含有摇摆装置的双层桥墩结构随机振动分析

基于一种下层含有摇摆装置的双层桥墩结构模型，建立了系统的随机动力学系统模型，其中随机激励选用高斯白噪声模型，自复位恢复力采用经典旗帜形模型。运用广义谐波平衡法将旗帜形滞回力近似分解为幅值依赖的等效拟线性弹性力和拟线性阻尼力，获得原系统的等效随机系统；采用标准随机平均法理论，将等效系统近似为关于幅值的平均伊藤随机微分方程，建立并求解与之对应的Fokker-Planck-Kolmogorov(FPK)方程获得系统的稳态响应。探讨系统参数，如能量耗散系数等对系统稳态响应的影响，并通过Monte Carlo数值模拟加以验证。另外，借助Laplace变换，得到等效系统的转换函数及条件功率谱密度，结合下层幅值的稳态概率密度，得到下层幅值响应的功率谱密度估计。     

导电梁在磁场中的磁弹性随机振动

根据磁弹性基本理论和连续体的随机振动理论，得到了在外加磁场中通有随机电流的梁的磁弹性随机振动方程。给出了导电梁在耦合场中的洛伦兹力及力矩，并将洛沦兹力的耦合项假设为梁的一种阻尼，另一部分假设为随机分布载荷，对梁的位移响应进行了分析。以简支梁为例，分别得到了外加磁场中通入平稳和非平稳随机电流时简支梁的位移响应的均值、自相关函数、功率谱密度函数等数字特征。最后以一个通有理想白噪声平稳随机电流的简支梁为实际算例，对其位移响应的功率谱密度函数进行了计算。图形形状的改变表明了耦合项对梁的功率谱密度函数的影响，据此可知通过控制随机电流和磁场可以达到控制梁的随机振动的目的。     

非线性成层地基下基岩平稳地震动输入

采用二次反演计算基岩的平稳地震动输入。首先根据《建筑抗震设计规范》（GB 50011—2001）的地震影响系数曲线，利用加速度峰值等效原则迭代计算得到地面加速度平稳功率谱密度；然后通过等价线性化法计算地基各土层的剪切模量和阻尼比，将地面加速度平稳功率谱密度作为地面边界上的虚拟稳态输入，运用半无限长剪切梁上的一维波动方程，通过非线性迭代分析求得非线性分层地基下基岩的加速度平稳功率谱密度。该方法可为缺乏地面或基岩地震加速度记录资料的地区的堤坝等大型上部结构及隧道、管道等建筑物的随机地震响应分析提供较为可靠准确的基岩地震动输入，并可得到地基中不同深度的加速度、剪应变等动力响应的稳态峰值期望值，为应用随机理论进行工程抗震计算提供了依据。     

非平稳地震动过程模拟的谱表示-随机函数方法

在Priestley演变谱理论的基础上,采用随机函数的思想,建议了一类新的全非平稳过程模拟的谱表示-随机函数方法。在谱表示-随机函数方法中,实现了用2个基本随机变量即可精确表达原随机过程的目的。通过选取基本随机变量的离散代表点集,可以直接由演变功率谱密度函数生成具有给定赋得概率的代表性样本集合。以全非平稳地震动加速度过程的演变功率谱为例,验证了本文方法的有效性和优越性。最后,结合概率密度演化方法,进行了Duffing振子的随机地震反应分析与抗震可靠度计算。     

导电薄板在磁场中的磁弹性随机振动

根据磁弹性基本理论和连续体的随机振动理论,得到了在外加磁场中通有随机电流的导电薄板的磁弹性随机振动方程。给出了导电薄板在耦合场中的洛伦兹力及力矩,并将洛伦兹力的耦合项假设为薄板一种阻尼,另一部分假设为随机分布载荷,对板的随机位移响应进行了分析。以四边简支矩形薄板为例,分别得到了外加磁场中通入平稳和非平稳随机电流时导电板的随机位移响应的均值、自相关函数、功率谱密度函数等数字特征。     

不同支撑形式的梁在地震作用下的随机响应分析

基于随机振动理论,建立了梁结构在随机地震动作用下的振动方程。将地震地面运动考虑为随机过程,利用地震响应谱方法,推导了结构在随机地震动作用下最大位移响应的计算过程。以某简支梁和悬臂梁为例,推导了其位移功率谱密度函数、峰值位移功率谱密度函数及最大位移反应的功率谱密度函数。     

平稳随机激励下非黏滞阻尼系统功率谱密度函数的灵敏度分析EI北大核心CSCD

功率谱密度(power spectral density,PSD)函数的灵敏度分析是实现结构系统在随机激励下梯度优化算法的基础。区别于黏性阻尼模型假设阻尼力正比于瞬时速度,非黏滞阻尼模型的阻尼力与质点的时间历程相关,因而能够更准确地描述黏弹性材料的耗能特性。针对卷积型非黏滞阻尼系统PSD函数的灵敏度求解问题,利用虚拟激励法(pseudo-excitation method,PEM)将平稳随机激励下非黏滞阻尼系统的随机响应问题等效转化为确定性的简谐响应问题;利用直接微分法推导出PSD函数的灵敏度表达式;分别引入基于复模态的一阶、二阶近似法和基于实模态的迭代法构建PSD函数的灵敏度算法;通过数值算例比较三种方法的计算精度和效率。结果表明,迭代法更适合大规模非黏滞阻尼系统PSD函数的灵敏度求解。     

单边碰撞分数阶Rayleigh振子的随机P-分岔EI北大核心CSCD

基于非光滑变换和随机平均法分析了随机激励下含有分数阶微分的Rayleigh振子碰撞振动系统的随机P-分岔问题。基于Caputo定义计算了分数阶导数,将分数阶微分等效为相应的阻尼力与恢复力,并用非光滑变换将原系统等效为一个新的不含速度跳的系统;基于随机平均法建立了随机伊藤方程,得到了随机响应的Markovian近似,进而计算出系统的概率密度函数及其稳态解;引入突变理论推导出随机P-分岔的临界参数条件表达式,并分析了分数阶系数、分数阶阶次、恢复系数等主要参数对分数阶Rayleigh振子碰撞系统发生分岔的影响。     

van der Pol振子对高斯白噪声的响应

在过去20多年中,若干作者曾用不同方法研究过van der Pol振子对高斯白噪声的稳态响应,得到了不同甚至相反的结论。本文对此问题作了更为深入的研究,结果表明,在非线性不大时,随机平均法与等效非线性微分方程法给出良好的近似,而等效线性化法与高斯截断法给出错误的结论。     

基于Melnikov方法的分数阶Duffing振子混沌阈值解析研究

研究了含有分数阶微分项的Duffing振子的分岔与混沌行为,利用等效刚度和等效阻尼的概念对分数阶微分项进行处理,将分数阶微分项等效成三角函数与指数函数的形式,用Melnikov方法分析了分数阶Duffing振子产生分岔与混沌的必要条件,得到了其解析结果。进行了解析解和数值解的比较,证明了解析结果的精确度,并通过仿真计算研究了分数阶的阶次和系数对系统产生混沌必要条件的影响。在数值模拟过程中,还发现分数阶Duffing振子中存在双稳态特性,从两个稳态解出发,随着外激励参数的变化都能通过倍周期分岔到达混沌的状态。通过分析系统的动力学响应验证了这一现象。     

Survival probability determination of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitation

An approximate analytical technique based on a combination of statistical linearization and stochastic averaging is developed for determining the survival probability of stochastically excited nonlinear/hysteretic oscillators with fractional derivative elements. Specifically, approximate closed form expressions are derived for the oscillator non-stationary marginal, transition, and joint response amplitude probability density functions (PDF) and, ultimately, for the time-dependent oscillator survival probability. Notably, the technique can treat a wide range of nonlinear/hysteretic response behaviors and can account even for evolutionary excitation power spectra with time-dependent frequency content. Further, the corresponding computational cost is kept at a minimum level since it relates, in essence, only to the numerical integration of a deterministic nonlinear differential equation governing approximately the evolution in time of the oscillator response variance. Overall, the developed technique can be construed as an extension of earlier efforts in the literature to account for fractional derivative terms in the equation of motion. The numerical examples include a hardening Duffing and a bilinear hysteretic nonlinear oscillators with fractional derivative terms. The accuracy degree of the technique is assessed by comparisons with pertinent Monte Carlo simulation data.     

Response spectral density determination for nonlinear systems endowed with fractional derivatives and subject to colored noise

A computationally efficient method for determining the response of non-linear stochastic dynamic systems endowed with fractional derivative elements subject to stochastic excitation is presented. The method relies on a spectral representation both for the system excitation and its response. Specifically, first the ordinary non-linear differential equation of motion is transferred into a set of non-linear algebra equations by employing the harmonic balance method. Next, the response Fourier coefficients are determined by solving these non-linear equations. Finally, repeated use of the proposed procedure yields the response power spectral density. Pertinent numerical examples, including a fractional Duffing and a bilinear oscillator, demonstrate the accuracy of the proposed method.     

Stochastic response of fractionally damped beams

This paper aims at introducing the governing equation of motion of a continuous fractionally damped system under generic input loads, no matter the order of the fractional derivative. Moreover, particularizing the excitation as a random noise, the evaluation of the power spectral density performed in frequency domain highlights relevant features of such a system.Numerical results have been carried out considering a cantilever beam under stochastic loads. The influence of the fractional derivative order on the power spectral density response has been investigated, underscoring the damping effect in reducing the power spectral density amplitude for higher values of the fractional derivative order. Finally, the fractional derivative term introduces in the system dynamics both effective damping and effective stiffness frequency dependent terms.     

On the determination of the power spectrum of randomly excited oscillators via stochastic averaging: An alternative perspective

An approximate formula which utilizes the concept of conditional power spectral density (PSD) has been employed by several investigators to determine the response PSD of stochastically excited nonlinear systems in numerous applications. However, its derivation has been treated to date in a rather heuristic, even “unnatural” manner, and its mathematical legitimacy has been based on loosely supported arguments. In this paper, a perspective on the veracity of the formula is provided by utilizing spectral representations both for the excitation and for the response processes of the nonlinear system; this is done in conjunction with a stochastic averaging treatment of the problem. Then, the orthogonality properties of the monochromatic functions which are involved in the representations are utilized. Further, not only stationarity but ergodicity of the system response are invoked. In this context, the nonlinear response PSD is construed as a sum of the PSDs which correspond to equivalent response amplitude dependent linear systems. Next, relying on classical excitation-response PSD relationships for these linear systems leads, readily, to the derivation of the formula for the determination of the PSD of the nonlinear system. Related numerical results are also included.     

Survival probability of nonlinear oscillators endowed with fractional derivative element and subjected to evolutionary excitation: A stochastic averaging treatment with path integral concepts

An analytical method for determining stochastic response and survival probability of nonlinear oscillators endowed with fractional element and subjected to evolutionary excitation is developed in this paper. This is achieved by the variational formulation of the recently developed analytical Wiener path integral (WPI) technique. Specifically, the stochastic average/linearization treatment of the fractional-order non-linear equation of motion yields an equivalent linear time-varying substitute with integer-order derivative. Next, relying on the path integral technique, a closed-form analytical approximation of the response joint transition probability density function (PDF) for small intervals is obtained. Further, a combination of the derived joint transition PDF and the discrete version of Chapman–Kolmogorov (C–K) equation, leads to analytical solution of the non-stationary response and survival probability of non-linear oscillator under the evolutionary excitation. Finally, pertinent numerical examples, including a hardening Duffing and a bi-linear hysteretic oscillator, are considered to demonstrate the reliability of the proposed technique.     

Response of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitations: A Path Integral approach based on Laplace’s method of integration

In this paper, an approximate analytical technique is developed for determining the non-stationary response amplitude probability density function (PDF) of nonlinear/hysteretic oscillators endowed with fractional element and subjected to evolutionary excitations. This is achieved by a novel formulation of the Path Integral (PI) approach. Specifically, a stochastic averaging/linearization treatment of the original fractional order governing equation of motion yields a first-order stochastic differential equation (SDE) for the oscillator response amplitude. Associated with this first-order SDE is the Chapman–Kolmogorov (CK) equation governing the evolution in time of the non-stationary response amplitude PDF. Next, the PI technique is employed, which is based on a discretized version of the CK equation solved in short time steps. This is done relying on the Laplace’s method of integration which yields an approximate analytical solution of the integral involved in the CK equation. In this manner, the repetitive integrations generally required in the classical numerical implementation of the procedure are avoided. Thus, the non-stationary response amplitude PDF is approximately determined in closed-form in a computationally efficient manner. Notably, the technique can also account for arbitrary excitation evolutionary power spectrum forms, even of the non-separable kind. Applications to oscillators with Van der Pol and Duffing type nonlinear restoring force models, and Preisach hysteretic models, are presented. Appropriate comparisons with Monte Carlo simulation data are shown, demonstrating the efficiency and accuracy of the proposed approach.     

非线性系统响应功率谱密度的小波-Galerkin方法

发展了广义谐和小波在确定非线性系统随机动力响应中的应用。首先,利用周期广义谐和小波展开非线性动力微分方程,并考虑小波的联系系数后,可将动力微分方程转化为一组非线性代数方程。其次,利用Newton迭代法数值解答了非线性代数方程,得到了非线性动力响应的小波变换。最后,根据响应时变功率谱与各阶小波变换之间的关系,计算求得了非线性动力响应的功率谱密度。数值模拟显示了本文建议方法与Monte Carlo模拟之间的吻合程度。     

A Wiener path integral technique for determining the stochastic response of nonlinear oscillators with fractional derivative elements: A constrained variational formulation with free boundaries

A technique based on the concept of Wiener path integral (WPI) is developed for determining approximately the joint response probability density function (PDF) of nonlinear oscillators endowed with fractional derivative elements. Specifically, first, the dependence of the state of the system on its history due to the fractional derivative terms is accounted for, alternatively, by augmenting the response vector and by considering additional auxiliary state variables and equations. In this regard, the original single-degree-of-freedom (SDOF) nonlinear system with fractional derivative terms is cast, equivalently, into a multi-degree-of-freedom (MDOF) nonlinear system involving integer-order derivatives only. From a mathematics perspective, the equations of motion referring to the latter can be interpreted as constrained. Second, to circumvent the challenge of increased dimensionality of the problem due to the augmentation of the response vector, a WPI formulation with mixed fixed/free boundary conditions is developed for determining directly any lower-dimensional joint PDF corresponding to a subset only of the response vector components. This can be construed as an approximation-free dimension reduction approach that renders the associated computational cost independent of the total number of stochastic dimensions of the problem. Thus, the original SDOF oscillator joint PDF corresponding to the response displacement and velocity is determined efficiently, while circumventing the computationally challenging task of treating directly equations of motion involving fractional derivatives. Two illustrative numerical examples are considered for demonstrating the reliability of the developed technique. These pertain to a nonlinear Duffing and a nonlinear vibro-impact oscillators with fractional derivative elements subjected to combined stochastic and deterministic periodic loading. Note that alternative standard approximate techniques, such as statistical linearization, need to be significantly modified and extended to account for such cases of combined loading. Remarkably, it is shown herein that the WPI technique exhibits the additional advantage of treating such types of excitation in a straightforward manner without the need for any ad hoc modifications. Comparisons with pertinent Monte Carlo simulation data are included as well.     

First passage failure of SDOF nonlinear oscillator with lightly fractional derivative damping under real noise excitations

The first passage failure of single-degree-of-freedom (SDOF) nonlinear oscillator with lightly fractional derivative damping under real noise excitations is investigated in this paper. First, the system state is approximately represented by one-dimensional time-homogeneous diffusive Markov process of amplitude through stochastic averaging. Then, the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the conditional mean of first passage time are established from the averaged Itô equation for Hamiltonian. The conditional reliability function, the conditional probability density and mean of the first passage time are obtained by solving these equations together with suitable initial condition and boundary conditions. Finally, two examples are worked out in detail and the analytical solutions are checked by those from the Monte Carlo simulation of original systems.     

Spectral response of asymmetrical random oscillators

An analytical procedure to estimate the power spectral density (PSD) response of a weakly damped oscillator with a nonlinear asymmetrical restoring force under external stochastic wide-band excitation is presented. An equivalent linear system with random parameters is derived, from which the PSD is deduced. The use of corrective terms yields an approximation of the PSD in the region of the second and third harmonic resonant frequencies (as well as at higher harmonics). In the same way, the PSD of the reaction force is also reported. The method is applied to an impact oscillator with asymmetrical clearances and asymmetrical elastic stops.