shape and morse theory of attractors we study the shape in the sense of borsuk of attractors of continuous semi dynamical systems on general metric spaces we show in particular that the natural inclusion of the global attractor into the state space is a shape equivalence this and other results of the paper are used to develop an elementary morse theory of an attractor 2000 john wiley sons inc

generic buckling curves for specially orthotropic rectangular plates using a double affine transformation the classical buckling equation for specially orthotropic plates and the corresponding virtual work theorem are presented in a particularly simple fashion these dual representations are characterized by a single material constant called the generalized rigidity ratio whose range is predicted to be the closed interval from 0 to 1 if this prediction is correct then the numerical results using a ratio greater than 1 in the specially orthotropic plate literature are incorrect when natural boundary conditions are considered a generalized poisson s ratio is introduced thus the buckling results are valid for any specially orthotropic material hence the curves presented in the text are generic rather than specific the solution trends are twofold the buckling coefficients decrease with decreasing generalized rigidity ratio and when applicable they decrease with increasing generalized poisson s ratio since the isotropic plate is one limiting case of the above analysis it is also true that isotropic buckling coefficients decrease with increasing poission s ratio

a cohomological conley index for maps on metric spaces we define a cohomological conley index of an isolated invariant set of a time discrete semidynamical system on a metric space we prove the homotopy invariance and the additivity properties of the index for applications to time periodic parabolic partial differential equations pdes the reader is referred to the authors paper nontrivial full bounded solutions of time periodic semilinear parabolic pdes proc roy soc edinburgh to appear

new result on pid controller design of lti systems via dominant eigenvalue assignment this note considers the problem of assigning the dominant eigenvalues of a linear time invariant lti system to the desired positions by using proportional integral derivative pid controllers the procedure is based on first setting some rightmost eigenvalues of the system at the desired positions and then guaranteeing the dominance of those rightmost eigenvalues by using the generalization of the hermite biehler theorem it is worth pointing out that this work aims to ascertain the gains of pid controllers in a straightforwardly computational way which plays an important role in practical applications

an index product formula for the study of elliptic resonance problems abstract let p u au f u be a semilinear parabolic equation if f 0 0 and f is of class c1 in a neighborhood of 0 then there exists a local center manifold m near zero containing all small invariant sets of p the purpose of this paper is to prove an index product formula relating the homotopy index h k of a small isolated invariant set k relative to p to the homotopy index hm k of the same set with respect to the equation induced by p on the center manifold m this formula can be applied to elliptic bvp with resonance at zero in particular earlier results of amann and zehnder ann scuola norm sup pisa iv 7 1980 534 603 can be obtained under less restrictive assumptions than those used in that paper further more the formula permits applications to cases not discussed in amann and zehnder s paper the applications of the index product formula are given in k p rybakowski nontrivial solutions of elliptic boundary value problems with resonance at zero ann mat pura appl to appear

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

the connection matrix theory for semiflows on not necessarily locally compact metric spaces abstract the index theory of rybakowski for isolated invariant sets and attractor repeller pairs in the setting of a semiflow on a not necessarily locally compact metric space is extended to include a connection matrix theory for morse decompositions partially ordered morse decompositions and attractor semifiltrations of invariant sets are defined and shown to be equivalent the definition and proof of existence of index filtrations for an ordered morse decomposition is provided via the index filtration the homology index braid and the connection matrices of the morse decomposition are defined

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

counterexamples to some conjectures on the number of solutions of nonlinear equations on construit trois contre exemples resolvant par la negative certains problemes d analyse non lineaire

divergence and flutter instability of elastic specially orthotropic plates subject to follower forces abstract divergence and flutter instabilities of a rectangular specially orthotropic plate simply supported on a pair of opposite sides subject to follower forces are computationally analyzed the effects of the tangential follower parameter aspect ratio boundary condition on the side opposite the loaded edge and material orthotropy on the magnitude of the critical load and the mode divergence or flutter of instability are detailed for the three boundary conditions considered built in simply supported and free free for the side opposite the loaded edge and given material orthotropy the plane is divided into regions corresponding to different governing instabilities both the number and size of these regions depend strongly upon the boundary condition considered as does the governing instability for a given boundary condition the configuration of the stability regions is independent of material orthotropy material orthotropy affects only the buckling loads corresponding to the instabilities and the size of the regions

global bifurcation of periodic solutions to ordinary differential equations abstract general homotopy continuation and bifurcation results are discussed and proved for a class of semiflows these results and further conley homotopy index methods are applied to obtain a type of global bifurcation for periodic solutions of second order ordinary differential equations

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

on global dynamics of reaction diffusion systems at resonance abstract in this paper we use the homotopy invariants methods to study the global dynamics of the reaction diffusion systems that are at resonance at infinity considering degrees of the resonance for the nonlinear perturbation we establish landesman lazer type conditions and use them to express the rybakowski conley index of the invariant set consisting of all bounded solutions obtained results are applied to study the existence of solutions connecting stationary points for the system of nonlinear heat equations

technical communique the schur stability via the hurwitz stability analysis using a biquadratic transformation in this note it is shown that instead of a bilinear transformation a biquadratic transformation can be used to determine the schur stability of a given discrete time polynomial by determining the hurwitz stability of the corresponding continuous time polynomial however using a biquadratic transformation the analysis is not limited by any restriction on the root positions several numerical examples are provided to illustrate the procedure

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

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

sturm attractors for quasilinear parabolic equations abstract the goal of this paper is to construct explicitly the global attractors of quasilinear parabolic equations as it was done for the semilinear case by brunovsk and fiedler 1986 8 and generalized by fiedler and rocha 1996 13 in particular we construct heteroclinic connections between hyperbolic equilibria stating necessary and sufficient conditions for heteroclinics to occur such conditions can be computed through a permutation of the equilibria lastly an example is computed yielding the well known chafee infante attractor

the morse index repeller attractor pairs and the connection index for semiflows on noncompact spaces abstract this paper extends the morse index theory of c c conley to semiflows on a noncompact meric space x is assumed to satisfy a hypothesis related to conditional contraction we collect background material define quasi index pairs and the morse index of a compact isolated invariant set k and prove that the morse index is a connected simple system we study repeller attractor pairs in k define index triples and prove their existence and several properties leading to the concepts of the connection index the connection map and the splitting class finally we consider paths continuous families of pairs k and study continuations of the morse and the connection indices along such paths the present paper is a sequel to the author s previous work on the homotopy index for infinite dimensional semiflows trans amer math soc 269 1982 351 382

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

a cohomological conley index in hilbert spaces and applications to strongly indefinite problems abstract a cohomological conley index is defined for flows on infinite dimensional real hilbert spaces generated by vector fields of the form f h h f x lx k x where l h h is a bounded linear operator satisfying certain technical assumptions and k is a completely continuous perturbation generalized morse inequalities for morse decompositions of isolated invariant sets are proved simple examples are presented to show how the theory can be applied to strongly indefinite problems

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

bifurcating continua in infinite dimensional dynamical systems and applications to differential equations abstract general homotopy continuation and bifurcation results are proved for a class of semiflows these results are applied to ordinary differential equations and to systems of reaction diffusion equations

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

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

computation of all stabilizing pid gains for digital control systems this note determines the set of all proportional integral derivative pid gains that can stabilize a given discrete time plant of arbitrary order utilizing earlier results obtained for the continuous time case and the bilinear transformation it is shown that a complete solution can be obtained using linear programming several numerical examples are presented to illustrate the procedure

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

nested sequences of index filtrations and continuation of the connection matrix abstract in this paper we prove the existence of nested sequences of index filtrations for convergent sequences of admissible semiflows on a metric space this result is new even in the context of flows on a locally compact space the nested index filtration theorem implies the continuation of homology index braids which in turn implies the continuation of connection matrices in the infinite dimensional conley index theory

design of digital pid controllers using the parameter space approach in this paper a parameter space approach is taken for designing digital pid controllers the stability domains of the coefficients of the controllers are computed the existing continuous time results are extended to the case of discrete time systems in this approach the stability region is obtained in the plane of two auxiliary controller coefficients by assuming a fixed value for a third auxiliary controller coefficient the stability region is defined by several line segments or equivalently by several linear equalities and inequalities then through mapping from the auxiliary coefficient space to the original controller coefficient space exact stability domain in the kp ki kd space is obtained the method is also extended for locating the closed loop poles of pid control systems inside the circles with arbitrary radii centred at the origin of the z plane the results can be used in the design of dead beat control systems

trajectories joining critical points of nonlinear parabolic and hyperbolic partial differential equations abstract consider the following nonlinear dirichlet boundary value problems d t u t x lu t x f u t x t 0 x u t x 0 t 0 x 1 1 d t d t u t x lu t x l d t u t x f u t x 0 t 0 x u t x 0 t 0 x 1 2 lu x f u x x u x 0 x 1 3 in all of these equations f r r is a locally lipschitzian asymptotically linear function with positive asymptotic slope f 0 0 and l is a self adjoint negativedefinite and strongly elliptic second order differential operator on a smooth domain in rn the solutions of 1 1 and 1 2 generate semiflows which are not pointdissipative and whose equilibria are determined by solutions of 1 3 in this paper using an extension due to the present author of conley s morse index theory to noncompact spaces we prove not only the existence of positive solutions of 1 3 a result shown earlier by peitgen and schmitt using different methods but also show the existence of nonconstant heteroclinic orbits of 1 1 and 1 2 joining two sets of equilibria

conley index and tubular neighborhoods ii abstract this paper is a sequel to our previous paper 11 we prove existence of nested index filtrations for tubular singular semiflow perturbations this generalizes a corresponding result from 7 and is applicable to cases in which the phase space of the perturbed semiflow is not necessarily homeomorphic to a product of metric spaces having as a factor the phase space of the limiting semiflow this result implies tubular homology index braid and connection matrix continuation principles

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

multiple periodic solutions for γ symmetric newtonian systems abstract the existence of periodic solutions in symmetric newtonian systems x f x can be effectively studied by means of the o 2 equivariant gradient degree with values in the euler ring u o 2 in this paper we show that in the case of being a finite group the euler ring u o 2 and the related basic degrees are effectively computable using euler ring homomorphisms the burnside ring a o 2 and the reduced o 2 degree with no free parameters we present several examples of newtonian systems with various symmetries for which we show existence of multiple periodic solutions we also provide exact value of the equivariant topological invariant for those problems

nontrivial full bounded solutions of time periodic semilinear parabolic pdes consider the semilinear evolution equation p u a u f t u where a is a sectorial operator on a banach space and f is periodic in t using a time discrete conley index developed in a previous paper 6 we prove a few existence results on bounded solutions of p defined for all t r more specific results are given for time periodic scalar parabolic equations

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

an efficient mixed methodology for free vibration and buckling analysis of orthotropic rectangular plates abstract in this paper a simple and efficient mixed ritz differential quadrature dq method is presented for free vibration and buckling analysis of orthotropic rectangular plates the mixed scheme combines the simplicity of the ritz method and high accuracy and efficiency of the dq method the accuracy of the proposed method is demonstrated by comparing the calculated results with those available in the literature it is shown that highly accurate results can be obtained using a small number of ritz terms and dq sample points the proposed method is suitable for the problem considered due to its simplicity and potential for further development

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

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

unified panel flutter theory with viscous damping effects the idea of a unified panel flutter theory investigated previously by the author is amplified to make the inclusion of damping effects possible the paper attempts to study the combined effects of the aerodynamic and structural types of damping on the flutter boundaries for simply supported panels in high supersonic mach number flows classical plate theory and two dimensional first order aerodynamics are used a closed form rather than modal approach is used in the solution of the partial differential equation in the analysis to avoid convergence problems the standing and traveling wave theories of panel flutter are compared the dual nature of damping stabilizing and destabilizing is exposed while the trends are similar to those already established for isotropic panels the variables compared are generically defined to account for both the isotropic as well as the orthotropic properties of the panels

unified aeroelastic flutter theory for very low aspect ratio panels a theory unifying the flutter analyses of orthotropic and isotropic panels having very low aspect ratios and exposed to an inviscid potential flow on the upper surfaces is developed the analysis considers an infinitely long panel of finite width simply supported on the side edges and resting on a spring foundation three dimensional linearized aerodynamic and classical plate theories are used traveling wave modes are utilized as solutions to the panel s aeroelastic equation of motion the resulting aerodynamic integral is evaluated approximately using both numerical and analytical methods the structural and aerodynamic generic variables which are instrumental in establishing this unified theory are derived using affine transformations subsonic as well as supersonic flutter and divergence boundaries are determined the effects of air to panel mass ratios and midplane forces are examined viscous structural damping is found to be destabilizing there is a general agreement between the results of this analysis and those obtained for isotropic panels by previous investigators

on the homotopy index for infinite dimensional semiflows absrac r in this paper we consider semiflows whose solution operator is eventually a conditional a contraction such semiflows include solutions of retarded and neutral functional differential equations of parabolic and certain other classes of partial differential equations we prove existence of nonsmooth isolating blocks and index pairs for such semiflows via the construction of special lyapunov functionals we show that index pairs enjoy all the properties needed to define the notion of a homotopy index thus generalizing earlier results of conley 2 finally using a result of maii 9 we prove that under additional smoothness assumptions on the semiflow the homotopy index is essentially a finite dimensional concept this gives a formal justification of the applicability of waiewski s principle to infinite dimensional problems several examples illustrate the theory

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

a mechanical model for the buckling of unilaterally constrained rectangular plates a mechanical model is described for the problem of buckling of unilaterally constrained finite rectangular plates due to the nature of the imposed constraint on the plate s lateral deflection w solving for the buckling load required the solution of a nonlinear partial differential equation in w while the plates were modeled along the lines of classical plate theory the nonlinearity arose from the fact that the plates were attached to nonlinear elastic foundations exhibiting a deformation sign dependent force displacement relationship this feature was introduced to model the unilateral constraint the influence of different boundary conditions material orthotropy and transverse load distributions was investigated for each case the weak form of the governing differential equation was solved via the galerkin s method investigations of the buckling loads of rectangular plates attached to such foundations and subjected to a uniform inplane stress field showed the validity of this approach for the cases investigated and compared to some previous exact results reported in the literature

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

connecting orbits for nonlinear differential equations at resonance abstract we study the existence of orbits connecting stationary points for the first order differential equations being at resonance at infinity where the right hand side is the perturbations of a sectorial operator our aim is to prove an index formula expressing the conley index of associated semiflow with respect to appropriately large ball in terms of special geometrical assumptions imposed on the nonlinearity we also prove that the geometrical assumptions are generalization of the well known in literature landesman lazer and strong resonance conditions obtained index formula will be used to derive the criteria determining the existence of orbits connecting stationary points for the heat equation being at resonance at infinity

stabilization of discrete time systems by first order controllers in this note we consider the problem of stabilizing a given but arbitrary linear time invariant discrete time system with transfer function p z by a first order discrete time feedback controller c z zx sub 1 x sub 2 z x sub 3 the complete set of stabilizing controllers is determined in the controller parameter space x sub 1 x sub 2 x sub 3 the solution involves the chebyshev representation of the characteristic equation on the unit circle the set is shown to be computable explicitly for fixed x sub 3 by solving linear equations involving the chebyshev polynomials in closed form and the three dimensional set is recovered by sweeping over the scalar parameter x sub 3 this result gives a constructive solution of 1 the problem of first order stabilizability of a given plant 2 simultaneous stabilization of a set of plants p sub i z and 3 stable or minimum phase first order stabilization of a plant the solution is facilitated by the fact that it is based on linear equations and the intersection of sets can be found by adding more equations in each case the complete set of solutions is found and this feature is essential and important for imposing further design requirements illustrative examples are included

averaging conley index continuation and recurrent dynamics in almost periodic parabolic equations abstract we study a non autonomous parabolic equation with almost periodic rapidly oscillating principal part and nonlinear interactions we associate to the equation a skew product semiflow and for a special class of nonlinearities we define the conley index of isolated compact invariant sets as the frequency of the oscillations tends to infinity we prove that every isolated compact invariant set of the averaged autonomous equation can be continued to an isolated compact invariant set of the skew product semiflow associated to the non autonomous equation finally we illustrate some examples in which the conley index can be explicitly computed and can be exploited to detect the existence of recurrent dynamics in the equation

leray functor and cohomological conley index for discrete dynamical systems we introduce the leray functor on the category of graded modules equipped with an endomorphism of degree zero and we use this functor to define the cohomological conley index of an isolated invariant set of a homeomorphism on a locally compact metric space we prove the homotopy and additivity properties for this index and compute the index in some examples as one of applications we prove the existence of nonconstant bounded solutions of the euler approximation of a certain system of ordinary differential equations

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

a global continuation theorem and bifurcation from infinity for infinite dimensional dynamical systems a general continuation theorem for isolated sets in infinite dimensional dynamical systems is proved for a class of semiflows this result is then used to prove the existence of continua of full bounded solutions bifurcating from infinity for systems of reaction diffusion equations

symmetric liapunov center theorem for minimal orbit abstract using the techniques of equivariant bifurcation theory we prove the existence of non stationary periodic solutions of symmetric systems q t u q t in any neighborhood of an isolated orbit of minima q 0 of the potential u we show the strength of our result by proving the existence of new families of periodic orbits in the lennard jones two and three body problems and in the schwarzschild three body problem

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

a new approach to digital pid controller design in this note we present a new approach to the problem of designing a digital proportional integral derivative pid controller for a given but arbitrary linear time invariant plant by using the tchebyshev representation of a discrete time transfer function and some new results on root counting with respect to the unit circle we show how the digital pid stabilizing gains can be determined by solving sets of linear inequalities in two unknowns for a fixed value of the third parameter by sweeping or gridding over this parameter the entire set of stabilizing gains can be recovered the precise admissible range of this parameter can be predetermined this solution is attractive because it answers the question of whether there exists a stabilizing solution or not and in case stabilization is possible the entire set of gains is determined constructively using this characterization of the stabilizing set we present solutions to two design problems 1 maximally deadbeat design where we determine for the given plant the smallest circle within the unit circle wherein the closed loop system characteristic roots may be placed by pid control and 2 maximal delay tolerance where we determine for the given plant the maximal loop delay that can be tolerated under pid control in each case the set of controllers attaining the specifications is calculated illustrative examples are included

singular dynamics of strongly damped beam equation abstract the paper is devoted to the dynamics of the model for a beam with strong damping pe e 2 u t t e u t u x x x x u t x x x x g 0 l u 2 d e 0 l u t u d u x x 0 x t 0 l 0 where g r r is continuously differentiable r and l e 0 subject to boundary conditions corresponding to hinged or clamped ends we show that for e 0 the dynamics of the equation is close to the dynamics of equation p0 u t u g 0 l u 2 d a 1 2 u where a u u x x x x with the domain determined by one of the above boundary conditions specifically we show that isolated invariant sets of p0 continue to isolated invariant sets of pe e 0 small having the same conley index moreover isolated morse decompositions with respect to p0 continue to isolated morse decompositions of pe e 0 small having isomorphic homology index braids under some additional assumptions we establish existence and upper semicontinuity results for attractors of e 0 small extending previous results by sevcovic

h sub spl infin design with first order controllers the problem of determining all first order controllers c s x 1s x2 s x3 which stabilize a given single input single output siso linear time invariant lti plant of arbitrary order has been recently solved in this note these results are extended to determine the subset of controllers which also satisfy various robustness and performance specifications which can be formulated as specific hinfin norm constraints the problem is solved by converting the hinfin problem into the simultaneous stabilization of the closed loop characteristic polynomial and a family of related complex polynomials the stability boundary of each of these polynomials can be computed explicitly for fixed x3 by solving linear equations the union of the resulting stability regions yields the set of all x1 and x2 which simultaneously satisfy the hinfin condition and closed loop stability for a fixed x3 the entire three dimensional set meeting specifications is obtained by sweeping x3 over the stabilizing range

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

h pid controller design for lur e systems and its application to a ball and wheel apparatus in this paper we consider the problem of synthesizing proportional integral derivative pid controllers for a given lur e system in the presence of exogenous energy bounded disturbance based on the circle criterion sufficient conditions for guaranteeing closed loop absolute stability and achieving disturbance attenuation are given in terms of the multi objective h specifications the results from the earlier work are then used to solve the resulting multi objective h pid design problem it is shown that for a fixed proportional gain and by sweeping over a variable the set of admissible integral and derivative gain values can be determined constructively using linear programming techniques the proposed synthesis method is used to design a pid controller for the ball and wheel system and the experimental results are presented to show effectiveness of the proposed method

generic karman rostovstev plate equations in an affine space the karman rostovstev plate equations are affinely transformed into equations that depend on the generic plate constants generalized rigidity generalized extensional rigidity and generalized poisson ratio associated with the well known linearized orthotropic plate slab equations furthermore since the generalized extensional rigidity is a function of generalized rigidity and generalized poisson ratio only with generalized poisson ratio being a weak parameter the only fundamental constant is generalized rigidity this means that regardless of what analysis technique is used the results can be cast in terms of only one strong material parameter whose range is from zero to one the affine stretching is performed on the dependent variables as well as the independent variables so that the dependence of all quantities of interest on generalized rigidity and generalized poisson ratio are easily calculated e g even auxiliary quantities such as stress strain all weighted stress resultants etc some typical results static side load response collapse load limited by rib buckling are presented by modifying an approximate technique due to donnell 1976

a new method for the computation of all stabilizing controllers of a given order a new method is given for computing the set of all stabilizing controllers of a given order for linear time invariant scalar plants the method is based on a generalized hermite biehler theorem and the successive application of a modified constant gain stabilizing algorithm to subsidiary plants it is applicable to both continuous and discrete time systems

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

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

symmetry breaking of solutions of non cooperative elliptic systems abstract in this article we study the symmetry breaking phenomenon of solutions of non cooperative elliptic systems we apply the degree for g invariant strongly indefinite functionals to obtain simultaneously a symmetry breaking and a global bifurcation phenomenon

morse type index theory for flows and periodic solutions for hamiltonian equations an index theory for flows is presented which extends the classical morse theory for gradient flows on compact manifolds the theory is used to prove a morse type existence statement for periodic solutions of a time dependent periodic in time and asymptotically linear hamiltonian equation

eigenvalue similarity rules for symmetric cross ply laminated plates as recently demonstrated the stability and vibration problems of orthotropic plates can be efficaciously treated in suitable affine spaces and similarity rules independent of individual problem boundary conditions can be formulated and very effectively used in analysis and design the present work extends these concepts to symmetric cross ply laminated plates by employing the affine stretching process utilizing some previous material constant definitions due to tsai and generalizing the existing similarity rules as examples similarity rules are employed to find the frequency spectrum of a cross ply symmetric plate from the corresponding isotropic solution and a similarity solution is presented for the vibration of an axially loaded simply supported cross ply symmetric plate buckling similarity rules are also presented