Dynamic modelling of a one-stage spur gear systemand vibration-based tooth crack detection analysisOmar D. Mohammeda,b,n, Matti Rantataloa, Jan-Olov AidanpcaDivision of Operation, Maintenance and Acoustics, Lulea University of Technology, SwedenbMechanical Engineering Department, College of Engineering, University of Mosul, IraqcDivision of Product and Production Development, Lulea University of Technology, Swedenarticle infoArticle history:Received 29 December 2013Received in revised form4 June 2014Accepted 2 September 2014Available online 22 September 2014Keywords:Tooth crackMesh stiffnessGear dynamic modelGyroscopic DOFVibration analysisCrack detectionabstractFor thepurposeof simulationandvibration-basedconditionmonitoringof agearedsystem, itisimportanttomodelthesystem withanappropriatenumberofdegreesoffreedom(DOF). Inearlierpapersseveral modelsweresuggestedanditisthereforeofinterest to evaluate their limitations. In the present study a 12 DOF gear dynamic modelincluding a gyroscopic effect was developed and the equations of motions were derived.A one-stage reduction gear was modelled using three different dynamic models (with 6, 8and 8 reduced to 6 DOF), as well as the developed model (with 12 DOF), which is referredasthefourthmodel inthispaper. Thetime-varyingmeshstiffnesswascalculated, anddynamicsimulationwasthenperformedfordifferent cracksizes. Timedomainscalarindicators (the RMS, kurtosis andthe crest factor) were appliedfor fault detectionanalysis. Theresultsof thefirstmodel showaclearlyvisibledifferencefromthoseofthe other studied models, which were made more realistic by including two more DOF todescribethemotorandload. Boththesymmetricandtheasymmetricdisccaseswerestudied using the fourth model. In the case of disc symmetry, the results of the obtainedresponse are close to those obtained from both the second and third models. Furthermore,the second model showed a slight influence from inter-tooth friction, and therefore thethirdmodel isadequateforsimulatingthepinion'sy-displacementinthecaseof thesymmetric disc. Inthe case of the asymmetric disc, the results deviate fromthoseobtained in the symmetric case. Therefore, for simulating the pinion's y-displacement, thefourth model can be considered for more accurate modelling in the case of theasymmetric disc.& 2014 Elsevier Ltd. All rights reserved.1. IntroductionThe vibration-basedconditionmonitoring technique has gaineda great deal of importance inthe maintenanceengineeringof industrial geartransmissions. Theroleof thistechniqueistodetect deterioration, onthebasisof theobtained vibration signal, before the occurrence of sudden breakage. Any undetected fault can result in a malfunction andContentslistsavailableatScienceDirectjournal homepage: www.elsevier.com/locate/ymsspMechanical Systems and Signal Processinghttp://dx.doi.org/10.1016/j.ymssp.2014.09.0010888-3270/& 2014 Elsevier Ltd. All rights reserved.nCorresponding author at: Division of Operation, Maintenance and Acoustics, Lulea University of Technology, Sweden. Tel.: 46 920 49 1840;fax: 46 920 49 2818.E-mail addresses: omar.mohammed@ltu.se, omar.mohammed@uomosul.edu.iq (O.D. Mohammed).Mechanical Systems and Signal Processing 54-55 (2015) 293305thenaffecttheavailabilityof thewholesystem. Therefore, earlyfaultdetectionisrequiredtoallowproperscheduledmaintenance to prevent catastrophic failure and consequently provide safer operation and higher cost savings.Vibrationresponse can be measured experimentally or modelled theoretically. The experimental approach is usually associated withhigher costs and problems in accessing the measurement nodes, and is often time-consuming. Furthermore, experimentalwork is usually restricted in terms of producing enough real faults of desired dimensions. In many cases, therefore, dynamiclumped-parametersmodellingcan provideuswithaclearunderstandingofthedynamicbehaviourofthestudiedgearsystem.Gear modelling can be considered as a fundamental problem which is still the object of much on-going research. A greatdeal of research has been conducted to study different dynamic models of gear systems [1,2]. Different mathematical gearmodelswereexaminedin[2], andgearmodellingwithbothtorsionalandtranslationalvibration wasadoptedin[3,4].The one-stage 8 DOF gear dynamic model was applied in [3], while the 6 DOF model was investigated in [4], ignoring theinter-tooth friction. A different 6 DOF gear dynamic model was applied in [510]; in this model the friction was consideredby simulating 3 DOF for each disc (one torsional and two translational degrees). A one-stage 16 DOF gear dynamic modelwas developed in [11] and then adopted in [12] for simulating the system dynamic behaviour.Among the above-mentioned research studies, different dynamic models have been presented for different gear systems.However, there is no study that has examined the influence of adding more DOF to describe the gyroscopic effect of the geardisc. Inthepresent study, a one-stage12DOF spur gear model was developedfor describingthegyroscopic DOF.This developed model was used to simulate the studied gear system to examine, from a fault detection perspective, if it isnecessary to consider the disc asymmetry effect for the studied system. This presented model and three other models wereused to simulate the same gear system for different crack sizes. In addition, the present paper explains gear mesh stiffnesscalculation with a cracked tooth and presents the results of fault detection analysis applied on the dynamic response of thefour studied models.2. Gear dynamic modellingThe modelling of a one-stage reduction gear system is presented in this paper. The main gear parameters were obtainedfrom a real spur gear transmission and are explained in Table 1. This gear transmission is a part of a machine which is usedasatest-rigintheConditionBasedMaintenance(CBM) Laboratoryat LuleaUniversityof Technology. Toperformthedynamic simulation, some more parameters need to be introduced in the studied dynamic models, and these parametersare explained in Table 2.The term pinion refers here to the smaller gear, which is a driver gear connected to the input shaft, and the term gearrefers to the larger gear, which is a driven gear connected to the output shaft. The following notation is used:mp/mg: mass of the pinion/gear;Table 1Parameters of the gearpinion set.Parameter Pinion GearMass (kg) 0.289 1.789Number of teeth 36 90Module (mm) 1.5Teeth width (mm) 15Pressure angle (deg) 20Contact ratio 1.76Gear ratio 2.5Young's modulus, E (N/mm2) 2105Poisson's ratio 0.3Table 2Parameters of the dynamic modelling.Parameter InputshaftOutputshaftRadial stiffness of the bearings in x and y direction (N/m) 6.01086.0108Radial damping of the bearings in x and y direction (N s/m) 1.81031.8 103Applied torque (N m) 50 125Torsional stiffness (N m/rad) 11041 104Torsional damping (N m s/rad) 10 10Rotational speed (Hz) 55.55 22.22Gear mesh frequency (Hz) 2000Coefficient of friction 0.06O.D. Mohammed et al. / Mechanical Systems and Signal Processing 54-55 (2015) 293305 294Ip/Ig: mass moment of inertia of the pinion/gear;Kxp/Kyp: radial stiffness in the x/y directions of the pinion;Kxg/Kyg: radial stiffness in the x/y directions of the gear;Cxp/Cyp: radial damping in the x/y directions of the pinion;Cxg/Cyg: radial damping in the x/y directions of the gear;Km: equivalent mesh stiffness;Cm: mesh damping coefficient;rbp/rbg: base circle radius of the pinion/gear;Tp/Tg: torque applied on the pinion/gear;Tm/Tb: torque applied on the motor/load;kt/ct: torsional stiffness/damping of the input and output shaft; andwp/wg: constant speed of the pinion/gear.Inthepresent work, four dynamicmodels wereusedtosimulatethedynamicresponse. Inter-toothfrictionwasintroduced in three of the studied models, and the effect of ignoring it was examined, as stated in Section 2.3.2.1. First model (6 DOF)For simplicity the one-stage gear system can be modelled without considering the motor and load. This model consists of6 DOF, is currently applied and was adopted in [510]. A schematic diagram of the 6 DOF model, which has 3 DOF (onerotational andtwotranslational)foreachgeardisc, isshowninFig. 1. Theequationsof motionforthismodel canbeexplained as follows.The equations of motion in thex direction for the pinion and gear arempxp KxpxpCxp_xpFp1mg xg KxgxgCxg _ xgFg2The equations of motion in they direction for the pinion and gear aremp yp NKypypCyp _ yp3mg ygNKygygCyg _ yg4The equations of motion in the direction for the pinion and gear areIp prpNTpMp5Ig g rgNTgMg6Fig. 1. Dynamic model of a reduction gear system with 6 DOF.O.D. Mohammed et al. / Mechanical Systems and Signal Processing 54-55 (2015) 293305 295In these equations,N Kmypyg_ _rpprgg_ __ _Cm_ yp_ yg_ _rp _prg _g_ _ _ _N N1N2Nn(n is the number of teeth in contact) Mp/Mg: the moments due to the friction forces Fp/FgFpFp1Fp2Fn; Fp1m N1and Fp2 m N2FgFg1Fg2Fn; Fg1 m N1and Fg2m N22.2. Second model (8 DOF)For more reality the one-stage gear system can be modelled taking the motor and load into consideration. This modelconsists of 8 DOF and was applied in [2,3]; it has 3 DOF (one rotational and two translational) for each gear disc, as well as 1DOF for each motor disc and load disc to describe the rotation. A schematic diagram of the 8 DOF model is shown in Fig. 2.The equations of motion for this model can be explained as follows.The equations of motion in thex direction for the pinion and gear arempxp KxpxpCxp_xpFp7mg xg KxgxgCxg _xgFg8The equations of motion in they direction for the pinion and gear aremp yp NKypypCyp _ yp9mg ygNKygygCyg _yg10The equations of motion in the direction for the pinion and gear areIpprp NMpktpm ct_p_m 11Ig g rgNMgktgbct_g_b 12The equations of motion in the direction for the motor and load areImm ktmp ct_m_pTm13Ibb ktbg ct_b_g Tb14Fig. 2. Dynamic model of a reduction gear system with 8 DOF.O.D. Mohammed et al. / Mechanical Systems and Signal Processing 54-55 (2015) 293305 2962.3. Third model (8 DOF reduced to 6 DOF)To examine the effect of including inter-tooth friction, the second model (with 8 DOF) can be used with the friction effectignored. Accordingly, the xp and xg displacements are then excluded. The 8 DOF are then reduced to 6 DOF, and this model isreferred to as having8 DOF reduced to 6 DOF. This model (with 6 DOF) was used previously in [2,4].2.4. Fourth model (12 DOF), the developed modelA one-stage gear dynamic model including a gyroscopic effect has been developed in this study. This model consists of 12DOF and has 5 DOF (three rotational and two translational) for each gear disc, as well as 1 DOF for each motor disc and loaddisc to describe the rotation. A schematic diagram of the 12 DOF model is shown in Fig. 3. The equations of motion for thismodel can be explained as follows.The equations of motion in thex direction for the pinion and gear aremp xp KxpT xpCxpT_ xpFpKxpCpCxpC_ p15mg xg KxgT xgCxgT_ xgFgKxgCgCxgC_ g16The equations of motion in they direction for the pinion and gear arempyp KypT ypCypT_ypNKypCpCypC_ p17mg yg KygT ygCygT_ ygNKygCgCygC_ g18The equations of motion in the direction for the pinion and gear areIpprp NMpktpm_ _ct_p_m_ _19Igg rg NMgktgb_ _ct_g_b_ _20The equations of motion in the direction for the pinion and gear areIdp pIpwp_ pKypC ypKypR pCypC _ ypCypR_ p21Idg g Igwg_ gKygC ygKygR gCygC _ygCygR_ g22The equations of motion in the direction for the pinion and gear areIdp p Ipwp_ pKxpC xpKxpR pCxpC _xpCxpR_ p23Idg gIgwg_ gKxgC xgKxgR gCxgC _ xgCxgR_ g24The equations of motion in the direction for the motor and load areImm ktmp_ _ct_m_p_ _Tm25Fig. 3. Dynamic model of a reduction gear system with 12 DOF.O.D. Mohammed et al. / Mechanical Systems and Signal Processing 54-55 (2015) 293305 297Ibb ktbg_ _ct_b_g_ _Tb26The subscript T in the stiffness and damping terms is used to distinguish the radial or translational terms from the otherrotational terms, which are denoted by R, and the coupling terms, which are denoted by C. These terms are explained inAppendix A.3. Vibration-based tooth fault detectionForgearfaultdetectionpurposes, thestatusof thetoothdeteriorationcanbeevaluatedmainlybyintroducingthereductioninthetime-varyinggear meshstiffness. Several researchstudies [35,913] haveintroducedthis stiffnessreductionindynamic modellingfor fault detectionpurposes. Inthepresent paper, themeshstiffness parameter isconsidered to assess the fault status, as the mesh stiffness affects the output dynamic response.3.1. Modelling of gear mesh stiffness with a crack in one toothTo describe how deep a crack in the tooth root is, the crack level (CL) can be defined as the crack depth percentage of thetotal tooth root thickness measured at the crack initiation point, which is 2.96 mm in the studied model. Different crackpropagationscenarios were discussedin[5,9], where their performances, froma fault detectionperspective, werecompared. However, for simplicity the crack is assumed in the present study to extend along the whole tooth width withauniformcrackdepthdistribution. Thirteencrackcasesof differentcrackdepthswereintroduced, withastepsizeof0.1 mm, as stated in Table 3.The mesh stiffness calculation presented in [5] was adopted in [9], as it is a more comprehensive approach and offers thepossibility of simulating a parabolic crack distribution. A modified method for stiffness evaluation was discussed in [10].One conclusion drawn was that the method presented in [5] had been proved valid for stiffness evaluation for crack levelsless than 30% based on the studied model. Another conclusion was that the modified method could be considered as analternativeforlargecracksizes, and, basedonthestudiedmodel, abetterresultagreementhasbeenshownwiththemodified method for crack sizes larger than 30%. In the present study the mesh stiffness evaluation was investigated usingboth methods, and, based on the present studied model, the difference was found to be insignificant (less than 2%) for thelargest studied crack size, which can be shown in Table 3. Therefore, the method presented in [5] was applied in the presentstudy and the equivalent mesh stiffness could be obtained as explained in the following sub-sections.3.1.1. Tooth stiffness with a constant crack depth along the tooth widthThe deflections under the action of the force can be determined, and then the stiffness can be calculated by consideringthe tooth as a non-uniform cantilever beam with an effective length of d, see Fig. 4a and b. The bending, shear, and axialcompressive stiffnesses act in the direction of the applied load and can be obtained as follows [5]:1Kb_d0ycos 1 hsin1 2EIxdy 271Ks_d01:2cos21 GAxdy 281Ka_d0sin21 EAxdy 29Kbis the bending stiffness, Ksis the shear stiffness and Kais the axial compressive stiffness.h, hq, hc, hx, y, dy, d, and 1are illustrated in Fig. 4b. 1varies with the gear tooth position.Table 3Crack propagation case data.Crack case qo (mm) CL (%) Crack case qo (mm) CL (%)1 0 0.00 8 0.7 23.642 0.1 3.37 9 0.8 27.023 0.2 6.75 10 0.9 30.404 0.3 10.13 11 1.0 33.785 0.4 13.51 12 1.1 37.166 0.5 16.89 13 1.2 40.547 0.6 20.27O.D. Mohammed et al. / Mechanical Systems and Signal Processing 54-55 (2015) 293305 298Moreover, the following notation is used: G: shear modulus, G E=21; Ix: area moment of inertia,Ix1=12hxhx3W; hxrhq1=12hxhq3W; hx4hq_Ax: area of the section of distancey measured from the load application point,Axhxhx W; hxrhqhxhq W; hx4hq_hqhcq0 sin(c);andq0andcarethecrackdepthandcrackangle, respectively, (seeFig. 4b). Thecrackanglecisconsidered to be 703.The total tooth stiffness resulting from the effect of all the stiffnesses calculated previously can be obtained as follows:Ktp1=1Kb 1Ks 1Ka_ _303.1.2. The effect of the fillet foundation deflection on the mesh stiffnessThe fillet foundation deflection can be calculated as follows [14]:f F cos2m WELnufSf_ _2MnufSf_ __Pn1Qntan2m_31mis the pressure angle, and uf and Sf are illustrated in Fig. 5.Ln; Mn; Pn; and Qncan be approximated using polynomial functions as follows [14]:Xnihf i; f_ _Ai=2f Bih2f iCihf i=fDi=fEihf iFi32Xnirepresents the coefficients Ln; Mn; Pn; and Qn. The coefficients Ai; Bi; Ci; Di; Ei and Fiare given in Table 4.hf irf =rint; rf ; rint; and fare illustrated in Fig. 5.Then the stiffness due to the fillet foundation deflection can be obtained as1KffF33For a pinion it can be denoted by Kfp.Fig. 4. Modelling of a gear tooth crack: (a) modelling of a cracked tooth, (b) tooth notation, and (c) uniform crack distribution.O.D. Mohammed et al. / Mechanical Systems and Signal Processing 54-55 (2015) 293305 2993.1.3. The effect of Hertzian contact on the mesh stiffnessThe Hertzian contact stiffness Kh can be calculated as stated in [15] as follows:1Kh412 E W343.1.4. Total mesh stiffnessAftercalculatingthestiffnessofacrackedpiniontooth, Ktp, duetobending, shear, andaxial compression, andthencalculating the stiffness due to the fillet foundation deflection, Kfp, the same calculations can be performed for an uncrackedmating gear tooth to find Ktg and Kfg. Thus, the total mesh stiffness for one meshing tooth pair isKt11=Ktp_ _1=Kf p_ _1=Ktg_ _1=Kf g_ _1=Kh_ _ Km135where Km1 is the total mesh stiffness for the first tooth pair.In cases where there are two tooth pairs in contact, the same calculations are repeated for the second tooth pair to findKm2. Then we can obtain the equivalent mesh stiffness as follows:KmKm1Km236Fig. 6 shows the varying mesh stiffness Km obtained for the studied crack sizes. Actually, the damping between meshingteeth is proportional to the equivalent mesh stiffness Km, and can be evaluated approximately using the following equation [16]:Cm2Km1=mp1=mg37In this study the mesh damping value Cm is considered to be 1147 N s/m for the case of two teeth in contact, and 869 N s/mfor the case of one tooth in contact.3.2. Dynamic simulation using the three studied dynamic modelsA dynamic simulation was performed for the healthy case, after which the simulation was repeated for the faulty cases.To introduce the tooth fault in dynamic simulation, the equivalent mesh stiffness corresponding to the crack sizes can beinputinthestudieddynamicmodel. Fourgeardynamicmodelswerestudied, andthesimulation wasrepeatedforthethirteen studied crack cases for each model.Fig. 5. Geometrical parameters for fillet foundation deflection [14].Table 4Values of the coefficients of Eq. (32).Ai ( 105) Bi ( 103) Ci ( 104) Di ( 103) EiFiLn(hfi,f) 5.574 1.9986 2.3015 4.7702 0.0271 6.8045Mn(hfi,f) 60.111 28.100 83.431 9.9256 0.1624 0.9086Pn(hfi,f) 50.952 185.50 0.0538 53.300 0.2895 0.9236Qn(hfi,f) 6.2042 9.0889 4.0964 7.8297 0.1472 0.6904O.D. Mohammed et al. / Mechanical Systems and Signal Processing 54-55 (2015) 293305 300AMatlabcomputersimulationusingtheODE45functionwascarriedouttomodel theequationsof motionwithasamplingfrequencyof 200 kHz toprevent aliasingof the highest detectable harmonics around20 kHz. Adynamicsimulation with timedomain analysis was performed to obtain thevibrationalsignals ofall thestudied cases. Normallydistributed noise was added with an SNR value of 30 dB to include the influence of measurement noise. In reality a randommanufacturingerrororanyothercontribution whichissynchronouswiththerotationalspeedwilladdenergiesinthespectra at multiple integers of the rotational speed. These signal types are theoretically removed from future signal contentbytheresidualsignalprocess. Thisworkwascarriedoutby studyingthepinionsy-displacement, whichwasthemostsensitive movement for crack propagation in the pinion tooth root.3.3. Residual signal and time domain statistical indicatorsThe performance of some of the statistical indicators was studied in [4]. When the second proposed method of generatingthe residual signal is applied, the RMS indicator shows the best performance. Kurtosis is the most robust indicator for all thesignalsused. Thesecond methodrecommendedbyWuetal. [4]forgeneratingtheresidualsignalinvolves removingthewhole original signal of the healthy case from the original signal of the faulty cases. The original signal obtained for the healthycaseisconsideredasaregularsignal, andbyremovingthisregularsignalfromtheoriginalsignalofthefaultycase, theinfluence of regular vibration can be removed and then the signal components generated due to crack propagation can behighlighted. This method for generating the residual signal was applied in the present study, after which the RMS, kurtosis andthe crest factor were applied to the obtained residual signal. These indicators can be explained as follows.The RMS is considered as one of the basic statistical indicators that measure the energy level of a signal. The RMS can bedefined as follows [4]:RMS 1NNn 1xn x 2; where x 1NNn 1xn 38Kurtosisis anindicator whichmeasures thedegreeof peakinessof adistributionanddescribes thesignal shapeascompared to the normal distribution. The kurtosis value depends on the distribution tail length, so that the kurtosis value ofthe residual signal is much higher than that of the original signal. The kurtosis indicator can be defined as follows [4,17]:Kurtosis 1=NNn 1xnx41=NNn 1xn x2

2_ 39The crest factor is the ratio between the maximum absolute value reached by the signal and the RMS of the signal. Thisindicator gives one an idea as to whether any impacting can exist in the signal [17].CF max xnRMS40Fig. 6. Gear mesh stiffness cycle with different crack sizes.O.D. Mohammed et al. / Mechanical Systems and Signal Processing 54-55 (2015) 293305 3014. Results and discussionDynamic simulations were performed for the four gear dynamic models studied. The fourth model (with 12 DOF) wasappliedtothesymmetricdisccaseinadditiontotheasymmetricdisccase, whichwas simulatedtostudythediscasymmetry effect on the obtained dynamic response. The response of the pinion's y-displacement was analysed, because itwas the most sensitive movement for crack propagation in the pinion tooth root. Some of the time signals obtained usingthe fourth model with the disc symmetry case are shown in Fig. 7. Moreover, Fig. 8 shows the percentage change in theperformance of the statistical indicators applied on the time signals obtained from the different dynamic models used inthis study.The results of the first model (with 6 DOF) show a clearly visible difference from those of the other studied models. It canbe observed in Fig. 8 that, with the higher studied crack level, the RMS change obtained from the first model was about 160%higher than the RMS change obtained from the other models. In the first model the input and output torques were applieddirectlyonthegearandpinion, andthereforethedisplacement responseshowsahighersensitiveresponsethantheresponseobtainedfromtheothermodels, whichweremademorerealisticbyincludingtwomoreDOFtodescribethemotor and load. The results of the third model (8 DOF reduced to 6 DOF), where inter-tooth friction is ignored, are very closeto those obtained from the second model (with 8 DOF), and it is shown that friction exerts a very slight influence. Moreover,thefourthmodel appliedto thesymmetric discshows aresponsethatisvery closeto theresponsesobtainedfromthesecond and third models, because the coupling terms which affect the obtained response yp are zero when the lengths a andb are equal. The slight difference between their results is due to a re-generation of the random signal accompanying theobtained simulated signal.There is a difference between the results for the 12 DOF model applied to the asymmetric disc case and the results for thesymmetric disc case, since the coupling terms start to contribute to the obtained dynamic response. These coupling termsdescribe the influence of the gyroscopic DOF on the dynamic response. For fault detection purposes, the disc asymmetryeffect is important to include, as the obtained response yp was affected by the asymmetry of the gear disc.Todemonstratethegyroscopiceffect, aCampbell diagramof thefourthmodel withthesymmetricdisccasewasobtainedfor constantaverage meshstiffness, seeFig. 9. Despitethesmallinertia valuesofthestudiedgearsystem, theinfluence of the gyroscopic effect can be seen as the fifth eigenfrequency varies with the running speed. This eigenfrequencyis much greater than the operating speed, but it is useful to examine the influence of the gyroscopic effect on the system'seigenfrequencies.Fig. 7. Original and residual signals of three selected crack cases obtained using the 12 DOF model with the disc symmetry case.O.D. Mohammed et al. / Mechanical Systems and Signal Processing 54-55 (2015) 293305 302Fig. 8. Performance of the statistical indicators applied on the residual signals obtained from the studied models.Fig. 9. Campbell diagram of the 12 DOF model with the symmetric disc case.O.D. Mohammed et al. / Mechanical Systems and Signal Processing 54-55 (2015) 293305 3035. ConclusionsIn this paper,a gear dynamic model (with 12 DOF) including a gyroscopic effect was developed and the equations ofmotionswerederived. Thestudiedgearsystemwasmodelledusingthreedifferentdynamicmodels(with6, 8and8reducedto6DOF), aswellasthedevelopedmodel(with12DOF), whichisreferredasthefourthmodelinthispaper.Dynamic simulation was performed for different crack sizes, and the pinion's y-displacement was analysed as it is the mostsensitive movement for crack propagation in the pinion tooth root. Time domain scalar indicators (the RMS, kurtosis and thecrest factor) were applied for fault detection analysis.The results of the first model show a clearly visible difference from those of the other studied models, which were mademore realistic by including two more DOF to describe the motor and load.Both the symmetric and the asymmetric disc cases were studied using the fourth model. In the case of disc symmetry,the results for the obtained response yp are close to those obtained from both the second and third models. Furthermore, thesecond model showed a slight influence from inter-tooth friction, and therefore the third model is adequate for simulatingthe pinion's y-displacement in the case of the symmetric disc.In the case of the asymmetric disc using the fourth model, the results deviate from those obtained in the symmetric case.Therefore, forsimulatingthepinion'sy-displacement, thismodel (with12DOF) canbeconsideredformoreaccuratemodelling in the case of the asymmetric disc.Appendix A. Terms used in the derived equations of motionIn the equations of motion derived in Section 2, the subscript T in the stiffness and damping terms is used to distinguishthe radial or translational terms from the other rotational terms, which are denoted by R, and the coupling terms, which aredenoted by C. These terms are explained as follows, see Fig. A1.The translational terms represent the total effect of the supports on both sides.KxpT K1xpK2xp ; KypT K1ypK2yp; KxgT K1xgK2xg; KygT K1ygK2ygCxpT C1xpC2xp; CypT C1ypC2yp ; CxgT C1xgC2xg; CygT C1ygC2ygTherotational termsareaffectedbythediscsymmetryandthenbytheparametersaandb. Inthisstudy, forthesymmetric disc case the gear shaft lengths are ab0.05 m, and for the asymmetric case a0.07 m and b0.03 m.KxpRa2K1xpb2K2xp; KypRa2K1ypb2K2yp; KxgRa2K1xgb2K2xg; KygRa2K1ygb2K2ygCxpRa2C1xpb2C2xp; CypRa2C1ypb2C2yp; CxgRa2C1xgb2C2xg; CygRa2C1ygb2C2ygThe coupling terms are also affected by the parameters a and b; note that these terms are zero in the case of the symmetricdisc.KxpCa K1xpb K2xp; KypCa K1ypb K2yp; KxgC a K1xgb K2xg; KygC a K1ygb K2ygCxpCa C1xpb C2xp; CypCa C1ypb C2yp; CxgC a C1xgb C2xg; CygC a C1ygb C2ygFig. A1. Definition of dynamic model terms .O.D. 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