Four-Point Method of Roundness and Spindle Error Measurements

A new three-point method for spindle roundness error measuring was introduced, three capacitive sensors were installed on the measured shaft circularly, and with a certain angle each other, while the spindle was rotating with a certain speed, the capacitive sensors had measurement on the circular profile simultaneously, then through linear combination for three-sensor output signals, the roundness error was effectively separated from the spindle error. There were no strict requirement to initial installation position by using capacitive sensors, as long as ensuring the change of measured rotor error within the linear range of sensors, we could realize effective measurement. The experiment results have verified the effectiveness of roundness error measuring system based on three capacitive sensors.

A four-point error separation technique integrating three-point roundness and two-point straightness error separations is proposed to measure and reconstruct a cylindrical profile for the accurate evaluation of cylindricity. Four probes mounted onto two sections in the probe carriage target the surface of the cylinder to detect the differential vector of the least-squares centers in two adjacent cross-sectional profiles of the cylinder. The vector of the least-squares center in each cross-sectional profile is extracted by accumulation. The spatial curved median line of the cylinder is determined by fitting the extracted least-squares centers. Theoretical analysis and numerical validation prove that radial error motions of the spindle and straightness error motions of the slider are eliminated during reconstruction of the cylindrical profile. The influence of tilt error motions of the spindle on the accuracy of error separation can be weakened by reducing the distance between the two sections in which the four probes are located. The spatial median line profile of the cylinder is accurately determined even if the error motions of the spindle are not repeatable in each revolution. An experimental system was constructed to perform comparison experiments. The results validate the fact that the technique has higher precision in reconstructing cylindrical profiles owing to its robust anti-interference ability. The technique is suitable for in situ measurement of cylindricity under unrepeatable radial error motions of the spindle and indeterminate straightness error motions of the slider.

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.

Modeling and error analysis for assessing spindle radial error motions

An LDV-based methodology for measuring axial and radial error motions when using miniature ultra-high-speed (UHS) micromachining spindles

The rotating accuracy of a machine tool spindle directly affects the roundness of machined parts. Commonly, a precision arbor and one or two probes are used to inspect the spindle axis error motion. When a reference precision arbor is not available or when the spindle error motion is in the same order of magnitude as the accuracy of the reference arbor, it is imperative to separate the roundness error of the reference arbor from the spindle error. One of the most widely used method is the three-probe method. This paper presents an exact model and in-depth error analysis for the conventional three-probe method. The results show that contrary to conventional wisdom that a precise arbor is not needed with the three-probe method, the arbor accuracy should be at least ten times better than that of the axis motion. To reduce inaccuracy in the converted axis motion and arbor contour, it is also suggested that the mounting error of the probes should be less than one-fiftieth of the size of the axis motion and the arbor size. The exact geometric model developed in this paper can also be extended to analyze the precision aspect of other methods.

In this study, an extended Reynolds equation incorporating the shape errors and the inertia effects of compressible fluid was formulated. The pressure-solving equation for the gas film was constructed using a second-order finite difference method. By combining the measurement data from the cylindricity of multiple shafts obtained from a cylindricity measuring instrument, the variation of the aerostatic bearing load capacity with the rotation of the shaft was calculated, both with and without considering the inertia effect. An experimental rig was also constructed to measure the error motion data of the corresponding shaft. The results of numerical analysis and experimental validation demonstrated that shafts with larger cylindricity error values exhibit greater deviations in the calculated bearing load capacity, which correspondingly leads to larger radial error motion values measured in the experiment. When the roundness error value of the shaft is small or the shaft speed is low, the influence of inertial effects on the calculation results of the load capacity deviation is minimal and can generally be disregarded.

Interference-enhanced micro-vision-based single-shot imaging of five degrees-of-freedom error motions for ultra-precision rotary axes

The modern industry has a constantly increasing demand toward vibration-free rotating machines. Large rotor systems are commonly operated in the subcritical speed range. In the subcritical speed range, the bearings can be a significant source of vibration excitation. The bearing inner ring excites the rotor system at a frequency, which is the rotating frequency multiplied by the waviness component number in the roundness profile of the bearing inner ring. Consequently, subcritical resonance peaks can be observed when the bearing excitation frequency coincides with the rotor natural frequency.The present study utilized a novel compensative 3D grinding to manufacture the bearing installation shaft into a geometry, which minimized the roundness error of the installed bearing inner ring. The decreased roundness error reduced the bearing based excitations to the rotor system. The successful grinding operation was confirmed with roundness measurements. The relevance of the method and the study was proven with rotor dynamic measurements.The results clearly suggest that the compensative 3D grinding reduced the roundness error in both bearing inner rings of the rotor system. The decreased roundness error led to a significantly improved rotor dynamic response, which was observed as reduced amplitudes of the subcritical resonance peaks in the typical operating rotating frequency range.The present study includes also a comparison between the proposed method and a previous steel strip method. The comparison shows that the proposed compensative 3D grinding method produced increasingly better roundness profiles and thus also better rotor dynamic responses.

This paper presents an on-line measurement method for the diameter and roundness error of balls. An easy-installation rotary scanning system, which integrates the principles of the diameter and the roundness measurements, is constructed. The rotary scanning system consists of a rotary stage, a linear stage, and two sensors with a flat probe. Two sensors are initially installed on the linear stage and contact each other. The outputs of two sensors are reset to zero at first. The ball is then mounted on the rotary stage and positioned between two flat probes. The variations of the diameter and the roundness error of the ball at each angular position can be directly recorded by two sensors when the ball is rotated by the rotary stage. Substituting the outputs of two sensors into the proposed mathematical models, the diameter and roundness error can be evaluated. The effects of the alignment error induced by the spindle error of the rotary stage and the titling error and the eccentric error of the ball on the measurement accuracy can be self-separated in the proposed on-line measurement method. A series of experiments are carried out to verify the effectiveness and the capability of the proposed on-line measurement method and the designed rotary scanning system. The designed system is easy to construct both in the laboratory environment and the factory field.

A novel method is proposed for measuring the six degrees-of-freedom (DOF) geometric motion errors of a rotary axis based on a polyhedral prism. An error-sensitive unit which consists of a polyhedral prism and a planar reflector, is designed to carry out measurement of all six DOF errors, including the angular positioning error, the tilt motion error around the Y axis, the tilt motion error around the X axis, the radial motion error along the X and Y axes, and the axial motion error along the Z axis. The mathematical error model, including the six DOF geometric motion errors of the rotary axis, the installation errors between the polyhedral prism and the rotary axis, the manufacturing errors of the polyhedral prism, and the position errors of the sensors, are established. The effectiveness of the proposed method and the compensation model was simulated and experimentally verified.

In this paper, an on-machine Fourier five-sensor (F5S) measurement method is developed using Fourier series and sensor integration techniques to determine the straightness and yawing motion errors of a linear slide. The profile of the slide is also determined in this error separation technique. The method is an extension of the previous Fourier three-sensor (F3S) method (Fung, E. H. K., and Yang, S. M., 2000, “An Error Separation Technique for Measuring Straightness Motion Error of a Linear Slide,” Meas. Sci. Technol., 11, pp. 1515–1521; Yang, S. M., Fung, E. H. K., and Chiu, W. M., 2002, “Uncertainty Analysis of On-Machine Motion and Profile Measurement With Sensor Reading Errors,” Meas. Sci. Technol., 13, pp. 1937–1945) by including the effects of yawing error in the straightness motion error and profile measurements. The principles and operation of the F5S measurement method are described. The uncertainty analysis of the method in the presence of a sensor reading error is studied both in the frequency domain and the spatial domain. The spatial domain parameter is first optimized to yield the 12 possible sensor configurations and the final configuration is chosen based on the frequency domain parameter values. The method is evaluated by computer simulation where the simulated sensor outputs are derived from the predefined profile, straightness, and yawing motion errors. By comparing the calculated results with the input data, the F5S method is found to be superior to the F3S method as far as accuracy is concerned. The results reported in this simulation study not only confirm the feasibility of the F5S method but also encourage the author to perform an experimental study in the near future.

The three-probe error separation technique uses three displacement sensors to separate the spindle radial error motion from the artefact roundness error. The choice of appropriate sensor angles is crucial for an accurate and complete separation. This work discusses the optimal arrangement of the sensors’ angular positions in detail, including the influence of angular position errors of the sensors on the separation results. The optimization goal is to find a suitable sensor arrangement to simultaneously maximize the minimum value and the average value of the transfer function, in order to improve the anti-interference ability and ensure measurement precision of the three-probe method without harmonic suppression. The result shows that there is a global optimum for the sensor angles for a specific range of harmonics. The obtained optimal measurement angles under different harmonic ranges should be useful for applications of the three-probe method.

The contouring accuracy of a multi-axis machine has a critical effect on the quality of many advanced technology products. One of the best approaches to assessing the contouring performance of machine tools is through double ball bar measurement. During circular interpolation motion, the machine traverses, with two axes at a time, a circular trajectory, with each axis subject to sinusoidal changes in acceleration, velocity, and position. The motion error is measured by detecting the relative distance between a point on the spindle nose and another point on the machine table and plotting this distance in polar coordinates. The present paper derives mathematical models and diagnosis procedures for first and second-order motion error resulting from the active degrees of freedom of a multi-axis machine. The theoretical results are verified by both computer simulation and double ball bar testing experiments.

Four-point method of roundness and spindle error measurements: Zhang, G. X.; Wang, R. K. CIRP Annals. 1993; 42(1): 593–596