Efficient spectral response of locally uncertain linear systems

It is difficult to model any real dynamical system with fully deterministic characteristics and yet, capture its behavior reasonably. Randomness arises from many sources, such as uncertain material properties, assumptions involved in structural modeling, and the stochastic nature of input forces. Thus, the random vibration analysis of systems with uncertain parameters is a crucial component of structural design and optimization procedures. In this paper, a new method is presented for fast spectral analysis of locally uncertain systems subjected to random inputs based on the response of one such system (called the nominal system). Unlike other methods, such as modal expansion methods, the proposed method is applicable to general uncertainties in the damping and stiffness matrices with the sole restriction that the system remains stable with probability one. Moreover, the proposed method yields exact responses for the perturbed systems and its accuracy is not affected by the size or magnitude of the uncertainties. However, the degree of locality of the uncertainty dictates the observed gains in computational efficiency when using the proposed method. When the uncertainty is extremely localized, one can expect gains in computational efficiency of two to three orders of magnitude while only modest gains of 2–3 times are observed when half the model is uncertain. Two numerical examples are presented to illustrate the accuracy and gains in computational efficiency of the proposed method.     

A two-step method for analysis of linear systems with uncertain parameters driven by Gaussian noise

A two-step method is proposed to find state properties for linear dynamic systems driven by Gaussian noise with uncertain parameters modeled as a random vector with known probability distribution. First, equations of linear random vibration are used to find the probability law of the state of a system with uncertain parameters conditional on this vector. Second, stochastic reduced order models (SROMs) are employed to calculate properties of the unconditional system state. Bayesian methods are applied to extend the proposed approach to the case when the probability law of the random vector is not available. Various examples are provided to demonstrate the usefulness of the method, including the random vibration response of a spacecraft with uncertain damping model.     

Stochastic response moments for linear systems

State space moment analysis is developed as a practical tool for investigating the response of a linear system subjected to stochastic excitation. General formulations are presented to show that the method can be used to evaluate response moments, or cumulants, of any order for both stationary and nonstationary response. The limitation is that the excitation of the linear system must be a generalized white noise called a delta-correlated process. This generalization of the Lyapunov method for finding response covariances gives a comparable matrix method for finding the higher order moments which are often important in predicting failure due to first-passage or fatigue. The technique used here involves rewriting the mth order tensor of mth order cumulants into a minimum length vector, making use of all inherent symmetry, in order to minimize the size of the resulting matrix. Easily implemented algorithms are presented for finding the terms in this matrix. General relationships are also given relating the eigenvalues and eigenvectors of this matrix for mth order cumulants to those of a much smaller matrix. This eigen solution is needed for evaluating nonstationary response cumulants, and the given relationships provide a particularly efficient method for evaluating the eigenvalues. The method is illustrated by evaluating the 35 fourth cumulants of nonstationary response for a class of two-degree-of-freedom oscillators.     

A very efficient moment calculation method for uncertain linear dynamic systems

The problem of calculating the uncertainty in the dynamic response of a structure due to uncertainties related to the modeling of its dynamic behavior, is addressed. Based on a Bayesian probabilistic approach, a new approximate numerical method is proposed to investigate the resulting uncertainties in the structural response. The proposed method provides a very efficient and accurate approach to the solution of stochastic finite-element models. It can be used to quantify the uncertainties in the predicted response of a structure during its design, where engineering judgement is used to quantify the uncertainties in the modeling process.     

不确定系统稳定性的仿射不等式分析

针对不确定系统的区间表示不能描述变量间的相关性,相应的区间算法容易导致误差爆炸的缺点,提出了不确定系统的仿射表示法及系统稳定性的仿射不等式判断方法.首先将系统中的不确定信息用仿射参数来表示,得到不确定控制系统传递函数的仿射形式,然后通过求解含仿射参数的不等式组求得了满足系统的稳定性条件时各噪声允许的范围.算例表明,由于考虑了变量间的相关性,相对于区间算法,所提方法可以在更大的不确定范围内判断出系统的稳定性,算例验证了该方法的有效性.     

Probabilistic analysis of linear elastic cracked structures with uncertain damage

This paper presents a simple and reliable method for the probabilistic characterization of the linear elastic response of cracked structures with uncertain damage. In particular, truss and frame structures with edge cracks of uncertain depth and location are considered. The method of analysis originates from an approach recently appeared in the literature, which is generalized to treat structures with cracks affected by uncertainty. According to this approach, the uncertainties are transformed into superimposed deformations depending on the distribution of internal forces and an iterative procedure is established to solve the resultant equations. The procedure is optimally tuned based on the convergence analysis. Several numerical tests evidence excellent accuracy and convergence qualities also in the case of multicracked structures with large fluctuation of damage.     

Solution of linear dynamic systems with uncertain properties by stochastic reduced order models

A novel method, referred to as the stochastic reduced order model (SROM) method, is proposed for finding statistics of the state of linear dynamic systems with random properties subjected to random noise. The method is conceptually simple, accurate, computationally efficient, and non-intrusive in the sense that it uses existing solvers for deterministic differential equations to find state properties.Bounds are developed on the discrepancy between the exact and the SROM solutions under some assumptions on system properties. The bounds show that the SROM solutions converge to the exact solutions as the SROM representation of the vector of random system parameters is refined. Numerical examples are presented to illustrate the implementation of the SROM method and demonstrate its accuracy and efficiency.     

Stability analysis and regularization of uncertain linear multi-objective integer optimization problems

This article presents a systematic approach to analysing linear integer multi-objective optimization problems with uncertainty in the input data. The goal of this approach is to provide decision makers with meaningful information to facilitate the selection of a solution that meets performance expectations and is robust to input parameter uncertainty. Standard regularization techniques often deal with global stability concepts. The concept presented here is based on local quasi-stability and includes a local regularization technique that may be used to increase the robustness of any given efficient solution or to transform efficient solutions that are not robust (i.e. unstable), into robust solutions. An application to a multi-objective problem drawn from the mining industry is also presented.     

Non-stationary random response analysis of structures with uncertain parameters

This paper proposes a non-stationary random response analysis method of structures with uncertain parameters. The structural physical parameters and the input parameters are considered as random variables or interval variables. By using the pseudo-excitation method and the direct differentiation method (DDM), the analytical expression of the time-varying power spectrum and the time-varying variance of the structure response can be obtained in the framework of first order perturbation approaches. In addition, the analytical expression of the first-order and second-order partial derivative (e.g., time-varying sensitivity coefficient) for the time-varying power spectrum and the time-varying variance of the structure response expressed via the uncertainty parameters can also be determined. Based on this and the perturbation technique, the probabilistic and non-probabilistic analysis methods to calculate the upper and lower bounds of the time-varying variance of the structure response are proposed. Finally the effectiveness of the proposed method is demonstrated by numerical examples compared with the Monte Carlo solutions and the vertex solutions.     

气动弹性系统的模型确认与鲁棒颤振分析

研究了气动弹性系统的不确定性建模和鲁棒颤振分析问题.将结构的不确定性考虑为参数形式,非定常气动力的不确定性考虑为参数和未建模动态两种形式,建立了不确定系统的线性分式变换模型.分别使用基于Carathe-dory-Fejer插值定理和Nevanlinna-Pick插值定理的模型集检验方法进行了模型确认,在时间域和频率域中对模型集的有效性进行了验证,确定了不确定性的幅值.对于模型确认得到的不确定气动弹性系统,使用μ分析方法进行了鲁棒颤振分析.计算中,飞行速度是作为给定参数而不再是作为摄动变量,由此得到的鲁棒稳定性边界是匹配点解.仿真数值结果给出了鲁棒颤振速度,表明了方法的有效性.     

Dynamic response and reliability analysis of structures with uncertain parameters

An original approach for dynamic response and reliability analysis of stochastic structures is proposed. The probability density evolution equation is established which implies that incremental rate of the probability density function is related to the structural response velocity. Therefore, the response analysis of stochastic structures becomes an initial‐value partial differential equation problem. For the dynamic reliability problem, the solution can be derived through solving the probability density evolution equation with an initial value condition and an absorbing boundary condition corresponding to specified failure criterion. The numerical algorithm for the proposed method is suggested by combining the precise time integration method and the finite difference method with TVD scheme. To verify and validate the proposed method, a SDOF system and an 8‐storey frame with random parameters are investigated in detail. In the SDOF system, the response obtained by the proposed method is compared with the counterparts by the exact solution. The responses and the reliabilities of a frame with random stiffness, subject to deterministic excitation or random excitation, are evaluated by the proposed method as well. The mean, the standard deviation and the reliabilities are compared, respectively, with the Monte Carlo simulation. The numerical examples verify that the proposed method is of high accuracy and efficiency. Moreover, it is found that the probability transition of structural responses is like water flowing in a river with many whirlpools, showing complexity of probability transition process of the stochastic dynamic responses. Copyright © 2004 John Wiley & Sons, Ltd.     

An efficient framework for the reliability-based design optimization of large-scale uncertain and stochastic linear systems

This paper is focused on the development of an efficient reliability-based design optimization algorithm for solving problems posed on uncertain linear dynamic systems characterized by large design variable vectors and driven by non-stationary stochastic excitation. The interest in such problems lies in the desire to define a new generation of tools that can efficiently solve practical problems, such as the design of high-rise buildings in seismic zones, characterized by numerous free parameters in a rigorously probabilistic setting. To this end a novel decoupling approach is developed based on defining and solving a limited sequence of deterministic optimization sub-problems. In particular, each sub-problem is formulated from information pertaining to a single simulation carried out exclusively in the current design point. This characteristic drastically limits the number of simulations necessary to find a solution to the original problem while making the proposed approach practically insensitive to the size of the design variable vector. To demonstrate the efficiency and strong convergence properties of the proposed approach, the structural system of a high-rise building defined by over three hundred free parameters is optimized under non-stationary stochastic earthquake excitation.     

Approximate methods of identifying dynamic systems

Translated from Izmeritel'naya Tekhnika, No. 7, pp. 8–9, July, 1992.     

相关激励作用下随机结构振动响应的统计分析

应用随机过程理论,以能量为变量,分析了随机结构振动响应的统计特性。结构受相关激励作用时,通过输入激励的解相关方法,将作用在结构上的相关激励转变为各个不相关激励的作用;分析结构的振动响应的统计特性时,计及响应特征频率的相关性,在响应特征频率满足高斯正交总体的假设下,推导出了随机结构振动响应分析的统计分析表达式。应用设计的实验件和试验验证了所提出的统计分析的正确性,通过和已存在的统计分析结果的比较,表明了统计分析具有更高的分析精度,能够定性和定量的给出随机结构振动响应的统计变化情况。     

Response analysis of nonlinear multi-degree-of-freedom systems with uncertain properties to non-Gaussian random excitations

The investigation reported in this paper is concerned with the development of an approach for response analysis of multi-degree-of-freedom (mdof) nonlinear systems with uncertain properties of large variations and under non-Gaussian nonstationary random excitations. The developed approach makes use of the stochastic central difference (SCD) method, time co-ordinate transformation (TCT), and adaptive time schemes (ATS). It is applicable to geometrically complicated systems idealized by the finite element method (FEM). For demonstration of its use and availability of results for direct comparison, a two-degree-of-freedom (tdof) nonlinear asymmetric system with uncertain natural frequencies and under Gaussian and non-Gaussian nonstationary random excitations is considered. Computed results obtained for the system with and without uncertain natural frequencies, and under Gaussian and non-Gaussian nonstationary random excitations are presented. It is concluded that the approach is relatively simple, accurate and efficient to apply.     

Probability density evolution method for dynamic response analysis of structures with uncertain parameters

Probability density evolution method is proposed for dynamic response analysis of structures with random parameters. In the present paper, a probability density evolution equation (PDEE) is derived according to the principle of preservation of probability. With the state equation expression, the PDEE is further reduced to a one-dimensional partial differential equation. The numerical algorithm is studied through combining the precise time integration method and the finite difference method with TVD schemes. The proposed method can provide the probability density function (PDF) and its evolution, rather than the second-order statistical quantities, of the stochastic responses. Numerical examples, including a SDOF system and an 8-story frame, are investigated. The results demonstrate that the proposed method is of high accuracy and efficiency. Some characteristics of the PDF and its evolution of the stochastic responses are observed. The PDFs evidence heavy variance against time. Usually, they are much irregular and far from well-known regular distribution types. Additionally, the coefficients of variation of the random parameters have significant influence on PDF and second-order statistical quantities of responses of the stochastic structure.The support of the Natural Science Funds for Distinguished Young Scholars of China (Grant No.59825105) and the Natural Science Funds for Innovative Research Groups of China (Grant No.50321803) are gratefully appreciated.     

Attenuating oscillations in uncertain dynamic systems

Two basic approaches exist for the design of robust H _∞-controllers used to optimally attenuate oscillations in uncertain dynamic systems. One of these is based on solving Riccati equations; the other approach involves a linear matrix-inequality (LMI) technique. It is shown that the Riccati equations associated with this problem, which contain additional parameters (scalings) as Lagrangian multipliers, are feasible only when the values of these parameters are within a parallelepiped whose boundaries are to be determined. A new algorithm for synthesizing a robust H _∞-controller, using the LMI technique, is suggested. The boundaries of the admissible values of the scalings are identified. An illustrative example is considered, which concerns the optimal attenuation of oscillations of a parametrically disturbed pendulum     

弹性-粘弹性复合结构系统的随机响应分析

本文建立在随机振动时域复模态分析的基础上,利用扩阶状态变量,将弹性－粘弹性复合结构系统的微分积分动力学方程变换成常规的状态方程,提出了一种分析弹性－粘弹性复合结构系统随机响应的方法,得到了弹性－粘弹性复合结构系统在平稳随机激励下响应相关函数矩阵的表达式,并对典型的平稳随机激励（平稳白噪声激励及平稳滤过白噪声激励）情形,进行了分析,得到了典型平稳随机激励下,弹性－粘弹性复合结构系统响应相关函数矩阵的复代数解析表达式。所提分析方法简便、易用,无论单自由度系统或多自由度系统均可适用。本文方法为粘弹性系统的随机响应分析提供了一条途径。     

Wind-excited response of structures with uncertain parameters

The wind-excited response of structures is classically evaluated by considering the model parameters as deterministic. Due to this assumption, the density function of the maximum response is so narrow and sharp as to make the expected maximum a suitable pseudo-deterministic representation of the maximum response. Based on Taylor series expansions retaining up to the first and second-order derivative terms, this paper provides closed form expressions of the first and second statistical moments of the maximum response taking the uncertainties of the parameters and the model error into account. It is shown that such uncertainties may spread and shift the density function of the maximum response to the point at which the classical value of the expected maximum is no longer representative of the structural behaviour.