direct parameter estimation for linear and non linear structures abstract a method is presented for directly estimating the physical parameter matrices i e mass stiffness and damping of linear and non linear structures by using measured time data the structure is modelled by an equivalent lumped parameter system with symmetric parameter matrices a least squares strategy which makes use of these symmetry relations is adopted to allow the estimation of all the parameters and requires excitation at only one location the validity of the method is demonstrated by using both computer simulation and experimental procedures

direct and inverse problems encountered in vibro impact oscillations of a discrete system abstract we study direct and inverse problems that arise in the vibro impact oscillations of a discrete system specifically we examine a class of systems with two coordinates undergoing single or double sided impacts however the presented techniques are sufficiently general to apply to systems with multiple impacts the analytical methods employed are a nonlinear normal mode nnm type analysis and a boundary value problem bvp formulation and enable the computation of various branches of bifurcating periodic solutions with different impacting characteristics additional insight on the dynamics of these systems is obtained by direct integrations of the equations of motion and by numerical poincare maps it is found that the vibro impact systems considered possess rich nonlinear dynamics including vibro impact localized and nonlocalized time periodic motions complicated bifurcation structures giving rise to new types of single and double sided impacting motions mode instabilities and chaotic responses we also formulate inverse vibro impact problems whereby we seek the class of dynamical systems that produce specified orbits in the configuration plane the solutions of the inverse problems are generally non unique since they can be reduced to underdetermined sets of algebraic equations with multiple infinities of unknowns numerical applications are provided to demonstrate the techniques and validate the analytical results

technical communique who needs qp for linear mpc anyway conventional mpc uses quadratic programming qp to minimise on line a cost over n linearly constrained control moves however stability constraints often require the use of large n thereby increasing the on line computation rendering the approach impracticable in the case of fast sampling here we explore an alternative that requires a fraction of the computational cost which increases only linearly with n and propose an extension which in all but a small class of models matches to within a fraction of a percent point the performance of the optimal solution obtained through qp the provocative title of the paper is intended to point out that the proposed approach offers a very attractive alternative to qp based mpc

non reciprocity in nonlinear elastodynamics abstract reciprocity is a fundamental property of linear time invariant lti acoustic waveguides governed by self adjoint operators with symmetric green s functions the break of reciprocity in lti elastodynamics is only possible through the break of time reversal symmetry on the micro level and this can be achieved by imposing external biases adding nonlinearities or allowing for time varying system properties we present a volterra series based asymptotic analysis for studying spatial non reciprocity in a class of one dimensional 1d time invariant elastic systems with weak stiffness nonlinearities we show that nonlinearity is neither necessary nor sufficient for breaking reciprocity in this class of systems rather it depends on the boundary conditions the symmetries of the governing linear and nonlinear operators and the choice of the spatial points where the non reciprocity criterion is tested extension of the analysis to higher dimensions and time varying systems is straightforward from a mathematical point of view but not in terms of new non reciprocal physical phenomena whereas the connection of non reciprocity and time irreversibility can be studied as well finally we show that suitably defined non reciprocity measures enable optimization and can provide physical understanding of the nonlinear effects in the dynamics enabling one to establish regimes of maximum nonlinearity we highlight the theoretical developments by means of a numerical example

forced harmonic response analysis of nonlinear structures using describing functions a semianalytical quasilinear method based on the describing function formulation is proposed for the harmonic response analysis of structures with symmetrical nonlinearities the equations of motion are converted to a set of nonlinear algebraic equations the linear and nonlinear parts of the structure are dealt with separately the former being represented by the constant linear receptance matrix and the latter by the generalized quasilinear matrix which is updated at each iteration several examples dealing with cubic stiffness piecewise linear stiffness and coulomb friction type of nonlinearities are presented

spectral analysis of non linear systems involving square law operations abstract this paper contains a number of useful theoretical formulas to analyze the frequency domain properties of gaussian input data passing through non linear square law systems special bispectral density functions are defined and applied that are functions of a single variable from measurements of input data and output data only results are obtained to identify the separate frequency response functions for two models of linear systems in parallel with non linear square law systems non linear coherence functions are defined from these models which determine the proportion of the output spectrum due to the non linear operations together with ordinary coherence functions a measured output spectrum for these models can be decomposed into three components representing the linear operations the non linear operations and the remaining uncorrelated noise effects this material indicates also how to analyze other types of non linear models by employing similar techniques

normal forms for periodic systems the theory of normal forms for autonomous differential equations based on the lie bracket has proved to be an important technique in determining the behaviour of solutions of nonlinear differential equations near a stationary solution it has been widely used in the bifurcation theory for autonomous ordinary differential equations see e g 1 3 6 9 11 i in its simplest form the theory applies to an equation of the form

the cauchy problem and a generalization of the hille yosida theorem on demontre une generalisation du theoreme de hille yosida en caracterisant les generateurs d une certaine classe de semi groupes d operateurs non bornes sur un espace de banach on resout le probleme de cauchy abstrait

singularities in vibro impact dynamics abstract the non differentiable nature of vibro impact dynamics can lead to breakdown of the global stable manifold theorem applicable to smooth dynamical systems in one dimensional periodically excited system the breakdown leads to the shredding of the stable manifolds and this process is analyzed in detail bodily translation of filaments of stable manifolds can lead to homoclinic intersections and this topic is discussed in the context of a new bifurcation due to re entrance shredding leads to a strong similarity between the geometries of the singular subspaces and the geometries of homoclinic tangles and strange attracting sets

recent advances in the application of functional series to non linear structures abstract the volterra and wiener kernels of non linear systems have been studied for some time now the multidimensional fourier transforms of volterra and wiener kernels are higher order frequency response functions frf s higher order frf s can successfully be measured for non linear structural systems as is demonstrated these can predict aspects of the system behavior unobtainable from measurements of first order frf s by considering the measured frf s of a structure as volterra kernel transforms it is possible to curve fit a non lindear multi degree of freedom parametric model to frf data analysis techniques for linear structures of measuring and curve fitting to first order frf s can thus be extended to non linear systems by the measurement of second and third order frf s

non similar normal oscillations in a strongly non linear discrete system abstract the free oscillations of a strongly non linear discrete oscillator are examined by computing its non similar non linear normal modes these are motions represented by curves in the configuration space of the system and they are not encountered in classical linear vibration theory or in existing non linear perturbation techniques the mikhlin manevich asymptotic methodology is used for solving the singular functional equation describing the non similar modes and approximate analytical expressions are derived for an oscillator with weak coupling stiffness and mistuning both localized and non localized modes are detected occurring in small neighborhoods of degenerate and global similar modes of the tuned system when strong coupling is considered only non localized modes were found to exist an interesting result of this work is the detection of mode localization in the tuned periodic system a result with no counterpart in existing theories on linear mode localization as a check of the analytical results numerical integrations of the equations of motion were carried out and the existence of the theoretically predicted non similar modes was verified

on the orthogonalised reverse path method for nonlinear system identification abstract the problem of obtaining the underlying linear dynamic compliance matrix in the presence of nonlinearities in a general multi degree of freedom mdof system can be solved using the conditioned reverse path crp method introduced by richards and singh 1998 journal of sound and vibration 213 4 pp 673 708 the crp method also provides a means of identifying the coefficients of any nonlinear terms which can be specified a priori in the candidate equations of motion although the crp has proved extremely useful in the context of nonlinear system identification it has a number of small issues associated with it one of these issues is the fact that the nonlinear coefficients are actually returned in the form of spectra which need to be averaged over frequency in order to generate parameter estimates the parameter spectra are typically polluted by artefacts from the identification of the underlying linear system which manifest themselves at the resonance and anti resonance frequencies a further problem is associated with the fact that the parameter estimates are extracted in a recursive fashion which leads to an accumulation of errors the first minor objective of this paper is to suggest ways to alleviate these problems without major modification to the algorithm the results are demonstrated on numerically simulated responses from mdof systems in the second part of the paper a more radical suggestion is made to replace the conditioned spectral analysis which is the basis of the crp method with an alternative time domain decorrelation method the suggested approach the orthogonalised reverse path orp method is illustrated here using data from simulated single degree of freedom sdof and mdof systems

the multiple load method for measuring the source characteristics of time variant sources abstract a new measurement method the multiple load method for determining the source characteristics of linear time variant sources is presented the results of this measurement method are compared to the results of the two load method for a linear time invariant source the measurement methods are tested on a modified compressor the source characteristics measured by using the multiple load method gave a good prediction of the pressure at the outlet of a compressor the result was better when using the multiple load method than when using the two load method different methods have been used to study and limit the errors occurring in the measurement and calculation of the source characteristics

optimizing prediction dynamics for robust mpc a convex formulation is derived for optimizing dynamic feedback laws for constrained linear systems with polytopic uncertainty we show that when it exists the maximal invariant ellipsoidal set for the plant state under a dynamic feedback law incorporating any chosen static feedback gain is equal to the maximal invariant ellipsoidal set under any linear feedback law the dynamic controller and its associated invariant set define a computationally efficient robust model predictive control mpc law with prediction dynamics belonging to a polytopic uncertainty set

a periodically forced piecewise linear oscillator a single degree of freedom non linear oscillator is considered the non linearity is in the restoring force and is piecewise linear with a single change in slope such oscillators provide models for mechanical systems in which components make intermittent contact a limiting case in which one slope approaches infinity an impact oscillator is also considered harmonic subharmonic and chaotic motions are found to exist and the bifurcations leading to them are analyzed

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

time optimal model predictive control for linear systems based on ellipsoidal and polytopic sets time optimal model predictive control mpc strategies are investigated in this paper we first derive a series of controllable sets described by ellipsoids or convex polytopes offline then obtain

analysis of critical and post critical behaviour of non linear dynamical systems by the normal form method part ii divergence and flutter abstract the normal form method nfm presented in part i 1 is applied to the analysis of systems near two basic bifurcation conditions divergence and flutter the reduced systems and the normalizing transformation obtained by means of a computer routine implementing the normalization formulae of part i are utilized to determine a the stability of the bifurcating singular point b the local equilibrium paths associated with divergence bifurcation and c the amplitude and frequency or more generally the waveform of post flutter oscillations

normal modes of vibration for non linear continuous systems a definition and a constructive methodology for normal nodes of motion are developed for a class of vibratory systems the dynamics of which are governed by non linear partial differential equations the definition for normal modes is given in terms of the dynamics on two dimensional invariant manifolds in the system phase space these manifolds are a continuation of the planes which represent the well known normal modes of the linearized system a local asymptotic approximation of the geometric structure of these manifolds can be obtained by using an approach which follows that used for generating center manifolds the procedure also provides the non linear ordinary differential equations which govern individual modal dynamics and a physical description of the system configuration when it is undergoing a modal motion in this paper the general theory is described for the application of vibrations of continuous media but it can easily be extended to other situations in order to demonstrate the power of the approach and its unique procedural aspects three examples involving beam vibrations are worked out in detail the examples are conservative simply supported beams the first demonstrates the methodology for a linear beam model the second is a beam on a non linear elastic foundation and the third example is a beam with non linear torsional springs attached at each end in these examples the simplicity of the mode shapes of the linearized model yields relatively simple calculations that do not obscure the important features of the procedure

on the use of the auto bispectral density for detecting quadratic nonlinearity in structural systems abstract higher order spectra appear often in the analysis and identification of nonlinear systems the auto bispectral density is one example of a higher order spectrum and may be used in the analysis of stationary structural response data to detect the presence of certain types of structural nonlinearities in this work a closed form expression for the auto bispectral density derived previously by the authors is used to find the bispectral frequency most sensitive to the nonlinearity the properties of nonlinearity detectors based on estimates of the magnitude of the auto bispectral density at this frequency are then explored estimates of the auto bispectral density are obtained using the direct method based on the discrete fourier transform the bias associated with this estimator is derived here and combined with previously derived expressions for the estimator variance to give both type i and type ii errors for the detector detector performance is quantified using a receiver operating characteristic roc curve illustrating the trade off between false positives type i error and power of detection 1 0 type ii error theoretically derived roc curves are compared to those obtained via numerical simulation and show excellent agreement results are presented for different levels of nonlinearity in both the stiffness and damping terms for a spring mass system possible consequences are discussed with regard to the detection of damage induced nonlinearities in structures

analysis of non linear dynamical systems by the normal form theory abstract a method is proposed for calculating the periodic solutions of non linear mechanical systems with analytical non linearities the jordan normalization procedure for the case of non linear autonomous systems is described and generalized to dampened harmonically excited oscillators non linear modes for hamiltonian systems are introduced normal forms simplify the analysis of bifurcation it is shown how to extend the modal synthesis procedure the proposed non linear modes obtained from free vibrations are used to construct a superposition technique to describe the forced response of harmonically excited systems these results are tested for one and two degrees of freedom systems with cubic non linearities the results are compared with expressions obtained by classical analytical methods averaging or multiple scales methods or runge kutta numerical methods

center manifolds normal forms and bifurcations of vector fields with application to coupling between periodic and steady motions abstract we study the instabilities known to aeronautical engineers as flutter and divergence mathematically these states correspond to bifurcations to limit cycles and multiple equilibrium points in a differential equation making use of the center manifold and normal form theorems we concentrate on the situation in which flutter and divergence become coupled and show that there are essentially two ways in which this is likely to occur in the first case the system can be reduced to an essential model which takes the form of a single degree of freedom nonlinear oscillator this system which may be analyzed by conventional phase plane techniques captures all the qualitative features of the full system we discuss the reduction and show how the nonlinear terms may be simplified and put into normal form invariant manifold theory and the normal form theorem play a major role in this work and this paper serves as an introduction to their application in mechanics repeating the approach in the second case we show that the essential model is now three dimensional and that far more complex behavior is possible including nonperiodic and chaotic motions throughout we take a two degree of freedom system as an example but the general methods are applicable to multi and even infinite degree of freedom problems

the degradation of higher order spectral detection using narrowband processing in certain scenarios higher order statistic hos based detectors exploiting the gaussian noise rejection property of hos have been shown to have performances superior to widely used second order sos techniques however a crucial limitation on these detectors that has not been addressed is a constraint on the processing bandwidth we study the effects of commonly used narrowband processing on these detectors as the processing bandwidth is decreased we characterize the tradeoff between hos and sos detectors and demonstrate that the performance of hos detectors degrades faster than that of sos ones we then consider distributed wideband detection by a bank of local narrowband detectors whose decisions are fused together given a fixed bandwidth input signal we show that as the number of local detectors increases corresponding to a decreasing amount of bandwidth for each local detector the performance of such a scheme using hos local detectors degrades more quickly than a scheme using sos ones

dynamic testing of nonlinear vibrating structures using nonlinear normal modes modal testing and analysis is well established for linear systems the objective of this paper is to progress toward a practical experimental modal analysis ema methodology of nonlinear mechanical structures in this context nonlinear normal modes nnms offer a solid theoretical and mathematical tool for interpreting a wide class of nonlinear dynamical phenomena yet they have a clear and simple conceptual relation to the classical linear normal modes lnms a nonlinear extension of force appropriation techniques is developed in this study in order to isolate one single nnm during the experiments with the help of time frequency analysis the energy dependence of nnm modal curves and their frequencies of oscillation are then extracted from the time series the proposed methodology is demonstrated using two numerical benchmarks a two degree of freedom system and a planar cantilever beam with a cubic spring at its free end

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

robust constrained model predictive control using linear matrix inequalities abstract the primary disadvantage of current design techniques for model predictive control mpc is their inability to deal explicitly with plant model uncertainty in this paper we present a new approach for robust mpc synthesis that allows explicit incorporation of the description of plant uncertainty in the problem formulation the uncertainty is expressed in both the time and frequency domains the goal is to design at each time step a state feedback control law that minimizes a worst case infinite horizon objective function subject to constraints on the control input and plant output using standard techniques the problem of minimizing an upper bound on the worst case objective function subject to input and output constraints is reduced to a convex optimization involving linear matrix inequalities lmis it is shown that the feasible receding horizon state feedback control design robustly stabilizes the set of uncertain plants several extensions such as application to systems with time delays problems involving constant set point tracking trajectory tracking and disturbance rejection which follow naturally from our formulation are discussed the controller design is illustrated with two examples

baseband volterra filters for implementing carrier based nonlinearities a diagonal coordinate representation for volterra filters is developed and exploited to derive efficient volterra filter implementations for processing carrier based input signals in the diagonal coordinate representation the output is expressed as a sum of linear filters applied to modified input signals hence linear filtering methods are employed to implement the nonlinear filter on a baseband version of the input downsampling is then used to reduce computational complexity the same approach is employed to develop efficient implementations for processing continuous time carrier based signals pulse amplitude modulated signals and frequency division multiplexed input signals

exponentially bounded c semigroups and generation of semigroups this paper concerns exponentially bounded c semigroups and semigroups of operators in a banach space x recently davies and pang l introduced the notion of an exponentially bounded c semigroup and characterized the generator of an exponentially bounded c semigroup on the other hand miyadera 7 discussed the complete infinitesimal generator of an exponentially bounded c semigroup and tanaka 14 has characterized the complete infinitesimal generator of an exponentially bounded c semigroup there are many important works on semigroups of linear operators for example see 4 5 12 161 extending the notion of c semigroups oharu 9 introduced the semigroups of classes cc k 0 1 2 and da prato 2 and sobolevskii 13 introduced the semigroups of growth order c 2 0 generation theorems for semigroups of these classes have been studied in 2 9 10 131 our first purpose is to clarify the relationship between semigroups and exponentially bounded c semigroups it is shown that if t t t 2 0 is a semigroup and c is an injective bounded linear operator with dense range satisfying r t c ct t for t 0 and r c c c xex lim r o t r x x then ct t t 2 0 is an exponentially bounded c semigroup whose complete infinitesimal generator is equal to that of t t t 2 0 in particular if r t t k 0 is a semigroup of class c or of growth order 1 then ct t t 2 o becomes an exponentially bounded c semigroup with a suitable c and its complete infinitesimal generator coincides with that of t t t 2 o our second and main purpose is to study the generation of semigroups via the theory of exponentially bounded c semigroups we shall characterize the complete infinitesimal generator of a semigroup r t t 2 0

decoupling nonlinear systems noncommutative generating power series and lie algebras the joint use of noncommutative generating power series and lie algebras gives an algebraic approach for decoupling problem of nonlinear systems we produce a coherent decoupling method using the additional notion of the immersion of a system into another a concept which makes clear the fact that two systems have a same input output behaviour thus a judicious choice of feedback which gives decoupling is able to immerse a system into a system defined on mathbb r n with eta depending on the characteristic numbers of the system

analysis of nonlinear oscillators using volterra series in the frequency domain the volterra series representation is a direct generalisation of the linear convolution integral and has been widely applied in the analysis and design of nonlinear systems both in the time and the frequency domain the volterra series is associated with the so called weakly nonlinear systems but even within the framework of weak nonlinearity there is a convergence limit for the existence of a valid volterra series representation for a given nonlinear differential equation barrett 1965 1 proposed a time domain criterion to prove that the volterra series converges within a given region for a class of nonlinear systems with cubic stiffness nonlinearity in this paper this time domain criterion is extended to the frequency domain to accommodate the analysis of nonlinear oscillators subject to harmonic excitation a common and severe nonlinear phenomenon called jump a behavior associated with the duffing oscillator and the multi valued properties of the response solution is investigated using the new frequency domain criterion of establishing the upper limits of the nonlinear oscillators to predict the onset point of the jump and the volterra time and frequency domain analysis of this phenomenon are carried out based on graphical and numerical techniques

separation of variables in the nonlinear wave equation we develop a technique making it possible to handle the problem of separation of variables in nonlinear differential equations using it we obtain a number of new two dimensional nonlinear wave equations admitting separation of variables and construct their exact solutions

matching of local expansions in the theory of non linear vibrations abstract linear systems are a generalization of normal principal vibrations of linear systems 1 3 in this case all position co ordinates can be defined well from any one of them r m rosenberg is credited with being the first to introduce broad classes of conservative systems allowing normal vibrations with rectilinear trajectories in a configurational space in systems of a more general type trajectories of normal vibrations are curvilinear assume that in a conservative system the potential energy is a positively definite polynomial in the co ordinates at small amplitudes a linear system is to be selected as the initial one while at large amplitudes a homogeneous non linear system allows normal vibrations with rectilinear trajectories in the vicinity of a linear system trajectories of normal vibrations can be determined as power series in the amplitude while in the vicinity of a homogeneous non linear system they can be determined as power series in the inverse amplitude in order to join together local expansions and to investigate the behavior of normal vibration trajectories at arbitrary amplitude values fractional rational diagonal pade approximants are used necessary conditions for the convergence of a succession of pade approximants have been obtained and that allows one to establish relations between quasi linear and essentially non linear expansions that is to decide which of them correspond to the same solution and which to different ones additional modes of vibrations exist only in a non linear systems as the amplitude decreases they vanish at a certain limiting point

nonlinear structural identification by the reverse path spectral method abstract when dealing with nonlinear dynamical systems it is important to have efficient accurate and reliable tools for estimating both the linear and nonlinear system parameters from measured data an approach for nonlinear system identification widely studied in recent years is reverse path this method is based on broad band excitation and treats the nonlinear terms as feedback forces acting on an underlying linear system parameter estimation is performed in the frequency domain using conventional multiple input multiple output or multiple input single output techniques this paper presents a generalized approach to apply the method of reverse path on continuous mechanical systems with multiple nonlinearities the method requires few spectral calculations and is therefore suitable for use in iterative processes to locate and estimate structural nonlinearities the proposed method is demonstrated in both simulations and experiments on continuous nonlinear mechanical structures the results show that the method is effective on both simulated as well as experimental data

the bispectrum and bicoherence for quadratically nonlinear systems subject to non gaussian inputs in the analysis of data from nonlinear systems both the bispectrum and the bicoherence have emerged as useful tools both are frequently used to detect the influence of a nonlinear system on the joint probability distribution of the system input previous work has provided an analytical expression for the bispectrum of a quadratically nonlinear system output if the input is stationary jointly gaussian distributed this work significantly generalizes the previous analysis by providing an analytical expression for the bispectrum of the response of quadratically nonlinear systems subject to stationary jointly non gaussian inputs possessing arbitrary auto correlation function the expression is then used to determine the optimal input probability density function for detecting a quadratic nonlinearity in a second order system it is also shown how the expression can be used to design an optimal nonlinear filter for detecting deviations from normality in the probability density of a signal

statistical errors for non linear system measurements involving square law operations abstract this paper develops normalized random error formulas for special bispectra estimates and associated frequency response function estimates in finite memory square law systems error formulas are also derived for output spectrum estimates from these non linear systems and for associated non linear coherence functions these formulas are useful to evaluate such measured non linear results as well as to design experimental programs

multiplicative perturbations of c regularized semigroups we present in this note two new theorems for multiplicative perturbations of c regularized semigroups which generalize the previous related ones for c0 semigroups

a comparison of optimized higher order spectral detection techniques for non gaussian signals using the gaussian noise rejection property of higher order spectra hos hos based detectors have been proposed that outperform conventional second order techniques in certain scenarios based on statistical tests proposed by suhha rao and gabr 1980 as well as hinich 1982 kletter and messer 1990 and hinich and wilson 1990 have developed similar bifrequency domain detectors that are dependent on bispectral estimates of the observation process formalizing the estimate consistency requirements and the asymptotics for these detectors we derive a new f test statistic we consider the detrimental effects of using spectral estimates in the denominator of hinich s test we determine refined conditional distributions for third and fourth order versions of his detector we also modify his test for colored scenarios extending the bispectral detectors to their kth order counterparts we calculate the optimal smoothing bandwidth to use in constructing the hos estimates producing the best detection performances for both our f test and hinich s test with our refined distributions these new bandwidths yield significant improvements in detector performance over previous results for the finite sample case our calculations characterize the tradeoff between the two detectors and demonstrate that a larger smoothing bandwidth than the one suggested by previous researchers should be used our calculations are verified using simulations for both white and colored cases

identification of multi degree of freedom non linear systems under random excitations by the reverse path spectral method abstract conventional frequency response estimation methods such as the h 1 and h 2 methods often yield measured frequency response functions which are contaminated by the presence of non linearities and hence make it difficult to extract underlying linear system properties to overcome this deficiency a new spectral approach for identifying multi degree of freedom non linear systems is introduced which is based on a reverse path formulation as available in the literature for single degree of freedom non linear systems certain modifications are made in this article for a multi degree of freedom reverse path formulation that utilizes multiple input multiple output data from non linear systems when excited by gaussian random excitations conditioned h c 1 and h c 2 frequency response estimates now yield the underlying linear properties without contaminating effects from the non linearities once the conditioned frequency response functions have been estimated the non linearities which are described by analytical functions are also identified by estimating the coefficients of these functions identification of the local or distributed non linearities which exist at or away from the excitation locations is possible the new spectral approach is successfully tested on several example systems which include a three degree of freedom system with an asymmetric non linearity a three degree of freedom system with distributed non linearities and a five degree of freedom system with multiple non linearities and multiple excitations

detection of non gaussian signals in non gaussian noise using the bispectrum the problem of detecting a non gaussian time series in the presence of additive gaussian or non gaussian noise is cast into a classical hypothesis testing framework using the sample bispectrum as the test statistic the power of the test is demonstrated as a function of signal to noise ratio the degree of skewness of the signal and processing parameters the results are compared to the power of a classical energy detection test it is concluded that the bispectrum can be used effectively to detect non gaussian signals in the presence of interfering noise and that it may perform better depending on the degree of non gaussianity than energy detection

a bayesian approach to identifying structural nonlinearity using free decay response application to damage detection in composites this work discusses a bayesian approach to approximating the distribution of parameters governing nonlinear structural systems specifically we use a markov chain monte carlo method for sampling the posterior parameter distributions thus producing both point and interval estimates for parameters the method is first used to identify both linear and nonlinear parameters in a multiple degree of freedom structural systems using free decay vibrations the approach is then applied to the problem of identifying the location size and depth of delamination in a model composite beam the influence of additive gaussian noise on the response data is explored with respect to the quality of the resulting parameter estimates

input output parametric models for non linear systems part ii stochastic non linear systems recursive input output models for non linear multivariate discrete time systems are derived and sufficient conditions for their existence are defined the paper is divided into two parts the first part introduces and defines concepts such as nerode realization multistructural forms and results from differential geometry which are then used to derive a recursive input output model for multivariable deterministic non linear systems the second part introduces several examples compares the derived model with other representations and extends the results to create prediction error or innovation input output models for non linear stochastic systems these latter models are the generalization of the multivariable arm ax models for linear systems and are referred to as narmax or non linear autoregressive moving average models with exogenous inputs

ellipsoidal techniques for reachability analysis internal approximation for the reach tube of a linear time varying system with ellipsoidal bounds on the control variable consider the following approximation problem find a tight ellipsoid valued tube inside the reach tube that touches it along any prespecified smooth curve on the boundary we construct this ellipsoidal tube and show that if the given curve is itself a system trajectory the tube can be calculated recursively with minimum computational burden

global dynamics of a vibro impacting linear oscillator abstract the steady state vibro impacting responses of one dimensional harmonically excited linear oscillators are studied by using a modern dynamical systems approach allied with numerical simulation the steady state motions are attracting sets in the system phase space and capture initial conditions in their domains of attraction unlike the free harmonically excited oscillator the phase space of a vibro impacting system may be inhabited by many attracting sets for example there are sub harmonic multi impact periodic orbits and chaotic steady state responses in order to build a qualitative understanding of vibro impact response an attempt is made to build generic topological models of their phase spaces for physically significant parameter ranges use is made of the poincare section or stroboscopic mapping technique essentially following an initial impact forwards or backwards in time to subsequent or previous impacts using a computer the qualitative understanding gained from the analysis and simulations is discussed in an engineering context

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

an energy balance approach for oscillator parameter identification an energy based approach for parametric nonlinear identification was investigated the presented method uses an energy balance on the oscillator governing equations for identification purposes thus requiring the availability of the position and velocity signals since it is rarely practical to measure every state variable in an experimental setting we describe an alternative procedure for estimating velocity from the measured displacement the presented approach uses cubic smoothing splines to avoid the noise amplification effect that occurs for numerical signal derivatives finally we investigate the identification of parameters from both numerical and experimental data for three nonlinear oscillators these studies demonstrate the effectiveness of the presented energy balance approach for transient periodic and chaotic response behavior

a numerical study of an impact oscillator with the addition of dry friction abstract this paper presents an investigation into the non linear behaviour of an impact oscillator with the addition of dry friction the equations that govern the relative motion of the impact oscillator including the effects of dry friction are formulated resulting in a non conservative piecewise linear ordinary differential equation of motion the system is excited by a single component sinusoidal base motion a computer model is used to numerically simulate the motion of the system and to identify periodic solutions and the stability of these solutions trends of the periodic solutions are then interpreted using bifurcation theory to identify the non linear behaviour of this system as a function of both the excitation amplitude and the excitation frequency for two levels of the dry friction force these results illustrate a rich behaviour that includes turning point bifurcations symmetry breaking pitchfork bifurcation period doubling bifurcation cascades and bifurcations that can be characterized as boundary grazing the results also include the existence of a subharmonic resonance motion that encompasses three periods of the forcing during each period of the solution also presented are samples of time histories and phase protraits and a plot of the stable response regions for clamping force as a function of excitation amplitude

feasibility of identifying non linear vibratory systems consisting of unknown polynomial forms abstract system identification techniques for non linear systems may require a priori knowledge of the nature and mathematical form of the non linearities however for practical systems this is not always possible as a result non linearities are often approximated and questions remain as to whether a reasonably accurate model can be determined concurrently under experimental conditions some means of quantifying the amount of measurement noise present in the identification process must also be obtained to resolve such issues a discrete non linear system problem is formulated in the presence of uncorrelated noise and critically examined from the standpoint of identification coherence functions are introduced which are based on a reverse path spectral approach recently developed by the authors for multi degree of freedom systems these coherence functions as calculated from conditioned spectra indicate the extent of uncorrelated noise present and the accuracy of assumed mathematical models employed for describing non linear systems using several example simulation systems including a system with a continuous non linearity described by a non integer exponent both temporal and spectral identification techniques are employed to study the issues described above

analysis of critical and post critical behaviour of non linear dynamical systems by the normal form method part i normalization formulae abstract a method based on normal form theory is presented to study the dynamical behaviour of a system in the neighbourhood of a nearly critical equilibrium state associated with a bifurcation condition explicit formulae for the normalization procedure are derived these formulae can be numerically programmed avoiding usual complicated algebraic calculations and making the method effectively applicable for n dimensional systems rather general bifurcations can be included e g non linear flutter hopf bifurcation divergence and internal resonance

a simple criterion for establishing an upper limit to the harmonic excitation level of the duffing oscillator using the volterra series abstract high order frequency response functions based on the volterra series are employed to represent the input output characteristics of the duffing oscillator subject to sinusoidal excitation from these a series representation of a first order frequency response function of the non linear system is obtained and used to derive a simple criterion which provides an estimate of the magnitude of the input sine wave above which the predicted response begins to diverge the criterion results in a small overestimate of the magnitude of the input force excitation at the undamped natural frequency condition although the derived criterion strictly applies only to the duffing oscillator the procedure can be applied generally and is extended to a two degree of freedom system with a non linear cubic stiffness producing a similar criterion to that of the single degree of freedom system

robust time optimal control of constrained linear systems a version of dynamic programming which computes level sets of the value function rather than the value function set itself is used to design robust non linear controllers for linear discrete time dynamical systems subject to hard constraints on controls and states the controller stabilizes the system and steers all trajectories emanating in a prescribed set to a control invariant set in minimum time for the robust regulator problem the control invariant terminal set is a neighborhood preferably small of the origin for the robust tracking problem the control invariant terminal set is a neighborhood of the invariant set in which the tracking error is zero two non linear controllers which utilize the level sets of the value function are described the first requires the controller to solve on line a modest linear program whose dimension is approximately the same as that of the control variable the second decomposes each level set into a set of simplices a piecewise linear control law affine in each simplex is then constructed

nonlinear system analysis and identification from random data 1991 nonlinear system analysis and identification from random data technometrics vol 33 no 4 pp 482 483

amplitude equations for systems with competing instabilities we describe how to derive ordinary differential equations to predict the time dependence of a system governed by partial differential equations when that system is near to the polycritical condition for the onset of several instabilities the method is illustrated for the general case of two competing instabilities and then a specific physical example of this case is worked out in detail the basic idea is to extend the krylov bogolyubov mitropolsky method of nonlinear mechanics to bifurcation problems

periodic nonlinear waves on a half line nontrivial solutions of the equationutt uxx g u which are 2 periodic int and which decay asx are shown to exist ifg a 0 andg 0 1 breather like solutions which also decay asx can be interpreted as homoclinic solutions in thex dynamics their existence is still in question for generalg

non linear normal modes and invariant manifolds small amplitude motions of dynamic systems structural fluid control etc about an equilibrium state are modeled by linear differential equations which have constant coefficients these are typically obtained by a taylor series expansion of the forces about the equilibrium point under quite general circumstances these equations admit a set of special solutions called normal mode motions in which each system component moves with the same frequency and with a fixed ratio amongst the displacements of the components for a conservative system for a non conservative system all displacements and velocities are linearly related to a single displacement velocity pair

optimal bispectral detection of weak quadratic nonlinearities in structural systems abstract this work derives the expressions for the bispectrum and bicoherence functions for multi degree of freedom spring mass systems with quadratic nonlinearities subject to inputs described by a wide class of random processes the derivation uses a volterra series model for the system response and yields expressions for both homogeneous spectra where the output of only one degree of freedom is used and mixed spectra where the bispectral density contains multiple response time series this expression is then used to determine the optimal probability distribution of the input and the optimal bispectrum to compute for the goal of maximizing the probability of detecting the nonlinearity

incomplete iterated cauchy problems abstract we treat numerous higher order incomplete abstract cauchy problems by writing them in the form 1 1 k 1 n d dt a k u t 0 t 0 2 2 ui 1 0 xi 1 i j 3 3a ui 1 t bounded 1 i n j or 4 3b limt u i 1 t 0 1 i n j the abstract cauchy problem 1 2 and 3a is wellposed in a certain sense when a1 aj generate bounded c semigroups and aj 1 an generate uniformly stable c semigroups for 1 2 and 3b we need a1 aj and aj 1 an to generate stable c semigroups

non periodic motion caused by grazing incidence in an impact oscillator the motion of a single degree of freedom periodically forced oscillator subjected to a rigid amplitude constraint is considered using analytical methods the singularities caused by grazing impact are studied it is shown that as a stable periodic orbit comes to grazing impact under the control of a single parameter a special type of bifurcation occurs the motion after the bifurcation may be non periodic and a criterion for this based on orientation and eigenvalues is given

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

efficient robust predictive control predictive constrained control of time varying and or uncertain linear systems has been effected through the use of ellipsoidal invariant sets kothare et al 1996 linear matrix inequalities lmis have been used to design a state dependent state feedback law that maintains the state vector inside invariant feasible sets for the purposes of prediction however at each time instant the state feedback law is assumed constant in addition due to the large number of lmis involved online computation becomes intractable for anything other than small dimensional systems here we propose an approach that deploys a fixed state feedback law but introduces extra degrees of freedom through the use of perturbations on the fixed state feedback law the problem is so formulated that all demanding computations can be performed offline leaving only a simple optimization problem to be solved online over and above the very significant reduction in computational cost the extra degrees of freedom allow for better performance and wider applicability

decomposition of wave forces into linear and non linear components abstract this paper details a methodology for analyzing non linear systems involving a square law operation with sign the analysis is applied specifically to the problem of decomposing random wave forces on an ocean structure into their inertial linear and drag non linear components a procedure is presented for identifying the individual inertial and drag force parameters based solely upon measurements of the input wave velocity and the output wave force where the wave velocity has an arbitrary spectral density function and mean value and the inertial and drag forces have an arbitrary frequency dependence and phase it is assumed only that the wave velocity is a gaussian random process experimental verifications of the analysis procedure are presented

aeroelastic response of nonlinear wing sections using a functional series technique this paper addresses the problem of the determination of the subcritical aeroelastic response and flutter instability of nonlinear two dimensional lifting surfaces in an incompressible flow field via indicial functions and volterra series approach the related aeroelastic governing equations are based upon the inclusion of structural and damping nonlinearities in plunging and pitching of the linear unsteady aerodynamics and consideration of an arbitrary time dependent external pressure pulse unsteady aeroelastic nonlinear kernels are determined and based on these frequency and time histories of the subcritical aeroelastic response are obtained and in this context the influence of the considered nonlinearities is emphasized conclusions and results displaying the implications of the considered effects are supplied

bifurcations and chaos in autonomous self excited oscillators with impact damping dynamics and bifurcations of a class of single degree of freedom self excited oscillators with an impact damper have been studied the basic motivation is to determine a persistent birfurcation structure of co dimension one which is independent of the exact functional form of the self excitation mechanism in the process an analytical method for studying the qualitative dynamical features is presented a new numerical scheme for analyzing the eigenproperties of a solution is developed a detailed numerical study revealing the superstructure of the bifurcation phenomena has been performed

modal analysis of mechanical systems with impact non linearities limitations to a modal superposition this paper presents an attempt to generalize the modal superposition formula to mechanical systems with impact type non linearities following the procedure introduced for smooth non linearities the study is restricted to simple one and two degree of freedom systems with a unilateral constraint on one of the degrees of freedom for which the response can be analytically determined generalized frequencies modes and masses are built in the procedure the results obtained for various sets of parameters point out some limitations to the validity of a general modal superposition formula

non linear parameter estimation with volterra series using the method of recursive iteration through harmonic probing volterra series provides a platform for non linear response representation and definition of higher order frequency response functions frfs it has been extensively used in non parametric system identification through measurement of first and higher order frfs a parametric system identification approach has been adopted in the present study the series response structure is explored for parameter estimation of polynomial form non linearity first and higher order frequency response functions are extracted from the measured response harmonic amplitudes through recursive iteration relationships between higher order frfs and first order frf are then employed to estimate the non linear parameters excitation levels are selected for minimum series approximation error and the number of terms in the series is controlled according to convergence requirement the problem of low signal strength of higher harmonics is investigated and a measurability criterion is proposed for selection of excitation level and range of excitation frequency the procedure is illustrated through numerical simulation for a duffing oscillator robustness of the estimation procedure in the presence of measurement noise is also investigated

impact oscillations linear theory of stability and bifurcations abstract a unified approach to analysis of impact oscillations is proposed its mathematical essence is the continuous representation of the impulsive motion in some auxiliary variables as a result an explicit formula for the fundamental matrix is obtained which allows one to construct the characteristic equation then the stability and bifurcations can be studied by the usual techniques the exceptional case of bifurcations related to grazing impacts is investigated by taking into account a non zero impact duration due to this approach linearization is also possible and the complex bifurcation becomes a sequence of ordinary ones some algebraic conditions are obtained which allow one to determine the type of the resulting motion a mechanical example is considered a disc with an offset centre of gravity bouncing on an oscillating surface