Roundness error measurement method based on equivalence of moving average filtering and Gaussian filtering

Abstract

The invention discloses a roundness error measurement method based on moving average filtering and Gaussian filtering equivalence, which belongs to the technical field of precision measurement, and is characterized in that in roundness error measurement of a rotary part, two filtering methods are respectively adopted to convert the roundness error into a frequency domain form through Fourier transform, and based on the amplitude transmission characteristic of a signal, the roundness error of the rotary part is measured. An approximate equivalent relation between the moving average filtering window width and the Gaussian filtering cut-off wave number is provided, a least square method is performed on output sequences of sampled original measurement signals after two kinds of filtering to fit a circle, the radial distance from each sampling point to the center of the fitted circle is calculated, and the maximum radius difference is taken as the roundness error of the rotary part. The method solves the problem of roundness error measurement deviation caused by difference of filtering characteristics of different instruments, remarkably improves data consistency and detection efficiency, is suitable for roundness error detection of high-precision parts such as crankshafts and bearings, is compatible with existing equipment and does not need hardware transformation.

Description

Technical Field

[0001] The present invention relates to a precise measurement method, and specifically discloses a roundness error measurement method based on the equivalence of moving average filtering and Gaussian filtering, which is applicable to the high-precision detection of roundness errors of rotary parts (such as crankshafts, bearings, etc.) and belongs to the technical field of precise measurement. Background Art

[0002] Roundness error is a basic form error and an important indicator reflecting the precision of rotary parts. The magnitude of roundness error directly affects the assembly precision, rotary precision, service life, etc. of parts. The detection process of roundness error includes five basic operations: separation, extraction, filtering, fitting, and evaluation. The original cross-sectional profile obtained from a roundness measuring instrument is a mixed signal, which contains not only the low-frequency roundness signal but also a part of medium-frequency waviness, high-frequency roughness, electrical noise, etc. Therefore, it is necessary to select an appropriate low-pass filtering method to separate the roundness error. In the early days, RC or 2RC analog filtering methods were mostly used for roundness measurement. Due to its non-linear phase distortion, the filtered cross-sectional profile is severely distorted. With the development of computer technology, existing roundness measuring instruments (such as roundness meters, coordinate measuring machines) mostly use digital filtering technology to suppress measurement signal noise, and moving average filtering and Gaussian filtering are two typical methods. The moving average filtering method achieves the effect of low-pass filtering by smoothing the signal, and has the characteristics of simple process, practicality, strong controllability, etc., especially suitable for occasions with high real-time requirements such as online data processing. The moving average filtering method was used on many early roundness measuring instruments. The Gaussian filtering method has the characteristic of zero phase shift, and it is also the filtering method with the smallest time-frequency product. By using the fast Fourier transform (FFT) to improve the calculation efficiency, it is directly applied to the filtering process of the sampling signal of the closed circular contour, and it is the mainstream filtering method adopted in current roundness measuring instruments. However, the filtering methods configured by different instruments are different, and there is no unified standard for filtering parameters (such as window width, cut-off wavelength), resulting in differences in measurement results and affecting data comparability. In addition, traditional methods have not fully studied the mathematical correlation between the two types of filtering methods, resulting in inefficient repeated modeling and parameter optimization. Therefore, there is an urgent need for a technical solution that can unify the characteristics of the two filtering methods and improve measurement consistency. Summary of the Invention

[0003] The purpose of the present invention is to solve the problems existing in the prior art, and propose a roundness error measurement method based on the equivalence of moving average filtering and Gaussian filtering. Through the establishment of a mathematical model, parameter mapping, and equivalence verification, the unified application of the two filtering methods in roundness measurement is realized, and the deviation problem caused by differences in filtering methods in measurement results is solved.

[0004] The present invention is implemented as follows: A roundness error measurement method based on the equivalence of moving average filtering and Gaussian filtering, comprising the following steps: S1. Rotate a roundness measuring instrument one week along the outer surface of the workpiece to be measured, and obtain cross-sectional profile data composed of M equally angularly spaced sampling points, denoted as the original measurement signal x[m], where 1 ≤ m ≤ M, and the standard number of sampling points M is taken as 1800 or 3600 points / week.

[0005] S2. Perform a filtering operation on the original measurement signal x[m] based on the equivalence of the moving average filtering method and the Gaussian filtering method: Taking x[m] obtained in step S1 as the input sequence, according to the cut-off wave number selected by the Gaussian filtering method By performing a convolution operation on the Gaussian filtering weight function and the original measurement signal, the output sequence w[m] after Gaussian filtering is obtained. Taking x[m] obtained in step S1 as the input sequence, determine the moving average filtering window width N corresponding to the cut-off wave number selected by the Gaussian filtering method according to the equivalence relationship between the moving average filtering method and the Gaussian filtering method. After moving average filtering, the output sequence is y[m], and each value in the output sequence is the arithmetic mean of N points (from the current index m back N - 1 points) in the input sequence. S3. Calculate the roundness error of the filtered workpiece: Perform the least squares fitting circle on the output sequences w[m] and y[m] filtered in step S2 respectively, calculate the radial distance from each sampling point to the center of the fitted circle, and take the maximum radius difference as the roundness error of the workpiece to be measured.

[0006] The determination process of the equivalence between the moving average filtering method and the Gaussian filtering method in step S2 is as follows: 1. Establish the equivalence relationship between the moving average filtering method and the Gaussian filtering method: 1.1 Perform a discrete-time Fourier transform on the impulse response of the moving average filtering method to obtain the relationship between the amplitude transfer characteristic and the filtering window width N, the number of waves per week and the number of sampling points M. .

[0007] 1.2 Perform a continuous Fourier transform on the weight function of the Gaussian filtering to obtain the relationship between the amplitude transfer characteristic and the cut-off wave number , the number of waves per week between, α is a constant. .

[0008] 1.3 In the processing of the roundness error measurement signal, make the two filtering methods have equivalent filtering characteristics, and at the cut-off wave number At each point, the original measurement signal is attenuated by 50% to establish the equivalent relationship between the two filtering methods. 。

[0009] 2. Determine the selection of filtering parameters for the moving average filtering method and the Gaussian filtering method: When using the Gaussian filtering method, select the cut-off wave number according to the measurement standard requirements ; when using the moving average filtering, calculate and obtain the filtering window width N (N is an integer) according to the equivalent relationship between the moving average filtering method and the Gaussian filtering method.

[0010] The beneficial effects of the present invention are: It is compatible with the filtering method configuration of existing equipment, without hardware modification, saving costs. By analyzing the amplitude transmission characteristics of the signal, the equivalent relationship between the moving average filtering and Gaussian filtering parameters is established, enabling roundness measuring instruments equipped with different filtering methods to output consistent measurement results, solving the problem of incomparability of data caused by differences in filtering methods. The present invention provides an efficient, economical and standardized solution for the roundness error measurement of rotating parts. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] Figure 1 The working step flow chart of the roundness error measurement method described in the present invention.

[0012] Figure 2 The waveform diagram of the original measurement signal in the example of verifying the equivalence between the moving average filtering and the Gaussian filtering.

[0013] Figure 3 The waveform diagram of the signal after Gaussian filtering in the example of verifying the equivalence between the moving average filtering and the Gaussian filtering.

[0014] Figure 4 The waveform diagram of the signal after moving average filtering in the example of verifying the equivalence between the moving average filtering and the Gaussian filtering. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0015] The present invention will be further described below in conjunction with the drawings and specific embodiments. This embodiment is based on the technical solution of the present invention and gives the detailed implementation manner and specific operation process, but the protection scope of the present invention is not limited to the following specific embodiments.

[0016] According to the attached Figure 1 , the present invention provides a roundness error measurement method based on the equivalence between the moving average filtering and the Gaussian filtering. The specific working steps are as follows: S1. Rotate a roundness measuring instrument once along the outer surface of the workpiece to be measured to obtain cross-sectional profile data consisting of M equally angularly spaced sampling points, denoted as the original measurement signal x[m] (1 ≤ m ≤ M), where the standard number of sampling points M is taken as 1800 or 3600 points / week.

[0017] S2. Perform a filtering operation on the original measurement signal x[m] based on the equivalence between the moving average filtering method and the Gaussian filtering method: Taking x[m] obtained in step S1 as the input sequence, according to the cut-off wave number selected by the Gaussian filtering method By performing a convolution operation on the Gaussian filtering weight function and the original measurement signal, the output sequence w[m] after Gaussian filtering is obtained. Taking x[m] obtained in step S1 as the input sequence, determine the moving average filtering window width N corresponding to the cut-off wave number selected by the Gaussian filtering method according to the equivalence relationship between the moving average filtering method and the Gaussian filtering method. After moving average filtering, the output sequence is y[m], and each value in the output sequence is the arithmetic mean of N points starting from the current index m and going back N - 1 points in the input sequence.

[0018] S3. Calculate the roundness error of the filtered workpiece: Perform a least squares fitting circle on the output sequences w[m] and y[m] filtered in step S2 respectively, calculate the radial distance of each sampling point to the center of the fitting circle, and take the maximum radius difference as the roundness error of the workpiece to be measured.

[0019] The methods of least squares fitting circle and calculating the radial distance of each sampling point to the center of the fitting circle are both common methods in existing technology measurements and calculations, and will not be elaborated in this application.

[0020] The specific process for determining the equivalence between the moving average filtering method and the Gaussian filtering method in step S2 is as follows: 1. Establish the equivalence relationship between the moving average filtering method and the Gaussian filtering method: 1.1 Perform a discrete-time Fourier transform on the impulse response of the moving average filtering method to obtain the relationship between the magnitude transfer characteristic and the filtering window width N, the number of waves per week and the number of sampling points M. Specifically: Perform a discrete-time Fourier transform on the moving average filtering: The moving average filtering is a finite impulse response (FIR) type of filtering method, and the impulse response h[i] is .

[0021] Perform a discrete-time Fourier transform on the impulse response of the moving average filtering to obtain the frequency response characteristic : , where \(i\) is the discrete-time index, \(N\) is the filter window width, \(j\) is the imaginary unit, is the normalized angular frequency, represents the phase delay of the moving average filter, which determines the filtering ability of the filter for the spectrum.

[0022] Therefore, the magnitude transfer characteristic of the moving average filter and the relationship with the filter window width \(N\) is .

[0023] Convert the frequency-domain model of the moving average filter into a function of the number of waves per cycle: When measuring roundness error, the number of waves per cycle is often used to represent the frequency characteristic of the filtering method, and its unit is waves per cycle (UPR). Convert the magnitude transfer characteristic of the moving average filtering method into a function of the number of waves per cycle . According to the normalized angular frequency , the number of waves per cycle , the number of sampling points \(M\) and the sampling frequency \(f\) s The relationship between them is

[0024] where \(f\) is the frequencies of various signals included in the original measurement signal, and \(D\) is the diameter of the cross-sectional profile. It can be obtained that , Convert the relationship between the magnitude transfer characteristic of the moving average filter and the filter window width \(N\), denoted as , to obtain the relationship between the magnitude transfer characteristic and the filter window width \(N\), the number of waves per cycle and the number of sampling points \(M\), .

[0025] 1.2 Perform a continuous Fourier transform on the weight function of the Gaussian filter to obtain the relationship between the magnitude transfer characteristic and the cut-off wave number , the number of waves per cycle The relationship is Specifically: Perform a continuous Fourier transform on the weight function of the Gaussian filter to obtain the magnitude transfer characteristic of the Gaussian filter method and the relationship with the cut-off wavelength ,

[0026] where \(S(x)\) is the weight function of Gaussian filtering, \(x\) is the distance from the center of the weight function, \(j\) is the imaginary unit, and \(\alpha\) is a constant.

[0027] Convert the frequency-domain model of Gaussian filtering into a function of wavenumber: Cutoff wavelength Corresponding cutoff wavenumber Denoted as , convert the amplitude transmission characteristic of the Gaussian filtering method into being represented by the cutoff wavenumber , denoted as ,

[0028] where \(\alpha\) is a constant.

[0029] 1.3 Establish the equivalent relationship between the two filtering methods Based on the characteristic that the amplitude transmission characteristic attenuates by 50% at the cutoff wavenumber , in the roundness error measurement signal processing, to make the two filtering methods have equivalent filtering characteristics, thus at the cutoff wavenumber , both make the original measurement signal attenuate by 50%, and obtain the equivalent relationship formula between the moving average filtering window width \(N\) and the Gaussian filtering cutoff wavenumber : .

[0030] The present invention converts the filtering processes of moving average filtering and Gaussian filtering into the form in the frequency domain through Fourier transform and the sampling characteristics of roundness signals, obtains the relationship between the amplitude transmission characteristic, the key filtering parameters, and the signal sampling wavenumber, thereby realizing high-precision detection of the roundness error of rotary parts by using the equivalence of the effects of the two filtering methods.

[0031] 2. Determine the selection of filtering parameters for the moving average filtering method and the Gaussian filtering method: When using the Gaussian filtering method, select the cutoff wavenumber according to the measurement standard requirements; when using moving average filtering, calculate the filtering window width \(N\) ( \(N\) is taken as an integer) according to the equivalent relationship between the moving average filtering method and the Gaussian filtering method.

[0032] Roundness error is a low-frequency shape error, and the original measurement signal obtained during measurement is a mixed signal. In addition to the low-frequency roundness signal, it also contains a part of medium-frequency waviness, high-frequency roughness, electrical noise, etc. Therefore, it is necessary to select an appropriate low-pass filtering method to separate the roundness error.

[0033] The Gaussian filtering method is a low-pass filtering method with a signal smoothing function. When the frequency of the input signal reaches the cut-off wave number, after being processed by the Gaussian filtering method, the signal amplitude decays to 50% of the original value. For signals with frequencies exceeding the cut-off wave number, the attenuation rate increases with the increase of frequency. In a preferred embodiment, for the Gaussian filtering operation of the cross-sectional circular contour, the working frequency range of the filtering method is selected from any one of the following groups: 1 - 15 UPR, 1 - 50 UPR, 1 - 150 UPR, or 1 - 500 UPR, where the upper limit value of the frequency range is the cut-off wave number of the Gaussian filtering method. By setting the cut-off wave number, the high-frequency signal components exceeding this cut-off wave number are effectively suppressed, and the precise filtering operation of the target signal is achieved.

[0034] The moving average filtering method realizes the low-pass filtering function by performing weighted average processing on the time-domain signal. N is the width of the filtering window, which determines the cut-off frequency characteristics of the filtering method. The width of the filtering window is adjusted according to the actual application requirements to obtain different filtering effects. When the value of N increases, the cut-off frequency of the filtering method decreases, and the smoothing effect is enhanced. When the value of N decreases, the cut-off frequency increases, and more signal details are retained.

[0035] The method for filtering the original measurement signal in step S2 is as follows: 1. The filtering operation using the moving average filtering method The moving average filter is a finite impulse response (FIR) type filter. The output signal y[m] is obtained by solving the convolution of the input original signal x[m] and the impulse response h[i], that is

[0036] where the impulse response h[i] is .

[0037] 2. The filtering operation using the Gaussian filtering method The implementation of the Gaussian filter is through the convolution operation of weighted averaging the Gaussian weight function and the original measurement signal. In actual measurement, the original measurement signal is a discrete signal. To improve the calculation efficiency, the convolution operation of the discrete-time signal is converted into a product operation through the discrete Fourier transform (DFT).

[0038] The original measurement signal is x[m], and the discrete form of the Gaussian filtering weight function is S[m]. After performing the discrete Fourier transform on both of them respectively, and then performing the inverse transform on the product of the two, the output signal w[m] after Gaussian filtering is obtained. The calculation process is .

[0039] According to Appendix Figures 2 to 4 , the equivalent verification example of the moving average filtering and Gaussian filtering of the present invention is as follows: 1. Simulation verification In this verification example, the cross-sectional circular contour is represented as a periodic signal composed of multiple harmonic components, denoted as ,

[0040] where r0 represents the nominal radius of the workpiece, represents the eccentricity of the workpiece relative to the rotation axis, a i , b i represent the coefficients of each harmonic. When n is large enough is a minimum value. The number of sampling points for the roundness measuring instrument to rotate one week along the cross-section of the workpiece is n, and the original measurement signal is expressed as , , Assume n = 3600; r0 = 25mm; A1 = 0.01mm; f1 = 10Hz; A2 = 0.02mm; f2 = 20Hz; A3 = 0.03mm; f3 = 60Hz; A4 = 0.02mm; f4 = 400Hz.

[0041] where f is the frequency of the harmonic signal in the original measurement signal, and A is the amplitude of the harmonic signal. The simulation waveform of the original measurement signal of Figure 2 is as shown. Set the filtering parameters. The cut-off frequency of the Gaussian filtering method is set to 50Hz, and the filtering window width N of the corresponding moving average filtering method is taken as an integer of 43. The original signal is filtered using the Gaussian filtering and moving average filtering methods respectively. The filtered simulation waveform diagrams are as shown in Figure 3 , Figure 4 respectively.

[0042] The circle is fitted by the least squares method to determine the center coordinates of the filtered contour, and the roundness error is calculated, which is defined as the difference between the maximum radial distance and the minimum radial distance from each point on the contour curve to the reference circle.

[0043] Comparing and analyzing the filtering effects, the roundness error of the original measurement signal is 0.1395mm, the roundness error after Gaussian filtering is 0.0606mm, and the roundness error after the moving average filtering method is 0.0604mm. The difference in roundness error obtained by the two filtering methods is less than 0.33%, indicating that the two filtering methods described in the present invention have good consistency and reliability. Based on the equivalence of the moving average filtering and Gaussian filtering, high-precision measurement of the roundness error can be achieved.

[0044] 2. Crankshaft Measurement Experiment In this verification example, a roundness measuring instrument was used to collect the roundness data of the same workpiece. A total of 10 groups of measurement samples were obtained: 2.0μm, 2.1μm, 2.2μm, 2.0μm, 2.0μm, 2.0μm, 2.1μm, 2.2μm, 2.0μm, 2.0μm. Filtering operations were respectively performed using two filtering methods. The cut-off wave number of Gaussian filtering was 50Hz. According to the equivalence of the two filtering methods proposed in the present invention, the width N of the moving average filtering window was taken as an integer of 43. After performing filtering operations on the 10 groups of collected data respectively, the roundness error was calculated using the method of fitting a circle by the least squares method described in step S3. The roundness errors obtained after Gaussian filtering were 0.3μm, 0.5μm, 0.6μm, 0.4μm, 0.4μm, 0.3μm, 0.3μm, 0.5μm, 0.6μm, 0.4μm respectively; the roundness errors obtained after moving average filtering were 0.2μm, 0.4μm, 0.6μm, 0.4μm, 0.3μm, 0.3μm, 0.4μm, 0.3μm, 0.5μm, 0.4μm respectively. The average value of each group of data before and after filtering was taken as the roundness error measurement result, that is, the roundness error of the original measurement signal was 2.1μm, the roundness error after Gaussian filtering was 0.4μm, and the roundness error after moving average filtering was 0.4μm. The roundness error results obtained by the two filtering methods were the same, further verifying the stability and consistency of the application of the two filtering methods in roundness error measurement, that is, the moving average filtering and Gaussian filtering methods described in the present invention have equivalence, and the approximate equivalence relationship holds. The two filtering methods can be uniformly applied in roundness error measurement, solving the deviation problem of measurement results caused by differences in filtering methods.

[0045] The preferred specific embodiments of the present invention have been described in detail above. It should be understood that those of ordinary skill in the art can make many modifications and variations based on the concept of the present invention without creative labor. Therefore, all technical solutions that can be obtained by those skilled in the art in the technical field based on the concept of the present invention through logical analysis, reasoning, or limited experiments on the basis of the prior art should be within the protection scope determined by the claims.

Claims

1. A roundness error measurement method based on the equivalence of moving average filtering and Gaussian filtering, characterized in that: It includes the following steps: S1. Use a roundness measuring instrument to rotate one week along the outer surface of the workpiece to be measured, and obtain cross-sectional profile data composed of M equally angularly spaced sampling points, denoted as the original measurement signal x[m], where 1 ≤ m ≤ M. The standard number of sampling points M is 1800 or 3600 points per week. S2. Perform a filtering operation on the original measurement signal x[m] based on the equivalence between the moving average filtering method and the Gaussian filtering method: Taking x[m] obtained in step S1 as the input sequence and the cut-off wave number selected according to the Gaussian filtering method ,by performing a convolution operation on the Gaussian filtering weight function and the original measurement signal, the output sequence w[m] after Gaussian filtering is obtained. Taking x[m] obtained in step S1 as the input sequence, determine the moving average filter window width N corresponding to the cut-off wave number selected by the Gaussian filter method according to the equivalent relationship between the moving average filtering method and the Gaussian filtering method. After performing moving average filtering, the output sequence is y[m]. Each value in the output sequence is the arithmetic mean of N points starting from the current index m and going back N−1 points in the input sequence. S3. Calculate the roundness error of the filtered workpiece: Perform the least squares fitting circle on the output sequences w[m] and y[m] after filtering in step S2 respectively, calculate the radial distance from each sampling point to the center of the fitting circle, and take the maximum radius difference as the roundness error of the workpiece to be measured.

2. The roundness error measurement method based on the equivalence of moving average filtering and Gaussian filtering according to claim 1, characterized in that The equivalence between the moving average filtering method and the Gaussian filtering method in step S2 includes establishing the equivalent relationship between the moving average filtering method and the Gaussian filtering method: (1) Perform discrete-time Fourier transform on the impulse response of the moving average filtering method to obtain the relationship between the amplitude transmission characteristic and the filter window width N, the number of cycles per week and the number of sampling points M. , (2) Perform a continuous Fourier transform on the weight function of Gaussian filtering to obtain the relationship between the amplitude transfer characteristic and the cut-off wave number , the wave number per cycle , where α is a constant , (3) In the processing of roundness error measurement signals, make the two filtering methods have equivalent filtering characteristics, and attenuate the original measurement signal by 50% at the cut-off wave number to establish the equivalent relationship between the two filtering methods. 。 3. The roundness error measurement method based on the equivalence of moving average filtering and Gaussian filtering according to claim 1 or 2, characterized in that The equivalence between the moving average filtering method and the Gaussian filtering method in step S2 also includes determining the selection of filtering parameters for the moving average filtering method and the Gaussian filtering method: When using the Gaussian filtering method, select the cut-off wave number according to the measurement standard requirements; when using the moving average filtering method, calculate the filtering window width according to the equivalent relationship between the moving average filtering method and the Gaussian filtering method.

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