the stiffness interaction between the in plane and bending degrees of freedom further the flexural motion of the plate is assumed to be harmonic and the in plane movement is assumed to be periodic the nonlinear matrix amplitude equation is obtained by employing galerkin s method the coupled nonlinear matrix amplitude equation in plane motion is coupled with flexural motion is solved to obtain 1 nonlinear free flexural vibration frequencies of isotropic and composite plates with different in plane boundary conditions 2 flexural vibration amplitudes of such plates under transverse harmonic pressure or periodic in plane load finally the time history analysis is carried out to understand the steady state or unsteady nature of the flexural vibration under different loading and boundary condition

two mode nonlinear vibration of orthotropic plates using method of multiple scales nonlinear forced oscillation of a rectangular orthotropic plate subjected to uniform harmonic excitation is solved using the method of multiple scales the governing equations are based on the von karman type geometrical nonlinearity and the effect of damping is included the general multimode solution is developed for simply supported boundary conditions and the solution is specialized for two symmetric modes analysis the primary resonances and the subharmonic and superharmonic secondary resonances are studied in detail hin laminated composite panels subjected to transverse periodic loadings can encounter deflections of the order of panel thickness or even higher the effect of these periodic excitations on the panel can be very severe responses of this kind cannot be predicted by linear theory consequently the need to study large deflections using nonlinear methods of analysis is of paramount importance the formulation of the equations governing the fundamen tal kinematic behavior of the laminated composite plates in the presence of the von karman geometrical nonlinearity is attributed to whitney and leissa 1 based on these equations various methods have been developed to solve nonlinear free and forced vibrations of composite panels a good survey on mainly nonlinear free and forced vibrations of isotropic plates is given in a book by nayfeh and mook 2 the most compre hensive work on geometrically nonlinear analysis of both static and dynamic behavior of the laminated panels through 1972 is collected in a book by chia 3 bert4 has conducted a survey on the dynamics of composite panels for the period of 1979 81 a review of literature on linear vibrations of plates can be found in a review paper by sathyamoorthy 5 relatively few investigations have been reported on the nonlinear forced vibration of isotropic or composite panels under harmonic excitations yamaki6 presented a one term solution for free and forced vibrations of the rectangular plates using galerkin s method lin7 studied the response of a nonlinear flat panel to periodic and randomly varying load ings nonlinear forced vibrations of beams and rectangular plates were studied by eisely8 using a single mode galerkin s method in conjunction with the linstedt duffing perturbation technique free and forced response of beams and plates undergoing large amplitude oscillations using the ritz averag ing method were studied by srinivasan 9 bennett10 studied the nonlinear vibration of simply supported angle ply laminated plates by considering the instability regions of the response of such plates subjected to harmonic excitations nonlinear free and forced vibration of a circular plate with clamped bound

a new numeric analytical principle for nonlinear deterministic and stochastic dynamical systems a new implicit local linearization method termed as the locally transversal linearization ltl is developed in the present paper for a numeric analytical integration of nonlinear dynamical systems under deterministic and or stochastic excitations the ltl procedure is developed in such a way that at a chosen time instant the ltl solution manifold transversally intersects the nonlinear solution manifold at that particular point or cross section in the state space where the solution vector needs to be obtained in general construction of the ltl equation is non unique and thus it is generally needed to substitute the conditionally linear ltl based solution into the given nonlinear ordinary differential equation ode at a specified time instant to ensure a transversal intersection in the present study however it is demonstrated that a transversal intersection at a given time instant may as well be ensured by just choosing a few specific forms of the ltl equation thereby eliminating the need for substitution this new variant of ltl is particularly suitable for discretization of stochastic differential equations sdes in that one simultaneously avoids dealing with formal derivatives of wiener processes while satisfying the nonlinear sde at discrete time points the stochastic ltl procedure is adaptable for obtaining strong pathwise solutions of a general nonlinear dynamical system with continuous but not necessarily differentiable vector fields under additive and or multiplicative excitations a specific advantage of the ltl method is that for a class of response regime herein referred to as the phase independent regime it can obtain an accurate description of response irrespective of the choice of the time step and without a propagation of local errors presently the method has been numerically illustrated for strong solutions of the hardening duffing oscillator under deterministic sinusoidal or additive stochastic white noise excitations or their combination limited numerical results on periodic solutions chaos with or without noise and stochastic resonance are included in this study the results presented are indicative of the sample path accuracy consistency stochastic numerical stability and a partial insensitivity to time step sizes of the proposed method

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

lower current large deviations for zero range processes on a ring we study lower large deviations for the current of totally asymmetric zero range processes on a ring with concave current density relation we use an approach by jensen and varadhan which has previously been applied to exclusion processes to realize current fluctuations by travelling wave density profiles corresponding to non entropic weak solutions of the hyperbolic scaling limit of the process we further establish a dynamic transition where large deviations of the current below a certain value are no longer typically attained by non entropic weak solutions but by condensed profiles where a non zero fraction of all the particles accumulates on a single fixed lattice site this leads to a general characterization of the rate function which is illustrated by providing detailed results for four generic examples of jump rates including constant rates decreasing rates unbounded sublinear rates and asymptotically linear rates our results on the dynamic transition are supported by numerical simulations using a cloning algorithm

on the deflected configuration of a slender elastic rod subject to parallel terminal forces and moments this paper investigates the three dimensional configurations of a slender elastic rod of uniform circular cross section subject to parallel terminal forces and moments the nonlinear equilibrium equations for the rod are established for a cartesian coordinate system and solved analytically without linearization consequently the results are applicable for large nonlinear elastic deformations

multi step tangential versus transversal linearizations in non linear dynamics a family of multi step tangential linearization mtnl techniques is developed for numeric analytic and geometric integration of non linear oscillators under deterministic loading a family of multi step transversal linearization mtrl schemes is also explored and a comparison between the two linearization families is provided with a view of bringing out their relative advantages and deficiencies the mtnl methodology essentially works through an application of the concept of mtrl to non linear rate equations different variants of the mtnl are obtained and it turns out that they are specific cases of a generalized mtnl method local error estimates are also provided moreover higher order mtrl and mtnl methods are successfully implemented through chebyshev nodes instead of equidistant nodes this helps to improve accuracy even with larger step sizes a comparison of relative errors through the mtrl and mtnl techniques consistently indicates a superior quality of approximations via the mtrl route whilst the usage of the tangential linearization is very common in non linear computational mechanics it is demonstrated that the transversal version offers in addition to higher numerical accuracy an easier implementation and avoids computation of jacobian matrices this allows the mtrl family to be applicable even to non linear oscillators having c0 vector fields copyright 2007 john wiley sons ltd

free vibration analysis of a flat plate using the hierarchical finite element method abstract the hierarchical finite element method is used to determined the natural frequencies and modes of a flat rectangular plate ten different boundary conditions including free edges and point supports are considered in this paper extensive results are presented for each case including the variation of frequency with the aspect ratio and the poisson ratio and these are shown to be in very good agreement with the work of other investigators this confirms both the applicability and accuracy of solution of the hfem to problem of this type

large amplitude axisymmetric vibrations of geometrically imperfect circular plates abstract this paper deals with the effects of geometric imperfections on the large amplitude vibrations of circular plates it is found that geometric imperfections of the order of a fraction of the plate thickness may significantly raise the linear vibration frequencies furthermore such imperfections may even change the inherent non linear hard spring character of the circular plates and cause them to exhibit soft spring behavior the effects of various boundary conditions are examined

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

finite element method for non linear forced vibrations of circular plates geometric non linearities for large amplitude free and forced vibrations of circular plates are investigated in plane displacement and in plane inertia are included in the formulation the finite element method is used an harmonic force matrix for non linear forced vibration analysis is introduced and derived various out of plane and in plane boundary conditions are considered the relations of amplitude and frequency ratio for different boundary conditions and various load conditions are presented

finite element analysis of the non linear vibrations of moderately thick unsymmetrically laminated composite plates abstract the influence of finite amplitudes on the free flexural vibration response of moderately thick laminated plates is investigated for this purpose a simple higher order theory involving only four unknowns and satisfying the stress free conditions at the top and bottom surface of the composite plate is proposed the proposed theory eliminates the use of shear correction factors which are otherwise required in mindlin s plate theory a rectangular four node formula continuous finite element is developed based on this theory the non linear finite element equations are reduced to two non linear ordinary differential equations governing the response of positive and negative deflection cycles direct numerical integration method is then employed to obtain the periods or non linear frequencies the finite element developed and the direct numerical integration method employed are validated for the case of isotropic rectangular plates it is found that unsymmetrically laminated rectangular plates with hinged hinged edge conditions oscillate with different amplitudes in the positive and negative deflection cycles furthermore such plates would oscillate with a frequency less than the fundamental frequency for finite small amplitudes of oscillation it is shown that this behaviour is strongly influenced by the boundary conditions results are presented for many configurations of composite plates

differential quadrature method for nonlinear vibration of orthotropic plates with finite deformation and transverse shear effect based on the reddy s theory of plates with the effect of higher order shear deformations the governing equations for nonlinear vibration of orthotropic plates with finite deformations are presented the nonlinear free vibration is analyzed by the differential quadrature method the differential quadrature approach suggested by wang and bert is extended to handle the multiple boundary conditions of the plate a new technique is also further extended to simplify nonlinear computations and the harmonic balance method is used in deriving the equation of motion the numerical convergence and comparison studies are carried out to validate the present solutions the results show that the presented differential quadrature method is fairly reliable and valid influences of geometric and material parameters transverse shear deformations and rotation inertia as well as vibration amplitudes on the nonlinear free vibration characteristics of orthotropic plates are studied

solving nonlinear differential equations of mechanics with the boundary element method and radial basis functions the boundary element method is a very effective method for solving linear differential equations to use it also in the consideration of non linear problems some different procedures were developed among them the dual reciprocity method and the particular integral method both procedures use interpolation conditions for the approximation with radial basis functions in this paper a method is presented which avoids problems connected with interpolation by means of quasi interpolation it is possible to solve differential equations of the kind mu p u approximately the application to two non linear problems of plate theory yield good results hints to a theoretical examination of the method including sufficient conditions for feasibility and convergence are given 1997 by john wiley sons ltd

the non linear free vibration of fully clamped rectangular plates second non linear mode for various plate aspect ratios abstract the theoretical model based on hamilton s principle and spectral analysis previously used to obtain the first three non linear modes of a clamped clamped beam 1 and the first non linear mode of a fully clamped rectangular plate 2 is used here in order to calculate the second non linear mode of a fully clamped rectangular plate the large vibration amplitude problem reduced to a set of non linear algebraic equations is solved numerically results are given for the second mode of fully clamped rectangular plates for various plate aspect ratios and vibration amplitudes the non linear mode shows a higher bending stress near to the clamps at large deflections compared with that predicted by linear theory

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

hydrodynamic limit of mean zero condensing zero range processes with sub critical initial profiles condensing zero range processes are interacting particle systems with zero range interaction exhibiting phase separation at densities above a finite critical density we prove the hydrodynamic limit of mean zero condensing zero range processes with bounded local jump rate for sub critical initial profiles i e for initial profiles that are everywhere strictly below the critical density the proof is based on h t yau s relative entropy method and is made possible by a generalisation of the one block estimate

numerical results from large deflection beam and frame problems analysed by means of elliptic integrals numerical evaluations of elliptic integral solutions of some large deflection beam and frame problems are presented the values are given in tabular form with up to six significant figures the numerical technique used for evaluating the elliptic integrals is described

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

large deviations of the current for driven periodic diffusions we study the large deviations of the time integrated current for a driven diffusion on the circle often used as a model of nonequilibrium systems we obtain the large deviation functions describing the current fluctuations using a fourier bloch decomposition of the so called tilted generator and we also construct from this decomposition the effective biased auxiliary or driven markov process describing the diffusion as current fluctuations are observed in time this effective process provides a clear physical explanation of the various fluctuation regimes observed it is used here to obtain an upper bound on the current large deviation function which we compare to a recently derived entropic bound and to study the low noise limit of large deviations

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

non linear vibrations of simply supported rectangular cross ply plates a method of direct numerical integration of the frequency ratio expression is proposed to study the non linear free vibration behaviour of rectangular cross ply laminates the proposed method even with single term approximations for the admissible functions yields results that agree very well with the existing perturbation solutions non linear behaviour of the cross ply laminates is also studied with the harmonic oscillations assumption by using the conservation of energy and the modal equation the results are found to be lower and upper bounds to those obtained from the direct numerical integration method it is also observed that the arithmetic mean of the two solutions with the harmonic oscillations assumption matches very well with that of the direct numerical integration method non linear vibration characteristics are obtained for several configurations of cross ply plates results for orthotropic and isotropic plates are also obtained as special cases

the effects of large vibration amplitudes on the axisymmetric mode shapes and natural frequencies of clamped thin isotropic circular plates part ii iterative and explicit analytical solution for non linear coupled transverse and in plane vibrations the objective of this paper is to present a more realistic and complete study of the geometrically non linear free vibrations of clamped immovable circular plates by taking into account the in plane deformation which has not been examined in part i of this series of papers j sound vibration 265 2003 123 the problem is solved by a numerical iterative procedure in order to obtain more accurate results for vibration amplitudes up to twice the plate thickness the numerical results thus obtained are presented and compared with the available published results and with the ones calculated when neglecting the in plane displacement for the first two non linear axisymmetric mode shapes an explicit analytical solution is then presented and its range of validity for the fundamental non linear mode is determined via a detailed comparison with the solution based on the iterative procedure the results obtained by the explicit method show the usefulness of the new approach in comparison with the single mode approach solution frequently used in non linear vibration analysis of structures and are expected to be easy to implement in fatigue models in order to make more realistic and secure predictions of the structural fatigue life

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

fully nonlinear model of cables in this paper an energy approach is used to develop a fully nonlinear cable model which contains most cable models as special cases the theory accounts for the effects of static and dynamic loads initial sags compressibility material nonuniformity poisson s effect and geometric nonlinearities also a cubic nonlinear equation of motion is derived for sagged cables taut strings and extensional bars and compare the model with several other models

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

nonlinear vibration analysis for a generic coupled laminated plate with surface bonded or embedded induced strain actuators abstract studies are made on the nonlinear vibrations of piezo laminated rectangular thin plates with induced strain actuation by following kirchoff s hypothesis and using strain displacement relations of von karman type the von karman s large deflection equations for generally laminated elastic plates are derived in terms of stress function and transverse deflection function a deflection function satisfying the geometric boundary conditions is assumed and a stress function is then obtained after solving the compatibility equation the modified galerkin s method is applied to the governing nonlinear partial differential equation to obtain the nonlinear ordinary differential equation of motion modal equation this is of duffing s type analytical expressions for the constants in the modal equation are provided to use for any lay up sequence procedure for exact integration of the modal equation is described numerical results of simply supported rectangular plates with immovable edges are presented

differential quadrature for static and free vibration analyses of anisotropic plates abstract the differential quadrature method is used to analyse the deflection buckling and free vibration behavior of anisotropic rectangular plates under various boundary conditions the roots of chebyshev polynomials are used to obtain grid point locations and a new approach is used to apply boundary conditions results compare very well with existing numerical data and less computational effort is required for the problems considered

a semi analytical approach to the non linear dynamic response problem of beams at large vibration amplitudes part ii multimode approach to the steady state forced periodic response abstract the semi analytical approach to the non linear dynamic response of beams based on multimode analysis has been presented in part i of this series of papers azrar et al 1999 journal of sound and vibration 224 183 207 1 the mathematical formulation of the problem and single mode analysis have been studied the objective of this paper is to take advantage of applying this semi analytical approach to the large amplitude forced vibrations of beams various types of excitation forces such as harmonic distributed and concentrated loads are considered the governing equation of motion is obtained and can be considered as a multi dimensional form of the duffing equation using the harmonic balance method the equation of motion is converted into non linear algebraic form techniques of solution based on iterative incremental procedures are presented the non linear frequency and the non linear modes are determined at large amplitudes of vibration the basic function contribution coefficients to the displacement response for various beam boundary conditions are calculated the percentage of participation for each mode in the response is presented in order to appraise the relation to higher modes contributing to the solution also the percentage contributions of the higher modes to the bending moment near to the clamps are given in order to determine accurately the error introduced in the non linear bending stress estimated by different approximations solutions obtained in the jump phenomena region have been determined by a careful selection of the initial iteration at each frequency the non linear deflection shapes in various regions of the solution the corresponding axial force ratios and the bending moments are presented in order to follow the behaviour of the beam at large vibration amplitudes the numerical results obtained here for the non linear forced response are compared with those from the linear theory with available non linear results based on various approaches and with the single mode analysis

a three node mindlin plate element with improved transverse shear abstract a displacement methodology for mindlin elements recently employed in the development of an efficient four node quadrilateral min4 is the basis for a three node explicitly integrated triangular element min3 the approach herein referred to as anisoparametric is founded upon two kinematic requirements that are auxiliary to the standard convergence criteria the initial complete quadratic deflection field is constrained by continuous shear edge constraints to achieve an isoparametric like three node configuration much success of the element is due to an innovative shear correction factor concept several numerical experiments are performed in the linear elasto static regime to ascertain the element performance

nonlinear curvature based model and resonant finite amplitude vibrations of symmetric cross ply laminates abstract moving from a general plate theory a modified general classical laminated plate theory mgclpt exhibiting nonlinear curvatures but still allowing for some worth features of linear curvature models von karman is formulated starting from mgclpt partial differential equations a minimal discretized model suitable for the analysis of resonant finite amplitude vibrations of symmetric cross ply laminates with immovable or movable supports is obtained via the galerkin procedure periodic responses of a single mode model and of a 3 1 internally resonant two mode model excited at primary resonance are obtained via the multiple time scale method the influence of various system parameters thickness ratio plate aspect number of laminae kind of material mode number is addressed and the comparison of nonlinear vibration results as obtained with the mgclpt and the von karman models for different boundary conditions shows some interesting differences

an implementation of the hp version of the finite element method for reissner mindlin plate problems reissner mindlin plate theory is still a topic of research in finite element analysis one reason for the continuous development of new plate elements is that it is still difficult to construct elements which are accurate and stable against the well known shear locking effect in this paper we suggest an approach which allows high order polynomial degrees of the shape functions for deflection and rotations a balanced adaptive mesh refinement and increase of the polynomial degree in an hp version finite element program is presented and it is shown in numerical examples that the results are highly accurate and that high order elements show virtually no shear locking even for very small plate thickness

comparison of two linearization schemes for the nonlinear bending problem of a beam pinned at both ends the nonlinear bending problem of a constant cross section simply supported beam pinned at both ends and subject to a uniformly distributed load q x is analyzed in detail the numerical integration of the two point boundary value problem bvp derived for the nonlinear timoshenko beam is tackled through two different linearization schemes the multi step transversal linearization mtrl and the multi step tangential linearization mtnl proposed by viswanath and roy 2007 the fundamentals of these linearization techniques are to replace the nonlinear part of the governing odes through a set of conditionally linearized ode systems at the nodal grid points along the neutral axis ensuring the intersection between the solution manifolds transversally in the mtrl and tangentially in the mtnl in this paper the solution values are determined at grid points by means of a centered finite differences method with multipoint linear constraints keller 1969 and a simple iterative strategy the analytical solution for this kind of bending problem including the extensional effects can be worked out by integration of the governing two point bvp equations monleon et al 2008 finally the comparison of analytical and numerical results shows the better ability of mtnl with the proposed iterative strategy to reproduce the theoretical behavior of the beam for each load step because the restraint of equating derivatives in mtnl leads to further closeness between solution paths of the governing odes and the linearized ones in comparison with mtrl this result is opposed to the conclusion reached in viswanath and roy 2007 where the relative errors produced by mtrl are said to be smaller than the mtnl ones for the simply supported beam and the tip loaded cantilever beam problems

multi step transversal and tangential linearization methods applied to a class of nonlinear beam equations a novel and continuously parameterized form of multi step transversal linearization mtrl method is developed and numerically explored for solving nonlinear ordinary differential equations governing a class of boundary value problems bvps of relevance in structural mechanics a similar family of multi step tangential linearization mtnl methods is also developed and applied to such bvp s within the framework of mtrl and mtnl a bvp is treated as a constrained dynamical system i e a constrained initial value problem ivp while the mtrl requires the linearized solution manifold to transversally intersect the nonlinear solution manifold at a chosen set of points across the axis of the independent variable the essential difference of the present mtrl method from its previous version roy d kumar r 2005 a multistep transversal linearization mtl method in nonlinear structural dynamics j sound vib 17 829 852 is that it has the flexibility of treating nonlinear damping and stiffness terms as time variant damping and stiffness terms in the linearized system the resulting time variant linearized system is then solved using magnus characterization magnus w 1954 on the exponential solution of differential equations for a linear operator commun pure appl math 7 649 673 towards numerical illustrations response of a tip loaded cantilever beam elastica is first obtained next the response of a simply supported nonlinear timoshenko beam is obtained using a variationally correct vc model for the beam marur s prathap g 2005 nonlinear beam vibration problems and simplification in finite element model comput mech 35 5 352 360 the new model does not involve any simplifications commonly employed in the finite element formulations in order to ease the computation of nonlinear stiffness terms from nonlinear strain energy terms a comparison of results through mtrl and mtnl techniques consistently indicate a superior quality of approximations via the transversal linearization technique while the usage of tangential system matrices is common in nonlinear finite element practices it is demonstrated that the transversal version of linearization offers an easier and more general implementation requires no computations of directional derivatives and leads to a consistently higher level of numerical accuracy it is also observed that higher order versions of mtrl mtnl with lagrangian interpolations may not work satisfactorily and hence spline interpolations are suggested to overcome this problem

sinc galerkin method for solving nonlinear boundary value problems the sinc galerkin method is used to approximate solutions of nonlinear problems involving nonlinear second fourth and sixth order differential equations with homogeneous and nonhomogeneous boundary conditions the scheme is tested on four nonlinear problems the results demonstrate the reliability and efficiency of the algorithm developed

first order shear deformation p version finite element for laminated plate nonlinear vibrations a p version hierarchical finite element for moderately thick composite laminated plates is presented where the effects of the rotary inertia transverse shear and geometrical nonlinearity are taken into account the time domain free vibration equations of motion are obtained by applying the principle of virtual work those equations are mapped into the frequency domain by the harmonic balance method and solved by a predictor corrector procedure the linear natural frequencies of vibration of several plates are determined and the convergence properties of the element are investigated it is shown that the element is not prone to shear locking and that a moderate number of degrees of freedom is sufficient for accuracy the influences of the plate s thickness of the width to length ratio and of the fiber orientation on nonlinear free vibrations are investigated

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

diffusions conditioned on occupation measures a markov process fluctuating away from its typical behavior can be represented in the long time limit by another markov process called the effective or driven process having the same stationary states as the original process conditioned on the fluctuation observed we construct here this driven process for diffusions spending an atypical fraction of their evolution in some region of state space corresponding mathematically to stochastic differential equations conditioned on occupation measures as an illustration we consider the langevin equation conditioned on staying for a fraction of time in different intervals of the real line including the positive half line which leads to a generalization of the brownian meander problem other applications related to quasi stationary distributions metastable states noisy chemical reactions queues and random walks are discussed

the influence of boundary conditions and transverse shear on the vibration of angle ply laminated plates circular cylinders and cylindrical panels abstract this paper investigates the influence of the edge boundary conditions on the vibration characteristics of transverse shear deformable composite plates closed cylindrical shells and open cylindrical panels having an arbitrary angle ply lay up the shell model employed for this purpose is equivalent to the unified shear deformable love type theory introduced in refs 6 7 whereas the analysis is based on the ritz method following some recent relevant developments 1 5 in which the method was applied successfully in connection with vibration problems of shear deformable cross ply laminates the ritz method is employed in conjunction with complete functional bases formed by appropriate orthonormal polynomials the present vibration study is based on angle ply laminates having different combinations of simply supported clamped and free edge boundaries apart from the convergence tests performed and the new results presented the success of the present analysis is verified through comparisons with the corresponding results based on classical plate and shell theories as well as through comparisons with corresponding numerical and experimental results that are available in the literature

reduced order models for microelectromechanical rectangular and circular plates incorporating the casimir force we consider the von karman nonlinearity and the casimir force to develop reduced order models for prestressed clamped rectangular and circular electrostatically actuated microplates reduced order models are derived by taking flexural vibration mode shapes as basis functions for the transverse displacement the in plane displacement vector is decomposed as the sum of displacements for irrotational and isochoric waves in a two dimensional medium each of these two displacement vector fields satisfies an eigenvalue problem analogous to that of transverse vibrations of a linear elastic membrane basis functions for the transverse and the in plane displacements are related by using the nonlinear equation governing the plate in plane motion the reduced order model is derived from the equation yielding the transverse deflection of a point for static deformations of a plate the pull in parameters are found by using the displacement iteration pull in extraction method reduced order models are also used to study linear vibrations about a predeformed configuration it is found that 9 basis functions for a rectangular plate give a converged solution while 3 basis functions give pull in parameters with an error of at most 4 for a circular plate 3 basis functions give a converged solution while the pull in parameters computed with 2 basis functions have an error of at most 3 the value of the casimir force at the onset of pull in instability is used to compute device size that can be safely fabricated

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

frequency analysis of symmetrically laminated plates with free edges abstract this paper is concerned with the estimation of natural frequencies of undamaged symmetrically laminated plates rectangular circular annular and elliptical having completely free edges the rayleigh ritz energy approach is employed here to obtain the approximate natural frequencies and the associated mode shapes the plates are considered to be thin and to undergo small deflections satisfying kirchhoff s hypothesis a complete power series is used as admissible functions for in plane and out of plane displacements in the numerical examples the accuracy of the present results is compared with the other analytical results available in the literature the effects of the shape and size of the plate number of plies and the fiber orientation on the natural frequencies flexural mode of symmetrically laminated plates are documented the mode shape corresponding to each frequency has been identified in this paper

analysis of vibrating thick rectangular plates with mixed boundary constraints using differential quadrature element method a free vibration analysis of moderately thick rectangular plates with mixed boundary conditions is presented on the basis of the first order shear deformation plate theory the differential quadrature element method a highly efficient and accurate hybrid approach has been employed to establish the numerical model the complex plate domain is first decomposed into small simple continuous sub domains elements and the differential quadrature method is then applied to each continuous sub domain to solve the problems compatibility conditions are developed for the conjunction nodes on the interface boundaries of elements in order to connect the elements convergence and comparison studies are carried out to establish the reliability of the solutions the first eight frequency parameters are predicted for various types of thick rectangular plates with mixed edge constraints

avoiding locking in reissner mindlin plates with high order hierarchic bubble plus finite elements abstract in this paper we prove that special finite elements of suitable degree do not exhibit any kind of locking phenomenon when used for the numerical solution of reissner mindlin plate bending problems despite its simple approach the discretization of the reissner mindlin model is not straightforward the inclusion of transverse shear strain effect in standard finite element models introduces undesirable numerical effects the approximate solution is very sensitive to the plate thickness and for small thickness it is very far from the true solution the phenomenon is known as locking of the numerical solution the finite element we present is constructed by adding bubble functions to standard serendipity elements the new elements referred to as bubble plus elements from the degree three onwards avoid the numerical locking the mathematical proof of such a good behavior is based on a deep relation between the c 0 bubble plus element of degree three for reissner mindlin plates and the c 1 element for kirchhoff plates known as bogner fox schmit element

thermodynamic limit for the invariant measures in supercritical zero range processes we prove a strong form of the equivalence of ensembles for the invariant measures of zero range processes conditioned to a supercritical density of particles it is known that in this case there is a single site that accomodates a macroscopically large number of the particles in the system we show that in the thermodynamic limit the rest of the sites have joint distribution equal to the grand canonical measure at the critical density this improves the result of groskinsky schutz and spohn where convergence is obtained for the finite dimensional marginals we obtain as corollaries limit theorems for the order statistics of the components and for the fluctuations of the bulk

free vibration of structures with cubic non linearity remarks on amplitude equation and rayleigh quotient in the paper several aspects concerning the formulation of and solution to amplitude equations for free vibration of systems with cubic non linearity are discussed in detail the use of several possible constraint equations to achieve the non trivial solution to the amplitude equation is also presented the well known concept of the rayleigh quotient is extended to the non linear case of interest some general remarks on the formulation of the non linear amplitude equation and the non linear rayleigh quotient are also presented moreover the introduced rayleigh quotient is used to develop a new and suitably efficient method to solve amplitude equations the accuracy of solutions obtained by means of some widely used methods of analysis of free vibration problems is also presented it is found that the galerkin method gives the most accurate results

non linear forced vibrations of plates by an asymptotic numerical method abstract non linear forced vibrations of thin elastic plates have been investigated by an asymptotic numerical method anm various types of harmonic excitation forces such as distributed and concentrated are considered using the harmonic balance method and hamilton s principle the equation of motion is converted into an operational formulation based on the finite element method a starting point corresponding to a non linear solution associated to a given frequency and amplitude of excitation is computed applying perturbation techniques in the vicinity of this solution the non linear governing equation obtained is transformed into a sequence of linear problems having the same stiffness matrix employing one matrix inversion a large number of terms of the perturbation series of the displacement and frequency can be easily computed with a small computation time iterations of this method lead to a powerful path following technique comprehensive numerical tests for forced vibrations of plates subjected to time harmonic lateral excitations are reported

a multi step transversal linearization mtl method in non linear structural dynamics an implicit family of multi step transversal linearization mtl methods is proposed for efficient and numerically stable integration of nonlinear oscillators of interest in structural dynamics the presently developed method is a multi step extension and further generalization of the locally transversal linearization ltl method proposed earlier by roy proceedings of the academy of the royal society london 457 2001 539 566 roy and ramachandra journal of sound and vibration 41 2001a 653 679 international journal for numerical methods in engineering 51 2001b 203 224 and roy international journal of numerical methods in engineering 61 2004 764 the mtl based linearization is achieved through a non unique replacement of the nonlinear part of the vector field by a conditionally linear interpolating expansion of known accuracy whose coefficients contain the discretized state variables defined at a set of grid points in the process the nonlinear part of the vector field becomes a conditionally determinable equivalent forcing function the mtl based linearized differential equations thus become explicitly integrable based on the linearized solution a set of algebraic constraint equations are so formed that transversal intersections of the linearized and nonlinearized solution manifolds occur at the multiple grid points the discretized state vectors are thus found as the zeros of the constraint equations simple error estimates for the displacement and velocity vectors are provided and in particular it is shown that the formal accuracy of the mtl methods as a function of the time step size depends only on the error of replacement of the nonlinear part of the vector field presently only two different polynomial based interpolation schemes are employed for transversal linearization viz the taylor like interpolation and the lagrangian interpolation while the taylor like interpolation leads to numerical ill conditioning as the order of interpolation increases the lagrangian interpolation is shown to overcome this numerical problem finally the family of mtl methods is illustrated through limited numerical results for a couple of harmonically driven workhorse oscillators viz the hardening duffing and the duffing holmes oscillators comparisons with results obtained via a sixth order runge kutta method with adaptive time step sizes are also provided

current large deviations for asymmetric exclusion processes with open boundaries we study the large deviation functional of the current for the weakly asymmetric simple exclusion process in contact with two reservoirs we compare this functional in the large drift limit to the one of the totally asymmetric simple exclusion process in particular to the jensen varadhan functional conjectures for generalizing the jensen varadhan functional to open systems are also stated

on continuous solutions of a generalized cauchy riemann system with more than one singularity abstract we study the solvability of the riemann hilbert problem for a generalized cauchy riemann system with several singularities and reveal several new phenomenon for the number of continuous solutions we shall show that it depends not only on the index but also on the location and type of the singularities moreover it does not depend continuously on the coefficients of the equation

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

theory of vibrating diaphragm type pressure sensor starting with the von karman equation the series solution of the problem of nonlinear bending of circular thin plates under the action of uniform pressure and uniform initial edge tension preloaded at the periphery is obtained based on this solution small vibrations in the vicinity of the static equilibrium configuration are discussed here a series method is suggested in which the characteristic value problem is reduced to a third order algebraic characteristic value problem with explicit expression and thus the characteristic value problem of axisymmetric free vibration is solved precisely

a coupled field study on the non linear dynamic characteristics of an electrostatic micropump abstract in this paper a micropump actuated by electrostatic forces is dynamically analyzed coupled electro mechanical effects are considered in the evaluation of the performance of the electrostatic micropump the boundary element method is employed here to solve the quasi 3 d laplace equation that the potential difference satisfies in order to obtain the surface charge density and corresponding electrostatic force first order shear deformation theory is used to model the electrode membrane of the pump geometric non linearity arises due to the inclusion of von karman strains the finite element method is employed to discretize the governing equations and newton s iteration method is employed to solve the discretized equations with the electro mechanical coupling effects considered within the framework of linear plate theory i e ignoring the von karman strains similar response trends are obtained for the 2 d plate analysis as that of 1 d analysis found in open literature the present study is extended to non linear analysis with von karman strains included and non linear load deflection relationship is demonstrated variation of the amplitude and frequency of the potential difference applied across the two electrodes are investigated and responses are compared with those of linear analysis qualitatively different responses are observed also the effects of the length to thickness ratio of the electrode plate are examined in detail

a refined nonlinear theory of plates with transverse shear deformation abstract a higher order shear deformation theory of plates accounting for the von karman strains is presented the theory contains the same dependent unknowns as in the hencky mindlin type first order shear deformation theory and accounts for parabolic distribution of the transverse shear strains through the thickness of the plate exact solutions of simply supported plates are obtained using the linear theory and the results are compared with the exact solutions of 3 d elasticity theory the first order shear deformation theory and the classical plate theory the present theory predicts the deflections stresses and frequencies more accurately when compared to the first order theory and the classical plate theory

condensation in stochastic particle systems with stationary product measures we study stochastic particle systems with stationary product measures that exhibit a condensation transition due to particle interactions or spatial inhomogeneities we review previous work on the stationary behaviour and put it in the context of the equivalence of ensembles providing a general characterization of the condensation transition for homogeneous and inhomogeneous systems in the thermodynamic limit this leads to strengthened results on weak convergence for subcritical systems and establishes the equivalence of ensembles for spatially inhomogeneous systems under very general conditions extending previous results which focused on attractive and finite systems we use relative entropy techniques which provide simple proofs making use of general versions of local limit theorems for independent random variables

application of the r function method to nonlinear vibrations of thin plates of arbitrary shape in this paper free large amplitude flexural vibrations of thin plates with various planforms and boundary conditions are studied by the r function method this method is based on the joint application of the r function theory and variational methods the main feature of the r function theory is the possibility to present all geometric information given in the boundary value problem in analytical form which allows one to seek a solution in the form of some formula called the solution structure a method of constructing the solution structures for the given nonlinear vibration plate bending problems is developed numerical examples of large amplitude flexural vibrations of thin plates with arbitrary shapes and boundary conditions for illustrating the aforementioned r function method and comparison against the other methods are made to demonstrate its merits and advantages

oscillations and instability of a shallow arch under two frequency excitation a shallow sinusoidal elastic arch with pinned ends is considered it is subjected to a distributed load with sinusoidal spatial distribution and an amplitude comprised of two harmonic components the equation for the response amplitude contains quadratic and cubic non linearities as well as damping and it is solved by numerical integration the effects of the arch rise and the excitation frequencies on the critical load are investigated including the influence of primary subharmonic superharmonic and combination resonances to study the response of the arch time histories phase plane portraits and power spectra are plotted for a number of loading conditions various types of limit cycles occur and in some cases the motion is chaotic

dynamics of the bilinear mindlin plate element plate and shell finite elements based upon mindlin plate theory and evaluated by one point quadrature are attractive because of their effectiveness for both thick and thin shells at minimal computing cost however these elements are rank deficient and must be stabilized to obtain reliable behaviour methods for achieving full rank of the stiffness and for stabilizing element behaviour in static analysis and in explicit dynamic calculations exist and are quite effective this paper addresses the remaining issue of controlling spurious modes of response in vibration analysis and implicit dynamic solutions several alternatives for the element mass formulation are examined in detail we show that non physical dynamic modes present a potential problem with most mass matrix formulations and that spurious modes other than the familiar hourglassing motion are possible a combination of projection methods and reduced quadrature is suggested which eliminates these deficiencies and produces accurate numerical results the remaining techniques investigated give rise to anomalous behaviour which make them unsuitable for general use