MODELING AND CONTACT ANALYSIS OF CROWNED SPUR GEAR TEETH

Engineering MECHANICS, Vol. 18, 2011, No. 1, p. 65 78 65 MODELING AND CONTACT ANALYSIS OF CROWNED SPUR GEAR TEETH Ramalingam Gurumani*, Subramaniam Shanmugam* This paper provides a novel method to model lead crowned spur gears. The teeth o circular and involute crowned external spur gears are modeled or the same crowning magnitude. Based on the theory o gearing, mathematical model o tooth generation and meshing are presented. Eect o major perormance characteristics o uncrowned spur gear teeth are studied at the pitch point and compared with longitudinally modiied spur gear teeth. The results o three dimensional FEM analyses rom ANSYS are presented. Contact ellipse patterns and other contact parameters are also studied to investigate the crowning eects. Keywords : proile modiication, crowning, tooth contact analysis, stress analysis, contact ellipse, tooth generation 1. Introduction The main purpose o gear mechanisms is to transmit rotation and torque between axes. The gear wheel is a machine element that has intrigued many engineers because o numerous technological problems arises in a complete mesh cycle. In order to achieve the need or high load carrying capacity with reduced weight o gear drives but with increasedstrengthingear transmission, design, gear tooth stress analysis, tooth modiications and optimum design o gear drives are becoming major research area. Gears with involute teeth have widely been used in industry because o the low cost o manuacturing. Transmission error occurs when a traditional non-modiied gear drive is operated under assembly errors. Transmission error is the rotation delay between driving and driven gear caused by the disturbances o inevitable random noise actors such as elastic deormation, manuacturing error, alignment error in assembly. It leads to very serious tooth impact at the tooth replacing point, which causes a high level o gear vibration and noise. At the same time, edge contact oten happens, which induces a signiicant concentration o stress at the tooth edge and reduces the lie time o a gear drive. Many researchers have proposed modiied shapes or traditional gears to localize the bearing contact thereby avoiding edge contact. In the present work, involute spur gear teeth or the selected module, gear ratio, centre distance and number o teeth is taken or analysis. Two more sets o dierently crowned involute spur gear teeth are also considered or stress and tooth contact analysis. The procedure ollowed to create the 3D models o teeth in mesh is described. Proportions o contact ellipses are determined. Their perormance behavior is studied by assuming loading at pitch point under static load and rictionless hypothesis. In order to handle critical proile variations, the analyses are done in three dimensions with ace contact * R. Gurumani (research scholar), S. Shanmugam (proessor), Department o Mechanical Engineering National Institute o Technology, Tiruchirappalli-620 015, India

66 Gurumani R. et al.: Modeling and Contact Analysis o Crowned Spur Gear Teeth model. Surace Contact Stress (SCS), Root Bending Stress (RBS), and Tooth Delection (TD) calculations o the pair o spur gears with and without lead crowning are carried out through FEM. Comparison o these parameters are also carried out and interpreted suitably. 2. Literature review It is necessary to choose a quicker way o evaluation o the perormance o gear teeth, regardless o the material o construction and manuacturing process adopted. Perormance parameters like tooth bending, surace distress and tooth delection are the basic modes o atigue ailure o any toothed gearing [1]. Kramberger et al. [2] applied inite element method and boundary element method or analyzing the bending atigue lie o thin-rim spur gear. A computational model or determining the service lie o gear teeth lanks with regard to surace pitting was presented by Glodez et al. [3] through which the probable service lie period o contacting suraces are also estimated or surace curvatures and loading that are most commonly encountered in engineering practice. Akata et al. [4] determined geometrical location o HPSTC rom the basic gear geometry and compared bending atigue behavior o the gear teeth by loading them rom the HPSTC, was chosen as a rapid way o evaluation o perormance o gear teeth produced by dierent manuacturing methods. Zeping Wei [5] used three dimensional inite element method to conduct surace contact stress and root bending stress calculations o a pair o spur gears with machining errors, assembly errors and tooth modiications. Traditional non-modiied gear drive is operated under assembly errors, which leads to very serious tooth impact at the tooth replacing point, which causes a high level o gear vibration and noise. Also, ideal spur gears with line contact are very sensitive to axial misalignment and manuacturing errors which cause the edge contact o gear drive. Concentration o contact load at one end o a gear tooth produces bending stress concentration and reduces lie time o a gear drive. Tooth breakage at one end o a gear associated with misalignment. The light scu marks towards the end o the teeth also indicate misalignment. The easiest way to avoid the edge contact is to localize the bearing contact by crowning gear tooth suraces [6]. Further crowning modiication is still required to transorm the meshing rom line contact to point contact. The most common way o crowning is the so-called lead crowning. The lead crowning o spur gear can be considered as the longitudinal deviation rom a straight line along the tooth width. This modiication prevents excessive loading at the ends due to misalignment, tooth inaccuracies, delection, or heat treatment. As the load increases, a smooth spreading o the contact occurs until the entire lank is loaded. There are dierent orms that can be chosen or the modiied tooth proile including linear and parabolic variation. Nowadays, in order to pursue the development o a higher quality o transmission, the ormer method o using parabolic unctions need improvement. The parabolic unction demands a shrinking o the tooth ace inwards near the top land and illet, which results in a decrease in bending strength o the tooth illet. But, or a ourth order polynomial unction shrinking near the tooth illet is smaller and thus the tooth illet is stronger [7]. An increased magnitude o crowning weakens the pitting durability, and too small crowning decreases the ability to stabilize the bearing contact. Although crowning is a very important manuacturing technique in gearing ield, a suicient analysis or the proper amount o crowning has not been provided [8]. It is preerable to crown the pinion tooth surace

Engineering MECHANICS 67 than the gear tooth surace since the number o pinion teeth is smaller. The conjugation o crowned pinion tooth surace and an ordinary gear tooth surace should be a subject o special investigation directed at minimization o transmission errors and avourable location o bearing contact [9]. 3. Methodology The major design parameters like center distance, module and number o teeth on pinion [10] are directly taken or analysis and tooth modiications or a given pinion speed, power and gear ratio. This paper represents longitudinal circular crowned gears in which the tooth suraces as the surace o revolution that is generated by rotation o the involute curve about a ixed axis. Further, an attempt is made to analyze longitudinal involute crowning. In the proposed involute crowning the amount o tooth ace shrinking is reduced. Crowning magnitude in both the cases are taken as equal. Crowned and uncrowned tooth suraces are generated and transerred into commercial inite element analysis sotware [11] or determining stresses and delections with proper element mesh. Loading is applied as input torques at the gear centres. Tab. 1 provides the design and operating parameters o the standard spur gear considered or contact analysis and crowning modiications. Description Pinion Wheel Material 40Ni2Cr1Mo28 (En24) high strength alloy steel Ultimate tensile strength, MPa 1550 Yield strength, MPa 1300 Modulus o elasticity, MPa 2.07 10 5 Density, kg/m 3 7840 Pitch circle diameter, mm 36 180 Root diameter, mm 31 175 Outside diameter, mm 40 184 Base diameter, mm 33.83 169.15 Number o teeth 18 90 Radius o curvature at pitch point, mm 6.1563 30.782 Fillet radius, mm 0.3module Toplandthickness,mm 0.4module Face width, mm 30 Addendum, mm 1module Dedendum, mm 1.25module Poisson s Ratio 0.25 Pressure angle, deg. 20 Tab.1: Design and operating parameters o standard spur gears 3.1. Tooth generation The idea o using the involute is that it generates a straight contact path when turning the gear wheels which is important since a non-straight contact path or line o action leads to vibration at higher rpm. In order to avoid high contact pressures in the outermost regions root relie, tip relie and tip rounding are employed. All these tooth proile modiications towards involute curve can o course, i not properly designed, lead to a non-straight contact path, which causes noise and vibration.

68 Gurumani R. et al.: Modeling and Contact Analysis o Crowned Spur Gear Teeth It is more suitable to use 3D modeling, because o tooth height, tooth thickness and tooth ace width o a gear tooth in three mutually perpendicular directions are comparable with each other. Further, an elliptical contact area ormed when two three-dimensional bodies, each described locally with orthogonal radii o curvature, come into contact. In order to consider the eect o adjacent teeth, three tooth model was taken into account, which can also be used to analyze simultaneous contact o single and double pair teeth. For the selected pitch diameter and number o teeth, tooth proile models have been developed by Pro/ENGINEER and then transerred to ANSYS or static stresses and delection analysis. The gears are provided with involute proile based on the ollowing geometric Eq. (1), described in parametric orm and it is highly lexible in terms o gear size and ace width. The innermost cord o involute is used in proile generation. x = r b sin θ r b θ cos θ, y = r b cos θ + r b θ sin θ, z =0. (1) Crowning towards the ace width is another eature taken or analysis. Crowning is the advancement o end relie. Edge o the end relieved gear results in edge contact again. To avoid such problems the end relie is made in the orm o curved suraces adopting any standard curve proiles. The point contact o Fig.1: Radius o curvature tooth suraces is achieved by the crowning o circular crowning the pinion and wheel tooth aces in the longitudinal direction. Two types o lead crowning are carried out in which the proile generating curve is an exact involute curve, which is the innermost portion o the involute curve generated by the base circle o respective pinion and wheel. In the irst case, lead circular crowning with a magnitude derived rom the outermost portion o an arc o involute o wheel is done and the radius o rotation is given accordingly. The magnitude is ound in such a way that outermost portion o the arc o the involute is positioned in the top land o the gear teeth giving equal magnitude on both the ends. This will be the only minimum magnitude and used in both o these models and avoids the problem o giving magnitude. The radius o curvature o the symmetrical circular crowning ϱ sym and amount o modiication e(f j ) reerring to Figure 1 are calculated using Eqs. (2) and (3) [12]. ϱ sym = 4 C2 c + F 2, (2) 8 C c e(f j )=ϱ sym ϱ 2 sym Fj 2, F 2 F j F 2. (3) Further, to overcome the eects shrinkage o tooth ace and root stress concentration, involute crowning towards ace width is attempted. Care is taken because improper crowning which could move the load path rom the centre o ace width towards the heel or toe regions. Excessive wear caused by this load shit could result in premature surace ailure or breakage.

Engineering MECHANICS 69 For the given ace width, the outermost portion o the involute curve, as shown in Figure 2, is generated by the selected base circle radius o wheel is used as trajectory to sweep the involute proile or the crowning modiication o pinion and gear wheel. Total length o the cord = 2 π 2 R b. (4) The outermost portion used or providing crowning or any selected ace width is calculated by the ollowing expression obtained using Figure 2. { [2 ( ) 2 πθ3 R b 2 πrb (θ 3 2 α) F = R b tan α + R b tan α cos 2α] + 360 360 [( ) ] } 1 2 2 2 πrb (θ 3 2 α) + + R b tan α sin 2α 360 { [2 ( ) 2 F πθ3 2 π = R b 360 tan α (θ3 2 α) +tanα cos 2α] + 360 [( ) ] } 1 2 2 2 π (θ3 2 α) + +tanα sin 2α 360 The outermost portion is so positioned as to give minimum crowning. So, the cross section o tooth suraces o the gear along both the planes is involute. Reerring to Figure 2, the geometrically determined magnitude o involute crowning C c is given in Eq. (6). C c = CI = 2 πr b θ M AO 2 AD 2. (6) 360 Reerring to Fig. 3, radius o curvature increases continuously, ultimately it lattens and its magnitude at point o contact, ϱ inv is given in Eq. (7),, (5) ϱ inv = EI = 2 πr b θ M 360. (7) Fig.2: Selection o crowning segment rom the involute proile Fig.3: Radius o curvature involutecrowning

70 Gurumani R. et al.: Modeling and Contact Analysis o Crowned Spur Gear Teeth The magnitude and radius o curvature at the contact points are then used in tooth contact analysis. Figure 4 presents the solid modeling o assembly models o standard gear, circular crowned and involute crowned gears rom Pro/ENGINEER. Fig.4: Assembly models: (a) standard gear, (b) circular crowned, (c) involute crowned Fig.5: Coordinate system applied or generating crowned tooth suraces Based on the theory o gearing, mathematical model o tooth generation and meshing are developed. Tooth surace is represented as the surace o revolution that is generated by rotation o the involute curve about a ixed axis. First, the involute proile is represented in X p -Y p plane in one-parameter orm as shown in Figure 5. r p = r p (θ p ). In auxiliary coordinate system S a, Y a and X a axes coincide with the tangential vector and normal vector at pitch point P, respectively. The origin O a is positioned at the distance o R p rom P measured along the direction o the normal. The surace o revolution is generated in another auxiliary coordinate system S b by rotating the involute proile about Y a axis. Finally, ater coordinate transormation, the crowned pinion tooth surace is represented in Σ p as ollows : r p (u p,θ p,r p )=M pb M ba M ap r p (θ p ). (8)

Engineering MECHANICS 71 Here, the radius o revolution R p controls the amount o crowning. I R p is given, the pinion tooth surace is represented in two-parameter orm where parameters θ p and u p are independent. Similar procedure can be applied or generation o gear tooth surace. The crowned gear tooth surace Σ g is obtained as r g (u g,θ g,r g ). For the second case, the crowned pinion tooth surace Σ p is represented in S p as ollows : r p (u p,θ p,u cp,θ cp,ϱ cp )=M pb M ba M ap r p (θ p ). (9) Similarly, crowned gear tooth surace Σ g is obtained as r g (u g,θ g,u cg,θ cg,ϱ cg ). 3.2. Analysis o meshing The contacting suraces must be in continuous tangency and this can be obtained i their position vectors and normals coincide at any instant. The tooth normals and their unit normalsarerepresentedinacommonixedcoordinatesystems. The contact o interacting suraces Σ i are localized and they are in point tangency. The conditions o continuous tangency o suraces are represented by the Eqs. (10) and (11). r (p) (u p,θ p,φ p ) r (g) (u g,θ g,φ g ) = 0 (10) n (p) (u p,θ p,φ p ) n (g) (u g,θ g,φ g ) = 0 (11) where, u i, θ i,(i =p, g) are the surace parameters, φ i are the parameter o motion, r (i) and n (i) are the position vector and the surace unit normal o the contact point on surace Σ i. Vector Eqs. (10) and (11) yield ive independent scalar equations in six unknowns, since = n (g) =1. Here n (p) i (u p,θ p,φ p,u g,θ g,φ g )=0, i C 1, i =1, 2, 3, 4, 5. (12) Considering φ p as the known input parameter, Eq. (12) can be solved with unctions o {u p (φ p ),θ p (φ p ),u g (φ p ),θ g (φ p ),φ g (φ p )}. The Jacobian o the scalar equations will dier rom zero as the precondition o point tangency o meshing suraces. Thereore, D( 1, 2, 3, 4, 5 ) D(u p,θ p,u g,θ g,φ g ) = 1 1 1 1 1 u p θ p u g θ g φ g..... 5 5 5 5 5 u p θ p u g θ g φ g 0. (13) The computational procedure provides the paths o contact on pinion and gear tooth suraces and unction o transmission errors. Function φ g (φ p )} represents the relation between the angles o gear rotation (the law o motion). Functions r p (u p,θ p ), u p (φ p ), θ p (φ p )

72 Gurumani R. et al.: Modeling and Contact Analysis o Crowned Spur Gear Teeth determine the path o contact points on surace Σ p. Similarly, unctions r g (u g,θ g ), u g (φ p ), θ g (φ p ) determine the path o contact points on surace Σ g. The unction o transmission errors is represented in Eq. (14) Δφ g (φ p )=φ g (φ p ) N p N g φ p. (14) Aligned ideal gear drives are in line contact. Due to misalignment, the line contact o meshing suraces is turned into point contact resulting in shiting o bearing contact and transmission errors. The simulation o meshing and analysis o contact or such drives is a diicult problem since Jacobian becomes equal to zero. When the path o contact on the gear tooth surace or one cycle o meshing exceeds the dimensions o the tooth, edge contact will occur. It is a curve-to-surace tangency and is represented by the Eqs. (15) and (16) r (p) (u p,θ p,φ p )=r (g) (u g,θ g,φ g ), (15) r (p) θ 1 n (p) =0, (16) where r (p) (u p (θ p ),θ p,φ p ) represents the edge E p on pinion surace and r (p) / θ 1 is the tangent to E p. 4. Tooth contact analysis The stresses on or beneath the surace o contact are the major causes o ailure. These contact stresses are created when suraces o two bodies are pressed together by external loads. H. Hertz replaced the complicated gear pair geometry with an equivalent model o two cylinders and obtained only principal stresses on the area o contact. The Hertz ormula can be applied to spur gears by considering the contact conditions o each gear to be equivalent to those o cylinders that have the same radius o curvature at the point o contact as the gears have. This is an approximation because the radius o curvature o an involute tooth will change while going across the contact zone [13]. The contact stress, by the Hertz equation is a nominal stress and assumes uniorm distribution o load [14]. But, the pitting durability o gear pair depends on the contact stress. With crowned gears this uniorm distribution will not be the case. The contact area becomes an ellipse, or in most cases, the centre portion o an ellipse, with highest contact stress at the centre o the ace width and progressively lower contact stress towards ends. Instantaneous contact ellipse determines the amount o contact stress. The determination o dimensions and orientation o instantaneous contact ellipse needs the details o (i) principal directions and curvatures o meshing suraces, and (ii) the elastic approach. Proportions o contact ellipses correspond to the pitch point o the crowned gears are calculated empirically [14, 15]. Radius o curvature o the selected gear drive along the lead or circular crowning rom the Eq. (2)is 535.82 mm. The radius o curvature along the lead or involute crowning is geometrically determined using Eq. (7) and the value is 528.99 mm. Also, the crowning magnitude or both the crowning models is taken as equal. The value o the crowning magnitude or a given ace width is determined rom the Eq. (2) or the circular crowning where as or involute crowning, it is determined geometrically using Eq. (6) and the value o crowning magnitude is 0.21 mm. Geometry actor related to radius

Engineering MECHANICS 73 o curvature at the pitch point, B = 0.09746. Geometry actor related to radius o curvature along the lead o the gears or circular crowning, A =1.866 10 3. Geometry actor related to radius o curvature along the lead o the gears or involute crowning, A =1.8904 10 3. Semi contact width o contact cylinder with line contact is given by the Eq. (17) and is computed as 0.1152 mm. [ 4 P 1 μ 2 P b = πf E P + 1 ] μ2 G E G 1 r P + 1 r G. (17) The elastic approach or line contact is given by the Eq. (18) and is computed as 0.00823 mm. δ = 2 P [ ( 1 μ 2 P ln 4 r P 1 ) + 1 ( μ2 G ln 4 r G 1 )] πf E P b 2 E G b 2. (18) Maximum pressure (Hertz stress)is given by the Eq. (19) and is computed as 1238 N/mm 2. p = 2 P πb = PEc, (19) πr c eective curvature and contact modulus 1 R c = 1 R P + 1 R G (20) 1 = 1 μ2 P + 1 μ2 G. (21) E c E P E G 4.1. Contact stress-elliptical contact An elliptical contact area is ormed when two spheres o dierent radius is in contact. With crowned gears contact area contracts towards the centre o ace width and becomes an ellipse. The highest contact stress occurs at the centre o the ace width and progressively lower contact stress towards the ends. At the point o contact, the contact spheres are taken as toroids. Hertz equation is used to calculate delection and stress between two elastic toroids. The angular orientation between the axes o pinion and gear is assumed to be zero. Dimensions o equivalent sphere and toroid and other contact parameters are given in Eqs. (22) to (28). Equivalent sphere in contact with a plane R c = R a R b. Equivalent toroids in contact with a plane is given as R a = 1 (A + B) (B A) and R b = 1 (A + B)+(B A), where A = 1 2 ( 1 R P + 1 ) R G and B = 1 2 ( 1 + 1 ) R P R G.

74 Gurumani R. et al.: Modeling and Contact Analysis o Crowned Spur Gear Teeth Eccentricity o contact ellipse Equivalent radius o contact Major and minor contact radii e 2 =1 ( b a ) 2 = 1 ( Rb R a ) 4 3. (22) c = ( ) 1 3 PRc 3 ab= F1. (23) 4 E c a = c (1 e 2 ) 1 4, b = c (1 e 2 ) 1 4. (24) Maximum pressure Normal displacement Normal stiness p = 3 P 2 πc 2 = 3 P 2 πab. (25) δ = c2 R c F 2 F 2 1 = ab R c F 2 F 2 1. (26) k = 2 E c c F 1 F 2. (27) [ ( ) ] 0.0602 1.456 [ ( ) ] 0.0684 1.531 F 1 Ra = 1 1, F 2 Ra = 1 1. (28) R b R b Considering same ormulae as that o circular crowned gears, replacing radius o curvature o crown along the lead, the contact stress (involute crowned) and other contact parameters are ound and tabulated. The dimensions o contact ellipses, induced contact stresses and contact parameters are presented in Tab. 2. Contact type Contact parameters Line contact Elliptical contact Circular crowned Involute crowned semi-major axis semi-major axis Semi-contact width a =3.4225 a =3.4 0.1152 or contact radius, mm semi-minor axis semi-minor axis b =0.1152 b =0.1152 Contact area, mm 2 6.912 1.239 1.23 Max. contact pressure (Hertz stress), N/mm 2 1238 7792 7851 Approaches o centres (Normal displacement), mm 0.0073 0.0134 0.0135 Normal stiness, N/μm 887.794 311.407 314.843 Tab.2: Contact parameters o crowned spur gear teeth

Engineering MECHANICS 75 4.2. Finite element analysis The root bending stress, contact compressive stress and gear tooth delection are calculated with load application at the pitch points location are presented in Table 1. The most powerul method o determining accurate stress and delection inormation is the inite element method. Gear geometry is a inite element replica o the simple cantilever beam model according to Lewis beam strength equation where one tooth is considered as a base and the root is ixed. The loads acting on the mating gears will act along the pressure line, which can be resolved into both tangential and radial components using the pressure angle. The radial component o orce is neglected and the eect o only tangential component is considered as uniormly distributed load at the line o contact. In the case o crowned gears, entire load is assumed to take place at the point o contact. Initially, it is assumed that at any point o time only one pair o teeth is in contact and takes the total load. The gear tooth contact stresses, tooth bending stresses and gear tooth delection o single tooth contact o spur gear are analyzed using FEM. A thick ull rim with three teeth FE model is used or stress analysis. It is not recommended to have a ine mesh everywhere in the model, in order to reduce the computational requirements. The iner mesh size is selected to ensure line contact and uniorm distribution o load over lank. The element is o the type SOLID45 eight-noded hexahedral is chosen. SOLID45 is used or the 3-D modeling o solid structures. The element is deined by eight nodes having three degrees o reedom at each node, viz. translations in the nodal x, y, and z directions. The element has plasticity, creep, swelling, stress stiening, large delection, and large strain capabilities [16]. It has compatible displacement behaviour and well suited to model curved boundaries. The material is assumed to be isotropic and homogeneous. The type o contact was surace to surace. The contact pair consists o a target surace and a contact surace. The contact surace moves into the target surace. The target surace was chosen in the gear tooth and meshed by TARGE170, 3D target segment while the contact surace was chosen in the pinion tooth with CONTA174, 3D, 8-node surace-to-surace contact element. The contact element is deined as a connection between a node o a contact body and a target segment o a target body. Areas around the contacting suraces have been meshed with larger density o inite elements mesh or results accuracy. The wheel nodes placed on inner rim radius have been constrained in global Cartesian coordinate system in all directions. The movements in directions o both axes have been disabled. The pinion nodes placed on inner rim radius have been constrained. Rotation o the pinion nodes placed on inner rim radius around the centre o the global cylindrical coordinate system has been enabled i.e. the rotation about Z axis enabled. The torque is applied at Z axis in clockwise direction. Nodes at the side portions and the bottom portion o rim o the gear are ixed during crowned gear analysis to avoid misalignment and to localize the load distribution The maximum principal stress at the root on the tensile side o tooth is used or evaluating the tooth bending strength o a gear. The Von Mises stress at the critical contact points was ound out. Theoretical values o these stresses are compared with the FEM results and agree well. Further gear tooth delection under load is o primary interest or proper operation o a gear set. Tooth tip delection is also considered or analysis. Also, by considering the act that the load is usually shared between two tooth pairs when the contact is away rom the pitch point, the compression stress at the pitch line is used as the critical contact stress [13].

76 Gurumani R. et al.: Modeling and Contact Analysis o Crowned Spur Gear Teeth Tab. 3 presents the values o these perormance parameters o the standard involute spur gear, circular crowned and involute crowned spur gears set at the pitch point by FEM analysis. Figures 6 9 represent the contact stress pattern, root bending stress pattern o standard, circular crowned and involute crowned gears through FEM using ANSYS. Tooth modiication Contact stress Tooth bending stress Gear tooth delection MPa at root, MPa mm Non-modiied 1021 393.38 0.0166 Circular crowned 6078 1025 0.0572 Involute crowned 5947 946 0.0577 Tab.3: Value o perormance parameters o standard spur gear teeth by FEM Fig.6: Contact stress pattern at the contact area o standard spur gear Fig.7: Root bending stress o standard spur gear

Engineering MECHANICS 77 Fig.8: Contact stress pattern o circular crowned gears Fig.9: Contact stress pattern o involute crowned gears 5. Conclusions Amount o crowning can be determined rom the given ace width or vice versa and it can be veriied analytically and geometrically. Eect o crown radius in crowning is studied and reveals that the area o contact o involute crowned gears is slightly lower than the circular crowned gears which show the existence o more point contact compared to circular crowned gears. Also, the area o contact and contact pattern depend on the radius o curvature at the point o interest and proile o curves used or lead crowning generation. The proposed method that the portion o involute curve deining the crowning magnitude can be applied to other types o crowning modiications done with standard curve proiles. The crowned

78 Gurumani R. et al.: Modeling and Contact Analysis o Crowned Spur Gear Teeth gears with elliptical contact are approximated as an equivalent sphere in contact or contact analysis. It is evident rom the results that the transmission error o spur gear which has large changes in mesh stiness can be reduced by applying the proposed crowning modiication. Reerences [1] Carl C Osgood: Fatigue Design, 1st edition, Wiley-Interscience, 1970, New York. [2] Kramberger J., Sraml M., Potrc I., Flasker J.: Numerical calculation o bending Fatigue Lie o Thin-rim Spur Gears, J. Engineering racture mechanics 71(4 6)(2004) 647 [3] Glodez S., Abersek B., Flasker J., Zen Z.: Evaluation o Service Lie o Gears in Regard to Pitting, J. Engineering racture mechanics, 71 (2004) 429 [4] Akata E., Altmbalik M.T., Can Y.: Three Point Load Application in Single Tooth Bending Fatigue Test or Evaluation o Gear Blank Manuacturing Methods, Int. J. Fatigue 26 (2004) 785 [5] Zeping Wei: Stresses and Delection in Involute Spur Gears by Finite Element Method, M.Sc. Thesis, University o Saskatchewan, Saskatchewan, 2004 [6] Litvin F.L., Kim D.H.: Computerized Design, Generation and Simulation o Meshing o Modiied Involute Spur Gears with Localized Bearing Contact and Reduced Level o Transmission Errors, ASME J. Mech. Des. 119 (1997) 96 [7] Cheng-Kang Lee: Manuacturing Process or a Cylindrical Crown Gear Drive with a Controllable Fourth Order Polynomial Function o Transmission Error, J. Mater. Processing Technology 209 (2009) 3 [8] Seol H.I., Kim D.H.: The Kinematic and Dynamic Analysis o Crowned Spur Gear Drive, Computer Methods in Applied Mechanics and Engineering, 167(1 2) (1998) 109 [9] Litvin F.L.: Gear Geometry and Applied Theory, Prentice Hall, 1994, Englewood Clis, NJ [10] Hunglin Wang, Hsu-Pin Wang: Optimal Engineering Design o Spur Gear Sets, Mechanisms and Machines theory, l29 (7) (1994) 1071 [11] ANSYS, Release11.0, Structural Analysis Manual, ANSYS, 2007, USA [12] Tae Hyong Chong, Jae Hyong Myong, Ki Tae Kim: Tooth Modiication o Helical gears or Minimization o Vibration and Noise, Int.J. Korean Society o Precision Engineering, 2 (4) (2001) 5 [13] Lin Liu, Pines D.J.: The Inluence o Gear Design Parameters on Gear Tooth Damage Detection Sensitivity, ASME J.Mechanical Design, 124 (2002) 794 [14] Davis J.R.: Gear Materials, Properties and Manuacture, ASM International, 2005, (oh) [15] Boresi A.P., Schmidt R.J.: Advanced Mechanics o Materials, 2003, John Wiley & Sons Inc. [16] Hyperworks, version 9.0, Altair Engineering Inc. Received in editor s oice : August 27, 2010 Approved or publishing : January 7, 2011