1

VARIATION OF ROOT BENDING STRESS WITH HOB NOSE RADIUS AND

PROTUBERANCE BASED ON ISO AND FEM METHODS FOR SPUR

GEARS

V. Sridhar*, R. Malik and Prakher Sharma

Product Design Development, Eicher Tractors (A Unit of TAFE Motors and Tractors Limited)

*Corresponding Author, Tel-91-7480-407680, E-Mail-sridhar6781@gmail.com

Abstract:-Tooth Bending Strength of Spur gears depends on the form factor YF and stress correction

factor YS in accordance with ISO 6336 for basic rack profiles with and without undercut. The chord

and radius at the critical section varies with the hob nose radius and affects the form factor YF and

stress correction factor YS. It is assumed that the root fillet is generated during the pre-treatment

itself and the flank is generated during the subsequent shaving process. Comparison between the

stresses calculated by the standard method and stresses calculated using FEM is depicted.

Keywords-: Theory of Gearing, Root Fillet, FEM, Fillet and Involute generation

INTRODUCTION

The tooth bending strength of spur gears is one of the basic criteria for the design of gears. ISO 6336

outlines the procedure for calculating the nominal and local tooth root stresses. The nominal tooth

root stress depends on YF (tooth form factor), YS (stress correction factor) and Y (Helix angle factor).

The basic relation of the nominal tooth root stress is given by equation (1) [5].

YYYmb

FSF

nf

tF 0 (1)

and the local tooth root stress is given by equation (2).

FFvAFF KKKK0 (2)

The geometry of the tooth root fillet determines YF and YS.

GEOMETRY OF TOOTH ROOT FILLET AND INVOLUTE

The geometry of the gear tooth root and the involute is based on the cutter path relative to the gear

during generation process. Consider the coordinate systems S1, S2, and Sf rigidly connected to the

hob cutter, the spur gear being generated, and the frame, respectively as depicted in figure 1. The

theory of gearing proposed by Litvin [1] provides the framework for calculating the gear tooth and

root shapes as the envelope to the family of cutter shapes in the coordinate system S2 rigidly

connected to the gear being generated.

2

FIGURE 1: Gear Blank, hob and Fixed Coordinate System

The cutter surface 1 is represented in parametric form in S1 as [1], [2].

1

1111 ,,,,,, Cuzuyuxur (3)

011

r

u

r

2121 , uuu

Where 1C denotes that ,1 ur is a regular surface which has continuous derivatives to at least the

first order. The relative motion of 1 with respect to coordinate system S2 generates the family 2 of

generating surface 1 [1], [2].

The family of surfaces 2 in coordinate system S2 is represented by

,,, 1212 rMur (4)

Where

21M is the coordinate transformation from coordinate system S1 to S2.

In addition to the above set of equations, there exists an equation of meshing which should be

satisfied. The equation of meshing ensures that for any the generating surface 1 should not lose

its contact with the generated surface 2 because the contact of the two surfaces must be continuous

[1], [2]. The equation of meshing in coordinate system Sf is given by

0)12(

11 vN

where 1N is the normal to the cutter profile and )12(

1v is the relative sliding velocity between the cutter

and the gear blank, both in coordinate system S1.

3

FIGURE 2: Protuberance Hob Geometry

Figure 2 shows the normal profile of a hob. The hob can be divided into four segments, straight line

segment EF, straight line segment DE, arc BD and straight line segment AB. The involute profile is

generated by the straight line segment EF. The root circle of gear is generated by the line segment

AB. Any point of line segment EF can be represented in coordinate system S1 as

sin2

1 us

x , cos1 uy , 21 uuu (5)

where the range of u is

coscos

aPprPfP hu

hh

(6)

Therefore

1

cos

sin2

1

, 1

1

1

u

us

y

x

ur (7)

The transformation matrix from coordinate system S1 to S2 is given as

100

cossincossin

sincossincos

21

r

r

M (8)

Where

r is the pitch circle radius of the gear.

The equation for meshing in the coordinate system S1 can be represented as

01

11

1

11

yx N

yY

N

xX (9)

0, 11 YrX are the coordinates of the instantaneous center of rotation in coordinate system S1

and TN 0sincos1 is the normal vector to the cutter surface.

4

The following system of equations gives the pre-shaved involute

1

cos

sin2

100

cossincossin

sincossincos

,,2

u

us

r

r

ur (10)

0sinsin2

, rs

uuf (11)

The set of equation becomes

sincossincos2

2 rus

x (12)

cossincossin2

2 rus

y (13)

0sinsin2

rs

u (14)

sin

cossin

2

sin

cossin

2

r

hs

r

hhsaP

prPfP

(15)

ROOT FILLET GENRATION

The envelope to the arc BD in the coordinate system S2 gives the root fillet of the spur gear. The

center of the arc in coordinate system S1 is denoted by ),( baC where

cos

1sintantantan

2prPfP hh

sa (16)

fPhb (17)

FIGURE 3: Hob Nose Radius

As shown in figure 3 any point at angle on arc BD in coordinate system S1 is given by

1

cos

sin

1

1

1

1

b

a

y

x

r f

f

f (18)

Transforming from coordinate system S1 to S2 gives the tooth root profile in S2.

ff rMr 1212 (19)

5

The normal vector to the generating cutter arc BD is given by T0cossin

The equation of meshing in coordinate system S1 becomes

01

11

1

11

y

f

x

f

N

yY

N

xX (20)

The resulting system of equations, which give the tooth root profile in coordinate system S2, is

sincossinsincos2 rbax f (21)

cossincoscossin2 rbay f (22)

0tan bar (23)

r

ba

r

a

tan

(24)

SECONDARY INVOLUTE

The straight line segment DE generates the secondary involute. The secondary involute will only

exist under non-undercut conditions [3]. The purpose of the protuberance is to provide the necessary

undercut and hence the secondary involute profile is not expected in pre-shaving process [3].

The equation of the secondary involute in coordinate system S2 using coordinate transformation is

given by

1

cossincossincos

sincossincossin

2

ruBA

ruBA

r s

s

s (25)

prPfP hhA (26)

tan2

As

B (27)

cos

1sin0

prP

s

hu (28)

and the equation of meshing gives

0sincossin BAur s (29)

The sufficient condition for singular point existence and hence the undercut on the generated surface

2 is [3]

0||||

*

2

d

rd f (30)

This gives the limiting value of * as

tantan*

r

AB (31)

Using the above two equations gives the singular point as

tansintansincos* rAAus (32)

The hob will undercut the gear shape 2 if the following inequality is true

cos

1sintansintansincos

prPhrAA (33)

6

At this value of the secondary profile angle, the root fillet is generated as the cutter „rolls

out‟ the root rather than as it „rolls into‟ the root [3].

r

hr fP

2

4sin

2

1

(34)

The relation, which governs the minimum value of secondary pressure angle

prP

prP

hh 1sinsin1 1 (35)

ROOT AND TIP DIAMETER

Line segment AB generates the root diameter. The generated root diameter is tangent to the root

fillet and theory of gearing gives the root profile. The hob used for the generation of the gear profile

is assumed to be a non-topping hob and hence the Tip diameter of the gear is determined by the gear

blank diameter. The determination of root and tip diameter profile also follows the same calculation

procedure as developed for the gear involute and the gear root profiles.

TOOTH FILLET AND INVOLUTE PROFILE

On the basis of the equations developed above, the root and involute profiles of the pre-shaved tooth

is developed. To ensure that the secondary involute is not generated the maximum value of the

secondary pressure angle has been kept below the value determined from equation (34). Shown in

figure 4 is the pre-shaved and the shaved gear tooth and fillet profile generated using the above

equations.

FIGURE 4: Gear Involute and Root profile, Shaved and Pre-Shaved

Simultaneous solution of equations (12) to (15) and (21) to (24) gives the point of intersection of the

pre-shaved involute and the root fillet. The shaved involute profile is assumed to start from the point

of intersection of the root fillet and the form circle diameter. The form circle diameter is calculated

assuming a finishing hob without protuberance and with the same hob nose radius and primary

profile angle as the pre-shaving protuberance hob and is given by [13].

7

2

2

2tan

222

c

fc

f

Nf hdh

dd

d

(36)

FIGURE 5: Root Fillet Profiles for various hob nose radii for same gear

)sin1( ch is the hob clearance

The form circle diameter should be lower than the SAP diameter by at least 0.25mm [4].

Different profiles were generated by varying the hob nose radius , and the protuberance spr. The

range of hob nose radius was from 0.2mn to 0.35mn, limited by the maximum hob nose radius *

given by [3]

tantantan

2sin1

cos*

prPfP hhs

(37)

Two values of hob parameter spr =0.05 and spr =0.03 were considered to ensure uniformity of the

results.

Since the variation of the secondary profile angle of the hob has no influence on the root fillet

geometry, so it was not varied.

The critical root chord sFn is not related to the finished tooth profile, so the shaving stock between

pre-shaved and shaved gears does not affect the root stress. Figure 5 shows the various root profiles

generated by varying the hob nose radius and protuberance.

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ROOT STRESS ANALYSIS FOR VARIOUS HOB NOSE RADIUS AND PROTUBERANCE

Based on the analysis developed above, gears made with different hob nose radii and protuberances

were compared for the magnitude of the root stresses. Both analytical predictions based on ISO 6336

and FEA analysis of the nominal tooth root stress was carried out. The ISO standard determines the

root stress point, which is tangent to the root fillet and subtends an angle of 30° with the tooth

centerline as depicted in figure 8. The parameters sFn, hFe and Fare calculated at this point. The

form factor represents the state of stress in the gear root assuming the gear tooth as a short

cantilever.

The calculation of form factor YF and stress correction factor YS as per ISO 6336 is outlined below.

n

n

Fn

Fen

n

Fe

F

m

s

m

h

Y

cos

cos6

2

(38)

n

n

n

pr

nfP

sh

saE

cos)sin1(

costan

2 (39)

where

qprspr

0prs where the gears are not undercut.

s Hob width at the generating line

nm

bG (40)

32

2

nn m

E

zH (41)

Hz

G

n

tan2

(42)

The critical root chord is given by

n

n

n

Fn

m

Gz

m

s

cos3

3sin (43)

Gz

G

mm nnn

F

2coscos

22

2

(44)

n

n

n

enFenee

n

Fe

m

Gz

m

d

m

h

cos3costansincos

2

1 (45)

Feh Bending Moment Arm

end Diameter at Highest Point of Single tooth Contact (HPSTC)

2

222

21

coscos

222

bn

nnbnan

en

d

z

ddd

z

zd

(46)

9

en

bnen

d

d1cos

e = angle subtended by chordal tooth thickness at HPSTC.

Fen Pressure Angle at HPSTC

coscos2

b

n

zz

nz Virtual Number of teeth

The stress correction factor YS is given by

a

sS qLY 13.02.1 (47)

where

L

as

qh

sL

F

Fns

Fe

Fn

3.221.1

1,

2,

1Y (48)

Given below are the details of the gear pairs analysed for various combination of protuberance and

hob nose radius. The gears are low contact ratio spur gears with contact ratios in the range of 1.2 to

1.5. The tangential load applied at HPSTC of gear corresponds to a pinion input torque of 12 Nm

and is uniformly distributed along the face width of the gear. The various pairs of gears are subjected

to the same input torque and operate at the same center distance.

YF and YS do not depend on the secondary pressure angle '' and hence the effect of '' has not

been considered

GEAR DATA Normal Module mn 3 3 3 3 4 4 4 4

Normal Pressure Angle n 22.5° 22.5° 22.5° 22.5° 20° 20° 20° 20°

Number of Teeth z 30 38 22 46 17 35 22 30

Pitch circle Diameter d 90 114 66 138 68 140 88 120

Base Circle Diameter 83.149 105.322 60.976 127.495 63.899 131.557 82.693 112.763

Facewidth bf 15 13 22 20 22 20 15 13

Tip Circle Dia da 99.2 123.45 74.76 146.66 77.6 148.33 97.47 128.45

Root Circle Dia df 85.17 108.83 60.66 132.56 59.67 130.4 79.54 110.53

HOB DATA Primary Profile Angle 22.5° 22.5° 22.5° 22.5° 20° 20° 20° 20°

Secondary Profile Angle 10° 8° 12° 9° 15° 12° 15° 12°

Addendum hfP 2.415 2.585 2.67 2.718 4.1648 4.8 4.228 4.7368

Dedendum haP 4.6 4.725 4.38 4.33 4.799 4.164 4.736 4.227

Generating Line Width s 3.4067 3.5451 3.3955 3.4477 5.7252 6.1876 5.7712 6.1416

Maximum Hob Nose

radius 1.116 1.112 0.955 0.959 1.947 1.918 1.947 1.918

Maximum Secondary

Profile Angle Max 10.732

°

9.995° 14.219° 9.728° 17.848° 13.334° 15.66° 14.346°

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Figures 6 and 7 show the variation of sFn, hFe, , YS*YF and the root bending stress F0 with the hob

nose radius for two gear pairs. It is quite clear that with increasing hob nose radius the nominal root

bending stress decreases and the radius of the root fillet at the critical chord increases. The product

of YF*YS also decreases with the hob nose radius. The effect of protuberance spr on these parameters

is not that significant. There is an almost linear relationship between the nominal tooth root stress

and the hob nose radius, irrespective of the module and the pressure angle. With the appropriate hob

nose radius limited by equation (37) and the secondary pressure angle governed by equations (34)

and (35), it is possible to reduce the nominal tooth root stress. There is also a very slight decrease in

the root bending stress with lower protuberance.

FIGURE 6: Variation of Nominal Root Bending Stress Parameters with Hob Nose Radius for

module 3 and Pressure Angle 22.5°

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FIGURE 7: Variation of Nominal root Bending Stress Parameters with Hob Nose Radius for module

4 and Pressure Angle 20°

FINITE ELEMENT ANALYSIS FOR DETERMINATION OF ROOT BENDING STRESS

The determination of Nominal Root Bending Stress based on ISO 6336 depends on YF, YS and Y in

accordance with equation (1). The change in the stress profile due to the transition from the involute

portion of the gear to the root fillet is accounted for by the stress correction factor YS. Finite element

analysis for determination of root bending stress in gears can be alternate method for accurate

determination of root bending stress in gears [6, 7, 8, and 9]. In order to determine the root bending

stress in gears, finite element analysis of the spur gear pairs in mesh was carried out for various

combinations of hob nose radius and protuberance. According to the research presented in [10], the

torsional stiffness estimated using plane stress assumption under estimated the torsional stiffness,

while the plane strain assumption over estimates the torsional stiffness. The torsional stiffness

evaluated using 3D elements predicted torsional stiffness in between those evaluated using plane

stresss and plane strain assumption. In addition the relative errors are less than 3% for the face

widths under consideration. Based on this 2D plane stress analysis with thickness was used for finite

element analysis. Accurate modeling using finite element method requires very accurate

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determination of the involute and fillet profiles of the gears. To achieve this gear tooth and root

profile generated using the above developed equations were used for modeling.

FIGURE 8: Comparison of locations of critical section and the angle between tangent to fillet and

tooth centerline computed according to ISO and FEM method [6].

The gears were meshed using SOLID 183, 2D 8 node solid elements in ANSYS [11]. The plane

stress with thickness assumption considered the face width of the meshing gears. The mesh in the

root fillet region had 60 elements. This was required for accurately determine the stress distribution

in the root fillet region. The torque used in both ISO and Finite element methods is 246 Nm. The

torque is applied on the inner diameter of the pinion and since the tooth root is located far away from

the location of load application, the root bending stresses should not be affected by the local

deformations due to the application of torque on the inner diameter of the pinion [6]. All degrees of

freedom of the nodes on the inner diameter of the gear are constrained, while the nodes on the inner

diameter of the pinion are unconstrained in the direction of rotation. Contact elements are defined on

the flanks of the meshing gears and the contact conditions are established on the HPSTC of the

pinion. Figure 9 shows the location of the stress contour results from FEM analysis. The location of

point of maximum bending stress using FEM method varies between 30.67° to 44.65° as against the

constant 30° in ISO calculations (figure 8) for the gears analysed. The variation of the point of

maximum root bending stress is consistent with the results presented in [6]. 8 cases were analysed

comprising of 4 gear and 4 pinion configurations, with the contact point on HPSTC of both the

pinion and the gear.

Figure 10 shows the variation of root stress with hob nose radius using ISO and FEM methods for 4

cases with pinion loaded at HPSTC. The reduction of root bending stress with increasing hob nose

radius is evident in both methods. Also the magnitude of stresses computed using FEM methods is

less than that computed using ISO standards [5]. The effect of protuberance on Nominal Root

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Bending Stress is not significant in both the methods, so the FEM analysis results do not consider the

effect of change in protuberance. The FEM method gives the nominal root bending stress. The effect

of the stress correction factor is considered by the geometry of the gear itself. Although FEM

method also shows that the root bending stress decreases with increasing hob nose radius, the

magnitude of the maximum and minimum stresses for different hob nose radii are lesser than

computed by ISO method.

FIGURE 9: Root Bending Stress computation using FEM method

FIGURE 10: Comparison of Variation of Root Bending Stress with Hob Nose Radius from ISO and

FEM Methods

The variation of Root Bending stress along the fillet profile can be used as measure for establishing

the accuracy of Finite element analysis as elaborated in the next section. Figure 11 shows the plot of

variation of the root bending stress as a function of the radius. The location of the maximum stress

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progressively moves towards the root circle as the hob nose radius decreases, leading to an increase

in the angle from 30° as outlined in ISO-6336.

FIGURE 11: Variation of Root Bending Stress with Radius a) Pinion with 17 teeth b) Gear with 35

teeth

VERIFICATION OF FINITE ELEMENT ANALYSIS RESULTS

The experimental verification of stresses predicted using finite element method was not carried out.

Instead in order to establish the accuracy of the finite element method, Finite element analysis of

SAE symmetric test gear with 34 teeth and full rounded fillet were compared to the theoretical

results derived using LDP (Load Distribution Program from The Ohio State University) and

experimental results given in [12]. Finite element analysis using 2D plane stress with thickness, for

the base line geometry given in [12], was carried out. Figure 12 shows the load and boundary

condition of the test gear used in [12]. The gear is rotationally free. The load anvil was positioned

such that the load radius was 75.15mm.

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FIGURE 12: Finite Element Analysis of SAE test Gear with 34 teeth, simulating the single tooth

bending fatigue test

FIGURE 13: Load versus Stress plot for SAE test

gear using FEA

FIGURE 14: Load versus Stress plot for SAE test

gear [12]

Figure 13 shows the load versus stress relation for the SAE test gear in [12] using finite element

method. Figure 14 shows the results from [12]. The finite element analysis results correlate with the

values corresponding to Measurement Gauge A. Since the FEA was done using 2D assumption, the

comparison with other strain gauge locations cannot be done.

Figures 15 and 16 show the variation of root bending stress with the radius and shows the

comparison of the root bending stress computed using FEA and that predicted using LDP along with

the measured stress values for a load of 31 kN. The comparison is for only one position of the strain

gauge as the FEA was done with 2D assumption. The comparison of the predicted root bending

stress using LDP correlates with the measured stress as in [12] and also shows correlation with the

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FEA results. There is a slight difference in the radial location where the maximum stress is predicted

using FEA and that from measurement and also with the LDP prediction with the actual

manufactured root profile. This could not be verified in FEA due to lack of experimentation with

actual root profiles, but the location with respect to the predicted theoretical profile using LDP

agrees quiet well with FEA. The variation in the location of the strain gauges with respect to the

theoretical geometry as well as the surface roughness in the root fillet region could be attributed to

the difference in location and magnitude of the stress results. The stress predicted using predicted

theoretical profile using LDP is higher than that predicted by FEA while the measured bending stress

is lower than that for predicted theoretical profile and is quiet close to the FEA results.

FIGURE 15: FEA Plot of Root Bending Stress

with Radius

FIGURE 16: Experimental, theoretical and

theoretical with actual root profile bending

stress [12]

SUMMARY AND CONCLUSIONS

This paper analysed the root bending stress computation in accordance with ISO 6336 for various

hob nose radii. The effect of hob nose radius on the root bending stress was analysed both using ISO

method as well as FEM method. Form factor and stress correction factor variation with hob nose

radius were determined. The limits of the maximum hob nose radius and the secondary profile angle

were determined. The results of both ISO method and FEM calculation of the root bending stress

confirm the fact that with increasing hob nose radius the nominal root bending stresses decreases. A

maximum difference between the root stress predicted by ISO method and FEA was 26% which

corresponds to the case with minimum hob nose radius for the gear with 35 teeth. The influence of

Hob nose radius on Nominal Root bending stress is much more pronounced in the ISO method. The

difference between the minimum Root bending stress corresponding to minimum hob nose radius

and maximum root bending stress is 28.6 MPa for the pinion with 17 teeth loaded at HPSTC and

33.16 MPa for the Gear with 35 teeth loaded at HPSTC, while the corresponding differences using

ISO method were 65.2 and 66.25 MPa respectively. The effect of Hob nose radius on Root Bending

stress is more pronounced in the ISO method than in FEA.

Based on this study it is possible to choose the parameters of the cutting hob in a way to minimise

the root bending stress and increase the root safety factor along with increase in the bending fatigue

life of the gears.

The root bending stress based on FEM method is lower than that predicted using the ISO

methodology. In addition, the location of the maximum stress as calculated using the FEM technique

17

varies from that of ISO method. The angle between the tangent at the point of maximum root

bending stress and the tooth symmetry line increases with the decrease in the hob nose radius.

Further experimental work would establish the accuracy of Finite element method for predicting the

nominal root bending stress in Spur gears.

SYMBOLS

a: x coordinate of the center of hob nose radius KA: application factor

b: y coordinate of the center of hob nose radius Kv: dynamic factor

bf: face width KF: transverse load factor (root stress)

C: center of hob nose radius KF: face load factor (root stress)

d: pitch circle diameter of gear u: parameter along the hob cutting edge

da: tip diameter of gear YF: tooth form factor

den: HPSTC diameter YS: stress correction factor

df: root diameter of gear Y: helix angle factor

dNf: root form circle diameter of gear z: number of teeth

Ft: tangential force zn: virtual number of teeth

haP: protuberance hob addendum n: normal pressure angle

hfP: protuberance hob dedendum : primary profile angle of hob

hprP: buckling height of hob : helix angle of gear

hc : hob clearance b: base helix angle of gear

hFe : bending lever arm F0: nominal tooth root stress

mn: normal module F: local tooth root stress

N1: normal vector in coordinate system S1 : angular displacement

r: pitch circle radius of gear F: gear root fillet radius at critical section

{r1}: position vector in coordinate system S1 : hob nose radius

{r2}: position vector in coordinate system S2 : angular location of point on hob nose radius

s: hob width at the generating line 1 : hob cutter shape

spr: protuberance of hob 2 :tooth shape

sFn: chord at critical section secondary hob profile angle

ACKNOWLEDGEMENT

The authors would like to thank the management of Eicher Tractors for their support and

encouragement in this endeavor.

REFERENCES

[1] Litvin, F.L., and Fuentes, A., “Gear Geometry and Applied Theory,” Cambridge University

Press, 2004.

[2] Yamada, S. and Lee, H., “Root Fillet Geometry of Spur, Helical, Spiral Bevel and Hypoid

Gears,” SAE Technical Paper 962230, 1996.

[3] Kuang, J., and Chen, W., “Determination of Tip parameters for the Protuberance Preshaving

Cutters,” Mech. and Mach. Theory, Vol. 31, No. 7, pp. 839-849, 1996.

[4]. Dudley, D.W., “Handbook of Practical Gear Design,” CRC, Boca Raton, FL, 2002.

[5] ISO 6336-2006, Standard, Calculation of Load Capacity of Spur and Helical Gears.

[6] Kawalec, A., Wiktor, J. and Ceglarek, D., “Comparative Analysis of Tooth-Root Strength using

ISO and AGMA Standards in Spur and Helical Gears with FEM-based Verification,” Journal of

Mechanical Design, Vol. 128, pp. 1141-1158, ASME, 2006.

18

[7] Andrews, J.D., “A Finite Element Analysis of Bending Stresses induced in External and Internal

Involute Spur Gears,” Journal of Strain Analysis Volume 26 No. 3, 1991.

[8] Rameshkumar, M., Venkatesan, G. and Sivakumar, P., “Finite Element Analysis of High Contact

Ratio Gear,” 10FTM06, AGMA Technical paper.

[9] Wang, J. and Howard, I., “Finite Element Analysis of High Contact Ratio Spur Gears in Mesh,”

Journal of Tribology, Vol 127, pp.469-482, ASME, 2005.

[10] Wang, J. and Howard, I., “Error Analysis on Finite Element Modeling of Involute Spur Gears,”

Journal of Mechanical Design, Vol. 128, ASME, 2006.

[11] ANSYS Users Guide. ANSYS Inc.

[12] Sanders, A.A., “An Experimental Investigation of the influence of Elliptical Root Shapes and

Asymmetric Teeth on Root Stresses and Bending fatigue Lives,” M.S. Thesis, The Ohio State

University, 2010.

[13] Buckingham, E., “Manual of Gear Design, Section Two,” Industrial Press Inc., 1973.