INVESTIGATION OF CONTACT STRESS IN SPUR GEAR USING · PDF fileGEAR USING LEWIS EQUATION AND FINITE ELEMENT METHOD ... verification by 2D FEM analyzer using ANSYS, ... solid modeler

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Int. J. Mech. Eng. & Rob. Res. 2013 Ashish V Kadu and Sanjay S Deshmukh, 2013

INVESTIGATION OF CONTACT STRESS IN SPURGEAR USING LEWIS EQUATION AND FINITE

ELEMENT METHOD

Ashish V Kadu1* and Sanjay S Deshmukh1

*Corresponding Author: Ashish V Kadu,[email protected]

This work deals with the characteristics of involutes gear system including contact stresses,bending stresses and the transmission error of the gear in mesh the gear system of the unityindustry for this work. To estimate the transmission error in the actual gear system which arisesbecause of a irregular tool geometry or imperfect geometry or imperfect mounting thecharacteristic of the involutes spur gear are analyzed by using finite element method. The contactstresses are examined by using 2D FEM Model. And the bending stresses in the tooth root areexamined by using 3D FEM Model. The conventional method of calculating gear contact stressusing Hertz’s theory for verification by 2D FEM analyzer using ANSYS, in later investigation thestiffness relationship between two contact area is usually established using a spring placebetween source and target surfaces for the contact generation between two gears. The stressesare compared with theoretical result. This work also considered static transmission error andanalysis of load shearing method using displacement vector and the effect of this error in theactual transmission power of the mesh gear.

Keywords: Component, Formatting, Style, Styling, Insert (key words)

INTRODUCTIONGears are essential to the global economyand are used in nearly all application wherethe power transmission is required such asautomobiles, industrial equipment airplaneshelicopters and marine vessels. Frequencyof product model changeover, also calledtime-based competition has become a

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Int. J. Mech. Eng. & Rob. Res. 2013

1 Department of Mechanical Engineering, Prof. Ram Meghe Institute of Technology & Research, Badnera, Amravati, India.

character feature of global manufacturing ofnew product development in automotiveaerospace and other industries. This forcesgear manufacturer to respond with improvedgear. Simultaneously, current trends inengineering globalization required researchto revisit various normalized standard todetermine their common fundamentals and

Research Paper

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those approaches needed to identify “bestpractices” in industries.

The increasing demand for quiet powertransmission in machines, vehicles, elevatorsand generators, has created a growingdemand for a more precise analysis of thecharacteristics of gear systems. In theautomobile industry, the largest manufacturerof gears, higher reliability and lighter weightgears are necessary as lighter automobilescontinue to be in demand. In addition, thesuccess in engine noise reduction promotesthe production of quieter gear pairs for furthernoise reduction. Noise reduction in gear pairsis especially critical in the rapidly growing fieldof office-automation equipment as the officeenvironment is adversely affected by noise,and machines are playing an ever-wideningrole in that environment. Ultimately, the onlyeffective way to achieve gear noise reductionis to reduce the vibration associated withthem& hence the transmission inaccuracy ofgear. The reduction of noise through vibrationcontrol can only be achieved through researchefforts by specialists in the field. However, ashortage of these specialists exists in thenewer, lightweight industries in Japan mainlybecause fewer young people are specializingin gear technology today and traditionally thespecialists employed in heavy industries tendto stay where they are.

The prime source of vibration and noise ina gear system is the transmission errorbetween meshing gears. Transmission erroris a term used to describe or is defined as thedifferences between the theoretical and actualpositions between a pinion (driving gear) anda driven gear. It has been recognized as amain source for mesh frequency excited noise

and vibration. With prior knowledge of theoperating conditions of the gear set it ispossible to design the gears such that thevibration and noise is minimized.

Transmission error is usually due to twomain factors. The first is caused bymanufacturing inaccuracy and mounting errors.Gear designers often attempt to compensatefor transmission error by modifying the gearteeth. The second type of error is caused byelastic deflections under load. Among thetypes of gearbox noise, one of the most difficultto control is gear noise generated at the toothmesh frequency.

Transmission error is considered to be oneof the main contributors to noise and vibrationin a gear set. This suggests that the gear noiseis closely related to transmission error. If apinion and gear have ideal involute profilesrunning with no loading torque they shouldtheoretically run with zero transmission error.However, when these same gears transmittorque, the combined torsional mesh stiffnessof each gear changes throughout the meshcycle as the teeth deflect, causing variationsin angular rotation of the gear body. Eventhough the transmission error is relatively small,these slight variations can cause noise at afrequency which matches a resonance of theshafts or the gear housing, causing the noiseto be enhanced. This phenomenon has beenactively studied in order to minimize the amountof transmission error in gears.

In this thesis, first, the finite element modelsand solution methods needed for the accuratecalculation of two dimensional spur gearcontact stresses and gear bending stresseswere determined by using ANSYS. Then, thecontact and bending stresses calculated using

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ANSYS were compared to the results obtainedfrom existing methods. The purpose of thisthesis is to develop a model to study andpredict the transmission error model includingthe contact stresses, and the torsional meshstiffness of gears in mesh using the ANSYSsoftware package based on numericalmethod. The aim is to reduce the amount oftransmission error in the gears, and therebyreduce the amount of noise generated.

Objectives of the Research

In spite of the number of investigations devotedto gear research and analysis there stillremains to be developed, a general numericalapproach capable of predicting the effects ofvariations in gear geometry, contact andbending stresses, tensional mesh stiffnessand transmission errors. The objectives of thisthesis are to use a numerical approach todevelop theoretical models of the behavior ofspur gears in mesh, to help to predict the effectof gear tooth stresses and transmission error.The main focus of the current research asdeveloped here is:

• To develop and to determine appropriatemodels of contact elements, to calculatecontact stresses using ANSYS andcompare the results with HERTZIAN theory.

• To generate the profile of spur gear teethand calculate of gear bending stress usingLewis equation and hence check thefeasibility of modified gear profile.

• To compare the static transmission errorsof slandered and modified profile of thegear teeth.

Focus of the Work

• Stress analysis such as prediction ofcontact stress and bending stress.

• Prediction of transmission efficiency.

• Finding the natural frequencies of thesystem before making the gears.

• Performing vibration analyses of gearsystems.

• Evaluating condition monitoring, faultdetection, diagnosis, and prognosis,reliability and fatigue life.

LITERATURE REVIEW ANDBACKGROUNDThere has been a great deal of research ongear analysis, and a large body of literatureon gear modeling has been published. Thegear stress analysis, the transmission errors,the prediction of gear dynamic loads, gearnoise, and the optimal design for gear setsare always major concerns in gear design.Errichello (1979) and Ozguven and Houser(1988) survey a great deal of literature on thedevelopment of a variety of simulation modelsfor both static and dynamic analysis ofdifferent types of gears. The first study oftransmission error was done by Harris(1958). He showed that the behavior of spurgears at low speeds can be summarized in aset of static transmission error curves. In lateryears, Mark (1978 and 1979) analyzed thevibratory exci tation of gear systemstheoretically. He derived an expression forstatic transmission error and used it to predictthe various components of the statictransmission error spectrum from a set ofmeasurements made on a mating pair of spurgears. Kohler and Regan (1985) discussedthe derivation of gear transmission error frompitch error transformed to the frequencydomain. Kubo et al. (1991) estimated thetransmission error of cylindrical involutes

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gears using a tooth contact pattern. Thecurrent literature reviews also attempt toclassify gear model into groupings withparticular relevance to the researchAbbreviations and Acronyms.

Spur Gear Failures

Bending Fatigue: This common type offailure which is a slow, progressive failurecaused by repeated loading.

Figure 1: Bending Fatigue

Contact Fatigue: In another failure mode,called contact or Hertzian fatigue, repeatedstresses cause surface cracks anddetachment of metal fragments from the toothcontact surface.

Wear in Gear: Gear tooth surface wearinvolves removal or displacement of materialdue to mechanical, chemical or electricalaction.

Scuffing in Gear: Scuffing is a transfer ofmetal from the surface of one tooth to that ofanother tooth.

Figure 2: Contact Fatigue

Figure 3: Wear in Gear

Figure 4: Scuffing in Gear

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FEM ANALYSISIntroduction

The finite element method is a numericaltechnique, well suited to digital computers,which can be applied to solve problems in solidmechanics, fluid mechanics, heat transfer andvibrations. The procedures to solve problemsin each of these fields is similar; however thisdiscussion will address the application of finiteelement methods to solid mechanicsproblems. In all finite element models thedomain (the sol id in solid mechanicsproblems) is divided into a finite number ofelements. These elements are connected atpoints called nodes. In solids models,displacements in each element are directlyrelated to the nodal displacements. The nodaldisplacements are then related to the strainsand the stresses in the elements. The finiteelement method tries to choose the nodaldisplacements so that the stresses are inequilibrium (approximately) with the appliedloads. The nodal displacements must also beconsistent with any constraints on the motionof structure.

The finite element method converts theconditions of equilibrium into a set of linearalgebraic equations for the nodaldisplacements. Once the equations are solved,one can find the actual strains and stresses inall the elements. By breaking the structure intoa larger number of smaller elements, thestresses become closer to achievingequilibrium with the applied loads. Thereforean important concept in the use of finiteelement methods is that, in general, a finiteelement model approaches the true solutionto the problem only as the element density isincreased.

There are a number of steps in the solutionprocedure using finite element methods. Allfinite element packages require the user to gothrough these steps in one form or another.

Specifying Geometry: First the geometry ofthe structure to be analyzed is defined. Thiscan be done either by entering the geometricinformation in the finite element packagethrough the keyboard or mouse, or by importingthe model from a solid modeler like Pro/ENGINEER.

Specify Element Type and MaterialProperties: Next, the material properties aredefined. In an elastic analysis of an isotropicsolid these consist of the Young’s modulus andthe Poisson’s ratio of the material.

Mesh the Object: Then, the structure is broken(or meshed) into small elements. This Involvesdefining the types of elements into which thestructure will be broken, as well as specifyinghow the structure will be subdivided intoelements (how it will be meshed). Thissubdivision into elements can either be inputby the user or, with some finite elementprograms (or add-ons) can be chosenautomatically by the computer based on thegeometry of the structure (this is called automeshing).

Apply Boundary Conditions and ExternalLoads: Next, the boundary conditions (e.g.,location of supports) and the external loads arespecified.

Generate a Solution: Then the solution isgenerated based on the previously inputparameters.

Postprocessing: Based on the initialconditions and applied loads, data isreturned after a solution is processed. This

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data can be viewed in a variety of graphs anddisplays.

Refine the Mesh: Finite element methods areapproximate methods and, in general, theaccuracy of the approximation increases withthe number of elements used. The number ofelements needed for an accurate modeldepends on the problem and the specificresults to be extracted from it. Thus, in orderto judge the accuracy of results from a singlefinite element run, you need to increase thenumber of elements in the object and see if orhow the results change.

Interpreting Results: This step is perhapsthe most critical step in the entire analysisbecause it requires that the modeler use hisor her fundamental knowledge of mechanicsto interpret and understand the output of themodel. This is critical for applying correctresults to solve real engineering problems andin identifying when modeling mistakes havebeen made (which can easily occur).

The eight steps mentioned above have tobe carried out before any meaningfulinformation can be obtained regardless of thesize and complexity of the problem to besolved. However, the specific commands andprocedures that must be used for each of thesteps will vary from one finite element packageto another. The solution procedure for ANSYSis described in this tutor. Note that ANSYS (likeany other FEM package) has numerouscapabilities out of which only a few would beused in simple beam problems.

Limitations of Finite ElementMethods

• Finite element methods are extremelyversatile and powerful and can enable

designers to obtain information about thebehavior of complicated structures withalmost arbitrary loading.

• In spite of the significant advances that havebeen made in developing finite elementpackages, the results obtained must becarefully examined before they can be used.This point cannot be overemphasized.

• The most significant limitation of finiteelement methods is that the accuracy of theobtained solution is usually a function of themesh resolution. Any regions of highlyconcentrated stress, such as aroundloading points and supports, must becarefully analyzed with the use of asufficiently refined mesh. In addition, thereare some problems which are inherentlysingular (the stresses are theoreticallyinfinite). Special efforts must be made toanalyze such problems.

• An additional concern for any user is thatbecause current packages can solve somany sophisticated problems, there is astrong temptation to “solve” problemswithout doing the hard work of thinkingthrough them and understanding theunderlying mechanics and physicalapplications. Modern finite elementpackages are powerful tools that havebecome increasingly indispensible tomechanical design and analysis. However,they also make it easy for users to makebig mistakes.

• Obtaining solutions with finite elementmethods often requires substantial amountsof computer and user time. Nevertheless,finite element packages have becomeincreasingly indispensable to mechanicaldesign and analysis.

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Tools in Finite Element Analysis

Pre-Processing: The user constructs amodel of the part to be analyzed in which thegeometry is divided into a number of discretesub regions, or elements, “connected atdiscrete points called nodes.” Certain of thesenodes will have fixed displacements, andothers will have prescribed loads. Thesemodels can be extremely time consuming toprepare, and commercial codes vie with oneanother to have the most user-friendlygraphical preprocessor to assist in this rathertedious chore. Some of these preprocessorscan overlay a mesh on a pre-existing CADmodel, so that finite element analysis can bedone conveniently as part of the computerizeddrafting-and-design process.

Solver: The dataset prepared by the pre-processor is used as input to the finite elementcode itself, which constructs and solves asystem of linear or nonlinear algebraicequations

Kiju

i = f

i

where, Kij = Structural Stiffness

ui and f

i = The displacements and externally

applied forces at the nodal points.

The formation of the K matrix is dependenton the type of problem being attacked, andthis module will outline the approach for trussand linear elastic stress analyses. The staticanalysis of gear can be carried out exactly, andthe equations of even complicated gear canbe assembled in a matrix form amenable tonumerical solution. This approach, sometimescalled matrix analysis, provided the foundationof early FEA development. Matrix analysis oftrusses operates by considering the stiffnessof each truss element one at a time, and then

using theseto determine the forces that are setup in the truss elements by the displacementsof the joints, usually called nodes infiniteelement analysis. Then noting that the sum ofthe forces contributed by each element to anode must equal the force that is externallyapplied to that node, we can assemble asequence of linear algebraic equations in whichthe nodal displacements are the unknowns andthe applied nodal forces are known quantities.These equations are conveniently written inmatrix form, which gives the method its name:

nnnnnn

n

n

f

f

f

u

u

u

KKK

KKK

KKK

2

1

2

1

21

22221

11211

The Kij coefficient array is called the global

stiffness matrix, with the ij component beingphysically the influence of the jth displacementon the ith force. The matrix equations can beabbreviated as:

Kiju

j = f

ior Ku = f

using either subscripts or boldface to indicatevector and matrix quantities. Either the forceexternally applied or the displacement is knownat the outset for each node, and it is impossibleto specify simultaneously both an arbitrarydisplacement and a force on a given node.These prescribed nodal forces anddisplacements are the boundary conditions ofthe problem. It is the task of analysis todetermine the forces that accompany theimposed displacements, and thedisplacements at the nodes where knownexternal forces are applied.

Solver Step Tooth Stiffness: An importantparameter that must be known in order todetermine tooth engagement is the stiffness

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of a pair of mating internal and external splineteeth. Equation shows the generalrelationship for a linear spring, where the forceis a function of deflection u, and the springrate, or stiffness K.

F = KU

Because a single spline tooth is an elasticbody, it may be treated as a linear spring. Theforce applied, then determines the deflectionuniquely. The stiffness of a single pair ofmating spline teeth can be represented asthe combination oftwo springs in series. Asshown in Figure 5, each spring represents onetooth.

Because the sum of K1 and K

2 is in the

denominator, Keqs will always be less than the

sum of K1 and K

2.

For Engagement of Tooth: As sequentialteeth engage in a spline coupling, theequivalent stiffness can be determined by thenumber of teeth that are in contact. Figure 6represents a pair of linear springs in a parallelconfiguration.

Both springs transmit the applied force tothe frame, since they are in series. The totaldeflection dtotal, is the sum of the two springdeflections, u

1 and u

2. Using Equation the

equivalent spring constant of the two teeth inseries may be derived from:

F = K1u

1 = K

2u

2

Substituting the equivalent stiffness for Kand the total deflection for gives

F = Keq u1 = Keq u

2

Solving for Keqs gives

21

21

2121

2111

1

KK

KK

KKKF

KF

FFKeqs

The equivalent stiffness of two springs inparallel is the sum of K

1 and K

2, which is

shown in Equation. This is a simplified modelof two pairs of spline teeth in contact, sharingthe load F:

Keq = K1 + K

2

K1 represents the Keqs of Tooth Pair No. 1

and K2 is the Keq

s of Tooth Pair No. 2.

By combining Figures 5 and 6, theengagement of two pairs of spline teeth wouldlook like Figure 7. The stiffness of each mating

Figure 7: Series and Parallel Combinationof the Spring

Figure 5: Series Combinationof Two Spring

Figure 6: Parallel Combinationof Two Springs

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Point-to-Point Contact: The exact locationof contact should be known beforehand. Thesetypes of contact problems usually only allowsmall amounts of relative sliding deformationbetween contact surfaces.

Point-to-Surface Contact: The exactlocation of the contacting area may notbeknown beforehand. These types of contactproblems allow large amounts of deformationand relative sliding. Also, opposing meshesdo not have to have the same discriminationor a compatible mesh. Point to surface contactwas used in this chapter.

Surface-to-surface contact is typically usedto model surface-to-surface contactappl ications of the rigid-to-flexibleclassification. There are some difficult whiledealing with contact problems manydifficulties. First, the actual region of contactbetween deformable bodies in contact is notknown until the solution has been obtained.Depending on the loads, materials, andboundary conditions, along with other factors,surfaces can come into and go out of contactwith each other in a largely unpredictablemanner. Secondly, most contact problemsneed to account for friction. The modeling offriction is very difficult as the friction dependson the surface smoothness, the physical andchemical properties of the material, theproperties of any lubricant that might bepresent in the motion, and the temperature ofthe contacting surfaces. There are severalfriction laws and models to choose from, andall are nonlinear. Frictional response can bechaotic, making solution convergence difficult(ANSYS).

The most general case of contact stressoccurs when each containing body has a

internal and external pair adds in series,resulting in an equivalent stiffness Keq

s. The

equivalent tooth stiffness for both tooth pairsare then added in parallel.

K1 and K

3 represent the stiffness of the

internal teeth, while K2 and K

4 represent the

stiffness of the mating external teeth. Thecorresponding equivalent stiffness of theseri,es-parallel combination, Keq

sp, is

calculated by the following equation

43

43

21

21

KK

KK

KK

KKKeqsp

Postprocessing: In the earlier days of finiteelement analysis, the user would pore throughreams of numbers generated by the code, listingdisplacements and stresses at discretepositions within the model. It is easy to missimportant trends and hot spots this way, andmodern codes use graphical displays to assistin visualizing the results. A typical postprocessordisplay overlay colored contours representingstress levels on the model, showing a full fieldpicture similar to that of photo elastic or moireexperimental results.

Contact AnalysisContact Stress: When two bodies havingcurved surfaces are pressed together, pointor line contact changes to area contact andthe stress developed in the two bodies arethree dimensional. Contact-stress problemarise in the contact of a wheel and a rail, inautomobile valve cams tappets, in the metingteeth, and in the action of rolling bearings.Typical failures are seen as crack, pits orflanking in the surface material.

In general, there are three basic types ofcontact modeling application as far aspracticalapplication is concerned.

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double radius of curvature, i.e., when the radiusin the plane of the rolling is different from theradius in a particular plane, both plane takenplanes through the axis of the contacting force.The stress determine from these are alsoknown as Hertzian stresses.

Contact Analysis of Two Cylinders:Consider two solid cylinder of diameter d

1 and

d2 of length l, press together with force F. As

shown in Figure 8 the area of contact is anarrow rectangular of width 2b and length l,and the pressure distribution is elliptical.

Along X axis

b

z

b

zpvx 2

2

max 12

Along Y axis

b

z

b

z

bz

py 211

12

2

2

2

2max

Along Z axis

2

2

max

1b

z

pz

these three equation are plotted as shownbelow up to distance of 3b below the surfacethough Tzy is not the largest of the three shearstress for all value of z/b, it is a maximum atabout z/b = 0.75 and larger at that point thanpoint either of the other two shear stress forany for value of z/b.

Figure 9 shows the variation of the stresscomponents along the z-axis. Note that themaximum shear stress is much less than themaximum contact pressure.

Figure 8: Contacts of Two Cylinder

The half width is given by the equation:

21

2221

21

/1/1

112

dd

EvEv

L

Fb

the maximum pressure is given by-

bL

Fp

2

max

These equation can be applied to a cylinderand plane surface such as rail by making d =infinity for the plane surface the equation canalso be applied to internal cylinder by makingd negative.

Figure 9: Variation of the StressComponents

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Next Figure 10 shows the variation of theoctahedral shear stress below the surface forthe contacting cylinder. This stress is usedsome time for failure stress instead ofmaximum shear stress, to define failure. Theoctahedral shear stress and maximum shearstress reach their highest value on the curveat the same depth z, but the location of appointis very sensitive to the value of Poisson’s ratio.

Figure 10: Variation of the OctahedralShear Stress

Von Mises stress variation along the z-axis.Note that the von Mises stress is much lessthan the maximum contact pressure.

Hertz (1981) provided the precedingmathematical model of the stress field whenthe contact zone is free from the shear stress.Another important contact stress case is lineof contact withthe friction providing shearingstress on the contact zone such stress arevery small with camsandroller follower but flatface follower wheel-rail contact and gearteeth, the stress are evaluated above theHertzian field.

ANSYS Model for Two Cylinders: In orderto verify the FEM contact model procedure,contact between two cylinders was modeled.

To reduce computer time, only half cylinderswere meshed in the model as shown in Figure9. The fine meshed rectangular shapedelements were generated near contact areasshown as Figure 10. The dimensions of theelements are based on the half width of thecontact area. The contact conditions aresensitive to the geometry of the contactingsurfaces, which means that the finite elementmesh near the contact zone needs to be highlyrefined. Finer meshing generally leads to amore accurate solution, but requires moretime and system resources. It isrecommended not to have a fine mesheverywhere in the model to reduce thecomputational requirements.

ANSYS Hertz. Difference

(Pmax), psi 62335 64788 2453

Table 1: ANSYS and HERTZ Results

CONCLUSIONIt was shown that an FEA model could be used

to simulate contact between two bodiesaccurately by verification of contact stresses

between two cylinders in contact andcomparison with the Hertzian equations.

Effective methods to estimate the toothcontact stress using a 2D contact stress model

and to estimate the root bending stresses

using 2D and 3D FEA model are proposed.The analysis of gear contact stress and the

investigation of 2D and 3D solid bendingstress.

REFERANCES1. Errichello R (1979), “State-of-Art Review:

Gear Dynamics”, Trans. ASME J. Mech.

Des., Vol. 101, No. 3, pp. 368-372.

321


2. Harris S L (1958), “Dynamic Load on theTeeth of Spur Gears”, Proc. Instn Meth.Engrs., Vol. 172, pp. 87-112.

3. Kohler H and Regan R (1985), “TheDerivation of Gear Transmission Errorfrom Pitch Error Records”, Proc. Instn.Mech. Engrs., Part C, Journal ofMechanical Engineering Science ,Vol. 199, No. C3, pp. 195-201.

4. Kubo A et al. (1991), “Estimation ofTransmission Error of Cylindrical InvoluteGears by Tooth Contact Pattern”, JSMEInt. J., Ser. III, Vol. 34, No. 2, pp. 252-259.

5. Mark W D (1978), “Analysis of theVibratory Excitation of Gear System:

Basic Theory”, J. Acoust. Soc. Am.,

Vol. 63, pp. 1409-1430.

6. Mark W D (1979), “Analysis of the

Vibratory Excitation of Gear System II:

Tooth Error Representations,

Approximations, and Application”, J.

Scouts Soc. Am., Vol. 66, pp. 1758-1787.

7. Ozguven H N and Houser D R (1988),

“Mathematical Models Used in Gear

Dynamics”, Sound Vibr., Vol. 121,

pp. 383-411.

8. Zeping Wei (1989), “Stresses and

Deformations in Involute Spur Gears by

Finite Element Method”.

INVESTIGATION OF CONTACT STRESS IN SPUR GEAR USING · PDF fileGEAR USING LEWIS EQUATION AND FINITE ELEMENT METHOD ... verification by 2D FEM analyzer using ANSYS, ... solid modeler

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