Circular Profile Error Separation Method Based on Optimal Selection of Rotation Angle by Time-Frequency Domain Transformation

Abstract

The present invention relates to a circular profile error separation method based on the optimization of the indexing angle by time-frequency domain transformation, belonging to the technical field of ultra-precision geometric quantity measurement. The method includes the following steps: applying a single indexing to cause a certain time delay of the profile error signal of the measured part in the time domain, and using the additional information obtained by this indexing to separate the spindle rotation error and the profile error of the measured part. The present invention establishes the relationship between the separation accuracy and the indexing angle error through time-frequency domain transformation. Thus, parameter optimization is carried out according to the requirements of error separation, realizing the roundness error separation that takes into account both the error separation accuracy and the harmonic range that can be accurately separated, so as to obtain more comprehensive, real and accurate circular profile information, providing an accurate theoretical basis for the metrology institutions and key laboratories to implement high-precision measurement of the circular cross-section of rotary parts.

Description

Technical Field

[0001] The present invention relates to a method for separating the profile error of a transposed angular circle based on time-frequency domain transformation, and belongs to the technical field of ultra-precision geometric measurement. Background Art

[0002] In the field of high-end intelligent manufacturing, the metrological high-precision measurement of circumferential surface profile data is the basis for the precision machining and precision manufacturing of rotating bodies. With the development of science and technology such as atomic energy, aerospace, microelectronics, information, and bioengineering, higher and higher requirements have been put forward for the measurement accuracy of the roundness error of workpieces. For example, in the process of ultra-precision machining and measurement, the roundness error of the round standard, which is usually used for comparison and calibration, is generally 10 nm to 50 nm. The roundness tolerance in the normal direction of any circular cross-section of the curved mirror in an X-ray microscope is 5 nm. In addition, roundness is the basis of the key quality of rotating parts in many fields, such as engine turbine shafts, motors, bearings, etc. At the same time, the spindle rotation error of various rotating machining instruments is also a very important performance, and the spindle rotation error is one of the main sources of workpiece size and shape errors.

[0003] To ensure high-precision measurement, the error brought by the measuring instrument can only account for a small part of the allowable error of the workpiece, about (1 / 3 to 1 / 10). However, in fact, the precision indexes of many workpieces now reach or are higher than the precision level of the roundness measuring instrument with the highest precision currently used for measuring roundness. The error separation technology is a measurement technology that separates the useful signal component from the error signal by means of information source transformation or model parameter estimation. It can obtain a precision level that is difficult to achieve by "hard technology" at a relatively small cost, so the error separation technology has developed rapidly.

[0004] The error separation technology is a measurement technology that separates the useful signal component from the error signal by means of information source transformation or model parameter estimation. It can obtain a precision level that is difficult to achieve by "hard technology" at a relatively small cost, so the error separation technology has been quickly accepted by scholars and experts from various countries. The error separation technology is mainly divided into two types: the transposition method and the multi-probe method. Both types of roundness error separation methods have the problem that the separation principle is not sensitive to some harmonic components of the error. The insensitive harmonic components are related to the number of probes selected or the number of transpositions, which results in the inability to separate the measured round profile error and the spindle rotation error at the corresponding harmonics. This phenomenon is also called harmonic suppression, which is a principle and fundamental problem that prevents the improvement of the roundness error separation accuracy.

[0005] In addition to the influence of harmonic suppression on the accuracy of the multi-probe method, factors such as the difference in characteristics between the probes and the angle error of the arrangement are also the reasons that limit its accuracy. This method is relatively more suitable for the needs of high-speed, efficient, and online measurement of spindle rotation errors, and it is difficult to achieve metrological-grade high-precision measurement of circular contours. Various studies on the harmonic suppression problem of the indexing method usually compensate for the harmonic suppression frequency by increasing the number of indexing times or combining the results of several indexings, but the above process is very time-consuming and introduces random and systematic errors, making it difficult to achieve accurate separation of a large harmonic range. For example: The publication number is CN117400060A, and the name of the invention is a machine tool spindle rotation error detection device and method. In its technical solution, in addition to the influence of harmonic suppression on the accuracy, factors such as the difference in characteristics between the probes and the angle error of the arrangement are also the reasons that limit its accuracy. Therefore, this type of method has low accuracy and is not suitable for metrological-grade roundness measurement.

[0006] In the prior art, Qiao Lingxiao, Chen Jiangning, Chen Wenhui, Zhang Li, Tian Jingzhi. A high-precision spindle rotation error separation algorithm based on multi-step method [J]. Journal of Metrology, 2018, 39(1): 6-11 mentioned that the multi-step method is a kind of indexing method and is also the most basic method for roundness error separation in this field. However, this method has the problem that the separation principle is insensitive to certain harmonic components of the error, namely harmonic suppression; as a result, the measured circular contour error and the spindle rotation error cannot be separated at some harmonics, affecting the roundness measurement accuracy.

[0007] Currently, there is still no method to achieve high-precision circle profile error separation within a large separation harmonic range. This has a great impact on our measurement and acquisition of the true circle profile, thereby further reducing the accuracy of value transfer and affecting measurement, processing, and assembly.

[0008] Therefore, it is urgent to propose a circular contour error separation method based on time-frequency domain transformation and transposition angle optimization to solve the above technical problems. Summary of the invention

[0009] The purpose of the present invention is to provide a circular profile error separation method based on time-frequency domain transformation and transposition angle optimization in order to solve the problems existing in the above-mentioned prior art. The method realizes parameter optimization through time-frequency domain transformation according to the error separation requirements, and realizes circular profile error separation that takes into account separation accuracy and separable harmonic order range by fully analyzing the additional data obtained by a single transposition. A brief overview of the present invention is given below to provide a basic understanding of certain aspects of the present invention. It should be understood that this overview is not an exhaustive overview of the present invention. It is not intended to identify the key or important parts of the present invention, nor is it intended to limit the scope of the present invention.

[0010] The technical solution of the present invention:

[0011] The circular contour error separation method based on time-frequency domain transformation and transposition angle optimization includes the following steps:

[0012] Applying a single indexing makes the contour error signal of the workpiece to be measured have a certain time delay in the time domain, and the additional information obtained by this indexing is used to separate the spindle rotation error and the contour error of the workpiece to be measured.

[0013] Preferably: the spindle, the error separation table, and the workpiece to be measured are coaxially arranged, the measuring direction of the probe is arranged radially, the probe is used for signal measurement and acquisition, and the probe adopts a capacitive sensor or a point laser sensor.

[0014] Preferably: Step 1: The method of applying a single indexing to make the contour error signal of the workpiece to be measured have a certain time delay in the time domain includes the following steps:

[0015] Step 1.1: Data acquisition;

[0016] Step 1.2: Convert the data acquired in the time domain to the frequency domain, and analyze the characteristics of each frequency component in the data by using Fourier transform; it is used to identify the frequency domain characteristics of the spindle rotation error and the roundness error of the workpiece.

[0017] Preferably: In Step 1.1, after completing the acquisition of the signal V1(θ i ) at the first indexing position before indexing by performing multiple measurements at the original position, control the error separation table to drive the workpiece to be measured to generate an indexing angle α relative to the turntable, that is, the spindle; perform multiple measurements to obtain the signal V2(θ i );

[0018] Let R(θ i ) be the measured circular contour signal, and M(θ i ) be the spindle rotation error signal. During the indexing process of the workpiece to be measured, each harmonic of the measured circular contour signal generates a certain time delay in the time domain relative to the spindle rotation error signal, and its phase changes in the corresponding frequency domain. The phase of the spindle rotation motion error component remains unchanged. Then, the signals collected by the probe before and after indexing can be expressed as:

[0019]

[0020] Preferably: In Step 1.2, expand the circular contour error signal R(θ i ) of the workpiece to be measured, that is, the standard hemisphere, and the spindle rotation error signal M(θ i ) into Fourier series forms respectively:

[0021]

[0022] In the formula, R0, a k , b k are the Fourier expansion coefficients of the workpiece circular contour error signal R(θ i ), M0, c k , d kis the Fourier expansion coefficient of the spindle rotation error M(θ i ); after indexing, the profile error signal of the measured part is:

[0023]

[0024] Take the difference of equation (1), and denote the difference as r(θ i ), then there is:

[0025]

[0026] Discretize the above equation by N-point sampling. Filters with different cut-off frequencies can be applied for filtering according to actual requirements to filter out signals higher than the cut-off frequency Nc; after completing the filtering and DC bias removal steps, the result of the discretization process can only take the 1st to Nc-1th harmonics. The angle corresponding to the nth sampling point is 2nπ / N, and its discretized form formula is as follows:

[0027]

[0028] In the above equation, r(n) can also be expanded in the form of a Fourier series:

[0029]

[0030] Among them, r0,e k ,f k are the Fourier coefficients obtained by expanding r(n), and each coefficient can be expressed as:

[0031]

[0032] Comparing the part where k≥2 in (7) and (8), there is:

[0033]

[0034] The Fourier expansion coefficients a k , b k of the roundness error of the measured part can be obtained as:

[0035]

[0036] Denote the coefficient sinkα / (1-coskα)=p, then there is:

[0037]

[0038] After obtaining the Fourier expansion coefficients a k , b k of the roundness error of the measured part, the round profile error signal R(θ i)Discretization processing; meanwhile, in roundness measurement and evaluation, the fundamental wave component and the first harmonic component (k = 0, 1) in the harmonic components do not belong to the category of roundness. Therefore, the time-domain discrete value R(n) can be expressed as:

[0039]

[0040] Then R(n) is the workpiece circular contour error value after removing the spindle rotation motion error. Subsequently, the spindle rotation motion error can be calculated by Equation (1):

[0041]

[0042] To reduce the influence of random errors in the measurement process on the calculation, the average of the spindle rotation motion error calculation results obtained from the above two equations is taken as the final spindle rotation motion error calculation result;

[0043] Analyzing and solving the information before and after the rotation of the rotation angle unit, the roundness error of the measured part is the synthesis result of the Fourier expansion terms of its various harmonic orders. From (10), it can be seen that the Fourier expansion coefficients a k , b k The calculation formula is:

[0044]

[0045] The variation relationship of the coefficient p with the rotation angle α is as follows:

[0046]

[0047] For a given rotation angle α, when k = 2π·z / α (z = 1, 2,...), k / coskα - 1 → ∞. At this time, any deviation Δα at the harmonics near the mutation point will significantly affect the coefficient p; thus affecting the coefficients a k and b k in the solution of Equation (12), resulting in the inability to separate the spindle rotation error from the original contour error signal of the corresponding frequency, and ultimately reducing the error separation accuracy and the effective separation range; it can be seen that when other conditions are the same, an inappropriate selection of the rotation angle has a greater impact on both the separation accuracy and the accurately separable range; therefore, the optimization of the rotation angle is extremely important for large-range and high-precision roundness error separation.

[0048] Preferably: Step 2: A method for separating the spindle rotation error and the contour error of the measured part by using the additional information obtained by this rotation includes the following steps:

[0049] Analyze the influence of the selection of the rotation angle α and its error Δα on the separation result error of this method and the range of accurately separable harmonic orders; from Equation (13), it can be seen that the separation error ΔR(n) of the circular contour signal of the measured cross-section caused by the rotation angle error Δα at the nth sampling point is:

[0050]

[0051] Roundness error R(θ of the device under test i ) Fourier expansion coefficient a k , b k Taking the partial derivative with respect to the indexing angle α, the results are as follows:

[0052]

[0053] Then the separation error ΔR(n) of the circular profile signal of the measured section can be expressed as:

[0054]

[0055] The separation error ΔR(n) of the circular profile signal of the measured section is positively correlated with the coefficient Q and the indexing angle error Δα, f k , e k and is only related to the signals collected by the sensor in the actual experiment; therefore, when the indexing angle error Δα is the same, ΔR(n) is the smallest when |Q| obtains the minimum value;

[0056] For different required precise separation harmonic ranges, the corresponding indexing angles are different when |Q| obtains the minimum value; plot the variation relationship of |Q| with the separation harmonic range N c and the indexing angle α;

[0057] So far, the optimal indexing angle values can be obtained through the above process for different required precise separation harmonic ranges; control the frequency corresponding to the harmonic singularity point of this method outside the required precise separation harmonic range, and at this time, through the optimization of the indexing angle, high-precision circular profile error separation over a large range can be achieved.

[0058] The present invention has the following beneficial effects:

[0059] The present invention establishes the relationship between the separation accuracy and the indexing angle error through time-frequency domain transformation. Thus, parameter optimization is carried out according to the error separation requirements, realizing circularity error separation that takes into account both the error separation accuracy and the harmonic range that can be precisely separated, thereby obtaining more comprehensive, real, and accurate circular profile information, providing an accurate theoretical basis for the metrology institutions and key laboratories to implement high-precision measurement of the circular cross-section of rotary parts. Brief Description of the Drawings

[0060] Figure 1 is the schematic diagram of the circular profile error separation method based on the optimization of the indexing angle by time-frequency domain transformation.

[0061] Figure 2 is the variation relationship diagram of the coefficient |Q| with the separation harmonic range N c and the indexing angle α.

[0062] Figure 3 are the signal graphs collected by the sensor before and after indexing;

[0063] Figure 3 (a) is the signal collected before indexing (after optimizing the indexing angle, α = 1.406°);

[0064] Figure 3 (b) is the signal collected after indexing (after optimizing the indexing angle, α = 1.406°);

[0065] Figure 3 (c) is the signal collected before indexing (without optimizing the indexing angle, α = 3.164°);

[0066] Figure 3 (d) is the signal collected after indexing (without optimizing the indexing angle, α = 3.164°).

[0067] Figure 4 are the roundness error separation result graphs before and after optimizing the indexing angle based on time-frequency domain transformation;

[0068] Figure 4 (a) is the circular profile error signal (after optimizing the indexing angle, α = 1.406°);

[0069] Figure 4 (b) is the spindle rotation error signal (after optimizing the indexing angle, α = 1.406°);

[0070] Figure 4 (c) is the circular profile error signal (without optimizing the indexing angle, α = 3.164°);

[0071] Figure 4 (d) is the spindle rotation error signal (without optimizing the indexing angle, α = 3.164°).

[0072] In the figure, 1 - spindle, 2 - error separation table, 3 - workpiece to be measured, 4 - probe. Detailed implementation manners

[0073] To make the objectives, technical solutions and advantages of the present invention clearer and more understandable, the present invention will be described below through specific embodiments shown in the accompanying drawings. However, it should be understood that these descriptions are only exemplary and do not intend to limit the scope of the present invention. In addition, in the following description, the descriptions of well-known structures and technologies are omitted to avoid unnecessarily confusing the concepts of the present invention.

[0074] Detailed implementation manner one: Combining Figure 1-2 To illustrate this implementation manner, the circular profile error separation method based on optimizing the indexing angle by time-frequency domain transformation in this implementation manner includes the following steps:

[0075] The single indexing makes the contour error signal of the measured object 3 produce a certain delay in the time domain. The additional information obtained by the indexing is used to separate the rotation error of the spindle 1 and the contour error of the measured object 3 (the workpiece circle or the measured circle contour). The parameter optimization is realized according to different error separation requirements, thereby taking into account the error separation accuracy and the harmonic range that can be accurately separated. The principle diagram is shown in the figure. Figure 1 As shown;

[0076] The present invention firstly reduces the number of harmonic suppression points in principle, and secondly optimizes the indexing angle by analyzing the relationship between separation accuracy and indexing angle error, thereby achieving high-precision circular profile error separation within a larger separation harmonic range;

[0077] The present invention aims at the problem that the existing high-precision roundness error separation methods have harmonic suppression, which limits the range of accurately separable harmonics and reduces the separation accuracy. A circular contour error separation method based on time-frequency domain transformation and inversion angle optimization is proposed. The relationship between separation accuracy and inversion angle error is established through time-frequency domain transformation. Therefore, parameter optimization is performed according to the error separation requirements to achieve roundness error separation that takes into account both error separation accuracy and the range of accurately separable harmonics. Thereby, more comprehensive, true and accurate circular contour information is obtained, which provides an accurate theoretical basis for national metrology institutions and key laboratories to implement high-precision measurement of circular sections of rotary parts.

[0078] Specific implementation method 2: Combination Figure 1-2 The present embodiment is described. In the present embodiment, the circular contour error separation method based on the optimization of the transposition angle by time-frequency domain transformation is as follows: the spindle 1, the error separation platform 2 and the measured object 3 are coaxially arranged, the measuring direction of the probe 4 is arranged radially, the probe 4 is used for signal measurement and acquisition, and the probe 4 is a capacitive sensor or a point laser sensor; the spindle 1 and the error separation platform 2 respectively have driving devices, which are the spindle 1, the error separation platform 2 and the measured object 3 from bottom to top, the error separation platform drives the circular contour to rotate α, and the spindle mainly rotates a full circle to realize the measurement of the full circle circular contour signal.

[0079] Specific implementation method three: Combination Figure 1-2 The present embodiment is described. The present embodiment is a circular contour error separation method based on time-frequency domain transformation indexing angle optimization. Step 1: a method for applying a single indexing to cause a certain time delay in the time domain of the contour error signal of the measured object, including the following steps:

[0080] Step 1.1: Data collection;

[0081] Step 1.2: Convert the data collected in the time domain to the frequency domain, and use Fourier transform to analyze the characteristics of each frequency component in the data; this is used to identify the frequency domain characteristics of the spindle rotation error and the workpiece roundness error.

[0082] Specific implementation method four: Combination Figure 1-2Describing this embodiment, for the circular profile error separation method based on the optimal indexing angle by time-frequency domain transformation in this embodiment, in step 1.1, after collecting the signal V1(θ i ) at the first indexing position before indexing through multiple measurements at the original position, the error separation stage 2 (i.e., the high-precision angle indexing turntable) is controlled to drive the measured part 3 to generate an indexing angle α relative to the turntable, i.e., the spindle 1; multiple measurements are carried out to obtain the signal V2(θ i ).

[0083] Denote R(θ i ) as the measured circular profile signal and M(θ i ) as the spindle rotation error signal. During the indexing process of the measured part, each harmonic of the measured circular profile signal generates a certain time delay relative to the spindle rotation error signal in the time domain, and its phase changes in the corresponding frequency domain. The phase of the spindle rotation motion error component remains unchanged. Then, the signals collected by the probe 4 before and after indexing can be expressed as:

[0084]

[0085] Specific embodiment five: Combining Figure 1-2 Describing this embodiment, for the circular profile error separation method based on the optimal indexing angle by time-frequency domain transformation in this embodiment, in step 1.2, the circular profile error signal R(θ i ) and the spindle rotation error signal M(θ i ) of the measured part, i.e., the standard hemisphere, are respectively expanded into Fourier series forms:

[0086]

[0087] In the formula, i represents the number of sampling points in the time domain, i = 1, 2,..., N; θ i represents the sampling angle corresponding to the sampling point in the time domain; R0,a k ,b k are the Fourier expansion coefficients of the workpiece circular profile error signal R(θ i ), and M0,c k ,d k are the Fourier expansion coefficients of the spindle rotation error M(θ i ); after indexing, the circular profile error signal of the measured part (workpiece circle) is:

[0088]

[0089] Taking the difference of formula (1) and denoting the difference as r(θ i ), then there is:

[0090]

[0091] Discretize the above formula by N-point sampling. According to ISO 12181-2, filters with different cut-off frequencies can be applied for filtering according to actual requirements to filter out signals higher than the cut-off frequency Nc. After completing the filtering and DC bias removal steps, the result of the discretization process can only take the 1st to (Nc - 1)th harmonics. The angle corresponding to the nth sampling point is 2nπ / N, and r(n) is the discretized form of r(θ i ) in the above formula, and its discretized form formula is as follows:

[0092]

[0093] In the formula, N is the number of sampling points in one circumference of the circular contour, n represents the nth sampling point, and its value ranges from 1 to N, representing the 1st to Nth sampling points in one circle; r(n) in the above formula can also be expanded into the form of a Fourier series:

[0094]

[0095] Among them, r0,e k ,f k are the Fourier coefficients obtained by expanding r(n), and each coefficient can be expressed as:

[0096]

[0097] Comparing the parts where k ≥ 2 in (7) and (8), we have:

[0098]

[0099] The Fourier expansion coefficients a k and b k of the roundness error of the measured part can be obtained as:

[0100]

[0101] Denote the coefficient sinkα / (1 - coskα) = p, then we have:

[0102]

[0103] After obtaining the Fourier expansion coefficients a k and b k of the roundness error of the measured part, discretize the circular contour error signal R(θ i ). At the same time, in the roundness measurement and evaluation, the fundamental wave component and the first harmonic component (k = 0, 1) in the harmonic components do not belong to the category of roundness. Therefore, the time-domain discrete value R(n) can be expressed as:

[0104]

[0105] Then R(n) is the workpiece circular contour error value after removing the spindle rotation error, and subsequently, the spindle rotation error can be calculated by Equation (1):

[0106]

[0107] To reduce the influence of random errors on the calculation during the measurement process, the average of the spindle rotation error calculation results obtained from the above two equations is taken as the final spindle rotation error calculation result;

[0108] The roundness error of the measured part obtained by analyzing and solving the information before and after the rotation of the rotation angle unit is the synthesis result of the Fourier expansion terms of its various harmonic orders. As can be seen from (10), the Fourier expansion coefficients a k and b k The calculation formula is:

[0109]

[0110] The variation relationship of the coefficient p with the rotation angle α is as follows:

[0111]

[0112] Among them, △ represents the rotation angle deviation, represents the partial derivative. For a given rotation angle α, when k = 2π·z / α (z = 1, 2,...), k / coskα - 1 → ∞. At this time, any deviation Δα at the harmonics near the mutation point will significantly affect the coefficient p; thus affecting the coefficients a k and b k in the solution, resulting in the inability to separate the spindle rotation error from the original contour error signal of the corresponding frequency, and ultimately reducing the error separation accuracy and the effective separation range; it can be seen that when other conditions are the same, an inappropriate selection of the rotation angle has a greater impact on both the separation accuracy and the accurately separable range; therefore, the optimization of the rotation angle is extremely important for large-range and high-precision roundness error separation.

[0113] Specific Embodiment 6: In combination with Figure 1-2 to illustrate this embodiment. The round contour error separation method based on the optimization of the rotation angle by time-frequency domain transformation in this embodiment, Step 2: The method for separating the rotation error of the spindle 1 and the contour (workpiece circle or measured circle contour) error of the measured part 3 by using the additional information obtained by this rotation includes the following steps:

[0114] Analyze the influence of the selection of the rotation angle α and its error Δα on the separation result error of this method and the range of accurately separable harmonic orders; as can be seen from Equation (13), the separation error ΔR(n) of the measured cross-section circular contour signal caused by the rotation angle error Δα at the nth sampling point is:

[0115]

[0116] Roundness error R(θ of the device under test i ) Fourier expansion coefficient a k , b k Taking the partial derivative with respect to the indexing angle α, the resulting outcomes are as follows:

[0117]

[0118] Then the separation error ΔR(n) of the circular profile signal of the measured section can be expressed as:

[0119]

[0120] The separation error ΔR(n) of the circular profile signal of the measured section is positively correlated with the coefficient Q and the indexing angle error Δα, f k , e k is only related to the signals collected by the sensor in the actual experiment; therefore, when the indexing angle error Δα is the same, ΔR(n) is minimized when |Q| reaches the minimum value;

[0121] For different required precise separation harmonic ranges, the corresponding indexing angles are different when |Q| reaches the minimum value; Plot the variation relationship of |Q| with the separation harmonic range N c and the indexing angle α as follows Figure 2 shown;

[0122] Thus, through the above process, the optimal indexing angle values can be obtained for different required precise separation harmonic ranges; By controlling the frequency corresponding to the harmonic singularity point of this method outside the required precise separation harmonic range, at this time, through the optimization of the indexing angle, high-precision circular profile error separation over a large range can be achieved.

[0123] Example 1:

[0124] To verify the separation effect of the circular profile error separation method based on the indexing angle optimization by time-frequency domain transformation, an experimental verification is carried out on this method. After setting the required precise separation harmonic range, the error separation results before and after the indexing angle optimization based on time-frequency domain transformation are compared; When the required precise separation harmonic range is [0, 200] upr, the optimized indexing angle is 1.406°; Multiple measurements of the circular profile are averaged to reduce the influence of random noise. The signals collected by the sensor before and after indexing are as Figure 3 shown;

[0125] Repeating the above process multiple times, the roundness error separation is performed on the collected signals each time. The workpiece circular profile error and the spindle rotation error obtained by the error separation method before and after the indexing angle optimization based on time-frequency domain transformation are as Figure 4As shown; for ease of observation, only the harmonic synthesis results within the range of [0, 50] upr are plotted here. The results obtained from each experiment are evaluated for roundness (RLSC, MLSC) by the least squares method as shown in Table 1;

[0126] Table 1 Separation results before and after the preferred indexing angle based on time-frequency domain transformation

[0127]

[0128] Compared with the separation results without the preferred indexing angle, the standard deviation of the results obtained by the roundness error separation method with the preferred indexing angle based on time-frequency domain transformation is reduced from about 23 nm to less than 7 nm, and the standard deviation is only about 30% of that without the preferred indexing angle. This method is less affected by the positioning error of the error separation table angle, etc. At the same time, the repeatability of the separation results is improved, and high-precision roundness error separation can be achieved over a wide range. Therefore, this method is of great significance for people to have a more accurate and comprehensive understanding of the circular profile.

[0129] It should be noted that in the above embodiments, as long as the technical solutions are not contradictory, they can be combined. Those skilled in the art can exhaust all possibilities according to the mathematical knowledge of permutation and combination. Therefore, the present invention will not describe the technical solutions after permutation and combination one by one, but it should be understood that the technical solutions after permutation and combination have been disclosed by the present invention.

[0130] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. For those skilled in the art, the present invention can have various changes and modifications. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A circular profile error separation method based on the optimal selection of the rotation angle by time-frequency domain transformation, characterized in that: Including the following steps: Applying indexing to cause a time delay of the profile error signal of the workpiece under test (3) in the time domain, and using indexing to separate the spindle (1) runout error and the profile error of the workpiece under test (3); The spindle (1), the error separation table (2), and the workpiece under test (3) are coaxially arranged, and the measuring direction of the probe (4) is arranged along the radial direction; Step 1: The method of applying indexing to cause a time delay of the profile error signal of the workpiece under test in the time domain includes the following steps: Step 1.1: Data acquisition; In Step 1.1, after the signal V1(θ i ) at the first indexing position before indexing is collected through multiple measurements at the original position, the error separation stage (2) is controlled to drive the workpiece under test (3) to generate an indexing angle α relative to the spindle (1); multiple measurements are carried out to obtain the signal V2(θ i ); Let \(R(\theta i )\) be the measured circular contour signal, and \(M(\theta i )\) be the spindle rotational error signal. During the indexing process of the measured component, each harmonic of the measured circular contour signal has a certain time delay relative to the spindle rotational error signal in the time domain, and its phase changes in the corresponding frequency domain. Since the phase of the spindle rotational error component remains unchanged, the signals collected by the probe (4) before and after indexing can be expressed as follows: Step 1.2: Convert the data collected in the time domain to the frequency domain, and use Fourier transform to analyze the characteristics of each frequency component in the data; In Step 1.2, the measured circular profile signal R(θ of the device under test (3) i ) and the spindle rotational error signal M(θ i ) are respectively expanded into Fourier series forms: where R0,a k ,b k are the Fourier expansion coefficients of the workpiece circular profile error signal R(θ i ), and M0,c k ,d k are the Fourier expansion coefficients of the spindle rotational error M(θ i ); after indexing, the profile error signal of the measured part is: Subtract Equation (1) and denote the difference as r(θ i ), then we have: Performing N-point sampling discretization on the above formula, filters with different cut-off frequencies can be applied for filtering according to actual requirements to filter out signals higher than the cut-off frequency Nc; after completing the filtering and DC bias removal steps, the result of the discretization process can only take the 1st to Nc-1th harmonics. The angle corresponding to the nth sampling point is 2nπ / N, and its discretized form formula is as follows: In the above formula, r(n) can also be expanded in the form of a Fourier series: where r0,e k , f k are the Fourier coefficients obtained by expanding r(n), and each coefficient can be expressed as: Comparing the parts where k≥2 in (7) and (8) gives: The Fourier expansion coefficients a k and b k of the roundness error of the measured part can be obtained as follows: Denote the coefficient sinkα / (1 - coskα) = p, then there is: Obtain the Fourier expansion coefficients a k and b k of the roundness error of the measured part. After that, discretize the round profile error signal R(θ i ). At the same time, in the roundness measurement and evaluation, the fundamental wave component and the first harmonic component in the harmonic components do not belong to the roundness category. Therefore, the time-domain discrete value R(n) can be expressed as: Then R(n) is the circular profile error value of the workpiece after removing the spindle runout error, and subsequently the spindle runout error can be calculated by Equation (1): To reduce the influence of random errors in the measurement process on the calculation, the average of the spindle runout error calculation results obtained from the above two equations is taken as the final spindle runout error calculation result; The roundness error of the measured part obtained by analyzing and solving the information before and after the rotation of the corner unit is the synthesis result of the Fourier expansion terms of each harmonic order, and it can be seen from (10) that the Fourier expansion coefficient a k 、b k The calculation formula is as follows: The variation relationship of the coefficient p with the indexing angle α is as follows: For a given indexing angle α, when k = 2π·z / α, k / coskα - 1 → ∞, and any deviation Δα at the harmonics near the mutation point will affect the coefficient p; Step 2: The method of using indexing to separate the spindle (1) runout error and the profile error of the workpiece under test (3) includes the following steps: Analyze the influence of the selection of the indexing angle α and its error Δα on the separation result error of this method and the range of harmonics that can be accurately separated; it can be seen from Equation (13) that the separation error ΔR(n) of the circular profile signal of the measured cross-section caused by the indexing angle error Δα at the nth sampling point is: Roundness error R(θ of the device under test i ) Fourier expansion coefficients a k , b k Taking the partial derivative with respect to the indexing angle α, the results are as follows: Then the separation error ΔR(n) of the circular profile signal of the measured cross-section can be expressed as: The separation error ΔR(n) of the measured cross-sectional circular contour signal is positively correlated with the coefficient Q and the indexing angle error Δα, f k ,e k and is only related to the signals collected by the sensor in the actual experiment; when the indexing angle error Δα is the same, ΔR(n) is minimized when |Q| reaches the minimum value.

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