Spindle Error Motion Research Articles

Influence of shape errors and inertia effects on the error motion of the aerostatic spindle

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.

An adaptive digital filter method for measuring the radial error motion of ultra-precision spindle

To ensure the nanometer-level running accuracy of the ultra-precision spindle, the radial error motion should be precisely measured. However, the asynchronous errors induced by the measurement instrument will mix in the measured data and significantly deteriorate the measurement precision. In this study, a novel adaptive digital filter method (ADFM) is proposed to suppress the asynchronous errors and lower requirements for the precision of measurement instruments based on the Donaldson reversal method. Specifically, the Gaussian mixture model is first used to investigate the characteristics of asynchronous errors and obtain appropriate initial filter parameters. The least mean square algorithm is utilized to adaptively optimize the filter parameters. Monte Carlo simulation results indicate that ADFM can suppress asynchronous errors by more than 90% compared with the traditional methods. Experimental results show that the standard deviation uncertainty is reduced by more than 60%. Besides, the measurement results of two sensors with different precision verify the feasibility of ADFM.

Simultaneous measurement of 5DOF spindle error motions in CNC machine tools.

We present a method for the simultaneous measurement of 5 degree-of-freedom (DOF) spindle error motions in computer numerical control (CNC) machine tools and develop a measurement system. The measurement system uses polarization-maintaining fiber coupled with a dual-frequency laser as the light source. The axial error motion of the spindle is measured by heterodyne interferometry, and other 4DOF error motions are obtained by collimation measurement. Based on the measurement system, a comprehensive measurement model of spindle error motion was established. The influence of system error on the measurement accuracy of the spindle tilt error motion and radial error motion was analyzed. According to the measurement model, a corresponding data-processing method is proposed, which removes some systematic errors. A series of experiments was conducted to verify the feasibility and effectiveness of the proposed measurement system and the corresponding measurement model.

Roundness error separation based on singularity detection and exact-stop of spindle in on-machine measurement of spindle rotation error

At present, non-contact sensors are being extensively used for the measurement of spindle rotation errors. However, as the measuring target of the sensors, the roundness error of the artifact inevitably mixes into the error motion of the spindle. Moreover, the challenging nature of manufacturing high-precision artifacts has resulted in the separation of the roundness error of artifacts from the spindle rotation error being considered an urgent research prospect. In this study, a roundness error separation method based on singularity detection and exact-stop of spindle was proposed to perform on-machine measurement of spindle rotation error, wherein compared to the existing methods, the proposed method implements the Donaldson reversal method. It is independent of the index signal of the spindle encoder, which is usually unavailable on actual machine tools. The singularity of measured motion was detected using the wavelet transform modulus maximum (WTMM) ridges and their Lipschitz exponents, to locate the exact-stop and rotating stages of spindle in the measured motion. Consequently, a WTMM ridge extraction method was established to extract the continuous optimal WTMM ridges of the measured motion. Further, the synchronous motion of the measured motion was used for roundness error separation to promote the separation accuracy. Finally, the effectiveness of the proposed method was verified using the measurement results of a coordinate measuring machine (CMM). Thus, the measurement error caused by the roundness error of the artifact can be reduced significantly by applying the proposed method to the on-machine measurement of the spindle error motion.

Effects of wheel spindle error motion on surface generation in grinding

In ultra-precision grinding, error motions of the aerostatic bearing wheel spindle lead to unexpected variations in the actual depth of cut, thereby directly affecting machining accuracy. However, few studies have focused on dynamic characteristics of the wheel spindle error motion and its effects on surface generation. In this paper, the ground surface generation mechanism under the wheel spindle error motion is explored by establishing a grinding kinematics-based surface topography model integrated with a wheel spindle dynamics model considering the wheel topography and the overlapping effect. Grinding experiments are performed to identify the theoretical results through spindle vibration signals and machined surface topographies. The results show that the spindle error motions include the drift of the axis average line which produces the surface form error, and the spindle vibration relative to the axis average line that contributes to the waviness circumferentially. It is also found that the increasing wheel speed leads to the decreasing waviness amplitude at a small radial distance of the workpiece while the trend of the waviness amplitude with the wheel speed is the same as that of the spindle vibration amplitude at a large radial distance. Furthermore, based on the ground surface simulation, the grinding mark pattern formation mechanism is found to include three types, identified by the machined surfaces. A novel and convenient method, where the ratio of the grinding contact width to the cross feed distance per workpiece revolution and the reduced fraction of rotational speed ratio are used, is proposed to predict the grinding mark patterns and spatial frequencies, and validated by the ground surface simulation and experimental results.

An advanced Fourier-based separation method for spindle error motion identification

Classical error separation methods (Donaldson reversal, multistep, and multiprobe methods) were developed by separating both the spindle synchronous radial error motions of precision measurement instruments and the roundness deviations of cylindrical or spherical material standards. The application of such methods improves the measurement uncertainty, even though some methods are time-consuming (multistep) or require a more complex set-up (e.g., the Donaldson reversal and multiprobe methods). In this study, an advanced Fourier-based method for error separation in roundness measurements is developed and implemented. It can be considered a fully stable method, achievable through a reduced number (two or three) of angular shifts, and uses at least one fixed probing system. The proposed Fourier-based method is optimised, tested, and validated on simulated datasets and scenarios combining synchronous and asynchronous spindle error motions and roundness deviations. The given results are accurate at the sub-nanometre level (<0.5 nm) and the effectiveness of the Fourier-based method for ultra-high accuracy applications is proven. Further, Monte Carlo simulations, performed to investigate the influence of artefact indexing errors on measurement uncertainties, confirmed the numerical investigation. Finally, an experiment was conducted on spherical material standards made of ceramic using a high-precision roundness measurement machine. The given results are more accurate (at nanometre level) than those of the ‘classical’ multistep methods.

Moiré fringes-based measurement of radial error motion of high-speed spindle

High-speed spindles (HSS) are widely employed in the fabrication of ultra-precision microstructures. However, the error motion of an HSS could seriously damage the produced microstructure. Therefore, measurement of radial error motion of HSS is vital. However, the measurement of the dynamic error motion of the spindle requires the utilization of a displacement sensor with a high sampling frequency. The sampling frequency of the applied displacement sensors cannot satisfy the requirement of measuring the dynamic error motion of an HSS. To overcome these limitations, the moiré fringes technology (MFT) is proposed to detect the dynamic radial error motion of an HSS. For that reason, an indicator grating and a ruler grating were used to measure the radial error motion of a spindle at speed of 6000 r/min. Our experimental results indicate that the improved MFT can successfully capture the radial error motion. The fluctuation of measurement results is less than 2% when the spindle speed increases from 600 to 6000 r/min, which corroborates the capability of the improved MFT to measure the dynamic radial error motion of the spindle at a higher speed.

Measurement of five-degrees-of-freedom spindle error motion using multi probe error separation

Herein, a precision measurement method is proposed to evaluate the radial, axial, and tilt error motions of rotating devices such as precision spindles. To improve the accuracy of the estimated multiple-degree-of-freedom error motion components, form error signals in the runout signals are separated and precompensated for before the calculation of the five-degrees-of-freedom error motion components. Fourier model-based multi probe error separation (MPES) techniques, which can prevent the occurrence of harmonic distortions, are applied to separate the form error signals. A three-probe method is applied to layers on the side surface of the cylindrical artifact, and a modified two-probe method is developed and applied to the upper surface. The radial, tilt, and axial error motions are calculated using the runout signals that do not contain the separated form error signals. The measurement system uses eight capacitive probes to detect the runout signals of the cylindrical artifact mounted at the center of the rotating device. To compare the proposed method with the five-probe-based conventional measuring method, an evaluation test simulation is conducted repeatedly five times. Results indicate that the proposed MPES method calculated the uncertainty using the deviation between the computed results from the existing and novel methods; additionally, the input signal in terms of the radial, tilt (layers 1 and 2), and axial error motions are [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text], respectively. It is confirmed that the undesirable effects of the form error signals are successfully removed and that the accuracies of the measured spindle error motion components improve.

The effect of spindle dynamics on tool-tip radial throw in micromachining

Abstract In this work, we present a model to predict the speed-dependent radial throw at the tool-tip during micromachining with ultra-high-speed (UHS) spindles. This speed-dependent nature of the radial throw arises from the interaction between the quasi-static radial throw (tool attachment errors, tool geometric errors, and spindle error motions) and the dynamic response of the tool-collet-spindle system. The radial throw causes the cutting edge trajectory to deviate from the ideal trajectory, critically affecting the attainable dimensional accuracy and surface quality, as well as the micromachining forces. Hence, accurate determination of radial throw at the micro-tool tip is important for both practical applications and process-modeling efforts. In the current work, the proposed model describes the radial throw of the tool-axis as a dynamic response to excitation from rotating unbalance of the spindle assembly. The model parameters are first calibrated using experimentally obtained spindle dynamics and the radial throw measurements, both at two speeds. The calibrated model is then used to predict the radial throw for any spindle speed. The presented model is used to predict radial throw at two different axial locations (microtool's shaft at 2 mm and its tool-tip at 15 mm). For both the axial locations, the spindle dynamics measured at 2 mm is used for model calibration. The average error is observed to be less than 2.4% at the tool-tip (15 mm). It is concluded that the speed-dependent spindle dynamics can be used in an analytical formulation to determine tool-tip radial throw at any speed.

Characterization and Evaluation of Rotation Accuracy of Hydrostatic Spindle under the Influence of Unbalance

The paper studies the characterization and evaluation technology of rotation accuracy of hydrostatic spindle under the influence of unbalance. The dynamic model of the motion error of the hydrostatic spindle is established based on the dynamic parameters. The variation law of motion error of spindle rotor is analyzed under the unbalanced mass. The paper finds that with the increase of the spindle speed, the amplitude of the spindle error motion will increase, and the inclination angleθerror is more sensitive to the change in the rotational speed. In the total synthesis accuracy, the proportion of synchronization error decreases with the increase of the rotational speed. Finally, the least squares evaluation algorithm is used to evaluate the rotation error of the hydrostatic spindle, and a method for evaluating the rotation accuracy of the hydrostatic spindle with high calculation accuracy and calculation efficiency is proposed.

Cosine Error Elimination Method for One-Dimensional Convex and Concave Surface Profile Measurements

AbstractThis paper presents a novel method to eliminate cosine error in precision concave and convex surface measurement by integrating a displacement probe in a precision spindle. Cosine error in surface profile measurement comes from an angular misalignment between the measurement axis and the axis of motion and negatively affects the measurement accuracy, especially in optical surface measurements. A corrective multiplier can solve this problem for spherical surface measurement, but cosine error cannot be eliminated in the case of complex optical surface measurement because current tools do not measure such surfaces along the direction normal to the measurement plane. Because the displacement probe is placed on the spindle axis, the spindle error motion will affect the shape precision and surface roughness measurement of optical components such as mirrors and lenses, and the displacement probe will measure a combination of the spindle error motion and the geometry of optical surfaces. Here, the one-dimensional concave, convex, and hollow measurement targets were used, and cosine error was fundamentally eliminated by aligning the probe on the spindle always normal to the measured surface, and compensation was made for the aerostatic bearing spindle rotational error obtained by the reversal method. The results show that this proposed measurement method cannot only eliminate cosine error but also scan the large area quickly and conveniently. In addition, measurement uncertainty and further consideration for future work were discussed.

Optimal measurement angles of the three-probe spindle error motion separation technique

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.

Comparison of Current Five-Point Cylindricity Error Separation Techniques

Cylindricity is a kind of three-dimensional form distortion of a cylinder. An accurate in situ measurement of cylindricity is relatively complex because measuring and reconstructing cylindrical profile and evaluating out-of-cylindricity should be involved. Any method of in situ measuring cylindricity must solve a common issue, i.e., to eliminate spindle error motions and carriage error motions during measurement and reconstruction. Thus, error separation techniques have played an important role in in situ cylindricity measurement through multipoint detections. Although several valuable five-point methods for in situ measurement of cylindrical profile have been proposed up to present, namely the parallel scan, spiral scan, and V-block scan, there are obvious differences in many aspects, such as the arrangement of probes, error separation model, reconstruction method, adaptability to service environment, accuracy and reliability in practical application, etc. This paper presents the evaluation of their advantages and disadvantages in theory and the actual measurement based on the standard ISO 12180. Suggestions for best meeting the requirements of modern manufacturing and the most prospective one for industrial applications are also given.

Study on the influence of machine tool spindle radial error motion resulted from bearing outer ring tilting assembly

To reveal the spindle radial error motion characteristics in condition of bearing outer ring tilting assembly, mathematical method on spindle radial error motion were analyzed. Then, in real operation condition the natural frequency of the test rig was investigated. Experimental system and methods were designed to test axial thermal displacement, radial error motion and modal characteristic of spindle in condition of bearing outer ring tilting assembly. Results show that axial thermal extension and radial vertical rising of spindle front-end occurs during thermal displacement test. With the same outer spacer nonparallelism, the synchronous error motion and total error motion generally increase with spindle rotation speed, and reach a peak at certain rotation speed.

A theoretical and experimental study of spindle imbalance induced forced vibration and its effect on surface generation in diamond turning

Abstract In diamond turning, spindle error motions exhibit dynamic characteristics under spindle imbalance, which significantly deteriorate form accuracy and surface roughness. However, in the spindle error motion community dynamic characteristics of spindle imbalance induced forced vibration have not attracted much attention, involving unbalanced magnetic forces induced forced spindle vibration, unbalanced air bearing pressure induced forced spindle vibration, and spindle whirling and spindle tilting under bearing hydrodynamics along with inertial effect. In this study, a theoretical and experimental investigation has been conducted to discuss dynamic characteristics of spindle imbalance induced forced vibration and its effect on surface generation in diamond turning. Firstly, an analysis model of magnetic forces for a spindle motor was built up to study dynamic characteristics of unbalanced magnetic forces induced forced spindle vibration. Secondly, a mathematical model of Reynolds equations for air bearings was established with a five-DOF dynamic model for spindle motion to discuss dynamic characteristics of unbalanced air bearing pressure induced forced spindle vibration, spindle whirling and spindle tilting. Finally, it has been verified by two kinds of designed face-turning tests for comparison carried out in diamond turning. The theoretical and experimental results revealed that bearing hydrodynamics coupled with inertial effect would jointly induce synchronous quasi-elliptical spindle whirling/tilting, accordingly producing a quasi-elliptical concave/convex surface in diamond turning, whose cyclic frequency is 2 rev.−1. Under spindle imbalance, unbalanced magnetic forces induced forced spindle vibration with the fundamental cyclic frequency of 2p rev.−1 (the spindle motor pole number) would occur as well as unbalanced air bearing pressure induced forced spindle vibration with the cyclic frequency of α rev.−1 (the air bearing orifice number) further to produce the corresponding straight radial patterns at the machined surface. This research work gives a deep insight into dynamic characteristics of spindle imbalance induced forced vibration and its effect on surface generation in diamond turning.

Four-point error separation technique for cylindricity

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.

Error Separation Method for Precision Measurement of the Run-Out of a Microdrill Bit by Using a Laser Scan Micrometer Measurement System

This paper describes an error separation method for a precision measurement of the run-out of a microdrill bit by using a measurement system consisting of a concentricity gauge and a laser scan micrometer (LSM). The proposed error separation method can achieve a sub-micrometric measurement accuracy of the run-out of the microdrill bit without the requirement of ultra-precision rotary drive devices. In the measurement, the spindle error motion of the concentricity gauge is firstly measured by using the LSM and a small-diameter artifact, instead of the conventionally used displacement probes and large-diameter artifact, in order to determine the fine position of the concentricity gauge when the spindle error motion is at its minimum. The microdrill bit is rotated at the fine position for the measurement of the run-out, so that the influence of the spindle error motion can thus be reduced, which could not be previously realized by commercial measurement systems. Experiments were carried out to verify the feasibility of the proposed error separation method for the measurement of the run-out of the microdrill bit. The measurement results and the measurement uncertainty confirmed that the proposed method is reliable for the run-out measurement with sub-micrometric accuracy.

Investigation on the position drift of the axis average line of the aerostatic bearing spindle in ultra-precision diamond turning

To evaluate the performance of the spindle, many techniques have been proposed to measure the spindle error motion. However, few studies have focused on the investigation of the position drift of the axis average line (AAL). In the present study, the AAL of the aerostatic bearing spindle is investigated both theoretically and experimentally. An error model is developed to analyze the errors which contribute to the error of depth of cut in slow tool servo assisted turning. Moreover, an experiment of microstructure fabrication is conducted to investigate the amplitude error of microstructures along both axial and radial direction of the cylindrical workpiece. The effects of spindle error motion, spindle unbalance induced eccentricity, thermal error and position drift of AAL are analyzed. The results indicates that the position drift of AAL varies significantly in terms of the variation of the spindle speed due to the hydrodynamic effect, and the relation between the drift and the spindle speed is nonlinear.

A multi-orientation error separation technique for spindle metrology of miniature ultra-high-speed spindles

This paper presents a multi-orientation error separation technique to remove the artifact form error from the radial measurements to obtain the radial spindle error motions of miniature ultra-high-speed (UHS) spindles. Unlike the existing approaches, the present technique neither relies on high-accuracy fixtures, nor necessitates measurements from specific orientations of the artifact. Rather, the spindle error motions are measured from a set of arbitrary artifact orientations using laser Doppler vibrometry (LDV). The angle of each artifact-setup orientation with respect to the spindle is determined with high precision through reflectivity measurement of the marks made on both the artifact and the spindle using another LDV. Although the presented approach can be applied by using different sensors (e.g., capacitance probes), we demonstrate the approach using LDVs. With the displacement measurement direction fixed, measurements are conducted from both LDVs for multiple orientations of the artifact. Using the unique implementation scheme developed in this paper, data from these orientations are post-processed to compute the artifact form error and further remove it from the radial motion measurements to obtain the synchronous radial spindle error motions. A thorough experimental evaluation is presented to quantify both the repeatability of the measured artifact form errors as well as the bandwidth of error separation for various number of artifact orientations. The spindle error motions measured from both the sphere and stem portions of a custom fabricated sphere-on-stem artifact mounted on a typical miniature UHS spindle, are seen to be similar in shape and within 5nm in magnitude across the revolution, thus demonstrating the effectiveness of the technique. Using this technique, spindle error motions at ultra-high speeds up to 150krpm were successfully quantified. Although the implementation scheme is demonstrated for miniature UHS spindles, it is readily applicable for error separation on macro-scale spindles without the need for any high-precision fixtures and precise setting of angles.

Investigation of the effects of spindle unbalance induced error motion on machining accuracy in ultra-precision diamond turning

In ultra-precision machining, error motions of the aerostatic bearing spindle (ABS) have significant effects on the machining accuracy. Spindle unbalance is a critical factor attributing to error motions of the ABS. Much work currently has been focused on the measurement of error motions and spindle balancing. However, the unbalance induced spindle error motion (UISEM) and the corresponding effects on machining accuracy are not well understood. In this paper, a dynamics model of the ABS was established to characterize the UISEM and its dynamic behavior with consideration of the unbalance effects. A series of groove turning experiments were especially designed to investigate the UISEM. Good agreement between theoretical and experimental results was achieved, demonstrating the low frequency enveloping phenomenon of the error motions of the ABS, identified as the unique superposition effects of two motion components at high frequency in the spindle vibration. In addition, the experimental result reveals that the relative distance between the rotational axis of the ABS and the tool tip varies with respect to the different spindle speeds, significantly degrading the machining accuracy.