CN103559550A - Milling stability domain prediction method under multi-modal coupling - Google Patents

Abstract

The invention discloses a milling stability domain prediction method under multi-modal coupling to solve a technical problem that the efficiency of an existing milling stability domain prediction method is low. The technical scheme is that firstly the transmission function of a process system is determined through a modal test experiment, then modal parameters are extracted from the transmission function and two orthogonal direction modal parameters are paired and combined, a milling test is carried out to calibrate a milling force coefficient, based on the milling force coefficient and each scale of modal parameters, an improved semi discrete method is employed to obtain a stability lobe diagram under each scale of modals, and finally lobe diagrams under obtained from each scale of modals are drawn in a same coordinate system to obtain a stable domain under multi modal coupling. Through a test, when a cutting cycle is divided into 40 sections, 80 sections and 120 sections, compared with a background technology method, the method can save time for 4949.6 seconds, 74200.4 seconds and 344699.5 seconds, and the efficiency is increased by 85.9%, 92% and 88.5%.

Description

Milling stability domain prediction method under multi-modal coupling

Technical Field

The invention relates to a milling stable domain prediction method, in particular to a milling stable domain prediction method under multi-mode coupling.

Background

Reference is made to fig. 1-2. Milling is one of the most common metal material forming processes, and is widely applied to the production and manufacture of various molds, automobile parts and aerospace parts thereof. The regenerative chatter phenomenon caused by unreasonable selection of cutting parameters in the milling process can seriously affect the machining precision and the service life of a cutter, and even lead to the premature failure of a machine tool. In order to reasonably select cutting parameters and avoid the occurrence of chatter, an effective stable region prediction method needs to be established. Considering the complexity of an actual process system, in order to accurately represent the dynamic behavior of the actual process system, a multi-degree-of-freedom model is required to be established to predict a milling process stability domain, so that a stability prediction method under multi-modal coupling is established.

Document 1 "t.inster, g.stepan, Updated semi-discrete method for periodic differential optimization with discrete delay, International Journal for Numerical Methods in engineering 61(2004) 117-.

Document 2 "m.wan, w.h.zhang, j.w.dang, y.yang, a unified stability prediction method for milling process with multiple deltaschemes, International Journal of Machine Tools and manual 50(2010) 29-41" discloses a method for predicting a stable domain (i.e., a stable lobe map) of a multi-delay milling system in a time domain by using an improved semi-discrete method, which can systematically consider the influence of phenomena such as tool eccentricity and unequal teeth on the milling stable domain.

The typical features of the above documents relating to predicting the milling stability domain in the time domain are: in the modeling process, in order to shorten the calculation time, only the single-order modal parameter with the worst rigidity is selected for solving, and the influence of multi-modal coupling is not considered, so that the prediction distortion in certain rotating speed ranges is caused.

Disclosure of Invention

In order to overcome the defect of low efficiency of the conventional milling stable domain prediction method, the invention provides a milling stable domain prediction method under multi-mode coupling. The method comprises the steps of firstly determining a transfer function of a process system through a modal test experiment, then extracting modal parameters from the transfer function, carrying out pairing combination on the two modal parameters in the orthogonal direction according to orders, then carrying out a cutting experiment to calibrate a milling force coefficient, then obtaining a stability lobe graph under each order of modes by adopting an improved semi-discrete method based on the milling force coefficient and each order of modal parameters, finally drawing the lobe graph obtained by each order of modes in the same coordinate system, and obtaining a stability domain under multi-mode coupling by taking the lowest envelope curve of the lobe graph, so that the prediction efficiency of the milling stability domain can be improved.

The technical scheme adopted by the invention for solving the technical problems is as follows: a milling stable domain prediction method under multi-mode coupling is characterized by comprising the following steps:

the first step,Installing the milling cutter and the cutter handle in a machine tool spindle system, testing the modal parameters of the milling cutter-cutter handle-machine tool spindle system by adopting a standard impact test, and recording the tested j-th order modal parameter as m_X,j、c_X,j、k_X,j，m_Y,j、c_Y,jAnd k_Y,jSubscripts X and Y denote test results in X and Y directions, subscript j denotes a j-th order modal parameter, and symbols m, c, and k denote modal mass, damping coefficient, and modal stiffness, respectively.

Step two, combining the j-th order modal parameter in the X direction and the j-th order modal parameter in the Y direction measured in the step one to form a combination, and marking the combination as a modal j;

step three, testing the milling force through a milling experiment, and calibrating a tangential milling force coefficient K_tRadial milling force coefficient K_rAnd tool eccentricity parameters ρ and λ. ρ represents the offset of the tool rotation center from the tool geometric center, and λ represents the angle between the direction of tool eccentricity and the nearest adjacent tooth head.

Step four, analyzing and determining the size and the number of delay quantities which may occur in the milling system according to the geometric parameters of the milling cutter and the eccentricity parameters rho and lambda obtained in the step three, and recording the delay quantities as the sequence from small to small

N_MThe number of delay amounts is indicated.

Step five, expressing a dynamic control equation of the milling system under the mode j as follows:

<math> <mrow> <msub> <mi>M</mi> <mi>j</mi> </msub> <mover> <mi>X</mi> <mrow> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> </mrow> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>C</mi> <mi>j</mi> </msub> <mover> <mi>X</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>K</mi> <mi>j</mi> </msub> <mi>X</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mi>&Sigma;</mi> <mrow> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>N</mi> <mi>M</mi> </msub> </munderover> <mrow> <mo>[</mo> <msub> <mi>H</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>l</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>X</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>X</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>]</mo> </mrow> </mrow> </math>

wherein,

$M_j=\left[\begin{array}{cc}{m}_{X,j}& 0\\ 0& m_{Y,j}\end{array}\right]$

$C_j=\left[\begin{array}{cc}{c}_{X,j}& 0\\ 0& C_{Y,j}\end{array}\right]$

$K_j=\left[\begin{array}{cc}{k}_{X,j}& 0\\ 0& k_{Y,j}\end{array}\right]$

x (t) represents a dynamic deformation vector of the tool, $H_l\left(t\right)=\left[\begin{array}{cc}{H}_{l,\mathrm{XX}}& H_{l,\mathrm{YY}}\\ H_{l,\mathrm{YX}}& H_{l,\mathrm{YY}}\end{array}\right]$ representing a matrix of directional coefficients, each element of which

The expression is as follows:

H_l,XX(t)＝∑[z_l,p,qsinθ_l,p,q(t)(K_tcosθ_l,p,q(t)+K_rsinθ_l,p,q(t))]

H_l,XY(t)＝∑[z_l,p,qcosθ_l,p,q(t)(K_tcosθ_l,p,q(t)+K_rsinθ_l,p,q(t))]

H_l,YX(t)＝∑[z_l,p,qsinθ_l,p,q(t)(-K_tsinθ_l,p,q(t)+K_rcosθ_l,p,q(t))]

H_l,YY(t)＝∑[z_l,p,qcosθ_l,p,q(t)(-K_tsinθ_l,p,q(t)+K_rcosθ_l,p,q(t))]

wherein each cutter tooth is axially dispersed into a plurality of cutter tooth blades with equal length, z_l,p,qAnd theta_l,p,q(t) represents the axial length and cutting angle corresponding to the qth unit on the pth cutter tooth; the subscript l indicates the amount of delay τ corresponding to the qth unit on the pth tooth at time t_l。

And step six, performing stability analysis on the control equation obtained in the step five, and solving a stability lobe graph corresponding to the mode j.

And step seven, repeating the step five and the step six from j =1 to j = n until a stability lobe graph corresponding to each order mode is obtained, wherein n represents the maximum number of the measured modes.

And step eight, drawing the lobe graphs obtained by the single-order modes under the same coordinate system, connecting the lowest edges of all the lobe graphs into a continuous curve, namely making a lowest envelope curve, and obtaining the stability lobe graph of the multi-mode coupling multi-delay milling system.

The invention has the beneficial effects that: the method comprises the steps of firstly determining a transfer function of a process system through a modal test experiment, then extracting modal parameters from the transfer function, carrying out pairing combination on the two modal parameters in the orthogonal direction according to orders, then carrying out a cutting experiment to calibrate a milling force coefficient, then obtaining a stability lobe graph under each order of modes by adopting an improved semi-discrete method based on the milling force coefficient and each order of modal parameters, finally drawing the lobe graph obtained by each order of modes in the same coordinate system, and taking the lowest envelope line of the lobe graph to obtain a stability domain under multi-modal coupling, thereby improving the prediction efficiency of the milling stability domain. Through tests, when the cutting period is dispersed into 40 sections, the method saves 4949.6 seconds compared with the method in the background art, and the efficiency is improved by 85.9 percent; when the cutting period is dispersed into 80 sections, the method saves 74200.4 seconds compared with the method of the background art, and the efficiency is improved by 92.0 percent; when the cutting period is dispersed into 120 sections, the method saves 344699.5 seconds compared with the method of the background art, and the efficiency is improved by 88.5 percent; it can be seen that, for the prediction of the multi-modal multi-delay stable domain with higher precision requirement, the high efficiency of the method of the invention is more obvious.

The present invention will be described in detail below with reference to the accompanying drawings and examples.

Drawings

FIG. 1 is a diagram of a multimodal model in a background art method.

FIG. 2 is a graph of stability lobes in each single order mode of the background art method.

FIG. 3 is a multi-modal lobe plot obtained by the method of the present invention.

In the figure, m_X,1、c_X,1、k_X,1And m_Y,n、c_Y,n、k_Y,nRespectively representing the first-order modal mass, modal damping and modal stiffness and the nth-order modal mass, modal damping and modal stiffness; theta_l,p,qShowing the cutting angle corresponding to the q unit on the p cutter tooth. 1-is a lobe graph obtained through calculation according to a first-order modal parameter, 2-is a lobe graph obtained through calculation according to a second-order modal parameter, and 3-is a lobe graph obtained through calculation according to a third-order modal parameter; a, B, C, D, E, F, G and H represent intersection points of lobe graphs obtained by calculating modal parameters of different orders. The black filled squares represent stable cutting processes as determined by actual experiments, and the open circles represent unstable cutting processes as determined by actual experiments.

Detailed Description

The milling stable domain prediction method under the multi-modal coupling comprises the following specific steps.

1) The method comprises the following steps of installing a milling cutter with the radius of 12mm, the helical angle of 30 degrees and the number of cutter teeth N of 3 on a cutter handle and then on a machine tool spindle system, testing modal parameters of the milling cutter-cutter handle-machine tool spindle system by adopting a standard impact test, and testing the obtained modal parameters of each order, wherein the table is as follows:

TABLE 1 Modal parameters

2) Carrying out pairing combination on the modal parameters in the X direction and the modal parameters in the Y direction measured in the step 1), namely combining the first-order modal parameters in the X direction and the first-order modal parameters in the Y direction into a group which is marked as a modal 1; similarly, the second-order modal parameters in the X direction and the second-order modal parameters in the Y direction are combined into a group which is marked as a modal 2; that is, the third-order modal parameter in the X direction and the third-order modal parameter in the Y direction are combined into a group, which is referred to as a modal state 3.

3) Installing a dynamometer on a Machine Tool workbench, installing an aluminum alloy 7050 on the dynamometer, machining the aluminum alloy 7050 by using the milling cutter-Tool holder-Machine Tool spindle system in the step 1), testing the milling force in the machining process, and calibrating a tangential milling force coefficient K by using a method disclosed in the documents 3 W.A.Kline, R.E.DeVor, J.R.Lindberg, the prediction of cutting force in end milling with application to machining purpose, International journal of Machine Tool Design and Research,22(1982),7-22 ″, according to the obtained milling force_tAnd coefficient of radial milling force K_rThe tool eccentricity parameters ρ and λ are calibrated by the method disclosed in the references 4 "M.Wan, M.S.LU, W.H.Zhang, Y.Yang, Y.Li, A new method for identifying the cutter runout parameters in flat milling process, Materials Science Forum,697 and 698(2011), 71-74". ρ represents the offset of the tool rotation center from the tool geometric center, and λ represents the angle between the direction of tool eccentricity and the nearest adjacent tooth head. The results of the calibration in this example are as follows：K_tIs 1110.6MPa, K_r554.45MPa, p and lambda are respectively 13.5 μm and-17.51 deg.

4) Analyzing and determining the total number of delay amount possibly occurring in the milling system to be 3 according to the geometric parameters of the milling cutter and the eccentric parameters rho and lambda obtained in the step 3), wherein the total number of the delay amount is T/3, 2T/3 and T respectively, and T is the rotation period of the cutter.

5) The dynamic control equation of the milling system under the mode j is expressed as follows:

<math> <mrow> <msub> <mi>M</mi> <mi>j</mi> </msub> <mover> <mi>X</mi> <mrow> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> </mrow> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>C</mi> <mi>j</mi> </msub> <mover> <mi>X</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>K</mi> <mi>j</mi> </msub> <mi>X</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mi>&Sigma;</mi> <mrow> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>3</mn> </munderover> <mrow> <mo>[</mo> <msub> <mi>H</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>l</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>X</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>X</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>]</mo> </mrow> </mrow> </math>

wherein,

$M_j=\left[\begin{array}{cc}{m}_{X,j}& 0\\ 0& m_{Y,j}\end{array}\right]$

$C_j=\left[\begin{array}{cc}{c}_{X,j}& 0\\ 0& c_{Y,j}\end{array}\right]$

$K_j=\left[\begin{array}{cc}{k}_{X,j}& 0\\ 0& k_{Y,j}\end{array}\right]$

x (t) represents a dynamic deformation vector of the tool, $H_l\left(t\right)=\left[\begin{array}{cc}{H}_{l,\mathrm{XX}}& H_{l,\mathrm{XY}}\\ H_{l,\mathrm{YX}}& H_{l,\mathrm{YY}}\end{array}\right]$ a directional coefficient matrix is represented, and the elements thereof are expressed as follows:

H_l,XX(t)＝∑[z_l,p,qsinθ_l,p,q(t)(K_tcosθ_l,p,q(t)+K_rsinθ_l,p,q(t))]

H_l,XY(t)＝∑[z_l,p,qcosθ_l,p,q(t)(K_tcosθ_l,p,q(t)+K_rsinθ_l,p,q(t))]

H_l,YX(t)＝∑[z_l,p,qsinθ_l,p,q(t)(-K_tsinθ_l,p,q(t)+K_rcosθ_l,p,q(t))]

H_l,YY(t)＝∑[z_l,p,qcosθ_l,p,q(t)(-K_tsinθ_l,p,q(t)+K_rcosθ_l,p,q(t))]

in the calculation process of the formula, each cutter tooth is dispersed into a plurality of cutter tooth blades with equal length along the axial direction, z_l,p,qAnd theta_l,p,q(t) represents the axial length and cutting angle corresponding to the qth unit on the pth cutter tooth; the subscript l indicates the amount of delay τ corresponding to the qth unit on the pth tooth at time t_l。

6) Stability analysis is performed on the control equation obtained in step 5) according to an improved semi-discrete method disclosed in document 2 "m.wan, w.h.zhang, j.w.dang, y.yang, a unified stability verification method for milling process with multiple defects, International Journal of machine Tools & manual 50(2010) 29-41", and a stability lobe graph corresponding to the mode j is solved;

7) and repeating the step 5) and the step 6) from j =1 to j = n until a stability lobe graph corresponding to each order mode is obtained, wherein n represents the measured maximum mode number and is 3.

8) And (3) drawing the lobe graphs obtained by calculating each single-order mode in the same coordinate system, and referring to fig. 2, wherein A, B, C, D, E and F represent intersection points of the lobe graphs obtained by different-order mode parameters. Connecting the lowest edges of all the lobe graphs into a continuous curve, namely sequentially connecting an AB section of a first-order modal parameter, a BC section of a third-order modal parameter, a CD section of the first-order modal parameter, a DE section of the third-order modal parameter, an EF section of the first-order modal parameter, an FG section of the third-order modal parameter and a GH section of the first-order modal parameter in the graph, and drawing the connected curves under the same coordinate system to obtain a stable lobe graph of the multi-delay milling system under multi-modal coupling, which is shown in the attached figure 3; to verify the effectiveness of the given method, four sets of cutting verification experiments were designed, the results of which are given in fig. 3: the solid black squares in fig. 3 represent the stable cutting process as determined by the actual experiment, and the open circles represent the unstable cutting process as determined by the actual experiment. The result of the verification experiment is consistent with the prediction result of the milling stable domain, and the method is feasible and effective.

As can be seen from comparison between fig. 2 and fig. 3, in the actual milling process, due to existence of multiple modes, the stable domain is a result of coupling of multiple modes, and if only a certain-order mode parameter is taken for solving, an error necessarily exists in a prediction result. For example, if the first-order simulation is performed, the obtained lobe graph is the lobe graph of the mode 1, see fig. 2, and the prediction result of the stable region is inaccurate when the rotating speed ranges from 15000 rpm to 21000 rpm.

On the other hand, the envelope method computation time vs. the multi-modal computation time considered in solving the kinetic equations is shown in table 2. In table 2, since the improved semi-discrete method disclosed in document 2 "m.wan, w.h.zhang, j.w.dang, y.yang, a unified stability reproduction method for milling process with multiple delays, International Journal of machine Tools & manual 50(2010) 29-41" requires a step processing of the rotation speed, axial cutting depth and cutting cycle in calculating the lobe map, the number of step processing will affect the calculation time. In the embodiment, the axial cutting depth is divided into 100 sections, namely each section is 0.4mm, the rotating speed calculation range is 5000-50000 r/min, the rotation speed is dispersed into 100 sections, namely each section is 450 r/min, the cutting period of the cutter teeth is respectively dispersed into 40 sections, 80 sections and 120 sections, and the calculation time is compared.

TABLE 2 calculation time comparison

As can be seen from the table, the method improves the prediction efficiency of the milling stable region, and the efficiency is improved more obviously along with the increase of the number of the delay time discrete sections. When the cutting period is dispersed into 40 sections, the method saves 4949.6 seconds compared with the method of the background art, and the efficiency is improved by 85.9 percent; when the cutting period is dispersed into 80 sections, the method saves 74200.4 seconds compared with the method of the background art, and the efficiency is improved by 92.0 percent; when the cutting period is dispersed into 120 sections, the method saves 344699.5 seconds compared with the method of the background art, and the efficiency is improved by 88.5 percent; it can be seen that, for the prediction of the multi-modal multi-delay stable domain with higher precision requirement, the high efficiency of the method of the invention is more obvious.

Claims (1)

1. A milling stable domain prediction method under multi-modal coupling is characterized by comprising the following steps:

step one, installing the milling cutter and the cutter handle in a machine tool spindle system, then testing modal parameters of the milling cutter-cutter handle-machine tool spindle system by adopting a standard impact test, and recording j-th order modal parameter obtained by testing as m_X,j、c_X,j、k_X,j，m_Y,j、c_Y,jAnd k_Y,jThe indices X and Y denote the test results in the X and Y directions, the index j denotes the j-th order modal parameter, and the symbols m, c and k denote the modal properties, respectivelyMagnitude, damping coefficient, and modal stiffness;

step two, combining the j-th order modal parameter in the X direction and the j-th order modal parameter in the Y direction measured in the step one to form a combination, and marking the combination as a modal j;

step three, testing the milling force through a milling experiment, and calibrating a tangential milling force coefficient K_tRadial milling force coefficient K_rTool eccentricity parameters rho and lambda; rho represents the offset of the rotation center of the cutter and the geometric center of the cutter, and lambda represents the included angle between the direction generated by the eccentricity of the cutter and the head of the adjacent nearest cutter tooth;

step four, analyzing and determining the size and the number of delay quantities which may occur in the milling system according to the geometric parameters of the milling cutter and the eccentricity parameters rho and lambda obtained in the step three, and recording the delay quantities as the sequence from small to small

N_MThe number of the delay amount is represented;

step five, expressing a dynamic control equation of the milling system under the mode j as follows:

<math> <mrow> <msub> <mi>M</mi> <mi>j</mi> </msub> <mover> <mi>X</mi> <mrow> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> </mrow> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>C</mi> <mi>j</mi> </msub> <mover> <mi>X</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>K</mi> <mi>j</mi> </msub> <mi>X</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mi>&Sigma;</mi> <mrow> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>N</mi> <mi>M</mi> </msub> </munderover> <mrow> <mo>[</mo> <msub> <mi>H</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>l</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>X</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>-</mo> <msub> <mi>&tau;</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>X</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>]</mo> </mrow> </mrow> </math>

wherein,

$M_j=\left[\begin{array}{cc}{m}_{X,j}& 0\\ 0& m_{Y,j}\end{array}\right]$

$C_j=\left[\begin{array}{cc}{c}_{X,j}& 0\\ 0& c_{Y,j}\end{array}\right]$

$K_j=\left[\begin{array}{cc}{k}_{X,j}& 0\\ 0& k_{Y,j}\end{array}\right]$

x (t) represents a dynamic deformation vector of the tool, $H_l\left(t\right)=\left[\begin{array}{cc}{H}_{l,\mathrm{XX}}& H_{l,\mathrm{XY}}\\ H_{l,\mathrm{YX}}& H_{l,\mathrm{YY}}\end{array}\right]$ a directional coefficient matrix is represented, and the elements thereof are expressed as follows:

H_l,XX(t)＝∑[z_l,p,qsinθ_l,p,q(t)(K_tcosθ_l,p,q(t)+K_rsinθ_l,p,q(t))]

H_l,XY(t)＝∑[z_l,p,qcosθ_l,p,q(t)(K_tcosθ_l,p,q(t)+K_rsinθ_l,p,q(t))]

H_l,YX(t)＝∑[z_l,p,qsinθ_l,p,q(t)(-K_tsinθ_l,p,q(t)+K_rcosθ_l,p,q(t))]

H_l,YY(t)＝∑[z_l,p,qcosθ_l,p,q(t)(-K_tsinθ_l,p,q(t)+K_rcosθ_l,p,q(t))]

wherein each cutter tooth is axially dispersed into a plurality of cutter tooth blades with equal length, z_l,p,qAnd theta_l,p,q(t) represents the axial length and cutting angle corresponding to the qth unit on the pth cutter tooth; the subscript l indicates the amount of delay τ corresponding to the qth unit on the pth tooth at time t_l；

Sixthly, performing stability analysis on the control equation obtained in the fifth step, and solving a stability lobe graph corresponding to the mode j;

step seven, repeating the step five and the step six from j =1 to j = n until a stability lobe graph corresponding to each order of mode is obtained, wherein n represents the measured maximum mode number;

and step eight, drawing the lobe graphs obtained by the single-order modes under the same coordinate system, connecting the lowest edges of all the lobe graphs into a continuous curve, namely making a lowest envelope curve, and obtaining the stability lobe graph of the multi-mode coupling multi-delay milling system.

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