random matrix eigenvalue problems in structural dynamics natural frequencies and mode shapes play a fundamental role in the dynamic characteristics of linear structural systems considering that the system parameters are known only probabilistically we obtain the moments and the probability density functions of the eigenvalues of discrete linear stochastic dynamic systems current methods to deal with such problems are dominated by mean centred perturbation based methods here two new approaches are proposed the first approach is based on a perturbation expansion of the eigenvalues about an optimal point which is best in some sense the second approach is based on an asymptotic approximation of multidimensional integrals a closed form expression is derived for a general rth order moment of the eigenvalues two approaches are presented to obtain the probability density functions of the eigenvalues the first is based on the maximum entropy method and the second is based on a chi square distribution both approaches result in simple closed form expressions which can be easily calculated the proposed methods are applied to two problems and the analytical results are compared with monte carlo simulations it is expected that the small randomness assumption usually employed in mean centred perturbation based methods can be relaxed considerably using these methods

probability distributions of natural frequencies of uncertain dynamic systems of random eigenvalues commonly encountered in dynamic systems the method involves a hierarchical decomposition of a multivariate function in terms of variables with increasing dimensions a broad range of orthonormal polynomial bases consistent with the probability measure for a fourier polynomial expansion of component functions and an innovative dimension reduction integration for calculating the expansion coefficients the new decomposition does not require sample points yet it generates a convergent sequence of lower variate estimates of theprobabilitydistributionsofeigensolutions numericalresults includingfrequencydistributionsofapiezoelectric transducer indicate that the decomposition method developed provides accurate convergent and computationally efficient estimates of the tail probabilistic characteristics of eigenvalues

computational procedure for a fast calculation of eigenvectors and eigenvalues of structures with random properties abstract the inherent uncertainties in geometry material properties etc of engineering structures can be represented by stochastic models where the parameters are described by probabilistic laws results from any analysis based on stochastic models inherit probabilistic information as well which can be used e g for reliability analysis particularly in linear dynamics of structures the calculation and analysis of random eigenvalues and eigenvectors is crucial a very flexible however computationally intensive way to analyze such systems is direct monte carlo simulation mcs in this paper procedures are shown which allow a significant reduction of computational efforts of the simulation using a subspace iteration scheme with optimally selected start vectors the procedures are not restricted to any kind of sampling technique

stochastic convergence acceleration through basis enrichment of polynomial chaos expansions given their mathematical structure methods for computational stochastic analysis based on orthogonal approximations and projection schemes are well positioned to draw on developments from deterministic approximation theory this is demonstrated in the present paper by extending basis enrichment from deterministic analysis to stochastic procedures involving the polynomial chaos decomposition this enrichment is observed to have a significant effect on the efficiency and performance of these stochastic approximations in the presence of non continuous dependence of the solution on the stochastic parameters in particular given the polynomial structure of these approximations the severe degradation in performance observed in the neighbourhood of such discontinuities is effectively mitigated an enrichment of the polynomial chaos decomposition is proposed in this paper that can capture the behaviour of such non smooth functions by integrating a priori knowledge about their behaviour the proposed enrichment scheme is applied to a random eigenvalue problem where the smoothness of the functional dependence between the random eigenvalues and the random system parameters is controlled by the spacing between the eigenvalues it is observed that through judicious selection of enrichment functions the spectrum of such a random system can be more efficiently characterized even for systems with closely spaced eigenvalues copyright 2007 john wiley sons ltd

maximum entropy approach for modeling random uncertainties in transient elastodynamics a new approach is presented for analyzing random uncertainties in dynamical systems this approach consists of modeling random uncertainties by a nonparametric model allowing transient responses of mechanical systems submitted to impulsive loads to be predicted in the context of linear structural dynamics the information used does not require the description of the local parameters of the mechanical model the probability model is deduced from the use of the entropy optimization principle whose available information is constituted of the algebraic properties related to the generalized mass damping and stiffness matrices which have to be positive definite symmetric matrices and the knowledge of these matrices for the mean reduced matrix model an explicit construction and representation of the probability model have been obtained and are very well suited to algebraic calculus and to monte carlo numerical simulation in order to compute the transient responses of structures submitted to impulsive loads the fundamental properties related to the convergence of the stochastic solution with respect to the dimension of the random reduced matrix model are analyzed finally an example is presented

probabilistic eigenvalue buckling analysis solved through the ratio of polynomial response surface an efficient procedure for the reliability analysis of frame structures with respect to the buckling limit state is proposed under the assumption that no imperfections are present and that the elastic parameters are uncertain and modeled as random variables the approach allows a deeper investigation of structures which are not sensitive to imperfections the procedure relies on a response surface method adopting simple ratio of polynomials without cross terms as performance function such a relationship approximates analytically the dependence between the buckling load and the basic variables furnishing a limit state equation which is very close to the exact one when a proper experimental design is adopted in this way a monte carlo simulation applied to the response surface leads to a good approximation with low computational effort several numerical examples show the accuracy and effectiveness of the method varying structural complexity correlation between basic variables and their distribution

on the absolute continuity of the blackwell measure in 1957 blackwell expressed the entropy of hidden markov chains using a measure which can be characterised as an invariant measure for an iterated function system with place dependent weights this measure called the blackwell measure plays a central role in understanding the entropy rate and other important characteristics of fundamental models in information theory we show that for a suitable set of parameter values the blackwell measure is absolutely continuous for almost every parameter in the case of binary symmetric channels

efficient modal analysis of systems with local stiffness uncertainties the characterization of the uncertainty in modal quantities of an uncertain linear structural system is essential to the rapid determination of its response to arbitrary loadings although the size of many computational structural models used is extremely large i e thousands of equations the uncertainty to be analyzed is oftentimes localized to very small regions of the model this paper addresses the development of an efficient computational methodology for the modal analysis of linear structural systems with local stiffness uncertainties the newly developed methodology utilizes an enriched basis that consists of the sub spectrum of a nominal structural system augmented with additional basis vectors generated from a knowledge of the structure of the stiffness uncertainty in addition methods for determining bounds on the approximate modal frequencies and mode shapes are discussed numerical results demonstrate that the algorithm produces highly accurate results with greatly reduced computational effort copyright 2009 john wiley sons ltd

efficient characterization of the random eigenvalue problem in a polynomial chaos decomposition a new procedure for characterizing the solution of the eigenvalue problem in the presence of uncertainty is presented the eigenvalues and eigenvectors are described through their projections on the polynomial chaos basis an efficient method for estimating the coefficients with respect to this basis is proposed the method uses a galerkin based approach by orthogonalizing the residual in the eigenvalue eigenvector equation to the subspace spanned by the basis functions used for approximation in this way the stochastic problem is framed as a system of deterministic non linear algebraic equations this system of equations is solved using a newton raphson algorithm although the proposed approach is not based on statistical sampling the efficiency of the proposed method can be significantly enhanced by initializing the non linear iterative process with a small statistical sample synthesized through a monte carlo sampling scheme the proposed method offers a number of advantages over existing methods based on statistical sampling first it provides an approximation to the complete probabilistic description of the eigensolution second it reduces the computational overhead associated with solving the statistical eigenvalue problem finally it circumvents the dependence of the statistical solution on the quality of the underlying random number generator copyright 2007 john wiley sons ltd

a galerkin finite element technique for stochastic field problems a new finite element method is developed to analyse the structures with more than one parameter behaving in a stochastic manner the galerkin weighted residual method is used as a generalization this paper treats the random eigenvalue problems arising when the material property values of the structural systems are having stochastic fluctuations i e the measurement errors can be accounted for when probabilistic modelling is used the free vibration problem of a stochastic beam whose young s modulus and mass density are distributed stochastically is considered using galerkin s weighted residual procedure a stochastic finite element method is developed and implemented to arrive at a random algebraic eigenvalue problem the stochastic characteristics of eigensolutions are derived in terms of the stochastic material property variations numerical examples are given it is demonstrated that through this formulation the finite element discretization need not be dependent on the characteristics of stochastic fields describing the fluctuations in the material property values further calculation of covariances between stiffness elements mass elements etc is carried out in a relatively simple manner and is more general in nature

hybrid uncertain natural frequency analysis for structures with random and interval fields abstract this paper presents a robust non deterministic free vibration analysis for engineering structures involving hybrid yet spatially dependent uncertain system parameters distinguished from the conventional hybrid uncertain eigenvalue problem the concept of interval field is enclosed with random field model such that both the stochastic and non stochastic representations of the spatial dependency of the uncertainties are simultaneously incorporated within a unified non deterministic free vibration analysis in order to determine the probabilistic characteristics i e means and standard deviations of the extremities of structural natural frequencies an extended unified interval stochastic sampling x uiss method is implemented for the purpose of effective hybrid uncertain free vibration analysis by meticulously blending sharpness promised interval eigenvalue analysis with stochastic sampling techniques the stochastic profiles i e probability density functions pdfs and the cumulative distribution functions cdfs of the extreme bounds of the structural natural frequencies can be rigorously established by utilizing the adequate statistical inference methods the applicability and effectiveness of the proposed computational framework are evidently demonstrated through the numerical investigations on various practically motivated engineering structures

an uncertainty principle for quaternion fourier transform we review the quaternionic fourier transform qft using the properties of the qft we establish an uncertainty principle for the right sided qft this uncertainty principle prescribes a lower bound on the product of the effective widths of quaternion valued signals in the spatial and frequency domains it is shown that only a gaussian quaternion signal minimizes the uncertainty

stationary measures for projective transformations the blackwell and furstenberg measures in this paper we study the blackwell and furstenberg measures which play an important role in information theory and the study of lyapunov exponents for the blackwell measure we determine parameter domains of singularity and give upper bounds for the hausdorff dimension for the furstenberg measure we establish absolute continuity for some parameter values our method is to analyze linear fractional iterated function schemes which are contracting on average have no separation properties that is we do not assume that the open set condition holds see hutchinson in indiana univ math j 30 713 747 1981 and in the case of the blackwell measure have place dependent probabilities in such a general setting even an effective upper bound on the dimension of the measure is difficult to achieve

explicit invariant measures for products of random matrices we construct explicit invariant measures for a family of infinite products of random independent identically distributed elements of sl 2 c the matrices in the product are such that one entry is gamma distributed along a ray in the complex plane when the ray is the positive real axis the products are those associated with a continued fraction studied by letac seshadri z wahr verw geb 62 1983 485 489 who showed that the distribution of the continued fraction is a generalised inverse gaussian we extend this result by finding the distribution for an arbitrary ray in the complex right half plane and thus compute the corresponding lyapunov exponent explicitly when the ray lies on the imaginary axis the matrices in the infinite product coincide with the transfer matrices associated with a one dimensional discrete schrodinger operator with a random gamma distributed potential hence the explicit knowledge of the lyapunov exponent may be used to estimate the exponential rate of localisation of the eigenstates

derivatives of complex eigenvectors using nelson s method 6shyy w an adaptive grid method for navier stokes flow computation applied mathematics and computation vol 21 no 3 1987 pp 201 219 7jeng y n andliou s c modi edmultipleone dimensional adaptive grid method numerical heat transfer part b vol 15 no 2 1989 pp 241 247 8jeng y n and lee z s revisit to the modi ed muliple onedimensional adaptive grid method numerical heat transfer part b vol 29 no 3 1996 pp 305 323 9bockelie m j and smith r e an adaptive grid method for computing the high speed 3d viscous flow about a re entry vehicle aiaa paper 92 2685 june 1992 10yee h c and harten a implicit tvd schemes for hyperbolic conservation laws in curvilinear coordinates aiaa journal vol 25 no 2 1987 pp 266 274 11jeng y n and liou y c adaptive grid generation by elliptic equations with grid control at all of the boundaries numerical heat transfer part b vol 23 no 2 1993 pp 135 151

a generalized dimension reduction method for multidimensional integration in stochastic mechanics summary a new generalized multivariate dimension reduction method is presented for calculating statistical moments of the response of mechanical systems subject to uncertainties in loads material properties and geometry the method involves an additive decomposition of an n dimensional response function into at most s dimensional functions where s n an approximation of response moments by moments of input random variables and a moment based quadrature rule for numerical integration a new theorem is presented which provides a convenient means to represent the taylor series up to a specific dimension without involving any partial derivatives a complete proof of the theorem is given using two lemmas also proved in this paper the proposed method requires neither the calculation of partial derivatives of response as in commonly used taylor expansion perturbation methods nor the inversion of random matrices as in the neumann expansion method eight numerical examples involving elementary mathematical functions and solid mechanics problems illustrate the proposed method results indicate that the multivariate dimension reduction method generates convergent solutions and provides more accurate estimates of statistical moments or multidimensional integration than existing methods such as first and second order taylor expansion methods statistically equivalent solutions quasi monte carlo simulation and the fully symmetric interpolatory rule while the accuracy of the dimension reduction method is comparable to that of the fourth order neumann expansion method a comparison of cpu time suggests that the former is computationally far more efficient than the latter copyright 2004 john wiley sons ltd

spectral power iterations for the random eigenvalue problem two computationally efficient algorithms are developed for solving the stochastic eigenvalue problem an algorithm based on the power iteration technique is proposed for the calculation of the dominant eigenpairs this algorithm is then extended to find other subdominant random eigenpairs the uncertainty in the operator is represented by a polynomial chaos expansion and a similar representation is considered for the random eigenvalues and eigenvectors the algorithms are distinguished due to their speed in converging to the true random eigenpairs and their ability to estimate a prescribed number of subdominant eigenpairs the algorithms are demonstrated on two examples with close agreement observed with the exact solution and a solution synthesized through monte carlo sampling

beyond the limitations of perturbation methods for real random eigenvalue problems using exceptional points and analytic continuation a numerical method is proposed to approximate the solution of parametric eigenvalue problem when the variability of the parameters exceed the radius of convergence of low order perturbation methods the radius of convergence of eigenvalue perturbation methods based on taylor series is known to decrease when eigenvalues are getting closer to each other this phenomenon knwon as veering in structural dynamics is a direct consequence of the existence of branch point singularity in the complex plane of the varying parameters where some eigenvalues are defective when this degeneracy referred to as exceptional point ep is close to the real axis the veering becomes stronger the main idea of the proposed approach is to combined a pair of eigenvalues to remove this singularity to do so two analytic auxiliary functions are introduced and are computed through high order derivatives of the eigenvalue pair with respect to the parameter this yields a new robust eigenvalue reconstruction scheme which is compared to taylor and puiseux series in all cases theoretical bounds are established and all approximations are compared numerically on a three degrees of freedom toy model this system illustrate the ability of the method to handle the vibrations of a structure with a randomly varying parameter computationally efficient the proposed algorithm could also be relevant for actual numerical models of large size arising from other applications involving parametric eigenvalue problems e g waveguides rotating machinery or instability problems such as squeal or flutter

on the dimension of furstenberg measure for sl _ 2 mathbb r random matrix products let mu be a measure on sl 2 mathbb r generating a non compact and totally irreducible subgroup and let nu be the associated stationary furstenberg measure for the action on the projective line we prove that if mu is supported on finitely many matrices with algebraic entries then begin aligned dim nu min big 1 frac h mathrm mathrm rw mu 2 chi mu big end aligned where h mathrm mathrm rw mu is the random walk entropy of mu chi mu is the lyapunov exponent for the random matrix product associated with mu and dim denotes pointwise dimension in particular for every delta 0 there is a neighborhood u of the identity in sl 2 mathbb r such that if a measure mu in mathcal p u is supported on algebraic matrices with all atoms of size at least delta and generates a group which is non compact and totally irreducible then its stationary measure nu satisfies dim nu 1

parametric instability of stochastic columns columns which have stochastically distributed young s modulus and mass density and are subjected to deterministic periodic axial loadings are considered the general case of a column supported on a winkler elastic foundation of random stiffness and also on discrete elastic supports which are also random is considered material property fluctuations are modeled as independent one dimensional univariate homogeneous real random fields in space in addition to autocorrelation functions or their equivalent power spectral density functions the input random fields are characterized by scale of fluctuations or variance functions for their second order properties the foundation stiffness coefficient and the stiffnesses of discrete elastic supports are treated to constitute independent random variables the system equations of boundary frequencies are obtained using bolotin s method for deterministic systems stochastic fem is used to obtain the discrete system with random as well as periodic coefficients statistical properties of boundary frequencies are derived in terms of input parameter statistics a complete covariance structure is obtained the equations developed are illustrated using a numerical example employing a practical correlation structure

analysis of discrete vibratory systems with parameter uncertainties part i eigensolution abstract a new analytical method is proposed for the estimation of eigensolutions for undamped and proportionally damped discrete vibratory systems when the system parameters are uncertain or random variables given several simplifying assumptions a direct product technique is used to estimate the first and second moments of eigensolutions in terms of the moments of system parameter matrices the proposed methodology is reasonably accurate and computationally fast when compared with existing methods such as the first perturbation method and the monte carlo simulation application of the new method is demonstrated through several single and multi degree of freedom systems it is seen that the standard deviation predictions match well with the results yielded by the monte carlo simulation unlike the first order perturbation method especially when moderately large random fluctuations are considered the impulse response issue is addressed in the companion paper part ii

efficient implementation of quaternion fourier transform convolution and correlation by 2 d complex fft the concepts of quaternion fourier transform qft quaternion convolution qcv and quaternion correlation which are based on quaternion algebra have been found to be useful for color image processing however the necessary computational algorithms and their complexity still need some attention we develop efficient algorithms for qft qcv and quaternion correlation the conventional complex two dimensional 2 d fourier transform ft is used to implement these quaternion operations very efficiently with these algorithms we only need two complex 2 d fts to implement a qft six complex 2 d fts to implement a one side qcv or a quaternion correlation and 12 complex 2 d fts to implement a two side qcv and the efficiency of these quaternion operations is much improved meanwhile we also discuss two additional topics the first one is about how to use qft and qcv for quaternion linear time invariant qlti system analysis this topic is important for quaternion filter design and color image processing besides we also develop the spectrum product qcv it is an improvement of the conventional form of qcv for any arbitrary input functions it always corresponds to the product operation in the frequency domain it is very useful for quaternion filter design

statistics of complex eigenvalues in friction induced vibration abstract self excited vibrations appear in many mechanical systems with sliding contacts there are several mechanisms whereby friction can cause the self excited vibration to become unstable of these mechanisms mode coupling is thought to be responsible for generating annoying high frequency noise and vibration in brakes conventionally in order to identify whether a system is stable or not complex eigenvalue analysis is performed however what has recently received much attention of researchers is the variability and uncertainty of input variables in the stability analysis of self excited vibrations for this purpose a second order perturbation method is extended and employed in the current study the moments of the output distribution along with its joint moment generating function are used for quantifying the statistics of the complex eigenvalues moreover the eigen derivatives required for the perturbation method are presented in a way that they can deal with the asymmetry of the stiffness matrix and non proportional damping since the eigen derivatives of such systems are complex valued numbers it is mathematically more informative and convenient to derive the statistics of the eigenvalues in a complex form without decomposing them into two real valued real and imaginary parts then the variance and pseudo variance of the complex eigenvalues are used for determining the statistics of the real and imaginary parts the reliability and robustness of the system in terms of stability can also be quantified by the approximated output distribution

polynomial chaos for random fractional order differential equations fractional order models provide a powerful instrument for description of memory and hereditary properties of systems in comparison to integer order models where such effects are difficult to incorporate and often neglected also many physical real world problems that involve uncertainties and errors can be best modeled with random differential equations thus to be able to deal with many real life problems it is important to develop mathematical methodologies to solve systems that include both memory effects and uncertainty the aim of this paper is to study the application of the generalized polynomial chaos gpc to random fractional ordinary differential equations the method of polynomial chaos has played an increasingly important role when dealing with uncertainties the main idea of the method is the projection of the random parameters and stochastic processes in the system onto the space of polynomial chaoses however to apply polynomial chaos to random fractional differential equations requires careful attention due to memory effects and the increasing of the computation time in respect to the classic random differential equations in order to avoid more complex numerical computations and obtain accurate solutions we rely on richardson extrapolation it is shown that the application of generalized polynomial chaos method in conjunction with richardson extrapolation is a reliable and accurate method to numerically solve random fractional ordinary differential equations

windowed fourier transform of two dimensional quaternionic signals in this paper we generalize the classical windowed fourier transform wft to quaternion valued signals called the quaternionic windowed fourier transform qwft using the spectral representation of the quaternionic fourier transform qft we derive several important properties such as reconstruction formula reproducing kernel isometry and orthogonality relation taking the gaussian function as window function we obtain quaternionic gabor filters which play the role of coefficient functions when decomposing the signal in the quaternionic gabor basis we apply the qwft properties and the right sided qft to establish a heisenberg type uncertainty principle for the qwft finally we briefly introduce an application of the qwft to a linear time varying system

random eigenvalues algebraic methods and structural dynamic models a review of the literature of the study of structural vibration problems with random parameters is provided there have been several approaches to this problem these have been encapsulated in this paper for the benefit of those who need to assess such possibilities in particular the algebraic theory of random variables is delineated here with the application in mind being the determination or estimation of the statistics of the eigenvalues of linear dynamical systems various transformation techniques are summarized and discussed with simple examples

wigner distributions and ambiguity functions of 2 d quaternionic and monogenic signals the paper presents new notions of wigner distributions and corresponding ambiguity functions defined by quaternionic fourier transforms of correlation products of recently defined quaternionic and monogenic two dimensional 2 d signals the properties of new defined wigner distributions are compared with wigner distributions of 2 d analytic signals with single quadrant spectra it is well known that wigner distributions of complex signals are real functions differently the wigner distributions of quaternionic and monogenic signals may be quaternionic valued functions however it may happen that some 2 d slices of 4 d wigner distributions are real functions

rates of change of eigenvalues and eigenvectors in damped dynamic system rates of change of eigenvalues and eigenvectors of a damped linear discrete dynamic system with respect to the system parameters are presented a nonproportional viscous damping model is assumed because of the nonproportional nature of the damping the mode shapes and natural frequencies become complex and as a consequence the sensitivities of eigenvalues and elgenvectors are also complex the results are presented in terms of the complex modes and frequencies of the second order system and the use of rather undesirable state space representation is avoided the usefulness of the derived expressions is demonstrated by considering an example of a nonproportionally damped two degree of freedom system

higher order eigenpair perturbations a matrix method is presented for the exact calculation of large eigenpair perturbations arising from design variable changes this method of forward analysis is presented that takes into account all order of the perturbation expansion

an approximate solution scheme for the algebraic random eigenvalue problem a reduced basis formulation is presented for the efficient solution of large scale algebraic random eigenvalue problems this formulation aims to improve the accuracy of the first order perturbation method and also allow the efficient computation of higher order statistical moments of the eigenparameters in the present method the two terms of the first order perturbation approximation for the eigenvector are used as basis vectors for ritz analysis of the governing random eigenvalue problem this leads to a sequence of reduced order random eigenvalue problems to be solved for each eigenmode of interest since only two basis vectors are used to represent each eigenvector explicit expressions for the random eigenvalues and eigenvectors can readily be derived this enables the statistics of the random eigenparameters and the forced response to be efficiently computed numerical studies are presented for free and forced vibration analysis of a linear stochastic structural system it is demonstrated that the reduced basis method gives better results as compared to the first order perturbation method

the stochastic finite element method past present and future a powerful tool in computational stochastic mechanics is the stochastic finite element method sfem sfem is an extension of the classical deterministic fe approach to the stochastic framework i e to the solution of static and dynamic problems with stochastic mechanical geometric and or loading properties the considerable attention that sfem received over the last decade can be mainly attributed to the spectacular growth of computing power rendering possible the efficient treatment of large scale problems this article aims at providing a state of the art review of past and recent developments in the sfem area and indicating future directions as well as some open issues to be examined by the computational mechanics community in the future

matrix models for beta ensembles this paper constructs tridiagonal random matrix models for general 0 hermite gaussian and laguerre wishart ensembles these generalize the well known gaussian and wishart models for 1 2 4 furthermore in the cases of the laguerre ensembles we eliminate the exponent quantization present in the previously known models we further discuss applications for the new matrix models and present some open problems

stochastic dynamic systems with complex valued eigensolutions a dimensional decomposition method is presented for calculating the probabilistic characteristics of complex valued eigenvalues and eigenvectors of linear stochastic dynamic systems the method involves a function decomposition allowing lower dimensional approximations of eigensolutions lagrange interpolation of lower dimensional component functions and monte carlo simulation compared with the commonly used perturbation method neither the assumption of small input variability nor the calculation of the derivatives of eigensolutions is required by the method developed results of numerical examples from linear stochastic dynamics indicate that the decomposition method provides excellent estimates of the moments and or probability densities of eigenvalues and eigenvectors for various cases including large statistical variations of input copyright 2007 john wiley sons ltd

a method for solving stochastic eigenvalue problems ii we present a novel approach for calculating the stochastic eigenvalues of a linear operator this is an extension of our earlier work on the subject and introduces a significant simplification that reduces the amount of numerical work required when evaluating the polynomial chaos expansions improves the accuracy and enables higher harmonics to be studied examples based on several types of problem involving differential equations and matrices have been analysed and detailed numerical results obtained

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

an invariant subspace based approach to the random eigenvalue problem of systems with clustered spectrum the repeated or closely spaced eigenvalues and corresponding eigenvectors of a matrix are usually very sensitive to a perturbation of the matrix which makes capturing the behavior of these eigenpairs very difficult similar difficulty is encountered in solving the random eigenvalue problem when a matrix with random elements has a set of clustered eigenvalues in its mean in addition the methods to solve the random eigenvalue problem often differ in characterizing the problem which leads to different interpretations of the solution thus the solutions obtained from different methods become mathematically incomparable these two issues the difficulty of solving and the non unique characterization are addressed here a different approach is used where instead of tracking a few individual eigenpairs the corresponding invariant subspace is tracked the spectral stochastic finite element method is used for analysis where the polynomial chaos expansion is used to represent the random eigenvalues and eigenvectors however the main concept of tracking the invariant subspace remains mostly independent of any such representation the approach is successfully implemented in response prediction of a system with repeated natural frequencies it is found that tracking only an invariant subspace could be sufficient to build a modal based reduced order model of the system copyright c 2012 john wiley sons ltd

model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics summary a priori model reduction methods based on separated representations are introduced for the prediction of the low frequency response of uncertain structures within a parametric stochastic framework the proper generalized decomposition method is used to construct a quasi optimal separated representation of the random solution at some frequency samples at each frequency an accurate representation of the solution is obtained on reduced bases of spatial functions and stochastic functions an extraction of the deterministic bases allows for the generation of a global reduced basis yielding a reduced order model of the uncertain structure which appears to be accurate on the whole frequency band under study and for all values of input random parameters this strategy can be seen as an alternative to traditional constructions of reduced order models in structural dynamics in the presence of parametric uncertainties this reduced order model can then be used for further analyses such as the computation of the response at unresolved frequencies or the computation of more accurate stochastic approximations at some frequencies of interest because the dynamic response is highly nonlinear with respect to the input random parameters a second level of separation of variables is introduced for the representation of functions of multiple random parameters thus allowing the introduction of very fine approximations in each parametric dimension even when dealing with high parametric dimension copyright 2011 john wiley sons ltd

analyticity of entropy rate of hidden markov chains we prove that under mild positivity assumptions the entropy rate of a hidden markov chain varies analytically as a function of the underlying markov chain parameters a general principle to determine the domain of analyticity is stated an example is given to estimate the radius of convergence for the entropy rate we then show that the positivity assumptions can be relaxed and examples are given for the relaxed conditions we study a special class of hidden markov chains in more detail binary hidden markov chains with an unambiguous symbol and we give necessary and sufficient conditions for analyticity of the entropy rate for this case finally we show that under the positivity assumptions the hidden markov chain itself varies analytically in a strong sense as a function of the underlying markov chain parameters

calculation of eigenvector derivatives for structures with repeated eigenvalues in structural optimization and system identification eigenvector derivatives provide important information for updating design model parameters when the current parameter values yield repeated eigenvalues it has not been possible previously to calculate unique eigenvector derivatives recent work has provided a method for determining unique eigenvalue derivatives for this case but methods for calculating the eigenvector sensitivities have been incomplete in this work a complete method for calculation of repeated root eigenvector derivatives is shown for the real symmetric structural eigenproblem the derivation is completed by using information from the second derivative of the eigen problem and is limited to the case of distinct eigenvalue sensitivities as an example the repeated root eigenvector sensitivities are calculated for a simple three degree of freedom beam grillage comparisons of linear approximations using these derivatives to the calculated eigenvectors support the accuracy of the formulation

sampling of closely spaced ordered set of uniformly distributed random variables summary when the individual pdfs of closely spaced random variables such as natural frequencies of a structure overlap generation of sample sets by assuming the frequencies to be independent random variables can lead to incorrect sets of frequencies in the sense that the frequencies do not remain as ordered sets rejection of such disordered sample sets results in individual density functions that are significantly different from the distributions initially assumed for sampling each random variable one way to overcome this constraint in the simulation of an ordered set of random variables is to consider them in an implicit manner using a joint pdf in this paper we present a formulation for a joint density function that is developed using fundamental probability approaches the formulation ensures that the sampled random variables always remain as ordered sets and maintain the individual density functions for each variable application of the proposed formulation is illustrated for cases with not just two closely spaced variables but also for a case with multiple closely spaced variables such that the pdfs of more than two random variables overlap with each other an expression is presented to determine the exact number of terms needed in the formulation however it is also illustrated that only two terms are sufficient in most applications even when the exact number of terms needed is very high copyright 2011 john wiley sons ltd

random matrix theory and non parametric model of random uncertainties in vibration analysis recently a new approach called a non parametric model of random uncertainties has been introduced for modelling random uncertainties in linear and non linear elastodynamics in the low frequency range this non parametric approach differs from the parametric methods for random uncertainties modelling and has been developed in introducing a new ensemble of random matrices constituted of symmetric positive definite real random matrices this ensemble differs from the gaussian orthogonal ensemble goe and from the other known ensembles of the random matrix theory the present paper has three main objectives the first one is to study the statistics of the random eigenvalues of random matrices belonging to this new ensemble and to compare with the goe the second one is to compare this new ensemble of random matrices with the goe in the context of the non parametric approach of random uncertainties in structural dynamics for the low frequency range the last objective is to give a new validation for the non parametric model of random uncertainties in structural dynamics in comparing in the low frequency range the dynamical response of a simple system having random uncertainties modelled by the parametric and the non parametric methods these three objectives will allow us to conclude about the validity of the different theories

hybrid perturbation polynomial chaos approaches to the random algebraic eigenvalue problem the analysis of structures is affected by uncertainty in the structure s material properties geometric parameters boundary conditions and applied loads these uncertainties can be modelled by random variables and random fields amongst the various problems affected by uncertainty the random eigenvalue problem is specially important when analyzing the dynamic behavior or the buckling of a structure the methods that stand out in dealing with the random eigenvalue problem are the perturbation method and methods based on monte carlo simulation in the past few years methods based on polynomial chaos pc have been developed for this problem where each eigenvalue and eigenvector are represented by a pc expansion in this paper four variants of a method hybridizing perturbation and pc expansion approaches are proposed and compared the methods use rayleigh quotient the power method the inverse power method and the eigenvalue equation pc expansions of eigenvalues and eigenvectors are obtained with the proposed methods the new methods are applied to the problem of an euler bernoulli beam and a thin plate with stochastic properties

derivatives of entropy rate in special families of hidden markov chains consider a hidden markov chain obtained as the observation process of an ordinary markov chain corrupted by noise recently zuk et al showed how in principle one can explicitly compute the derivatives of the entropy rate of at extreme values of the noise namely they showed that the derivatives of standard upper approximations to the entropy rate actually stabilize at an explicit finite time we generalize this result to a natural class of hidden markov chains called black holes we also discuss in depth special cases of binary markov chains observed in binary symmetric noise and give an abstract formula for the first derivative in terms of a measure on the simplex due to blackwell

analysis of eigenvalues and modal interaction of stochastic systems an eigenvalue spectral analysis of stochastic engineering systems is presented a comparative numerical study between approximations based on monte carlo sampling a taylor series based perturbation approach and the polynomial chaos representation is conducted it is observed that the polynomial chaos representation gives more accurate estimates of the statistical moments than the perturbation method especially for the higher modes the differences of accuracy in the two methods are more pronounced as the system variability increases moreover the chaos expansion gives a more detailed probabilistic description of the eigenvalues and the eigenvectors in addition a method for representing the statistical modal overlapping is presented that characterizes the statistical interaction between the various modes

on the optimality of symbol by symbol filtering and denoising we consider the problem of optimally recovering a finite alphabet discrete time stochastic process x sub t from its noise corrupted observation process z sub t in general the optimal estimate of x sub t will depend on all the components of z sub t on which it can be based we characterize nontrivial situations i e beyond the case where x sub t z sub t are independent for which optimum performance is attained using symbol by symbol operations a k a singlet decoding meaning that the optimum estimate of x sub t depends solely on z sub t for the case where x sub t is a stationary binary markov process corrupted by a memoryless channel we characterize the necessary and sufficient condition for optimality of symbol by symbol operations both for the filtering problem where the estimate of x sub t is allowed to depend only on z sub t sub t spl les t and the denoising problem where the estimate of x sub t is allowed dependence on the entire noisy process it is then illustrated how our approach which consists of characterizing the support of the conditional distribution of the noise free symbol given the observations can be used for characterizing the entropy rate of the binary markov process corrupted by the binary symmetric channel bsc in various asymptotic regimes for general noise free processes not necessarily markov general noise processes not necessarily memoryless and general index sets random fields we obtain an easily verifiable sufficient condition for the optimality of symbol by symbol operations and illustrate its use in a few special cases for example for binary processes corrupted by a bsc we establish under mild conditions the existence of a spl delta sup 0 such that the say what you see scheme is optimal provided the channel crossover probability is less than spl delta sup finally we show how for the case of a memoryless channel the large deviations ld performance of a symbol by symbol filter is easy to obtain thus characterizing the ld behavior of the optimal schemes when these are singlet decoders and constituting the only known cases where such explicit characterization is available

eigenvalue curve veering in stressed structures an experimental study there have been extensive research works on the veering phenomenon in dynamic systems as eigenvalues change under parametric variation converging loci are commonly seen to veer away suddenly in a small region of the graph such that the modes swap trajectories all of the modal properties are swapped in the process leading to some curious behaviour in the transition zone theoretical studies of this behaviour have been reported for half a century but despite this heritage explicit references to experimental results are scarce in this paper detailed experimental and numerical investigations on veering are reported the parameter varied is the internal pre load of a redundant frame an fe model is presented and stress stiffening approximations are employed to obtain a tangent stiffness for a nonlinear static solution which is then used in a linear dynamic analysis experimental results are given and the behaviour is found to correspond well with the analytical results in particular the mode shape variation is found to be consistent and the implications of this finding with regard to modal correlation and model validation are noted analyses of the mode shapes in terms of eigenvector rotations are presented and are found to form a valuable tool for the interpretation of experimental results

an interpolation scheme for the approximation of dynamical systems abstract the demand of many industries for highly accurate predictions of the structural response of dynamical systems leads to a rapidly growing complexity of the established numerical models due to the thereby arising large computational efforts the use of model reduction techniques becomes an essential task model reduction procedures in the context of deterministic structural analysis are already well developed and widely used such as guyan reduction and component mode synthesis techniques however the assumption of deterministically known structural properties is in most cases not realistic hence the rational propagation of uncertainties becomes an indispensable task in the robust prediction of the structural response the associated multiple simulation runs of complex models bring the topic of model reduction to the foreground of technical and scientific interest in this work a novel method which approximates eigenfrequencies and mode shape vectors needed for the evaluation of frequency response functions will be presented this meta model of structural matrices is established by employing interpolation between the matrices evaluated at some supporting points the technique has no difficulties with closely spaced eigenfrequencies and mode switching and it can be employed over the entire frequency range on the full model but also on the substructures and it is hence applicable together with deterministic model reduction techniques

solutions of integral and integro differential equation systems by using differential transform method in this study differential transform method dtm is applied to both integro differential and integral equation systems the method is further expanded with a formulation to treat fredholm integrals if the system considered has a solution in terms of the series expansion of known functions this powerful method catches the exact solution so as to show this capability and robustness some systems of integral and integro differential equations are solved as numerical examples

window functions represented by b spline functions characteristics of window functions provided by b spline functions are analyzed in comparison to other known windows for the same highest sidelobe level those windows have a mainlobe bandwidth between those of gaussian and cos sup a x windows are as good as those of cos sup a x ones in the sidelobe falloff

a hybrid approach for the time domain analysis of linear stochastic structures abstract a new hybrid approach for the time domain analysis of linear stochastic structures with uncorrelated or correlated random variables is proposed this new hybrid approach combines the modal approach the second order perturbation technique and the number theoretical method ntm in the hybrid approach an approximate stochastic model of the finite element fe model is developed in the sense of first order accuracy which provides second order estimations of the mean and covariance matrices of structural responses compared with the fe model the proposed model is more convenient in terms of the computation effort it requires by employing the ntm to evaluate the statistical moments of solutions of the approximate model the secular terms contained in the results of the perturbation stochastic finite element method are eliminated two numerical examples are presented to demonstrate the accuracy and efficiency of the method proposed

modal approaches for the stochastic finite element analysis of structures with material and geometric uncertainties this paper presents two efficient modal approaches as an alternative to direct formulations for the time harmonic dynamic analysis of structures with random material and shape parameters in both approaches the structural eigenproblem is solved and complemented by a sensitivity analysis to the random parameters the modal perturbation stochastic finite element method sfem then condenses the response sensitivities to assess the response variability the mixed perturbation monte carlo sfem assesses the response statistics by sampling the structural eigenmodes according to the perturbation estimation of their probability distribution functions pdfs geometric uncertainties are handled through an appropriate shape parameterisation and a shape design sensitivity analysis two numerical applications examine both approaches in terms of accuracy variability level and computational requirements the applications involve a plate bending problem with random young modulus or edge length and a plate with a random flatness default particular observations related to the influence of the parameter pdfs in simulation based methods are also provided

harmonic measures for distributions with finite support on the mapping class group are singular kaimanovich and masur showed that a random walk on the mapping class group for an initial distribution whose support generates a nonelementary subgroup when projected into teichmuller space converges almost surely to a point in the space pmf of projective measured foliations on the surface this defines a harmonic measure on pmf here we show that when the initial distribution has finite support the corresponding harmonic measure is singular with respect to the natural lebesgue measure class on pmf

a solution of the random eigenvalue problem by crossing theory abstract a method is developed for finding probabilistic descriptors of the eigenvalues of stochastic real valued symmetric matrices the method is based on the mean rate at which the characteristic polynomial of a stochastic matrix crosses level zero the proposed approach can also be applied when additional information is available on the value of the first or the first few eigenvalues several examples are presented in the paper to demonstrate the application and usefulness of the method

on the hausdorff dimension of invariant measures of weakly contracting on average measurable ifs we consider measures which are invariant under a measurable iterated function system with positive place dependent probabilities in a separable metric space we provide an upper bound of the hausdorff dimension of such a measure if it is ergodic we also prove that it is ergodic iff the related skew product is

a solution of the random eigenvalue problem by a dimensional decomposition method summary this paper presents a dimensional decomposition method for obtaining probabilistic descriptors of realvalued eigenvalues of positive semi definite random matrices the method involves a novel function decomposition allowing lower variate approximations of eigenvalues lower dimensional numerical integration for statistical moments and lagrange interpolation facilitating efficient monte carlo simulation for probability density functions compared with commonly used perturbation and recently developed asymptotic methods no derivatives of eigenvalues are required by the new method developed results of numerical examples from structural dynamics indicate that the decomposition method provides excellent estimates of moments and probability densities of eigenvalues for various cases including closely spaced modes and large statistical variations of input copyright 2006 john wiley sons ltd

quadratic forms in random variables theory and applications textbook for a one semester graduate course for students specializing in mathematical statistics or in multivariate analysis or reference for theoretical as well as applied statisticians confines its discussion to quadratic forms and second degree polynomials in real normal random vectors and matr

random eigenvalue problems in structural dynamics experimental investigations an experimental investigation of uncertainty in natural frequencies of linear structures estimated from measured frequency response function frf under dynamic loading is presented experiments were conducted on one hundred nominally identical realizations of a fixed fixed beam each realisation was obtained by placin g twelve identical masses at random spatial locations generated by a computer along the length of the beam the total random mass is about 2 of the total mass of the beam these experiments contain parametric uncertainty in the mass matrix and hence may be useful for the validation of many random eigenvalue analysis and prediction methods currently available to structural dynamicist predictions from monte carlo simulation of deterministic finite elements model are compared with experiments it is co ncluded that the method of estimation of natural frequencies from frfs and the spatial location of the measurements has significant influence upon the first two moments mean and standard deviation of the natural frequency ensemble furthermore whilst the monte carlo simulation estimates of the mean and standard deviation are in reasonable agreement with experiments at higher frequencies the probability density function differ appreciably within the limits of the sample size investigated in this study

eigenvalue density of linear stochastic dynamical systems a random matrix approach abstract eigenvalue problems play an important role in the dynamic analysis of engineering systems modeled using the theory of linear structural mechanics when uncertainties are considered the eigenvalue problem becomes a random eigenvalue problem in this paper the density of the eigenvalues of a discretized continuous system with uncertainty is discussed by considering the model where the system matrices are the wishart random matrices an analytical expression involving the stieltjes transform is derived for the density of the eigenvalues when the dimension of the corresponding random matrix becomes asymptotically large the mean matrices and the dispersion parameters associated with the mass and stiffness matrices are necessary to obtain the density of the eigenvalues in the frameworks of the proposed approach the applicability of a simple eigenvalue density function known as the mar enko pastur mp density is investigated the analytical results are demonstrated by numerical examples involving a plate and the tail boom of a helicopter with uncertain properties the new results are validated using an experiment on a vibrating plate with randomly attached spring mass oscillators where 100 nominally identical samples are physically created and individually tested within a laboratory framework

two dimensional quaternion wavelet transform abstract in this paper we introduce the continuous quaternion wavelet transform cqwt we express the admissibility condition in terms of the right sided quaternion fourier transform we show that its fundamental properties such as inner product norm relation and inversion formula can be established whenever the quaternion wavelets satisfy a particular admissibility condition we present several examples of the cqwt as an application we derive a heisenberg type uncertainty principle for these extended wavelets

absolutely continuous convolutions of singular measures and an application to the square fibonacci hamiltonian author s damanik d gorodetski a solomyak b abstract 2015 we prove for the square fibonacci hamiltonian that the density of states measure is absolutely continuous for almost all pairs of small coupling constants this is obtained from a new result we establish about the absolute continuity of convolutions of measures arising in hyperbolic dynamics with exact dimensional measures

iterative solution of the random eigenvalue problem with application to spectral stochastic finite element systems a new algorithm for the computation of the spectral expansion of the eigenvalues and eigenvectors of a random non symmetric matrix is proposed the algorithm extends the deterministic inverse power method using a spectral discretization approach the convergence and accuracy of the algorithm is studied for both symmetric and non symmetric matrices the method turns out to be efficient and robust compared to existing methods for the computation of the spectral expansion of random eigenvalues and eigenvectors