Design optimization for dynamic response of vibration mechanical system with uncertain parameters using convex model

The concept of uncertainty plays an important role in the design of practical mechanical system. The most common methods for solving uncertainty problems are to model the parameters as a random vector. A natural way to handle the randomness is to admit that a given probability density function represents the uncertainty distribution. However, the drawback of this approach is that the probability distribution is difficult to obtain. In this paper, we use the non-probabilistic convex model to deal with the uncertain parameters in which there is no need for probability density functions. Using the convex model theory, a new method to optimize the dynamic response of mechanical system with uncertain parameters is derived. Because the uncertain parameters can be selected as the optimization parameters, the present method can provide more information about the optimization results than those obtained by the deterministic optimization. The present method is implemented for a torsional vibration system. The numerical results show that the method is effective.     

Non-probabilistic interval analysis method for dynamic response analysis of nonlinear systems with uncertainty

Effects of uncertainties on the dynamic response of the nonlinear vibration systems with general form are investigated. Based on interval mathematics, modeling the uncertain parameters as interval numbers, a non-probabilistic interval analysis method, which estimates the range of the nonlinear dynamic response with the help of Taylor series expansion, is presented, where the partial derivatives of the dynamic response with respect to uncertain parameters are considered to be interval numbers. The sensitivity matrices of dynamic response with the uncertain parameters are derived. For the presented method, only the bounds on uncertain parameters are needed, instead of probabilistic density distribution or statistical quantities. Numerical examples are used to illustrate the validity and feasibility of the presented method.     

基于观测器的模型不确定的耦合时空混沌H∞跟踪控制

考虑子系统的时空耦合作用及模型的不确定性,实现模型不确定的耦合时空混沌的 跟踪控制非常困难.然而耦合时空混沌的每个子系统用一系列模糊逻辑模型逼近,同时考虑子 系统状态的不可测性,采用模糊观测器来估计子系统的状态.由于混沌模型的很多参数和动态 特性很难准确地确定即模型具有不确定性,因此在用模糊模型逼近的同时定会产生建模误差. 基于模糊模型及状态观测器,考虑混沌模型的不确定性,提出一种Ｈ∞ 模糊跟踪控制方法,实现模型不确定性的耦合时空混沌的鲁棒跟踪控制.将控制方 案表征为求解线性矩阵不等式问题,并用凸优化方法 关键词： 耦合时空混沌 模糊模型 模糊观测器 Ｈ∞模糊跟踪控 制 线性矩阵不等式     

Assessment of uncertainty on structural dynamic responses with the short transformation method

The predictive capability of finite element models is limited by their deterministic nature: typically, not all model parameters are exactly known, while even small deviations may have significant effects on the predicted response. Parameter uncertainty should therefore be taken into account, e.g. with fuzzy arithmetic. The absence of fuzzy solvers led to interval arithmetic as a numerical alternative. The Transformation Method (TM), presented by M. Hanss, replaces interval arithmetic with a set of deterministic computations: for each interval, all parameter extrema are combined in every possible way. In a Design of Experiments terminology, the TM is a so-called Full Factorial design.The TM is applicable if the output is monotonic in the inputs. Unlike interval arithmetic, it does not overestimate the response uncertainty, as only parameter combinations are evaluated that actually occur. In this paper, the TM has been applied to visualise uncertain frequency response functions (FRFs), obtained with modal superposition. This yields accurate results when validated against Monte Carlo data, but the computation time is rather high. The Short Transformation Method (STM) is proposed as an attractive alternative to the original TM. A full set of deterministic computations, combining all interval extrema, is only performed at the lowest interval. For higher levels, a smaller set is evaluated. This allows reconstructing the fuzzy FRF from a much lower number of deterministic computations, with only a small reduction in the accuracy of FRFs. Both methods are demonstrated on a clamped plate and a car front cradle with uncertain design parameters.     

Hybrid uncertainty propagation of coupled structural–acoustic system with large fuzzy and interval parameters

Based on the finite element framework and uncertainty analysis theory, this paper proposes a first-order subinterval perturbation finite element method (FSPFEM) and a modified subinterval perturbation finite element method (MSPFEM) to solve the uncertain structural–acoustic problem with large fuzzy and interval parameters. Fuzzy variables are used to represent the subjective uncertainties associated with the expert opinions; whereas, interval variables are adopted to quantify the objective uncertainties with limited information. By using the level-cut strategy and subinterval methodology, the original large fuzzy and interval parameters are decomposed into several subintervals with small uncertainty level. In both FSPFEM and MSPFEM, the subinterval matrix and vector are expanded by the Taylor series. The inversion of subinterval matrix in FSPFEM is approximated by the first-order Neumann series, while the modified Neumann series with higher order terms is adopted in MSPFEM. The eventual fuzzy interval frequency responses are reconstructed by the interval union operation and fuzzy decomposition theorem. A numerical example evidences the remarkable accuracy and effectiveness of the proposed methods to solve engineering structural–acoustic problems with hybrid uncertain parameters.     

Structural damage measure index based on non-probabilistic reliability model

Uncertainties in the structural model and measurement data affect structural condition assessment in practice. As the probabilistic information of these uncertainties lacks, the non-probabilistic interval analysis framework is developed to quantify the interval of the structural element stiffness parameters. According to the interval intersection of the element stiffness parameters in the undamaged and damaged states, the possibility of damage existence is defined based on the reliability theory. A damage measure index is then proposed as the product of the nominal stiffness reduction and the defined possibility of damage existence. This new index simultaneously reflects the damage severity and possibility of damage at each structural component. Numerical and experimental examples are presented to illustrate the validity and applicability of the method. The results show that the proposed method can improve the accuracy of damage diagnosis compared with the deterministic damage identification method.     

Interval analysis of transient temperature field with uncertain-but-bounded parameters

Based on the traditional finite volume method, a new numerical technique is presented for the transient temperature field prediction with interval uncertainties in both the physical parameters and initial/boundary conditions. New stability theory applicable to interval discrete schemes is developed. Interval ranges of the uncertain temperature field can be approximately yielded by two kinds of parameter perturbation methods. Different order Neumann series are adopted to approximate the interval matrix inverse. By comparing the results with traditional Monte Carlo simulation, a numerical example is given to demonstrate the feasibility and effectiveness of the proposed model and methods.     

Robust adaptive synchronization for a general class of uncertain chaotic systems with application to Chua's circuit

The synchronization problem for a general class of uncertain chaotic systems is addressed. The underlying systems may be perturbed by unknown time-varying parameters, unstructured uncertainties, and external disturbances. Meanwhile, the time-varying parameters and disturbances are neither required to be periodic nor to have known bounds. Assuming the disturbances are L(2) signals, an adaptive control incorporated with H(∞) control technique is employed to construct a robust adaptive synchronization algorithm. Then, removing such assumption, a novel adaptive-based method is developed to achieve the goal of synchronization. In order to demonstrate the effectiveness of the proposed algorithms, such methods are applied to solve the synchronization problem of uncertain chaotic Chua's circuits.     

Brake squeal reduction of vehicle disc brake system with interval parameters by uncertain optimization

An uncertain optimization method for brake squeal reduction of vehicle disc brake system with interval parameters is presented in this paper. In the proposed method, the parameters of frictional coefficient, material properties and the thicknesses of wearing components are treated as uncertain parameters, which are described as interval variables. Attention is focused on the stability analysis of a brake system in squeal, and the stability of brake system is investigated via the complex eigenvalue analysis (CEA) method. The dominant unstable mode is extracted by performing CEA based on a linear finite element (FE) model, and the negative damping ratio corresponding to the dominant unstable mode is selected as the indicator of instability. The response surface method (RSM) is applied to approximate the implicit relationship between the unstable mode and the system parameters. A reliability-based optimization model for improving the stability of the vehicle disc brake system with interval parameters is constructed based on RSM, interval analysis and reliability analysis. The Genetic Algorithm is used to get the optimal values of design parameters from the optimization model. The stability analysis and optimization of a disc brake system are carried out, and the results show that brake squeal propensity can be reduced by using stiffer back plates. The proposed approach can be used to improve the stability of the vehicle disc brake system with uncertain parameters effectively.     

A fuzzy finite element procedure for the calculation of uncertain frequency response functions of damped structures: Part 2—Numerical case studies

This work introduces a numerical algorithm to calculate frequency response functions of damped finite element models with fuzzy uncertain parameters. Part 1 of this paper focusses on the numerical procedure for the solution of the underlying interval finite element problem, based on the undamped procedure and the principle of Rayleigh damping. Part 2 of this paper illustrates the applicability of the methodology through four case studies. The concepts of the interval and the fuzzy finite element frequency response function analysis are illustrated for different types of uncertainties. The obtained results are compared with the results of Monte Carlo simulations.     

A dimension-wise method for the static analysis of structures with interval parameters

A novel method for the static analysis of structures with interval parameters under uncertain loads is proposed, which overcomes the inherent conservatism introduced by the conventional interval analysis due to ignoring the dependency phenomenon. Instead of capturing the extremum of the structural static responses in the entire space spanned by uncertain parameters, their lower and upper bounds are calculated at the minimal and maximal point vectors obtained dimension by dimension with respect to uncertain parameters based on the Legend orthogonal polynomial approximation, overcoming the potential engineering insignificance caused by the optimization strategy. After performing its theoretical analysis, both the accuracy and applicability of the proposed method are verified.     

不确定声场分析的二阶区间摄动有限元法

针对一阶区间摄动有限元法在声场参数不确定程度增大时误差过大的缺陷,在二阶Taylor展开的基础上推导了声学二阶区间摄动有限元法,并将其应用于区间不确定声场的声压响应分析。该方法先对声学区间有限元方程的声压响应向量进行二阶Taylor展开,获取声压响应的二阶近似响应向量;再根据二次函数极值定理获得声压响应向量的上下界。二维管道声场与轿车声腔模型的数值分析算例表明,与一阶区间摄动有限元法相比,二阶区间摄动有限元法有效提高了计算精度。因此二阶区间摄动有限元适合不确定度更大的区间不确定声场声压响应分析,具有良好的工程应用前景。     

Modified perturbation method for eigenvalues of structure with interval parameters

In overcoming the drawbacks of traditional interval perturbation method due to the unpredictable effect of ignoring higher order terms,a modified parameter perturbation method is presented to predict the eigenvalue intervals of the uncertain structures with interval parameters.In the proposed method,interval variables are used to quantitatively describe all the uncertain parameters.Different order perturbations in both eigenvalues and eigenvectors are fully considered.By retaining higher order terms,the original dynamic eigenvalue equations are transformed into interval linear equations based on the orthogonality and regularization conditions of eigenvectors.The eigenvalue ranges and corresponding eigenvectors can be approximately predicted by the parameter combinatorial approach.Compared with the Monte Carlo method,two numerical examples are given to demonstrate the accuracy and efficiency of the proposed algorithm to solve both the real eigenvalue problem and complex eigenvalue problem.     

时变不确定时滞连续系统的鲁棒H∞保成本控制

针对一类同时具有状态时滞和输入时滞的时变不确定连续系统,研究了H_∞保成本状态反馈控制器的设计,假定其中的时变不确定性项是范数有界的,但不需要满足匹配条件.通过构造广义Lyapunov函数和线性矩阵不等式(LMI)方法,给出了系统可H_∞鲁棒镇定同时满足保性能指标的一个充分条件,仅通过求解一个相应的线性矩阵不等式,就可得到鲁棒H_∞保性能控制器使得闭环系统的一个保成本函数对所有允许的不确定参数有上界,并经过迭代,通过求解凸优化问题得到最优鲁棒H_∞保性能控制器.最后用示例说明了该方法的有效性. 关键词： 连续系统 时滞 H_∞鲁棒控制')" href="#">H_∞鲁棒控制 保成本控制     

Uncertainty identification by the maximum likelihood method

To incorporate uncertainty in structural analysis, a knowledge of the uncertainty in the model parameters is required. This paper describes efficient techniques to identify and quantify variability in the parameters from experimental data by maximising the likelihood of the measurements, using the well-established Monte Carlo or perturbation methods for the likelihood computation. These techniques are validated numerically and experimentally on a cantilever beam with a point mass at an uncertain location. Results show that sufficient accuracy is attainable without a prohibitive computational effort. The perturbation approach requires less computation but is less accurate when the response is a highly nonlinear function of the parameters.     

Structural Reliability Analysis by Using Non-Probabilistic Multi-Cluster Ellipsoidal Model

Uncertainties are normally unavoidable in engineering practice, which should be taken into account in the structural design and optimization so as to reduce the relevant risks. Yet, the probabilistic models of the uncertainties are often unavailable in the problems due to the lack of samples, and the precision of the conventional non-probabilistic models are not satisfactory when the samples are of multi-cluster distribution. In view of this, an improved method by using a non-probabilistic multi-cluster ellipsoidal model (multi-CEM) for the critical structural reliability analysis is proposed in this paper, which describes the samples in a more accurate and compact way and helps to acquire more satisfactory reliability analysis results. Firstly, a Gaussian mixture model (GMM) is built for the multi-cluster samples with performing expectation maximization (EM) algorithm, based on which the multi-CEM can be constructed. In the structural reliability analysis, two cases, respectively, considering whether the components of the multi-CEM are intersected or not are researched in detail. The non-probabilistic reliability (NPR) indexes for each component of the multi-CEM are computed using the Hasofer–Lind–Rackwitz–Fiessler (HL-RF) algorithm, and then the multidimensional volume ratios of the safe domain to the whole uncertainty domain are computed based on these indexes, indicating the structural NPR. In the end, two numerical examples and a practical application are conducted and analyzed to testify the effectiveness of the method.     

Calculating frequency response functions for uncertain systems using complex affine analysis

This paper is concerned with the effect of parametric uncertainty upon the response of mechanical systems. Two uncertainty propagation techniques are employed and compared against a benchmark based upon randomly scanning input parameter ranges. The first technique, complex interval analysis, is computationally very efficient but suffers from the issue of dependency which may result in large overestimation of the system response. The second technique, namely complex affine analysis, is presented here for the first time and it keeps track of variable dependencies throughout the calculation thus limiting overestimation, albeit at some computational expense. The methods are demonstrated on the problem of calculation of the frequency response functions of a simple lumped mass system with uncertain parameters for purposes of transparency of the presented ideas, although these ideas are equally applicable to more complex systems. It transpired that the complex affine analysis performed significantly better than its interval counterpart.     

Uncertainty propagation in SEA for structural–acoustic coupled systems with non-deterministic parameters

Considering limited available information on uncertainties in structural - acoustic coupled systems, two methods namely the vertex method and the Legendre orthogonal polynomial based method for predicting their dynamic behavior are developed based on the Statistical Energy Analysis (SEA) approach. For the vertex method, an efficient program for determining coordinates of all vertices of the rectangular spanned by entries of the involved interval input vector is coded, which is well suited for an interval input vector in arbitrary dimension. Instead of calculating the extremum of the response of interest, a method for determining its minimal and maximal point vectors dimension by dimension with respect to uncertain parameters is proposed based on the Legendre orthogonal polynomial approximation. Following the theoretical analysis of the accuracy and efficiency of the proposed methods, their validation is performed by one numerical example and two applications.     

认知不确定二维声场的响应特性数值分析

针对声学参数存在认知不确定性的问题,为实现认知不确定声场声压响应的预测。提出了解决二维认知不确定声场的有限元法(Evidence Theory-based Finite Element Method,ETFEM),引入证据理论,采用焦元和基本可信度的概念来描述认知不确定参数,基于摄动法的区间分析技术,推导了认知不确定声场声压响应的标准差、期望求解公式。为验证本文方法的可行性。以认知不确定参数下的二维管道声场模型和某轿车二维声腔模型为例进行了数值计算,对比离散随机变量得到认知不确定参数的声场分析结果和相应的随机声场所得分析结果,研究表明:该方法能够有效的处理认知不确定参数下的二维声场,为工程问题中噪声预测提供可靠的分析模型。     

Numerical solution of the imprecisely defined inverse heat conduction problem

This paper investigates the numerical solution of the uncertain inverse heat conduction problem. Uncertainties present in the system parameters are modelled through triangular convex normalized fuzzy sets. In the solution process, double parametric forms of fuzzy numbers are used with the variational iteration method(VIM). This problem first computes the uncertain temperature distribution in the domain. Next, when the uncertain temperature measurements in the domain are known, the functions describing the uncertain temperature and heat flux on the boundary are reconstructed. Related example problems are solved using the present procedure. We have also compared the present results with those in [Inf.Sci.(2008) 178 1917] along with homotopy perturbation method(HPM) and [Int. Commun. Heat Mass Transfer(2012) 3930] in the special cases to demonstrate the validity and applicability.