interpreting proper orthogonal modes of randomly excited vibration systems proper orthogonal modes poms of displacements are interpreted for linear vibration systems under random excitation excitations are considered for which the fourier transform is convergent meaning that the input must have zero mean and no sustained sinusoidal component in such a case the poms in undamped discrete linear symmetric systems can represent linear natural modes if the mass distribution is known poms in one dimensional distributed parameter self adjoint systems can approximately represent the linear normal modes if the mass distribution is known simulation examples are presented simulations show that these ideas are also applicable under light modal damping

limits of hypercyclic and supercyclic operators abstract an operator t on a complex separable infinite dimensional hilbert space h is hypercyclic if there is a vector y in h such that the orbit y ty t2y is dense in h it is shown that if t is hypercyclic then t does not have any eigenvalue and that the union of the spectrum of t and the unit circle is a connected set this result is used to obtain a spectral characterization of the norm closure of the class hc h of all hypercyclic operators acting on h as well as a formula for the distance from a given operator to hc h analogous results are obtained for the closely related class sc h of all supercyclic operators this paper includes some information about the structure of the set of all hypercyclic supercyclic vectors of a given hypercyclic supercyclic resp operator t

continuous time covariance approaches for modal analysis proper orthogonal decomposition pod can be used to obtain complete information about the linear normal modes of structural systems only when special conditions are met principally related to the knowledge of mass distribution and to the absence of damping the main advantage of pod is the possibility to directly use covariance matrices estimated from time responses for the modal identification of the structure under investigation this paper proposes a method to extend this covariance based identification to general cases where significant damping is present and when no data is available on mass distribution a least square based decomposition is used to infer information about modal characteristics of the system this decomposition is applied to three different cases the first one uses records related to the free response of the system while the remaining two use records of the system under persistent white noise excitations in those latter cases the information on the excitation input time history can either be used or treated as unknown basic yet complete examples are presented to illustrate the properties of the proposed technique

on the physical interpretation of proper orthogonal modes in vibrations abstract the authors relate the proper orthogonal modes as applied in discrete vibration systems to normal modes of vibration in systems with a known mass matrix in the case of undamped free vibration the proper orthogonal modes converge to the linear normal modes as the amount of data increases this interpretation is also practical for lightly modally damped systems forced resonances lead to proper orthogonal modes which approximate vibration modes a particular case of non linear normal modes are looked at in which the motion of a single mode follows a curve in the co ordinate space in this case the proper orthogonal modes represent the principal axes of inertia formed by the distribution of data on the modal co ordinate curve more generally the proper orthogonal modes represent the principal axes of inertia of the data

dynamics of the differentiation operator on weighted spaces of entire functions the continuity of the differentiation operator on weighted banach spaces of entire functions with sup norm has been characterized recently by harutyunyan and lusky we give necessary and sufficient conditions to ensure that the differentiation operator on these weighted banach spaces of entire functions is hypercyclic or chaotic when it is continuous

experiments on chaotic vibrations of a post buckled beam with an axial elastic constraint abstract to clarify chaotic behavior of thin walled beams detailed experimental results are presented on chaotic vibrations of a post buckled beam subjected to periodic lateral acceleration a thin steel beam of thickness 0 198 mm breadth 12 7 mm and length 106 mm is used as a test beam both ends of the beam are clamped for deflection one end of the beam is elastically constrained by an axial spring the beam is compressed to the post buckled configuration by the axial spring first characteristics of restoring force and natural frequency of the beam are obtained dynamic nonlinear responses of the beam are measured under periodic acceleration in specific frequency regions chaotic responses are generated the chaotic responses are examined carefully with the poincare maps the fourier spectra the maximum lyapunov exponents and the principal component analysis the post buckled beam shows the soften and hardening characteristics of restoring force the dominant chaotic responses of the beam are bifurcated from the sub harmonic resonances of 1 2 and 1 3 orders with the lowest mode of vibration changing the exciting frequency gradually dynamical transition behaviors from these steady state sub harmonic response to the chaotic responses are precisely inspected by the poincare projection the maximum lyapunov exponent of the former chaotic response of 1 2 order is larger than that of the latter chaotic response of 1 3 order the principal component analysis predicts that the contribution of the lowest mode of vibration to the chaos is dominant among other contributions of multiple vibration modes

bilinear modal representations for reduced order modeling of localized piecewise linear oscillators abstract a novel reduced order modeling method for vibration problems of elastic structures with localized piecewise linearity is proposed the focus is placed upon solving nonlinear forced response problems of elastic media with contact nonlinearity such as cracked structures and delaminated plates the modeling framework is based on observations of the proper orthogonal modes computed from nonlinear forced responses and their approximation by a truncated set of linear normal modes with special boundary conditions first it is shown that a set of proper orthogonal modes can form a good basis for constructing a reduced order model that can well capture the nonlinear normal modes next it is shown that the subspace spanned by the set of dominant proper orthogonal modes can be well approximated by a slightly larger set of linear normal modes with special boundary conditions these linear modes are referred to as bi linear modes and are selected by an elaborate methodology which utilizes certain similarities between the bi linear modes and approximations for the dominant proper orthogonal modes these approximations are obtained using interpolated proper orthogonal modes of smaller dimensional models the proposed method is compared with traditional reduced order modeling methods such as component mode synthesis and its advantages are discussed forced response analyses of cracked structures and delaminated plates are provided for demonstrating the accuracy and efficiency of the proposed methodology

smooth decomposition of random fields abstract the smooth decomposition sd method was introduced to analyze discrete time signals and generalized to continuous time vector valued random processes the sd is obtained solving a generalized eigenproblem defined from the covariance matrix of the random process and the covariance matrix of the associated time derivative random process which defines the decomposition basis this paper presents a new extension of the sd to continuous time and continuous space vector valued random processes classically named random fields this generalization is a major step since one now deals with operators in infinite dimensional spaces and not matrices it is shown that in this new context the main properties of the sd are preserved applied to the responses of randomly excited continuous mechanical systems the sd can be considered as an output only analysis tool moreover two natural orderings are defined to classify the decomposition terms which permit to interpret the sd in terms of modal analysis or in terms of karhunen loeve analysis

on the exploitation of chaos to build reduced order models the present study focuses on the model reduction of non linear systems the proper orthogonal decomposition is exploited to compute eigenmodes from time series of displacement these eigenmodes called the proper orthogonal modes are optimal with respect to energy content and are used to build a low dimensional model of the non linear system for this purpose the proper orthogonal modes obtained from a chaotic orbit are considered indeed such an orbit is assumed to cover a portion of the phase space of higher dimension and hence of greater measure this higher dimensional data is further assumed to contain more information about the system dynamics than data of a lower dimensional periodic orbit in an example it is shown that the modes for this particular behaviour are more representative of the system dynamics than any other set of modes extracted from a non chaotic response this is applied to a buckled beam with two permanent magnets and the reduced order model is validated using both qualitative and quantitative comparisons

two stage lotteries without the reduction axiom preference relations over two stage lotteries are analyzed empirical evidence indicates that decisionmakers do not always behave in accordance with the reduction of compound lotteries axiom but they seem to satisfy a compound independence axiom although the reduction and the compound independence axioms together with continuity imply expected utility theory each of them by itself is compatible with all possible preference relations over simple lotteries using these axioms the author analyzes three different versions of expected utility for two stage lotteries the author suggests several different compound dominance axioms as possible replacements of the reduction axiom which are strictly weaker than the reduction of compound lotteries axiom copyright 1990 by the econometric society

expected uncertain utility theory we introduce and analyze expected uncertain utility euu theory a prior and an interval utility characterize an euu decision maker the decision maker transforms each uncertain prospect into an interval valued prospect that assigns an interval of prizes to each state she then ranks prospects according to their expected interval utilities we define uncertainty aversion for euu use the euu model to address the ellsberg paradox and other ambiguity evidence and relate euu theory to existing models

modelling of stochastic dynamic systems using hamilton s law of varying action abstract this paper presents a model based on hamilton s law of varying action for stochastic dynamic systems in this model the state variables are approximated as a linear sum of orthogonal polynomials for deterministic systems the coefficients of the polynomials are constant but for stochastic systems the coefficients are random variables the initial conditions are treated as gaussian random variables with specified mean values and covariance matrix and the external forcing functions input processes are treated as stationary gaussian processes with specified mean and correlation functions using the karhunen loeve expansion the random input processes are represented in terms of truncated linear sums of orthonormal eigenfunctions with uncorrelated random variables as coefficients this method reduces the infinite dimensional input force vector to one with finite dimensions a method to compute the mean and the variance functions for both the singular and the regular components of the state processes is developed a single degree of freedom spring mass damper systems subjected to random initial conditions and a random forcing function is considered to show the feasibility and effectiveness of the formulation the response is obtained without any reference to the differential equations associated with the dynamics of the system the study shows that the results of this formulation agree well with those obtained using other schemes

poms analysis of randomly vibrating systems obtained from karhunen loeve expansion abstract the karhunen loeve kl decomposition establishes that a 2d random field can be expanded as a series involving a sequence of deterministic orthogonal functions with orthogonal random coefficients the proper orthogonal decomposition pod method consists in detecting spatially coherent modes in the dynamics of a spatio temporally varying system by diagonalizing the spatial correlation function given by an averaging operator the kl expansion is applied here to the responses of randomly excited vibrating systems with a view to performing a pod in separated variables time and space form discrete and continuous mechanical systems are considered in this study as well as stationary and transient non stationary responses an averaging operator involving time and ensemble averages is proposed to draw up the pod in separated variables form from the associated kl expansion the result obtained using this approach agrees with the classical pod in the case of deterministic or ergodic random signals the associated proper orthogonal modes are interpreted in case of linear and nonlinear vibrating systems subjected to white noise excitation in terms of normal modes

a smooth model of decision making under ambiguity we propose and axiomatize a model of preferences over acts such that the decision maker prefers act f to act g if and only if e e u f e e u g where e is the expectation operator u is a vn m utility function is an increasing transformation and is a subjective probability over the set of probability measures that the decision maker thinks are relevant given his subjective information a key feature of our model is that it achieves a separation between ambiguity identified as a characteristic of the decision maker s subjective information and ambiguity attitude a characteristic of the decision maker s tastes we show that attitudes towards risk are characterized by the shape of u as usual while attitudes towards ambiguity are characterized by the shape of we also derive x 1 e x as the special case of constant ambiguity aversion ambiguity itself is defined behaviorally and is shown to be characterized by properties of the subjective set of measures this characterization of ambiguity is formally related to the definitions of subjective ambiguity advanced by epstein zhang 2001 and ghirardato marinacci 2002 one advantage of this model is that the welldeveloped machinery for dealing with risk attitudes can be applied as well to ambiguity attitudes the model is also distinct from many in the literature on ambiguity in that allows smooth rather than kinked indifference curves this leads to different behavior and improved tractability while still sharing the main features e g ellsberg s paradox etc the maxmin eu model e g gilboa and schmeidler 1989 with a given set of measures may be seen as an extreme case of our model with infinite ambiguity aversion two illustrative applications to portfolio choice are offered

hypercyclic operators failing the hypercyclicity criterion on classical banach spaces abstract by a recent result of m de la rosa and c read there exist hypercyclic banach space operators which do not satisfy the hypercyclicity criterion in the present paper we prove that such operators can be constructed on a large class of banach spaces including c 0 n or l p n

vector expected utility and attitudes toward variation this paper proposes a model of decision under ambiguity deemed vector expected utility or veu in this model an uncertain prospect or savage act is assessed according to a a baseline expected utility evaluation a nd b anadjustment that reflects the individual s perception of ambiguity and her attitudes toward it the adjustment is itself a function of the act s exposure to distinct sources of ambiguity as well as its variability the key elements of the veu model are a baseline probability and a collection of random variables or adjustment factors which represent acts exposed to distinct ambiguity sources and also reflect complementarities among ambiguous events the adjustment to the baseline expected utility evaluation of an act is a function of the covariance of its utility profile with each adjustment factor which reflects exposure to the corresponding ambiguity source a behavioral characterization of the veu model is provided furthermore an updating rule for veu preferences is proposed and characterized the suggested updating rule facilitates the analysis of sophisticated dynamic choice with veu preferences

risk ambiguity and the separation of utility and beliefs we introduce a general model of static choice under uncertainty arguably the weakest model achieving a separation of cardinal utility and a unique representation of beliefs most of the nonexpected utility models existing in the literature are special cases of it such separation is motivated by the view that tastes are constant whereas beliefs change with new information the model has a simple and natural axiomatization elsewhere forthcoming we show that it can be very helpful in the characterization of a notion of ambiguity aversion as separating utility and beliefs allows us to identify and remove aspects of risk attitude from the decision maker s behavior here we show that the model allows us to generalize several results on the characterization of risk aversion in betting behavior these generalizations are of independent interest as they show that some traditional results for subjective expected utility preferences can be formulated only in terms of binary acts

smooth karhunen loeve decomposition to analyze randomly vibrating systems abstract this paper deals with the application of smooth karhunen loeve decomposition skld procedure applied to random fields the skld is obtained by solving a generalized eigenproblem defined by combining the covariance matrix of the random field with that of the associated time derivative random field the main properties of the skld procedure thus obtained are described and compared with the classical karhunen loeve decomposition the skld is then applied to the responses of randomly excited vibrating systems in order to perform modal analysis the resulting skld characteristics are discussed in the case of linear vibrating systems subjected to white noise excitation in terms of normal modes

a semi continuous formulation for goal oriented reduced order models 1d problems a semi continuous formulation is introduced for finding bases that minimise the error in a specific output functional of a reduced order model the formulation is advantageous in that it can be used with arbitrary reference data and can be easily applied to nonlinear models and functionals a general description of the approach is given then explicit formulations are derived for the advection diffusion and burgers equations numerical results are given for both linear and nonlinear functionals these show substantial reductions in error when compared with pod modes depending on the functional considered the optimisation of bases for a reduced order model using an approximated governing equation is also described for which large increases in accuracy are obtained relative to pod modes copyright c 2015 john wiley sons ltd

a nonsymmetric state variable decomposition for modal analysis a modal decomposition strategy based on state variable ensembles is formulated a nonsymmetric generalized eigenvalue problem is constructed the data based eigenvalue problem is related to the generalized eigenvalue problem associated with free vibration solutions of the state variable formulation of linear multi degree of freedom systems for linear free response data the inverse transpose of the eigenvector matrix converges to the state variable modal eigenvectors and the eigenvalues of the nonsymmetric eigenvalue problem approximate those of the state variable model as such the eigenvalues lead to estimates of frequencies and modal damping the interpretation holds for linear systems with multi modal free responses whether damping is small or somewhat large modal or nonmodal and without the need of input data

non linear generalization of principal component analysis from a global to a local approach abstract principal component analysis pca also known as proper orthogonal decomposition or karhunen loeve transform is commonly used to reduce the dimensionality of a data set with a large number of interdependent variables pca is the optimal linear transformation with respect to minimizing the mean square reconstruction error but it only considers second order statistics if the data have non linear dependencies an important issue is to develop a technique which takes higher order statistics into account and which can eliminate dependencies not removed by pca recognizing the shortcomings of pca researchers in the field of statistics and neural networks have developed non linear extensions of pca the purpose of this paper is to present a non linear generalization of pca called vqpca this algorithm builds local linear models by combining pca with clustering of the input space this paper concludes by observing from two illustrative examples that vqpca is potentially a more effective tool than conventional pca

methodology for nonlinear quantification of a flexible beam with a local strong nonlinearity abstract this study presents a methodology for nonlinear quantification i e the identification of the linear and nonlinear regimes and estimation of the degree of nonlinearity for a cantilever beam with a local strongly nonlinear stiffness element the interesting feature of this system is that it behaves linearly in the limits of extreme values of the nonlinear stiffness an euler bernoulli cantilever beam with two nonlinear configurations is used to develop and demonstrate the methodology one configuration considers a cubic spring attached at a distance from the beam root to achieve a smooth nonlinear effect the other configuration considers a vibro impact element that generates non smooth effects both systems have the property that in the limit of small and large values of a configuration parameter the system is almost linear and can be modeled as such with negligible error for the beam with a cubic spring attachment the forcing amplitude is the varied parameter while for the vibro impact beam this parameter is the clearance between the very stiff stops and the beam at static equilibrium proper orthogonal decomposition is employed to obtain an optimal orthogonal basis used to describe the nonlinear system dynamics for varying parameter values the frequencies of the modes that compose the basis are then estimated using the rayleigh quotient the variations of these frequencies are studied to identify parameter values for which the system behaves approximately linearly and those for which the dynamical response is highly nonlinear moreover a criterion based on the betti maxwell reciprocity theorem is used to verify the existence of nonlinear behavior for the set of parameter values suggested by the described methodology the developed methodology is general and applicable to discrete or continuous systems with smooth or nonsmooth nonlinearities

numerical computation of wave localization in large disordered beam like lattice trusses wave localization in randomly disordered multiwave structures is investigated the smallest localization factor is of particular interest and is related to the smallest positive lyapunov exponent the numerical algorithm by wolf et al is modified to determine all of the lyapunov exponents the wave localization in a large beamlike lattice truss modeled as an equivalent continuous timoshenko beam is studied any engineering structures are designed to be composed of identically constructed elements assembled end to end to form a spatially periodic structure such as long space antennae or large beamlike lattice trusses used in space solar power stations periodic structures behave like bandpass filters when propagating stress waves if damping is neglected waves are classified as traveling waves that propagate without attenuation and nontraveling or attenuating waves whose amplitudes attenuate as the waves propagate for single wave periodic structures the frequency axis is divided into alternating passbands and stopbands in the frequency passbands the wave is a traveling wave whereas in the frequency stopbands the wave is an attenuating wave for multiwave structures in a given frequency band some of the waves may be traveling waves and the others nontraveling waves however due to defects in manufacture and assembly no structure designed as a periodic structure can be perfectly periodic disorder can occur in the geometry of configuration s and material properties of the structure in a disordered periodic structure wave amplitudes of all waves will be attenuated even those that are traveling waves in the perfectly periodic counterpart this means that the vibrational energy imparted to the structure by an external source cannot propagate to arbitrarily long distances but is instead substantially confined to a region close to the source this phenomenon is known as wave localization in disordered periodic structures it is therefore of great practical importance to study the localization behavior and evaluate the so called localization factors namely the exponential rates at which the amplitudes of the waves propagating in the structures decay the reciprocal of the localization factor gives the localization length which characterizes the distance that the propagating wave extends in the structure as an application the range of damage that is spread in a structure due to impact at some location can be estimated there are d waves in a d wave disordered periodic structure each wave attenuates at a certain exponential rate or corresponds to a certain localization factor which implies that each wave will extend to a certain localization length the smallest localization factor or the largest localization length is of particular interest for multiwave structures since it corresponds to the wave that has potentially the least amount of decay or that carries energy along the multiwave structure farther than any other waves the study of the localization phenomenon has been and remains an active research area in solid state physics after the celebrated work of anderson 1 hodges2 was the first to recognize the relevance of localization theory to dynamical behavior of periodic engineering structures since then there have been several studies on the localization of vibration of structures in a series of publications pierre and dowell 3 pierre 4 and cha and pierre5 studied the localization phenomenon for monocoupled disordered structural systems cai and lin6 developed a perturbation scheme to calculate the localization factor based on a generic periodic structure kissel7 and ariaratnam and xie8 9 used a traveling wave approach to investigate the localization effects in one dimension al periodic engineering structures a transfer matrix formulation including wave transfer matrices was used to model disordered periodic structures furstenberg s theorem10 for products of random matrices was applied to calculate the localization effect as a function of frequency the localization factor was related to the largest lyapunov exponent in these publications only one dimensional or monocoupled periodic structures were considered in which the transfer matrices are of dimension 2 few studies have been done on the localization behavior of multiwave periodic structures kissel11 derived the localization factor as a function of the transmission matrix for multiwave disordered systems using the multiplicative ergodic theorem of oseledec 12 the difficulty in the study of multiwave periodic structures is that the localization factor is related to the smallest positive lyapunov exponent and not the largest lyapunov exponent as in the singlewave case the numerical algorithms used previously for evaluating the largest lyapunov exponent for single wave periodic structures cannot be employed for multiwave structures in this paper a modification of an algorithm due to wolf et al 13 is used to determine all of the lyapunov exponents for randomly disordered multiwave periodic structures the localization factor of a large beamlike lattice truss modeled as an equivalent continuous timoshenko beam is evaluated

free response simulation via the proper orthogonal decomposition the proper orthogonal decomposition is applied to the response of linear and nonlinear beam models to construct a reduced order model for predicting the response to various initial conditions without requiring the equations of motion for the structure the proper orthogonal decomposition is interpreted as a modal sum and the structural response to an initial condition is expressed as a weighted summation of proper orthogonal modes and corresponding proper orthogonal coordinate histories a method for calculating the weights of each mode in the response to an altered initial condition is presented this method is applied to predict the responses of a linear undamped beam model and a damped beam model with a nonlinear spring to various initial displacement and velocity profiles the results obtained are compared with those from respective finite element models for each beam

a new measure of efficiency for model reduction application to a vibroimpact system the aim of this paper is to propose a new way to measure the efficiency of the proper orthogonal decomposition pod to construct a reduced order model in structural dynamics it investigates the efficiency of three reduced order models for a vibroimpact problem 1 poddir basis which is the basis constructed using the direct method of the proper orthogonal decomposition 2 podsnap basis which is the basis constructed using the snapshot method of the pod and 3 lin basis which is the basis composed by the normal modes of the associated linear lin conservative system the efficiency is measured in terms of 1 number of elements to represent the dynamics with a given precision and 2 computational cost to simulate the time response within a given precision

drill string horizontal dynamics with uncertainty on the frictional force abstract this paper analyzes the dynamics of a horizontal drill string in this dynamics the frictional forces between the column and the borehole are relevant and uncertain a stochastic model is proposed for the frictional coefficient a random field with exponential autocorrelation function in the numerical analysis the drill string is modeled using the bar model tension compression and is discretized by means of the finite element method an oscillatory force due to mud motor is imposed on the system and there is a bit rock interaction as the column moves forward we propose a new way to measure the efficiency of the process an output input power ratio the resultant random ratio is analyzed and it turns out that it presents a bimodal characteristic an unexpected result

existence of disjoint weakly mixing operators that fail to satisfy the disjoint hypercyclicity criterion abstract recently bes martin and sanders 11 provided examples of disjoint hypercyclic operators which fail to satisfy the disjoint hypercyclicity criterion however their operators also fail to be disjoint weakly mixing we show that every separable infinite dimensional banach space admits operators t 1 t 2 t n with n 2 which are disjoint weakly mixing and still fail to satisfy the disjoint hypercyclicity criterion answering a question posed in 11 moreover we provide examples of disjoint hypercyclic operators t 1 t 2 whose corresponding set of disjoint hypercyclic vectors is nowhere dense answering another question posed in 11 in fact we explicitly describe their set of disjoint hypercyclic vectors those same disjoint hypercyclic operators fail to be disjoint topologically transitive lastly we create examples of two families of d hypercyclic operators which fail to have any d hypercyclic vectors in common

hereditarily hypercyclic operators abstract we show that a continuous linear operator t on a frechet space satisfies the so called hypercyclicity criterion if and only if it is hereditarily hypercyclic and if and only if the direct sum t t is hypercyclic in particular hypercyclic operators with either a dense generalized kernel or a dense set of periodic points i e chaotic in the sense of r l devaney 1989 an introduction to chaotic dynamical systems addison wesley reading ma must satisfy the criterion finally we provide a characterization of those weighted shifts t that are hereditarily hyper cyclic with respect to a given sequence n k of positive integers as well as conditions under which t and t n k k 1 share the same set of hypercyclic vectors

assessing the local stability of periodic motions for large multibody non linear systems using proper orthogonal decomposition the eigenvalues of the monodromy matrix known as floquet characteristic multipliers are used to study the local stability of periodic motions of a non linear system of differential algebraic equations dae when the size of the underlying system is large the cost of computing the monodromy matrix and its eigenvalues may be too high in addition for non minimal set equations such as those of a dae system there is a certain number of spurious eigenvalues associated with the algebraic constraint equations which are meaningless for the assessment of the stability of motions an approach to extract the dominant eigenvalues of the transition matrix without explicitly computing it is presented the selection of the eigenvalues is based on a proper orthogonal decomposition pod which extracts the minimal set of dominant local modes of the transient dynamics on an energy contents basis to make the procedure applicable to both numerical and experimental tests a unifying experimental philosophy is pursued for the analysis of complex multidisciplinary multibody models some applications are outlined and comparisons with other techniques are presented to demonstrate the accuracy of the proposed procedure

system identification guided basis selection for reduced order nonlinear response analysis reduced order nonlinear simulation is often times the only computationally efficient means of calculating the extended time response of large and complex structures under severe dynamic loading this is because the structure may respond in a geometrically nonlinear manner making the computational expense of direct numerical integration in physical degrees of freedom prohibitive as for any type of modal reduction scheme the quality of the reduced order solution is dictated by the modal basis selection the techniques for modal basis selection currently employed for nonlinear simulation are ad hoc and are strongly influenced by the analyst s subjective judgment this work develops a reliable and rigorous procedure through which an efficient modal basis can be chosen the method employs proper orthogonal decomposition to identify nonlinear system dynamics and the modal assurance criterion to relate proper orthogonal modes to the normal modes that are eventually used as the basis functions the method is successfully applied to the analysis of a planar beam and a shallow arch over a wide range of nonlinear dynamic response regimes the error associated with the reduced order simulation is quantified and related to the computational cost

prediction condensed models adapted to the nonlinear structures in time domain abstract this paper deals with a method to study closely the stationary solution of nonlinear dynamic systems in time domain this method is based on the exploitation of karhunen loeve decomposition with or without parametric modifications as well as on the characteristics of localized nonlinearities with the application of this method at first on linear models initially condensed by karhunen loeve the predictions of nonlinear responses can be obtained rapidly this method is adapted to a condensed linear model used in the first optimization procedure of the nonlinear dynamic behaviour this robust basis will be used as condensation basis of the modified model local per zone which leads to a prediction of vibratory responses of complex structures modified and affected by localized nonlinearities

existence of hypercyclic operators on topological vector spaces the present paper introduces a very simple but very useful notion of the so called quasi extension ofl1 operators and proves that a large class of topological vector spaces admit continuous hypercyclic operators in particular it answers in the affirmative a question of s rolewicz posed in 1969 whether or not every infinite dimensional separable banach space admits a continuous hypercyclic operator

low dimensional models for the nonlinear vibration analysis of cylindrical shells based on a perturbation procedure and proper orthogonal decomposition in formulating mathematical models for dynamical systems obtaining a high degree of qualitative correctness i e predictive capability may not be the only objective the model must be useful for its intended application and models of reduced complexity are attractive in many cases where time consuming numerical procedures are required this paper discusses the derivation of discrete low dimensional models for the nonlinear vibration analysis of thin cylindrical shells in order to understand the peculiarities inherent to this class of structural problems the nonlinear vibrations and dynamic stability of a circular cylindrical shell subjected to static and dynamic loads are analyzed this choice is based on the fact that cylindrical shells exhibit a highly nonlinear behavior under both static and dynamic loads geometric nonlinearities due to finite amplitude shell motions are considered by using donnell s nonlinear shallow shell theory a perturbation procedure validated in previous studies is used to derive a general expression for the nonlinear vibration modes and the discretized equations of motion are obtained by the galerkin method using modal expansions for the displacements that

independent component analysis to enhance performances of karhunen loeve expansions for non gaussian stochastic processes application to uncertain systems abstract in the context of engineering systems an essential step in uncertainty quantification is the development of accurate and efficient representation of the random input parameters for such input parameters modeled as stochastic processes karhunen loeve expansion is a classical approach providing efficient representations using a set of uncorrelated but generally statistically dependent random variables the dependence structure among these random variables may be difficult to estimate statistically and is thus ignored in many practical applications this simplifying assumption of independence may lead to considerable errors in estimating the variability in the system state thus limiting the effectiveness of karhunen loeve expansion in certain cases in this paper independent component analysis is exploited to linearly transform the random variables used in karhunen loeve expansion resulting into a set of random variables exhibiting higher order decorrelation the stochastic wave equation is investigated for numerical illustration whereby the random stiffness coefficient is modeled as a non gaussian stochastic process under the assumption of independence among the random variables used in the karhunen loeve expansion and independent component analysis representations the latter provides more accurate statistical characterization of the output process for the specific cases examined

karhunen loeve decomposition of the transient dynamics of a multibay truss we compute the transient dynamics of an impulsively loaded multibay truss in computing the transient responses we avoid numerical instabilities due to exponential dichotomy of the bay transfer matrices by employing the direct global matrix approach this method eliminates the need of transfer matrix multiplications that are prone to numerical instabilities and instead requires a numerically stable inversion of a global transfer matrix of large dimension the transient responses at multiple points of the truss are used to perform system identification of the dynamics using karhunen loeve k l decomposition this analysis computes dominant coherent structures proper orthogonal modes of the truss under specific transient excitations together with the energy of each identified coherent structure the identified k l modes can be used to develop accurate low order models of the truss dynamics for control or substructure synthesis purposes

chaotic banach algebras we construct an infinite dimensional non unital banach algebra a and a a such that the sets zan z c n n and 1 a na n n are both dense in a where 1 is the unity in the unitalization a a span 1 of a as a byproduct we get a hypercyclic operator t on a banach space such that t t is non cyclic and t 1 msc 47a16 46j45

linear subsets of nonlinear sets in topological vector spaces for the last decade there has been a generalized trend in mathematics on the search for large algebraic structures linear spaces closed subspaces or infinitely generated algebras composed of mathematical objects enjoying certain special properties this trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis operator theory summability theory polynomials in banach spaces hypercyclicity and chaos and general functional analysis this expository paper is devoted to providing an account on the advances and on the state of the art of this trend nowadays known as lineability and spaceability

reduced bases for nonlinear structural dynamic systems a comparative study abstract the presented work provides an overview of some commonly used approaches for generating reduced bases for discrete nonlinear dynamic systems it investigates the performance and the robustness of these bases if they are applied in a reduction by projection procedure on different test cases the bases are created from the linear normal modes the ritz vectors the proper and the smooth orthogonal decomposition method the a priori reduction the centroidal voronoi tessellation and the local equivalent linear stiffness method second order terms and an enhanced proper orthogonal decomposition formulation are included as variants the test cases are small dimensional locally or entirely nonlinear system subjected to a harmonic or an impulse force excitation the double objective of this numerical study is first to determine which bases are most adequate for a given combination of nonlinearity and excitation and second to which extend the bases exhibit an inherent robustness if the parameterisation of the excitation is changed a specific multicriteria decision analysis score is developed to assess the bases performance as a major result a strong dependence of the performance of the bases on the type of excitation is established and thus some bases become more adequate for a certain situation than others also a lack of robustness for all considered bases can be observed this situation improves in most cases if the basis is generated with the most critical values of the parameter

uncertainty aversion with second order utilities and probabilities subjective expected utility theory does not distinguish between attitudes toward uncertainty ambiguous probabilities and attitudes toward risk unambiguous probabilities both are explained in terms of nonlinear utility for money rather than properties of events per se hence the decision maker displays the same attitude toward all sources of risk and uncertainty there is ample evidence that real decision makers do not always behave or even wish to behave in this way and instead they often distinguish between risk and uncertainty as in ellsbergs 1961 paradox this paper presents a simple axiomatic model of nonneutral attitudes toward uncertainty and a behavioral test for uncertainty aversion that is applicable even if utility is state dependent the decision maker may display different degrees of aversion toward gambles on different kinds of events e g being systematically more averse toward gambles on events whose probabilities are more ambiguous for such a decision maker the elicitation of preferences among objective gambles may not yield the correct measure of risk aversion for modeling real world decisions

modal interaction in chaotic vibrations of a shallow double curved shell panel abstract experimental results and analytical results are presented on chaotic vibrations of a shallow double curved shell panel subjected to gravity and periodic excitation modal interactions in the chaotic responses are discussed the shell panel with square boundary is simply supported for deflection in plane displacement at the boundary is elastically constrained in the experiment time histories of the chaotic responses at the spatial multiple positions of the shell panel are measured for the inspection of modal interaction in the analysis the shallow shell panel is assumed to have constant curvatures along to orthogonal directions and geometric initial imperfection the donnell mushtari vlasov type equation is used as governing equation with lateral inertia force assuming deflection with multiple modes of vibration the governing equation is reduced to a set of nonlinear ordinary differential equations by the bubnov galerkin procedure chaotic responses are integrated numerically the chaotic responses which are obtained by the experiment and the analysis are inspected with the fourier spectra the poincare projections the maximum lyapunov exponents and the lyapunov dimension contribution of modes of vibration to the chaotic responses is analyzed by the principal component analysis i e karhunen loeve transformation

common hypercyclic vectors for the conjugate class of a hypercyclic operator abstract given a separable in nite dimensional hilbert space it was recently shown by the authors that there is a path of chaotic operators which is dense in the operator algebra with the strong operator topology and along which every operator has the exact same dense set of hypercyclic vectors in the present work we show that the conjugate set of any hypercyclic operator on a separable in nite dimensional banach space always contains a path of operators which is dense with the strong operator topology and yet the set of common hypercyclic vectors for the entire path is a dense set as a corollary the hypercyclic operators on such a banach space form a connected subset of the operator algebra with the strong operator topology 1 introduction throughout the present paper let be a separable in nite dimensional banach space over the scalar eld or and let denote the algebra of bounded linear operators an operator in is hypercyclic if there is a vector in for which its orbit 0 is dense in such a vector is called a hypercyclic vector for an operator in is hypercyclic if and only if the set of hypercyclic vectors for denoted by is a dense set see kitai 23 for a countable family of hypercyclic operators a direct application of the baire category theorem implies that the set

a robust bi orthogonal dynamically orthogonal method using the covariance pseudo inverse with application to stochastic flow problems abstract we develop a new robust methodology for the stochastic navier stokes equations based on the dynamically orthogonal do and bi orthogonal bo methods 1 2 3 both approaches are variants of a generalized karhunen loeve kl expansion in which both the stochastic coefficients and the spatial basis evolve according to system dynamics hence capturing the low dimensional structure of the solution the do and bo formulations are mathematically equivalent 3 but they exhibit computationally complimentary properties specifically the bo formulation may fail due to crossing of the eigenvalues of the covariance matrix while both bo and do become unstable when there is a high condition number of the covariance matrix or zero eigenvalues to this end we combine the two methods into a robust hybrid framework and in addition we employ a pseudo inverse technique to invert the covariance matrix the robustness of the proposed method stems from addressing the following issues in the do bo formulation i eigenvalue crossing we resolve the issue of eigenvalue crossing in the bo formulation by switching to the do near eigenvalue crossing using the equivalence theorem and switching back to bo when the distance between eigenvalues is larger than a threshold value ii ill conditioned covariance matrix we utilize a pseudo inverse strategy to invert the covariance matrix iii adaptivity we utilize an adaptive strategy to add remove modes to resolve the covariance matrix up to a threshold value in particular we introduce a soft threshold criterion to allow the system to adapt to the newly added removed mode and therefore avoid repetitive and unnecessary mode addition removal when the total variance approaches zero we show that the do bo formulation becomes equivalent to the evolution equation of the optimally time dependent modes 4 we demonstrate the capability of the proposed methodology with several numerical examples namely i stochastic burgers equation we analyze the performance of the method in the presence of eigenvalue crossing and zero eigenvalues ii stochastic kovasznay flow we examine the method in the presence of a singular covariance matrix and iii we examine the adaptivity of the method for an incompressible flow over a cylinder where for large stochastic forcing thirteen do bo modes are active

the reduced model multiscale method r3m for the non linear homogenization of hyperelastic media at finite strains this paper presents a new multi scale method for the homogenization analysis of hyperelastic solids undergoing finite strains the key contribution is to use an incremental nonlinear homogenization technique in tandem with a model reduction method in order to alleviate the complexity of multiscale procedures which usually involve a large number of nonlinear nested problems to be solved the problem associated with the representative volume element rve is solved via a model reduction method proper orthogonal decomposition the reduced basis is obtained through pre computations on the rve the technique coined as reduced model multiscale method r3m allows reducing significantly the computation times as no large matrix needs to be inverted and as the convergence of both macro and micro problems is enhanced furthermore the r3m drastically reduces the size of the data base describing the history of the micro problems in order to validate the technique in the context of porous elastomers at finite strains a comparison between a full and a reduced multiscale analysis is performed through numerical examples involving different micro and macro structures as well as different nonlinear models neo hookean mooney rivlin it is shown that the r3m gives good agreement with the full simulations at lower computational and data storage requirements

subjective probabilities on subjectively unambiguous events this paper suggests a behavioral definition of subjective ambiguity in an abstract setting where objects of choice are savage style acts then axioms are described that deliver probabilistic sophistication of preference on the set of unambiguous acts in particular both the domain and the values of the decision maker s probability measure are derived from preference it is argued that the noted result also provides a decision theoretic foundation for the knightian distinction between risk and ambiguity

a complex orthogonal decomposition for wave motion analysis a method is presented for decomposing wave motion into its principle components the basic idea is a complex generalization of proper orthogonal decomposition the method involves the representation of real oscillatory signals as complex analytic signals the relationship between complex modes and wave motion is explored from an ensemble of complex signals a complex correlation matrix is formed and its complex eigensolution is the basis of the decomposition like a complex singular value decomposition the complex eigenvectors contain standing and traveling characteristics a traveling index is proposed to quantify the relative degree of traveling and standing in a waveform a method of dissecting a wave mode into its traveling and standing parts is also proposed from the complex modes and modal coordinates frequencies wavelengths and characteristic wave speeds can be obtained the method is applied to traveling and standing wave examples

karhunen loeve decomposition of coupled axial bending vibrations of beams subject to impacts abstract this work presents a study of the oscillations of a vertical slender beam clamped in its upper extreme pinned in its lower one and constrained inside an outer cylinder in its lower portion the beam is subject to distributed axial loads due to its own weight leading to geometric softening of its lower portion and thus yielding a large number of vibroimpacts with the outer cylinder this is due to the axial bending coupling often called geometric stiffening and largely discussed in the last two decades here it is accounted for by using a non linear finite element model proposed in a previous work in which non linear strain displacement relations are considered to help understand this non linear coupled vibro impact problem the karhunen loeve decomposition also known as the proper orthogonal decomposition is applied to its simulated dynamics the results show that the micro impacts accompanying the beam hole impacts and mainly due to the beam compressive softening and the reaction forces at the top and bottom positions are well represented only when using a non linear axial bending coupling it is also shown that 15 proper orthogonal modes are sufficient to reconstruct the dynamics of the impacting beam under a 3 error margin

proper orthogonal decomposition and its applications part ii model reduction for mems dynamical analysis abstract proper orthogonal decomposition pod methods are popular tools for data analysis aimed at obtaining low dimensional approximate descriptions of a high dimensional process in many engineering fields the applications of pod methods to model reduction for microelectromechanical systems mems are reviewed in this paper in view of the fact that existing pod methods in the model reduction for dynamic simulation of mems dealt with only noise free data this paper proposes a neural network based method that combines robust principal component analysis pca neural network model with galerkin procedure for dynamic simulation and analysis of non linear mems with noisy data simulations are given to show the performance of the proposed method in comparison with the existing method compared with the standard pca neural network model the robust pca neural network model has a number of numerical advantages such as the stability and robustness to noise injected data and the faster convergence of iterations in the training stages than the existing neural network technique the macro model generated by using the eigenvectors extracted from the proposed method as basis functions shows its flexibility and efficiency in the representation and simulation of the original non linear partial differential equations

on proper orthogonal co ordinates as indicators of modal activity proper orthogonal decomposition pod is applied for examining modal activity the extraction of proper orthogonal modal co ordinates pocs is outlined the proper orthogonal values povs are the mean squared values of the pocs the number of povs above the noise floor provides a bound on the number of significant modes based on povs above the noise floor the ideas are illustrated on a linear cantilevered beam experiment the displacement ensembles are obtained by processing six strain measurements coherent proper orthogonal modes poms and povs above the noise floor together confirm that six active modes are detected in the system the distribution of modal components in pocs is discussed the characteristics of the poms povs and pocs are then examined in the presence of added noise

space time proper generalized decompositions for the resolution of transient elastodynamic models in this paper we investigate ability of proper generalized decomposition pgd to solve transient elastodynamic models in space time domain classical methods use time integration schemes and an incremental resolution process we propose here to use standard time integration methods in a non incremental strategy as a result pgd gives a separated representation of the space time solution as a sum of tensorial products of space and time vectors that we interpret as space time modes recent time integration schemes are based on multi field formulations in this case separated representation can be constructed using state vectors in space and same vectors in time however we have experienced bad convergence order using this decomposition furthermore temporal approximation must be the same for all fields thus we propose an extension of classical separated representation for multi field problems this multi field pgd mf pgd uses space and time vectors that are different for each field calculation of decomposition is done using a monolithic approach in space and time potentially allowing the use of different approximations in space and time finally several simulations are performed with the transient elastodynamic problem with one dimension in space different approximations in time are investigated newmark scheme single field time discontinuous galerkin method and two fields time continuous and discontinuous galerkin methods

hybrid proper orthogonal decomposition formulation for linear structural dynamics abstract hybrid proper orthogonal decomposition podh formulation is a pod based reduced order modeling method where the continuous equation of the physical system is projected on the pod modes obtained from a discrete model of the system the aim of this paper is to evaluate the hybrid pod formulation and to compare it with other pod formulations on the simple case of a linear elastic rod subject to prescribed displacements in the perspective of building reduced order models for coupled fluid structure systems in the future in the first part of the paper the hybrid pod is compared to two other formulations for the response to an initial condition an approach based on the discrete finite elements equation of the rod called the discrete pod podd and an analytical approach using the exact solution of the problem and consequently called the analytical pod poda this first step is useful to ensure that the podh performs well with respect to the other formulations the podh is therefore used afterwards for the forced motion response where a displacement is imposed at the free end of the rod the main contribution of this paper lies in the comparison of three techniques used to take into account the non homogeneous dirichlet boundary condition with the hybrid pod the first method relies on control functions the second on the penalty method and the third on lagrange multipliers finally the robustness of the hybrid pod is investigated on two examples involving firstly the introduction of structural damping and secondly a nonlinear force applied at the free end of the rod

sliding window proper orthogonal decomposition application to linear and nonlinear modal identification abstract the proper orthogonal decomposition pod method is a tool well adapted to analyze vector signals whereas continuous gabor transform cgt is suitable for scalar signal with multi frequency components in this paper a method named sliding window proper orthogonal decomposition swpod combining pod and cgt to analyze multi degrees of freedom mdof vibration system responses is presented swpod gives access to the evolution of the coherent spacial structures and their frequency components versus time the method is of principal interest in the case of swept sine excitation of linear or nonlinear systems to access the resonance frequencies mode shapes and modal damping ratios a procedure is proposed to extract the linear and nonlinear normal modes of weakly damped mdof mechanical systems and illustrated using numerical examples

modal parameter estimation for a wetted plate under flow excitation a challenging case in using pod abstract in this paper some issues related to the use of the proper orthogonal decomposition pod for the identification of the modal parameters of 2d structures are highlighted with reference to the experimental identification of the vibration modes of a plate excited on one side by a turbulent boundary layer the poor convergence of the pod modes to the linear normal modes is explained analytically and provides also the theoretical framework to avoid these issues by applying band pass filtering bp pod the application of the pod with data pre filtering has allowed for the evaluation of the modal damping from the proper orthogonal coordinates by using the random decrement technique to obtain the corresponding free decay function though the emphasis is about methodological aspects related to the pod as a byproduct of the performed analysis some more physical insights concerning the damping sensitivity and the hydrodynamic added mass are finally carried out

proper orthogonal decomposition and its applications part i theory abstract in view of the increasing popularity of the application of proper orthogonal decomposition pod methods in engineering fields and the loose description of connections among the pod methods the purpose of this paper is to give a summary of the pod methods and to show the connections among these methods firstly the derivation and the performance of the three pod methods karhunen loeve decomposition kld principal component analysis pca and singular value decomposition svd are summarized then the equivalence problem is discussed via a theoretical comparison among the three methods the equivalence of the matrices for processing the objective functions the optimal basis vectors the mean square errors and the asymptotic connections of the three methods are demonstrated and proved when the methods are used to handle the pod of discrete random vectors

on convex cyclic operators abstract we give a hahn banach characterization for convex cyclicity we also obtain an example of a bounded linear operator s on a banach space with p s such that s is convex cyclic but s is not weakly hypercyclic and s 2 is not convex cyclic this solved two questions of rezaei in 25 when p s we also characterize the diagonalizable normal operators that are convex cyclic and give a condition on the eigenvalues of an arbitrary operator for it to be convex cyclic we show that certain adjoint multiplication operators are convex cyclic and show that some are convex cyclic but no convex polynomial of the operator is hypercyclic also some adjoint multiplication operators are convex cyclic but not 1 weakly hypercyclic

acceleration of the spectral stochastic fem using pod and element based discrete empirical approximation for a micromechanical model of heterogeneous materials with random geometry abstract the spectral stochastic fem with local basis functions in the stochastic domain sl fem is one of the most flexible and accurate stochastic methods however also the most computationally expensive these expenses are traditionally associated with the extra large tangent stiffness matrix and a huge number of elements which need to be re integrated in every iteration in this work we incorporate the proper orthogonal decomposition pod into the sl fem thus performing a drastic reduction of the stiffness matrix in order to reduce the integration costs by hyperreduction a novel element based modification of the discrete empirical interpolation the so called element based empirical approximation method edeam is developed and combined with the pod particular advantages of the sl fem for order reduction and hyperreduction compared to other stochastic techniques are discussed the new reduced order sl fem is applied to the computational homogenization of materials with random geometry of the microstructure i e to a general class of problems exhibiting strongly nonlinear non smooth and sometimes discontinuous dependency of the solution on some random parameters the reduced order sl fem demonstrates a high accuracy and a high solution speed whereby the solution time for the reduced order sl fem is comparable to the solution time of only one single monte carlo sample

dynamics of large scale coupled structural mechanical systems a singular perturbation proper orthogonal decomposition approach we have combined the theories of geometric singular perturbation and proper orthogonal decomposition to study systematically the dynamics of coupled systems in mechanics involving coupling between continuous structures and nonlinear oscillators here we analyze a prototypical structural mechanical system consisting of a planar nonlinear pendulum coupled to a flexible rod made of linear viscoelastic material we cast the equations of motion in a singularly perturbed set of oscillators and compute analytic approximations to an attractive global invariant manifold in phase space of the coupled system the invariant manifold two dimensional for the unforced system and three dimensional for the forced system carries a continuum of slow motions for a sufficiently stiff rod a proper orthogonal decomposition of any long time motion extracts a single structure for the spatial coherence of the dynamics which is a realization of the slow invariant manifold as the flexibility of the rod increases the energy of

karhunen loeve modes of a truss transient response reconstruction and experimental verification karhunen loeve k l or proper orthogonal decomposition modes are used to discretize the dynamics of a four bay linear truss this is achieved by defining global k l modal amplitudes and employing the orthogonality relations between k l modes that are inherent in the k l decomposition it is found that the k l based low order models can capture satisfactory the transient dynamics of the truss even when only a limited number of them is used for the order reduction a comparison between the exact and low order dynamics in the frequency domain reveals that the low order models capture the leading resonances of the truss a series of experiments with a practical three bay truss is then performed to validate the theoretical k l decomposition a comparison between theory and experiment indicates agreement between the predicted and realized dominant k l mode shapes but less so in the higher order modes the reasons for this discrepancy between theory and experiment are discussed and possible applications of the k l based order reduction to passive and active control of practical large scale flexible systems are outlined

analytical sensitivities of principal components in time series analysis of dynamical systems original method for calculating the analytical sensitivities of the principal components of dynamical systems is developed methods for analytical sensitivity calculations are developed for both the singular value decomposition and eigenanalysis based approaches for principal component calculation sensitivities with respect to state initial conditions and system parameters are enabled by state transition matrix calculations for augmented state and parameter differential equations a novel approach to compute principal component sensitivities with respect to transient forcing function parameters isintroduced bytransforming nonhomogenous differential equations forced system into homogenous differential equations unforced system these new developments are applied to several example problems in dynamics analysis with an emphasis on structural dynamics analysis analytical sensitivities provide the necessary derivatives for gradient based optimization algorithms and provide an analytical framework for evaluating structural modifications based on principal components analysis

a time varying identification method for mixed response measurements the proper orthogonal decomposition is a method that may be applied to linear and nonlinear structures for extracting important information from a measured structural response this method is often applied for model reduction of linear and nonlinear systems and has been applied recently for time varying system identification although methods have previously been developed to identify time varying models for simple linear and nonlinear structures using the proper orthogonal decomposition of a measured structural response the application of these methods has been limited to cases where the excitation is either an initial condition or an applied load but not a combination of the two this paper presents a method for combining previously published proper orthogonal decomposition based identification techniques for strictly free or strictly forced systems to identify predictive models for a system when only mixed response data are available i e response data resulting from initial conditions and loads that are applied together this method extends the applicability of the previous proper orthogonal decomposition based identification techniques to operational data acquired outside of a controlled laboratory setting the method is applied to response data generated by finite element models of simple linear time invariant time varying and nonlinear beams and the strengths and weaknesses of the method are discussed

smooth orthogonal decomposition for modal analysis of randomly excited systems modal parameter estimation in terms of natural frequencies and mode shapes is studied using smooth orthogonal decomposition for randomly excited vibration systems this work shows that under certain conditions the smooth orthogonal decomposition eigenvalue problem formulated from white noise induced response data can be tied to the unforced structural eigenvalue problem and thus can be used for modal parameter estimation using output response ensembles only the generalized eigenvalue problem is formed to estimate eigenfrequencies and modal vectors for an eight degree of freedom lightly damped vibratory system the estimated frequencies are compared against system frequencies obtained from structural eigenvalue problem and estimated modal vectors are checked using the modal assurance criterion for lightly damped simulations satisfactory results are obtained for estimating both system frequencies and modal vectors

physical interpretation of the proper orthogonal modes using the singular value decomposition abstract proper orthogonal decomposition is a statistical pattern analysis technique for finding the dominant structures called the proper orthogonal modes in an ensemble of spatially distributed data while the proper orthogonal modes are obtained through a statistical formulation they can be physically interpreted in the field of structural dynamics the purpose of this paper is thus to provide some insights into the physical interpretation of the proper orthogonal modes using the singular value decomposition

subjective probability and expected utility without additivity comonotic acts are defined and an axiom of comonotomic independence 19 introduced that weakens the von newmann morgenstern axiom of independence if a nondegenerate continuous and monotonic state independent weak order over acts satisfies comonotonic independence then it induces a unique non necessarily additive probability and a von neumann morgenstern utility an extension of theexpected utility theory covers situations as the ellsberg paradox which are inconsistent with additive expected utility the concept of uncertainty aversion and interpretation of comonotonic independence in the context of social welfare functions are included

modal reduction of mathematical models of biological molecules this paper reports a detailed study of modal reduction based on either linear normal mode lnm analysis or proper orthogonal decomposition pod for modeling a single a d glucopyranose monomer as well as a chain of monomers attached to a moving atomic force microscope afm under harmonic excitations also a modal reduction method combining pod and component modal synthesis is developed the accuracy and efficiency of these methods are reported the focus of this study is to determine to what extent these methods can reduce the time and cost of molecular modeling and simultaneously provide the required accuracy it has been demonstrated that a linear reduced order model is valid for small amplitude excitation and low frequency excitation it is found that a nonlinear reduced order model based on pod modes provides a good approximation even for large excitation while the nonlinear reduced order model using linear eigenmodes as the basis vectors is less effective for modeling molecules with a strong nonlinearity the reduced order model based on component modal synthesis using pod modes for each component also gives a good approximation with the reduction in the dimension of the system using these methods the computational time and cost can be reduced significantly

remarks on the efficiency of pod for model reduction in non linear dynamics of continuous elastic systems the first objective of this paper is to analyse the efficiency of the reduced models constructed using the proper orthogonal decomposition pod basis and the lin basis in non linear dynamics for continuous elastic systems the pod basis is the hilbertian basis constructed with the pod method while the lin basis is the hilbertian basis derived from the generalized continuous eigenvalue problem associated with the underlying conservative part of the continuous elastic system and usually called the eigenmodes of vibration the efficiency of the pod basis or the lin basis is related to the rate of convergence in the frequency domain of the solution constructed with the reduced model with respect to its dimension a basis will be more efficient than another if the reduced order solution of the galerkin projection converges to the solution of the dynamical system more rapidly than the reduced order solution of the other

smooth orthogonal decomposition based vibration mode identification a new multivariate data analysis method called smooth orthogonal decomposition sod is proposed to extract linear normal modes and natural frequencies of multi degree of freedom and distributed parameter vibration systems it is demonstrated that for an undamped free vibration of a multi degree of freedom system the computed smooth orthogonal modes are in direct correspondence with the actual normal vibration modes and the smooth orthogonal values are related to the corresponding natural frequencies the same is also shown to be true for lightly damped free vibrations of both lumped and distributed parameter systems in contrast to the intrinsic limitations of the proper orthogonal decomposition pod analysis which requires the knowledge of system s mass matrix to extract normal modes and cannot uniquely identify modal subspaces that have similar proper orthogonal values the sod is shown to overcome both of these deficiencies numerical examples are provided to compare the performances of the pod and sod based modal identification in various types of vibration environment