Cumulant-neglect Closure Research Articles

Stochastic cumulant analysis of MDOF systems with polynomial-type nonlinearities

A new methodology is presented for formulating the equations governing the evolution of the response cumulants of MDOF nonlinear dynamic systems subjected to external delta-correlated processes. The system nonlinearities are represented by polynomial terms involving the system variables. Kronecker algebra and matrix calculus are used as efficient mathematical tools to organize the information and present the cumulant equations in a compact form. Appropriate recursive relationships are developed to relate the joint cumulants involving Kronecker powers of the response vector variables to the joint cumulants involving the original response vector variables. It is found that the differential equations governing the system of cumulants has a form similar to the state space form of equations for the original dynamic system. The state and excitation matrices describing the system of cumulants are obtained, respectively, from the state and excitation matrices describing the original dynamic system. Cumulant-neglect closure techniques, in which the joint cumulants of the response variables with order higher than a specified order are neglected, can be directly incorporated and efficiently used in the analysis to truncate the infinite hierarchy of the resulting system of cumulant equations. Examples are presented to illustrate the use of the formulation, as well as to investigate convergence and accuracy issues related to the higher-order cumulant-neglect closure scheme.

Solution of random structural system subject to nonstationary excitation: transforming the equation with random coefficients to one with deterministic coefficients and random initial conditions

This paper deals with the lower order (first four) nonstationary statistical moments of the response of linear systems with random stiffness and random damping properties subject to random nonstationary excitation modeled as white noise multiplied by an envelope function. The method of analysis is based on a Markov approach using stochastic differential equations (SDE). The linear SDE with random coefficients subject to random excitation with deterministic initial conditions are transformed to an equivalent nonlinear SDE with deterministic coefficients and random initial conditions subject to random excitation. In this procedure, new SDE with random initial conditions, deterministic coefficients and zero forcing functions are introduced to represent the random variables. The joint statistical moments of the response are determined by considering an augmented dynamic system with state variables made up of the displacement and velocity vectors and the random variables of the structural system. The zero time-lag joint statistical moment equations for the augmented state vector are derived from the Itô differential formula. The statistical moment equations are ordinary nonlinear differential equations where hierarchy of moments appear. The hierarchy is closed by the cumulant neglect closure method applied at the fourth order statistical moment level. General formulation is given for multi-degree-of-freedom (MDOF) systems and the performance of the method in problems with nonstationary excitations and large variabilities is illustrated for a single-degree-of-freedom (SDOF) oscillator.

Dynamic response of non-linear systems to poisson-distributed random impulses

The dynamic response of non-linear systems to external excitations in form of a Poisson-distributed train of random impulses is considered. The state vector of the dynamical system under consideration is a Poisson-driven Markov vector process. The equations governing its joint statistical moments are obtained by making use of a generalized Itô's differential rule. The resulting infinite hierarchy of equations is truncated with the help of a cumulant-neglect closure of different orders. For a Duffing oscillator the time histories of the mean value, the variance and the skewness coefficient of the response are evaluated analytically and compared with those obtained from Monte Carlo simulations. It is found that if the departure of the Poisson-driven excitation process from Gaussianity ranges from small to moderately large, i.e., the expected arrival rate of impulses is not very low, the application of the fourth order cumulant-neglect closure technique, i.e., neglecting the cumulants above the fourth order, yields very good estimates of the response mean values and variances.

Application of an It�-based approximation method to random vibration of a pinching hysteretic system

In this paper, an extension of the Cumulant-Neglect closure scheme is utilized for the random vibration analysis of a single degree of freedom system with a general pinching hysteresis restoring force. The hysteresis element used in the system model can simulate commonly observed forms of stiffness, strength and pinching degradations. The second order statistics of the system response to a stationary Gaussian white noise input are derived using an Ito-based approximation technique. The validity of these response statistics are then verified by comparing them to Monte Carlo simulation results. The numerical studies performed for different combinations of degradation parameters and excitation levels show that the response estimates obtained by this solution method are in good agreement with Monte Carlo simulation. These studies also indicate the applicability of this technique for response analysis of complicated forms of non-linearities.

Dynamic response of hysteretic systems to Poisson-distributed pulse trains

A single-degree-of-freedom hysteretic system subjected to a specific non-Gaussian random excitation in the form of a Poisson-distributed train of random pulses is considered. The hysteretic behaviour is described by the Bouc-Wen model of smooth hysteresis. The total hysteretic energy dissipation is assumed as a cumulative damage indicator. The state variables of the system together with the damage indicator form in that case a Poisson-driven, non-diffusive Markov vector process. Two equivalent systems are introduced by substituting the original non-analytical, non-algebraic non-linearities by equivalent linear and cubic forms in the pertinent state variables. Equations for mean responses are obtained by direct averaging of the governing equations, whereas the equations for second- and higher-order joint central moments are derived from the equivalent systems with the help of a generalized Ito's differential rule. Appearing in the equations for mean responses and for equivalent coefficients, expectations of non-algebraic functions of state variables are performed with respect to a non-Gaussian joint probability density function assumed in the form of a generalized, bivariate Gram-Charlier expansion. In the case of equivalent cubic non-linearities, the equations for moments form an infinite hierarchy which is truncated with the help of a cumulant-neglect closure technique. Mean values and variances of the response variables are evaluated by the numerical integration of equations for moments for two equivalent systems. For the sake of comparison the excitation process is also substituted by a Gaussian white noise and the usual equivalent linearization technique, combined with the Gaussian closure, is applied. The cases of non-zero-mean as well as zero-mean excitation processes are included. The case of general pulses is dealt with by a suitable augmentation of the state vector. The accuracy of the analytical techniques developed is verified against Monte Carlo simulations.

Exact and approximate response of non-linear stochastic systems

ABSTRACT The method of finding the exact solution and a new approximation method for a class of non-linear stochastic systems is presented. In both methods a non-linear transformation of the original system is applied. The idea of this method is presented for a non-linear system with stochastic parametric excitations. If the transformed system is a linear one then one can find an exact solution, otherwise the cumulant neglect closure technique is applied to this new system. The resuts obtained are illustrated by a numerical example.

Stochastic non-linear flutter of a panel subjected to random in-plane forces

—An analysis of non-linear flutter of a simply-supported panel exposed to supersonic gas flow and random in-plane forces is presented for two- and three-mode interactions. A first order quasi-steady state aerodynamic piston theory is used to model the aerodynamic loading. The Fokker-Planck equation is used to derive a general moment equation for two- and three-mode interactions. For stability analysis the moment equation is consistent and the mean square stability boundaries of the equilibrium are obtained in terms of the system parameters. The stability boundaries reveal common features to those predicted by the deterministic theory of panel nutter. For the non-linear response the moment equation is found inconsistent and a cumulant-neglect closure is used by setting cumulants of fifth and sixth orders to zero. This first order non-Gaussian closure is carried out to solve for the response statistics in terms of the air-to-plate mass ratio, aerodynamic pressure, modal damping, and in-plane random force spectral density. It is found that the non-Gaussian solution yields higher levels for the response statistics than those obtained by the Gaussian solution. The inclusion of more modes results in a reduction of the response levels and expands the stability region.

Stochastic response of hysteretic systems

This paper deals with approximate stochastic response of hyseretic structures under white noise excitation, based on Itǒ stochastic differential equations. Instead of the original system an equivalent nonlinear system is considered, in which the drift vector is given by a series expansion of the order n ⩾ 1 in terms of the state variables. n = 1 represents the well-known case of equivalent linearization. Components in the drift vector representing the nonanalytical constitutive equations are replaced by a cubic polynomial expansion. The hierarchy of statistical moment equations is closed by a cumulant neglect closure scheme. The method has been applied to a bilinear single degree-of-freedom system subjected to white noise excitation, for which an equivalent system with a cubic series expansion to the constitutive equation is considered. The results obtained are compared with those of numerical simulation and alternative methods, and they provide substantial improvements compared to equivalent linearization.

Cumulant Neglect Closure of Nonstationary Solutions of Stochastic System

The comparison of the exact nonstationary solution of stochastic dynamic system with the corresponding one obtained by the cumulant closure technique is presented in this paper. The solutions of linear system with colored noise will be considered.

Dynamic response of non-linear systems to poisson-distributed pulse trains: Markov approach

The dynamic response of non-linear systems with algebraic non-linearities to Poisson-distributed trains of impulses and general pulses is considered. The displacement and velocity response of the system form in that case a Poisson-driven Markov vector process. The differential equations governing the joint response moments are obtained by making use of a generalized Ito's differential rule which is valid for this kind of problems. Two closure techniques are used to truncate the hierarchy of moment equations: an ordinary and a modified cumulant-neglect closure technique. Transient response statistics such as the mean value and the variance are evaluated numerically. Verification of the obtained approximate analytical results against Monte Carlo simulations shows that the ordinary cumulant-neglect closure technique is appropriate in the case of non-linear systems subject to Poisson-distributed impulses and general pulses if the mean arrival rate of impulses is not very low, i.e. if the departure of the excitation from the Gaussian process is not very large. Otherwise, i.e. in the case of a low mean arrival rate of impulses, the modified cumulant-neglect closure scheme provides better results.

Reliability of hysteretic systems subjected to white noise excitation

A single-degree-of-freedom bilinear hysteretic system driven by filtered white noise is analysed by means of stochastic differential equations. Failure of the system takes place by first passages at critical levels of so-called damage indicators, which are non-decreasing response quantities measuring at a macroscopic scale the strength and stiffness deterioration. For illustration the cumulative plastic deformation ratio has been used as damage indicator. The white noise excitation is filtered through a time-invariant rational second-order filter. The analytical technique applied is based on a replacement of the original system with non-linear and non-analytical right hand sides for the constitutive equation and damage indicator differential equation by an equivalent system, where these equations are given by a cubic polynomial expansion. The expectations appearing in the equations for the coefficients of the equivalent polynomials are calculated by a multi-dimensional Gram-Charlier expansion. In order to close the infinite hierarchy of moment equations the conventional cumulant-neglect closure scheme and a modification with due consideration to the finite probability of yielding are considered. The analytical results obtained are compared to those obtained by Monte Carlo simulation, from which it is concluded that substantial improvements are obtained by applying the modified closure scheme.

On loss of accuracy and non-uniqueness of solutions generated by equivalent linearization and cumulant-neglect methods

The equivalent linearization, the Gaussian closure and the non-Gaussian cumulant-neglect closure schemes are used to analyze responses of a non-linear system with multiple potential wells under random external excitations. The resulting response statistics are compared with those obtained from Monte-Carlo simulations and exact stationary solutions to the corresponding Fokker-Planck-Kolmogorov equation. The question of uniqueness of mean-square responses for different approximation methods is also examined and discussed. The results presented show that accuracies of these approximation techniques vary depending on the nature and strength of non-linearity of the system and the intensity of excitation. For certain conditions, the exact mean-square responses are underestimated by a factor of ten or more. The Gaussian closure technique and the equivalent linearization method lead to identical results which are somewhat less accurate than those obtained by the non-Gaussian cumulant-neglect closure scheme. It is also shown that the solution generated by these techniques may not be unique.

Comparison between equivalent linearization and Gaussian closure for random vibration analysis of several nonlinear systems

Method of Equivalent Linearization as extended by Atalik and Utku, Spanos and others and the Cumulant-Neglect closure scheme originally developed by Ibrahim and Wu-Lin are utilized for the random vibration analysis of three nonlinear systems: A Duffing oscillator; A system with a set-up spring and a general hysteretic system. It is shown that the differential equations for the moments derived by this closure scheme are identical to the covariance matrix equations obtained from equivalent linearization. This comparison is made both analytically and also through numerical studies. Response statistics obtained by these two techniques are then compared with Monte Carlo simulation and/or the available exact solutions. A comparison is also made with other closure techniques. The advantages of each of the two techniques are discussed. Suggestions are made for further investigation in this area.

Response analysis of hysteretic multi‐storey frames under earthquake excitation

AbstractLocal incremental stiffness relations are formulated for a class of elasto‐plastic beam elements. The earthquake acceleration is modelled as a filtered white noise process. The Itǒ differential equations of the integrated system made up of the structural system and the excitation process are then formulated. Instead of the original system an equivalent nonlinear system is considered, in which the drift vector is given by a series expansion of order n ≧ 1, where n = 1 represents the well‐known case of equivalent linearization. Only components of the drift vector representing the non‐analytieal constitutive equations are replaced by a polynomial expansion. The coefficients of this expansion are determined from a least mean square criterion, and are sequentially updated. Especially an equivalent system with a cubic expansion to the drift vector is investigated. The hierarchy of statistical moment equations is closed by a cumulant neglect closure scheme. The method has been applied to a two‐storey frame. The results are compared to those of numerical simulation, and provide substantial improvements compared to equivalent linearization.

An Itô-based general approximation method for random vibration of hysteretic systems, part I: Gaussian analysis

The cumulant-neglect closure scheme independently developed by Ibrahim and Lin is extended for determining the stationary and non-stationary response of non-linear systems with hysteric restoring force characteristics. The method is applied to the analysis of a hysteresis and model with strength and/or stiffness degradation capabilities. This model has been studied in the past by Baber and Wen for the analysis of hysterically degrading systems using equivalent linearization. The same model has also been used for stochastic seismic performance evaluation of reinforced concrete buildings. Response statistics obtained for the model by using this closure scheme are compared with results of equivalent linearization via Monte Carlo simulation. The study performed, for a wide range of degradation parameters and input power spectral density levels, shows that the Gaussian responses obtained by this approach are identical with the linearized results. This general approximation technique, however, can provide information on higher order statistics for hysteretic systems. These non-Gaussian statistics have not been made available so far by the existing approximation techniques. In this paper the Gaussian statistics are presented.

Cumulant-Neglect Closure Method for Nonlinear Systems Under Random Excitations

The validity of the cumulant-neglect closure method is examined by applying it to a system for which an exact solution is available. A comparison of the results indicates that the Gaussian closure technique usually leads to a mean-square versus excitation strength curve which follows the same general shape as that of the exact solution but has substantial errors in some cases. The 4th order cumulant-neglect method is found to be inapplicable and to predict erroneous behavior for systems in certain parameter ranges, including a faulty prediction of a jump in response as the excitation varies through a certain critical value. On the other hand, for systems in other ranges the 4th order cumulant-neglect closure method predicts the mean square response quite well. These two parameter ranges are delineated in the paper. The 6th order cumulant-neglect closure method is also examined, leading to similar conclusions.

Cumulant-neglect closure for non-linear oscillators under random parametric and external excitations

The statistical moments of a non-linear system responding to random excitations are governed by an infinite hierarchy of equations; therefore, suitable closure schemes are needed to compute the more important lower order moments approximately. One easily implemented and versatile scheme is to set the cumulants of response variables higher than a given order to zero. This is applied to three non-linear oscillators with very different dynamic properties, and with Gaussian white noises acting as external and/or parametric excitations. It is found that the accuracy of computed second moments can be improved greatly by extending from the second order closure (Gaussian closure) to the fourth order closure, and that further refinement is unnecessary for practical purposes. Treatment of nonstationary transient response is also illustrated.