Analysis of Gear Geometry and Durability With Asymmetric

INTRODUCTIONIn conventional involute spur and helical gears, both drive

and coast flanks experience the same bending and contactstrength. However, in most of the cases for automotiveapplications drive side and coast side are not equally loadedwhile transmitting power, the drive side of gear tooth isloaded for longer period of time as compared to coast side.Therefore an asymmetric gear tooth is preferable for suchapplications. In conventional symmetric gear design, bendingand contact strength of a particular gear pair can be increasedby an increase in pressure angle. But there is a limitation oftop land thickness of tooth which is reduced significantlywith increasing pressure angle. With an asymmetric geardesign, keeping same pressure angle on the drive side and ahigher pressure angle on coast side, the bending strength canbe improved without improving contact strength for driveside. Hence gear mass, noise and vibration can be reducedsignificantly. Similarly, by keeping the same pressure angleon the coast side and a higher pressure angle on the drive

side, both bending and contact strength can be significantlyincreased for the drive side significantly. Additionally, theasymmetric gear with a high pressure angle on drive side willreduce the vibration level due to a reduction in sliding and anincrease in mesh stiffness. A gear pair with asymmetricpressure angle is shown in Fig.1.

Fig 1. Asymmetric Pressure Angle Gear Pair

2012-01-1995Published 09/24/2012

Copyright © 2012 SAE Internationaldoi:10.4271/2012-01-1995saecomveh.saejournals.org

Analysis of Gear Geometry and Durability with AsymmetricPressure Angle

Amit SandoojaMahindra Engineering Services

Sagar JadhavMahindra & Mahindra Ltd

AbstractGear design is one of the most critical components in the Mechanical Power Transmission industry. Among all the gear

design parameters pressure angle is the most critical parameter, which mainly affects the load carrying capacity of the gear.Generally gears are designed with a symmetric pressure angle for drive and coast side. It means that both flank side of gearare able to have same load carrying capacity. In some applications, such as in wind turbines, the gears experience only uni-directional loading. In such cases, the geometry of the drive side need not be symmetric to the coast side. This allows forthe design of gears with asymmetric teeth. Therefore new gear designs are needed because of the increasing performancerequirements, such as high load capacity, high endurance, long life, and high speed. These gears provide flexibility todesigners due to their non-standard design. If they are correctly designed, they can make important contributions to theimprovement of designs in aerospace industry, automobile industry, and wind turbine industry.

In this paper we present a mathematical model of helical gear pair with an asymmetric pressure angle. We haveincreased the pressure angle of gear on the drive side to increase the load capacity and performance of the gear pair interms of noise and mesh stiffness while transmitting power. Also we have analyzed the results of bending and contactstresses generated on gear pair with the asymmetric pressure angle.

CITATION: Sandooja, A. and Jadhav, S., "Analysis of Gear Geometry and Durability with Asymmetric Pressure Angle,"SAE Int. J. Commer. Veh. 5(2):2012, doi:10.4271/2012-01-1995.

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INFLUENCE OF PRESSURE ANGLEIn the design of an asymmetric gear pair, proper selection

of pressure angle on drive side and coast side is veryimportant. Although the increase in pressure angle willincrease the bending strength, there is a limit to the increasein pressure angle because it reduces the top land thickness oftooth which makes the tooth sharp at top which is known as‘Tooth Peaking’. Also increasing the pressure angledecreases the contact and overlap ratio which affects theperformance in terms of NVH. By increasing the pressureangle the sliding velocity of the gear reduces because ofincrease in flank curvature, this in turn reduces the noise andvibration and increases the stiffness of the gear tooth inloaded condition.

An asymmetric gear tooth design effects the followingterms and factors related to gear geometry which will bediscussed in detail.

1. Tooth Top land thickness

2. Contact Ratio

3. Sliding velocity

4. Bending strength

5. Contact Strength

TOOTH TOP LAND THICKNESSA major issue or limitation with increasing the pressure

angle is the reduction of gear tooth width at the addendumcircle. Tooth shape becomes more and more pointed as thepressure angle increases and the top land becomes smallerwhich results in permanent deflection at the tooth tip. Thisphenomenon is termed as “Tooth Peaking “. The peakinglimit sets the limit of the pressure angle. The effect on thetooth flank and top land thickness with the increase inpressure angle is shown in Fig. 2a and 2b.

The thickness of the gear tooth at the addendum circledecreases as the pressure angle increases, and the tooth tipbecomes more pointed. Gear Standards such as DIN, ISO andAGMA recommend that the thickness at the addendum circlebe a minimum 0.4 times of module. Tooth thickness onaddendum circle is calculated by:

……(1)

……… (2)

Fig. 2a. Change in Tooth Flank and Top land Thickness w.r.t Pressure Angle

Fig. 2b. Top land thickness Vs pressure angle

Sandooja et al / SAE Int. J. Commer. Veh. / Volume 5, Issue 2(October 2012) 547


………(3)

CONTACT RATIOContact ratio is the measure of the average number of

teeth in contact along the total path of contact. It means thecontact period in which a tooth comes and goes in and out ofcontact with the mating gear. As the pressure angle increasesthe contact ratio reduces and load increases on gear tooth. Ingeneral the contact ratio of a gear pair should be a minimum1.10, below this limit the loading on gear tooth increases,which results in tooth breakage during cyclic loading. Alsothe highest point of single tooth contact (HPSTC) reachesnear the tooth tip causes permanent deflection in gear teethand affects the gear performance in terms of noise andvibration. Hence it is important to the keep maximumpressure angle on drive side so as to maintain a minimumcontact ratio of 1.10 and keeping same pressure angle oncoast side for better performance. The behavior of contactratio with increase in pressure angle is shown in Fig. 3

Total contact ratio of gear pair is calculated by:

………………… (4)

………………… (5)

………………… (6)

SLIDING VELOCITYGear tooth sliding velocity is defined as a difference

between rolling velocities of teeth in mesh. For two involuteprofile teeth in mesh, at a given point on the line of action, aproduct of radius of curvature and the rotational speed are notequal; therefore, the resultant rolling velocities are different.The pitch point is the only contact point for the meshing teethwhere there is a pure rolling. At any other location during theinteraction of tooth profile there is some sliding. Fig. 4a

shows the contact of gear and pinion at the tip of the tooth.The velocities at contact point of gear and pinion are denotedby Vg and Vp. The individual velocities of the gear andpinion can be resolved into two components normal andtangent to tooth surface as Vgn and Vgt for gears and Vpnand Vpt for pinion. If the teeth in meshing contact at the pitchpoint ‘P’, then Vpn and Vgn are same. But under loadingconditions there is some misalignment and deflection on gearthe tooth resulting in sliding. Therefore the tangentialvelocity components Vpt and Vgt are different and thedifference between two is known as sliding velocity. Thesliding velocity is directly proportional to the distancebetween the pitch point and point of contact. The maximumsliding velocity occurs with contact at tooth tip.

The sliding velocity at tip vga and at root vgf of gear teethis calculated by following equation:

…………… (7)

…………… (8)

…………… (9)

…………… (10)As the pressure angle increases, the sliding velocity

decreases due to, an increase in tooth thickness at the root andmesh stiffness, the deflection of gear teeth reduces. Henceperformance of the gear pair increases in terms of reductionin noise and vibration. Fig. 4b shows the behavior of slidingvelocity at the tooth tip as the pressure angle increases on thedrive side.

GEAR TOOTH BENDING STRENGTHDetermination of the capacity of the gear to transfer the

required torque for the desired operating life is completed by

Fig. 3. Contact Ratio Vs Pressure Angle

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determining the strength of the gear teeth in bending anddurability of teeth. Gear tooth bending strength increases asthe stresses at the tooth root decreases. As the pressure angleincreases the tooth root thickness at the critical sectionincreases hence reducing stress at the tooth root. Fig. 5shows, the direction of load applied on gear tooth whiletransferring torque.

According to ISO and DIN standards the maximumbending stress on a gear tooth is given by

…………… (11)

Fig. 4a. Single Tooth Contact

Fig. 4b. Sliding Velocity Vs Pressure angle



Overload factors KA, KV, KFß, KFα and correctivefactors Yε, Yß are not affected by variable pressure angle ondrive and coast side.

Yε is the overlap ratio factor and is expressed as:

…………… (12)Yß is the Helix angle factor and is expressed as:

……………(13)Bending stress with an asymmetric pressure angle,

keeping same module, torque and face width variessignificantly from the bending stress with a symmetricpressure angle because of the changes in tooth form factorYFa and stress concentration factor Ysa.

Tooth form factor YFa is expressed as:

………… (14)Tooth stress concentration factor Ysa is expressed as:

…………(15)

Tooth thickness at the critical section ‘SFn’ increases withan increase in pressure angle. As ‘SFn’ increases, the distancebetween the critical section and point of intersection betweentooth axis and direction of force of contact i.e. bending leverarm ‘hFa’, is also reduced, which decreases the stress at toothroot. Therefore, the bending strength and durability of thegear pair increases. Tooth bending stress and the pressureangle relation is shown in Fig. 6.

GEAR TOOTH CONTACTSTRENGTH

When a gear pair meshes, the region of contacttheoretically is a line. The curvature of the individual matingsurfaces at the point of contact will vary according to theposition of the point of contact on the line of action, as thegear tooth surface rolls and slides during this action. Thoughtheoretically it is a line contact, the line actually develops intothe band of certain width along the length of the tooth due tomutual comprehensive pressure. As the tooth surfaces moverelative to each other with a combination of rolling andsliding, this band also continuously moves. Since the toothsurface undergoes fluctuating, repeated and cyclic stresses ofall kinds during the course of this action, fatigue failure of the

Fig. 5. Schematic representation of load applied on Gear tooth

Fig. 6. Pressure Angle Vs Bending Stress



tooth surface is likely. The failure of the tooth surface due tofatigue is termed as ‘Pitting’.

Contact fatigue strength of a gear material plays a majorrole in determining the service life of gears. It has beenobserved that pitting mostly occurs in the area surroundingthe pitch line. This may be due to the fact that the direction ofsliding velocity changes at the pitch point, resulting in thecomprehensive stress increasing in that region. HeinrichHertz, a German physicist, developed the expression for thestresses generated when two curved surface are in contact.These surface stresses are generally known as HertzianStresses.

Fig. 7a shows the equivalent cylinders of the gear tooth incontact at pitch point with an asymmetric pressure angle.Since the radius of curvature of gear tooth decreases towardsthe base circle, the Lowest Point of Single Tooth ContactLPSTC will be at where the highest stress occurs (Fig. 7b).Hence the area below the pitch point or at the tooth rootshould be strong enough to withstand the stresses developedin that area. Besides the contact pressure, sliding velocity,lubricant viscosity and frictional forces also influence thestress distribution.

The Nominal surface pressure developed at the toothsurface is expressed as:

……………(16)

Fig. 7b. Stress Zone band below Pitch point at LPSTC

The actual contact stress is calculated by multiplying thenominal stress by certain factors that increase the stress ontooth flank, which is expressed as:

……………(17)

As the pressure angle increases at drive side of gear pair,the sliding velocity decreases (Fig. 3) and tooth thickness atcritical section increases (Fig. 4), therefore the shear stress inthe section below the pitch line decreases, hence improvingthe contact strength and fatigue life. The gear pair contactstress at the critical section is shown graphically with theinput data in Fig. 8.

Fig. 7a. Equivalent cylinders from the Pair of Asymmetric tooth at Pitch Point



GEOMETRY DESIGNA mathematical model of an asymmetric pressure angle

gear pair geometry has been designed, consideration of thekey parameters of gear geometry; module, helix angle,number of teeth, face width and center distance, and theresults of the bending and contact strength is compared to asymmetric pressure angle gear pair with the same parameters.It has been observed that the bending stress at the gear rootand the contact stress on gear flank are reduced in theasymmetric pressure angle gear pair hence increasing loadcarrying capacity approximately 20% when compared with asymmetric pressure angle gear pair.

Relevant parameters shown in Table.1 are considerationswhen designing a symmetric and asymmetric pressure anglegear pair.

Symmetric pressure angle geometry is formed by twosymmetrical involutes with one base circle, but asymmetricpressure angle gear geometry is formed by two differentinvolutes of different base circles. Symmetric gear toothgeometry with pressure angle 20° and 35° shown in Fig. 9aand 9b.

Tooth geometry with a 20° pressure angle shown in Fig.9a, have sufficient tip thickness and sufficient root radius, butthe thickness at critical section of tooth root is less whichcauses high bending stress and low load bearing capacity.Also the radius of curvature of the involute is low whichincreases the sliding velocity during mesh.

Tooth geometry with a 30° pressure angle shown in Fig.9b, thickness at the critical section of the tooth root increaseswhich reduces the bending stress and increases the loadbearing capacity. Also the radius of curvature of the involutecurve increases which reduces the sliding velocity hencedecreases contact stress on the tooth flank. But as shown,there is not sufficient thickness at the tip which results in tipbreakage or permanent deflection at the tip during torquetransfer. Also there is not sufficient space between two teethat the root which results in the tooth tip contacting the matinggear in the root radius.

To increase the load bearing capacity and contact strengthwhile maintaining sufficient thickness at the tip to preventpermanent deflection and sufficient space at the root toprovide root radius clearance, a gear pair with a 35° pressureangle on drive side and a 20° pressure angle on coast side has

Fig. 8. Pressure Angle Vs Contact Stress

Tab 1. Input Parameters for Gear Pair Design



been designed and analyzed. The analysis of a gear pair withCAE (Computer Aided Engineering) and FEA (FiniteElement Analysis) in terms of bending strength, contactstrength and tooth deflection during torque transfer werecompleted. Fig. 10 shows the geometry of gear tooth withasymmetric pressure angle.

Asymmetric pressure angle gear geometry is verified bymeasuring the diameter over pin ‘DOP’ considering suitablediameter of pin. DOP shown in Fig. 11, is calculated byequations given below:

……………(18)

Fig. 9a. Involute Curve And Tooth Profile With 20° Pressure Angle

Fig. 9b. Involute Curve And Tooth Profile With 35° Pressure Angle



……………(19)

……………(20)

……………(21)

……………(22)

……………(23)

……………(24)

……………(25)

Tip thickness or top land thickness of the tooth with anasymmetric pressure angle is calculated by Equation1.Minimum tip tooth thickness as per DIN and ISO GearStandards should be 0.4 times of module, to prevent tipbreakage and permanent deflection at tip of teeth.

Total contact ratio (transverse and overlap) for anasymmetric pressure angle gear pair is calculated by equation4. The standards require the minimum contact ratio should be1.1, but it is better to maintain a contact ratio of more than2.0, so that the load is always shared by two teeth to reducethe contact stress on gear flank.

Fig. 11. Measurement Of Diameter Over Pin

Sliding velocity vg, of gear pair with symmetric andasymmetric pressure angle is calculated from equations 7 and8, and it is observed that as the contact ratio and length ofpath of contact decreases, the sliding velocity and specificsliding of gear pair decreases, reducing the noise andvibration.

Fig. 10. Involute Curve And Tooth Profile With Asymmetric Pressure Angle



FEA ANALYSIS & RESULTSA mathematical model of the asymmetric gear pair with a

35° pressure angle on the drive side and a 20° pressure angleon the coast side is compared with math model of symmetricgear pair with 20° pressure angle on both sides. The analysisis done on CAE using NX Nastran. The bending stress,contact stress and deflection at the tooth root and tooth flankis analyzed by applying a load of 8560 N normal to toothflank as shown in Fig. 12a.

Bending stress of 391.15 MPa, contact stress of 1839.28MPa and tooth deflection of 45 micron is found with asymmetric pressure angle gear tooth, as shown in Fig. 12b.

Similarly, a load of 8560N is applied normal to gear toothwith a 35° pressure angle on drive side and a 20° pressureangle on coast side as shown in Fig. 13a.

Bending stress of 306.0 MPa, contact stress of 1759.27MPa and tooth deflection of 37.2 micron is found with theasymmetric pressure angle gear tooth, as shown in Fig. 13b.

Similarly, with a load of 8560N on the pinion tooth,bending stress of 408.39MPa, contact stress of 1839.28 Mpaand tooth deflection of 24 micron is observed with thesymmetric pressure angle of 20°, as shown in Fig. 14a.

With the same load of 8560N on the pinion tooth, bendingstress of 342.70MPa, contact stress of 1759.27MPa and toothdeflection of 17 micron is observed with the asymmetricpressure angle of 35° on drive side and 20° on coast side, asshown in Fig. 14b.

The comparison between the symmetric pressure anglegear pair and the asymmetric pressure angle gear pair areshown in Tab. 2 given below.

Fig. 12a. Load Applied Normal To Symmetric Gear Flank Face

Fig.12b. Tooth Stress And Deflection On Gear With Symmetric Pressure Angle (20°- 20°)



Fig 13a. Load Applied Normal To Asymmetric Gear Flank Face

Fig 13b. Tooth Stress And Deflection On Gear With Asymmetric Pressure Angle (35° - 20°)

Sandooja et al / SAE Int. J. Commer. Veh. / Volume 5, Issue 2(October 2012)

Fig 14a. Tooth Stress And Deflection On Pinion With Symmetric Pressure Angle (20°- 20°)

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CONCLUSIONAs the pressure angle of a spur and helical gear pair

increases, it will show improvement in the aspect of bendingand contact strength, but there are limitations which limit theincrease in the pressure angle desired. An asymmetricpressure angle gear design overcomes many of theselimitations, resulting in better life of the gear pair withreduced weight and a decrease in sliding velocity, this resultsin reduced noise and vibration and an increase in the loadcarrying capacity of gear and an increase in durability underdynamic conditions.

REFERENCES1. Maitra, G.M. “Handbook of Gear Design”2. Deng, G. and Nakanishi, T., “Bending Load capacity Enhancement

using an Asymmetric Tooth Profile”3. Gunay, Durmus, Ozar, Halil and Aydemir, Alpay “The effect of

Addundum Modification coefficient on Tooth Stresses of Spur Gear”4. ISO 6336 and DIN 3990 Gear standard for tooth bending and Contact

stress calculation.5. Dudley, Darle W. “Handbook of Practical Gear Design”

Fig 14b. Tooth Stress And Deflection On Pinion With Asymmetric Pressure Angle (35° - 20°)

Tab 2. Comparison of Symmetric And Asymmetric Pressure Angle

Sandooja et al / SAE Int. J. Commer. Veh. / Volume 5, Issue 2(October 2012)

CONTACT INFORMATIONAmit Sandooja and Sagar Jadhav can be reached via email

[email protected]@gmail.com

[email protected]@gmail.com

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mailto:[email protected]




ABBREVIATIONSa - Top land thicknessDo - Tip circle diameterSna - Tooth thickness at Pitch circle diameterd - Pitch circle diameterαnd - Normal pressure angle at Drive side

αnc - Normal pressure angle at Coast side

αd - Transverse Pressure angle of Drive side

αc - Transverse Pressure angle of Coast side

αtcd - Pressure angle at Tip circle diameter Drive side

αtcc - Pressure angle at Tip circle diameter Coast side

αwtd - Working Transverse Pressure Angle Drive side

εges - Total Contact Ratio

εαd - Transverse contact Ratio Drive side

εßd - Overlap Ratio drive side Drive side

gadp - Length Of Path of contact drive side of Piniongadg - Length of Path of Contact drive side of GearPbt - Transverse base Pitchb - Face widthß - Helix anglemt - Transverse Modulevga - Sliding velocity at tip (m/s)vgf - Sliding Velocity at root (m/s)ωp - Angular speed of pinion (rad/s)ga - Length of path of Addundum of piniongf - Length of path of Addundum of GeardNa1 - Tip form Diameter of PiniondNa2 - Tip Form Diameter of Geardb1 - Base diameter of Piniondb2 - Base Diameter of GearσFb - Bending Stress at RootFt - Tangential Load on Gear Pairmn - Normal ModuleYFa - Tooth Form FactorYSa - Stress Concentration Factor

Yε - Overlap Ratio FactorYß - Helix Angle Factorßb - Base Helix AngleSFn - Root Tooth thickness at critical sectionhFa - Length of bending lever Armρf - Profile curvature radius at critical sectionαFan - Angle of Load appliedσH0 - Nominal surface pressure at tooth surface

ZH - Contact zone Factor

ZE - Tooth Elasticity Factor

Zε - Tooth overlap ratio factor (for contact stress)Zß - Helix angle Factor

u - Gear RatioσH - Actual contact stress at tooth surface

KA - Application Factor

KV - Dynamic Factor

KHß - Tooth Misalignment Factor

KHα - Tooth Pitch Error

DOP - Diameter Over PindP - Diameter at Pin CenterD - Pin Diameterdbd - Base diameter of Drive sidedbc - Base diameter of Coast side



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Analysis of Gear Geometry and Durability With Asymmetric