Winkler地基上环扇形板的Fourier—Bessel级数解

本文应用加补充项的富里叶级数理论，对弹性地基环扇形板的弯曲问题，提出了一种应用范围比较广的、便于计算的、解析形式的解法，从而进一步推广了富里叶级数的应用范围。     

回转对称结构半解析有限元法动力分析

回转对称结构半解析有限元法动力分析，是利用数学上对环向坐标θ进行富里叶级数展开的方法，把原问题转换为ｎ个调和项的谐波，再对之进行有限单元法的动力分析，从而大大降低了分析花费。特别是对工程上的回转对称结构进行地震响应分析，这种方法往往能取得明显的经济效益。     

基于傅里叶级数的缠绕复合材料刚度预测方法

采用傅里叶级数有关理论,根据缠绕复合材料内部细观结构的特点,建立了缠绕复合材料刚度预测模型。该模型将缠绕复合材料特征单元的刚度作为一个刚度场,利用二维傅里叶级数展开得到描述该刚度场的函数。算例的结果表明,本文中所建立的模型可以预测缠绕复合材料整体刚度特性,同时通过傅里叶级数各阶展开项的叠加可以反映材料内部细观结构对缠绕复合材料刚度特性的影响。      

复合矩形薄板交界面上的Ⅲ型双边裂纹

本文借助富立叶变换与富立叶级数的基本理论,对四种介质所组成的复合矩形薄板在介质交界面处内含Ⅲ型双边裂缝的问题,找到了它的一般分析解。令人感兴趣的是:分析还发现本问题的应力强度因子与复合板的材料性质无关。这个结论无疑将对设计工作者和实验工作者是很重要的,因为根据它可以避免多余的试验。      

非对称复合材料层板在面内载荷作用下的非线性横向变形或屈曲

本文利用广义富里叶级数法对非对称复合材料层板在面内载荷作用下的非线性横向变形或屈曲问题进行了分析。文中主要研究了四边简支边界条件下,非对称层板的铺层方式和面内载荷形式对横向变形的影响。通过求解非线性控制方程得到了层板的载荷—挠度曲线或特殊情况下的后屈曲平衡曲线。计算结果表明,对于大多数非对称复合材料层板在面内载荷作用下所产生的是横向变形问题,而不是分叉屈曲问题。     

热释电材料层合板柱形弯曲解析解

由热机电耦合微分方程出发, 用富里叶级数展开方法, 探讨了含有热释电材料铺层的复合材料层合板的柱形弯曲问题。由于不作任何简化, 所得结果为精确解且适用于任何厚跨比的层合板分析。用复指数形式, 使公式推导简洁统一且极大方便了程序实施。分析中考虑了各种可能的热学、力学和静电学边界条件, 对各种耦合元素相互关系和各自特点进行了分析, 发现结构对热载荷与电载荷的响应有明显的不同。      

单位冲激抽样序列频谱与傅里叶级数收敛性分析

由傅里叶变换的时移和频移特性,单位冲激抽样序列有两种频谱函数:周期型频谱和级数型频谱,其中周期型频谱函数的推导并不严谨,缺少傅里叶级数收敛性分析。对此提出通过证明两频谱函数等价来验证周期冲激信号傅里叶级数的收敛性。根据脉冲函数定义,运用极限和积分思想,利用抽样函数性质,证明了级数型频谱函数本质是强度和周期均为圆频率的频域冲激序列,验证了冲激抽样序列傅里叶级数的收敛性,周期型频谱函数的傅里叶级数与级数型频谱函数的分析也再次验证了级数的收敛性,但不能验证冲激点不存在吉布斯现象的观点。     

各种支承的正交异性矩形厚板的自由振动

本文利用富里级数法，得到了各种边界条件下正交各向异性矩形板在点支承、线支承和局部面支承下的自由振动解，由于是以ＡＭБａｐПУМЯＨ理论为基础，所以包含了剪切变形影响。     

反对称角铺层矩形板振动和屈曲分析

本文利用富里哀级数法,把求层板振动与屈曲问题控制偏微分方程特征函数和特征根的问题转化成求矩阵特征向量和特征值的问题。并给出了一对边有转动弹性约束,另两边有横向弹性约束的正方形板的数值结果。     

有限长板条弹性力学平面问题的研究

本文研究有限长板条的弹性力学平面问题。把Stockes变换推广到双重付里叶级数,从而求得平面问题的一般解析解。文章结合具体问题给出数字结果。     

Analytical solutions of functionally graded piezoelectric circular plates subjected to axisymmetric loads

Based on the three-dimensional theory of piezoelectricity, this work analytically investigates the axisymmetric bending of circular plates whose material properties vary along the thickness. The transverse loads are expanded in terms of the Fourier–Bessel series, and the solutions corresponding to each item of the series are derived by a semi-inverse method. The overall results are then obtained through the superposition principle. The exact solutions are obtained for two unusual boundary conditions if the material properties obey an exponential law. Meanwhile, a layerwise model is employed for the case of arbitrary and independent distribution of each component of material properties and the approximate solutions are proposed for simply supported and clamped boundaries. Finally, the numerical examples are illustrated, and the results are compared with those of the finite element method to demonstrate the present method.     

四边弹性梁支承的矩形板非线性弯曲

本文给出四边弹性梁支承的矩形板在横向荷载作用下的非线性弯曲的重级数解。通过把板内的横向位移展成重傅里哀级数,在边界上的位移展成单傅里哀级数以及把应力函数展成广义傅里哀级数,使边界条件转化成了无穷线性代数方程,控制微分方程转化成了无穷非线性代数方程。文后给出了四边弹性梁支承和对边简支对边弹性梁支承的矩彤板非线性弯曲的计算结果。     

A higher-order theory for functionally graded beams based on Legendre's polynomial expansion

In this article a higher-order theory for functionally graded beams based on the expansion of the two-dimensional (2D) equations of elasticity for functionally graded materials into Fourier series in terms of Legendre's polynomials is presented. Starting from the 2D equations of elasticity, the stress and strain tensors, displacement, traction, and body force vectors are expanded into Fourier series in terms of Legendre's polynomials in the thickness coordinate. In the same way, the material parameters that describe the functionally graded material properties are also expanded into Fourier series. All equations of the linear elasticity including Hooke's law are transformed into the corresponding equations for the Fourier series expansion coefficients. Then a system of differential equations in terms of the displacements and the boundary conditions for the Fourier series expansion coefficients are obtained. In particular, the first- and second-order approximations of the exact infinite dimensional beam theory are considered in more detail. The obtained boundary-value problems are solved by the finite element method with MATHEMATICA, MATLAB, and COMSOL multiphysics software. Numerical results are presented and discussed.     

弹性地基上各向异性平行四边形板

推导出弹性地基上各向异性平行四边形板的控制微分方程和内力方程。并利用对称性将其分解成中心对称和中心反对称两个问题,进而对每个问题给出了两个齐次解和一个特解,通过叠加而得到用重傅立叶级数表示的全解,该解克服了过去人们用重傅立叶级数仅在一个方向用一种级数(正弦或是余弦函数)表示导致的弊端。另外还给出了加快傅立叶级数收敛速度的计算公式。     

Application of discrete Fourier series to the finite element stress analysis of axi-symmetric solids

This paper describes an efficient formulation, based on a discrete Fourier series expansion, for the analysis of axi-symmetric solids subjected to non-symmetric loading. In contrast to the commonly adopted continuous Fourier series approach, the discrete Fourier series representation has largely overcome problems such as the presence of Gibb's phenomenon and the lack of conformity of elements. The computational advantage over a fully three-dimensional formulation is illustrated. Finally, the method is demonstrated through a finite element study of rigid caissons founded in both elastic and elasto-plastic soils.     

First-order PMD compensation using optical FIR filter

We present a synthesis algorithm to design an optical finite impulse response (FIR) filter for compensating a first-order polarization mode dispersion (PMD) by minimizing the differential group delay (DGD). The desired frequency response was approximated using two widely used methods in designing digital FIR filters: the Fourier series expansion method and the frequency sampling method. A numerical simulation was performed for an eighth-order filter to demonstrate the difference between the two methods. The simulation results produced a sharper cutoff for the Fourier series expansion and higher stopband attenuation for the frequency sampling method. The Fourier series method produced better results in reducing the DGD.     

Fourier series based reliability analysis of aeroengine turbine blade under linear fuzzy safety state

A Fourier series method is established to compute fuzzy reliability of aeroengine turbine blade. Based on Fourier series of Normal probability density function (pdf), reliability analysis is carried out under linear fuzzy safety states defined by forms I–III. For aeroengine turbine blade root flange subjected to Normal random variables using form I, fuzzy reliability (P_r) evaluated by Fourier series integration is 0.999787. Moreover in blade edge using form II, Fourier series P_r is 0.999684 by balancing between the fluctuation level and the series number. Furthermore for blade connection under Normal distributed stress using form III, Fourier series P_r is 0.918523. Caused by undetermined factors, error ratios of numerical results range from 1.31 × 10⁻⁵ to 2.38 × 10⁻⁴. As the exactness, convergence and computational capability of Fourier series method are validated, it can be used in fuzzy reliability of aeroengine turbine blade.     

Incompressible impulsive wall impact of liquid cylinders

Analytical solutions are given for cylindrical fluid bodies in translational motion that hit a plane wall in impulsive impact at constant velocity. We consider impact of fluid bodies with different rectangular and circular cross sections, and derive analytical solutions in terms of Fourier series or Fourier–Bessel series. The initial velocity fields, the corresponding pressure impulse fields, and the associated virtual masses for each fluid body are calculated.     

The time-dependent Green's function of the transverse vibration of a composite rectangular membrane

A new method for the approximate computation of the time-dependent Green's function for the equations of the transverse vibration of a multi stepped membrane is suggested. This method is based on generalization of the Fourier series expansion method and consists of the following steps. The first step is finding eigenvalues and an orthogonal set of eigenfunctions corresponding to an ordinary differential operator with boundary and matching conditions. The second step is a regularization (approximation) of the Dirac delta function in the form of the Fourier series with a finite number of terms, using the orthogonal set of eigenfunctions. The third step is an approximate computation of the Green's function in the form of the Fourier series with a finite number of terms relative to the orthogonal set of eigenfunctions. The computational experiment confirms the robustness of the method.     

Numerical analysis of the strained state of cellular pipes in inhomogeneous soil

With the help of Fourier series, we propose a method for the evaluation of the strained state of cellular underground pipes with regard for the variations of the reaction coefficient of the surrounding soil. As a result, we write, in recurrent form, the coefficients of Fourier series for the averaged components of the vector of displacements of a cellular structure. On this basis, we find the radial displacements of a polymeric pipe along its contour.