Stress analysis of a helical gear set with localized bearing contact

Transcription

1 Finite Elements in Analysis and Design 38 (00) Stress analysis o a helial gear set with loalized bearing ontat Yi-Cheng Chen, Chung-Biau Tsay Department o Mehanial Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinhu 30010, Taiwan, ROC Abstrat This study investigates the ontat stress and bending stress o a helial gear set with loalized bearing ontat, by means o nite element analysis (FEA). The proposed helial gear set omprises an involute pinion and a double rowned gear. Mathematial models o the omplete tooth geometry o the pinion and the gear have been derived based on the theory o gearing. Aordingly, a mesh-generation program was also developed or nite element stress analysis. The gear stress distribution is investigated using the ommerial FEA pakage, ABAQUS=Standard. Furthermore, several examples are presented to demonstrate the inuenes o the gear s design parameters and the ontat positions on the stress distribution.? 00 Elsevier Siene B.V. All rights reserved. Keywords: Finite element stress analysis; Modied helial gear; Von-Mises stress; Bending stress; Hertzian ontat stress 1. Introdution Helial gears are widely used in power transmission between parallel shats. Conventional parallelaxes helial gears with involute teeth are insensitive to enter-distane assembly errors and possess line ontats under an ideal assembly ondition. However, involute helial gears are very sensitive to axial misalignments, ausing disontinuous transmission errors (TE) and edge ontats, resulting in noise and vibration [1]. Thereore, the teeth o helial gears are usually modied to attain a loalized point ontat and to avoid edge ontats. Reently, Litvin [] proposed the onept o tooth surae modiation to obtain a pre-designed paraboli TE as well as a loalized bearing ontat o the gear set. This onept o tooth modiation has been applied to the generation o various kinds o gearing, suh as spur gears, helial gears and worm gear drives [3 6]. Corresponding author. Tel.: ; ax: address: btsay@.ntu.edu.tw (C.-B. Tsay) X/0/$ - see ront matter? 00 Elsevier Siene B.V. All rights reserved. PII: S X(01)

2 708 Y.-C. Chen, C.-B. Tsay / Finite Elements in Analysis and Design 38 (00) The ontat stress and llet stress on gears, whih are losely related to pitting ailure, bending ailure and the gear s servie lie, have attrated muh attention [7,8]. Nevertheless, the alulation ormulae or gears with speial prole modiations are rarely available in handbooks [9,10]. Thereore, nite element analysis (FEA), whih an involve ompliated tooth geometry, is now a popular and powerul analysis tool to determine tooth deetions and stress distributions. Many researhers have applied FEA to tooth deetion and stress distribution or various gear drives. Several researhers have analyzed line-ontat involute helial gears using three-dimensional (3-D) nite element (FE) stress analysis [7,8]. However, these researhers applied loads diretly to the ontat ellipses and ontat lines obtained rom tooth ontat analysis (TCA). Nevertheless, FE ontat analysis or deormable bodies is omplex and non-linear. Most early 3-D FE ontat analyses were perormed using gap elements [11]. Now, due to the progress o omputer tehnology and omputational tehniques, some FEA pakages an deal with ontat analysis without using gap elements. Some researhers have begun to apply these FEA sotwares to ontat problems o gear suraes [1,5]. This study adopts FEA to evaluate the stress distribution o a helial gear set with loalized point ontat. The gear set is omposed o an involute pinion and a modied helial gear. The authors have presented a generation method or the modied helial gear, possessing double rowning eets in the prole and lengthwise diretions [13]. This novel modied helial gear has been generated by adopting a generating tool with irular-ar normal setions instead o the onventional straight-edged setions, to attain the rowning eet on the gear prole diretion. The generating tool moves along a urved-template guide on a hobbing mahine to produe the rowning eet on the gear in the lengthwise diretion. This study also derives the omplete mathematial models or the pinion and the gear, inluding the working suraes and the llets, based on the theory o gearing and the generation mehanism. A omputer program or the FE mesh generation o a 3-D tooth model is developed rom the derived tooth geometry. An FEA pakage, ABAQUS, apable o ontat analysis or two 3-D deormable bodies was employed to determine the stress distribution o a pair o ontat gear teeth in point ontat [14,15]. Finally, some numerial examples are presented to demonstrate the FE stress analyses under various design parameters and dierent ontat positions.. Mathematial model o the modied helial gear set The proposed helial gear set is omposed o an involute pinion and a modied helial gear. The modied helial gear possesses both prole rowning and lengthwise rowning. Mathematial models o the pinion and the gear have been developed aording to the theory o gearing [16,] and the proposed generation mehanism [13,1]. For brevity, the equations are not derived in detail here..1. Geometry o the involute helial pinion 1 Gear generation by hob utters an be simulated using an imaginary rak utter [16,]. Aording to Fig. 1(a), the normal setion o the rak utter surae P used to generate the involute pinion

3 Y.-C. Chen, C.-B. Tsay / Finite Elements in Analysis and Design 38 (00) Fig. 1. Formation shema o rak utter surae P. surae, ontains our major regions: two straight-edges (regions 1 and 3) and two irular urves (regions and 4). Regions 1 and 3 generate the let-side and right-side involute srew suraes o the helial pinion, while regions and 4 generate the let-side and right-side llets. Regions 1 and are symmetri with regions 3 and 4, respetively, with respet to the X r (P) -axis. For simpliity, only the parameters o regions 1 and are indiated in Fig. 1(a) Working suraes o the involute helial pinion 1 Fig. 1(b) illustrates the relationship between oordinate systems S (P) and S r (P), and the ormation o the 3-D rak utter P or the generation o an involute helial pinion. The working suraes o the pinion are involute srew suraes generated by straight utting edges (regions 1 and 3) o rak utter P. In Fig. 1, symbols P and u P stand or the parameters o the tool surae. Parameter A represents the pinion s dedendum, while S P denotes the tooth spae. Angles n (P) and P represent the normal pressure angle and the lead angle o the pinion, respetively. The position vetor R (i) 1 o the working suraes o 1 an be represented as ollows [1]: x (i) 1 =( P os n (P) A + r 1 ) os 1 ± ( P os n (P) A) ot n (P) sin P sin 1 ; y (i) 1 =( P os n (P) A + r 1 ) sin 1 ( P os n (P) A) ot n (P) sin P os 1 ; and ( z (i) 1 = ±(A tan n (P) P sin n (P) A ) os P ± os n (P) sin n (P) ) P sin n (P) tan P sin P ± S P os P + r 1 1 tan P i = 1 and 3: (1) The upper and lower signs reer to the let-side and right-side working suraes, respetively. Parameter r 1 denotes the pinion s pith radius and 1 is the pinion s rotational angle during its generation.

4 710 Y.-C. Chen, C.-B. Tsay / Finite Elements in Analysis and Design 38 (00) Fig.. Formation shema o rak utter surae G..1.. Fillets o the involute helial pinion 1 The llets o the involute helial pinion are generated by regions and 4 (irular urves) o rak utter P. The equation o the llets o the involute helial pinion are as ollows: r (P) R (i) 1 = (O (i;p) rx (O (i;p) rx (O (i;p) ry i = and 4: r (P) sin (i;p) + r 1 ) os 1 ± (O rx (i;p) r (P) sin (i;p) ± r (P) + r 1 ) sin 1 (O rx (i;p) ( O ry (i;p) os (i;p) ) os P r (P) r (P) sin (i;p) ) ot (i;p) sin P sin 1 sin (i;p) ± O rx (i;p) ot (i;p) r 1 1 sin P ) ot (i;p) sin P sin 1 ) tan P sin P Similarly, the upper and lower signs reer to the let-side and right-side llets, respetively. Here, denotes the radius o the llet, and (i;p) and u P are parameters o the tool surae. ().. Geometry o the modied irular-ar helial gear G Fig. (a) depits the normal setion o the rak utter G applied to the generation o the modied helial gear, whih omprises our major regions. Regions 1 and 3 generate the let-side and right-side working suraes o the gear, while regions and 4 generate the let-side and right-side llets, respetively. Regions 1 and are symmetri with regions 3 and 4, respetively, with respet to the X (G) r -axis. For simpliity, only the design parameters o regions 1 and are shown in Fig. (a). In pratie, a urved-template guide an be employed on a onventional hobbing mahine to produe a varied plunge o the hob utter during gear generation. Fig. (b) illustrates the ormation o the imaginary rak utter surae G when a hob utter moves with a varied plunge during the gear generation proess. An auxiliary oordinate system S a (G) (X a (G) ;Y a (G) ;Z a (G) ) whih translates along the line O (G) O a (G) (i.e. axis Z a (G) ) is set up rst. Line O (G) O a (G) orms an angle G with axis Z (G) o the oordinate system S (G) (X (G) ;Y (G) ;Z (G) ). The normal setion o the irular-ar rak utter is rigidly attahed to oordinate system S r (G) (X r (G) ;Y r (G) ;Z r (G) ) with its origin O r (G) moving

5 Y.-C. Chen, C.-B. Tsay / Finite Elements in Analysis and Design 38 (00) along a urve o radius R (G). This urve has the same shape as the urved-template guide. The oordinate system S (G) (X (G) ;Y (G) ;Z (G) ) is rigidly attahed to the transverse setion o the rak utter. Thereore, oordinate system S r (G) shits by a variable amount, E G, with respet to oordinate system S (G). Parameter G indiates the position o point O r (G) on the urved-template guide and E G is the orresponding shit o the hob. Parameter (G) max denotes the extreme value o (G) at whih the parameter E G reahes its maximum value E (G). Parameters W and G represent the ae width and the helix angle o the gear, respetively. The signiant dierenes between the normal setions o P (Fig. 1(a)) and G (Fig. (a)) are the shapes o regions 1 and 3 whih generate the working suraes o tooth proles. Regions 1 and 3 o the normal setion o rak utter G are irular ars rather than straight lines, to produe tooth rowning in the prole diretion o the generated gear. The deviation between the irular-ar and the straight line results in a built-in paraboli TE on the generated tooth surae. A urved-template guide is employed on a onventional hobbing mahine to produe a varied plunge o the hob utter during the gear generation proess. Consequently, the varied shit-amount o the hob utter auses a lengthwise rowning eet on the tooth ank to indue loalized bearing ontats. Double rownings on the prole and lengthwise diretions o the gear tooth surae are thus ahieved on the modied helial gear...1. Working suraes o the modied irular-ar helial gear The let-side and right-side working suraes o the modied irular-ar helial gear are generated by regions 1 and 3 o rak utter surae G, respetively. The equations o the working suraes an be expressed as ollows [13]: R (i) = x (i) y (i) z (i) = (x (i;g) (x (i;g) z (i;g) r ) os +(y (i;g) r ) sin r ) sin +(y (i;g) r ) os ; i= 1 and 3; (3) and { [ (i) ( ; G ; G )= ±r + R G (os n (G) os G )+ S ] G os G } R (G) (sin (G) max sin G ) sin G os G sin G +[R G (sin (G) n sin G )+R (G) (1 os G )] (± sin G sin G sin G os G os G os G )=0; i= 1 and 3: (4) where x (i;g) ;y (i;g) and z (i;g) are expressed in the ollowing: x (i;g) y (i;g) = R G (sin n (G) sin G )+R (G) (1 os G ); [ = R G (os n (G) os G )+ S G ] os G + R (G) (sin (G) max sin G ) sin G ;

6 71 Y.-C. Chen, C.-B. Tsay / Finite Elements in Analysis and Design 38 (00) and [ z (i;g) = ± R G (os n (G) os G )+ S ] G sin G + R (G) (sin (G) max sin G ) os G : (5) The upper and lower signs represent the let-side and right-side working regions o, respetively. Eq. (4) is the equation o meshing in the theory o gearing [16,]. G and G are the surae parameters o rak utter G : n (G) is the normal pressure angle o the gear, while G represents the helix angle o the gear. S G denotes the tooth thikness, R G is the radius o the irular-ar utting edges and R (G) denotes the radius o the urved-template guide used or lengthwise rowning. is the gear s rotational angle during the generation proess, and r denotes the pith radius o the gear.... Fillets o the modied irular-ar helial gear The let-side and right-side llets o the modied irular-ar helial gear are generated by regions and 4 o rak utter G, respetively. Similarly, the position vetor o the llets an be represented by the ollowing equations: and (i) where and R (i) = x (i) y (i) z (i) = (x (i;g) (x (i;g) z (i;g) ( ; (i;g) ; G )= {r [(O ry (i;g) x (i;g) y (i;g) z (i;g) = O (i;g) rx =(O (i;g) ry = (O (i;g) ry r ) os +(y (i;g) r ) sin r ) sin +(y (i;g) r ) os r (G) sin G ) sin G ]} os G sin (i;g) os (i;g) ) os G + R (sin (G) max +[(O (i;g) rx i = and 4 (6) + r (G) sin (i;g) + R (1 os G )] ( sin G sin (i;g) sin G + os G os (i;g) os G )=0 i = and 4; (7) + r (G) sin (i;g) + R (1 os G ); r (G) r (G) os (i;g) ) os G + R (sin (G) max sin G ) sin G ; os (i;g) ) sin G + R (sin (G) max sin G ) os G : (8) The upper and lower signs indiate the let-side and right-side llets o the modied irular-ar gear, respetively. (i;g) and G are the surae parameters o regions and 4 o rak utter G. Eq. (7) is the equation o meshing or the gear s llets.

7 Y.-C. Chen, C.-B. Tsay / Finite Elements in Analysis and Design 38 (00) Finite element stress analysis 3.1. Finite element models and mesh generation program This study adopts the general-purpose FEA sotware, ABAQUS=Standard operating on an HP workstation to evaluate the stress distribution o the proposed helial gear set. Sine the ommerial FEA pakage, ABAQUS=Standard, does not provide an interative preproessor, we have developed a mesh-generation program to establish FEA models or the pinion and the gear aording to the tooth geometry given in the preeding setions. A linear brik element, C3D8, having eight nodes and six aes, is employed to disretize the geometri models o the pinion and the gear tooth suraes [14,15]. The developed mesh-generation program allows the mesh density and the number o elements to be adjusted to meet spei requirements. The mesh-generation program an be applied to onstrut FEA models or other types o gear tooth suraes by modiying the subroutine related to tooth geometry. In general, a FEA model with a larger number o elements or FE stress analysis may lead to more aurate results. However, an FEA model o the whole gear drive is not preerred, espeially onsidering the limit o omputer memories and the need or saving omputational time. This study establishes an FEA model o one pair o ontat teeth or the helial gear set. Fig. 3 displays the mesh system o the pinion and the gear. Eah FE tooth model is staked by 34 unequally-spaed transverse setions in the tooth lengthwise diretion. The regions where stress onentration may our, suh as the llets and possible ontat areas, are disretized by a ner mesh. Moreover, the ontat points on the tooth suraes under light load an be predited aurately by TCA [16,]. Fig. 3. Finite element model and boundary onditions o one pair o ontat teeth.

8 714 Y.-C. Chen, C.-B. Tsay / Finite Elements in Analysis and Design 38 (00) Hene, the mesh density o the gear tooth middle setions is inreased as shown in Fig. 3. In sum, 6188 elements and 7840 nodes are used or the pinion and the gear FE model, respetively. 3.. Material properties and boundary onditions Carbon steel has been hosen or the FEA model. Its basi mehanial properties are Young s Modulus E = 07 GPa and Poisson s Ratio =0:9. Fig. 3 displays the FE model o one pair o ontat teeth and the applied boundary onditions. Aording to the FEA sotware, a linear brik C3D8 element is hosen, and eah node has six degrees-o-reedom (DOF), inluding translations along the nodal x-, y- and z-diretions and rotations about the nodal x-, y- and z-axes. In this study, all the six DOF o the nodes loated on the two lateral sides o the pinion s base are xed, as depited in Fig. 3. On the other hand, rigid beam elements onnet the nodes on the bottom o the gear s base with those on the gear s rotational axis. Furthermore, the nodes on the gear s rotational axis are onstrained suh that the gear an rotate only about its rotational axis. Consequently, the pinion is statially xed and a torque is applied at the gear s rotational axis to make the gear and pinion tooth suraes ontat with eah other Preliminary onsiderations and assumptions In the ontat stress analysis, the user must dene the ontat pair (the suraes whih may ontat eah other during the analysis) as the master and slave suraes. Here, the master and slave suraes are identied as the gear and the pinion tooth suraes, respetively. During the analysis, the slave nodes annot penetrate the master surae segments, but the nodes on the master surae may penetrate the slave surae segments. Additional ontat elements are generated automatially during the analysis. Two other options, small sliding and rition, should be speied to dene the interation between the ontat suraes. Small sliding is hosen in this study sine it is omputationally less expensive, espeially in 3-D ontat analyses. Coulomb rition is onsidered and the rition oeient an be speied. This study assumes the gears to mesh under onditions o good lubriation, and the rition oeient is given as zero. Initially, the models are statially loaded by xing the pinion and then applying a small torque to the gear member whih makes the gear tooth ontat the pinion tooth. The analyses proeed inrementally, and the ontat between the two deormable bodies is handled automatially by imposing non-penetration onstraints between the pinion and gear tooth suraes. In the FEA, a single pair o ontat teeth is onstruted to perorm the stress analysis, and the ollowing assumptions have been made: (1) the stress is in the elasti range o the material; () the material is isotropi; and (3) heat generation and thermal stress are ignored. 4. Illustrative examples Table 1 summarized the design parameters o the proposed modied helial gear pair, omposed o an involute pinion and a modied helial gear. In the FEA, a torque o 150 N m was applied to the gear s axis.

9 Y.-C. Chen, C.-B. Tsay / Finite Elements in Analysis and Design 38 (00) Table 1 Major design parameters o the proposed modied helial gear pair Design values Pinion Gear Parameters Number o teeth Helix angle 15 (RH) 15 (LH) Pressure angle, normal 0 Module, normal 4 mm Radius o urved-template guide R (G) Straight-edged 00 mm Radius o rak utter normal setion R G Straight-edged 1000 mm Fae width 40 mm Fig. 4. Stress distribution on the gear Example 1: ontat stress Aording to the FE stress analysis simulation, Fig. 4 illustrates the distribution o von-mises stress on the gear s tooth surae when the pinion s rotational angle is 0. The maximum stress ours at the ontat position near the middle o the tooth ank. Based on the FEA results, the maximum prinipal stress is 1059:4 MPa, whih is very lose to the Hertzian ontat stress, H = 104:34 MPa (alulated rom Appendix A). Thereore, the proposed FEA method an be used to evaluate the ontat stress. 4.. Example : ontat stress under dierent design parameters o gear rowning Reall that or the double-rowned gear generation, parameter R G indiates the radius o the rak utter s normal setion, while parameter R (G) denotes the radius o the urved-template guide

10 716 Y.-C. Chen, C.-B. Tsay / Finite Elements in Analysis and Design 38 (00) Fig. 5. The inuene o parameters, R G and R (G), on the maximum von-mises stress o the gear. along whih the hob utter moves during the generation proess, as shown in Fig.. Thereore, R G is related to the deviation o the generated tooth prole rom the standard involute urve. The deviation results in a pre-designed paraboli TE o this helial gear set. On the other hand, R (G) aets the ontat areas and the degree o lengthwise rowning is inversely proportional to R (G). Consequently, inreasing the design parameter R (G) inreases the ontat area as well as a redued ontat stress. Aording to the gear tooth mathematial model and the FEA results, Fig. 5 displays the maximum von-mises stress on the gear under dierent design parameters o R G and R (G). Aording to Fig. 5, when R G is xed at 1000 mm, the maximum von-mises stress dereases as R (G) inreases. Nevertheless, the inuene o R G on the ontat stress is insigniant when ompared with that o R (G) Example 3: bending stress alulations The llet stresses are determined at our pinion s rotational angles, 1 = 5 ; 0 ; 5 and 9.As mentioned earlier, the FE models o the pinion and the gear have eah been divided into 34 transverse setions, with the interae o the 17th and 18th transverse setions passing through the middle o the tooth ank. The theoretial ontat point is at the 18th transverse setion o the pinion and the gear tooth models, based on the TCA results. Aordingly, Figs. 6(a) (d) demonstrate the stress

11 Y.-C. Chen, C.-B. Tsay / Finite Elements in Analysis and Design 38 (00) Fig. 6. Stress distribution on the 18th transverse setion o the ontat teeth under dierent ontat positions. distributions o the 18th transverse setions o the pinion and the gear, under these our ontat positions. The variation o the bending stress is small beause the bending stress in the llet is muh smaller than the ontat stress. Generally, the bending stresses in the llets o the two ontating tooth sides are onsidered tensile stresses, and those in the llets o the opposite, unloaded tooth side, are onsidered ompressive stresses. Figs. 7(a) and (b) depit the tensile and ompressive bending stresses along the pinion s llet or the our ontat positions. The bending stress is the average o von-mises stresses at the eight integration points o the th element ounted rom the dedendum. As Fig. 7(a) shows, the

12 718 Y.-C. Chen, C.-B. Tsay / Finite Elements in Analysis and Design 38 (00) Fig. 7. Bending stress along the pinion s llet. maximum tensile bending stress ours around the hal-ae width (W= = 0 mm), that is below the ontat point, or eah ontat position. In the our ontat positions, the maximum tensile bending stresses are 59.65, 50.64, and 56:73 MPa. When 1 = 5, as shown in Fig. 6(a), the ontat point is lose to the llet and the tensile bending stress is high due to stress onentration. Furthermore, at 1 =9, the ontat position is near the addendum o the pinion, exerting a large bending moment and a high bending stress on the tooth root. Thereore, the maximum tensile bending stresses under the two ontat positions ( 1 = 5 and 9 ) are higher than other positions ( 1 =0 and 5 ). Figs. 6 (a) (d) also illustrate that as the pinion rotates rom 5 to 9, the ontat position moves upward rom the dedendum to the addendum on the pinion, yielding a larger ompressive bending stress on the opposite and unloaded sides. Thereore, the maximum ompressive bending stresses inreases as the pinion rotates rom 5 to 9, as is lear in Fig. 6(b). Furthermore, the respetive peak values o the ompressive llet stresses under the our ontat positions are 35.99, 41.8, 49.1 and 6:11 MPa Example 4: stress analysis o a onventional involute helial gear pair In this example, the stress o a onventional involute helial gear pair is studied via FEA and Amerian Gear Manuaturers Assoiation (AGMA) stress ormulae (please reer to Appendix B). Table summarizes the major design parameters o the involute helial gear pair, and Fig. 8 displays the stress distribution on the pinion aording to FEA results. Based on FEA results and Fig. 8, the von-mises stress in the llet o the pinion is 8:93 MPa or the involute helial gear pair. In addition, the maximum prinipal stress on the gear is 34 MPa based on the FEA results. On the other hand, the ontat and bending stress numbers o the involute helial gear pair are alulated based on AGMA standard, AGMA 101-C95 [17]. Aording to the AGMA stress ormulae, the

13 Y.-C. Chen, C.-B. Tsay / Finite Elements in Analysis and Design 38 (00) Table Major design parameters o the involute helial gear pair Design values Pinion Gear Parameters Number o teeth Helix angle 0 (RH) 0 (LH) Pressure angle, normal 0 Module, normal 4 mm Fae width 40 mm Fig. 8. Stress distribution on the pinion o the involute helial gear pair. ontat stress is 394:99 MPa and the bending stress on the pinion is 8:34 MPa. Thereore, the proposed FE stress analysis model or the modied helial gear pair yields reasonable results. 5. Conlusions In this study, nite element stress analysis was perormed to investigate the ontat stress and the bending stress o a modied helial gear set omprising an involute pinion and a modied helial gear. The FEA tooth models inluding the working suraes and the llets o the pinion and the gear were developed. Commerial FEA sotware, ABAQUS=Standard, apable o ontat analysis was applied to evaluate the stress distribution on the tooth suraes. The analysis results leads to the ollowing onlusions: (1) The proposed helial gear set exhibits loalized bearing ontats due to double rowning on the gear s tooth suraes. () The ontat stress alulated by FEA is lose to the Hertzian ontat stress obtained rom the Hertzian stress ormulae and urvature analysis.

14 70 Y.-C. Chen, C.-B. Tsay / Finite Elements in Analysis and Design 38 (00) (3) Inreasing R (G) results in an inrease in the ontat area and a redution in ontat stress, due to a smaller lengthwise rowning eet on the gear s tooth suraes. Although a larger R G auses a smaller prole rowning eet, the redution o ontat stress is less signiant than the inuene o R (G). (4) The tensile and ompressive bending stresses along the pinion s llets under dierent ontat positions were investigated. The maximum llet stress ours near the middle setion o the tooth ank (below the ontat points). (5) Although this study investigated a modied helial gear set, the developed mesh generation program an also be applied to disretize FEA models or other types o gearing. (6) The proposed FEA method an aurately alulate the ontat and bending stresses. This model an be extended urther to investigate the load share and transmission errors under load. Aknowledgements The authors would like to thank the National Siene Counil o the ROC or nanially supporting this researh under Contrat No. NSC 89-1-E Appendix A. Determination o Hertzian ontat stress The instantaneous ontat point o the gear tooth suraes is spread over an elliptial area with the enter o symmetry loated at the theoretial ontat point, due to the elastiity. Tooth ontat analysis an determine aurately the theoretial ontat point under a light load [16,]. Assume that a torque T is applied at the gear s rotational axis. The ontat ore F ating on the ontat point an be determined by [18] T F = (R n ) ; (A.1) a() where R and n denote the position vetor and the unit normal vetor o the ontat point represented in the gear s oordinate system, and a () represents the unit vetor o the gear s rotational axis. Aording to Hertzian ontat stress ormulae, the semi-axis b o the ontat ellipse and the maximum Hertzian stress H an be estimated by b = C b 3 F H = C ( b ) : (A.) (A.3) Coeients C b and C an be determined using Figs. 14 6:7 and in Res. [19]. The auxiliary parameter is dened as = (1 ) (A + B)E ; (A.4)

15 Y.-C. Chen, C.-B. Tsay / Finite Elements in Analysis and Design 38 (00) Fig. 9. Denition or orientation and dimension o a ontat ellipse. where is the Poisson s ratio and E denotes the Young s modulus. A and B are determined by A = 1 4 [(1) B = 1 4 [(1) where (1) () (g 1 g 1 g os + g ) 1= ]; (A.5) () +(g 1 g 1 g os + g ) 1= ]; (A.6) = (1) I + (1) II ; (A.7) () = () I + () II ; (A.8) and g 1 = (1) I (1) II ; (A.9) g = () I () II : (A.10) Here, (1) I and (1) II represent the rst and seond prinipal urvatures o the pinion surae 1, while () I and () II represent the rst and seond prinipal urvatures o the gear surae, respetively. As Fig. 9 shows, angle is measured ounterlokwise rom i () I to i (1) I and an be evaluated by ( ) = tan 1 i (1) I i () II ; (A.11) i (1) I i () I where i (1) I and i (1) II denote the unit vetors o the rst and seond prinipal diretions or the pinion, while i () I and i () II represent the unit vetors o the rst and seond prinipal diretions or the gear, respetively. Furthermore, the prinipal diretions and the prinipal urvatures o the pinion and the gear tooth suraes, and the orientation o the ontat ellipses an be determined aording to the dierential geometry and Litvin s approah [16,].

16 7 Y.-C. Chen, C.-B. Tsay / Finite Elements in Analysis and Design 38 (00) Appendix B. AGMA stress ormulae The ontat stress and bending stress o gears are alled ontat stress number and bending stress number in AGMA standards. Aording to AGMA 101-C95, the ontat stress number H and the bending stress number F or involute helial gears an be determined as ollows [17]: K H Z R H = Z E F t K o K v K s ; (B.1) r 1 W Z I and 1 K H K B F = F t K o K v K s ; (B.) Wm t Y J where F t is the transmitted tangential load, K o is the overload ator, K v is the dynami ator, K s is the size ator, K H is the load distribution ator, K B is the rim thikness ator, Z E is the elasti oeient, Z R is the surae ondition ator, r 1 denotes the pith radius o the pinion, W is the ae width and m t is the transverse module. Parameters Z I and Y J denote the geometry ators or pitting resistane and or bending strength, respetively. The detailed derivations and Tables o geometry ators Z I and Y J are inluded in AGMA 908-B89 [0]. Aording to the design parameters o the involute helial gear pair listed in Table, the geometry ator Z I is 0.177, and the values o geometry ator Y J are 0.46 and 0.51 or the pinion and the gear, respetively. Reerenes [1] C.B. Tsay, Helial gears with involute shaped teeth: geometry, omputer simulation, tooth ontat analysis, and stress analysis, ASME J. Meh. Transmissions Automation Des. 110 (1988) [] F.L. Litvin, Gear Geometry and Applied Theory, Prentie-Hall, U.S.A., New Jersey, [3] F.L. Litvin, D.H. Kim, Computerized design, generation and simulation o modied involute spur gears with loalized bearing ontat and redued level o transmission errors, ASME J. Meh. Des. 119 (1997) [4] F.L. Litvin, N.X. Chen, J. Lu, R.F. Handshuh, Computerized design and generation o low-noise helial gears with modied surae topology, ASME J. Meh. Des. 117 (1995) [5] F.L. Litvin, Q. Lian, A.L. Kapelevih, Asymmetri modied spur gear drives: redution o noise, loalization o ontat, simulation o meshing and stress analysis, Comput. Method Appl. Meh. Eng. 188 (000) [6] Y. Zhang, Z. Fang, Analysis o transmission errors under load o helial gears with modied tooth gears, ASME J. Meh. Des. 119 (1997) [7] M.A.S. Arikan, M. Tamar, Tooth ontat and 3-D stress analysis o involute helial gears, ASME, International Power Transmission and Gearing Conerene, De-Vol. 43, No., 199, pp [8] C.R.M. Roa, G. Muthuveerappan, Finite element modelling and stress analysis o helial gear teeth, Comput. Strut. 49(6) (1993) [9] E. Bukingham, Analytial Mehanis o Gears, Dover, U.S.A., New York, [10] D.W. Dudley, Dudley s Gear Handbook, nd Edition, MGraw-Hill, U.S.A., New York, 199. [11] I.H. Filiz, O. Eyerioglu, Evaluation o gear tooth stresses by nite element method, ASME J. Meh. Des. 117 (1995) [1] R.F. Handshuh, G.D. Bibel, Experimental and analytial study o aerospae spiral bevel gear tooth llet stresses, ASME J. Meh. Des. 11 (1999) [13] Y.C. Chen, C.B. Tsay, Mathematial model and underutting analysis o modied irular-ar helial gears, J. Chinese So. Meh. Eng. (1) (000) [14] ABAQUS=Standard 5.8, User s Manual, Hibbitt, Karlsson & Sorensen U.S.A, 1998.

17 Y.-C. Chen, C.-B. Tsay / Finite Elements in Analysis and Design 38 (00) [15] R.D. Cook, D.S. Malkus, M.E. Plesha, Conepts and Appliations o Finite Element Analysis, 3rd Edition, Wiley, U.S.A., New York, [16] F.L. Litvin, Theory o Gearing, NASA Publiation RP-11 U.S.A., Washington, DC, [17] Amerian Gear Manuaturers Assoiation, AGMA 101-C95, Fundamental rating ators and alulation methods or involute spur and helial gears, U.S.A, [18] F.L. Litvin, J.S. Chen, J. Lu, R.F. Handshuh, Appliation o nite element analysis or determination o load share, real ontat ratio, preision o motion, and stress analysis, ASME J. Meh. Des. 118 (1996) [19] A.P. Boresi, O.M. Sidebottom, Advaned Mehanis o Materials, 4th Edition, Wiley, U.S.A., New York, [0] Amerian Gear Manuaturers Assoiation, AGMA 908-B89, Geometry ators or determining the pitting resistane and bending strength o spur, helial and herringbone gear teeth, U.S.A, 1989.