A machine tool spindle rotation signal denoising method based on VMD-PE-MWSTD
By employing the VMD-PE-MWSTD method for decomposition and denoising, the problem of noise interference in spindle rotation error measurement was solved, achieving high-precision signal separation and feature preservation, thereby improving measurement accuracy and signal-to-noise ratio.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- SHENYANG UNIVERSITY OF TECHNOLOGY
- Filing Date
- 2026-03-26
- Publication Date
- 2026-06-26
Smart Images
Figure CN122286089A_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The present application relates to the field of precision measurement, and in particular to a machine tool spindle rotation signal denoising method based on VMD-PE-MWSTD. BACKGROUND
[0002] Spindle rotation error is a key indicator of spindle dynamic performance, directly affecting the machining quality and efficiency of the machine tool. Spindle rotation error not only affects the dimensional accuracy of the workpiece, but also seriously affects the surface roughness, geometric error and other quality requirements. Related studies have shown that the error caused by spindle rotation error accounts for about 30% to 70% of the spindle accuracy error.
[0003] However, the existing spindle rotation error measurement method still faces problems such as noise, error source interference in high speed and high precision applications, especially under fast rotation conditions. Traditional sensors and measurement methods are difficult to meet the real-time and accurate rotation error measurement requirements. Therefore, how to effectively remove noise components and improve the accuracy of rotation error measurement has become a major challenge in current technology. SUMMARY
[0004] The present application provides a machine tool spindle rotation signal denoising method based on VMD-PE-MWSTD to solve the problem of modal aliasing, large noise interference and easy loss of feature information of spindle rotation signal in high speed and strong noise background in the prior art.
[0005] The present application provides a machine tool spindle rotation signal denoising method based on VMD-PE-MWSTD, which specifically includes the following steps:
[0006] Step S1: Collect the rotation signal of the machine tool spindle and pre-process the rotation signal;
[0007] Step S2: decompose the pre-processed rotation signal using a variational mode decomposition VMD method to obtain a plurality of intrinsic mode function IMF components and a residual component;
[0008] Step S3: perform permutation entropy PE analysis on each intrinsic mode function IMF component obtained, and divide each intrinsic mode function IMF component into a useful signal dominant intrinsic mode function IMF and a noise dominant intrinsic mode function IMF according to a preset permutation entropy threshold;
[0009] Step S4: for the selected noise dominant intrinsic mode function IMF, perform denoising processing using an improved wavelet soft threshold denoising MWSTD method to obtain a denoised intrinsic mode function IMF component;
[0010] Step S5: Reconstruct the obtained useful signal dominant intrinsic mode function (IMF) and the obtained denoised IMF components to obtain the denoised machine tool spindle rotation signal.
[0011] Furthermore, the variational mode decomposition (VMD) method described in step S2 includes:
[0012] For each modal component to be decomposed The Hilbert transform is expressed by the following formula:
[0013]
[0014] In the formula, It is the unit impulse function, j is the imaginary unit, and * represents convolution. It is the first One modal component to be decomposed, It is a time variable.
[0015] Introduction The modal frequencies are modulated onto the fundamental frequency band, as expressed by the following formula:
[0016]
[0017] In the formula, It is the unit impulse function, j is the imaginary unit, and * represents convolution. It is the first One modal component to be decomposed, It is the time derivative. It is a time variable. It is a complex exponential modulation term, used to modulate the first... The spectrum of each modal component is shifted to the fundamental frequency band;
[0018] According to the square norm The sum of the bandwidths of the modal variables is estimated by the following formula:
[0019]
[0020] In the formula, These are constraints, indicating that the sum of all decomposed modal components should equal the original signal. , It is the first The center frequency of each modal component It is a unit impulse function. It is a time variable, j is the imaginary unit, and * represents convolution. It is a complex exponent;
[0021] By introducing the Lagrange operator, we obtain the optimal solution in the equation, resulting in the augmented Lagrange expression as follows: In the formula It is the first One modal component to be decomposed, It is the first The center frequencies of each modal component, where is the penalty factor. It is the time derivative. It is a unit impulse function. It is the original signal. It is a time variable, j is the imaginary unit, and * represents convolution. It is a compound exponent. For the Lagrange penalty operator.
[0022] The frequency center and bandwidth of the IMF component separated from the original signal satisfy the iterative condition, and the iteration stops. The iterative formula is expressed as follows:
[0023]
[0024] In the formula It is the first The first modal component Frequency domain representation of the next iteration It is the first The first modal component Frequency domain representation of the next iteration It is a frequency variable. It is the convergence threshold;
[0025] right The mode function is obtained by inverse transformation.
[0026] Furthermore, the permutation entropy (PE) analysis process in step S3 is as follows:
[0027] Permutation entropy (PE) analysis was performed on each obtained intrinsic mode function (IMF) component to assess the disorder level of each IMF component. The larger the PE value, the higher the disorder level and the more noise components the IMF contains. Based on a preset PE threshold of 0.45, IMFs with lower PE values (PE < 0.45) were selected, indicating that the IMF mainly contains useful signal components. IMFs with higher PE values (PE > 0.45) were identified as noise-dominated IMFs.
[0028] Given discrete time series
[0029]
[0030] Construct the embedding vector:
[0031]
[0032] In the formula, t=1,2,…,N-(m-1)τ, m is the number of data points in the subsequence, and τ is the time step between adjacent data points when constructing the embedding vector;
[0033] Count the occurrences of all embedded vector patterns The probability of obtaining each permutation pattern is:
[0034]
[0035] In the formula, N is the length of the time series. τ is the number of times the i-th sorting pattern appears, m is the number of data points in the subsequence, and τ is the time step between adjacent data points when constructing the embedding vector.
[0036] Calculate the permutation entropy PE:
[0037]
[0038] In the formula It is the first The probability of a sorting pattern.
[0039] Furthermore, the improved soft-threshold wavelet denoising process described in step S4 is as follows:
[0040] Soft threshold wavelet denoising:
[0041]
[0042] In the formula These are wavelet coefficients after soft thresholding. This indicates the translation position index within this layer.
[0043] Represents the original wavelet decomposition coefficients. Indicates the decomposition scale. It is a threshold.
[0044] Improved soft-threshold wavelet denoising:
[0045]
[0046] These are the coefficients after processing with the improved threshold function. Represents the original wavelet decomposition coefficients. For threshold adjustment parameters, It is a threshold.
[0047] Furthermore, the signal reconstruction process described in step S5 is as follows: The reconstruction formula is as follows:
[0048]
[0049] In the formula These are the coefficients after processing with the improved threshold function. It is the reconstructed signal after denoising. It is the first Low-frequency coefficient of layer, It is a scaling function. It is the first The detail factor after layer processing It is the basis function after the wavelet function has been translated and scaled.
[0050] Compared with the prior art, the present invention has the following advantages:
[0051] 1. Effectively solves the modal aliasing problem:
[0052] By employing the variational mode decomposition (VMD) method, the spindle rotation error signal can be effectively decomposed into multiple intrinsic mode functions (IMF) components. The VMD method avoids the mode aliasing phenomenon in the traditional EMD method, giving each IMF component a clear physical meaning. It can effectively separate signal components in different frequency bands, which helps to extract spindle rotation error characteristics more accurately.
[0053] 2. Adaptive screening of IMF components:
[0054] By using permutation entropy (PE) analysis, each IMF is automatically filtered based on its PE value to remove noise-dominated IMF components. IMFs with lower PE values (PE < 0.45) represent effective signal components, exhibiting high adaptability. This avoids the subjectivity of manually setting thresholds in traditional methods and improves the accuracy of noise reduction.
[0055] 3. Wavelet denoising and layered processing:
[0056] The wavelet soft thresholding denoising method (MWSTD) is adopted, which only performs thresholding on the noise-dominated IMF component. By processing the wavelet detail coefficients in a layered manner, the influence of high-frequency random noise such as cutting noise and environmental noise can be effectively removed, while preserving the effective characteristics of spindle rotation to the maximum extent and maintaining the original characteristics of the signal.
[0057] 4. Improved signal-to-noise ratio and reduced root mean square error:
[0058] The signal denoised using the method of this invention has a higher signal-to-noise ratio (SNR) and a significantly lower root mean square error (RMSE) than traditional methods, indicating that this method can more accurately preserve the characteristics of the gyration error signal.
[0059] Therefore, compared with the prior art, the present invention shows significant advantages in terms of adaptability, noise removal effect and preservation of rotation error characteristics.
[0060] Based on the implementation methods provided in the above aspects, this application can be further combined to provide more implementation methods. Attached Figure Description
[0061] The above and other objects, features, and advantages of exemplary embodiments of the present invention will become readily apparent upon reading the following detailed description with reference to the accompanying drawings. In the drawings, several embodiments of the invention are illustrated by way of example and not limitation, with the same or corresponding reference numerals denoteing the same or corresponding parts, wherein:
[0062] Figure 1 This is a flowchart of the method of the present invention;
[0063] Figure 2 These are comparison images before and after noise reduction using the VMD-PE-MWSTD method;
[0064] Figure 3 This is the time-domain waveform diagram of the IMF components obtained from VMD decomposition;
[0065] Figure 4 This is the frequency domain waveform of the IMF component obtained from VMD decomposition;
[0066] Figure 5 This is a comparison chart of the noise reduction effects of the VMD-PE-MWSTD noise reduction method and the CEEMDAN method. Detailed Implementation
[0067] The exemplary embodiments disclosed in this application will now be described in more detail with reference to the accompanying drawings. While exemplary embodiments of this application are shown in the drawings, it should be understood that this application can be implemented in various forms and should not be limited to the embodiments set forth herein. Rather, these embodiments are provided to enable a more thorough understanding of this application and to fully convey the scope of this application to those skilled in the art. Unless otherwise specified, the technical means used in the embodiments are conventional means well known to those skilled in the art.
[0068] This invention proposes a noise reduction method based on variational mode decomposition (VMD) and permutation entropy (PE) analysis. By effectively removing noise components and retaining useful signals, it greatly improves the measurement accuracy of spindle rotation error. Specifically, this invention combines the signal decomposition capability of VMD and the noise identification advantage of PE with an improved wavelet soft threshold denoising (MWSTD) technique to effectively handle interference caused by environmental noise, mechanical errors, etc.
[0069] Compared with other noise reduction methods (such as CEEMDAN), the method of this invention shows superior noise reduction performance under noise of different intensities. In particular, under high noise backgrounds, the VMD-PE-MWSTD method can better separate noise and useful signal components in the gyratory signal, thereby ensuring the quality of the measurement signal. Experiments show that this method has significant advantages in terms of improving signal-to-noise ratio (SNR), reducing root mean square error (RMSE), and increasing correlation coefficient (CC).
[0070] The purpose of this invention is to overcome the shortcomings of the prior art and provide an efficient method for denoising rotational signals suitable for measuring the rotational error of machine tool spindles. This method targets the characteristics of rotational signals of machine tool spindles under high speed, non-stationarity and strong noise backgrounds, and combines variational mode decomposition (VMD), permutation entropy (PE) analysis and improved wavelet soft threshold denoising (MWSTD) method to optimize the signal decomposition and denoising effect.
[0071] like Figure 1 As shown, the noise reduction method for machine tool spindle rotation signals based on VMD-PE-MWSTD of the present invention includes the following steps:
[0072] S1: Use a laser displacement sensor to collect the rotation signal of the machine tool spindle, and set the signal sampling parameters and machine tool operating status.
[0073] S2: The preprocessed vibration signal is decomposed using the variational mode decomposition (VMD) method to obtain several intrinsic mode function (IMF) components and residual components.
[0074] S3: Perform permutation entropy (PE) analysis on each obtained IMF component to assess the disorder level of each IMF component.
[0075] S4: For the selected noise-dominant IMF, an improved wavelet soft threshold denoising (MWSTD) method is used for denoising.
[0076] S5: Reconstruct the denoised effective IMF component with the useful signal dominant IMF obtained from the screening to obtain the denoised cyclic signal.
[0077] Optionally, the specific steps of S2 are as follows:
[0078] For the decomposed modal components The Hilbert transform is expressed by the following formula:
[0079]
[0080] In the formula It is the unit impulse function, j is the imaginary unit, and * represents convolution. It is the first One modal component to be decomposed, It is a time variable.
[0081] Introduction The modal frequencies are modulated onto the fundamental frequency band, as expressed by the following formula:
[0082]
[0083] In the formula It is the unit impulse function, j is the imaginary unit, and * represents convolution. It is the first One modal component to be decomposed, It is the time derivative. It is a time variable. It is a complex exponential modulation term, used to modulate the first... The spectrum of each modal component is shifted to the fundamental frequency band.
[0084] According to the square norm The sum of the bandwidths of the modal variables is estimated by the following formula:
[0085]
[0086] In the formula These are constraints, indicating that the sum of all decomposed modal components should equal the original signal. , It is the first The center frequency of each modal component It is a unit impulse function. It is a time variable, j is the imaginary unit, and * represents convolution. It is a compound exponent.
[0087] By introducing the Lagrange operator, we obtain the optimal solution in the equation, resulting in the augmented Lagrange expression as follows:
[0088] In the formula It is the first One modal component to be decomposed, It is the first The center frequency of each modal component As a penalty factor, It is the time derivative. It is a unit impulse function. It is the original signal. It is a time variable, j is the imaginary unit, and * represents convolution. It is a compound exponent. For the Lagrange penalty operator.
[0089] about The minimum value problem is expressed by the following formula:
[0090]
[0091] In the formula It is the first The modal component in the ... The update result at the next iteration It is the first The center frequency of each modal component As a penalty factor, It is the time derivative. It is the original signal, and j is the imaginary unit. These are the modal components to be decomposed, and * represents convolution. It is a compound exponent. For the Lagrange penalty operator.
[0092] Converted to the frequency domain, the formula is as follows:
[0093]
[0094] In the formula It is the first The mode in the th ... Frequency domain representation at the next iteration It is the first The center frequency of each modal component The penalty factor is j, where j is the imaginary unit. It is the modal spectrum. It is the original input signal Fourier transform, It is an angular frequency variable. It is a Lagrange multiplier The frequency domain representation of .
[0095] Replace with Afterwards, the solution was obtained. The formula is expressed as follows:
[0096]
[0097] In the formula For when Wiener filtering, It is the first The center frequency of each modal component The penalty factor is j, where j is the imaginary unit. It is the modal spectrum. It is an angular frequency variable. It is a Lagrange multiplier The frequency domain representation of .
[0098] Similarly, the center frequency can be obtained. and Lagrange operators The formula is expressed as follows:
[0099]
[0100] In the formula It is the first The mode in the th ... The center angular frequency at the next iteration It is the first Frequency domain Lagrange multipliers at the next iteration It is the angular frequency variable in the frequency domain. It is the first Modal components Fourier transform, It is the first The mode in the th ... Frequency domain representation at the next iteration To ensure fidelity, It is the original input signal Fourier transform.
[0101] The frequency center and bandwidth of the IMF component separated from the original signal satisfy the iterative condition, and the iteration stops. The iterative formula is expressed as follows:
[0102]
[0103] In the formula It is the first The first modal component Frequency domain representation of the next iteration It is the first The first modal component Frequency domain representation of the next iteration It is a frequency variable. It is the convergence threshold.
[0104] right The mode function is obtained by inverse transformation.
[0105] Optionally, the specific steps in S3 are as follows:
[0106] Permutation entropy (PE) analysis was performed on each obtained intrinsic mode function (IMF) component to assess the disorder level of each IMF component. The larger the PE value, the higher the disorder level and the more noise components the IMF contains. Based on a preset PE threshold of 0.45, IMFs with lower PE values (PE < 0.45) were selected, indicating that the IMF mainly contains useful signal components. IMFs with higher PE values (PE > 0.45) were identified as noise-dominated IMFs.
[0107] The normalized permutation entropy threshold is set to 0.45. This is because permutation entropy measures the randomness and complexity of a time series; a larger value indicates a more disordered sequence, closer to random noise, while a smaller value indicates a more ordered sequence, containing more deterministic structural information. The effective components in the spindle rotation error signal typically exhibit periodicity, harmonicity, or slowly varying characteristics, corresponding to lower permutation entropy in the IMF (Integrated Motion Filter). Noise-dominated IMFs, on the other hand, have permutation patterns closer to random distributions, resulting in higher permutation entropy. Therefore, using 0.45 as the threshold effectively distinguishes between IMFs with strong ordered structures and those dominated by randomness, thus providing a basis for subsequent wavelet thresholding denoising and signal reconstruction.
[0108] Given discrete time series
[0109]
[0110] Construct the embedding vector:
[0111]
[0112] In the formula, t=1,2,…,N-(m-1)τ, m is the number of data points in the subsequence, and τ is the time step between adjacent data points when constructing the embedding vector.
[0113] Count the occurrences of all embedded vector patterns The probability of obtaining each permutation pattern is:
[0114]
[0115] In the formula, N is the length of the time series. τ is the number of times the i-th sorting pattern appears, m is the number of data points in the subsequence, and τ is the time step between adjacent data points when constructing the embedding vector.
[0116] Calculate permutation entropy (PE)
[0117]
[0118] In the formula It is the first The probability of a sorting pattern.
[0119] Optionally, the specific steps in S4 are as follows:
[0120] Soft threshold wavelet denoising
[0121]
[0122] In the formula These are wavelet coefficients after soft thresholding. This represents the translation position index in this layer, where... It is the first The probability of a sorting pattern. Represents the original wavelet decomposition coefficients. Indicates the decomposition scale. It is a threshold.
[0123] Improved soft-threshold wavelet denoising
[0124]
[0125] These are the coefficients after processing with the improved threshold function. Represents the original wavelet decomposition coefficients. For threshold adjustment parameters, It is a threshold.
[0126] An improved wavelet soft-thresholding denoising method (MWSTD) utilizes the multi-scale time-frequency local analysis capability of wavelet transform to separate the effective structural components from high-frequency random noise in the spindle rotation error signal. Cutting noise, environmental noise, and measurement noise typically manifest as high-frequency, random, and locally discontinuous wavelet detail coefficients, while the periodic terms, harmonic terms, eccentricity errors, and slowly varying modulation components in the spindle rotation error appear as structural coefficients with certain amplitudes and correlations in the wavelet domain. By applying an improved soft-thresholding function to the high-frequency detail coefficients, small-amplitude random noise coefficients can be strongly suppressed, while large-amplitude effective feature coefficients are smoothly contracted, thus avoiding the discontinuity distortion caused by traditional hard thresholding and the excessive shift of large coefficients by traditional soft thresholding. Combined with prior VMD decomposition and PE screening, MWSTD processing is applied only to the noise-dominant IMF component, further improving the targeting of noise suppression. This effectively removes high-frequency random interference such as cutting noise and environmental noise while preserving the true feature information in the spindle rotation error to the maximum extent.
[0127] Optionally, the specific steps in S5 are as follows:
[0128] The useful signal obtained from the component filtering module is used to reconstruct the dominant IMF and the useful components obtained from the component denoising module to obtain the denoised signal.
[0129] The reconstruction formula is as follows:
[0130]
[0131] In the formula These are the coefficients after processing with the improved threshold function. It is the reconstructed signal after denoising. It is the first Low-frequency coefficient of layer, It is a scaling function. It is the first The detail factor after layer processing It is the basis function after the wavelet function has been translated and scaled.
[0132] The purpose of signal reconstruction is to recombine the intrinsic mode components (IMFs) after VMD decomposition, PE discrimination, and improved wavelet soft thresholding denoising to restore a complete time-domain signal capable of characterizing the actual spindle rotation state. Since the original spindle rotation error signal is represented as multiple IMF components in different frequency bands after decomposition, and the results obtained after permutation entropy filtering and wavelet thresholding are still at the component level, without reconstruction, a holistic denoised signal that can be directly used for error assessment and state analysis cannot be formed. Therefore, reconstruction can unify the retained effective mode components with the modal components to restore a complete signal. This effectively suppresses cutting noise, environmental noise, and measurement random noise while maximizing the preservation of periodic terms, harmonic terms, eccentricity error terms, and slowly varying structural information in the spindle rotation error. This improves the signal-to-noise ratio, temporal continuity, and physical interpretability, providing a reliable basis for subsequent quantitative evaluation and engineering applications of the spindle rotation error.
[0133] In a specific example of the present invention, Figure 2 This is a comparison chart of the spindle rotation error signal before and after noise reduction using the VMD-PE-MWSTD method. By comparing the signal waveforms before and after noise reduction, it is clear that the signal after noise reduction using the VMD-PE-MWSTD method has significantly smoother characteristics, removing higher-frequency noise while maintaining the periodic structure of the signal and the main features of the spindle rotation error. The noise-reduced signal not only removes random noise components but also retains the effective characteristics of the rotation error, making the signal more stable and suitable for subsequent analysis and machining quality evaluation. Figure 3 This is the waveform of each IMF component in the time domain after decomposing the spindle rotation error signal using VMD. Each IMF component represents a part of the signal within a different frequency range. Lower-frequency IMF components reflect the long-term trend of the spindle rotation error, while higher-frequency IMF components mainly correspond to high-frequency disturbances and noise in the signal. These components provide a foundation for subsequent noise analysis, feature extraction, and error analysis. Figure 4This figure shows the frequency domain waveforms of each IMF component obtained after decomposing the spindle rotation error signal using VMD. Frequency domain analysis of each IMF component clearly reveals the energy distribution across different frequency bands. This figure demonstrates how the VMD method efficiently decomposes the signal in the frequency domain, showcasing the important frequency components. The frequency domain waveforms provide further insight into the signal's spectral characteristics, allow for the assessment of noise components in different frequency bands, and provide a basis for subsequent noise reduction processing. Figure 5 This comparison examines the noise reduction performance of the VMD-PE-MWSTD and CEEMDAN methods. By comparing the signal waveforms before and after noise reduction, it's clear that the VMD-PE-MWSTD method has a significant advantage in noise reduction. Specifically, the VMD-PE-MWSTD method is more effective at removing high-frequency noise while preserving the main characteristics of the signal, especially in removing high-frequency random noise while maintaining the main periodic characteristics and structure of the hysteresis error. While the CEEMDAN method can also reduce noise, its noise reduction effect is less pronounced than that of the VMD-PE-MWSTD method in certain frequency bands.
[0134] Table 1 shows a comparison of the noise reduction effects of the method of the present invention and the CEEMDAN method.
[0135] Table 1 Comparison of Noise Reduction Effects of Different Methods
[0136]
[0137] The comparison results in Table 1 show that the VMD-PE-MWSTD method outperforms the CEEMDAN method in overall noise reduction. On the one hand, the signal-to-noise ratio improvement index of VMD-PE-MWSTD is... The mean square error (MSE) of VMD-PE-MWSTD is 15.1426, significantly higher than CEEMDAN's 7.2856, indicating that this method can more effectively suppress noise and increase the proportion of effective signal components. On the other hand, the MSE of VMD-PE-MWSTD is significantly higher than CEEMDAN's 7.2856. The value is 0.2425, significantly lower than CEEMDAN's 0.5435, indicating that the method introduces less distortion during noise reduction and preserves the original effective features more completely. In summary, the VMD-PE-MWSTD method proposed in this invention outperforms the CEEMDAN method in both noise suppression and feature fidelity, and can more accurately extract effective information from the spindle rotation error signal. From the above embodiments and comparative experimental results, it can be seen that the machine tool spindle rotation error noise reduction method proposed in this invention, under the same experimental conditions, can significantly improve the signal-to-noise ratio and effectively reduce the root mean square error compared to other methods, exhibiting superior noise reduction effect and signal reconstruction accuracy.
[0138] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.
Claims
1. A method for noise reduction of machine tool spindle rotation signals based on VMD-PE-MWSTD, characterized in that, Specifically, the steps include the following: Step S1: Acquire the rotation signal of the machine tool spindle and preprocess the rotation signal; Step S2: The preprocessed gyratory signal is decomposed using the variational mode decomposition (VMD) method to obtain several intrinsic mode function (IMF) components and residual components. Step S3: Perform permutation entropy (PE) analysis on the obtained intrinsic mode function (IMF) components, and divide the IMF components into useful signal-dominated IMFs and noise-dominated IMFs according to the preset permutation entropy threshold; Step S4: For the selected noise-dominant intrinsic mode functions (IMFs), the improved wavelet soft thresholding denoising method (MWSTD) is used for denoising to obtain the denoised IMF components. Step S5: Reconstruct the obtained useful signal dominant intrinsic mode function (IMF) and the obtained denoised IMF components to obtain the denoised machine tool spindle rotation signal.
2. The method for noise reduction of machine tool spindle rotation signals based on VMD-PE-MWSTD according to claim 1, characterized in that, The variational mode decomposition (VMD) method described in step S2 includes: For each modal component to be decomposed The Hilbert transform is expressed by the following formula: In the formula It is the unit impulse function, j is the imaginary unit, and * represents convolution. It is the first One modal component to be decomposed, It is a time variable; Introduction The modal frequencies are modulated onto the fundamental frequency band, as expressed by the following formula: In the formula, It is the unit impulse function, j is the imaginary unit, and * represents convolution. It is the first One modal component to be decomposed, It is the time derivative. It is a time variable. It is a complex exponential modulation term, used to modulate the first... The spectrum of each modal component is shifted to the fundamental frequency band; According to the square norm The sum of the bandwidths of the modal variables is estimated by the following formula: In the formula These are constraints, indicating that the sum of all decomposed modal components should equal the original signal. , It is the first The center frequency of each modal component It is a unit impulse function. It is a time variable, j is the imaginary unit, and * represents convolution. It is a complex exponent; By introducing the Lagrange operator, we obtain the optimal solution in the equation, resulting in the augmented Lagrange expression as follows: In the formula, It is the first One modal component to be decomposed, It is the first The center frequency of each modal component As a penalty factor, It is the time derivative. It is a unit impulse function. It is the original signal. It is a time variable, j is the imaginary unit, and * represents convolution. It is a compound exponent. Lagrange's punishment operators; The optimal solution of the constrained variational mode is obtained by iterating using the alternating direction multiplier method. The frequency center and bandwidth of the IMF component separated from the original signal satisfy the iterative condition, and the iteration stops. The iterative formula is expressed as follows: In the formula It is the first The first modal component Frequency domain representation of the next iteration It is the first The first modal component Frequency domain representation of the next iteration It is a frequency variable. It is the convergence threshold; right The mode function is obtained by inverse transformation.
3. The method for noise reduction of machine tool spindle rotation signals based on VMD-PE-MWSTD according to claim 1, characterized in that, The permutation entropy (PE) analysis process in step S3 is as follows: Permutation entropy (PE) analysis was performed on each obtained intrinsic mode function (IMF) component to assess the disorder level of each IMF component. The larger the PE value, the higher the disorder level and the more noise components the IMF contains. Based on a preset PE threshold of 0.45, IMFs with lower PE values (PE < 0.45) were selected, indicating that the IMF mainly contains useful signal components. IMFs with higher PE values (PE > 0.45) were identified as noise-dominated IMFs. Given discrete time series Construct the embedding vector: In the formula, t=1,2,…,N-(m-1)τ, m is the number of data points in the subsequence, and τ is the time step between adjacent data points when constructing the embedding vector; Count the occurrences of all embedded vector patterns The probability of obtaining each permutation pattern is: In the formula, N is the length of the time series. is the number of times the i-th sorting pattern appears, m is the number of data points in the subsequence, and τ is the time step between adjacent data points when constructing the embedding vector; Calculate the permutation entropy PE: In the formula, It is the first The probability of a sorting pattern.
4. A method for noise reduction of machine tool spindle rotation signals based on VMD-PE-MWSTD according to claim 1, characterized in that, The improved soft-threshold wavelet denoising process described in step S4 is as follows: Soft threshold wavelet denoising: In the formula, These are wavelet coefficients after soft thresholding. This indicates the translation position index within this layer. Represents the original wavelet decomposition coefficients. Indicates the decomposition scale. It is a threshold; Improved soft-threshold wavelet denoising: These are the coefficients after processing with the improved threshold function. Represents the original wavelet decomposition coefficients. For threshold adjustment parameters, It is a threshold.
5. A method for noise reduction of machine tool spindle rotation signals based on VMD-PE-MWSTD according to claim 1, characterized in that, The signal reconstruction process described in step S5 is as follows: The reconstruction formula is as follows: in, These are the coefficients after processing with the improved threshold function. It is the reconstructed signal after denoising. It is the first Low-frequency coefficient of layer, It is a scaling function. It is the first The detail factor after layer processing It is the basis function after the wavelet function has been translated and scaled.