Application of stochastic averaging to non-linear ecosystems

Stochastic models for the prey–predator type ecosystems in a random environment are proposed and investigated. They are the stochastic versions of the classical Lotka–Volterra model, the predator-saturation model, and the time-delay model. Variations in the birth rate of the preys and the death rate of the predators are modeled as random processes. For the three stochastic models, the stochastic averaging procedure of Stratonovich and Khasminskii is applied to obtain the probability distributions of the prey and predator populations at the state of statistical stationarity. Effects of different system parameters on the ecosystem behaviors are evaluated. The analytical results are substantiated by results obtained from Monte Carlo simulations.     

Stochastic averaging: An approximate method of solving random vibration problems

Results obtained by applying the method of stochastic averaging to random vibration problems are discussed. This method is applicable to a variety of problems involving the response of lightly damped systems to broad-band random excitations. Solutions pertaining to both linear and non-linear vibrations are reviewed, and it is shown that the technique enables, in the case of parametric excitation, stability criteria to be established. Some results which have been obtained relating to the first-passage reliability problems are also surveyed. Various applications of the theory to engineering problems are outlined.     

Generalization of the equivalent linearization method for non-linear random vibration problems

The method of equivalent linearization is generalized such that the response of a non-linear oscillator subject to both parametric and external random white-noise excitations can be determined approximately. The main objective of the new method is to obtain a closed system of differential equations for certain statistical moments of the response. This can be achieved by linearizing the drift term of the corresponding Itô-equations and by replacing the square of the diffusion term by a second degree polynomial. It is found that the accuracy of the mean square amplitudes is improved considerably compared with the original equivalent linearization.     

A stochastic averaging approach for printed circuit boards with nonlinear damping characteristics subjected to random vibration loads

Today many electronic devices consist of plate-like printed circuit boards carrying electric and electronic components mounted in some sort of housing. Depending on the application, these electronic devices may be subjected to severe random vibration loads over their lifetime, e.g. in automotive or aerospace environments. The present work shows that although state-of-the-art printed circuit boards are typically structurally linear with respect to stiffness properties, they are usually characterized by strongly nonlinear damping characteristics that have to be taken into account for proper vibration modelling and testing of corresponding devices. An approach allowing the nonlinear damping characteristics observed to be properly taken into account for testing and life-time prediction purposes is presented.     

Higher order linearization in non-linear random vibration

In this paper a higher order linearization method for analyzing non-linear random vibration problems is presented. The non-linear terms of the given equation are replaced by unknown linear terms. These are in turn described by extra non-linear differential equations. The combined system of equations is then linearized to arrive at a higher degree-of-freedom equation for the original system. The method is illustrated by considering the Duffing oscillator under white noise input. The equivalent two d.o.f linear system is derived by the present method. Numerical results on steady state variance and PSD functions are obtained. These are found to be better than the simple linearization results.     

An exact solution to a certain non-linear random vibration problem

A single-degree-of-freedom system with a special type of non-linear damping and both external and parametric white-noise excitations is considered. For the special case, when the intensities of coordinates and velocity modulation satisfy a certain condition an exact analytical solution is obtained to the corresponding stationary Fokker-Planck-Kolmogorov equation yielding an expression for joint probability density of coordinate and velocity. This solution is analyzed particularly in connection with stochastic stability problem for the corresponding linear system; certain implications are illustrated for the system, which is stable with respect to probability but unstable in the mean square. The solution obtained may be used to check different approximate methods for analysis of systems with randomly varying parameters.     

A consistent closure method for non-linear random vibration

—A closure method is proposed for calculating moments of the state vector of a non-linear system satisfying an Itô stochastic differential equation. It is based on a finite set of moment equations and a sequence of probabilities converging to the exact distribution of the state vector that is obtained by the minimization of an objective function. Numerical results for one-dimensional diffusion processes show that the proposed closure technique is robust in the sense that resultant moments are satisfactory even when crude approximations are used for the probability of the state vector.     

Stochastic averaging of n-dimensional non-linear dynamical systems subject to non-Gaussian wide-band random excitations

The averaged generalized Fokker-Planck-Kolmogorov (GFPK) equation for response of n-dimensional (n-d) non-linear dynamical systems to non-Gaussian wide-band stationary random excitation is derived from the standard form of equation of motion. The explicit expressions for coefficients of the fourth-order approximation of the averaged GFPK equation are given in series form. Conditions for convergences of these series are pointed out. The averaged GFPK equation is then reduced to that for 1-d dynamical systems derived by Stratonovich and compared with the closed form of GFPK equation for n-d dynamical systems subject to Poisson white noise derived by Di Paola and Falsone. Finally, this averaged GFPK equation is further reduced to that for quasi linear system subject to non-Gaussian wide-band stationary random excitation. Stationary probability density for quasi linear system subject to filtered Poisson white noise is obtained. Theoretical results for an example are confirmed by using Monte-Carlo simulation for different parameter values.     

Wide-band random vibration in members of complex structures

Bolotin's method of integral estimates, Statistical Energy Analysis and the theory of high-frequency vibration are combined in order to predict the vibrations of structural members in complex systems. The concept allows us to consider any driven component separately, where the presence of other structural members and coupling are taken into account by means of a frequency drift and an effective damping factor. The utility of the concept is demonstrated by solving two practical problems: (i) vibration of a solid-propellant engine caused by its vibrating combustion; (ii) vibrations of a thin-walled cover of an engine head. The approach enables one to indicate a frequency domain wherein a localization of vibration within structural members occurs.     

Application of non-linear damping to vibration isolation: an experimental study

In a previous study, the authors have proved that in theory the introduction of a cubic non-linear damping can produce ideal vibration isolation such that the system force transmissibility over the resonant frequency region is modified, but the transmissibility over the non-resonant regions remain unaffected. The present study is concerned with both an experimental verification of this theoretical finding and the selection of the cubic damping characteristic parameter required to achieve a desired performance for a single degree of freedom vibration isolation system. These results provide an important basis for the design and practical application of non-linearly damped vibration isolation systems in engineering practice.     

A stochastic complex model with random imaginary noise

In this paper, we study a stochastic complex beam–beam interaction model subjected to random imaginary noise. The general procedure is presented to obtain the Fokker–Planck–Kolmogorov equation (FPK) using stochastic averaging method in the case of a special example. The exact stationary probability of FPK is examined theoretically under certain conditions and then the first and second moments for the amplitude are expressed analytically. Finally, a numerical simulation is performed to verify the theoretical results of moments and excellent agreement can be observed between these two results.     

Probability density function for stochastic response of non-linear oscillation system under random excitation

Probability density function (PDF) of stochastic responses is a critical topic in uncertainty analysis. In this paper, orthogonal decomposition technique was extended to discuss non-stationary response of non-linear oscillator under random excitation. The PDF of stochastic reponses is represented by a set of standardized multivariable orthogonal polynomials. According to the Galerkin scheme, the original problem, which has to solve the Fokker-Planck-Kolmogorov (FPK) equation, was converted to a first-order linear ordinary differential equation, in terms of unknown time-dependent coefficients. Then, stationary and non-stationary PDFs of uncertainty responses were obtained. In numerical examples, first-order and second-order non-linear systems exposed to the Gaussian white noise were considered. Finally, the accuracy of the proposed method was demonstrated through appropriate comparisons to Monte-Carlo simulation and analytical results.     

An extended stochastic response surface method for random field problems

An efficient and accurate uncertainty propagation methodology for mechanics problems with random fields is developed in this paper. This methodology is based on the stochastic response surface method (SRSM) which has been previously proposed for problems dealing with random variables only. This paper extends SRSM to problems involving random fields or random processes fields. The favorable property of SRSM lies in that the deterministic computational model can be treated as a black box, as in the case of commercial finite element codes. Numerical examples are used to highlight the features of this technique and to demonstrate the accuracy and efficiency of the proposed method. A comparison with Monte Carlo simulation shows that the proposed method can achieve numerical results close to those from Monte Carlo simulation while dramatically reducing the number of deterministic finite element runs. The project supported by the National Natural Science Foundation of China (10602036). The English text was polished by Yunming Chen.     

Stochastic averaging using elliptic functions to study nonlinear stochastic systems

In this paper, a new scheme of stochastic averaging using elliptic functions is presented that approximates nonlinear dynamical systems with strong cubic nonlinearities in the presence of noise by a set of Itô differential equations. This is an extension of some recent results presented in deterministic dynamical systems. The second order nonlinear differential equation that is examined in this work can be expressed as % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qeguuDJXwAKbacfiGaf8hEaGNbamaacqGHRaWkcaWGJbadcaaIXaGc% cqWF4baEcqGHRaWkcaWGJbadcaaIZaGccqWF4baEdaahaaWcbeqaai% aaiodaaaGccqGHRaWkcqaH1oqzcaWGMbGaaiikaiab-Hha4jaacYca% cqWFGaaicuWF4baEgaGaaiaacMcacqGHRaWkcqaH1oqzdaahaaWcbe% qaaiaaigdacaGGVaGaaGOmaaaaruWrL9MCNLwyaGGbcOGaa43zaiaa% cIcacqWF4baEcaGGSaGae8hiaaIaf8hEaGNbaiaacaGGSaGae8hiaa% IaeqOVdGNaaeikaiaadshacaqGPaGaaiykaiabg2da9iaaicdaaaa!645D!\[\ddot x + c1x + c3x^3 + \varepsilon f(x, \dot x) + \varepsilon ^{1/2} g(x, \dot x, \xi {\text{(}}t{\text{)}}) = 0\] where c ₁ and c ₃ are given constants, (t) is stationary stochastic process with zero mean and 1 is a small parameter. This method involves the laborious manipulation of Jacobian elliptic functions such as cn, dn and sn rather than the usual trigonometric functions. The use of a symbolic language such as Mathematica reduces the computational effort and allows us to express the results in a convenient form. The resulting equations are Markov approximations of amplitude and phase involving integrals of elliptic functions. Finally, this method was applied to study some standard second order systems.     

Similarity solutions to some non-linear impact problems

Impact and wave propagation problems are considered for nonlinearly viscous and nonlinearly elastic materials. The governing partial differential equations are reduced to ordinary differential equations by means of similarity transformations. The resulting non-linear two point boundary value problems are then, in general, integrated numerically, although some closed form solutions are presented.     

Thermomechanically coupled non-linear vibration of plates

An analytical method is presented for the analysis of large amplitude thermomechanically coupled vibrations of rectangular elastic thin plates with various boundary conditions. The field of temperature and deformation are assumed to be coupled, and the transverse and longitudinal deformations are mutually dependent. The fundamental equations of non-linear flexural vibration of a plate stemming from Berger's analysis are coupled with the energy equation. Based on one-term approximate solution technique, the system of non-linear equations is solved by employing the methods of Galerkin and successive approximations. The analytical solutions are compared with those for the linear case and from the numerical analysis to investigate the influence of thermomechanical coupling and large amplitude on the period of plate lateral vibration.     

Parametric vibration of a non-linear system

Summary An analysis is presented of a non-linear system with one degree of freedom, in which the restoring force is expressed by the product of a periodic function of time and a non-linear function of deflection. In such a system there can occur not only the expected parametric resonances of the ordern (n=1,2, ...) but resonances of the order 1/N (N=2,3, ...) as well.

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übersicht Ein nichtlineares System mit einem Freiheitsgrad, in dem die Rückstellkraft das Produkt einer periodischen Funktion der Zeit und einer nichtlinearen Funktion der Abweichung ist, wird analysiert. In diesem Fall k?nnen neben den erwarteten Parameterresonanzen der Ordnungn (n=1,2, ...) noch die Resonanzen der Ordnung 1/N (N=2,3, ...) erscheinen.

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Herrn Professor Dr. K. Klotter zum 75. Geburtstag gewidmet.     

Survival probability of non-linear oscillators subjected to broad-band random disturbances

An analytical method is developed for examining the first-passage problem formulated in context with the response of a class of lightly damped non-linear oscillators to broad-band random excitations. A circular (E-type) barrier is considered. The amplitude of the oscillator response is modeled as a Markovian process. This modeling leads to a backward Kolmogorov equation which governs the evolution of the survival probability of the oscillator. The Kolmogorov equation is solved approximately by using the Galerkin technique and a perturbation technique. A set of confluent hypergeometric functions are used as an orthogonal basis for the expansions which are involved in the application of the Galerkin technique and the perturbation technique. The proposed method is exemplified by considering the response of the classical Van der Pol oscillator to white noise excitation. The reliability of the derived analytical solution is assessed by comparison with digital data obtained by a Monte Carlo simulation.     

Response statistics of two-degree-of-freedom non-linear system to narrow-band random excitation

The principal resonance of two-degree-of-freedom non-linear system to narrow-band random external excitation is investigated. The method of multiple scales is used to determine the equations of modulation of amplitude and phase. The behavior, stability and bifurcation of steady state response are studied by means of qualitative analysis. The effects of damping, detuning, bandwidth, and magnitudes of deterministic and random excitations are analyzed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that when the intensity of the random excitation increases, the nontrivial steady state solution may be changed from a limit cycle to a diffused limit cycle. Under some conditions the system may have two steady state solutions, saturation and jumps may exist.     

An approach to the problem of closure in non-linear stochastic mechanics

An approach combining the method of moment equations and the statistical linearization technique is proposed for analysis of the response of non-linear mechanical systems to random excitation. The adaptive statistical linearization procedure is developed for obtaining a more accurate mean square of responses. For these, a Duffing oscillator and an oscillator with cubic non-linear damping subject to white noise excitation are considered. It is shown that the adaptive statistical linearization proposed yields good accurate results for both weak and strong non-linear stochastic systems.Presented at the First European Solid Mechanics Conference, September 9–13, 1991. Munich, Germany