转轴系统平稳随机地震响应的变异性分析

本文对于具有随机参数的线性转轴系统受平稳随机地面激励的问题,给出了求解其随机响应方差变异性的计算方法。由于先用虚拟激励法将平稳随机激励转化为确定性的简谐激励,使双随机问题精确地转化为单随机问题,使求解过程简便有效。     

Uncertainty quantification of multidimensional dynamical systems based on adaptive numerical solutions of the Liouville equation

Propagation of uncertainty in multidimensional dynamical systems, in the presence of parametric uncertainties, can be quantified by the solution of the underlying Liouville equation that governs the evolution of a multivariate joint probability density function of random variables associated with states and parameters. In this paper we propose an efficient numerical solution of the Liouville equation that involves (a) sampling at Gauss-quadrature nodes of random variables corresponding to uncertain parameters and (b) evolution of the associated conditional probability density functions using a finite difference method with time-adaptive computational domains in multiple dimensions. The proposed approach is designed to accurately predict long-time statistics of random variables corresponding to system states, including moments and probability density function, for dynamical systems of moderate dimension. The proposed approach is applied to four different dynamical systems, including (i) single spring–mass system, (ii) Van der Pol oscillator, (iii) double spring–mass system and (iv) a typical section nonlinear aeroelastic model. When compared to a conventional finite difference based numerical solution on a fixed grid, the solution obtained from the proposed adaptive grid based approach involves a considerable reduction in the required number of grid points for equivalent accuracy. For the single spring–mass system, for which an analytical solution is found, comparison with Monte Carlo simulation results indicates that the proposed adaptive numerical solution approach is generally one to two orders of magnitude more computationally efficient for a given level of accuracy.     

MIMO随机振动试验频响估计中激励和响应的同步方法

频响函数估计是多输入多输出随机振动试验控制算法中的必要环节。在振动环境试验中,通常不采集振动台的实际激励信号,而用实测的响应信号和计算机内存中的激励信号来进行系统频响函数估计。由于计算机内存中的激励信号与实际振动台上的激励信号之间在相位上存在着差异,从而导致实测的响应信号和计算机内存中的激励信号不同步。这给系统频响函数的估计带来了较大困难。为此,根据线性系统中激励和响应之间的关系,结合随机减量法,提出了一种二次相关法用于系统频响函数的估计。用该方法进行频响函数估计只需要采集响应信号。在三轴振动台上进行了对比试验,结果验证了本文所提出的二次相关法的正确性。     

Estimation of the Dynamic Properties of Non‐linear Packaging Materials Using a Reverse Multiple Input/Single Output Based Approach

During the distribution phase, packaged consignments are exposed to a variety of environmental hazards (such as vibrations) that, if excessively severe, may cause damage to or even destroy the product. Structural deterioration can be tracked by monitoring variations in the packaging system's modal parameters, particularly its natural frequency (stiffness). Natural frequency estimates are often extracted using a least‐squares regression curve fit, applied to an estimate of the system's frequency response function (FRF). FRF estimates are generally obtained using the Fourier transform with a single input and single output (SISO). This approach is suitable for many applications; however, as the non‐linearity of the system under analysis increases, the ability of this technique to accurately monitor changes in the system will decrease. In addition, when the excitation to the non‐linear system is varied (increased or decreased amplitude), a SISO‐based approach may indicate a shift in natural frequency (as a result of the varied input) even though no change in the condition of the system has occurred. This paper discusses an approach that is designed to separate the linear component of the system's FRF using a reverse multiple input/single output (RMISO) algorithm. Such separation will allow traditional modal parameter extraction (curve fitting) techniques to be used to monitor the condition of non‐linear systems. The paper presents the results of experiments in which expanded polystyrene samples were subjected to broad‐band random base excitation with a free‐moving load placed atop the cushion sample. Continuous acceleration measurements of the vibration table and the free‐moving load were used to compute the FRFs of the cushions, and the differences between a conventional (SISO) approach and the proposed RMISO‐based parameter extraction technique were evaluated. Copyright © 2014 John Wiley & Sons, Ltd.     

A Study of Nonlinear Effects in a Cushion‐Product System on its Vibration Transmissibility Estimates with the Reverse Multiple Input–Single Output Technique

The Reverse Multiple Input–Single Output (R‐MISO) identification technique was implemented to determine vibration transmissibility frequency response functions (FRFs) of a cushion‐product system that accounts for nonlinearities known to exist in such systems. The signed quadratic x|x| nonlinear term was used. The results were compared with the classical linear approximation FRFs calculated with the single input–single output identification method. Comparisons were made for five types of expanded polystyrene (EPS) cushions with varying density and two levels of band‐limited random excitation. It was shown that vibration transmissibility FRFs obtained with the R‐MISO method and with the inclusion of a signed quadratic x|x| nonlinear term significantly (up to ~70%) improved the total coherence of the nonlinear vibration transmissibility FRFs for the EPS cushions subjected to the static load selected to coincide with the optimum of their respective cushion curves. The R‐MISO technique was successful in accounting for some of the nonlinearities in the EPS cushion‐product system. However, its ability to deal with nonlinearities is not specific to the EPS but rather common to cushion‐product systems subjected to vertical vibrations. Copyright © 2012 John Wiley & Sons, Ltd.     

Global sensitivity analysis using polynomial chaos expansions

Global sensitivity analysis (SA) aims at quantifying the respective effects of input random variables (or combinations thereof) onto the variance of the response of a physical or mathematical model. Among the abundant literature on sensitivity measures, the Sobol’ indices have received much attention since they provide accurate information for most models. The paper introduces generalized polynomial chaos expansions (PCE) to build surrogate models that allow one to compute the Sobol’ indices analytically as a post-processing of the PCE coefficients. Thus the computational cost of the sensitivity indices practically reduces to that of estimating the PCE coefficients. An original non intrusive regression-based approach is proposed, together with an experimental design of minimal size. Various application examples illustrate the approach, both from the field of global SA (i.e. well-known benchmark problems) and from the field of stochastic mechanics. The proposed method gives accurate results for various examples that involve up to eight input random variables, at a computational cost which is 2–3 orders of magnitude smaller than the traditional Monte Carlo-based evaluation of the Sobol’ indices.     

Inverse Sub‐Structuring Theory for Coupled Product Transport System Based On the Dummy Masses Method

It is of high importance to predict the components frequency response functions (FRFs) for obtaining the coupled product transport system's response. However, the components behaves much differently when coupled with another components compared with that in free state. Inverse sub‐structuring method has been recently proposed and applied for inverse analysis of the dynamical response of coupled product transport system. The component‐level FRFs and the coupling dynamic stiffness are all predicted from only the system‐level FRFs, facilitating the engineering design for product transport system. However, in most engineering application practices, the system‐level FRFs from coupling degrees of freedom may not be measured accurately because of the difficulties of vibration excitation and/or response measurement for the coupled interface between components within the limited accessible space. The aim of this paper is to develop a new FRF‐based indirect inverse sub‐structuring method for the analysis of the dynamic characteristics of a two‐component coupled product transport system without measuring system‐level FRFs at the coupling degrees of freedom. A so‐called dummy masses method is developed and applied for predicting the unmeasured FRFs at the coupling degrees of freedom, and the inverse sub‐structuring approach based on the dummy mass method is derived for inverse analysis of coupled product transport system, which is further verified by a lumped‐mass model, showing exact agreement. Finally, the experiment on a physical prototype of two‐substructure coupled product transport system is performed to further check the accuracy of the suggested method. The new method shows its great application prospect in coupled product transport system.     

Indirect Inverse Substructuring Theory for Coupling Dynamic Stiffness Identification of Complex Interface Between Packaged Product and Vehicle Transport System

Inverse substructuring method has been recently proposed and applied for inverse analysis of the dynamical response of product transport system. The component‐level frequency response functions (FRFs) and the coupling dynamic stiffness for facilitating the cushioning packaging design are all predicted from only the system‐level FRFs. However, the system‐level FRFs from coupling degree of freedoms may not be measured accurately because of the difficulties of vibration excitation and response measurement for the coupled interface between packaged product and vehicle within the limited accessible space. The aim of this paper is to develop a new FRF‐based indirect inverse substructuring method for the analysis of the dynamic characteristics of a three‐substructure coupled product transport system without measuring system‐level FRFs at the coupling degree of freedoms. By enforcing the dynamic equilibrium conditions at the coupling coordinates and the displacement compatibility conditions, a closed‐form analytical solution to inverse sub‐structuring analysis of multi‐substructure coupled product transport system is derived based on the relationship of easy‐to‐monitor component‐level FRFs and the system‐level FRFs at the coupling coordinates.. The proposed method is validated by a lumped mass‐spring‐damper model, and the predicted coupling dynamic stiffness is compared with the direct computation, showing exact agreement. Then, the FRF tests of a physical prototype of multi‐substructure coupled product transport system are performed to further check the accuracy of the suggested method. The method developed offers an approach to predict the unknown coupling dynamic stiffness from measured FRFs purely. The proposed method may help to obtain the main controlling factors and contributions from the various structure‐borne paths for product transport system. Copyright © 2014 John Wiley & Sons, Ltd.     

多级刚性包装运输系统间接逆向子结构分析

目的针对各子部件耦合界面之间的系统水平频响函数难以测量,刚性耦合系统逆向子结构分析方法无法顺利应用的情况,提出多级系统间接分析方法。方法基于子结构理论,提出单点耦合和多点耦合系统的多级刚性耦合系统间接逆向子结构分析方法,然后建立相对应的集总参数模型,利用已知参数和公式获得部件频响函数直接计算值和预测值,最后将两者进行对比验证。结果部件频响函数直接计算值与预测值相吻合,验证了方法的准确性。结论提出的方法可为逆子结构理论在解决耦合界面频响函数难测问题时提供新思路,以及为在运输包装领域更广泛的应用提供更多的可能性。     

基于频响函数的逆子结构方法误差分析

目的 通过分析随机误差在基于试验频响函数(FRFs)的逆子结构分析方法中的传递,得出随机误差对预测结果的影响规律,为基于逆子结构方法分析复杂结构的动态特性提供参考价值.方法 对获得的系统频响函数施加不同程度(1％,5％,10％)的随机误差,对比分析各个耦合系统频响函数对预测子结构频响函数的影响.结果 对耦合系统频响函数施加随机误差后,采用逆子结构方法对耦合系统解耦后预测的子结构频响函数严重偏离真实值,尤其是共振频率附近,所施加的随机误差在预测子结构频响函数中甚至被放大了数十倍,导致预测结果不可靠;且耦合系统耦合点处的频响函数对预测结果的影响最大.结论 通过分析明确了系统频响函数所携带的随机误差对预测结果的影响规律,且这些误差将随着矩阵的求逆运算被放大,且交叉耦合系统频响函数对预测结果的影响最为显著.     

A solution method for linear and geometrically nonlinear MDOF systems with random properties subject to random excitation

A method for computing the lower-order moments of response of randomly excited multi-degree-of-freedom (MDOF) systems with random structural properties is proposed. The method is grounded in the techniques of stochastic calculus, utilizing a Markov diffusion process to model the structural system with random structural properties. The resulting state-space formulation is a system of ordinary stochastic differential equations with random coefficients and deterministic initial conditions which are subsequently transformed into ordinary stochastic differential equations with deterministic coefficients and random initial conditions. This transformation facilitates the derivation of differential equations which govern the evolution of the unconditional statistical moments of response. Primary consideration is given to linear systems and systems with odd polynomial nonlinearities, for in these cases there is a significant reduction in the number of equations to be solved. The method is illustrated for a five-story shear-frame structure with nonlinear interstory restoring forces and random damping and stiffness properties. The results of the proposed method are compared to those estimated by extensive Monte-Carlo simulation.     

Static condensation based reduced order modelling of stochastically parametered large ordered systems

A surrogate stochastic reduced order model is developed for the analysis of randomly parametered structural systems with complex geometries. It is assumed that the mathematical model is available in terms of large ordered finite element (FE) matrices. The structure material properties are assumed to have spatial random inhomogeneities and are modelled as non-Gaussian random fields. A polynomial chaos expansion (PCE) based framework is developed for modelling the random fields directly from measurements and for uncertainty quantification of the response. Difficulties in implementing PCE due to geometrical complexities are circumvented by adopting PCE on a geometrically regular domain that bounds the physical domain and are shown to lead to mathematically equivalent representation. The static condensation technique is subsequently extended for stochastic cases based on PCE formalism to obtain reduced order stochastic FE models. The efficacy of the method is illustrated through two numerical examples.     

基于不完备频响函数辨识结合部等效参数的研究

研究了输出信息不完备情况下结合部等效动力学参数的辨识问题,阐述了一种较为通用的结合部等效动力学参数辨识方法.首先采用实验与有限元法相结合,辨识出试验中难以获取的频响函数,构造出完备频响函数.然后通过完备频响函数,运用迭代算法辨识出结合部等效动力学参数.在辨识过程中,建立损失函数模型,将问题转化为求解损失函数的最小值问题.该方法不需要求出模态参数,所以同样适用于大阻尼系统或者模态密集系统的结合部参数辨识.由于只需部分易测的频响函数,所以具有很好的工程应用性.最后通过算例验证了本方法具有很高的辨识精度.     

Inverse Sub‐structuring Method for Multi‐coordinate Rigidly Coupled Product Transport System based on a Novel Shearing Probe Technique

The inverse sub‐structuring method can predict the component‐level frequency response functions (FRFs) of product (critical component) for product transport system from only measured system‐level FRFs, facilitating the cushioning packaging design. However, the FRFs of the coupling interface between product and vehicle are usually of extreme difficulty to be measured due to the limited accessible space. To overcome this difficulty, the authors suggested a so‐called FRF probe technique method in the previous study, which may be more suitable for the single‐coordinate coupled system. In practice, most of the product transport systems should be treated as multi‐coordinate coupled system. The aim of this paper is to derive a new FRF‐based inverse sub‐structuring method for multi‐coordinate rigidly coupled product transport system and develop a new shearing probe technique to obtain the difficult‐to‐monitor FRFs at the coupling interface, which will be validated by a lumped mass model and finite element models, respectively, showing perfect agreement. Finally, the experiment on a physical prototype of multi‐coordinate rigidly coupled product transport system is performed to further check the feasibility of the application prospect of the shearing probe technique for inverse analysis of product transport system. The method proposed in this study will provide the packaging designers an alternative method to monitor the integrity of product transport system. Copyright © 2017 John Wiley & Sons, Ltd.     

Nonlinear random vibration analysis: A Bayesian nonparametric approach

Random vibration analysis aims to estimate the response statistics of dynamical systems subject to stochastic excitations. Stochastic differential equations (SDEs) that govern the response of general nonlinear systems are often complicated, and their analytical solutions are scarce. Thus, a range of approximate methods and simulation techniques have been developed. This paper develops a hybrid approach that approximates the governing SDE of nonlinear systems using a small number of response simulations and information available a priori. The main idea is to identify a set of surrogate linear systems such that their response probability distributions collectively estimate the response probability distribution of the original nonlinear system. To identify the surrogate linear systems, the proposed method integrates the simulated responses of the original nonlinear system with information available a priori about the number and parameters of the surrogate linear systems. There will be epistemic uncertainty in the number and parameters of the surrogate linear systems because of the limited data. This paper proposes a Bayesian nonparametric approach, called a Dirichlet Process Mixture Model, to capture these uncertainties. The Dirichlet process models the uncertainty over an infinite-dimensional parameter space, representing an infinite number of potential surrogate linear systems. Specifically, the proposed method allows the number of surrogate linear systems to grow indefinitely as the nonlinear system observed dynamic unveil new patterns. The quantified uncertainty in the estimates of the unknown model parameters propagates into the response probability distribution. The paper then shows that, under some mild conditions, the estimated probability distribution approaches, as close as desired, to the original nonlinear system’s response probability distribution. As a measure of model accuracy, the paper provides the convergence rate of the response probability distribution. Because the posterior distribution of the unknown model parameters is often not analytically tractable, a Gibbs sampling algorithm is presented to draw samples from the posterior distribution. Variational Bayesian inference is also introduced to derive an approximate closed-form expression for the posterior distribution. The paper illustrates the proposed method through the random vibration analysis of a nonlinear elastic and a nonlinear hysteretic system.     

Timing and frequency offset estimation scheme for the uplink of OFDMA systems

Timing and frequency synchronisation for the uplink of OFDMA systems is discussed. The uplink synchronisation procedure is presented and a novel timing and frequency offset estimation scheme is proposed. The timing and frequency offsets are estimated by identifying the differential phases of the training subcarriers in frequency and time dimensions, respectively. The frequency offset is estimated ahead of the timing offset, after which intercarrier interference compensation is carried out based on the estimated frequency offset. Finally, the timing offset is estimated after eliminating the frequency offset's influence. The principle of best linear estimation is applied. Both the case of a single user and that of multiple users simultaneously accessing the network are considered. In contrast to other methods, the proposed scheme has moderate complexity and allows flexible subcarrier assignment schemes. The analyses and simulation results demonstrate the accuracy of the proposed scheme in the uplink channels.     

基于频响函数的结构健康监测主成分分析法

摘 要：基于测试频响函数,提出一种简单而有效的结构健康监测主成分分析（PCA）新方法。以结构的频响函数作为基本数据,首先将结构健康状态下的频响函数数据作为基本训练样本,通过PCA技术提取结构健康状态特征,并获得结构健康特征变换矩阵,即协方差的特征向量矩阵;然后再对损伤结构的测试频响函数数据进行转换以提取结构相应损伤状态特征;最后在二维PCA空间比较两次提取的结构状态特征分布图即可判断结构是否发生损伤并评估其损伤程度。两个数值算例表明基于频响函数的结构健康监测主成分分析新方法正确有效。该方法基于结构振动响应,与模型无关且诊断前无需大量的训练样本、计算量小、抗噪性能好,具有良好的应用前景。     

Inverse Sub‐structuring Method for Rigidly Coupled Product Transport System based on Frequency Response Function Testing Probe Technique

The inverse sub‐structuring method has been recently proposed and applied for inverse analysis of product transport system, to predict the component‐level frequency response functions (FRFs) and the coupling dynamic stiffness from only the system‐level FRFs. However, previous applications of this method were all developed based on the assumption that the components were coupled by flexible couplings. Actually, increasing more components are welded or bolted to construct a coupled system, which should be treated as rigidly coupled system. The aim of this paper is to derive a new FRF‐based inverse sub‐structuring method for the analysis of the dynamic characteristics of a two‐component coupled product transport system with rigid couplings. And then a so‐called FRF testing probe technique is proposed and applied to measure the difficult‐to‐monitor FRFs at the coupling interface. The developed method is verified by a lumped‐mass model, showing exact agreement. Finally, the experiment on a physical prototype of two‐substructure coupled product transport system is performed to further check the accuracy of the suggested method. The proposed method is an extension of previous inverse sub‐structuring method and may help to obtain the main controlling factors and contributions from the various structure‐borne paths for product transport system. Copyright © 2016 John Wiley & Sons, Ltd.     

A Dendrite Net based decoupled framework for the reliability-based control co-design problem

To improve the reliability of the dynamic system including physical and control design, the reliability-based control co-design (RB-CCD) problem has been studied to account for the uncertainty stemming from the random physical design. However, when encountering RB-CCD in the sophisticated system in which the dynamic model simulation is time-consuming or the state equation is expressed implicitly, the available RB-CCD methods will consume significant computational effort to perform numerous system simulations for the reliability analysis and deterministic optimization. Therefore, this work proposes a Dendrite Net-based decoupled framework for RB-CCD to alleviate the computational burden. Specifically, the Dendrite (DD) model constructed by the suggested training scheme integrated with an adaptive sampling strategy is used to approximate the state equation in the dynamic system. After that, the sequential optimization and reliability assessment method decouples RB-CCD into the control co-design (CCD) problem and time-dependent reliability assessment problem, which are solved sequentially based on the cheap estimations of DD model, rather than the expensive simulations of the original system. Furthermore, two numerical examples and an engineering example of 3-DOF robot system are applied to demonstrate the feasibility and efficiency of the proposed framework.     

MIMO线性系统输入输出信号统计特征的双谱评定方法

基于高阶统计量具备处理随机信号的特性,提出了一种利用三阶谱(双谱)评定MIMO线性系统时域输入输出信号统计特征的新方法。通过建立线性系统双谱数学模型,根据系统响应、所测得的频响函数以及离散信号的双谱数值估计算法,经逆运算获得系统的双谱驱动信号,随后利用高阶谱对高斯随机信号的盲性判定其输入信号的高斯性。将上述方法与采用传统相位随机化法(对功率谱添加随机相位)所获得的驱动信号分别应用于一悬臂梁模拟控制系统中,通过对输入信号的分析及控制结果的比较,发现基于双谱所生成的时域随机驱动信号呈现出较强的非高斯性且收敛速度更快。对于输出信号统计特征的评定,提出从输入信号与系统频带接近的程度入手,再次利用高阶统计量对高斯随机信号的盲性进行定性判定,对于无法判别满足何种非高斯统计分布特征的,不管是对于输入信号还是输出信号,一律采用绘制信号的概率分布特征曲线进行定量评定。