A method for obtaining a process frequency response function of a milling tool shaft system considering the influence of measurement noise

Abstract

This invention specifically relates to a method for obtaining the process frequency response function of a milling cutter spindle system considering the influence of measurement noise. The method includes: first, based on the assumption that the mode shape is independent of the cutting state, a modal impact experiment is performed on the milling cutter tip to obtain the frequency response function and mode shape of the milling cutter tip in a static spindle state; second, cutting experiments are conducted at different speeds, and an accelerometer mounted on the spindle head is used to measure the noisy response signal during the cutting process. Utilizing the statistical independence of the noise-free response signal and the measurement noise, a covariance matrix consisting only of the noise-free response signal is separated from the noisy response signal, and the unbiased transitivity is calculated from the covariance matrix; then, the cutting process poles are obtained based on TOMA with mode shape compensation; finally, the milling process frequency response function is obtained based on the static mode shape and the process poles.

Description

Technical Field

[0001] This invention relates to the field of milling technology, specifically to a method for predicting the process frequency response function of a milling cutter spindle system, and particularly to a method for obtaining the process frequency response function of a milling cutter spindle system considering the influence of measurement noise. Background Technology

[0002] The TOMA (Transmissibility Function-Based Operational Modal Analysis) method, based on the characteristics of transmissibility, can suppress harmonic components in the excitation and extract the poles of the process frequency response function. However, in practical applications, due to the presence of measurement noise, the transmissibility estimation results have systematic biases, which in turn leads to a decrease in the accuracy of pole identification. Therefore, in order to improve the accuracy of process frequency response function identification, it is necessary to consider the impact of measurement noise on transmissibility estimation.

[0003] Reference 1, “Y. Liu, Y. Altintas, In-process identification of machine tool dynamics, CIRP Journal of Manufacturing Science and Technology 32 (2021) 322-337”, discloses a method for obtaining the frequency response function of the milling process based on mode shape compensation using TOMA. This method uses acceleration signals from multiple locations on the spindle head during machining. The transmissibility was estimated. The results showed that the transmissibility still contained a large amount of noise, requiring the use of a filter. However, the article did not provide a criterion for selecting the filter cutoff order, which led to a decrease in modality recognition accuracy.

[0004] Reference 2, "Mao Z, Todd M. A model for quantifying uncertainty in the estimation of noise-contaminated measurements of transmissibility, Mechanical Systems and Signal Processing 28 (2012) 470-481," discloses a method for quantifying transmissibility uncertainty in noisy environments. This method assumes the noise is white noise and calculates the uncertainty based on the statistical characteristics of the noise. The paper presents the confidence intervals for the transmission rate obtained under the class estimation. However, it does not provide a solution for eliminating the influence of noise.

[0005] The typical characteristics of the above literature are: when estimating the transmissibility, the impact of noise on the transmissibility is not specifically analyzed, and the obtained transmissibility and process frequency response function have low accuracy. Summary of the Invention

[0006] To address the issue that existing TOMA-type methods do not adequately consider the impact of measurement noise, resulting in low modal recognition accuracy, this invention proposes a method for obtaining a high-precision process frequency response function of a milling cutter axis system, which is based on an unbiased estimation of the transmissivity under noisy conditions.

[0007] Other features and advantages of the invention will become apparent from the following detailed description, or may be learned in part by practice of the invention.

[0008] According to a first aspect of the present invention, a method for obtaining the process frequency response function of a milling cutter spindle system considering the influence of measurement noise is provided, the method comprising: Based on the principle of modal superposition, the frequency response function of the tool shaft system is expressed as a function of modal constants and process poles; Modal impact tests were conducted on the tip of the milling cutter to obtain the frequency response function of the tip in the static state of the spindle, and the static mode shape and static mode constant were extracted. Cutting experiments were conducted at different speeds, and the noisy response signal was measured using an accelerometer mounted on the spindle head. Based on the statistical independence of the noiseless response signal and the measurement noise, the covariance matrix consisting only of the noiseless response signal is separated from the noisy response signal, and the unbiased transmissibility is calculated accordingly. The operational modal analysis method based on modal shape compensation extracts the cutting process poles from the unbiased transmissivity; The process frequency response function of the milling cutter axis system is reconstructed using the static mode constant and the cutting process poles.

[0009] In some exemplary embodiments, the specific method for calculating the unbiased transitivity includes: The measurement noise is explicitly modeled, and the noisy response is represented as the sum of the noise-free response and additive white noise; Construct the relationship between the covariance matrix of the noisy response and the noise covariance matrix; Construct solution set constraints using the nonnegative definite property of the noiseless response covariance matrix; Higher-order Yule-Walker equations are introduced as additional constraints. Unbiased estimates of the coefficients of the transitivity polynomial are obtained by solving an optimization problem.

[0010] In some exemplary embodiments, the relationship between the covariance matrix of the constructed noisy response and the noise covariance matrix is ​​specifically as follows:

[0011] in, Let covariance be the noisy response. The noise covariance matrix is... Let be the coefficient vector to be estimated.

[0012] In some exemplary embodiments, the construction of solution set constraints using the non-negative definite property of the noiseless response covariance matrix specifically involves:

[0013] in, It is by The determined surface.

[0014] In some exemplary embodiments, the higher-order Yule-Walker equation is specifically:

[0015] in, It is a high-order cross-covariance matrix. This is the lag order.

[0016] In some exemplary embodiments, the optimization problem is expressed as:

[0017]

[0018] in, This is the cost function for estimating the parameter vector.

[0019] In some exemplary embodiments, the specific method for extracting the poles of the cutting process includes: Calculate the transmissibility after mode shape compensation based on the relationship between unbiased transmissibility and mode shape. Construct a modal indication function based on the transmissibility after modal mode compensation; The Polymax method is used to identify the poles of the cutting process from the modal indicator function.

[0020] In some exemplary embodiments, the calculation of the transmissibility after modal compensation specifically includes:

[0021]

[0022] in, and For the estimated polynomial coefficients, and These are the polynomial orders, z It is a discrete variable. l It is a constant. and Representing the first and second reference measuring points respectively. l First-order mode shape.

[0023] In some exemplary embodiments, the construction of the modal indication function based on the transmissibility after modal mode compensation specifically involves:

[0024] in, This is a modal indicator function.

[0025] In some exemplary embodiments, the process frequency response function of the milling cutter axis system is reconstructed using the static mode constant and the cutting process poles, specifically as follows:

[0026] in, The extreme point of the cutting process. is the static mode constant.

[0027] The method for obtaining the process frequency response function of a milling cutter spindle system considering the influence of measurement noise, provided by embodiments of the present invention, firstly explicitly models the measurement noise, considering its impact on transmissibility estimation. Then, based on the statistical independence of the noise-free response signal and the measurement noise, it separates the signal containing system modal information and calculates the transmissibility. Finally, it extracts the system poles and reconstructs the process frequency response function. Compared to Reference 1, it eliminates the need for filters and determining truncation accuracy; compared to Reference 2, it obtains an accurate transmissibility, rather than a confidence interval for the transmissibility. The present invention will be described in detail below with reference to the accompanying drawings and embodiments.

[0028] It should be understood that the above general description and the following detailed description are exemplary and explanatory only, and are not intended to limit the invention. Attached Figure Description

[0029] The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments consistent with the invention and, together with the description, serve to explain the principles of the invention. It is obvious that the drawings described below are merely some embodiments of the invention, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort.

[0030] Figure 1 This is a schematic diagram of the method flow of the present invention; Figure 2 It is the amplitude of the tool tip frequency response function under the static state of the spindle obtained by this invention.

[0031] Figure 3 This is a comparison of the modal indication functions calculated using a non-parametric method and the method proposed in this paper for the acceleration signal of the spindle speed of 3700 rpm, axial cutting depth of 0.6 mm, and feed per tooth of 0.1 mm in this embodiment of the invention.

[0032] Figure 4 It is the amplitude of the process frequency response function of the micro-milling cutter when the spindle speed is 3700 rpm and 6000 rpm, the axial cutting depth is 0.6 mm and 1.6 mm, and the feed per tooth is 0.1. Detailed Implementation

[0033] Exemplary embodiments will now be described more fully with reference to the accompanying drawings. However, these exemplary embodiments can be implemented in many forms and should not be construed as limited to the examples set forth herein; rather, they are provided so that the invention will be more comprehensive and complete, and will fully convey the concept of the exemplary embodiments to those skilled in the art. The described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments.

[0034] Furthermore, the accompanying drawings are merely illustrative of the invention and are not necessarily drawn to scale. The same reference numerals in the drawings denote the same or similar parts, and therefore repeated descriptions of them will be omitted. Some block diagrams shown in the drawings are functional entities and do not necessarily correspond to physically or logically independent entities. These functional entities can be implemented in software, in one or more hardware modules or integrated circuits, or in different network and / or processor devices and / or microcontroller devices.

[0035] To address the shortcomings and deficiencies of existing technologies, this exemplary embodiment provides a method for obtaining the process frequency response function of a milling cutter spindle system, considering the influence of measurement noise. First, based on the assumption that the mode shape is independent of the cutting state, a modal impact experiment is performed on the milling cutter tip to obtain the frequency response function and mode shape of the milling cutter tip in a stationary spindle state. Second, cutting experiments are conducted at different speeds, and an accelerometer mounted on the spindle head is used to measure the noisy response signal during the cutting process. Utilizing the statistical independence of the noise-free response signal and the measurement noise, a covariance matrix consisting only of the noise-free response signal is separated from the noisy response signal, and the unbiased transitivity is calculated from the covariance matrix. Then, the cutting process poles are obtained based on TOMA with mode shape compensation. Finally, the milling process frequency response function is obtained based on the stationary mode shape and the process poles.

[0036] Expected technical effect: The method for calculating the process frequency response function of the milling cutter axis system provided by this invention can accurately obtain the frequency response function during the milling process in a noisy environment.

[0037] refer to Figure 1 As shown, the specific steps may include: Based on the principle of modal superposition, the frequency response function of the tool shaft system is expressed as a function of modal constants and process poles; Modal impact tests were conducted on the tip of the milling cutter to obtain the frequency response function of the tip in the static state of the spindle, and the static mode shape and static mode constant were extracted. Cutting experiments were conducted at different speeds, and the noisy response signal was measured using an accelerometer mounted on the spindle head. Based on the statistical independence of the noiseless response signal and the measurement noise, the covariance matrix consisting only of the noiseless response signal is separated from the noisy response signal, and the unbiased transmissibility is calculated accordingly. The operational modal analysis method based on modal shape compensation extracts the cutting process poles from the unbiased transmissivity; The process frequency response function of the milling cutter axis system is reconstructed using the static mode constant and the cutting process poles.

[0038] The steps in this exemplary embodiment will now be described in more detail with reference to the accompanying drawings and examples.

[0039] Step 1: According to the principle of modal superposition, the frequency response function of the tool shaft system can be expressed as:

[0040] In the formula, It is the process frequency response function. These are process modal constants, which often depend on the mode shape. It is the extreme point of the process. l It is a constant. j It is the imaginary unit. n The system order is denoted by .

[0041] Step 2: Conduct modal impact tests to obtain the mode shapes and modal constants of the principal shaft in its static state, and let:

[0042] In the formula, and These are the process and static mode shapes, respectively. is the static mode constant.

[0043] Step 3: Pole finding uses a transitivity-based operational modal analysis method. Transitivity is defined as the ratio of the responses at different measurement points of the system, and can be expressed in the z-domain as:

[0044] In the formula, This is the noise-free non-reference acceleration response, and it is the numerator term in the transmissivity. The noise-free reference acceleration response is the denominator term in the transmissivity. and These are the polynomial orders, and is the polynomial coefficient, and z is the discrete variable. Step 4: Considering the truncation error, the transitivity in the time domain can be expressed as:

[0045] In the formula, To compensate for process noise such as truncation accuracy, it is modeled as white noise and is incoherent with the response; This is the sampling sequence number.

[0046] Step 5: Considering measurement noise, the transmissibility is expressed as:

[0047] In the formula, and These are the noisy non-reference acceleration response and the reference acceleration response, respectively. and The measurement noise in the non-reference acceleration response and the reference acceleration response are respectively modeled as white noise and are independent of the response.

[0048] Step 6: The transmission relationship between response signals is written in matrix form:

[0049] In the formula, The covariance matrix of the noise-free response can be expressed as the difference between the covariance matrix of the noisy response and the covariance matrix of the noise. Let be the coefficient vector to be estimated. According to the Frisch scheme, the transitivity can be expressed in matrix form as:

[0050] In the formula, Let covariance be the noisy response. The noise covariance matrix is ​​expressed as follows:

[0051] In the formula, E In order to achieve expectations, T For transpose, N For data length, , and These are the variances of the non-reference response measurement noise, the reference response measurement noise, and the process noise, respectively.

[0052] Step 7: For the covariance matrix of the noiseless response If it satisfies the non-negative definite property, then the solution set satisfying the condition can be obtained:

[0053] In the formula, It is by The determined surface, where the coefficients correspond to a solution set, is used to obtain a unique solution. Higher-order Yule-Walker equations are introduced as additional constraints.

[0054] Step 8: Calculate the higher-order Yule-Walker equations for the response data:

[0055] In the formula, The lag order is... It is a high-order cross-covariance matrix, calculated using noisy response data, and referenced from " New estimation method for periodicautoregressive time series of order 1 with additive noise, International Journal of Advances in Engineering Sciences and Applied Mathematics 13(2021)163–176" public method acquisition.

[0056] Step 9: Calculate the cost function with respect to the estimated parameter vector:

[0057] In the formula, This represents finding the 2-norm.

[0058] Step 10: Combining Steps 7 and 9, express it as an optimization problem:

[0059] Step 11: Solve the optimization problem in Step 10 using the interior-point method to obtain the estimated polynomial coefficients. and .

[0060] Step 12: Calculate the transfer rate based on Step 3:

[0061] Step 13: Calculate the transmissibility of modal compensation:

[0062] In the formula, and Representing the first and second reference measuring points respectively. l First-order mode shape.

[0063] Step Fourteen: Calculate the modal indication function:

[0064] The modal indication function and process frequency response function obtained in steps 15 and 14 have the same poles. The Polymax method is used for process pole identification.

[0065] Step 16: Substitute the pole information obtained in Step 15 and the modal constants obtained in Step 2 into Step 1 to obtain the process frequency response function of the milling cutter axis system.

[0066]

[0067] Example 1: Experimental selection of tooth number Tooth, radius A flat-bottomed carbide end mill with a length of 50 mm was subjected to modal impact testing on a Hardinge machine tool. The tool density was 14,500 kg / m², and the Young's modulus was 580,000 MPa. The cutting parameters used were radial depth of cut. millimeters, feed per tooth The spindle speeds are 3700 rpm and 6000 rpm, respectively, and the axial cutting depths are 0.6 mm and 1.6 mm, respectively.

[0068] Step 1: According to the principle of modal superposition, the frequency response function of the tool shaft system can be expressed as:

[0069] In the formula, It is the process frequency response function. The mode constants of the first-order process depend on the mode array. It is the pole of the first-order process. r It is the first order. j It is the imaginary unit.

[0070] Step 2: Conduct modal impact experiments to obtain the frequency response function of the knife tip at rest, such as... Figure 2As shown. Modal analysis is used to calculate the mode shapes and modal constants of the principal shaft in its static state, and let:

[0071] In the formula, and These are the process and static first-order mode shapes, respectively. is the static mode constant.

[0072] Step 3: Pole finding uses a transitivity-based operational modal analysis method. Transitivity is defined as the ratio of the responses at different measurement points of the system, and can be expressed in the z-domain as:

[0073] In the formula, and These are the noiseless non-reference acceleration response and the reference acceleration response, respectively. and These are the polynomial orders, both of which are taken as 6 in this example; and is the polynomial coefficient, and z is the discrete variable. Step 4: Considering the truncation error, the transitivity in the time domain can be expressed as:

[0074] In the formula, To compensate for process noise such as truncation accuracy, it is modeled as white noise and is incoherent with the response; This is the sampling sequence number.

[0075] Step 5: Considering measurement noise, the transmissibility is expressed as:

[0076] In the formula, and These are the noisy non-reference acceleration response and the reference acceleration response, respectively. and The measurement noise in the non-reference acceleration response and the reference acceleration response are respectively modeled as white noise and are independent of the response.

[0077] Step 6: The transmission relationship between response signals is written in matrix form:

[0078] In the formula, The covariance matrix of the noise-free response can be expressed as the difference between the covariance matrix of the noisy response and the covariance matrix of the noise. Let be the coefficient vector to be estimated. According to the Frisch scheme, the transitivity can be expressed in matrix form as:

[0079] In the formula, Let covariance be the noisy response. The noise covariance matrix is ​​expressed as follows:

[0080] In the formula, E represents the expected value, T represents the transpose, and N represents the data length. , and These are the variances of the non-reference response measurement noise, the reference response measurement noise, and the process noise, respectively.

[0081] Step 7: For the covariance matrix of the noiseless response If it satisfies the non-negative definite property, then the solution set satisfying the condition can be obtained:

[0082] In the formula, It is by The determined surface, where the coefficients correspond to a solution set, is used to obtain a unique solution. Higher-order Yule-Walker equations are introduced as additional constraints.

[0083] Step 8: Calculate the higher-order Yule-Walker equations for the response data:

[0084] In the formula, The lag order is set to 12; It is a high-order cross-covariance matrix, calculated using noisy response data, and referenced from " New estimation method for periodicautoregressive time series of order 1 with additive noise, International Journal of Advances in Engineering Sciences and Applied Mathematics 13(2021)163–176" public method acquisition.

[0085] Step 9: Calculate the cost function with respect to the estimated parameter vector:

[0086] In the formula, This represents finding the 2-norm.

[0087] Step 10: Combining Steps 7 and 9, express it as an optimization problem:

[0088] Step 11: Solve the optimization problem in Step 10 using the interior-point method to obtain the estimated polynomial coefficients. and .

[0089] Step 12: Calculate the transfer rate based on Step 3:

[0090] Step 13: Calculate the transmissibility of modal compensation:

[0091] In the formula, and These represent the first-order mode shapes at the reference measuring point and the non-reference measuring point, respectively.

[0092] Step 14: Calculate the modal indication function, the result is as follows Figure 3 As shown.

[0093]

[0094] The modal indication function and process frequency response function obtained in steps 15 and 14 have the same poles. The Polymax method is used for process pole identification.

[0095] Step 16: Substitute the pole information obtained in Step 15 and the modal constants obtained in Step 2 into Step 1 to obtain the process frequency response function of the milling cutter axis system. The result is as follows: Figure 4 As shown.

[0096]

[0097] Furthermore, the above figures are merely illustrative of the processes included in the method according to exemplary embodiments of the present invention, and are not intended to be limiting. It is readily understood that the processes shown in the above figures do not indicate or limit the temporal order of these processes. Additionally, it is readily understood that these processes may be executed synchronously or asynchronously, for example, in multiple modules.

[0098] Other embodiments of the invention will readily occur to those skilled in the art upon consideration of the specification and practice of the invention herein. This application is intended to cover any variations, uses, or adaptations of the invention that follow the general principles of the invention and include common knowledge or customary techniques in the art not disclosed herein. The specification and embodiments are to be considered exemplary only, and the true scope and spirit of the invention are indicated by the claims.

[0099] It should be understood that the present invention is not limited to the precise structure described above and shown in the accompanying drawings, and various modifications and changes can be made without departing from its scope. The scope of the invention is defined only by the appended claims.

Claims

1. A method for obtaining the process frequency response function of a milling cutter spindle system considering the influence of measurement noise, characterized in that, The method includes: Based on the principle of modal superposition, the frequency response function of the tool shaft system is expressed as a function of modal constants and process poles; Modal impact tests were conducted on the tip of the milling cutter to obtain the frequency response function of the tip in the static state of the spindle, and the static mode shape and static mode constant were extracted. Cutting experiments were conducted at different speeds, and the noisy response signal was measured using an accelerometer mounted on the spindle head. Based on the statistical independence of the noiseless response signal and the measurement noise, the covariance matrix consisting only of the noiseless response signal is separated from the noisy response signal, and the unbiased transmissibility is calculated accordingly. The operational modal analysis method based on modal shape compensation extracts the cutting process poles from the unbiased transmissivity; The process frequency response function of the milling cutter axis system is reconstructed using the static mode constant and the cutting process poles.

2. The method according to claim 1, characterized in that, The specific method for calculating the unbiased transitivity includes: The measurement noise is explicitly modeled, and the noisy response is represented as the sum of the noise-free response and additive white noise; Construct the relationship between the covariance matrix of the noisy response and the noise covariance matrix; Construct solution set constraints using the nonnegative definite property of the noiseless response covariance matrix; Higher-order Yule-Walker equations are introduced as additional constraints. Unbiased estimates of the coefficients of the transitivity polynomial are obtained by solving an optimization problem.

3. The method according to claim 2, characterized in that, The relationship between the covariance matrix of the constructed noisy response and the noise covariance matrix is ​​as follows: in, Let covariance matrix be the noisy response. The noise covariance matrix is... Let be the coefficient vector to be estimated.

4. The method according to claim 3, characterized in that, The method of constructing solution set constraints using the non-negative definite property of the noiseless response covariance matrix is ​​as follows: in, It is by The determined surface.

5. The method according to claim 4, characterized in that, The higher-order Yule-Walker equations are specifically as follows: in, It is a high-order cross-covariance matrix. This is the lag order.

6. The method according to claim 5, characterized in that, The optimization problem is expressed as: in, This is the cost function for estimating the parameter vector.

7. The method according to claim 1, characterized in that, The specific method for extracting the poles during the cutting process includes: Calculate the transmissibility after mode shape compensation based on the relationship between unbiased transmissibility and mode shape. Construct a modal indication function based on the transmissibility after modal mode compensation; The Polymax method is used to identify the poles of the cutting process from the modal indicator function.

8. The method according to claim 7, characterized in that, The calculation of the transmissibility after modal compensation is specifically as follows: in, and For the estimated polynomial coefficients, and These are the polynomial orders, z It is a discrete variable. l It is a constant. and Representing the first and second reference measuring points respectively. l First-order mode shape.

9. The method according to claim 8, characterized in that, The modal indication function is constructed based on the transmissivity after modal mode compensation, specifically as follows: in, This is a modal indicator function.

10. The method according to claim 1, characterized in that, Using the static mode constant and the cutting process poles, the process frequency response function of the milling cutter axis system is reconstructed as follows: in, The extreme point of the cutting process. is the static mode constant.

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