TOOTH SURFACE FUNDAMENTAL FORMS IN GEAR TECHNOLOGY UDC:

Transcription

1 NIRSIT O NIŠ The scietific joual ACTA NIRSITATIS Seies: Mechaical gieeig ol1 N o pp dito of seies: Nead Radojković adojkovic@iacyu Addess: ivezitetski tg Niš Tel: ax: TOOTH SRAC NDAMNTAL ORMS IN AR TCHNOLO DC: 6190 Stepa P Radzevich 1 ik D oodma ikto A Palaguta 3 1 Depatmet «Cuttig Tool Desig ad Poductio» (10) Mechaical gieeig aculty Natioal Techical ivesity of kaie «Kyiv Polytechic Istitute» Peemohy Ave 37 Kyiv 506 kaie College of gieeig Michiga State ivesity ast Lag MI 4884 SA 3 Mechaical gieeig aculty Diepodzeghisk State Techical ivesity Diepostoyevskaya Ave Diepodzeghisk 3618 kaie Astact A ew appoach to ivestigate the local topology of the ivolute gea tooth suface has ee developed The appoach is ased o the fudametal esults i diffeetial geomety of sufaces The method descied hee may always e applied ut is less efficiet tha othe methods if calculatio of oly oe gle paamete is equied Howeve whe multiple paametes of the local geomety (itisic o exteal) of the gea tooth suface must e calculated this method is moe efficiet tha othe methods i use Key wods: gea tooth suface gea techology 1 INTRODCTION Desig ad maufactue of geas equies fequet calculatio of the paametes of the wokig suface of a gea tooth I cases whe it is ecessay to calculate may paametes of that suface applicatio of the fist ad secod fudametal foms of the suface (the aussia fudametal foms of the suface) is efficiet Applicatio of the fist Φ 1 ad the secod Φ fudametal foms of the wokig suface of a gea tooth allows simplifyig the fomulas ecessay fo calculatios ad is coveiet fo detailed study of the geomety of suface Received Apil

2 516 S P RADICH D OODMAN A PALATA LOCAL TOPOLO O TH AR TOOTH SRAC Let us stat fom the detemiatio of suface As is well kow [ p53] a kiematic method of ceatig a ivolute cuve is to use a cicula disk of adius with a stig wapped aoud it widig the that stig poduces a seies of lies taget to this ase cicle (ie to the cicula disk) ad tacig the eds of these taget lies geeates a ivolute cuve (ig 1) I a simila mae the scew ivolute suface may e geeated y scew motio of a staight lie that maitais its diectio taget to the helix o the ase cylide duig this motio ig 1 shows the scew ivolute suface that is coveed with staight lies (staight geeatix) - taget to the helix o the ase cylide [ p ] ig 1 eeatio of a ivolute cuve

3 Tooth Suface udametal oms i ea Techology 517 As with the ivolute cuve (ig 1) the positio vecto of a aitay poit o the ivolute suface i coodiate system ca e witte as (ig ): A B C (1) whee ig eeatio of suface I (1) vectos A B ad C espectively ae equal to (ig): A i j () B k ( ta ) (3) C i j (4) - adius of the ase cylide of the gea ad - cuviliea (aussia) coodiates o suface - is the ase helix agle B

4 S P RADICH D OODMAN A PALATA 518 Sustitutig () (3) ad (4) to the (1) oe ca otai: ) ta ( ) ( ) ( k j i (5) To calculate the local geometical paametes of suface : omal ad piciple adii of cuvatue piciple diectios etc it is coveiet to apply the fist ad the secod fudametal foms Φ 1 ad Φ espectively of suface om defiitio [7] the fist fudametal fom Φ 1 of suface is equal to: 1 d d d d Ѓ Φ (6) whee the aussia coefficiets ad ae give y: (7) All deivatives ecessay to calculate the aussia coefficiets ad ug (7) ca e calculated fom (5) as: (8) ad ta (9) Accodig to (8) ad (9) the aussia coefficiets ad of the fist fudametal fom Φ 1 of suface ae equal to: 1 (10) The secod fudametal fom Φ of suface is detemied as follows [7]: d N d d M d L Φ (11) whee the aussia coefficiets L M ad N of the fudametal fom Φ ae give

5 Tooth Suface udametal oms i ea Techology 519 y: L (1) M (13) N (14) The secod deivatives of suface ecessay to calculate the aussia coefficiets L M ad N of the fudametal fom Φ ae: 0 (15) ad 0 (16) ad

6 50 S P RADICH D OODMAN A PALATA 0 (17) Accodig to (15)(17) the aussia coefficiets L M ad N of the fudametal fom Φ ae: L 0 M 0 N (18) 3 CALCLATIONS O TH RADII O TH PRINCIPL CRATR O SRAC The adii of the piciple cuvatue R 1 ad R of suface ae calculated as a oots of the quadatic equatio [1]: 1 ( L N M ) R ( N M L ) R ( ) 0 (19) which i this case simplifies to: 1 o i tems of the paametes of the gea desig 1 N R 0 (0) 1 R 0 (1) Because ( L N M ) 0 the adii of the piciple cuvatue R 1 ad R of suface ae equal to: R1 R cot () 4 CALCLATIONS O TH RADII O NORMAL CRATR R O TH WORKIN SRAC O AR TOOTH Accodig to ule s fomula the adius of omal cuvatue R of ay aitay sectio y a omal plae of suface is equal to: R R ϕ R ϕ (3) g the esult aove (R 1 ) (3) simplifies to 1 R R ϕ (4) o that easo the adius of cuvatue of the cuved lie of itesectio of suface y a aitay omal plae sectio ca e calculated y the fomula: R1 R (5) ϕ whee the value of the agle ϕ is ukow

7 Tooth Suface udametal oms i ea Techology 51 To detemie the value of the cetal agle ϕ coside ig 3 ist of all it is ovious that QK o ig3 is equal to i () So: QS QK ψ whee D is the pitch diamete of the gea D is the ase diamete of the gea D D ψ (6) ig 3 Detemiatio of the cetal agle ϕ

8 5 S P RADICH D OODMAN A PALATA 5 AMPL It is ecessay to detemie the adius of omal cuvatue of the wokig suface of the gea tooth i a give diectio taget to the helix o the gea pitch cylide Accodig to a dawig of gea the ase diamete of the gea is equal to D pitch diamete D pessue agle (omal) φ 0 0 helix agle ψ 5 0 Accodig to a equatio i [6 p30]: o that easo at poit K ψ ψ (7) φ D D (8) o o ψ φ 5 0 The 90 0 ψ So i () sot ta ψ ad R taψ As is well kow µ ta µ (9) 1 µ fo ay agle µ Accodig to (9) oe ca wite: ad R taψ ψ 1 ψ ψ 1 ψ o o o ψ φ o ψ φ Coside a plae though poit K taget to suface The staight geeatix QK of suface lies i this taget plae A staight lie though poit K taget to the helix o the pitch cylide coicides with this taget plae as well The cetal agle ϕ sought is equal to the agle etwee the staight geeatix of suface (o staight lie though the poit K taget to the helix o the ase diamete cylide) ad the staight lie taget to the helix o the pitch cylide These staight lies ae ecessay i ode to deive the equatio of the aove-metioed staight lies i a commo coodiate system I coodiate system attached to the gea at poit K the staight lie is detemied y the equatio D d D * p i ta θ j k ta θcot ψ (3) whee θ - is a paamete desciig the locatio of a aitay poit o the staight lie though poits Q ad K I matix fom (3) is (30) (31)

9 Tooth Suface udametal oms i ea Techology 53 ta θ * Dd 1 p (33) ta θc cot ψ Hee i (33) at poit K (ig 3) θ ta θ θ 1 θ θ D 1 D D D (34) ig 4 Deivatio of the equatio of a staight lie taget to the helix o the ase cylide I coodiate system with oigi at poit K o suface the staight lie taget to the helix o the gea pitch cylide is equal to (ig 4): p il ψ kl ψ (35) o i matix fom l ψ p 1 (36) l ψ Now it is possile to wite (33) i coodiate system o to wite (36) i coodiate system Choose the fist I this case matix [M 1 ] of the coodiate system tasfomatio is equal to θ θ 0 [ M 1 ] θ θ 0 (37) 0 0 1

10 54 S P RADICH D OODMAN A PALATA I (37) we have eglected the taslatios of the coodiate systems alog the coodiate axes ecause the cetal agle ϕ sought depeds o coodiate system otatios oly ad is idepedet of taslatios of the oigi of the coodiate systems Takig ito accout (37) the staight lie taget to the helix o the ase cylide (see (33)) is detemied y: θ θ 0 ta θ θ * p[ M1 ] p θ θ 0 1 θ ta θ θ (38) ta θcot ψ θ ψ ta cot om (36) ad (38) ad i accodace with [1 p69] it follows that: * p p ϕ * [4 θ (θ θtaθ) l θ ψ ta θtaψ p p ta θcot ψ ψ ] ( l At poit K the paamete l 0 o that easo (39) simplifies to ψ ψ) (39) ta θ ta ψ ϕ (40) 4 θ ( θ θ ta θ) ta θcot ψ Takig ito cosideatio (6) ad (30) oe ca otai: ϕ (1 ψ φ ta θ ψ φ 1 ψ φ ) [4 θ (θ θ ta θ) ] ta θ ψ φ o o o (1 5 0 ) ( ) o (41) It follows that R This esult shows the omal adius of cuvatue of suface i a give diectio taget to the helix o the gea pitch cylide which is eeded i ode to calculate the paametes of the desig of the cuttig tool fo maufactuig the gea Thee is also aothe way to solve the polem I that method it is ecessay to coside the cuved lie of itesectio of suface of the gea tooth with a plae taget to the pitch cylide of the gea ad to calculate the cuvatue of the lie of itesectio The method descied hee may always e applied ut is less efficiet tha othe methods if calculatio of oly oe gle paamete is equied Howeve whe multiple paametes of the local geomety (iteal o exteal) of the gea tooth suface must e calculated this method is moe efficiet tha othe methods i use RRNCS 1 Bostei IN ad Semedyayev KA A uide-book to Mathemetics elag Kai Deutsch

11 Tooth Suface udametal oms i ea Techology 55 akfut/mai ud uich 1971 Dudley DW ea Hadook Maufactue ad Applicatio of eas d ed Mcaw-Hill N Litvi L ea eomety ad Applied Theoy Petice Hall glwood Cliffs NJ Litvi L Theoy of eaig d ed (i Russia) Nauka Mascow Litvi L Theoy of eaig NASA Refeece Pulicatio 11 NASA Lewis Reseach Cete Clevelad OH Mode Methods of ea Maufactue 4 th ed Natioal Boach & Machie Div Stuik JD Lectues o Classical Diffeetial eomety Addiso-Wesley Camidge MA 1950 OSNONI OBLICI PORŠIN BACA THNOLOIJI PČANIKA Stepa P Radzevich ik D oodma ikto A Palaguta adu je pikaza jeda ovi pistup istaživaja lokale topologije ivolute povšie zuaca Pistup je zasova a fudametalim ezultatima difeecijale geometije povšia Ovaj metod se može uvek pimeiti ali je maje efikasa ako se poačuavaja vše samo po jedom paametu Međutim ako se poaču vši pema većem oju paametaa lokale geometije povšie zuaca ovaj pistup daje olje ezultate u oosu a duge metode