The Gear Hub Profiling for Machining Surfaces with Discreetly Expressed Surfaces

Size: px

Start display at page:

Download "The Gear Hub Profiling for Machining Surfaces with Discreetly Expressed Surfaces"

Arron Parks
5 years ago
Views:

1 he Ger Hub Profiling for Mchining Surfces with iscreetly Exressed Surfces NILE NE, VIRGIL ER, INUŢ PP, GBRIEL UR Mnufcturing Science nd Engineering ertment unăre de Jos University of Glţi omnescă street, no.. RMNI bstrct: - hey re know nd used the vrious methods for the rofiling of the ger hub reciroclly enwring with rofiles whirl, ssocited with rolling centrodes. In this er, is roosed rofiling method of the ger hub rimry eriherl surfce, reciroclly enwring with n ordered rofiles whirl bsed on the rinciles of helicl motion decomosing nd on the surfces enwring theory. he roosed method is lied for surfces known in discreetly form nd roximted by nd or rd degree Bezier olynomils. Bsed on dedicted softwre re resented numericl exmles nd the results re shown versus results obtined by clssicl nlyticl methods, exmles which show the reduced error level of roosed method. Key-Words: - Ger hub, Bezier olynomil roximtion, iscreetly exressed surfces Introduction hey re known modlities for ger hub rimry eriherl surfce rofiling [], [4], [5], [6], reciroclly enwring with n ordered surfces whirl, ssocited with n xoid in rolling with rck ger s xoid, common with the helicl surfce s rck of the ger hub. Princiled, the fundmentls theorems [4], [5], [6], of the surfces enwring resume to know in nlyticlly form the enwring surfces, the contct conditions between the conjugted surfces or nlyticl exression which ssocited with the generted surfces fmily in the reltive motions between the surfces to be generted nd the rimry eriherl surfces of tools, determine the form of the tools. From some resons my be imortnt to know in discreetly form the surfce to be generted [5]. hese kinds of roblem need secific lgorithm. he surfce descrition, known in numericl form, by Bezier roximtion olynomils, my constitute n lterntive for the ger hub rofiling when the generting recision is stisfctory. bviously, this kind of solving is designted for the surfces ordered whirl genertion, firstly for the non-involute rofiles, for which lwys er the necessity of tool s rofiling, when the tool s rofile is not known. Reference systems. Generting motion In figure, is resented the system of rolling xoid: the surfces to be generted whirl xoid the rck-ger reciroclly enwring xoid the ger hub rimry eriherl surfces xis osition nd the globl motions of the reference systems ssocited with these xoids. Fig.. he rolling xis nd coordinte systems hey re defined the reference systems: xyz is the globl reference system with z revolving xis of the xoid ssocited with the whirl of surfce to be generted x y z globl reference system, with y xis overled to the rimry eriherl surfce xis of the ger hub Z reltive reference system joined with the surfce to be generted whirl, nd xoid reltive reference system joined with the rck ger xis (in-lne surfce overled with lne ), nd xoid

2 Z reltive reference system ssocited with rimry eriherl surfce of ger hub. Is known the kinemtics of the genertion rocess: x, () the xoid rottion, joined with the Z reference system, with ngulr rmeter R r x,, () the xoid trnsltion, joined with the reference system nd the movement rmeter x, () the Z system rottion round the y xis, with ngulr rmeter. lso, re known the conditions:, (4) R r rolling condition of nd xoid cos (5) deendency of ger hub rimry eriherl surfce, worm with known itch ( helicl rmeter) nd the trnsformtion between globl reference systems: x x cos sin y, (6) sin cos z distnce between the xis of xoid nd helicl surfce s xis V. In rincile, the ngulr velocities for the revolution motions re even motions. he reltive motion of the reference system joined with xis of the surfce to be generted, Z, regrding the reference system ssocited with the rck ger sce,, is give by the trnsformtion: (7) with the current oint s definition on the surfce to be generted, s cylindricl surfce with genertrix rllel with the direction Z t, Z k : u u (8) for u discreetly known vrible with reduced vlues number ( or 4 oints), s element of comlex rofile which will be generte by enwring. he crossing rofile of the cylindricl surfce (8) my be stright lined segment, circle rc, n involute rc etc. he t rmeter is mesured long the cylindricl surfce genertrix. In following will be nlyzed the roblem of ger hub rofiling using the exression of the cylindricl surfce genertrix by limited number of oints. he genertion kinemtics is described by the trnsformtion (7) nd one of the fundmentls theorem of the enwring genertion. Rck ger surfce form determintion From (7) nd (8) re determined the surfces fmily in the reference system of the rck-ger tool with vrible rmeter: cos sin u Rr sin cos u Rr (9) t which is ssocited with the enwring condition: u u u u () where, Rr cos () Rr sin reresenting the condition of normls. Fig.. he surfce of the whirl to be generted, known by four oints on genertrix In rincile, the surfces fmily described by equtions (9) is on form: B () B. he identifiction of the, B,,,, B,, is resented in tble, for cylindricl surfce with circulr genertrix regrding the exmle resented in subsection 5. nd, similrly for nother genertrix tyes. long the directrix of this surfce tye re defined limited oint number ( or 4), which describe Bezier olynomil relcing this curve.

3 ble. he rck-ger coefficients identifiction Points on tool s rofile Identifiction of the olynomils constnts R R / / B r r R B (/ ) B B cosb sinb Rr (/ ) (5 / 6) B sinb cosb Rr B (/ ) B B (/ ) (5 / 6) B rcsin Rr B r R r R r cos sin Rr sin cos R rcsin Rr c r cos sin R r sin co s R rcsin Rr r (5 / 6) (/ ) (/ ) (5 / 6) (/ ) B (/ ) B By the identifiction of the olynomil coefficients is defined the discreetly form of the rck-ger reciroclly enwring with the whirl to be generted. 4 Ger Hub Primry Periherl Surfce Profiling Being know the rck-ger flnk surfce is roosed the determintion of the chrcteristiclly (the contct curve) t the contct with the future rimry eriherl surfce of the ger hub using the method of the helicl movement decomosition []. Is cceted tht the helicl motion of the ger hub rimry eriherl surfce V, is decomosing in sum of equivlents motions: trnsltion movement long the t direction of the unitry vector of the cylindricl surfce genertrix nd revolution round xis rllel with V nd t distnce tn () to the helicl surfce xis V, see figure. In this wy, the S surfce s chrcteristiclly curve, in the comosed motion, don t deend to the motion comonent in which the surfce is uto-generted, being fulfilled the identity: NS t, (4) (the norml t the S surfce is lwys erendiculrly on the own genertrix), nd the condition to determine the chrcteristiclly curve, in the helicl motion V,, will deend only to the revolution round xis. Fig.. Helicl movements decomosition method. Reference systems So, the chrcteristiclly curve of the S cylindricl surfce is defined s the rojection of the xis on the S surfce. he rojection will be the geometric locus on

4 the S surfce where the normls t this intersect the xis. hey re defined, see figure : - xis, in the x y z reference system, cos j sin k (5) - the norml t S surfce, in form N N i N j N k, (6) S x y z with N x, N, y N z directrix rmeters of the norml t S surfce, roximted by n Bezier olynomil, x y z NS i j k x y z (7) i j k t t t - the r vector, r i r (8) where r is the current oint vector on the surfce reresented in the discreetly form, surfce S. From (), result the coordintes trnsformtion in the reference system x y z, x R rs y (9) z leding t the S in the x y z reference system: x B Rrs y B z t, () with R rs itch rdius of the ger hub. he rmeter vlue is determined from condition tht the helix on the cylinder with R rs rdius to be rllel with the rck ger genertrix cylindricl flnk, see figure 4: tn () Rrs Rrs with helicl rmeter of the rimry eriherl surfce of ger hub. In this wy, the condition to determine the chrcteristiclly curve become:, N, r. () S In rincile, the () eqution reresent deendency q, t, with between nd t vrible rmeters:. he () nd () equtions ssembly reresent geometric locus on the S surfce with significnce of the S surfce s chrcteristiclly curve. he coule of rmetricl vlues nd t for which is stisfied the () condition by relcing the rck-ger flnk determine the mtrix Z S i Z i () i n Z n n reresenting the S chrcteristiclly curve s coordintes. Fig. 4. Unfold of the helicl line Is mde the coordinte trnsformtion from Z reference system to system Z with xis overled with the ger hub tool s xis, see figure : (4) so, in the helicl movement j, (5) i of the s curve it rrive t form:,, Z,,, Z, Z Z,, Z,, (6) reresenting the ger hub s rimry eriherl surfce equtions. ssociting with surfce the condition Z, (7) is obtined the xil section of the ger hub, in rincile, in form:,,, Z,, Z, with corresonding to the xil section Z. 5 Numericl exmles (8) 5. Shft with squre crossing section s first exmle is roosed shft with squre crossing section, see figure 5.

5 Fig. 5. Shft s crossing section nd reference systems he surfce to be generted hs equtions: u Z t, (9) with side of squre, u nd t vribles. he surfce equtions in the reference system of the rck-ger tools will be: cos sin Rr sin cos u Rr, () t or cos u sin Rr sin u cos Rr () t. In figure 6, is resented screenshot of the softwre used to rofile the ger hub lso the chrcteristiclly curve nd xil section re resented there. In tble, re resented the ger hub rofile coordintes clculted for the tool which rofile the shft with dimensions: R r = mm = 4.4 mm R rs =4 mm e =47. mm (helicl itch), versus the results obtined by using n nlyticl method (Gohmn [4]), for the sme surfce. he roximte Bezier olynomil for the surfce s genertrix (), is rd degree olynomil. he error level of the ger hub is mm regrding the theoreticlly rofile. Fig. 6. Softwre for ger hub rofiling xil section nd chrcteristiclly curve ble. Ger hub rofile coordintes roximted Rel rofile [mm] Error rofile [mm] [mm], 48,79, 48,79,, 48,769, 48,78,, 48,9 7,775 48,5 7,775,6, ,68 47,855 8,67, ,5 47,754 9,49, 45,77 6,4 45,7 6,49,5,666 44,994 6,85 44,99 6,86, 44,589 7,596 44,589 7,598, 4,97,999 4,95,7,, 4,,967 4,4,948,6 5. hin wheel In following, is resented nother exmle of the roosed lgorithm for the helicl surfce reciroclly enwring with chin wheel. he dimensionl elements of the wheel re: w =5 mm (wheel itch) dividing rdius, R rs =6.77 mm z=5 teeth, [-5 ] B [-6.5.6] [ ] R =.5 mm R =7.5 mm R rs =4 mm =.84 rd helicl rmeter =6.87 mm. he wheel s rofile is resented in figure 7, nd the ger hub coordintes re given in tble nd 4. he error level regrding the tool s rofile determined by

6 nlyticl methods is on level [mm]. Fig. 7. hin wheel s rofile ble. Ger hub rofile for B rc roximted heoreticlly Error rofile [mm] rofile [mm] [mm], 9,69,4 9,69,9,5 7,7,4 7,698,9,4, 7,5,496 7,5,499, 6,4 5,4 6,4 5,47,,666 5,976 5,58 5,975 5,55,, 5,4 7,96 5,5 7,94, In figure 8, is resented screenshot of the softwre used for ger hub rofiling 4 onclusion he rofiling method of the ger hub which genertes n ordered rofiles whirl is bsed on the rinciles of the helicl motion decomosition. he method use the Bezier roximtion olynomils for the rck-ger reciroclly enwring with rofiles whirl nd is rigorous enough, for usully rofiles, s is roved by the resented numericl exmles. he method hs generl chrcter nd the method recision my be incresed by the incresing the roximte Bezier olynomils degree, for chrcteristiclly curves roximtion. For the olynomils with reduced degree re defined re-clculted forms of the coefficients. Fig. 8. Softwre screenshot xil section nd chrcteristiclly curve ble. Ger hub rofile for B rc roximted heoreticlly Error rofile [mm] rofile [mm] [mm], 4,, 4,,, 4,884,84 4,885,85,, 4,856,94 4,857,96, 4,84,97 4,84,97, 4,47,74 4,46,74,,666 4,97,769 4,97,769, 4,,84 4,4,84, 9,89,98 9,89,99,, 9,7,4 9,75,, cknowledgement he uthors grtefully cknowledge the finncil suort of the Romnin Ministry of Eduction, Reserch nd Innovtion through grnt PN_II_I_79/8. References: [] Bll, R.S., retise on the heory of Screws, mbridge University Press, mbridge, 99 [] Lukshin, V.S., heory of Screw Surfces in utting ool esign, Mshinstroyenie, Moscov, 968 [] Rdzevich, S.P., Kinemtic Geometry of Surfce Mchining, R Press, London, ISBN , 8 [4] Litvin, F.. heory of Gering, Reference Publiction,, NS, Scientific nd echnicl Informtion ivision, Wshington,., 989 [5] nce, N., Surfce genertion through Winding, Glţi, 4, ISBN [6] eodor, V., im, M., nce, N., Profilre sculelor rin metode nlitice, Glţi, 5, ISBN

KINEMATICS OF RIGID BODIES

KINEMATICS OF RIGID BODIES KINEMTICS OF RIGID ODIES Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. Description