Design-point excitation for non-linear random vibrations

It has been shown in recent years that certain non-linear random vibration problems can be solved by well established methods of time-invariant structural reliability, such as FORM and importance sampling. A key step in this approach is finding the design-point excitation, which is that realization of the input process that is most likely to give rise to the event of interest. It is shown in this paper that for a non-linear, elastic single-degree-of-freedom oscillator subjected to white noise, the design-point excitation is identical to the excitation that generates the mirror image of the free-vibration response when the oscillator is released from a target threshold. This allows determining the design-point excitation with a single non-linear dynamic analysis. With a slight modification, this idea is extended to non-white and non-stationary excitations and to hysteretic oscillators. In these cases, an approximate solution of the design-point excitation is obtained, which, if necessary, can be used as a ‘warm’ starting point to find the exact design point using an iterative optimization algorithm. The paper also offers a simple method for computing the mean out-crossing rate of a response process. Several examples are provided to demonstrate the application and accuracy of the proposed methods. The methods proposed in this paper enhance the feasibility of approximately solving non-linear random vibration problems by use of time-invariant structural reliability techniques.     

一种非高斯随机振动过程数值模拟方法

目的 考虑真实随机振动的非高斯特性,提出一种根据已知信息生成与其相符的非高斯随机振动过程的数值模拟方法。方法 基于均值、方差、偏斜度、峭度及功率谱密度函数（或自相关函数）等约束条件,对非高斯随机振动进行模拟。根据功率谱获取非高斯过程的自相关矩阵;通过Hermite多项式的正交性质和多项式混沌展开方法推导出的公式,构造满足标准正态分布随机过程的协方差矩阵,并对其进行谱分解和主成分分析;最后,利用Karhunen-Loeve展开和多项式混沌展开来表示所模拟的非高斯振动过程。结果 随着采样点个数的增加,实测数据与模拟数据之间的误差越来越小,该方法具有较好的模拟精度。结论 应用多项式混沌展开、Karhunen-Loeve展开以及蒙特卡洛等方法,可生成非高斯随机振动过程,并得到准确有效的各项统计参数模拟值。     

非高斯随机振动下包装件时变振动可靠性分析

包装件在流通过程中经常受到非高斯随机振动激励的作用,提出了一种包装件在非高斯随机振动激励条件下的时变可靠性的分析方法。结合多项式混沌扩展和Karhunen-Loeve扩展,提出了基于功率谱(或自相关函数)、均值、方差、偏斜度和峭度信息的非高斯随机振动激励的模拟方法;为减小数值分析量,应用拟蒙特卡洛法,在随机变量空间中合理控制变量的分布模拟非高斯随机振动激励,通过四阶龙格库塔法分析,用较少的随机振动模拟样本准确得到了包装件加速度响应的前四阶矩和自相关函数。基于响应的统计信息,应用该研究提出的多项式混沌扩展、Karhunen-Loeve扩展和拟蒙特卡洛分析,获得包装件加速度响应样本,计算包装件的时变可靠性,用原始蒙特卡洛法验证了计算的准确性;该方法在包装件的可靠性分析、包装系统优化等方面具有重要意义。     

Extreme value distributions for nonlinear transformations of vector Gaussian processes

Approximations are developed for the marginal and joint probability distributions for the extreme values, associated with a vector of non-Gaussian random processes. The component non-Gaussian processes are obtained as nonlinear transformations of a vector of stationary, mutually correlated, Gaussian random processes and are thus, mutually dependent. The multivariate counting process, associated with the number of level crossings by the component non-Gaussian processes, is modelled as a multivariate Poisson point process. An analytical formulation is developed for determining the parameters of the multivariate Poisson process. This, in turn, leads to the joint probability distribution of the extreme values of the non-Gaussian processes, over a given time duration. For problems not amenable for analytical solutions, an algorithm is developed to determine these parameters numerically. The proposed extreme value distributions have applications in time-variant reliability analysis of randomly vibrating structural systems. The method is illustrated through three numerical examples and their accuracy is examined with respect to estimates from full scale Monte Carlo simulations of vector non-Gaussian processes.     

Random response of Duffing oscillator to periodic excitation with random disturbance

This paper discusses the random response of a non-linear Duffing oscillator subjected to a periodic excitation with random phase modulation. Effects of uncertainty in the periodic excitation and level of the system non-linearity on the response moments and non-Gaussian nature of the response caused by both the system non-linearity and the non-Gaussian loading are investigated. Results are presented in terms of the second- and the fourth-order moments as well as the excess factor of the response and some results are compared with those from the Monte Carlo simulation. An iterated linearisation technique is proposed to improve the accuracy of the numerical results for strongly non-linear systems.     

铁路非高斯随机振动的数字模拟与包装件响应分析

目的研究铁路振动环境的非高斯特性,并分析包装件在非高斯随机振动环境条件下的响应情况。方法结合离散傅里叶变换与EARPG(1)模型,模拟了铁路随机振动信号。根据采集的数据的PSD曲线计算幅值,利用EARPG(1)模型生成了具有尖峰特征的模拟信号,计算了相位并进行了相位整体平移,根据幅值和相位,合成了所需的非高斯随机振动信号。将包装件简化为单自由度系统,分析了包装件在非高斯振动条件下的响应情况。结果铁路随机振动的峭度大于3,偏斜度为0,属于对称超高斯随机振动,提出的模型可准确模拟出铁路振动的非高斯特性,峭度和偏斜度的误差均小于3%,包装系统的固有频率、阻尼比、激励峭度对系统的响应的峭度、均方根均有较大的影响。结论通过合理地选择包装系统的固有频率和阻尼比,可有效减小系统的响应峭度和均方根,提高包装系统的可靠性。     

Non-Gaussian solution for random rocking of slender rigid block

This paper adopts a random vibration approach to study the response of the slender rigid block to seismic action. The problem is strongly non-linear because of (i) the restoring term and (ii) the quadratic dissipation of energy due to the inelastic impacts, modeled as an impulsive process. The excitation process is firstly assumed to be a Gaussian white noise; secondly, a non-stationary filtered Gaussian white noise is assumed to simulate seismic shaking more accurately. The solution of the associated Fokker-Planck equation in terms of moments of the response is obtained by means of a non-Gaussian closure technique, that enables the complete statistical definition of the approximated transient response process to be achieved. The mean upcrossing rates and the response spectra in terms of displacement are evaluated. The reliability of the solutions derived is assessed by comparing them with Monte Carlo simulations.     

高斯与非高斯随机振动对洗护产品包装的影响

目的 研究高斯激振力和非高斯激振力的随机振动对商品包装的影响差异，提供随机振动试验方法和参数，并对商品运输包装设计方案提供优化建议。方法 利用科学计量方法研究随机振动技术的发展态势。通过增大峭度的方法进行非高斯激振模拟随机振动试验。研究随机振动强度通过功率谱密度和疲劳损伤的表达，以及随机振动试验时间与疲劳损伤的表达关系。结果 电商渠道销售通过快递配送的洗洁精商品包裹优先采用ISTA 3A—2018 Packaged-Products for Parcel Delivery System Shipment 70 kg(150 lb) or less。使用Over-The-Road Trailer图谱和Pick-up and Delivery Vehicle图谱进行随机振动试验，通过加载峭度进行加速模拟试验和加强模拟试验。结论 在洗洁精商品包裹加载峭度为5、7、9条件下，能够复现实际配送中商品的货损。基于试验结果提出的包装设计修改方案在加强模拟试验条件下可以实现商品防护的功能。全渠道供应链配送商品包裹应采用高效、科学的实验室测试方法，为运输包装设计和验证提供方法和数据。     

泊松白噪声激励下斜拉索的面内随机振动

目前针对斜拉索非线性随机振动的研究已广泛开展,但仅限于高斯随机激励情形。然而,现实中大部分的随机扰动都是非高斯的。若使用高斯激励模型将产生较大误差。假设拉索所受非高斯激励为泊松白噪声,研究了泊松白噪声激励下斜拉索面内随机振动。推导了受泊松白噪声激励的斜拉索面内振动的随机微分方程,建立了支配系统平稳响应概率密度函数的广义FPK方程。提出迭代加权残值法求解了四阶广义FPK方程,得到了系统响应概率密度函数的近似稳态闭合解。考察了垂跨比、阻尼系数以及脉冲到达率对拉索面内随机振动响应的影响。结果表明:拉索的响应随着垂跨比的增大,响应呈现不对称现象愈加明显;随阻尼比增加,系统响应得到显著抑制;当脉冲到达率增大,拉索的响应也随之增大,并逐渐接近于高斯白噪声激励的情形。另外,获得的理论结果与蒙特卡罗模拟的结果吻合地非常好。     

窄带随机噪声参激作用下强非线性系统的响应

将改进的L-P(ModifiedLindstedt-Poincare)方法和随机多尺度法结合起来,研究了窄带随机噪声参激作用下强非线性VanderPol-Duffing系统的响应、稳定性和分叉问题。文中首先由改进的L-P方法引入变换参数,然后用多尺度法分离了系统的快变量,求出了最大Lyapunov指数的解析表达式,得到了系统几乎必然稳定的充分必要条件,讨论了系统的参数对稳定性的影响,分析了系统在1/2亚谐共振区的性态。理论分析与数值计算表明:在一定条件下系统将发生随机跳跃,且随着随机激励带宽的增大,扩散的极限环的宽度将逐渐增大。数值模拟结果表明:MLP方法结合随机多尺度法研究强非线性随机系统的响应、稳定性和分叉问题是有效的。     

随机激励作用下硬涂层薄板振动有限元分析及减振预估

硬涂层减振是一项新兴的阻尼减振技术,硬涂层复合结构在随机激励载荷作用下的振动特性建模与分析方法还未开展研究.将有限元建模方法和虚拟激励法(pseudo excitation method,PEM)相结合,实现了随机激励作用下硬涂层悬臂薄板结构振动建模及减振性能预估.首先,基于等效单层法创建了双面涂敷硬涂层的薄板结构振动...     

复合随机振动系统的动力可靠度分析

建议了一类新的复合随机振动系统动力可靠度分析方法。基于复合随机振动系统反应分析的密度演化方法,根据首次超越破坏准则,对密度演化方程施加相应的边界条件,进而求解密度演化方程,在安全域内积分给出结构的动力可靠度。结合精细时程积分方法与具有TVD性质的差分格式,研究了基于密度演化方法求解结构动力可靠度问题的数值方法。以受到随机地震作用、具有随机参数的八层层间剪切型结构为例,进行了结构动力可靠度分析并与随机模拟结果进行了比较。研究表明,建议的方法具有较高的精度和效率。     

Probability distributions of peaks and troughs of non-Gaussian random processes

This paper deals with the development of probability density functions applicable for peaks, troughs and peak-to-trough excursions of a non-Gaussian random process where the response of a non-linear system is represented in the form of Volterra's second-order functional series. The density functions of peaks and troughs are derived in closed form and presented separately. It is found that the probability density function applicable to peaks (and troughs) is equivalent to the density function of the envelope of a random process consisting of the sum of a narrow-band Gaussian process and sine wave having the same frequency. Furthermore, for a non-Gaussian random process for which the skewness of the distribution is less than 1.2, the density function of peaks (and troughs) can be approximately presented in the form of a Rayleigh distribution. The parameter of the Rayleigh distribution is given as a function of parameters representing the non-Gaussian characteristics. The results of comparisons between newly derived density functions and histograms of peaks, troughs and peak-to-trough excursions constructed from data with strong non-linear characteristics show that the distributions well represent the histograms for all cases.     

Numerical and experimental studies of linear systems subjected to non-Gaussian random excitations

This paper presents a numerical simulation scheme for generating symmetric non-Gaussian random processes governed by prescribed kurtosis and spectral density. The generated process is represented as a continuous stationary random signal with occasional spikes superimposed on a Gaussian random background. The generated time history data records are used to simulate random excitations acting on linear single-degree-of-freedom systems. The results of the numerical simulation are compared with those measured experimentally. For a wide-band random excitation with kurtosis close to 3, the response kurtosis is found to be very sensitive to small changes in the excitation kurtosis. This is manifested by the appearance of significant spikes in the time history records when the excitation records do not display any significant spikes. The influence of the system damping is also examined for narrow-band and wide-band random excitations, and some differences are reported in the results.     

Stochastic and chaotic sub- and superharmonic response of shallow cables due to chord elongations

The paper deals with the non-linear response of shallow cables driven by stochastically varying chord elongations caused by random vibrations of the supported structure. The chord elongation introduces parametric excitation in the linear stiffness terms of the modal coordinate equations, which are responsible for significant internal subharmonic and superharmonic resonances. Under harmonically varying support motions coupled ordered or chaotic in-plane and out-of-plane subharmonic and superharmonic periodic motions may take place. If the harmonically varying chord elongation is replaced by a zero-mean, stationary narrow-band random excitation with the same standard deviation and center frequency, qualitatively and quantitatively completely different modes of vibration are registered no matter how small the bandwidth of the excitation process is. Additionally, the stochastic excitation process tends to enhance chaotic behavior. Based on Monte Carlo simulation on a reduced non-linear two-degree-of freedom system the indicated effects have been investigated for stochastic subharmonic resonance of order 2:1, and stochastic superharmonic resonances of orders 1:2 and 2:3. By analyzing the responses for two chord elongation processes with almost identical auto-spectral density function, but completely different amplitudes, it is shown that the indicated qualitative and quantitative changes of the subharmonic resonance primarily are caused by the slowly varying phase of the stochastic excitation. The superharmonic stochastic responses are dominated by random jumps between a single mode in-plane and a coupled mode attractor, which are caused by the variation of the amplitude of the random excitation. Such jumps do not occur in the subharmonic response, because the single mode in-plane attractor is unstable.     

分裂导线扭转舞动分析的动力学建模

高压输电线路导线覆冰舞动分析的主要困难,是由导线的大幅运动与随导线几何位形而变化的非线性气动力之间的相互耦合产生的。对伴有大幅扭转运动的分裂导线的舞动,由于安装了间隔棒和防舞器,必须考虑导线结构产生的约束非线性和重力非线性。本文应用虚功原理建立了整个分裂导线结构总值形的基本虚功方程,分析推导了间隔棒和防舞器的非线性动态特性与非线性约束特性,在此基础上,最后得到了分裂导线结构舞动分析的动力学模型．所得结果为进一步分析计算分裂导线的舞动打下一个基础。     

The Green’s functions of a vertically inhomogeneous soil with a random dynamic shear modulus

This paper deals with the study of the Green’s functions of a layered soil with random characteristics. The dynamic shear modulus of the soil is modelled as a non-Gaussian random process that varies in the vertical direction and is characterized by a marginal probability density function and a correlation function. The stochastic finite element method is applied to a hybrid thin layer — direct stiffness formulation in order to obtain the stochastic system equations, which are solved by means of a Monte Carlo simulation. The influence of the variations of the dynamic shear modulus on the Green’s functions is illustrated for different excitation frequencies and receiver positions.     

Approximate solutions for non-linear random vibration problems

Two methods are described for obtaining approximate solutions to non-linear random vibration problems. The first method approximates the non-linear system with the linear system whose corresponding probability density function best solves the Fokker-Planck equation associated with the non-linear system. In the second method, a class of non-linear systems with known solutions to the Fokker-Planck equation is used to best approximate the non-linear system of interest. Two illustrative examples are presented and the results are compared with existing methods.     

模糊随机桁架结构在模糊随机激励下的动力响应

研究了模糊随机参数桁架结构在模糊随机荷载激励下的复合模糊随机振动动力响应的问题。同时考虑结构的物理参数、几何尺寸和外载荷幅值的模糊随机性,从Duham e l积分式出发,利用振型迭加法求出了结构动力响应模糊随机变量的表达式;再由随机函数的矩法推导出结构模糊随机动力响应的模糊数字特征。最后,通过算例考察了结构参数和作用荷载的模糊随机性对结构动力响应的影响,并用M on te C arlo数值法对算例进行模拟,验证了文中模型和分析方法是可行有效的。     

Linear models for non-Gaussian processes and applications to linear random vibration

Linear models are finite sums of specified deterministic, continuous functions of time with random coefficients. It is shown that linear models provide (i) accurate approximations for real-valued non-Gaussian processes with continuous samples defined on bounded time intervals, (ii) simple solutions for linear random vibration problems with non-Gaussian input, and (iii) efficient techniques for selecting optimal designs from collections of proposed alternatives. Theoretical arguments and numerical examples are presented to establish properties of linear models, illustrate the construction of linear models, solve linear random vibration with non-Gaussian input, and propose an approach for optimal design of linear dynamic systems. It is shown that the proposed linear model provides an efficient tool for analyzing linear systems in non-Gaussian environment.