Milling chatter stability domain prediction method and system based on approximate analysis method

Abstract

The invention discloses a milling chatter stability domain prediction method and system based on an approximate analysis method, and the method comprises the steps: employing a Fourier approximation method to represent a milling steady-state response based on a milling direction coefficient and a time delay characteristic in a milling dynamic model; applying small disturbance to the steady-state response, substituting the steady-state response into the milling dynamic model, and obtaining a milling stability analysis expression by applying a Galerkin process; substituting the bifurcation condition into the stability analysis expression based on the characteristics of the stability domain boundary corresponding to the milling stability bifurcation condition, and deducing an approximate implicit analytical expression of the milling stability boundary; solving the implicit analytical expression by adopting a numerical continuation method to obtain a stability boundary expressed by rotating speed-cutting depth; according to the method, the stability lobe graph is obtained, the problems of initial continuation point selection and transition between lobes with different stability in the solving process are solved, complete and accurate construction of the stability lobe graph is achieved, theoretical guidance of flutter-free working conditions is provided for milling, and the method has important engineering application value.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of milling chatter stability prediction, and particularly relates to a milling chatter stability domain prediction method and system based on an approximate analytical method. TECHNICAL BACKGROUND Milling is one of the key cutting processes in modern manufacturing industry, and is widely used in aerospace, mold manufacturing and precision equipment fields. With the popularity of high-performance numerical control machine tools and the increasing demand for high efficiency processing, how to improve the processing efficiency while ensuring the surface quality and dimensional accuracy of the workpiece has become a key technical problem to be solved. In the milling process, chatter is a regenerative vibration phenomenon caused by system self-excitation, and is one of the main factors affecting the processing quality and production efficiency. The occurrence of chatter will lead to deterioration of workpiece surface roughness, decrease of dimensional accuracy, and aggravation of tool wear, and even cause tool breakage or damage to machine tool parts, thereby limiting the cutting depth and spindle speed, significantly reducing the production efficiency and increasing the manufacturing cost. Therefore, it is of great theoretical and engineering significance to carry out research on the prediction and control of milling chatter stability.

[0002] At present, the research methods for milling stability prediction mainly include analytical method and discretization method. The analytical method is widely used due to its simple mathematical form and high calculation efficiency, among which the zero-order approximation method is the most representative. However, this method ignores the high harmonic components in the direction coefficient, and is only applicable to low radial depth of cut conditions, making it difficult to accurately predict the stability lobe caused by period-doubling bifurcation. To improve the prediction accuracy, researchers have proposed a multi-frequency method, which improves the model accuracy by introducing high-order harmonics, but this method has large calculation amount and low efficiency. The semi-analytical method appeared in recent years improves the prediction accuracy to some extent, but it still cannot effectively consider the influence of high-order harmonics on the Neimark-Sacker (NS) bifurcation stability boundary. In contrast, the discretization method has good generality and can handle complex situations such as variable pitch tools, multiple delays and nonlinear systems. This method usually discretizes the delay differential equation into a state space form and judges the system stability based on the Floquet theory. However, the calculation accuracy depends on the refinement of the discrete step, and too small step size will significantly increase the matrix dimension, resulting in rapid increase of calculation amount and low efficiency. In addition, the discretization method usually lacks explicit analytical expression, which is not conducive to the optimization design of processing parameters.

[0003] The patent with publication number CN106934170A provides a method for modeling the chatter stability lobe diagram of the contact area between the ball-end mill and the workpiece. The method is based on the full discrete time domain solution of the second-order dynamic equation of the milling system. In the single-tooth cutting period of the ball-end mill, the contact area between the ball-end mill and the workpiece is projected based on the boundary equation and the projection equation of the cutting edge at different times. The number of cutting edges involved in cutting at different times and the actual cutting position of the cutting edge involved in cutting in a time lag period are extracted. The chatter stability lobe diagram of the ball-end mill in the cutting process is constructed. However, this method is essentially a discretization method, and its calculation accuracy depends heavily on the grid density of time domain discretization. When predicting high-frequency chatter, there is a contradiction between calculation efficiency and accuracy.

[0004] In summary, the existing milling stability prediction methods generally have the problem of difficult to balance the calculation efficiency and prediction accuracy. Analytical methods have advantages in efficiency, but the accuracy is limited. Discretization methods have higher applicability, but the calculation cost is too high. Therefore, it is urgent to develop a milling stability prediction method that balances calculation accuracy and efficiency, which can realize efficient calculation based on full consideration of high-order harmonic influence, and provide theoretical support and parameter guidance for high-performance milling machining. SUMMARY

[0005] To overcome the shortcomings of the prior art, the purpose of the present application is to provide a milling chatter stability domain prediction method based on approximate analytical method. By Fourier approximation of the milling direction coefficient matrix, based on the characteristic of time lag elimination in the steady-state response stage of milling dynamics, the steady-state response expressed by Fourier series is derived. Then, based on the Hill method, the milling stability analysis expression is established, and combined with the bifurcation condition, the implicit analytical expression of the stability boundary is derived. Further, the numerical continuation method is used to solve the implicit analytical expression, and specific solving strategies are obtained for the initial point selection and stability lobe switching problems in the solving process. The method of the present application significantly improves the prediction efficiency while ensuring the prediction accuracy.

[0006] To achieve the above-mentioned objectives, in a first aspect, the present application provides a milling chatter stability domain prediction method based on approximate analytical method, comprising the following steps: Step 1: Based on the milling direction coefficient in milling dynamics and the time lag characteristic of the regenerative effect in milling dynamics, the steady-state response of milling is represented by Fourier approximation method; Step 2: Based on the Hill stability analysis theory, the steady-state response represented by Fourier approximation is analyzed for stability to obtain the stability analysis expression of the milling system; Step 3: Substitute the bifurcation condition into the stability analysis expression, and introduce the normalization condition to obtain the approximate implicit analytical expression of the milling chatter stability domain boundary; Step 4, the approximate implicit analytical expression is solved by using the continuation method to obtain the stability boundary expressed in the form of rotating speed-depth of cut; Step 5, the lowest critical depth of cut of the stability boundary is selected as the initial point for continuation solution, the eigenvalues are monitored based on the transition between the stability lobes caused by different bifurcations, and the bifurcations are marked after being found, the stability boundary calculation is carried out based on the bifurcations as the initial points, the different stability lobes are continued based on the initial points of different lobes, and the complete stability lobe diagram is obtained by combining the stability lobes.

[0007] Further, in the step 1, the milling dynamics equation describing the milling dynamics has the following characteristics: the milling force direction coefficient is a periodic function with the milling interval as the period, and can be approximated by Fourier series; in the steady-state response stage, the system response period is the same as the time delay, the time delay effect is eliminated, the system behaves as a linear system, and the steady-state response of the milling dynamics is also approximated by Fourier series:

[0008] wherein, are the coefficients of the Fourier approximation of the steady-state response, respectively; is the order of the Fourier approximation of the steady-state response.

[0009] Further, the milling dynamics equation is converted to obtain a normalized milling dynamics equation,

[0010] wherein, are the normalized mass, damping and stiffness matrices, respectively; are the normalized time and time delay variables, respectively; is the normalized milling force direction coefficient matrix, and the milling force direction coefficient is approximated by Fourier series as:

[0011] wherein, are the coefficients of the Fourier approximation of the milling force direction coefficient matrix, respectively; is the order of the Fourier approximation of the milling force direction coefficient matrix, and the time delay term is the time delay in the middle is consistent with the steady-state response period, and the regenerative effect in the steady-state response stage of the milling dynamics will be eliminated to become a linear system.

[0012] Further, in the step 2, based on the steady-state response expression obtained in the step 1, the system is subjected to stability analysis by using the Hill stability analysis method, and the stability analysis analytical expression of the milling dynamics is established under the Hill theory framework,

[0013] The expression is used to determine the stability of a system under given milling conditions, where, For eigenvalues, These are the Fourier coefficients for small perturbations within the Hill theory stability framework. These are the coefficients corresponding to the quadratic eigenvalues, linear eigenvalues, and remainder eigenvalues, respectively. The eigenvectors corresponding to the eigenvalues.

[0014] Furthermore, in step 3, the boundary of the milling chatter stability domain is generated by the combined action of the NS bifurcation and the PD bifurcation. Substituting the stability critical conditions corresponding to the NS bifurcation and the PD bifurcation into the stability analysis expression, and introducing a normalization condition to make the number of equations describing the milling chatter stability domain boundary equal to the number of variables, an implicit analytical expression for the milling chatter stability domain boundary is obtained:

[0015] in, For the eigenvector Imaginary part of eigenvalues Rotation speed N The variables that constitute These are the normalized system mass and damping matrix, respectively. These are the real and imaginary parts of the eigenvector, respectively. These are the zero-order eigenvalues ​​and the Fourier coefficients of the time delay term, respectively. The derivative of For the real part of the eigenvector With the imaginary part The basis vectors of Zhang Cheng's vector space.

[0016] Furthermore, in step 4, the implicit analytical expression is solved using a numerical continuation method, specifically including: A known point on the boundary of the stability region is selected as the initial continuation point. The prediction-correction method is used to solve the problem point by point along the boundary of the stability region, and the prediction point is obtained using the prediction method.

[0017] Introducing normalization conditions The predicted point is obtained by the following formula:

[0018] After obtaining the predicted points, multiple correction iterations are used to obtain points on the stability region boundary, employing arc-length correction.

[0019] in, Analytical expression of the boundary of the stability domain for milling chatter with respect to variables exist The derivative value.

[0020] Furthermore, in step 5, the following strategy is adopted to select an initial continuation point and solve the boundary problem corresponding to the current initial point using the numerical continuation method: Using the depth of cut and rotational speed corresponding to the lowest point of each stable domain lobe as initial continuation points, the stable domain boundary corresponding to each lobe is solved separately. During the solution process, the changes in eigenvalues ​​are monitored, period-doubling bifurcation points are identified, and the corresponding rotational speed and depth of cut are recorded. These are used as initial values ​​to construct the stable lobe diagram boundary caused by PD bifurcation. The stable boundaries caused by PD bifurcation and NS bifurcation are merged to form the complete stable boundary of a single lobe. This process is repeated, and the lowest point of the remaining lobes is selected as the initial continuation point in turn to finally realize the construction of the complete stable domain boundary.

[0021] Secondly, the present invention provides a milling chatter stability domain prediction system based on an approximate analytical method, including a Fourier approximation module, a stability analysis module, a stability domain boundary analysis module, and a solution module; The Fourier approximation module uses the Fourier approximation method to represent the steady-state response of milling based on the time-delay characteristics of the milling direction coefficient and the regenerative effect of milling dynamics in milling dynamics. The stability analysis module, based on Hill's stability analysis theory, performs stability analysis on the steady-state response expressed by the Fourier approximation to obtain the stability analysis expression of the milling system. The stability domain boundary analysis module is used to substitute the bifurcation condition into the stability analysis expression and introduce the normalization condition to obtain an approximate implicit analytical expression for the stability domain boundary of milling chatter. The solution module uses a numerical continuation method to solve the approximate implicit analytical expression, obtaining a stability boundary represented by rotational speed and depth of cut. The lowest critical depth of cut of the stability boundary is selected as the initial point for continuation solution. Based on the transition between stability lobes caused by different bifurcations, the eigenvalues ​​are monitored. After bifurcations are detected, they are marked. The stability boundary is calculated with the bifurcations as the initial points. Different stability lobes are continued with different initial points of different lobes. The stability lobes are combined to obtain a complete stability lobe diagram.

[0022] Thirdly, the present invention provides a computer device, including a processor and a memory, wherein the memory is used to store a computer executable program, the processor reads part or all of the computer executable program from the memory and executes it, and the processor can realize the above-mentioned milling chatter stability domain prediction method based on approximate analytical method when executing part or all of the executable program.

[0023] Finally, a computer-readable storage medium may be provided, in which a computer program is stored, which, when executed by a processor, can implement the above-mentioned milling chatter stability domain prediction method based on an approximate analytical method.

[0024] Compared with the prior art, the present invention has the following beneficial effects: This invention, based on nonlinear dynamics theory, combines the Fourier approximation of the cutting direction coefficient and the time delay characteristics of the steady-state response stage of milling dynamics to derive the steady-state response expression of milling dynamics. Furthermore, based on Hill theory, a small perturbation is applied to the steady-state response and stability analysis is performed. The critical conditions for NS bifurcation and PD bifurcation are substituted into the stability equation, and a regularization condition is introduced to obtain an implicit analytical expression of the stability boundary. The numerical continuation method is used to solve this implicit analytical expression, thereby constructing the milling stability boundary. The implicit analytical expression proposed in this invention is based on the Fourier series approximation of the milling direction coefficient, comprehensively considering the influence of NS bifurcation and PD bifurcation on the stability boundary, resulting in higher prediction accuracy. The numerical continuation method is directly used for accurate solution of the stability boundary, eliminating the need for exhaustive calculation of the speed-depth of cut combination, significantly improving prediction efficiency. This method provides an efficient and accurate prediction framework for chatter stability prediction. The approximate analytical rule proposed in this application obtains an explicit expression of the stability boundary through analytical derivation, effectively overcoming the computational bottleneck of the discretization method and providing a more practical technical approach for stability prediction under high-speed conditions. Attached Figure Description

[0025] Figure 1 This is a flowchart of the present invention.

[0026] Figure 2 This is a flowchart illustrating the implementation of the present invention.

[0027] Figure 3 This is the dynamic model of the milling system, the subject of study in this embodiment of the invention.

[0028] Figure 4 The following are the leaf lobe diagrams of the chatter stability domain obtained by the method proposed in this invention under the given milling system parameters in the embodiments of this invention: (a) is the leaf lobe diagram of the stability domain obtained by the zero-order mean method, (b) is the leaf lobe diagram of the stability domain obtained by the fully discrete method, and (c) is the leaf lobe diagram of the chatter stability domain obtained by the method of this invention. Detailed Implementation

[0029] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings, but the scope of protection of the present invention is not limited to the embodiments described.

[0030] refer to Figure 1 The present invention provides a milling chatter stability domain prediction method based on an approximate analytical method, the specific operation of which is as follows: Step 1, the milling dynamics equation is: (1) in, These are the mass, damping, and stiffness matrices of the milling dynamics system; These are the acceleration, velocity, displacement, and time-delayed displacement response vectors, respectively. These are time and time-delay variables, respectively. This refers to the axial depth of cut. This is the matrix of milling force direction coefficients; This is the static axial depth of cut, which is related to the feed rate. The milling dynamics equations are referenced below. Figure 3 .

[0031] Frequency conversion of milling dynamics equations and quality Normalization, For rotational speed, Given the number of cutting teeth, the normalized milling dynamics equations are obtained. (2) in, These are the normalized mass, damping, and stiffness matrices, respectively. These are the normalized time and time-delay variables, respectively; This is the normalized milling force direction coefficient matrix.

[0032] Based on the periodic characteristics of the milling force direction coefficient matrix in milling dynamics, the milling force direction coefficient is approximated using a Fourier series as follows: (3) in, These are the coefficients of the Fourier approximation of the milling force direction coefficient matrix; Let be the order of the Fourier approximation of the milling force direction coefficient matrix.

[0033] Based on time delay Medium time delay This characteristic of having the same period as the steady-state response eliminates the regenerative effect in the steady-state response stage of milling dynamics, transforming the system into a linear system, resulting in the following steady-state response: (4) in, These are the coefficients of the Fourier approximation of the steady-state response; Let be the order of the Fourier approximation of the steady-state response.

[0034] Step 2, the steady-state response in Step 1 Add perturbation achievable Substituting this into equation (2), and applying the Galerkin process, we can obtain the stability analysis equation: (5) in, For eigenvalues, For feature vectors, For disturbance The coefficients represented by the Fourier series, ; , , , ; Represents the Kronecker product. Operator is defined as .

[0035] In the stability analysis equation (5), all coefficients except the time delay term have been analytically represented. The product of the time delay term and the milling direction coefficient can be obtained by using the product-to-sum method of trigonometric functions. (6) In equation (6) right Taking the derivative yields the stability analysis equation: (7) in, represent The first of the matrix m-1 OK, n-1 Column elements; represent The first of the matrix 1 OK, n-1 Column elements; represent The first of the matrix m-1 OK, 1 The elements of the column.

[0036] Step 3, set the critical conditions for NS bifurcation. Substituting into the stability analysis equation (5) and simplifying it, we obtain the stable boundary conditions: (8) in, The imaginary part of the eigenvalues; These are the real and imaginary parts of the eigenvector, respectively.

[0037] Introducing normalization conditions yields the implicit analytical expression of the stability boundary: (9) in, For the real part of the eigenvector With the imaginary part The basis vectors of Zhang Cheng's vector space.

[0038] The condition for the stability boundary caused by PD bifurcation is to let Soon Substituting 0.5 into (9) yields the implicit analytical expression of the stability boundary caused by PD bifurcation.

[0039] (10) Step 4: Release the spindle speed as a free parameter to obtain the extended variable: (11) The implicit analytical expression (9) is solved using the numerical continuation method to obtain the stability boundary. A point on the stability boundary is selected as the initial point, and the prediction method is first used to obtain the prediction point: (12) Introducing normalization conditions The prediction point can be obtained by the following formula: (13) After obtaining the predicted points, multiple correction iterations are used to obtain the points on the boundary of the stable region. The correction process adopts the arc length correction method, and the specific operation of arc length correction is as follows: (14) in, Analytical expression of the boundary of the stability domain for milling chatter with respect to variables exist The derivative value.

[0040] Step 5, use Step 4 to solve the implicit analytical expression of the stability boundary derived in Step 3 (9), select the initial continuation point, and compare it with the stability boundary of the average milling direction coefficient after introducing higher-order harmonics. It is found that the stability boundary under the real milling direction coefficient still appears as a stability lobe. In this application, the lowest critical depth of cut of the stability boundary is selected as the initial point for continuation solution. The transition between stability lobes caused by different bifurcations is to monitor the eigenvalues, mark the PD bifurcation or NS bifurcation after discovery, and use this as the initial point for calculation of the stability boundary. Different stability lobes are continued from the initial point of different lobes. The minimum depth of cut corresponding to the stability boundary at the same rotation speed is taken among the stability boundaries caused by PD bifurcation and NS bifurcation as the final stability boundary, thus forming the complete boundary of a single lobe. The stability lobes obtained above are combined to obtain the complete stability lobe diagram. The specific process is as follows. Figure 2 As shown, the chatter stability domain lobe diagram obtained using the method proposed in this invention under given milling system parameters is as follows:Figure 4 As shown, (a) is the leaf lobe diagram of the stability domain obtained by the zero-order mean method, (b) is the leaf lobe diagram of the stability domain obtained by the fully discrete method, and (c) is the leaf lobe diagram of the flutter stability domain obtained by the method of the present invention.

[0041] Based on the concept of the analytical method, this invention provides a milling chatter stability domain prediction system based on an approximate analytical method, including a Fourier approximation module, a stability analysis module, a stability domain boundary analysis module, and a solution module. The Fourier approximation module uses the Fourier approximation method to represent the steady-state response of milling based on the time-delay characteristics of the milling direction coefficient and the regenerative effect of milling dynamics in milling dynamics. The stability analysis module, based on Hill's stability analysis theory, performs stability analysis on the steady-state response expressed by the Fourier approximation to obtain the stability analysis expression of the milling system. The stability domain boundary analysis module is used to substitute the bifurcation condition into the stability analysis expression and introduce the normalization condition to obtain an approximate implicit analytical expression for the stability domain boundary of milling chatter. The solution module uses a numerical continuation method to solve the approximate implicit analytical expression, obtaining a stability boundary represented by rotational speed and depth of cut. The lowest critical depth of cut of the stability boundary is selected as the initial point for continuation solution. Based on the transition between stability lobes caused by different bifurcations, the eigenvalues ​​are monitored. After bifurcations are detected, they are marked. The stability boundary is calculated with the bifurcations as the initial points. Different stability lobes are continued with different initial points of different lobes. The stability lobes are combined to obtain a complete stability lobe diagram.

[0042] In addition, the present invention also provides a computer-readable storage medium storing a computer program, which, when executed by a processor, can implement the milling chatter stability domain prediction method based on an approximate analytical method described in the present invention.

[0043] The present invention can also provide a computer device, including a processor and a memory, wherein the memory is used to store a computer executable program, the processor reads the computer executable program from the memory and executes it, and the processor can implement the milling chatter stability domain prediction method based on the approximate analytical method described in the present invention when executing the computer executable program.

[0044] The computer device may be a laptop, a desktop computer, or a workstation.

[0045] The processor can be a central processing unit (CPU), a graphics processing unit (GPU), a digital signal processor (DSP), an application-specific integrated circuit (ASIC), or an off-the-shelf programmable gate array (FPGA).

[0046] The memory described in this invention can be an internal storage unit of a laptop, desktop computer, or workstation, such as memory or hard disk; or it can be an external storage unit, such as a portable hard disk or flash memory card.

[0047] Computer-readable storage media can include computer storage media and communication media. Computer storage media includes volatile and non-volatile, removable and non-removable media implemented using any method or technology for storing information such as computer-readable instructions, data structures, program modules, or other data. Computer-readable storage media can include: read-only memory (ROM), random access memory (RAM), solid-state drives (SSDs), or optical discs, etc. Random access memory can include resistive random access memory (ReRAM) and dynamic random access memory (DRAM).

[0048] The above content is only for illustrating the technical concept of the present invention and should not be construed as limiting the scope of protection of the present invention. Any modifications made to the technical solution based on the technical concept proposed in this invention shall fall within the scope of protection of the claims of this invention.

Claims

1. A method for predicting the chatter stability domain of milling based on an approximate analytical method, characterized by, The method comprises the following steps: Step 1, based on the milling direction coefficient in milling dynamics and the time delay characteristics of the regenerative effect in milling dynamics, the steady-state response of milling is represented by using the Fourier approximation method; Step 2, based on the Hill stability analysis theory, the stability of the steady-state response represented by the Fourier approximation method is analyzed to obtain a stability analysis expression of the milling system; Step 3, the bifurcation condition is substituted into the stability analysis expression, and a normalized condition is introduced to obtain an approximate implicit analytical expression of the milling chatter stability domain boundary; Step 4, the approximate implicit analytical expression is solved by using a numerical continuation method to obtain a stability boundary expressed in terms of rotational speed and depth of cut; Step 5, the lowest critical depth of cut of the stability boundary is selected as an initial point for continuation solving, the characteristic values are monitored based on the transition between stability lobes caused by different bifurcations, the bifurcations are marked after being found, the stability boundary is calculated based on the bifurcations as initial points, different stability lobes are continued based on the initial points of different lobes, and the complete stability lobe diagram is obtained by combining the stability lobes.

2. The method of predicting the milling chatter stability domain based on the approximate analytical method according to claim 1, characterized in that, In step 1, the milling dynamics equation describing the milling dynamics has the following characteristics: the milling force direction coefficient is a periodic function with a milling interval as a period, and can be approximated by a Fourier series; in the steady-state response stage, the system response period is the same as the time delay, the time delay effect is eliminated, the system behaves as a linear system, and the steady-state response of the milling dynamics is also approximated by a Fourier series: wherein are respectively the coefficients of the Fourier approximation representation of the steady-state response; is the order of the Fourier approximation representation of the steady-state response.

3. The method of predicting the milling chatter stability domain based on the approximate analytical method according to claim 2, characterized in that, The milling dynamics equation is converted to obtain a normalized milling dynamics equation, where, are the normalized mass, damping and stiffness matrices, respectively; are the normalized time and time-lag variables, respectively; is the normalized milling force direction coefficient matrix, the milling force direction coefficients are approximated using Fourier series as wherein, are the coefficients of the Fourier approximation representation of the milling force direction coefficient matrix, respectively; is the order of the Fourier approximation representation of the milling force direction coefficient matrix, and is the time delay The regenerative effect will be eliminated in the steady state response phase of the milling dynamics, which becomes a linear system, consistent with the steady state response period.

4. The method for predicting the milling chatter stability domain based on the approximate analytical method according to claim 1, characterized in that, In step 2, based on the steady-state response expression obtained in step 1, the system is analyzed for stability by using the Hill stability analysis method, and a stability analysis analytical expression of the milling dynamics is established under the Hill theory framework, The expression is used to judge the stability of the system under given milling conditions, wherein, is the eigenvalue, is the Fourier coefficient of small perturbation under the Hill theory stability framework, are the coefficients corresponding to the quadratic eigenvalue, the linear eigenvalue and the residual eigenvalue, respectively, is the eigenvector corresponding to the eigenvalue.

5. The method for predicting the milling chatter stability domain based on the approximate analytical method according to claim 1, wherein, In step 3, the milling chatter stability domain boundary is generated by the joint action of NS bifurcation and PD bifurcation, the stability critical conditions corresponding to the NS bifurcation and the PD bifurcation are substituted into the stability analysis expression, and a normalized condition is introduced to make the number of equations and the number of variables equal to describe the milling chatter stability domain boundary, thereby obtaining an implicit analytical expression of the milling chatter stability domain boundary: in, For the eigenvector Imaginary part of eigenvalues Rotation speed N The variables that constitute These are the normalized system mass and damping matrix, respectively. These are the real and imaginary parts of the eigenvector, respectively. These are the zero-order eigenvalues ​​and the Fourier coefficients of the time delay term, respectively. The derivative, For the real part of the eigenvector With the imaginary part The basis vectors of Zhang Cheng's vector space.

6. The method for predicting the milling chatter stability domain based on the approximate analytical method according to claim 1, wherein, In step 4, the implicit analytical expression is solved by using a numerical continuation method, specifically including: A known point on the stability domain boundary is selected as an initial continuation point, a prediction-correction method is used to solve points on the stability domain boundary, a prediction method is used to obtain a predicted point: Introducing the normalization condition The prediction point is given by After obtaining the predicted point, an arc length correction is used to obtain a point on the stability domain boundary, and an arc length correction is used: wherein Analytical expression for the milling chatter stability boundary as a function of variables In derivative values.

7. The method for predicting the milling chatter stability domain based on the approximate analytical method according to claim 1, wherein In step 5, the initial continuation point is selected, and the numerical continuation method is used to solve the problem of the boundary corresponding to the current initial point, and the following strategies are used: The minimum points of each stability domain lobe are taken as initial continuation points to solve the stability domain boundary of each lobe; in the solving process, the characteristic value is monitored to identify the period-doubling bifurcation point and record the corresponding rotational speed and depth, which are taken as initial values to construct the stability lobe diagram boundary caused by PD bifurcation; the stability boundaries caused by PD bifurcation and NS bifurcation are fused to form the complete stability boundary of a single lobe; the process is repeated to select the minimum points of the remaining lobes as initial continuation points, and finally the complete stability domain boundary is constructed.

8. A system for predicting the chatter stability domain of milling based on an approximate analytical method, characterized by, The method comprises a Fourier approximation module, a stability analysis module, a stability domain boundary analysis module and a solving module; The Fourier approximation module represents the steady-state response of milling based on the milling direction coefficient in milling dynamics and the time delay characteristics of the regenerative effect in milling dynamics by using the Fourier approximation method; The stability analysis module performs stability analysis on the steady-state response represented by the Fourier approximation method based on the Hill stability analysis theory to obtain a stability analysis expression of the milling system; The stability domain boundary analysis module is used to substitute the bifurcation condition into the stability analysis expression and introduce a normalization condition to obtain an approximate implicit analytical expression of the milling chatter stability domain boundary; The solving module solves the approximate implicit analytical expression by using a numerical continuation method to obtain the stability boundary expressed in terms of rotational speed and depth; the lowest critical depth of the stability boundary is selected as an initial point for continuation solving, the characteristic value is monitored based on the transition between stability lobes caused by different bifurcations, the bifurcation is marked after the bifurcation, the stability boundary is calculated based on the initial point of the bifurcation, and the complete stability lobe diagram is obtained by combining the stability lobes.

9. A computer device, comprising: The method comprises a processor and a memory, the memory is used to store computer executable programs, the processor reads part or all of the computer executable programs from the memory and executes, and the processor can implement the method according to any one of claims 1-7 when executing part or all of the computer executable programs.

10. A computer-readable storage medium, characterized in that, A computer program is stored in a computer readable storage medium, and the computer program can implement the method according to any one of claims 1-7 when executed by a processor.

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