Measurement of directly designed gears with symmetric and asymmetric teeth

Transcription

1 Mesurement of irectly esigne gers with symmetric n symmetric teeth Dr. Alexner L. Kelevich, AKGers, LLC, Shoreview, MN, USA Toic: Mnufcturing ABSTRACT In comrison with the tritionl ger esign roch se on reselecte, tyiclly stnr generting rck rmeters, the Direct Ger Design metho rovies certin vntges for custom high-erformnce ger rives tht inclue: increse lo ccity, efficiency n lifetime; reuce size n weight, noise n virtions, cost, etc. However, mnufcturing of such irectly esigne gers requires not only custom tooling, ut lso customiztion of the ger mesurement methoology. This er resents efinitions of min insection imensions n rmeters for irectly esigne sur n helicl, externl n internl gers with symmetric n symmetric teeth. MEASUREMENT OVER (BETWEEN) BALLS OR PINS Sur gers Direct Ger Design metho [1,2] resents the ger tooth y two involutes of two se circles with the ngulr istnce etween them n tooth ti circle rc (Fig.1). The eqully sce n teeth form the ger. The fillet etween teeth is esigne ineenently, roviing minimum ening stress concentrtion n sufficient clernce with the mting tooth ti in mesh. If the two se circles re ienticl, the ger teeth re symmetric; in they re ifferent, the teeth re symmetric. Fig 1. Ger tooth rofile; externl ger; internl ger; tooth ti circle imeter; se circle imeter; f form circle imeter; reference circle imeter; S circulr tooth thickness t the reference imeter; involute rofile (or ressure) ngle t the reference imeter; - involute

2 intersection rofile ngle; n numer of teeth; suscrits n c re for the rive n cost flnks of the symmetric tooth. Mesurement over (etween) lls or ins for sur gers is efine se on the given: - numer of teeth n; - reference circle imeter ; - involute rofile ngles t the reference imeter n c ; for symmetric gers involute rofile ngle t the reference imeter = c. - circulr tooth thickness t the reference imeter S; - ger tooth ti imeter. Initilly selecte ll or in imeter D cn e juste se on the clcultion results. Reltion etween ngles n c, n n c is where: cos cos, (1) cosc cosc c cos n cos. Angles n c re efine from equtions: for externl ger for internl ger where: inv( x) tn( x) x c c 2 S inv( c c), (2) S inv( c c) 2 ( ), (3) n is involute function n x is involute rofile ngle in rins. The centers of the ll or the in re locte on the imeter (see Fig. 2), which is where the ngles n c re efine y equtions [3]: for externl ger for internl ger c cos cos, (4) D D 2 inv( c c), (5) n D D inv( c c). (6) The ll or in touches the ger tooth in the oints T n T c. They shoul e lwys locte on the involute flnks. This conition is escrie y the following eqution: for externl gers n rccos f c c t rccos (7) c

3 for internl gers n rccos c fc rccos rccos c c tc rccos ; (8) t rccos (9) f c tc rccos. (10) fc Fig 2. Bll or in osition; externl ger; internl ger; D ll or in imeter; P center of the ll or in; n c involute rofile ngles t the center of the ll or in; ll or in center loction imeter; T n T c contct oints of the ll or in with the tooth rive n cost tooth flnks; t n tc involute rofile ngles t the contct oints. The mesurement over two lls or ins for the externl ger is for even numer of teeth (Fig. 3) for o numer of teeth (Fig. 3) M D ; (11) M cos D. (12) 2 n

4 Fig 3. Mesurement over lls or ins for externl gers; even numer of teeth; o numer of teeth The mesurement etween two lls or ins for the internl ger is for even numer of teeth (Fig. 4) for o numer of teeth (Fig. 4) M D ; (13) M cos D. (14) 2 n Fig 4. Mesurement etween lls or ins for internl gers; even numer of teeth; o numer of teeth

5 For insection convenience the mesurement over lls or ins for externl gers shoul e M n the mesurement etween lls or ins for internl gers shoul e M. These n conitions (7-10) efine the ll or in imeter. Helicl gers Mesurement over (etween) lls or over ins for helicl gers is efine se on the given: - numer of teeth n; - reference circle imeter ; - norml involute rofile ngles t the reference imeter n n nc ; for symmetric gers n n = nc. - norml circulr tooth thickness t the reference imeter S n ; - helix ngle t the reference imeter ; - ger tooth ti imeter. Cylinricl ins cnnot e use to mesure the internl helicl gers, ecuse the in surfce cnnot e tngent to the internl helicl ger flnks. The trnsverse tooth thickness t the reference imeter S is S Sn / cos. (14) The trnsverse involute rofile ngles t the reference imeter n c re c tnn rctn cos tnnc rctn cos, (15). (16) The helix ngles t the rive n cost se imeters n c re rctn(tn cos ), (17) rctn(tn cos ). (18) c The centers of the ll or the in (for externl ger with even numer of teeth) re locte on the imeter tht efine y the eqution (4), where the ngles n c re efine y: for externl helicl ger D D 2 inv( ) inv( c c) cos cos n c c c, (19) for internl helicl ger (for mesurement over lls) D inv( c c) cos D cos c c. (20) The ll or in imeters shoul lso stisfy equtions (7 10). Mesurements over two lls for externl helicl gers (Fig. 5) n etween two lls for internl helicl gers (Fig. 6) re efine y equtions (11), (12), n (13), (14) ccoringly.

6 Fig. 5. Mesurement over lls of the externl helicl ger Fig. 6. Mesurement etween lls of the internl helicl ger Mesurement over two ins for externl helicl gers with even numer of teeth is lso efine y eqution (11). For externl helicl gers with o numer of teeth, the shortest istnce L etween the in centers oes not ly in the trnsverse section of the circle imeter. This istnce n mesurement over two ins for externl helicl gers with o numer of teeth efinition is escrie in [4]. The trnsverse istnce Lt etween the ll centers, in cse of the o numer of teeth, is lwys greter thn the istnce L tht is (see Fig.7) L 4 (tn cos( )) 2 tn 2n 2 where the helix ngle t the in center imeter is n the ngle is solution of the eqution 2 2, (21) rctn( tn ) (22)

7 sin( ) 0 tn n. (23) Fig. 7. Definition of the istnce etween the in centers for the helicl gers with o numer of teeth Then the mesurement over two ins for externl helicl gers with o numer of teeth (Fig. 8) is M L D. (24) Fig. 8. Mesurement over ins of the externl helicl ger with o numer of teeth SPAN MEASUREMENT

8 Sn mesurement is the mesurement of the istnce cross severl teeth, long line tngent to the se cyliner [5]. This kin of insection is use for gers with externl teeth. It is lso lie only for gers with symmetric teeth, ecuse it is imossile to hve common tngent line to two concentric se cyliners of symmetric tooth flnks.. Fig. 9. Sn mesurement; for sur ger; for helicl ger Sn mesurement over n w teeth (Fig. 9) is where S is the tooth thickness t the se imeter is the circulr itch t the se imeter W ( S ( n 1) ) cos, (25) w S S cos inv( ), (26) n, (27) n w is numer of teeth for sn mesurement n wmx is mximum numer of teeth n 2 nw nwmx, (28) S 2 2 wmx. (29)

9 Cliers, micrometers, or secil gges re use for sn mesurement. CMM GEAR INSPECTION Fig.10. CMM mesurement of symmetric ger CMM ger insection (Fig. 10) llows ming the whole surfce of ll teeth incluing the fillet rofiles. However, it is tyiclly use to control the involute ccurcy. Although the ger tooth fillet is n re of mximum ening stress concentrtion, its rofile n ccurcy re mrginlly efine on the ger rwing y tyiclly very generous root imeter tolernce n, in some cses, y the minimum fillet rius. Direct Ger Design metho otimises the ger tooth fillet rofile for minimum ening stress concentrtion [6]. For such criticl liction gers the tooth fillet rofile must e cler secifie, tolernce, n then insecte. The whole tooth (incluing the fillet) CAD rofile t the verge mteril conition resente s the B-sline or the tngent rcs ccomnies the ger rwing for the CMM insection. The t set lso inclues the involute flnk n fillet rofile tolernces tht re estlishe y the esigner eening on the ger ccurcy n lso the mnufcturing technology. The CMM is rogrmme to inicte if the insecte tooth rofile oints ly within the corrior efine y the CAD tooth rofile +/- rofile tolernce. Similr insection technique is use to insect curve surfces, for exmle, of the irfoil ir comressor or gs turine les. SUMMARY AND CONCLUSION This er hs covere the mesurement secifics of the symmetric n symmetric gers tht re esigne using the Direct Ger Design metho. - Define the mesurement over (or etween) lls n ins for externl n internl gers; - Define the sn mesurement for externl with symmetric teeth; - Descrie some CMM insection issues for irectly esigne gers. Presente mterils shoul e helful for mnufcturing custom gers with symmetric n symmetric teeth. REFERENCES

10 1. A. L. Kelevich, Geometry n esign of involute sur gers with symmetric teeth, Mechnism n Mchine Theory, 35 (2000), A. L. Kelevich, R. E. Kleiss, Direct Ger Design for Sur n Helicl Gers, Ger Technology, Setemer/Octoer 2002, A. L. Kelevich, Mesurement over ins of the gers with symmetric teeth (in Russin), Mshinoveenie, (6) (1986) I. P. Nezhurin, Clcultion of the mesurement over ins of the helicl gers with o numer of teeth, Vestnik Mshinostroeniy, (2) 1961, (in Russin). 5. Stnr ANSI/AGMA 1012-F90, Ger Nomenclture, Definitions of Terms with Symols. 6. A. L. Kelevich, Y. V. Shekhtmn, Tooth Fillet Profile Otimiztion for Gers with Symmetric n Asymmetric Teeth, AGMA Fll Technicl Meeting, Sn Antonio, Texs, Octoer 12-14, 2008, (08FTM06), ulishe in the Ger Technology, Setemer/Octoer 2009,