CN113297696B - Modeling method for static milling force of ball end mill based on semi-analytic method - Google Patents

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

A modeling method for static milling force of ball-end milling cutter based on semi-analytical method

Technical Field

The invention belongs to the technical field of mechanical processing and relates to a modeling method of static milling force of a ball-end milling cutter based on a semi-analytical method.

Background Art

Ball end mills are widely used in the milling of important surfaces of related parts in industries such as molds, automobiles, and aerospace. In-depth research on the milling mechanism of ball end mills is of great significance to improving product quality. However, the modeling of its static milling force is the focus of cutting mechanism research, the basis and key to subsequent dynamic modeling, and the key basis for cutting parameter selection and optimization.

The identification of the tool-work contact area is a key link in the static milling force modeling. Its accuracy and computational efficiency directly affect the accuracy and efficiency of the static milling force prediction. However, the blade shape of the ball-end milling cutter is complex. In addition, due to the influence of factors such as posture adjustment and runout error, the identification of the tool-work contact area is difficult. The solid modeling and Boolean operation algorithms in the commonly used tool-work contact area identification methods use the swept envelope surface of the tool sweep body to simplify the real swept body of the tool teeth, ignoring the trochoidal motion of the cutting point on the tool teeth, which has certain principle errors. The Z-MAP discrete method can better judge the contact state of the tool teeth and improve the identification accuracy of the tool-work contact area through the idea of micro-element discreteness, but there is a problem of balancing accuracy and efficiency, which affects the application of subsequent research. Some scholars used a semi-analytical method to identify the tool-work contact area during ball-end milling. In the case of five-axis milling, the swept surface is always equivalent to a spherical surface with the radius of the tool ball head, and the actual effective radius change caused by eccentricity is not considered, which leads to certain errors.

The methods for calculating instantaneous undeformed chip thickness mainly include tool translation method and analytical calculation method. When the tool posture is adjusted, the modeling difficulty of the analytical calculation method increases. In order to simplify the calculation, the arc approximate trochoidal sweep trajectory is often used, which increases the model error.

Summary of the invention

The purpose of the present invention is to provide a modeling method for static milling force of a ball-end milling cutter based on a semi-analytical method, which can reduce model errors.

The technical solution adopted by the present invention is a modeling method of static milling force of a ball end milling cutter based on a semi-analytical method, comprising the following steps:

Step 1, respectively establish the local coordinate system of tooth j, the ball end milling cutter coordinate system, the spindle follower coordinate system, the tool instantaneous feed coordinate system, and the workpiece coordinate system, and obtain the trajectory equation of any point on tooth j in the workpiece coordinate system during the ball end milling cutter machining process based on the principle of homogeneous coordinate transformation;

Step 2, dividing the cutter tooth into a plurality of cutter tooth micro-elements with equal cutter tooth axial position angle increments, and establishing a micro-element cutting force model of the cutter tooth micro-element;

Step 3, identifying the knife-worker contact zone;

Step 4: Take the line from the sweep point _QC of the discrete point i on the cutter tooth j at time t to the cutter position _OCL as the reference line, calculate the distance between _QC and the intersection point _QL of the sweep surface of the previous cutter tooth and the reference line, and obtain the instantaneous undeformed chip thickness;

Step 5: Express the cutting force coefficient as a polynomial of the tool axial position angle, calculate the unknown coefficients in the polynomial of the tool axial position angle according to the average milling force, and then identify the cutting force coefficient.

The present invention is also characterized in that:

Step 1 specifically includes the following steps:

Step 1.1, taking the ball head center of the ball end milling cutter as the coordinate origin O _j , establish the local coordinate system O _j -X _j Y _j Z _j of the cutter tooth j, referred to as {j}; obtain the coordinates of any point P on any cutter tooth j of the ball end milling cutter in the local coordinate system {j};

Step 1.2, taking the ball head center of the ball end milling cutter as the coordinate origin O _C , establish the ball end milling cutter coordinate system O _C -X _C Y _C Z _C , referred to as {C}; obtain the homogeneous coordinate transformation matrix of the local coordinate system {j} relative to the ball end milling cutter coordinate system {C};

与主轴轴线重合；得到球头铣刀坐标系{C}相对于主轴随动坐标系{A}的齐次坐标变换矩阵；Step 1.3, take the spindle center as the coordinate origin _OA , and establish the spindle follower coordinate system _OA -X _A Y _A Z _A on the machine tool spindle, referred to as {A}, the coordinate axis

Coincident with the spindle axis; obtain the homogeneous coordinate transformation matrix of the ball end milling cutter coordinate system {C} relative to the spindle follower coordinate system {A};

Step 1.4, establish the tool instantaneous feed coordinate system O _CL -X _CL Y _CL Z _CL , referred to as {CL}, and obtain the homogeneous coordinate transformation matrix of the spindle follower coordinate system {A} relative to the tool instantaneous feed coordinate system {CL};

Step 1.5, establish a global coordinate system O _W -X _W Y _W Z _W on the workpiece, referred to as {W}, and obtain the homogeneous coordinate transformation matrix of {CL} relative to {W};

Combined with steps 1.1-1.5, the trajectory equation of any point P on the cutter tooth j under {W} during ball-end milling can be obtained through homogeneous coordinate matrix transformation:

Step 2 specifically includes the following steps:

Step 2.1, divide the cutter tooth into many cutter tooth micro-elements with equal cutter tooth axial position angle increments, use the characteristic information of the cutter tooth discrete point i to represent the information of the cutter tooth micro-element i between (i-1) and i on the cutter tooth, and decompose the cutting force on the cutter tooth micro-element i on the cutter tooth j at time t into tangential unit force cutting force dF _t (j,i,t), radial unit force cutting force dF _r (j,i,t), and axial unit force cutting force dF _a (j,i,t). According to the mechanical modeling method of cutting force, it can be obtained:

Wherein, g(j,i,t) is a unit step function. When the tooth element i on the tooth j contacts the workpiece at time t, g(j,i,t) = 1, otherwise, g(j,i,t) = 0; h(j,i,t) is the instantaneous undeformed chip thickness of the tooth element i on the tooth j at time t; _Kt , _Kr and _Ka are the tangential, radial and axial force coefficients respectively;

Step 2.2: Convert the tangential unit force cutting force dF _t (j,i,t), radial unit force cutting force dF _r (j,i,t), and axial unit force cutting force dF _a (j,i,t) of the tooth microelement i at time t to {A}. Then, the instantaneous cutting force of the ball end milling cutter at time t is expressed in the spindle follower coordinate system {A} as follows:

Where n _i is the total number of blade tooth elements;

Through the principle of homogeneous coordinate transformation, the instantaneous cutting force of the ball-end milling cutter at time t is expressed in the workpiece coordinate system {W} as:

Step 3.1 specifically includes the following steps:

Step 3.1.1, solve the boundary line I;

The intersection line of the sphere swept by the blade tooth and the sphere swept by the previous blade tooth, i.e., the boundary line I, is expressed as:

The surface formed by the last feed process is simplified into a cylindrical surface, which can be expressed in the coordinate system {CL} as follows:

(y ^CL +f _p ) ² +(z ^CL ) ² =R ² (25);

Combining (24) and (25), we can get the coordinates of point S under {CL}:

The equation of the workpiece top surface in the coordinate system {CL} is:

z ^CL = -(Ra _p ) (27);

Combining (24) and (27), we can get the coordinates of point M in the coordinate system {CL}:

The coordinates of the boundary line I, endpoint S, and endpoint M in the coordinate system {A} are obtained through homogeneous transformation:

Step 3.1.2, solve boundary line II;

Under {CL}, by combining (22) and (27), we can obtain the equation of the intersection line between the swept surface of the current cutter tooth and the surface to be machined, i.e., boundary line II:

Combining (25) and (30) we can obtain the coordinates of point N in the coordinate system {CL}:

The coordinates of boundary line II and endpoint N are transformed to {A} through homogeneous coordinate transformation:

Step 3.1.3, solve boundary line III;

By combining (22) and (25), we can obtain the equation of the intersection line between the swept surface of the current cutter tooth and the machined surface completed by the last feed under {CL}, i.e., boundary line III:

The equation of boundary line III is transformed to {A} by homogeneous coordinate transformation:

Step 3.2 specifically includes the following steps:

Step 3.2.1, assuming that the discrete accuracy of the axial position angle of the cutter tooth is Δθ, select discrete points on each boundary line whose maximum distance between discrete points is less than πΔθRcosγ/180, and substitute into (29), (32) and (34) to obtain the coordinate values of the discrete points on each boundary line under {A};

径向位置角

找出每条边界线所对应的当前刀齿起切触作用的最大、最小轴向位置角

并从三条边界线中找出最大、最小轴向位置角

即得到当前刀齿在主轴一转范围内切触工件的轴向位置角范围

Step 3.2.2: Use equations (35) and (36) to calculate the axial position angle of the cutter tooth corresponding to each discrete point on the boundary line obtained in step 3.2.1:

Radial position angle

Find the maximum and minimum axial position angles of the current cutter teeth corresponding to each boundary line.

And find the maximum and minimum axial position angles from the three boundary lines

That is, the axial position angle range of the current cutter tooth in contact with the workpiece within one spindle rotation range is obtained.

Where, mm∈(I,II,III), n is the number of the discrete points on the boundary line, nn＝1, 2,… _Nnn , _Nnn is the total number of discrete points on the boundary line;

为

的反正切函数，其主值域为(-180°,180°)；In the formula,

for

The inverse tangent function of has a main range of (-180°, 180°);

内的所有刀齿离散点所对应的径向位置角，按照第一切入角、第一切出角、第二切入角、第二切出角……顺序确定刀齿j上的当前轴向位置角θ的刀-工切触区间，即得到每个刀齿在主轴每一转范围内的刀-工切触区域。Step 3.2.3: Search the axial position angle range

The radial position angles corresponding to all the discrete points of the cutter teeth in the matrix are used to determine the tool-worker cutting contact interval of the current axial position angle θ on the cutter tooth j in the order of the first cutting-in angle, the first cutting-out angle, the second cutting-in angle, the second cutting-out angle, etc., that is, the tool-worker cutting contact area of each cutter tooth within each rotation range of the spindle is obtained.

Step 4 specifically includes the following steps:

Step 4.1: According to formula (9), the coordinates of the sweep point Q _C of the discrete point i on the current tooth j at time t can be obtained;

Step 4.2, ignore the feed motion of the previous tooth, simplify the front swept surface to a sphere, assume that the intersection of the reference line and the sphere is Q ^* , and solve the spherical equation and the reference line equation under {CL}:

为点Q^*在坐标系{CL}中的坐标值，

为点Q_C在坐标系{CL}中的坐标值；In the formula,

is the coordinate value of point Q ^* in the coordinate system {CL},

is the coordinate value of point Q _C in the coordinate system {CL};

已知，求解式(37)，利用齐次坐标变换原理，获取Q^*在机床主轴随动坐标系{A}中的坐标：because

It is known that by solving equation (37), the coordinates of Q ^* in the machine tool spindle follower coordinate system {A} can be obtained by using the principle of homogeneous coordinate transformation:

Then the axial position angle and radial position angle of point Q ^* are as follows:

近似求出被切削点Q^*所对应的切削时刻

同时，近似认为点Q_C、Q_L所对应刀位点之间的距离为每齿进给量f_z，根据正弦定理近似求出Q_L的轴向位置角

According to equations (40) and (41), the axial position angle θ _C and radial position angle φ _C of Q _C are obtained, and then the spiral lag angles ψ _{C and ψ C} corresponding to Q _C and Q ^* are calculated by the spiral lag angle calculation formula.

Approximately calculate the cutting time corresponding to the cut point Q ^*

At the same time, it is approximately assumed that the distance between the tool positions corresponding to points Q _C and Q _L is the feed per tooth f _z , and the axial position angle of Q _L is approximately calculated according to the sine theorem

Since Q _L is on the blade action line O _CL Q _L , the equation group is established according to the straight line formula:

为Q_C在工件坐标系{W}中的坐标，

为刀位点O_CL在工件坐标系{W}中的坐标；In the formula,

is the coordinate of Q _C in the workpiece coordinate system {W},

is the coordinate of the tool position point O _CL in the workpiece coordinate system {W};

为初值点，即

应用Newton-Raphson方法求得方程组(43)的解，如下式所示：by

is the initial value point, that is

The Newton-Raphson method is used to obtain the solution of equation (43), as shown below:

Wherein, 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 ;

Substituting the result obtained from equation (44) into equation (9), we can obtain the coordinates of Q _L in the workpiece coordinate system {W}:

Finally, the undeformed chip thickness is calculated according to the following formula:

Step 5 specifically includes the following steps:

Step 5.1, express the cutting force coefficient as the following polynomial of the tool axial position angle:

In the formula, a ₀ , a ₁ , a ₂ , a ₃ , b ₀ , b ₁ , b ₂ , b ₃ , c ₀ , c ₁ , c ₂ and c ₃ are unknown coefficients;

Step 5.2: Calculate the maximum axial position angle corresponding to the cutting depth a _p

Step 5.3, calculate the undeformed chip thickness according to the following formula:

h(j,θ,t)＝f _z sinφ(j,t)sinθ (48)

顺时针方向旋转所形成的夹角为正，φ(j,t)计算公式如下：Where φ(j,t) is the radial position angle of the plane blade tooth j at time t, and the specified rotation angle around the vector

The angle formed by clockwise rotation is positive, and the calculation formula of φ(j,t) is as follows:

Where, φ ₀ is the radial position angle of the reference cutter tooth in the initial state;

If φ(j,t)∈[-90,90], the tooth element touches the workpiece, g(j,θ,t)＝1; otherwise, g(j,θ,t)＝0;

Step 5.4. Express g(j,i,t), dF _t (j,i,t), dF _r (j,i,t), and dF _a (j,i,t) in formula (10) by g(j,θ,t), dF _t (j,θ,t), dF _r (j,θ,t), and dF _a (j,θ,t). Combining formulas (10), (48), and (49), transform dF _t (j,θ,t), dF _r (j,θ,t), and dF _a (j,θ,t) to the directions of coordinate axes O _A X _A , O _A Y _A , and O _A Z _A. The formulas are as follows:

Step 5.5, at a certain cutting depth, sum the milling forces of all tooth micro-elements involved in milling on tooth j at time t, and you can get the milling force on tooth j at time t. Then sum the milling forces on all teeth at that time, and finally get the total instantaneous milling force on the tool at time t, as shown in the following formula:

Using formula (48), the time variable t in (51) is converted into the tool tooth position angle variable φ, and then the average milling force on the tool in the directions of coordinate axes O _A X _A , O _A Y _A and O _A Z _A within one spindle rotation can be obtained:

和

代入公式(52)，然后利用最小二乘法回归出公式(47)所示的切削力系数公式中的待定系数a₀、a₁、a₂、a₃、b₀、b₁、b₂、b₃、c₀、c₁、c₂和c₃，从而，辨识出了切削力系数K_t、K_r和K_a。The average milling force within one rotation of the spindle is obtained through experiments

and

Substituting into formula (52), the least squares method is then used to regress the unknown coefficients a ₀ , a ₁ , a 2 , a ₃ , b ₀ , b ₁ , b ₂ , b ₃ , c ₀ , c ₁ , c ₂ and c ₃ in the cutting force coefficient formula shown in formula ( ₄₇ ), thereby identifying the cutting force coefficients K _t , K _r and _Ka .

The beneficial effects of the present invention are:

The present invention discloses a modeling method for the static milling force of a ball-end milling cutter based on a semi-analytical method, establishes the motion trajectory of the cutter teeth during the ball-end milling process based on the principle of homogeneous coordinate transformation, and proposes a cutting force coefficient identification method, a semi-analytical identification method for the cutter-workpiece contact area, and a solution method for the undeformed cutting thickness according to the actual milling conditions of the ball-end milling cutter, in order to provide a basis for subsequent research and a reference for the selection of machining parameters in the actual machining process.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG1 is a reference coordinate system diagram of the milling motion of a ball-end milling cutter according to a modeling method of static milling force of a ball-end milling cutter based on a semi-analytical method of the present invention;

FIG2 a is an axonometric diagram of the milling trajectory of a spiral-edged ball-end milling cutter according to a method for modeling static milling force of a ball-end milling cutter based on a semi-analytical method of the present invention;

FIG2 b is a top view of a milling trajectory of a method for modeling static milling force of a ball-end milling cutter based on a semi-analytical method of the present invention;

FIG3 a is an axonometric diagram of a coordinate system taking into account tool runout in a modeling method for static milling force of a ball-end milling cutter based on a semi-analytical method according to the present invention;

FIG3 b is a top view of a coordinate system taking into account tool runout in a modeling method for static milling force of a ball-end milling cutter based on a semi-analytical method of the present invention;

4 is a diagram of tool posture adjustment and tool path in a modeling method for static milling force of a ball-end milling cutter based on a semi-analytical method of the present invention;

FIG5 is a diagram showing the cutter-workpiece contact area during tilt milling of a modeling method for static milling force of a ball-end milling cutter based on a semi-analytical method of the present invention;

6 is a micro-element force diagram of a cutter tooth of a modeling method for static milling force of a ball-end milling cutter based on a semi-analytical method of the present invention;

7 is a schematic diagram of the instantaneous state of ball-end milling cutter milling according to a method for modeling static milling force of a ball-end milling cutter based on a semi-analytical method of the present invention;

8 is a schematic diagram of milling force coefficient identification of a modeling method for static milling force of a ball-end milling cutter based on a semi-analytical method of the present invention.

DETAILED DESCRIPTION

The present invention is described in detail below with reference to the accompanying drawings and specific embodiments.

A method for modeling static milling force of a ball end milling cutter based on a semi-analytical method comprises the following steps:

Step 1, as shown in FIG1 , respectively establish the local coordinate system of tooth j, the ball end milling cutter coordinate system, the spindle follower coordinate system, the tool instantaneous feed coordinate system, and the workpiece coordinate system, and obtain the trajectory equation of any point on tooth j in the workpiece coordinate system during the ball end milling cutter machining process based on the principle of homogeneous coordinate transformation;

与刀齿j的刃线在坐标平面

上投影线起点的切线方向重合；Step 1.1, take the ball head center of the ball end milling _cutter as the coordinate origin _Oj , establish the local coordinate system of tooth j _Oj - _XjYjZj , referred to as {j}, _and the coordinate axis

The edge line of tooth j is in the coordinate plane

The tangent directions of the starting points of the upper projection lines coincide;

As shown in Figure 2, the milling of a fixed-pitch helical-edge ball-end milling cutter widely used in actual production is taken as the research object. The coordinates of any point P on any tooth j of the ball-end milling cutter in the local coordinate system {j} are:

Where θ is the axial position angle of point P, R is the tool radius, ψ is the helical hysteresis angle corresponding to point P, ψ＝180tanγ ₀ (1-cosθ)/π, where γ ₀ is the helix angle of the tooth edge curve on the cylindrical surface;

与

完全一致，

与刀具的理论轴线重合，且与

始终保持平行，

与基准刀齿(第一个刀齿)刃线在坐标平面O_CX_CY_C上投影线起点的切线方向重合；Step 1.2, take the ball head center of the ball end milling cutter as the coordinate origin O _C , establish the ball end milling cutter coordinate system O _C -X _C Y _C Z _C , referred to as {C}, and the coordinate axis

and

Completely consistent,

coincides with the theoretical axis of the tool and

Always keep parallel,

It coincides with the tangent direction of the starting point of the projection line of the reference tooth (the first tooth) edge line on the coordinate plane O _C X _C Y _C ;

The included angle between tooth j and the reference tooth is φ _j = 360(j-1)/n _t , where n _t is the total number of teeth. Then the homogeneous coordinate transformation matrix of the local coordinate system {j} relative to the ball end milling cutter coordinate system {C} is:

与主轴轴线重合，坐标轴

与

之间的夹角为μ₀+φ_C(μ₀为主轴未开始旋转的初始状态下两者之间的夹角，φ_C为t时刻主轴旋转过的角度，φ_C＝ωt)；Step 1.3, take the spindle center as the coordinate origin _OA , and establish the spindle follower coordinate system _OA -X _A Y _A Z _A on the machine tool spindle, referred to as {A}, the coordinate axis

Coincident with the main axis, the coordinate axis

and

The angle between them is μ ₀ +φ _C (μ ₀ is the angle between them in the initial state when the spindle has not started to rotate, φ _C is the angle that the spindle has rotated at time t, φ _C =ωt);

相对于坐标轴

的夹角为μ，且规定绕坐标轴

顺时针旋转方向为正，主轴顺时针方向旋转，其转速为N，则角速度ω＝πN/30，t时刻旋转过的角度φ_C＝180ωt/π，则球头铣刀坐标系{C}相对于主轴随动坐标系{A}的齐次坐标变换矩阵为：Due to the influence of factors such as manufacturing and clamping errors, there is always eccentricity between the center axis of the tool and the center axis of the spindle, as shown in Figure 3. Assuming that the eccentric distance between the coordinate origin _OC and the coordinate origin _OA is ρ, the vector

Relative to the axis

The angle between them is μ, and it is stipulated that

The clockwise rotation direction is positive, the spindle rotates clockwise, and its speed is N, then the angular velocity ω＝πN/30, the angle rotated at time t φ _C ＝180ωt/π, then the homogeneous coordinate transformation matrix of the ball end milling cutter coordinate system {C} relative to the spindle follower coordinate system {A} is:

与

的初始夹角；本实施例中设定μ₀＝0；In the formula, μ＝μ ₀ +φ _C , where μ ₀ is the initial state.

and

The initial angle is: μ ₀ = 0 in this embodiment;

与进给速度方向平行且同向，

为理想的被加工表面的法线方向，指向实体外，

为

与

的叉乘；当

与

完全重合时，该坐标系的另外两个坐标轴及其方向与{CL}的完全重合，但是，实际工况当刀具姿态调整时，

与

之间存在夹角，体现为刀具相对于工件被加工表面的侧倾和前倾。如图4所示，所以通过使{A}通过相对于

和

的旋转实现主轴姿态的调整，进而实现刀具姿态的调整，从而获得不同的铣削方式，具体如下：Step 1.4, establish the tool instantaneous feed coordinate system O _CL -X _CL Y _CL Z _CL , referred to as {CL}, the coordinate axis vector

Parallel and in the same direction as the feed speed,

is the normal direction of the ideal machined surface, pointing outside the solid.

for

and

The cross product of

and

When the coordinate system is completely coincident, the other two coordinate axes and their directions of the coordinate system are completely coincident with {CL}. However, in actual working conditions, when the tool posture is adjusted,

and

There is an angle between them, which is reflected as the side tilt and forward tilt of the tool relative to the workpiece surface. As shown in Figure 4, by making {A} pass relative to

and

The rotation of the spindle can adjust the spindle posture, and then adjust the tool posture, so as to obtain different milling methods, as follows:

方向为刀具进给方向，

为刀具间歇进给方向，主轴随动坐标系{A}分别绕这两个坐标轴矢量旋转实现主轴姿态的调整。主轴姿态调整后坐标系{A}的坐标轴矢量

在坐标平面Y_CLO_CLZ_CL上的投影线与坐标轴矢量

间的夹角，称为侧倾角，用α表示；坐标轴矢量

在坐标平面X_CLO_CLZ_CL上的投影与坐标轴矢量

之间的夹角，称为前倾角，用β表示。先使{A}绕

旋转角度β'，使β'＝arctan(tanβcosα)，再使{A}绕

旋转角度α，且定义绕各自参考方向的正方向逆时针旋转为正，则刀具侧倾和前倾的齐次坐标变换矩阵分别为Coordinate axis vector

Direction is the tool feed direction,

The intermittent feed direction of the tool, the spindle follower coordinate system {A} rotates around these two coordinate axis vectors to adjust the spindle posture. The coordinate axis vector of the coordinate system {A} after the spindle posture is adjusted

Projection line and coordinate axis vector on coordinate plane Y _CL O _CL Z _CL

The angle between the two is called the roll angle and is represented by α; the coordinate axis vector

Projection on the coordinate plane X _CL O _CL Z _CL and the coordinate axis vector

The angle between them is called the forward tilt angle and is represented by β.

Rotate the angle β' to make β' = arctan (tanβcosα), and then make {A} rotate around

The rotation angle is α, and the counterclockwise rotation around the positive direction of each reference direction is defined as positive. Then the homogeneous coordinate transformation matrices of tool tilt and forward tilt are

Then the homogeneous coordinate transformation matrix of the spindle follower coordinate system {A} relative to the tool instantaneous feed coordinate system {CL} is:

Step 1.5, establish the global coordinate system O _W -X _W Y _W Z _W on the workpiece, referred to as {W}. Assuming that the coordinates of O _CL in {W} during feeding are (x _CL , y _CL , z _CL ), the homogeneous coordinate transformation matrix of {CL} relative to {W} is:

和

分别表示坐标轴

和

上的单位矢量，下标x、y和z表示各矢量在

和

上的投影矢量；In the formula,

and

Respectively represent the coordinate axes

and

The subscripts x, y, and z denote the unit vectors on

and

The projection vector on ;

This embodiment takes the unidirectional linear feed milling plane as the research object, and the homogeneous coordinate transformation matrix of {CL} relative to {W} is:

Wherein, (x ₀ ,y ₀ ) is the starting position of O _CL in {W} at the first feed, q is the number of tool feeds (q＝1,2,3…), t is the time taken by the tool from the first pass to the current position, f _z is the feed amount per tooth, f _p is the feed line spacing, L is the length of a single pass, R is the tool radius, w _h is the blank height, and a _p is the cutting depth;

Combining formulas (1)-(6) and (8), the trajectory equation of any point P on the cutter tooth j under {W} during ball-end milling can be obtained through homogeneous coordinate matrix transformation:

Step 2, as shown in FIG5 , the cutter tooth is divided into a plurality of cutter tooth micro-elements with equal cutter tooth axial position angle increments, and a micro-element cutting force model of the cutter tooth micro-element is established;

Step 2.1, divide the cutter tooth into many cutter tooth micro-elements with equal cutter tooth axial position angle increments, use the characteristic information of the cutter tooth discrete point i to represent the information of the cutter tooth micro-element i between (i-1) and i on the cutter tooth, and decompose the cutting force on the cutter tooth micro-element i on the cutter tooth j at time t into tangential unit force cutting force dF _t (j,i,t), radial unit force cutting force dF _r (j,i,t), and axial unit force cutting force dF _a (j,i,t). According to the mechanical modeling method of cutting force, it can be obtained:

Wherein, g(j,i,t) is a unit step function. When the tooth element i on the tooth j contacts the workpiece at time t, g(j,i,t) = 1, otherwise, g(j,i,t) = 0; h(j,i,t) is the instantaneous undeformed chip thickness of the tooth element i on the tooth j at time t; _Kt , _Kr and _Ka are the tangential, radial and axial force coefficients respectively;

Step 2.2: The tangential unit force cutting force dF _t (j,i,t), radial unit force cutting force dF _r (j,i,t), and axial unit force cutting force dF _a (j,i,t) of the tooth element i at time t are converted into {A} by formula (11):

顺时针转过的角度，

为刀齿j上离散点i和坐标原点O_A的连线与O_AZ_A的锐角夹角；Where φ(j,i, _t ) is the projection of the line connecting the origin _OA and the position of discrete point i on tooth j at time t on the plane _XA0AYA relative to the coordinate axis _vector

The angle of clockwise rotation,

is the acute angle between the line connecting the discrete point i on the tooth j and the coordinate origin _{OA and OAZA} ;

Then the instantaneous cutting force on the ball end mill at time t is expressed in the spindle follower coordinate system {A} as:

Where n _i is the total number of blade tooth elements;

Through the principle of homogeneous coordinate transformation, the instantaneous cutting force of the ball-end milling cutter at time t is expressed in the workpiece coordinate system {W} as:

搜寻轴向位置角范围

内的所有刀齿j上离散点i所对应的径向位置角，根据径向位置角对刀齿j上离散点i进行筛选，确定刀齿j上的当前轴向位置角θ的刀-工切触区间。Step 3. In actual processing, the spindle posture is often adjusted through program design to achieve the adjustment of the tool posture. This is necessary to prevent the tool from interfering with the workpiece being processed, and it is also necessary to avoid the ball-end milling cutter head to achieve high-quality and efficient cutting. However, the tool posture adjustment makes it difficult to identify the tool-work contact area. According to the trajectory of any point on the tooth j during the ball-end milling process, the tool-work contact area is determined to be the yellow part shown in Figure 6, that is, the area formed by boundary lines I, II, and III. The boundary lines I, II, III, and the intersection of the three boundary lines are solved; boundary line I is the intersection between the current tooth and the swept surface of the previous tooth, boundary line II is the intersection of the swept surface of the current tooth and the surface to be processed, and boundary line III is the intersection of the swept surface of the current tooth and the processed surface completed by the last feed; from boundary lines I, II, and III, find the maximum and minimum axial position angles of the tooth for contact.

Search axial position angle range

The radial position angles corresponding to the discrete points i on all the cutter teeth j in the image are obtained, and the discrete points i on the cutter tooth j are screened according to the radial position angles to determine the tool-worker contact interval of the current axial position angle θ on the cutter tooth j.

Step 3.1, solve the boundary line I;

以角速度ω旋转时，不同刀齿上具有相同轴向位置角的切削点的回转半径是不同的，相邻两齿间的容屑角(如图3所示的η_P)也随着刀齿轴向位置角的变化而变化。根据步骤1的分析可知，在不考虑主轴旋转时，刀齿j上的离散点i在{A}中的坐标为：In order to simplify the calculation, only the rotational motion of the cutter teeth is considered, the continuous feed motion between two adjacent cutter teeth is ignored, and the swept surface of the previous cutter teeth is simplified to 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. The boundary line I can be obtained by solving the intersection line between the current cutter tooth rotation swept surface and the spherical surface. However, in actual production, due to the effect of tool eccentricity, when the tool rotates around the coordinate axis, the boundary line I can be obtained.

When rotating at an angular velocity ω, the gyration radius of the cutting points with the same axial position angle on different teeth is different, and the chip angle between two adjacent teeth (η _P as shown in Figure 3) also changes with the change of the axial position angle of the teeth. According to the analysis in step 1, when the spindle rotation is not considered, the coordinates of the discrete point i on the tooth j in {A} are:

表示刀齿j上的离散点i在{j}中的坐标；Wherein, M _AC | _{φC = 0} is the transformation matrix of {C} relative to {A} when only the tool eccentricity is considered without considering the spindle rotation.

represents the coordinates of discrete point i on tooth j in {j};

的回转半径，即实际切削半径

在μ₀＝0的情况下，由式(14)可得：The discrete point i on the tooth j is relative to the coordinate axis

The radius of gyration, i.e. the actual cutting radius

When μ ₀ = 0, we can get from formula (14):

为：Similarly, its actual axial position angle

for:

Then the actual spiral lag angle of discrete point i on the reference tooth is:

Where, ψ _i and θ _i are the spiral hysteresis angle and axial position angle of the ideal discrete point i of the tooth, respectively;

Then the actual cutting radius vector of discrete point i on tooth j is:

The angle between the spindle axis and the normal of the machined surface is

等于γ，将其带入式(16)得到刀齿j离散点i的位置，进而得到理论轴向位置角θ_i，即可获知该刀齿j上与被加工表面法线方向一致的切削点；然后，由式(15)求出切削点的实际切削半径

并根据下式求出相邻两齿的切削点的半径矢量

和

之间的径向夹角：The actual axial position angle of discrete point i on tooth j is

=γ, and it is substituted into formula (16) to obtain the position of the discrete point i of the cutter tooth j, and then the theoretical axial position angle θ _i is obtained, and the cutting point on the cutter tooth j that is consistent with the normal direction of the machined surface can be obtained; then, the actual cutting radius of the cutting point is calculated by formula (15):

And the radius vector of the cutting point of two adjacent teeth is calculated according to the following formula:

and

The radial angle between:

When the radius vectors of the characteristic cutting points of two adjacent teeth are consistent with the normal line of the workpiece surface, the distance between the two cutting points left on the workpiece in the feed direction is:

处刀齿做旋转运动时，在{CL}下，当前刀齿扫掠面、上一刀齿扫掠面的方程分别如式(22)、(23)：The sweep surface of the current tooth is simplified to a spherical surface, the feed motion of the current tooth is ignored, and only the distance from the center of the sweep surface of the previous tooth to O _A is considered.

When the cutter tooth rotates, under {CL}, the equations of the current cutter tooth swept surface and the previous cutter tooth swept surface are as follows:

表示刀齿j上的离散点i至O_A点的距离；In the formula,

Represents the distance from discrete point i on tooth j to point _OA ;

According to equations (22) and (23), the intersection line between the swept sphere of the blade tooth and the swept sphere of the previous blade tooth, i.e., the boundary line I, can be obtained:

The surface formed by the last feed process is simplified into a cylindrical surface, which can be expressed in the coordinate system {CL} as follows:

(y ^CL +f _p ) ² +(z ^CL ) ² =R ² (25);

Combining (24) and (25), we can get the coordinates of point S under {CL}:

The equation of the workpiece top surface in the coordinate system {CL} is:

z ^CL = -(Ra _p ) (27);

Combining (24) and (27), we can get the coordinates of point M in the coordinate system {CL}:

The coordinates of the boundary line I, endpoint S, and endpoint M in the coordinate system {A} are obtained through homogeneous transformation:

Solve for boundary line II;

Under {CL}, by combining (22) and (27), we can obtain the equation of the intersection line between the swept surface of the current cutter tooth and the surface to be machined, i.e., boundary line II:

Combining (25) and (30) we can obtain the coordinates of point N in the coordinate system {CL}:

The coordinates of boundary line II and endpoint N are transformed to {A} through homogeneous coordinate transformation:

Solve for boundary line III;

By combining (22) and (25), we can obtain the equation of the intersection line between the swept surface of the current cutter tooth and the machined surface completed by the last feed under {CL}, i.e., boundary line III:

The equation of boundary line III is transformed to {A} by homogeneous coordinate transformation:

Step 3.2. In order to simplify the complex calculation, the boundary lines I, II, and 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 Δθ, in order to ensure that the maximum distance between the discrete points on the boundary line after transformation does not exceed πΔθR ₀ /180, the discrete points whose maximum distance between the discrete points on each boundary line is less than πΔθRcosγ/180 are selected, and the coordinate values of the discrete points on each boundary line under {A} are obtained by substituting (29), (32), and (34);

径向位置角

找出每条边界线所对应的当前刀齿起切触作用的最大、最小轴向位置角

并从三条边界线中找出最大、最小轴向位置角

即得到当前刀齿在主轴一转范围内切触工件的轴向位置角范围

The axial position angle of the cutter tooth corresponding to each discrete point on the boundary line obtained in step 3.2.1 is obtained by equations (35) and (36):

Radial position angle

Find the maximum and minimum axial position angles of the current cutter teeth corresponding to each boundary line.

And find the maximum and minimum axial position angles from the three boundary lines

That is, the axial position angle range of the current cutter tooth in contact with the workpiece within one spindle rotation range is obtained.

Where, mm∈(I,II,III), n is the number of the discrete points on the boundary line, nn＝1,2,… _Nnn , _Nnn is the total number of discrete points on the boundary line;

为

的反正切函数，其主值域为(-180°,180°)；In the formula,

for

The inverse tangent function of has a main range of (-180°, 180°);

内的所有刀齿离散点所对应的径向位置角，大多数情况下，一刀齿离散点的切入和切出发生在不同的边界线上，但是，也存在一条边界线切入并切出的少数情况，同时，考虑到某一刀齿离散点可能存在两次切入和切出的情况，因此，用结构数组存储切入切出角。具体过程如下：a.从

开始，以Δθ为增量，判断当前刀齿j的轴向位置角θ所属边界线区间

b.找到的每条边界线中轴向位置角接近θ的10个离散点，并按相对于θ绝对差值的升序排列；c.对排列之后的每条边界线中的离散点，从第二个开始，剔除与相邻的上一个离散点的径向位置角绝对差值小于3°的离散点；d.将筛选之后的所有边界线上的离散点放在一起，按径向位置角升序排列，同样，从第二个开始，剔除与相邻的上一个离散点的径向位置角绝对差值小于3°的离散点，完成二次筛选；如果本次筛选完之后只剩下一个离散点，则需要将最后一个剔除的离散点重新添加；e.将步骤d得到的按径向位置角升序排列的边界线离散点，按照第一切入角、第一切出角、第二切入角、第二切出角……顺序确定刀齿j上的当前轴向位置角θ的刀-工切触区间，即得到每个刀齿在主轴每一转范围内的刀-工切触区域。Search axial position angle range

The radial position angles corresponding to all the discrete points of the cutter teeth in the cutter tooth. In most cases, the cut-in and cut-out of a discrete point of a cutter tooth occur on different boundary lines. However, there are a few cases where a boundary line cuts in and cuts out. At the same time, considering that a discrete point of a cutter tooth may cut in and out twice, a structure array is used to store the cut-in and cut-out angles. The specific process is as follows: a. From

At the beginning, with Δθ as the increment, determine the boundary line interval to which the axial position angle θ of the current cutter tooth j belongs

b. Find 10 discrete points in each boundary line whose axial position angle is close to θ, and arrange them in ascending order of absolute difference relative to θ; c. For the discrete points in each boundary line after arrangement, starting from the second one, remove the discrete points whose absolute difference in radial position angle with the adjacent previous discrete point is less than 3°; d. Put all the discrete points on the boundary lines after screening together, and arrange them in ascending order of radial position angle. Similarly, starting from the second one, remove the discrete points whose absolute difference in radial position angle with the adjacent previous discrete point is less than 3° to complete the secondary screening; if only one discrete point is left after this screening, the last removed discrete point needs to be added back; e. Determine the tool-worker contact interval of the current axial position angle θ on the cutter tooth j in the order of the first cutting-in angle, the first cutting-out angle, the second cutting-in angle, the second cutting-out angle... for the discrete points of the boundary lines obtained in step d that are arranged in ascending order of radial position angle, that is, obtain the tool-worker contact area of each cutter tooth within each rotation range of the spindle.

Step 4: Take the line from the sweep point _QC of the discrete point i on the cutter tooth j at time t to the cutter position _OCL as the reference line, as shown in Figure 7. The distance between the two points _QL and _QC on the reference line is the undeformed chip thickness h(j,i,t). Calculate the distance between _QC and the intersection point _QL of the sweep surface of the previous cutter tooth and the reference line to obtain the instantaneous undeformed chip thickness.

Step 4.1: According to formula (9), the coordinates of the sweep point Q _C of the discrete point i on the current tooth j at time t can be obtained;

Step 4.2, ignore the feed motion of the previous tooth, simplify the front swept surface to a sphere, assume that the intersection of the reference line and the sphere is Q ^* , and solve the spherical equation and the reference line equation under {CL}:

为点Q^*在坐标系{CL}中的坐标值，

为点Q_C在坐标系{CL}中的坐标值；In the formula,

is the coordinate value of point Q ^* in the coordinate system {CL},

is the coordinate value of point Q _C in the coordinate system {CL};

已知，求解式(37)，根据实际加工情况，舍弃

的大的取值，得because

It is known that, solving equation (37), according to the actual processing situation, discard

For a large value of

Using the principle of homogeneous coordinate transformation, obtain the coordinates of Q ^* in the machine tool spindle follow-up coordinate system {A}:

Then the axial position angle and radial position angle of point Q ^* are as follows:

近似求出被切削点Q^*所对应的切削时刻

同时，近似认为点Q_C、Q_L所对应刀位点之间的距离为每齿进给量f_z，根据正弦定理近似求出Q_L的轴向位置角

According to equations (40) and (41), the axial position angle θ _C and radial position angle φ _C of Q _C are obtained, and then the spiral lag angles ψ _{C and ψ C} corresponding to Q _C and Q ^* are calculated by the spiral lag angle calculation formula.

Approximately calculate the cutting time corresponding to the cut point Q ^*

At the same time, it is approximately assumed that the distance between the tool positions corresponding to points Q _C and Q _L is the feed per tooth f _z , and the axial position angle of Q _L is approximately calculated according to the sine theorem

Since Q _L is on the blade action line O _CL Q _L , the equation group is established according to the straight line formula:

为Q_C在工件坐标系{W}中的坐标，

为刀位点O_CL在工件坐标系{W}中的坐标；In the formula,

is the coordinate of Q _C in the workpiece coordinate system {W},

is the coordinate of the tool position point O _CL in the workpiece coordinate system {W};

为初值点，即

应用Newton-Raphson方法求得方程组(43)的解，如下式所示：by

is the initial value point, that is

The Newton-Raphson method is used to obtain the solution of equation (43), as shown below:

Wherein, 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 ;

Substituting the result obtained from equation (44) into equation (9), we can obtain the coordinates of Q _L in the workpiece coordinate system {W}:

Finally, the undeformed chip thickness is calculated according to the following formula:

Step 5, expressing the cutting force coefficient as a polynomial of the tool axial position angle, calculating the undetermined coefficient in the polynomial of the tool axial position angle according to the average milling force, and then identifying and obtaining the cutting force coefficient;

Step 5.1, the cutting force coefficient is the proportional relationship between the cross-sectional area of the instantaneous undeformed chip and the micro-element forces in all directions. The cutting force coefficient directly affects the prediction accuracy of the micro-element milling force and is one of the key factors in cutting force modeling. However, the cutting force coefficient varies with factors such as the tool and workpiece material, cutting parameters, etc., which adds a certain degree of difficulty to the identification of the cutting force coefficient. When the ball-end of the ball-end milling cutter is cutting, the cutting speed, radial cutting depth, etc. of the tooth micro-elements at different axial positions in the actual cutting are different, which makes the cutting mechanism different. Therefore, the cutting force coefficient is expressed as the following polynomial of the tool axial position angle:

In the formula, _a0 , _a1 , _a2 , _a3 , _b0 , _b1 , _b2 , _b3 , _c0 , _c1 , _c2 and _c3 are unknown coefficients.

因此，可以建立切削力系数和刀具轴向位置角的关系；Step 5.2, adopt the slot milling method shown in Figure 8 to facilitate the determination of the cutter teeth's cut-in and cut-out angles. At the same time, the average milling force method is used to eliminate the influence of the helical angle on the identification accuracy, and the flat-edge ball-end milling cutter model is used to replace the complex helical edge to achieve the purpose of simplifying the calculation. Since the minimum axial position angle of the cutter teeth contacting the workpiece in vertical milling is zero, changing the cutting depth means changing the maximum axial position angle of the cutter teeth contacting the workpiece. Calculate the maximum axial position angle corresponding to the cutting depth a _p

Therefore, the relationship between the cutting force coefficient and the tool axial position angle can be established;

Step 5.3: Since the vertical slot milling method is adopted and the influence of runout is eliminated by the average milling force method, the undeformed chip thickness is calculated as follows:

h(j,θ,t)=f _z sinφ(j,t)sinθ (48);

顺时针方向旋转所形成的夹角为正，φ(j,t)计算公式如下：Where φ(j,t) is the radial position angle of the plane blade tooth j at time t, and the specified rotation angle around the vector

The angle formed by clockwise rotation is positive, and the calculation formula of φ(j,t) is as follows:

Where, φ ₀ is the radial position angle of the reference cutter tooth in the initial state;

If φ(j,t)∈[-90,90], the tooth element touches the workpiece, g(j,θ,t)＝1; otherwise, g(j,θ,t)＝0;

Step 5.4. Express g(j,i,t), dF _t (j,i,t), dF _r (j,i,t), and dF _a (j,i,t) in formula (10) by g(j,θ,t), dF _t (j,θ,t), dF _r (j,θ,t), and dF _a (j,θ,t). Combining formulas (10), (48), and (49), transform dF _t (j,θ,t), dF _r (j,θ,t), and dF _a (j,θ,t) to the directions of coordinate axes O _A X _A , O _A Y _A , and O _A Z _A. The formulas are as follows:

Step 5.5: In the case of vertical milling, change the cutting depth to conduct experiments and measure the average cutting force in the tooth action cycle at different cutting depths. Since the total amount of material removed in a tooth cycle is a constant and has nothing to do with the presence or absence of a helix angle, the average cutting force is also independent of the helix angle. In order to reduce the influence of eccentricity caused by factors such as tool installation and force, first measure the total cutting force in the spindle rotation cycle with a dynamometer, then divide it by the number of teeth of the tool to calculate the average cutting force.

At a certain cutting depth, the milling force of all tooth micro-elements involved in milling on tooth j at time t is summed to obtain the milling force on tooth j at time t. Then the milling force on all teeth at that moment is summed to finally obtain the total instantaneous milling force on the tool at time t, as shown in the following formula:

Using formula (48), the time variable t in (51) is converted into the tool tooth position angle variable φ, and then the average milling force on the tool in the directions of coordinate axes O _A X _A , O _A Y _A and O _A Z _A within one spindle rotation can be obtained:

和

代入公式(52)，然后利用最小二乘法回归出式公式(47)所示的切削力系数公式中的待定系数a₀、a₁、a₂、a₃、b₀、b₁、b₂、b₃、c₀、c₁、c₂和c₃，从而，辨识出了切削力系数K_t、K_r和K_a。The average milling force within one rotation of the spindle is obtained through experiments

and

Substituting into formula (52), the least squares method is then used to regress the unknown coefficients a ₀ , a ₁ , a 2 , a ₃ , b _{0 , b 1 ,} b ₂ , b ₃ , c ₀ , c ₁ , c ₂ and c ₃ in the cutting force coefficient formula shown in formula ( ₄₇ ), thereby identifying the cutting force coefficients K _t , K _r and _Ka .

Through the above methods, the present invention provides a modeling method for the static milling force of a ball-end milling cutter based on a semi-analytical method, establishes the motion trajectory of the cutter teeth during the ball-end milling process based on the principle of homogeneous coordinate transformation, and proposes a cutting force coefficient identification method, a semi-analytical identification method for the cutter-workpiece contact area, and a solution method for the undeformed cutting thickness according to the actual situation of the ball-end milling cutter, in order to provide a basis for subsequent research and a reference basis for the selection of machining parameters in the actual machining process; under the premise of ensuring the recognition accuracy, a semi-analytical identification method for the cutter-workpiece contact area is obtained based on the spherical assumption and the principle of inverse transformation of homogeneous coordinates, which can improve the recognition efficiency of the cutter-workpiece contact area; the cutting force coefficient of the ball-end milling cutter is identified by the average milling force method, and the mechanical identification method that combines the theory and experiment of the milling force coefficient of the milling cutter is used to eliminate the influence of the tool helix angle and offset the influence of periodic chatter on the measurement data.

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 。/>

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