带位移偶次方项非线性随机振子在位移参数激励下的概率解

论文用指数多项式闭合或EPC(Exponential Polynomial Closure)法分析了具非零均值响应的带位移偶次方项非线性随机振子在参数激励下响应的概率密度函数解.给出了求解过程并通过算例分析验证了指数多项式闭合法在此情况下的有效性.数值结果显示,指数多项式闭合法得到的响应概率密度结果与蒙特卡洛模拟的结果符合较好,尤其是在对系统可靠性分析起主要作用的概率密度函数尾部区域符合很好.     

Prediction of the response of non-linear oscillators under stochastic parametric and external excitations

The method of equivalent external excitation is derived to predict the stationary variances of the states of non-linear oscillators subjected to both stochastic parametric and external excitations. The oscillator is interpreted as one which is excited solely by an external zero-mean stochastic process. The Fokker-Planck-Kolmogorov equation is then applied to solve for the density functions and match the stationary variances of the states. Four examples which include polynomial, non-polynomial, and Duffing type non-linear oscillators are used to illustrate this approach. The validity of the present approach is compared with some exact solutions and with Monte Carlo simulations.     

EPC procedure for PDF solution of non-linear oscillators excited by Poisson white noise

The stationary probability density function (PDF) solution of the responses of non-linear stochastic oscillators subjected to Poisson pulses is analyzed. The PDF solutions are obtained by the exponential-polynomial closure (EPC) method. To assess the effectiveness of the solution procedure numerically, non-linear oscillators are analyzed with different impulse arrival rates, degree of oscillator non-linearity and excitation intensity. Numerical results show that the PDFs obtained with the EPC method yield good agreement with those obtained from Monte Carlo simulation when the polynomial order is 4 or 6. It is also observed that the EPC procedure is the same as the equivalent linearization procedure under Gaussian white noise in the case of the polynomial order being 2.     

PDF solution of nonlinear oscillators subject to multiplicative Poisson pulse excitation on displacement

A solution procedure for the stationary probability density function (PDF) of the responses of nonlinear oscillators subjected to Poisson white noises is formulated with exponential-polynomial closure (EPC) method. The effectiveness of the solution procedure is investigated with nonlinear oscillators subjected to both external and multiplicative Poisson white noises at different levels of system nonlinearity, excitation intensity, and impulse arrival rates. Numerical results show that the PDFs obtained with the EPC procedure are in good agreement with those from Monte Carlo simulation.     

Stochastic averaging of strongly nonlinear oscillators with small fractional derivative damping under combined harmonic and white noise excitations

A stochastic averaging method for strongly nonlinear oscillators with lightly fractional derivative damping of order α (0<α<1) under combined harmonic and white noise external and (or) parametric excitations is proposed and then applied to study the first passage failure of Duffing oscillator with lightly fractional derivative damping of order 1/2 under combined harmonic and white noise excitations in the case of primary parametric resonance. Numerical results show that the proposed method works very well.     

Response and stability of strongly non-linear oscillators under wide-band random excitation

A new stochastic averaging procedure for single-degree-of-freedom strongly non-linear oscillators with lightly linear and (or) non-linear dampings subject to weakly external and (or) parametric excitations of wide-band random processes is developed by using the so-called generalized harmonic functions. The procedure is applied to predict the response of Duffing–van der Pol oscillator under both external and parametric excitations of wide-band stationary random processes. The analytical stationary probability density is verified by digital simulation and the factors affecting the accuracy of the procedure are analyzed. The proposed procedure is also applied to study the asymptotic stability in probability and stochastic Hopf bifurcation of Duffing–van der Pol oscillator under parametric excitations of wide-band stationary random processes in both stiffness and damping terms. The stability conditions and bifurcation parameter are simply determined by examining the asymptotic behaviors of averaged square-root of total energy and averaged total energy, respectively, at its boundaries. It is shown that the stability analysis using linearized equation is correct only if the linear stiffness term does not vanish.     

The stationary solution of a random dynamical model

This paper studies the stationary probability density function (PDF) solution of a nonlinear business cycle model subjected to random shocks of Gaussian white-noise type. The PDF solution is controlled by a Fokker–Planck–Kolmogorov (FPK) equation, and we use exponential polynomial closure (EPC) method to derive an approximate solution for the FPK equation. Numerical results obtained from EPC method, better than those from Gaussian closure method, show good agreement with the probability distribution obtained with Monte Carlo simulation including the tail regions.     

Nonzero mean PDF solution of nonlinear oscillators under external Gaussian white noise

This paper is concerned with the nonzero mean stationary probability density function (PDF) solution for nonlinear oscillators under external Gaussian white noise. The PDF solution is governed by the well-known Fokker–Planck–Kolmogorov (FPK) equation and this equation is numerically solved by the exponential-polynomial closure (EPC) method. Different types of oscillators are further investigated in the case of nonzero mean response. Either weak or strong nonlinearity is considered to show the effectiveness of the EPC method. When the polynomial order equals 2, the results of the EPC method are identical with those given by equivalent linearization (EQL) method. These results obtained with the EQL method differ significantly from exact solution or simulated results. When the polynomial order is 4 or 6, the PDFs obtained with the EPC method present a good agreement with the exact solution or simulated results, especially in the tail regions. The numerical analysis also shows that the nonzero mean PDF of the response is nonsymmetrically distributed about its mean unlike the case of the zero mean PDF reported in the references.     

On statistical quasi-linearization

In carrying out the statistical linearization procedure to a non-linear system subjected to an external random excitation, a Gaussian probability distribution is assumed for the system response. If the random excitation is non-Gaussian, however, the procedure may lead to a large error since the response of bother the original non-linear system and the replacement linear system are not Gaussian distributed. It is found that in some cases such a system can be transformed to one under parametric excitations of Gaussian white noises. Then the quasi-linearization procedure, proposed originally for non-linear systems under both external and parametric excitations of Gaussian white noises, can be applied to these cases. In the procedure, exact statistical moments of the replacing quasi-linear system are used to calculate the linearization parameters. Since the assumption of a Gaussian probability distribution is avoided, the accuracy of the approximation method is improved. The approach is applied to non-linear systems under two types of non-Gaussian excitations: randomized sinusoidal process and polynomials of a filtered process. Numerical examples are investigated, and the calculated results show that the proposed method has higher accuracy than the conventional linearization, as compared with the results obtained from Monte Carlo simulations.     

Constructing transient response probability density of non-linear system through complex fractional moments

The probability density function for transient response of non-linear stochastic system is investigated through the stochastic averaging and Mellin transform. The stochastic averaging based on the generalized harmonic functions is adopted to reduce the system dimension and derive the one-dimensional Itô stochastic differential equation with respect to amplitude response. To solve the Fokker–Plank–Kolmogorov equation governing the amplitude response probability density, the Mellin transform is first implemented to obtain the differential relation of complex fractional moments. Combining the expansion form of transient probability density with respect to complex fractional moments and the differential relations at different transform parameters yields a set of closed-form first-order ordinary differential equations. The complex fractional moments which are determined by the solution of the above equations can be used to directly construct the probability density function of system response. Numerical results for a van der Pol oscillator subject to stochastically external and parametric excitations are given to illustrate the application, the convergence and the precision of the proposed procedure.     

A new approximate solution technique for randomly excited non-linear oscillators

A new technique is proposed to obtain an approximate probability density for the response of a non-linear oscillator under Gaussian white noise excitations. The random excitations may be either multiplicative (also known as parametric) or additive (also known as external), or both. In this new technique, the original non-linear oscillator is replaced by another oscillator belonging to the class of generalized stationary potential for which the exact solution is obtainable. The replacement oscillator is selected on the basis that the average energy dissipation remains unchanged. Examples are given to illustrate the application of the new procedure. In one of the examples, the new procedure leads to a better approximation than that obtained by stochastic averaging.     

Stationary response of strongly non-linear oscillator with fractional derivative damping under bounded noise excitation

The stochastic averaging method for strongly non-linear oscillators with lightly fractional derivative damping of order α (0<α≤1) subject to bounded noise excitations is proposed by using the generalized harmonic function. The system state is approximated by a two-dimensional time-homogeneous diffusion Markov process of amplitude and phase difference using the proposed stochastic averaging method. The approximate stationary probability density of response is obtained by solving the reduced Fokker–Planck–Kolmogorov (FPK) equation using the finite difference method and successive over relaxation method. A Duffing oscillator is taken as an example to show the application and validity of the method. In the case of primary resonance, the stochastic jump of the Duffing oscillator with fractional derivative damping and its P-bifurcation as the system parameters change are examined for the first time using the stationary probability density of amplitude.     

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.     

An equivalent polynomial approximation technique in nonlinear structural dynamics

Summary A new technique is proposed to obtain an approximate probability density for the response of a general nonlinear system under Gaussian white noise excitations. In this new technique, the original nonlinear system is replaced by another equivalent nonlinear system, structured by the polynomial formula, for which the exact solution of stationary probability density function is obtainable. Since the equivalent nonlinear system structured in this paper originates directly from certain classes of real nonlinear mechanical systems, the technique is applied to some very challenging nonlinear systems in order to show its power and efficiency. The calculated results show that applying the technique presented here can yield exact stationary solutions for the nonlinear oscillators. This is obtained by using an energy-dependent system, and for a nonlinearity of a more complex type. A more accurate approximate solution is then available, and is compared with the approximation. Application of the technique is illustrated by examples.     

Quasi-Periodic Oscillations,Chaos and Suppression of Chaos in a Nonlinear Oscillator Driven by Parametric and External Excitations

An analysis is given of the dynamic of a one-degree-of-freedom oscillator with quadratic and cubic nonlinearities subjected to parametric and external excitations having incommensurate frequencies. A new method is given for constructing an asymptotic expansion of the quasi-periodic solutions. The generalized averaging method is first applied to reduce the original quasi-periodically driven system to a periodically driven one. This method can be viewed as an adaptation to quasi-periodic systems of the technique developed by Bogolioubov and Mitropolsky for periodically driven ones. To approximate the periodic solutions of the reduced periodically driven system, corresponding to the quasi-periodic solution of the original one, multiple-scale perturbation is applied in a second step. These periodic solutions are obtained by determining the steady-state response of the resulting autonomous amplitude-phase differential system. To study the onset of the chaotic dynamic of the original system, the Melnikov method is applied to the reduced periodically driven one. We also investigate the possibility of achieving a suitable system for the control of chaos by introducing a third harmonic parametric component into the cubic term of the system.     

Analysis of period-doubling bifurcation in double-well stochastic Duffing system via Laguerre polynomial approximation

The Laguerre polynomial approximation method is applied to study the stochastic period-doubling bifurcation of a double-well stochastic Duffing system with a random parameter of exponential probability density function subjected to a harmonic excitation. First, the stochastic Duffing system is reduced into its equivalent deterministic one, solvable by suitable numerical methods. Then nonlinear dynamical behavior about stochastic period-doubling bifurcation can be fully explored. Numerical simulations show that similar to the conventional period-doubling phenomenon in the deterministic Duffing system, stochastic period-doubling bifurcation may also occur in the stochastic Duffing system, but with its own stochastic modifications. Also, unlike the deterministic case, in the stochastic case the intensity of the random parameter should also be taken as a new bifurcation parameter in addition to the conventional bifurcation parameters, i.e. the amplitude and the frequency of harmonic excitation.     

几类非线性系统平稳随机响应的概率密度

本文考虑非保守力依赖于系统能量的非线性系统,构造了四类这种系统对白噪声外激与/或参激的平稳响应的精确概率密度,讨论了存在平稳响应的条件。同时指出,迄今为止已有的非线性系统平稳随机响应的精确解皆属本文给出一般结果的特殊情形。最后还给出几个例子说明一般结果。     

On the design of external excitations in order to make nonlinear oscillators respond as free oscillators of the same or different type

This study deals with nonlinear oscillators whose restoring force has a polynomial nonlinearity of the cubic or quadratic type. Conservative unforced oscillators with such a restoring force have closed-form exact solutions in terms of Jacobi elliptic functions. This fact can be used to design the form of the external elliptic-type excitation so that the resulting forced oscillators also have closed-form exact steady-state solutions in terms of these functions. It is shown how one can use the amplitude of such excitations to change the way in which oscillators behave, making them respond as free oscillators of the same or different type. Thus, in cubic oscillators, a supercritical or subcritical pitchfork bifurcation can appear, whilst in quadratic oscillators, a transcritical bifurcation can take place.     

A finite-element method for analysis of a non-linear system under stochastic parametric and external excitation

A finite-element method approach is developed to solve the Fokker-Planck-Kol-mogorov equation for the probability density function of stationary response of a general non-linear system subject to both parametric and external Gaussian white noise excitation. N-dimensional shape functions are expanded by a one-dimensional shape function in global coordinate. As the domain of the density function is usually infinite, adaptive grid generation is adopted for the estimation of a finite range of the finite-element mesh. Several examples with existing exact solutions are used to illustrate the validity of the method. Results are also compared with those obtained by the Gaussian closure method, the cumulant-neglect method, equivalent external excitation and Monte Carlo simulation.     

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.