Cumulant-neglect Closure Research Articles

Probability density function of roll amplitude for parametric rolling using moment equation

For the ship stability estimation and ship design, it is helpful to know the probability of the roll amplitude exceeding a certain threshold. Therefore, it is necessary to obtain the probability density function of the roll amplitude. In this study, first, we derive the moment values of roll amplitude by combining the moment equations and the linearity of expectation. With this, we propose a method to estimate the probability density function of the roll amplitude by using the obtained moment values. The results of our proposed method are compared to those obtained from Monte Carlo simulations. When the higher-order cumulant neglect closure method is used, the moment values resulting from the moment equations are close to the results of corresponding Monte Carlo simulations. In addition, our proposed method for deriving the probability density function of the roll amplitude is validated by comparison with Monte Carlo simulation results. In conclusion, we can state that the proposed methods for deriving the moments and the probability density function of the roll amplitude are available for practical use cases.

Application of linear filter and moment equation for parametric rolling in irregular longitudinal waves

Parametric rolling is one of the dangerous dynamic phenomena. In order to discuss the safety of a vessel when a dangerous phenomenon occurs, it is important to estimate the probability of certain dynamical behavior of the ship with respect to a certain threshold level. In this paper, the moment values are obtained by solving the moment equations. Since the stochastic differential equation(SDE) is needed to obtain the moment equations, the Autoregressive Moving Average(ARMA) filter is used. The effective wave is modeled by using the 6th-order ARMA filter. In addition, the parametric excitation process is modeled by using a non-memory transformation obtained from the relationship between GM and wave elevation. The resulting system of equations is represented by the 8th-order It\^o stochastic differential equation, which consists of a second-order SDE for the ship motion and a 6th-order SDE for the effective wave. This system has nonlinear components. Therefore, the cumulant neglect closure method is used as higher-order moments need to be truncated. Furthermore, the probability density function of roll angle is determined by using moment values obtained from the SDE and the moment equation. Here, two types of the probability density function are suggested and have a good agreement.

Moment equations and cumulant-neglect closure techniques for non-linear dynamic systems under renewal impulse process excitations

The moment equations technique together with modified cumulant-neglect closure techniques is developed for a non-linear dynamic system subjected to a random train of impulses driven by an Erlang renewal counting process. The original non-Markov problem is converted into a Markov one by recasting the excitation process with the aid of an auxiliary, pure-jump stochastic process characterized by a Markov chain. Hence the conversion is carried out at the expense of augmentation of the state space of the dynamic system by Markov states of the auxiliary jump process. The problem is characterized by the set of joint probability density — discrete distribution function, which are the joint probabilities of the original state vector and of the Markov chain being in the particular jth state. Accordingly the statistical moments of the state variables are defined as integrals with respect to the mixed-type, probability density — discrete distribution function. The differential equations for moments are obtained with the aid of the forward integro-differential Chapman–Kolmogorov operator (Iwankiewicz, 2014). Two modified closure techniques are developed. The first one is the result of application of cumulant-neglect closure directly to unconditional moments. The second modified closure approximation technique is based on the representation of the joint probability density — discrete distribution function by conditioning it on two mutually exclusive and exhaustive events: that the Markov chain is in the jth state while the system is at rest (it is in the jth state for the first time) and that the Markov chain is in the jth state while the system is not at rest (it is in the jth state for any subsequent time). Thus the joint probability density function consists of a Dirac-delta spike and of the continuous part (cf Iwankiewicz et al. (1990)). The cumulant-neglect closure approximations are first formulated for the conditional moments resulting from the continuous part of the probability density function and next for the unconditional moments. As an example of a non-linear system the oscillator with cubic restoring force term is considered. The equations for moments up to the fourth-order are derived. Hence the modified cumulant-neglect closure approximations are derived for redundant fifth- and sixth order moments, both centralized and ordinary. The developed moment equations with modified cumulant-neglect closure techniques are verified against Monte Carlo simulations.

Stochastic failure process of railway vehicle dampers and the effects on suspension and vehicle dynamics

Railway vehicle dampers undergo degradation and failure after long-term service. Bench tests in this paper indicate a stochastic failure process, in which damping characteristics follow bimodal random distributions between normal and failure; failure probability increases in the ageing process until it becomes deterministic. This failure mode may be neglected due to the probabilistic absence of failure, which poses great risk to reliability. Then, nonlinear suspension dynamics with stochastic failure is modelled; Itô stochastic differential rule and cumulant-neglect closure are adopted to derive moment equations. The scheme of progressive iteration is proposed to obtain fast-convergent and stable solutions for analysing the effect of stochastic failure process on vibration response. Further, numerical simulation of a vehicle with stochastic failure is performed for failure effects analysis on dynamical performance. The results demonstrate that the deterioration of failure probability and damping reduction amplitude leads to larger vibrations. The stochastic failure of secondary lateral damper and anti-yaw damper is harmful for lateral vehicle dynamics, while that of secondary vertical damper has negative effects on vertical ride quality. It is of great importance to pay full attention to the stochastic failure process of dampers for reliability and security of railway vehicles.

Mean square stability of a second-order parametric linear system excited by a colored Gaussian noise

The stochastic stability of a second-order linear parametric oscillator whose stiffness is perturbed by a stationary zero-mean colored Gaussian process is investigated in this paper referring to the stability in the second response moments (mean square stability). The stochastic perturbation is the output of a linear filter excited by a stationary Gaussian white noise stochastic process, and the hypothesis of weak excitation is not formulated. The first assumption allows using Itô׳s stochastic differential calculus. The moment equations of the response are written till the second order by means of Itô׳s differential rule. They form an infinite hierarchy, that is the equations for the moments of order r contain moments of order larger than r. In order to close the hierarchy, the cumulant neglect closure method is applied herein. The moment equation set is closed by neglecting the cumulants of third order. Nonlinear moment equations result: they are linearized, and the limit of stability is searched by studying the eigenvalues of the matrix of the coefficients of the linearized moment equations. Several numerical analyses are performed to evaluate the critical mean square value of the excitation when a system parameter is varied for the cases of both first and second-order parametric excitation. A first-order excitation, an Ornstein–Uhlenbeck process, does not cause stochastic resonance, while a second-order excitation does. The first result was not known previously.

On intentional introduction of stiffness nonlinearities for energy harvesting under white Gaussian excitations

A significant body of the open literature on vibratory energy harvesting is currently focused on the concept of purposeful inclusion of stiffness nonlinearities for broadband transduction. When compared to their linear resonant counterparts, nonlinear energy harvesters have a wider steady-state frequency bandwidth, leading to the idea that they can be utilized to improve performance especially in random and non-stationary vibratory environments. To further investigate this common belief, this paper studies the response of vibratory energy harvesters to white Gaussian excitations. Both mono- and bi-stable piezoelectric Duffing-type harvesters are considered. The Fokker–Plank–Kolmogorov equation governing the evolution of the system’s transition probability density function is formulated and used to generate the moment differential equations governing the response statistics. The moment equations are then closed using a fourth-order cumulant-neglect closure scheme and the relevant steady-state response statistics are obtained. It is demonstrated that the energy harvester’s time constant ratio, i.e., the ratio between the nominal period of the mechanical subsystem and the time constant of the harvesting circuit, plays a critical role in characterizing the performance of nonlinear harvesters in a random environment. When the time constant ratio is large, stiffness-type nonlinearities have very little influence on the voltage response. In such a case, no matter how the potential function of the harvester is altered, it does not affect the average output power of the device. When the time constant ratio is small, the influence of the nonlinearity on the voltage output becomes more prevalent. In this case, a Duffing-type mono-stable harvester can never outperform its linear counterpart. A bi-stable harvester, on the other hand, can outperform a linear harvester only when it is designed with the proper potential energy function based on the known noise intensity of the excitation. Such conclusions hold for harvesters with nonlinearities appearing in the restoring force.

Continuous stirred tank coalescence/redispersion reactor: A simulation study

This paper presents an analysis and detailed numerical study of the continuous stirred tank coalescence/redispersion reactor (CSTCRR), developed as a multidimensional population balance model combining a well-stirred reactor with the generalized coalescence/redispersion (gCR) model of micromixing. The gCR micromixing model is based on the concept of interactive population of fluid elements, identified as Kolmogorov microscale eddies under the given turbulence conditions, and represents mixing of chemical species on microscale by collision induced mass exchange processes. This model allows modelling diverse micromixing rates for different chemical species depending on their molecular diffusion properties. An infinite order multivariate moment equation model was elaborated by applying the joint moments of concentrations of chemical species from which a second order moment equation reduction was obtained by means of the cumulant-neglect closure method. The simulation results obtained by the second order moment equation model for the irreversible quasi-linear bimolecular and quadratic autocatalytic with linear decay reactions, and for the Van de Vusse reaction network revealed that the CSTCRR spans the whole range of variation of the micromixing intensity between the two micromixing extremes, i.e. perfect mixing and complete segregation, and it produces a smooth monotone sequence of product yields in the mixing space as the micromixing intensity is varied between these two extremes by modulating the rate determining parameters.

Stochastic Response of an Inclined Shallow Cable with Linear Viscous Dampers under Stochastic Excitation

Considering the coupling between the in-plane and out-of-plane vibration, the stochastic response of an inclined shallow cable with linear viscous dampers subjected to Gaussian white noise excitation is investigated in this paper. Selecting the static deflection shape due to a concentrated force at the dampers location and the first sine term as shape functions, a reduced four-degree-of-freedom system of nonlinear stochastic ordinary differential equations are derived to describe dynamic response of the cable. Since only polynomial-type terms are contained, the fourth-order cumulant-neglect closure together with the C-type Gram-Charlier expansion with a fourth-order closure are applied to obtain statistical moments, power spectral density and probabilistic density function of the cable response, whose availability is verified by Monte Carlo method. Taking a typical cable as an example, the influence of several factors, which include excitation level and direction as well as damper size, on the dynamic response of the cable is extensively investigated. It is found that the sum of mean square in-plane and out-of-plane displacement is primarily independent of the load direction when the excitation level and viscous coefficient of the damper are fixed. Moreover, the peak frequency and half-band width of the spectra of both the in-plane and the out-of-plane displacements are increasing with excitation level when the damper size is constant. It is also observed that, even though the actual optimal damper size is slightly greater than the one obtained by the complex modal theory, the difference of statistical moment of the cable caused by these two damper size is negligible, so the vibration reduction effect provided by the theoretical optimal viscous coefficient is satisfactory.

Population balance model for mixing in continuous flow systems

A population balance model is developed for describing macromixing–micromixing hierarchy of continuous flow systems based on the concept of interactive populations of fluid elements. Macromixing is represented as motion of fluid elements in the physical space described by convection–dispersion models while micromixing is described as motion in the composition space induced by randomised mass exchange interactions of fluid elements. A generalised coalescence/redispersion model is formulated, characterised by the intensity of coalescence/re-dispersion interactions and a set of random variables describing the transfer rates of species between the fluid elements. The model allows defining different micromixing rates for different species. The axial dispersion-coalescence/redispersion (ADCR) model, formulated as a special case of the general population balance model describes macromixing–micromixing hierarchy for multicomponent processes in chemical reactors. An infinite system of moment equations is derived for the joint moments of concentrations of species which is closed by applying a cumulant-neglect closure for higher order chemical reactions. Verification of the ADCR model carried out by comparing the computational results of the second-order moment equation reduction with Vassilatos’ experimental data in the case of a tubular reactor with bimolecular quasilinear chemical reaction shows good correspondence. The properties of macromixing–micromixing interactions are examined by numerical experiments using the second-order moment equation reduction. Predictions of the model in maximally micromixed states provide excellent qualitative and quantitative agreement with the results of the continuum balance equation model, and appears to be a good tool for direct modelling of macromixing–micromixing hierarchy in chemical reactors.

Population Balance Modelling of Crystallisation Processes

The work presents and discusses the basics of the population balance approach and multi-dimensional population balance models for describing different aspects of solution crystallisation. A two dimensional model is presented for describing evolution of needle-shape crystals characterised by two size dimensions in cooling crystallisation, and a two-population model is developed for describing micromixing of solution, and its effects on the size distribution of crystals and their aggregates in reaction crystallization. Moment equation reductions for joint moments of internal variables are developed in both cases which are closed by means of cumulant neglect closure models. The properties and behaviour of crystallization processes described by the models are investigated by simulation.

Diffusion-induced growth of compositional heterogeneity in polymer blends containing random copolymers

The compositional relaxation in random copolymer systems on a macroscopic scale is considered in theory. A set of diffusion equations is derived that describes the motion of chains of different composition and then converted into coupled equations for statistical moments of the compositional distribution. Several ways to solve the closure problem for these equations are discussed. The simplest is the situation when the shape of the transient compositional distribution can be predicted a priori, for example, a bimodal distribution is kept during interdiffusion of two copolymers that are not very close in composition. For a general case, it is shown that the cumulant-neglect closure based on the truncation of high-order cumulants is an effective method to get an approximate solution in terms of the time-dependent local mean composition and its dispersion. This method is applied to non-homogeneous compatible polymer systems, such as a random copolymer AB of a composition varying in space, a bilayer of Bernoullian copolymers AB of different composition, and a bilayer of homopolymers A and B, in which an autocatalytic polymer-analogous reaction A --> B takes place, with possibility of the neighbor group effect. It is found that the interdiffusion can lead to a substantial broadening of the local compositional distribution, which, in turn, accelerates the system dynamics and promotes chemical reactions.

A NUMERICAL APPROACH TO STOCHASTIC OPTIMAL CONTROL VIA DYNAMIC PROGRAMMING

This paper presents a strategy for finding optimal controls of nonlinear systems subject to random excitations. The method is capable to generate global control solutions when state and control constraints are present. The solution is global in the sense that controls for all initial conditions in a region of the state space are obtained. The approach is based on the Bellman's Principle of optimality the cumulant neglect closure method and the Short-time Gaussian approximation. Nonlinear problems with non-smooth terms and range control bounds are considered in the examples. The controlled system responses derived are simulated and sucessfully validated by using the Generalized Cell Mapping method.

Stochastic optimal control via Bellman's principle

This paper presents a strategy for finding optimal controls of non-linear systems subject to random excitations. The method is capable to generate global control solutions when state and control constraints are present. The solution is global in the sense that controls for all initial conditions in a region of the state space are obtained. The approach is based on Bellman's principle of optimality, the cumulant neglect closure method and the short-time Gaussian approximation. Problems with state-dependent diffusion terms, non-closeable hierarchies of moment equations for the states and singular state boundary condition are considered in the examples. The uncontrolled and controlled system responses are evaluated by creating a Markov chain with a control dependent transition probability matrix via the generalized cell mapping method. In all numerical examples, excellent controlled performances were obtained.

Stochastic Optimal Control of Nonlinear Systems via Short-Time Gaussian Approximation and Cell Mapping

A novel strategy to obtain global solutions of stochasticoptimal control problems with fixed state terminal conditions and controlbounds is proposed in this paper. The solution is global in the sense that theoptimal control solutions for all the initial conditions in a region of thestate space are obtained. The method makes use of Bellman's principle ofoptimality, the cumulant neglect closure method and the short-time Gaussianapproximation. A Markov chain with a control dependent transition probabilitymatrix is built using the generalized cell mapping method. This allows toevaluate the transient and steady state response of the controlled system. Themethod is applied to several linear and nonlinear systems leading to excellentcontrol performances.

NON-CHAOTIC RESPONSE OF NON-LINEAR OSCILLATORS UNDER COMBINED DETERMINISTIC AND WEAK STOCHASTIC EXCITATIONS

Non-linear oscillators under harmonic and/or weak stochastic excitations are considered in this paper. Under harmonic excitations alone, an analytical technique based on a set of exponential transformations followed by harmonic balancing is proposed to solve for a variety of one-periodic orbits. The stability boundaries for such orbits in the associated parameter space are constructed using the Floquet theory. Under a combination of harmonic and weak stochastic excitations, a stochastic perturbation approach around the deterministic orbit is adopted to obtain response statistics in terms of the evolving moment functions. In the present study, the stochastic perturbation is assumed to be an additive white noise process and equations for the evolving moments are derived using Ito differential rule. A fifth order cumulant neglect closure is used to close the infinite hierarchy of moment equations. Limited numerical results are presented to illustrate the implementation of the proposed scheme. The method is found to be quite versatile and admits ready extensions to Md.o.f. systems under combined harmonic and white or non-white, multiplicative or additive random excitations.

Response cumulants of nonlinear systems subject to external and multiplicative excitations

A general framework is presented for deriving the differential equations governing the evolution of the response cumulants of linear and nonlinear dynamical systems subjected to external and multiplicative non-Gaussian delta-correlated processes. Significant simplifications of these equations are given based on using appropriate recursive relationships for joint cumulants involving products of one or more variables. A compact form of the equations for the response cumulants is presented which provides insight into the structure of the cumulant equations for specific types of dynamical systems. The procedure developed can easily be implemented in computer software to derive symbolic cumulant equations and to estimate numerically the response cumulants of systems with power-law nonlinearities using approximate cumulant-neglect closure schemes. Comparison between the equations for cumulants and the equations for moments are also presented, with particular emphasis on the advantages and disadvantages of each formulation. Suggestions are given regarding the choice to use cumulant or moment equations for analysing the stochastic response of dynamical systems. The preferred formulation is shown to depend on the type of system analysed (linear or nonlinear), the type of system nonlinearity (polynomial or non-polynomial), and the type of excitation (external or multiplicative, delta-correlated or filtered).

Response spectral densities of strongly nonlinear systems under random excitation

A new analytical procedure is developed to evaluate the response spectral densities for nonlinear systems excited by Gaussian white noises or filtered Gaussian white noises. The cumulant-neglect closure scheme is applied to truncate the governing differential equations for statistical moments of the response variables at two different times. The truncated equations in the time domain are transformed to a set of linear algebraic equations in the frequency domain, which include the response spectral densities as unknowns. This new procedure is illustrated in the example of a Duffing oscillator, and analytical results are compared with those obtained from Volterra series method and Monte Carlo simulations.

Random Response to Periodic Excitation with Correlated Disturbances

The paper addresses non-Gaussian stationary response of linear single-degree-of-freedom (SDOF) systems subject to a periodic excitation with correlated random amplitude and phase disturbances that are modeled as correlated Gaussian white noise processes. Correlation between amplitude and phase modulation is specified by the cross-correlation coefficient. Numerical results for the second and fourth moment responses are presented. The probability density function of the response is calculated based on the cumulant-neglect closure method. Non-Gaussian nature of the response is discussed in terms of the excess factor. The results show that the moment responses generally increase with larger random amplitude disturbance and may decrease with larger random phase modulation for a lightly damped system at resonance. The cross correlation between amplitude and phase disturbances plays an important role in the system moment response. Larger system damping results in smaller system moment responses. The moment response may approach a limiting value, depending on the intensity of the amplitude disturbance, as the relative detuning or phase modulation increases. For the case of the phase modulation alone, the response may become Gaussian in the sense of up to the fourth-order moment for sufficiently large relative detuning or random phase disturbances.

A stationary model for periodic excitation with uncorrelated random disturbances

The paper presents a stationary model for periodic excitations with random amplitude and phase disturbances for linear and nonlinear random vibration analysis. The disturbances are modeled as uncorrelated stationary white noise processes. Application of the model is demonstrated by stationary moment response of a linear single-degree-of-freedom system subject to such excitations. To find moment responses, an equivalent augmented system subject to parametric white noise excitations under certain constraint conditions is studied. Numerical results for the second and fourth-order moment responses are presented. The probability density function of the response is calculated based on thhe cumulant-neglect closure method. NonGaussianity of the response is discussed in terms of the excess factor. The results show that the random amplitude disturbance can significantly increase system moment response. The random phase modulation may increase or reduce the system moment response, depending on the value of relative detuning berween the system natural frequency and the mean excitation frequency. The response may become Gaussian in the sense of up to the fourth-order moment for sufficiently large random phase or relative detuning.

On the cumulant-neglect closure method in stochastic dynamics

This paper examines the efficacy of cumulant-neglect closure methods for complex dynamical systems. Previous efforts to employ this method were limited by the complicated and lengthy nature of the governing moment equations. To address this difficulty, an algorithm was developed and is presented herein for the efficient and automated generation of the moment equations for dynamical systems with an arbitrary number of states and closed at an arbitrary level (limited only by available computational resources). Both stationary and non-stationary moment responses were obtained through solution of the closed set of moment equations. The Duffing oscillator subjected to additive Gaussian white noise and the linear oscillator subjected to the square of a Gaussian process were examined in order to demonstrate the effectiveness and accuracy of the algorithm. A number of additional systems were examined to determine the range of applicability of the cumulant neglect closure method. Some general observations regarding stability and performance of cumulant-neglect closure methods are given.