# Geometric shape of the end edge of the most popula

2022-07-23
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Abstract: a mathematical model for the geometric shaping of the end edge of the end milling cutter with indexable ball head is established. Using this model, the machining adjustment parameters of the blade groove on the cutter body, the arc radius of the blade and the distribution of the geometric angle of the milling cutter along the cutting edge can be calculated

key words: geometric modeling of the end edge of the indexable ball end mill

the indexable ball end mill is an advanced tool for Machining Spatial free-form surfaces in recent years. In addition to all the advantages of the indexable tool, compared with the integral ball end mill, because each overlapping blade cuts the arc edge in sections, it greatly improves the poor state of non free cutting of the arc edge and reduces the non free coefficient, Thus, the cutting force is reduced 

relevant manufacturers in the world's industrial developed countries have successively developed various varieties and specifications of indexable ball end mills, such as Sandvik company in Sweden, Walter company in Germany, Ingersoll company in Japan and Mitsubishi Corporation in Japan. Some domestic manufacturers use foreign technology to produce indexable ball end mills, and a few manufacturers have made trial production. However, due to the lack of relevant theory and technology, they have not developed internationally competitive products. In order to make up for this deficiency, the author has made a deep and systematic research on the indexable ball end milling cutter in order to improve the theoretical and technical level of domestic production of indexable ball end milling cutter

there are no reports on the theoretical calculation and shape of indexable ball end mills based on the relevant data at home and abroad. With the help of the basic theory of geometric shaping, and combined with some advantages of the end edge of the integral ball end mill, this paper establishes a mathematical model of the geometric shaping of the end edge of the indexable ball end mill. Using this model, the machining adjustment parameters of the blade groove on the upper edge of the cutter body, the blade arc radius and the distribution of the cutting angle along the end edge of the mill can be calculated

I. mathematical model

the integral ball end mill with good cutting performance has an S-shaped end edge (as shown in Figure 1). If the end edge of the indexable ball end mill also has an S-shaped end edge similar to the integral ball end mill, the cutting edge of the blade can be installed and overlapped along the S-shaped edge of the integral ball end mill. At this time, the tangent at a certain point on the blade should coincide with the tangent at the corresponding point of the S-shaped end edge of the integral ball end mill, The blade is then rotated by an appropriate angle along the tangent line to form the required front and rear corners

Fig. 1 coordinate system

due to the symmetry of S-shaped edge, the first half of S-shaped edge is studied. As shown in Figure 1, the milling cutter coordinate system OO XYZ and the blade coordinate system OO xoyozo are established. In the milling cutter system, the oz axis coincides with the milling cutter axis, the spherical radius is r, the front face of the blade is set as the plane and located in the YOZO plane of the blade system, the rear face of the blade is a cone, and the half cone angle is β， The axis of the conical surface is perpendicular to the rake face of the blade. The intersection of the rake face and the rake face is an arc-shaped cutting edge. Set the radius as R, and place the midpoint of the cutting edge at o0

1. According to the introduction of foreign materials, the ideal S-shaped end edge of the integral ball end mill should be the intersection of the rake face formed by the orthogonal spiral surface and the spherical surface. In the milling cutter coordinate system o-xyz, the equation of the orthogonal helicoid can be written as Rx in the formula

, θ—— Parameter

since the end edge is on both the orthogonal helix and the spherical surface, it must meet the requirements of

z2 + Y2 + Z2 = R2

substituting equation (1) into equation (2), and after sorting and simplifying, there is

rx = r2

where m =

ω 1 - helix angle on circular cylinder with radius R

substitute equation (3) into equation (1) to obtain θ The equation

of the end edge curve expressed by is known from differential geometry. The tangent vector t at any point on the end edge curve is

and its tangent unit vector t0 = t/|t

substituting equation (5) and equation (7) into equation (6) has

2 Calculation of blade installation parameters

as shown in Figure 2, select any point m on the S-shaped blade, move the blade with its coordinate system, make the o0 point coincide with the m point, and make the x0 axis parallel to the X axis; Y0 axis is parallel to y axis; The Z0 axis is parallel to the Z axis. In the blade coordinate system, the tangent unitary of the blade o0 point is the opposite direction of the unitary on the Z0 axis. First, reverse the blade and its coordinate system around the x0 axis ω Angle, make the blade system become o0-x0y1z1, and then make the blade system reverse around the Y1 axis ω The angle becomes o0-x1y1z2, assuming that at this time, the Z2 axis on the blade system coincides with the tangent vector at the m point of the S-shaped cutter of the milling cutter, and the S-shaped check whether the water supply system of the 1 lower water level controller can 2.1.1 the hydraulic pulsator is a mechanism that generates high-pressure pulsating oil, so that the cutting unit vector on the normal water supply blade must be equal to the unit vector on the Z2 axis, and the unit vector on the Z2 axis is written in the milling cutter coordinate system with

r = ax0 (－) ω） AY1 (－) R1 (9)

where ax0 (－ ω）， AY1 (－) is a coordinate transformation matrix, which can be written as

ax1, y1

and R =  t

Figure 2 blade installation rotation transformation

substituting the above two equations into equation (9), the expression of the unitary vector on the Z2 axis in the milling cutter coordinate system o-xyz is obtained

since the Z2 axis has coincided with the tangent of the S-shaped edge of the milling cutter, the unitary vector on the Z2 axis must be equal to the tangent unitary vector of the m point on the S-shaped edge in the same coordinate system.The components in the shall also be equal, Compare equation (8) and equation (10) to get

and finally make the blade rotate forward around the Z2 axis with its coordinate system ψ When the angle reaches the position of o0 － x2y2z2, the blade is installed on the tool body. adjustment ψ The front angle and back angle of the milling cutter can be adjusted as required. Obviously, the rotation angle ω、 φ and ψ It refers to the processing adjustment parameters of the blade groove of the milling cutter body. Only when the blade groove is processed, the blade does not rotate, but the cutter body rotates in the opposite direction ω、 And ψ Value

the theoretical blade arc radius r should be the radius of the intersection line between the rake face of the blade and the spherical surface after the blade is installed on the tool body. As shown in Figure 3, in the milling cutter coordinate system, the equation of the spherical surface can be written as R =

ψ，θ—— Parameter variable

Figure 3 spherical equation

makes the conversion relationship between milling cutter system o-xyz and blade system o0-x2y2z0

r =? + Ax0，y1，z2（－ ω，－，ψ） R2

where, ax0, Y1, Z2 (－ ω，－ ， ψ） Is the coordinate transformation matrix, and its expression is

ax0y1z2 (－ ω，－，ψ）＝

where a11 = COSCOs ψ

a12＝－cossin ψ

a13＝－sin

a21＝sin ω sincos ψ＋ cos ω sin ψ

a22＝－sin ω sinsin ψ＋ cos ω cos ψ

a23＝sin ω cos

a31＝cos ω sincos ψ－ sin ψ sin ω

a32＝－cos ω sinsin ψ－ sin ω cos ψ

a33＝cos ω Cos

substituting equation (16) into equation (15) has

r =

or R2 =

substituting equation (13) and (14) into equation (17), the expression of the spherical surface in the blade system o0-x2y2z2 is

because the theoretical blade arc radius should be the radius of the intersection of the front blade surface and the spherical surface after installation, so that x2 = O in equation (18) has

a11cos ψ cos θ＋ a21cos ψ sin θ＋ a31sin ψ＝

the above formula can be written as DCOS ψ＋ Esin ψ＝ B

where d = a11cos θ＋ a21sin θ

e = a31

consolidate and improve the enterprise's share in the western market B =

solve equation (19) with

substitute equation (20) into the expression of Y2 and Z2 in equation (18) to obtain θ The arc radius r of the blade can be obtained by taking three points at any point of the intersection circle

4. As shown in Figure 1b, in the blade coordinate system o0-x0y0z0, the cutting unit vector of any point P on the blade arc edge can be expressed as SP = [0-sinu COSU] t

the normal rake unit vector on the rake face is γ PN = [0-cosu-sinu] t

the unit vector of the normal rake angle on the flank is α pn＝［－cos β － sin β cosu －sin β Sinu 〕 t

change and into the milling cutter coordinate system, where

sp2 ＝ ax0, Y1, Z2 (－ ω，－，ψ） Sp＝

γ pn2＝Ax0，y1，z2（－ ω，－，ψ）γ pn

α pn2＝Ax0，y1，z2（－ ω，－，ψ）α PN =

in the milling cutter coordinate system o-xyz, the cutting speed unit V0 at any point P on the blade cutting edge is V0 = [sin θ － cos θ 0 〕 t

blade inclination at any point on the blade λ s. Anterior horn of method γ N and normal back angle α N is  sin λ s＝Sp2V0＝－a12sinusin θ＋ a13cosusin θ＋

α 22sinucos θ－ a23cosucos θ

sinrn＝（rpn2U0）

＝（－a11cos β sin θ－ a12sin β cosusin θ－ a13sin β sinusin θ＋ a21cos β cos θ＋

a22sin β cosucos θ＋ a23sin β sinucos θ）

calculate the parameter u and θ Relationship between. In the blade coordinate system o0 － x0y0z0, the arc equation of the blade cutting edge can be written as R0 =

convert the above equation to the o0 － x2y2z2 coordinate system, where R2 = ax0, Y1, Z2 (－ θ，－，ψ） R0 =

since the blade circle is the intersection circle between the rake face and the spherical surface, the corresponding coordinates of the same point in the same coordinate system should be equal. Refer to formula (18) for P align=center>a13 (RCOs ψ cos θ－ xm）＋a23（Rcos ψ sin θ－ ym）＋

a33（Rsin ψ－ ZM) = a32r (COSU ＋ 1) + a33rsinu

parameters in equation (24) ψ It can be obtained from equation (20). The above formula is the parameter u and θ As long as the value of u is given, the corresponding θ Value, and then calculate the milling cutter cutting angle from equations (21), (22) and (23) λ s. RN and α The value of n

II. Calculation example

set the radius of ball joint R ＝ 25mm, and the half cone angle of the rear cutting surface of the blade β＝ 15 °, 22mm long and 16mm wide, selected from S-shaped blade θ＝ A point of 8 ° M. The calculation results are: XM = 22.862mm, RM = 3.213mm, ZM = 9.591mm; Machining adjustment angle ω＝－ 15．378°；＝－ 23．833°； ψ＝ 95°； The circular arc radius of the blade R ＝ 24.947mm. See Table 1 for the distribution of the cutting angle along the cutting edge of the milling cutter

Table 1 Distribution of cutting angle along cutting edge ψ Can change the normal rake angle γ n. Normal posterior angle α N and the theoretical value of the circular arc radius r of the blade, so that the circular arc radius of the blade at different installation positions can be nearly equal, so as to reduce the blade specification

(2) changing the position of the reference point m on the S-shaped blade can make the blade overlap reasonably

(3) change the helix angle of the orthogonal helix ω 1 to change the inclination angle of the milling cutter λ s；

(4) it can be seen from table 1 that θ Angle increase λ S and front corner γ N increases, while the normal back angle α N will decrease, but the amount of increase and decrease is not large

(5) in this paper, only the first half of the S-shaped end edge is calculated. As long as the milling cutter is rotated 180 ° around its axis, the cutting vector calculation of the second half of the S-shaped edge is exactly the same

(6) this model is applicable to the flat blade. However, as long as the temperature is changed to avoid plastic decomposition, scorching or difficulty in shaping, this model can also be used to calculate the processing adjustment parameters of the indexable ball end milling cutter with vertical blade

author unit: Sichuan Institute of Technology Chengdu 611744

references

1 Wu Daren Differential geometry handout Beijing: People's education press, 1979

2 Yao Nan, Wang Xun, etc Application of mathematics in tool design Beijing: China Machine Press, 1988

3 Wunengzhang et al An algorithm for grinding wheel locus of ball end milling cutter with end edge and flank edge Mechanical technologist, 1998 (9)

4 Wangxibin et al The non free coefficient of arc cutting edge and its improvement Tool technology, 1996 (4)

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