Mathematical and Computer Modelling 57 (2013) 40–49 Contents lists available at SciVerse ScienceDirect Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm Contact stress analysis for a pair of mating gears✩ Seok-Chul Hwang a , Jin-Hwan Lee b , Dong-Hyung Lee c , Seung-Ho Han a , Kwon-Hee Lee a,∗ a Department of Mechanical Engineering, Dong-A University, 840 Hadan2-dong, Saha-gu, Busan 604-714, Republic of Korea b Design Analysis Team, R & D Center, Nexen Tire Corporation, 30 Yusan-Dong, Yangsan-Si, Republic of Korea c Railway System Research Department, Korea Railroad Research Institute, Uiwang, 360-1, Republic of Korea article info Article history: Received 22 March 2011 Received in revised form 21 June 2011 Accepted 22 June 2011 Keywords: Spur gear Helical gear Contact stress Lowest point of single-tooth contact (LPSTC) Contact ratio abstract This paper presents a contact stress analysis for a pair of mating gears during rotation Contact stress analyses for spur and helical gears are performed between two gear teeth at different contact positions during rotation Two examples of spur and helical gears are presented to investigate the respective variations of the contact stress in a pair of mating gears with the contact position The variation of the contact stress during rotation is compared with the contact stress at the lowest point of single-tooth contact (LPSTC) and the AGMA (American Gear Manufacturers Association) equation for the contact stress In this study, we can see that the gear design that considers the contact stress in a pair of mating gears is more severe than that of the AGMA standard © 2011 Elsevier Ltd All rights reserved Introduction Gears are used to change the speed, magnitude, and direction of a power source Gears are being most widely used as the mechanical elements of power transmission When two gears with unequal numbers of teeth are combined, a productive output is realized with both the angular speeds and the torques of the two gears differing through a simple relationship AGMA [1] and ISO [2] standards generally are being used as the strength standard for the design of spur, helical, and worm gears The strength determined from the AGMA and ISO standards is valid under the assumption that the load is uniformly distributed along the line of contact However, in actuality, the load per unit length varies with the point of contact [3] In practice, in gear transmission, sudden load changes occur from the viewpoint of load transmission That is, the load acting on a pair of teeth depends on the meshing stiffness of that pair This leads to a variation in the load distribution across contact points [3,4] In Ref [3], a mathematical model of load distribution along the contact line is suggested This research utilizes the finite element method to investigate the variation of contact stress along the line of contact Stress analysis for gear teeth is regarded as a limiting factor for designers Stress analysis focuses on the determination of the regions of stress concentration where failure or fracture may be initiated [5] At this point, the change in the load is one of the elements that is involved in surface pitting A preexisting pit aggravates the operating condition with noise and vibration, and if this pit is left as it is, an adventitious crack can be induced However, though surface pitting by contact fatigue has been a concern of researchers for a long time, no universal theory has been defined as yet [6] Therefore, in order to take the contact fatigue into consideration, it is necessary to investigate the change in the contact stress following the change in the transmitted load in gear transmission The present work concentrates on the change in the contact stress that is generated in meshing gear teeth The change in the contact stress at any point of the line of contact is analyzed through the finite element method [7] Dynamic finite element ✩ This paper was presented at the PCO’2010 conference in Kuching, Malaysia on 2–4th December 2010 This paper was recommended for publication in revised form ∗ Corresponding author Tel.: +82 51 200 7638; fax: +82 51 200 7656 E-mail address: leekh@dau.ac.kr (K.-H Lee) 0895-7177/$ – see front matter © 2011 Elsevier Ltd All rights reserved doi:10.1016/j.mcm.2011.06.055 S.-C Hwang et al / Mathematical and Computer Modelling 57 (2013) 40–49 41 Nomenclature contact ratio pitch of the base circle circular pitch of the base circle normal pitch tooth width module radius of the addendum circle radius of the pitch circle approaching angle roll angle recessing angle pressure angle helix angle of the helical gear contact ratio transverse contact ratio overlap ratio CR Pb Pg Pn b m r0 rp ΦA ΦC ΦR α0 β ε εα εβ analysis allows the load at any point of the line of contact to be changed automatically In this process, two-dimensional finite element analysis based on statics is performed for spur gears, while the three-dimensional finite element analysis based on dynamics is undertaken for helical gears The results obtained from finite element analysis are compared with the stresses yielded through AGMA standards Contact stress analysis of spur gears There are two methods for analyzing the stress generated in meshing gear teeth through finite element analysis The first is to apply the concentrated load at the load position directly Then, the bending stress of the gear can be calculated This method is being widely used owing to its simplicity, but the contact stress cannot be calculated The second method is used for the meshing gears by applying a torque on the gear or pinion after modeling both the gear and the pinion This study utilizes a two-dimensional finite element model to calculate the contact stress in a pair of mating spur gears In order to perform contact analysis while meshing the gear and the pinion, the geometrical shape of the meshing gear and pinion needs to be modeled 2.1 Highest and lowest points of single-tooth contact in spur gears If the involute gear is revolving as shown in Fig 1, the length of the action AB is calculated as follows [6,8] AB = ε × Pg = AC + CB (1) The lengths AC and CB are represented as AC = CB = − (rp2 × cos(αo ))2 − rp2 × sin(αo ) ro2 (2) ro1 − (rp1 × cos(αo ))2 − rp1 × sin(αo ) (3) Then, the length AB can be derived as AB = ro2 − (rp2 × cos(αo ))2 + ro1 − (rp1 × cos(αo ))2 − (rp1 + rp2 ) × sin(αo ) (4) ′ ′ The teeth of the gear and the pinion contact between A and B along the line AB In Fig 1, B is the HPSTC, and A is the LPSTC Thus, the load is distributed across the two teeth over the intervals of AA′ and BB′ On the other hand, the load is applied to a single tooth for the interval of A′ B′ [6] That is, though the load is transmitted through two pairs of contacted teeth in AA′ and B′ B, the entire load is being transmitted through only one pair of contacted teeth in A′ B′ The lengths of AB′ and A′ B are the same as the circular pitch Pg · Therefore, the length CB′ and the radius rB′ are determined by the following equation CB′ = AB′ − AC = Pg · AC = π × m × cos(α0 ) · AC rB′ = rp1 + (CB′ )2 − × rp1 × CB′ × cos(αo + 90◦ ) Using the same method, the LPSTC of one pair of contact locations can be calculated as well (5) (6) 42 S.-C Hwang et al / Mathematical and Computer Modelling 57 (2013) 40–49 Fig Geometric determination of HPSTC and LPSTC 2.2 Contact ratio of spur gears In order for spur gears to transmit revolutions continuously, surface contact between at least one pair of subsequent teeth should commence before the present tooth-surface contact is complete The contact ratio means the ratio that represents the average number of gear tooth pairs in contact for a pair of meshing gears A greater contact ratio can create a smoother operation The contact ratio CR is the average number of teeth that are in contact when the gears are meshed and revolved That is, the contact ratio is defined as a number of teeth in contact as these teeth pass through the contact area It can be calculated from the definition of the contact ratio given in Eq (7) CR = l Pb (7) In Eq (7), l is the length of the line of action AB in Fig The length l is represented as l= 2 ro2 − rb2 + 2 ro1 − rb1 − (rp1 + rp2 ) × sin αo , (8) and the pitch of the base circle Pb is equal to the normal pitch Pn Thus, Pb = Pn = π × m × cos(α0 ) (9) The applied load on the pinion changes with the contact ratio This means that when the spur gear is meshed with the pinion and revolves, contact between gear teeth can be distinguished as the position with one pair of contacted gear teeth and the position with two pairs of contacted gear teeth In general, the highest contact ratio leads to the least stress generation by distributing the load over the teeth Suppose that gears are transmitting a constant torque Then, since the sum of the contact forces that are working on the teeth surfaces should be constant as well, the contact force that is working on two pairs of contacting teeth will be approximately half the force on one pair of contacting teeth Fig is a diagram that shows the change in the contact force for each contact location [9] The figure means that the entire load will be transferred to one tooth as only one pair of teeth are in contact in sections A′ and B, as shown in Figs and The roll angle corresponding to the length of the contact path can be defined as follows [8,9] Φc = ΦA + ΦR (10) The angles ΦA and ΦR are represented as ΦA = ΦR = −SA rb1 SB rb1 , , SA = −rb1 tan(α0 ) + SB = −rb2 tan(α0 ) + (ro1 )2 − (rb1 )2 (11) (ro2 )2 − (rb2 )2 (12) The symbols rb , ro , and α0 are the same as those in Fig S.-C Hwang et al / Mathematical and Computer Modelling 57 (2013) 40–49 43 Fig Variation of the tooth load with the position of the contact point Table Meaning of the symbols used in Eq (13) for spur gears Meaning Symbol Value Transmitted tangential load (N) Elastic coefficient (MPa) Overload factor Dynamic factor Size factor Load distribution factor Face width (mm) Geometry factor for pitting resistance Pitch radius of the pinion (mm) Ft ZE Ko Kv Ks KH W ZI r1 9032.0 187.0 1.0 1.0 1.0 1.0 120.0 0.115 88.0 Table Data for the contact analysis of mating spur gears Parameter Pinion Gear Number of teeth Pitch diameter (mm) Base diameter (mm) Torque (N mm) Modulus of elasticity (GPa) Module, m Pressure angle (°) 22 176.0 165.39 55 440.0 413.46 1987 × 103 200.0 8.0 20.0 2.3 The AGMA standard In this study, the contact stress of an involute spur gear pair is studied through the finite element method and AGMA stress formulas The contact stress σH based on the AGMA 2101-C95 standard is represented as follows [1] σH = ZE Ft Ko Kv Ks KH 2r1 WZI (13) The meanings and the values of the factors that are used in Eq (13) are shown in Table 2.4 Finite element analysis of spur gears The gear tooth profile can be distinguished as involute and trochoid fillet curves First of all, it is necessary to embody the gear tooth that has the involute gear tooth profile If the basic specifications such as the pressure angle, module, number of teeth and shift coefficient are given, the involute gear tooth profile can be embodied using the gear specification and involute function [8] In this research, the number of gear teeth is 55, the number of teeth in the pinion is 22, the module is 8, and the pressure angle is 20° The material is isotropically homogeneous with an elastic ratio of 200 GPa and a Poisson’s ratio of 0.3; further, the meshing gears are regarded as the plane stress problem that has a unit tooth thickness [9] Table shows the specifications of the gear and the pinion and the material properties Abaqus 6.9 [7] is used to perform stress analysis considering the LPSTC 44 S.-C Hwang et al / Mathematical and Computer Modelling 57 (2013) 40–49 Table Positions of the LPSTC and the HPSTC Position (mm) Pinion Gear Radius to LPSTC Radius to HPSTC 86.43 88.96 221.74 219.09 (a) Boundary condition (b) FE model of Case (c) FE model of Case 10 Fig Two-dimensional finite element model of mating spur gears The roll angle Φc represented in Eq (10) is calculated as 27.4° That is, the contact starts from 0° and ends at 27.4° The contact stress on the pinion at the contact location is observed by increasing the roll angle from 2.7° to 27.4° in 10 equal increments The load of a torque is applied to the center of gear The contact ratio is determined from Eq (7) as 1.68 A contact ratio of 2.0 indicates that two pairs always are in contact, which means that as soon as one pair goes out of contact, a new pair comes into contact [10,11] Thus, a contact ratio of 1.68 means that two pairs of teeth are contacting for 68% of a revolution, and one pair of teeth are contacting for the rest 32% Therefore, in the case when two pairs of teeth are contacting, the load is the same as half the transferred torque From this point of view, 10 finite element models for each roll angle and the finite element model at the LPSTC for one pair of contact locations are generated The LPSTC for one pair of contact locations is calculated as shown in Table 3, which is derived from Eq (6) The finite element model is constructed as shown in Fig Three teeth of pinion and one tooth of gear are included in the finite element model The finite element model was determined as Fig through the mesh convergence tests A number of nodes and elements in Fig are 2800 and 2240, respectively The gear is considered as the master surface and the pinion as the slave The boundary condition is represented as Fig 3(a) A torque is applied directly to the ends of the rigid bars that are connected with the teeth of the gear The pinion rim is fixed, and a torque is applied on the gear The finite element models of Case 1, which is just after the start of gear contact with a roll angle of 2.7°, and of Case 10 that represents the end of contact under a roll angle of 27.4° are shown in Fig 3(b) and (c), respectively A torque of 987 × 103 N mm is applied on the gear when a pair of teeth are in contact with each other On the contrary, a torque of 993.5 × 103 N mm, which is half of 987 × 103 N mm, is applied on the gear when two pairs of teeth are in contact Shell elements with four nodes are used, and the contact surface is densely meshed S.-C Hwang et al / Mathematical and Computer Modelling 57 (2013) 40–49 45 Table Data for the contact analysis of mating helical gears Parameter Pinion Gear Number of teeth Helix angle (°) Profile shift coefficient Addendum (mm) Tooth width (mm) Center distance (mm) Normal module, m 51 12.5 (left) 0.1573 190.93 118.0 134 12.5 (left) −0.0456 487.06 110.0 330.0 3.5 Table Position of the LPSTC Position Pinion Gear Radius to LPSTC (mm) 93.28 241.25 Table Meanings of the symbol used in Eq (13) for helical gears Meaning Symbol Value Transmitted tangential load (N) Elastic coefficient (MPa) Overload factor Dynamic factor Size factor Load distribution factor Face width (mm) Geometry factor for pitting resistance Pitch radius of the pinion (mm) Ft ZE Ko Kv Ks KH W ZI r1 13254.0 187.0 1.0 1.0 1.0 1.0 118.0 0.115 92.0 Contact stress analysis of helical gears Helical gears are used in machines that have fast rotating parts; they are employed for reducing vibration and noise Because of the angle of the teeth, a helical gear tends to meet and move with other gears more smoothly, allowing them to be used at faster speeds than a standard spur gear [12] Spur gear strength analysis that is based on the static finite element analysis generally is conducted by using a two-dimensional model This is valid under the assumptions that changes in shape not occur in the width-direction of the tooth and the stress variation generated in the width-direction of the tooth is negligible However, helical gears not satisfy these assumptions Thus, the strength analysis of helical gears uses threedimensional dynamic finite element analysis The model allows for changes in shape due to the helix angle in the width-direction of the tooth The specifications of the helical gears are shown in Table The following is the procedure for analyzing the strength of helical gears 3.1 LPSTC, contact ratio, and the AGMA standard for helical gears The position of LPSTC in the helical gear and pinion is listed in Table The contact ratio of helical gears is calculated through Eqs (14)–(16) ε = εα + εβ 2 ro2 − (rb2 )2 + ro1 − (rb1 )2 − (rp1 + rp2 ) × sin(α0 ) εα = π × m × cos(α0 ) b sin β εβ = π ×m (14) (15) (16) The transverse contract ratio of a helical gear is computed in the same manner as for the spur gear contact ratio However, for helical gears, the overlap ratio owing to the helix angle is calculated, and the sum of the transverse contact ratio and the overlap ratio becomes the contact ratio of the helical gear Since the calculated contact ratio is 1.97, when a helical gear tooth is in complete contact, the next tooth will be in contact for approximately 97% of the time The change in the contact force that is applied to a pair of gear teeth is the same as that for spur gears The contact stress for the helical gear can be calculated using Eq (14) The description of each coefficient and the values used for analysis are presented in Table 46 S.-C Hwang et al / Mathematical and Computer Modelling 57 (2013) 40–49 Fig FE model of helical gears Fig Boundary and loading conditions for contact analysis for helical gears 3.2 Finite element analysis of helical gears The gear used in this section is a helical gear, and its tooth form takes the shape of an involute curve The finite element modeling process is as follows First, the geometric shape of a pair of helical gears that mesh with each other is modeled by using CATIA V5 While three or more teeth generally are included in the finite element modeling of a gear, the finite element model in this study includes six teeth for the gear and four for the pinion A commercial program, HyperMesh, is used for modeling The finite element model of a pair of helical gears is displayed in Fig The elements used here are 8-node hexahedrons Through the mesh convergence tests, the number of nodes and elements in Fig were determined as 10 380 and 7680, respectively The gear is considered as the master surface and the pinion as the slave For the boundary condition, the inner rim of the pinion and all the degrees of freedom in the vertical section of the pinion are constrained, and the inner side of the rim of the gear is connected to the center of the gear by using beam elements Also, the degrees of freedom in the center of the gear are constrained, excluding the z-directional rotation Then, a torque of 227.9 N m is applied to the center of the gear This is shown in Fig Results The results of finite element analysis for each contact location of spur gears are summarized in Table The results for the LPSTC corresponding to one pair of contacting teeth and the contact stress from the AGMA standard are listed in Table The diagram of Fig represents the change in the maximum contact stress according to the case While the gears are meshed and revolved, we can see that the stress becomes the maximum around the LPSTC point corresponding to one pair of contacting teeth and then reduces The AGMA contact stress is calculated as 361 MPa from Eq (13) The stresses in Cases and 10 should be excluded from the analysis since they are generated from the application of a drastic load Furthermore, tip relief is not considered in the analysis Then, the maximum stress at the contact location in Table is 415.2 MPa in Case 7, and the stress distribution in Case is as shown in Fig From Fig 7, it can be seen that the maximum stress is generated at the contact point and the stress is propagated along diagonally If the AGMA stress is assumed to be the true value, the errors between the stress in Case and the AGMA stress, and between the stress at LPSTC and the AGMA stress are represented in Table Compared with the AGMA contact stresses, the computed stresses are slightly higher S.-C Hwang et al / Mathematical and Computer Modelling 57 (2013) 40–49 Table Maximum stress obtained through contact stress analysis Roll angle (°) Load type Case Half load 0.00 2.74 5.48 8.22 802.9 273.2 250.0 255.8 Maximum stress (MPa) Full load 10.96 13.70 16.44 19.18 360.0 376.4 396.7 415.2 Half load 10 21.92 24.66 27.40 321.1 358.8 788.8 Full load Full load LPSTC AGMA 16.35 395.3 361.0 Fig Maximum stress for spur gears (10 cases) Fig Contact regions of a pair of mating Table Error in contact stress analysis Case Error (%) Case LPSTC 14.6 9.5 47 48 S.-C Hwang et al / Mathematical and Computer Modelling 57 (2013) 40–49 Table Maximum stress generated from the contact stress analysis of helical gears Stress FE analysis AGMA standard Maximum stress 683.2 MPa 537.0 MPa Fig Contact stress in mating helical gears Fig Stress generated in the line of contact Table shows the contact stresses obtained from dynamic finite element analysis at the LPSTC and calculated from the AGMA standard for mating helical gears The contact stress based on the AGMA standard is 537.0 MPa On the contrary, the stress contour via finite element analysis is shown in Fig 8, and the stress value is 683.2 MPa Similarly to the spur gear case, the maximum stress is generated at the contact point and the stress is propagated along with the line of action If the strength based on the standard is assumed as the true value, the result of finite element analysis would have an error of approximately 27% The strength based on the standard is the largest value among the stress values on the line of contact On the other hand, strength evaluation based on finite element analysis has the advantage of calculating the stress at each position on the line of contact, and it yields the largest stress value among the calculated values The error could have resulted from the difficulty in coping with the variety of coefficients that are considered in the standard Fig shows the change in stress along the line of contact in the helical gear It is obvious that the highest contact stress appears on the surface The material of the spur and helical gears used in this study shows a surface fatigue strength of approximately 700 MPa or higher when the smallest surface hardness is 240 HB This implies that the spur and helical gears analyzed in this study are safe in terms of the contact fatigue strength Then, tooth modification considering misalignment must be investigated to prevent surface pitting due to repeated use S.-C Hwang et al / Mathematical and Computer Modelling 57 (2013) 40–49 49 Conclusions This study presents the change in the contact stress of spur and helical gears in relation to the contact position Regarding changes in the contact stress, the maximum value measured at the lowest point single-tooth contact is compared with the contact stress calculated based on the AGMA standard According to the analysis, the design that considers the contact stress is stricter than the AGMA standard The values calculated by using finite element analysis are below the contact fatigue strength of the material; hence, they yield the appropriate strength and safety Acknowledgment This study was supported by research funds from Dong-A University References [1] AGMA standards, http://www.agma.org/ [2] ISO gear standards, http://www.iso.org/ [3] J.I Pedrero, M Pleguezuelos, M Munoz, Critical stress and load conditions for pitting calculations of involute spur and helical gear teeth, Mechanism and Machine Theory 46 (2011) 425–437 [4] S Li, Effect of addendum on contact strength, bending strength and basic performance parameters of a pair of spur gears, Mechanism and Machine Theory 43 (2008) 1557–1584 [5] I Atanasovska, V Nikoli-Stanojlovi, D Dimitrijevi, D Momăcilovi, Finite element model for stress analysis and nonlinear contact analysis of helical gears, Scientific Technical Review (Serbia J.) LVIX (2009) 61–68 [6] G.K Raptis, N.T Costopoulos, A.G Papadopulos, D.A Tsolakis, Rating of spur gear strength using photoelasticity and the finite element method, American Journal of Engineering and Applied Sciences (2010) 222–231 [7] Dassault Systemes, Abaqus analysis user’s manual ver 6.9 2009 [8] F.L Litvin, A Fuentes, Gear Geometry and Applied Theory, Cambridge University Press, 2004, pp 287–303 [9] R.A Hassan, Contact stress analysis of spur gear teeth pair, World Academy of Science, Engineering and Technology 58 (2009) 611–616 [10] Y.C Chen, C.B Tsay, Stress analysis of a helical gear set with localized bearing contact, Finite Elements in Analysis and Design 38 (2002) 707–723 [11] R.L Norton, Design of Machinery, McGraw-hill Companies, 2001 [12] http://www.ehow.com/how_6245632_draw-helical-gear.html, 2011 ... stress analysis of helical gears Stress FE analysis AGMA standard Maximum stress 683.2 MPa 537.0 MPa Fig Contact stress in mating helical gears Fig Stress generated in the line of contact Table... (7) as 1.68 A contact ratio of 2.0 indicates that two pairs always are in contact, which means that as soon as one pair goes out of contact, a new pair comes into contact [10,11] Thus, a contact. .. the ratio that represents the average number of gear tooth pairs in contact for a pair of meshing gears A greater contact ratio can create a smoother operation The contact ratio CR is the average