finite element analysis and fatigue analysis of spur

533

Int. J. Mech. Eng. & Rob. Res. 2014 Sameer Chakravarthy N C and B Subbaratnam, 2014

FINITE ELEMENT ANALYSIS AND FATIGUEANALYSIS OF SPUR GEAR UNDER RANDOM

LOADINGSameer Chakravarthy N C1* and B Subbaratnam2

*Corresponding Author: Sameer Chakravarthy N C, [email protected]

The gear fitted in the gearbox Armored tracked vehicle is vulnerable to considerable fatiguedamage over its life period due to the dynamic excitations caused by the terrain undulations, therotating wheel and track assemblies. For this purpose, initially static analysis of the model wascarried out to validate the model and the boundary conditions correctness. Further Modal analysisis carried out to determine the dynamic characteristics of the gear model. The random load timehistory is transformed in to frequency domain using Fast Fourier transform to obtain load PowerSpectral Density (PSD). Then the stress PSD response is obtained at critical node from therandom vibration analysis. Once the spectrum of stress variation is obtained given input to thefatigue analysis and fatigue life is determined by FE package ANSYS 11.0.

Keywords: Spur gear, Static analysis, Modal analysis, PSD analysis

INTRODUCTIONGears are the most common means oftransmitting power in the modern mechanicalengineering world. They vary from tiny size usedin the watches to the large gears used inmarine speed reducers; bridge liftingmechanism and railroad turn table drivers.They form vital elements of main and ancillarymechanism in many machines such asautomobiles, tractors, metal cutting machinetools, rolling mills, hoisting and transmittingmachinery and marine engines, etc.

ISSN 2278 – 0149 www.ijmerr.comVol. 3, No. 4, October 2014

© 2014 IJMERR. All Rights Reserved

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

1 M.Tech Student, Department of Mechanical Engineering, Krishnachaitanya Institute of Technology & Sciences, Markapur, AndhraPradesh, India.

2 Professor, Department of Mechanical Engineering, Krishnachaitanya Institute of Technology & Sciences, Markapur, Andhra Pradesh,India.

The four major failure modes in gearsystems are tooth bending fatigue, contactfatigue, surface wear and scoring. Two kindsof teeth damage can occur on gears underrepeated loading due to fatigue; namely thepitting of gear teeth flanks and tooth breakagein the tooth root. Tooth breakage is clearly theworst damage case, since the gear could haveseriously hampered operating condition oreven be destroyed. Because of this, the stressin the tooth should always be carefully studiedin all practical gear application. The fatigue

Research Paper

534


process leading to tooth breakage is dividedinto crack initiation and crack propagationperiod. However, the crack initiation periodgenerally account for the most of service life,especially in high cycle fatigue.

The initial crack can be formed due tovarious reasons. The most common reasonsare short-term overload, material defects,defects due to mechanical or thermal treatmentand material fatigue. The initial crack thenpropagates under impulsive loading until somecritical length is reached, when a completetooth breakage occurs. The service life of agear with a crack in the tooth root can bedetermined experimentally or numerically (e.g.,with finite element method). The fatigue life ofcomponents subjected to sinusoidal loadingcan be estimated by using cumulative damagetheories. Their extension to random loadfatigue, through straightforward, may not bevery accurate owing to inherent scatterexhibition by the fatigue phenomena. Due tothe complexity in geometry and loading on thestructure, the finite element method ispreferably adopted.

Major contribution being the terrainundulations, the rotating wheel, the trackassemblies, the drive line and the engine andalso influenced by the incidental variations inthe vehicle components, firing ammunition fromthe vehicle body and hitting of ammunition onthe vehicle body.

PROBLEM STATEMENTThe present work deals with the calculation ofstatic and dynamic analysis, and fatigue lifeestimation of test gear, which is contactingmaster gear and assuming loading on the gearis random or constant amplitude by Finite

Element package ANSYS

AIMS AND OBJECTIVES1. To carry out the detail finite element analysis

like static analysis to validate the model andboundary conditions correctness, Modalanalysis to determine dynamiccharacteristics of the model and transientdynamic analysis to determine dynamicload factor.

2. Simulation of loads of deterministic andnon-deterministic nature causingfluctuations bending stresses in gear.

3. To carry out dynamic analysis of gear bothunder random loading and constantamplitude loading conditions and obtain thestress PSD response at vulnerablelocations.

4. To carry out fatigue analysis in order todetermine the fatigue life of the test gearwith help of above stress PSD responsesat vulnerable locations.

3D Modeling of Spur GearsModeling of spur gear using NX-cad software

Figure 1: Test Gear

535


Assembly of Spur Gears

ASSUMPTIONSThe following conditions have been assumed.

1. The material has been assumed to behaveas linear elastic material.

2. The material is homogeneous andisotropic.

3. Heat generation and thermal stress areignored.

4. The gear to mesh under conditions of goodlubrication and the friction is taken as zero.

5. No contact behavior is considered.

6. There are no residual stresses present inthe gears.

Figure 2: Master Gear

Figure 3: Model of Spur Gearsin Unigraphics N x 7.5

FINITE ELEMENT ANALYSISIn FEM, complex structure or continuum isdivided into finite number of small regionscalled as elements. The material propertiesand governing equations are considered overthese elements and the field quantity isexpresses as unknown values at corners ofthese elements called as the nodes. Due tothis the infinite degree of freedom is convertedinto finite degree of freedom problem. Anassembly process, duly considering theloading and constraints, results in a set ofequations. Solution of these equations givesus approximate behavior of the continuum.

Element Type Shell

Total number of node at each element 8

Number of degrees of freedom per node 6

Total number of elements 2265

Total number of nodes 7648

Largest node number 7648

Total number of degrees of freedom inmodel 43735

Table 1: Finite Element Model Properties

Figure 4: Boundary Condition Model

536


Static AnalysisA static load is a stationary force or momentacting on a member. To be stationary the forceor moment must have an unchangingmagnitude, unchanging point or points ofapplication, and an unchanging direction.

Carbon 0.12 0.43

Silicon 0.22 0.19

Manganese 0.51 0.73

Nickel 3.42 –

Chromium 0.94 1.30

Molybdenum 0.16 0.27

Table 2: Composition of Alloys En-36 andEn-19

ConstituentMaterial

Percentage Content

Master GearEn-36 Steel

Test Gear En-19 Steel

Material name En-36c En-19

Young’s modulus ofelasticity (MPa) 2E 5 2E 5

Poisson’s ratio 0.29 0.29

Yield strength (MPa) 1230 1030

Table 3: Mechanical Properties

Properties Master Gear Test Gear

Fatigue strengthcoefficient (SF) 2415.92 (N/mm2) 1944.32 (N/mm2)

Fatigue strengthexponent (B) –0.07 –0.25

Fatigue ductilitycoefficient (EF) 0.002 1.22

Fatigue ductilityexponent (C) –0.47 –0.73

Cyclic strengthcoefficient (KP) 4263.71 (N/mm2) 1862.96 (N/mm2)

Cyclic strainhardeningexponent (NP) 0.1 0.14

Table 4: Cyclic Properties

Properties Master Gear Test Gear

Sxx 540.1 7311

Syy 267.8 7315

1 756.3 7315

2 89.64 7261

Von misses stress 792.4 7311

Table 5: Static Analysis Results

Type of Stress MaximumStress (N/mm)

CorrespondingNode Number

Figure 5: (Bending) Stress Contour

Type of Stress Analytical Ansys Software % Error

Sxx(N/mm2) 532.3 540.1 1.5

Table 6: Comparison of Analytical andAnsys Result

We can observe that the values formaximum bending stress obtained by Ansysare very closer to analytical value and the erroris 1.5%, which confirms that the model andboundary conditions applied to the gear, iscorrect. The stresses induced in the adjacenttooth are about 1.13% of loaded tooth.

DYNAMIC ANALYSISDynamic analysis involves the computation ofresponse of a linear system subjected to timedependent loads.

537


Results of Transient DynamicAnalysis1. The displacement of node which has

maximum deformation corresponding tofirst mode shape (Node number 363) isshown in Figure 7.

2. The stress histories of critical node whichhas maximum bending stress in case ofstatic analysis (Node number 7311) isshown in Figure 8.

3. Transient dynamic analysis is predominanton static analysis because at same peakload condition the stress is higher ascompared with static analysis and thedynamic load factor is 1.2. It is due to massand acceleration effects induced in dynamicloading.

1 1.753E+03 2.78 E+02

2 4.322E+03 6.87 E+02

3 4.599E+03 7.31E+02

4 5.474E+03 8.71E+02

5 6.557E+03 1.043E+03

6 6.729E+03 1.071E+03

7 1.1928E+04 1.898E+03

8 1.3300E+04 2.116E+03

9 1.3709E+04 2.181E+03

10 1.6830E+04 2.678E+03

11 1.7022E+04 2.092E+03

12 1.9077E+04 3.036E+03

13 1.9135E+04 3.045E+03

14 1.9718E+04 3.138E+03

15 2.0082E+04 3.196E+03

16 2.0707E+04 3.295E+03

17 2.1285E+04 3.387E+03

18 2.1382E+04 3.403E+03

19 2.1508E+04 3.423E+03

20 2.230E+04 3.549E+03

Table 7: Frequency Correspondingto Mode Shape

Mode Shape Frequency(rad/sec)

Frequency(cycles/sec)

TRANSIENT DYNAMICANALYSISTransient dynamic analysis may be used todetermine the response of structuressubjected to arbitrary time-varying loads.

Loading DetailGeometric model, material properties and theboundary conditions are same as the staticanalysis except that the contact force isvariable in nature but peak load is same as instatic analysis. And the time step taken as0.001 sec.

Figure 6: Graph of Time and Load

Figure 7: Ux (Displacement) ResponseHistory Plot for Node 363

538


RANDOM VIBRATIONANALYSISThe load history is converted to load PSD withthe help of fast Fourier transform. This loadPSD is used in ANSYS package for randomvibration analyses. The load PSD is shown inthe Figure 9.

are very small. This means that careful analysisis need in the region of first mode frequencyrather than at other modes.

CONSTANT AMPLITUDELOAD ANALYSISDeterministic data are those that can bedescribed by an explicit mathematicalrelationship. Data representing a physicalphenomenon can be categorized has beingeither periodic or non periodic. Periodic datacan be further categorized as being bothsinusoidal and complex periodic. Non-periodic data can be further categorized asbeing both almost periodic and transient.

Figure 8: Sxx (Bending) Stress ResponseHistory Plot for Node 7311

Figure 9: Load PSD

Results of Random VibrationAnalysisThe output of random analysis is stress PSDresponse of node no. 7311 which is shown inFigure 10. As we see that the maximum powercontent is at the frequency 278.99 Hzcorresponding to first mode shape and thepower content at other modes are negligibleas compared to the first mode because they

Figure 10: Stress PSD Responseof Node No. 7311

Figure 11: Load PSD

539


The load history is converted to load PSDwith the help of fast Fourier transform. This loadPSD is used in ANSYS package for randomvibration analyses. The load PSD is shown inthe Figure 11.

Results of Constant AmplitudeLoad AnalysisThe output of constant amplitude analysis isstress PSD response of node no. 7311 whichis shown in Figure. As we see that themaximum power content is at the frequency278.99 Hz corresponding to first mode shapeand there is very little power content at othermodes. This means that careful analysis isneed in the region of first mode frequencyrather than at other modes.

And for constant amplitude loading, it isobserved that energy content is more ascompared to random loading. So the stressesinduced due to constant amplitude loading areconsiderably higher than that random loading.

other major industries where the structuralfailures are mainly due to fatigue.

Fatigue PropertiesThe properties used in crack initiation analysis.

AnalysisThere are two types of analysis done in thisproject.

1. Fatigue analysis for random loading

2. Fatigue analysis for constant amplitudeloading

Fatigue Analysis for RandomLoadingTaking the same stress PSD as input in Ansysfor crack initiation analysis. Geometric model,boundary conditions and material propertiesare same.

Figure 12: Sxx (Bending) Stress PSDat Node 7311

FATIGUE ANALYSISFatigue life evaluation has almost becomemandatory for structural components underrepeated loading in aerospace, nuclear and

Properties Test Gear

Fatigue strength coefficient (SF) 1944.32 N/mm2

Fatigue strength exponent (B) –0.25

Cyclic strength coefficient (KP) 1862.96 N/mm2

Cyclic strain hardening exponent (NP) 0.14

Table 8: Table Cyclic Properties

Type of Analysis Crack Initiation Analysis

Type of correlation Stress-life (S-N)

Basic FE analysis Random vibration

Material properties Table cycle properties

Table 9: Input to Fatigue Analysis

Results of Fatigue Analysis forRandom LoadingDue to the stress PSD response given inputat critical node 7311, life obtained by FEM is5.8 x 105 cycles where as life obtained byexperiment is 6.5 x 105 cycles (Hanumannaet al., 2001). And the variation is less. Thisleads to the validation of the results. Life

540


obtained by FEM is less as compared to theexperimental life, this is due to clampedcondition of test gear.

Life Obtained by Experiment Life Obtained by FEM(cycles) (cycles)

6.5 x 105 5.8 x 105

Table 10: Fatigue Life Under RandomLoading

Fatigue Analysis for ConstantAmplitude LoadingConstant amplitude loading pattern has beenshown in Figure 11. Its stress PSD responseat critical node 7311 has been obtained inconstant amplitude load analysis, which isshown in Figure 10. Taking the same stressPSD as input in Ansys for crack initiationanalysis. Geometric model, boundarycondition and material properties are same.

to stresses induced in constant amplitudeloading is higher for same peak loadcondition.

Type of Analysis Crack Initiation Analysis

Type of correlation Stress-life (S-N)

Basic FE Analysis Random Vibration

Material properties Tablecycle properties

Approach Miner-Palmgren Damage Rule

Table 11: Input to Fatigue Analysis

Results of Fatigue Analysis forConstant Amplitude LoadingDue to stress PSD response given input atcritical node 7311, life obtained by FEM is 2.9x 10 cycles where as life obtained byexperiment is 3.25 x 10 cycles (Hanumannaet al., 2001). And the variation is less. Thisleads to the validation of the results. Lifeobtained by FEM is less as compared to theexperimental life, this is due to the clampedcondition of test gear. And the life obtained byconstant amplitude loading is much lesser thanthat obtained by random loading, this is due

Life Obtained by Experiment Life Obtained by FEM(cycles) (cycles)

3.25 x 105 2.9 x 105

Table 12: Fatigue Life Under ConstantAmplitude Loading

Sxx 540.1 7311

Syy 267.8 7315

1 756.3 7315

2 89.64 7261

Von misses stress 792.4 7311

Table 13: Static Analysis Results

Type of Stress MaximumStress (N/mm)

CorrespondingNode Number

CONCLUSIONThe gear in a gear box fitted an armouredtracked vehicle for the purpose of powertransmission and positioned of heavy massto the desired angle with high accuracy aresubjected to fluctuating loads that are randomin nature. Therefore, it is analyzed for randomloading and also under constant amplitudeloading conditions for fatigue analysis and lifeof gear has been obtained by finite elementpackage ANSYS.

Following are the conclusions:

1. The stresses obtained for static analysiswere presented in Table 13.

2. Comparison of analytical and Ansys resultwas presented in Table 14.

Type of Stress Analytical Ansys Software % Error

Sxx(N/mm2) 532.3 540.1 1.5

Table 14: Comparison of Analytical andAnsys Result

541


3. The maximum power content is at thefrequency 278.99 Hz corresponding to firstmode shape and the power content at othermodes are negligible as compared to thefirst mode because they are very small.

4. Due to the stress PSD response given inputat critical node 7311, life obtained by FEMis 5.8 x 10 cycles where as life obtained byexperiment is 6.5 x 10 cycles.

5. Due to stress PSD response given input atcritical node 7311, life obtained by FEM is2.9 x 10 cycles where as life obtained byexperiment is 3.25 x 10 cycles.

SCOPE FOR FUTURE WORK1. 3-D contact analysis of gears including heat

generation and frictional component of loadmay be investigated.

2. Surface wear is one of the major failuremode in gear systems, therefore surfacewear prediction methodology of gear pairmay be carried out.

3. Present study may be extended to differentcomplex cases of multi axial fatigue loadingof proportional and non-proportional nature.

4. Carryout fatigue reliability assessment oftest gear under random loading both by S-N curve approach and Fracture mechanicsapproach.

5. Fatigue and reliability analysis under non-stationary random loading may be carriedout.

REFERENCES1. Abersek B, Flasker J and Glodez S

(2004), “Review of Mathematical andExper imenta l Models forDetermination of Service Life of

Gears” , Engineer ing FractureMechanics, Vol. 71, pp. 439-453.

2. Alexander L Kapelevich and Roderick EKleiss (2002), “Direct Gear Design forSpur and Helical Involute Gears ”, GearTechnology, September/October.

3. Glodez S, Sraml M and Kramberger J(2202), “A Computational Model forDetermination of Service Life of Gears”,International Journal of Fatigue, Vol. 24,No. 10, pp. 1010-1020.

4. Glodez S, Abersek B, Flasker J and RenZ (2004), “Evaluation of the Service Lifeof Gears in Regard to Surface Pitting”,Engineering Fracture Mechanics ,Vol. 71, Nos. 4-6, pp. 429-438.

5. Hanumanna D, Narayanana S andKrishnamurthy S (2001), “BendingFatigue Testing of Gear Teeth UnderRandom Loading”, Proc. Instn. Mech.Engrs, Vol. 215, Part C, pp. 773-784.

6. Julius S Bendat and Allan G Piersol(1966), “Measurement and Analysis ofRandom Data”, John Wiley & Sons, Inc.,USA.

7. Statistical Considerations in Fatigue(1996), “Fatigue and Fracture”, ASMHandbook, Vol. 19, pp. 295-302.

8. Structural Life Assessment Methods(2001), “Failure Analysis and Prevention”,ASM Handbook, Vol. 11, pp. 225-289.

9. Yang Q J (1996), “Fatigue Test andReliability of TLP Tethers Under RandomLoading”, Marine Structures, Vol. 14,pp. 331-352.

finite element analysis and fatigue analysis of spur

Documents