Basics of Cutting Tool Geometry

Transcription

1 D. Vikto P. Astakhov, Tool Geomet: Basics Basics of Cutting Tool Geomet Vikto P. Astakhov Fo man eas thee wee diffeent sstems used to define a geat vaiet of angles of faces and edges of cutting tools. Although ISO Standad ISO Geomet of the Active Pat of Cutting Tools - Geneal Tems, Refeence Sstems, Tool and Woking Angles has atiall esolved this situation b definition of a sstem of lanes and a numbe of angles, a numbe of deficiencies in definition of cetain angles have been noted. The man angles defined b the standad ae not necessail indeendent, and man tigonometic elations descibing the vaious angles ae diffeent fom those usuall develoed fo an acute angle of oientation of the cutting edge. The following sstems fo identifing cutting tool geomet wee intoduced in []:. Tool-in-Hand Sstem: Definitions of the basic efeence lanes (the main efeence lane; the assumed woking lane; the tool cutting edge lane; the tool back lane; the othogonal lane; the cutting edge nomal lane) fo each of the cutting edges. A sstem of tool angles (the tool cutting edge angle; the tool mino (end) cutting edge angle; the tool aoach angle; the ake angles; the cleaance (flank) angles; the wedge angles; the cutting edge inclination angle), thei definitions, meaning, and inteelationshis among them.. Tool-in-Machine Sstem (Setting Sstem) of Angles and Planes. 3. Tool-in-Use Sstem. The following is to ovide the definitions of some basic lane and angles in the Tool-in-Hand sstem.. Planes The woking at of the cutting tool basicall consists of two sufaces intesecting to fom the cutting edge. The suface along which the chi flows is known as the ake face o moe siml as the face, and that suface which is gound back to clea the new o machined suface is known as the flank suface o siml as the flank. In the simlest et common case the ake and flank sufaces ae lanes. Figue shows the definition of the main efeence lane P as eendicula to the assumed diection of ima motion and the tool-in-hand coodinate sstem. In this figue, v f is the assumed diection of the cutting feed. Because angles of the cutting tool ae defined in a seies of efeence lanes, the standad defines a sstem of these lanes in the tool-in-hand sstem, as shown in figue. The sstem consists of five basic lanes defined elative to the efeence lane P. Peendicula to the efeence lane P and containing the assumed diection of feed motion is the assumed woking lane P f. The tool cutting edge lane P s is eendicula to P, and contains the side (main) cutting edge (- in Fig. ). The tool back lane P is eendicula to P and P f. Peendicula to the ojection of the cutting edge into the efeence lane is the othogonal lane P o. The cutting edge nomal lane P n is eendicula to the cutting edge.

2 D. Vikto P. Astakhov, Tool Geomet: Basics Figue. Definition of the main efeence lane P. Figue : Standad sstem of efeence lanes in the tool-in-hand sstem (majo cutting edge). Similal, an additional sstem of lanes can be attibuted to the mino cutting edge and contains the following lanes: P s, P o, P n as shown in figue 3.. Angles The geomet of a cutting element is defined b cetain basic tool angles and thus ecise definitions of these angles ae essential []. A sstem of tool angles is shown in figue 4. Rake, wedge and cleaance (flank) angles ae secified b γ, β, and

3 D. Vikto P. Astakhov, Tool Geomet: Basics Figue 3: Standad sstem of efeence lanes in the tool-in-hand sstem (mino cutting edge). α, esectivel, and these ae identified b the subscit of the lane of intesection. The definitions of basic tool angles in the tool-in-hand sstem ae as follows: κ is the tool cutting edge angle; it is the acute angle that P s makes with P f and is measued in the efeence lane P. It can be also defined as the acute angle between the ojection of the main cutting edge into the efeence lane and the x-diection (Fig. ). κ is alwas ositive and it is measued in a counteclockwise diection fom the osition of P. κ is the tool mino (end) cutting edge angle; it is the acute angle that P s makes with P f and is measued in the efeence lane P. It can be also defined as the acute angle between the ojection of the mino (end) cutting edge into the efeence lane and the x-diection (Fig.). κ is alwas ositive (including zeo) and it is measued in a clockwise diection fom the osition of P. ψ is the tool aoach angle; it is acute angle that P s makes with P and is measued in the efeence lane P as shown in figue 4. The ake angles ae defined in the coesonding lanes of measuement. The ake angle is the angle between the efeence lane (the tace of which in the consideed lane of measuement aeas as the nomal to the diection of ima motion) and the intesection line fomed b the consideed lane of measuement and the tool ake lane. The ake angle is defined as alwas being acute and ositive when looking acoss the ake face fom the selected oint and along the line of intesection of the face and lane of measuement. The viewed line of intesection lies on the oosite side of the tool efeence lane fom the diection of ima motion in the measuement lane fo γ f, γ, γ o, o a majo 3

4 D. Vikto P. Astakhov, Tool Geomet: Basics Figue 4: Tool angles in the tool-in-hand sstem. comonent of it aeas in the nomal lane fo γ n. The sign of the ake angles is well defined (Fig. 4). The cleaance (flank) angles ae defined in a wa simila to the ake angles, though hee if the viewed line of intesection lies on the oosite side of the cutting edge lane P s fom the diection of feed motion, assumed o actual as the case ma be, then the cleaance angle is ositive. Angles α f, α, α o, α n ae cleal defined in the coesonding lanes. The cleaance angle is the angle between the tool cutting edge lane P s and the intesection line fomed b the tool flank lane and the consideed lane of measuement as shown in figue 4. The wedge angles β f, β, β o, β n ae defined in the lanes of measuements. The wedge angle is the angle between the two intesection lines fomed as the 4

5 D. Vikto P. Astakhov, Tool Geomet: Basics coesonding lane of measuement intesects with the ake and flank lanes. Fo all cases, the sum of the ake, wedge and cleaance angles is 9 o, i.e. o + β + α = γ n + β n + α n= γ o + β o + α o = γ f + β f + α f = 9 γ () Fo the mino (side) cutting edge, the flank angle α o is secified as the angle between the tool mino (side) cutting edge lane P s and the intesection line fomed b the tool mino flank lane and the lane of measuement P o as shown in figue 4. The oientation and inclination of the cutting edge ae secified in the tool cutting edge lane P s. In this lane, the cutting edge inclination angle λ s is the angle between the cutting edge and the efeence lane. This angle is defines as alwas being acute and ositive if the cutting edge, when viewed in a diection awa fom the selected oint at the tool cone being consideed, lies on the oosite side of the efeence lane fom the diection of ima motion. This angle can be defined at an oint of the cutting edge. The sign of the inclination angle is well defined in figue 4. Simle elationshis exist among the consideed angles in the tool-in-hand sstem. These elationshis have been deived assuming that the tool side ake angle γ f, the tool back ake angle γ, and the tool cutting edge angle κ ae the basic angles fo the tool face, and the tool side cleaance angle α f, the tool back cleaance angle α, and the tool cutting edge angle κ ae the basic angles fo the tool flank [,3]: tanλ s sinκ cosκ = () tan γ n = cosλs o (3) tan γ cosκ + sinκ o = (4) cos αn = cosλs cotαo (5) cot α cosκ cotα + sinκ cotα o = (6) It must be stated, howeve, that some of these elationshis al onl when the cutting edge angle κ is less than 9 o. Nowadas, it is becoming common actice to use cutting tools having κ geate than 9 o. Fo these tools, the following elationshis ae valid: tan λ = sin κ tan γ cos κ tan γ (7) s tan γ o = cosλs o (8) tan γ = cosκ + sinκ (9) o cot αn = cosλs cotαo () cot α cosκ cotα + sinκ cotα o = () f f f f f f 5

6 D. Vikto P. Astakhov, Tool Geomet: Basics 3. The Alication Of Vecto Analsis To The Stud Of Cutting Tool Geomet As a fist ste in the alication of vecto analsis to cutting tool geomet oblems, conside the detemination of the cutting edge inclination angle in the tool-inmachine sstem. Figues 5 shows a single-oint tool having a zeo inclination angle (λ s = ) in the tool-in-hand sstem (i.e., its side cutting edge- is hoizontal). In tool-inmachine sstem, the tool in installed so that its ti is shifted elative to the efeence lane on distance h. The oblem is to detemine the esultant cutting edge inclination angle due to this shift. R C h v x N z Figue 5: Geomet of a single-oint tool when the cutting edge, having a zeo inclination angle in the tool-in-hand sstem, is shifted above the efeence lane though the axis of otation. Note that the tool cutting edge angle κ <π/. In figue 5, a oint is selected on the side cutting edge - to be a cuent oint in ou consideation. The ight-hand (x,, z) coodinate sstem with the oigin in oint is set u as follows:. The x-axis along with the feed motion.. The -axis is chosen to be eendicula to the x-axis with sense as shown in Fig The z-axis is eendicula to the x- and -axes, with sense as shown in Fig. 5. Let be a vecto along the cutting edge - then in the selected coodinate sstem this vecto is eesented as: = i + j tanκ () Let h be a distance between oint and the hoizontal lane assing though the cente of otation ; R c be the adius of otation of oint then the angle µ between the z-axis and the vecto of the cutting seed at oint is calculated as µ = h sin (3) R c 6

7 D. Vikto P. Astakhov, Tool Geomet: Basics Let v be a vecto along the diection of the cutting seed. This vecto can be detemined easil because it is eendicula to, hence v = j µ + k tan (4) The angle between vectos and v is then calculated as π ν ) = cos + λ s = sin λ s ν = ν cos( (5) Exessing the scala oduct and vecto modules though the vectos coodinate (Eqs. and 5), we obtain sin λ tan µ tanκ s = = sinκ sin µ (6) ± ( + tan µ )( + tan κ ) Using Eq. (6) one can calculate the inclination angle fo an given oint of the cutting edge -. Two imotant conclusions ma be dawn fom Eq. (6). Fist, the inclination angle is negative (as exected accoding to Fig. 4) and it vaies along the cutting edge due to the vaiation of angle µ. As seen, the maximum λ s is in oint and the minimum is in oint. Second, if the tool would be installed below the discussed efeence lane then angle µ would be negative. As such, λ s is ositive changing fom its maximum in oint to its minimum at oint. It was shown in [3] that the ake and cleaance angles, defined in the othogonal lane in the tool-in-hand sstem, change in the tool-in-machine sstem if a tool is shifted with esect to the efeence lane. This is due to the fact that the lane of cut (which is defined as to be tangent to the suface of cut at the consideed oint of the cutting edge) ceases to be eendicula to the efeence lane. The angle between this lane and the cutting edge lane is denoted as τ and measued in the othogonal lane. When λ s = and a cutting tool is installed as shown in Fig. 5, the ake and elief angles in the tool-in-machine sstem ae calculated as [3] α o = α o (7) τ γ γ + = o τ o (8) To detemine τ, conside nomal N, which is eendicula to the cutting edge in oint and lies in the efeence lane though the cutting edge (Fig. 5). As seen N = i tan κ j (9) + Nomal N to the lane of cut is detemined as the vecto oduct of vectos v and located in this lane 7

8 D. Vikto P. Astakhov, Tool Geomet: Basics i j k N v = tan µ = i( tanκ ) + j + k tan µ = () tanκ Because angle τ is the angle between the lane of cut and the cutting edge lane, it can be calculated as the angle between nomals to these lanes as follows tan N = ( N N N τ () The vecto oduct of N and N is calculated as i N N = tanκ = i( tan µ ) + tanκ tan µ ) j tanκ j k tan µ () and so its module is equal to N N = tan κ + tan κ tan µ = tan µ cosκ (3) The scala oduct of N and N is calculated as N N = tan κ + = cos κ (4) Substituting Eqs. (3) and (4) into (), one can obtain tanτ tan µ cosκ = (5) = cos κ tan µ cosκ Analsis of Eq. (5) shows that angle τ vaies along the cutting edge because it deends on angle µ, which is a function of the adius of otation (Eq. (3)). As a esult, the ake and elief angles also va along the cutting edge (Eqs. (7) and (9)). The maximum γ o and the minimum α o ae in oint while oosite is tue in oint. Moeove, γ o > γ o while α o < α o. If the tool would be installed below the discussed efeence lane, angle µ is negative. In effect, γ o < γ o while α o > α o and the maximum γ o and the minimum α o ae in oint while oosite is tue in oint. Conside Fig. 6, which shows a single-oint tool having angle κ > π/ while othe aametes and designations ae ket the same. This model eesents the oute 8

9 D. Vikto P. Astakhov, Tool Geomet: Basics h R C v N z Figue 6: Geomet of a single-oint tool when the cutting edge, having a zeo inclination angle in the tool-in-hand sstem, is shifted above the efeence lane though the axis of otation. The tool cutting edge angle κ > π/. cutting edge of a gundill having angle ϕ = κ 9 o. This angle will be used in ou consideations. Following the same methodolog as discussed fo Fig. 5, we can wite: Nomal N, which is eendicula to the cutting edge in oint and lies in the efeence lane though the cutting edge, is eesented as N = i tanϕ (6) Vectos along the cutting seed and the cutting edge, esectivel v = j µ + k j tan (7) = i + j tanϕ (8) As befoe, nomal N to the lane of cut is detemined as the vecto oduct of vectos v and located in this lane i N v = tan µ = i j tanϕ k tan µ tanϕ tanϕ j k = (9) The vecto oduct of N and N and its module ae calculated as 9

10 D. Vikto P. Astakhov, Tool Geomet: Basics N N = i tanϕ j tanϕ k tan µ tanϕ = (3) = i tan µ tan ϕ j tanϕ tan µ tan µ tanϕ N N = tan µ tan ϕ + tan ϕ tan µ = (3) cosϕ The scala oduct of N and N is calculated as Finall N N = + tan ϕ = (3) cos ϕ tan µ tanϕ cosϕ tanϕ = = tan µ sinϕ (33) cos ϕ Analsis of the second case (Fig..4) esults in the following conclusions:. Equations (7) and (8) ae no longe valid when κ > π/, i.e., when κ = π/ + ϕ. This follows fom Eq. (5) whee cos κ = cos (π/ + ϕ ) = - sin ϕ. It can also be seen fom Eq. (9) accoding to which nomal N to the lane of cut goes down (since k is negative) with esect to the hoizontal nomal N (comae with the fist case whee this nomal goes u (Eq. ) since k is ositive). As a esult, Eqs. (7) and (8) should be e-witten fo the consideed case as α o = α o + (34) γ = γ o τ τ o (35) Equation (34) and (35) ae of exteme imotance in the consideations of the geomet of all kinds of dills because cuentl the oosite esult (as e Eqs. (7) and (8) is used in the analsis of the ake and elief angle of twist and gun dills. The location of the cutting edge above the efeence lane though the dill otation axis leads to the inceased ake and deceased flank angles if and onl if κ < π/. When κ > π/ (and this is the common case fo most dills) such a location leads to the deceased ake and inceased elief angles. When the dill s cutting edge is located below the mentioned efeence lane, the oosite is tue. Fo gundills, it leads to the inceased ake and deceased elief angles on the oute cutting edge and to the deceased ake and inceased elief angles on the inne cutting edge.. Because angle τ vaies along the cutting, the ake and elief angles also va along the cutting edge. The maximum γ o and the minimum α o ae in oint while the oosite is tue in oint.

11 D. Vikto P. Astakhov, Tool Geomet: Basics To exlain the obtained esults gahicall, figue 7 shows a gahical summa of the consideed cases. As seen fom figue 7a, when the suface of cut is convex, an shift of the tool above the efeence lane esults in incease ake and deceased elief (flank) angles. Howeve, the oosite is tue when the suface of cut is concave. The latte is the case in all kinds of dilling and boing oeations while the fome is the case fo common tuning oeations. As seen fom figue 7b, when the suface of cut is convex, an shift of the tool below the efeence lane esults in deceased ake and inceased elief (flank) angles. Howeve, the oosite is tue when the suface of cut is concave. When the suface of cut is lane, thee is no influence of the tool vetical location on the ake and elief angles. Suface of cut z Plane of cut Nomal to the lane of cut Suface of cut z Plane of cut R R Nomal to the lane of cut o o o o (a) (b) Figue 7. Influence of the tool location on the ake and elief (flank) angles: (a) tool is shifted above the efeence lane though the axis of otation, (b) tool is shifted below the efeence lane though the axis of otation. To comlete the analsis, conside a secial case when κ = π/ as shown in figue 8. Following the same methodolog as discussed fo figues 5 and 6, we can wite: Nomal N, which is eendicula to the cutting edge in oint and lies in the efeence lane though the cutting edge (Fig. 8), is eesented as N = i (36) Vectos along the cutting seed and the cutting edge, esectivel v = j tan µ + k (37) = j (38) As befoe, nomal N to the lane of cut is detemined as the vecto oduct of vectos v and located in this lane N i = v = j tan k = i µ (39) The vecto oduct of N and N and its module ae calculated as

12 D. Vikto P. Astakhov, Tool Geomet: Basics R C h v x N z 9 Figue 8: Geomet of a single-oint tool when the cutting edge, having a zeo inclination angle in the tool-in-hand sstem, is shifted above the efeence lane though the axis of otation. The tool cutting edge angle κ = π/. i j k N = = N (4) N N = (4) The scala oduct of N and N is calculated as Finall N N = (4) tanτ = = thus = τ (43) It follows fom Eq. (43) that when κ = π/, the vetical shift of a tool (with esect to the efeence lane though the wokiece axis of otation) does not affect the ake and flank angles of the cutting edge. The eason fo this is athe simle and follows fom figue 8. As seen, the suface of cut is a lane that coincides with the lane of cut and it does not change it oientation when a tool shifts along this lane. Refeences:. Astakhov V. Mechanics of Metal Cutting, CRC Pess, 998/999.. Watson A.R.,, Geomet of Dill Elements, Int. J. Mach. Tool. Des. Res., Vol. 5, No. 3, 985, Ganovski G.I. and Ganovski V.G., Metal Cutting, (in Russian), Vishaja Shcola, Moscow, 985.