the nilpotent centralizer method for spectrally arbitrary patterns this paper introduces a new criterion for an n n sign pattern respectively zero nonzero pattern a to be spectrally arbitrary that is to have the property that for each monic real polynomial r x of degree n there exists a matrix with sign pattern respectively zero nonzero pattern a that has r x as its characteristic polynomial to date analytic properties of a certain polynomial map associated with a have been used to prove that a is spectrally arbitrary in this paper we derive a method that uses the algebraic structure of a certain nilpotent matrix to show that a sign pattern respectively zero nonzero pattern is spectrally arbitrary

stochastic joint time frequency response analysis of nonlinear structural systems abstract a novel approximate analytical approach for determining the response evolutionary power spectrum eps of nonlinear hysteretic structural systems subject to stochastic excitation is developed specifically relying on the theory of locally stationary processes and utilizing a recently proposed representation of non stationary stochastic processes via wavelets a versatile formula for determining the nonlinear system response eps is derived this is done in conjunction with a stochastic averaging treatment of the problem and by resorting to the orthogonality properties of harmonic wavelets further the nonlinear system non stationary response amplitude probability density function pdf which is required as input for the developed approach is determined either by utilizing a numerical path integral scheme or by employing a time dependent rayleigh pdf approximation technique a significant advantage of the approach relates to the fact that it is readily applicable for treating not only separable but non separable in time and frequency eps as well the hardening duffing and the versatile preisach hysteretic oscillators are considered in the numerical examples section comparisons with pertinent monte carlo simulations demonstrate the reliability of the approach

group theoretic exploitations of symmetry in computational solid and structural mechanics the use of group theory in simplifying the study of problems involving symmetry is a well established approach in various branches of physics and chemistry and major applications in these areas date back more than 70 years within the engineering disciplines the search for more systematic and more efficient strategies for exploiting symmetry in the computational problems of solid and structural mechanics has led to the development of group theoretic methods over the past 40 years this paper reviews the advances made in the application of group theory in areas such as bifurcation analysis vibration analysis and finite element analysis and summarizes the various implementation procedures currently available illustrative examples of typical solution procedures are drawn from recent work of the author it is shown how the group theoretic approach through the characteristic vector space decomposition enables considerable simplifications and reductions in computational effort to be achieved in many cases group theoretic considerations also allow valuable insights on the behaviour or properties of a system to be gained before any actual calculations are carried out copyright 2009 john wiley sons ltd

an exact and efficient time domain method for random vibration analysis of linear structures subjected to uniformly modulated or fully non stationary excitations abstract this paper presents a novel time domain method for non stationary random vibration analysis of linear structures specifically using the well known spectral representation method the stochastic excitation process is first decomposed into a set of trigonometric basis functions tbfs modulated by a series of orthogonal basis random variables brvs next each tbf is used as a deterministic excitation input for the structural model and the induced structural responses are derived analytically using the modal superposition method after superposition of each tbf induced deterministic responses an explicit closed form expression between the structural stochastic responses and the brvs is derived based on which the response statistics of structures at any time instant can be evaluated easily furthermore the accuracy of the solution is independent of the time step used finally several numerical examples in civil engineering including a euler beam under stochastic moving load and a shear frame structure subjected to uniformly modulated or fully non stationary seismic motions are studied to illustrate the performance of the proposed method the obtained structural response statistics are compared with the solutions obtained by the monte carlo and evolutionary spectral methods the results verified the high accuracy and efficiency of the proposed method

regulating the vibratory motion of beams using shape optimization abstract in this paper shape optimization is used to regulate the vibrations of an euler bernoulli beam having constant material volume this is achieved by varying appropriately the beam cross section and thus its stiffness and mass properties along its length so that the beam vibrates with its minimum maximum or a prescribed eigenfrequency as well as with the minimum or maximum difference between two successive eigenfrequencies the problem is reduced to a nonlinear optimization problem under equality and inequality constraints as well as specified lower and upper bounds the evaluation of the objective function requires the solution of the free vibration problem of a beam with variable mass and stiffness properties this problem is solved using the analog equation method aem for hyperbolic differential equations with variable coefficients besides its accuracy this method overcomes the shortcoming of a fem solution which would require resizing of the elements and re computation of their stiffness and mass properties during the optimization process certain example problems are presented which illustrate the method and demonstrate its efficiency

nested sequences of principal minors and potential stability abstract sufficient conditions are determined for an n by n zero nonzero pattern to allow a nested sequence of nonzero principal minors a method is then given to sign such a pattern so that it allows a nested sequence of k by k principal minors with sign 1 k for 1 k n it is a classical result that a sign pattern with such a sequence is potentially stable thus classes of potentially stable sign patterns can be identified a necessary and sufficient condition for a certain tree sign pattern class to be potentially stable is also given

a digital method of modeling quadratically nonlinear systems with a general random input without assuming particular statistics of the input a practical digital method of estimating linear and quadratic transfer functions of a nonlinear time invariant system that can be described by volterra series of up to second order is presented the method is tested and validated by analyzing input output data of a known quadratically nonlinear system it is used when there is little knowledge about the input statistics or the input is non gaussian it is also noted that the ordinary coherence functions cannot be used in explaining the input output power transfer relationship of a quadratic system excited by a non gaussian input signal with respect to the practical application of the method the relationship between the mean square errors involved in the transfer function estimates and the number of averages taken from the spectral estimation is qualitatively discussed

spectrally and inertially arbitrary sign patterns we introduce some n by n sign patterns which allow for arbitrary spectrum and hence also arbitrary inertia consequently we demonstrate that some known inertially arbitrary patterns are in fact spectrally arbitrary we demonstrate that all inertially arbitrary patterns of order 3 are spectrally arbitrary and classify all spectrally arbitrary patterns of order 3 we illustrate that in general the class of spectrally arbitrary patterns is distinct from the inertially arbitrary patterns and present some observations about inertially arbitrary patterns

sets of refined inertias of zero nonzero patterns abstract a set h n of refined inertias for zero nonzero patterns is introduced that is analogous to the set h n previously considered for sign patterns for n 3 and 4 a complete characterization of irreducible zero nonzero patterns that allow or require h n is given and each zero nonzero pattern that allows h n has a signing that allows h n in contrast for n 5 a family of irreducible zero nonzero patterns is given that allows h n but for which no one signing allows h n

stochastic averaging of energy envelope of preisach hysteretic systems the method of stochastic averaging of energy envelope is applied to predict the response of bilinear hysteretic systems to white noise excitation the statistics of stationary response for displacement velocity and energy envelope as functions of excitation level secondary slope and viscous damping are analyzed in detail comparisons with digital simulation show that the method is superior to other approximate methods have been applied to random response of hysteretic systems

wavelet processes and adaptive estimation of the evolutionary wavelet spectrum this article defines and studies a new class of non stationary random processes constructed from discrete non decimated wavelets which generalizes the cramer fourier representation of stationary time series we define an evolutionary wavelet spectrum ews which quantifies how process power varies locally over time and scale we show how the ews may be rigorously estimated by a smoothed wavelet periodogram and how both these quantities may be inverted to provide an estimable time localized autocovariance we illustrate our theory with a pedagogical example based on discrete non decimated haar wavelets and also a real medical time series example

spectral response of a bilinear oscillator abstract an approximate analytical procedure is presented to estimate the response power spectral density of a randomly excited spring mass damper system having a bilinear spring the approximate expression for the response spectrum is developed by representing the non linear oscillator as a linear system having a natural frequency that depends on the envelope of the random response this approximate representation of the system leads to estimates of the response spectrum that agree extremely well with those obtained by direct numerical simulation of the governing equation

a volterra series approximation to the coherence of the duffing oscillator following previous work on computing approximate frequency response functions for the duffing oscillator under white gaussian excitation an approximation is obtained here for the coherence function a pade approximation of order 1 1 is made for the asymmetric duffing oscillator i e with non zero quadratic term and an approximation of order 2 2 is made for the symmetric no quadratic term oscillator the analytical results are shown to give excellent qualitative agreement with numerical simulation however in quantitative terms the approximations underpredict the coherence distortion as is consistent with the low order truncations of the volterra series

stochastic processes evolutionary spectrum estimation via harmonic wavelets the problem of estimating the power spectrum of non stationary stochastic processes using dyadic generalized and filtered harmonic wavelets is addressed explicit relationships between the statistical moments of the coefficients of the wavelets based representation of the process and of its evolving spectral values are given it is shown that the popular concept of a separable evolutionary spectrum involving a deterministic modulating envelope is subject to interpretation further mathematical expressions elucidating the analogy between the wavelets based spectral representation and the traditional one involving trigonometric functions are derived finally the applicability and the physical soundness of the developed procedure are demonstrated by applying it to records of the kocaeli turkey earthquake 8 17 1999

continuous time non linear system identification in the frequency domain a new algorithm for identifying non linear differential equation models is introduced the algorithm avoids the computation of derivatives by using the generalized frequency response functions to reconstruct the model it is shown that the model can be constructed sequentially by building in first the linear terms then the quadratic terms and so on and provides unbiased estimates in the presence of noise a simulated example and the identification of a model relating the action of waveforces on an offshore structure are included to demonstrate the procedure

mapping non linear integro differential equations into the frequency domain a recursive algorithm is derived to compute the generalized frequency response functions for a large class of non linear integro differential equations applications to duffing s equation and a modified van der pol model are discussed

recursive algorithm for computing the frequency response of a class of non linear difference equation models abstract a recursive algorithm is developed which is both computationally compact and enables the nth order frequency response functions of a large class of non linear difference equation models to be found without restriction on the order n

effect of damping and stiffness on the random vibration of non linear periodic plates abstract sonic fatigue analysis of large complex structures requires accurate prediction of the response of plates and shells life predictions based on linear plate theory have been shown to deviate significantly from experimental data when the structure is subjected to high acoustic loading the accuracy of the analysis can be improved by considering the non linearities of the structure it is well known that the geometrical non linearities of the plate yield a stiffness which increases non linearly with the amplitude of vibration the effect of this non linearity becomes significant as the deflections approach the thickness of the plate in addition to the non linear stiffness it has been argued that non linear damping also could play an important role in the vibration of plates the argument is based on the observation that during random excitation of plates the power spectra resonance peaks broaden as the excitation level is increased in this paper we present an investigation of this broadening phenomenon for generality both unstiffened and orthogonally stiffened plates are considered it is shown that widening of the resonance peaks can occur simply by the existence of the non linear stiffness and without the introduction of non linear damping it is also shown that the method of equivalent linearization is not suitable for the study of the power spectra of the response of non linear plates

zero nonzero patterns of order n 4 do not require hn abstract in 1 the set of refined inertias h n for sign patterns is expanded to h n for zero nonzero patterns also it is proved that there are no zero nonzero patterns of even order 2m that require h 2 m in this paper it is shown that there are no zero nonzero patterns of odd order n 5 that require h n which together with the previous result prove that there are no zero nonzero patterns of order n 4 that require h n

comparisons between harmonic balance and nonlinear output frequency response function in nonlinear system analysis by using the duffing oscillator as a case study this paper shows that the harmonic components in the nonlinear system response to a sinusoidal input calculated using the nonlinear output frequency response functions nofrfs are one of the solutions obtained using the harmonic balance method hbm a comparison of the performances of the two methods shows that the hbm can capture the well known jump phenomenon but is restricted by computational limits for some strongly nonlinear systems and can fail to provide accurate predictions for some harmonic components although the nofrfs cannot capture the jump phenomenon the method has few computational restrictions for the nonlinear damping systems the nofrfs can give better predictions for all the harmonic components in the system response than the hbm even when the damping system is strongly nonlinear

a note on sign patterns of order 3 that require particular refined inertias abstract for a real n n matrix a having n n eigenvalues with positive resp negative real part n z zero eigenvalues and 2 n p nonzero pure imaginary eigenvalues the refined inertia of a is ri a n n n z 2 n p when n 3 let h 3 0 3 0 0 0 1 0 2 2 1 0 0 a 3 3 sign pattern a requires refined inertia h 3 if ri a a has sign pattern a h 3 necessary and sufficient conditions for an irreducible sign pattern to require h 3 are given and used to determine all such sign patterns up to equivalence these remove equivalences and complete the list of patterns given in 1 appendix a

spectrally arbitrary patterns abstract an n n sign pattern matrix s is an inertially arbitrary pattern iap if each nonnegative triple n 1 n 2 n 3 with n 1 n 2 n 3 n is the inertia of a matrix with sign pattern s analogously s is a spectrally arbitrary pattern sap if for any given real monic polynomial r x of order n there is a matrix with sign pattern s and characteristic polynomial r x focusing on tree sign patterns consider the n n tridiagonal sign pattern t n with each superdiagonal entry positive each subdiagonal entry negative the 1 1 entry negative the n n entry positive and every other entry zero it is conjectured that t n is an iap by constructing matrices a n with pattern t n it is proved that t n allows any inertia with n 3 0 1 2 n 1 n for all n 2 this leads to a proof of the conjecture for n 5 the truth of the conjecture is extended to n 7 by showing the stronger result that t n is a sap the proof of this latter statement involves finding a matrix a n with pattern t n that is nilpotent further questions about patterns that are saps and iaps are considered

approximation of the duffing oscillator frequency response function using the fpk equation abstract although a great deal of work has been carried out on nonlinear structural dynamic systems under random excitation there has been a comparatively small amount of this work concentrating on the calculation of the quantities commonly measured in structural dynamic tests perhaps the most fundamental of these quantities is the frequency response function frf a number of years ago yar and hammond took an interesting approach to estimating the frf of a duffing oscillator system which was based on an approximate solution of the fokker planck kolmogorov equation despite reproducing the general features of the statistical linearisation estimate the approximation failed to show the presence of the poles at odd multiples of the primary resonance which are known to occur experimentally the current paper simply extends the work of yar and hammond to a higher order of approximation and is thus able to show the existence of a third harmonic in the frf a comparison is made with previous work where an approximation to the frf was computed using the volterra series

application of lie transformation group methods to classical linear theories of rods and plates in the present paper a class of partial differential equations governing various rod and plate theories of bernoulli euler and poisson kirchhoff type is studied by lie transformation group methods a system of equations determining the generators of the admitted point lie groups symmetries is derived and the general statement of the associated group classification problem is given a simple relation is deduced allowing to recognize easily the variational symmetries among the ordinary symmetries of a self adjoint equation of the class examined explicit formulae for the conserved currents of the corresponding via bessel hagens extension of noethers theorem conservation laws are suggested solutions of group classification problems are given for subclasses of equations of the foregoing type governing stability and vibration of rods fluid conveying pipes and plates resting on variable elastic foundations the obtained group classification results are used to derive conservation laws and group invariant solutions readily applicable in rod dynamics and plate statics and dynamics new generalized symmetries and conservation laws for the theories of timoshenko beams reissner mindlin plates and three dimensional elastostatics are presented 2002 elsevier science ltd all rights reserved

truncation of nonlinear system expansions in the frequency domain the truncation of volterra series expansions is studied using the output frequency characteristics of nonlinear systems to develop a new algorithm for determining the terms to include in a volterra series expansion the results show the influence of both the generalized frequency response functions and properties of the input spectra on the significance of individual terms in the series the effectiveness of the proposed method is demonstrated using simulation studies including the analysis of a single degree of freedom mechanical oscillator nonlinear system analyses using volterra series theory must always be based on a truncated volterra series description the present study provides an effective strategy for determining which terms to include in the analysis of practical nonlinear systems based on volterra series models

fitting time series models to nonstationary processes a general minimum distance estimation procedure is presented for nonstationary time series models that have an evolutionary spectral representation the asymptotic properties of the estimate are derived under the assumption of possible model misspecification for autoregressive processes with time varying coefficients the estimate is compared to the least squares estimate furthermore the behavior of estimates is explained when a stationary model is fitted to a nonstationary process

identification of mimo non linear systems using a forward regression orthogonal estimator abstract an orthogonal least squares estimator which was originally derived for single input single output systems is extended to multi input multi output non linear systems the estimator can provide information about the structure or which terms to include in the model and final parameter estimates in a very simple and efficient manner multivariable non linear model validity tests are also discussed and the results of applying the orthogonal estimator to both simulated and real data are included

the power spectral density of response for a strongly non linear random oscillator abstract an oscillator with a non linear restoring force and a small linear damping under wide band random excitation is considered a modified van der pol transformation with a suitable amplitude dependent frequency is used to transform the original system into a first order vector system to which the stochastic averaging method sam and its higher approximations apply despite the presence of a large non linear term the averaged equations are derived in the appendix by using a two timescale approach after performing a random time change which depends on the solution an efficient equivalent linear system with random coefficients is then proposed from which the power spectral density psd is deduced the use of corrective terms yields to an approximation of the psd in the region of the third harmonic resonant frequency as well as at higher harmonics the analytical results are in excellent agreement with numerical simulations

characterization of star sign patterns that require hn abstract the refined inertia of a square real matrix b denoted ri b is the ordered 4 tuple n b n b n z b 2 n p b where n b resp n b is the number of eigenvalues of b with positive resp negative real part n z b is the number of zero eigenvalues of b and 2 n p b is the number of pure imaginary eigenvalues of b for n 3 the set of refined inertias h n 0 n 0 0 0 n 2 0 2 2 n 2 0 0 is important for the onset of hopf bifurcation in dynamical systems an n n sign pattern a is said to require h n if h n ri b b q a the star sign patterns of order n 5 that require h n are characterized more specifically it is shown that for each n 5 a star sign pattern requires h n if and only if it is equivalent to one of the five sign patterns identified in the paper

convergence analysis of volterra series response of nonlinear systems subjected to harmonic excitation volterra series provides a strong platform for non linear analysis and higher order frequency response functions however limited convergence is an inherent difficulty associated with the series and needs to be addressed rigorously prior to its application to a physical system the power series representation of the response of non linear systems subjected to harmonic excitation is investigated in this study the problem of convergence is addressed in terms of the convergence of individual frequency harmonics of the non linear response though the procedure is applicable to general polynomial form non linearity it is illustrated for a duffing oscillator subjected to harmonic excitation a general and structured series expression is obtained for amplitudes of all the response harmonics and convergence is investigated in terms of a non dimensional non linear parameter critical values of this parameter representing the upper limit of excitation level for the convergence are defined for a wide range of excitation frequencies zones of convergence and divergence of the response series are presented graphically for a range of the non dimensional non linear parameter and the number of terms included in the approximation of a response harmonic an algorithm based on ratio test is presented to compute the critical value of the non dimensional non linear parameter results obtained from the suggested algorithm are found to be in close agreement with the exact values the method gives better results compared to previous methods and has wider application in terms of excitation frequency the procedure is also investigated for a two degree of freedom system

sign patterns that require hn exist for each n 4 abstract the refined inertia of a square real matrix a is the ordered 4 tuple n n n z 2 n p where n resp n is the number of eigenvalues of a with positive resp negative real part n z is the number of zero eigenvalues of a and 2 n p is the number of nonzero pure imaginary eigenvalues of a for n 3 the set of refined inertias h n 0 n 0 0 0 n 2 0 2 2 n 2 0 0 is important for the onset of hopf bifurcation in dynamical systems we say that an n n sign pattern a requires h n if h n ri b b q a bodine et al conjectured that no n n irreducible sign pattern that requires h n exists for n sufficiently large possibly n 8 however for each n 4 we identify three n n irreducible sign patterns that require h n which resolves this conjecture

an approximate solution for the spectral response of duffing s oscillator with random input abstract an approximate method is presented for estimating the power spectral density of the response of duffing s oscillator when it is driven with gaussian white noise the approach involves calculating the expected value of the spectral response of an equivalent linear system where the equivalent natural frequency is assumed to be a random variable comparisons of the response power spectral density obtained by the present method with results obtained by numerical simulations show remarkably close agreement even for large non linearities it is found that as the amplitude of the random excitation is increased the resonant response peak in the response spectrum tends to broaden this effect results from the non linear stiffness which causes the effective natural frequency to be non deterministic when the system is subjected to random excitation

harmonic and musical wavelets the concept of a harmonic wavelet is generalized to describe a family of mixed wavelets with the structure w m n x exp i n 2 x exp i m 2 x i n m 2 x it is shown that this family provides a complete set of orthogonal basis functions for signal analysis by choosing the real numbers m and n not necessarily integers appropriately wavelets whose frequency content ascends according to the musical scale can be generated these musical wavelets provide greater frequency discrimination than is possible with harmonic wavelets whose frequency interval is always an octave an example of the wavelet analysis of music illustrates possible applications

frequency domain truncation of volterra series for weakly non linear systems subject to harmonic excitation new results for model order reduction for weakly non linear systems in the frequency domain are derived based on a parametric modelling approach

sign patterns that require or allow particular refined inertias abstract the refined inertia ri a of a real n n matrix a is the ordered 4 tuple n n n z 2 n p where n resp n n z 2 n p is the number of positive resp negative zero nonzero pure imaginary eigenvalues of a let h n 0 n 0 0 0 n 2 0 2 2 n 2 0 0 an n n sign pattern s n requires h n if h n ri a a has sign pattern s n and allows h n if h n ri a a has sign pattern s n sign patterns that require or allow h n are investigated for small values of n and for patterns with all diagonal entries negative examples are given relating these concepts to hopf bifurcation in dynamical systems

non linear systems of multiple degrees of freedom under both additive and multiplicative random excitations a quasi linearization procedure is developed for non linear systems under both additive and multiplicative white noise excitations to obtain response statistical properties including moments correlation functions and spectral densities in the proposed procedure only the system properties are linearized while all excitations are kept unchanged in particular the basic characteristics associated with the multiplicative excitations are preserved which is the key factor for the accuracy of the approach numerical examples are given and the accuracy of the procedure is substantiated by comparing the analytical results with those obtained from monte carlo simulations