Random excitation of coupled oscillators with single and simultaneous internal resonances

The primary objective of this paper is to examine the random response characteristics of coupled nonlinear oscillators in the presence of single and simultaneous internal resonances. A model of two coupled beams with nonlinear inertia interaction is considered. The primary beam is directly excited by a random support motion, while the coupled beam is indirectly excited through autoparametric coupling and parametric excitation. For a single one-to-two internal resonance, we used Gaussian and non-Gaussian closures, Monte Carlo simulation, and experimental testing to predict and measure response statistics and stochastic bifurcation in the mean square. The mean square stability boundaries of the coupled beam equilibrium position are obtained by a Gaussian closure scheme. The stochastic bifurcation of the coupled beam is predicted theoretically and experimentally. The stochastic bifurcation predicted by non-Gaussian closure is found to take place at a lower excitation level than the one predicted by Gaussian closure and Monte Carlo simulation. It is also found that above a certain excitation level, the solution obtained by non-Gaussian closure reveals numerical instability at much lower excitation levels than those obtained by Gaussian and Monte Carlo approaches. The experimental observations reveal that the coupled beam does not reach a stationary state, as reflected by the time evolution of the mean square response. For the case of simultaneous internal resonances, both Gaussian and non-Gaussian closures fail to predict useful results, and attention is focused on Monte Carlo simulation and experimental testing. The effects of nonlinear coupling parameters, internal detuning ratios, and excitation spectral density level are considered in both investigations. It is found that both studies reveal common nonlinear features such as bifurcations in the mean square responses of the coupled beam and modal interaction in the neighborhood of internal resonances. Furthermore, there is an upper limit for the excitation level above which the system experiences unbounded response in the neighborhood of simultaneous internal resonances.     

Nonlinear Random Response of Ocean Structures Using First- and Second-Order Stochastic Averaging

This paper deals with the dynamic response of nonlinear elastic structure subjected to random hydrodynamic forces and parametric excitation using a first- and second-order stochastic averaging method. The governing equation of motion is derived by using Hamilton's principle, taking into account inertia and curvature nonlinearities and work done due to hydrodynamic forces. Within the framework of first-order stochastic averaging, the system response statistics and stability boundaries are obtained. Unfortunately, the effects of nonlinear inertia and curvature are not reflected in the final results, since the contribution of these nonlinearities is lost during the averaging process. In the absence of hydrodynamic forces, the method fails to give bounded response statistics, and the analysis yields stability conditions. It is the second-order stochastic averaging which can capture the influence of stiffness and inertia nonlinearities that were lost in the first-order averaging process. The results of the second-order averaging are compared with those predicted by Gaussian and non-Gaussian closures and by Monte Carlo simulation. In the absence of parametric excitation, the non-Gaussian closure solutions are in good agreement with Monte Carlo simulation. On the other hand, in the absence of hydrodynamic forces, second-order averaging gives more reliable results in the neighborhood of stochastic bifurcation. However, under pure parametric random excitation, the stochastic averaging and Monte Carlo simulation predict the on-off intermittency phenomenon near bifurcation point, in addition to stochastic bifurcation in probability.     

Multiple Internal Resonance in Suspended Cables under Random In-Plane Loading

The random excitation of a suspended cable with simultaneous internal resonances is considered. The internal resonances can take place among the first in-plane and the first two out-of-plane modes. The external loading is represented by a wide-band random process. The response statistics are estimated using the Fokker-Planck-Kolmogorov (FPK) equation, together with Gaussian and non-Gaussian closures. Monte Carlo simulation is also used for numerical verification. The unimodal in-plane motion exists in regions away from the internal resonance condition. The mixed mode interaction is manifested within a limited range of internal detuning parameters, depending on the excitation power spectrum density and damping ratios. The Gaussian closure scheme failed to predict bounded solutions of mixed mode interaction. The non-Gaussian closure results are in good agreement with the Monte Carlo simulation. The on-off intermittency of the autoparametrically excited modes is observed in the Monte Carlo simulation over a small range of excitation levels. The influence of the cable parameters, such as damping ratios, sag-to-span ratio, internal detuning parameters, and excitation level on the autoparametric interaction, is studied. It is found that the internal detuning and excitation level are the two main parameters which affect the autoparametric interaction among the three modes. Due to the system's nonlinearity, the response of the three modes is strongly non-Gaussian and the coupled modes experience irregular modulation.     

Planar and non-planar non-linear dynamics of suspended cables under random in-plane loading-I. Single internal resonance

The non-linear interaction of the in-plane and out-of-plane motions of a suspended cable in the neighbourhood of 2:1 internal resonance under random loading is studied. The random loading acts externally on the in-plane mode, while the out-of-plane mode is non-linearly coupled with the in-plane mode. Any non-trivial motion of the out-of-plane mode is mainly due to this non-linear coupling, which becomes significant in the neighbourhood of internal resonance. The response statistics are estimated by employing the Fokker-Planck equation together with Gaussian and non-Gaussian closures. Monte-Carlo simulation is also used for numerical verification. Away from the internal resonance condition, the response is governed by the inplane motion, and the non-Gaussian closure solution is found to be in good agreement with numerical simulation results. The stochastic bifurcation of the out-of-plane mode is predicted by Gaussian and non-Gaussian closures, and by Monte-Carlo simulation. The non-Gaussian closure can only predict bounded solutions within a limited region. The on-off intermittency of the second mode is observed in the Monte-Carlo simulation over a small range of excitation level. The influence on response statistics of excitation level and cable parameters, such as internal detuning, damping ratios, and sag-to-span ratio, is reported.     

An improved approach for random parametric response of dynamic systems with non-linear inertia

A non-Gaussian closure scheme is developed for determining the stationary response of dynamic systems including non-linear inertia and stochastic coefficients. Numerical solutions are obtained and examined for their validity based on the preservation of moments properties. The method predicts the jump phenomenon, for all response statistics at an excitation level very close to the threshold level of the condition of almost sure stability. In view of the increased degree of non-linearity, resulting from the non-Gaussian closure scheme, the mean square of the response displacement is found to be less than those values predicted by other methods such as the Gaussian closure or the first order stochastic averaging.     

Structural first failure times under non-Gaussian stochastic behavior

An analytical moment-based method for calculating structural first failure times under non-Gaussian stochastic behavior is proposed.In the method,a power series that constants can be obtained from response moments (skewness,kurtosis,etc.) is used firstly to map a non-Gaussian structural response into a standard Gaussian process,then mean up-crossing rates,mean clump size and the initial passage probability of a critical barrier level by the original structural response are estimated,and finally,the formula for calculating first failure times is established on the assumption that corrected up-crossing rates are independent.An analysis of a nonlinear single-degree-of-freedom dynamical system excited by a Gaussian model of load not only demonstrates the usage of the proposed method but also shows the accuracy and efficiency of the proposed method by comparisons between the present method and other methods such as Monte Carlo simulation and the traditional Ganssian model.     

考虑非高斯特性的结构首次失效时间计算

本文发展了一个计算具有非高斯特性的结构首次失效时间的解析方法.该方法利用Hemite矩模型将非高斯结构反应映射为标准高斯过程,由此计算反应的平均超越率、成群超越以及初始状态的影响,并最终给出结构的首次失效时问概率.二次力函数激励下线性单自由度系统的首次失效时间分析说明了该方法的使用过程,同时对该方法的计算结果与Monte Carlo模拟结果进行了对比.     

Deterministic and Stochastic Response of Nonlinear Coupled Bending-Torsion Modes in a Cantilever Beam

The purpose of this study is to understand the main differences between the deterministic and random response characteristics of an inextensible cantilever beam (with a tip mass) in the neighborhood of combination parametric resonance. The excitation is applied in the plane of largest rigidity such that the bending and torsion modes are cross-coupled through the excitation. In the absence of excitation, the two modes are also coupled due to inertia nonlinearities. For sinusoidal parametric excitation, the beam experiences instability in the neighborhood of the combination parametric resonance of the summed type, i.e., when the excitation frequency is in the neighborhood of the sum of the first bending and torsion natural frequencies. The dependence of the response amplitude on the excitation level reveals three distinct regions: nearly linear behavior, jump phenomena, and energy transfer. In the absence of nonlinear coupling, the stochastic stability boundaries are obtained in terms of sample Lyapunov exponent. The response statistics are estimated using Monte Carlo simulation, and measured experimentally. The excitation center frequency is selected to be close to the sum of the bending and torsion mode frequencies. The beam is found to experience a single response, two possible responses, or non-stationary responses, depending on excitation level. Experimentally, it is possible to obtain two different responses for the same excitation level by providing a small perturbation to the beam during the test.     

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.     

Influence of the mode number on the stochastic stability regions of the elastic beam

The moment Lyapunov exponents and Lyapunov exponent of a two-dimensional system under stochastic parametric excitation are studied. A perturbation approach is used to obtain explicit expressions for these exponents in the presence of small intensity noises. Approximate analytical results for the pth moment Lyapunov exponents are compared with the numerical values obtained by the Monte Carlo simulation approach. The results are applied to the study of the almost-sure and the moment stability of the stationary solutions of the elastic beam subjected to the stochastic axial load. The boundaries of the almost-sure and the moment stability of the elastic beam as the function of the damping coefficient, spectral density of the stochastic force and mode number are obtained.     

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.     

Responses of Duffing oscillator with fractional damping and random phase

This paper aims to investigate dynamic responses of stochastic Duffing oscillator with fractional-order damping term, where random excitation is modeled as a harmonic function with random phase. Combining with Lindstedt–Poincaré (L–P) method and the multiple-scale approach, we propose a new technique to theoretically derive the second-order approximate solution of the stochastic fractional Duffing oscillator. Later, the frequency–amplitude response equation in deterministic case and the first- and second-order steady-state moments for the steady state in stochastic case are presented analytically. We also carry out numerical simulations to verify the effectiveness of the proposed method with good agreement. Stochastic jump and bifurcation can be found in the figures of random responses, and then we apply Monte Carlo simulations directly to obtain the probability density functions and time response diagrams to find the stochastic jump and bifurcation. The results intuitively show that the intensity of the noise can lead to stochastic jump and bifurcation.     

Parametric resonance,stability and heteroclinic bifurcation in a nonlinear oscillator with time-delay: Application to a quarter-car model

This work examines dynamical behavior of a nonlinear oscillator with symmetric potential that models a quarter-car forced by the road profile under parametric excitation. The parametric resonance of a harmonically excited nonlinear quarter-car model with position and velocity time-delayed active control are investigated. We focus on the influence of delay and parametric excitation in the system. The influence of parametric excitation, time-delay and feedback gain parameters on the stability of the steady state response are investigated. By means of Melnikov's method, conditions for onset of chaos resulting from heteroclinic bifurcation is derived analytically and numerically.     

Bifurcation in most probable phase portraits for a bistable kinetic model with coupling Gaussian and non-Gaussian noises

The phenomenon of stochastic bifurcation driven by the correlated non-Gaussian colored noise and the Gaussian white noise is investigated by the qualitative changes of steady states with the most probable phase portraits. To arrive at the Markovian approximation of the original non-Markovian stochastic process and derive the general approximate Fokker-Planck equation (FPE), we deal with the non-Gaussian colored noise and then adopt the uni¯ed colored noise approximation (UCNA). Subsequently, the theoretical equation concerning the most probable steady states is obtained by the maximum of the stationary probability density function (SPDF). The parameter of the uncorrelated additive noise intensity does enter the governing equation as a non-Markovian effect, which is in contrast to that of the uncorrelated Gaussian white noise case, where the parameter is absent from the governing equation, i.e., the most probable steady states are mainly controlled by the uncorrelated multiplicative noise. Additionally, in comparison with the deterministic counterpart, some peculiar bifurcation behaviors with regard to the most probable steady states induced by the correlation time of non-Gaussian colored noise, the noise intensity, and the non-Gaussian noise deviation parameter are discussed. Moreover, the symmetry of the stochastic bifurcation diagrams is destroyed when the correlation between noises is concerned. Furthermore, the feasibility and accuracy of the analytical predictions are verified compared with those of the Monte Carlo (MC) simulations of the original system.

     

A direct probabilistic approach to solve state equations for nonlinear systems under random excitation

In this paper, a direct probabilistic approach (DPA) is presented to formulate and solve moment equations for nonlinear systems excited by environmental loads that can be either a stationary or nonstationary random process. The proposed method has the advantage of obtaining the response’s moments directly from the initial conditions and statistical characteristics of the corresponding external exci-tations. First, the response’s moment equations are directly derived based on a DPA, which is completely independent of the It?/filtering approach since no specific assumptions regarding the correlation structure of excitation are made. By solving them under Gaussian closure, the response’s moments can be obtained. Subsequently, a multiscale algo-rithm for the numerical solution of moment equations is exploited to improve computational efficiency and avoid much wall-clock time. Finally, a comparison of the results with Monte Carlo (MC) simulation gives good agreement. Furthermore, the advantage of the multiscale algorithm in terms of efficiency is also demonstrated by an engineering example.     

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.     

非高斯随机激励下滞迟系统响应与首次穿越失效

针对由有界噪声、泊松白噪声和高斯白噪声共同构成的非高斯随机激励,通过Monte Carlo数值模拟方法研究了此激励作用下双线性滞迟系统和Bouc-Wen滞迟系统这两类经典滞迟系统的稳态响应与首次穿越失效时间。一方面,分析了有界噪声和泊松白噪声这两种分别具有连续样本函数和非连续样本函数的非高斯随机激励,在不同激励参数条件下对双线性滞迟系统和Bouc-Wen滞迟系统的稳态响应概率密度、首次穿越失效时间概率密度及其均值的不同影响;另一方面,揭示了在这类非高斯随机激励荷载作用下,双线性滞迟系统的首次穿越失效时间概率密度将出现与Bouc-Wen滞迟系统的单峰首次穿越失效时间概率密度截然不同的双峰形式。     

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.     

随机干扰与随机参数激励联合作用下的Hopf分叉

本文研究van der Pol-Duffing型的非线性振子在随机干扰和随机参数联合作用下的Hopf分叉现象。本文所得结果证实了当系统处在于Hopf分叉点附近时,对系统的参数的变化具有敏感性。在研究过程中,我们利用Markov扩散过程逼近系统的随机响应,得到了沿稳定矩的概率1稳定和矩稳定的条件。对于非线性振子,我们得到了振幅过程的稳态概论密度函数。研究发现,确定性系统的Hopf分叉点在随机参数作用下具有漂移现象,这种漂移是由系统的性质所决定的,当分叉点为超临界的,分叉点向前漂移;而当分叉点为亚临界时,这种漂移是向后的。当系统处在外部随机干扰作用下时,系统出现非零响应。另外我们发现,稳态矩的分叉与其阶数无关。     

耦合Duffing-van der Pol系统的首次穿越问题

利用拟不可积Hamilton系统随机平均法，研究了高斯白噪声激励下耦 合Duffing-van der Pol系统的首次穿越问题. 首先给出了条件可靠性函数满足的后向 Kolmogorov 方程以及首次穿越时间条件矩满足的广义Pontryagin方程. 然后根据 这两类偏微分方程的边界条件和初始条件，详细分析了在外激与参激共 同作用以及纯外激作用等情况下系统的可靠性与首次穿越时间的各阶矩. 最后以图表形式给 出了可靠性函数、首次穿越时间的概率密度以及平均首次穿越时间的数值结果.