On the prediction of the roundness error in BTA deep hole machining based on the stochastic description of the cutting tool motion

This paper presents a methodology of predicting the maximum possible out-of-roundness of the hole produced, in BTA deep hole machining, as a function of certain machining parameters. Based on the solutions of the stochastic differential equations representing the machine tool-workpiece system in BTA deep hole machining, and the true cutting tool motion, an index describing the upper bound of the roundness error is defined. A parametric analysis of the out-of-roundness index is carried out. This analysis points out that at a low length-to-diameter ratio of the cutting tool, the axial force is the predominant factor causing the tool-tip deviation from the ideal motion, and that the radial and tangential forces are the major causes for the tool tip deviation at high length-to-diameter ratio of the cutting tool. The experimental measurement of out-of-roundness of the specimens under different machining conditions shows that the roundness error obtained lies within the zone described by the theoretical prediction.     

微圆弧金刚石刀具刀尖圆弧的测量及评价

为实现对微圆弧金刚石刀具刀尖圆弧的评价,提出了一种基于高分辨率扫描电镜图像的评价方法,并对刀尖圆弧轮廓提取、轮廓曲线拟合、圆弧度评价等算法进行了研究.首先,运用Canny边缘检测算子提取刀尖圆弧图像的二维轮廓数据,并用移动最小二乘法对该数据进行拟合,使所提取轮廓光滑化;接着,建立了基于最小二乘准则的刀尖圆弧评价模型,并采用二次序列规划法对模型进行求解;最后,分析了轮廓拟合误差、测量不垂直度误差对刀尖圆弧评价结果的影响,并计算了刀尖圆弧半径及圆弧度不确定度值.实验结果表明所评价微圆弧金刚石刀具的刀尖圆弧半径为30.213μm,圆弧半径不确定度为351 nm,圆弧度为0.114μm,圆弧度不确定度为24 nm.由评价结果可以看出,本文所提出的方法可以实现微圆弧金刚石刀具刀尖圆弧纳米级精度的测量及评价.     

Analysis of geometrical error in the stereolithography process using a stochastic approach

The geometrical error in the stereolithography process is analysed using a stochastic approach. This approach is based on a unified methodology, developed by the authors, for studying the mechanical error in different rapid prototyping processes. The tolerances and clearances have been assumed to be random variables. The coordinates of a point on the resin surface, traced by the laser beam, are expressed as a function of random variables. In a numerical example, the geometrical error has been found for a grid of points traced by the laser beam. The three-sigma error bands are plotted when tracing example curves. This is the band in which the laser beams of 99.73% of machines, produced on a mass scale, lie on the work surface for the given tolerances and clearances. Stringent values of tolerances and clearances reduce the error at the tool tip, but the cost of manufacturing and assembling the machines may become prohibitive.     

基于多源融合理论的机床圆度误差测量不确定度评定

圆度误差是评价机床加工精度的重要指标。为实现机床圆度误差测量不确定度的评定,对基于球杆仪测量的机床圆度误差的贡献因素及不确定度评定方法进行研究。首先,采用最小二乘法（least sqaure method,LSM）对圆度误差进行评定。然后,基于黑箱理论提出了多源融合误差测量不确定度评定方法。接着,根据球杆仪测量机床圆度误差的基本原理,建立了机床圆度误差测量不确定度评定模型。最后,利用球杆仪对某加工中心ZX平面的圆度误差进行多组重复测量实验,采用所提出的方法对圆度误差测量不确定度进行计算,并与基于蒙特卡罗法（Monte Carlo method,MCM）的计算结果进行对比验证。结果表明,基于所提出方法和MCM的机床圆度误差测量不确定度的评定结果分别为1.190 0 和1.160 0 μm,两者的相对偏差约为2.5%,由此验证了所提出方法的可行性。所提出方法规避了繁杂的输入量与输出量之间函数关系的求解过程,简化了机床圆度误差测量不确定度的评定过程,具有较强的实用性。     

Maximum entropy principle and non-stationary distributions of stochastic systems

The principle of maximum entropy (in its classical form), successfully applied in many fields (e.g. statistics, reliability, estimation), has recently been extended to analyze the systems governed by stochastic differential equations and especially to determining the stationary probability distribution of the solution process. In this paper we develop the maximum entropy approach to characterize non-stationary probability distributions of the solutions of stochastic systems. The variational problem for the entropy functional includes time-dependent constraints in the form of differential equations for moments. The general scheme of the method is given along with the effective treatment of a number of first and second order stochastic systems. The maximum entropy probability distributions are compared with the exact solutions and with the simulation results.     

微分进化算法在圆度误差评定中的应用

为了精确快速计算圆度误差,提出了基于微分进化智能优化算法的最小区域圆度误差评定方法。介绍了微分进化算法的基本原理及种群初始化、变异、交叉、选择实现步骤,建立了该算法求解最小区域圆度误差的数学模型。为验证算法的有效性,进行了大量实验并与多种算法进行对比,证实了方法的评定结果不仅小于最小二乘法及标准遗传算法评定结果,精度高,而且计算结果稳定,运算速度快。实验表明:微分进化算法用于最小区域圆度误差评定有较强的自适应能力、快速全局收敛性和高稳定性,适于对高精度圆度误差的快速评定。     

Analysis of mechanical error in a fused deposition process using a stochastic approach

A stochastic model has been developed for studying the mechanical error in different rapid prototyping (RP) processes. Tolerances and clearances, which cause mechanical error, have been assumed to be random variables. The coordinates of a point on the work surface traced by the laser beam or the tip of the extruder head is expressed as a function of the random variables involved in the process. Using a unified approach for the RP processes, the mechanical error in the fused deposition process is analysed. In a numerical example, the mechanical error has been found for a grid of points traced by the nozzle tip. The three-sigma bands of the error in tracing example curves are plotted. This is the band in which the nozzle tips of 99.73% of machines, produced on a mass scale, lie for the given tolerances and clearances. Stringent values of tolerances and clearances reduce the error at the nozzle tip, but the cost of manufacturing and assembling the machines may become prohibitive.     

Stochastic uniform observability of general linear differential equations

The aim of this article is to discuss the uniform observability property of general linear differential equations with multiplicative white noise in Hilbert spaces. Based on perturbation theory for evolution operators on Hilbert Schmidt spaces and on the space of nuclear operators, new representations of the covariance operators associated to the mild solutions of the investigated stochastic differential equations are given. Using these results we obtain deterministic characterizations of the stochastic uniform observability property. We also identify an entire class of stochastic differential equations which are never stochastic uniformly observable. Some examples will illustrate the theory.     

考虑桥面随机不平顺的桥梁动态响应研究

基于随机振动理论研究了桥面不平顺影响下,车桥耦合振动作用时的中小桥梁动态响应曲线。通过建立车桥耦合振动方程,基于虚拟激励法对重力引起的确定性激励和桥面不平顺引起的随机激励求解,得到了桥梁跨中挠度和应力响应的均值和标准差。运用 法则定义随机激励的确定值值域,分析了桥梁跨中位移和应力响应在不同车速和桥面不平顺等级作用下的特性,并讨论了动态响应曲线与准静态影响线的差异。结果表明：桥梁跨中挠度和应力标准差受车速和桥面不平顺等级变化的影响很大;桥梁动态响应值域范围很大,具有较强的随机性;相比准静态影响线,动态响应曲线更能体现车桥之间激励的耦合随机作用。     

基于机器视觉的非均匀分布点圆度误差评定

对圆度误差的4种评定方法进行比较和改进,改进后的最小二乘法适用于非均匀分布圆或圆弧的计算,同时研究最小区域法在离散点非均匀分布圆的运用。用解析的方法求解最小区域下的圆度误差,改善了优化迭代法的缺陷,经过数次计算准确得到最小区域,以最小区域圆心作为圆心、平均半径作为半径,可将其应用于不完整圆和分布点非均匀分布的圆中。引出圆度误差评定的相对误差率ξk ,可说明圆度误差、加工精度、加工方法及微观占比例等情况。在圆孔视觉检测实验中,4种评定方法的计算结果分别为ξk1=0.019、ξk2=0.017、ξk3=0.018、ξk4=0.019。离散非均匀分布圆的检测方法可广泛应用于实际生产检测中。     

随机Kadomtsev-petviashvili方程的精确解

利用厄尔米特变换和F-展开法,得到了随机Kadomtsev-petviashvili方程由Jacobi椭圆函数表示的精确解,此显示了F-展开法也可以用来求解随机微分方程。     

Error estimation in a stochastic finite element method in electrokinetics

Input data to a numerical model are not necessarily well known. Uncertainties may exist both in material properties and in the geometry of the device. They can be due, for instance, to ageing or imperfections in the manufacturing process. Input data can be modelled as random variables leading to a stochastic model. In electromagnetism, this leads to solution of a stochastic partial differential equation system. The solution can be approximated by a linear combination of basis functions rising from the tensorial product of the basis functions used to discretize the space (nodal shape function for example) and basis functions used to discretize the random dimension (a polynomial chaos expansion for example). Some methods (SSFEM, collocation) have been proposed in the literature to calculate such approximation. The issue is then how to compare the different approaches in an objective way. One solution is to use an appropriate a posteriori numerical error estimator. In this paper, we present an error estimator based on the constitutive relation error in electrokinetics, which allows the calculation of the distance between an average solution and the unknown exact solution. The method of calculation of the error is detailed in this paper from two solutions that satisfy the two equilibrium equations. In an example, we compare two different approximations (Legendre and Hermite polynomial chaos expansions) for the random dimension using the proposed error estimator. In addition, we show how to choose the appropriate order for the polynomial chaos expansion for the proposed error estimator. Copyright © 2009 John Wiley & Sons, Ltd.     

Boundary integral equations for notch problems in an anisotropic half-space

A crack or a hole embedded in an anisotropic half-plane space subjected to a concentrated force at its surface is analyzed. Based on the Stroh formalism and the fundamental solutions to the half-plane solid due to point dislocations, the problem can be formulated by a system of boundary integral equations for the unknown dislocation densities defined on the crack or hole border. These integral equations are then reduced to algebraic equations by using the properties of the Chebyshev polynomials in conjunction with the appropriate transformations. Numerical results have been carried out for both crack problems and hole problems to elucidate the effect of geometric configurations on the stress intensity factors and the stress concentration.     

全非平稳地震动过程的概率模型及反应谱拟合

在强度调制函数基础上,发展了一类全非平稳地震动过程的时-频调制函数。结合平稳地震动过程的功率谱密度函数,建立了全非平稳地震动过程的演变功率谱模型,并根据建筑抗震设计规范确定了模型参数的取值。应用非平稳过程模拟的谱表示-随机函数方法,生成具有完备概率的非平稳地震动加速度过程的代表性时程集合,进而计算代表性时程的平均反应谱。为保证计算平均反应谱与建筑抗震设计规范反应谱的一致性,建议了反应谱容许误差的双重控制准则,即平均相对误差和最大相对误差。研究表明,通过对演变功率谱的3次迭代修正,即可实现计算平均反应谱与规范反应谱拟合的目的,为应用结构随机动力学进行实际工程结构抗震设计提供了依据。     

基于混合有限差分格式的非线性奇异摄动问题的最大范数的后验误差估计

自适应移动网格算法在奇异摄动微分方程的数值解法中占有非常重要的地位,其关键技术是构造出有效的离散格式和相应的后验误差估计。基于此,对一类带参数的一阶非线性奇异摄动初值问题,给出了其连续解的稳定性估计及相关推论。然后,在任意非均匀网格上,利用向后欧拉公式和一阶中心有限差分格式建立了一个混合有限差分格式,并严格分析了离散解的稳定性。同时,基于连续解的稳定性估计和分段线性插值技术,推导出混合有限差分格式的最大范数的后验误差估计。利用该后验误差估计选择了一个最优的网格控制函数,并结合网格等分布原理设计了一个自适应网格生成算法。最后的数值实验验证了自适应移动网格算法的有效性,且算法的平均收敛阶可达到二阶。数值结果进一步表明自适应移动网格的误差明显小于 Shishkin 网格的误差,且其收敛阶也高于 Shishkin 网格计算得到的收敛阶。     

基于单缝衍射原理的圆度误差测量方法

提出了一种全新的圆度误差光学自动测量方法.新方法基于单缝衍射原理,利用工件的圆度误差来改变衍射单缝的宽度,进而改变衍射条纹的间距.在测量单缝衍射中央明纹的宽度时,提出了用最小二乘法对测量到的衍射图像的光强分布进行二次曲线拟合,并由拟合得到的数学表达式,确定衍射暗条纹的精确位置.由导出的工件圆度误差与单缝衍射中央明纹宽度之间的关系,用最小二乘法对工件的圆度误差进行了评定.测量结果表明,新的圆度误差测量方法是可行的,圆度误差的相对不确定度小于1.4%,并且测量系统具有操作方便、精度高的优点,新方法也可应用于锥角、直线度误差等其它几何量的精密测量中.     

结构在水平与竖向随机地震同时作用下的相关函数和谱密度

对单自由度结构在水平与竖向地震同时作用下的随机稳定性、响应及其相关函数和谱密度函数进行系统研究。首先利用Stratonovich和It随机微分方程与响应矩微分方程的互相关转化关系,建立了结构响应矩方程;然后根据Hurwitz随机稳定准则,获得了结构一阶和二阶响应矩渐近稳定的解析判别式;继而,利用复模态法获得了结构响应二阶矩的解析瞬态解和平稳解;最后利用It随机微分方程解具有的非可料函数性质,获得了结构位移、速度响应的自相关函数、互相关函数以及谱密度函数、互谱密度函数的解析解,给出了算例,并综合分析了各种参数对结构响应、稳定性以及相关函数和谱密度函数的影响。     

Experimental Investigation into the Hole Quality in Ultrasonic Machining of WC-Co Composite

This study has focused on the evaluation of the influence of different input variables (cobalt content, thickness of workpiece, tool profile, tool material, size of abrasive grains, and power rating) on the hole quality (dimensional and form accuracy) obtained in ultrasonic machining of WC-Co composite material. Taguchi's approach has been employed for planning the experiments and optimization of the experimental results. Three measures of hole quality (hole oversize, out of roundness, and conicity) have been investigated under controlled experimental conditions. The experimental results showed that abrasive grit size and power rating were most influential for the hole quality. Hole quality has been found to be improved at higher cobalt content, whereas it is found to be degraded with the use of coarse grit size.     

基于量子遗传算法的圆度误差测量研究

提出采用量子遗传算法,以提高圆度测量精度。首先用最小二乘法拟合获得建模数据中圆度图像的圆心坐标和半径;再通过圆度计算剔除不符合要求的圆度;然后用量子遗传算法进行多进制编码,量子旋转门非固定步长调整更新;最后给出圆度误差测量流程。实验仿真显示该算法获得了精确的测量数值,与三坐标测量机测量结果误差相差小于0.005 8 mm,半径相对误差小于0.19%,测量最大误差均在0.01%以内,同时最大误差波动比较平稳,测量不确定度比其它方法值较低。     

Distribution of stresses near V-shaped notches in the complex stressed state

By the method of singular integral equations, we obtain the solution of a two-dimensional problem of the elasticity theory for a plane containing a semiinfinite rounded V-notch under complex loading. On the basis on this solution, we establish the relationships between the stress intensity factors at the tip of the sharp V-notch and the maximum stresses and their gradients at the tip of the corresponding rounded notch. For finite bodies with V-notches, the presented solutions are obtained as asymptotic dependences for small radii of rounding of the tips. The deduced relationships can be used to perform the limit transitions and find the stress intensity factors at the tips of sharp V-notches on the basis of the solutions obtained for the corresponding rounded stress concentrators. The efficiency of the method is illustrated for the problem of determination of the stress intensity factors at the vertices of a rectangular hole in the elastic plane.