Contact Stress Analysis for a Pair of Mating Gears

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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. 1. 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: [email protected] (K.-H. Lee). 0895-7177/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.06.055

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Contact Stress Analysis for a Pair of Mating Gears

Transcript of Contact Stress Analysis for a Pair of Mating Gears

Page 1: Contact Stress Analysis for a Pair of Mating Gears

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 Koreab Design Analysis Team, R & D Center, Nexen Tire Corporation, 30 Yusan-Dong, Yangsan-Si, Republic of Koreac Railway System Research Department, Korea Railroad Research Institute, Uiwang, 360-1, Republic of Korea

a r t i c l e i n f o

Article history:Received 22 March 2011Received in revised form 21 June 2011Accepted 22 June 2011

Keywords:Spur gearHelical gearContact stressLowest point of single-tooth contact(LPSTC)

Contact ratio

a b s t r a c t

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 teethat different contact positions during rotation. Two examples of spur and helical gearsare presented to investigate the respective variations of the contact stress in a pair ofmating gears with the contact position. The variation of the contact stress during rotationis compared with the contact stress at the lowest point of single-tooth contact (LPSTC) andthe AGMA (American Gear Manufacturers Association) equation for the contact stress. Inthis study, we can see that the gear design that considers the contact stress in a pair ofmating gears is more severe than that of the AGMA standard.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Gears are used to change the speed, magnitude, and direction of a power source. Gears are beingmost widely used as themechanical elements of power transmission. When two gears with unequal numbers of teeth are combined, a productiveoutput 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 wormgears. The strength determined from the AGMA and ISO standards is valid under the assumption that the load is uniformlydistributed 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 loadacting on a pair of teeth depends on themeshing stiffness of that pair. This leads to a variation in the load distribution acrosscontact points [3,4]. In Ref. [3], a mathematical model of load distribution along the contact line is suggested. This researchutilizes 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 determinationof the regions of stress concentration where failure or fracture may be initiated [5]. At this point, the change in the load isone of the elements that is involved in surface pitting. A preexisting pit aggravates the operating condition with noise andvibration, and if this pit is left as it is, an adventitious crack can be induced. However, though surface pitting by contactfatigue has been a concern of researchers for a long time, no universal theory has been defined as yet [6]. Therefore, in orderto take the contact fatigue into consideration, it is necessary to investigate the change in the contact stress following thechange 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 inthe contact stress at anypoint of the line of contact is analyzed through the finite elementmethod [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 inrevised form.∗ Corresponding author. Tel.: +82 51 200 7638; fax: +82 51 200 7656.

E-mail address: [email protected] (K.-H. Lee).

0895-7177/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.mcm.2011.06.055

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Nomenclature

CR contact ratioPb pitch of the base circlePg circular pitch of the base circlePn normal pitchb tooth widthm moduler0 radius of the addendum circlerp radius of the pitch circleΦA approaching angleΦC roll angleΦR recessing angleα0 pressure angleβ helix angle of the helical gearε contact ratioεα transverse contact ratioεβ overlap ratio

analysis allows the load at any point of the line of contact to be changed automatically. In this process, two-dimensionalfinite element analysis based on statics is performed for spur gears, while the three-dimensional finite element analysisbased on dynamics is undertaken for helical gears. The results obtained from finite element analysis are compared with thestresses yielded through AGMA standards.

2. Contact stress analysis of spur gears

There are two methods for analyzing the stress generated in meshing gear teeth through finite element analysis. Thefirst 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 methodis used for the meshing gears by applying a torque on the gear or pinion after modeling both the gear and the pinion. Thisstudy utilizes a two-dimensional finite element model to calculate the contact stress in a pair of mating spur gears. In orderto perform contact analysis while meshing the gear and the pinion, the geometrical shape of the meshing gear and pinionneeds 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 =

r2o2 − (rp2 × cos(αo))2 − rp2 × sin(αo) (2)

CB =

r2o1 − (rp1 × cos(αo))2 − rp1 × sin(αo). (3)

Then, the length AB can be derived as

AB =

r2o2 − (rp2 × cos(αo))2 +

r2o1 − (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 theLPSTC. Thus, the load is distributed across the two teeth over the intervals of AA′ and BB′. On the other hand, the load isapplied to a single tooth for the interval of A′B′ [6]. That is, though the load is transmitted through two pairs of contactedteeth 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 followingequation.

CB′= AB′

− AC = Pg · AC = π × m × cos(α0) · AC (5)

rB′ =

r2p1 + (CB′)2 − 2 × rp1 × CB′ × cos(αo + 90◦). (6)

Using the same method, the LPSTC of one pair of contact locations can be calculated as well.

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Fig. 1. 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 teethshould commence before the present tooth-surface contact is complete. The contact ratio means the ratio that representsthe average number of gear tooth pairs in contact for a pair of meshing gears. A greater contact ratio can create a smootheroperation.

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 calculatedfrom the definition of the contact ratio given in Eq. (7).

CR =lPb

. (7)

In Eq. (7), l is the length of the line of action AB in Fig. 1. The length l is represented as

l =r2o2 − r2b2 +

r2o1 − r2b1 − (rp1 + rp2) × sinαo, (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 withthe pinion and revolves, contact between gear teeth can be distinguished as the position with one pair of contacted gearteeth and the position with two pairs of contacted gear teeth. In general, the highest contact ratio leads to the least stressgeneration by distributing the load over the teeth. Suppose that gears are transmitting a constant torque. Then, since thesum of the contact forces that are working on the teeth surfaces should be constant as well, the contact force that is workingon two pairs of contacting teeth will be approximately half the force on one pair of contacting teeth. Fig. 2 is a diagram thatshows the change in the contact force for each contact location [9]. The figure means that the entire load will be transferredto one tooth as only one pair of teeth are in contact in sections A′ and B, as shown in Figs. 1 and 2.

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 =−SArb1

, SA = −rb1 tan(α0) +

(ro1)2 − (rb1)2 (11)

ΦR =SBrb1

, SB = −rb2 tan(α0) +

(ro2)2 − (rb2)2. (12)

The symbols rb, ro, and α0 are the same as those in Fig. 1.

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Fig. 2. Variation of the tooth load with the position of the contact point.

Table 1Meaning of the symbols used in Eq. (13) for spur gears.

Meaning Symbol Value

Transmitted tangential load (N) Ft 9032.0Elastic coefficient (MPa) ZE 187.0Overload factor Ko 1.0Dynamic factor Kv 1.0Size factor Ks 1.0Load distribution factor KH 1.0Face width (mm) W 120.0Geometry factor for pitting resistance ZI 0.115Pitch radius of the pinion (mm) r1 88.0

Table 2Data for the contact analysis of mating spur gears.

Parameter Pinion Gear

Number of teeth 22 55Pitch diameter (mm) 176.0 440.0Base diameter (mm) 165.39 413.46Torque (N mm) 1987 × 103

Modulus of elasticity (GPa) 200.0Module,m 8.0Pressure angle (°) 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 AGMAstress formulas. The contact stress σH based on the AGMA 2101-C95 standard is represented as follows [1].

σH = ZE

FtKoKvKs

KH

2r1WZI. (13)

The meanings and the values of the factors that are used in Eq. (13) are shown in Table 1.

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 embodythe gear tooth that has the involute gear tooth profile. If the basic specifications such as the pressure angle, module, numberof teeth and shift coefficient are given, the involute gear tooth profile can be embodied using the gear specification andinvolute 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 pressureangle is 20°. Thematerial is isotropically homogeneouswith an elastic ratio of 200GPa and a Poisson’s ratio of 0.3; further, themeshing gears are regarded as the plane stress problem that has a unit tooth thickness [9]. Table 2 shows the specificationsof the gear and the pinion and the material properties. Abaqus 6.9 [7] is used to perform stress analysis considering theLPSTC.

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Table 3Positions of the LPSTC and the HPSTC.

Position (mm) Pinion Gear

Radius to LPSTC 86.43 221.74Radius to HPSTC 88.96 219.09

(a) Boundary condition. (b) FE model of Case 1.

(c) FE model of Case 10.

Fig. 3. 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°. Thecontact stress on the pinion at the contact location is observed by increasing the roll angle from 2.7° to 27.4° in 10 equalincrements. The load of a torque is applied to the center of gear. The contact ratio is determined from Eq. (7) as 1.68. Acontact 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 arevolution, and one pair of teeth are contacting for the rest 32%.

Therefore, in the casewhen two pairs of teeth are contacting, the load is the same as half the transferred torque. From thispoint of view, 10 finite element models for each roll angle and the finite element model at the LPSTC for one pair of contactlocations are generated. The LPSTC for one pair of contact locations is calculated as shown in Table 3, which is derived fromEq. (6).

The finite elementmodel is constructed as shown in Fig. 3. Three teeth of pinion and one tooth of gear are included in thefinite element model. The finite element model was determined as Fig. 3 through the mesh convergence tests. A number ofnodes and elements in Fig. 3 are 2800 and 2240, respectively. The gear is considered as the master surface and the pinionas the slave. The boundary condition is represented as Fig. 3(a). A torque is applied directly to the ends of the rigid bars thatare connected with the teeth of the gear. The pinion rim is fixed, and a torque is applied on the gear. The finite elementmodels 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 endof contact under a roll angle of 27.4° are shown in Fig. 3(b) and (c), respectively.

A torque of 1 987× 103 Nmm 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 1 987 × 103 N mm, is applied on the gear when two pairs of teeth are incontact. Shell elements with four nodes are used, and the contact surface is densely meshed.

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Table 4Data for the contact analysis of mating helical gears.

Parameter Pinion Gear

Number of teeth 51 134Helix angle (°) 12.5 (left) 12.5 (left)Profile shift coefficient 0.1573 −0.0456Addendum (mm) 190.93 487.06Tooth width (mm) 118.0 110.0Center distance (mm) 330.0Normal module, m 3.5

Table 5Position of the LPSTC.

Position Pinion Gear

Radius to LPSTC (mm) 93.28 241.25

Table 6Meanings of the symbol used in Eq. (13) for helical gears.

Meaning Symbol Value

Transmitted tangential load (N) Ft 13254.0Elastic coefficient (MPa) ZE 187.0Overload factor Ko 1.0Dynamic factor Kv 1.0Size factor Ks 1.0Load distribution factor KH 1.0Face width (mm) W 118.0Geometry factor for pitting resistance ZI 0.115Pitch radius of the pinion (mm) r1 92.0

3. 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 tobe used at faster speeds than a standard spur gear [12]. Spur gear strength analysis that is based on the static finite elementanalysis generally is conducted by using a two-dimensional model. This is valid under the assumptions that changes inshape do not occur in the width-direction of the tooth and the stress variation generated in the width-direction of the toothis negligible. However, helical gears do not satisfy these assumptions. Thus, the strength analysis of helical gears uses three-dimensional 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 ofthe helical gears are shown in Table 4. 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 5.The contact ratio of helical gears is calculated through Eqs. (14)–(16).

ε = εα + εβ (14)

εα =

r2o2 − (rb2)2 +

r2o1 − (rb1)2 − (rp1 + rp2) × sin(α0)

π × m × cos(α0)(15)

εβ =b sinβ

π × m. (16)

The transverse contract ratio of a helical gear is computed in the samemanner 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 theoverlap ratio becomes the contact ratio of the helical gear. Since the calculated contact ratio is 1.97, when a helical geartooth is in complete contact, the next tooth will be in contact for approximately 97% of the time. The change in the contactforce 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 valuesused for analysis are presented in Table 6.

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Fig. 4. FE model of helical gears.

Fig. 5. 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 elementmodeling process is as follows. First, the geometric shape of a pair of helical gears that mesh with each other is modeledby using CATIA V5. While three or more teeth generally are included in the finite element modeling of a gear, the finiteelement model in this study includes six teeth for the gear and four for the pinion. A commercial program, HyperMesh, isused formodeling. The finite elementmodel of a pair of helical gears is displayed in Fig. 4. The elements used here are 8-nodehexahedrons. Through the mesh convergence tests, the number of nodes and elements in Fig. 4 were determined as 10 380and 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 pinionare 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 of3 227.9 N m is applied to the center of the gear. This is shown in Fig. 5.

4. Results

The results of finite element analysis for each contact location of spur gears are summarized in Table 7. The results for theLPSTC corresponding to one pair of contacting teeth and the contact stress from the AGMA standard are listed in Table 7. Thediagramof Fig. 6 represents the change in themaximumcontact stress according to the case.While the gears aremeshed andrevolved, we can see that the stress becomes the maximum around the LPSTC point corresponding to one pair of contactingteeth and then reduces.

The AGMA contact stress is calculated as 361 MPa from Eq. (13). The stresses in Cases 0 and 10 should be excluded fromthe analysis since they are generated from the application of a drastic load. Furthermore, tip relief is not considered in theanalysis. Then, the maximum stress at the contact location in Table 7 is 415.2 MPa in Case 7, and the stress distribution inCase 7 is as shown in Fig. 7. From Fig. 7, it can be seen that the maximum stress is generated at the contact point and thestress is propagated along diagonally.

If the AGMA stress is assumed to be the true value, the errors between the stress in Case 7 and the AGMA stress, andbetween the stress at LPSTC and the AGMA stress are represented in Table 8. Compared with the AGMA contact stresses, thecomputed stresses are slightly higher.

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Table 7Maximum stress obtained through contact stress analysis.

Load type Case Roll angle (°) Maximum stress (MPa)

Half load

0 0.00 802.91 2.74 273.22 5.48 250.03 8.22 255.8

Full load

4 10.96 360.05 13.70 376.46 16.44 396.77 19.18 415.2

Half load8 21.92 321.19 24.66 358.810 27.40 788.8

Full load LPSTC 16.35 395.3Full load AGMA 361.0

Fig. 6. Maximum stress for spur gears (10 cases).

Fig. 7. Contact regions of a pair of mating.

Table 8Error in contact stress analysis.

Case Error (%)

Case 7 14.6LPSTC 9.5

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Table 9Maximum stress generated from the contact stress analysis of helical gears.

Stress FE analysis AGMA standard

Maximum stress 683.2 MPa 537.0 MPa

Fig. 8. Contact stress in mating helical gears.

Fig. 9. Stress generated in the line of contact.

Table 9 shows the contact stresses obtained from dynamic finite element analysis at the LPSTC and calculated from theAGMA standard for mating helical gears. The contact stress based on the AGMA standard is 537.0 MPa. On the contrary, thestress contour via finite element analysis is shown in Fig. 8, and the stress value is 683.2 MPa. Similarly to the spur gearcase, the maximum stress is generated at the contact point and the stress is propagated along with the line of action. If thestrength based on the standard is assumed as the true value, the result of finite element analysis would have an error ofapproximately 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 eachposition on the line of contact, and it yields the largest stress value among the calculated values. The error could have resultedfrom the difficulty in coping with the variety of coefficients that are considered in the standard.

Fig. 9 shows the change in stress along the line of contact in the helical gear. It is obvious that the highest contact stressappears on the surface. The material of the spur and helical gears used in this study shows a surface fatigue strength ofapproximately 700MPa or higher when the smallest surface hardness is 240 HB. This implies that the spur and helical gearsanalyzed in this study are safe in terms of the contact fatigue strength. Then, tooth modification considering misalignmentmust be investigated to prevent surface pitting due to repeated use.

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5. Conclusions

This study presents the change in the contact stress of spur and helical gears in relation to the contact position. Regardingchanges in the contact stress, the maximum value measured at the lowest point single-tooth contact is compared with thecontact stress calculated based on the AGMA standard. According to the analysis, the design that considers the contact stressis stricter than the AGMA standard. The values calculated by using finite element analysis are below the contact fatiguestrength of the material; hence, they yield the appropriate strength and safety.

Acknowledgment

This study was supported by research funds from Dong-A University.

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