ESTIMATION OF MACHINING ERRORS ON GLEASON BEVEL

Transcription

1 5 th INTERNATIONAL MEETING OF THE CARPATHIAN REGION SPECIALISTS IN THE FIELD OF GEARS ESTIMATION OF MACHINING ERRORS ON GLEASON BEVEL GEAR CUTTING BOB, Daila UNIO SA Satu Mare - 35, Luia Blaga Blvd, 39 Satu Mare, Romaia daila.bob@uio.ro Abstrat: I order to alulate the mahiig errors it is eessary to develope a mathematial model for the alulatio of mahiig errors. The gears ut by Gleaso gear uttig method has measured with CMM. The results of oordiate easuremets must be trasformed ito deviatios of the real surfae represeted i the diretio of the surfae ormal. Key words: mahiig error model, bevel gear uttig, measurig o CMM. INTRODUCTION Gleaso gear uttig mahie has bee very popular i the bevel gear maufaturers. Eve the mahies settigs for obtaiig desirable tooth bearig gears are give by Gleaso s istrutio sheets, a gear pair with a desirable tooth bearig is diffiult to obtai at the first ut beause of the iheret errors. The tehologial aspets of the problems are as follows[]: a) The deviatios of the real toothe surfaes are ievitable due to the surfae distortio by heat treatmet, errors of iitial mahie tool setigs, defletio by maufaturig ad so o. b) Apliatio of additioal fiishig operatio for elimiatio of the deviatios would be too expesive i omparisio with the approah based o orretios of iitially applied mahie-tool settigs.the advatage of this approah is the possibility of usig the same equipmet to orret the the

2 deviatios. The disadvatage is that the approah will be suesful oly if the deviatios are repetable. ) The oordiate maesuremets must be performed with high preisio, wih urretly prohibits them from beig performed simultaeously with the maufaturig. Therefore, the oordiate measuremets are performed after maufaturig, but oly the first gear of the whole gear set to be maufatured is tested. d) I some ases, master-gears are used ad the oordiate measuremets provide the iformatios about deviatios from the master-surfae for the surfae beig tested.this approah is osidered less efetiveas ompared to omputerised determiatio of surfae deviatios ad orretios of mahie-tool settigs. Miimizig the deviatios of real tooth surfaes results i a redutio i the level of trasmissio errors that ause gear oise ad vibratio. 2. MATHEMATICAL MODEL OF THE MACHINING ERORS The piio ad gear tooth surfae a be expressed matematially whe the geeratio proess is desribed aordig to the kiemati theory. I ase of the Gleaso gear uttig method, the tooth surfae of the gear ad of the piio is able to see the trasriptio of the tool surfaes. That meas that the umerial formula of the tooth surfae a be expressed by the umerial formula of the tool surfaes. Geeratig surfaes ad oordiate systems [,2.3] Fig. shows a head utter that is used for the geeratio of Gleaso s spiral bevel gears. This tool is provided with blades havig straight-lied profiles. These profiles beig rotated about C-C form two oes that ut both sides of the tooth. Thus the geeratig surfae is a oe surfae. I the proess of geeratio the followigs motios are performed (Fig.2): () a rotatioal motio of the head utter about axis C-C, that provide the desired veloity of uttig ad a rotatioal motio of the head utter about axis O-a g while the gear to br geerated rotates about axis O-a i. This geeratio of gear-tooth surfaes is based o appliatio of two tool surfaes, Σ f ad Σ p, whih geerate gears ad 2, respetively. The (f) geeratig surfaes (geeratig oes do ot oiide; they have differet oe agled, Ψ

3 ad Ψ (p) ad differet mea radii, r (f) ad r (p) (Fig.)). Speial mahie-tool settigs, ΔE ad ΔL must be used for the geeratio of the piio. ] O m, ( i=,2 ) ;( =P,F ) a.) b.) Fig.. Coordiate system [2] Cradle Cutter Blades Blades Cradle Gear Cutter Gear Fig.2 Cutter axis. Geeratig of the gear [2].

4 Cosiderig the geeratio of the gear 2 tooth surfae we use the followigs oordiate systems: () S (p) whih is rigidly oeted to the geeratig surfae Σ p (Fig..b). The fixed oordiate system S m, that is rigidly oeted to the frame of the uttig mahie (3) The oordiate system S 2, whih is rigidly oeted to gear 2 (Fig.3.b) x f x m O m O f O f Ω (P) Ω (P) M Δ2 γ 2 z f z m O C (P), O m φ P z m z C (P) Gear 2 Ω z 2 y C (P) y m a) b) Fig.3. Coordiates of the gear2 I the proess of geeratio, the geeratig surfae rotatea about the X m axis with agular veloity Ω (P), while the gear blak rotates about the Z 2 axis with the agular veloity Ω. Axes X m ad Z 2 iterset eah other ad form the agle 9 + ϒ 2 - Δ 2, where Δ 2 is the dededum agle for gear 2. Axis X m is perpediular to the geeratrix of the root oe of gear 2. The oordiate system S f show i Fig.3 is rigidly oeted to the housig of the gears ad will be used for aalysis of oditios of meshig of the gears. Cosiderig the geeratio of the piio, we use the followig oordiate systems: () S (F) C, that is rigidly oeted to the geeratig surfae Σ F S () m, that is rigidly oeted to the frame of the uttig mahie (3) S, that is rigidly oeted to the piio (gear ) (Fig. 4)

5 X h X m () Piio Z ΔL O h O m () Ω () Δ N γ Z m () O h O m () ΔΕ Z h ; Z φ P () Z m (F) Z C Z h ΔL osδ ΔL O h N = l Y C () Y m () Ω (F) Fig.4 Coordiate system of the piio Axes X m () şi Z do ot iterset but ross eah other; ΔE ad ΔL are the orretios of mahie-tool settigs that are used for the improvemet of the meshig of the gears. I the proess of geeratio, the geeratig surfae rotates about the X m () with agular veloity Ω (F), while the gear blak rotates about the axis Z with agular veloity Ω (). Axes X m () ad Z form the agle 9 -ϒ +Δ, where Δ is the dededum agle of gear ad axis X m () (Fig..a): is perpediular to the geeratrix of the root oe of gear. Geeratig tool surfae The tool surfae is a oe ad is represeted i the oordiate system S S () as follows xs ( ). r ( ) tg Ψ ( ) u. os Ψ ( ) y S ( ) zs ( ) = u.. si Ψ ( ) si θ ( ( = = F, F, P P ) ) () ( 3.. ) u.. si Ψ ( ) os θ where u ad θ are the surfae oordiates.. The oordiate system S () (= F,P) is a auxiliary oordiate system that is also rigidly oeted to the tool (Fig..b). To represet the geeratig surfae Σ F ad Σ P i

6 oordiate system S () we use the followig matrix equatio (a left-had geeratig gear is osidered): x ( ) x ( ) S x ( ) S y ( ) z ( ) = M. ( ) S y ( ) S z ( ) S = osq siq siq osq b. siq b. osq y ( ) S z ( ) S ( 3.2. ) Here b ad q are parameters that determie the loatio of the tool i oordiate system S () C. Euatios 3.. ad 3.2. yield: x ( ) y ( ) = r. ( ) tg Ψ ( ) u. os Ψ ( ) = u.. siψ ( ) si θ q. b siq ( 3.3 ) (3) z ( ) = u.. siψ ( ) os θ q. b osq (3.)(3 where = (F,P). The uit ormal to the geeratig surfae Σ (=F;P) is represeted by: ( ) = N ( ) N ( ), ude N ( ) = d r ( ) X d where r ( ) ( 3.4. ) (4) d θ du Usig euatios (3.3) şi (3.4.) ( provided u siψ C () ) we obtai: ( ) = siψ. ( ). i ( ) osψ ( ) si θ. q. ( ) os θ q k ( ) ((5) 3.5. ) Estimatio method of the mahiig errors The tooth surfae a be expressed mathematially as explaied i the previous hapter usig variable parameters u ad θ. This tooth surfae equatio, mahiig errors suh as uttig errors, wear of utter head, heat treatmet distortios, et. are iluded. Geerally the tooth surfae is represeted as follows, usig ostat mahie settigs, i order to elimiate the mahiig errors, values C to C respetively [4]:

7 X(u,θ; C +ΔC, C 2 +ΔC 2,, C +ΔC ). (6) where ΔC, ΔC 2,, ΔC idiates small differet value from the give value by summary of eah mahie settigs respetively: Next the tooth surfae seleted arbitrary is measured by CMM. The equatio (6) is represeted as follows by measured poit louds M o the tooth surfae: X M= X(u,θ; C, C 2,, C ) + Δ C X Δ + C2 2 X + + Δ C (7 ) By the traspositio of matrix X, followig is represeted: X M - X(u,θ; C, C 2,, C ) = Δ C X X Δ C + + Δ C As the differet values from summary, ΔC, ΔC 2,, ΔC are very small ad the relatioship of eah is idepedet ad liear eah other, the equatio (8) a be represeted as follows by the method of superpositio: X M X = Δ C X M X = Δ C2.. X M X = Δ C 2 (8) ΔC 2 = ΔC 3 = =ΔC = ΔC = ΔC 3 = =ΔC = (9) ΔC = ΔC 2 = =ΔC - = The differet value ΔC at tur a be alulated by eah measured poit data whih ostrut the louds M i the equatio (9). I pratie, however, eah differet value at tur is ot always equal, that is there is dispersio i the differet values. Therefore firstly, i the equatio about C i the equatio (9), alulate the ΔC whih miimize the sum of the square of eah residual ad stadard deviatio at the alulatio of tur. Lastly fid out the parameter C whih shows the smallest stadard deviatio amog eah parameter ad estimate the real mahie settig parameter as C k +ΔC k. I this ase, the theoretial tooth surfae X(u,θ; C, C 2,, C k +ΔC k,, C ) is the best fitted the measured poits louds M. Ad otiuig the same way o the remaiig - equatios, fid out the real mahie settig parameter whih shows the miimum stadard deviatio by utilizig the fouded real

8 mahie settig parameters before. Whe smaller stadard deviatio is ot foud, the remaiig parameters are estimated as the same value as i the summary. 3.MEASURING POINT DATA BY CMM The gear is set o the table of the oordiate measurig mahie (CMM) arbitrarily. The positio of the gear axis ad the datum plae must be determied by measuremet idepedet of the tooth surfae measuremet beause the gear is set arbitrarily [5]. The results of oordiate easuremets must be trasformed ito deviatios of the real surfae represeted i the diretio of the surfae ormal CONCLUSIONS Fig.5. Measurig sheme by CMM A mathematial model for estimatio of mahiig errors o Gleaso gear uttig was proposed. The gears have measured with CMM ad the deviatios of the real surfae a be determied. 5. REFERENCES. Litvi, F. L. Gear Geometry ad Applied Theory, PTR Pretie Hall 994, Eglewood Cliffs, New Jersey 2. Litvi, F. L. Theory of gearig, 989, NASA Referee Publiatio Litvi, F. L., Tsug W.-J., Coy, J.J., Heie, C. Method for geeratio of spiral bevel gears with ougate gear tooth surfaes, ASME J. Meh. Tras., Auto. Desig 987, 9, N2, pages Mihiwaki, H., Tamura, H., Kawasaki, K. Estimatio of real mahie settig o Helixform hypoid gear uttig, MPT2-Fukuoka The JSME Iteratioal Coferee o Motio ad Power Trasmissios, 5-7 Nov. 2, Pages ******** Brow&Sharpe, Quidos. Measuremet of Bevel Gears. Referee Maual, Wetzlar, Germay, 992