Numerical Simulation of Stresses in Tin-rimmed Spur Gears wit Keyway B. Brůžek, E. Leidic Tis paper contains an investigation of te key on a stress distribution in a tin-rimmed spur gear. A stress analysis was carried out by means of te Finite Element Metod (FEM). Te 2D-FE analysis as elped to find te influence of turning te gearing towards te keyway on te stress in te loaded root of te toot and in te keyway. 2D and 3D numerical analysis as been used to find mutual influence of every single notc (root of toot and keyway), influence of tickness of te ub, lengt of te key and te form of loading. Verification as been carried out troug eperimental metod. Keywords: gear, keyway, rim tickness. Introduction Tese days tere is no generally accepted calculation for design of tin-rimmed asymmetric loaded ub wit gearing wicwouldtakeintoaccountbot impactoftenotcof root of te toot and impact of te notc of te keyway. Terefore a designer prefers ticker wall of ub in te critical places, tat means over dimensioning. Growing competition and increasing costs demand te measures wic suppose to optimise constructional parts. However, tere are still reserves in te ub wit te keyway. Tin-rimmed ub wit gearing is used in many driving macine sets as a pinion in transmission, couplers and as a part of te braking mecanisms. Saft-key-ub system is used very often, because of simple assembly and disassembly. During te testing of tooted weels it was discovered tat in te area of keyway crack initiation frequently occurs (Fig. ). Te cause are te notc stresses produced in te root of te toot andintekeyway. course of a crack F 2 Dimension assignation of te ub wit gearing Geometrical dimensions of te saft-key-ub system are standardized in DIN 6885 [2] and DIN 6892 [3]. Fig. 2 sows a valid designation according to tese standards and fundamental geometrical dimensions for a description of te gearing as well. Tese are standardized according to DIN 399 []. Important parameter for an analysis is minimum wall tickness s k. It is defined as te sortest distance between te fillet of te keyway r 2 and te dedendum circle d f of te gearing, counted by equation [4]: d 2 f d b sk w t 2 r 2 2 2 2 2 2. () For a description of te loaded root of te toot towards to tekeywayofteubwasdefinedanangle. Teangle means, tat te toot Z 4 lies symmetrical over te keyway of te ub. For eac of te investigations was used A Form of key (rounded ends). Eiter tere were te key and te ub flused or te load bearing lengt of te key corresponded to ub widt. First case occurs often in te practice, te second is important to a comparison of te results of 2D-FE-analysis wit eperimental acquired results. 3 Design of te FE-Model, boundary conditions Fig. : Pinion wit saft and key Geometric dimensions used in numerical calculations are introduced in Table. For absolute description of all aspects of investigated parameters sould be designed a 3D-FE-Model of te saft- -key-ub system. For te reason tat a evaluation of te 3D-Model is difficult and time demanding, most of te calculations were carried out wit 2D-FE-Model (plane stress). Tere will be sown a sufficient agreement of te results of te 2D- and 3D-Models on te one geometrical variant. Czec Tecnical University Publising House ttp://ctn.cvut.cz/ap/ 47
d w Diameter of te saft d Pitc circle d f Dedendum circle d f Dedendum circle Heigt of te key d a Addendum circle b Widt of te key d b Base circle t Dept of te keyway in saft Radius of te root of te toot t tr Load bearing dept of te keyway in saft s k Minimum tickness of te ub t 2 Dept of te keyway in ub Position from toot Z 4 to middle of te keyway t 2tr Load bearing dept of te keyway in ub s Camfer or radius of te keyway in saft s 2 Camfer or radius of te keyway in ub r Camfer or radius of te key r 2 Fillet of te keyway Fig. 2: Geometric dimensions of te saft-key-ub system (left) and te fundamental geometrical dimensions of te gearing Table : Geometric dimensions Calculation variants Module m 2mm 2.5mm 4mm Number of teet z 29 24 7 Diameter of te saft d w 4 mm Widt of te ub b n 4 mm Cross section of te key DIN 6885 A 2 8 mm mm Fillet of te keyway r 2, mm Pitc circle d 58. mm 6 mm 68. mm Minimum tickness of te ub (Eq. ) s k 7 m.6 m 6 m Toot root radius of te of te reference profile fp.75 mm.94 mm.5 mm For an eact description of te investigated variant is necessary to model all components wic are in mutual contact (saft, key and ub). Because of a reduction of a computational time tere were modeled only eigt teet along circumference in region of te keyway and tree teet lying opposite (as non-attenuate teet, tat means te teet aren t influenced by te keyway). Te oter teet ave no influence on investigated problem. On te oter and it would be insufficient to model only a loaded toot, since adjacent teet influence stiffness of te researced zone. 48 Czec Tecnical University Publising House ttp://ctn.cvut.cz/ap/
. Z 6... Z Z 7 Z8 u =u y = for saft y Fig. 3: 2D-FE-Model of te saft-key-ub system wit boundary conditions; normal force (eample for toot Z 3 ) Tereisdisplayed2D-Modelandteboundaryconditions in Fig. 3. 4 2D-FE-Model At first te most loaded geometry was detected. Te maimum stress was calculated in te root of te toot and in te keyway for various geometrical variants. Net te eventuality of a recrimination of bot opposite notces (root of toot and keyway) was investigated. Fig. 4 (left) sows te influence of te angle on te maimum loading in te root of te toot ( ma )alwaysforte loaded toot and te module 2 mm, tat means te influence of te position of toot Z 4 on te symmetry plane (middle of te keyway). Resulting from te same calculations tere is corresponding stress in te keyway displayed (Fig. 4 rigt). It is evident from te course of stress, as for te root of toot tat te maimum stress is at te angle 2.5 and by loading of te toot Z 3. By furter turn and consecutive loading of te toot Z 4 te stress in te root of te toot decreases again and goes even under basic stress []. Te angle as only a small influence upon te place of te maimum stress in te root of te toot. However, tere is anoter situation by te loading of te keyway. During te sequential loading of te toot wit normal force rise two maimums of stress on different places in dependence on angle. Fig. 4 also sows tat tere are always iger stresses in te root of te toot. From wence it follows tat te root of te toot is ever a critical location and not te keyway. It is necessary to confirm tis assumption by eperiment. It will focus mainly on te root of te toot by furter calculations. It is possible to eplain te stress distribution bot in te root of te toot and in te keyway from te deformation of a ub wit eigt teet in te region of te keyway under rotary normal force loading. In Fig. 5 is a deformation displayed by s ma / s F.3....9 j = j = 2.5 j =3,7 j =6 j =9 Z Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 8 loaded toot Z 4.... j.. Z Z 8 s ma / s F j = j = 2.5 j =3 j =6 j =9 Z Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 8 loaded toot Fig. 4: Cange of te maimum first principal stress in te loaded root of te toot (left) and in te keyway (rigt) by rotating loaded toot and various angles for te ub wit module 2 mm; F caracteristic value of stress Czec Tecnical University Publising House ttp://ctn.cvut.cz/ap/ 49
a) b) d l d r d l d r loaded toot Z 3 d l < 9 (compression) d r < 9 (compression) loaded toot Z 4 d l < 9 (compression) d r < 9 (compression) c) d) d l d r d l d r loaded toot Z 5 d l > 9 (etension) d r > 9 (etension) loaded toot Z 8 d l» 9 d r» 9 Fig. 5: Deformation of te ub wit teet in te region of te keyway loading of te tird, fourt, fift and eigt toot. It is obvious, tat te maimum stress in te root of te toot is always on te loaded toot. However, tere is quite different situation in te region of te keyway. Te location of maimum stress varies. Te angle between te bottom and te side of te keyway is smaller tan 9 by te loading of te teet Z 3 and Z 4 and te maimum stress lies on te bottom of te keyway. By te loading of te toot Z 5 te angle between te bottom and te side of te keyway is bigger tan 9 and te maimum stress is in a fillet of te keyway. By te loading of te toot Z 8 te angle between te bottom and te side of te keyway is just about constant and tere is no significant maimum stress in te region of te keyway. 5 3D-FE-Model Te 3D-calculations were carried out only for te critical case, tat means for = 2,5 by te loading of toot Z 3.To save a computational time te number of elements in unloaded zones was more reduced in comparison wit te 2D-Model. Te basic boundary conditions for te 3D-Model are sown in Fig. 6. By te double-sided torque distribution bot ends of te saft are steadily supported. By te one- -sided torque distribution te left end of te saft is steadily supported and te rigt end of te saft is supported only in a radial direction. 6 Double-sided torque distribution Fig. 7 is determined to compare te 2D and 3D calculations. In te picture it is evident, tat calculated stresses for 2D and 3D-Model are almost identical. Terefore it is possible to use 2D-Model for symmetric boundary conditions (double-sided torque distribution). Te first principal stress was evaluated in loaded area of te root of te toot, tat means between te toot Z 2 and te toot Z 3, and in te keyway. Fig. 8 sows evaluated areas and te stress distribution for double-sided torque distribution. u r =u j =u z = double-sided u r =u j =u z = one-sided u r =u j =u z = u r = z Fig. 6: Boundary conditions for 3D-Model, 2.5 ; Modul 2 mm; normal force (toot Z 3 ) 5 Czec Tecnical University Publising House ttp://ctn.cvut.cz/ap/
s / s FO Z2 Z 3 Z4-2D 3D 3D 2D s / s FO - - - 6. 2D 3D - 8... 2. 4. 6 8.. Fig. 7: Comparison of te results of te 2D and 3D-Model for module 2 mm, 2.5 andloadontootz 3. Left: stress distributions in loaded toot radius; rigt: stress distribution in keyway; F caracteristic value of stress. 7 One-sided torque distribution In te Fig. 9 tere is displayed an analog evaluation of te stress in te loaded root of te toot and in te keyway for one-sided torque distribution. In te Fig. 9 is possible to identify only small influence of te asymmetrical loading. Te result is, tat te difference between double-sided and one- -sided distribution of te torque is imponderable. Tis presumption is confirmed by eperimental analysis. Z 2 Z 3 Z 4 Z 2 Z 3 Z 4 Z5 Z 6 y z z y s [MPa] s [MPa] 8 6 4 2-2 5 5 2 3 4,2,4,6,8 8 6 4 2-2 8 6 4 2-2 -4-6 -8 5 5 2 3 4,8,6 8 6 4 2-2 -4-6 -8,2,4 Fig. 8: Stress distribution in te loaded root of te toot (left) and in te keyway (rigt) for double-sided torque distribution; 2.5 ; module 2 mm; normal force on toot Z 3 ;, : seefig.7. Czec Tecnical University Publising House ttp://ctn.cvut.cz/ap/ 5
s [MPa] s [MPa] 8 6 4 2-2 5 5 2 3 4,2,4,6,8 8 6 4 2-2 8 6 4 2-2 -4-6 -8 5 5 2 3 4,8,6,2,4 8 6 4 2-2 -4-6 -8 Fig. 9: Stress distribution in te loaded root of te toot (left) and in te keyway (rigt) for one-sided torque distribution; 2.5 ; module 2 mm; normal force on toot Z 3 ;, : seefig.7 8 Lengt of te key Intepracticetevariantwitteflusubandkeyoften occurs. Boundary conditions and force application are displayed in Fig. 6, te same as for an overang key. Te evaluation is analogue as well. Te first principal stress was evaluated in te loaded root of te toot and in te keyway. Stress distributions is sown in Fig.. From te stress course in te keyway te influence of te lengt of te key is obvious. Te stresses in te area of te s [MPa] keyway decrease on bot ends of te ub, owever tey are bigger due to sorter supporting lengt of te key tan by te overang key. On te oter and stress distribution in te root of te toot (Fig. left) is similar to tat one at te overang key (Fig. 8 left). Since te first principal stress is in te root of te toot bigger tan in te keyway, terefore te lengt of te key as no influence on a crack initiation for te ub wit module 2 mm. Te crack initiation is always in te root of te toot. Tis assumption was also successfully verified by te eperimental metod. s [MPa] 8 6 4 2-2 5 5 2 3 4,2,4,6,8 8 6 4 2-2 8 6 4 2-2 -4-6 -8 5 5 2 3 4,8,6,2,4 8 6 4 2-2 -4-6 -8 Fig. : Stress distribution in te loaded root of te toot (left) and in te keyway (rigt) for double-sided torque distribution; 2.5 ; module 2 mm; normal force on toot Z 3 ; ub and key flused 52 Czec Tecnical University Publising House ttp://ctn.cvut.cz/ap/
9 Numerical investigated influences Te influence of te keyway on stress increase in te root of te toot is mainly due to small tickness of te ub s k (see Fig. 2). Te stress distribution was evaluated in te root of te toot and in te keyway for various ticknesses of te ub s k always for te critical geometry of te ub wit module 2 mm. Fig. sows eemplary maimum values of te first principal stress in te root of te toot and in te keyway depending on dimensionless tickness of te ub. From te picture it is evident tat from definite ub tickness te keyway as no influence on te stress in gearing. For assessment of strengt beaviour of tin rimmed spur gears wit keyway is important to consider interaction bot notces (root of te toot and fillet of te keyway) wic lie close togeter. From te calculations in [5] results tat for various fillets of te keyway te stress distribution in te root of te toot stays identical. As well as for various radiuses of te toot root te stress distribution in te keyway stays similar. Tat means tat by te used minimum tickness of te ub (see Table ) bot notces do not influence eac oter. s ma / s FO.8.6 s k /m root of toot keyway Fig. : Maimum first principal stress in te loaded root of te toot and in te keyway depending on dimensionless tickness of te ub wit module 2 mm; F caracteristic value of stress Beyond calculations for te ub wit module 2 mm was also counted wit modules 2.5 mm and 4 mm. For eac module te influence of te angle on te maimum loading in te root of te toot and in te keyway was investigated. s ma / s FO.8.6 root of toot keyway Modul [mm] Fig. 2 represents maimum values of te first principal stress depending on module, always for critical angle in te loaded root of te toot and in te keyway for te constant minimum tickness of te ub s k. From te picture is evident tat until definite module te maimum first principal stress in te root of te toot is bigger tan in te keyway, wile for te greater module te maimum loaded place moves towards te keyway. Depending on te stress gradient in bot notces te initiation of te crack moves accordingly, also from te root of te toot towards te keyway. Conclusion In tis paper te stress distribution in te tin-rimmed spur gears wit te keyway is investigated. Te results sow tat not only te ub tickness between te root of te toot and te keyway as te big influence on te stress increase but also te position of te gearing towards te keyway. On te oter and te form of te loading (one-sided and double-sided torque distribution) and te lengt of te key (ub andkeyflusedandoverangkey)aveonlysmallinfluence. From te calculations it is also evident tat for small module te maimum first principal stress is bigger in te root of te toot tan in te keyway wile for te greater module te maimum loaded place moves towards te keyway. Verification of te numerical calculations as been carried out troug te eperimental metod. References [] DIN 399, Tragfäigkeitsberecnung von Stirnrädern. Beut Verlag, 987. [2] DIN 6885, Mitnemerverbindungen one Anzug; Passfedern, Nuten, Hoe Form; Abmessungen und Anwendung. Beut Verlag, 968. [3] DIN 6892, Mitnemerverbindungen one Anzug Passfedern Berecnung und Gestaltung. Beut, 998. [4] Floer, M.: Beansprucungsanalyse an torsionsbelasteten Passfedernabe. Aacen: Saker, 2. [5] Leidic, E., Brůžek, B.: Verzante dünnwandige Naben mit Passfederverbindung. AiF - Absclussberict, 22. Dipl.Ing.BoumilBrůžek pone: +493 75 34 572 fa: +493 75 34 56 e-mail: boumil.bruzek@mb.tu-cemnitz.de Prof. Dr. Ing. Erard Leidic pone: +493 75 34 66 fa: +493 75 34 56 e-mail: erard.leidic@mb.tu-cemnitz.de Department of Engineering Design Cemnitz University of Tecnology Reicenainer Straße 7 926 Cemnitz, Germany Fig. 2: Maimum first principal stress in te loaded root of te toot and in te keyway depending on module always for te critical position of te keyway; F caracteristic value of stress Czec Tecnical University Publising House ttp://ctn.cvut.cz/ap/ 53