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

signatures and higher signatures of s 1 quotients we define and study the signature widehat a genus and higher signatures of the quotient space of an s 1 action on a closed oriented manifold we give applications to questions of positive scalar curvature and to an equivariant novikov conjecture

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

the stable rank of topological algebras and a problem of r g swan abstract various notions of stable ranks are studied for topological algebras some partial answers to r g swan s problem have two banach or good frechet algebras as in the density theorem in k theory the same stable rank are obtained for example a frechet dense subalgebra a of a c algebra b closed under c functional calculus of self adjoint elements has the same bass stable rank as b

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

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

sieve empirical likelihood ratio tests for nonparametric functions generalized likelihood ratio statistics have been proposed in fan zhang and zhang ann statist 29 2001 153 193 as a generally applicable method for testing norparametic hypotheses about nonparametric functions the likelihood ratio statistics are constructed based on the assumption that the distributions of stochastic errors are in a certain parametric family we extend their work to the case where the error distribution is completely unspecified via newly proposed sieve empirical likelihood ratio selr tests the approach is also applied to test conditional estimating equations on the distributions of stochastic errors it is shown that the proposed selr statistics follow asymptotically resealed chi 2 distributions with the scale constants and the degrees of freedom being independent of the nuisance parameters this demonstrates that the wilks phenomenon observed in fan zhang and zhang ann statist 29 2001 153 193 continues to hold under more relaxed models and a larger class of techniques the asymptotic power of the proposed test is also derived which achieves the optimal rate for nonparametric hypothesis testing the proposed approach has two advantages over the generalized likelihood ratio method it requires one only to specify some conditional estimating equations rather than the entire distribution of the stochastic error and the procedure adapts automatically to the unknown error distribution including heteroscedasticity a simulation study is conducted to evaluate our proposed procedure empirically

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

a note on maximizing a special concave function subject to simultaneous loewner order constraints maximization of functions with multivariate arguments can be computationally difficult we show that for a special function which is proportional to the density of a wishart distribution reparametrization can lead to maximization of a concave function we use this fact to produce an algorithm which maximizes the function over a restricted parameter space of the form 0 l u where l u means that u l is a nonnegative definite matrix and l u means that u l is positive definite this restriction is often referred to as the loewner ordering

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

an adaptive proper orthogonal decomposition method for model order reduction of multi disc rotor system abstract the proper orthogonal decomposition pod method is a main and efficient tool for order reduction of high dimensional complex systems in many research fields however the robustness problem of this method is always unsolved although there are some modified pod methods which were proposed to solve this problem in this paper a new adaptive pod method called the interpolation grassmann manifold igm method is proposed to address the weakness of local property of the interpolation tangent space of grassmann manifold itgm method in a wider parametric region this method is demonstrated here by a nonlinear rotor system of 33 degrees of freedom dofs with a pair of liquid film bearings and a pedestal looseness fault the motion region of the rotor system is divided into two parts simple motion region and complex motion region the adaptive pod method is compared with the itgm method for the large and small spans of parameter in the two parametric regions to present the advantage of this method and disadvantage of the itgm method the comparisons of the responses are applied to verify the accuracy and robustness of the adaptive pod method as well as the computational efficiency is also analyzed as a result the new adaptive pod method has a strong robustness and high computational efficiency and accuracy in a wide scope of parameter

rapidly decreasing functions in reduced algebras of groups let r be a group we associate to any length function l on r the space h r of rapidly decreasing functions on r with respect to l which coincides with the space of smooth functions on the k dimensional torus when r zk we say that r has property rd if there exists a length function l on r such that h r is contained in the reduced c algebra c r of r we study the stability of property rd with respect to some constructions of groups such as subgroups over groups of finite index semidirect and amalgamated products finally we show that the following groups have property rd 1 finitely generated groups of polynomial growth 2 discrete cocompact subgroups of the group of all isometries of any hyperbolic space

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

deconvolution density estimation on so n this paper develops nonparametric deconvolution density estimation over so n the group of n x n orthogonal matrices of determinant 1 the methodology is to use the group and manifold structures to adapt the euclidean deconvolution techniques to this lie group environment this is achieved by employing the theory of group representations explicit to so n general consistency results are obtained with specific rates of convergence achieved under sufficient smoothness conditions application to empirical bayes prior estimation and inference is also discussed

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

normal modes for non linear vibratory systems a methodology is presented which extends to non linear systems the concept of normal modes of motion which is well developed for linear systems the method is constructive for weakly non linear systems and provides the physical nature of the normal modes along with the non linear differential equations which govern their dynamics it also provides the non linear co ordinate transformation which relates the original system co ordinates to the modal co ordinates using this transformation we demonstrate how an approximate non linear version of superposition can be employed to reconstruct the overall motion from the individual non linear modal dynamics the results presented herein for non linear systems reduce to modal analysis for the linearized system when non linearities are neglected even though the approach is entirely different from the traditional one the tools employed are from the theory of invariant manifolds for dynamical systems and were inspired by the center manifold reduction technique in this paper the basic ideas are outlined a few examples are presented and some natural extensions and applications of the method are briefly described in the conclusions

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

model order reduction and domain decomposition strategies for the solution of the dynamic elastic plastic structural problem abstract a new strategy for the efficient solution of highly nonlinear structural problems is proposed based on the combined use of domain decomposition dd and snapshots version of the proper orthogonal decomposition pod techniques the formulation here presented is tailored for applications in elastic plastic structural dynamics in this context the pod is applied to domains that remain elastic and a double strategy to update the reduced basis is adopted first the singular value decomposition svd proposed allows to update the reduced basis as soon as a new snapshot is stored secondly an online adaptation technique of the reduced space is performed through a plastic check during the reduced analysis the applications show that the computation time necessary for solving elastic plastic problems can be reduced of approximately 50 while keeping accuracy comparable to that obtained for the full model with a classical monolithic method

an optimal modal reduction of a system with frictional excitation abstract the proper orthogonal decomposition theorem is used in an optimal modal reduction of a frictionally excited system the chaotic dynamics are exploited in obtaining a basis that broadly represents the system s dynamics this basis is used in a galerkin s approximation to reduce the system order the reduced model is validated using qualitative and quantitative means and also from the occurrence of bifurcations the results confirm the validity of the method and expose the benefits of using proper orthogonal modes rather than the linear natural modes of the system

a reduced spectral function approach for the stochastic finite element analysis the stochastic finite element analysis of elliptic type partial differential equations with non gaussian random fields are considered a novel approach by projecting the solution of the discretized equation into a reduced finite dimensional orthonormal vector basis is investigated it is shown that the solution can be obtained using a finite series comprising functions of random variables and orthonormal vectors these functions called as the spectral functions can be expressed in terms of the spectral properties of the deterministic coefficient matrices arising due to the discretization of the governing partial differential equation based on the projection in a reduced orthonormal vector basis a galerkin error minimization approach is proposed the constants appearing in the galerkin method are solved from a system of linear equations which has much smaller dimension compared to the original discretized equation a hybrid analytical and simulation based computational approach is proposed to obtain the moments and probability density function of the solution the method is illustrated using the stochastic nanomechanics of a zinc oxide zno nanowire deflected under the atomic force microscope afm tip the results are compared with the results obtained using direct monte carlo simulation classical neumann expansion and polynomial chaos approach for different correlation lengths and strengths of randomness

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

spectral invariant subalgebras of reduced crossed product c algebras abstractletg beadiscretegroupwithaproperlengthfunctionl inthispaper weprovethattheschwartz space s l2 g a is a spectral invariant dense subalgebra of the reduced crossedproduct c nr g a for all commutative c n algebra a with a g action if and only if g haspolynomialgrowthwithrespecttol r 2002elsevierscience usa allrightsreserved keywords reducedcrossedproductc n algebras spectralinvariantsubalgebras property mrd 0 introductionletg beadiscretegroupwithalengthfunctionl in 9 jolissaintprovedthatthe schwartzspace s l2 g is a spectralinvariant dense subalgebra of the reducedgroupc n algebrac n r g ifg haspolynomialgrowthorg isthefundamentalgroupofanegativelycurvedcompactmanifold in 4 delaharpegeneralizedjolissaint stheorem to all hyperbolic groups jolissaint and de la harpe s result plays animportantroleintheworkofconnes moscovicionthenovikovconjecture 2 andtheworkoflafforgueonthebaum connesconjecture 10 inthispaper wegiveanecessaryandsuf cientconditionfortheschwartzspaces

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

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

good wannier bases in hilbert modules associated to topological insulators for a large class of physically relevant operators on a manifold with discrete group action we prove general results on the non existence of a basis of well localized wannier functions for their spectral subspaces this turns out to be equivalent to the freeness of a certain hilbert module over the group c algebra canonically associated with the spectral subspace this brings into play k theoretic methods and justifies their importance as invariants of topological insulators in physics

the isocohomological property higher dehn functions and relatively hyperbolic groups abstract the property that the polynomial cohomology with coefficients of a finitely generated discrete group is canonically isomorphic to the group cohomology is called the weak isocohomological property for the group in the case when a group is of type hf i e that has a classifying space with the homotopy type of a polyhedral complex with finitely many cells in each dimension we show that the isocohomological property is geometric and is equivalent to the property that the universal cover of the classifying space has polynomially bounded higher dehn functions if a group is hyperbolic relative to a collection of subgroups each of which is polynomially combable respectively hf and isocohomological then we show that the group itself has these respective properties combining with the results of connes moscovici and dru u sapir we conclude that a group satisfies the strong novikov conjecture if it is hyperbolic relative to subgroups which are of property rd of type hf and isocohomological

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

morita equivalence of smooth noncommutative tori we show that in the generic case the smooth noncommutative tori associated with two n n real skew symmetric matrices are morita equivalent if and only if the matrices are in the same orbit of the natural so n n z action

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

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

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

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

n4sid subspace algorithms for the identification of combined deterministic stochastic systems abstract recently a great deal of attention has been given to numerical algorithms for subspace state space system identification n4sid in this paper we derive two new n4sid algorithms to identify mixed deterministic stochastic systems both algorithms determine state sequences through the projection of input and output data these state sequences are shown to be outputs of non steady state kalman filter banks from these it is easy to determine the state space system matrices the n4sid algorithms are always convergent non iterative and numerically stable since they only make use of qr and singular value decompositions both n4sid algorithms are similar but the second one trades off accuracy for simplicity these new algorithms are compared with existing subspace algorithms in theory and in practice

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

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

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

differentiable transformations preserving a cone preordering abstract a necessary and sufficient condition for a differentiable transformation to be isotone w r t cone preorderings on a convex set is given the result is applied to characterize monotone matrix functions a a p with integer p 2 and isotone differentiable transformations w r t g majorizations

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

admissibility as a touchstone on considere le probleme de l estimation simultanee de moyennes i de variables aleatoires normales independantes x i de variance unite on etudie le concept d admissibilite

smooth dense subalgebras of reduced group c algebras schwartz cohomology of groups and cyclic cohomology abstract we introduce the notion of schwartz co homology of a discrete group with a length function which is the natural generalization of the bounded cohomology of a discrete group we prove that if the length function is chosen to be a word length on a group of polynomial growth then its schwartz cohomology is canonically isomorphic to the usual group cohomology with complex coefficients using this result we will show that the group cohomology of a discrete group of polynomial growth is of polynomial growth in the sense of connes and moscovici and that there are no nontrivial idempotents in the group c algebra of a torsion free group of polynomial growth by the method of cyclic cohomology of connes

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

quantum hall effect on the hyperbolic plane in this paper we study both the continuous model and the discrete model of the quantum hall effect qhe on the hyperbolic plane the hall conductivity is identified as a geometric invariant associated to an imprimitivity algebra of observables we define a twisted analogue of the kasparov map which enables us to use the pairing between k theory and cyclic cohomology theory to identify this geometric invariant with a topological index thereby proving the integrality of the hall conductivity in this case

posterior propriety and admissibility of hyperpriors in normal hierarchical models hierarchical modeling is wonderful and here to stay but hyperparameter priors are often chosen in a casual fashion unfortunately as the number of hyperparameters grows the effects of casual choices can multiply leading to considerably inferior performance as an extreme but not uncommon example use of the wrong hyperparameter priors can even lead to impropriety of the posterior for exchangeable hierarchical multivariate normal models we first determine when a standard class of hierarchical priors results in proper or improper posteriors we next determine which elements of this class lead to admissible estimators of the mean under quadratic loss such considerations provide one useful guideline for choice among hierarchical priors finally computational issues with the resulting posterior distributions are addressed

realizing the analytic surgery group of higson and roe geometrically part iii higher invariants we construct an isomorphism between the geometric model and higson roe s analytic surgery group reconciling the constructions in the previous papers in the series on realizing the analytic surgery group of higson and roe geometrically with their analytic counterparts following work of lott and wahl we construct a chern character on the geometric model for the surgery group it is a delocalized chern character from which lott s higher delocalized rho invariants can be retrieved following work of piazza and schick we construct a geometric map from stolz positive scalar curvature sequence to the geometric model of higson roe s analytic surgery exact sequence

inequalities theory of majorization and its applications by albert w marshall and ingram olkin abstract in the years since the appearance of marshall and olkin s masterful survey the influence of this work has been so great that some comments however incomplete and idiosyncratic on subsequent developments will be of value figure 1 an appendix lists minor errata to the first printing of the book

group cohomology and the cyclic cohomology of crossed products definite results are obtained in the conditions the group g is torsion free and the class h of the extension 0 zh g h n h 0 in h2 nh k is niipotent here gh for h g stands for the centralizer of h the cyclic homology groups hc a g decompose naturally as a direct sum of two subgroups called the homogeneous and the inhomogeneous parts ofhc a g the homogeneous part can be obtained from a spectral sequence converging to it with e2 q hp g hcq a the inhomogeneous part vanishes after inverting s for g z and in the topological situation we thus reobtain the result of nest 17 for general g but a k and trivial action thus for group rings k g we reobtain results of burghelea 4 the free groups satisfy our conditions see 10 for other classes of groups satisfying these conditions in this case our results are compatible with the results of pimsner and voiculescu 18 19 and suggest that for a large class of groups the part of ktor a r d g is obtained from a convergent spectral sequence with e q hp g k here r176 denotes the topological k theory functors see 2 a is a c algebra and a d g the reduced crossed product c algebra is a suitable completion of a g kasparov 13 succeeded to prove this for groups having a special manifold as classifying space we expect that our spectral sequence will give more insight in kasparov s spectral sequence we also mention

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

a spectral approach for fuzzy uncertainty propagation in finite element analysis abstract uncertainty propagation in complex engineering systems with fuzzy variables constitutes a significant challenge this paper proposes a polynomial chaos type spectral approach based on orthogonal function expansion a fuzzy variable is represented as a set of interval variables via the membership function the interval variables are further transformed into the standard interval 1 1 smooth nonlinear functions of standard interval variables are projected in the basis of legendre polynomials by exploiting its orthogonal properties over the interval 1 1 the coefficients associated with the basis functions are obtained by a galerkin type of error minimisation the method is first illustrated using scalar functions of multiple fuzzy variables later the method is proposed for elliptic type finite element problems where the technique is extended to vector valued functions with multiple fuzzy variables the response of such systems can be expressed in the complete basis of multivariate legendre polynomials the coefficients obtained by galerkin type of error minimisation can be calculated from the solution of an extended set of linear algebraic equations an eigenfunction based model reduction technique is proposed to obtain the coefficient vectors in an efficient way a numerical example of axial deformation of a rod with fuzzy axial stiffness is considered to illustrate the proposed methods linear and nonlinear membership functions are used and the results are compared with direct numerical simulation results

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

on the homotopy invariance of higher signatures for manifolds with boundary if m is a compact oriented manifold with boundary whose fundamental group is virtually nilpotent or gromov hyperbolic we show that the higher signatures of m are oriented homotopy invariants we give applications to the question of when higher signatures of closed manifolds are cut and paste invariant

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

an exact decomposition theorem and a unified view of some related distributions for a class of exponential transformation models on symmetric cones a class of exponential transformation models is defined on symmetric cones q with the group of automorphisms on q as the acting group we show that these models are reproductive and the exponent of their joint distribution for a given sample of size q can be split into q independent components introducing one sample point at a time the automorphism group can be factorized into the group of positive dilation and another group accordingly the symmetric cone q can be seen as the direct product of r and a unit orbit and every x in q can be identified by its orbital decomposition we derive the distributions of the independent components of the exponent of the length of x the direction of x the conditional distribution of the direction given the length and other distributions for a given sample the wishart distribution and the hyperboloid distribution are two special cases in the class we define we also give a unified view of several distributions which are usually treated separately

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

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