Sphericity Error Research Articles

Precision Shaping Experiment of Ceramic Ball Blank with Oscillating-Plate Lapping Method

An oscillating-plane lapping method, by which pre-sintered ceramic ball blanks are shaped before full sintering, was proposed to improve the precision of the ceramic ball blanks. For good sphercity and diameter consistency of ceramic ball blanks obtained after this shaping process, much less workmaterial is needed to be removed in the following lapping process of the full sintered ceramic ball blanks. Shaping experiments were carried out on a newly developed oscillating-plane lapping device. Results show that material removal rate of the pre-sintered Si3N4 ball blanks with nominal diameter Ø13mm reaches 0.92mm/h in average. The sphericity error is reduced from 0.07-0.196mm to 0.002-0.006mm, and the consistency error of diameter is reduced from 0.309 to 0.019mm after 45min's shaping. Compared with the processing time of the unshaped full-sintered blanks in the following lapping process, it takes just a half time for the shaped full-sintered blanks to be semi-finished in this study. It indicates that the oscillating-plane lapping is a promising high efficient method to improve the precision and consistency of the ceramic ball blanks, and the efficiency of the followed lapping process can be improved notably.

Minimum centroid neighborhood for minimum zone sphericity

Abstract The minimum zone sphericity tolerance is derived from the ANSI and ISO standards for roundness and has extensive applications in the tribology of ball bearings, hip joints and other lubricated pairs. The worst-case proposed in this paper provides theoretical evidence that the minimum zone center of the two (circumscribed and inscribed reference) spheres with minimum radial separation containing the sampled spherical surface is included in a spherical neighborhood centered in the centroid of radius 2 π −2 E C , where E C is the sphericity error related to the centroid, which can be determined in closed form. Such linear estimating (about 20% of E C from the centroid, i.e., about one order of magnitude lower than the sphericity tolerance to be assessed) can be used to locate the sphere center with a given tolerance and as a search neighborhood for minimum zone center-based algorithms, such as metaheuristics (genetic algorithms, particle swarm optimization, etc.). The proposed upper bound has been experimentally assessed, using a genetic algorithm (GA) with parameters previously optimized for roundness and extended to three dimensions, which has overcome most of all available datasets from the literature that have been tested with center-based minimum zone algorithms by different authors. The optimum dataset size on artificially generated datasets is also discussed and it is speculated to allow the extension of the proposed upper bound to partial (or incomplete) spherical features.

Evaluation of Sphericity Error Using Differential Evolution Method

The mathematical model of sphericity error under the condition of minimum zone is derived. An algorithm based on the differential evolution (DE) is proposed to evaluate the sphericity error. Its principle and realization approach are discussed to solve the minimum zone sphericity error. The proposed method was verified by an example. Compared with other methods, the results show that the proposed method makes the sphericity error evaluation more accurate.

Adaptive Monte Carlo and GUM methods for the evaluation of measurement uncertainty of cylindricity error

Abstract Measurement uncertainty is one of the most important concepts in geometrical product specification (GPS). The “Guide to the expression of uncertainty in measurement (GUM)” is the internationally accepted master document for the evaluation of uncertainty. The GUM method (GUMM) requires the use of a first-order Taylor series expansion for propagating uncertainties. However, when the mathematical model of measurand is strongly non-linear the use of this linear approximation may be inadequate. Supplement 1 to GUM (GUM S1) has recently been proposed based on the basis of probability density functions (PDFs) using the Monte Carlo method (MCM). In order to solve the problem that the number of Monte Carlo trials needs to be selected priori, adaptive Monte Carlo method (AMCM) described in GUM S1 is recommended to control over the quality of the numerical results provided by MCM. The measurement and evaluation of cylindricity errors are essential to ensure proper assembly and good performance. In this paper, the mathematical model of cylindricity error based on the minimum zone condition is established and a quasi particle swarm optimization algorithm (QPSO) is proposed for searching the cylindricity error. Because the model is non-linear, it is necessary to verify whether GUMM is valid for the evaluation of measurement uncertainty of cylindricity error. Then, AMCM and GUMM are developed to estimate the uncertainty. The procedure of AMCM scheme and the validation of GUMM using AMCM are given in detail. Practical example is illustrated and the result shows that GUMM is not completely valid for high-precision evaluation of the measurement uncertainty of cylindricity error if only the first-order terms in the Taylor series approximation are taken into account. Compared with conventional methods, not only the proposed QPSO method can search the minimum zone cylindricity error precisely and rapidly, but also the Monte Carlo simulation is adaptive and AMCM can provide control variables (i.e. expected value, standard uncertainty and lower and higher coverage interval endpoints) with an expected numerical tolerance. The methods can be extended to the evaluation of measurement uncertainty of other form errors such as roundness and sphericity errors.

An Approach for the Evaluation of Sphericity Error and Its Uncertainty

The verification uncertainty theory in new generation geometrical product specification (GPS) has been applied in the whole process of geometrical product specification and verification. According to the uncertainty theory, a novel method for the uncertainty evaluation of sphericity errors was proposed. The association operator in new generation GPS was applied. The mathematical model of the sphericity error and the optimal objective function were developed under the condition of minimum zone. Particle swarm optimization (PSO) algorithm was used to search for the optimal solution of sphericity error. The coefficients of the correlation element and propagation of uncertainty were calculated so that the sphericity error and the uncertainty of the result can be obtained. The evaluation results are accurate and consistent with requirements of a new generation GPS standard. It also indicates that this method can improve the integrity and veracity of verification.

The Relationship between Minimum Zone Sphere and Minimum Circumscribed Sphere and Maximum Inscribed Sphere

Among the four methods (minimum zone sphere, minimum circumscribed sphere, maximum inscribed sphere, and least square sphere), only the minimum zone sphere complies with ANSI and ISO standards and has the minimum sphericity error value. Evaluation of sphericity error is formulated as a non-differentiable unconstrained optimization problem and hard to handle. The minimum circumscribed sphere and the maximum inscribed sphere are all easily solved by iterative comparisons, so the relationship between the minimum zone sphere and the minimum circumscribed sphere, the maximum inscribed sphere is proposed to solve efficiently the minimum zone problem. The relationship is implemented and validated with the data available in the literature.

The Minimum Zone Evaluation for Sphericity Error Based on the Dichotomy Approximating

A new algorithm which is “the minimum zone evaluation for sphericity error based on the dichotomy approximating” has been presented with the geometric characteristics and research situation of sphericity error and through the understanding of the definition and the evaluation problem of the sphericity. The principle of the algorithm is that based on finding the initial point of the ball as the base, around which the side decorates cube by certain length, calculating the distance of the points on the sphere to the cube measuring points of the eight vertex and the initial point, by comparing and judging to find a new beginning starting point, and arrange new cube repeatedly until meets the requirement. The sphericity error value of corresponding evaluation method (Minimum Zone Sphere method (MZS)) was obtained according to that. The mathematical formula、program flowchart and example verification were given. The example verification results show that the sphericity error can be evaluated effectively and exactly with this algorithm.

Error analysis and in-process compensation on cup wheel grinding of hard sphere

Grinding using cup wheel has been widely used in hard sphere machining process. In conventional sphere grinding (CSG), the linear velocity of the point with the same latitude on the sphere varies, which causes the sphericity error. The misalignment of the machine tool is another source of sphericity error. In this paper, these two types of sphericity errors are analyzed, and an adjustable depth of cut sphere grinding (ADCSG) is proposed to reduce the sphericity error. CSG and ADCSG were carried out on a home-made MD6050 NC precision sphere grinding machine, and the effectiveness of ADCSG is demonstrated with grinding tests. The results showed that ADCSG can reduce the sphericity error effectively compared with CSG. The sphericity error can reach less than 5 μm.

Sphericity Evaluation Using Maximum Inscribed Sphere Method

The maximum inscribed sphere (MIS) method is an important method to evaluate sphericity error suitable for the sphere with the maximum material condition of a concave. It is proven that the MIS of a data point set is decided only by four points in the paper. An algorithm of sphericity evaluation based on the MIS is proposed. Six points with extreme coordinates are selected as an initial subset and the MIS is constructed by four of them. Another point which is the farthest from the center among the set replaces one of the former four points each time. According with the minimum criterion, the final sphere is just the MIS of the data point set when all data points is outside the sphere. The validated results show that the proposed strategy offers an effective way to identify the control data points at few iterative turns and gives an efficient approach to solve the sphericity problems.

激光CCD复合测量方法在ICF靶丸球度误差检测中的应用

激光CCD复合测量方法在ICF靶丸球度误差检测中的应用

Processing of WC-Co Cemented Carbide Balls by Rotated Dual-Plates Lapping Machine

In this paper, the batch processing of Φ35mm WC-Co cemented carbide balls used in one-way value is studied by rotated dual-plates (RDP) lapping machine. On the basis of this innovated equipment, the processing parameters, including abrasive grain, rotating velocity and load, are changed legitimately in experiment. The experimental results show that the abrasive grain is used from big to small, while the rotating velocity and load are gradually reduced in order to improve the uniformity of lapping trace distribution on the ball surface in batch processing. Finally the rational course and parameters are defined, these WC-Co cemented carbide balls satisfy the products’ final demand which means the sphericity error (Rnd) is less than 1μm and the roughness of surface (Ra) is better than 50nm. Because of the feasibility, the precision of final WC-Co cemented carbide balls improves and can be applied in one-way value used in oil exactor.

Novel three‐dimensional MRI technique for study of cartilaginous hip surfaces in Legg‐Calvé‐Perthes disease

Treatment of Legg-Calvé-Perthes disease (LCPD) may improve if new knowledge can be obtained regarding how articular cartilage changes shape during the course of this disorder. A new technique is presented showing how analyses of magnetic resonance images can be used to quantify the three-dimensional changes in the femoral and acetabular articulating cartilage surfaces of children with LCPD. Ten male subjects (8 +/- 1 years) with unilateral LCPD were enrolled in this IRB approved study. Sets of magnetic resonance images of both hips were obtained at three different times. Three-dimensional virtual models of the cartilage were created from these images, and mathematical spheres were fit to the articulating surfaces. Five parameters (size, shape deformity (sphericity error), radial growth rate, joint fit, and joint incongruity) were used to quantify cartilage surface shape. Data were analyzed by using a linear mixed-model. Joint incongruity, i.e., the distance between the centers of the femoral and acetabular spheres, was slightly more than 2.5 times larger (p = 0.001) in LCPD hips than the contralateral normal hips. Cartilage shape deformity was 65% larger in hips with LCPD than in normal hips. Growth rates of the femoral head and the opposing acetabular surface showed that distortion of the femoral surface occurred first and the opposing acetabular surface followed. Mean radial difference (acetabular surface radius minus femoral surface radius) in LCPD hips was less than half (p < 0.01) the value of normal hips. Interobserver variability was approximately 10% of the value attributable to LCPD. This is the first known report presenting a technique that quantifies the three-dimensional size, deformity, growth, fit. and incongruity of the femoral and acetabular articulating cartilaginous surfaces of LCPD and contralateral normal hips. The data obtained support the use of this technique and provide pilot data for a future clinical study of LCPD. Objective assessment of cartilage shape enabled by this technique may aid future diagnoses, enable monitoring of three-dimensional femoral and acetabular remodeling, and permit quantitative assessment of treatment efficacy.

Evaluation of Sphericity Error by Using Minimum Circumscribed Sphere

Aiming at evaluation of sphericity form error in cartesian coordinates,an evaluation algorithm of sphericity error by using the minimum circumscribed sphere(MCS) is introduced.In order to evaluate sphericity error accurately,a 3D model of measuring and computing based on cartesian coordinates is established,and geometric curved surface character of circumscribed sphere is studied.As a result,the theory of using geometrical symmetry of intersecting chord to quickly evaluate sphericity error is obtained.The evaluation entia of "2+1","3+1","4+1"modes for MCS by intersecting chord relation is constructed.The virtual center location produced through intersecting chord twice can determine the position of evaluation point accurately,which means making the evaluation more accurate and reliable.The analysis data and results indicate that the algorithm has high computing speed and evaluating precision,thus providing a new strategy and means for sphericity form error evaluation.

Measurement and Evaluation of the Geometric Error of Precision Machined Spherical Surface

This paper is principally concerned with the form error measurement and evaluation of the spherical surface after precision machining. According to the minimum zone condition, the definition of sphericity error is first analyzed. To obtain the minimum sphericity error from the form data, an approximations technique is devised. First the method is modeled, then the process of approximations is constructed, and finally software for data processing is programmed. Six steel bolls of φ50 are measured on a coordinate measurement machine (CMM) in the Measurement experiment. The experiment results indicate that the geometric approximations technique can give an accurate solution with high efficiency.

Measurement of Roundness and Sphericity of the Micro Sphere Based on Atomic Force Microscope

The measuring system was developed based on a reconstructed atomic force microscope (AFM) combined with the precision rotating air-bearing and assistant transform shaft. By this system the experiment was carried out according to the principle of three orthogonal orientations towards a micro sphere about 0.46 mm in diameter, and nine one-dimensional traces around the sphere were obtained. Analyses on roundness and sphericity are carried out from the measured data.The results show that the maximal roundness is 0.3895 μm, and the sphericity error is 0.3880 μm.These evaluations are significant references to evaluate its fabrication accuracy or to reform its machining processes.

Algorithm for sphericity error and the number of measured points

处理决定测量的点的最佳的数字的技术和方法的数据被学习瞄准在一台并列测量机器(CMM ) 上测量的球状错误。为球形的表面的最小的地区的完美的标准首先被分析，然后从表格数据寻找最小的球状错误的一种近似技术被学习。以便获得球形的表面的最小的地区，光线的分离被与某些步沿着某些方向移动同心范围的中心逐渐地减少。因此算法精确、有效。在为近似技术的适当数学模型被创造以后，处理节目的一个数据因此被开发。由用发达程序处理韵律的数据，当测量的点的不同数字从一样的样品被拿时，球形的错误被评估，然后相应散布图和合适的曲线因为样品图形地被代表。测量的点的最佳的数字通过回归分析被决定。实验两个都显示出那为决定测量的点的最佳的数字处理技术和方法的数据是有效的。平均，获得的球状错误是比最少的方形的答案小的 5.78 亩 m，其精确性被 8.63% 增加；测量的点的获得的最佳的数字是通常测量的数字的一半。

Novel 3D sphericity evaluation based on SFS-NDT

Sphericity evaluation is an important work in modern industry. Because the former 3D measurement is complex and with low precision, a lot of sphericity evaluation is studied only based on 2D measurement. But the 2D sphericity evaluation method is reported unreasonable because sphere is a 3D object. In order to evaluate sphericity error accurately and effectively, based on SFS-NDT 3D measurement method studied in reference [Research on 3D recovery based on only one online measuring image. TianJin University, Doctor thesis (2004)], this paper proposes a novel 3D sphericity evaluation method, which use all the 3D data in the evaluation procedure. Minimum zone solution using all the 3D data is adopted to make the evaluation more accurate and reliable. Compared with the conventional sphericity evaluation based on 2D measurement, the proposed 3D evaluation has the merits of non-contact, non-destructive, low cost, high speed, and high reliable. 3D shape recovery of a sphere based on SFS and its 3D sphericity evaluation procedure will be analyzed in this paper.

An immune evolutionary algorithm for sphericity error evaluation

An immune evolutionary algorithm (IEA) is proposed by imitating the defense process of the immune system and the ideas of mutation in evolutionary biology. In the IEA, a self-adaptive mutation operator is constructed to decide the mutation step size of every antibody according to its environment, and an affinity calculation process is also embedded to maintain diversity. Application of the proposed algorithm to the evaluation of sphericity error is investigated. The method takes the sphericity error objective function and constraints as antigen. Through genetic evolution, an antibody that most fits the antigen becomes the solution. Finally, the experimental results evaluated by different methods confirm the effectiveness of the proposed IEA. Furthermore, the method has good flexibility and is also suitable for other evaluations of form errors such as circularity, flatness etc.

An immune evolutionary algorithm for sphericity error evaluation

An immune evolutionary algorithm (IEA) is proposed by imitating the defense process of the immune system and the ideas of mutation in evolutionary biology. In the IEA, a self-adaptive mutation operator is constructed to decide the mutation step size of every antibody according to its environment, and an affinity calculation process is also embedded to maintain diversity. Application of the proposed algorithm to the evaluation of sphericity error is investigated. The method takes the sphericity error objective function and constraints as antigen. Through genetic evolution, an antibody that most fits the antigen becomes the solution. Finally, the experimental results evaluated by different methods confirm the effectiveness of the proposed IEA. Furthermore, the method has good flexibility and is also suitable for other evaluations of form errors such as circularity, flatness etc.

Evaluation of circularity and sphericity from coordinate measurement data

The coordinate measuring machines (CMMs) have proven to be reliable, flexible and very much suitable for determining the acceptability of manufactured parts. In this paper, techniques for evaluating circularity and sphericity error from CMM data are presented. The form error can be evaluated directly from CMM data by employing circle/sphere as assessment features and using normal deviations. The CMM data can also be transformed by applying appropriate methods that not only suppress the size but also introduce distortion. The form error is evaluated from the transformed data by employing limacon/limacoid as assessment features and using linear deviations. The methods for handling CMM and transformed data are given in this paper. The proposed methods are validated using the data available in literature.