exponential stability of variable coefficients rayleigh beams under boundary feedback controls a riesz basis approach in this paper we study the boundary stabilizing feedback control problem of rayleigh beams that have non homogeneous spatial parameters we show that no matter how non homogeneous the rayleigh beam is as long as it has positive mass density stiffness and mass moment of inertia it can always be exponentially stabilized when the control parameters are properly chosen the main steps are a detail asymptotic analysis of the spectrum of the system and the proving of that the generalized eigenfunctions of the feedback control system form a riesz basis in the state hilbert space as a by product a conjecture in guo j optim theory appl 112 3 2002 529 is answered

on the stability and instability regions of non conservative continuous system under partially follower forces abstract the transition between stability and instability of a uniform cantilever beam subjected to a partially follower load is discussed by means of the finite element method the stability diagram in terms of the load versus the non conservativeness loading parameter is obtained both in the hypothesis of linearized small displacements and non linearized large displacements analysis it is found that the regions of divergence and flutter instability of the finite element model of continuous system are quite different from the corresponding ones obtained for the classical 2 degrees of freedom ziegler s model i e the inverted double pendulum both quantitatively and qualitatively the transition from the 2 degrees of freedom model to the continuous model is investigated through the study of some multi degrees of freedom approximation models i e the 3 4 5 and 10 degrees of freedom models of the continuous column finally the effect of damping in the stability diagram of the finite element model of continuous system is discussed and the destabilizing effect of small damping is emphasized

on the failure of static stability analyses of nonconservative systems in regions of divergence instability abstract the stability of perfect bifurcational discrete dissipative systems under follower loads in regions of existence non existence of adjacent equilibria is thoroughly re examined in the light of recent progress in nonlinear dynamics a general theory for such nongradient systems described by autonomous ordinary differential equations is developed conditions for the existence of adjacent equilibria the stability of precritical critical and postcritical states as well as for different types of local bifurcations are established focusing attention on the interaction of geometric nonlinearities and vanishing damping new findings contradicting widely accepted results of the classical linear analysis are discovered in a small region of adjacent equilibria near a compound branching point which is explicitly determined an interaction of two consecutive postbuckling modes occurs related to the following phenomena in case of vanishing damping loss of stability may occur via a hopf dynamic bifurcation prior to static divergence buckling moreover the critical states of divergence instability may be associated with a double zero jacobian eigenvalue satisfying also the conditions of a hopf local bifurcation besides local dynamic bifurcations global bifurcations are also found an example is used to illustrate the qualitative findings

on the work theorems for finite and incremental elastic deformations with discontinuous fields a unified treatment of different versions abstract starting from the statical and kinematical relations in three different versions characterized by the use of piola s kirchhoff s or biot s jaumann s stresses respectively and the corresponding deformation quantities a unified abstract formulation of the basic equations is given because of certain properties of the statical and geometrical operators various material independent work theorems follow including the generalized and modified principles of virtual displacements and virtual forces for a hyperelastic body under conservative loading these are transformed into generalized variational principles and strengthened to complementary extremum principles finally the abstract formulation is applied to the incremental equations the admissible functions are allowed to have relaxed continuity properties therefore the presented work theorems may be used as a basis for a consistent finite element approximation according to the mixed version or to the classical displacement formulation

how to get a conservative well posed linear system out of thin air part ii controllability and stability let a0 be a possibly unbounded positive operator on the hilbert space h which is boundedly invertible let c0 be a bounded operator from cal d a 0 1 2 with the norm z 1 2 2 langle

extended variational formulations and f e models for nonlinear beams under nonconservative loading abstract for nonconservative mechanical systems classical variational principles do not hold true hence the principle of incremental virtual work is usually taken as a convenient foundation for finite element solution methods 2 8 9 recent results on the so called inverse problem of calculus of variations however lead to extended variational formulations which can be provided even for nonconservative systems in the present paper this approach is used for the particular case of a nonlinear beam subjected to a co planar follower static loading suitable finite element models are deduced from the extended variational principle and their numerical performance is assessed by means of some meaningful examples

on the parametrization of finite rotations in computational mechanics a classification of concepts with application to smooth shells this paper concerns the computational treatment of large rotations with application to the finite element discretisation of smooth shells formulations based on various rotational parametrizations are reviewed and classified with respect to their update structure category ii is based on total rotational degrees of freedom leading to an additive update structure whereas category i relies on linearized rotational degrees of freedom leading to a multiplicative update structure based on this classification new formulations are developed among them are the rodrigues formula applied within category ii with additive update structure for the two components of the rotation vector and two successive elementary rotations in the sense of euler angles applied within category i yielding a singularity free formulation the numerical examples confirm that every rotational parametrization when applied within category ii leads to singularities whereas the application within category i enables the calculation of overall rotations unrestricted in size without any singularity

exponential stability of nondissipative linear system in banach space and application to unbounded bilinear systems abstract we consider linear control systems of the form d z t d t a z t b z t where is a real parameter a is the infinitesimal generator of a linear c 0 semigroup of contractions s 0 t on a banach space x and the linear perturbation b is a relatively bounded operator with respect to a we provide necessary and sufficient conditions for exponential stability of the above system our stability results are then applied to establish uniform exponential stabilization of linear and bilinear systems with unbounded control operators applications to hyperbolic like equations are considered

critical load analysis of undamped nonconservative systems using bieigenvalue curves to study the instability of an undamped nonconservative system using the finite element method an asymmetric load matrix has to be included to account for the path dependent nature of the applied loads in addition to the mass matrix elastic stiffness matrix and geometric stiffness matrix before the critical loads can be determined for a structure one basic problem in research of this sort has been the construction of load frequency relationships from the eigenvalue equations traditionally this requires solution of complex eigenvalues from the characteristic equations at many load levels which in practice is very time consuming in this paper an efficient approach based on a fourth order hyperbolic curve will be proposed for predicting the critical loads this curve also knoww ad the bieigenvalue curve can be uniquely determined once the first and second derivatives of the frequency with respect to the load parameter have been calculated for the structure under a preset load level based on the eigensolutions for the first few or all modes of vibration effectiveness and accuracy of the procedure based on the bieigenvalue curve is demonstrated in the numerical study

the damped string problem revisited abstract we revisit the damped string equation on a compact interval with a variety of boundary conditions and derive an infinite sequence of trace formulas associated with it employing methods familiar from supersymmetric quantum mechanics we also derive completeness and riesz basis results with parentheses for the associated root functions under less smoothness assumptions on the coefficients than usual using operator theoretic methods rather than detailed eigenvalue and root function asymptotics only

small displacements about equilibrium of a beam subjected to large static loads small vibrations about the equilibrium configuration of a beam structure are treated the beam is subjected to large static loads causing large displacements the influence of these static loads on the small vibrations is investigated in the stationary harmonic case the symmetry of the dynamic stiffness matrix is discussed a semifollower static moment is introduced to maintain the symmetry of the dynamic stiffness matrix and to create conservative subsystems the beam is deformed only in bending and torsion i e no extension of the geometric centerline is permitted and no transverse shear of the beam lamina is taken into account the euler bernoulli beam theory is applied to bending and the saint venant theory to torsion an euler axis transformation is used for handling the large rotations in the nonlinear case the material of the beam is assumed to be linearly elastic a continuous model is made of the beam i e the governing differential equations are derived and solved without recourse to a spatial discretization

a hybrid mixed model for non linear shell analysis and its applications to large rotation problems adopting an updated lagrange approach the general framework for the fully non linear analysis of curved shells is developed using a simple quadrilateral c0 model hmsh5 the governing equations are derived based on a consistent linearization of an incremental mixed variational principle of the modified hellinger reissner type with independent assumptions for displacement and strain fields emphasis is placed on devising effective solution procedures to deal with large rotations in space finite stretches and generalized rate type material models in particular a geometrically exact scheme for configuration update is developed by making use of the so called exponential mapping algorithm and the resulting element was shown to exhibit a quadratic rate of asymptotic convergence in solving practical shell problems with newton raphson type iterative schemes for the purpose of updating the spatial stress field of the element an objective generalized midpoint integration rule is utilized which relies crucially on the concept of polar decomposition for the deformation gradient and is in keeping with the underlying mixed method finally the effectiveness and practical usefulness of the hmsh5 element are demonstrated through a number of test cases involving beams plates and shells undergoing very large displacements and rotations

spectral theory of sparse non hermitian random matrices sparse non hermitian random matrices arise in the study of disordereda 13 physical systems with asymmetric local interactions and have applications ranginga 13 from neural networks to ecosystem dynamics the spectral characteristics of thesea 13 matrices provide crucial information on system stability and susceptibility however a 13 their study is greatly complicated by the twin challenges of a lack of symmetry anda 13 a sparse interaction structure in this review we provide a concise and systematica 13 introduction to the main tools and results in this field we show how the spectraa 13 of sparse non hermitian matrices can be computed via an analogy with infinitea 13 dimensional operators obeying certain recursion relations with reference to threea 13 illustrative examples adjacency matrices of regular oriented graphs adjacencya 13 matrices of oriented erd s renyi graphs and adjacency matrices of weighted orienteda 13 erd s renyi graphs we demonstrate the use of these methods to obtain both analytica 13 and numerical results for the spectrum the spectral distribution the location of outliera 13 eigenvalues and the statistical properties of eigenvectors a 13

weighted l2 estimates for dissipative wave equations with variable coefficients abstract we establish weighted l 2 estimates for the wave equation with variable damping u t t u a u t 0 in r n where a x a 0 1 x with a 0 0 and 0 1 in particular we show that the energy of solutions decays at a polynomial rate t n 2 1 if a x a 0 x for large x we derive these results by strengthening significantly the multiplier method this approach can be adapted to other hyperbolic equations with damping

energy decay rate of wave equations with indefinite damping abstract we consider the one dimensional wave equation with an indefinite sign damping and a zero order potential term using a shooting method we establish the asymptotic expansion of eigenvalues and eigenvectors of the damped wave equation for a large class of coefficients in addition if the damping coefficient is more positive than negative we prove that the energy of system decays uniformly exponentially to zero this sharp result generalizes a previous work of freitas and zuazua 1996

index statistical properties of sparse random graphs using the replica method we develop an analytical approach to compute the characteristic function for the probability p n k that a large n n adjacency matrix of sparse random graphs has k eigenvalues below a threshold the method allows to determine in principle all moments of p n k from which the typical sample to sample fluctuations can be fully characterized for random graph models with localized eigenvectors we show that the index variance scales linearly with n 1 for 0 with a model dependent prefactor that can be exactly calculated explicit results are discussed for erdos renyi and regular random graphs both exhibiting a prefactor with a nonmonotonic behavior as a function of these results contrast with rotationally invariant random matrices where the index variance scales only as lnn with an universal prefactor that is independent of numerical diagonalization results confirm the exactness of our approach and in addition strongly support the gaussian nature of the index fluctuations

a co rotational finite element formulation for buckling and postbuckling analyses of spatial beams abstract a consistent co rotational finite element formulation and numerical procedure for the buckling and postbuckling analyses of three dimensional elastic euler beam is presented all coupling among bending twisting and stretching deformations for a beam element is considered by consistent second order linearization of the fully geometrically nonlinear beam theory however the third order terms which are relevant to the twist rate and curvature of the beam axis are also considered an incremental iterative method based on the newton raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations the zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state a bisection method of the arc length is proposed to find the buckling load numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method and to investigate the effect of third order terms on the buckling load and postbuckling behavior of three dimensional beams

generalized mixed finite element model for pre and post quasistatic buckling response of thin walled framed structures adopting an updated lagrangian approach the general framework for the fully non linear analysis of thin walled framed structures is developed using a simple two node c0 model hmb2 the governing equations are derived based on a consistent linearization of an incremental mixed variational principle of modified hellinger reissner type with independent assumptions for displacement and strain fields all coupled significant modes of deformations i e stretching bending shear torsion and warping are accounted for in the generalized beam theory employed emphasis is placed on devising effective solution procedures to deal with finite rotations in space particularly with regard to their effect on the derivations of load correction matrices corresponding to configuration dependent externally applied forces moments the effectiveness and practical usefulness of present model are demonstrated through a number of test problems involving beam assemblages undergoing large displacements and rotations in space

hautus condition on the exact controllability of conservative systems we consider the exact controllability of a linear conservative system a b associated with hilbert spaces h and u we get a necessary and sufficient controllability condition this condition is related to the hautus lemma from the finite dimensional systems theory it is an estimate in terms of operators a and b alone

boundary stabilization of a 1 d wave equation with in domain antidamping we consider the problem of boundary stabilization of a 1 d one dimensional wave equation with an internal spatially varying antidamping term this term puts all the eigenvalues of the open loop system in the right half of the complex plane we design a feedback law based on the backstepping method and prove exponential stability of the closed loop system with a desired decay rate for plants with constant parameters the control gains are found in closed form our design also produces a new lyapunov function for the classical wave equation with passive boundary damping

a spectral approach for the exact observability of infinite dimensional systems with skew adjoint generator abstract let a be a possibly unbounded skew adjoint operator on the hilbert space x with compact resolvent let c be a bounded operator from d a to another hilbert space y we consider the system governed by the state equation z t az t with the output y t cz t we characterize the exact observability of this system only in terms of c and of the spectral elements of the operator a the starting point in the proof of this result is a hautus type test recently obtained in burq and zworski j amer soc 17 2004 443 471 and miller j funct anal 218 2 2005 425 444 we then apply this result to various systems governed by partial differential equations with observation on the boundary of the domain the schrodinger equation the bernoulli euler plate equation and the wave equation in a square are considered for the plate and schrodinger equations the main novelty brought in by our results is that we prove the exact boundary observability for an arbitrarily small observed part of the boundary this is done by combining our spectral observability test to a theorem of beurling on nonharmonic fourier series and to a new number theoretic result on shifted squares

exponential stability of an abstract nondissipative linear system in this paper we consider an abstract linear system with perturbation of the form frac dy dt ay varepsilon by on a hilbert space cal h where a is skew adjoint b is bounded and varepsilon is a positive parameter motivated by a work of freitas and zuazua on the one dimensional wave equation with indefinite viscous damping p freitas and e zuazua j differential equations 132 1996 pp 338 352 we obtain a sufficient condition for exponential stability of the above system when b is not a dissipative operator we also obtain a hautus type criterion for exact controllability of system a g where g is a bounded linear operator from another hilbert space to cal h our result about the stability is then applied to establish the exponential stability of several elastic systems with indefinite viscous damping as well as the exponential stabilization of the elastic systems with noncolocated observation and control

on the role of frame invariance in structural mechanics models at finite rotations in this work we re examine so called geometrically exact models for structures such as beams shells or solids with independent rotation field with respect to invariance under superposed rigid body motion a special attention is given to clarifying the issues pertaining to the finite element implementation which guarantees that the invariance of the continuum problem is preserved by its discrete approximation several numerical simulations dealing with finite rotations of structural models are presented in order to further illustrate and confirm the given theoretical considerations

a spillover phenomenon in the optimal location of actuators in this paper we are interested in finding the optimal location and shape of the actuators in a stabilization problem namely we consider the one dimensional wave equation damped by an internal feedback supported on a subdomain omega of given length the criterion we want to optimize represents the rate of decay of the total energy of the system it theoretically involves all the eigenmodes of the operator from an engineering point of view it seems more realistic to consider only a finite number of modes say the n first ones in that context we are able to prove existence and uniqueness of an optimal domain omega n it is the better possible location for the actuators we characterize this optimal domain and we point out the following strange phenomenon at least for small lengths the optimal domain omega n which is the better one for the n first modes is actually the worse one for the n 1th mode this looks like the well known spillover phenomenon in control theory at last we will give some possible extension and open problems in higher dimension

on the differentiation of the rodrigues formula and its significance for the vector like parameterization of reissner simo beam theory in this paper we present a systematic way of differentiating up to the second directional derivative i the rodrigues formula and ii the spin rotation vector variation relationship to achieve this goal several trigonometric functions are grouped into a family of scalar quantities which can be expressed in terms of a single power series these results are then applied to the vector like parameterization of reissner simo beam theory enabling a straightforward derivation and leading to a clearer formulation in particular and in contrast with previous formulations a relatively compact and obviously symmetric form of the tangent operator is obtained the paper also discusses several relevant issues concerning a beam finite element implementation and concludes with the presentation of a few selected illustrative examples copyright 2002 john wiley sons ltd

computational aspects of vector like parametrization of three dimensional finite rotations theoretical and computational aspects of vector like parametrization of three dimensional finite rotations which uses only three rotation parameters are examined in detail in this work the relationship of the proposed parametrization with the intrinsic representation of finite rotations via an orthogonal matrix is clearly identified careful considerations of the consistent linearization procedure pertinent to the proposed parametrization of finite rotations are presented for the chosen model problem of reissner s non linear beam theory pertaining details of numerical implementation are discussed for the simplest choice of the finite element interpolations for a 2 node three dimensional beam element a number of numerical simulations in three dimensional finite rotation analysis are presented in order to illustrate the proposed approach

asymptotic stability for intermittently controlled second order evolution equations motivated by several works on ordinary differential equations we are interested in the asymptotic stability of intermittently controlled partial differential equations we give a condition of asymptotic stability for second order evolution equations uniformly damped by an on off feedback this result extends to the case of partial differential equations a previous result of r a smith concerning ordinary differential equations

simplified conditions of initial observability for infinite dimensional second order damped dynamical systems abstract in the paper the initial observability of abstract second order infinite dimensional dynamical systems is considered two cases of observation operator are explored first the observation operator is assumed to be bounded then results are extended to more general class of observations related to the observation operators that are assumed to be bounded with respect to square root of elasticity operator which covers the case of some unboundedness in observation using the frequency domain method it is proved that initial observability of second order system under discussion can be verified by the initial observability conditions for the corresponding simplified first order system general results are then applied for initial observability investigation of infinite dimensional systems described by partial differential equations modelling flexible mechanical structures some special cases are also discussed and practical comments and remarks are given the paper extends earlier results on initial observability for a class of infinite dimensional second order abstract dynamical systems

error on the constitutive relation in the nonlinear formulation of elastic beams the plane case the error measure we propose in this study is based on the mechanics concept of error on the constitutive relation the error on the constitutive relation has already been used for the updating of elastic structures in dynamics under the small perturbation assumption here we extend this concept to the context of geometrically nonlinear elastic beams large rotations and finite strains it is shown that the proposed strategy which uses measured experimental data enables us to locate damaged substructures or if the quality of the model is to be checked the method provides the location of the zones of modelling errors the formulation and its derivation are developed and a complete algorithmic treatment is presented the implications of the formulation of the errors location discussed in this paper are illustrated through a set of numerical examples

stability results for the wave equation with indefinite damping abstract the one dimensional wave equation with damping of indefinite sign in a bounded interval with dirichlet boundary conditions is considered it is proved that solutions decay uniformly exponentially to zero provided the damping potential is in thebv class has positive average is small enough and satisfies a finite number of further constraints guaranteeing that the derivative of the real part of the eigenvalues is negative when the damping vanishes this sharp result completes a previous one by the first author showing that an indefinite sign damping always produces unstable solutions if it is large enough and it answers by the afirmative to a conjecture concerning small damping terms the method of proof relies on the developments by s cox and the second author on the high frequency asymptotic expansion of the spectrum for damped wave equations and on the characterization of the decay rate in terms of the spectral abscissa

co rotational formulation for geometric nonlinear analysis of doubly symmetric thin walled beams abstract a doubly symmetric thin walled beam element with open section is derived using co rotational cr total lagrangian tl formulation the effects of deformation dependent third order terms of element nodal forces on the buckling load and post buckling behavior are investigated all coupling among bending twisting and stretching deformations for beam element is considered by consistent second order linearization of the fully geometrically nonlinear beam theory however all third order terms of nodal forces which are relevant to the twist rate rate of twist rate and curvature of the beam axis are also considered an incremental iterative method based on the newton raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations the zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state a parabolic interpolation method of the arc length is used to find the buckling load numerical examples are presented to demonstrate the accuracy and efficiency of the proposed element and to investigate the effect of third order terms of element nodal forces on the buckling load and post buckling behavior of doubly symmetric thin walled beams

exponential stability of the plate equations with potential of second order and indefinite damping we study the exponential stability of the plate equations with potential of second order and indefinite sign damping term by means of a global carleman type estimate and the usual perturbation method we show that the energy of the system decays exponentially provided that the negative damping is sufficiently small both the energy decay rate and the upper bound estimate on the negative damping are given explicitly

stabilization of the wave equations with potential and indefinite damping by means of global carleman type estimate we study the stabilization problem of the wave equations with potential and indefinite damping the energy decay rate of the system is given explicitly also we obtain an upper bound estimate on the negative damping to guarantee the energy of the system decays exponentially

boundary eigenvalue problems for differential equations nη λpη with λ polynomial boundary conditions abstract the present paper deals with the spectral properties of boundary eigenvalue problems for differential equations of the form n p on a compact interval with boundary conditions which depend on the spectral parameter polynomially here n as well as p are regular differential operators of order n and p respectively with n p 0 the main results concern the completeness minimality and riesz basis properties of the corresponding eigenfunctions and associated functions they are obtained after a suitable linearization of the problem and by means of a detailed asymptotic analysis of the green s function the function spaces where the above properties hold are described by independent boundary conditions an application to a problem from elasticity theory is given

the influence of shear deformations on the stability of thin walled beams under non conservative loading abstract this paper analyses the influence of shear deformations on the stability of thin walled beams of open section subjected to follower type loads the warping of the cross sections is assumed to depend on the shear forces as well as on both primary and secondary torsion the internal stresses are consistently deduced from the kinematic field an algorithm is presented which works in the framework of the least squares method making use of a vector iteration technique an extensive comparison is performed with the results offered by the classical models

nonlinear finite element analysis of elastic systems under nonconservative loading natural formulation part ii dynamic problems abstract the paper presents a nonlinear finite element analysis of elastic structures subject to nonconservative forces attention is focused on the stability behaviour of such systems this leads mathematically to non self adjoint boundary value problems which are of great theoretical and practical interest in particular in connection with alternative modes of instability like divergence of flutter only quasistatic effects are however considered in the present part the methodology of our theory is general but the specific thrust of the present research is directed towards the analysis of structures acted upon by displacement dependent nonconservative follower forces in a finite element formulation the analysis of geometrically nonlinear elastic systems subject to such forces gives in general rise to a contributory nonsymmetric stiffness matrix known as the load correction matrix as a result the total tangent stiffness matrix becomes unsymmetric an indication of the non self adjoint character of the problem our theory is based on the natural mode technique 1 2 3 and permits i a a simple but elegant derivation of the load correction matrix the application of the general theory as evolved in this paper is demonstrated on the beam element in space a number of numerical examples are considered including divergence and flutter types of instability for which exact analytic solutions are known the problems demonstrate the efficiency of the present finite element formulation the paper furnishes also a novel and concise formulation of finite rotations in space which may be considered as a conceptual generalization of the theory presented in 2 3

stabilization of high eigenfrequencies of a beam equation with generalized viscous damping in this paper using the one dimensional vibrating beam equation with generalized viscous damping as a model of vibration of flexible robot arms it is shown that for such a system the high eigenmodes decay at a uniform rate the proof is obtained by perturbation theory of linear operators y h luo acta math sinica 32 1991 pp 556 563 and asymptotic estimates of eigenvectors based on an earlier work of g b birkhoff and m h stone feedback control for this class of system is investigated and a finite dimensional controller is presented for an exponentially stable closed loop system our method may be used to study nondissipative systems

computing finite rotations of shells by an asymptotic numerical method abstract we present an asymptotic numerical algorithm for the computation of elastic shells with large rotations the theoretical formulation involves a three field hu washizu functional which allows us to put the problem into a quadratic framework the spatial discretization is based on geometrically exact element recently presented by buchter et al buchter et al three dimensional extension of non linear shell formulation based on the enhanced assumed strain concept int j numer methods engrg 37 1994 2551 1568 several classical benchmarks are discussed to define the best strategy and to assess the validity and the efficiency of the present method as compared to more classical iterative algorithms

a mixed finite element for three dimensional nonlinear analysis of steel frames abstract this paper develops and demonstrates a mixed beam finite element that can represent the inelastic three dimensional stability behavior of frames composed of open walled cross section members the kinematics of deformation of the element include finite rotation and warping of the cross section due to torsion the material inelasticity is based on a two space model that includes the effect of shear stresses due to uniform torsion in addition to normal stresses due to axial force biaxial bending and bimoment initial geometric imperfections and residual stresses are accommodated the formulation is based on a two field displacement and generalized stress hellinger reissner hr variational principle the interpolation of the generalized stresses i e cross section stress resultants along the element length is based on the geometrically exact nonlinear governing differential equations of equilibrium this leads to significant improvements in the accuracy over conventional displacement based elements in problems involving highly nonlinear variations in the curvature along the member lengths due to geometric nonlinearity and or distributed plasticity effects several numerical examples involving two and three dimensional elastic and inelastic stability behavior are presented to illustrate the capabilities of the formulation

on exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback abstract the uniform stabilization of an originally regarded nondissipative system described by a semilinear wave equation with variable coefficients under the nonlinear boundary feedback is considered the existence of both weak and strong solutions to the system is proven by the galerkin method the exponential stability of the system is obtained by introducing an equivalent energy function and using the energy multiplier method on the riemannian manifold this equivalent energy function shows particularly that the system is essentially a dissipative system this result not only generalizes the result from constant coefficients to variable coefficients for these kinds of semilinear wave equations but also simplifies significantly the proof for constant coefficients case considered in a guesmia a new approach of stabilization of nondissipative distributed systems siam j control optim 42 2003 24 52 where the system is claimed to be nondissipative

sampling expansions associated with kamke problems the present paper is devoted to the derivation of sampling expansions for entire functions which are represented as integral transforms where a differential operator is acting on the kernels the situation generalizes the results obtained in sampling theory associated with boundary value problems to the case when the differential equation has the form n y lambda p y where n and p are two differential expressions of orders n and p respectively n p and lambda is the eigenvalue parameter both self adjoint and non self adjoint cases will be considered with examples in which the boundary conditions are strongly regular

large deflections of a cantilever beam subjected to a follower force abstract the large deflection problem of a non uniform spring hinged cantilever beam under a tip concentrated follower force is considered the angle of inclination of the force with respect to the deformed axis of the beam remains unchanged during deformation the mathematical formulation of this problem yields a nonlinear two point boundary value problem which is reduced to an initial value problem by change of variables the resulting problem can be solved without iterations it is shown that there exist no critical loads in the euler sense divergence for any flexural stiffness distribution and angle of inclination of the follower force the load displacement characteristics of a uniform cantilever under a follower force normal to the deformed beam axis are presented

exponential decay of energy of evolution equations with locally distributed damping consider an evolution equation with energy dissipation frac partial partial t y x t ay x t by x t on a bounded x domain omega where by signifies a dissipative perturbation to an otherwise energy conserving system this dissipation may be due to medium impurities viscous effects or artificially imposed dampers and stabilizers it is distributed over only part of the domain omega the question of when the dissipation is effective enough to cause uniform exponential decay of energy is examined because of the locally distributed nature of energy dissipation the problem lacks coercivity and is not directly solvable by energy identities thus to get conditions sufficient for uniform exponential decay a different approach needs to be taken provided here is a set of tight sufficient conditions in terms of the influence of the dissipative operatorbon the separated eigenmodes or clustered eigenmodes ofa the main theorems are general enough to treat the wave and bea

optimal energy decay rate for partially damped systems by spectral compensation we study the stability of weakly coupled and partially damped systems by means of the riesz basis approach in higher dimensional spaces we propose a weaker structural damping that compensates for the behavior of the eigenvalues of the system therefore giving the optimal polynomial energy decay rate for smooth initial data

instability of lightly damped linear nonconservative systems in computing the critical load for a linear nonconservative system one needs to construct the load frequency curve from the characteristic equation that contains asymmetric matrices based on the finite element formulation traditionally this requires repeated solution of complex eigenvalues from the characteristic equation at many load levels which is extremely time consuming in practice small damping of the rayleigh type is assumed a bieigenvalue curve is adopted to approximate the load frequency curve for the first few modes of interest such a curve can be uniquely determined once the derivatives of the frequency with respect to the load parameter have been calculated for a preset load level after the bieigenvalue curve is established the critical load can be computed by setting its first derivative equal to zero the effectiveness of the present method is demonstrated in the numerical study

bresse system with indefinite damping abstract we consider the bresse system in a bounded domain 0 l r 1 the system has an indefinite damping mechanism i e with a damping function a a x possibly changing sign presented only in the equation for the rotation angle we shall prove that the system is still exponentially stable under the same conditions as in the positive constant damping case and provided a 1 l 0 l a x d x 0 and a a l 2 for small enough in the arguments we shall also give a new proof of exponential stability for the constant case a a

timoshenko systems with indefinite damping abstract we consider the timoshenko system in a bounded domain 0 l r 1 the system has an indefinite damping mechanism i e with a damping function a a x possibly changing sign present only in the equation for the rotation angle we shall prove that the system is still exponentially stable under the same conditions as in the positive constant damping case and provided a 0 l a x d x 0 and a a l 2 for small enough the decay rate will be described explicitly in the arguments we shall also give a new proof of exponential stability for the constant case a a moreover we give a precise description of the decay rate and demonstrate that the system has the spectrum determined growth sdg property i e the type of the induced semigroup coincides with the spectral bound for its generator

asymptotic stability for bresse systems abstract in this paper consider a vibrating system of bresse type in a one dimensional bounded domain with a nonlinear damping acting in the equation of the shear angle displacement and nonlinear localized damping in other equations we show the asymptotic stability without imposing conditions about the equal speed wave propagation using a method developed by lasiecka and tataru 9

on the spectrum of euler bernoulli beam equation with kelvin voigt damping abstract the spectral property of an euler bernoulli beam equation with clamped boundary conditions and internal kelvin voigt damping is considered the essential spectrum of the system operator is rigorously identified to be an interval on the left real axis under some assumptions on the coefficients it is shown that the essential spectrum contains continuous spectrum only and the point spectrum consists of isolated eigenvalues of finite algebraic multiplicity the asymptotic behavior of eigenvalues is presented

objectivity of incremental constitutive relations over finite time steps in computational finite deformation analyses abstract computational techniques which preserve the objectivity of incremental constitutive relations for finite elastic or inelastic materials for finite time steps during which the material elements may undergo finite rotations are presented these techniques are useful in finite deformation finite rotations and finite strain computational analysis of two as well as three dimensional solids and structures

exponential stability for wave equations with non dissipative damping abstract we consider the nonlinear wave equation u t t u x x a x u t 0 in a bounded interval 0 l r 1 the function a is allowed to change sign but has to satisfy a 1 l 0 l a x d x 0 for this non dissipative situation we prove the exponential stability of the corresponding linearized system for i possibly large a l with small a a l 2 and ii a class of pairs a l with possibly negative moment 0 l a x sin 2 x l d x estimates for the decay rate are also given in terms of a moreover we show the global existence of smooth small solutions to the corresponding nonlinear system if additionally the negative part of a is small enough

exponential stability of wave equations with potential and indefinite damping abstract first we consider the linear wave equation u t t u x x a x u t b x u 0 on a bounded interval 0 l r the damping function a is allowed to change its sign if a 1 l 0 l a x d x is positive and the spectrum of the operator x x b is negative exponential stability is proved for small a a l 2 explicit estimates of the decay rate are given in terms of a and the largest eigenvalue of x x b second we show the existence of a global small smooth solution of the corresponding nonlinear wave equation u t t u x x a x u t b x u 0 if additionally the negative part of a is small enough compared with

effective modelling of spatial buckling of beam assemblages accounting for warping constraints and rotation dependency of moments a simple two noded finite element model for the three dimensional buckling analysis of beam assemblages is developed the underlying generalized beam theory employed accounts for all coupled significant modes of deformations including stretching bending shear and torsion as well as warping its mixed formulation is of the two field type utilizing linear and constant interpolants for displacements and strain fields due considerations are given to the effect off inite rotations in space on the non linear kinematic descriptions as well as the configuration dependent behaviour of externally applied moment vectors of the conservative and non conservative types the performance of the model and particularly the accuracy of its geometric and load correction stiffnesses is assessed in a fairly complete set of numerical simulations

static and dynamic stability of nonlinear elastic systems under nonconservative forces natural approach abstract the paper develops a nonlinear large displacement small strain finite element analysis of elastic structures subject to arbitrary nonconservative external forces both in the static and dynamic realms special attention is focused on the stability behaviour of such systems the methodology of the theory is general covering displacement dependent circulatory velocity dependent dissipative and acceleration dependent stationary as well as instationary explicitly on time dependent nonconservative forces the finite element formulation selected here is based on the natural mode technique permitting i a a simple but elegant as well as concise derivation of the contributory nonsymmetric matrices in conjunction with a clear physical interpretation some typical examples of coalescence and single mode flutter are presented most attention is focused on the evolution of highly nonlinear post flutter response due to various kinds of disturbances in both the stable and the unstable domains the discussion demonstrates the efficiency of the present finite element formulation

a mixed hybrid finite beam element with an interface to arbitrary three dimensional material models abstract a theory for spatial beams and an associated mixed finite element formulation are presented the basic kinematic assumption allows for finite rotations and transverse shear strains the relation of the green lagrangean strains to the beam strains is derived a three field variational formulation with independent displacements rotations stress resultants and beam strains is developed using the kinematic and static field equations and the constitutive equations higher order stress resultants are set to zero within the euler lagrange equations the formulation in this paper is restricted to rectangular cross sections the interpolation of the beam strains consists of two parts the first part is identical to the stress resultant interpolation the second part describes cross section warping and allows for transverse normal strains within this approach an interface to arbitrary three dimensional constitutive laws can easily be realized due to the mixed interpolation technique full gauss integration in length direction of the beam element can be applied to obtain a locking free element formulation the effectiveness of the method is demonstrated by means of several numerical examples we present results for elasticity inelasticity and stability problems and evaluate the load carrying capacities of spatial beam structures the essential feature of the new element is the robustness in nonlinear applications it allows very large load steps in comparison to other element formulations

an excursion into large rotations the present discourse develops an enlarged exploration of the matrix formulation of finite rotations in space initiated in 1 it is shown how a consistent but subtle matrix calculus inevitably leads to a number of elegant expressions for the transformation or rotation matrix t appertaining to a rotation about an arbitrary axis also analysed is the case of multiple rotations about fixed or follower axes particular attention is paid to an explicit derivation of a single compound rotation vector equivalent to two consecutive arbitrary rotations this theme is discussed in some detail for a number of cases semitangential rotations for which commutativity holds first proposed in 2 3 are also considered furthermore an elementary geometrical analysis of large rotations is also given finally we deduce in an appendix using a judicious reformulation of quarternions the compound pseudovector representing the combined effect of n rotations in the author s opinion the present approach appears preferable to a pure vectorial scheme and even more so to an indicial formulation and is computationally more convenient

secondary bifurcations and global instability of an aeroelastic non linear system in the divergence domain abstract the behaviour of dissipative non linear systems is studied in the instability region of the trivial zero solution the study is motivated by the classical problem of stability of an initially flat elastic panel subjected to the combination of a supersonic gas flow and a quasistatic compression that may produce buckling this allows one to investigate various types of instabilities such as primary flutter and divergence as well as secondary tertiary etc instability phenomena three items are analyzed in detail in this paper 1 the local lyapunov stability of a buckled panel with special attention paid to the influence of the damping distribution between particular modes 2 the stable flutter like vibrations of the buckled panel in the domains of its lyapunov instability 3 the stable flutter like vibrations in the domain where the buckled mode of the panel is stable in lyapunov sense the case of damping distributed non uniformly among the vibration modes is studies in detail it is shown that a region of secondary flutter exist in the divergence domain the secondary flutter responses are asymmetric with respect to the initial plane of the panel and the structure of their attraction basins is rather complicated global flutter like responses in the divergence domain are observed the hysteretic properties of these responses are studied by numerical simulation

generalization of the c1 tuba plate finite elements to the geometrically exact kirchhoff love shell model abstract the finite element implementation of a geometrically exact thin shell model is reported in the present paper the shell kinematics is based on the kirchhoff love assumption and is characterized by the deformation gradient which yielded the generalized cross section strain measures stretches and curvatures written in terms of first and second order derivatives of displacements the energetically conjugate quantity the first piola kirchhoff stress tensor is used to define the generalized stresses the shell s initial geometry is exactly represented using a mapping from a reference configuration a neo hookean material functional supplemented by the plane stress condition is incorporated in the constitutive level of the model both configuration dependent and independent forces are considered for the domain and boundary loading the consistent linearization of the rendered weak form is addressed the tuba family of plate finite elements is adopted for the discretization of the displacement vector field these triangular elements with necessarily straight sides in the reference configuration provide c 1 continuous approximations a sine qua non property for a consistent discretization of the kirchhoff love model the family contains 3 pairs of elements primary and reduced which provide polynomial approximants of degrees k 5 6 and 7 respectively the reduced elements have a simpler structure and a smaller number of nodes but still guarantee c 1 continuity the boundary normal derivative of the approximation in these elements is constrained such that its order is reduced by one unit from k 1 at the primary elements to k 2 due to complexity of the elements degrees of freedom which are the function value its normal first and second order derivatives the imposition of the essential boundary conditions requires some preliminary analysis of the boundary behavior such study is presented for the most common practical cases six benchmark examples are presented to demonstrate the model usefulness and efficiency

boundary control of an anti stable wave equation with anti damping on the uncontrolled boundary abstract much of the boundary control of wave equations in one dimension is based on a single principle passivity under the assumption that control is applied through neumann actuation on one boundary and the other boundary satisfies a homogeneous dirichlet boundary condition we have recently expanded the scope of tractable problems by allowing destabilizing anti stiffness a robin type condition on the uncontrolled boundary where the uncontrolled system has a finite number of positive real eigenvalues in this paper we go further and develop a methodology for the case where the uncontrolled boundary condition has anti damping which makes the real parts of all the eigenvalues of the uncontrolled system positive and arbitrarily high i e the system is anti stable exponentially stable in negative time using a conceptually novel integral transformation we obtain extremely simple explicit formulae for the gain functions for the case with only boundary sensing available at the same end with actuation we design backstepping observers which are dual to the backstepping controllers and have explicit output injection gains we then combine the control and observer designs into an output feedback compensator and prove the exponential stability of the closed loop system

riesz basis property and exponential stability of controlled euler bernoulli beam equations with variable coefficients this paper studies the basis property and the stability of a distributed system described by a nonuniform euler bernoulli beam equation under linear boundary feedback control it is shown that there is a sequence of generalized eigenfunctions of the system which forms a riesz basis for the state hilbert space the asymptotic distribution of eigenvalues the spectrum determined growth condition and the exponential stability are concluded the results are applied to a nonuniform beam equation with viscous damping of variable coefficient as a generalization of existing results for the uniform beam

the damped wave equation with unbounded damping abstract we analyze new phenomena arising in linear damped wave equations on unbounded domains when the damping is allowed to become unbounded at infinity we prove the generation of a contraction semigroup study the relation between the spectra of the semigroup generator and the associated quadratic operator function the convergence of non real eigenvalues in the asymptotic regime of diverging damping on a subdomain and we investigate the appearance of essential spectrum on the negative real axis we further show that the presence of the latter prevents exponential estimates for the semigroup and turns out to be a robust effect that cannot be easily canceled by adding a positive potential these analytic results are illustrated by examples

locally distributed control and damping for the conservative systems in this paper we note the equivalence between exact controllability and exponential stabilizability for an abstract conservative system with bounded control this enables us to establish a frequency domain characterization for the exact controllability uniform exponential decay property of second order elastic systems such as the wave equation and the petrovsky equation with locally distributed control damping a piecewise multiplier method for frequency domain is introduced for several classes of pdes on regions which are not necessarily smooth we obtain a sufficient condition for the subregion on which the application of control damping will yield the exact controllability uniform exponential decay property this result provides useful information for designing the location of controllers dampers for distributed systems with a law of conservation

finite element theory for curved and twisted beams based on exact solutions for three dimensional solids part 1 beam concept and geometrically exact nonlinear formulation a geometrically exact and completely consistent finite element theory for curved and twisted beams is proposed it is based on the kinematical hypothesis generally formulated for large deformation and accounts for all kinds of deformation of a three dimensional solid translational and rotational displacements of the cross sections warping of their plane and distortion of their contours the principle of virtual work is applied in a straightforward manner to all non zero six components of the strain and stress tensors expressions are given for tangent matrices of elastic inertia and external forces and specific techniques for discretization and updating are developed for the analysis of beams in inertial and non inertial frames finally the numerical properties of the finite element models are demonstrated through examples c 1998 elsevier science s a all rights reserved

instability results for the damped wave equation in unbounded domains abstract we extend some previous results for the damped wave equation in bounded domains in r d to the unbounded case in particular we show that if the damping term is of the form a with bounded a taking on negative values on a set of positive measure then there will always exist unbounded solutions for sufficiently large positive in order to prove these results we generalize some existing results on the asymptotic behaviour of eigencurves of one parameter families of schrodinger operators to the unbounded case which we believe to be of interest in their own right

stability of a nonuniform rayleigh beam with indefinite damping abstract this is a continuation of our earlier work j m wang g q xu s p yung exponential stability for variable coefficients rayleigh beams under boundary feedback control a riesz basis approach systems control lett 51 1 2004 33 50 on the study of a nonhomogeneous rayleigh beam and this time the stabilization is achieved via an internal damping instead of the boundary feedbacks we continue to address a conjecture of guo basis property of a rayleigh beam with boundary stabilization j optim theory appl 112 3 2002 529 547 in this paper and demonstrate how the damping term can affect the decay rate asymptotically by a detailed spectral analysis we obtain a necessary condition for the stability and establish the riesz basis property as well as the spectrum determined growth condition for the system furthermore when the damping is indefinite we provide a condition on how negative the damping can be without destroying the exponential stability

exponential stabilization of laminated beams with structural damping and boundary feedback controls we study the boundary stabilization of laminated beams with structural damping which describes the slip occurring at the interface of two layered objects by using an invertible matrix function with an eigenvalue parameter and an asymptotic technique for the first order matrix differential equation we find out an explicit asymptotic formula for the matrix fundamental solutions and then carry out the asymptotic analyses for the eigenpairs furthermore we prove that there is a sequence of generalized eigenfunctions that forms a riesz basis in the state hilbert space and hence the spectrum determined growth condition holds furthermore exponential stability of the closed loop system can be deduced from the eigenvalue expressions in particular the semigroup generated by the system operator is a c 0 group due to the fact that the three asymptotes of the spectrum are parallel to the imaginary axis

admissibility and exact observability of observation operators for micro beam model time and frequency domain approaches this study focuses on the exact observability of a non classical euler bernoulli micro beam equation this non classical model was derived based on the strain gradient elasticity theory which is intended to explain the phenomenon of size effect at the micron scale spectral properties of the corresponding state operator are studied an asymptotic expression for eigenvalues is calculated and eigenfunctions are analyzed in order to check the necessary conditions for the exact observability of the system by examining the eigenfunctions it is shown that among non collocated boundary outputs only measurement of the non classical moment at the root of the beam yields an admissible observation operator and also defines an exactly observable system an alternative proof based on the multiplier method which is commonly employed in the literature on the observability and controllability of infinite dimensional dynamical systems is presented to provide a comparison between the time and frequency domain approaches

optimal shape and position of the actuators for the stabilization of a string abstract the energy in a string subject to constant viscous damping k on a subset of length l 0 decays exponentially in time we consider the problem of optimizing the decay rate for the which are the unions of at most n intervals this rate is given by the spectral abscissa of the linear operator associated to the wave equation we are interested in small values of k therefore we consider the derivative of the spectral abscissa at k 0 we prove that except for the case l 1 2 when the number of intervals is not fixed a priori an optimal domain does not exist we study numerically the case of one or two intervals using a genetic algorithm these numerical results are not intuitive in particular the optimal position of one interval is never at the middle of the string

replica symmetric approach to the typical eigenvalue fluctuations of gaussian random matrices we discuss an approach to compute the first and second moments of the number of eigenvalues i n that lie in an arbitrary interval of the real line for gaussian random matrices the method combines the standard replica symmetric theory with a perturbative expansion of the saddle point action up to leading to the correct logarithmic scaling of the variance as well as to an analytical expression for the correction to the average standard results for the number variance at the local scaling regime are recovered in the limit of a vanishing interval the limitations of the replica symmetric method are unveiled by comparing our results with those derived through exact methods the present work represents an important step to study the fluctuations of i n in non invariant random matrix ensembles where the joint distribution of eigenvalues is not known