基于位移频响函数灵敏度分析的模型修正

针对频响函数灵敏度分析法进行模型修正时,由于一阶近似固有的局限性,很难直接迭代得到正确收敛结果的问题,提出了频响函数灵敏度拟合函数进行基于模型修正的结构损伤识别方法。首先,采用模态参与变异系数准则选取激励点位置、频率响应有效独立法结合距离系数-有效独立法选取测点位置;然后,综合考虑频响函数对修正参数变化的灵敏度和频响函数的相关性,选择一个合理的频率点;最后,拟合随参数变化的频响函数灵敏度函数曲线,引入新的中间设计参数,从而建立中间设计参数的频响函数灵敏度方程组,通过求解中间设计参数的改变量来求解修正参数的改变量。三自由度质量—弹簧和二维桁架系统的数值算例验证了所提出方法的可行性;数值算例表明,该方法具有较高的计算效率和识别精度,也对随机噪声具有鲁棒性,同时有效地避免了灵敏度方程组数值平衡导致的收敛问题。     

多项式拟合振型函数法求解多跨传动轴的弯曲振动响应

为求解边界条件为弹支的多跨传动轴弯曲振动响应,基于第二类拉格朗日方程,建立了瑞利连续轴动力学模型,并使用伽辽金法将偏微分方程转化为常微分方程;提出了多跨传动轴振型函数的多项式拟合方法,通过传递矩阵法计算得到了传动轴有限截面的振型值,据此用多项式分别拟合振型函数及其高阶导数。以直升机尾传动轴为算例,将拟合得到的前三阶振型函数及相应的高阶导数代入到动力学方程组中,通过振型叠加法计算得到了尾传动轴在偏心激励下的弯曲振动响应。算例结果表明,该方法具有较好的收敛性和可靠性。     

考虑螺栓连接刚度不确定性的带法兰-圆柱壳结构频响函数分析

提出了一种利用Chebyshev多项式代理模型来分析螺栓连接带法兰-圆柱壳结构频响函数不确定性的区间分析法。首先,利用8节点退化壳单元,通过有限元方法建立了带法兰-圆柱壳结构的动力学模型,从而求解系统的频响函数,并与实验测试的频响函数进行对比,验证了所建模型的有效性。然后,基于区间分析法建立了含区间参数的频响函数Chebyshev多项式代理模型。最后,考虑法兰处螺栓连接刚度不确定性,得到了单方向和多方向的连接刚度存在不确定性时的频响函数区间范围,并通过Monte-Carlo抽样法进行了验证。研究结果表明:Chebyshev多项式代理模型具有较高的求解精度和计算效率,轴向连接刚度不确定性对系统的频响函数影响最大;螺栓连接刚度不确定性主要导致频响函数在系统第1阶和第3阶固有频率处形成较宽的共振带。     

非线性系统参数识别的一种频域方法

在非线性振动系统的控制微分方程已知的前提下,一般能够用近似方法求出弱非线性系统的频率响应函数。本文提出了一种新的频域方法,即利用测量得到的一组频率、振幅值拟合频响曲线来识别非线性系统的参数。该问题实际是一个优化设计问题,本文用优化原理中的直接寻优方法和解非线性方程组的迭代法,对杜芬方程控制的系统进行参数识别,两者的结果是非常接近的。     

舰用燃气轮机框架模型的模态参数识别

本文采用一次有理正交多项式－多模态曲线合法对一舰用燃气轮机框架模型进行了实验模态分析，以Ｆｏｒｓｙｔｈｅ复合交多项式作为频响函数的理论值，用实测的频响函数建立系数矩阵，来寻找使目标函数最小的最优值，本文在ＨＰ９８２６微机上建立了建立了测试，数据采集及参数识别等系统识别软件，给出了具有工程实用意义的前五阶振动模态参数，识别结果和有限元计算相比较，令人满意。     

英国可再生能源技术管理与开发调研

管内湍动流体摩擦因数是雷诺数和相对粗糙度的二元非线性函数，由Colebrook隐式方程计算摩擦因数要用迭代的方法求解，很不方便。为了得到形式简单、精度高的计算摩擦因数的显式方程，提出了二元非线性多项式智能拟合法。该法将二元非线性多项式转化成多元线性多项式并建立线性最小二乘法标准矩阵，用遗传算法结合矩阵法对多项式的项数、项型式项指数及项系数进行搜索得到最优的拟合函数式。用该法拟合了Colebrook方程解的数据，得到一个计算管内湍动流体摩擦因数的显式新方程。在雷诺数3.000≤Re≤10.8、相对粗糙度0≤e/d≤0.05的范围内，该方程计算结果与Colebrook方程的平均偏差为0.5%，最大偏差不超过1.8%，与实验数据偏差为23%。新方程具有形式简单、精度高、适用范围广的优点，且便于简化成光滑管或阻力平方区等情况下的计算摩擦因数的方程。     

用Padé多项式拟合法辨识动力学系统的物理参数

提出了一种动力学系统的物理参数辨识方法。应用Padé多项式对动力学系统的动刚度曲线进行拟合,通过最小二乘法确定Padé多项式中的系数矩阵,利用遗传算法对Padé拟合式中的参数进行优化,从而得到系统的质量矩阵、阻尼矩阵和刚度矩阵。数值算例表明该方法具有较高的辨识精度且适用于黏性阻尼系统和非黏性阻尼系统。     

传递路径分析中计算子系统频响函数的方法

传递路径分析是在振动噪声控制领域被广泛应用的一种有效方法。传递路径分析中将振动系统分成主动部分、被动部分以及连接主、被动部分的若干传递路径。在传递路径分析中需要对被动部分的频响函数进行测量。传统的传递路径分析需要先拆除子结构然后再测量频响函数,测试过程十分繁琐。提出了一种全新的方法来计算子系统的频响函数,直接由整个系统的频响函数矩阵推导得到子系统的频响函数矩阵的计算公式。该方法不需要对子系统进行物理解耦,大大缩短了测量子系统频响函数所需要的时间。数值算例和实验均验证了该方法的正确性和有效性。     

基于谐波平衡法的摆线钢球行星传动等速输出机构非线性动态特性研究

为揭示摆线钢球行星传动等速输出机构的非线性动力学行为,建立考虑机构钢球数目、输入激励、啮合副啮合状态及啮合刚度的纯扭转强非线性动力学模型。将啮合副预紧函数表现为多项式的形式,将啮合副间隙函数表达为描述函数的形式,通过谐波平衡法将微分方程组转化为非线性代数方程组,利用MATLAB进行求解,得到系统的基频稳态响应。通过改变钢球数、轴向压缩量与啮合刚度,分析参数变化对系统非线性特性的影响。结果表明,预紧系统只有两阶频率激发共振,系统非线性程度随钢球数、啮合刚度和预紧量的增加而减弱,预紧量是影响系统非线性程度的主要因素;间隙系统激发共振频率的阶数与钢球数目有关,幅频响应曲线出现典型非线性特征,出现单边冲击与双边冲击现象。基于多项式函数的谐波平衡法为深入研究摆线钢球行星传动系统的动态特性提供了一种有效方法。     

参数振动系统频响特性研究

采用矩阵谱分解中常用的Sylvester理论和Fourier级数展开法,推导了单自由度参数振动系统的频响函数,并得到了系统外激励共振条件。在此基础上,以直齿轮副参数振动系统为例仿真了系统的频响特性,并讨论了系统参数稳定性、时变参数以及阻尼的影响。结果表明,参数振动系统的频响特性主要有以下一些特点:(1)系统具有多个频响函数,分别对应于多频响应中的各个频率成分;(2)系统存在多个外激励共振区。除了外激励频率等于系统固有频率的共振区外,当激励频率等于系统固有频率与参数激励频率的组合值时,同样存在共振现象;(3)参数振动系统共振响应时,主导频率成分为系统固有频率;(4)阻尼使得频响函数峰值有明显下降,而对非共振区的频响曲线影响不大。     

基于实测频响函数主成分的在役网架损伤识别方法

鉴于从实际网架动测中得到的频响函数已受到噪声的污染,会使模态分析出现较大的误差,提出了基于实测频响函数和主成分分析的网架损伤识别方法:用网架实测的频响函数数据作为损伤识别的基本变量,建立损伤识别矩阵,通过主成分分析和变量重构对频响函数进行减消噪处理,利用重构的频响函数前几阶主成分,在低维空间中对损伤信息进行分析、提取,并通过多元控制图,来识别网架的损伤。该方法不需要模态参数,避开了模态参数误差所引起的损伤识别不准问题,不需建立力学模型,对网架边界约束条件和结构型式没有特别的限制。为了验证该方法的可靠性,在试验室完成了足尺网架模型在不同损伤情况下动测试验。结果表明,所提出的损伤识别方法简便可行,结果可靠,尤其对噪声环境下和具有一定非线性网架的损伤识别有良好的适应性。     

Analysis of first‐order partial differential equations via shifted Chebyshev polynomials

Abstract

The method of Chebyshev polynomials is introduced to represent approximate solutions of first‐order partial differential equations consisting of two independent variables. A set of linear algebraic equations is obtained by using the properties of Chebyshev polynomials and Kronecker product to analyse first‐order partial differential equations. The coefficient vector of Chebyshev polynomials of the first‐order partial differential equations can be obtained directly from Kronecker product formulas, which are suitable for computer computation. A numerical example for a set of first‐order partial differential equations is solved by a Chebyshev polynomials approximation and the results are satisfactory.     

A Technique for Synthesizing Digital Filters

In designing digital systems, one often faces the task of replacing a given analog filter by an equivalent digital filter. This paper proposes a method for synthesizing such digital filters in the time domain. It is assumed that the pulsed transfer function of the digital filter is a ratio of two rational polynomials. The coefficients are then determined by least-square fitting the digital filter to the analog filter's sampled input and output data. The resulting equations for computing the coefficients are linear. It is shown that the digital filter is essentially related to the analog filter, the sampling time, and the power spectrum of the signal being processed. If the signal is band-limited and the sampling frequency is sufficiently high, the digital filter can then be simply approximated by the Z transform of the analog filter multiplied by the sampling period.     

An improved continued‐fraction‐based high‐order transmitting boundary for time‐domain analyses in unbounded domains

A high‐order local transmitting boundary to model the propagation of acoustic or elastic, scalar or vector‐valued waves in unbounded domains of arbitrary geometry is proposed. It is based on an improved continued‐fraction solution of the dynamic stiffness matrix of an unbounded medium. The coefficient matrices of the continued‐fraction expansion are determined recursively from the scaled boundary finite element equation in dynamic stiffness. They are normalised using a matrix‐valued scaling factor, which is chosen such that the robustness of the numerical procedure is improved. The resulting continued‐fraction solution is suitable for systems with many DOFs. It converges over the whole frequency range with increasing order of expansion and leads to numerically more robust formulations in the frequency domain and time domain for arbitrarily high orders of approximation and large‐scale systems. Introducing auxiliary variables, the continued‐fraction solution is expressed as a system of linear equations in iω in the frequency domain. In the time domain, this corresponds to an equation of motion with symmetric, banded and frequency‐independent coefficient matrices. It can be coupled seamlessly with finite elements. Standard procedures in structural dynamics are directly applicable in the frequency and time domains. Analytical and numerical examples demonstrate the superiority of the proposed method to an existing approach and its suitability for time‐domain simulations of large‐scale systems. Copyright © 2011 John Wiley & Sons, Ltd.     

Three-step multi-domain BEM solver for nonhomogeneous material problems

A three-step solution technique is presented for solving two-dimensional (2D) and three-dimensional (3D) nonhomogeneous material problems using the multi-domain boundary element method. The discretized boundary element formulation expressed in terms of normalized displacements and tractions is written for each sub-domain. The first step is to eliminate internal variables at the individual domain level. The second step is to eliminate boundary unknowns defined over nodes used only by the domain itself. And the third step is to establish the system of equations according to the compatibility of displacements and equilibrium of tractions at common interface nodes. Discontinuous elements are utilized to model the traction discontinuity across corner nodes. The distinct feature of the three-step solver is that only interface displacements are unknowns in the final system of equations and the coefficient matrix is blocked sparse. As a result, large-scale 3D problems can be solved efficiently. Three numerical examples for 2D and 3D problems are given to demonstrate the effectiveness of the presented technique.     

Analytical frequency response functions for contact of multilayered materials

Multilayer coatings are often seen in surface engineering for surface modifications. Optimal design of the multilayered materials requires the understanding of their mechanical behaviors based on deformation and stress analyses. The frequency response functions (FRFs) of the displacement and stress fields in multilayered materials under unit normal and shear loadings are the analytical cores for solving the contact of such materials. The authors have successfully derived these functions by utilizing the Papkovich–Neuber potentials and appropriate boundary conditions. Two matrix equations containing unknown coefficients in the FRFs are established by following the structure rules, and then the closed-form FRFs written in a recurrence format are established. A fast numerical semi-analytical model based on the derived FRFs is further developed for investigating the elastic contact of multilayered materials with any desired material design.     

Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons

We construct efficient quadratures for the integration of polynomials over irregular convex polygons and polyhedrons based on moment fitting equations. The quadrature construction scheme involves the integration of monomial basis functions, which is performed using homogeneous quadratures with minimal number of integration points, and the solution of a small linear system of equations. The construction of homogeneous quadratures is based on Lasserre’s method for the integration of homogeneous functions over convex polytopes. We also construct quadratures for the integration of discontinuous functions without the need to partition the domain into triangles or tetrahedrons. Several examples in two and three dimensions are presented that demonstrate the accuracy and versatility of the proposed method.     

Linear equality constraints in finite element approximation

The typical numerical problem associated with finite element approximations is a quadratic programming problem with linear equality constraints. When nodal variables are employed, the coefficient matrix of the constraint equations, [ A ], acquires a block-diagonal structure. The transformation from polynomial coefficients to nodal variables involves finding a basis for [ A ] and computing its inverse. Simultaneous satisfaction of completeness and C¹ (or higher) continuity requirements establishes linear relationships among the nodal variables and precludes inversion of the basis by exclusively element-level operations. Linear dependencies among the constraint equations and among the nodal variables can be evaluated by the simplex method. The computational procedure is outlined.     

Study on the elastic‐plastic buckling strength of plate structures

Abstract

Shifted Legendre polynomials are applied to solve the state equations of linear system. The computation procedure is greatly simplified by introducing the operational matrix for the integration of shifted Legendre vectors whose elements are shifted Legendre polynomials. The key of the method is that the state and forcing functions are expressed in terms of a series of shifted Legendre polynomials with expansion coefficients. Ordinary differential equations of state system are transformed into a series of algebraic equations of the shifted Legendre expansion coefficients and then are solved by employing the technique of matrix inverse. The methods of the computational algorithms are also investigated in order to simplify the calculation procedure and make the calculation convergent.     

A coupled symmetric BE–FE method for acoustic fluid–structure interaction

A coupled symmetric BE–FE method for the calculation of linear acoustic fluid–structure interaction in time and frequency domain is presented. In the coupling formulation a newly developed hybrid boundary element method (HBEM) will be used to describe the behaviour of the compressible fluid. The HBEM is based on Hamilton's principle formulated with the velocity potential. The state variables are separated into boundary variables which are approximated by piecewise polynomial functions and domain variables which are approximated by a superposition of static fundamental solutions. The domain integrals are eliminated, respectively, replaced by boundary integrals and a boundary element formulation with a symmetric mass and stiffness matrix is obtained as result. The structure is discretized by FEM. The coupling conditions fulfil C¹-continuity on the interface. The coupled formulation can also be used for eigenfrequency analyses by transforming it from time domain into frequency domain.