# Crop of free curved surface establishs milling cutter the computational method of contrail of 5 axes CNC Machining

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As The Algorithm Considers The Radius Of The Curvature Difference Between Different Cutter Contact Points, so It Can Improve Machining Efficiency And The Accuracy Of The Finished Surface.

This Algorithm Is Suitable For Machining Sculptured Surface.

A Method Of Calcul Ating Cutter Location Point Is Also Presented In This Paper.

Key Words:fLat-end Milling Cutter, seulptured Surface, 5-axis NC Machining, curved surface of Cutter Path1 introductive freedom applies in the mould very extensive, wait like forging die of mould of car automobile body, plastic pattern, lamina, mould, include free curved surface mostly (curved surface of the following abbreviation) . The treatment of curved surface is mixed by spherical knife normally be not spherical knife (if crop establishs knife of shape of milling cutter, awl, bosomy form knife,wait) finish, because crop establishs milling cutter (the following abbreviation establishs milling cutter) quality of cutting efficiency, treatment, service life excel is spherical knife [1] , should first choice. In last few years, establish milling cutter domestic and internationally about curved surface the algorithmic research of contrail of 5 axes CNC Machining is more [4 ～ 6] , these algorithm use the method of differential geometry to grow to undertake forecasting with row spacing to the pace, formulary and simple, computational amount is small, but the change that they contact the curvature between the dot without cutting tool of consideration photograph adjacent, small to machining curvature to change curved surface is more effective, do not suit the curved surface with curvature big change. The article puts forward a kind to get used to a pace to grow the computational method with row spacing oneself below parameter coordinate department. This algorithm is satisfying the premise that machines precision and surface roughness to fall, improve treatment efficiency effectively again, suit the treatment of the curved surface with curvature big change. 5 axes establish the effective radius of 2 cutting tool when mill machines curved surface, because establish the cutting blade of milling cutter to be on the periphery that establishs milling cutter, buy ought to slant between the normal of the axes that establishs milling cutter so and curved surface radius of a cutting tool, curved surface of cutting of square can effective ground. Wait for an element considering cutting tool and curved surface interference, establish milling cutter to be in slant buy while, one angle φ still should be placed with normal inside the normal plane that its axes is being ordered by treatment (graph 1) , n is be measured to arrow by law of unit of cutting curved surface, tax is vector of axial of cutting tool unit, r is cutting tool radius, so, the definition of effective cutting radius of cutting tool is: Re=Rsin φ . In the graph a shown state falls, on the secret tangent plane that its end panel is being ordered by treatment umbriferous the ellipse that is Re to grow axle shaft R, short axle shaft. If coordinate origin is on the centre of a circle of cutting tool end panel, it is reference axis with long, short axle shaft, the equation of the end panel that establishs milling cutter is: (1) pursues 1 cutting tool is this effectively, establish curved surface of milling cutter cutting to be able to regard knife of an elliptical figuration machining curved surface. The 3 computation that get used to a pace to grow oneself are in process of 5 coordinate CNC Machining, as a result of curved surface everywhere Xiang Shi measures the way is change, inevitable meeting causes the change of vector of cutting tool axial, namely of cutter shaft swing can making the contact of cutting tool and curved surface nods orbit is not linear, however curve. So, 5 coordinate establish mill machining error δ includes to approach error δ T and cutter shaft to swing point-blank error δ N (graph 2) . Direction of this parameter of some of algorithmic choice curved surface (like U to) as pace progress travel error is controlled (graph 3) , dot R(wi, 0) , R(wi, 1) is curvilinear R(wi, 0)r(wi, on 1) at 2 o'clock, join nods R(wi, 0) , R(wi, 1) Cheng Yixian, dot R(wi, u) arrives for the curve bowstring R(wi, 0)r(wi, the largest space of 1) is nodded, what computation is in at this o'clock is linear approach cutter shaft of sum of errors to swing error, compare machining error δ and the size that allow differs ε , be like δ ＞ ε , the end points R(wi with new link, u) , R(wi, 0) forms Xin Xian R(wi, 0)r(wi, u) , the interval of will new parameter curve [0, u] changeover arrives [0, 1] interval, calculate again machining error δ , go down so, till δ < ε till. Note cost of U of every Δ , as treatment the pace grows, it is new start with R(u) again, repeat calculation, can pace of each bits grows cipher out. Graph the control of 2 paces chief error 3.

The 1 computation that approachs an error point-blank pursues the 3 computation that approach an error point-blank are shown 3 times like the graph, curvilinear R(wi, 0)r(wi, the treatment of 1) is effective it is to pass interpolation multistage inside receive bowstring to approach it. Considering treatment efficiency, hope chord as far as possible big, paragraph of number of bowstring as far as possible little, take the biggest chord to approach below the circumstance of contented precision namely, normally method is the one aspect of the matter from the curve begins to use iteration to search a law to beg the another end points that takes spring. Join nods R(wi, 0) , R(wi, 1) Cheng Yixian, be like bowstring R(wi, 0)r(wi, 1) expresses with vector C, d states music on-line nods bowstring R(wi, 0)r(wi, the largest space of 1) , namely ｜ of D of δ T= ｜ . So exist: D=r(wi, u)-r(wi, of of of 0)- λ C (in 2) type: λ -- coefficient, λ ∈ [0, 1] . Because C and D are perpendicular, attainable: of Cd=0 (vertical of 3) couplet (2) , (3) solves: (4) (the type on 5) can write into: D=P [R(wi, u)-r(wi, 0) ] (6) among them, , call umbriferous matrix, I is unit matrix. Because be in bowstring to deviation the largest part, the curve cuts arrow perpendicular Yu Shi measures D, namely R ′ ((Wi, of U)d=0 (′ of 7) reason R ((Wi, u)P [R(wi, u) - R(wi, 0) ] of =0 (outside the root is except two end points in the type on 8) , still have many, because be in bowstring to deviation the largest part, exist: R ″ (Wi, 0 of U)d ＜ (9) so skill (8) and U ∈ (0, 1) use at each root place, type (type is satisfied in 7) (8) and U ∈ (0, 1) the root manages corresponding U cost for the biggest error namely, accordingly, can beg T giving δ . In afore-mentioned algorithmic processes, must have inside examination music line segment above all without inflection point, calculate the seat that gives this a little bit. If music line segment has inflection point, nod with this abduct curvilinear one divides into two, the interval that enters two paragraphs several curves [U1, u2] changeover arrives [0, 1] interval, use dichotomy to have iteration to two paragraphs of curves respectively. 3.

2 cutter shaft swing the computational cutter shaft of the error swings the error is treatment in the process, what because vector of cutting tool axial swings,cause is nonlinear error (2) seeing a picture. Can prove [6] : Re(kf of ≤ of ｜ of ｜ δ N.

of of Δ Su) (in 10) type: Kf -- approach direction of paragraph of feed of incurvate face edge to be in point-blank the biggest the law curvature that approachs error point point point-blank; Δ Su -- approach Duan Hu to grow. ′ of R of ｜ of U2u1 of Δ Su= ∫ (Wi, u) of of ｜ Du (11) is become Kf ＜ 0 when, treatment surface is protruding curve along direction taking a knife, cutting tool contacts the contrail of the dot to be sunken curve, accordingly, machining error swings to approach cutter shaft of sum of errors point-blank the sum of error absolute value, namely: ｜ of N of δ of δ = ｜ + ｜ of ｜ δ T. When Kf ＜ 0 when, treatment surface is sunken curve along direction taking a knife, cutting tool contacts the contrail of the dot to also be sunken curve, and cutter shaft swings δ N ｜ always is less than error ｜ to approach ｜ of T of error ｜ δ point-blank, accordingly, can inspect approach ｜ of T of error ｜ δ to be machining error δ point-blank, namely: ｜ of T of δ of δ = ｜ . The 4 computation that get used to row spacing oneself although the appearance of curved surface each different, but when cutting tool is machining these curved surface, walk along knife treatment to give pieces of whole curved surface according to certain curve. To dot of cutting tool contact, the curvature size that is in a curve to be in in this bits according to its can divide it is 3 kinds: Protruding dot, hollow site, inflection point, also put in the dot that goes up point-blank 's charge here for inflection point kind. Protruding dot, hollow site, inflection point is OK according to its the size of curvature Kf tries differentiate: When Kf ＞ 0 when, for protruding dot; When Kf ＜ 0 when, for hollow site; When Kf ＝ 0 when, for inflection point. Answer relatively with protruding dot, hollow site, inflection point, these curves that choose a location can be divided the curve that it is protruding, sunken curve, linear. The osculatory dot when cutting tool is protruding when dot, hollow site or inflection point, inside close face, the curve of these 3 kinds of region that nod adjacent can regard protruding as respectively circular arc, sunken circular arc and linear. Accordingly, when the row spacing of computational freedom curved surface, can mix in circular arc directly go up point-blank try to calculate. Graph 4a machines the case when protruding curved surface to establish milling cutter, ρ contacts the curvature radius of bit of place for cutting tool, Δ Sw is row spacing, elliptical delegate establishs milling cutter, dot of cutting tool contact is B, C respectively at 2 o'clock, nod R(w1 namely, ui) , R(w2, ui) , set coordinate origin to be on the center that establishs milling cutter, so, the coordinate that A nods is: X=(ρ + ρ of Y=(of of of H)sin α + H)cos α - (ρ + Rsin ) of of curve of graph 4(a) protruding ( of of B) sunken curve (C) is nodded as a result of A point-blank go up in ellipse, satisfy type surely (the elliptical equation of 1) , namely: This is a yuan of quadratic equation, can solve a H: (12) to graph 4b, (13) type (12) , (Sw/ of Δ of = of the α in 13) (2 ρ ) (14) among them the arc length: that Δ Sw is arc BC? ′ of R of ｜ of W2w1 of ∫ of W ＝ (W, ui) of of ｜ Dw (15) to graph 4c, h=Rsin φ - Sin φ (R2-L2/4)0.

of 5 (16) right now, the row spacing of cutting tool is line segment L, namely B, C the distance between 2 o'clock. So far, had chosen the computational formula that derived height remains below 3 kinds of case, can use formula respectively (12) , (13) , (16) the rudimental height that computation gives these 3 kinds of case to fall. The size of row spacing is a basis of H and ε compare a result and decide. Be like H ≤ ε , so Sw of right now Δ of computational row spacing can satisfy a requirement; Be like H ＞ ε , need to reduce Δ W so, till till H ≤ ε . To get the Δ Wmin of contented condition, grow algorithm by the pace attainable value of a group of Ui, seek the Δ W value that gives correspondence, take among them the smallest Δ W to be the row spacing on parameter axis namely. The computation of the contrail of computational cutting tool of contrail of 5 cutting tool includes the consideration of the center of cutting tool and vector of cutting tool axial. If the graph is shown 5 times, dot A contacts a dot for cutting tool, n is nodding the unit law vector that A is in for curved surface, tax is vector of axial of cutting tool unit, the unit that A orders to point to cutting tool center for Tc slants buy vector. Graph contrail of 5 cutting tool all is because of N, Tax, Tc on same plane, tc ⊥ Tax, so vector N, Tax1, Tc1 forms right-angled triangle of a vector. Tax of here Tax1 ∥ , tc1 ∥ Tc, reason: Tc1=n - Tax1=n - Tax of ｜ of ｜ Ncos φ Cos φ =nTax, reason: Tc1=n - (NTax)Tax unit changes vector Tc: of of of ｜ of Tc=Tc1/ ｜ Tc1 (17) among them Tax of vector of cutting tool axis can nod N of unit law vector normal plane inward turning to turn in A φ horn gets, namely: of Tax=nTR (the TR in the type on 18) is change coming back to change matrix: In type: N1, N2, N3 -- the directional cosine of vector of unit of runner shaft coming back. Then, the contrail of cutting tool center is: Ce=r(w, of U)+RTc (the article collects 19)6 conclusion to differ a law to calculate with bowstring what 5 axes establish milling cutter to machine curved surface is linear approach an error, according to differential geometry relation computation is in the biggest approach an error to be in cutter shaft to swing point-blank error, by both the machining error that make will control treatment pace to grow jointly, this algorithm considered different cutting tool to contact the curvature difference of bit of place. Formula of pair of row spacing computation were returned to undertake derivation in article, gave out knife computation is formulary. This algorithm suits to machine the curved surface with curvature big change. CNC Milling CNC Machining