Calculation of the Combined Torsional Mesh Stiffness of Spur Gears ...

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)11, 810-818 Paper received: 07.12.2010DOI: 10.5545/sv-jme.2010.248 Paper accepted: 02.08.2011Calculation of the Combined Torsional Mesh Stiffness of SpurGears with Two- and Three-Dimensional Parametrical FEModelsKiekbusch, T. ‒ Sappok, D. ‒ Sauer, B. ‒ Howard, I.Timo Kiekbusch 1,* ‒ Daniel Sappok 1 ‒ Bernd Sauer 1 ‒ Ian Howard 21 University of Kaiserslautern, Institute for Machine Elements, Gears and Transmissions, Germany2 Curtin University Perth, Department of Mechanical Engineering, AustraliaThe torsional mesh stiffness is one of the most important characteristics of spur gears. This paperpresents the development of detailed two- and three-dimensional finite element models which can be usedto calculate the torsional mesh stiffness. Using the parametrical design language of the FE softwareANSYS the models offer the possibility to generate various different pairs of spur gears and include anadaptive meshing algorithm for the contact zones. Due to the short computation times the 2D model iswell suited to simulate a variety of different gear pairs in a short time period. The more complex 3D modelfeatures more options in terms of investigating tooth face modifications for further studies. The resultingvalues of the torsional stiffness can be used – for example – in multi body simulations of gearboxes.The results from the 2D FEA are used to derive a simple formula for the combined torsionalstiffness of spur gears in mesh. The results presented are based on the individual stiffness of the three maincomponents – body, teeth and contact. Hence, the introduced formula uses these three parts to determinethe overall stiffness for a wide range of gears and gear ratio combinations.Finally, the results from both the two- and three-dimensional finite element model and the derivedformula are compared and the results from the 3D model are checked against results obtained by analyticalequations.©2011 Journal of Mechanical Engineering. All rights reserved.Keywords: gear dynamic modelling, spur gear, finite element modelling, torsional mesh stiffness,contact stiffness0 INTRODUCTIONGears are the main component of manydifferent kinds of rotating machinery and theyare often a critical part to the function of themachinery. There have been many attempts inrecent years to understand and describe the processof the meshing of spur gears. As the process ofmeshing is very complex and difficult to describe,the finite element analysis is the method of choiceto investigate the underlying relationships.The approach used in this paper todetermine the stiffness of spurs gears in mesh isthe development of a FE model of the completegear arrangement. Recent studies on this topic[1] to [4] have shown that these models producethe best results in comparison with single-toothmodels or partial teeth models. As computerhardware and FE software advances, workingwith these rather complex models is now feasible.1 THE FINITE ELEMENT GEAR MODELSIn the following sections the developmentof the two- and three-dimensional finite elementmodels is described. This involves the fullyparametrical creation of the gear shape, themeshing and the simulation process including theadaptive meshing.1.1 2D FE ModelFully parametrical APDL-scripts (ANSYSParametric Design Language) are used to describethe geometry of the gears and to control thesimulation process. The first step is the generationof the geometry consisting of lines and areas,which in the next step are meshed with a relativelycoarse FE-mesh. The result is a FE model ofpinion and gear which is shown in Fig. 1.The process of creating various differentpairs of gears is automated with the powerful810*Corr. Author’s Address: Institute for Machine Elements, Gears and Transmissons,Gottlieb-Daimler-Str., 67663 Kaiserslautern, Germany, timo.kiekbusch@mv.uni-kl.de

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)11, 810-818design language of ANSYS. Without changingthe model itself many parameters like modulus,number of teeth, shaft radius etc. can easilybe varied as all parameters are provided in adata input file. This file is loaded during thepreprocessing.different angular positions of the gears. Therefore,the gears have to be rotated to the correspondingposition before the model is solved. At everyroll angle the contact point(s) change so that themesh refinement at the contact point(s) requirean adaptive remesh algorithm which takes intoaccount the actual contact situation.Fig. 1. Result from the first APDL-script; premeshedgeometryAs the contact between pinion and gear isthe most important and critical part of the gearsimulation the description of the tooth involutesand feet require the highest possible precisionduring the modelling process. Therefore, bothgeometric details are generated by using a veryhigh number of keypoints. The location of thesepoints is calculated on the basis of the data inputfile. The keypoints are connected with splinesto represent the outline of the tooth geometry.After adding the gear body the areas created aremeshed with a coarse mesh. This model (see Fig.1) is the basis for the next steps of the modellingprocess namely the mesh refinement at the contactpoint(s). This refinement is realised using anotherAPDL-Script. The result is shown in Fig. 2. Thisscript also adds the constraints, contact elementsand different torque loads to the model, solvesand post processes the job. The constraints usedin the model are as follows: the hub of the gear iscompletely constrained from motion; the nodes atthe hub of the pinion can only rotate around thecentre of the pinion. The torque is applied usinga force on every node at the driving gear’s hub,adding up to the specified torque.A quasi-static simulation method is usedfor the determination of the mesh stiffness. Thismeans that the stiffness is calculated at severala) b)Fig. 2. Adaptive refining of the mesh at thecontact point(s); a) remeshed contact for singlecontact, b) remeshed contact for double contactDuring the automated postprocessing thefollowing results are extracted from the model:combined torsional mesh stiffness, deformation ofgear body, teeth and contact zone for both pinionand gear. This data is written to a text file forfurther processing.When analysing gears with tooth shapeerrors or profile correction, an initial gapbetween the teeth can occur. In order to createa very flexible model which can be used infurther studies, the model was built to be able tosolve problems including an initial gap betweenmeshing teeth. Typically a static FE model cannotbe solved if some parts of the model do not haveenough constraints as the stiffness matrix becomessingular [5]. This can happen, for example, ifa contact is not initially closed and in this casethe pinion can rotate a small angle without anyresistance or stiffness.To avoid rigid body motion, a weak springis attached to the pinion. This small stiffnessprevents the unintentional rotational motion. Thisspring is only used for the initial simulation stepwith a very small torque just to get the teeth incontact. When the actual load torque is applied,the spring is disabled using the birth/death ofelements command [6]. This approach can handlemuch bigger gaps (rigid body motion) than theCalculation of the Combined Torsional Mesh Stiffness of Spur Gears with Two- and Three-DimensionalParametrical FE Models811

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)11, 810-818automatic adjustment which tends to be used bydefault.1.2 3D FE ModelThere are some restrictions when usinga two dimensional FE model for the analysis ofspur gears in mesh. For example, simulatinghelical gears or applying tooth face modificationsin the direction of the rotation axis of the gearlike crowning or face angle correction is notpossible. By using a three dimensional FE modelthese restrictions can be evaded. Furthermore, theinfluence of misalignment between the rotationaxes of the gears as well as shaft centre distancechanges on the mesh stiffness can be investigated.The disadvantages of a three dimensional versusa two dimensional model are higher complexityand hardware requirements as well as computationtimes.In order to overcome the restrictions thatare given by the 2D model a three dimensional FEmodel built on the latter was developed. Again afully parametrical approach was used to create theshape of the gears and an initial coarse mesh. Inaddition the model was designed to support themodelling of helical gears and the application oftooth face modifications for further studies.To create the gear model, each gear wheelis sliced into multiple layers along the rotationaxis. For each layer, the two dimensional shapeof the gear is modelled by keypoints which areconnected with straights, arcs and splines. Thetooth involutes and roots are described with ahigh number of keypoints to provide a very highaccuracy like in the 2D model. As each 2D shapeof the gear is built separately, the tooth shape canbe changed along the rotation axis. This procedureis a prerequisite to model helical gears and to applythe above mentioned tooth face modifications.Only the teeth which are involved inone mesh cycle are created for each gear. Thesurrounding teeth have only a subsidiary influenceon the mesh stiffness but demand a higher amountof elements which leads to higher computationtimes and hardware requirements. The gear bodiesare completely modelled.During the modelling process the gearsare completely meshed using a mapped meshingwith hexahedral elements. In this stage, the meshin the area of contact, that is on the tooth faceson the loaded side of the teeth, is comparativelycoarse. Mesh transitions between the body areasadjacent to modelled teeth and the rest of the gearbody are used to reduce the amount of elements.A completely meshed gear pair is shown inFig. 3.Fig. 3. Meshed gear pair created during themodel build processA relatively fine mesh is needed toaccurately simulate the non-linear contactdeformation between the tooth faces. Thereare two options to refine the mesh in the areaof contact during the simulation process. Thefirst one is to adaptively refine the mesh at theposition(s) were the tooth face contact(s) occurs.The second one is to refine the complete toothfaces on the loaded side of the teeth. Fig. 4 showsthe mesh refinement for two meshing angles withthe first option.The modelling process generates a modeldatabase file and a parameter file which comprisesall necessary parameters. These files are used byan APDL-script in which the boundary conditionsand the contact elements are created, the solvingprocess is started and finally a fully automatedpostprocessing is conducted.In order to close an initial gap betweenthe tooth faces and avoid rigid body motion, theusage of a small stiffness spring attached to thedriving gear’s hub is adopted from the 2D model.This procedure is extended with a method tocompute the spring stiffness and the initial torqueload according to the actual gap size. Thereby theamount of simulation steps to close the initial gapcan be reduced to a minimum which consequentlyreduces the simulation time.During the postprocessing the sameresults as within the 2D model are extracted.812 Kiekbusch, T. ‒ Sappok, D. ‒ Sauer, B. ‒ Howard, I.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)11, 810-818a) b)Fig. 4. Adaptive refining of the mesh at the contact point(s) for the 3D model; a) remeshed contact forsingle contact, b) remeshed contact for double contactThat is the combined torsional mesh stiffness, thedeformation of gear body, teeth and contact zonefor pinion and gear.2 THE COMBINED TORSIONAL MESHSTIFFNESSThe definition of the combined torsionalmesh stiffness K m used in this paper is the quotientof input load T [Nm] and driving gear hub rotationunder load T.E. [rad] [6] to [8]:TKm = TE . . . (1)This definition can be used for the dynamicsimulation of spur gear systems as it directlydescribes the relation between load torque andrelative motion of the gears.In 2001, Jia [2] introduced a commonformula describing the combined torsional meshstiffness by body and tooth bending stiffness. Thissimplification neglects the effect of the appliedtorque on the gearing's stiffness. The results fromthe FE model, however, show that there is aninfluence of the torque on the resulting combinedtorsional mesh stiffness as shown in Fig. 5.The studies of the deformational behaviourof the gear wheels show that the stiffness of gearbody and teeth are almost load-independent whilethe contact deformation is non-linear, as a Hertziancontact occurs between the teeth of the gears. Theinfluence of the applied torque on the component’sstiffness is shown in Fig. 6. The position and widthof the handover region between single and doublecontact zone are also torque-dependent which hasalso been shown previously [1].Fig. 5. Torsional mesh stiffness for a complete mesh cycle with different torque loads (model with 1:1 gearratio, 23 teeth, modulus 6 mm, steel)Calculation of the Combined Torsional Mesh Stiffness of Spur Gears with Two- and Three-DimensionalParametrical FE Models813

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)11, 810-818Fig. 6. Influence of the applied torque on body, teeth and contact stiffness (model with 1:1 gear ratio, 23teeth, modulus 6 mm, steel)3 TORSIONAL STIFFNESS OF A SINGLEGEARThe description of the torsional meshstiffness is based on the assumption that thestiffness of body, teeth and contact zone can beconsidered to act like three springs in a row, whichmeans that the combined stiffness K i [Nm/rad] foreach pinion and gear can be calculated as:( )−1−1 −1 −1i B, i T, i C, i ,K = K + K + K(2)where K B,i is the gear body stiffness, K T,i the toothstiffness and K C,i the contact stiffness with i beingP or G for pinion resp. gear.Fig. 7. Nodes selected to determine thecomponent’s deformationIt has to be mentioned that these valuesare not the actual stiffness values of the particularcomponent, but items which help to develop thecommon formula for the mesh stiffness. In thefollowing pages the combined gear stiffness K iis used to calculate the stiffness for the two gearsin mesh. The values of K B,i , K T,i and K C,i can beobtained from the FE model. For this purposethe deformations of body, teeth and contact zonehave to be separated. In the FE model differentnodes are used to read out their displacements.These displacements are put into relation with theapplied torque which results in the component’sstiffness. The nodes which are chosen to receivethe deformation data are placed at the shaft radius,the dedendum radius and at the radius of contact;in each case in the middle of the tooth in contactand in addition one node at the contact point. Fig.7 shows the placement of the selected nodes for asingle tooth pair in contact.3.1 Stiffness for the Driving Gear with a SinglePair of Teeth in ContactThe description of the torsional stiffnessof the different components requires an analysiswhich gearing parameters affect the particularstiffness. The relevant parameters for eachcomponent are pointed out in the correspondingsections.The range of the gear model parameterswhich have been used for this research is shownin Table 1.Table 1. Range of parameters for the models usedin this researchParameter Unit Min MaxNumber of teeth z [-] 7 50Modulus m [mm] 3 15Torque load T [Nm] 0.1 1000Gear ratio u [-] 0.34 2.94814 Kiekbusch, T. ‒ Sappok, D. ‒ Sauer, B. ‒ Howard, I.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)11, 810-8183.1.1 Stiffness of the Gear BodyThe stiffness of the gear body is assumedto only depend on the following parameters: shaftradius r s dedendum radius r d , face width w andYoung’s modulus E.A simplified model (see Fig. 8) is usedto analyse the influence of the above mentionedparameters on the body’s stiffness. Differentcombinations of parameters were used to come tothe following Eq.:( ) ⋅16 . 16 .BP , B d s sK = c ⋅E⋅w⋅ln r − r r ,(3)where c B is a coefficient which was found out tobe 9.555e -4 .The results from this equation reproducethe results from the FE model within an accuracyof about 5% for gear bodies with a outer radiusbetween 10 and 200 mm and various inner radii.where c T is a coefficient which was found out tobe 3.2e -5 . The results from this Eq. are within 7%of the results from the FE model for the parameterrange analysed.3.1.3 Stiffness of the Teeth ContactIn contrast to the stiffness mentionedabove, the stiffness of the contact between themeshing teeth is highly non-linear with the loadas it is a Hertzian contact between two curvedsurfaces. Therefore, the load torque T needs to beconsidered in the equation describing the contactstiffness. In addition, the following parametershave been found out to have a significantinfluence: modulus m and number of teeth z,contact radius, Young’s modulus E and face widthw. The contact stiffness is approximated by:K = c ⋅E⋅w⋅m ⋅z ⋅TCP ,C185 . 2 0.105,(5)where the coefficient c C is 7.937e -5 . The resultsfor the contact stiffness were found to be within10% of the results from the FE model.3.2 Stiffness for the Driven Gear with a SinglePair of Teeth in ContactFig. 8. Simplified model of the gear body withapplied constraints and forces3.1.2 Bending Stiffness of the TeethAs the teeth basically bend under load,the assumption is made that the parameters thatinfluence the stiffness of the teeth are the sameas those of a bending beam. These parametersare: height and width w of the teeth and Young’smodulus E. The influence of the radius at whichthe tooth is located and the tooth height was takeninto account with the parameters modulus m andnumber of teeth z. This results in the stiffness ofthe tooth K T,P :K = c ⋅E⋅w⋅m ⋅ zTP ,T2 22 .,(4)The stiffness values determined aboveare related to the driving gear (index P). Withthe given gear ratio u, the driven gear’s stiffness(index G) can be directly deduced taking intoaccount that both load torque and radius of contactare u-times the respective torque and contactradius of the driving gear. So the body, teeth andcontact stiffnesses are calculated by:2 zPKiG , = KiP , ⋅ u = KiP, ⋅ ⎛ ,⎝ ⎜ ⎞⎟ (6)zG⎠with i being the indices B, T and C for body,teeth and contact. It has to be mentioned that theparameters of the driven gear have to be used forthe calculation of its stiffness (e.g. z G , E G ).3.3 Stiffness in the Double Contact ZoneThe body’s stiffness value in the doublecontact zone is obtained by adding the factor f B .The only difference between single and doublecontact for the body stiffness is the width of the2Calculation of the Combined Torsional Mesh Stiffness of Spur Gears with Two- and Three-DimensionalParametrical FE Models815

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)11, 810-818area which is affected by the load. This influenceThe results from the formula are withinneeds further investigation which can be done−1 −1 −1K = K + K . (10) using the FE model.is considered to be rather small which wasconfirmed by the FE model. Hence, the factor f Bwas found to be equal to 1.1:10% from the 2D FE model’s results for mostinput parameter sets. This accuracy seems tobe good enough for most of the cases where theformula can be used as there are many otherKiB , , double = fB ⋅ Ki, B, single = 11 . ⋅Ki, B,single. , (7) different factors which influence the value of themesh stiffness, like the shaft to collar connectionThe description of the double contact zone and the lubrication of the gears.teeth stiffness assumes that both teeth pairs sharethe load equally. This means that each tooth’sdeformation is half the deformation as in the singlecontact zone. This results in a stiffness value twiceas high as for the single contact:KiT, , double = 2 ⋅KiT, , single.(8) Fig. 9. Simplified model of the combined torsionalmesh stiffness of two gears in meshThe evaluation of the contact stiffnessshows that the difference between single andA simple modification during thedouble contact cannot only be described by asimple factor but by an additional change of theexponent for the applied load (cf. Eq. (5)):calculation can be made to take into account thesmall variation of the stiffness inside the singleor double contact zone. Adding a quadraticcorrection term will result in a lower difference. .KiC, , double = cCE w 185m 2z 0 068T , (9) between the simulated and the calculated valuesover the whole mesh cycle (see Fig. 10). Thewhere c C is 1.1905e -4 .modified stiffness K m * against the distance fromthe actual relative roll angle to the centre of the3.4 The Formula for the Combined Torsional single resp. double contact zone ΔΩ is:Mesh Stiffness2Km* = Km⋅( 1−c⋅∆© ), (11)The stiffness for the single gear (K P , K G )is calculated by assuming all three springs in arow as mentioned above. The combined torsionalmesh stiffness K m can be derived by consideringthe two gears as two springs in series (see Fig. 9):with the factor c which has to be adapted on basisof the FE model.The determination of the position andwidth of the hand-over region, however, still( )m P GFig. 10. Comparison of the FE results with quadratic correction term (Eq. (11))816 Kiekbusch, T. ‒ Sappok, D. ‒ Sauer, B. ‒ Howard, I.

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)11, 810-8184 RESULTSDue to the fact that the FEA of a spur gearpair is complex and therefore prone to errorsthe results have to be checked for plausibility.Hence the results from the 2D and 3D model aswell as the mesh stiffness formula are comparedwith each other. Additionally, the results from the3D model are checked against analytical resultsaccording to DIN 3990 [9]. A variety of gear pairshas been used for this comparison. Table 2 showsan excerpt from the compared gear pairs.Table 2. Investigated gear pairs (excerpt)Variant-No. 1 2 3 4z 1 23 25 36 13z 2 23 25 43 31m [mm] 6 2 11 4α [°] 20 20 20 20w 1 [mm] 16 28 58 16w 2 [mm] 15 24 50 15R s1 [mm] 15 8 80 10R s2 [mm] 15 8 100 20E-Modulus [GPa] 210 210 210 2104.1 2D – 3D – Mesh Stiffness FormulaThe torsional mesh stiffness has beencalculated for the single contact zone (SCZ) andthe double contact zone (DCZ) to check if the twoFE models and the mesh stiffness formula produceconsistent results.Table 3 shows the stiffness results and theproportion between FE and mesh stiffness formulafor each FE model.The results show a maximum deviation of10% that implies that the FEA and mesh stiffnessformula produce consistent results for the torsionalmesh stiffness.4.2 3D – DIN 3990DIN 3990 provides methods to calculatethe load capacity of cylindrical gears whichincludes the determination of the tooth springstiffness. That is the normal tooth load which isneeded to deform a meshing tooth pair with 1 mmtooth width perpendicular to the tooth involute for1 mm. This deformation corresponds to the basecircle arc length and thus a rotation angle whichcan be converted into the before defined torsionalmesh stiffness. These results are compared toTable 3. Comparison of the torsional mesh stiffness (K m_MSF ) between 2D (K m_2D ), 3D FE simulation(K m_3D ) and the mesh stiffness formula (italic: proportion values)Variant-No. 1 2 3 4Torque Load [Nm] 1000 75 1000 100K m_MSF [10 6 Nm/rad] 0.562 1.00 0.152 1.00 22.8 1.00 0.100 1.00SCZ K m_2D [10 6 Nm/rad] 0.573 1.02 0.146 0.96 20.2 0.89 0.098 0.98K m_3D [10 6 Nm/rad] 0.594 1.06 0.154 1.01 22.1 0.97 0.102 1.02K m_MSF [10 6 Nm/rad] 0.722 1.00 0.206 1.00 31.4 1.00 0.138 1.00DCZ K m_2D [10 6 Nm/rad] 0.751 1.04 0.207 1.00 29.7 0.95 0.141 1.02K m_3D [10 6 Nm/rad] 0.781 1.08 0.215 1.04 30.9 0.98 0.145 1.05Table 4. Comparison of the torsional mesh stiffness between 3D FE simulation (K m_3D ) and DIN 3990(K m_DIN )Variant-No. 1 2 3 4Torque Load [Nm] 1000 75 1000 100Tooth Spring Stiffness [N/(mm∙μm)] 15.1 15.0 17.4 14.4Linear Distributed Load [N/mm] 966 125 101 256K m_3D [10 6 Nm/rad] 0.594 0.154 22.1 0.102SCZ K m_DIN [10 6 Nm/rad] 0.652 0.159 24.0 0.103K m_3D / K m_DIN 0.91 0.97 0.92 0.99Calculation of the Combined Torsional Mesh Stiffness of Spur Gears with Two- and Three-DimensionalParametrical FE Models817

Strojniški vestnik - Journal of Mechanical Engineering 57(2011)11, 810-818results obtained by the 3D FE model in the singlecontact zone (SCZ) which is shown in Table 4.The results from the 3D FE modelreproduce the results from DIN 3990 withinan accuracy of about 10%. This shows that theresults from the FEA are good. In particular if thereal contact situation is taken into account whichincludes effects like friction, lubrication andtolerances a deviation of 10% has to be regardedlittle.5 CONCLUSIONThis paper presents detailed two- and threedimensionalFE models to create a set of gears andsimulate the torsional mesh stiffness for one meshcycle. By using the ANSYS parametric designlanguage the models are fully parametric and bothmodels feature an adaptive meshing algorithm forthe contact zones.The 2D FE model is used to derive asimple formula for the combined torsional meshstiffness of spur gears in mesh. This formula usesthe three main parts of a gear – body, teeth andcontact – to calculate the overall stiffness of thegear pair. The 3D FE model is introduced as thebasis for further studies. With a 3D model it willbe possible to simulate helical gears or applytooth face modifications like crowning or faceangle corrections. Furthermore, the influence ofmeshing interferences like misalignment betweenthe gears axes due to shaft, bearing or housingdeformation and tolerances can be investigated.The resulting values from the FE modelsand the mesh stiffness formula can be used indynamic simulations such as multi body simulationof gearboxes. A benefit of the developed formulais the fact that only the basic gearing parametersare needed to derive the torsional mesh stiffness.Still the FE models feature the option to analysestresses in critical areas of the gears – for examplethe tooth root ‒ as well as the contact pressure onthe tooth faces.A comparison of the results with eachother and with analytical equations shows theplausibility of both FE models and the meshstiffness formula. When compared to the realcontact situation the conducted simulations andcalculations involve some simplifications – thatis for example lubrication, friction and tolerances.Due to this the obtained results and the determineddeviations are completely satisfying.6 REFERENCES[1] Wang, J., Howard, I. (2004). The torsionalstiffness of involute spur gears. Proceedingsof the Institution of Mechanical Engineers,Part C: Journal of Mechanical EngineeringScience, vol. 218, no. 1, p. 131-142.[2] Jia, S., Howard, I., Wang, J. (2001). Acommon formula for spur gear meshstiffness. Proceedings of the JSMEInternational Conference on Motion andPower Transmissions, p. 1-4.[3] Wang, J., Howard, I. (2005). Finite elementanalysis of high contact ratio spur gears inmesh. Journal of Tribology, vol. 127, no. 3,p. 469-483.[4] Jia, S., Howard, I. (2006). Comparison oflocalised spalling and crack damage fromdynamic modelling of spur gear vibrations.Mechanical Systems and Signal Processing,vol. 20, no. 2, p. 332-349.[5] Madenci, E., Guven, I. (2006). The finiteelement method and applications inengineering using ANSYS. Springer-Verlag,New York.[6] Wang, J., Howard, I. (2006). Comprehensiveanalysis of spur gears in mesh with varioustypes of profile modifications. ProceedingsInternational Conference on MechanicalTransmissions, p. 42-47.[7] Sirichai, S. (1999). Torsional propertiesof spur gears in mesh using nonlinearfinite element analysis. PhD Thesis. CurtinUniversity, Perth.[8] Wang, J. (2003). Numerical andexperimental analysis of spur gears in mesh.Curtin University, Perth.[9] DIN 3990 T1 (1987). Tragfähigkeitsberechnungvon Stirnrädern - Calculationof load capacity of cylindrical gears.Deutsches Institut für Normung. Berlin.818 Kiekbusch, T. ‒ Sappok, D. ‒ Sauer, B. ‒ Howard, I.