Uncertainty evaluation and reduction in three-probe roundness profile measurement based on the system transfer function
Introduction
Rotary components are applied in every possible engine. The surface topography, especially the out-of-roundness, is always an essential performance factor. Previous studies show that the manufacturing error of the surface of aerostatic/hydrostatic bearing affects not only the static stiffness, the load-carrying capacity, the service life, but also the running accuracy [1]. Especially, with the decrease of roundness error in a journal bearing, the rotating accuracy of the spindle improves to the nanometer level whilst achieving nearly-frictionless property [2]. In a paper machine, the roundness profile of its roller decides the even thickness and the surface quality of the produced paper [[3], [4], [5]]. In an axial piston pump, improving the roundness profiles of the piston and the cylinder bore is beneficial to enhance the outlet pressure limit [6].
To be able to manufacture the component with a tight roundness tolerance, next to ultra-precision machining, ultra-precision roundness metrology is of vital importance. Up to now, several ISO standards have been proposed regarding the topic of roundness measurement [[7], [8], [9]]. To calibrate the roundness standards at the highest level of accuracy, at least three international comparisons of roundness measurements have been carried out between the national laboratories in Europe, respectively, in 1990, 1993, and 2019 [[10], [11], [12], [13]]. These calibration results showed that the measurement differences were mainly due to the rotation error of the spindle on the used roundness testers [10].
In this context, on one hand, ultra-precision spindles, whose error motion is at the level of 10 nm, have been developed and applied in the most precise roundness measuring instruments [14]. On the other hand, various error separation (also termed self-calibrated) techniques, including the reversal method [[15], [16], [17]], the multi-step method [17,18], the three-probe method (3-PM) [[15], [16], [17]], the two-step method [15,19], and their varieties have been proposed, with the aim of removing the adverse effects of the spindle error motion.
Among these error separation techniques, the 3-PM has been studied intensively, as it can be applied for on-machine and in-process roundness inspection. Still, it has long been recognized that the uncertainty in the three-probe measurement is limited due to the harmonic suppression problem (HSP). This refers to the phenomenon that when the determinant of the transfer matrix drops to zero, the harmonics are suppressed and cannot be estimated; when is around zero, the harmonics are extremely susceptible to the error sources [20]. In other papers, different but similar parameters have been derived: in Ref. [21] where the angle-dependent signals are expanded into a complex Fourier series, rather than the trigonometric series, is replaced by the transfer function ; in Refs. [22,23], by calculating the partial derivative of the expression of roundness Fourier coefficient, Zhang derived the indicator as a measure of the deviation in the harmonic estimates. Based on these criteria, Zhang introduced the four/multi-probe method [22,23] to solve the HSP; Shi put forward the hybrid 3-PM in which the optimal harmonic estimates are selected out according to the value of [20]; Cappa optimized measurement angles through maximizing the minimal value of [24].
However, in experiments applying the 3-PM method, Shi found that the sensitivities in harmonic estimates are rigorously correlated with neither [20] nor [20,25]. As a result of this, the best harmonic estimates actually cannot be correctly picked out using these criteria. In Ref. [20], Shi considered that the best harmonic estimates refer to these with minimal harmonic uncertainty/maximal precision. Therefore, they can be reliably recognized and selected once the harmonic uncertainty is quantified. Additionally, the optimal angle arrangement can be determined by minimizing the total roundness uncertainty rather than by maximizing the minimal value of [24].
Hence, the basis of error separation using the 3-PM method, meaning that the HSP problem must be overcome, comes down to evaluating the uncertainty in the 3-PM, which itself is indispensable for the measurement. Fortunately, in recent years, major advances have been made regarding this issue. From 2017, Widmaier et al. evaluated the uncertainty in the four-probe roundness measurements through Monte Carlo simulation (MCS) [[3], [4], [5]]. However, performing the MCS generally takes a long time and it cannot give a clear view of the hidden uncertainty propagation rules. From 2018, by utilizing the system theory, Shi analytically derived the laws of propagation of uncertainties in the two-step roundness measurements [[26], [27], [28]]. These derived laws bring to light the propagation mechanisms of the input uncertainties, especially in the harmonic domain. Also, the nature of HSP could be well explained.
Based on this previous work, this paper will consider the uncertainty in the three-probe roundness measurements. The paper is structured as follows. In Section 2, the algorithm for solving the error separation of the 3-PM will be derived by using the Laplace transform. In Section 3, the system model of the measurement will be established and expressed by the system function (i.e. the S-function); propagation laws of individual input uncertainties will be derived separately; based on the uncertainty propagation laws, the measurement uncertainty will be evaluated. In Section 4, several approaches to reduce the uncertainty will be explained and their result is shown. In Section 5, practical measurements will be performed. Conclusions will be summarized in Section 6.
Section snippets
Algorithm of the 3-PM
To implement the three-probe measurements to estimate the roundness profile of a specimen, three displacement sensors are required. The first sensor is mounted in line with the X-axis. The other two sensors are, respectively, spaced and apart from the first sensor, as depicted in Fig. 1.
When the specimen rotates together with the spindle anticlockwise, the outputs , , and of the three sensors can, respectively, be expressed as
Law of propagation of measurement uncertainty
In Section 2.2, it has been proven that by properly choosing the angles, in theory, all harmonics can be recovered suffering no suppressed harmonics. Thus, the roundness profile can be perfectly estimated suffering no systematic deviation caused by the spindle error. Still, the three-probe measurements inevitably suffer from random uncertainty, due to various error sources. In this section, the following three steps will be taken for quantification of this uncertainty.
- (1)
The system function for
Quantitative enhancement of measurement precision
In Section 3, the uncertainty propagation laws in the 3-PM were deduced and validated. Based on this, the roundness uncertainty, including the harmonic uncertainty and the total uncertainty, can be conveniently predicted. Fig. 8 display the predicted combined harmonic uncertainties of roundness profile in two cases where the spindle error, the roundness, and the input uncertainties are as supposed in Table 1, Table 2 while the measurement angles are different: these are and
Test setup
To verify the theory proposed above, a master ball is produced and its roundness is calibrated on a Haozhi motorized spindle (Model DGZ-06260), as shown in Fig. 15. Three capacitive sensors, together with the NI 9125 digitizer, are adopted to collect the runout signals. The runout signals are firstly sampled at evenly spaced time increments, and then, resampled at evenly spaced angular intervals. During resampling, the rotation angle of the spindle is determined from the fundamental frequency
Conclusions
Evaluation of the measurement uncertainty is an integral part of a complete measurement, and also essential for a reliable enhancement of measurement precision. To achieve the highest precision of the 3-PM, this paper presents a methodology to quantify, as well as to reduce, its uncertainty. First, by utilizing the Laplace transform, the system model of the 3-PM is deduced, expressed by a Laplace equation, and also depicted by the system block diagram. Based on this, the input uncertainties
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This research is supported by the National Key Research and Development Program of China (Grant no. 2018YFB1701202), the National Natural Science Foundation of China (Grant no. 51805174), and the Fundamental Research Funds for the Central Universities (Grant no. 2019MS059), which are highly appreciated by the authors. The authors wish to acknowledge Professor Richard Leach at University of Nottingham and Professor Han Haitjema at KU Leuven for their help during the germination of the idea.
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