Method for analyzing machining stability in five-axis nc small radial cutting depth of ball end mill

Abstract

The present application belongs to the field of mechanical manufacturing, and proposes a machining stability analysis method for five-axis NC small radial cutting depth of ball end mill, establishes a five-axis NC machining dynamics model of ball end mill considering runout, develops a high-efficiency construction model of tool-workpiece contact area boundary under the influence of runout, develops a CWE separation technology for accurately and efficiently determining the time delay period corresponding to each cutting microelement, realizes efficient and high-precision prediction of machining stability under five-axis NC small radial cutting depth of ball end mill, and has great significance for realizing high-performance NC milling.

Description

Technical Field

[0001] This invention belongs to the field of mechanical manufacturing and relates to metal milling, especially a method for analyzing the stability of machining in small radial depth of cut with a five-axis CNC ball end mill. Background Technology

[0002] As a typical highly flexible machining method, five-axis CNC machining of ball end mills is widely used in the processing of key components of equipment such as aero-engines and compressors. Achieving chatter-free five-axis CNC machining of ball end mills is of great significance.

[0003] By using time-delay differential equations to describe the dynamic characteristics of the machining system, the critical stability curve, also known as the machining stability lobe diagram (SLD), can be plotted. This diagram divides the spindle speed and depth-of-cut plane into stable and unstable regions. By selecting parameters below the critical stability curve, chatter caused by self-excited vibrations can be effectively avoided. Therefore, it is of great significance to conduct efficient and high-precision dynamic analysis of five-axis CNC machining of ball end mills and to accurately construct the machining stability lobe diagram.

[0004] Numerous scholars both domestically and internationally have proposed various methods for analyzing the stability of five-axis CNC ball end mill machining. However, these methods primarily target ideal machining conditions. During small radial depth of cut, runout, especially radial runout, significantly impacts the machining process. It directly disrupts the cutting state between the cutter teeth, causing deformation in the contact area between the tool and the workpiece. Severe runout can even result in some cutter teeth not cutting, and the cutting system transforms from a single-delay ideal system into a multi-delay system. Therefore, developing a machining stability analysis method that considers tool runout for five-axis CNC small radial depth of cut is a critical issue that urgently needs to be addressed.

[0005] This invention focuses on the research of machining dynamics modeling and stability analysis methods for ball end mills with small radial depth of cut in five-axis CNC machining. It establishes a five-axis CNC machining dynamics model for ball end mills considering radial runout, develops an efficient model for constructing the boundary of the ball end mill-workpiece engagement (BWE) region under the influence of radial runout, and develops a technology for accurately and efficiently identifying the time delay period corresponding to each cutting micro-element and the cutting limit axial angle of the cutting teeth. This enables efficient and high-precision prediction of machining stability under small radial depth of cut in five-axis CNC machining with ball end mills. Summary of the Invention

[0006] This invention focuses on the research of machining dynamics modeling and stability analysis methods for ball end mills with small radial depth of cut in five-axis CNC machining. A five-axis CNC machining dynamics model for ball end mills considering runout is established. An efficient model for constructing the ball end mill-workpiece engagement (BWE) region under the influence of runout is developed. A technique for accurately and efficiently identifying the time delay period corresponding to each cutting micro-element and the cutting limit axial angle of the cutting teeth is developed, achieving efficient and high-precision prediction of machining stability under small radial depth of cut with ball end mills in five-axis CNC machining. Details are as follows:

[0007] 1. Dynamics modeling of five-axis CNC machining

[0008] 1.1 Mathematical Model of Tool Motion Trajectory

[0009] To clarify the tool condition under ideal machining conditions, a tooling configuration is established based on path planning information, as shown in the attached figure. Figure 1 The three coordinate systems shown are, firstly, the feed coordinate system OX. f Y f Z f The second coordinate system is the main axis coordinate system OX. s Y s Z s The third coordinate system is the workpiece coordinate system O. w -X w Y w Z w Point O is the center of the ball end mill in the path planning. In the coordinate system OX... f Y f Z f In the middle, axis Z f Perpendicular to the workpiece machining plane, axis X f Coinciding with the feed direction of the path planning; principal axis coordinate system OX s Y s Z s The X-axis can be rotated by the tool's tilt angle and yaw angle. f and Y f Obtain; axis Z t Coincident with the planned tool axis, coordinate system O w -X w Y w Z w By translating OX f Y f Z f From O to O w Obtain. Establish the tool coordinate system OX. t Y t Z t Tool coordinate system OX t Y tZ t and principal axis coordinate system OX s Y s Z s They are considered to overlap.

[0010] The i-th cutting element on the j-th cutting tooth has an axial angle of k, named P. Point P lies in the coordinate system OX. t Y t Z t The definition is as follows:

[0011]

[0012] in,

[0013] Where R represents the milling cutter radius, μ represents the helix angle of the cutter teeth, t is the spindle rotation time, k is the axial angle, ranging from [0, π / 2], θ is the spindle rotation angle, and ψ ji The radial hysteresis angle, φ ji (t) represents the rotation angle of the cutting element, N. f This refers to the number of cutting teeth.

[0014] 1.2 Dynamic Modeling Considering Radial Runout

[0015] (1) Dynamic cutting force modeling

[0016] Feed coordinate system OX f Y f Z f and tool coordinate system OX t Y t Z t The relationship is defined as follows:

[0017]

[0018] Where [x] f ,y f ,z f ] T and [x t ,y t ,z t ] T They are coordinate systems OX f Y f Z f and OX t Y t Z t The coordinates of the interior point, T t and T l Defined in equation (4), where α and β are the tool rake angle and the tool side rake angle, respectively.

[0019]

[0020] Tangential cutting force dF acting at point P t (φ ji (t)), radial cutting force dF r (φ ji (t) and axial cutting force dF a (φ ji (t) is represented as

[0021]

[0022] Among them, K tc K rc and K ac These are the tangential shear force coefficient, radial shear force coefficient, and axial shear force coefficient, respectively, and db is the cutting width, which is equal to db = Rdk.

[0023] In the feed coordinate system OX f Y f Z f Within the range, the dynamic displacement of point P (with an axial angle of k) is expressed as x(t)-x(tT(j,k)), y(t)-y(tT(j,k)) and z(t)-z(tT(j,k)), where T(j,k) is the time delay period corresponding to point P, abbreviated as T. j,k Through coordinate system transformation, the instantaneous undeformed cutting thickness h(j,k,t) corresponding to point P is expressed as:

[0024]

[0025] For equally spaced toothed cutting tools, T j,k = g(j,k)T, where T is the fundamental time delay period, and its magnitude is equal to 60 / Ω / N. f Ω is the spindle speed, N f Let g(j,k) be the number of cutting teeth, belonging to {1,2,…,N}. f} indicates that point P is cutting the material left by the g(j,k)th rake tooth, where g(j,k) simplifies to g j,k V is defined as

[0026] V=[sin k sinφ ji (t),sin k cosφ ji (t),-cosk] (7)

[0027] In the tool coordinate system OX t Y t Z t The cutting force acting on point P is expressed as

[0028]

[0029] Where the coordinate transformation matrix T m Defined as

[0030]

[0031] The transformation from the tool coordinate system to the workpiece coordinate system occurs in the workpiece coordinate system O. w -X w Y w Z w The cutting force acting on the tool is obtained internally.

[0032]

[0033] Where, k max ,j and k min ,j represents the maximum and minimum axial angles at time t where the j-th cutting tooth participates in the cutting.

[0034] 1.3 Multi-delay dynamic equations

[0035] In the workpiece coordinate system O w -X w Y w Z w The dynamic cutting force is expressed as

[0036]

[0037] Equation (10) simplifies to

[0038]

[0039] in

[0040]

[0041] Based on equation (11), the dynamic equation is established as follows:

[0042]

[0043] Where M, C, and K represent the modal mass, damping, and stiffness of the system, respectively.

[0044] 2. BWE analytical model considering radial runout

[0045] To determine the time delay period corresponding to the cutting micro-element and the limiting position angle of the cutting edge participating in the cutting, a BWE analytical model construction method considering radial runout is proposed, as shown in the appendix. Figure 2 As shown, the BWE boundary is mainly composed of lines a, b, c, and d; the tool coordinate system OX t Y t Z tand principal axis coordinate system OX s Y s Z s Overlapping together, the ball end mill's ball end portion is monotonically distributed along the cutter axis. The 3D BWE model is formed by this ball end mill in a plane perpendicular to the cutter axis, i.e., the coordinate system OX. t Y t The projection within the interior is used to replace the 3D modeling, transforming it into a 2D modeling, where the coordinate system OX... t Y t Tool coordinate system OX t Y t Z t In Z t The two-dimensional coordinate system when = 0; the relationship shown in equation (3) clarifies the BWE in the tool coordinate system OX. t Y t Rather than in the feed coordinate system OX f Y f Z f Mutual mapping, when BWE is constructed in OX f Y f When calculating the projection equation within the coordinate system OX, the BWE projection equation in the coordinate system OX can be constructed based on equation (3). t Y t The projection equation in, where the coordinate system OX f Y f For the feed coordinate system OX f Y f Z f In Z f A two-dimensional coordinate system when = 0; taking the reverse milling direction as an example, in coordinate system OX t Y t The method for constructing the boundary projection equations of BWE is as follows:

[0046] 2.1 BWE Model Construction Method Considering Radial Jump

[0047] Line A

[0048] Coordinate system OX f Y f Z f and OX t Y t Z t The origin O is established at the center of the tool sphere, and line a is in the coordinate system OX. f Y f The inner part is a circle, and point a is in the coordinate system OX. t Y t Z t The coordinates within are defined as (x) t ,y t ,z tLine a lies in coordinate system OX. t Y t The projection equation is

[0049] Ps = T l -1 ·T t -1 ·[x t y t z t ] T Ps(1) 2 +Ps(2) 2 =R² - (Rw) 2

[0050] Where w is the depth of cut.

[0051] Line B

[0052] The intersection of line a and line b is named point A, and in the coordinate system OX... t Y t Inside, line b is a curve; however, in the coordinate system OX... f Y f Inside, it is related to Y f A straight line whose axes coincide. Therefore, in coordinate system OX... f Y f Z f Within the defined space, determining the equation of line b will be very convenient, as shown in the attached diagram. Figure 3 As shown, the method for determining the projection equation of line b is as follows:

[0053] [1] The tool rotation angle corresponding to point A is determined by the transient cutting force signal;

[0054] [2] Project the cutting edge onto the coordinate system OX t Y t Based on the angle determined in [1], the rotation angle of the cutting edge is changed to determine the coordinate system OX of line a and the cutting edge. t Y t The intersection point is the point where point A is located in the coordinate system OX. t Y t Projection within;

[0055] [3] Based on the relationship shown in equation (3), the intersection point determined in [2] is transformed into the feed coordinate system OX. f Y f Z f In coordinate system OX f Y f Z f Within, determine the equation of the spatial line OA;

[0056] [4] The equation of OA is derived from the feed coordinate system OX by equation (3). f Y f Z f Transform to tool coordinate system OX t Y t Z t Obtain the coordinate system OX of line b. t Y t Projection equations within;

[0057] [5] Determine the projection equations of line b corresponding to other cutting edges by using [1]-[4];

[0058] Line D

[0059] The intersection of lines a and d is named point D. Under the condition of no runout, the point on line d is related to the tool radius R, the depth of cut w, and the cutting distance a under the condition of no runout. l Related; under runout conditions, the cutting distance is altered by the runout. (See attached image) Figure 3 As shown, the method for determining the projection equation of line d is as follows:

[0060] [1] The tool rotation angle corresponding to point D is determined by the transient cutting force signal;

[0061] [2] Based on the angle determined in [1], the rotation angle of the cutting edge is changed to determine the relationship between line a and the cutting edge in the coordinate system OX. t Y t The intersection point within the coordinate system is the point D in the coordinate system OX. t Y t Projection within;

[0062] [3] Based on the relationship shown in equation (3), the intersection point determined in [2] is transformed into the feed coordinate system OX. f Y f Z f The point is in coordinate system OX f Y f The internal projection coordinates are defined as (x′) d ,y′ d The cutting line spacing is disrupted by radial runout; after this disruption, the cutting line spacing is considered to be L = a. l -y′ d +y d , where y d When there is no jump, point D is in coordinate system OX f Y f The corresponding Y f The coordinate values ​​of the axes; combining lines a and e, we can determine the coordinate system of point D in coordinate system Ox. f y fProjected coordinates, line e represents the machining mark left by the previous toolpath on the workpiece surface, in coordinate system Ox f y f The projected coordinates of point D are shown in equation (14), where x is the projection of point D onto the coordinate system OX. f Y f The corresponding X f The y-axis coordinate value is the projection of point D onto the coordinate system OX. f Y f The corresponding Y f The axis coordinate values; by changing the cutting depth w, the points on line d can be determined, and then the coordinates of line d in coordinate system Ox can be obtained. f y f The projection equation;

[0063]

[0064] [4] The equation of line d is obtained from equation (3) in the feed coordinate system OX. f Y f Z f Transform to tool coordinate system OX t Y t Z t Obtain the coordinate system OX of line d. t Y t Projection equations within;

[0065] [5] By using [1]-[4], the projection equations of line d corresponding to other cutting edges are determined.

[0066] Line C

[0067] Line c in coordinate system OX t Y t The projection equation is obtained by connecting points B and C, where points B and C are in the coordinate system OX, respectively. t Y t Internal Y t negative direction of the axis towards Y t Looking along the positive axis, the first point projected from lines b and d;

[0068] The stability analysis of multi-time-delay processing is as follows.

[0069] 3. Stability Analysis Methods for Multi-Time-Delay Dynamic Systems

[0070] Equation (13) is expressed in the state-space format as follows:

[0071]

[0072] in,

[0073]

[0074] definition The output of the machining system at time t is obtained as follows

[0075]

[0076] Where t p For t=t p At time t, δ is a variable;

[0077] To establish the discrete output of the machining system, a numerical algorithm is introduced for discretization. First, the spindle rotation period T is... s =60 / Ω is divided into m parts, each with a length of τ. Then, g(δ) is at the interval [t] p ,t p+1 Decomposed into

[0078] g(δ)=r0+r1(δ-t p+1 (18)

[0079] Among them, t p+1 For t=t p+1 At that moment, [t] p ,t p+1 The length is τ; r0 and r1 are represented as

[0080] r0=g(t p+1 ), r1=[g(t p+1 )-g(t p )] / τ

[0081] r0 and r1 are simplified to

[0082] r0 = g p+1 , r1=(g p+1 -g p ) / τ (19)

[0083] Based on equations (17) and (18), v(t) p+1 ) represents

[0084] v(t p+1 )= T1v(t p )+M0r0+M1r1 (20)

[0085] in,

[0086] r in equation (20) i (i=0,1,2) is replaced by equation (19):

[0087] v(t p+1 )= T1v(tp )+G0g p+1 +G1g p (twenty one)

[0088] Among them, G0=M0+M1 / τ, G1=-M1 / τ;

[0089] because Equation (21) is converted to

[0090]

[0091] Among them, v p Represents v(t) p ), v p+1 Represents v(t) p+1 );

[0092] Equation (22) is converted to

[0093]

[0094] Among them, v p+1-i and They represent v(t) p+1-i ) and v(t) p+1-i -T j,k ), l j,k =fix(T j,k / τ)+1, fix is ​​a function that can determine the integer part of a number;

[0095] Equation (23) is converted to

[0096]

[0097] in,

[0098] When I-F0 is invertible, v p+1 Represented as

[0099]

[0100] Among them, F -1 =(I-F0) -1 When I-F0 is not invertible, the extended Moore-Penrose matrix is ​​used as an alternative.

[0101] To determine the stability of the machining system, the state transition matrix Φ is defined as follows:

[0102]

[0103] in,

[0104] For the j-th cutting tooth, C and Ej The dimension is equal to (2l) max +4), where l max =max(l j,k (k∈[0,π / 2]); F1 is v p coefficient; F i,j yes The coefficient.

[0105] By comparing the relationship between the maximum value of the eigenvalue modulus of matrix Φ and 1, the processing state can be determined. If I-F0 is not invertible, the extended Moore-Penrose matrix can be used as a substitute.

[0106] 4. Methods for determining the time delay period corresponding to the cutting micro-element and the limiting axial angle of the cutting teeth participating in the cutting.

[0107] To construct the state transition matrix shown in equation (26), the time delay period corresponding to the cutting micro-element required by equation (13) and the limit position angle of the cutting edge participating in the cutting need to be determined. The time delay period corresponding to the cutting micro-element is determined, the maximum and minimum axial angles of each cutting tooth participating in the cutting are identified, and a dynamic equation considering runout is established, with N... f Taking 2 as an example, the method for determining the time delay period corresponding to the cutting micro-element and the limiting axial angle of the cutting teeth participating in the cutting is as follows;

[0108] All cutting edges are projected onto the coordinate system OX. t Y t The cutting edge rotates around point O. For the j-th cutting tooth and the i-th cutting element, at each moment, it is determined whether the cutting element falls within the BWE boundary established by the corresponding cutting tooth. When the cutting element falls within the BWE boundary established by the corresponding cutting tooth, the cutting element is participating in cutting; when the cutting element does not fall within the BWE boundary established by the corresponding cutting tooth, the cutting element is not participating in cutting, and the coefficient g corresponding to the cutting element is... j,k =0;

[0109] When the cutting element is participating in cutting, determine whether the cutting element falls into the coupling region of the BWE model constructed by the first and second cutting teeth. If the cutting element falls into the coupling region, the coefficient g... j,k =1, record the axial angle corresponding to the cutting micro-element; when the cutting micro-element does not fall into the coupling region, the coefficient g j,k =2, record the axial angle corresponding to the cutting micro-element;

[0110] Using the above method, it is determined whether each cutting micro-element of the j-th cutting tooth participates in cutting, and the coefficient g corresponding to the cutting micro-element that participates in cutting. j,k and axial angle;

[0111] Through formula T j,k =gj,k T determines the time delay period corresponding to the i-th cutting micro-element of the j-th cutter tooth, where T is the basic time delay period, with a magnitude equal to 60 / Ω / N. f Ω represents the machine tool spindle speed, N f The number of cutting teeth;

[0112] When the coefficient g j,k =1, determine the j-th cutting tooth, g j,k The maximum and minimum values ​​of the axial angle recorded when = 1, the maximum value of the axial angle is the j-th cutter tooth time delay cycle g. j,k The maximum axial angle k at time T max The minimum value of the axial angle is the j-th cutter tooth delay period g. j,k The minimum axial angle k at time T min ,j;

[0113] When the coefficient g j,k =2, determine the j-th cutting tooth, g j,k The maximum and minimum values ​​of the axial angle recorded when = 2, the maximum value of the axial angle is the j-th cutter tooth time delay cycle g. j,k The maximum axial angle k at time T max The minimum value of the axial angle is the j-th cutter tooth delay period g. j,k The minimum axial angle k at time T min ,j.

[0114] The effects and benefits of this invention are as follows: Research on machining dynamics modeling and stability analysis methods for ball end mills with small radial depth of cut in five-axis CNC machining is conducted. A machining dynamics model for ball end mills with small radial depth of cut considering runout is established. An efficient model for constructing the tool-workpiece contact area boundary under the influence of runout is developed. A technique for accurately and efficiently determining the time delay period corresponding to each cutting micro-element is developed. This enables efficient prediction of machining stability under small radial depth of cut in five-axis CNC machining of ball end mills, which is of great significance for achieving high-performance CNC milling. Attached Figure Description

[0115] Figure 1 These are the machine tool spindle coordinate system, workpiece coordinate system, feed coordinate system, and tool coordinate system.

[0116] Figure 2 It consists of the boundary of the contact area between the ball end mill and the workpiece.

[0117] Figure 3 This describes the process of solving the projection equations of lines b and d, which define the boundary of the ball end mill-workpiece contact area.

[0118] Figure 4 It is a method for determining the time delay period corresponding to the cutting micro-element and the instantaneous cutting limit position of the cutting edge. Detailed Implementation

[0119] The specific embodiments of the present invention are described in detail below with reference to the technical solutions and accompanying drawings. The implementation process is as follows:

[0120] 1. Dynamics modeling of five-axis CNC machining

[0121] 1.1 Establishing a mathematical model of the tool motion trajectory

[0122] To clarify the tool condition under ideal machining conditions, a tooling configuration is established based on path planning information, as shown in the attached figure. Figure 1 The three coordinate systems shown are, firstly, the feed coordinate system OX. f Y f Z f The second coordinate system is the main axis coordinate system OX. s Y s Z s The third coordinate system is the workpiece coordinate system O. w -X w Y w Z w Point O is the center of the ball end mill in the plan. In OX... f Y f Z f In the middle, axis Z f Perpendicular to the workpiece machining plane, axis X f Coinciding with the feed direction of the path planning; principal axis coordinate system OX s Y s Z s The X-axis can be rotated by the tool tilt angle and the tool sidesweep angle. f and Y f Obtain; axis Z t The tool axis coincides with the path planning coordinate system O. w -X w Y w Z w By translating OX f Y f Z f From O to O w Obtain. Establish the tool coordinate system OX. t Y t Z t Tool coordinate system OX t Y t Z t and principal axis coordinate system OX s Y s Z s They are considered to overlap.

[0123] The i-th cutting element on the j-th cutting tooth has an axial angle of k, named P. Point P lies in the coordinate system OX. t Y tZ t The definition is as follows:

[0124]

[0125] in,

[0126] Where R represents the milling cutter radius, μ represents the helix angle of the cutter teeth, t is the spindle rotation time, k is the axial angle, ranging from [0, π / 2], θ is the spindle rotation angle, and ψ ji The radial hysteresis angle, φ ji (t) represents the rotation angle of the cutting element, N. f This refers to the number of cutting teeth.

[0127] 1.2 Dynamic Modeling Considering Radial Runout

[0128] (1) Dynamic cutting force modeling

[0129] Coordinate system OX f Y f Z f and coordinate system OX t Y t Z t The relationship is defined as follows:

[0130]

[0131] Where [x] f ,y f ,z f ] T and [x t ,y t ,z t ] T They are coordinate systems OX f Y f Z f and OX t Y t Z t The coordinates of the interior point, T t and T l Defined in equation (4), where α and β are the tool rake angle and the tool side rake angle, respectively.

[0132]

[0133] Tangential cutting force dF acting at point P t (φ ji (t)), radial cutting force dF r (φ ji (t) and axial cutting force dF a (φ ji (t) is represented as

[0134]

[0135] Among them, K tc K rc and K ac These are the tangential shear force coefficient, radial shear force coefficient, and axial shear force coefficient, respectively, and db is the cutting width, which is equal to db = Rdk.

[0136] In coordinate system OX f Y f Z f Within the range, the dynamic displacement of point P (with an axial angle of k) is expressed as x(t)-x(tT(j,k)), y(t)-y(tT(j,k)) and z(t)-z(tT(j,k)), where T(j,k) is the time delay period corresponding to point P, abbreviated as T. j,k By coordinate system transformation, the instantaneous undeformed cutting thickness h(j,k,t) corresponding to point P is expressed as:

[0137]

[0138] For equally spaced toothed cutting tools, T j,k = g(j,k)T, where T is the fundamental time delay period, and its magnitude is equal to 60 / Ω / N. f Ω is the spindle speed, N f Let g(j,k) be the number of cutting teeth, belonging to {1,2,…,N}. f} indicates that point P is cutting the material left by the g(j,k)th rake tooth, where g(j,k) simplifies to g j,k V is defined as

[0139] V=[sin k sinφ ji (t),sin k cosφ ji (t),-cosk] (33)

[0140] In the tool coordinate system OX t Y t Z t The cutting force acting on point P is expressed as

[0141]

[0142] Where the coordinate transformation matrix T m Defined as

[0143]

[0144] The transformation from the tool coordinate system to the workpiece coordinate system occurs in the workpiece coordinate system O. w -Xw Y w Z w The cutting force acting on the tool is obtained internally.

[0145]

[0146] Where, k max ,j and k min ,j represents the maximum and minimum axial angles at time t where the j-th cutting tooth participates in the cutting.

[0147] 1.3 Multi-delay dynamic equations

[0148] In coordinate system O w -X w Y w Z w The dynamic cutting force is expressed as

[0149]

[0150] Equation (36) simplifies to

[0151]

[0152] in

[0153]

[0154] Based on equation (38), the dynamic equation is established as follows:

[0155]

[0156] Where M, C, and K represent the modal mass, damping, and stiffness of the system, respectively.

[0157] 2. Considering the BWE analytical model with hopping.

[0158] A method for constructing a BWE analytical model of the tool-workpiece contact area considering runout is proposed, as shown in the appendix. Figure 2 As shown, the BWE boundary is mainly composed of lines a, b, c, and d; the tool coordinate system OX t Y t Z t and principal axis coordinate system OX s Y s Z s Overlapping together, the ball end mill's ball end portion is monotonically distributed along the cutter axis. The 3D BWE model is formed by this ball end mill in a plane perpendicular to the cutter axis, i.e., the coordinate system OX. t Y t The projection within the interior is used to replace the 3D modeling, transforming it into a 2D modeling, where the coordinate system OX... t Y tTool coordinate system OX t Y t Z t In Z t The two-dimensional coordinate system when = 0; the relationship shown in equation (3) clarifies the BWE in the tool coordinate system OX. t Y t Z t Rather than in the feed coordinate system OX f Y f Z f Mutual mapping, when BWE is constructed in OX f Y f When calculating the projection equation within the coordinate system OX, the BWE projection equation in the coordinate system OX can be constructed based on equation (3). t Y t The projection equation in, where the coordinate system OX f Y f For the feed coordinate system OX f Y f Z f In Z f A two-dimensional coordinate system when = 0; taking the reverse milling direction as the object, in the coordinate system OX t Y t The method for constructing the boundary projection equations of BWE is as follows:

[0159] 2.1 BWE Analytical Model Considering Radial Runout

[0160] Line A

[0161] Coordinate system OX f Y f Z f and OX t Y t Z t The origin O is established at the center of the tool sphere, and line a is in the coordinate system OX. f Y f The inner part is a circle, and point a is in the coordinate system OX. t Y t Z t The coordinates within are defined as (x) t ,y t ,z t Line a lies in coordinate system OX. t Y t The projection equation is

[0162] Ps = T l -1 ·T t -1 ·[x t y t z t ]T Ps(1) 2 +Ps(2) 2 =R² - (Rw) 2

[0163] Where w is the depth of cut.

[0164] Line B

[0165] The intersection of line a and line b is named point A, and in the coordinate system OX... t Y t Inside, line a is a curve; however, in the coordinate system OX... f Y f Inside, it is related to Y f A straight line whose axes coincide. Therefore, in the coordinate system OX... f Y f Z f Within the defined space, determining the equation of line b is very convenient. Based on this viewpoint, as shown in the appendix... Figure 3 As shown, the solution process for the projection equation of line b is as follows:

[0166] [1] The tool rotation angle corresponding to point A is determined by the transient cutting force signal;

[0167] [2] Project the cutting edge onto the coordinate system OX t Y t Based on the angle determined in [1], the rotation angle of the cutting edge is changed to determine the coordinate system OX of line a and the cutting edge. t Y t The intersection point is the point where point A is located in the coordinate system OX. t Y t Projection within;

[0168] [3] Based on the relationship shown in equation (3), the intersection point determined in [2] is transformed into the feed coordinate system OX. f Y f Z f In coordinate system OX f Y f Z f Within, determine the equation of the spatial line OA;

[0169] [4] The equation of OA is derived from the feed coordinate system OX by equation (3). f Y f Z f Transform to tool coordinate system OX t Y t Z t Obtain the coordinate system OX of line b. t Y t Projection equations within;

[0170] [5] Determine the projection equations of line b corresponding to other cutting edges by using [1]-[4];

[0171] Line D

[0172] The intersection of lines a and d is named point D. Under the condition of no runout, the point on line d is related to the tool radius R, the depth of cut w, and the cutting distance a under the condition of no runout. l Related; under runout conditions, the cutting distance is altered by the runout. (See attached image) Figure 3 As shown, the solution process for the projection equation of line d is as follows:

[0173] [1] The tool rotation angle corresponding to point D is determined by the transient cutting force signal;

[0174] [2] Based on the angle determined in [1], the rotation angle of the cutting edge is changed to determine the relationship between line a and the cutting edge in the coordinate system OX. t Y t The intersection point within the coordinate system is the point D in the coordinate system OX. t Y t Projection within;

[0175] [3] Based on the relationship shown in equation (3), the intersection point determined in [2] is transformed into the feed coordinate system OX. f Y f Z f The point is in coordinate system OX f Y f The internal projection coordinates are defined as (x′) d ,y′ d The cutting line spacing is disrupted by radial runout; after this disruption, the cutting line spacing is considered to be L = a. l -y′ d +y d , where y d When there is no jump, point D is in coordinate system OX f Y f The corresponding Y f The coordinate values ​​of the axes; combining lines a and e, we can determine the coordinate system of point D in coordinate system Ox. f y f Projected coordinates, line e represents the machining mark left by the previous toolpath on the workpiece surface, in coordinate system Ox f y f The projected coordinates of point D are shown in equation (14), where x is the projection of point D onto the coordinate system OX. f Y f The corresponding X f The y-axis coordinate value is the projection of point D onto the coordinate system OX. f Y f The corresponding Y fThe axis coordinate values; by changing the cutting depth w, the points on line d can be determined, and then the coordinates of line d in coordinate system Ox can be obtained. f y f The projection equation;

[0176]

[0177] [4] The equation of line d is obtained from equation (3) in the feed coordinate system OX. f Y f Z f Transform to tool coordinate system OX t Y t Z t Obtain the coordinate system OX of line d. t Y t Projection equations within;

[0178] [5] By using [1]-[4], the projection equations of line d corresponding to other cutting edges are determined.

[0179] Line C

[0180] Line c in coordinate system OX t Y t The projection equation is obtained by connecting points B and C, where points B and C are in the coordinate system OX, respectively. t Y t Internal Y t negative direction of the axis towards Y t Looking along the positive axis, the first point projected from lines b and d;

[0181] The time delay period corresponding to the cutting micro-element and the cutting limit axial angle of the cutting tooth are obtained from BWE, as shown below.

[0182] 4. Methods for determining the time delay period corresponding to the cutting micro-element and the limiting axial angle of the cutting teeth participating in the cutting.

[0183] Equation (39) requires the limiting axial angle of the cutting teeth and g to participate in the cutting. j,k It needs to be confirmed that, below, N f Taking 2 as an example, the method for determining the time delay period corresponding to the cutting micro-element and the limiting axial angle of the cutting teeth participating in the cutting is as follows;

[0184] First, as attached Figure 4 As shown, all cutting edges are projected onto the coordinate system OX. t Y tThe cutting edge rotates around point O. For the j-th tooth and the i-th cutting element, at each moment, it is determined whether the cutting element falls into the BWE model established for the corresponding tooth. When the cutting element falls into the BWE model established for the corresponding tooth, the cutting element is participating in cutting; when the cutting element does not fall into the BWE model established for the corresponding tooth, the cutting element is not participating in cutting, and the coefficient g corresponding to the cutting element is... j,k =0;

[0185] When the cutting element is participating in cutting, determine whether the cutting element falls into the coupling region of the BWE model constructed by the first and second cutting teeth. If the cutting element falls into the coupling region, the coefficient g... j,k =1, record the axial angle corresponding to the cutting micro-element; when the cutting micro-element does not fall into the coupling region, the coefficient g j,k =2, record the axial angle corresponding to the cutting micro-element;

[0186] Using the above method, it is determined whether each cutting micro-element of the j-th cutting tooth participates in cutting, and the coefficient g corresponding to the cutting micro-element that participates in cutting. j,k and axial angle;

[0187] Through formula T j,k =g j,k T determines the time delay period corresponding to the i-th cutting micro-element of the j-th cutter tooth, where T is the basic time delay period, with a magnitude equal to 60 / Ω / N. f Ω represents the machine tool spindle speed, N f The number of cutting teeth;

[0188] When the coefficient g j,k =1, determine the j-th cutting tooth, g j,k The maximum and minimum values ​​of the axial angle recorded when = 1, the maximum value of the axial angle is the j-th cutter tooth time delay cycle g. j,k The maximum axial angle k at time T max The minimum value of the axial angle is the j-th cutter tooth delay period g. j,k The minimum axial angle k at time T min ,j;

[0189] When the coefficient g j,k =2, determine the j-th cutting tooth, g j,k The maximum and minimum values ​​of the axial angle recorded when = 2, the maximum value of the axial angle is the j-th cutter tooth time delay cycle g. j,k The maximum axial angle k at time T max The minimum value of the axial angle is the j-th cutter tooth delay period g. j,k The minimum axial angle k at time T min ,j.

[0190] Determine the time delay period corresponding to the cutting micro-element, identify the maximum and minimum axial angles of each cutting tooth participating in the cutting, establish dynamic equations considering runout, and conduct stability analysis of the multi-time-delay dynamic system.

[0191] 3. Stability Analysis Methods for Multi-Time-Delay Dynamic Systems

[0192] Equation (39) is expressed in the state-space format as follows:

[0193]

[0194] in,

[0195]

[0196] definition The system output is obtained as follows

[0197]

[0198] Where t p For t=t p At time t, δ is a variable;

[0199] To establish the discrete output of the machining system, a numerical algorithm was introduced for discretization. First, the spindle rotation period T was... s =60 / Ω is divided into m parts, each with a length of τ. Then, g(δ) is at the interval [t] p ,t p+1 Decomposed into

[0200] g(δ)=r0+r1(δ-t p+1 (44)

[0201] Among them, [t p ,t p+1 The length is τ. r0 and r1 are represented as...

[0202] r0=g(t p+1 ), r1=[g(t p+1 )-g(t p )] / τ

[0203] r0 and r1 are simplified to

[0204] r0 = g p+1 , r1=(g p+1 -g p ) / τ (45)

[0205] Based on equations (43) and (44), v(t) p+1 ) represents

[0206] v(tp+1 )=T1v(t p )+M0r0+M1r1 (46)

[0207] in,

[0208] r in equation (46) i (i=0,1,2) is replaced by equation (45):

[0209] v(t p+1 )=T1v(t p )+G0g p+1 +G1g p (47)

[0210] Among them, G0=M0+M1 / τ, G1=-M1 / τ.

[0211] because Equation (47) is converted to

[0212]

[0213] Among them, v p Represents v(t) p ), v p+1 Represents v(t) p+1 );

[0214] Equation (48) is converted to

[0215]

[0216] Among them, v p+1-i and They represent v(t) p+1-i ) and v(t) p+1-i -T j,k ), l j,k =fix(T j,k / τ)+1, fix is ​​a function that can determine the integer part of a number.

[0217] Equation (49) is converted to

[0218]

[0219] in,

[0220] If I-F0 is invertible, v p+1 Represented as

[0221]

[0222] Among them, F -1 =(I-F0) -1.

[0223] To determine the stability of the machining system, the transition matrix Φ is defined as follows:

[0224]

[0225] in,

[0226] For the j-th cutting tooth, C and E j The dimension is equal to (2l) max +4), where l max =max(l j,k (k∈[0,π / 2]); F1 is v p coefficient; F i,j yes The coefficient.

[0227] By comparing the maximum value of the eigenvalue modulus of the state transition matrix Φ with 1, the processing state can be determined. If I-F0 is not invertible, the extended Moore-Penrose matrix can be used as a substitute.

[0228] By discretizing the spindle speed and depth of cut range, a state transition matrix Φ corresponding to each combination of speed and depth of cut is constructed. By comparing the maximum value of the eigenvalue modulus of the state transition matrix Φ with 1, the machining state corresponding to the process parameter is determined, thereby constructing a machining stability lobe diagram to achieve chatter-free milling process parameter optimization. The provided method for machining stability analysis of small radial depth of cut in five-axis CNC ball end mills has the advantages of high efficiency and high precision, and is of great significance for realizing high-performance CNC milling.

Claims

1. A method for stability analysis of small radial depth of cut machining using a five-axis CNC ball end mill, characterized by comprising the following steps: Step 1.1: Five-axis CNC machining dynamics modeling Step 1.1.1) Establish a mathematical model of the ball end mill's motion trajectory. Establish three coordinate systems, the first of which is the feed coordinate system. O - X f Y f Z f The second coordinate system is the principal coordinate system. O - X s Y s Z s The third coordinate system is the workpiece coordinate system. O w - X w Y w Z w ,point O The ball center of the ball end mill for path planning; in the feed coordinate system O - X f Y f Z f In the middle, axis Z f Perpendicular to the machined surface of the workpiece, axis X f Coinciding with the feed direction of the path planning; principal axis coordinate system O - X s Y s Z s Based on the tool rake angle and tool sideslip angle, the feed coordinate system is rotated. O - X f Y f Z f axis X f and shaft Y f Obtain; workpiece coordinate system O w - X wY w Z w By translating and feeding the coordinate system O - X f Y f Z f From point O Time O w Obtain; Establish tool coordinate system O - X t Y t Z t Tool coordinate system O - X t Y t Z t and principal coordinate system O - X s Y s Z s Coincident, axis Z t The tool axis coincides with the path planning; Ball end mill # j The first on the blade tooth i A cutting micro-element, named P The corresponding axial angle is k , P Point in tool coordinate system O - X t Y t Z t The definition is as follows: (1) in, (2) in, R Represents the radius of the milling cutter. μ Represents the helix angle of the cutting teeth. t The spindle rotation time of the machine tool. k The axial angle varies from [0, π / 2]. The rotation angle of the machine tool spindle. ψ ji It is the radial hysteresis angle. The rotation angle of the cutting micro-element.N f The number of cutting teeth; Step 1.1.2) Modeling milling dynamics considering runout Feed coordinate system O - X f Y f Z f and tool coordinate system O - X t Y t Z t The relationship is defined as follows: (3) in[ x f , y f , z f ] T and[ x t , y t , z t ] T feed coordinate system O - X f Y f Z f and tool coordinate system O - X t Y t Z t The coordinates of the interior point T t and T l Defined in equation (4), where α and β These are the tool tilt angle and the side tilt angle, respectively. , (4) Acting on P Tangential cutting force d at the point F t Radial cutting force d F r and axial cutting force dF a Represented as (5) in, K tc , K rc and K ac These are the tangential cutting force coefficient, radial cutting force coefficient, and axial shear force coefficient, respectively. b d is the cutting width. b=R d k ; In the feed coordinate system O - X f Y f Z f Inside, P The dynamic displacement of a point is expressed as x ( t )- x ( t - T ( j , k )), y ( t )- y ( t - T ( j , k ))and z ( t )- z ( t - T ( j , k )),in T ( j , k ) for P The time delay period corresponding to the point is abbreviated as T j,k Through the transformation between the feed coordinate system and the tool coordinate system, P The instantaneous undeformed cutting thickness corresponding to the point h ( j , k , t ) represents (6) For cutting tools with equally spaced teeth, T j, k = g ( j ,k ) T ,in, T The base time delay period is equal to 60 / Ω / N f , Ω The spindle speed of the machine tool. N f The number of teeth. g ( j , k ), belonging to {1, 2,…, N f},express P The point is cutting the first g ( j , k The material left by the front cutter teeth g ( j , k Simplified to g j, k ; V Defined as (7) In the tool coordinate system O - X t Y t Z t , acting on P The cutting force at a point is expressed as (8) in, T m Defined as Transformation from tool coordinate system to workpiece coordinate system, in workpiece coordinate system O w - X w Y w Z w The internal dynamic cutting force acting on the ball end mill is obtained. for (9) in, k max , j and k min , j They are respectively t Time of the first jThe maximum and minimum axial angles of each cutting tooth involved in the cutting process; In the workpiece coordinate system O w - X w Y w Z w middle, Represented as (10) Equation (10) simplifies to (11) in , (12) Based on equation (12), the dynamic equation considering the jumping is established as follows: (13) in, M , C and K These represent the modal mass, damping, and stiffness of the processing system, respectively. Step 1.2: Analytical model of the tool-workpiece contact area considering runout. The tool-workpiece contact area model (CWE) mainly consists of... a Line 1, b Line 1, c Line 1 and d Composition of lines; Tool coordinate system O - X t Y t Z t and principal coordinate system O - X s Y s Z s When they overlap, the ball end mill's ball end portion is monotonically distributed along the cutter axis. The 3D CWE model is based on this in a coordinate system perpendicular to the cutter axis. O - X t Y t The projection within the coordinate system is used to replace the 3D modeling, transforming it into a 2D modeling, where the coordinate system... O - X t Y t Tool coordinate system O- X t Y t Z t exist Z t The two-dimensional coordinate system when =0; the relationship shown in equation (3) clarifies the CWE in the tool coordinate system O - X t Y t Z t The projection within and its position in the feed coordinate system O - X f Y f Z f The projections within the CWE are mutually mapped, when the CWE is constructed in O - X f Y f When calculating the projection equation within the coordinate system, the CWE can be constructed based on equation (3). O - X t Y t The projection equations in the coordinate system, where O - X f Y f For the feed coordinate system O - X f Y f Z f exist Z f A two-dimensional coordinate system when =0; the CWE boundary in the coordinate system. O - X t Y t The method for constructing the projection equations in the code is as follows: 1.2.1) a Line 1 a The point on line 6 is in the tool coordinate system O - X t Y t Z t Internal definition is ( x t , y t , z t ), a Line 1 in coordinate system O - X t Y t The projection equation is in, w This refers to the depth of cut. 1.2.2) b Line 1 and d Line 1 b Line 1 and d Line 1 in coordinate system O - X t Y t The projection equation is determined by the transient cutting force signal obtained from the test; 1.2.3) c Line 1 c Line 1 in coordinate system O - X t Y t The projection equation is connected by B Point and C Points to obtain, among which B Point and C The points are respectively in the coordinate system O - X t Y t Internally Y t negative direction of axis Y t Looking at the positive axis, b Line 1 and d The first point of the line projection; Step 1.3: Method for determining the time delay period corresponding to the cutting micro-element and the limiting axial angle of the cutting teeth participating in the cutting. The time delay period corresponding to the cutting micro-element is determined, the maximum and minimum axial angles of each cutting tooth participating in the cutting are identified, and a dynamic equation considering runout is established. The methods for determining the time delay period corresponding to the cutting micro-element and the limiting axial angles of the cutting tooth participating in the cutting are as follows; All cutting edges are projected onto the coordinate system. O - X t Y t The cutting edge is around O Rotate the point, for the th j The first blade tooth i Each cutting micro-element is used to determine whether it falls into the CWE model established by the corresponding cutting tooth at each time step. When a cutting element falls into the CWE model established by its corresponding cutting tooth, the cutting element is participating in cutting; when a cutting element does not fall into the CWE model established by its corresponding cutting tooth, the cutting element is not participating in cutting, and the coefficient corresponding to the cutting element... g j,k =0; When the cutting element is participating in cutting, determine whether it falls into the coupling region of the CWE model constructed by the first and second cutting teeth. If the cutting element falls into the coupling region, the coefficient... g j,k =1, record the axial angle corresponding to the cutting element; when the cutting element does not fall into the coupling region, the coefficient is... g j,k =2, record the axial angle corresponding to the cutting micro-element; Using the above method, determine the first j Whether each cutting micro-element of each cutting tooth participates in cutting, and the coefficient corresponding to the cutting micro-element that participates in cutting. g j,k and axial angle; Through T j, k = g j,k T Determine the first j The first blade tooth i The time delay period corresponding to each cutting micro-element, where... T The base time delay period is equal to 60 / Ω / N f , Ω The spindle speed of the machine tool. N f The number of cutting teeth; When the coefficient g j,k =1, determine the first j One blade tooth, g j,k The maximum and minimum values ​​of the axial angle recorded when =1, the maximum value of the axial angle is the first... j Each cutting tooth delay cycle g j,k T Maximum axial angle at time k max , j The minimum value of the axial angle is the first... j Each cutting tooth delay cycle g j,k T Minimum axial angle at time k min , j ; When the coefficient g j,k =2, determine the first j One blade tooth, g j,k The maximum and minimum values ​​of the axial angle recorded when =2, the maximum value of the axial angle is the first... j Each cutting tooth delay cycle g j,k T Maximum axial angle at time k max , j The minimum value of the axial angle is the first... j Each cutting tooth delay cycle g j,k T Minimum axial angle at time k min , j ; Step 1.4, Method for constructing processing stability leaflet diagrams Discretize the spindle speed and depth of cut range to construct the state transition matrix corresponding to each combination of spindle speed and depth of cut. Φ By comparing the state transition matrices Φ By determining the relationship between the maximum value of the eigenvalue modulus and 1, the machining state corresponding to the milling process parameter is determined, and a machining stability lobe diagram is constructed to achieve chatter-free milling process parameter optimization.

2. The method for analyzing the machining stability of a ball end mill in a five-axis CNC small radial depth of cut according to claim 1, characterized in that, in step 1.2... b Line 1 and d The specific method for determining the projection equation of line number is as follows: 2.1) b Line 1 a Line 1 and b The intersection of Line 1 is named A Points, obtain b Line 1 in coordinate system O - X t Y t The specific process of the projection equation within is as follows: 2.1.1) Determined from the transient cutting force signal obtained by testing A The tool rotation angle corresponding to the point; 2.1.2) Project the cutting edge onto the coordinate system O - X t Y t Based on the angle determined in 2.1.1), the blade rotation angle is changed to determine... a The line and the cutting edge in the coordinate system O - X t Y t The intersection point, that intersection point is A Point in coordinate system O - X t Y t Projection within; 2.1.3) Using the relationship shown in equation (3), the intersection point determined in 2.1.2) is transformed into the feed coordinate system O- X f Y f Z f In the feed coordinate system O- X f Y f Z f Within, determine the equation of the spatial line OA; 2.1.4) The equation of the spatial line OA is derived from equation (3) by the feed coordinate system O- X f Y f Z f Transform to tool coordinate system O - X t Y t Z t ,get b Line 1 in coordinate system O - X t Y t Projection equations within; 2.1.5) Using 2.1.1)-2.1.4), determine the corresponding cutting edges of other cutting edges. b Projection equation of line number; 2.2) d Line 1 a Line 1 and d The intersection of Line 1 is named D Point, without jumping, d Line and tool radius R Cutting depth w Cutting distance when there is no runout a l Regarding the issue of runout, the cutting distance is altered by the runout; solution d Line 1 in coordinate system O - X t Y t The projection equation within is as follows: 2.2.1) Determined from the transient cutting force signal obtained by testing D The tool rotation angle corresponding to the point; 2.2.2) Based on the angle determined in 2.2.1), the blade rotation angle is changed to determine... a The line and the cutting edge in the coordinate system O - X t Y t The intersection point within, that intersection point is D Point in coordinate system O - X t Y t Projection within; 2.2.3) Using the relationship shown in equation (3), the intersection point determined in 2.2.2) is transformed into the feed coordinate system O- X f Y f Z f The intersection point is in coordinate system O- X f Y f Internal projection coordinates are defined as The cutting line spacing is disrupted by radial runout; after this disruption, it is considered the cutting line spacing and determined as follows. L = a l - + ,in y d When there is no jumping D Point in coordinate system O - X f Y f Corresponding Y f Axis coordinate values; a Line 1 and e The combination of line numbers can determine D Point in coordinate system O - x f y f Projected coordinates e Line 1 represents the machining mark left by the previous toolpath on the workpiece surface, in the coordinate system. O - x f y f Inside D The projected coordinates of the point are shown in equation (14), where x for D Point projection in coordinate system O - X f Y f Corresponding X f axis coordinate values, y for D Point projection on coordinate system O - X f Y f Corresponding Y f Axis coordinate values; by changing the depth of cut w ,Sure d The points on line number, thus obtaining d Line 1 in coordinate system O - x f y f The projection equation; (14) 2.2.4) From equation (3) d The equation of line number is derived from the feed coordinate system O- X f Y f Z f Transform to tool coordinate system O - X t Y t Z t ,get d Line 1 in coordinate system O - X t Y t Projection equations within; 2.2.5) Using 2.2.1)-2.2.4), determine the corresponding cutting edges of other cutting edges. d Line 1 projection equation.

3. The method for analyzing the machining stability of a ball end mill in a small radial depth of cut under five-axis CNC machining according to claim 1, characterized in that, In step 1.3), the number of cutting teeth N f =2.

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