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

nondegenerate homoclinic tangency and hyperbolic sets in this paper we show that a nondegenerate homoclinic tangency on a surface is accumulated by a sequence of uniformly hyperbolic sets and in fact it is a boundary point of a nonuniformly hyperbolic set

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

an energy decaying scheme for nonlinear dynamics of shells a novel integration scheme for nonlinear dynamics of geometrically exact shells is developed based on the inextensible director assumption the new algorithm is designed so as to imply the strict decay of the system total mechanical energy at each time step and consequently unconditional stability is achieved in the nonlinear regime furthermore the scheme features tunable high frequency numerical damping and it is therefore stiffly accurate the method is tested for a finite element spatial formulation of shells based on mixed interpolations of strain tensorial components and on a two parameter representation of director rotations the robustness of the scheme is illustrated with the help of numerical examples

growth distortion and coefficient bounds for caratheodory families in cn and complex banach spaces abstract let x be a complex banach space with the unit ball b the family m is a natural generalization to complex banach spaces of the well known caratheodory family of functions with positive real part on the unit disc we consider subfamilies m g of m depending on a univalent function g we obtain growth theorems and coefficient bounds for holomorphic mappings in m g including some sharp improvements of existing results when g is convex we study the family r g consisting of holomorphic mappings f b x which have the property that the mapping d f z z belongs to m g further we consider radius problems related to the family r g when x is a complex hilbert space in particular if x is the euclidean space c n we obtain some quasiconformal extension results for mappings in r g we also obtain some sufficient conditions for univalence and starlikeness in complex banach spaces

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

linear invariance of locally biholomorphic mappings in the unit ball of a jb triple abstract we study the notion of linear invariance on the unit ball of a j b triple x and we obtain some connection between the norm order of a linear invariant family and the starlikeness of order 1 2 also we give some result concerning the radius of univalence of some linear invariant families finally if the dimension of x is finite and if the norm order of a linear invariant family is finite then we prove the normality of the linear invariant family and we also obtain upper bounds on the distortion and the growth of mappings in a linear invariant family with specified norm order in particular our results are valid for the classical cartan domains and the unit polydisc

an intrinsic beam model based on a helicoidal approximation part i formulation the focus of the present work is directed towards the one dimensional non linear analysis of space curved and twisted beams undergoing large displacements and finite rotations according to cosserat s model a beam is intended here as a continuum generated by the rigid motion of a cross section along a curve the novelty of the proposed methodology is implied in the assumption that the reference line of the beam both in the undeformed and in the deformed configurations is a helicoid in space this formulation radically departs from the classical beam models based on polynomial interpolations of the independent fields in fact within the framework of helicoidal geometry a powerful description of space curved beam kinematics with large displacements and finite rotations can be developed furthermore the beam model developed herein can be proved to enjoy several invariance properties that make it particularly attractive for non linear analysis the formulation discussed in this work adheres to a mixed variational principle the independent fields being the generalized displacements and the sectional stress resultants in a companion paper we address the linearization of the governing weak form the finite element implementation of the proposed methodology with a novel incremental treatment of finite rotations and the development of a helicoidal model for rigid body dynamics using finite elements in time

on the similarity of the coleman and lyapunov floquet transformations for modal analysis of bladed rotor structures abstract structures with isotropic bladed rotors can be modally analyzed by eigenvalue analysis of time invariant coleman transformed equations of motion related to the inertial frame or by floquet analysis of the periodic equations of motion the coleman transformation is here shown to be a special case of the lyapunov floquet l f transformation which transforms system equations of structures with anisotropic bladed rotors into a time invariant system using the transition matrix and floquet eigenvectors as a basis the l f transformation is not unique whereby eigensolutions of the time invariant system are not directly related to the modal frequencies and mode shapes observed in the inertial frame this modal frequency indeterminacy is resolved by requiring the periodic mode shapes from the l f approach to be as similar as possible to the mode shapes from the coleman approach for an anisotropic rotor the floquet analysis yields a periodic mode shape that contains harmonics of integer multiples of the rotor speed for inertial state variables these harmonic components show up as resonance frequencies on the sides of the corresponding modal frequency in a computed frequency response function of a simple three bladed turbine with an anisotropic rotor

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

integration of elastic multibody systems by invariant conserving dissipating algorithms ii numerical schemes and applications abstract this work presents a novel methodology for the dynamic analysis of general non linear multibody systems composed of rigid and deformable bodies the latter under the small strain assumption in part i we developed the 6 d compact representation and parameterization of motion for constrained bodies part ii is devoted to the design of a class of modified runge kutta rk methods dedicated to non linear dynamics these are capable of integrating on the configuration manifold and of preserving linear and angular momenta within this class of methods two second order algorithms are designed under the requirement of attaining non linear unconditional stability the energy preserving ep and energy decaying ed methods these schemes are associated with an algorithmic law of conservation and dissipation respectively of the total mechanical energy of the system together with the vanishing of the algorithmic work done by ideal time independent constraints their performances are assessed with the aid of some representative numerical applications which confirm the non conventional properties predicted in the analysis

integrating finite rotations abstract we study the integration of problems of evolution in the rotation group instead of attacking the problem in the nonlinear differential manifold so 3 pure rotational dynamics as is usually done we derive equations for the complete problem of motion translational and rotational dynamics on an extended manifold we develop a generalization of runge kutta methods that by design ensures that the solution will remain on the manifold for any choice of the tableau this is obtained through configuration updates performed via the exponential map we show how certain terms can be approximated while retaining the order of accuracy of the scheme and how the method conserves the total momentum of the system within this framework we develop two nonlinearly unconditionally stable time integration schemes that are associated with discrete laws of conservation dissipation of the total energy the dissipating algorithm generalizes to the nonlinear case the high frequency damping characteristics provided by some well known conventional methods we present numerical results to support our analysis and we develop a complete application of this methodology to the nonlinear dynamics of three dimensional rods undergoing large displacements and finite rotations under the assumption of small strains

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

loewner chains associated with the generalized roper suffridge extension operator on some domains in this paper we consider the generalized roper suffridge extension operator defined by n 2 2 n n f z f z1 f z1 z1 2 f z1 2z2 f z1 z1 n f z1 nzn for z z1 z2 zn p1 p2 pn where 0 j 1 0 j 1 j pj 1 and we choose the branch of the power functions such that f z1 z1 j z1 0 1 and f z1 j z1 0 1 j 1 2 n p1 p2 pn z1 z2 zn cn j 1n zj pj 1 we prove that the set n 2 2 n n s u can be embedded in loewner chains and give the answer to the problem of liu taishun we also obtain that the operator n 2 2 n n f preserves starlikeness or spirallikeness of type on p1 p2 pn for some suitable constants j j where s u is the class of all univalent analytic functions on the unit disc u in the complex plane c with f 0 0 and f 0 1

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

coefficient bounds for biholomorphic mappings which have a parametric representation abstract let b be the unit ball in c n with respect to an arbitrary norm and let f z t be a g loewner chain such that z 0 is a zero of order k 1 of e t f z t z for each t 0 in this paper the authors obtain coefficient bounds of mappings in s g k 1 0 b these results generalize the related works of hamada honda and kohr

accuracy of one step integration schemes for damped forced linear structural dynamics summary this paper proposes an energy based measure for the evaluation of the local truncation error of two level one step integration schemes the measure applies to multiple degree of freedom systems and does not necessarily require modal reduction to a scalar model it naturally handles the structural damping and external forcing terms that are generally and mistakenly neglected in error analyses and it segregates the error associated with the free and forced response components of the problem to illustrate the approach two examples associated with the application of the trapezoidal scheme and of a high order scheme proposed in the literature are analyzed the latter reveals the shortcomings of the standard approach that is based on the undamped unforced linear oscillator and therefore highlights the need for the proposed framework indeed the scheme order of accuracy is below expectation when structural damping or external forcing is considered in the numerically dissipative setting developments on the basis of the time discontinuous galerkin tdg method are then proposed to recover the scheme high order accuracy additionally they show the similarity that exists between schemes related to the tdg method and the ones obtained by integration by parts of the equation of motion copyright 2014 john wiley sons ltd

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

sharp growth and distortion theorems for a subclass of biholomorphic mappings let x be a complex banach space with norm and b be the unit ball in x in this paper we introduce a class of holomorphic mappings m g on b let f x be a normalized locally biholomorphic mapping on b such that df x 1f x m g we investigate the sharp growth theorem for f x as applications the sharp distortion theorems for a subclass of starlike mappings are obtained

on the precise growth covering and distortion theorems for normalized biholomorphic mappings abstract in this paper we consider a normalized biholomorphic mapping f x defined on the unit ball in a complex banach space where the origin 0 is a zero of order k 1 of f x x the precise growth and covering theorem for f x is obtained when f x is a starlike mapping or a starlike mapping of order especially the precise growth and covering theorem for f x is also established when f x is a quasi convex mapping moreover the precise distortion theorem for f x is given when f x is a convex mapping our result includes many known results

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

the dynamics of the henon map the purpose of this paper is to develop machinery which is suitable for a study of the long time behavior of systems such as the henon map with expansion combined with strong contraction our study is modeled on the treatment of the one dimensional system x 1 ax 2 and we study the perturbation of a from the value a 2 and b small a surprising number of complications arise when b 0 the computer simulations give some indication of the complicated structure of the attractor but it can safely be stated that the real difficulties do not show up the central aim of the paper is to introduce what we consider the relevant concepts and techniques needed to study mappings such as the henon one

a self stabilized algorithm for enforcing constraints in multibody systems a new algorithm is developed for the enforcement of constraints within the framework of nonlinear flexible multibody system modeled with the finite element approach the proposed algorithm exactly satisfies the constraints at the displacement and velocity levels and furthermore it achieves nonlinear unconditional stability by imposing the vanishing of the work done by the constraint forces when combined with specific discretizations of the inertial and elastic forces identical convergence rates are observed for the displacements velocities and lagrange multipliers the proposed algorithm is closely related to the stabilized index 2 or ggl method although no additional multipliers are introduced in the present approach these desirable characteristics are obtained without resorting to numerically dissipative algorithms if high frequencies are present in the system i e the system is physically stiff dissipative schemes become necessary the proposed algorithm is extended to deal with this situation

robust integration schemes for flexible multibody systems a framework for the development of robust time integration schemes for nonlinear flexible multibody systems is presented the proposed schemes are designed to meet four specific requirements nonlinear unconditional stability of the scheme a rigorous treatment of both geometric and material nonlinearities exact satisfaction of the constraints and the presence of high frequency numerical dissipation specific algorithms are presented for rigid bodies cables beams shells and elasto dynamics the kinematic nonlinearities are treated in a rigorous manner for all elements and the material nonlinearities can be handled when the constitutive laws stem from the existence of a strain energy density function the treatment of the constraint equations associated with the six lower pair joints are presented as well the efficiency and robustness of the proposed approach is demonstrated with specific numerical examples

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

dynamic stability of periodic solutions of large scale nonlinear systems abstract a method suitable for the stability analysis of periodic solutions of large nonlinear systems is presented the stability behaviour is inferred from a galerkin approximation of the associated poincare map the practical value of the method is shown by applying it to the nonlinear finite element model of a large scale wind energy convertor

an input output based procedure for fully evaluating and monitoring dynamic properties of structural systems via a subspace identification method abstract dynamic behaviour of complex structural systems may be modelled by a system of second order linear ordinary differential equations i e m t d t sw t f t by means of either structural analysis for finite degree of freedom systems or discretization procedures e g fe methods for continuous systems here w t and f t are the displacement vector and the force vector owing to erosion friction and internal damage and cracks etc a working process of a system always accompanies gradual degradation of the performance of this system the stiffness of the system weakens whereas the damping of the system strengthens to evaluate such degradation the usual way is to model the evolution of property of a system obtain system property parameters trace the history of motion and loading carry out complicated analysis and computation under prescribed initial and boundary value conditions and finally derive the degraded property and responses of the system this traditional way however might be cumbersome and unsatisfactory in some cases due to the lack of adequate experimental data and well founded theoretical basis etc another way is to apply inverse methods such as modal analysis methods with fft and a subspace identification method etc developed in the theory of system identification which extracts information about system properties directly from experimental input output measurement data and hence do not involve the foregoing traditional analysis the latter method however could not supply full information about system properties due to the assumption of the black box viewpoint in this work with suitable experimental input output measurement data a simple effective procedure is described by which the stiffness matrix s and the damping matrix d may be determined in a complete unique manner using a subspace identification method the possibility of such a procedure arises from the observation of the self evident fact the conservation of mass of any part of a structural system implies that the mass matrix m of this system is constant and hence is given by its initial value the stiffness and damping matrices s and d determined by the proposed procedure may be used to evaluate and monitor in a full sense the degradation of dynamic properties of structural systems further with the information about the stiffness distribution of constituent elements of a structural system it is shown that it may be possible to estimate the locations of the damaged or faulty elements in this system an example is given to illustrate the application of the proposed procedure

conservation properties of a time fe method part iv higher order energy and momentum conserving schemes in the present paper a systematic development of higher order accurate time stepping schemes which exactly conserve total energy as well as momentum maps of underlying finite dimensional hamiltonian systems with symmetry is shown the result of this development is the enhanced galerkin eg finite element method in time the conservation of the eg method is generally related to its collocation property total energy conservation in particular is obtained by a new projection technique the eg method is moreover based on objective time discretization of the used strain measure this paper is concerned with particle dynamics and semi discrete non linear elastodynamics the related numerical examples show good performance in presence of stiffness as well as for calculating large strain motions copyright 2005 john wiley sons ltd

a stable energy conserving approach for frictional contact problems based on quadrature formulas a common approach for the numerical simulation of non linear multi body contact problems is the use of lagrange multipliers to model the contact conditions the stability of standard algorithms is improved by introducing a modified mass matrix which assigns no mass to the potential contact nodes by this the spurious algorithmic oscillations in the multiplier do not occur any more which facilitates the application of the primal dual active set strategy to dynamical contact problems the new mass matrix is calculated via a modified quadrature formula that needs no extra computational cost in addition the conservation properties of the underlying algorithm are transferred to the modified mass version different numerical examples for frictional two body contact problems illustrate the improvement in the results for the contact stresses copyright 2007 john wiley sons ltd

biholomorphic mappings on bounded starlike circular domains let cn be a bounded starlike circular domain with 0 in this paper we introduce a class of holomorphic mappings mg on let f z be a normalized locally biholomorphic mapping on such that jf 1 z f z mg and z 0 is the zero of order k 1 of f z z we obtain a sharp growth theorem and sharp coefficient bounds for f z as applications sharp distortion theorems for a subclass of starlike mappings are obtained these results unify and generalize many known results

energy decaying scheme for non linear beam models abstract this paper is concerned with the time integration of the equations of motion describing the non linear dynamic response of beams desirable characteristics of integration schemes for structural dynamics problems include unconditional stability accuracy and high frequency numerical dissipation several schemes exist that present all these features when applied to linear problems though the application of those schemes to non linear problems is often successful proofs of unconditional stability are rarely derived a finite difference integration scheme is derived in this paper for the non linear dynamic response of beams though of a finite difference nature the proposed scheme mimics the integration scheme obtained by applying the time discontinuous galerkin methodology to a single degree of freedom linear oscillator a formal proof of unconditional stability for the non linear problem is derived based on an energy decay argument numerical examples using the proposed scheme are given and the results are compared with the predictions of other available schemes the accuracy of the proposed scheme and its high frequency numerical dissipation characteristics are demonstrated in these examples

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

loewner chains and parametric representation in several complex variables abstract let b be the unit ball of c n with respect to an arbitrary norm we study certain properties of loewner chains and their transition mappings on the unit ball b we show that any loewner chain f z t and the transition mapping v z s t associated to f z t satisfy locally lipschitz conditions in t locally uniformly with respect to z b moreover we prove that a mapping f h b has parametric representation if and only if there exists a loewner chain f z t such that the family e tf z t t 0 is a normal family on b and f z f z 0 for z b also we show that univalent solutions f z t of the generalized loewner differential equation in higher dimensions are unique when e tf z t t 0 is a normal family on b finally we show that the set s0 b of mappings which have parametric representation on b is compact

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

using svd to detect damage in structures with different operational conditions the aim of this article is to present a method based on the use of singular value decomposition svd for detecting damage in structures at an early stage the main advantage of the proposed technique is firstly the capability to deal with structures with different operating conditions i e with dynamic characteristics which may change during the lifetime secondly no mathematical model of the analyzed structure is required a priori such that it can be applied to systems of arbitrary complexity application of this method to simulated data for a truss type structure is illustrated with results from an experimental investigation on a cantilever beam

growth theorems and coefficient bounds for univalent holomorphic mappings which have parametric representation abstract let b be the unit ball in c n with respect to an arbitrary norm and let f z t be a g loewner chain such that e t f z t z has a zero of order k 1 at z 0 in this paper we obtain growth and covering theorems for f 0 moreover we consider coefficient bounds and examples of mappings in s g k 1 0 b

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

energy conserving time integration scheme for geometrically exact beam an energy conserving finite element formulation for the dynamic analysis of geometrically non linear beam like structures undergoing large overall motions has been developed the formulation uses classical displacement based planar beam finite elements described in an inertial frame it takes into account finite axial bending and shear strains a theoretically consistent approach is used to derive a novel and simple energy conserving scheme using the unconventional incremental strain update rather than the standard strong form numerical examples demonstrate perfect energy and momenta conservation stability and robustness of the scheme and good convergence properties in terms of both the newton raphson method and time step size c 2006 elsevier b v all rights reserved

solutions for the generalized loewner differential equation in several complex variables we generalize a one variable result of j becker to several complex variables we determine the form of arbitrary solutions of the loewner differential equation that is satisfied by univalent subordination chains of the form f z t e ta z cdots where a in l mathbb c n mathbb c n has the property k a 2m a here k a max re lambda lambda in sigma a and m a min re langle a z z rangle z 1 the notion of parametric representation has a useful generalization under these conditions so that one has a canonical solution of the loewner differential equation in particular we determine the form of the univalent solutions the results are applied to subordination chains generated by spirallike mappings on the unit ball in mathbb c n finally we determine the form of the solutions in the presence of certain coefficient bounds

subordination chains and biholomorphic mappings on bounded balanced pseudoconvex domains let w be a bounded balanced pseudoconvex domain with c 1 plurisubharmonic defining functions in c n and let f z t be a univalent subordination chain such that z 0 is a zero of order k 1 of e tf z t z for each t 0 in this paper we study several properties for f 0 concerning the growth covering theorems and coefficient bounds these results generalize the related works of hamada and kohr

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

energy preserving decaying schemes for non linear beam dynamics using the helicoidal approximation abstract we develop two unconditionally stable displacement based time stepping schemes for the non linear dynamic response of beams the first algorithm guarantees the exact discrete conservation of energy and momentum the second is associated with an energy decay inequality that achieves control of the unresolved frequencies by means of a numerical dissipation mechanism both schemes emanate from a weak form of the equations of dynamic equilibrium referred to a fixed pole space and time discretizations are based on the exponential parameterization of motion this implies that the beam reference line and the trajectories of the beam nodes are helicoids in space the exponential mapping approach allows a unified treatment of translations and rotations greatly simplifying the derivation of the algorithms and their analysis the capabilities and performance of the new schemes are demonstrated and discussed with the aid of numerical simulations

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

the embedded projection method a general index reduction procedure for constrained system dynamics abstract the embedded projection method epm is an attempt to eliminate most of the difficulties related to the solution of high index differential algebraic equations daes as those typical of the dynamics of constrained systems the intimate coupling between the algebraic and the differential parts lies at the origin of these difficulties therefore we uncouple these components introduce an unconstrained modified state variable and complement the resulting ordinary differential equation ode by additional algebraic variables as a result a general consistent index reduction from arbitrarily high values to one is obtained allowing the use of any suitable ode integration algorithm without the need for a specialized dae solver the epm solution is intrinsically endowed with greater accuracy and stability than conventional counterparts thus balancing the higher complexity of the procedure we illustrate the epm including some applications of holonomic and non holonomic mechanical systems

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

optimal tracking of parameter drift in a chaotic system experiment and theory we present a method of optimal tracking for chaotic dynamical systems with a slowly drifting parameter the net drift in the parameter is assumed to be small this makes detecting and tracking the drift more difficult the method relies on the existence of underlying deterministic behavior in the dynamical system yet neither requires a system model nor develops one we begin by describing an experimental study where a heuristic optimality criterion gave good tracking performance the tracking method there was based on maximizing smoothness and overall variation in the drift observer which was found by solving an igenvalue problem we then develop a theory based on simplifying assumptions about the chaotic dynamics to explain the success of the tracking method for chaotic systems for signals from deterministic systems that are sufficiently complex in a sense that we make precise typical drift observers provide poor tracking performance and require the drift to be particularly slow in contrast our theory shows that the optimality criterion seeks out a special drift observer that both provides better tracking performance and allows the drift to be appreciably faster for periodic or quasiperiodic systems no chaos good tracking is easily achievable and the present method is irrelevant for stochastic systems no determinism the optimal tracking method does not asymptotically improve tracking performance exhaustive numerical simulations of a simple drifting chaotic map first without and then with stochastic forcing show agreement with theoretical predictions of tracking performance and validate the theory

polynomially bounded solutions to the loewner differential equation in several complex variables abstract we determine the form of polynomially bounded solutions to the loewner differential equation that is satisfied by univalent subordination chains of the form f z t e t a z where a l c n c n has the property m a 0 here m a min r a z z z 1 we also give sufficient conditions for g z t l f z t to be polynomially bounded where f z t is an a normalized polynomially bounded loewner chain solution to the loewner differential equation

an intrinsic beam model based on a helicoidal approximation part ii linearization and finite element implementation the helicoidal beam model developed in the first part of this work is applied here to the development of a mixed finite element for space curved and twisted beams undergoing large displacements and finite rotations starting from the governing weak form expressed by the principle of virtual work a consistent linearization is obtained in the following and a novel updated lagrangian finite element implementation is thoroughly discussed the unique features and the distinguishing properties previously claimed for the helicoidal model are shown here to imply remarkable numerical consequences for this purpose meaningful example problems regarding the non linear static response of beams are addressed in the following and the results are compared with those available from the literature furthermore a finite element in time for the rigid body dynamic problem is developed within the framework of the helicoidal geometry the underlying philosophy of this novel finite element for dynamics is the realization of the helicoidal decomposition of the rigid body motion within a time step

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

on the use of geometrically exact shells in a conserving framework for flexible multibody dynamics abstract the present work deals with the design of energy momentum conserving schemes for flexible multibody dynamics the proposed approach is based on nonlinear finite element methods for the space discretization of flexible bodies in particular the focus is on geometrically exact shells relying on reissner mindlin kinematics it is shown that the equations of motion pertaining to the semi discrete shell formulation can be written in the form of differential algebraic equations daes the daes provide a uniform framework for a rotationless description of flexible multibody dynamics the use of rotational parameters is circumvented throughout the discretization process in space and time the rotationless description facilitates the straightforward incorporation of geometrically exact shells and beams into a multibody framework in addition to that the advocated approach makes possible the design of a uniform energy momentum conserving time stepping scheme for general multibody systems numerical examples demonstrate the excellent numerical stability properties of the present scheme moreover comparison is made with more traditional formulations based on rotational parameters

the sharp estimates of homogeneous expansions for the generalized class of close to quasi convex mappings abstract in this paper the class of strongly close to quasi convex mappings of type and order is introduced in the unit ball of complex banach space or unit polydisk in c n and we obtain the sharp estimates of homogeneous expansions for this class these results generalize many known results

multibody implementation of finite volume c 0 beams a c 0 beam discretization based on the finite volume concept is described in the linear case this approach leads to a collocated evaluation of the stiffness matrix of the beam it proves to be intrinsically free from shear locking in the nonlinear formulation only a collocated evaluation of the elastic forces is required which dramatically simplifies the computation of the elastic contribution to the equilibrium equations the formulation is here developed for the general geometrically nonlinear case and implemented in multibody formulation the proposed approach proved to be consistent its major drawback lies in the loss of symmetry of both the linear and the linearized beam matrices for this reason the method is particularly suitable for dynamic problems like a nonlinear implicit multibody numerical approximation in which the symmetry of the matrices is not so important as it is already lost whereas the ease in the generation of the contributions to the equations can lead to faster and cheaper analyses some applications are outlined and the most relevant results are discussed

a consistent time fe method for large strain elasto plasto dynamics in the present paper energy consistent momentum conserving time stepping schemes for geometrically nonlinear multiplicative elasto plasto dynamics referred to as ecmc cg methods are developed in this context the entire discretisation procedure of the global equations of motion is based on galerkin methods in space and time whereby special emphasis is placed on the desired conservation properties related to the approximation of time integrals to guarantee energy consistency also for higher order finite elements in time a thermodynamically motivated enhanced stress tensor which defines a non standard quadrature rule is derived for elasto plastic material behaviour concerning the integration of the local evolution equations in time exemplarily a well established first order accurate exponential update is applied and adapted according to the global time stepping scheme based on linear finite elements in time the performance of the resulting scheme is demonstrated by means of several representative numerical examples whereby it is shown that global energy consistency is guaranteed exactly within the calculation accuracy by applying the proposed concepts

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

energy dissipative time finite elements for classical mechanics abstract based upon the temporal discretisation of hamilton s canonical equations by means of the continuous galerkin method a petrov galerkin time finite element formulation is presented to analyse non linear mechanical systems for long time integration with the presence of dissipation the time stepping scheme features a correct estimation of physical dissipative energy for mechanical systems with weak dissipation this enables to introduce a moderate numerical dissipation to damp out undesired spurious high frequency modes in a controllable manner numerical examples are performed to demonstrate that the proposed scheme allows us to obtain a correct amount by which the energy changes over the integration run

on the design of energy preserving and decaying schemes for flexible nonlinear multi body systems abstract energy preserving schemes achieve unconditional stability for nonlinear systems by establishing discrete energy preservation statements several schemes have been presented by various authors within drastically different frameworks finite difference schemes based on a mid point approximation galerkin and time discontinuous galerkin approximations of the equations of motion written in the symmetric hyperbolic form finite elements in time and 2 stage fsal runge kutta methods furthermore different types of parameterization of finite rotations were used in the various formulations this paper presents a unified finite difference framework which readily allows comparing the various schemes and their respective properties numerical examples are presented and show that the predictions of two of these schemes are in very close agreement with each other

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

a robust singular value decomposition for damage detection under changing operating conditions and structural uncertainties a technique is proposed to detect damage in structures from measurements taken under different conditions i e different operational excitation levels geometrical uncertainties and surface treatments of the structure the method is based on a robust singular value decomposition rsvd which will be introduced in this article using the rsvd the distance of an observation to the subspace spanned by the intact measurements can be computed furthermore from statistics a threshold can be determined to automatically decide based on the observation s distance to the subspace if the observation comes from a damaged or intact sample the proposed rsvd method is compared with an existing method based on the classical least squares ls svd the damage detection method is validated on an aluminium beam with different damage scenarios a saw cut and a fatigue crack under several conditions different beams with small dimensional changes beams covered with damping material and different operating levels

energy controlling time integration methods for nonlinear elastodynamics and low velocity impact it is now well established that discrete energy conservation dissipation plays a key role for the unconditional stability of time integration schemes in nonlinear elastodynamics in this paper from a rigorous conservation analysis of the hilber hughes taylor time integration scheme h hilber t hughes r taylor improved numerical dissipation for time integration algorithms in structural dynamics earthquake engrg struct dynam 5 1977 283 292 we propose an original way of introducing a controllable energy dissipation while conserving momenta in conservative strategies like j simo n tarnow the discrete energy momentum method conserving algorithms for nonlinear elastodynamics z angew math phys 43 1992 757 792 moreover we extend the technique proposed in o gonzalez exact energy and momentum conserving algorithms for general models in nonlinear elasticity comput methods appl mech engrg 190 13 14 2000 1763 1783 to provide energy controlling time integration schemes for frictionless contact problems enforcing the standard kuhn tucker conditions at time discretization points we also extend this technique to viscoelastic models numerical tests involving the impact of incompressible elastic or viscoelastic bodies in large deformation are proposed to confirm the theoretical analysis

bifurcation to infinitely many sinks this paper considers one parameter families of diffeomorphisms ft in two dimensions which have a curve of dissipative saddle periodic pointspt i e ftn pt pt and detdftn pt 1 the family is also assumed to create new homoclinic intersections of the stable and unstable manifolds ofpt as the parameter varies throught0 gavirlov and silnikov proved that if the new homoclinic intersections are created nondegenerately att0 then there is an infinite cascade of periodic sinks i e there are parameter valuestn accumulating att0 for which there is a sink of periodn gs2 sect 4 we show that this result is true for real analytic diffeomorphisms even if the homoclinic intersection is created degenerately we give computer evidence to show that this latter result is probably applicable to the henon map fora near 1 392 andb equal 0 3