CN1186157C - Drum taper tool and method for side milling complex cambered centrifugal impeller using drum taper tool - Google Patents

Abstract

The present invention provides a drum-tapered cutter which is a high-efficiency cutter for laterally milling centrifugal impellers with any curved surface, and also provides a method for calculating cutting feed space, cutting feed step length and cutter spacing. The cutting part of the cutter is a rotative surface, and a bus bar is a section of circular arc of which the radius is R; the rotative surface is tangental to a cutter handle, and the bottom of the cutter has no cutting edge. The cutter has higher machining efficiency than a spherical cutter and simultaneously overcomes the defect of small rigidity of a drum-shaped cutter; the present invention also has larger rotating flexibility than a tapered cutter and can conveniently avoid over-cutting and collision which possibly appear in the process of machining. The present invention is an ideal machining cutter for three-element impellers with any curved surface, and the number of cutter tracks of the drum-tapered cutter of the present invention for machining impellers is far smaller than that of the spherical cutter; the proportion of the two numbers of cutter tracks is approximately 1 to 3 in general. Under the same machining condition, the present invention has the advantages of short length of cutter tracks, short time required for machining and obvious economic effect.

Description

Drum-taper type cutter tool and with the method for the complex-curved centrifugal impeller of drum-taper type cutter tool side milling

One, technical field

The present invention relates to the complex-curved Computerized Numerical Control processing technology such as complex-curved impeller in machining and Computerized Numerical Control processing technology field, particularly a kind of drum-taper type cutter tool and utilize the complex-curved method of drum-taper type cutter tool side milling.

Two, background technology

Turbomachine is widely used for industries such as energy source and power, Aero-Space, petrochemical industry, metallurgy, is bringing into play huge effect in national economy and Defence business, all is typical turbomachine as compressor, air blast, ventilation blower and water pump etc.Impeller is the critical component in the turbomachine, rotates at a high speed by it, and the work done of blade convection cell obtains at a high speed, the fluid of high pressure.According to fluid flow direction, can be divided into axial wheel and centrifugal impeller, in axial wheel, fluid flows vertically; In centrifugal impeller, the main direction of motion of fluid be perpendicular to rotating shaft radially (Jiang Peizheng. turbomachine. Xi'an: publishing house of Xi'an Communications University, 1991.8).Impeller shown in Fig. 1 is for being a kind of centrifugal impeller, and it includes linear leaf 1, short blade 2, and runners 3 etc. have also schematically shown process tool 4 among the figure.

According to the geometry of blade, impeller can be divided into impeller with ruled surface blade and impeller with complex-curved blade.Ruled surface can be considered as by straight line at the formed curved surface of spatial movement; The complex-curved free form surface that yet claims, geometry does not have certain rules, represents that by data point (Mei Xiangming is after Huang is respected.Differential Geometry.Beijing: Higher Education Publishing House, 1981).The impeller that this patent relates to be exactly blade be complex-curved centrifugal impeller, Fig. 1 has shown that the position of impeller and cutter 4 concerns that impeller wherein is exactly complex-curved centrifugal impeller in the processing.

Because the impeller shape complexity, machine tool is difficult to be competent at, and adopting Digit Control Machine Tool processing is the best way.The function of Digit Control Machine Tool depends on its number of degrees of freedom, to a great extent, and the free degree of Digit Control Machine Tool is meant the motion of cutter with respect to workpiece.During lathe work, tool-holder is held on the main shaft, and workpiece is fixed on the workbench.The coordinate system of Digit Control Machine Tool adopts right hand rectangular coordinate system, and three rectilinear motion reference axis are X, Y, Z, and three gyration coordinate A, B, C represent respectively around the rotatablely moving of X, Y, Z axle, as shown in Figure 2.Press the number of degrees of freedom, of Digit Control Machine Tool, can be divided into three-dimensional, 4-coordinate and five-coordinate numerally controlled machine tool, the free degree of three-dimensional lathe has only X, Y, three translation freedoms of Z, the 4-coordinate lathe increases a rotary freedom on this basis, five coordinate lathes have three translation freedoms, (Japanese plum is bright, Gao Dongqiang for two rotary freedoms.The CAD/CAM technology.Xi'an: Shaanxi science tech publishing house, 1994).

For complex-curved centrifugal impeller shown in Figure 1, must adopt five-coordinate numerally controlled machine tool processing, otherwise cutter and blade will interfere, and be difficult to produce satisfactory impeller.Figure 3 shows that the schematic diagram that utilizes five number of coordinates control milling machines processing impeller 20, wherein, the gyration of the translational motion of X, Y-axis and C axle realizes that by platen 30 swing of the translation of Z axle and A axle is realized by machine tool chief axis.

The processing of impeller mainly contains fluting roughing and fine finishining, and in the whole process of complex-curved centrifugal impeller, the used time of fine finishining is occupied sizable ratio, and wherein, the shape and the working (machining) efficiency of cutter have much relations.Fig. 4 has shown the relation of effective radius of clean-up of cutter and working width.This figure was the contact point place of cutter and curved surface, perpendicular to the section of cutting direction.Among the figure, f is a cutting direction, and h is given residual height, and d is the distance between the adjacent tool track, r _cBe effective radius of clean-up of cutter, different cutters has different effective radiuss of clean-up, and effective radius of clean-up of spherical cutter is the radius of ball.From figure as can be seen, under identical residual height h, effective radius of clean-up r _cBig more, feed spacing d is big more; For given curved surface, under the identical situation of cutting data, d is big more, the feed number of times is just few, working (machining) efficiency then high (Vickers G W, Quan K W, Ball-mills versus end-mills for curved surfacemachining.Journal of Engineering for Industry, Vol.111,1989:22～26).In the Impeller Machining, because runner is narrow, tool diameter must be restricted, referring to Fig. 1, and under same tool radius, r _cIt is high more to be worth big more then working (machining) efficiency, and the tapered cutter of the drum among the present invention just has such characteristics, and its effective radius of clean-up has high stock-removing efficiency greater than the spherical cutter of same tool diameter.

In the digital control processing of part, key is to work out out numerical control program, and Digit Control Machine Tool is realized processing according to numerical control program.During programming, at first to determine the attitude of cutter, i.e. the direction of the contact point of cutter and curved surface and cutter, the set of cutter and curved surface contact point is exactly a cutter path, determines that cutter path will calculate orbit interval d exactly, and cutting step length; Determined tool orientation will guarantee that lathe can realize.After cutter contact point and tool orientation were determined, cutter had just been determined in the space uniquely with respect to the attitude of workpiece.Then, at different Digit Control Machine Tools, by rearmounted processing links, with the cutter spacing data be transformed into the machine tool instructions that lathe can discern process (Yao Yingxue, Cai Ying. CAD and manufacturing.Beijing: Higher Education Publishing House, 2002.1).

At present, the cutter that adopts usually in centrifugal impeller fine finishining is spherical cutter, and the effective radius of clean-up of spherical cutter is little, and working (machining) efficiency is low.

C.Y.Wu. (Arbitrary surface flank milling of fan, compressor, andimpeller blades.Transactions of the ASME, Journal of Engineering forGas Turbines and Power, Vol.117,1995, p 534～p 539), proposed to utilize the method for tapered knife side milling arbitrary surface impeller blade, its cutting edge is a taper seat, referring to Fig. 5.He has broken through the processing arbitrary surface and must adopt spherical cutter or slotting cutter to carry out this mode of thinking of contact processing, and the processing mode of line contact efficiently is applied in the processing of arbitrary surface, and this method working (machining) efficiency is far longer than spherical cutter.

Find in the practice that adopt tapered knife processing arbitrary surface to have following defective: 1) this method requires relatively harshness to the direction of cutter in processing blade shape and the processing, and leaf curling can not be too big, otherwise can not find suitable cutter Working position.For given position on the processing curve, the feasible direction territory of cutter is very little, and rotation leeway is little; 2) if cutter and blade bump, because the feasible direction territory of cutter is little, the cutter slewing area is little, interferes cutter spacing to be difficult to revise.Therefore, mill in the system process in arbitrary surface ternary centrifugal impeller integral body, this method may be failed.

Liu Xiongwei, and Zhang Dinghua etc. (digital control processing theory and programming technique. Beijing: China Machine Press, 1994) introduced the thinking of utilizing drum knife side milling curved surface, referring to Fig. 6, adopt this method, the contact length of cutter and finished surface is bigger.Just utilize drum knife to introduce the basic principle of side milling in the document, do not relate to the calculating of drum knife curved sides milling cutter position.

Because impeller channel is a slype, adopt the drum type cutter, the ratio of rigidity of cutter is less, influences machining accuracy.

Three, summary of the invention

Defective or deficiency at above-mentioned prior art existence, an object of the present invention is to provide a kind of drum-taper type cutter tool, the characteristics of this drum-taper type cutter tool are, Tool in Cutting partly is the surface of revolution, bus is one section circular arc that radius is R, and the surface of revolution and handle of a knife are tangent, the no cutting edge in cutter bottom, principle not take place to be cut to curved surface in the value of R in this cutter, and promptly the radius of curvature of curved surface contact point should be greater than the R value; Drum-taper type cutter tool of the present invention is a kind of cutting tool efficiently, kept drum knife and tapered knife working (machining) efficiency with respect to spherical knife up, overcome the little weakness of drum knife rigidity simultaneously, than tapered knife greater flexibility is arranged again, can avoid easily cutting and colliding, be the whole process tool of a kind of desirable arbitrary surface 3 d impeller.

Another object of the present invention provides a kind of method of utilizing the complex-curved centrifugal impeller of drum-taper type cutter tool side milling, may further comprise the steps:

1) calculating of cutter path spacing and cutting step length

(1) calculating of cutter path spacing

(2) calculating of cutting step length

2) cutter spacing data computing

Comprise the calculating of the calculating of generating tool axis vector and cutter computer center.

The present invention adopts drum-taper type cutter processing impeller, and under same processing conditions, cutter path length is short, and required process time is few, promptly adopts this cutter fine finishining arbitrary surface impeller, and working (machining) efficiency will be 3 times of conventional method.Simultaneously, it has greater flexibility than tapered knife, is easy to revise the interference of cutter and blade in the processing; And rigidity is higher than drum knife.

Four, description of drawings

Fig. 1 has shown that the position of impeller and cutter concerns in the processing;

Fig. 2 is the Digit Control Machine Tool coordinate system;

Fig. 3 is the schematic diagram that utilizes five number of coordinates control milling machines processing impeller;

Fig. 4 is the schematic diagram of the relation of effective radius of clean-up of cutter and working width, and wherein 5 is the contact-making surface of cutter and curved surface, and 6 are piece surface before the processing, and 7 representation theory machined surfaces;

Fig. 5 is conical cutter schematic diagram;

Fig. 6 is the drum knife schematic diagram;

Fig. 7 is a drum-taper type cutter schematic diagram of the present invention;

Fig. 8 utilizes drum-taper type cutter side milling curved surface schematic diagram;

Fig. 9 is the calculating schematic diagram of orbit interval d;

The schematic cross-section of Figure 10 drum-taper type cutter cutting tip;

Figure 11 is the calculating schematic diagram of cutting step length, the 8 expression circles of curvature wherein, and 9 be theory locus;

Figure 12 is definite schematic diagram of cutter computer center;

Figure 13 is the cutter path that adopts drum-taper type cutter to form;

Figure 14 is the cutter path that adopts spherical cutter to form.

Five, the specific embodiment

The present invention is described in further detail below in conjunction with accompanying drawing and inventor's specific embodiment.

5.1 the proposition of drum-taper type cutter

The present invention proposes a kind of cutting tool of the curved surface of processed complex efficiently centrifugal impeller, claim that this cutter is a drum-taper type cutter, referring to Fig. 7.Tool in Cutting partly is the surface of revolution, and bus is one section circular arc that radius is R, and the surface of revolution and handle of a knife are tangent, the no cutting edge in cutter bottom.Drum-taper type cutter has kept drum knife and the tapered knife working (machining) efficiency with respect to spherical knife up, overcome the little weakness of drum knife rigidity simultaneously, than tapered knife greater flexibility being arranged again, can avoid easily cutting and colliding, is the whole process tool of the centrifugal 3 d impeller of a kind of desirable arbitrary surface.The choosing with curved surface principle not to take place to be cut to of R in this cutter, promptly the radius of curvature of curved surface contact point should be greater than the R value.

5.2 the calculating of cutter path spacing and cutting step length

The situation of utilizing drum-taper type cutter side milling curved surface as shown in Figure 8, r (u, w) expression curved surface, cutter contact point is some C, some O is the computer center of cutter, l is the cutter axis orientation vector, unit vector f is a cutting direction, and n is that the per unit system that C is ordered is vowed vector b=n * f, these three vectors have been formed a rectangular coordinate system, and initial point is at a C.In the processing of five coordinates, cutter has two rotary freedoms.The inceptive direction of getting cutter shaft is consistent with vector b, and at first initial generating tool axis vector is rotated an angle [alpha] around vector f; Then, cutter rotates one angle-β around vector n again, and positive direction determines that with right-handed system this is because during Tool in Cutting, cutter shaft must be inclined to one side to cutting direction f, drags the cutter cutting with formation.As we know from the figure, cutter shaft when initial position, α=0, β=0.

In the curved surface numerical control programming, must calculate cutter path spacing and cutting step length.

(1) calculating of cutter path spacing

As shown in Figure 9, the C point is a contact point on the known initial track, initial cutter path is generally a border of curved surface, C ' point is the contact point on the adjacent track that need find the solution, vow that perpendicular to method f does a cross section at the C point, vector b and CC ' are in this plane, and the curved surface section line that obtains can be approx with one section arc representation, radius is that the C point is designated as R along the method transversal radius of curvature of direction b on the curved surface _bEqually, the Tool in Cutting face is along the also available one section circular arc approximate representation of the section line of direction b, and its radius also is effective radius of clean-up of cutter for the method transversal radius of curvature of cutter C point along direction b, is designated as r _c

For given R _b, r _cAnd residual height h, cutter path spacing d can calculate with following formula

$d=\sqrt{\frac{{8R}_b\·r_c\·h}{R_b\±r_c}}---\left(1\right)$

1) curved surface is along the method transversal radius of curvature R of vector b _bCalculating

According to Differential Geometry knowledge, the tangent line b that curved surface C is ordered can be expressed as

b＝r _ud _u+r _wd _w

Wherein, d _u, d _wBe respectively vector b at tangent vector r _u, r _wOn component.

Then the method transversal radius of curvature along direction b is

$R_b=\frac{I}{\mathrm{II}}=\frac{{\mathrm{<figure-callout\; id="2"\; label="Edu"\; filenames="C0213958200161.png,C0213958200192.png"\; state="\{\{state\}\}">Edu</figure-callout>}}^2+2\mathrm{Fdudw}+\mathrm{<figure-callout\; id="2"\; label="G\; dw"\; filenames="C0213958200161.png,C0213958200192.png"\; state="\{\{state\}\}">G</figure-callout>}{\mathrm{dw}}^{}{}^2}{{\mathrm{<figure-callout\; id="2"\; label="Ldu"\; filenames="C0213958200161.png,C0213958200192.png"\; state="\{\{state\}\}">Ldu</figure-callout>}}^2+2\mathrm{Mdudw}+\mathrm{Nd}{w}^2}---\left(2\right)$

Wherein, I is a first fundamental form of surface, and E, F, G are respectively the first kind fundamental quantity of curved surface, and II is a second fundamental form of a surface, and L, M, N are respectively the second class fundamental quantity of curved surface.

If d _w=0, then $R_b=\frac{I}{\mathrm{II}}=\frac{E}{L}$

If d _w≠ 0, then formula (2) is done a conversion, have

$R_b=\frac{I}{\mathrm{II}}=\frac{E{\left(\frac{\mathrm{du}}{\mathrm{dw}}\right)}^2+2F\frac{\mathrm{du}}{\mathrm{dw}}+G}{L{\left(\frac{\mathrm{du}}{\mathrm{dw}}\right)}^2+2M\frac{\mathrm{du}}{\mathrm{dw}}+N}---\left(3\right)$

As shown in Figure 8

b·f＝0

I.e. (r _uDu+r _wDw) f=0, thus

$\frac{\mathrm{du}}{\mathrm{dw}}=-\frac{r_w\&CenterDot;f}{r_u\&CenterDot;f}---\left(4\right)$

Formula (4) substitution formula (3) can be obtained the radius of curvature R of Surface Method transversal _bValue.

2) the Tool in Cutting face is along effective radius of clean-up r of vector b _cCalculating

The Tool in Cutting face is dissectd along the plane of crossing cutter shaft, shown partial cross section as Figure 10, wherein heavy line is a cutting edge, and some O is the computer center of cutter, and some O ' is the center of cutter revolution bus, and R is the bus radius, and C is a cutter contact point.With an O is the origin of coordinates, cutter shaft to radially respectively as reference axis z and r.α is the angle of O ' C and r axle.From figure as can be known, have point of contact, promptly put C and be in the cutting tip that heavy line is represented among Figure 10, the maximum deflection value α of angle [alpha] in order to make cutter and curved surface _MaxShould satisfy:

${\mathrm{\&alpha;}}_{\max }<\arcsin \frac{h}{R}$

By the geometrical relationship shown in Figure 10, can obtain the coordinate that C orders and be:

By Fig. 8 and Figure 10, can get according to Euler's formula, behind cutter tilt angle alpha and the β, at Tool in Cutting face contact point place, along the normal curvature of vector b direction be

k _c＝k ₁cos ²β+k ₂sin ²β

Wherein, k ₁And k ₂Be respectively the principal curvatures of this contact point, k ₁=1/R, k ₂=1/r=1/ (Rcos α-(R-d _c/ 2))

So effective radius of clean-up r of cutter _cFor

r _c＝1/k _c (5)

With formula (5) and formula (3) substitution formula (1),, can try to achieve cutter path spacing d because h is given.

(2) calculating of cutting step length

For curved surface, its theoretical cutter path is a space curve, in the multi-coordinate digital control processing of curved surface, the linear interpolation mode is generally adopted in the motion of cutter, be about to theory locus is separated into some under assigned error line segment, the actual motion track of cutter is a series of straightways (practical Computerized Numerical Control processing technology editorial board, practical Computerized Numerical Control processing technology, weapon industry publishing house, 1995:177 ~ 179).As shown in figure 11, establishing the straight line approximate error that exists between the movement locus of cutter reality and the theory locus is δ, and then cutting step length AB can be calculated as follows:

$\mathrm{AB}\&ap;\sqrt{\frac{8\mathrm{\&delta;}}{k_f}}$

Wherein, k _fCurvature for tool track contact point place.

5.3 cutter spacing data computing

The cutter spacing data comprise the calculating central position O and cutter axis orientation (unit vector) l of cutter, and the calculating of cutter spacing is relevant with the geomery of cutter.For the drum-taper type cutter tool, cutter spacing is calculated as follows.

(1) calculating of generating tool axis vector l

As shown in Figure 8, the initial position of generating tool axis vector is a vector b, at this moment, and α=β=0.At first making cutter is the center with an O ', around vector f anglec of rotation α, referring to Figure 10.Making cutter again is the center with a C, around the vector n anglec of rotation-β.If generating tool axis vector is from the vector l of inceptive direction b behind vector f anglec of rotation α ₁, then

l ₁＝b·cosα+n·sinα

Generating tool axis vector l ₁Around vector n rotation-β angle, can obtain the final direction vector l of cutter shaft, have

l＝l ₁·cosβ-n×l ₁·sinβ+n·(l ₁·n)·(1-cosβ) (6)

(2) calculating of cutter computer center

Figure 12 has shown cutter before and after vector f anglec of rotation α, the situation of change of computer center.Wherein, some O ' is the center of circle of Tool in Cutting surface of revolution bus, some O ₁Be the computer center of cutter initial position, some O ₂Computer center for cutter behind the anglec of rotation α.

If vector O ' O ₁Behind vector f anglec of rotation α is O ' O ₂, vector O ' O ₂After vector n rotation-β angle for vector O ' O (not expressing among the figure), then pass through rotation transformation after, the final computer center's point O of cutter is

O＝O′+O′O (7)

By Figure 10 and Figure 12 as can be known

O '=C+nR, O ₁=C+n (d _c/ 2) then

O′O ₁＝n·(d _c/2-R) (8)

Vector O ' O ₂O ' O is respectively with vector

O′O ₂＝O′O ₁·cosα+f×O′O ₁·sinα (9)

O′O＝O′O ₂·cosβ-n×O′O ₂·sinβ+n·(O′O ₂·n)·(1-cosβ) (10)

With formula (8) substitution formula (9), formula (9) substitution formula (10) can be tried to achieve O ' O,, can try to achieve final computer center's point O of cutter again with formula (10) substitution formula (7).

Formula (6) and formula (7) have been determined the O of computer center of cutter axis orientation l and cutter respectively, and (O l) determines the cutter spacing data thus.

Adopt drum-taper type cutter processing blade of the present invention, the result shows, its cutter path number is far smaller than the track number when adopting spherical cutter, the ratio of the two is generally about 1: 3, under same processing conditions, cutter path length is short, and required process time is few, promptly adopt this cutter fine finishining arbitrary surface impeller, working (machining) efficiency will be 3 times of conventional method.Simultaneously, it has greater flexibility than tapered knife, is easy to revise the interference of cutter and blade in the processing; And tool stiffness is higher than drum knife.

Embodiment:

Utilize spherical cutter and drum-taper type cutter respectively, the cutter path in the processing of centrifugal impeller blade five coordinates is generated discuss.Blade is an arbitrary surface, represents with the cubic B batten.Used spherical cutter diameter is d _c=32mm, drum-taper type cutter knife bar diameter d _c=32mm, R=70mm, h=25mm is referring to Fig. 7.Allowing residual height is 0.1mm, respectively at the cutter path in above-mentioned two kinds of cutters calculating blade processing.

Result of calculation such as Figure 13 and shown in Figure 14 adopt drum-taper type cutter, and the track number of processing this blade is 16, and the track total length is 6631mm; And adopt spherical cutter track number to reach 39 more than, and the track total length is 16453mm, the ratio of the two is 1: 2.5.Under same cutting speed, the working (machining) efficiency that adopts drum-taper type cutter is 2.5 times of spherical cutter.

Claims (3)

1. A drum-conical cutter, comprising a handle and a cutting part, characterized in that the cutting part of the cutter is a revolving surface, the generatrix is an arc with a radius of R, the revolving surface is tangent to the handle, and there is no cutting edge at the bottom of the cutter . 2. The drum-cone cutter according to claim 1, wherein the value of R in the cutter should be smaller than the radius of curvature of the contact point of the curved surface. 3. A method for side milling a complex curved surface centrifugal impeller utilizing a drum cone cutter, is characterized in that, comprising the following steps: 1) Calculation of tool path spacing and tool step length A. Establish a coordinate system Let the curved surface be r(u,w), the contact point between the tool and the curved surface r(u,w) is point C, the calculation center point of the tool is O, the tool axis direction vector is l, the unit vector f is the cutting direction, and n is C The unit normal vector of the point, vector b=n×f, these three vectors form a rectangular coordinate system, the origin is at point C; in five-coordinate machining, the tool has two rotational degrees of freedom; take the initial direction of the tool axis Consistent with the vector b, first the initial tool axis vector rotates an angle α around the vector f; then, the tool rotates around the vector n for an angle -β, and the positive direction is determined by the right-hand system, because when the tool is cutting, the tool axis must The cutting direction f is shifted to form dragging tool cutting; when the tool axis is at the initial position, α=0, β=0; B. Calculation of tool path spacing Make a section perpendicular to the normal vector f at point C, then the vector b and CC' must be in the plane, and the obtained surface section line can be approximately expressed by a section of arc, and the radius is the normal section of point C on the surface along the direction b The radius of curvature of the line is denoted as R _b , similarly, the section line of the cutting surface of the tool along the direction b can also be approximated by a circular arc, and its radius is the radius of curvature of the normal section line of the tool point C along the direction b, denoted as r _c ; For a given R _b , _rc and residual height h, the tool path distance d can be calculated by the following formula $\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; =</span>\sqrt{\frac{{\mathrm{<span\; class="notranslate">\; 8</span>}\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>\mathrm{<span\; class="notranslate">\; h</span>}}{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; \&PlusMinus;</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$ ① Calculation of the radius of curvature R _b of the normal section of the surface along the vector b According to the knowledge of differential geometry, the tangent line b of point C on the surface can be expressed as b＝r _u d _u +r _w d _w Among them, d _u and d _w are the components of vector b on the tangent vector r _u and r _w respectively; Then the radius of curvature of the normal section along the direction b is ${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; I</span>}}{\mathrm{<span\; class="notranslate">\; II</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; Edu</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; Fdudw</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; Gdw</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; Ldu</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; Mdudw</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; Ndw</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$ Among them, I is the first basic form of the curved surface, E, F, G are the first basic quantities of the curved surface, II is the second basic form of the curved surface, L, M, N are the second basic quantities of the curved surface respectively; If d _w =0, then $R_b=\frac{I}{\mathrm{II}}=\frac{\mathrm{E.}}{L}$ If d _w ≠0, then make a transformation to formula (2), we have ${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; I</span>}}{\mathrm{<span\; class="notranslate">\; II</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; du</span>}}{\mathrm{<span\; class="notranslate">\; dw</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; f</span>}\frac{\mathrm{<span\; class="notranslate">\; du</span>}}{\mathrm{<span\; class="notranslate">\; dw</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; G</span>}}{{\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; du</span>}}{\mathrm{<span\; class="notranslate">\; dw</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; m</span>}\frac{\mathrm{<span\; class="notranslate">\; du</span>}}{\mathrm{<span\; class="notranslate">\; dw</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$ Since b·f=0 That is (r _u du+r _w dw)·f=0 $\frac{\mathrm{<span\; class="notranslate">\; du</span>}}{\mathrm{<span\; class="notranslate">\; dw</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}}<span\; class="notranslate">\; \&CenterDot;</span>\mathrm{<span\; class="notranslate">\; f</span>}}{{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}<span\; class="notranslate">\; \&CenterDot;</span>\mathrm{<span\; class="notranslate">\; f</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$ Substituting formula (4) into formula (3) can obtain the value of the radius of curvature R _b of the surface normal section line; ② Calculation of the effective cutting radius r _c of the cutting surface of the tool along the vector b Cut the cutting surface of the tool along the plane passing the tool axis, point O is the calculation center of the tool, point O' is the center of the tool's rotary generatrix, R is the radius of the generatrix, and C is the tool contact point; Taking point O as the origin of coordinates, the axial and radial directions of the tool are used as coordinate axes z and r respectively; α is the angle between O′C and r-axis; in order to make the tool and the curved surface have a contact point, that is, point C is in the cutting part, the angle The maximum deflection value α _max of α should satisfy: ${\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; max</span>}}<span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; arcsin</span>}\frac{\mathrm{<span\; class="notranslate">\; h</span>}}{\mathrm{<span\; class="notranslate">\; R</span>}}$ The coordinates of point C can be obtained as: According to Euler's formula, after the inclination angles α and β of the tool, at the contact point of the cutting surface of the tool, the normal curvature along the direction of vector b is k _c =k ₁ cos ² β+k ₂ sin ² β Among them, k ₁ and k ₂ are the principal curvatures of the contact point respectively, k ₁ =1/R, k ₂ =1/r=1/(Rcosα-(Rd _c /2)) Therefore, the effective cutting radius r _c of the tool is r _c =1/k _c (5) Substituting formula (5) and formula (3) into formula (1), since h is given, the tool track distance d can be obtained C. Calculation of cutting step length In the multi-coordinate CNC machining of curved surfaces, the tool movement generally adopts linear interpolation. Assuming that the linear approximation error between the actual tool movement track and the theoretical CC track is δ, the step length AB of the tool can be calculated by the following formula : $\mathrm{<span\; class="notranslate">\; AB</span>}<span\; class="notranslate">\; \&ap;</span>\sqrt{\frac{\mathrm{<span\; class="notranslate">\; 8</span>}\mathrm{<span\; class="notranslate">\; \&delta;</span>}}{{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}}}$ Among them, k _f is the curvature at the contact point of the tool path; 2) Calculation of tool location data The tool position data includes the calculation center position O of the tool and the direction of the tool axis (unit vector) l. The calculation of the tool position is related to the shape and size of the tool. For the drum-conical tool, the tool position is calculated as follows: ① Calculation of the tool axis vector l The initial position of the tool axis vector is the vector b, at this time, α=β=0, first make the tool center on the point O', rotate the angle α around the vector f, and then make the tool center on the point C, rotate around the vector n by the angle - β; if the tool axis vector is rotated from the initial direction b around the vector f by an angle α to the vector l ₁ , then l ₁ ＝b·cosα+n·sinα The tool axis vector l ₁ rotates around the vector n by an angle of -β, and the final direction vector l of the tool axis can be obtained. l＝l ₁ ·cosβ-n×l ₁ ·sinβ+n·(l ₁ ·n)·(1-cosβ) (6) ②Calculation of tool calculation center Before and after the rotation angle α of the tool around the vector f, the transformation of the tool center is calculated; among them, the point O′ is the center of the generatrix of the tool cutting surface of revolution, the point O ₁ is the calculation center of the initial position of the tool, and the point O ₂ is the tool after the rotation angle α computing center; Let the vector O′O ₁ rotate around the vector f by an angle α to be O′O ₂ , and the vector O′O ₂ rotate around the vector n by an angle of -β to be the vector O′O, then after the rotation transformation, the final calculation center point of the tool O for O＝O′+O′O (7) Right now: O'=C+n·R, O ₁ =C+n·(d _c /2) but O'O ₁ =n·(d _c /2-R) (8) The vector O′O ₂ and the vector O′O are respectively O′O ₂ ＝O′O ₁ ·cosα+f×O′O ₁ ·sinα (9) O′O=O′O ₂ ·cosβ-n×O′O ₂ ·sinβ+n·(O′O ₂ ·n)·(1-cosβ) (10) Substituting formula (8) into formula (9), substituting formula (9) into formula (10) can obtain O′O, and then substituting formula (10) into formula (7), the final calculation center point of the tool can be obtained O; Formula (6) and formula (7) respectively determine the tool axis direction l and the calculation center O of the tool, and the tool position data (O, l) is determined accordingly.

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