Effects of excitation probability distribution on system responses

For a system subjected to a random excitation, the probability distribution of the excitation may affect behaviors of the system responses. Such effects are investigated for a variety of dynamical systems, including a linear oscillator, an oscillator of cubic non-linearity in both damping and stiffness, and a non-linear oscillator of the van der Pol type. The random excitations are assumed to be stationary stochastic processes, sharing the same spectral density, but with different probability distributions. Each excitation process is generated by passing a Brownian motion process through a non-linear filter, which is governed by an Ito stochastic differential equation. Monte Carlo simulations are carried out to obtain the transient and stationary properties of the system response in each case. It is shown that, under different excitations, the transient behaviors of the system response can be markedly different. The differences tend to reduce, however, as time of exposure to the excitations increases and the system reaches the stationary state.     

Jump and bifurcation of Duffing oscillator under narrow-band excitation

The jump and bifurcation of Duffing oscillator with hardening spring subject to narrow-band random excitation are systematically and comprehensively examined. It is shown that, in a certain domain of the space of the oscillator and excitation parameters, there are two types of more probable motions in the stationary response of the Duffing oscillator and jumps may occur. The jump is a transition of the response from one more probable motion to another or vise versa. Outside the domain the stationary response is either nearly Gaussian or like a diffused limit cycle. As the parameters change across the boundary of the domain the qualitative behavior of the stationary response changes and it is a special kind of bifurcation. It is also shown that, for a set of specified parameters, the statistics are unique and they are independent of initial condition. It is pointed out that some previous results and interpretations on this problem are incorrect. The project supported by National Natural Science Foundation of China     

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.     

ON DOUBLE PEAK PROBABILITY DENSITY FUNCTIONS OF DUFFING OSCILLATOR TO COMBINED DETERMINISTIC AND RANDOM EXCITATIONS

The principal resonance of Duffing oscillator to combined deterministic and random external excitation was investigated. The random excitation was taken to be white noise or harmonic with separable random amplitude and phase. The method of multiple scales was used to determine the equations of modulation of amplitude and phase. The one peak probability density function of each of the two stable stationary solutions was calculated by the linearization method. These two one-peak-density functions were combined using the probability of realization of the two stable stationary solutions to obtain the double peak probability density function. The theoretical analysis are verified by numerical results.     

Numerical path integration of a non-homogeneous Markov process

The numerical path integration method, based on Gauss-Legendre integration scheme, is applied to a Duffing oscillator subject to both sinusoidal and white noise excitations. The response of the system is a Markov process with one of the drift coefficients being periodic. It is a non-homogeneous Markov process that does not have a stationary probability distribution. When applying the numerical procedure, the values of transition probability density at the Gaussian-Legendre quadrature points need only be calculated for time steps of the first period of the sinusoidal excitation, and they can be saved for use in all subsequent periods. The numerical procedure is capable of capturing the evolution of the probability density from an initial distribution to one that is changing and rotating periodically in the phase space.     

Identification of Nonlinear Oscillator with Parametric White Noise Excitation

An identification technique is proposed for a nonlinear oscillator excited by response-dependent white noise. Stiffness, damping and excitation are estimated from records of the stationary stochastic response. The estimation of the stiffness is based on a nonparametric procedure in which the potential energy at the displacement extremes is obtained from the kinetic energy at the previous mean-level-crossing. A nonparametric estimate is obtained by an iterative averaging, in which the increased knowledge of the potential energy in each step is used to avoid bias. The second step in the procedure is to estimate the stationary probability potential in a nonparametric form from a histogram of the kinetic energy at mean-level-crossings. The damping is also estimated in a nonparametric way from approximate expressions of the covariance functions of a set of modified phase plane variables at a given energy level. Finally, the excitation is estimated from a relation between the stationary probability potential, the damping and the excitation. The separation of damping and excitation requires a parametric representation. The system identification technique is investigated by application to response records obtained by stochastic simulation. The stiffness estimation generally gives excellent results, while the damping and excitation estimation tend to be slightly biased for systems with strongly nonlinear stiffness.     

Non-Gaussianclosure techniques for stationary random vibration

The classical method of statistical linearization when applied to a non-linear oscillator excited by stationary wide-band random excitation, can be considered as a procedure in which the unknown parameters in a Gaussian distribution are evaluated by means of moment identities derived from the dynamic equation of the oscillator. A systematic extension of this procedure is the method of non-Gaussian closure in which an increasing number of moment identities are used to evaluate additional parameters in a family of non-Gaussian response distributions. The method is described and illustrated by means of examples. Attention is given to the choice of representations of non-Gaussian distributions and to techniques for generating independent moment identities directly from the differential equation of the non-linear oscillator. Some shortcomings of the method are pointed out.     

Quasi-periodic oscillation envelopes and frequency locking in rapidly vibrated nonlinear systems with time delay

The effect of time-delayed feedback and fast harmonic excitation (FHE) on stationary periodic vibration and quasi-periodic responses in a parametric and self-excited weakly nonlinear oscillator is analyzed in this paper. The method of direct partition of motion and two stages of multiple scales analysis are conducted to obtain analytical approximation for quasi-periodic oscillation envelopes and frequency-locking area near primary resonance. A parameter study shows that, in the absence or the presence of high-frequency excitation, time-delayed feedback may reduce significantly the amplitude and the envelopes of quasi-periodic oscillations leading to a quasi synchronization of the response over the whole frequency range around the resonance. The results presented for the parameters tested agree well with results obtained by numerical simulation.     

The nonstationary effects on a softening duffing oscillator

This paper presents a numerical study for the bifurcations of a softening Duffing oscillator subjected to stationary and nonstationary excitation. The nonstationary inputs used are linear functions of time. The bifurcations are the results of either a single control parameter or two control parameters that are constrained to vary in a selected direction on the plane of forcing amplitude and forcing frequency. The results indicate: 1. Delay (memory, penetration) of nonstationary bifurcations relative to stationary bifurcations may occur. 2. The nonstationary trajectories jump into the neighboring stationary trajectories with possible overshoots, while the stationary trajectories transit smoothly. 3. The nonstationary penetrations (delays) are compressed to zero with an increasing number of iterations. 4. The nonstationary responses converge through a period-doubling sequence to a nonstationary limit motion that has the characteristics of chaotic motion. The Duffing oscillator has been used as an example of the existence of broad effects of nonstationary (time dependent) and codimensional (control parameter variations in the bifurcation region) inputs which markedly modify the dynamical behavior of dynamical systems.     

On Double Crater-Like Probability Density Functions of a Duffing Oscillator Subjected to Harmonic and Stochastic Excitation

It is a well-known phenomenon of the Duffing oscillator under harmonic excitation,that there is a frequency range, where two stable and one unstable stationarysolution coexist. If the Duffing oscillator is harmonically excited in thisfrequency range and additionally excited, e.g. by white noise, a double crater-likeprobability density function can be observed, if the noise intensity is smallcompared to the harmonic excitation. The aim of this paper is to calculate thisprobability density function approximately using perturbation techniques. Thestationary solutions in the deterministic case are calculated using theperturbation technique for the resonance case. In a second step, the probabilitydensity function of the perturbation of each of those stationary solutions iscalculated using the perturbation technique for the nonresonance case. This resultsin two crater-like probability density functions which are superimposed by usingthe probability of realization of each of the stationary solutions in thedeterministic case. The probability is calculated using numerical integration orthe method of slowly changing phase and amplitude. Finally, probability densityfunctions obtained in this manner are compared to Monte Carlo simulations.     

A kind of noise-induced transition to noisy chaos in stochastically perturbed dynamical system

We investigate a kind of noise-induced transition to noisy chaos in dynamical systems. Due to similar phenomenological structures of stable hyperbolic attractors excited by various physical realizations from a given stationary random process, a specific Poincar map is established for stochastically perturbed quasi-Hamiltonian system. Based on this kind of map, various point sets in the Poincar's cross-section and dynamical transitions can be analyzed. Results from the customary Duffing oscillator show that, the point sets in the Poincar's global cross-section will be highly compressed in one direction, and extend slowly along the deterministic period-doubling bifurcation trail in another direction when the strength of the harmonic excitation is fixed while the strength of the stochastic excitation is slowly increased. This kind of transition is called the noise-induced point-overspreading route to noisy chaos.     

高斯色噪声和谐波激励共同作用下耦合SD振子的混沌研究

耦合SD振子作为一种典型的负刚度振子, 在工程设计中有广泛应用. 同时高斯色噪声广泛存在于外界环境中, 并可能诱发系统产生复杂的非线性动力学行为, 因此其随机动力学是非线性动力学研究的热点和难点问题. 本文研究了高斯色噪声和谐波激励共同作用下双稳态耦合SD振子的混沌动力学, 由于耦合SD振子的刚度项为超越函数形式, 无法直接给出系统同宿轨道的解析表达式, 给混沌阈值的分析造成了很大的困难. 为此, 本文首先采用分段线性近似拟合该振子的刚度项, 发展了高斯色噪声和谐波激励共同作用下的非光滑系统的随机梅尔尼科夫方法. 其次, 基于随机梅尔尼科夫过程, 利用均方准则和相流函数理论分别得到了弱噪声和强噪声情况下该振子混沌阈值的解析表达式, 讨论了噪声强度对混沌动力学的影响. 研究结果表明, 随着噪声强度的增大混沌区域增大, 即增大噪声强度更容易诱发耦合SD振子产生混沌. 当阻尼一定时, 弱噪声情况下混沌阈值随噪声强度的增加而减小; 但是强噪声情况下噪声强度对混沌阈值的影响正好相反. 最后, 数值结果表明, 利用文中的方法研究高斯色噪声和谐波激励共同作用下耦合SD振子的混沌是有效的.本文的结果为随机非光滑系统的混沌动力学研究提供了一定的理论指导.      

Gauss白噪声外激下Rayleigh振子的平稳响应与首次穿越

研究了Rayleigh振子在Gauss白噪声外激下的平稳响应和首次穿越。首先利用随机平均法给出了系统随机平均It^o微分方程,对平均方程的稳态概率密度做了数值分析;然后建立了条件可靠性函数的后向Kolmogorov方程及首次穿越时间条件矩的Pontragin方程;最后对三组不同的参数值分析了首次穿越的概率统计特性。     

Non-stationary response of a van der Pol-Duffing oscillator under Gaussian white noise

This paper investigates the probability density evolution process of a van der Pol-Duffing oscillator under Gaussian white noise. A path integration method is employed with the Gauss–Legendre integration scheme. In the path integration method, the short-time Gaussian approximation scheme is used for computing the one-step transition probability density. Two cases are considered with slight nonlinearity or strong nonlinearity in displacement. The stationary and non-stationary responses of the oscillator are studied. Compared with the simulation result, the path integration method can present a satisfactory probability density function (PDF) solution for each case. Different probability density evolution processes are observed correspondingly. In the case of slight nonlinearity, the PDF undergoes a clockwise motion around the origin. The peak region gradually expands and the PDF eventually forms a circle. By contrast, the strong nonlinearity drives the oscillator to oscillate around the limit cycle. In such a case, the PDF rapidly forms a circle. The circle keeps its shape and develops until the oscillator becomes stationary. More complicated phenomena can be studied by the adopted path integration method.     

Stochastic stability of parametrically excited random systems

Multidegree-of-freedom dynamic systems subjected to parametric excitation are analyzed for stochastic stability. The variation of excitation intensity with time is described by the sum of a harmonic function and a stationary random process. The stability boundaries are determined by the stochastic averaging method. The effect of random parametric excitation on the stability of trivial solutions of systems of differential equations for the moments of phase variables is studied. It is assumed that the frequency of harmonic component falls within the region of combination resonances. Stability conditions for the first and second moments are obtained. It turns out that additional parametric excitation may have a stabilizing or destabilizing effect, depending on the values of certain parameters of random excitation. As an example, the stability of a beam in plane bending is analyzed.Published in Prikladnaya Mekhanika, Vol. 40, No. 10, pp. 135–144, October 2004.     

Spectral Response of a Stochastic Oscillator under Impacts

An approximate analytical procedure is presented to estimate theresponse spectrum of an oscillator with elastic impacts under a Gaussian whitenoise excitation. The proposed approach is based on a perturbation analysis ofthe problem and on the use of the stochastic averaging principle. The basicidea is to replace the initial system by a more regular system obtained byapproximating the nonlinear restoring force by a Chebychev polynomial, and thento construct for this regular system two approximations: one for the flowand one for the stationary distribution of the response amplitude. Ananalytical approximation of the response spectrum can then be derived fromthese results. Predictions from this analytical approximation are compared with corresponding digital simulation estimates and with the ones obtained from theconventional equivalent linearization method.     

Nonstationary effects on safe basins of a forced softening Duffing oscillator

The safe basin of a forced softening Duffing oscillator is studied numerically. The changes of safe basins are observed under both stationary and nonstationary variations of the external excitation frequency. The kind of nonstationary variations of the excitation frequency can greatly change the erosion rate and the shape of the safe basin. The other effects of nonstationary variations on the safe basin are also discussed. Supported by the National Natural Science Foundation, the Aviation Science Foundation and the Doctoral Training Foundation of China.     

Stochastic response of fractional-order van der Pol oscillator

We studied the response of fractional-order van de Pol oscillator to Gaussian white noise excitation in this letter. An equivalent integral-order nonlinear stochastic system is obtained to replace the given system based on the principle of minimum mean-square error. Through stochastic averaging, an averaged Itô equation is deduced. We obtained the Fokker-Planck-Kolmogorov equation connected to the averaged Itô equation and solved it to yield the approximate stationary response of the system. The analytical solution is confirmed by using Monte Carlo simulation.     

Drifting response of elastic perfectly plastic oscillators under zero mean random load

The displacement response of an elastic perfectly plastic oscillator under a zero mean, stationary, broad band random load is known not to reach stationarity: asymptotically, its mean is zero but its variance linearly increases with time. Thus, as time passes the oscillator gradually drifts away from its initial position. A method is presented for estimating the time asymptotic behavior of this drifting. Developed within the context of stochastic averaging, the method is based on a generalized van der Pol transformation that differs from its classical counterpart by an extra term that is meant to capture the drifting. The introduction of this term makes it possible to successfully address the drifting by using a linearization technique, even when the excitation power spectrum vanishes at zero frequency. The results obtained with the method are in good agreement with Monte Carlo simulation estimates.     

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.