effect of pressure angle and back up ratio on … · ratio on asymmetric spur gear tooth bending...

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Int. J. Mech. Eng. & Rob. Res. 2014 Sharatchandrai R Karagi and P S Patil, 2014

EFFECT OF PRESSURE ANGLE AND BACK UPRATIO ON ASYMMETRIC SPUR GEAR TOOTH

BENDING STRESS

Sharatchandrai R Karagi1* and Prashant S Patil1

*Corresponding Author: Sharatchandrai R Karagi, [email protected]

Gearing is one of the most critical components in the mechanical power transmission systemand in most industrial rotating machinery. It is possible that gears will be the most effectivemeans of transmitting power in future machines due to their high degree of reliability andcompactness. In 18th century saw an explosion in metal gearing. Gear design and manufacturingrapidly developed in the 19th century. Now a day, the gear design has become a highly complicatedand comprehensive subject. An asymmetric spur gear tooth means that the two profiles (sides)of a gear tooth are functionally different for many gears. The workload on one profile is significantlyhigher and is applied for longer periods of time than for the opposite one. The design of theasymmetric tooth shape reflects this functional difference. The main objective of the presentwork is to estimate the stress across the critical section for different backup ratios and theresults obtained. Developed programme is used to create a finite element model for asymmetricspur gear tooth to study the effect of bending stress at the critical section for different backupratios. To study the effect of above parameter ANSYS was used. The rim thickness and thepressure angle was varied and the location and magnitude of the maximum bending stresseswere reported and results obtained and compared.

Keywords: Rim thickness, Pressure angle, Asymmetric spur gear tooth, Finite element analysis,Bending stress

INTRODUCTIONGears have long been widely used inmachines of all kinds with increasingrequirements in recent times with smaller andlighter designs. Gearing is one of the criticalcomponents in a mechanical power

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Int. J. Mech. Eng. & Rob. Res. 2014

1 Department of Mechanical Engineering, BLDEA’s College of Engineering & Technology, Bijapur, Karnataka, India.

transmission system and most industrialrotating machinery. It is possible that gearswill be the most effective means oftransmitting power in future machines due totheir high degree of reliabil ity andcompactness.

Research Paper

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It is possible that gears will be the mosteffective means of transmitting power in futuremachines due to their high degree of reliabilityand compactness. In 18th century saw anexplosion in metal gearing. Gear design andmanufacturing rapidly developed in the 19th

century. Now a day, the gear design hasbecome a highly complicated andcomprehensive subject.

Thin rim gears find application in high-power, lightweight aircraft transmissions.Bending stresses in thin rim spur gear toothfillets and root areas differ from the stressesin solid gears due to rim deformations. Rimthickness is a significant design parameter forthese gears. The rim thickness factor is usedin the situations in which a gear is made witha rim and spokes rather than a solid disc.

Presently, gears are suffered by backlashundercut and interference. Interference is aserious defect in the involute system of gearingwhen no of teeth less than the minimumrequired number of teeth and can be avoidedby undercutting the tooth. One major cause ofgear failure is fracture at the base of gear toothdue to bending fatigue. The bending strengthis influenced by gear size, described by thedaimetral pitch; the shape of the tooth,described by the number of teeth on the gear;the location of the highest load, described bythe number of teeth on the mating gear. Rimdeflections increase the bending stress intooth fillet and root area. Therefore in aircraftapplications rim thickness and allowablestress are optimized to achieve light weight.

In this paper a single teeth gear segmentwith 25 number of teeth and with module 6,asymmetric spur gear for different pressureangles on drive side and coast side of the gear

with different back up ratios were studied. Tostudy the effect of above parameters ANSYSwas used. Back up ratios and pressure anglesare varied and location and magnitude ofmaximum bending stress were reported.

ASYMMETRIC SPUR GEARTOOTHThe two profiles (sides) of a gear tooth arefunctionally different for many gears. Theworkload on one profile is significantly higherand is applied for longer periods of time thanfor the opposite one. The design of theasymmetric tooth shape reflects this functionaldifference.

Figure 1: Asymmetric Spur Gear

Figure 2: Asymmetric Spur Gear withDifferent Base Circles

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The difference between symmetric andasymmetric tooth is defined by two involutesof two different base circles Dbd and Dbc. Thecommon base tooth thickness does not existin the asymmetric tooth. The circular distance(tooth thickness) Sp between involute profilesis defined at some reference circle diameterDp that should be bigger than the largest basediameter.

Asymmetric gears simultaneously allowan increase in the transverse contact ratioand operating pressure angle beyond theconventional gear limits. Asymmetric gearprofiles also make it possible to managetooth stiffness and load sharing whilekeeping a desirable pressure angle andcontact ratio on the drive profiles bychanging the coast side profiles. Thisprovides higher load capacity and lowernoise and vibration levels compared withconventional symmetric gears.

The design intent of asymmetric gear teethis to improve the performance of the primarycontacting profile. The opposite profile istypically unloaded or lightly loaded duringrelatively short work periods. The degree ofasymmetry and drive profile selection for thesegears depends on the application.

RIM THICKNESS ANDBACKUP RATIOThin rim gears find application in high-power,lightweight aircraft transmissions and used inan airplane to meet the lightweight need. Inrecent years thin rim gear has been a widerapplication in common machines such as inprototype cars for weight and space reduction.But stress calculation of this gear has notsolved so far because this gear has very

complicated thin rim structure. In Figure showsa gear segment. The radial distance, betweenthe tip circle and root circle is termed as toothheight (ht) and distance between the root circleand inner surface of the rim is called the rimthickness (tr). The angle subtended at theg-ear centre by the two radial lines joininglowest point on the root fillet to the gear centreis termed as tooth root angle. The ratio of toothheight (ht) to the rim thickness (tr) is calledbackup ratio.

httr

Figure 3: Spur Gear Tooth with RimThickness and Tooth Height

PARAMETERS USED FORINVOLUTE GEAR TOOTHPROFILE GENERATIONWith the emergence of computers, engineeringmodeling and analysis is getting moredependent on computers day by day. Heredifferent parameters listed in table are usedto generate gear tooth profile using CProgramming.

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Equations used to generate spur gear toothwith back up ratio

yx

r

X

cossin2cos2

sin2 N

xN

MN n

Y

sinsin2cos2

cos2 N

xN

MN n

maxmin

cot2min XVU

N

22max cos22

cos1

NXNN

Nnx

2tan21

cos

tan4

U

V

yx

r

sincos QPMX n

cossin QPMY n

maxmin

cot2min XVU

N

NU2

max

2

nul

P

XNVnukv

LQ

22

2

httrmB

Parameter Notation Values Taken

Pressure angle 20°, 25°, 30°

Backup ratio 0.6, 0.8, 1.0, 1.2, 1.4, 1.6

No. of teeth z 25 and 47

Profile shift factor sf 0

Module m 6 mm

Table 1: Parameters Used for Analysis

Figure 4: Generated Involute Profileand Fillet Radius

FINITE ELEMENT ANALYSISMETHODFinite element analysis has been carried outfor different sets of asymmetric gears aslisted in Talbe 1, subjected to load at highestpoint of single tooth of contact. Key points

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are generated using c programme forgenerating involute gear tooth and is usedfor generating the model and is shown inbelow Figure 5.

The PLANE182 (plane stress) element ischosen for analyses and material propertyof steel is used for gear. The f irstinvestigation involved a two dimensionalplane stress analysis for 6mm module and20º pressure angles on both sides of thegear and 25 teeth and with back up ratio 0.6.The gear tooth is loaded at HPSTC.andanalysis is done.

Similar investigations are done for differentpressure angles and back up ratio.

Figure 5: Finite Element Analysisof Single Teeth Gear Segment Subjected

to Load and Boundary Conditions

RESULTSResults mainly contain bending stress atcritical section and von mises stress anddisplacements

Figure 6: Maximum Bending Stressfor Different Pressure Angle and Back Up

Ratios

For BP 0.6

For BP 0.8

For BP 1.0

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CONCLUSIONFrom the above results following conclusionsare made

• The bending stress obtained at the criticalsection is minimum for back up ratio 1.6 inasymmetric spur gear tooth.

• For back up ratio 0.6 and 0.7 there isincrease in the deflection takes place atgear tooth root which indicates there is achance of rim fracture.

• The bending stress is maximum for backup ratio more than 1.6.

Figure 6 (Cont.)

For BP 1.2

For BP 1.4

For BP 1.6

Figure 7: Bending Stress at the CriticalSection for Different Back Up Ratio

Figure 8: Bending Stress at the CriticalSection for Different Pressure Angles

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• These types of gears find application in lightweight high power aircraft transmission.

REFERENCES1. Bibel G D, Reddy S K, Saavage M and

Handschuh R E (1991), “Effect of RimThickness on Spur Gear Bending Stress”,NASA Technical Memorandum.

2. Chang S H, Huston R L and Coy J J(1983), “A Finite Element StressAnalysis of Spur Gears Including FilletRadii and Rim Thickness Effects”,ASME, Journal of Mechanisms,Transmissions, and Automation inDesign, Vol. 105, pp. 327-330.

3. Chong Tae Hyong, Katayam N and KuboA (1984), “Tooth Fillet Stresses of Gearwith Thin Rim (5th Report, Tooth Fillet andRoot Stresses of Internal Spur GearSupported by Pinned Coupling)”, Bulletinof JSME, Vol. 27, pp. 815-822.

4. David G Lewicki (1995), “CrackPropagation Studies to Determine Beginor Catastrophic Failure Modes forAerospace Thin-Rim Gears”, May,Department of Mechanical andAerospace Engineering, Case WesternReserve University.

5. David G Lewicki and Roberto Ballarini(1996), “Effect of Rim Thickness on GearCrack”, NASA Technical Memorandum.

6. David G Lewicki and Roberto Ballarini(1997), “Gear Crack PropagationInvestigations”, International Journal ofFatigue, Vol. 19, No. 10, p. 731.

7. Kramberger J, Sraml M, Potrc I andFlasker J (2004), “NumericalCalculation of Bending Fatigue Life ofThin-Rim Spur Gears”, EngineeringFracture Mechanics, Vol. 71, Nos. 4-6,pp. 647-656.

effect of pressure angle and back up ratio on … · ratio on asymmetric spur gear tooth bending...

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