Dynamic response of structures with uncertain parameters: A comparative study of probabilistic and fuzzy sets models

This paper presents the results of a study on the response of structures with uncertain properties such as mass, stiffness and damping. The effect of the uncertain parameters on the response and the effect of the modelling of the uncertainties on the response are investigated. In particular, two types of uncertainties are distinguished: random and fuzzy uncertainties. Two kinds of models are studied: probabilistic and fuzzy set models. The two approaches to uncertainty modelling are compared with regard to their impacts on the analysis and on the uncertain structural response obtained. The study considers free vibration, forced vibration with deterministic excitation, and forced vibration with Gaussian white noise excitation. It is concluded that, in general, fuzzy models are much easier to implement and the associated analysis easier to perform than their probabilistic counterparts. When the available data on the structural parameters are crude and do not support a rigorous probabilistic model, the fuzzy set approach should be considered in view of its simplicity.     

Transient responses of dynamical systems with random uncertainties

A new approach is presented for modeling random uncertainties by a nonparametric model allowing transient responses of mechanical systems submitted to impulsive loads to be predicted in the context of linear structural dynamics. The probability model is deduced from the use of the entropy optimization principle whose available information involves the algebraic properties related to the generalized mass, damping and stiffness matrices which have to be positive-definite symmetric matrices, and the knowledge of these matrices for the mean reduced matrix model. An explicit construction and representation of the probability model have been obtained and are very well suited to algebraic calculus and to Monte Carlo numerical simulation in order to compute the transient responses of structures submitted to impulsive loads. Finally, a simple example is presented.     

Entropy propagation analysis in stochastic structural dynamics: application to a beam with uncertain cross sectional area

This paper investigates the impact of different probabilistic models of uncertain parameters on the response of a dynamical structure. The probabilistic models of the uncertain parameters are constructed using the maximum entropy principle, where different information is considered, such as bounds, mean value, etc. Nested probabilistic models are constructed with increasing information; as the information given increases, the level of entropy of the input model decreases. The response of the linear dynamical model is given in the frequency domain, and the propagation of the input uncertainty throughout the computational model is analyzed in terms of Shannon’s entropy. Low and high frequencies are analyzed because uncertainties propagate differently depending on the frequency band. A beam discretized by means of the finite element method with random cross sectional area (random field) is the application analyzed.     

Generalized probabilistic approach of uncertainties in computational dynamics using random matrices and polynomial chaos decompositions

A new generalized probabilistic approach of uncertainties is proposed for computational model in structural linear dynamics and can be extended without difficulty to computational linear vibroacoustics and to computational non‐linear structural dynamics. This method allows the prior probability model of each type of uncertainties (model‐parameter uncertainties and modeling errors) to be separately constructed and identified. The modeling errors are not taken into account with the usual output‐prediction‐error method, but with the nonparametric probabilistic approach of modeling errors recently introduced and based on the use of the random matrix theory. The theory, an identification procedure and a numerical validation are presented. Then a chaos decomposition with random coefficients is proposed to represent the prior probabilistic model of random responses. The random germ is related to the prior probability model of model‐parameter uncertainties. The random coefficients are related to the prior probability model of modeling errors and then depends on the random matrices introduced by the nonparametric probabilistic approach of modeling errors. A validation is presented. Finally, a future perspective is introduced when experimental data are available. The prior probability model of the random coefficients can be improved in constructing a posterior probability model using the Bayesian approach. Copyright © 2009 John Wiley & Sons, Ltd.     

Computational stochastic statics of an uncertain curved structure with geometrical nonlinearity in three-dimensional elasticity

A methodology for analyzing the large static deformations of geometrically nonlinear structural systems in the presence of both system parameters uncertainties and model uncertainties is presented. It is carried out in the context of the identification of stochastic nonlinear reduced-order computational models using simulated experiments. This methodology requires the knowledge of a reference calculation issued from the mean nonlinear computational model in order to determine the POD basis (Proper Orthogonal Decomposition) used for the mean nonlinear reduced-order computational model. The construction of such mean reduced-order nonlinear computational model is explicitly carried out in the context of three-dimensional solid finite elements. It allows the stochastic nonlinear reduced-order computational model to be constructed in any general case with the nonparametric probabilistic approach. A numerical example is then presented for a curved beam in which the various steps are presented in details.     

Explicit finite element artificial boundary scheme for transient scalar waves in two‐dimensional unbounded waveguide

To simulate the transient scalar wave propagation in a two‐dimensional unbounded waveguide, an explicit finite element artificial boundary scheme is proposed, which couples the standard dynamic finite element method for complex near field and a high‐order accurate artificial boundary condition (ABC) for simple far field. An exact dynamic‐stiffness ABC that is global in space and time is constructed. A temporal localization method is developed, which consists of the rational function approximation in the frequency domain and the auxiliary variable realization into time domain. This method is applied to the dynamic‐stiffness ABC to result in a high‐order accurate ABC that is local in time but global in space. By discretizing the high‐order accurate ABC along artificial boundary and coupling the result with the standard lumped‐mass finite element equation of near field, a coupled dynamic equation is obtained, which is a symmetric system of purely second‐order ordinary differential equations in time with the diagonal mass and non‐diagonal damping matrices. A new explicit time integration algorithm in structural dynamics is used to solve this equation. Numerical examples are given to demonstrate the effectiveness of the proposed scheme. Copyright © 2011 John Wiley & Sons, Ltd.     

计算模型修正的频率响应函数方法

本文提出一种有限元动力分析计算模型修正的频率响应函数方法。这个方法综合考虑了系统的质量、刚度,特别是阻尼矩阵的修正,由位置矩阵将修正量集中体现在测量自由度上,利用结构系统的频率响应函数有限元分析值和实测值,计算出模型的质量、阻尼和刚度矩阵的修正量。文中算例结果表明,本方法有较好的修正结果。     

一类非线性不确定结构系统的鲁棒饱和主动控制

对一类非线性不确定结构系统研究了在外部干扰下的鲁棒饱和主动控制策略.将质量、阻尼、刚度矩阵都存在不确定的结构系统描述为一种不确定中立系统,采用鲁棒饱和主动控制策略,提出基于线性矩阵不等式的代数解.通过提出的控制方法同时保证了结构系统在模型和参数摄动下以及在控制变量和扰动存在非线性不确定性时的鲁棒稳定.最后通过对一个4层建筑大楼在地震波作用下的主动振动控制仿真证明了该方法的有效性.     

阻尼矩阵与刚度矩阵的一种直接修正方法

提出了一种利用复模态测量数据同时修正有限元阻尼与刚度矩阵的有效方法。借助于矩阵的奇异值分解得到了满足动力方程的最小修正矩阵。该方法有一个简洁的表达式,修正过程简单而且容易实现,数值试验表明修改后的结构参数能准确地同试验值吻合。     

Probabilistic analysis of a pull-out test

This paper presents a sensitivity analysis of the pull-out strength of reinforcement embedded in concrete. Considering both European and French design codes, this failure strength depends on the variability of uncertain parameters such as Young’s modulus of concrete and yield stresses of materials (concrete and steel); moreover, two failure modes can be observed in the studied experimental test. A methodology allowing the characterization of the sensitivity of the pull-out strength to these uncertain parameters is derived. These parameters are modeled by Lognormal random variables. Results show the evolution of the pull-out strength for different anchorage lengths. Probability density functions of the random variable modeling the failure strength are computed using probabilistic methods. A finite element model is also built to quantify uncertainties concerning failure modes, computing 95% confidence intervals.     

Generalized stochastic approach for constitutive equation in linear elasticity: a random matrix model

This work is concerned with the construction of stochastic models for random elasticity matrices, allowing either for the generation of elasticity tensors exhibiting some material symmetry properties almost surely (integrating the statistical dependence between the random stiffness components) or for the modeling of random media that requires the mean of a stochastic anisotropy measure to be controlled apart from the level of statistical fluctuations. To this aim, we first introduced a decomposition of the stochastic elasticity tensor on a deterministic tensor basis and considered the probabilistic modeling of the random components, having recourse to the MaxEnt principle. Strategies for random generation and estimation were further reviewed, and the approach was exemplified in the case of a material that was transversely isotropic almost surely. In a second stage, we made use of such derivations to propose a generalized model for random elasticity matrices that took into account, almost separately, constraints on both the level of stochastic anisotropy and the level of statistical fluctuations. An example was finally provided and showed the efficiency of the approach. Copyright © 2011 John Wiley & Sons, Ltd.     

Vibration analysis of beams with non‐local foundations using the finite element method

In this paper, a non‐local viscoelastic foundation model is proposed and used to analyse the dynamics of beams with different boundary conditions using the finite element method. Unlike local foundation models the reaction of the non‐local model is obtained as a weighted average of state variables over a spatial domain via convolution integrals with spatial kernel functions that depend on a distance measure. In the finite element analysis, the interpolating shape functions of the element displacement field are identical to those of standard two‐node beam elements. However, for non‐local elasticity or damping, nodes remote from the element do have an effect on the energy expressions, and hence the damping and stiffness matrices. The expressions of these direct and cross‐matrices for stiffness and damping may be obtained explicitly for some common spatial kernel functions. Alternatively numerical integration may be applied to obtain solutions. Numerical results for eigenvalues and associated eigenmodes of Euler–Bernoulli beams are presented and compared (where possible) with results in literature using exact solutions and Galerkin approximations. The examples demonstrate that the finite element technique is efficient for the dynamic analysis of beams with non‐local viscoelastic foundations. Copyright © 2007 John Wiley & Sons, Ltd.     

Prediction of material coupling effect on structural damping of composite beams and blades

The present paper investigates the effect of material coupling on static and modal characteristics of composite structures. Incorporation of stiffness and damping coupling terms into a beam formulation yields equivalent section stiffness and damping properties. Building upon the damping mechanics, an extended beam finite element is developed capable of providing the stiffness and damping matrices of the structure. Validation cases on beams and blades demonstrate the importance of all stiffness and damping terms. Numerical results validate the predicted effect of material coupling on static characteristics of composite box-section beams. The effect of the full coupling damping matrices on modal frequencies and structural modal damping of composite beams is investigated. Box-section beams and small blade models with various ply angle laminations at the girder segments are considered. Finally, the developed finite element is applied to the prediction of the modal characteristics of a 19 m realistic wind-turbine model blade.     

Non‐Gaussian positive‐definite matrix‐valued random fields with constrained eigenvalues: Application to random elasticity tensors with uncertain material symmetries

This paper is devoted to the construction of a class of prior stochastic models for non‐Gaussian positive‐definite matrix‐valued random fields. The proposed class allows the variances of selected random eigenvalues to be specified and exhibits a larger number of parameters than the other classes previously derived within a nonparametric framework. Having recourse to a particular characterization of material symmetry classes, we then propose a mechanical interpretation of the constraints and subsequently show that the probabilistic model may allow prescribing higher statistical fluctuations in given directions. Such stochastic fields turn out to be especially suitable for experimental identification under material symmetry uncertainties, as well as for the development of computational multi‐scale approaches where the randomness induced by fine‐scale features may be taken into account. We further present a possible strategy for inverse identification, relying on the sequential solving of least‐square optimization problems. An application is finally provided. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

独立矩形截面超高层建筑的顺风向气动阻尼风洞试验研究

通过37个超高层建筑气动弹性模型的风洞试验,利用随机减量法从模型的风致加速度响应中识别了气动阻尼,并通过与前人相关研究成果及基于准定常理论的计算结果的比较验证了识别结果的正确性。在此基础上,研究了独立矩形截面超高层建筑顺风向气动阻尼的变化规律,考察了质量密度比、广义刚度、结构阻尼比、高宽比、宽厚比及风场类型对建筑结构气动阻尼比的影响。研究结果表明：超高层建筑顺风向气动阻尼比随折减风速变化的曲线近似一条单调增加的直线;结构阻尼比、质量密度比、宽厚比、折减风速、高宽比是影响顺风气动阻尼的比较重要的参数;广义刚度、风场类型相对影响较小。基于这些研究数据,拟合了超高层建筑顺风向气动阻尼比的经验公式,供工程设计人员参考。     

Karhunen–Loève decomposition of random fields based on a hierarchical matrix approach

The simulation of the behavior of structures with uncertain properties is a challenging issue, because it requires suitable probabilistic models and adequate numerical tools. Nowadays, it is possible to perform probabilistic investigations of the structural performance, which take into account a space‐variant uncertainty characterization of the structures. Given a structural solver and the probabilistic models, the reliability analysis of the structural response depends on the continuous random fields approximation, which is carried out by means of a finite set of random variables. The paper analyzes the main aspects of discretization in the case of 2D problems. The combination of the well‐known Karhunen–Loève series expansion, the finite element method and the hierarchical matrices approach is proposed in the paper. Copyright © 2013 John Wiley & Sons, Ltd.     

Model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics

A priori model reduction methods based on separated representations are introduced for the prediction of the low frequency response of uncertain structures within a parametric stochastic framework. The proper generalized decomposition method is used to construct a quasi‐optimal separated representation of the random solution at some frequency samples. At each frequency, an accurate representation of the solution is obtained on reduced bases of spatial functions and stochastic functions. An extraction of the deterministic bases allows for the generation of a global reduced basis yielding a reduced order model of the uncertain structure, which appears to be accurate on the whole frequency band under study and for all values of input random parameters. This strategy can be seen as an alternative to traditional constructions of reduced order models in structural dynamics in the presence of parametric uncertainties. This reduced order model can then be used for further analyses such as the computation of the response at unresolved frequencies or the computation of more accurate stochastic approximations at some frequencies of interest. Because the dynamic response is highly nonlinear with respect to the input random parameters, a second level of separation of variables is introduced for the representation of functions of multiple random parameters, thus allowing the introduction of very fine approximations in each parametric dimension even when dealing with high parametric dimension. Copyright © 2011 John Wiley & Sons, Ltd.     

基于随机激励响应的时变系统物理参数子空间识别方法研究

该文提出了一种基于任意组合的DVA(displacement,velocity and acceleration)及输入激励识别时变系统物理参数的子空间方法。该方法以任意组合的位移、速度、加速度(DVA)随机响应信号为测量信息,仅利用一组输入、输出信号组成Hankel矩阵,通过奇异值分解的方法识别出等效状态的系统矩阵,然后运用推导出来的通用时变系统的转换矩阵,将等效状态系统矩阵转换成为实际物理状态下的系统矩阵,从而识别出实际系统的质量矩阵、刚度矩阵、阻尼矩阵。以二自由度弹簧-质量-阻尼模型为算例,研究了突变、线性变化和周期变化三种变化形式下物理参数的识别,并讨论了不同噪信比下噪声对识别结果的影响,仿真算例验证了该方法的正确性和有效性。