The Forming Theory and the NC Machining for The Rotary Burs with the Spectral Edge Distribution

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1 oden Appled Scence The Fomn Theoy and the NC achnn fo The Rotay us wth the Spectal Ede Dstbuton Huan Lu Depatment of echancal Enneen, Zhejan Unvesty of Scence and Technoloy Hanzhou, c.y. chan, , Chna E-mal: Ths poject s supposed by the natual scentfc foundaton of Chna, No Abstact Ths pape eseached the otay bus wth specal cuttn edes, whch s the newest lay out of such tools, and epesents the ntenatonal decton of development and advanced level. So that mpoved the cuttn condton of the cutte. Keywods: Acoss ede, Rotay bus, NC machnn Intoductons Fo the ball ended o tpped otay bus, the cuttn edes must jont toethe at the tp, whee the depth of the oove of cuttn ede s zeo n theoy. In the vcnty of the tp, the ooves ae vey shadow, whle the edes ae vey densely concentated. So that the space s vey lmted, the cuttn condton s vey bed. In ode to mpove the cuttn condton at the tp, the otay bus wth the spectal ede dstbuton s pesented n ths pape. In the vcnty of the tp, thee ae 2 nds of cuttn edes: one s called the man ede, t passn thouh the tp as the odnay ede; anothe nd s called the banch ede, whch dd not passn thouh the tp. In the vcnty of the tp the man edes tac pat nto the cuttn, whle the banch edes dd not tac pat nto the cuttn, so that nceased the space n the vcnty of the tp, and thee fo mpoved the cuttn condton damatcally. 1. The specal cuttn ede The odnay cuttn ede, s defned as the ntesect cuve of the nclned plane wth the otay suface of the cutte. The nclned anel of the plane s β. We have follown elatonshp: L x snϕ = t β In the equaton: φ : The anel between the adate lne of any pont on the ede and the xoy plane L : The dstance fom the cente of the ball to the ntesect pont of plane and the axs : The otay adus of any pont at the suface β : The anel between the plane and the axs The ane of the x:x L. The specal cuttn edes, howeve, s defned as a sees of ntesect cuves of a sees of nclned planes wth the otay suface of the cutte. The nclned anels of the plane ae β. We have follown elatonshp: L x snϕ = t β (2) 112

2 oden Appled Scence ach, 2008 Fue 1. A sees of ntesect cuves of a sees of nclned planes wth the otay suface of the cutte In the equaton: φ : The poston anel β : The anel between the plane and the axs. L : The dstance fom the cente of the ball to the ntesect pont of plane and the axs. mx a β Fue 2. Rotay bus wth the spectal ede dstbuton 2. The composed cuttn ede Fo the composed cuttn ede, at the connectn pont of two nd edes(x= x ), the nehbon edes must have same poston, and the two edes must satsfy the condton of smooth connecton. That mnes the plans to cut the ball should not be paallel wth each othe. Let the anel between the planes and the axs s β, than we have: L x snϕ = t β In the equaton: φ : The poston anel : The adus of the ball. β : The anel between the plane and the axs. L : The dstance fom the cente of the ball to the ntesect pont of plane and the axs. d + ( L x) dϕ cosϕ = 2 t β dϕ 1 d = [ tβ sn ϕ] cosϕ d tβ snϕ dϕ tβ = = d 2 d ( ) 1 + ( ) cosϕ (2) 113

3 oden Appled Scence β : The helcal anel of the ede. In ode that, at the connectn pont of two nd edes, the nehbon edes have the same helcal anel, and equal to the ven helcal anelβ, thee must have the follown equaton: d d 2 t β = t β o 1 + ( ) cos sn ϕ + ϕ (3) In the equaton: β0 : The helcal anel of the odnay ede : The adus of the cutte at the pont of connecton φ : The poston paamete of the ede at the pont of connecton Tac the β nto the equaton, we can fnd L L = ctβsnϕ + x (4) Whle the equaton (4) can be ewtten nto: x (1 + t β ) 2 L t β x + ( L t β R ) = 0 (5) 3. The fomn theoy fo the otay bus wth the spectal ede dstbuton Fue 3. The coodnate system fo the machnn The machnn of the otay bus wth the spectal ede dstbuton s shown n f.3. Thee ae fou systems: the fxed system S : ( 0) ( 0) In the system, the onal pont s at the cente of the ball, the axes x, y ae alon the lontudnal and tansvese decton of the machne tool. (2) The system fxed to the Swnn base In the system, the onal pont O concdes wth the pont O, the anel between z and z s τ. (3) The system fxed to the cutte (d ) In the system (d ) x otate an anel ϕ about S (4) The system fxed to the sand wheel (s) S In the system, the onal pont s at the cente of the sand wheel, and moved wth the sand wheel. y and y ae paallel wth each othe. The anel between y and y s π / Z. The tansfomaton matxes ae as follown: os sn 0 cos xc yc = cos 0 sn z c (6) 114

4 oden Appled Scence ach, 2008 o 1 cosτ snτ 0 0 snτ cosτ 0 0 = (7) Refe to the Fue 3, the coodnate of the any pont at the suface of sand wheel s: x = ( R ) tα m m Fue 4. Sand wheel y = cosθ (8) z = msnθ In the equaton, R : The adus of the sand wheel. : The adus of the f the any pont at the suface of sand wheel mb θ : The anula poston paamete of the any pont at the suface of sand wheel α : The anel of the cone of the sand wheel The nomal vecto of the any pont at the suface of sand wheel : n v = ( cos α, snα cos θ,snα sn θ ) (9) When tansfomed nto the fxed system: x = ( R ) tα sn + snθ cos + x m m c y = m cosθ + y (10) c z = ( R ) tα cos + snθ sn + z (11) m m c v n = ( cosα sn + snα snθ cos, snα cos θ, cosα cos + snα snθ sn ) The coodnate of the any pont at the bottom of sand wheel n the system (s) x = 0 y = macosθ A z = masnθ (12) A And the nomal vecto: v n = (1,0,0) (13) When tansfomed nto the fxed system: x = snθ cos + x y = snθ + y z = snθ sn + z (14) ma A c n = ma A c ma A c v (sn, 0, cos ) (15) The movement n manufactue: 115

oden Appled Scence The swnn anel of the Swnn base: d tτ = The otaton of the wo: ϕ = ϕ The coodnaton of the cente of the sand wheel: xc = xp( + 1) + ( R m) tαsn msnθcos yc = yp( + 1) + mcosθ z = z (

5 oden Appled Scence The swnn anel of the Swnn base: d tτ = The otaton of the wo: ϕ = ϕ The coodnaton of the cente of the sand wheel: xc = xp( + 1) + ( R m) tαsn msnθcos yc = yp( + 1) + mcosθ z = z ( 1) ( R ) tχ cos snθ cos c p + m m 4. Examples A ball end otay bus wth the spectal ede dstbuton, the basc paametes ae as follown: The damete of the ball: d = 13mm The lenth of the cutte: l=10mm The helcal anel: β= 20 o The numbe of the tooth: z= 24 When dvded nto 6 dstcts, one dstct contans 4edes. The adus of the sand wheel: D= 100mm The anel of the cone of the sand wheel: α = 50 o The machned otay bus wth the spectal ede dstbuton s as shown n F.5 Refeences Fue 5. The otay bus wth the spectal ede dstbuton Lu, Huan. (1994).The cuttn ede of the otay tool wth complex suface. Jounal of Cuttn Tool. 8 Lu, Huan. (1998). The Fomn Theoy and the NC achnn fo the Rotay us on the 5-axs NC machne tool. Jounal of achne tool and manufactuen. 1 Zhou, J. (2002). The molden and desn of the tanstonal cuttn edes. Jounal of Chnese echancal Enneen.No(5) 116

Chapter Fifiteen. Surfaces Revisited

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