CN113297696B - Modeling method for static milling force of ball end mill based on semi-analytic method - Google Patents
Modeling method for static milling force of ball end mill based on semi-analytic method Download PDFInfo
- Publication number
- CN113297696B CN113297696B CN202110578192.4A CN202110578192A CN113297696B CN 113297696 B CN113297696 B CN 113297696B CN 202110578192 A CN202110578192 A CN 202110578192A CN 113297696 B CN113297696 B CN 113297696B
- Authority
- CN
- China
- Prior art keywords
- cutter
- cutter tooth
- cutting
- point
- coordinate system
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Active
Links
Images
Classifications
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/10—Geometric CAD
- G06F30/17—Mechanical parametric or variational design
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- 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
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
- G06F17/10—Complex mathematical operations
- G06F17/16—Matrix or vector computation, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
- G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces
Landscapes
- Engineering & Computer Science (AREA)
- Physics & Mathematics (AREA)
- General Physics & Mathematics (AREA)
- Mathematical Physics (AREA)
- Theoretical Computer Science (AREA)
- Computational Mathematics (AREA)
- Mathematical Analysis (AREA)
- Mathematical Optimization (AREA)
- Pure & Applied Mathematics (AREA)
- Data Mining & Analysis (AREA)
- General Engineering & Computer Science (AREA)
- Geometry (AREA)
- Algebra (AREA)
- Databases & Information Systems (AREA)
- Software Systems (AREA)
- Operations Research (AREA)
- Computing Systems (AREA)
- Computer Hardware Design (AREA)
- Evolutionary Computation (AREA)
- Numerical Control (AREA)
Abstract
The invention discloses a modeling method of static milling force of a ball end mill based on a semi-analytic method, which comprises the following steps: respectively establishing a local coordinate system of the cutter tooth j, a ball-end milling cutter coordinate system, a main shaft follow-up coordinate system, a cutter instantaneous feed coordinate system and a workpiece coordinate system, and obtaining a track equation of any point on the cutter tooth j in the machining process of the ball-end milling cutter under the workpiece coordinate system based on a homogeneous coordinate transformation principle; establishing a micro-element cutting force model of the cutter tooth micro-element; identifying a knife-tool contact area; calculating the instantaneous undeformed chip thickness; and identifying to obtain the cutting force coefficient. The method is characterized in that a motion track of a cutter tooth in the machining process of the ball-end mill is established based on a homogeneous coordinate transformation principle, and a cutting force coefficient identification method, a semi-analytic identification method of a cutter-tool cutting contact area and a solving method of an undeformed cutting thickness are provided according to the actual milling condition of the ball-end mill so as to provide a basis for subsequent research and provide a reference basis for selection of machining parameters in the actual machining process.
Description
Technical Field
The invention belongs to the technical field of machining, and relates to a modeling method of static milling force of a ball end mill based on a semi-analytic method.
Background
The ball end mill is widely applied to milling of important surfaces of related parts in the industries of molds, automobiles, aerospace and the like, the deep research of the milling mechanism of the ball end mill has important significance for improving the quality of products, however, the modeling of static milling force is the key point of the research of the milling mechanism, is the basis and key of the subsequent dynamic modeling, and is also the key basis of the selection and optimization of the cutting parameters.
The identification of the knife-tool cutting contact area is a key link of static milling force modeling, the accuracy and the calculation efficiency of the method directly influence the accuracy and the efficiency of static milling force prediction, however, the knife tooth edge shape of the ball end milling cutter is complex, the influence of factors such as posture adjustment and jumping errors and the like is also caused, the difficulty of the knife-tool cutting contact area identification is high, the entity modeling and Boolean operation method in the current common knife-tool cutting contact area identification method uses the real swept body of the knife scanning body to simplify the knife tooth by using the swept envelope surface of the knife scanning body, and the trochoid movement of the cutting point on the knife tooth is ignored, so that the method has a certain principle error. The Z-MAP discrete method can better judge the cutting contact state of the cutter teeth through the idea of infinitesimal discrete, improves the identification precision of the cutter-tool cutting contact area, but has the problem of precision and efficiency balance, and influences the application of subsequent researches. The student uses a semi-analytic method to identify the knife-tool contact area when the ball end milling cutter mills, and in the case of five-axis milling, the sweep surface is always equivalent to a spherical surface taking the radius of the ball end of the cutter as the radius, and the actual action radius change caused by eccentricity is not considered, so that a certain error is caused.
The method for calculating the instantaneous undeformed chip thickness mainly comprises a cutter translation method and a parsing method. When the tool posture is adjusted, modeling difficulty of an analytical calculation method is increased, and a circular arc is often adopted to approximate a trochoid sweep track for simplifying calculation, so that model errors are increased.
Disclosure of Invention
The invention aims to provide a modeling method for static milling force of a ball end mill based on a semi-analytic method, which can reduce model errors.
The technical scheme adopted by the invention is that the modeling method of the static milling force of the ball end mill based on the semi-analytic method comprises the following steps:
step 2, dividing the cutter tooth into a plurality of cutter tooth infinitesimal cutter tooth angular increments of axial position of the cutter tooth, and establishing a infinitesimal cutter tooth infinitesimal cutting force model;
step 4, sweep point Q at time t with discrete point i on cutter tooth j C To tool position point O CL Is used as a reference line to calculate Q C Intersection point Q of sweep surface of front cutter tooth and reference line L The distance between the two layers is used for obtaining the instantaneous thickness of the undeformed chip;
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 and obtaining 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 center of the ball end milling cutter as the origin of coordinates O j Establishing a local coordinate system O of the cutter tooth j j -X j Y j Z j Simply { j }; obtaining the coordinates of any point P on any cutter tooth j of the ball end mill in a local coordinate system { j };
step 1.2, taking the ball center of the ball end milling cutter as the origin of coordinates O C Establishing a ball end mill coordinate system O C -X C Y C Z C Simply referred to as { C }; obtaining a homogeneous coordinate transformation matrix of the local coordinate system { j } relative to the ball end mill coordinate system { C };
step 1.3, taking the center of the main shaft as the origin of coordinates O A Establishing a main shaft follow-up coordinate system O on a main shaft of a machine tool A -X A Y A Z A Abbreviated as { A }, coordinate axis
Is coincident with the axis of the main shaft; obtaining a homogeneous coordinate transformation matrix of a ball end mill coordinate system { C } relative to a spindle follow-up coordinate system { A };
step 1.4,Establishing a tool instantaneous feed coordinate system O CL -X CL Y CL Z CL For short, { CL } obtains a homogeneous coordinate transformation matrix of the main shaft follow-up coordinate system { A } relative to the instantaneous feeding coordinate system { CL } of the cutter;
step 1.5, establishing a Global coordinate System O on the workpiece W -X W Y W Z W Abbreviated as { W }, to obtain a homogeneous coordinate transformation matrix of { CL } relative to { W };
combining the steps 1.1-1.5, and obtaining a track equation of any point P on a cutter tooth j under { W } in the machining process of the ball end mill through homogeneous coordinate matrix transformation, wherein the track equation is as follows:
the step 2 specifically comprises the following steps:
step 2.1, dividing the cutter tooth into a plurality of cutter tooth micro-elements with equal axial position angle increment of the cutter tooth, representing cutter tooth micro-element i information between points (i-1) and i on the cutter tooth by using characteristic information of a cutter tooth discrete point i, and decomposing cutting force applied by the cutter tooth micro-element i on the cutter tooth j at a moment t into tangential unit force cutting force dF t (j, i, t), radial unit force cutting force dF r (j, i, t), axial unit force cutting force dF a (j, i, t) from the mechanical modeling of the cutting forces, it is possible to:
wherein g (j, i, t) is a unit step function, when a cutter tooth infinitesimal i on the cutter tooth j is in tangential contact with a workpiece at a moment t, g (j, i, t) =1, otherwise, g (j, i, t) =0; h (j, i, t) is the instantaneous undeformed chip thickness of cutter tooth infinitesimal i on cutter tooth j cut at time t; k (K) t 、K r And K a Tangential, radial and axial force coefficients, respectively;
step 2.2, cutting force dF of tangential unit force applied to cutter tooth infinitesimal i at time t t (j, i, t), radial unit force cutting force dF r (j, i, t), axial cell force cutCutting force dF a (j, i, t) is converted to { A }, the instantaneous cutting force of the ball end mill at time t is expressed in the spindle-following coordinate system { A }:
wherein n is i The total number of cutter teeth is the total number of cutter teeth infinitesimal;
the instantaneous cutting force of the ball end mill at the moment t is obtained through the homogeneous coordinate transformation principle and expressed as:
step 3.1 specifically comprises the following steps:
step 3.1.1, solving a boundary line I;
the intersection line of the cutter tooth sweeping sphere and the previous cutter tooth sweeping sphere, namely the boundary line I, is expressed as follows:
the surface formed by the last feeding process is simplified to be a columnar surface, and can be expressed as:
(y CL +f p ) 2 +(z CL ) 2 ＝R 2 (25)；
simultaneously (24) and (25), the coordinates of the obtainable point S under { CL }, are
The equation for the top surface of the workpiece in the coordinate system { CL } is:
z CL ＝-(R-a p ) (27)；
simultaneously (24) and (27), the coordinates of the obtainable point M in the coordinate system { CL } are:
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:
step 3.1.2, solving a boundary line II;
at { CL }, the equation of the intersection of the swept surface of the current tooth with the surface to be machined, boundary line II, is obtained by combining (22) and (27):
The coordinates of point N in the coordinate system { CL } are obtained by combining (25) and (30):
the coordinates of the boundary line II and the endpoint N are converted into { A } by homogeneous coordinate transformation:
step 3.1.3, solving a boundary line III;
the equation of intersection of the swept surface of the current tooth with the finished surface of the last feed at { CL } is obtained by combining (22) and (25), namely boundary line III:
the equation for boundary line III is transformed to { A } by homogeneous coordinate transformation:
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 with the maximum distance between the discrete points on each boundary line smaller than pi delta theta Rcos gamma/180, and carrying out (29), (32) and (34) to obtain the coordinate value of the discrete point on each boundary line under { A };
step 3.2.2, obtaining the axial position angle of the cutter tooth corresponding to the discrete point on each boundary line obtained in step 3.2.1 through (35) and (36)
Radial position angle->
Finding out the maximum and minimum axial position angles of the current cutter tooth corresponding to each boundary line for cutting and touching>
And find the maximum and minimum axial position angles from the three boundary lines
Obtaining the axial position angle range of the current cutter tooth in the workpiece within the spindle rotation range >
In the formula, mm epsilon (I, II, III), N is the mark number of discrete points on the boundary line, and nn=1, 2, … N nn ，N nn Is the total number of discrete points on the boundary line;
in the method, in the process of the invention,
is->
The main value range of the arc tangent function of (a) is (-180 DEG, 180 DEG); />
Step 3.2.3 searching for axial position Angle Range
And determining the knife-work cutting contact area of the current axial position angle theta on the knife tooth j according to the radial position angles corresponding to all the discrete points of the knife tooth in the spindle according to the first cutting angle, the second cutting angle and the second cutting angle … …, and obtaining the knife-work cutting contact area of each knife tooth in each rotation range of the spindle.
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 time t according to the formula (9) C Coordinates of (c);
step 4.2, neglecting the feeding movement of the last cutter tooth, simplifying the front sweep 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 }:
in the method, in the process of the invention,
for point Q * Coordinate values in the coordinate system { CL }, -, and>
for point Q C Coordinate values in the coordinate system { CL };
due to
Solving equation (37) to obtain Q using homogeneous coordinate transformation principle * Coordinates in machine tool spindle follower coordinate system { a }:
Point Q * The axial position angle and the radial position angle of (a) are respectively shown as formulas (40) and (41):
q is obtained from formulas (40) and (41) C Axial position angle theta C And a radial position angle phi C Further, Q is calculated from the spiral lag angle calculation formula C 、Q * Corresponding spiral lag angle psi C 、
Approximate determination of the point to be cut Q * Corresponding cutting time ∈ ->
At the same time, approximate point Q C 、Q L The distance between the corresponding knife sites is the feeding quantity f of each tooth z Approximation of Q according to sine theorem L Axial position angle>
Due to Q L In the action line O of cutter teeth CL Q L And establishing an equation set according to a linear formula:
in the method, in the process of the invention,
is Q C Coordinates in the object coordinate system { W }, }>
Is the tool position point O CL Coordinates in the object coordinate system { W };
to be used for
Is the initial point, i.e.)>
The solution to equation set (43) is found using the Newton-Raphson method, as shown below:
where k is the number of iterations, k=0, 1,2, …; the iteration termination condition is [ t ] k -t k-1 θ k -θ k-1 ] T ＝[0.05λ t 0.05λ θ ] T ；
By introducing the result obtained in the formula (44) into the formula (9), Q can be obtained L Coordinates in the object coordinate system { W }:
finally, the thickness of the undeformed chip is determined according to the following formula:
the step 5 specifically comprises the following steps:
step 5.1, expressing the cutting force coefficient as the following polynomial of the axial position angle of the cutter:
wherein a is 0 、a 1 、a 2 、a 3 、b 0 、b 1 、b 2 、b 3 、c 0 、c 1 、c 2 And c 3 Is a coefficient to be determined; 
Step 5.2, calculating the cutting depth a p Corresponding maximum axial position angle
And 5.3, calculating the thickness of the undeformed chip according to the following steps:
h(j,θ,t)＝f z sinφ(j,t)sinθ (48)
wherein phi (j, t) is the radial position angle of the plane blade tooth j at time t, and the winding vector is defined
The included angle formed by clockwise rotation is positive, and the calculation formula of phi (j, t) is as follows:
in phi 0 The radial position angle of the reference cutter tooth in the initial state is set;
if phi (j, t) epsilon [ -90,90], then the cutter tooth infinitesimal cuts the workpiece, g (j, theta, t) =1; otherwise, g (j, θ, t) =0;
step 5.4, g (j, i, t), dF in the formula (10) t (j,i,t)、dF r (j,i,t)、dF a (j, i, t) g (j, θ, t), dF t (j,θ,t)、dF r (j,θ,t)、dF a (j, θ, t) represents, by combining equations (10), (48) and (49), dF t (j,θ,t)、dF r (j,θ,t)、dF a (j, θ, t) to coordinate axis O A X A 、O A Y A 、O A Z A In the direction, the formula is as follows:
and 5.5, summing milling forces of all cutter tooth microelements involved in milling on the cutter tooth j at a moment t under a certain cutting depth to obtain milling forces born by the cutter tooth j at the moment t, and summing the milling forces born by all cutter teeth at the moment, so that the total instantaneous milling forces born by the cutter at the moment t can be finally obtained, wherein the total instantaneous milling forces are shown in the following formula:
using the formula (48) to change the time variable t in (51) into the cutter tooth position angle variable phi so as to obtain the coordinate axis O of the cutter in the spindle rotation range A X A 、O A Y A And O A Z A Average milling force applied in direction:
obtaining average milling force in a spindle rotation range through experiments
And->
Substituting the coefficient 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 method 0 、a 1 、a 2 、a 3 、b 0 、b 1 、b 2 、b 3 、c 0 、c 1 、c 2 And c 3 Thereby, the cutting force coefficient K is identified t 、K r And K a 。
The beneficial effects of the invention are as follows:
the invention discloses a modeling method of static milling force of a ball end mill based on a semi-analytic method, which establishes a motion track of a cutter tooth in the machining process of the ball end mill based on a homogeneous coordinate transformation principle, and provides a cutting force coefficient identification method, a semi-analytic identification method of a cutter-tool cutting contact area and a solving method of an undeformed cutting thickness according to the actual milling situation of the ball end mill so as to provide a basis for subsequent research and provide a reference basis for selection of machining parameters in the actual machining process.
Drawings
FIG. 1 is a reference coordinate system diagram of a ball end mill milling motion of a modeling method of a ball end mill static milling force based on a semi-analytical method;
FIG. 2a is an isometric view of a milling track of a helical blade ball end mill of the modeling method of ball end mill static milling force based on a semi-analytical method of the present invention;
FIG. 2b is a top view of a milling track of a modeling method of static milling force of a ball nose milling cutter based on a semi-analytical method according to the present invention;
FIG. 3a is an isometric view of a coordinate system considering tool runout of a modeling method of ball nose milling cutter static milling force based on a semi-analytical method of the present invention;
FIG. 3b is a top view of a coordinate system considering tool runout for a modeling method of ball nose milling cutter static milling force based on a semi-analytical method of the present invention;
FIG. 4 is a diagram of the attitude adjustment and feed path of a tool of a modeling method of the static milling force of a ball end mill based on a semi-analytical method;
FIG. 5 is a graph of the tool-to-tool contact area during ramping of a modeling method of ball nose mill static milling force based on a semi-analytical method in accordance with the present invention;
FIG. 6 is a cutter tooth infinitesimal force diagram of a modeling method of static milling force of a ball end mill based on a semi-analytic method;
FIG. 7 is a schematic diagram of a milling transient state of a ball end mill according to a modeling method of static milling force of the ball end mill based on a semi-analytical method;
fig. 8 is a schematic diagram of milling force coefficient identification of a modeling method of static milling force of a ball end mill based on a semi-analytic method.
Detailed Description
The invention will be described in detail below with reference to the drawings and the detailed description.
A modeling method of static milling force of a ball end mill based on a semi-analytic method comprises the following steps:
step 1.1, taking the ball center of the ball end milling cutter as the origin of coordinates O j Establishing a local coordinate system O of the cutter tooth j j -X j Y j Z j Simply called { j }, coordinate axis
In the coordinate plane +.>
The tangential directions of the starting points of the upper projection lines coincide;
as shown in fig. 2, milling of a ball end mill with a constant-lead helical blade widely used in actual production is taken as a research object, and coordinates of any point P on any cutter tooth j of the ball end mill in a local coordinate system { j } are as follows:
where θ is the axial position angle of point P, R is the tool radius, ψ is the helical lag angle corresponding to point P, ψ=180 tan γ 0 (1-cos θ)/pi, wherein γ 0 The helical angle of the cutter tooth cutting edge curve on the cylindrical surface;
step 1.2, taking the ball center of the ball end milling cutter as the origin of coordinates O C Establishing a ball end mill coordinate system O C -X C Y C Z C Abbreviated as { C }, and coordinate axes
And->
Completely coincide with (I) a->
Coinciding with the theoretical axis of the tool and with +.>
Always keep parallel +.>
In the coordinate plane O with the edge line of the reference cutter tooth (the first cutter tooth) C X C Y C The tangential directions of the starting points of the upper projection lines coincide;
included angle phi between cutter tooth j and reference cutter tooth j ＝360(j-1)/n t Wherein n is t For the total number of cutter teeth, the homogeneous coordinate transformation matrix of the local coordinate system { j } relative to the ball end mill coordinate system { C } is:
step 1.3, taking the center of the main shaft as the origin of coordinates O A Setting up spindle follower on machine tool spindleCoordinate system O A -X A Y A Z A Abbreviated as { A }, coordinate axis
Coincides with the axis of the main shaft, coordinate axis +.>
And->
The included angle between them is mu 0 +φ C (μ 0 Is the included angle phi between the spindle and the spindle in the initial state of not starting to rotate C Is the angle phi of the rotation of the main shaft at the moment t C ＝ωt)；
Due to factors such as manufacturing and clamping errors, there is always an eccentricity between the central axis of the tool and the central axis of the spindle, as shown in fig. 3. Let the origin of coordinates O C And origin of coordinates O A The eccentric distance between the two is ρ, the vector
Relative to the coordinate axis->
Is μ, and specifies about the axis +.>
Clockwise rotation direction Xiang Wei is positive, the main shaft rotates clockwise, the rotating speed is N, the angular speed omega=pi N/30, and the rotating angle phi is the rotation time t C =180ωt/pi, the homogeneous coordinate transformation matrix of the ball nose milling coordinate system { C } with respect to the spindle follower coordinate system { a } is:
wherein μ=μ 0 +φ C Wherein μ is 0 Is in an initial state
And->
Is included in the first part; mu is set in this embodiment 0 ＝0；
Step 1.4, establishing a tool instantaneous feed coordinate system O CL -X CL Y CL Z CL For short { CL }, coordinate axis vector
Parallel and in the same direction as the feed speed direction, +.>
Is the ideal normal direction of the processed surface and points to the outside of the body, +.>
Is->
And->
Is multiplied by (a); when->
And->
When the coordinate system is completely coincident, the other two coordinate axes of the coordinate system and the directions thereof are completely coincident with { CL }, however, when the tool posture is adjusted in the actual working condition, the user is in the state of being in the position of being out of the position of the tool>
And->
An included angle is formed between the cutting tool and the processed surface of the workpieceRoll and pitch. As shown in fig. 4, by passing { a }, relative to
And->
The adjustment of the main shaft posture is realized, and then the adjustment of the tool posture is realized, so that different milling modes are obtained, and the method comprises the following steps:
coordinate axis vector
The direction is the feeding direction of the cutter, and the direction is->
For the intermittent feeding direction of the cutter, the main shaft follow-up coordinate system { A } rotates around the vectors of the two coordinate axes respectively to realize the adjustment of the main shaft posture. Coordinate axis vector of coordinate system { A } after main axis posture adjustment >
In the coordinate plane Y CL O CL Z CL Upper projection line and coordinate axis vector +.>
The included angle between the two is called a roll angle and is denoted by alpha; coordinate axis vector->
In the coordinate plane X CL O CL Z CL Projection and coordinate axis vector on the plane->
The included angle between them, called the rake angle, is denoted by beta. The { A } is first wound around }>
Rotating by an angle beta ', making beta' =arctan (tan beta cos alpha), and winding { A } around +.>
Rotation angle alpha and defining positive counter-clockwise rotation about the respective reference direction, the homogeneous coordinate transformation matrices for tool roll and rake are respectively
The homogeneous coordinate transformation matrix of the spindle follower coordinate system { A } with respect to the tool instantaneous feed coordinate system { CL } is:
step 1.5, establishing a Global coordinate System O on the workpiece W -X W Y W Z W Simply called { W }, let us assume O at the time of feeding CL The { W } coordinate is (x) CL ,y CL ,z CL ) The homogeneous coordinate transformation matrix of { CL } relative to { W } is:
in the method, in the process of the invention,
and->
Respectively represent coordinate axes +>
And->
The unit vectors above, subscripts x, y and z denote the respective vector at +.>
And->
Projection vectors on;
in this embodiment, the unidirectional linear feed milling plane is taken as a study object, and the homogeneous coordinate transformation matrix of { CL } relative to { W } is:
in (x) 0 ,y 0 ) For the first feeding O CL In { W }, q is the number of tool feeds (q=1, 2,3 …), t is the time taken for the tool to start from the 1 st feed to the current position, f z For each tooth feed amount, f p For feeding line spacing, L is single feed length, R is cutter radius, w h Height of blank, a p Is the cutting depth;
by combining formulas (1) - (6) and (8), the trajectory equation of any point P on the cutter tooth j under { W } in the machining process of the ball end mill can be obtained through homogeneous coordinate matrix transformation:
step 2, as shown in fig. 5, dividing the cutter tooth into a plurality of cutter tooth infinitesimal cutter tooth units with equal cutter tooth axial position angle increment, and establishing a infinitesimal cutter tooth infinitesimal cutting force model;
step 2.1, dividing the cutter tooth into a plurality of cutter tooth micro-elements with equal axial position angle increment of the cutter tooth, representing cutter tooth micro-element i information between points (i-1) and i on the cutter tooth by using characteristic information of a cutter tooth discrete point i, and decomposing cutting force applied by the cutter tooth micro-element i on the cutter tooth j at a moment t into tangential unit force cutting force dF t (j, i, t), radial unit force cutting force dF r (j, i, t), axial unit force cutting force dF a (j, i, t) from the mechanical modeling of the cutting forces, it is possible to:
wherein g (j, i, t) is a unit step function, when a cutter tooth infinitesimal i on the cutter tooth j is in tangential contact with a workpiece at a moment t, g (j, i, t) =1, otherwise, g (j, i, t) =0; h (j, i, t) is the instantaneous undeformed chip thickness of cutter tooth infinitesimal i on cutter tooth j cut at time t; k (K) t 、K r And K a Tangential, radial and axial force coefficients, respectively;
step 2.2, cutting force dF of tangential unit force applied to cutter tooth infinitesimal i at time t t (j, i, t), radial unit force cutting force dF r (j, i, t), axial unit force cutting force dF a (j, i, t) is converted to { A } by equation (11):
wherein phi (j, i, t) is the origin of coordinates O A The line connecting the position of the discrete point i on the cutter tooth j at the moment t is arranged on the plane X A O A Y A Projection onto a coordinate axis vector
Clockwise rotation angle, +.>
For discrete point i and origin of coordinates O on cutter tooth j A Is connected with O A Z A An acute included angle;
the instantaneous cutting force to which the ball nose milling cutter is subjected at time t is expressed in the spindle follower coordinate system { a }:
wherein n is i The total number of cutter teeth is the total number of cutter teeth infinitesimal;
the instantaneous cutting force of the ball end mill at the moment t is obtained through the homogeneous coordinate transformation principle and expressed as:
in the actual machining, the main shaft posture is usually adjusted through program design, so that the posture of a cutter is adjusted, the requirement of preventing interference between the cutter and a machined workpiece is met, and the requirement of avoiding the head part of the ball end milling cutter to realize high-quality and high-efficiency cutting is met. However, tool pose adjustment makes recognition of the tool-to-tool contact area difficult. Determining a knife-tool cutting contact area as a yellow part shown in fig. 6, namely an area formed by boundary lines I, II and III according to the track of any point on a cutter tooth j in the machining process of the ball end mill, and solving the boundary lines I, II and III and the intersection point of the three boundary lines; the boundary line I is the intersection line between the swept surfaces of the current cutter tooth and the previous cutter tooth, the boundary line II is the intersection line between the swept surfaces of the current cutter tooth and the surface to be processed, and the boundary line III is the intersection line between the swept surfaces of the current cutter tooth and the processed surface after the last feeding is completed; 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 
Search for axial position angular range +.>
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 in the cutter teeth, and determining the cutter-tool cutting contact section of the current axial position angle theta on the cutter teeth j.
Step 3.1, solving a boundary line I;
to simplify the calculation, only the rotary motion of the cutter teeth is considered, and the continuous feeding motion between two adjacent cutter teeth is ignoredThe sweep surface of the last cutter tooth is simplified to be 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 rotating sweep surface and the spherical surface. However, in practice, due to the eccentric effect of the tool, when the tool is rotated around the axis
When rotated at angular velocity ω, the turning radii of cutting points having the same axial position angle on different cutter teeth are different, and the chip receiving angle between adjacent teeth (η shown in fig. 3 P ) And also with the axial position angle of the cutter teeth. From the analysis of step 1, the coordinates of the discrete point i on the tooth j in { A } are:
wherein M is AC | φC＝0 In order to consider only the transformation matrix of C with respect to a in the case of tool eccentricity without considering spindle rotation, 
Representing coordinates of a discrete point i on tooth j in { j };
discrete point i on cutter tooth j relative to coordinate axis
Is defined as the actual cutting radius +.>
At mu 0 In the case of=0, it is obtainable by formula (14):
similarly, the actual axial position angle
The method comprises the following steps:
the actual spiral lag angle for discrete point i on the reference tooth is:
in the psi- i 、θ i The spiral lag angle and the axial position angle of the ideal cutter tooth discrete point i are respectively;
the actual cutting radius vector for discrete point i on tooth j is:
the angle of the axis of the main shaft relative to the normal line of the processing surface is
Making the actual axial position angle of discrete point i on cutter tooth j
Is equal to gamma, and is brought into (16) to obtain the position of the discrete point i of the cutter tooth j, thereby obtaining the theoretical axial position angle theta i The cutting point on the cutter tooth j consistent with the normal direction of the processed surface can be obtained; then, the actual cutting radius of the cutting point is determined from equation (15)>
And the radius vector of the cutting point of the adjacent two teeth is calculated according to the following formula>
And->
Radial included angle between:
when the radius vector of the characteristic cutting point of the adjacent two teeth is consistent with the normal line of the surface of the workpiece, the distance between the two cutting points left on the workpiece in the feeding direction is as follows:
simplifying the sweep surface of the current cutter tooth into a spherical surface, neglecting the feeding movement of the current cutter tooth, and only considering O A At a central distance from the sweep surface of the last cutter tooth
When the cutter teeth do rotary motion, under the condition of { CL }, equations of a current cutter tooth sweeping surface and a previous cutter tooth sweeping surface are respectively shown as formulas (22) and (23):
in the method, in the process of the invention,
representing discrete points i through O on tooth j A The distance of the points;
the intersection of the tooth swept sphere and the previous tooth swept sphere, i.e., the boundary line I, can be obtained according to equations (22), (23):
the surface formed by the last feeding process is simplified to be a columnar surface, and can be expressed as:
(y CL +f p ) 2 +(z CL ) 2 ＝R 2 (25)；
simultaneously (24) and (25), the coordinates of the obtainable point S under { CL }, are
The equation for the top surface of the workpiece in the coordinate system { CL } is:
z CL ＝-(R-a p ) (27)；
simultaneously (24) and (27), the coordinates of the obtainable point M in the coordinate system { CL } are:
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:
solving a boundary line II;
at { CL }, the equation of the intersection of the swept surface of the current tooth with the surface to be machined, boundary line II, is obtained by combining (22) and (27):
the coordinates of point N in the coordinate system { CL } are obtained by combining (25) and (30):
the coordinates of the boundary line II and the endpoint N are converted into { A } by homogeneous coordinate transformation:
solving a boundary line III;
the equation of intersection of the swept surface of the current tooth with the finished surface of the last feed at { CL } is obtained by combining (22) and (25), namely boundary line III:
The equation for boundary line III is transformed to { A } by homogeneous coordinate transformation:
step 3.2. To simplify the complex calculations, the boundary lines I, II, III are discretized before they are transformed from the coordinate system CL to a. Assuming that the discrete accuracy of the axial position angle of the cutter tooth is delta theta, the maximum distance between discrete points on the boundary line after conversion is ensured not to exceed pi delta theta R 0 And/180, so that discrete points with the maximum distance between the discrete points on each boundary line being smaller than pi delta theta Rcos gamma/180 are selected, and the discrete points on each boundary line are brought into (29), (32) and (34) to calculate the coordinate value of the discrete point under { A };
obtaining 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 steps (35) and (36)
Radial position angle->
Finding out the current cutter tooth position corresponding to each boundary lineMaximum, minimum axial position angle of tangential contact action ∈>
And finding the maximum and minimum axial position angle from the three boundary lines>
Obtaining the axial position angle range of the current cutter tooth in the workpiece within the spindle rotation range>
In the formula, mm epsilon (I, II, III), N is the mark number of discrete points on the boundary line, and nn=1, 2, … N nn ，N nn Is the total number of discrete points on the boundary line;
in the method, in the process of the invention,
is->
The main value range of the arc tangent function of (a) is (-180 DEG, 180 DEG);
Searching for axial position angular range
The radial position angles corresponding to all the discrete points of the cutter teeth in the cutter tooth are mostly that the cutting in and cutting out of one discrete point of the cutter tooth occur on different boundary lines, but there are few cases that one boundary line cuts in and cuts out, and meanwhile, the situation that two cutting in and cutting out of one discrete point of the cutter tooth can exist is considered, so the structure array is used for storingStoring the cut-in and cut-out angles. The specific process is as follows: a. from->
Initially, the boundary line section ++where the axial position angle θ of the current tooth j belongs is determined with Δθ as an increment>
b. The found 10 discrete points with the axial angles close to theta in each boundary line are arranged in ascending order relative to the absolute difference value of theta; c. for discrete points in each boundary line after arrangement, eliminating discrete points with radial position angle absolute difference smaller than 3 degrees from the adjacent last discrete point from the second one; 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 also, removing the discrete points with the absolute difference of the radial position angles of the adjacent last discrete point smaller than 3 degrees from the second one to finish secondary screening; if only one discrete point is left after the screening is finished, the last rejected discrete point needs to be added again; e. and d, sequentially determining the cutter-tool cutting contact regions of the current axial position angle theta on the cutter teeth j according to the first cutting angle, the second cutting angle and the second cutting angle … … by the boundary line discrete points which are obtained in the step d and are arranged in an ascending order of the radial position angles, and obtaining the cutter-tool cutting contact regions of each cutter tooth in each rotation range of the main shaft.
Step 4, sweep point Q at time t with discrete point i on cutter tooth j C To tool position point O CL As shown in FIG. 7, two points Q on the reference line L And Q C The distance between the two is the thickness h (j, i, t) of the undeformed chip, and Q is calculated C Intersection point Q of sweep surface of front cutter tooth and reference line L The distance between the two layers is used for obtaining the instantaneous thickness of the undeformed chip;
step 4.1, obtaining the sweep point Q of the discrete point i on the current cutter tooth j at the time t according to the formula (9) C Coordinates of (c);
step 4.2, neglecting the feeding movement of the last cutter tooth, simplifying the front sweep surface into a spherical surface, and assuming the intersection of the reference line and the spherical surfacePoint is Q * The spherical equation and the reference line equation are combined under { CL }:
in the method, in the process of the invention,
for point Q * Coordinate values in the coordinate system { CL }, -, and>
for point Q C Coordinate values in the coordinate system { CL };
due to
It is known that solving equation (37) eliminates +.>
Is large to obtain
Acquiring Q by using homogeneous coordinate transformation principle * Coordinates in machine tool spindle follower coordinate system { a }:
point Q * The axial position angle and the radial position angle of (a) are respectively shown as formulas (40) and (41):
q is obtained from formulas (40) and (41) C Axial position angle theta C And a radial position angle phi C Further, Q is calculated from the spiral lag angle calculation formula C 、Q * Corresponding spiral lag angle psi C 、
Approximate determination of the point to be cut Q * Corresponding cutting time ∈ ->
At the same time, approximate point Q C 、Q L The distance between the corresponding knife sites is the feeding quantity f of each tooth z Approximation of Q according to sine theorem L Axial position angle>
/>
Due to Q L In the action line O of cutter teeth CL Q L And establishing an equation set according to a linear formula:
in the method, in the process of the invention,
is Q C Coordinates in the object coordinate system { W }, }>
Is the tool position point O CL Coordinates in the object coordinate system { W };
to be used for
Is the initial point, i.e.)>
The solution to equation set (43) is found using the Newton-Raphson method, as shown below:
where k is the number of iterations, k=0, 1,2, …; the iteration termination condition is [ t ] k -t k-1 θ k -θ k-1 ] T ＝[0.05λ t 0.05λ θ ] T ；
By introducing the result obtained in the formula (44) into the formula (9), Q can be obtained L Coordinates in the object coordinate system { W }:
finally, the thickness of the undeformed chip is determined according to the following formula:
step 5, representing 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 and obtaining the cutting force coefficient;
and 5.1, the cutting force coefficient is the proportional relation between the cross-sectional area of the instant undeformed chip and the infinitesimal force in each direction. The cutting force coefficient directly influences the prediction accuracy of the infinitesimal milling force and is one of key factors of cutting force modeling. However, the cutting force coefficient varies with the cutter, the workpiece material, the cutting parameters and other factors, and adds a certain difficulty to the identification of the cutting force coefficient. When the ball end edge of the ball end milling cutter cuts, the cutting speeds, radial cutting depths and the like of cutter tooth micro-elements at different axial positions in actual cutting are different, so that the cutting mechanisms are also different, and therefore, the cutting force coefficients are expressed as the following polynomials of the cutter axial position angles:
Wherein a is 0 、a 1 、a 2 、a 3 、b 0 、b 1 、b 2 、b 3 、c 0 、c 1 、c 2 And c 3 For the coefficients to be determined,
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 force method, and replacing a complex spiral blade with a planar blade ball-end milling cutter model so as to achieve the aim of simplifying calculation. Since the minimum axial position angle of the cutting tool tooth for cutting the workpiece in vertical milling is zero, changing the cutting tool depth means changing the maximum axial position angle of the cutting tool tooth for cutting the workpiece, and calculating the cutting tool depth a p Corresponding maximum axial position angle
Thus, a relationship of the cutting force coefficient and the tool axial position angle can be established;
step 5.3, since a vertical slotting method is adopted and the influence of runout and the like is eliminated by an average milling force method, the undeformed chip thickness is calculated as follows:
h(j,θ,t)＝f z sinφ(j,t)sinθ (48)；
wherein phi (j, t) is the radial position angle of the plane blade tooth j at time t, and the winding vector is defined
The included angle formed by clockwise rotation is positive, and the calculation formula of phi (j, t) is as follows:
in phi 0 The radial position angle of the reference cutter tooth in the initial state is set;
if phi (j, t) epsilon [ -90,90], then the cutter tooth infinitesimal cuts the workpiece, g (j, theta, t) =1; otherwise, g (j, θ, t) =0;
Step 5.4, g (j, i, t), dF in the formula (10) t (j,i,t)、dF r (j,i,t)、dF a (j, i, t) g (j, θ, t), dF t (j,θ,t)、dF r (j,θ,t)、dF a (j, θ, t) represents, by combining equations (10), (48) and (49), dF t (j,θ,t)、dF r (j,θ,t)、dF a (j, θ, t) to coordinate axis O A X A 、O A Y A 、O A Z A In the direction, the formula is as follows:
and 5.5, under the condition of vertical milling, changing the cutting depth to test, and measuring the average cutting force in the action period of the cutter teeth under different cutting depths, wherein the total material removal amount in one cutter tooth period is a constant and is irrelevant to the existence of a helix angle, so that the average cutting force is irrelevant to the helix angle. In order to reduce the influence of eccentricity caused by factors such as cutter installation and stress, the total cutting force in the rotation period of the main shaft is measured through the dynamometer, and then divided by the number of teeth of the cutter, so that the average cutting force is calculated.
And under a certain cutting depth, the milling forces of all the cutter tooth microelements involved in milling on the cutter tooth j at the moment t are summed to obtain the milling force born by the cutter tooth j at the moment t, and then the milling forces born by all the cutter teeth at the moment t are summed to finally obtain the total instantaneous milling force born by the cutter at the moment t, wherein the total instantaneous milling force is represented by the following formula:
using the formula (48) to change the time variable t in (51) into the cutter tooth position angle variable phi so as to obtain the coordinate axis O of the cutter in the spindle rotation range A X A 、O A Y A And O A Z A Average milling force applied in direction:
obtaining average milling force in a spindle rotation range through experiments
And->
Substituting the coefficient 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 method 0 、a 1 、a 2 、a 3 、b 0 、b 1 、b 2 、b 3 、c 0 、c 1 、c 2 And c 3 Thereby, the cutting force coefficient K is identified t 、K r And K a 。
Through the mode, the modeling method of the static milling force of the ball end mill based on the semi-analytic method establishes the motion track of the cutter tooth in the machining process of the ball end mill based on the homogeneous coordinate transformation principle, and provides a cutting force coefficient identification method, a semi-analytic identification method of a cutter-tool cutting contact area and a solving method of the undeformed cutting thickness according to the actual milling situation of the ball end mill so as to provide a basis for subsequent research and provide a reference basis for selection of machining parameters in the actual machining process; on the premise of ensuring the recognition accuracy, the semi-analytic recognition method of the knife-tool cutting contact area is obtained based on the spherical surface hypothesis and the homogeneous coordinate inverse transformation principle, so that the recognition efficiency of the knife-tool cutting contact area can be improved; the cutting force coefficient of the ball end mill is identified based on an average milling force method, and the influence of the helical angle of the cutter can be eliminated and the influence of periodic chatter on measured data can be counteracted by a mechanical identification method combining the theory and the experiment of rapidly calibrating the milling force coefficient of the ball end mill.
Claims (7)
1. The modeling method of the static milling force of the ball end mill based on the semi-analytic method is characterized by comprising the following steps of:
step 1, respectively establishing a local coordinate system of a cutter tooth j, a ball end mill coordinate system, a spindle follow-up coordinate system, a cutter instantaneous feed coordinate system and a workpiece coordinate system, and obtaining a track equation of any point on the cutter tooth j in the machining process of the ball end mill under the workpiece coordinate system based on a homogeneous coordinate transformation principle;
step 2, dividing the cutter tooth into a plurality of cutter tooth infinitesimal cutter tooth angular increments of axial position of the cutter tooth, and establishing a infinitesimal cutter tooth infinitesimal cutting force model;
step 3, identifying a knife-tool cutting contact area;
step 4, sweep point Q at time t with discrete point i on cutter tooth j C To tool position point O CL Is used as a reference line to calculate Q C Intersection point Q of sweep surface of front cutter tooth and reference line L The distance between the two layers is used for obtaining the instantaneous thickness of the undeformed chip;
step 5, representing 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 identifying to obtain the cutting force coefficient;
the step 3 specifically comprises the following steps:
step 3.1, determining a cutter-tool cutting contact area as an area formed by boundary lines I, II and III according to the track of any point on a cutter tooth j in the machining process of the ball end mill, and solving intersection points of the boundary lines I, II and III and three boundary lines; the boundary line I is an intersection line between the sweeping surfaces of the current cutter tooth and the previous cutter tooth, the boundary line II is an intersection line between the sweeping surface of the current cutter tooth and the surface to be processed, and the boundary line III is an intersection line between the sweeping surface of the current cutter tooth and the processed surface of which the 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 lines I, II and III
Search for axial position angular range +.>
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 in the cutter teeth, and determining the cutter-tool cutting contact section of the current axial position angle theta on the cutter teeth j.
2. The modeling method of a ball end mill static milling force based on a semi-analytical method according to claim 1, wherein the step 1 specifically comprises the following steps:
step 1.1, taking the ball center of the ball end milling cutter as the origin of coordinates O j Establishing a local coordinate system O of the cutter tooth j j -X j Y j Z j Simply { j };
the coordinates of any point P on any cutter tooth j of the ball end mill in a local coordinate system { j }, are as follows:
where θ is the axial position angle of point P, R is the tool radius, ψ is the helical lag angle corresponding to point P, ψ=180 tan γ 0 (1-cos θ)/pi, wherein γ 0 The helical angle of the cutter tooth cutting edge curve on the cylindrical surface;
step 1.2, taking the ball center of the ball end milling cutter as the origin of coordinates O C Establishing a ball end mill coordinate system O C -X C Y C Z C Simply referred to as { C };
the included angle phi between the cutter tooth j and the reference cutter tooth j ＝360(j-1)/n t Wherein n is t For the total number of cutter teeth, the local coordinate system { j } is milled relative to the ball headThe homogeneous coordinate transformation matrix of the knife coordinate system { C } is:
step 1.3, taking the center of the main shaft as the origin of coordinates O A Establishing a main shaft follow-up coordinate system O on a main shaft of a machine tool A -X A Y A Z A Abbreviated as { A }, coordinate axis
Is coincident with the axis of the main shaft;
let the origin of coordinates O C And origin of coordinates O A The eccentric distance between the two is ρ, the vector
Relative to the coordinate axis->
Is μ, and specifies about the axis +.>
Clockwise rotation Xiang Wei is positive, the main shaft rotates clockwise, and the angle phi rotated at time t is the same C =180ωt/pi, the homogeneous coordinate transformation matrix of the ball nose milling coordinate system { C } with respect to the spindle follower coordinate system { a } is:
wherein μ=μ 0+ φ C Wherein μ is 0 Is in an initial state
And->
Is included in the first part;
step 1.4,Establishing a tool instantaneous feed coordinate system O CL -X CL Y CL Z CL Abbreviated as { CL }, coordinate axis vector
Parallel and in the same direction as the feed speed direction, +.>
Is the ideal normal direction of the processed surface and points to the outside of the body, +.>
Is that
And->
Is multiplied by (a);
the { A } is wound first
Rotating by an angle beta ', making beta' =arctan (tan beta cos alpha), and winding { A } around +.>
Rotation angle alpha and defining positive counter-clockwise rotation about the respective reference direction, the homogeneous coordinate transformation matrices for tool roll and rake are respectively
The homogeneous coordinate transformation matrix of the spindle follower coordinate system { A } with respect to the tool instantaneous feed coordinate system { CL } is:
step 1.5, establishing a Global coordinate System O on the workpiece W -X W Y W Z W Simply called { W }, let us assume O at the time of feeding CL The { W } coordinate is (x) CL ,y CL ,z CL ) Taking a unidirectional straight-line feed milling plane as a study object, the homogeneous coordinate transformation matrix of { CL } relative to { W } is:
in (x) 0 ,y 0 ) For the first feeding O CL In { W }, q is the number of tool feeds (q=1, 2,3 …), T is the time taken for the tool to start from the 1 st feed to the current position, f z For each tooth feed amount, f p For feeding line spacing, L is single feed length, R is cutter radius, w h Height of blank, a p Is the cutting depth;
by combining formulas (1) - (6) and (8), the trajectory equation of any point P on the cutter tooth j under { W } in the machining process of the ball end mill can be obtained through homogeneous coordinate matrix transformation:
3. the modeling method of a ball end mill static milling force based on a semi-analytical method according to claim 1, wherein the step 2 specifically comprises the following steps:
step 2.1, dividing the cutter tooth into a plurality of cutter tooth micro-elements with equal axial position angle increment of the cutter tooth, representing cutter tooth micro-element i information between points (i-1) and i on the cutter tooth by using characteristic information of a cutter tooth discrete point i, and decomposing cutting force applied by the cutter tooth micro-element i on the cutter tooth j at a moment t into tangential unit force cutting force dF t (j, i, t), radial unit force cutting force dF r (j, i, t), axial unit force cutting forcedF a (j, i, t) from the mechanical modeling of the cutting forces, it is possible to:
wherein g (j, i, t) is a unit step function, when a cutter tooth infinitesimal i on the cutter tooth j is in tangential contact with a workpiece at a moment t, g (j, i, t) =1, otherwise, g (j, i, t) =0; h (j, i, t) is the instantaneous undeformed chip thickness of cutter tooth infinitesimal i on cutter tooth j cut at time t; k (K) t 、K r And K a Tangential, radial and axial force coefficients, respectively;
step 2.2, cutting force dF of tangential unit force applied to the cutter tooth element i at time t t (j, i, t), radial unit force cutting force dF r (j, i, t), axial unit force cutting force dF a (j, i, t) is converted to { A } by equation (11):
wherein phi (j, i, t) is the origin of coordinates O A The line connecting the position of the discrete point i on the cutter tooth j at the moment t is arranged on the plane X A O A Y A Projection onto a coordinate axis vector
Clockwise rotation angle, +.>
For discrete point i and origin of coordinates O on cutter tooth j A Is connected with O A Z A An acute included angle;
the instantaneous cutting force to which the ball nose milling cutter is subjected at time t is expressed in the spindle follower coordinate system { a }:
in the method, in the process of the invention,n i the total number of cutter teeth is the total number of cutter teeth infinitesimal;
the instantaneous cutting force of the ball end mill at the moment t is obtained through the homogeneous coordinate transformation principle and expressed as:
4. The modeling method of a ball end mill static milling force based on a semi-analytical method according to claim 1, 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:
in the method, in the process of the invention,
in order to consider only the transformation matrix of { C } relative to { A } in the case of tool eccentricity without considering spindle rotation, +.>
Representing coordinates of a discrete point i on tooth j in { j };
discrete point i on cutter tooth j relative to coordinate axis
Is defined as the actual cutting radius +.>
At mu 0 In the case of=0, it is obtainable by formula (14): />
Similarly, the actual axial position angle
The method comprises the following steps:
the actual spiral lag angle for discrete point i on the reference tooth is:
in the psi- i 、θ i The spiral lag angle and the axial position angle of the ideal cutter tooth discrete point i are respectively;
the actual cutting radius vector for discrete point i on tooth j is:
the angle of the axis of the main shaft relative to the normal line of the processing surface is
γ＝arccos(cosαcosβ) (19)；
Making the actual axial position angle of discrete point i on cutter tooth j
Is equal to gamma, and is brought into (16) to obtain the position of the discrete point i of the cutter tooth j, thereby obtaining the theoretical axial position angle theta i The cutting point on the cutter tooth j consistent with the normal direction of the processed surface can be obtained; then, the actual cutting radius ++of the cutting point is obtained from equation (15) >
And the radius vector of the cutting point of the adjacent two teeth is calculated according to the following formula>
And->
Radial included angle between:
when the radius vector of the characteristic cutting point of the adjacent two teeth is consistent with the normal line of the surface of the workpiece, the distance between the two cutting points left on the workpiece in the feeding direction is as follows:
simplifying the sweep surface of the current cutter tooth into a spherical surface, neglecting the feeding movement of the current cutter tooth, and only considering O A At a central distance from the sweep surface of the last cutter tooth
When the cutter teeth do rotary motion, under the condition of { CL }, equations of a current cutter tooth sweeping surface and a previous cutter tooth sweeping surface are respectively shown as formulas (22) and (23):
in the method, in the process of the invention,
representing discrete points i through O on tooth j A The distance of the points;
the intersection of the tooth swept sphere and the previous tooth swept sphere, i.e., the boundary line I, can be obtained according to equations (22), (23):
the surface formed by the last feeding process is simplified to be a columnar surface, and can be expressed as:
(y CL +f p ) 2 +(z CL ) 2 ＝R 2 (25)；
simultaneously (24) and (25), the coordinates of the obtainable point S under { CL }, are
The equation for the top surface of the workpiece in the coordinate system { CL } is:
z CL ＝-(R-a p ) (27)；
simultaneously (24) and (27), the coordinates of the obtainable point M in the coordinate system { CL } are:
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:
step 3.1.2, solving a boundary line II;
At { CL }, the equation of the intersection of the swept surface of the current tooth with the surface to be machined, boundary line II, is obtained by combining (22) and (27):
the coordinates of point N in the coordinate system { CL } are obtained by combining (25) and (30):
the coordinates of the boundary line II and the endpoint N are converted into { A } by homogeneous coordinate transformation:
step 3.1.3, solving a boundary line III;
the equation of intersection of the swept surface of the current tooth with the finished surface of the last feed at { CL } is obtained by combining (22) and (25), namely boundary line III:
the equation for boundary line III is transformed to { A } by homogeneous coordinate transformation:
5. the modeling method of static milling force of ball end mill based on semi-analytical method according to claim 4, wherein 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 with the maximum distance between the discrete points on each boundary line smaller than pi delta theta Rcos gamma/180, and carrying out (29), (32) and (34) to obtain the coordinate value of the discrete point on each boundary line under { A };
step 3.2.2, obtaining the axial position angle of the cutter tooth corresponding to the discrete point on each boundary line obtained in step 3.2.1 through (35) and (36)
Radial position angle- >
Finding out the maximum and minimum axial position angles of the current cutter tooth corresponding to each boundary line for cutting and touching>
And find the maximum and minimum axial position angles from the three boundary lines
Obtaining the axial position angle range of the current cutter tooth in the workpiece within the spindle rotation range>
In the formula, mm epsilon (I, II, III), N is the mark number of discrete points on the boundary line, and nn=1, 2, … N nn ，N nn Is the total number of discrete points on the boundary line;
in the method, in the process of the invention,
is->
The main value range of the arc tangent function of (a) is (-180 DEG, 180 DEG);
step 3.2.3 searching for axial position Angle Range
The radial position angles corresponding to all the discrete points of the cutter teeth in the cutter are as follows: a. from->
Beginning withDelta theta is an increment, and the boundary line section of the axial position angle theta of the current cutter tooth j is judged
b. The found 10 discrete points with the axial angles close to theta in each boundary line are arranged in ascending order relative to the absolute difference value of theta; c. for discrete points in each boundary line after arrangement, eliminating discrete points with radial position angle absolute difference smaller than 3 degrees from the adjacent last discrete point from the second one; 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 also, removing the discrete points with the absolute difference of the radial position angles of the adjacent last discrete point smaller than 3 degrees from the second one to finish secondary screening; if only one discrete point is left after the screening is finished, the last rejected discrete point needs to be added again; e. and d, sequentially determining the cutter-tool cutting contact regions of the current axial position angle theta on the cutter teeth j according to the first cutting angle, the second cutting angle and the second cutting angle … … by the boundary line discrete points which are obtained in the step d and are arranged in an ascending order of the radial position angles, and obtaining the cutter-tool cutting contact regions of each cutter tooth in each rotation range of the main shaft.
6. The modeling method of a ball end mill static milling force based on a semi-analytical method according to claim 2, 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 time t according to the formula (9) C Coordinates of (c);
step 4.2, neglecting the feeding movement of the last cutter tooth, simplifying the front sweep 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 }:
in the method, in the process of the invention,
for point Q * Coordinate values in the coordinate system { CL }, -, and>
for point Q C Coordinate values in the coordinate system { CL };
due to
It is known that solving equation (37) eliminates +.>
Is large to obtain
Acquiring Q by using homogeneous coordinate transformation principle * Coordinates in machine tool spindle follower coordinate system { a }:
point Q * The axial position angle and the radial position angle of (a) are respectively shown as formulas (40) and (41):
q is obtained from formulas (40) and (41) C Axial position angle theta C And a radial position angle phi C Further, Q is calculated from the spiral lag angle calculation formula C 、Q * Corresponding spiral lag angle psi C 、
Approximate determination of the point to be cut Q * Corresponding cutting time ∈ ->
At the same time, approximate point Q C 、Q L The distance between the corresponding knife sites is the feeding quantity f of each tooth z Q is obtained according to sine theorem L Axial position angle>
Due to Q L In the action line O of cutter teeth CL Q L And establishing an equation set according to a linear formula:
in the method, in the process of the invention,
is Q C Coordinates in the object coordinate system { W }, }>
Is the tool position point O CL Coordinates in the object coordinate system { W };
to be used for
Is the initial point, i.e.)>
The solution to equation set (43) is found using the Newton-Raphson method, as shown below:
where k is the number of iterations, k=0, 1,2, …; the iteration termination condition is [ t ] k -t k-1 θ k -θ k-1 ] T ＝[0.05λ t 0.05λ θ ] T ；
By introducing the result obtained in the formula (44) into the formula (9), Q can be obtained L Coordinates in the object coordinate system { W }:
finally, the thickness of the undeformed chip is determined according to the following formula:
7. a method for modeling a static milling force of a ball end mill based on a semi-analytical method according to claim 3, wherein step 5 comprises the steps of:
step 5.1, expressing the cutting force coefficient as the following polynomial of the axial position angle of the cutter:
wherein a is 0 、a 1 、a 2 、a 3 、b 0 、b 1 、b 2 、b 3 、c 0 、c 1 、c 2 And c 3 Is a coefficient to be determined;
step 5.2, calculating the cutting depth a p Corresponding maximum axial position angle
And 5.3, calculating the thickness of the undeformed chip according to the following steps:
h(j,θ,t)＝f z sinφ(j,t)sinθ (48)
wherein phi (j, t) is the radial position angle of the plane blade tooth j at time t, and the winding vector is defined
The included angle formed by clockwise rotation is positive, and the calculation formula of phi (j, t) is as follows:
In phi 0 The radial position angle of the reference cutter tooth in the initial state is set;
if phi (j, t) epsilon [ -90,90], then the cutter tooth infinitesimal cuts the workpiece, g (j, theta, t) =1; otherwise, g (j, θ, t) =0;
step 5.4, g (j, i, t), dF in the formula (10) t (j,i,t)、dF r (j,i,t)、dF a (j, i, t) g (j, θ, t), dF t (j,θ,t)、dF r (j,θ,t)、dF a (j, θ, t) represents, by combining equations (10), (48) and (49), dF t (j,θ,t)、dF r (j,θ,t)、dF a (j, θ, t) to coordinate axis O A X A 、O A Y A 、O A Z A In the direction, the formula is as follows:
and 5.5, summing milling forces of all cutter tooth microelements involved in milling on the cutter tooth j at a moment t under a certain cutting depth to obtain milling forces born by the cutter tooth j at the moment t, and summing the milling forces born by all cutter teeth at the moment, so that the total instantaneous milling forces born by the cutter at the moment t can be finally obtained, wherein the total instantaneous milling forces are shown in the following formula:
using the formula (48) to change the time variable t in (51) into the cutter tooth position angle variable phi so as to obtain the coordinate axis O of the cutter in the spindle rotation range A X A 、O A Y A And O A Z A Average milling force applied in direction:
obtaining average milling force in a spindle rotation range through experiments
And->
Substituting the coefficient 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 method 0 、a 1 、a 2 、a 3 、b 0 、b 1 、b 2 、b 3 、c 0 、c 1 、c 2 And c 3 Thereby, the cutting force coefficient K is identified t 、K r And K a 。/>
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202110578192.4A CN113297696B (en) | 2021-05-26 | 2021-05-26 | Modeling method for static milling force of ball end mill based on semi-analytic method |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202110578192.4A CN113297696B (en) | 2021-05-26 | 2021-05-26 | Modeling method for static milling force of ball end mill based on semi-analytic method |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN113297696A CN113297696A (en) | 2021-08-24 |
| CN113297696B true CN113297696B (en) | 2023-05-26 |
Family
ID=77325294
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN202110578192.4A Active CN113297696B (en) | 2021-05-26 | 2021-05-26 | Modeling method for static milling force of ball end mill based on semi-analytic method |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN113297696B (en) |
Families Citing this family (6)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN114119501B (en) | 2021-11-05 | 2023-03-17 | 苏州大学 | Method and system for measuring non-deformed cutting thickness of micro-milling |
| CN114065427A (en) * | 2021-11-16 | 2022-02-18 | 哈尔滨工程大学 | Cycloid parameter optimization method based on cutting force modeling in cycloid milling |
| CN114004112B (en) * | 2021-12-02 | 2024-06-25 | 湖南工业大学 | Elliptical ultrasonic vibration assisted milling cutting force prediction method |
| CN115365892B (en) * | 2022-08-22 | 2023-11-10 | 成都飞机工业（集团）有限责任公司 | Milling load parameter optimization method under allowance time variation |
| CN116061004B (en) * | 2022-10-14 | 2023-08-08 | 哈尔滨理工大学 | Identification method for change characteristics of transient cutting energy efficiency distribution of milling cutter under vibration effect |
| CN116150910B (en) * | 2023-02-28 | 2023-10-20 | 哈尔滨理工大学 | Micro-texture design method and parameter prediction method for milling titanium alloy ball end mill |
Family Cites Families (6)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| DE10144932B4 (en) * | 2001-09-12 | 2014-07-31 | Siemens Aktiengesellschaft | Visualization of workpieces in the simulation of milling processes |
| JP5804367B2 (en) * | 2011-09-01 | 2015-11-04 | 広島県 | Machining error prediction method, machining error prediction apparatus, tool path correction method, and tool path correction apparatus |
| CN105955195B (en) * | 2016-05-16 | 2018-05-22 | 哈尔滨理工大学 | A kind of Machining of Curved Surface orbit generation method based on Milling Force prediction |
| CN110270705B (en) * | 2019-06-26 | 2020-07-28 | 东北大学 | Micro-milling machining cutting force simulation prediction method considering cutter bounce influence |
| CN111459097A (en) * | 2020-04-07 | 2020-07-28 | 西南交通大学 | Calculation method for curved surface machining contact area of ball-end milling cutter |
| CN112100810B (en) * | 2020-08-11 | 2022-05-20 | 清华大学 | Flat-bottom milling cutter multi-axis milling force prediction method |
-
2021
- 2021-05-26 CN CN202110578192.4A patent/CN113297696B/en active Active
Non-Patent Citations (1)
| Title |
|---|
| An analytical force model for ball-end milling based on a predictive machining theory considering cutter runout;FU Zhongtao，YANG Wenyu，WANG Xuelin;International Journal of Advanced Manufacturing Technology;第84卷;2449-2460 * |
Also Published As
| Publication number | Publication date |
|---|---|
| CN113297696A (en) | 2021-08-24 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| CN113297696B (en) | Modeling method for static milling force of ball end mill based on semi-analytic method | |
| US9696707B2 (en) | Method of controlling tool orientation and step-over distance in face milling of curvilinear surfaces | |
| Tsay et al. | Accurate 5-axis machining of twisted ruled surfaces | |
| Pham et al. | A manufacturing model of an end mill using a five-axis CNC grinding machine | |
| Tournier et al. | Iso-scallop tool path generation in 5-axis milling | |
| CN109597357B (en) | Numerical control programming method and device for blade rotary milling process | |
| CN111069642B (en) | Three-dimensional space inclined hole machining process | |
| CN109375580A (en) | A kind of geometric error recognition methods of the five-axis machine tool yaw based on double ball bars | |
| CN106312850A (en) | Design method for screw grinding tool | |
| Wei et al. | Force predictive model for five-axis ball end milling of sculptured surface | |
| Chen et al. | A novel CNC grinding method for the rake face of a taper ball-end mill with a CBN spherical grinding wheel | |
| CN103197601A (en) | Cutter-shaft-swing five-coordinate numerical control machine tool pendulum length determination method | |
| CN110727246B (en) | Tool and workpiece instantaneous contact contour extraction method based on tool position file | |
| CN112799299A (en) | Robot multi-axis milling stability model and construction method thereof | |
| Liu et al. | A novel CNC machining method for enveloping surface | |
| Chen et al. | Design and NC machining of concave-arc ball-end milling cutters | |
| CN103745098A (en) | Numerical control machine tool single-shaft geometrical angle motion error separation method | |
| Yip-Hoi et al. | Finding the maximum turnable state for mill/turn parts | |
| CN114169095A (en) | Analysis method for milling stability of weak-rigidity ball-end milling cutter | |
| CN112347585A (en) | Analytical calculation method for contact area between ball-end milling cutter and workpiece | |
| Lee et al. | Toolpath generation method for four-axis NC machining of helical rotor | |
| CN113255076B (en) | Method for identifying cutter-tool cutting contact area during vertical machining of ball end mill | |
| CN113065205A (en) | Track solving method for grinding rear cutter face of arc head by adopting parallel grinding wheel | |
| CN102566509A (en) | Envelope theory-based universal cutter scanning body generating method in numerically controlled processing simulation process | |
| Guo et al. | New mathematical method for the determination of cutter runout parameters in flat-end milling |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |