ORIGINAL ARTICLE

Profile modification—a design approach for increasingthe tooth strength in spur gear

Shanmugasundaram Sankar & Muthusamy Nataraj

Received: 6 August 2010 /Accepted: 8 November 2010# Springer-Verlag London Limited 2010

Abstract This research paper discusses a novel method toprevent the tooth failure in the spur gear by introducingcircular root filet instead of standard trochoidal root filet inthe gear. In general, gears with less than 17 teeth had theproblem of undercutting during gear manufacturing processdepending on the tip radius of the hob (cutter) whichminimizes the strength of the gear at the root. Analysis iscarried out using ANSYS software version 11.0 for theexisting standard design gear teeth as well as the proposeddesign gear teeth to evaluate the performance. In order tofacilitate the analysis, the gear tooth was considered as acantilever beam and the tooth force was applied normal tothe profile at the highest point of single tooth contact. Thestudy reveals that the proposed design (circular root filet)exhibits higher bending strength rather than the standardtrochoidal root filet design.

Keywords Bending stress . Contact shear stress . Circularroot filet . Deflection . FEA . Spur gear . Trochoidal root filet

Nomenclature list of variablesZ Number of teethMn Normal moduleα Normal pressure angleβ Helix angle

b Gear face widthd Pitch circle of the gearha Addendum of the gearhf Dedendum of the gearh Tooth heightt Tooth thickness of gear at the rootx1 Profile shift co efficient of gearu Gear ratioeps_g Total contact ration Speed of the gearT Transmitted torqueP Rated powerv The pitch line velocityFn Normal forceFt Tangential force on toothFr Radial forceσ Tooth bending stressY Lewis form factorKV Velocity factorRp Yield strengthE Young’s modulusNy Poison’s ratio

AcronymsFEA Finite element analysisHPSTC Highest point of single tooth contactBEM Boundary element methodCD Centre distance

1 Introduction

The objective of the gear drive is to transmit high powerwith comparatively smaller dimensions of the drivingsystem and to run reasonably free of noise and vibration

S. Sankar (*)Department of Mechanical Engineering, Anna University,Coimbatore 641047 Tamil Nadu, Indiae-mail: shanmugasundaramsankar@yahoo.com

M. NatarajDepartment of Engineering Design,Government College of Technology,Coimbatore, Indiae-mail: m_natanuragct@yahoo.com

Int J Adv Manuf TechnolDOI 10.1007/s00170-010-3034-3

with least manufacturing and maintenance cost. There is ademand for gears with higher load-carrying capacity andincreased fatigue life. In order to attain this, researchers inthe gear field have been working on the development ofadvanced materials, new heat treatment methods, designingthe gear with stronger teeth, and methods of gearmanufacturing to tackle the problem of failure of gears.However, in modern gear practice and manufacturing, themajority of gear applications are covered by the standard20° involute teeth generated by rack, hob, and CNC cuttingprocess. This has many advantages such as that ofinterchangeability, insensitivity to change in nominal centerdistance, commercial availability, and easy manufacturingby conventional methods (i.e., hobbing). However, the gearmade of standard involute profile with less than 17 teethwas liable to undercutting phenomenon in nature. Inundercutting, the tooth filet is generated as the tip of thecutter removes material from the involute profile, thusresulting in teeth that have less tooth thickness at the root,where the critical section is usually located. This reducesthe tooth strength and leads for the crack initiation andpropagation at the root filet area. In order to manage thisweakening of the gear teeth, many solutions have beenproposed. Spitas and Costopoulos [1] have introduced one-sided involute asymmetric spur gear teeth to increase theload-carrying capacity and to combine the meshing prop-erties. Tesfahunegn and Rosa [2] have investigated theinfluence of the shape of profile modifications on trans-mission error, root stress, and contact pressure through non-linear finite element approach. Spitas and Costopoulos [3]have carried out an analysis by introducing circular rootfilet instead of the standard trochoidal root filet in the spurgear and investigated numerically using boundary elementmethod. The analysis infers that the teeth with circular filetdesign exhibits higher bending strength in certain caseswithout affecting the pitting resistance, since, the geometryof the load-carrying involute is not changed. He also hasdetermined the geometry of the generating tool in order tobe able to cut the teeth using a generating method. Fredetteand Brown [4] have increased the fatigue life of the gearteeth by reducing the root tensile stress through stressrelieving method in critical areas. Ciavarella and Demelio[5] concluded that the fatigue life is lower on gears havingless number of teeth and also recommended to use thecomplete crack propagation analysis from the early stage ofthe gear design process. He also has optimized the specificsliding, stress concentration, and fatigue life of the gearthrough numerical methods. Hebbal and Math [6] havereduced the root filet stress in the spur gear using internalstress relieving features of different shapes. Senthilvelanand Gnanamoorthy [7] have analyzed the effect of geartooth filet radius on the performance of injection moldednylon 6/6 gears. Tae Hyong Chong and Jae Hyong Myong

[8] conducted a study to calculate the optimum amounts oftooth profile modification for minimization of vibration andnoise in helical gears.

Beghini [9] proposed a simple method to reduce thetransmission error for a given spur gear set, at the nominaltorque, by means of the profile modification parameters. Toimprove the gear tooth strength, a lot of work has beencarried out, but all mostly employed positive profileshifting. These contribution exhibits lower pitting andscoring resistance and lower contact ratio resulting in morenoise and vibration during the power transmission [10]. Thenovelty of the work in this paper is that a profilemodification is being made on the spur gear such that thegear teeth comprising a standard involute working profilefrom the tip circle diameter of the gear to the base circlediameter of the gear and of a circular root filet from thebase circle to the root circle replacing the conventionaltrochoidal root filet to increase the tooth strength and tominimize the failure of the tooth in the spur gear. Anattempt has been made to find the non-linear contact shearstress in gears using ANSYS software for predicting theperformance evaluation.

2 Geometrical modeling

Consider the involute spur gear tooth of circular root filetillustrated in Fig. 1 where point O′ is the center of the gear,axis Oy′ is the axis of symmetry of the tooth and point B′ isthe point where the involute profile starts (from the formcircle rs). Point A′ is the point of tangency of the circularfilet with the root circle rf. Point D′ lying on (ε2) identicalto OA′ represents the center of the circular filet. Line (ε3) istangent to the root circle at A′ and intersects with line (ε1)at C′. The filet is tangent to the line (ε1) at point E′. Sinceit is always rs>rf, the proposed circular filet can be

Fig. 1 Geometry of the circular filet

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implemented without exceptions on all spur gears irrelevant ofthe number of gear teeth or other manufacturing parameters. Acomparison of the geometrical shape of a tooth of circular rootfilet with that of the standard trochoidal root filet is presentedin Fig. 2. The geometry of the circular root filet whichcoordinates (points A, B, C, D, and E) in Fig. 1 are obtainedusing the following formulas:

XB ¼ rf SinΩs ð1Þ

YB ¼ rfCosΩs ð2Þ

YC ¼ XC

TanΩSð3Þ

XA ¼ rf Sin z þΩsð Þ ð4Þ

YA ¼ rfCos z þΩsð Þ ð5Þ

XE ¼ OCþ CEð ÞsinΩs ð6Þ

YE ¼ OCþ CEð ÞcosΩs ð7Þ

XD ¼ rf þ AD� �

Sin z þΩSð Þ ð8Þ

YD ¼ rf þ AD� �

Cos z þΩSð Þ ð9Þ

XC ¼ rfTanΩS

Sin z þΩSð ÞTanΩS þ Cos z þΩSð Þ ð10Þ

The remaining portion of the tooth profile betweenpoints B′ and E′ is a straight line.

Angle ωs /2 that corresponds to the arc SS /2 (Fig. 1) isgiven by the Eq. 11:

ws=2 ¼ SS=2

rS¼ ΩS ð11Þ

Angle ζ (Fig. 1) takes values between o and ζmax (Eq. 12):

zmax ¼pZ�ΩS ð12Þ

2.1 Part modeling

In actual practice, the trochoidal root filet is formed in gearsduring manufacturing process depending on the tip radiusof the hob. It was proved that the bending stress decreasesgradually in gears as the number of teeth increases and thetotal contact ratio increases [3]. In order to overcome theabove problem, a novel method namely, circular root filetinstead of standard trochoidal root filet is introduced ingears having less than 17 teeth to decrease the bendingstress at the root and gear tooth failure due to undercutting.

Fig. 2 Superposition of circular filet on a standard tooth

Table 1 Specifications of spur gear

Gear tooth type Standard involute full depth

Number of gear teeth (Z) 13

Normal module (Mn) 2 mm

Gear face width (b) 12.85 mm

Pressure angle (α) 20°

Helix angle (β) Spur gear

Tooth root filet Trochoidal and circular (proposed)

Center distance (CD) 26.0 mm

Profile shift coefficient (x1)

0.05 mm

Total contact ration(eps_g)

1.442

Gear ratio (u) 1.0

Addendum (ha) 2.10 mm

Dedendum (hf) 2.40 mm

Material and heattreatment

18CrNiMo7, case hardened andtempered

Method of finishing teeth Profile grinding

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According to Gitin Maitra [11], if a gear is undercut for onereason or another, it may become sometimes necessary toknow the magnitude of the undercutting radius. Under suchcircumferences, Gitin Maitra proposed a formula (Eq. 13)to find out the minimum number of teeth to avoidundercutting which is as follows:

Zmin ¼ 2

Sin2að13Þ

This expression is valid for standard gear tooth with theaddendum of the rack being equal to the module Mn′.However, the undercut–free minimum number of teeth isgiven by Eq. 14:

Zmin ¼ 2hcaMn Sin2a

ð14Þ

Where, hca is=the addendum of the rack cutter withouttip filet rounding

Table 1 gives the specifications of the 13 teeth spur gearused in this investigation. These design specifications havebeen arrived from the KISS soft [12] a calculation programsfor machine design according to DIN 3990 method B′standards for the given center distance. The law of gearing[13] requires that the mating gear should have the samenormal pressure angle α′ and the same module Mn′ in orderto be able to mesh properly. The above points wereconsidered in this analysis. The virtual model of the spur

gear with 13 teeth having circular as well as trochoidal rootfilet are modeled in Pro-E wildfire 3.0 software andanalyzed in ANSYS software version 11.0 to evaluate theperformance.

3 Force analyses

The load transmitting capability of the gear tooth isanalyzed and checked for designing a gear system. Theeffective circumferential force on the tooth at the pitchcircle of the gear while in meshing is estimated. Two kindsof stresses are induced in gear pair during the powertransmission from one shaft to another shaft. They are: (1)bending stress—induced on gear teeth due to the tangentialforce developed by the power and (2) surface contact stressor compressive stress. The load is assumed as uniformlydistributed along the face width of the tooth.

3.1 Components of forces

When the mating gears are engaged, the line of contactstarts from the bottom of the tooth to the tip of the toothalong the tooth profile for the pinion and from the tip of the

Table 2 Force components

Speed (r/min) Torque (N-mm) Force components (Newton)

Ft Fn Fr

1,500 47,480 3,652.3 3,886.7 1,329.3

2,000 35,600 2,739.0 2,914.7 996.9

2,500 28,490 2,191.2 2,331.8 797.5

3,000 23,740 1,826.1 1,943.3 664.6Fig. 4 FEA meshed model of tooth with trochoidal root filet

Table 3 Material properties

Gear material Alloy steel (18Cr NiMo7)

Young’s modulus (E) 2.1×105 N/mm2

Density 7.85×10−6 kg/m

Poison’s ratio (ny) 0.3

Yield strength (Rp) 366–1,798 MPa

Fig. 3 Force diagram of spur gear

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tooth to the bottom of the tooth for the gear. While the forceis acting at the tip of the tooth, the long distance of actionfrom the root causes maximum stress at the bottom of thetooth. Hence, the tangential force was applied at the tip ofthe tooth along the face width during bending stressanalysis. Referring now to Fig. 3, the normal force F

0n acts

along the pressure line. The normal force F0n is resolved

into two components, namely, (1) Tangential force (Ft) and(2) radial force (Fr). This normal force produces an equaland an opposite reaction at the gear tooth. As the gear ismounted on a shaft, the radial force F

0n acts at the center of

the shaft and is equal in magnitude but an opposite indirection to the normal force F

0n. For the given data, the

force F0n known as the tangential force or transmitting load

was derived from the standard Eq. 15:

Ft ¼ 2;000 x T

dð15Þ

Fig. 5 FEA meshed model of tooth with circular root filet

Fig. 6 ANSYS results

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Where,

T ¼ 9;550 x P

n

The tangential force F0t constitutes a couple which

produces the torque on the pinion which in turn drives themating gears. The tangential force bends the tooth and theradial force compresses it. The magnitude of the Fr isarrived using Eq. 16.

Fr ¼ Fn x Sin a ð16ÞWhere,

Fn ¼ Ft

cos a x cos b

As far as the transmission power is concerned, thetangential force is really the useful component, because theradial component serves no useful purpose. Irrespective ofthe value of the contact ratio, the gear forces are taken to beeffective on a single pair of teeth in the mesh. Thecomponents of force, i.e., Ft, Fr, and Fn are computed(Table 2) for a power value of 10 hp at 1,500, 2,000, 2,500,and 3,000 r/min.

4 Finite element analyses

A finite element model with a single tooth is considered forbending stress analysis and a gear pair is chosen for contactshear stress analysis. The gear material strength is a majorconsideration for the operational loading and environment.Generally, cast iron is used in normal loading and higher wearresisting conditions. In modern practice, the heat treated alloy

steels are used to overcome the wear resistance. In this work,heat-treated alloy steel is taken for analysis. The materialproperties chosen for finite element analysis are presented inTable 3. The fatigue and yielding of the gear tooth as a resultof excessive bending stresses are the two important geardesign considerations. In order to predict the fatigue and theyielding, the maximum stresses on the tensile and compres-sive sides of the tooth are essential. The gear tooth surface isnonlinear and in this analysis, the tooth forces are appliednormal to the profile at the highest point of single toothcontact to estimate the bending stress during single toothanalysis and the contact shear stress of a gear pair duringnonlinear contact analysis. The following are the conditionsassumed during the analysis:

➢ There is no sliding in the contact zone between thetwo bodies.

➢ The contact surface is continuous and smooth.

Since the solid 92 element has a quadratic displacementbehavior and is well suited to model irregular meshes suchas produced from various CAD/CAM systems, this Solid92 element type with 10 nodes is selected to describe thegear and the tooth deflection in ANSYS software version

Table 5 Lewis maximum bending stress values

Speed r/min Maximum bending stress (N/mm2)

1,500 602.904

2,000 459.69

2,500 372.679

3,000 314.261Fig. 7 FEA meshed model of a gear pair with circular root filet

Table 4 FEA results at 2,000 r/min—bending stress

Speed(r/min)

Deflection (mm) Bending stress (N/mm2) Stiffness (N/mm)For 13 teeth For 13 teeth For 13 teeth

Trochoidal root filet Circular root filet Trochoidal root filet Circular root filet Trochoidal root filet Circular root filet

2000 0.013089 0.010323 447.435 442.457 279,035.83 353,802.18

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11. This element also has plasticity, creep, swelling, stressstiffening, large deflection, and large strain capabilities. The10 node 3-D solid elements with 3° of freedom per node(UX, UY, and UZ) are stacked to model through thethickness discontinuities. As the gears are made of heat-treated alloy steel, carburized and case hardened alloy steel(18CrNiMo7) is taken for analyzing the root stresses. Fornon-linear contact analysis, CONTA174 element wasintroduced between the involute profiles of mating gear tohave surface-to-surface contact. This element has 3° offreedom at each node; translations in the nodal x, y, and zdirections.

4.1 Bending stress analysis

In order to facilitate the finite element analysis, the geartooth was considered as a cantilever beam. All degrees offreedom were constrained at the root circle and the toothforce was applied normal to the profile at their highest pointof single tooth contact that is at the tip of the gear tooth tothe entire face width. The total force was distributed to anindividual node on the line of contact. The line of contact inspur gear is described in detail in Fig. 3. The meshed modelof gear tooth having trochoidal and circular root filet is

Fig. 9 Contact shear stress results at 1,500 r/min

Fig. 10 Contact shear stress results at 1,500 r/min

Fig. 11 Contact shear stress results at 2,000 r/min

Fig. 8 FEA meshed model of a gear pair with trochoidal root filet

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depicted in Figs. 4 and 5. The finite element analysis iscarried out for gear speed of 2,000 r/min and the inducedbending stress and the corresponding tooth deflection(Fig. 6) in 13 teeth gear having different root filet arepresented in Table 4. Further, the maximum tooth bendingstress (σ) is calculated using the Lewis formula Eq. 17 [14]for all the four speeds and are compared with the ANSYSresult.

s ¼ Ky x Ft

b x Mn x Yð17Þ

Where, Ky ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi5:56þ ffiffiffi

Vp

5:56

qand V ¼ dx w

2 m=sð Þ:

The induced bending stress (Table 4) in both circularfilet and trochoidal filet design were found within thepermissible limit (Table 5).

4.2 Non-linear contact shear stress analysis for single gearpair

During contact shear stress analysis, all degrees of freedomare constrained at the root circle but for analysis purpose,the constrained degree of freedoms is transferred to the gearhub surface. Four speeds 1,500, 2,000, 2,500, and 3,000 r/minare selected for this analysis and the rotation of the gear is

Fig. 13 Contact shear stress results at 2,500 r/min

Fig. 12 Contact shear stress results at 2,000 r/min Fig. 14 Contact shear stress results at 2,500 r/min

Fig. 15 Contact shear stress results at 3,000 r/min

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limited to 3,000 r/min. The tooth force is applied on the geartooth profile at their highest point of single tooth contact that isat the pitch circle to the entire face width. The total force wasdistributed to an individual node on the line of contact.Figures 7 and 8 show the meshed model of both the circularand trochoidal root filet gear pair. Figure 9 shows the contactshear stress obtained for circular root filet gear pair at1,500 r/min. Similarly, Fig. 10 shows the contact shear stressfor trochoidal root filet gear pair for the same speed.Figures 11 and 12 show the contact shear stress for both,the circular root filet gear pair and trochoidal root filet gearpair at 2,000 r/min. Also, the results obtained at 2,500 and3,000 r/min are given in Figs. 13, 14, 15, and 16. Theinduced contact shear stress, tooth deflection over the line ofcontact, and derived tooth stiffness are presented in Table 6.

5 Results and discussion

The induced bending stress (Von Mises) at 2,000 r/min,corresponding tooth deflection and stiffness of 13 teeth gear

provided with circular and trochoidal root filet arepresented in Table 4. It is observed from Table 4 that thedeflection of both the circular and the trochoidal root filetgear is more or less the same at 2,000 r/min, but lookinginto the bending stress (Von Mises) it was 442.457 N/mm2

for circular root filet gear and it is found to be447.435 N/mm2 for trochoidal root filet gear. Table 6 givesthe contact shear stress and corresponding deflectionobtained using ANSYS corresponding to various root filetsconsidered for analysis. It is evident from Table 6 that thedeflection (0.005441 mm) for 13 teeth gear having circularroot filet is lesser than that of the gear having trochoidalroot filet (0.010903 mm) at 3,000 r/min. The contact shearstress for the given load is less in circular root filet designirrespective of the speeds when compared to the trochoidalroot filet. The contact shear stress for the gear teethprovided with circular root filet was 206.085 N/mm2 at3000 r/min and it was found to be 250.904 N/mm2 fortrochoidal root filet design. It is well understood fromANSYS results (Tables 4 and 6) that the obtained bendingstress (Von Mises) and contact shear stress values are theleast for circular root filet gear irrespective of the speedthan trochoidal root filet gear. Figure 17 shows thedeflection judgment for circular and trochoidal root filetgear at various speeds. Similarly, Fig. 18 enables to predicthow the contact shear stress is varying for the change inroot filet at various speeds. Contact shear stress decreases

Table 6 FEA results—contact shear stress

Speed(r/min)

Deflection (mm) Contact shear stress (N/mm2) Stiffness (N/mm)for 13 Teeth for 13 Teeth for 13 Teeth

Trochoidal root filet Circular root filet Trochoidal root filet Circular root filet Trochoidal root filet Circular root filet

1,500 0.019601 0.010894 451.594 412.089 186,332 335,258

2,000 0.014161 0.007904 326.113 299.253 193,418 346,533

2,500 0.014204 0.006527 287.021 247.243 192,480 335,714

3,000 0.010903 0.005441 250.904 206.085 203,046 335,627

Fig. 16 Contact shear stress results at 3,000 r/min

Fig. 17 Speed vs. deflection comparison

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gradually with increase in speed for circular root filetdesign.

6 Conclusions

The investigation result infers that the gear tooth deflectionin the circular root filet is less when compared to thetrochoidal root filet. Further, there is appreciable reductionin bending stress and contact shear stress for circular rootfilet design in comparison to that of trochoidal root filetdesign. From the foregoing analysis, it is found that thecircular root filet design is more apt for less number of teethand whatever may be the pinion speed. ANSYS resultsindicate that the gears made of circular root filet yield betterstrength (reduced bending and contact shear stress) therebyimprove the fatigue life of the gear material.

Acknowledgment The authors are grateful to Ms. Surya gears andMs. Gee-dee Technical Training Institute, Coimbatore, India for theirtechnical expertise and the support rendered for the successfulcompletion of this analysis and investigation.

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2. Tesfahunegn YA, Rosa F (2010) The effects of the shape of toothprofile modification on the transmission error, bending, andcontact stress of spur gears. Int J Mech Eng Sci 224:1749–1758.doi:10.1243/09544062JMES1844

3. Spitas V, Costopoulos Th, Spitas C (2005) Increasing the strengthof standard involute gear teeth with novel circular root filletdesign. American J of Applied Sci 2(6):1058–1064

4. Fredette L, Brown M (1997) Gear stress reduction using internalstress relief features. J Mech Des 119:518–521. doi:10.1115/1.2826398

5. Ciavarella M, Demelio G (1999) Numerical methods for theoptimisation of specific sliding, stress concentration and fatiguelife of gears. Int J Fatigue 21:465–474. doi:10.1016/S0142-1123(98)00089-9

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12. KISSsoft software release 10 (2008), www.KISSsoft.ch13. Dennis Townsend, Townsend Dennis P (1991) Dudley’s gear

handbook: design, manufacture, and application of gears. TheMcGraw-Hill Professional Publishing. ISBN: 0070179034

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Fig. 18 Speed vs. contact shear stress comparison

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