CN111158315B - Milling stability prediction method based on spline-Newton formula - Google Patents

Milling stability prediction method based on spline-Newton formula Download PDF

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CN111158315B
CN111158315B CN201911157273.6A CN201911157273A CN111158315B CN 111158315 B CN111158315 B CN 111158315B CN 201911157273 A CN201911157273 A CN 201911157273A CN 111158315 B CN111158315 B CN 111158315B
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CN111158315A (en
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马俊金
李云飞
郭强
赵波
焦锋
王楚
张定华
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Henan University of Technology
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    • G05B19/00—Programme-control systems
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    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
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Abstract

The invention discloses a milling stability prediction method based on a spline-Newton formula, which comprises the following steps: (1) establishing a system dynamic model considering regeneration flutter; (2) dispersing the time-lapse differential equation; (3) interpolating the equation discrete interval; (4) establishing a milling system transfer matrix; (5) performing stability analysis on the milling system based on the Floque theory; (6) and optimizing the prediction method. The method greatly improves the precision and the calculation efficiency, and saves about 70 percent of calculation time compared with the prior calculation method under the same calculation condition; meanwhile, the algorithm is optimized, so that the accuracy of the algorithm is not lost under the condition of ensuring high efficiency.

Description

Milling stability prediction method based on spline-Newton formula
Technical Field
The invention belongs to the technical field of manufacturing of complex thin-wall parts, and particularly relates to a spline-Newton formula-based milling stability prediction method which is mainly applied to stability prediction of complex thin-wall parts machined by numerical control multiple shafts.
Background
With the rapid development of the modern aviation and aerospace industries, various high-performance products need to be developed to meet the market demands. In order to better reduce the self weight of the airplane and meet the structural strength requirement, a large number of thin-wall parts are applied to the aerospace field, such as integral blade discs, blades and the like of the airplane. When the thin-wall parts are milled, the thin-wall parts have extremely poor processing manufacturability due to the reasons of complex structure, thin wall, poor rigidity, complex time-varying dynamic characteristics in the processing process and the like, and are very easy to generate flutter and processing deformation, so that the requirements on the position precision and the surface quality of the parts are finally difficult to meet. Therefore, important engineering application value is achieved in order to better inhibit the vibration in the machining process, accurately predict the milling machining stability and then preferably select reasonable cutting parameters to avoid milling vibration.
In general, in the milling process, a milling process dynamic model considering the regeneration effect is expressed by a multi-dimensional time-lag differential equation, the differential equation is solved by different methods, a system stable lobe graph is calculated, and high system stability prediction efficiency and accuracy are obtained. At present, the most widely used method is a time domain stability analysis method based on a differential equation, in which a milling system is dispersed in a time domain, and then the stability of a control equation is directly analyzed to judge whether the milling process is stable. See for details the references [ Y. Ding, L.M. Zhu, X.J. Zhang, H. Ding, A full-differentiation method for prediction of milking stability, international Journal of Machine Tools and Manual 50 (2010) 502-509]. Because the matrix index involved in the method is calculated only in the outer ring of the scanning spindle speed range and does not need to be updated in the inner ring of the scanning cutting depth range, the efficiency and the accuracy of the method are remarkably improved. However, when the method performs interpolation processing on the state term and the time lag term of the time lag differential equation, linear interpolation is adopted, the precision of the method only depends on two points of boundary values, and meanwhile, the derivative term is not considered, so that the precision of the method can be further improved.
Disclosure of Invention
The invention provides a spline-Newton formula-based milling stability prediction method capable of improving milling efficiency and accuracy, aiming at overcoming the defects in the prior art.
In order to solve the technical problems, the invention adopts the following technical scheme: a milling stability prediction method based on a spline-Newton formula comprises the following steps,
(1) establishing a system dynamic model considering regeneration flutter;
(2) dispersing the time-lapse differential equation;
(3) interpolating the equation discrete interval;
(4) establishing a milling system transfer matrix;
(5) performing stability analysis on the milling system based on the Floquet theory;
(6) and optimizing the prediction method.
The specific process of the step (1) is that,
first establishing formula (1)
Figure DEST_PATH_IMAGE001
（1）
Wherein, the first and the second end of the pipe are connected with each other,M、C、Krespectively representing the modal mass, relative damping and stiffness matrices of the tool,
Figure 61029DEST_PATH_IMAGE002
、
Figure DEST_PATH_IMAGE003
、
Figure 623597DEST_PATH_IMAGE004
respectively representing the acceleration, velocity and displacement vectors of the tool,
Figure DEST_PATH_IMAGE005
is a matrix of periodic coefficients and is,Tfor a single tooth transfer cycle, and
Figure 425241DEST_PATH_IMAGE006
，Nthe number of the cutter teeth of the cutter is,
Figure DEST_PATH_IMAGE007
the main shaft rotating speed;
order to
Figure 22445DEST_PATH_IMAGE008
，
Figure DEST_PATH_IMAGE009
By simplification, the formula (1) can be reduced to the following form:
Figure 761993DEST_PATH_IMAGE010
(2)
a in the formula (2) 0 Is a constant coefficient matrix of the cutter teeth, A (t) is a periodic coefficient matrix of the cutter teeth, and has:
Figure DEST_PATH_IMAGE011
,
Figure 903868DEST_PATH_IMAGE012
(3)。
the step (2) is specifically that,
in one tooth revolution periodTInterior, willTAre equally divided intomBetween cells, and the time length of each cell ishThen there isT=mhTaking the periodTWithin a certain time periodkh≤t≤(k+1)h，(k=0,1…,m) The initial condition of this period is known as
Figure DEST_PATH_IMAGE013
Record of
Figure 410067DEST_PATH_IMAGE014
For equation (2), the direct integration over this interval is:
Figure DEST_PATH_IMAGE015
(4)
order tot=kh+hComprises the following steps:
Figure 658514DEST_PATH_IMAGE016
(5)
the simplification has:
Figure DEST_PATH_IMAGE017
(6)。
the step (3) is specifically that,
by using
Figure 300455DEST_PATH_IMAGE018
,
Figure DEST_PATH_IMAGE019
,
Figure 650665DEST_PATH_IMAGE020
,
Figure DEST_PATH_IMAGE021
Four point to state items
Figure 174269DEST_PATH_IMAGE022
Carrying out cubic spline interpolation; in addition, when spline interpolation is adopted, two additional constraint conditions are required to be added, and the constraint conditions can be selected
Figure DEST_PATH_IMAGE023
And
Figure 651124DEST_PATH_IMAGE024
two known conditions are as follows
Figure DEST_PATH_IMAGE025
（7）
Figure 231010DEST_PATH_IMAGE026
（8）
The status item can be expressed as:
Figure DEST_PATH_IMAGE027
（9）
wherein:
Figure 334096DEST_PATH_IMAGE028
(10)
Iis an identity matrix;
by using
Figure DEST_PATH_IMAGE029
,
Figure 995146DEST_PATH_IMAGE030
,
Figure DEST_PATH_IMAGE031
,
Figure 483765DEST_PATH_IMAGE032
Four point-to-time lag terms
Figure DEST_PATH_IMAGE033
Performing Newton interpolation; then there are:
Figure 467508DEST_PATH_IMAGE034
（11）
wherein:
Figure DEST_PATH_IMAGE035
（12）
the linear interpolation used for the periodic coefficient matrix is:
Figure 792311DEST_PATH_IMAGE036
（13）
wherein:
Figure DEST_PATH_IMAGE037
（14）
and (3) bringing the interpolated state item, the interpolated time lag item and the interpolated periodic coefficient matrix item into a formula (6), wherein the time lag differential equation becomes an ordinary differential equation, and the expression is as follows:
Figure 286746DEST_PATH_IMAGE038
（15）
wherein:
Figure DEST_PATH_IMAGE039
（16）
meanwhile, define:
Figure 865757DEST_PATH_IMAGE040
（17）
the integral transform formula (17) includes:
Figure DEST_PATH_IMAGE041
（18）
the coefficient matrix term in equation (16) can be expressed as follows:
Figure 542374DEST_PATH_IMAGE042
Figure DEST_PATH_IMAGE043
（19）。
the concrete step (4) is that,
if matrix
Figure 774379DEST_PATH_IMAGE044
Is singular, then equation (15) can be:
Figure DEST_PATH_IMAGE045
（20）
defining a mapping relation:
Figure 901866DEST_PATH_IMAGE046
（21）
wherein:
Figure DEST_PATH_IMAGE047
（22）
then there are:
Figure 630656DEST_PATH_IMAGE048
（23）
therefore, the method comprises the following steps:
Figure DEST_PATH_IMAGE049
（24）
solving a system transfer matrix as follows:
Figure 536296DEST_PATH_IMAGE050
（25）。
the concrete step (5) is that,
calculating all characteristic values of transfer matrixQ(V) If the moduli of all the characteristic values are less than 1, the system is stable, otherwise, the system is unstable, that is:
Figure DEST_PATH_IMAGE051
。
the concrete step (6) is that,
in the method, during calculation, 7 items of numerical values in the transfer matrix need to be calculated in each discrete interval, and the method in the reference only needs to calculate 4 items of numerical values, so theoretically, the method in the reference is slightly higher than the method in the aspect of calculation efficiency; in order to solve the problem, a simple optimization method is proposed to improve the calculation speed; when calculating the eigenvalue of the transfer matrix, the traditional method needs to calculate the corresponding eigenvalue of all the main shaft rotating speeds and cutting depths, and then finds out the critical point with the corresponding eigenvalue of 1 to make a contour line, thereby drawing a milling stability lobe graph; the optimized prediction method comprises the steps of adding a judgment instruction into an algorithm, sequentially calculating a characteristic value of each cutting depth from the minimum cutting depth under the condition of a certain spindle rotation speed, judging whether the characteristic value is less than or equal to 1, continuing to calculate downwards if the characteristic value is less than or equal to 1, jumping out of the spindle rotation speed if the characteristic value is greater than 1, entering the next cycle, and calculating the characteristic value of another spindle rotation speed state; the method only calculates the characteristic values of the transfer matrixes in the stable area and the critical stable area, and omits the calculation of the characteristic values of the unstable area; and finally, drawing a milling stability lobe graph according to a calculation result of the critical stability region.
By adopting the technical scheme, the invention has the following beneficial effects:
(1) In the aspect of precision, a cubic spline and a Newton formula are adopted to carry out interpolation processing on a state term and a time lag term of a dynamic time lag differential equation respectively, a plurality of known terms are utilized to approximate an approximate unknown term, and meanwhile, the cubic spline uses the derivative term, so that the result is more accurate. Compared with the first-order fully-discrete method in the reference, the precision and the convergence rate of the method are obviously improved.
(2) In terms of calculation efficiency, the calculation of the unstable region is omitted by calculating the characteristic values of the stable region and the critical stable region, and the contour line can be drawn only by knowing the position of the critical stable region. In the experiment, under the same computing environment, the time required by the first-order full discretization method is 48s, the prediction method only needs 15s, the computing time is saved by about 70%, and the accuracy of the algorithm is not lost under the condition of ensuring high efficiency.
Drawings
FIG. 1 is a flow chart of the steps of the present invention;
FIG. 2 is a graph of the stability lobe of the present invention at a dip ratio of 0.05 for a single degree of freedom;
FIG. 3 is a graph of the stability lobe of the present invention at a dip ratio of 0.1 for a single degree of freedom;
FIG. 4 is a graph of the stability lobe of the present invention at an immersion ratio of 1 for a single degree of freedom.
Detailed Description
The feasibility and the effectiveness of the method are illustrated by taking a single-degree-of-freedom system as an example, as shown in fig. 1, the method for predicting the milling stability based on the spline-newton formula comprises the following operation steps:
(1) establishing a system dynamics model considering regenerative flutter
Figure 317913DEST_PATH_IMAGE052
（1）
In the formula (I), the compound is shown in the specification,q(t) is the system vibration response,m t 、z、w n respectively representing modal mass, relative damping factor and angular natural frequency,wthe depth of cut is indicated by a scale,h(t) is the cutting force coefficient equation, having:
Figure DEST_PATH_IMAGE053
（2）
in the formula (I), the compound is shown in the specification,k t 、k n all the coefficients of the cutting force are the coefficients of the cutting force,Nthe number of the cutter teeth is shown,
Figure 717933DEST_PATH_IMAGE054
is a firstjThe angular position of each cutter tooth, and having:
Figure DEST_PATH_IMAGE055
(3)
Figure 862082DEST_PATH_IMAGE056
is defined as follows:
Figure DEST_PATH_IMAGE057
(4)
wherein the content of the first and second substances,
Figure 469781DEST_PATH_IMAGE058
、
Figure DEST_PATH_IMAGE059
respectively represent the firstjThe angular position at which each tooth begins and ends cutting.
The forward milling comprises the following steps:
Figure 443422DEST_PATH_IMAGE060
(5)
the reverse milling is carried out by:
Figure DEST_PATH_IMAGE061
(6)
wherein a/D is the radial immersion ratio.
Order to
Figure 53657DEST_PATH_IMAGE062
，
Figure DEST_PATH_IMAGE063
Then, equation (1) can be as follows:
Figure 317892DEST_PATH_IMAGE064
(7)
a in the formula (7) 0 Is a constant coefficient matrix of the cutter teeth, A (t) is a periodic coefficient matrix of the cutter teeth, and has:
Figure DEST_PATH_IMAGE065
(8)
(2) discretizing time-lapse differential equations
In one tooth revolution periodTIn the interior, willTAre equally divided intomBetween cells, and the time length of each cell ishThen there isT=mhTaking a periodTWithin a certain time periodkh≤t≤(k+1)h，(k=0,1…, m) The initial condition of this period is known as
Figure 830913DEST_PATH_IMAGE066
Memory for recording
Figure DEST_PATH_IMAGE067
For equation (7), the direct integration over this interval is:
Figure 88588DEST_PATH_IMAGE068
(9)
let t = kh + h have:
Figure DEST_PATH_IMAGE069
(10)
the simplification is as follows:
Figure 79677DEST_PATH_IMAGE070
(11)
(3) interpolating equation discrete interval
By using
Figure DEST_PATH_IMAGE071
Four point to state items
Figure 607873DEST_PATH_IMAGE072
Cubic spline interpolation is performed. In addition, when spline interpolation is adopted, two additional constraint conditions are required to be added, and the constraint conditions can be selected
Figure DEST_PATH_IMAGE073
And
Figure 26216DEST_PATH_IMAGE074
two known conditions are as follows
Figure DEST_PATH_IMAGE075
（12）
Figure 505608DEST_PATH_IMAGE076
（13）
The status item can be expressed as:
Figure DEST_PATH_IMAGE077
（14）
wherein:
Figure 300388DEST_PATH_IMAGE078
(15)
Iis an identity matrix.
By using
Figure DEST_PATH_IMAGE079
Four point-to-time lag terms
Figure 680161DEST_PATH_IMAGE080
Newton interpolation is performed. Then there are:
Figure DEST_PATH_IMAGE081
（16）
wherein:
Figure 269405DEST_PATH_IMAGE082
（17）
the linear interpolation used for the periodic coefficient matrix is:
Figure DEST_PATH_IMAGE083
（18）
wherein:
Figure 501672DEST_PATH_IMAGE084
（19）
the interpolated state item, the interpolated time lag item and the interpolated periodic coefficient matrix item are brought into a formula (11), and the time lag differential equation becomes an ordinary differential equation at the moment, wherein the expression is as follows:
Figure DEST_PATH_IMAGE085
（20）
wherein:
Figure 100144DEST_PATH_IMAGE086
（21）
meanwhile, define:
Figure DEST_PATH_IMAGE087
（22）
the integral transformation formula (22) includes:
Figure 602931DEST_PATH_IMAGE088
（23）
the coefficient matrix term in equation (16) can be expressed as follows:
Figure DEST_PATH_IMAGE089
Figure 628656DEST_PATH_IMAGE090
（24）
(4) establishing a system transfer matrix
If matrix
Figure DEST_PATH_IMAGE091
Is singular, then equation (20) can be:
Figure 82640DEST_PATH_IMAGE092
（25）
defining a mapping relation:
Figure DEST_PATH_IMAGE093
（26）
wherein:
Figure 107972DEST_PATH_IMAGE094
（27）
then there are:
Figure DEST_PATH_IMAGE095
（28）
therefore, the method comprises the following steps:
Figure 448954DEST_PATH_IMAGE096
（29）
solving a system transfer matrix as follows:
Figure DEST_PATH_IMAGE097
（30）
(5) stability analysis of system based on Floquet theory
Calculating all characteristic values of transfer matrixQ(V) If the moduli of all the characteristic values are less than 1, the system is stable, otherwise, the system is unstable, that is:
Figure 894848DEST_PATH_IMAGE098
。
(6) optimization prediction method
In the calculation of the method, the values of 7 items in the transfer matrix need to be calculated in each discrete interval, and the method in the reference only needs to calculate 4 items, so theoretically, the method in the reference is slightly higher than the method in the aspect of calculation efficiency. To solve this problem, a simple optimization method is proposed to increase the calculation speed. When the eigenvalue of the transfer matrix is calculated in the traditional method, the corresponding eigenvalue of all the main shaft rotating speeds and cutting depths needs to be calculated, and then the critical point with the corresponding eigenvalue of 1 is found out to be taken as a contour line, so that a milling stability lobe graph is drawn. The optimization prediction method comprises the steps of adding a judgment instruction into an algorithm, sequentially calculating a characteristic value of each cutting depth from the minimum cutting depth under the condition of a certain spindle rotation speed, judging whether the characteristic value is less than or equal to 1, continuing to calculate downwards if the characteristic value is less than or equal to 1, jumping out of the spindle rotation speed if the characteristic value is greater than 1, entering the next cycle, and calculating the characteristic value of the other spindle rotation speed state. That is, the proposed prediction method calculates only the eigenvalues of the transfer matrix in the stable region and the critical stable region, and omits the eigenvalue calculation of the unstable region. And finally, drawing a milling stability lobe graph according to the calculation result of the critical stability region.
The process parameters were chosen in accordance with the data in the reference, where: the forward milling is carried out, the number of the cutter teeth is 2,
Figure DEST_PATH_IMAGE099
0.03993kg, 0.11 kg and,
Figure 852440DEST_PATH_IMAGE100
The tangential force coefficient Kt and the normal phase force coefficient Kn are respectively
Figure DEST_PATH_IMAGE101
And
Figure 277867DEST_PATH_IMAGE102
the rotation speed of the tool spindle is from 5000rpm to 25000rpm, and the cutting depth is from 0 to 10mm.
The formula and the data are used for calculating a milling stability lobe graph, the radial immersion ratio a/D is respectively selected to be 0.05, 0.1 and 1, and the milling stability lobe graph is obtained and is shown in figures 2, 3 and 4.
The present embodiment is not intended to limit the shape, material, structure, etc. of the present invention in any way, and any simple modification, equivalent change and modification made to the above embodiments according to the technical spirit of the present invention are within the scope of the technical solution of the present invention.

Claims (3)

1. A milling stability prediction method based on a spline-Newton formula is characterized in that: comprises the following steps of (a) preparing a solution,
(1) establishing a system dynamic model considering regeneration flutter;
(2) dispersing the time-lapse differential equation;
(3) interpolating the equation discrete interval;
(4) establishing a milling system transfer matrix;
(5) performing stability analysis on the milling system based on the Floque theory;
(6) optimizing the prediction method;
the specific process of the step (1) is that,
first establishing formula (1)
Figure FDA0004067360610000011
Wherein M, C and K respectively represent modal mass, relative damping and rigidity matrix of the cutter,
Figure FDA0004067360610000012
q (t) represents the acceleration, velocity and displacement vector of the tool, respectively, K c (T) is a periodic coefficient matrix, T is a single cutter tooth transmission period, T =60/N Ω, N is the number of cutter teeth of the cutter, and Ω is the rotation speed of the main shaft;
let X (t) = [ q (t) p (t)] T ，
Figure FDA0004067360610000013
By simplification, the formula (1) can be reduced to the following form:
Figure FDA0004067360610000014
a in the formula (2) 0 Is a constant coefficient matrix of the cutter teeth, A (t) is a periodic coefficient matrix of the cutter teeth, and has:
Figure FDA0004067360610000015
the step (2) is specifically that,
in the rotation period T of one cutter tooth, dividing T into m small partsInterval, and the time length of each small interval is h, then T = mh, a certain time period kh in the period T is not less than T and not more than (k + 1) h, (k =0,1 \8230;, m), and the initial condition of the time period is X k Record X k = X (kh), and the direct integral over this interval for equation (2) is:
Figure FDA0004067360610000021
let t = kh + h have:
Figure FDA0004067360610000022
the simplification is as follows:
Figure FDA0004067360610000023
the concrete step (3) is that,
by X k+1 ，X k ，X k-1 ，X k-2 Carrying out cubic spline interpolation on the state item X (kh + h-epsilon) by four points; in addition, when spline interpolation is adopted, two additional constraint conditions are required to be added, and the constraint conditions can be selected
Figure FDA0004067360610000024
And
Figure FDA0004067360610000025
two known conditions are as follows
Figure FDA0004067360610000026
Figure FDA0004067360610000027
The status item can be expressed as:
X(kh+h-ε)＝μ 1 X k+1 +μ 2 X k +μ 3 X k-1 +μ 4 X k-2 (9)
wherein:
Figure FDA0004067360610000028
Figure FDA0004067360610000029
i is an identity matrix;
by X k-m ，X k-m+1 ，X k-m+2 ，X k-m+3 Carrying out Newton interpolation on the time lag term X (kh + h-epsilon-T) by four points; then there are:
X(kh+h-ε-T)＝λ 1 X k-m +λ 2 X k-m+1 +λ 3 X k-m+2 +λ 4 X k-m+3 (11)
wherein:
Figure FDA00040673606100000210
Figure FDA00040673606100000211
the linear interpolation used for the periodic coefficient matrix is:
A(kh+h-ε)＝A u +A v ε (13)
wherein:
A u ＝A k+1 A v ＝(A k -A k+1 )/h (14)
and (3) bringing the interpolated state term, the interpolated time lag term and the interpolated periodic coefficient matrix term into an equation (6), wherein the time lag differential equation becomes an ordinary differential equation, and the expression is as follows:
Figure FDA0004067360610000031
wherein:
Figure FDA0004067360610000032
meanwhile, define:
Figure FDA0004067360610000033
the integral transformation formula (17) includes:
Figure FDA0004067360610000034
the coefficient matrix term in equation (16) can be expressed as follows:
Figure FDA0004067360610000035
Figure FDA0004067360610000036
Figure FDA0004067360610000037
Figure FDA0004067360610000038
Figure FDA0004067360610000039
Figure FDA00040673606100000310
Figure FDA0004067360610000041
the step (4) is specifically that,
if the matrix P k+1 Is singular, then equation (15) can be:
Figure FDA0004067360610000042
defining a mapping relation:
N k+1 ＝D k N k (21)
wherein:
N k ＝[X k ，X k-1 ，…，X k-m+1 ，X k-m ] T (22)
then there are:
Figure FDA0004067360610000043
therefore, the method comprises the following steps:
N m ＝VN 0 (24)
solving a system transfer matrix as follows:
V＝D m-1 D m-2 …D 0 (25)。
2. the spline-Newton formula-based milling stability prediction method of claim 1, wherein: the concrete step (5) is that,
calculating the modulus of all eigenvalues Q (V) of the transfer matrix, if the modulus of all eigenvalues is less than 1, the system is stable, otherwise, the system is unstable, namely:
Figure FDA0004067360610000051
3. the milling stability prediction method based on the spline-Newton formula according to claim 2, wherein: the concrete step (6) is that,
in the method, during calculation, 7 items of numerical values in the transfer matrix need to be calculated in each discrete interval, and the method in the reference only needs to calculate 4 items of numerical values, so theoretically, the method in the reference is slightly higher than the method in the aspect of calculation efficiency; in order to solve the problem, a simple optimization method is proposed to improve the calculation speed; when calculating the eigenvalue of the transfer matrix, the traditional method needs to calculate the corresponding eigenvalue of all the main shaft rotating speeds and cutting depths, and then finds out the critical point with the corresponding eigenvalue of 1 to make a contour line, thereby drawing a milling stability lobe graph; the optimization prediction method comprises the steps of adding a judgment instruction into an algorithm, sequentially calculating a characteristic value of each cutting depth from the minimum cutting depth under the condition of a certain spindle rotation speed, judging whether the characteristic value is less than or equal to 1, continuing to calculate downwards if the characteristic value is less than or equal to 1, jumping out of the spindle rotation speed if the characteristic value is greater than 1, entering the next cycle, and calculating the characteristic value of the other spindle rotation speed state; the method only calculates the characteristic values of the transfer matrixes in the stable area and the critical stable area, and omits the calculation of the characteristic values of the unstable area; and finally, drawing a milling stability lobe graph according to a calculation result of the critical stability region.
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