Cylindricity modeling and tolerance analysis for cylindrical components

The circular and cylindrical features are fundamental geometric features in machines. Cylindricity error affects the fitting conditions of cylindrical components and impacts the performance of the precision products. In this paper, the cylindricity error was modeled using L-F functions and evaluated by particle swarm optimization algorithm. Then the contact method is developed to determine the position accuracy through the virtual assembling of the bore and shaft using Monte Carlo simulation. The effects of the cylindricity error and the number of lobes on the position error were analyzed in detail. The results indicate that the cylindricity error has more significant influence on the position accuracy between the cylindrical parts than the roundness error. Using the suggested method in the paper, the position accuracy can be rapidly predicted after the design tolerances are allocated or the geometrical errors are measured on manufactured parts.     

High precision lapping for polygonal holes

Lapping has been used as a finishing method to improve the roundness of shafts and holes. This paper introduces two methods for the finishing of polygonal holes whose shapes are ellipsoidal, three-lobed, four-lobed, and so on. One method is the “regular polygonal lap” in which the amplitude of the desired Fourier component of the out-of-roundness is kept constant and the other components are decreased. The other is the “tripod lap,” which can increase the amplitude of any Fourier component using special types of lapping tools produced by applying the “tripod roundness measuring method.” Based on a theoretical analysis, this study shows that any component of the out-of-roundness can be increased by using the new tripod lap method.     

Roundness modeling of machined parts for tolerance analysis

Out-of-roundness of circular and cylindrical parts can greatly affect assembly accuracy. At present, the ASME Y14.5M-1994 specifies circularity tolerance based on two extreme circular boundaries to confine the highest peak and the lowest valley of a roundness profile; however, the profile variations within the two extreme boundaries are not accounted for. In this paper, we propose a harmonic roundness model using Fourier series expansion. In addition, a cutting profile model has been developed to illustrate the relationship between the radial error motion of a spindle and the resultant part profile. This relationship is then used to provide physical meanings to the harmonics generated by the proposed roundness model for a part profile, in particular those caused by fractional frequency spindle error motions. The proposed harmonic model has been verified statistically by a large number of real profiles produced by both turning and cylindrical grinding processes. The proposed roundness model is expected to provide insights into the advanced tolerance analysis of circular and cylindrical parts.     

Lapping high precision polygonal shafts

Lapping has been used as a finishing method to improve roundness accuracy of cylindrical shafts and holes. However, little has been published on the theoretical analysis of the finishing mechanism for high precision polygonal cross sections. This paper offers a method for finishing polygonal shafts, whose shapes are ellipsoidal, triangular, four-lobed and so on. By this method, the amplitude can be increased for any Fourier component of out-of-roundness. Special types of lapping tools were produced by applying the ‘V-block’ roundness measuring method. Using one of the V-block lapping tools, the magnitude for the four-lobed component was increased form 8.4 μm to 9.9 μm, while the other components were decreased; out-of-roundness improved from 35 μm to 25 μm. By setting the angle between the supporting teeth at an appropriate value, any polygonal shaft can be finished with these laps. This method is also applicable to finishing polygonal holes     

Correcting capacitive displacement measurements in metrology applications with cylindrical artifacts

Metrology applications commonly require non-contact, capacitive sensors for displacement measurements due to their nanometer resolution. In some metrology applications, for example, the measurement of roundness and spindle error motion, the displacements of stationary and rotating cylindrical artifacts are measured. Error from using a conventionally calibrated sensor with a non-flat (e.g., cylindrical) target is typically neglected, but these errors cannot be ignored for nanometer-level accuracy. The capacitance between a sensor and a cylindrical target is less than that of a sensor with a flat target, which causes four effects. As the diameter of the target shrinks, the sensitivity of the sensor increases, the sensing range decreases, the sensing range shifts towards the target, and the nonlinearity increases. These errors can be greatly reduced by either calibrating sensors with the correct target surface or by determining corrections for post-processing data. This paper quantifies and experimentally verifies these errors for a commonly used sensor, and a simulation of a nanometer-level measurement of out-of-roundness and spindle error motion demonstrates that measurement accuracy is improved with corrected sensitivities.     

基于学习的2x/y直线进给轴圆轮廓误差精密补偿方法

为了提高龙门2x/y直线进给轴联动的圆轮廓精度,对进给轴联动圆轮廓误差的测量、评价和补偿方法进行研究。分析直线电动机驱动的进给轴联动过程存在圆轮廓偏差的原因及其补偿的复杂性,给出一种基于学习的联动轴圆轮廓误差在线精密补偿方法,此方法通过双光束激光干涉仪动态精密测量x/y联动轴的实时坐标值,应用最小二乘圆方法评价确定理想圆,接着通过与理想圆轴坐标位置的比较,计算得到轴向偏差学习样本,建立基于最小二乘支持矢量回归机(Least square support vector regression,LS-SVR)方法的轴向偏差离线识别模型,通过模型的在线回归计算确定联动进给过程的偏差补偿量,给出补偿量控制输出策略与补偿系统构建方案,在自构建的直线进给轴平台上进行在线补偿试验。结果表明应用该方法对2x/y直线进给轴联动的轴向偏差进行在线补偿,可使圆轮廓精度提高68.7%。     

<Emphasis Type="Bold">Roundness Error Prediction with a Volumetric Error Model Including Spindle Error Motions of a Machine Tool</Emphasis>

This paper proposes a modified volumetric error model that includes spindle error motions as well as geometric errors. The model is constructed using rigid-body kinematics and homogeneous transformation matrices and an additional error matrix describing spindle error motions is included. The suggested model predicts the positioning errors at a given axis position as a function of both the axis position and the engaged spindle rotation angle. Two circular interpolation tests (inner and outer circle of the same radius) are simulated and the machined part profiles are predicted. To verify the simulation results, machining tests are performed according to the ISO 10791-7 standard. The error model with spindle errors shows a better agreement, between the simulated and measured roundness errors, than the simple geometric model. It can be seen that the geometric errors determine the basic part profiles and the spindle errors change the basic profiles according to the magnitude of the errors and the spindle rotation angle.     

Roundness error evaluation algorithm based on polar coordinate transform

A new method for roundness error evaluation using polar coordinate system, named as polar coordinate transform algorithm (PCTA), was presented in this paper. The algorithm first allocates a circular region around the least square circle center following certain rules, then calculates the polar radius for all measured points by translating polar coordinate system to each point in the region in turn, and finally obtains minimum circumscribed center point, maximum inscribed center point and minimum zone center point from comparing each polar radius relative to each polar coordinate system. With accurate center point, the algorithm could give more accurate roundness evaluation. In the paper, the process of PCTA was described in detail including the algorithm formula and flowchart. Theoretical calculation and testing results show that PCTA can evaluate roundness error effectually and accurately.     

基于网格模型的圆度在线检测系统开发

提出一种基于三角网格模型的圆度在线检测方法.根据圆柱的空间位置、分布的测点和测点的矢量来进行测量路径的设计.利用最小二乘法,计算圆柱测量截面的圆度误差.最后以CMM检测结果作为理想的标准,对圆度在线检测结果进行评估,结果得出该方法所带来的圆度误差非常小.     

Metrological characterization of interior circular features using digital optical machines: Calculation models and application scope

The objective of this work is to analyze and compare common methods used for indirect metrological characterization of circular geometrical features: least square circle, minimum zone circle, maximum inscribed circle and minimum circumscribing circle, based on images obtained using digital optical machines. The self-developed calculation algorithms used were implemented with Matlab^® software considering the following study variables: the amplitude of the angular sector of the circular feature, its nominal radius and the magnification used by the optical machine. Under different measuring conditions the radius and roundness error of an angular sector of a piece of LEGO and of a ring gauge were determined. That is, the proposed algorithms allow the characterization of both full circumferences as angular sectors. We were able to determine the degree of precision of each method and its scope of application by comparing the results obtained with different adjustments to values certified by an accredited laboratory.     

Chucking compliance compensation with a linear motor-driven tool system

This paper describes a system that compensates for machining errors resulting from chucking with separated jaws. In machining a chucked cylindrical workpiece, degradation in machining accuracy, such as out-of-roundness, is inevitable due to variation in the radial compliance of the chuck workpiece system caused by the position of jaws with respect to the direction of the applied force. To compensate for the machining error induced by chucking compliance, the roundness profile of the workpiece due to chucking compliance after machining should be predicted first. Then, using this prediction profile, a compensating tool feed trajectory can be generated. By synchronizing the cutting tool drive system with the rotation of the workpiece, machining errors induced by chucking compliance can be reduced. To satisfy the condition that the cutting tool feed system must provide both high speed and high position accuracy, a brushless linear DC motor is used. In this study, first, the variation in the radial compliance of the chuck workpiece system is obtained through a force-deflection experiment with a workpiece-chucked lathe. Next, using mathematical equations and the results of the cutting experiment, a workpiece profile prediction and its compensating tool trajectory are generated. Then, the configuration of the compensation system based on a linear motor is described, and a proportional integral derivative (PID) controller is designed to improve the system performance. Finally, the tracking performance of the system is confirmed by experiment. A real cutting experiment shows significant improvement in roundness.     

内燃机轴承失圆对润滑特性的影响

为研究内燃机主轴承失圆对轴承润滑特性的影响规律,以某V型12缸柴油机为研究对象,建立曲柄连杆机构多柔体动力学模型,结合基于质量守恒边界条件的广义雷诺方程模型,分析失主轴承波形失圆误差对轴承润滑特性的影响规律。研究表明：随主轴承波形失圆误差增大,轴承峰值总压力显著增加,轴承发生表面疲劳及黏接失效概率较高,同时轴承润滑状态明显变差,润滑稳定性降低;但主轴承波形失圆误差对平均总摩擦功率损失影响不显著。     

基于小波变换软测量技术在大型回转体中的圆度误差方法研究

针对大型回转体圆度误差难以检测的问题,采用非接触式软测量技术以相对测量方法采集大型回转体零件回转面的实测信号,基于小波变换消噪滤波的方法,快速分解提取实测信号中的圆度误差信号,通过圆度误差的评定计算大型回转体的圆度误差。结果表明:该方法能够有效提取大型回转体实测信号中的圆度误差信号,通过实例计算,某2000 mm直径回转体的圆度误差为0.32 mm,该方法为大型回转体圆度误差计算提供一定的理论依据,具有一定的实际应用性。     

Batch circular form error characterization and evaluation

In practice, it is necessary to understand the general quality status of a batch of circular features machined under the same or similar conditions. In particular, it is necessary to understand the possible worst case among such a batch of circular features. To this end, a brand-new concept called “batch circular form error” is presented in this paper. Detailed definitions of batch circular form error including batch circular form error with ideal conditions and batch circular form error with realistic conditions are provided. Accordingly, the evaluation algorithms of batch circular form error are developed. The algorithms include the characterization of the deterministic profile of circular features and the evaluation of batch circular form error based on a given profile confidence level. Case studies with simulation and experimental data are used for demonstration. The results show that the batch circular form error can be estimated with the data measured from any circular feature in the batch, as long as the sample size is large enough and random ratio is not too large.     

Evaluation of roundness error from coordinate data using curvature technique

Workpieces like a shaft or hole make contact on their functional boundaries and its circularity is obtained based on the functional boundaries by enveloping features. Data for evaluating roundness error can be obtained from coordinate measuring machines. A technique for roundness error based on the curvature has been proposed to deal with coordinate measurement data. The computational geometric concepts of convex hulls are used to reduce the computation. The method developed is implemented and validated with the data available in the literature.     

Experimental investigation of surface and geometrical characteristics of rotary ultrasonic drilling of soda glass

Edge-chipping, surface roughness and dimensional accuracy are crucial quality aspects of drilled holes in hard-to-cut material such as glass, ceramics and carbon fiber reinforced plastics. In this article, an experimental study was conducted to investigate the quality measures of holes produced by rotary ultrasonic drilling (RUD) and conventional drilling. Edge-chipping width at tool exit side, the surface roughness (Ra and Rz), out-of-roundness, cylindricity error and hole conicity were the main responses when drilling soda glass using diamond abrasive tools and a cutting fluid. Statistically designed experiments were carried out for rotary ultrasonic and conventional drilling (CD) at two levels of tool feed rate (0.6 and 6?mm/min), spindle speed (3,000 and 8,000?rpm) and tool particles-concentration. Analysis of variance was used to define the significant factors and their interactions and build models for predicting the responses. The results showed that reducing the chipping, surface roughness and roundness error. The normal tool concentration showed a substantial effect in improving the surface quality and reducing the hole-geometrical errors.     

圆度误差测量及误差分析

介绍了圆度误差的最小二乘法及其数学模型的建立 ,并分析了影响圆度误差测量的几种误差因素。     

Precision evaluation of surface form error of a large-scale roll workpiece on a drum roll lathe

This paper presents on-machine evaluation of surface form error components of a large-scale roll workpiece, including the out-of-roundness, the out-of-straightness, the taper angle and the diameter deviation, on a drum roll lathe. A pair of capacitive-type displacement probes is mounted on the carriage slide to target the two sides of the roll workpiece mounted on the spindle, which has a length of 2000 mm, a diameter of 320 mm and a mass of 350 kg. The outputs of the probes are employed to accurately evaluate the surface form error components through separating the influences of the motion errors of the spindle and the carriage slide of the lathe. It has been difficult to apply the reversal error-separation method for measurement of the out-of-roundness component of such a large workpiece because it is difficult to reverse the workpiece with respect to the spindle on the lathe due to its large size and heavy mass. An improved reversal operation technique, in which the spindle is rotated with respect to the stationary roll workpiece being held by a crane, is therefore proposed to solve this problem. Measurement uncertainty analysis is carried out to verify the reliability of the new technique for on-machine evaluation of the out-of-roundness component. The out-of-straightness component and the taper angle component of the roll workpiece are then evaluated by using a previously developed method. A simple and effective algorithm for evaluation of the diameter deviation component is also presented.     

Assessing Roundness Errors Using Discrete Voronoi Diagrams

In this paper, we propose anew procedure for measuring roundness errors by computong apaor of concentric corcles with the minimum tsdial separation for assessing the srrors by constructing the discrete farthest and nearest Voronoi diagrams. The peoperties of Voronoi diagrams are exploited to determine a pair of concentric circles with minimum radical separation such that all the data points are bounded by these two circles. A pixel map is used to construct a discrete Voronou diagram. The convergence of the proposed procedure to the true global minimum roundness error is guaranteed. This procedure based on discrete Voronoi diagrams is robust even for a large number of data points. The proposed procedure is simple in implementation and effective in computation.     

A new multiprobe method of roundness measurements

This paper presents a new multiprobe method for roundness measurements called the mixed method. In this method, displacements at two points on a cylindrical workpiece and an angle at one of the two points are simultaneously monitored by two probes. The differential output of the probes cancels the effect of the spindle error, and deconvolving the differential data yields the correct roundness error. The mixed method is compared to the traditional 3-point method with respect to the transfer function and resolution. Unlike the 3-point method, the mixed method can completely separate the roundness error and the spindle error, and can measure high-frequency components regardless of the probe distance. Resolution can also be improved throughout the entire frequency domain by increasing angular separation of the probes. An optical sensor specifically suited to the mixed method is designed and used to make roundness measurements. A fiber coupler and single-mode fibers are used in the sensor to divide a light beam from a laser diode into two beams, resulting in a compact sensor with good thermal drift characteristics. The displacement meter of the sensor is based on the imaging system principle and has a resolution of 0.1 μm. The angle meter is based on the principle of autocollimation and has a resolution of 0.5 in. A measurement system is constructed to realize measurements of roundness by using the optical sensor. Experimental results confirming the effectiveness of the mixed method for roundness measurements are also presented in this paper.