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

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

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

the liapunov dimension of strange attractors abstract many papers have been published recently on studies of dynamical processes in which the attracting sets appear quite strange in this paper the question of estimating the dimension of the attractor is addressed while more general conjectures are made here particular attention is paid to the idea that if the jacobian determinant of a map is greater than one and a ball is mapped into itself then generically the attractor will have positive two dimensional measure and most of this paper is devoted to presenting cases with such jacobians for which the attractors are proved to have non empty interior

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

proper orthogonal decomposition pod of a class of vibroimpact oscillations abstract this study employs the method of proper orthogonal decomposition pod also known as the karhunen loeve k l method to extract dominant coherent structures modes that approximate the system behavior from time series data these mode shapes can be used in a galerkin reconstruction process to obtain lower dimensional models for the structural systems under consideration the mode shapes and the energies obtained by this process can also be used to predict certain kinds of periodic or non periodic non linear motions the k l method has been applied successfully to fluid dynamical thermal processes and signal processing however only a handful of works exist in the area of vibration analysis and structural mechanics an extensive k l analysis is performed numerically for two vibroimpacting systems a beam and a rotor lower dimensional models are created and used to study non linear energy transmission from low to higher k l modes extensive reconstructions are also performed to prove the efficacy of the k l method to provide accurate low dimensional dynamical models in addition experimental investigations are presented for the case of the overhung impacting rotor and qualitative comparisons with the theoretical analysis are presented

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

post buckling nonlinear static and dynamical analyses of uncertain cylindrical shells and experimental validation the paper presents a complete experimental validation of an advanced computational methodology adapted to the nonlinear post buckling analysis of geometrically nonlinear structures in presence of uncertainty a mean nonlinear reduced order computational model is first obtained using an adapted projection basis the stochastic nonlinear computational model is then constructed as a function of a scalar dispersion parameter which has to be identified with respect to the nonlinear static experimental response of a very thin cylindrical shell submitted to a static shear load the identified stochastic computational model is finally used for predicting the nonlinear dynamical post buckling behavior of the structure submitted to a stochastic ground motion

a note on coefficient of restitution models including the effects of impact induced vibration in this work multi modal systems subject to impact are considered using energy balance techniques for an arbitrary contact interval the effects of modal vibration can be included the energy balance is used to obtain a relationship between the coefficient of restitution and the modal energy during the contact period this allows the effects of impact induced vibration to be considered the subsequent analytical relationships demonstrate that increasing contact duration and excitation of higher modes can reduce the effective value of the coefficient of restitution it is also shown how this approach can be related to work on energetically consistent impacts

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

experiments and analysis on chaotic vibrations of a shallow cylindrical shell panel abstract detailed experimental results and analytical results are presented on chaotic vibrations of a shallow cylindrical shell panel subjected to gravity and periodic excitation the shallow shell panel with square boundary is simply supported for deflection in plane displacement at the boundary is elastically constrained by in plain springs in the experiment the cylindrical shallow shell panel with thickness 0 24 mm square form of length 140 mm and mean radius 5150 mm is used for the test specimen all edges around the shell boundary are simply supported by adhesive flexible films first to find fundamental properties of the shell panel linear natural frequencies and characteristics of restoring force of the shell panel are measured these results are compared with the relevant analytical results then geometrical parameters of the shell panel are identified exciting the shell panel with lateral periodic acceleration nonlinear frequency responses of the shell panel are obtained by sweeping the frequency of periodic acceleration in typical ranges of the exciting frequency predominant chaotic responses are generated time histories of the responses are recorded for inspection of the chaos in the analysis the donnell equation with lateral inertia force is introduced assuming mode functions the governing equation is reduced to a set of nonlinear ordinary differential equations by the galerkin procedure periodic responses are calculated by the harmonic balance method chaotic responses are integrated numerically by the runge kutta gill method 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 it is found that the dominant chaotic responses of the shell panel are generated from the responses of the sub harmonic resonance of 1 2 order and of the ultra sub harmonic resonance of 2 3 order by the convergence of the maximum lyapunov exponent to the embedding dimension the number of predominant vibration modes which contribute to the chaos is found to be three or four fairly good agreements are obtained between the experimental results and the analytical results

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

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

some relations between dimension and lyapounov exponents we consider differentiable maps and compact invariant sets we introduce dimensional quantities related to the ergodic invariant measures and prove some simple relations

observations of order and chaos in nonlinear systems experiments on nonlinear electrical oscillators the belousov zhabotinskii reaction rayleigh benard convection and couette taylor flow have revealed several common routes to chaos that have also been found in numerical studies of models with a few degrees of freedom experimental results are presented illustrating the following transition sequences period doubling and the u sequence intermittency the periodic quasiperiodic chaotic sequence frequency locking and an alternating periodic chaotic sequence

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

nonlinear perturbations of a class of holomorphic semigroups of growth order α by comparison theorems for volterra equations abstract this paper studies nonlinear perturbation of a holomorphic semigroup of growth order in a banach space x the generator a of a holomorphic semigroup is assumed to be almost sectorial and it is also assumed that the part a 0 in the closure x 0 of the domain d a is sectorial a nonlinear perturbing operator b is assumed to be locally continuous from a subset of a real interpolation space between x 0 and d a 0 into x existence and uniqueness of mild solutions to the associated cauchy problems are proved by using comparison theorems for integral equations of volterra type the result obtained is applied to show the global well posedness of drift diffusion systems with no flux boundary conditions in the two dimensional case

existence and asymptotics of solutions for a parabolic elliptic system with nonlinear no flux boundary conditions u x 0 u c 2 0 3 from which d x 0 is determined via la and 2b the systems of this type appear in the theory of semiconductors l 4 or in electrochemistry where u or rather ut u2 for more general systems involving several unknown functions u are the densities of charge carriers electrons holes ions and in groundwater flow problems 5 61 where u is the density of a fluid in a porous medium the parabolic equation la is flux conservative and the boundary condition 2a is the physically relevant no flux condition hence the total charge or mass according to the physical interpretation

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

integrated semigroups and integrated solutions to abstract cauchy problems abstract concepts which have been developed in the theory of weakly continuous semigroups on dual banach spaces are extended to integrated semigroups the relation to integrated solutions of homogeneous and inhomogeneous abstract cauchy problems is established the results are illustrated for the wave equation in l2 rm

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

small overlap monoids i the word problem abstract we develop a combinatorial approach to the study of semigroups and monoids with finite presentations satisfying small overlap conditions in contrast to existing geometric methods our approach facilitates a sequential left right analysis of words which lends itself to the development of practical efficient computational algorithms in particular we obtain a highly practical linear time solution to the word problem for monoids and semigroups with finite presentations satisfying the condition c 4 and a polynomial time solution to the uniform word problem for presentations satisfying the same condition

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

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

which kleene semigroups are finite the set of idempotents in any given d class of a kleene semigroup is rational we use simon s theorem to obtain some finitude results

chaotic vibrations of circular cylindrical shells galerkin versus reduced order models via the proper orthogonal decomposition method abstract the geometric nonlinear response of a water filled simply supported circular cylindrical shell to harmonic excitation in the spectral neighbourhood of the fundamental natural frequency is investigated the response is investigated for a fixed excitation frequency by using the excitation amplitude as bifurcation parameter for a wide range of variation bifurcation diagrams of poincare maps obtained from direct time integration and calculation of the lyapunov exponents and lyapunov dimension have been used to study the system by increasing the excitation amplitude the response undergoes i a period doubling bifurcation ii subharmonic response iii quasi periodic response and iv chaotic behaviour with up to 16 positive lyapunov exponents hyperchaos the model is based on donnell s nonlinear shallow shell theory and the reference solution is obtained by the galerkin method the proper orthogonal decomposition pod method is used to extract proper orthogonal modes that describe the system behaviour from time series response data these time series have been obtained via the conventional galerkin approach using normal modes as a projection basis with an accurate model involving 16 degrees of freedom dofs validated in previous studies the pod method in conjunction with the galerkin approach permits to build a lower dimensional model as compared to those obtainable via the conventional galerkin approach periodic and quasi periodic response around the fundamental resonance for fixed excitation amplitude can be very successfully simulated with a 3 dof reduced order model however in the case of large variation of the excitation even a 5 dof reduced order model is not fully accurate results show that the pod methodology is not as robust as the galerkin method

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

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

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

structural damage detection based on proper orthogonal decomposition experimental verification the present paper describes a new structural damage detection method based on the monitoring of vibrational properties of the structure sensors record the accelerations of several points of the structure the recorded data are then used to compute the proper orthogonal modes the comparison between the proper orthogonal modes of the undamaged structure and those of the damaged structure provides information on damage location experimental results are presented

low dimensional dynamics for the complex ginzburg landau equation abstract a method is presented which results in low dimensional dynamical systems for situations in which low dimensional attractors are known to exist the method is based on the use of the karhunen loeve procedure for the determination of an optimal basis and the subsequent use of the galerkin procedure to generate the dynamical system the method is applied to two problems for the ginzburg landau equation for which large databases have been obtained in each instance a dynamical system is generated which has roughly twice the number of degrees of freedom as the hausdorff dimension of the exact case it is also demonstrated that the approximations are robust in that they are accurate over a wide range of parameter space

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

small overlap monoids ii automatic structures and normal forms abstract we show that any finite monoid or semigroup presentation satisfying the small overlap condition c 4 has word problem which is a deterministic rational relation it follows that the set of lexicographically minimal words forms a regular language of normal forms and that these normal forms can be computed in linear time we also deduce that c 4 monoids and semigroups are rational in the sense of sakarovitch asynchronous automatic and word hyperbolic in the sense of duncan and gilman from this it follows that c 4 monoids satisfy analogues of kleene s theorem and admit decision algorithms for the rational subset and finitely generated submonoid membership problems we also prove some automata theoretic results which may be of independent interest

identification of a continuous structure with a geometrical non linearity part ii proper orthogonal decomposition particular effort has been spent in the field of identification of multi degree of freedom non linear systems the newly developed methods permit the structural analyst to consider increasingly complex systems the aim of this paper and a companion paper is to study by means of two methods a continuous non linear system consisting of an experimental cantilever beam with a geometrical non linearity in the companion paper part i 1 the ability of the conditioned reverse path method which is a frequency domain technique to identify the behaviour of this structure is assessed this paper part ii is devoted to the application of proper orthogonal decomposition which is an updating technique to the test example

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

easy multiplications i the realm of kleene s theorem abstract we study here the properties of a family of monoids which we call the rational monoids and which are monoids with a multiplication of low complexity a monoid is rational if its multiplication may be described by a rational function from a free monoid into itself the main results are that rational monoids like finite ones have the properties that green s relations d and f are equal and that kleene s theorem holds in rational monoids as in free ones every monoid described so far in which kleene s theorem holds is a rational monoid the closure of the family of rational monoids under rees quotient direct product and free product is then studied extensions of rational monoids will be considered in a forthcoming paper

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

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

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

proper orthogonal decomposition reduced order model for nonlinear aeroelastic oscillations in this study the proper orthogonal decomposition method in conjunction with a galerkin projection scheme is employed to solve the title problem of a fluttering plate in both two and three dimensions undergoing supersonic flow using von karman s large deflection plate theory and quasi steady supersonic aerodynamic theory proper orthogonal decomposition based reduced order models are constructed by using the responses as snapshots from the conventional galerkin method which also plays a role as a reference method in this paper results for the buckled limit cycle oscillation and chaotic responses of the simply supported plate are presented and compared with the galerkin solutions numerical examples demonstrate that the proper orthogonal decomposition reduced order model permits a much lower dimensional model as compared to that obtainable via the galerkin approach for example for a plate length to width ratio equal to 4 only two proper orthogonal decomposition modes are required to describe the pa

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

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

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

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

easy multiplications ii extensions of rational semigroups abstract in our first part of this paper we defined and studied the family of rational monoids which are monoids with an easy multiplication by this we mean indeed that the multiplication can be realized by a finite automaton in this second part of our work we consider rational semigroups instead of monoids and investigate the properties of two operations on these semigroups the redei extensions and the ideal extensions we give necessary and sufficient conditions for such extensions of rational semigroups to be rational semigroups the theory of ideal extensions is also used to build a non rational monoid in which kleene s theorem holds which answers a question asked in the first part a last application of our work on redei and ideal extensions is the construction of a family of monoids too simple to be syntactic monoids of context free languages

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

global existence uniqueness and continuous dependence for a semilinear initial value problem abstract in this paper we consider the semilinear initial value problem associated with an operator a whose spectrum lies in a sector of the complex plane and whose resolvent satisfies z a 1 m z for some 1 z outside the sector the properties of existence and uniqueness of global mild solutions and continuous dependence on the initial data are investigated

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

karhunen loeve decomposition approach to analyzing complex network synchronization in this paper the approach of the karhunen lo ve decomposition known also as the proper orthogonal modes poms is taken to analyze phase synchronization of various complex networks with different topologies namely the classic kuramoto model coupled chaotic maps with gaussian delays and a chain of diffusively coupled bistable oscillators in the case of the kuramoto model the poms reveal the tendency and the level of synchronization with the increase of the coupling strength for globally coupled networks and scale free networks while periodic poms are found in nearest neighbor coupled networks furthermore for cluster networks on the kuramoto model the first leading poms based on different time intervals reveal that different sub groups of nodes synchronize gradually to different levels eventually leading to the complete phase synchronization in the case of coupled chaotic maps some properties of phase synchronization change with the coupling strength value in the case of the chain of diffusively coupled bistable oscillators several main poms not only determine the network phase synchronization but also provide good reconstruction of the network responses

application of non smooth modelling techniques to the dynamics of a flexible impacting beam non smooth modelling techniques have been successfully applied to lumped mass type structures for modelling phenomena such as vibro impact and friction oscillators in this paper the application of these techniques to continuous elements using the example of a cantilever beam is considered employing a galerkin reduction to form an n degree of freedom modal model a technique for modelling impact phenomena using a non smooth dynamics approach is demonstrated numerical simulations computed using the non smooth model are compared with experimentally recorded data for a flexible beam constrained to impact on one side a method for dealing with sticking motions when numerically simulating the beam motion is presented in addition choosing the dimension of the model based on power spectra of experimentally recorded time series is discussed

observation of a strange attractor abstract phase space portraits have been constructed and analyzed for noisy nonperiodic data obtained in an experiment on a nonequilibrium homogeneous chemical reaction the phase space trajectories define a limit set that is an attractor following a perturbation the trajectory quickly returns to the attracting set this attracting set is shown to be strange nearby trajectories separate exponentially on the average moreover the poincare sections exhibit the stretching and folding that is characteristic of strange attractors

symbolic dynamics of noisy chaos abstract one model of randomness observed in physical systems is that low dimensional deterministic chaotic attractors underly the observations a phenomenological theory of chaotic dynamics requires an accounting of the information flow from the observed system to the observer the amount of information available in observations and just how this information affects predictions of the system s future behavior in an effort to develop such a description we discuss the information theory of highly discretized observations of random behavior metric entropy and topological entropy are well defined invariant measures of such an attractor s level of chaos and are computable using symbolic dynamics real physical systems that display low dimensional dynamics are however inevitably coupled to high dimensional randomless e g thermal noise we investigate the effects of such fluctuations coupled to deterministic chaotic systems in particular the metric entropy s response to the fluctuations we find that the entropy increases with a power law in the noise level and that the convergence of the entropy and the effect of fluctuations can be cast as a scaling theory we also argue that in addition to the metric entropy there is a second scaling invariant quantity that characterizes a deterministic system with added fluctuations i0 the maximum average information obtainable about the initial condition that produces a particular sequence of measurements or symbols

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

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

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

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

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