CN113297696A - Modeling method of ball end mill static milling force based on semi-analytical method - Google Patents

Modeling method of ball end mill static milling force based on semi-analytical method Download PDF

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CN113297696A
CN113297696A CN202110578192.4A CN202110578192A CN113297696A CN 113297696 A CN113297696 A CN 113297696A CN 202110578192 A CN202110578192 A CN 202110578192A CN 113297696 A CN113297696 A CN 113297696A
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cutter
cutter tooth
coordinate system
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tooth
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CN113297696B (en
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董永亨
李淑娟
张倩
洪贤涛
李鹏阳
李旗
李言
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Xian University of Technology
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    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design
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    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
    • G—PHYSICS
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    • G06F—ELECTRIC DIGITAL DATA PROCESSING
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    • G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces

Abstract

The invention discloses a modeling method of a ball end mill static milling force based on a semi-analytical method, which comprises the following steps: respectively establishing a local coordinate system of cutter teeth j, a ball end mill coordinate system, a main shaft follow-up coordinate system, a cutter instantaneous feed coordinate system and a workpiece coordinate system, and obtaining a trajectory equation of any point on the cutter teeth j in the workpiece coordinate system in the ball end mill machining process based on a homogeneous coordinate transformation principle; establishing a infinitesimal cutting force model of the knife tooth infinitesimal; identifying a cutter-working contact area; calculating the instantaneous undeformed chip thickness; identifying to obtain the coefficient of cutting force. The method comprises the steps of establishing a motion track of cutter teeth in the ball-end milling cutter machining process based on a homogeneous coordinate transformation principle, providing a cutting force coefficient identification method, a semi-analytic identification method of a cutter-cutter contact area and a method for solving the undeformed cutting thickness according to the actual milling condition of the ball-end milling cutter, and providing a basis for follow-up research and a reference basis for selection of machining parameters in the actual machining process.

Description

Modeling method of ball end mill static milling force based on semi-analytical method
Technical Field
The invention belongs to the technical field of machining, and relates to a modeling method of a ball end mill static milling force based on a semi-analytical method.
Background
The ball-end milling cutter is widely applied to milling of important surfaces of related parts in industries such as molds, automobiles, aerospace and the like, deep research on a milling mechanism of the ball-end milling cutter is of great significance for improving product quality, however, modeling of static milling force is a key point of research on a cutting mechanism, is a basis and a key of subsequent dynamic modeling, and is a key basis for selection and optimization of cutting parameters.
The identification of the cutter-cutter contact area is a key link of static milling force modeling, the accuracy and the calculation efficiency of the identification directly influence the accuracy and the efficiency of static milling force prediction, however, the edge shape of the cutter tooth of the ball end mill is complex, and the identification difficulty of the cutter-cutter contact area is high due to the influence of factors such as attitude adjustment and run-out error. The Z-MAP discrete method can better judge the contact state of the cutter teeth through the idea of infinitesimal dispersion, improves the identification precision of the cutter-working contact area, but has the problem of balance of precision and efficiency, and influences the application of follow-up research. The learners identify the cutter-tool contact area during milling of the ball-end milling cutter by using a semi-analytical method, and in the case of five-axis milling, a swept surface is always equivalent to a spherical surface with the radius of the ball end of the cutter as the radius, and the change of the actual acting radius caused by eccentricity is not considered, so that certain errors are caused.
The method for calculating the thickness of the instantaneous undeformed chip mainly comprises a tool translation method and an analytical calculation method. When the tool posture is adjusted, the modeling difficulty of the analytic calculation method is increased, and the arc approximation trochoid swept locus is often adopted for simplifying calculation, so that the model error is increased.
Disclosure of Invention
The invention aims to provide a modeling method of the static milling force of a ball-end milling cutter based on a semi-analytical method, which can reduce model errors.
The technical scheme adopted by the invention is that a ball end mill static milling force modeling method based on a semi-analytical method comprises the following steps:
step 1, respectively establishing a local coordinate system of cutter teeth j, a ball end mill coordinate system, a main shaft follow-up coordinate system, a cutter instantaneous feed coordinate system and a workpiece coordinate system, and obtaining a trajectory equation of any point on the cutter teeth j under the workpiece coordinate system in the ball end mill machining process based on a homogeneous coordinate transformation principle;
step 2, dividing the cutter teeth into a plurality of cutter tooth infinitesimal equal to the increment of the axial position angle of the cutter teeth, and establishing a infinitesimal cutting force model of the cutter tooth infinitesimal;
step 3, identifying a cutter-working contact area;
step 4, using the sweep point Q of the discrete point i on the cutter tooth j at the time tCTo the knife location point OCLUsing the connecting line as a reference line, and calculating QCThe intersection point Q of the swept surface of the front cutter tooth and the reference lineLThe instantaneous undeformed chip thickness is obtained;
and 5, expressing the cutting force coefficient as a polynomial of the axial position angle of the cutter, calculating an undetermined coefficient in the polynomial of the axial position angle of the cutter according to the average milling force, and identifying to obtain the cutting force coefficient.
The invention is also characterized in that:
the step 1 specifically comprises the following steps:
step 1.1, taking the ball head center of the ball head milling cutter as a coordinate origin OjEstablishing a local coordinate system O of the cutter tooth jj-XjYjZjAbbreviated as { j }; obtaining any cutter tooth j of the ball end millThe coordinates of point P in the local coordinate system { j };
step 1.2, taking the ball head center of the ball head milling cutter as a coordinate origin OCEstablishing a ball end mill coordinate system OC-XCYCZC{ C } for short; obtaining a homogeneous coordinate transformation matrix of a local coordinate system { j } relative to a ball end mill coordinate system { C };
step 1.3, taking the center of the main shaft as a coordinate origin OAEstablishing a main shaft following coordinate system O on a main shaft of a machine toolA-XAYAZAAbbreviated as { A }, coordinate axes
Figure BDA0003085064710000031
Coinciding with the axis of the main shaft; obtaining a homogeneous coordinate transformation matrix of a ball end mill coordinate system { C } relative to a main shaft follow-up coordinate system { A };
step 1.4, establishing a cutter instantaneous feeding coordinate system OCL-XCLYCLZCLFor short, { CL } obtains a homogeneous coordinate transformation matrix of a main shaft follow-up coordinate system { A } relative to a cutter instantaneous feeding coordinate system { CL };
step 1.5, establishing a global coordinate system O on the workpieceW-XWYWZWThe method is abbreviated as { W }, and a homogeneous coordinate transformation matrix of { CL } relative to { W } is obtained;
combining the steps 1.1-1.5, obtaining a trajectory equation of any point P under { W } on the cutter tooth j in the ball end mill processing process through homogeneous coordinate matrix transformation, wherein the trajectory equation is as follows:
Figure BDA0003085064710000032
the step 2 specifically comprises the following steps:
step 2.1, the cutter tooth is divided into a plurality of cutter tooth infinitesimal with equal cutter tooth axial position angle increment, the characteristic information of a cutter tooth discrete point i represents the cutter tooth infinitesimal i information between points (i-1) -i on the cutter tooth, and the cutting force borne by the cutter tooth infinitesimal i on the cutter tooth j at the moment t is decomposed into tangential unit force cutting force dFt(j, i, t), radial Unit force cutting force dFr(j, i, t), axial Unit force cutting force dFa(j, i, t) according to a mechanical molding method using a cutting force, there can be obtained:
Figure BDA0003085064710000041
wherein g (j, i, t) is a unit step function, and when a cutter tooth infinitesimal i on a cutter tooth j is in contact with a workpiece at a time t, g (j, i, t) is 1, otherwise, g (j, i, t) is 0; h (j, i, t) is the instantaneous undeformed chip thickness of the cutting of the cutter tooth infinitesimal i on the cutter tooth j at the time t; kt、KrAnd KaTangential, radial and axial force coefficients, respectively;
step 2.2, subjecting the tangential unit force cutting force dF borne by the cutter tooth infinitesimal i at the moment tt(j, i, t), radial Unit force cutting force dFr(j, i, t), axial Unit force cutting force dFaAnd (j, i, t) is converted to { A }, the instantaneous cutting force applied to the ball end mill at the moment t is expressed as follows in the main shaft following coordinate system { A }:
Figure BDA0003085064710000042
in the formula, niThe total number of cutter tooth infinitesimal;
the instantaneous cutting force borne by the ball end mill at the moment t is obtained through the homogeneous coordinate transformation principle and expressed in a workpiece coordinate system { W }:
Figure BDA0003085064710000043
step 3.1 specifically comprises the following steps:
step 3.1.1, solving a boundary line I;
the intersection line of the cutter tooth swept spherical surface and the previous cutter tooth swept spherical surface, namely the boundary line I, is represented in the following way:
Figure BDA0003085064710000051
the surface formed by the last feeding process is simplified into a columnar surface, and can be expressed as follows under a coordinate system { CL }:
(yCL+fp)2+(zCL)2＝R2 (25)；
the coordinates of the available point S under { CL } are shown in the simultaneous relationship of (24) and (25)
Figure BDA0003085064710000052
The equation for the top surface of the workpiece in the coordinate system { CL } is:
zCL＝-(R-ap) (27)；
in conjunction with (24) and (27), the coordinates of the available point M in the coordinate system { CL } are:
Figure BDA0003085064710000053
the coordinates of the boundary line I, the end point S, and the end point M in the coordinate system { a } are obtained by homogeneous transformation:
Figure BDA0003085064710000061
step 3.1.2, solving a boundary line II;
under the condition of { CL }, an equation of an intersection line of a swept surface of the current cutter tooth and a surface to be machined, namely a boundary line II, is obtained through simultaneous operations (22) and (27):
Figure BDA0003085064710000062
simultaneous (25) and (30) can yield the coordinates of point N under coordinate system { CL }:
Figure BDA0003085064710000063
and (3) converting the coordinates of the boundary line II and the endpoint N to be below { A } through homogeneous coordinate transformation:
Figure BDA0003085064710000064
step 3.1.3, solving a boundary line III;
and (5) obtaining an equation of an intersection line of the swept surface of the current cutter tooth and the processed surface finished by the last feeding under { CL }, namely a boundary line III:
Figure BDA0003085064710000065
the equation of the boundary line III is converted to { a } by homogeneous coordinate transformation:
Figure BDA0003085064710000071
step 3.2 specifically comprises the following steps:
step 3.2.1, assuming that the discrete precision of the axial position angle of the cutter tooth is delta theta, selecting discrete points of which the maximum distance between the discrete points on each boundary line is less than pi delta theta Rcos gamma/180, and substituting (29), (32) and (34) to calculate the coordinate value of the discrete points on each boundary line under { A };
step 3.2.2, calculating the cutter tooth axial position angle corresponding to the discrete point on each boundary line obtained in the step 3.2.1 through the formulas (35) and (36)
Figure BDA0003085064710000072
Angle of radial position
Figure BDA0003085064710000073
Finding out the maximum and minimum axial position angles of the current cutter tooth for cutting and contacting corresponding to each boundary line
Figure BDA0003085064710000074
And from three boundariesFinding the maximum and minimum axial position angles in the line
Figure BDA0003085064710000075
That is to say, the angular range of the axial position of the current cutter tooth contacting the workpiece in one rotation range of the main shaft
Figure BDA0003085064710000076
Figure BDA0003085064710000077
Wherein mm ∈ (I, II, III), N is the mark number of discrete point on the boundary line, and nn is 1,2, … Nnn，NnnThe total number of discrete points on the boundary line;
Figure BDA0003085064710000078
in the formula (I), the compound is shown in the specification,
Figure BDA0003085064710000079
is composed of
Figure BDA00030850647100000710
The arctangent function of (1), the principal value range of which is (-180 °,180 °);
step 3.2.3, search axial position angle Range
Figure BDA00030850647100000711
And determining the cutter-cutter contact interval of the current axial position angle theta on the cutter tooth j according to the radial position angles corresponding to all the cutter tooth discrete points in the cutter tooth, namely determining the cutter-cutter contact area of each cutter tooth in each rotation range of the spindle according to the first cut-in angle, the first cut-out angle, the second cut-in angle and the second cut-out angle … ….
The step 4 specifically comprises the following steps:
step 4.1, obtaining the sweep point Q of the discrete point i on the current cutter tooth j at the moment t according to the formula (9)CThe coordinates of (a);
step 4.2, neglecting the feed motion of the previous cutter tooth, simplifying the front swept surface into a spherical surface, assuming that the intersection point of the reference line and the spherical surface is Q, and establishing a spherical surface equation and a reference line equation under { CL }:
Figure BDA0003085064710000081
in the formula (I), the compound is shown in the specification,
Figure BDA0003085064710000082
is a point Q*The coordinate values in the coordinate system { CL },
Figure BDA0003085064710000083
is a point QCCoordinate values in the coordinate system { CL };
due to the fact that
Figure BDA0003085064710000084
As is known, solving equation (37) utilizes the homogeneous coordinate transformation principle to obtain Q*Coordinates in a machine tool spindle following coordinate system { a }:
Figure BDA0003085064710000085
then point Q*The axial position angle and the radial position angle of (2) are respectively expressed in the formulas (40) and (41):
Figure BDA0003085064710000086
Figure BDA0003085064710000087
q is obtained from equations (40) and (41)CAxial position angle thetaCAnd radial position angle phiCFurther, Q is calculated from a calculation formula of the helical lag angleC、Q*Corresponding helical clearance angle psiC、
Figure BDA0003085064710000088
Approximate calculation of the point Q to be cut*Corresponding cutting time
Figure BDA0003085064710000089
At the same time, approximately consider point QC、QLThe distance between the corresponding cutter location points is the feed amount f of each toothzApproximately finding Q according to sine theoremLAngle of axial position of
Figure BDA00030850647100000810
Figure BDA0003085064710000091
Due to QLAt the action line O of the cutter teethCLQLAnd establishing an equation set according to a straight line formula:
Figure BDA0003085064710000092
in the formula (I), the compound is shown in the specification,
Figure BDA0003085064710000093
is QCThe coordinates in the object coordinate system W,
Figure BDA0003085064710000094
is knife location point OCLCoordinates in the workpiece coordinate system { W };
to be provided with
Figure BDA0003085064710000095
At an initial value, i.e.
Figure BDA0003085064710000096
Applying the Newton-Raphson method to solve equation set (43), as shown in the following formula:
Figure BDA0003085064710000097
wherein k is the number of iterations, k is 0,1,2, …; the iteration end condition is [ t ]k-tk-1 θk-θk-1]T＝[0.05λt0.05λθ]T；
Q can be obtained by bringing the result obtained by the formula (44) into the formula (9)LCoordinates in the object coordinate system { W }:
Figure BDA0003085064710000098
finally, the thickness of the undeformed chip is calculated according to the following formula:
Figure BDA0003085064710000101
the step 5 specifically comprises the following steps:
and 5.1, expressing the cutting force coefficient as the following polynomial of the axial position angle of the cutter:
Figure BDA0003085064710000102
in the formula, a0、a1、a2、a3、b0、b1、b2、b3、c0、c1、c2And c3Is the undetermined coefficient;
Figure BDA0003085064710000103
step 5.2, calculating the cutting depth apCorresponding maximum axial position angle
Figure BDA0003085064710000104
And 5.3, calculating the thickness of the undeformed chip according to the following formula:
h(j,θ,t)＝fzsinφ(j,t)sinθ (48)
where φ (j, t) is a radial position angle of the planar cutting edge tooth j at time t, and a winding vector is defined
Figure BDA0003085064710000105
The included angle formed by clockwise rotation is positive, and the calculation formula of phi (j, t) is as follows:
Figure BDA0003085064710000106
in the formula, phi0Is the radial position angle of the reference cutter tooth in the initial state;
if phi (j, t) is epsilon-90, 90', the knife tooth infinitesimal contacts the workpiece, and g (j, theta, t) is 1; otherwise, g (j, θ, t) ═ 0;
step 5.4, g (j, i, t) and dF in the formula (10)t(j,i,t)、dFr(j,i,t)、dFaG (j, theta, t) and dF for (j, i, t)t(j,θ,t)、dFr(j,θ,t)、dFa(j, θ, t) represents by combining the equations (10), (48) and (49) and converting dFt(j,θ,t)、dFr(j,θ,t)、dFa(j, θ, t) to coordinate axis OAXA、OAYA、OAZAIn the direction, the formula is as follows:
Figure BDA0003085064710000111
step 5.5, summing the milling forces of all the cutter tooth infinitesimal participating in milling on the cutter tooth j at the moment t under a certain depth of attack to obtain the milling force borne by the cutter tooth j at the moment t, and summing the milling forces borne by all the cutter teeth at the moment to finally obtain the total instantaneous milling force borne by the cutter at the moment t, wherein the total instantaneous milling force is shown as the following formula:
Figure BDA0003085064710000112
the time variable t in (51) is changed into a cutter tooth position angle variable phi by using a formula (48), and then the coordinate axis O of the cutter in the range of one rotation of the spindle can be obtainedAXA、OAYAAnd OAZAAverage milling force in the direction:
Figure BDA0003085064710000113
the average milling force within one rotation range of the main shaft is obtained through tests
Figure BDA0003085064710000114
And
Figure BDA0003085064710000115
substituting the formula (52) into the formula (52), and then regressing the undetermined coefficient a in the cutting force coefficient formula shown in the formula (47) by using a least square method0、a1、a2、a3、b0、b1、b2、b3、c0、c1、c2And c3Thereby, the coefficient of cutting force K is identifiedt、KrAnd Ka。
The invention has the beneficial effects that:
the invention relates to a modeling method of a static milling force of a ball-end milling cutter based on a semi-analytical method, which is characterized in that a motion track of cutter teeth in the ball-end milling cutter machining process is established based on a homogeneous coordinate transformation principle, and according to the actual milling condition of the ball-end milling cutter, a cutting force coefficient identification method, a semi-analytical identification method of a cutter-cutter contact area and a solving method of undeformed cutting thickness are provided so as to provide a basis for subsequent research and also provide a reference basis for the selection of machining parameters in the actual machining process.
Drawings
FIG. 1 is a reference coordinate system diagram of milling movement of a ball end mill of the modeling method of the static milling force of the ball end mill based on the semi-analytical method;
FIG. 2a is an axonometric view of milling tracks of a helical-edge ball-end mill of the modeling method of the static milling force of the ball-end mill based on the semi-analytical method;
FIG. 2b is a milling track top view of a modeling method of the static milling force of the ball end mill based on the semi-analytical method;
FIG. 3a is an axonometric view of a coordinate system considering tool runout of the modeling method of the static milling force of the ball end mill based on the semi-analytical method;
FIG. 3b is a top view of a coordinate system considering tool runout of the modeling method of the ball end mill static milling force based on the semi-analytic method of the present invention;
FIG. 4 is a diagram of the tool attitude adjustment and feed trajectory of the modeling method of the ball end mill static milling force based on the semi-analytical method;
FIG. 5 is a tool-to-tool cutting contact area during inclined milling of the modeling method of the static milling force of the ball-end mill based on the semi-analytical method;
FIG. 6 is a cutter tooth infinitesimal force diagram of the modeling method of the static milling force of the ball-end milling cutter based on the semi-analytical method;
FIG. 7 is a schematic diagram of milling instantaneous state of the ball end mill of the modeling method of the static milling force of the ball end mill based on the semi-analytical method;
FIG. 8 is a schematic diagram of milling force coefficient identification of a modeling method of ball end mill static milling force based on a semi-analytical method.
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.
A modeling method of ball end mill static milling force based on a semi-analytical method comprises the following steps:
step 1, as shown in fig. 1, respectively establishing a local coordinate system of a cutter tooth j, a ball end mill coordinate system, a main shaft follow-up coordinate system, a cutter instantaneous feeding coordinate system and a workpiece coordinate system, and obtaining a trajectory equation of any point on the cutter tooth j in the workpiece coordinate system in the ball end mill machining process based on a homogeneous coordinate transformation principle;
step 1.1, milling by using a ball headThe center of the ball head of the knife is the origin of coordinates OjEstablishing a local coordinate system O of the cutter tooth jj-XjYjZjJ, coordinate axes for short
Figure BDA0003085064710000131
Is in a coordinate plane with the edge line of the cutter tooth j
Figure BDA0003085064710000132
The tangential directions of the starting points of the upper projection lines are overlapped;
as shown in fig. 2, the milling of a ball end mill with a fixed lead helical edge, which is widely used in practical production, is considered as a research object, and the coordinate of any point P on any cutter tooth j of the ball end mill in a local coordinate system { j } is as follows:
Figure BDA0003085064710000133
where θ is the axial position angle of point P, R is the tool radius, ψ is the helical relief angle corresponding to point P, ψ is 180tan γ0(1-cos theta)/pi, wherein gamma0The spiral angle is the cutting edge curve of the cutter teeth on the cylindrical surface;
step 1.2, taking the ball head center of the ball head milling cutter as a coordinate origin OCEstablishing a ball end mill coordinate system OC-XCYCZCAbbreviated as { C }, and coordinate axes
Figure BDA0003085064710000134
And
Figure BDA0003085064710000135
the temperature of the molten steel is completely consistent,
Figure BDA0003085064710000136
coinciding with the theoretical axis of the tool and with
Figure BDA0003085064710000137
The parallel state is always kept, and the parallel state is always kept,
Figure BDA0003085064710000138
on the coordinate plane O with the edge line of the reference tooth (the first tooth)CXCYCThe tangential directions of the starting points of the upper projection lines are overlapped;
the included angle phi between the cutter tooth j and the reference cutter toothj＝360(j-1)/ntWherein n istAnd if the total number of the cutter teeth is, the homogeneous coordinate transformation matrix of the local coordinate system { j } relative to the ball end mill coordinate system { C } is as follows:
Figure BDA0003085064710000141
step 1.3, taking the center of the main shaft as a coordinate origin OAEstablishing a main shaft following coordinate system O on a main shaft of a machine toolA-XAYAZAAbbreviated as { A }, coordinate axes
Figure BDA0003085064710000142
Coincident with the axis of the spindle, axis
Figure BDA0003085064710000143
And
Figure BDA0003085064710000144
the included angle between is mu0+φC(μ0Is the angle between the spindle and the spindle in the initial state without starting rotationCIs the angle phi of the main shaft rotation at the moment tC＝ωt)；
Due to the influence of factors such as manufacturing and clamping errors, eccentricity always exists between the central axis of the cutter and the central axis of the spindle, as shown in fig. 3. Assumed origin of coordinates OCAnd origin of coordinates OAThe eccentricity distance between is rho, vector
Figure BDA0003085064710000145
Relative to the coordinate axis
Figure BDA0003085064710000146
Is mu, and defines a coordinate axis
Figure BDA0003085064710000147
When the clockwise rotation direction is positive, the spindle rotates clockwise and the rotation speed is N, the angular velocity ω is pi N/30, and the rotating angle phi is rotated at time tCAnd 180 ω t/pi, the homogeneous coordinate transformation matrix of the ball end mill coordinate system { C } relative to the main shaft following coordinate system { a } is as follows:
Figure BDA0003085064710000148
wherein μ ═ μ0+φCWherein, mu0Is in an initial state
Figure BDA0003085064710000149
And
Figure BDA00030850647100001410
the initial included angle of (a); in this embodiment, μ is set0＝0；
Step 1.4, establishing a cutter instantaneous feeding coordinate system OCL-XCLYCLZCLFor short, { CL }, coordinate axis vector
Figure BDA00030850647100001411
Parallel and in the same direction as the direction of the feed speed,
Figure BDA00030850647100001412
the normal direction of the ideal processed surface is directed to the outside of the entity,
Figure BDA00030850647100001413
is composed of
Figure BDA00030850647100001414
And
Figure BDA00030850647100001415
cross multiplication of (1); when in use
Figure BDA00030850647100001416
And
Figure BDA00030850647100001417
when the coordinate system is completely overlapped, the other two coordinate axes and the directions thereof are completely overlapped with those of the CL, however, when the tool posture is adjusted in the actual working condition,
Figure BDA0003085064710000151
and
Figure BDA0003085064710000152
an included angle exists between the two, which is represented by the lateral inclination and the forward inclination of the tool relative to the surface to be processed of the workpiece. As shown in fig. 4, the pass phase of { A } is opposite to that of { A }
Figure BDA0003085064710000153
And
Figure BDA0003085064710000154
the rotation of (2) realizes the adjustment of the main shaft posture, and then realizes the adjustment of the tool posture, thereby obtaining different milling modes, as follows:
coordinate axis vector
Figure BDA0003085064710000155
The direction is the feeding direction of the cutter,
Figure BDA0003085064710000156
and the main shaft follow-up coordinate system { A } respectively rotates around the two coordinate axis vectors to realize the adjustment of the main shaft posture in the intermittent feeding direction of the cutter. Coordinate axis vector of coordinate system { A } after main shaft attitude adjustment
Figure BDA0003085064710000157
In the coordinate plane YCLOCLZCLProjection line on and coordinate axis vector
Figure BDA0003085064710000158
Angle between, called roll angleDenoted by α; coordinate axis vector
Figure BDA0003085064710000159
In the coordinate plane XCLOCLZCLProjection on and coordinate axis vector
Figure BDA00030850647100001510
The angle between them, called anteversion, is denoted by β. First winding { A } around
Figure BDA00030850647100001511
Rotating the angle beta ', making beta' ═ arctan (tan beta cos alpha), and making { A } wind
Figure BDA00030850647100001512
The rotation angle alpha is defined as being positive when the rotation is counterclockwise around the positive direction of the respective reference direction, then the homogeneous coordinate transformation matrices for the tool roll and the tool tilt are respectively
Figure BDA00030850647100001513
Figure BDA00030850647100001514
Then the homogeneous coordinate transformation matrix of the main shaft follow coordinate system { A } relative to the tool instantaneous feed coordinate system { CL } is:
Figure BDA0003085064710000161
step 1.5, establishing a global coordinate system O on the workpieceW-XWYWZWAbbreviated as { W }, assuming O at feedCLThe coordinate at { W } is (x)CL,yCL,zCL) Then the homogeneous coordinate transformation matrix of { CL } with respect to { W } is:
Figure BDA0003085064710000162
in the formula (I), the compound is shown in the specification,
Figure BDA0003085064710000163
and
Figure BDA0003085064710000164
respectively representing coordinate axes
Figure BDA0003085064710000165
And
Figure BDA0003085064710000166
the subscripts x, y, and z indicate that each vector is in
Figure BDA0003085064710000167
And
Figure BDA0003085064710000168
a projection vector of (a);
in the present embodiment, the one-way straight-line feed milling plane is taken as the study object, and then the homogeneous coordinate transformation matrix of { CL } relative to { W } is:
Figure BDA0003085064710000169
in the formula (x)0,y0) For the first feed OCLThe start position in { W }, q is the number of times the tool is fed (q is 1,2,3 …), t is the time taken for the tool to reach the current position from the 1 st pass, fzFor feed per tooth, fpFor feed row spacing, L is the length of single pass, R is the radius of the tool, whHeight of the blank, apThe depth of the knife is taken;
by combining the formulas (1) - (6) and (8), the trajectory equation of any point P under { W } on the cutter tooth j in the ball-end milling cutter machining process can be obtained through homogeneous coordinate matrix transformation:
Figure BDA0003085064710000171
step 2, as shown in fig. 5, dividing the cutter tooth into a plurality of cutter tooth infinitesimal equal to the increment of the axial position angle of the cutter tooth, and establishing a infinitesimal cutting force model of the cutter tooth infinitesimal;
step 2.1, the cutter tooth is divided into a plurality of cutter tooth infinitesimal with equal cutter tooth axial position angle increment, the characteristic information of a cutter tooth discrete point i represents the cutter tooth infinitesimal i information between points (i-1) -i on the cutter tooth, and the cutting force borne by the cutter tooth infinitesimal i on the cutter tooth j at the moment t is decomposed into tangential unit force cutting force dFt(j, i, t), radial Unit force cutting force dFr(j, i, t), axial Unit force cutting force dFa(j, i, t) according to a mechanical molding method using a cutting force, there can be obtained:
Figure BDA0003085064710000172
wherein g (j, i, t) is a unit step function, and when a cutter tooth infinitesimal i on a cutter tooth j is in contact with a workpiece at a time t, g (j, i, t) is 1, otherwise, g (j, i, t) is 0; h (j, i, t) is the instantaneous undeformed chip thickness of the cutting of the cutter tooth infinitesimal i on the cutter tooth j at the time t; kt、KrAnd KaTangential, radial and axial force coefficients, respectively;
step 2.2, subjecting the tangential unit force cutting force dF borne by the cutter tooth infinitesimal i at the moment tt(j, i, t), radial Unit force cutting force dFr(j, i, t), axial Unit force cutting force dFa(j, i, t) is converted by equation (11) to { A } below:
Figure BDA0003085064710000173
where phi (j, i, t) is the origin of coordinates OAThe line connecting the positions of the discrete points i on the cutter tooth j at the moment t is on the plane XAOAYAProjection on with respect to coordinate axis vector
Figure BDA0003085064710000174
The angle of the clockwise rotation is the same as the angle of the clockwise rotation,
Figure BDA0003085064710000175
for a discrete point i on a cutter tooth j and a coordinate origin OAIs connected to OAZAThe acute included angle of the angle;
the instantaneous cutting force applied to the ball end mill at the moment t is expressed as follows in the main shaft following coordinate system { A }:
Figure BDA0003085064710000181
in the formula, niThe total number of cutter tooth infinitesimal;
the instantaneous cutting force borne by the ball end mill at the moment t is obtained through the homogeneous coordinate transformation principle and expressed in a workpiece coordinate system { W }:
Figure BDA0003085064710000182
step 3, in actual processing, the posture of the spindle is adjusted through program design, and then the posture of the cutter is adjusted, so that the requirement of preventing the cutter from interfering with a processed workpiece is met, and the requirement of realizing high-quality and high-efficiency cutting by avoiding the cutter head part of the ball-end milling cutter is met. However, tool pose adjustment makes recognition of the tool-to-tool contact area more difficult. Determining that a cutter-cutter contact area is a yellow part shown in fig. 6 according to a track of any point on a cutter tooth j in the ball-end milling cutter machining process, namely an area formed by a boundary line I, a boundary line II and a boundary line III, and solving the boundary line I, the boundary line II, the boundary line III and an intersection point of the three boundary lines; the boundary line I is an intersection line between the current cutter tooth and each swept surface of the previous cutter tooth, the boundary line II is an intersection line between the swept surface of the current cutter tooth and the surface to be machined, and the boundary line III is an intersection line between the swept surface of the current cutter tooth and the surface to be machined after last feeding is completed; finding out the maximum and minimum axial position angles of cutter teeth for cutting contact from boundary line I, boundary line II and boundary line III
Figure BDA0003085064710000183
Searching for axial position angle range
Figure BDA0003085064710000184
And screening the discrete points i on the cutter teeth j according to the radial position angles corresponding to the discrete points i on all the cutter teeth j to determine the cutter-cutter contact interval of the current axial position angle theta on the cutter teeth j.
Step 3.1, solving a boundary line I;
in order to simplify the calculation, only the rotary motion of the cutter teeth is considered, the continuous feed motion between two adjacent cutter teeth is ignored, the swept surface of the previous cutter tooth is simplified into a spherical surface, the radius of the spherical surface is equal to the actual working radius of the cutter tooth cutting point consistent with the surface normal direction, and the boundary line I can be obtained by solving the intersection line of the current cutter tooth rotary swept surface and the spherical surface. However, in practice, the tool is eccentric when it is rotated about a coordinate axis
Figure BDA0003085064710000191
When rotating at an angular velocity ω, the turning radii of the cutting points with the same axial position angle on different cutter teeth are different, and the chip-holding angle (shown as eta in fig. 3) between two adjacent teeth is differentP) And also with the change in the angle of the axial position of the cutter teeth. From the analysis of step 1, the coordinates of discrete point i on tooth j in { A } are:
Figure BDA0003085064710000192
in the formula (I), the compound is shown in the specification,
Figure BDA0003085064710000193
to consider only the transformation matrix of C with respect to a in the case of tool eccentricity without considering spindle rotation,
Figure BDA0003085064710000194
representing the coordinates of a discrete point i on cutter tooth j in { j };
discrete point i on cutter tooth j relative to coordinate axis
Figure BDA0003085064710000195
Radius of revolution of, i.e. actual cutting radius
Figure BDA0003085064710000196
At mu0When 0, the following formula (14) can be used:
Figure BDA0003085064710000197
in the same way, the actual axial position angle
Figure BDA0003085064710000198
Comprises the following steps:
Figure BDA0003085064710000199
the actual helical lag angle of the discrete point i on the reference tooth is:
Figure BDA0003085064710000201
in the formula, #i、θiRespectively being the spiral lag angle and the axial position angle of an ideal cutter tooth discrete point i;
the actual cutting radius vector for discrete point i on tooth j is:
Figure BDA0003085064710000202
the axis of the spindle forms an angle with respect to the normal of the machining surface
γ＝arccos(cosαcosβ) (19)；
Angle the actual axial position of discrete point i on tooth j
Figure BDA0003085064710000203
Is equal to gamma, the gamma is substituted into the formula (16) to obtain the position of the discrete point i of the cutter tooth j, and further obtain the theoretical axial position angle thetaiThen the cutting point on the cutter tooth j, which is consistent with the normal direction of the processed surface, can be obtained; then, the actual cutting radius of the cutting point is obtained from the equation (15)
Figure BDA0003085064710000204
And the radius vector of the cutting point of two adjacent teeth is obtained according to the following formula
Figure BDA0003085064710000205
And
Figure BDA0003085064710000206
the radial included angle between:
Figure BDA0003085064710000207
when the radius vector of the characteristic cutting point of two adjacent teeth is consistent with the surface normal of the workpiece, the distance between the two cutting points left on the workpiece in the feeding direction is as follows:
Figure BDA0003085064710000208
simplifying the swept surface of the current cutter tooth into a spherical surface, neglecting the feed motion of the current cutter tooth and only considering OAAt a distance of from the center of the swept surface of the upper knife tooth
Figure BDA0003085064710000209
When the cutter teeth do rotary motion, under { CL }, the equations of the swept surface of the current cutter tooth and the swept surface of the previous cutter tooth are respectively expressed as formulas (22) and (23):
Figure BDA00030850647100002010
in the formula (I), the compound is shown in the specification,
Figure BDA0003085064710000211
representing discrete points i to O on tooth jAThe distance of the points;
Figure BDA0003085064710000212
according to the formulas (22) and (23), the intersection line of the cutter tooth swept spherical surface and the previous cutter tooth swept spherical surface, namely a boundary line I, can be obtained:
Figure BDA0003085064710000213
the surface formed by the last feeding process is simplified into a columnar surface, and can be expressed as follows under a coordinate system { CL }:
(yCL+fp)2+(zCL)2＝R2 (25)；
the coordinates of the available point S under { CL } are shown in the simultaneous relationship of (24) and (25)
Figure BDA0003085064710000214
The equation for the top surface of the workpiece in the coordinate system { CL } is:
zCL＝-(R-ap) (27)；
in conjunction with (24) and (27), the coordinates of the available point M in the coordinate system { CL } are:
Figure BDA0003085064710000221
the coordinates of the boundary line I, the end point S, and the end point M in the coordinate system { a } are obtained by homogeneous transformation:
Figure BDA0003085064710000222
solving a boundary line II;
under the condition of { CL }, an equation of an intersection line of a swept surface of the current cutter tooth and a surface to be machined, namely a boundary line II, is obtained through simultaneous operations (22) and (27):
Figure BDA0003085064710000223
simultaneous (25) and (30) can yield the coordinates of point N under coordinate system { CL }:
Figure BDA0003085064710000224
and (3) converting the coordinates of the boundary line II and the endpoint N to be below { A } through homogeneous coordinate transformation:
Figure BDA0003085064710000225
solving a boundary line III;
and (5) obtaining an equation of an intersection line of the swept surface of the current cutter tooth and the processed surface finished by the last feeding under { CL }, namely a boundary line III:
Figure BDA0003085064710000231
the equation of the boundary line III is converted to { a } by homogeneous coordinate transformation:
Figure BDA0003085064710000232
and 3.2, in order to simplify complex calculation, the boundary lines I, II and III are dispersed before being converted from the coordinate system { CL } to { A }. Assuming that the discrete accuracy of the axial position angle of the cutter teeth is delta theta, the maximum distance between discrete points on the boundary line after transformation is ensured not to exceed pi delta theta R 0180, so that the maximum distance between discrete points on each boundary line is selected to be less thanSubstituting (29), (32) and (34) for the pi delta theta Rcos gamma/180 discrete point to obtain the coordinate value of the discrete point under the { A } on each boundary line;
calculating the axial position angle of the cutter tooth corresponding to the discrete point on each boundary line obtained in the step 3.2.1 through the formulas (35) and (36)
Figure BDA0003085064710000233
Angle of radial position
Figure BDA0003085064710000234
Finding out the maximum and minimum axial position angles of the current cutter tooth for cutting and contacting corresponding to each boundary line
Figure BDA0003085064710000235
And finding out the maximum and minimum axial position angles from the three boundary lines
Figure BDA0003085064710000236
That is to say, the angular range of the axial position of the current cutter tooth contacting the workpiece in one rotation range of the main shaft
Figure BDA0003085064710000237
Figure BDA0003085064710000238
Wherein mm ∈ (I, II, III), N is the mark number of discrete point on the boundary line, and nn is 1,2, … Nnn，NnnThe total number of discrete points on the boundary line;
Figure BDA0003085064710000239
in the formula (I), the compound is shown in the specification,
Figure BDA0003085064710000241
is composed of
Figure BDA0003085064710000242
Is positive and negativeA tangent function with a main value range of (-180 °,180 °);
searching for axial position angle range
Figure BDA0003085064710000243
In most cases, the cut-in and cut-out of a discrete point of a cutter tooth occur on different boundary lines, but a few cases of cut-in and cut-out of one boundary line exist, and meanwhile, the situation that a certain discrete point of a cutter tooth is cut in and cut out twice is considered, so the cut-in and cut-out angle is stored by using a structure array. The specific process is as follows: a. from
Figure BDA0003085064710000244
At the beginning, the boundary line interval to which the axial position angle theta of the current cutter tooth j belongs is judged by taking delta theta as increment
Figure BDA0003085064710000245
b. Finding 10 discrete points with axial position angles close to theta in each boundary line, and arranging the discrete points in ascending order relative to the absolute difference value of theta; c. for the discrete points in each boundary line after arrangement, removing the discrete points with the absolute difference value of the radial position angle of the adjacent previous discrete point being less than 3 degrees from the second discrete point; d. placing the discrete points on all the boundary lines after screening together, arranging the discrete points in ascending order of radial position angles, and similarly, removing the discrete points with the absolute difference value of the radial position angle of the adjacent previous discrete point being less than 3 degrees from the second discrete point to finish secondary screening; if only one discrete point is left after the screening, the last removed discrete point needs to be added again; e. and d, sequentially determining the cutter-cutter contact interval of the current axial position angle theta on the cutter tooth j according to the first cut-in angle, the first cut-out angle, the second cut-in angle and the second cut-out angle … … of the boundary line discrete points which are obtained in the step d and are arranged in the ascending order of the radial position angles, and obtaining the cutter-cutter contact area of each cutter tooth in each rotation range of the spindle.
Step 4, using the sweep point Q of the discrete point i on the cutter tooth j at the time tCTo the knife location point OCLIs taken as a referenceLine, as shown in FIG. 7, two points Q on the reference lineLAnd QCThe distance between the two is the undeformed chip thickness h (j, i, t), and Q is calculatedCThe intersection point Q of the swept surface of the front cutter tooth and the reference lineLThe instantaneous undeformed chip thickness is obtained;
step 4.1, obtaining the sweep point Q of the discrete point i on the current cutter tooth j at the moment t according to the formula (9)CThe coordinates of (a);
step 4.2, neglecting the feed motion of the previous cutter tooth, simplifying the front swept surface into a spherical surface, assuming that the intersection point of the reference line and the spherical surface is Q, and establishing a spherical surface equation and a reference line equation under { CL }:
Figure BDA0003085064710000251
in the formula (I), the compound is shown in the specification,
Figure BDA0003085064710000252
is a point Q*The coordinate values in the coordinate system { CL },
Figure BDA0003085064710000253
is a point QCCoordinate values in the coordinate system { CL };
due to the fact that
Figure BDA0003085064710000254
As is known, equation (37) is solved and discarded according to the actual processing conditions
Figure BDA0003085064710000255
A large value of (A) is obtained
Figure BDA0003085064710000256
Q is obtained by utilizing the principle of homogeneous coordinate transformation*Coordinates in a machine tool spindle following coordinate system { a }:
Figure BDA0003085064710000257
then point Q*The axial position angle and the radial position angle of (2) are respectively expressed in the formulas (40) and (41):
Figure BDA0003085064710000258
Figure BDA0003085064710000259
q is obtained from equations (40) and (41)CAxial position angle thetaCAnd radial position angle phiCFurther, Q is calculated from a calculation formula of the helical lag angleC、Q*Corresponding helical clearance angle psiC、
Figure BDA0003085064710000261
Approximate calculation of the point Q to be cut*Corresponding cutting time
Figure BDA0003085064710000262
At the same time, approximately consider point QC、QLThe distance between the corresponding cutter location points is the feed amount f of each toothzApproximately finding Q according to sine theoremLAngle of axial position of
Figure BDA0003085064710000263
Figure BDA0003085064710000264
Due to QLAt the action line O of the cutter teethCLQLAnd establishing an equation set according to a straight line formula:
Figure BDA0003085064710000265
in the formula (I), the compound is shown in the specification,
Figure BDA0003085064710000266
is QCThe coordinates in the object coordinate system W,
Figure BDA0003085064710000267
is knife location point OCLCoordinates in the workpiece coordinate system { W };
to be provided with
Figure BDA0003085064710000268
At an initial value, i.e.
Figure BDA0003085064710000269
Applying the Newton-Raphson method to solve equation set (43), as shown in the following formula:
Figure BDA00030850647100002610
wherein k is the number of iterations, k is 0,1,2, …; the iteration end condition is [ t ]k-tk-1 θk-θk-1]T＝[0.05λt0.05λθ]T；
Q can be obtained by bringing the result obtained by the formula (44) into the formula (9)LCoordinates in the object coordinate system { W }:
Figure BDA0003085064710000271
finally, the thickness of the undeformed chip is calculated according to the following formula:
Figure BDA0003085064710000272
step 5, expressing the cutting force coefficient as a polynomial of the axial position angle of the cutter, calculating an undetermined coefficient in the polynomial of the axial position angle of the cutter according to the average milling force, and further identifying to obtain the cutting force coefficient;
and 5.1, the cutting force coefficient is the proportional relation between the cross section area of the instantaneous undeformed chips and the micro-element force in each direction. The cutting force coefficient directly influences the prediction precision of the infinitesimal milling force and is one of the key factors of the cutting force modeling. However, the cutting force coefficient varies with the material of the tool and the workpiece, the cutting parameters and other factors, and thus, a certain difficulty is added to the identification of the cutting force coefficient. When a ball-end edge of the ball-end milling cutter is used for cutting, the cutting speeds, the radial cutting depths and the like of cutter tooth infinitesimals at different axial positions in actual cutting are different, so that the cutting mechanisms are different, and therefore, the cutting force coefficient is expressed as the following polynomial of the axial position angle of the cutter:
Figure BDA0003085064710000273
in the formula, a0、a1、a2、a3、b0、b1、b2、b3、c0、c1、c2And c3In order to determine the coefficient to be determined,
Figure BDA0003085064710000274
and 5.2, adopting a groove milling method as shown in fig. 8 to conveniently determine the cutting-in and cutting-out angles of the cutter teeth, eliminating the influence of the spiral angle on the identification accuracy by adopting an average milling method, and replacing a complex spiral blade with a planar blade ball-end milling cutter model so as to achieve the purpose of simplifying the calculation. Since the minimum axial position angle of the cutter tooth for cutting and contacting the workpiece is zero in the vertical milling, changing the depth of cut means changing the maximum axial position angle of the cutter tooth for cutting and contacting the workpiece, and calculating the depth of cut apCorresponding maximum axial position angle
Figure BDA0003085064710000281
Therefore, the relation between the cutting force coefficient and the axial position angle of the cutter can be established;
and 5.3, because a vertical groove milling method is adopted and the influence of jumping and the like is eliminated by an average milling force method, calculating the thickness of the undeformed chip according to the following formula:
h(j,θ,t)＝fzsinφ(j,t)sinθ (48)；
where φ (j, t) is a radial position angle of the planar cutting edge tooth j at time t, and a winding vector is defined
Figure BDA0003085064710000282
The included angle formed by clockwise rotation is positive, and the calculation formula of phi (j, t) is as follows:
Figure BDA0003085064710000283
in the formula, phi0Is the radial position angle of the reference cutter tooth in the initial state;
if phi (j, t) is epsilon-90, 90', the knife tooth infinitesimal contacts the workpiece, and g (j, theta, t) is 1; otherwise, g (j, θ, t) ═ 0;
step 5.4, g (j, i, t) and dF in the formula (10)t(j,i,t)、dFr(j,i,t)、dFaG (j, theta, t) and dF for (j, i, t)t(j,θ,t)、dFr(j,θ,t)、dFa(j, θ, t) represents by combining the equations (10), (48) and (49) and converting dFt(j,θ,t)、dFr(j,θ,t)、dFa(j, θ, t) to coordinate axis OAXA、OAYA、OAZAIn the direction, the formula is as follows:
Figure BDA0003085064710000284
and 5.5, changing the cutting depth to perform a test under the condition of vertical milling, and measuring the average cutting force in the action period of the cutter teeth under different cutting depths. In order to reduce the eccentric influence caused by factors such as tool installation and stress, the total cutting force in the rotation period of the main shaft is measured by a dynamometer, and then is divided by the number of teeth of the tool, so that the average cutting force is calculated.
Under a certain depth of attack, the milling forces of all the cutter tooth infinitesimal participating in milling on the cutter tooth j at the moment t are summed, so that the milling force borne by the cutter tooth j at the moment t can be obtained, then the milling forces borne by all the cutter teeth at the moment are summed, and finally the total instantaneous milling force borne by the cutter at the moment t can be obtained, as shown in the following formula:
Figure BDA0003085064710000291
the time variable t in (51) is changed into a cutter tooth position angle variable phi by using a formula (48), and then the coordinate axis O of the cutter in the range of one rotation of the spindle can be obtainedAXA、OAYAAnd OAZAAverage milling force in the direction:
Figure BDA0003085064710000292
the average milling force within one rotation range of the main shaft is obtained through tests
Figure BDA0003085064710000293
And
Figure BDA0003085064710000294
substituting the formula (52) into the formula (47), and then using the least square method to regress the undetermined coefficient a in the formula of the cutting force coefficient0、a1、a2、a3、b0、b1、b2、b3、c0、c1、c2And c3Thereby, the coefficient of cutting force K is identifiedt、KrAnd Ka。
Through the mode, the invention discloses a modeling method of the static milling force of a ball end mill based on a semi-analytical method, which is characterized in that the motion trail of cutter teeth in the ball end mill machining process is established based on the homogeneous coordinate transformation principle, and according to the actual milling condition of the ball end mill, a cutting force coefficient identification method, a semi-analytical identification method of a cutter-cutter contact area and a solving method of undeformed cutting thickness are provided, so that a foundation is provided for subsequent research, and a reference basis is provided for the selection of machining parameters in the actual machining process; on the premise of ensuring the identification precision, the semi-analytic identification method of the cutter-working contact area is obtained based on the spherical assumption and the homogeneous coordinate inverse transformation principle, so that the identification efficiency of the cutter-working contact area can be improved; the ball end milling cutter cutting force coefficient is identified by adopting an average milling force method, and the influence of the helical angle of the cutter can be eliminated and the influence of periodic flutter on measurement data can be counteracted by a mechanical identification method combining a theory and a test for rapidly calibrating the milling force coefficient of the milling cutter.

Claims (8)

1. A modeling method of ball end mill static milling force based on a semi-analytical method is characterized by comprising the following steps:
step 1, respectively establishing a local coordinate system of cutter teeth j, a ball end mill coordinate system, a main shaft follow-up coordinate system, a cutter instantaneous feed coordinate system and a workpiece coordinate system, and obtaining a trajectory equation of any point on the cutter teeth j under the workpiece coordinate system in the ball end mill machining process based on a homogeneous coordinate transformation principle;
step 2, dividing the cutter teeth into a plurality of cutter tooth infinitesimal equal to the increment of the axial position angle of the cutter teeth, and establishing a infinitesimal cutting force model of the cutter tooth infinitesimal;
step 3, identifying a cutter-working contact area;
step 4, using the sweep point Q of the discrete point i on the cutter tooth j at the time tCTo the knife location point OCLUsing the connecting line as a reference line, and calculating QCThe intersection point Q of the swept surface of the front cutter tooth and the reference lineLThe instantaneous undeformed chip thickness is obtained;
and 5, expressing the cutting force coefficient as a polynomial of the axial position angle of the cutter, calculating undetermined coefficients in the polynomial of the axial position angle of the cutter according to the average milling force, and further identifying to obtain the cutting force coefficient.
2. The modeling method for the static milling force of the ball nose milling cutter based on the semi-analytical method as claimed in claim 1, wherein the step 1 specifically comprises the following steps:
step 1.1, taking the ball head center of the ball head milling cutter as a coordinate origin OjEstablishing a local coordinate system O of the cutter tooth jj-XjYjZjAbbreviated as { j };
the coordinate of any point P on any cutter tooth j of the ball end mill in a local coordinate system { j } is as follows:
Figure FDA0003085064700000011
where θ is the axial position angle of point P, R is the tool radius, ψ is the helical relief angle corresponding to point P, ψ is 180tan γ0(1-cos theta)/pi, wherein gamma0The spiral angle is the cutting edge curve of the cutter teeth on the cylindrical surface;
step 1.2, taking the ball head center of the ball head milling cutter as a coordinate origin OCEstablishing a ball end mill coordinate system OC-XCYCZC{ C } for short;
the included angle phi between the cutter tooth j and the reference cutter toothj＝360(j-1)/ntWherein n istAnd if the total number of the cutter teeth is, the homogeneous coordinate transformation matrix of the local coordinate system { j } relative to the ball end mill coordinate system { C } is as follows:
Figure FDA0003085064700000021
step 1.3, taking the center of the main shaft as a coordinate origin OAEstablishing a main shaft following coordinate system O on a main shaft of a machine toolA-XAYAZAAbbreviated as { A }, coordinate axes
Figure FDA0003085064700000022
Coinciding with the axis of the main shaft;
assumed origin of coordinates OCAnd origin of coordinates OAThe eccentricity distance between is rho, vector
Figure FDA0003085064700000023
Relative to the coordinate axis
Figure FDA0003085064700000024
Is mu, and defines a coordinate axis
Figure FDA0003085064700000025
The clockwise rotation direction is positive, the main shaft rotates clockwise, and the rotated angle phi at the moment tCAnd 180 ω t/pi, the homogeneous coordinate transformation matrix of the ball end mill coordinate system { C } relative to the main shaft following coordinate system { a } is as follows:
Figure FDA0003085064700000026
wherein μ ═ μ0+φCWherein, mu0Is in an initial state
Figure FDA0003085064700000027
And
Figure FDA0003085064700000028
the initial included angle of (a);
step 1.4, establishing a cutter instantaneous feeding coordinate system OCL-XCLYCLZCLFor short, { CL }, coordinate axis vector
Figure FDA0003085064700000029
Parallel and in the same direction as the direction of the feed speed,
Figure FDA00030850647000000210
the normal direction of the ideal processed surface is directed to the outside of the entity,
Figure FDA00030850647000000211
is composed of
Figure FDA00030850647000000212
And
Figure FDA00030850647000000213
cross multiplication of (1);
first winding { A } around
Figure FDA00030850647000000214
Rotating the angle beta ', making beta' ═ arctan (tan beta cos alpha), and making { A } wind
Figure FDA00030850647000000215
The rotation angle alpha is defined as being positive when the rotation is counterclockwise around the positive direction of the respective reference direction, then the homogeneous coordinate transformation matrices for the tool roll and the tool tilt are respectively
Figure FDA0003085064700000031
Figure FDA0003085064700000032
Then the homogeneous coordinate transformation matrix of the main shaft follow coordinate system { A } relative to the tool instantaneous feed coordinate system { CL } is:
Figure FDA0003085064700000033
step 1.5, establishing a global coordinate system O on the workpieceW-XWYWZWAbbreviated as { W }, assuming O at feedCLThe coordinate at { W } is (x)CL,yCL,zCL) If the milling plane is fed in a unidirectional straight line as a study object, the homogeneous coordinate transformation matrix of { CL } relative to { W } is:
Figure FDA0003085064700000034
in the formula (x)0,y0) For the first feed OCLThe start position in { W }, q is the number of times the tool is fed (q is 1,2,3 …), t is the time taken for the tool to reach the current position from the 1 st pass, fzFor feed per tooth, fpFor feed row spacing, L is the length of single pass, R is the radius of the tool, whHeight of the blank, apThe depth of the knife is taken;
by combining the formulas (1) - (6) and (8), the trajectory equation of any point P under { W } on the cutter tooth j in the ball-end milling cutter machining process can be obtained through homogeneous coordinate matrix transformation:
Figure FDA0003085064700000041
3. the modeling method for the static milling force of the ball nose milling cutter based on the semi-analytical method as claimed in claim 1, wherein the step 2 specifically comprises the following steps:
step 2.1, the cutter tooth is divided into a plurality of cutter tooth infinitesimal with equal cutter tooth axial position angle increment, the characteristic information of a cutter tooth discrete point i represents the cutter tooth infinitesimal i information between points (i-1) -i on the cutter tooth, and the cutting force borne by the cutter tooth infinitesimal i on the cutter tooth j at the moment t is decomposed into tangential unit force cutting force dFt(j, i, t), radial Unit force cutting force dFr(j, i, t), axial Unit force cutting force dFa(j, i, t) according to a mechanical molding method using a cutting force, there can be obtained:
Figure FDA0003085064700000042
wherein g (j, i, t) is a unit step function, and when a cutter tooth infinitesimal i on a cutter tooth j is in contact with a workpiece at a time t, g (j, i, t) is 1, otherwise, g (j, i, t) is 0; h (j, i, t) isThe instantaneous undeformed chip thickness of the cutting tooth infinitesimal i on the cutting tooth j at the time t; kt、KrAnd KaTangential, radial and axial force coefficients, respectively;
step 2.2, subjecting the cutting force dF of the tangential unit force borne by the cutter tooth infinitesimal i at the moment tt(j, i, t), radial Unit force cutting force dFr(j, i, t), axial Unit force cutting force dFa(j, i, t) is converted by equation (11) to { A } below:
Figure FDA0003085064700000043
where phi (j, i, t) is the origin of coordinates OAThe line connecting the positions of the discrete points i on the cutter tooth j at the moment t is on the plane XAOAYAProjection on with respect to coordinate axis vector
Figure FDA0003085064700000044
The angle of the clockwise rotation is the same as the angle of the clockwise rotation,
Figure FDA0003085064700000045
for a discrete point i on a cutter tooth j and a coordinate origin OAIs connected to OAZAThe acute included angle of the angle;
the instantaneous cutting force applied to the ball end mill at the moment t is expressed as follows in the main shaft following coordinate system { A }:
Figure FDA0003085064700000051
in the formula, niThe total number of cutter tooth infinitesimal;
the instantaneous cutting force borne by the ball end mill at the moment t is obtained through the homogeneous coordinate transformation principle and expressed in a workpiece coordinate system { W }:
Figure FDA0003085064700000052
4. the modeling method for the static milling force of the ball nose milling cutter based on the semi-analytical method as claimed in claim 1, wherein the step 3 specifically comprises the following steps:
step 3.1, determining a cutter-cutter contact area as an area formed by a boundary line I, a boundary line II and a boundary line III according to the track of any point on the cutter tooth j in the ball-end milling cutter machining process, and solving the boundary line I, the boundary line II, the boundary line III and the intersection point of the three boundary lines; the boundary line I is an intersection line between the current cutter tooth and each swept surface of the previous cutter tooth, the boundary line II is an intersection line between the swept surface of the current cutter tooth and the surface to be machined, and the boundary line III is an intersection line between the swept surface of the current cutter tooth and the surface to be machined after last feeding is completed;
step 3.2, finding out the maximum and minimum axial position angles of the cutter teeth for cutting contact from the boundary line I, the boundary line II and the boundary line III
Figure FDA0003085064700000053
Searching for axial position angle range
Figure FDA0003085064700000054
And screening the discrete points i on the cutter teeth j according to the radial position angles corresponding to the discrete points i on all the cutter teeth j, and determining the cutter-tool contact interval of the current axial position angle theta on the cutter teeth j.
5. The modeling method for the static milling force of the ball nose milling cutter based on the semi-analytical method as claimed in claim 4, wherein the step 3.1 specifically comprises the following steps:
step 3.1.1, solving a boundary line I;
when the spindle rotation is not considered, the coordinates of the discrete point i on the cutter tooth j in { A } are:
Figure FDA0003085064700000061
in the formula (I), the compound is shown in the specification,
Figure FDA0003085064700000069
to consider only the transformation matrix of C with respect to a in the case of tool eccentricity without considering spindle rotation,
Figure FDA0003085064700000062
representing the coordinates of a discrete point i on cutter tooth j in { j };
discrete point i on cutter tooth j relative to coordinate axis
Figure FDA0003085064700000063
Radius of revolution of, i.e. actual cutting radius
Figure FDA0003085064700000064
At mu0When 0, the following formula (14) can be used:
Figure FDA0003085064700000065
in the same way, the actual axial position angle
Figure FDA0003085064700000066
Comprises the following steps:
Figure FDA0003085064700000067
the actual helical lag angle of the discrete point i on the reference tooth is:
Figure FDA0003085064700000068
in the formula, #i、θiRespectively being the spiral lag angle and the axial position angle of an ideal cutter tooth discrete point i;
the actual cutting radius vector for discrete point i on tooth j is:
Figure FDA0003085064700000071
the axis of the spindle forms an angle with respect to the normal of the machining surface
γ＝arccos(cosαcosβ) (19)；
Angle the actual axial position of discrete point i on tooth j
Figure FDA0003085064700000072
Is equal to gamma, the gamma is substituted into the formula (16) to obtain the position of the discrete point i of the cutter tooth j, and further obtain the theoretical axial position angle thetaiThen the cutting point on the cutter tooth j, which is consistent with the normal direction of the processed surface, can be obtained; then, the actual cutting radius of the cutting point is obtained from the equation (15)
Figure FDA0003085064700000073
And the radius vector of the cutting point of two adjacent teeth is obtained according to the following formula
Figure FDA0003085064700000074
And
Figure FDA0003085064700000075
the radial included angle between:
Figure FDA0003085064700000076
when the radius vector of the characteristic cutting point of two adjacent teeth is consistent with the surface normal of the workpiece, the distance between the two cutting points left on the workpiece in the feeding direction is as follows:
Figure FDA0003085064700000077
simplifying the swept surface of the current cutter tooth into a spherical surface, neglecting the feed motion of the current cutter tooth and only considering OAAt a distance of from the center of the swept surface of the upper knife tooth
Figure FDA0003085064700000078
When the cutter teeth do rotary motion, under { CL }, the equations of the swept surface of the current cutter tooth and the swept surface of the previous cutter tooth are respectively expressed as formulas (22) and (23):
Figure FDA0003085064700000079
in the formula (I), the compound is shown in the specification,
Figure FDA00030850647000000710
representing discrete points i to O on tooth jAThe distance of the points;
Figure FDA00030850647000000711
according to the formulas (22) and (23), the intersection line of the cutter tooth swept spherical surface and the previous cutter tooth swept spherical surface, namely a boundary line I, can be obtained:
Figure FDA0003085064700000081
the surface formed by the last feeding process is simplified into a columnar surface, and can be expressed as follows under a coordinate system { CL }:
(yCL+fp)2+(zCL)2＝R2(25)；
the coordinates of the available point S under { CL } are shown in the simultaneous relationship of (24) and (25)
Figure FDA0003085064700000082
The equation for the top surface of the workpiece in the coordinate system { CL } is:
zCL＝-(R-ap) (27)；
in conjunction with (24) and (27), the coordinates of the available point M in the coordinate system { CL } are:
Figure FDA0003085064700000083
the coordinates of the boundary line I, the end point S, and the end point M in the coordinate system { a } are obtained by homogeneous transformation:
Figure FDA0003085064700000091
step 3.1.2, solving a boundary line II;
under the condition of { CL }, an equation of an intersection line of a swept surface of the current cutter tooth and a surface to be machined, namely a boundary line II, is obtained through simultaneous operations (22) and (27):
Figure FDA0003085064700000092
simultaneous (25) and (30) can yield the coordinates of point N under coordinate system { CL }:
Figure FDA0003085064700000093
and (3) converting the coordinates of the boundary line II and the endpoint N to be below { A } through homogeneous coordinate transformation:
Figure FDA0003085064700000094
step 3.1.3, solving a boundary line III;
and (5) obtaining an equation of an intersection line of the swept surface of the current cutter tooth and the processed surface finished by the last feeding under { CL }, namely a boundary line III:
Figure FDA0003085064700000095
the equation of the boundary line III is converted to { a } by homogeneous coordinate transformation:
Figure 2
6. the modeling method for the static milling force of the ball nose milling cutter based on the semi-analytical method as recited in claim 5, wherein the step 3.2 specifically comprises the following steps:
step 3.2.1, assuming that the discrete precision of the axial position angle of the cutter tooth is delta theta, selecting discrete points of which the maximum distance between the discrete points on each boundary line is less than pi delta theta Rcos gamma/180, and substituting (29), (32) and (34) to calculate the coordinate value of the discrete points on each boundary line under { A };
step 3.2.2, calculating the cutter tooth axial position angle corresponding to the discrete point on each boundary line obtained in the step 3.2.1 through the formulas (35) and (36)
Figure FDA0003085064700000102
Angle of radial position
Figure FDA0003085064700000103
Finding out the maximum and minimum axial position angles of the current cutter tooth for cutting and contacting corresponding to each boundary line
Figure FDA0003085064700000104
And finding out the maximum and minimum axial position angles from the three boundary lines
Figure FDA0003085064700000105
That is to say, the angular range of the axial position of the current cutter tooth contacting the workpiece in one rotation range of the main shaft
Figure FDA0003085064700000106
Figure FDA0003085064700000107
Wherein mm ∈ (I, II, III), N is the mark number of discrete point on the boundary line, and nn is 1,2, … Nnn，NnnThe total number of discrete points on the boundary line;
Figure FDA0003085064700000108
in the formula (I), the compound is shown in the specification,
Figure FDA0003085064700000109
is composed of
Figure FDA00030850647000001010
The arctangent function of (1), the principal value range of which is (-180 °,180 °);
step 3.2.3, search axial position angle Range
Figure FDA00030850647000001011
The radial position angle corresponding to all the discrete points of the cutter teeth in the cutter is specifically processed as follows: a. from
Figure FDA00030850647000001012
At the beginning, the boundary line interval to which the axial position angle theta of the current cutter tooth j belongs is judged by taking delta theta as increment
Figure FDA00030850647000001013
b. Finding 10 discrete points with axial position angles close to theta in each boundary line, and arranging the discrete points in ascending order relative to the absolute difference value of theta; c. for the discrete points in each boundary line after arrangement, removing the discrete points with the absolute difference value of the radial position angle of the adjacent previous discrete point being less than 3 degrees from the second discrete point; d. to be screenedThe discrete points on all the subsequent boundary lines are put together and arranged in ascending order according to the radial position angle, and similarly, the discrete points with the absolute difference value of the radial position angle smaller than 3 degrees with the adjacent previous discrete point are removed from the second discrete point, so that secondary screening is completed; if only one discrete point is left after the screening, the last removed discrete point needs to be added again; e. and d, sequentially determining the cutter-cutter contact interval of the current axial position angle theta on the cutter tooth j according to the first cut-in angle, the first cut-out angle, the second cut-in angle and the second cut-out angle … … of the boundary line discrete points which are obtained in the step d and are arranged in the ascending order of the radial position angles, and obtaining the cutter-cutter contact area of each cutter tooth in each rotation range of the spindle.
7. The modeling method for the static milling force of the ball nose milling cutter based on the semi-analytical method as claimed in claim 1, wherein the step 4 specifically comprises the following steps:
step 4.1, obtaining the sweep point Q of the discrete point i on the current cutter tooth j at the moment t according to the formula (9)CThe coordinates of (a);
step 4.2, neglecting the feed motion of the previous cutter tooth, simplifying the front swept surface into a spherical surface, and assuming that the intersection point of the reference line and the spherical surface is Q*The spherical equation and the reference line equation are combined under { CL }:
Figure FDA0003085064700000111
in the formula (I), the compound is shown in the specification,
Figure FDA0003085064700000112
is a point Q*The coordinate values in the coordinate system { CL },
Figure FDA0003085064700000113
is a point QCCoordinate values in the coordinate system { CL };
due to the fact that
Figure FDA0003085064700000121
As is known, equation (37) is solved and discarded according to the actual processing conditions
Figure FDA0003085064700000122
A large value of (A) is obtained
Figure FDA0003085064700000123
Q is obtained by utilizing the principle of homogeneous coordinate transformation*Coordinates in a machine tool spindle following coordinate system { a }:
Figure FDA0003085064700000124
then point Q*The axial position angle and the radial position angle of (2) are respectively expressed in the formulas (40) and (41):
Figure FDA0003085064700000125
Figure FDA0003085064700000126
q is obtained from equations (40) and (41)CAxial position angle thetaCAnd radial position angle phiCFurther, Q is calculated from a calculation formula of the helical lag angleC、Q*Corresponding helical clearance angle psiC、
Figure FDA00030850647000001210
Approximate calculation of the point Q to be cut*Corresponding cutting time
Figure FDA0003085064700000127
At the same time, approximately consider point QC、QLThe distance between the corresponding cutter location points is the feed amount f of each toothzApproximately finding Q according to sine theoremLAngle of axial position of
Figure FDA0003085064700000128
Figure FDA0003085064700000129
Due to QLAt the action line O of the cutter teethCLQLAnd establishing an equation set according to a straight line formula:
Figure FDA0003085064700000131
in the formula (I), the compound is shown in the specification,
Figure FDA0003085064700000132
is QCThe coordinates in the object coordinate system W,
Figure FDA0003085064700000133
is knife location point OCLCoordinates in the workpiece coordinate system { W };
to be provided with
Figure FDA0003085064700000134
At an initial value, i.e.
Figure FDA0003085064700000135
Applying the Newton-Raphson method to solve equation set (43), as shown in the following formula:
Figure FDA0003085064700000136
wherein k is the number of iterations, k is 0,1,2, …; the iteration end condition is [ t ]k-tk-1 θk-θk-1]T＝[0.05λt 0.05λθ]T；
Q can be obtained by bringing the result obtained by the formula (44) into the formula (9)LCoordinates in the object coordinate system { W }:
Figure FDA0003085064700000137
finally, the thickness of the undeformed chip is calculated according to the following formula:
Figure FDA0003085064700000138
8. the modeling method for the static milling force of the ball nose milling cutter based on the semi-analytical method as claimed in claim 1, wherein the step 5 specifically comprises the following steps:
and 5.1, expressing the cutting force coefficient as the following polynomial of the axial position angle of the cutter:
Figure FDA0003085064700000141
in the formula, a0、a1、a2、a3、b0、b1、b2、b3、c0、c1、c2And c3Is the undetermined coefficient;
Figure FDA0003085064700000142
step 5.2, calculating the cutting depth apCorresponding maximum axial position angle
Figure FDA0003085064700000143
And 5.3, calculating the thickness of the undeformed chip according to the following formula:
h(j,θ,t)＝fzsinφ(j,t)sinθ (48)
where φ (j, t) is a radial position angle of the planar cutting edge tooth j at time t, and a winding vector is defined
Figure FDA0003085064700000144
The included angle formed by clockwise rotation is positive, and the calculation formula of phi (j, t) is as follows:
Figure FDA0003085064700000145
in the formula, phi0Is the radial position angle of the reference cutter tooth in the initial state;
if phi (j, t) is epsilon-90, 90', the knife tooth infinitesimal contacts the workpiece, and g (j, theta, t) is 1; otherwise, g (j, θ, t) ═ 0;
step 5.4, g (j, i, t) and dF in the formula (10)t(j,i,t)、dFr(j,i,t)、dFaG (j, theta, t) and dF for (j, i, t)t(j,θ,t)、dFr(j,θ,t)、dFa(j, θ, t) represents by combining the equations (10), (48) and (49) and converting dFt(j,θ,t)、dFr(j,θ,t)、dFa(j, θ, t) to coordinate axis OAXA、OAYA、OAZAIn the direction, the formula is as follows:
Figure FDA0003085064700000146
step 5.5, summing the milling forces of all the cutter tooth infinitesimal participating in milling on the cutter tooth j at the moment t under a certain depth of attack to obtain the milling force borne by the cutter tooth j at the moment t, and summing the milling forces borne by all the cutter teeth at the moment to finally obtain the total instantaneous milling force borne by the cutter at the moment t, wherein the total instantaneous milling force is shown as the following formula:
Figure FDA0003085064700000151
the time variable t in (51) is changed into a cutter tooth position angle variable phi by using a formula (48), and then the coordinate axis O of the cutter in the range of one rotation of the spindle can be obtainedAXA、OAYAAnd OAZAAverage milling force in the direction:
Figure FDA0003085064700000152
the average milling force within one rotation range of the main shaft is obtained through tests
Figure FDA0003085064700000153
And
Figure FDA0003085064700000154
substituting the formula (52) into the formula (47), and then using the least square method to regress the undetermined coefficient a in the formula of the cutting force coefficient0、a1、a2、a3、b0、b1、b2、b3、c0、c1、c2And c3Thereby, the coefficient of cutting force K is identifiedt、KrAnd Ka。
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