Advantages and drawbacks of applying periodic time-variant modalanalysistospurgeardynamicsRunePedersena,IlmarF. Santosb,,Ivan A.HedeaaSiemensWindPowerA/S,Borupvej16,DK-7330Brande,DenmarkbTechnicalUniversityofDenmark,DepartmentofMechanicalEngineering,NilsKoppelsAlle,Bygning403,DK-2800Kgs.Lyngby,Denmarkarticle infoArticlehistory:Received18 August 2009Receivedin revisedform18 November2009Accepted22December 2009Availableonline 7January2010Keywords:ModalanalysisGeardynamicsParametricvibrationTime-variantsystemsabstractAsimpliedtorsionalmodelwithareducednumberofdegrees-of-freedomisusedinorder to investigate the potential of the technique. Atime-dependent gear meshstiffness function is introduced and expanded in a Fourier series. The necessary numberof Fourier terms is determined in order to ensure sufcient accuracy of the results. Themethod of time-variant modal analysis is applied, and the changes in the fundamentalandtheparametricresonancefrequenciesasafunctionoftherotationalspeedofthegears, are found. By obtaining the stationary and parametric parts of the time-dependent modes shapes, the importance of the time-varying component relative to thestationarycomponentisinvestigatedandquantied. ThemethodusedforcalculationandsubsequentsortingoftheleftandrighteigenvectorsbasedonarstorderTaylorexpansion is explained. The advantages and drawbacks of applying the methodology towindturbinegearboxesareaddressedandelucidated.&2010ElsevierLtd.Allrightsreserved.1. IntroductionModal analysis method is very frequently used with the aim of solving time-invariant linear equations of motion. Theeigenvalues and eigenvectors obtained from the solution of eigenvalue problems allow engineers to interpret and visualizethe dynamic behavior of different mechanical systems. For time-variant equations of motion the use of modal analysis inits well-known form is not possible. In this case a stability analysis normally relies on Floquet theory. It does not deliver thecomplete homogenous solution, but gives only information about the stability of the system represented by equations ofmotion withtime-variant coefcients [1].In many cases as exible rotating blades, exible rotating discs, rotating shafts with non-symmetrical cross section, andgeardynamics, thecoefcientsof theequationsof motionvaryinaperiodicway. InXuandGasch[2] thecompletehomogenous solution of periodic time-variant linear equations of motions is presented, based on Hills approximation. Theperiodic time-variant matrix systems are expanded in Fourier series. Assuming that the solution also can be expanded inFourier series, it is possible to obtain a general homogenous solution by solving a hyper-eigenvalue problem, i.e. when thenumber of Fourier coefcients is not innite. The solution of the hyper-eigenvalue problem leads also to eigenvalues andeigenvectors, nevertheless, the eigenvectors become also periodic time-variant and the eigenvalues become dependent onthe periodicity of the parameter variation. Different contributions to the problem of periodic time-variant modal analysisare presented in the literature, i.e. non-symmetric rotors [1,3], exible rotating discs [4,5], rotor-blade dynamics [2,6] andContentslistsavailableatScienceDirectjournal homepage: www.elsevier.com/locate/jnlabr/ymsspMechanicalSystemsandSignal ProcessingARTICLEINPRESS0888-3270/$ - seefrontmatter&2010ElsevierLtd. Allrightsreserved.doi:10.1016/j.ymssp.2009.12.009Correspondingauthor.Tel.: +4545256269;fax: +4545931577.E-mailaddress:ifs@mek.dtu.dk (I.F.Santos).MechanicalSystemsandSignalProcessing24 (2010)14951508lateractivecontrol of rotor-bladedynamics[710]. Themathematical foundationsof modal analysisfortime-varyinglinearsystemsisclearlyand verynicelypresentedby Irretier in[11],and BucherandEwins in[12].In [1] the dynamics of a simple exible shaft with non-symmetric cross section, supported by anisotropic bearings, istheoretically investigatedusing Hills approach. In[3] suchaninvestigationis carriedout theoretically as well asexperimentally.Flexible rotating discs arealso an example of period time-variant structure. Their dynamics are carefully investigatedusing periodic-time variant modal analysis in [4,5]. The theoretical work presented in [4], and the experimental validationin[5],illustrates thecontinuation ofIrretiers work [13].Rotatingexiblebladesarealsoanexampleof aperiodictime-variant system. Their dynamicsarealsocarefullyinvestigatedusingperiodictime-variantmodal analysis, asitcanbeseenin[2,14,6,15]. In[15] acontributiontotheexperimental validationof linear andnon-linear dynamic models for representing rotor-blades parametric coupledvibrationsisgiven. Therotor-bladedynamicsisdescribedbyusingthreemodelswithdifferent levelsof complexityfollowedbyexperimentalvalidationofsuchmodels. Adeeperphysicalunderstandingofthedynamiccouplingandthebehavior of the parametric vibrations are achieved. Such an understanding is of fundamental importance while developingactivecontrol strategies. In[7] thedesign of time-variant modal controllers isinfocus.Time-variant modal analysisandmoderncontroltheoryareintegratedinanelegantwayallowingthedevelopmentofnewcontrolstrategies.Themodalcontrollability and observability of bladed discs are strongly dependent on the angular velocity, a detailed analysis of suchadependencyispresentedin[8]. Tocontrol rotorandbladevibrationusingonlyshaftactuationisaverychallengingproblem. In [9] such a problem is investigated theoretically as well as experimentally using different control strategies. Theelectromagnetic actuators are used to control a horizontal rotor-blade system (blades periodically excited by the gravity).In[10] thesameproblemistheoreticallyaswell asexperimentallyinvestigatedandnewstrategiesaredevelopedtocontrolvertical rotor-blade systems.Theexistingdynamicgearmodelscanbeclassiedbasedontheexcitationsource:transmissionerror(TE)-excitedmodels and parametrically excited models. As shown by Blankenship and Singh [16], the TE and the mesh stiffness dependon each other, making the dynamic gear mesh modeling very complicated. To simplify the calculations, it is common to usethe TE as the only external excitation source, and/or use the varying stiffness as a parametric excitation. The validity of thisprocedure is examined by Velex and Ajmi [17], who mention the problem of dening TE for helical gears as a limitation.Also the assumption of quasi-stationarity often used in TE-excited models is a disadvantage, as it excludes the possibility ofusingthemodel tocorrectlypredict dynamicbehavior inaresonant region. This problemis thenpartlysolvedbyKahramanandSinghin[1821], whousethemethodsofnon-lineardynamicsforsolvingtheequationsofmotioninaresonant regionof a gear pair witha clearance-type non-linearity. The effect of the non-linearity is ampliedbyintroducingthevarying mesh stiffness.In the non-TE-excited models, no knowledge about the TE is needed before the calculation is performed. Peeken et al.[2225] present models which are parametrically excited. It is shown that the equation of motion for the torsional 1-DOF(degree-of-freedom) systemreduces totheMathieuequation, whenthegear meshstiffness is replacedbyacosinefunction. MorecomplicatedmeshstiffnessfunctionsareintroducedviathelowesttermsoftheirFourierseries. In[26],VelexandBerthe solvetheequationsof motionafter splitting these inastationary part(resultsfrom themean externalload) and a dynamic part (from the varying part of the external load). The mesh stiffness is described by a Fourier series inthe time domain. Velex and Maatar [27] and Ajmi and Velex [28] extend the model by further investigations concerningthemeshstiffness.The goal of this work is to apply the theory of time-variant modal analysis to spur gear dynamics. The advantages anddrawbacksof usingthetechnique foranalyzing vibrations inspurgears areinvestigated.2. Mathematical modeling2.1. GearmeshstiffnessFindingthegearmeshstiffnessisaniterativeprocess, whichinvolvescalculationoftheloaddistributionacrossthetooth and the load sharing between the contacting tooth pairs. The gear mesh model is shown schematicallyand slightlysimpliedin Fig. 1(a). In this example, three tooth pairs are taken into account, marked by 1, 2, and 3. Each tooth pair isdivided intothree sections inthe longitudinaldirection ofthe tooth, asshown inFig. 1(b).Theelastic coupling betweensectionswithinonetoothishandledbyapplyingbeamtheorytosimplifytheplatedeectionproblem, asshownbySchmidt [32]. The goal for the stiffness model is to nd the stiffness matrices KHertz for the Hertzian deformation, and Ktoothforthebendingandshearinthegeartooth. ForalowloadPasshowninFig. 1(a), onlyonesectionisincontact. Withincreasing load, more sections will be in contact with the mating gear ank, and thus contribute to the gear mesh stiffness.The stiffness model is based on the true involute tooth form, which is calculated from basic gear data using the modeldescribed by Padieth [29]. The stiffness calculations are performed using the methods of Weber and Banaschek [30], Ziegler[31], and Schmidt [32]. Following these references, the gear body deformation is included in Ktoothby regarding the gearbodyasanelastic, semi-innitespace. Whenapplyingaforcetoatooth, theresultingstressesaretransferredtotheunderlyinggearbody,which isassumedto deforminazone thatstretches 23 toothwidthsfrom theloaded tooth.ARTICLEINPRESSR.Pedersenetal./MechanicalSystemsandSignalProcessing24(2010)14951508 1496Theinclusionofgrindingcorrectionsisanimportantaspectofthecontactmodeling.InFig.1(a), thetopologytotheright shows the grinding corrections. These will affect the number of contacting sections for a given load, which will thenaffectthegearmeshstiffness.The gear mesh stiffness is calculated for several relative positions of the meshing gears. This gives a load- and positiondependent stiffness. Under the high loads considered in the following dynamic analysis, it is assumed that tooth separationwill not occur. After completing the stiffness calculation, a constant rotational speed is assumed. By these assumptions, thegear mesh stiffness becomes a periodically varying parameter in the time domain, that is, the non-linearity is removed. Itmustalsobenotedthatallteethareconsideredidentical,i.e.pitcherrorsarenotincluded.Theresultingsystemcanbeinvestigated usingthetheory formodal analysisof lineartime-variant systems.2.2. Modalanalysisoftime-variantsystemWhen the only time-variant parameter in the system is the gear mesh stiffness, the equation of motion can be writtenasMqC_qKtq f 1{_qq" #0 IM1Kt M1C" #q_q" #0 00 M1 0f 2or_zAtz p; z q_q" #3TherstpartoftheanalysisofEq. (3)istondthesolutionstothehomogeneouspartoftheequation, thatis, solve_zAtz 0 for the unknown z. This is done by expanding the theory of modal analysis of time-invariant systems [33] toinclude periodically time-varying parameters, as shown in [6,2]. By the substitutions zt rteltand p 0, thehomogeneouspart ofEq.(3)canbe writtenastheeigenvalueproblem_rtlIAtrt 0 4Under the assumption of constant rotational speed, the gear mesh frequency in units rad/s is called O. The stiffness matrixKt and therefore the state matrix At are both periodic with period T 2p=O and can be expanded into innite, complexFourierseries. Alsothetime-varianteigenvectorrjt, belongingtothejtheigenvalue, isassumedtobeperiodic. Inallequations,i 1p:Kt X1k 1KkeikOt5At X1a 1AaeiaOt6rjt X1r 1rj;reirOtforj 1. . . 2N 7ARTICLEINPRESSFig.1. (a) Gearmeshstiffnessmodel;(b)tooth pairdiscretization.R.Pedersenetal./MechanicalSystemsandSignalProcessing24(2010)14951508 1497Itcanbeshownthat thecomponents oftheeigenvectors rjtcanbefound byre-writing Eq.(4):lj^I^A rj;2rj;1rj;0rj;1rj;2 26666666666643777777777775 00000 266666666666437777777777758where^A & 2iOIA0A1A2A3A4 A1iOIA0A1A2A3 A2A1A0A1A2 A3A2A1iOIA0A1 A4A3A2A12iOIA0 &266666666666437777777777759In an exact solution to Eq. (8),^A is innitely large. However, the magnitude of the Fourier components of At depend directlyon the Fourier components of Kt. Therefore, if the gear mesh stiffness function included in Kt is smooth, only a few of theFourier components of At will have a magnitude that will signicantly inuence the resulting displacements and velocitiesinthevectorzt.Inalatersection, itwillbeshownhowtocalculatethenecessarynumberofFouriercomponentstobeincluded in^A, in order to obtain accurate results. If the number of included Fourier components is n, in the sense thatKt Xnk nKkeikOt10and the number of degrees of freedom in the model isN, the size of matrix^Awill be 2N2n1 2N2n1.Thereisagreat amount of redundant informationinthesolutionof Eq. (8). Onlythebasiseigenvaluesandbasiseigenvectors, whichareidentiedas describedina later section, are neededinthefurther analysis. The2Nbasiseigenvalues are stored in a diagonal matrix K, and the Fourier components of the basis eigenvectors rj;0are stored in thethree-dimensional array R. This array can be visualized as 2n1 layers of matrices of size 2N 2N. The k th layer of Rwillhavethestructure Rkr1;kr2;k r2N1;kr2N;kfork n; . . . ; nTondthesolutionqofEq.(1), alsothelefteigenvectorsLt, whicharethesolutiontotheequationRtLt Iareneeded.These canbefoundby solvingthematrix equation:RtLt I 11{X1r 1RreirOt X1l 1LleilOtI 12{ R0R1R2R3R4 R1R0R1R2R3 R2R1R0R1R2 R3R2R1R0R1 R4R3R2R1R0 26666666666643777777777775 L2L1L0L1L2 26666666666643777777777775 00I00 2666666666664377777777777513In most practical cases, the magnitudes of the submatrices Rn will decrease by increasing n. Therefore, a solution based onthe central rows and columns of the innitely large system of Eqs. (13) will be sufciently accurate. For n 2, Eq. (13) canbeapproximated byR0R1R20 0R1R0R1R20R2R1R0R1R20 R2R1R0R10 0 R2R1R02666666437777775L2L1L0L1L2266666643777777500I00266666643777777514ARTICLEINPRESSR.Pedersenetal./MechanicalSystemsandSignalProcessing24(2010)14951508 1498With Llfound for l n n, the solutions to the homogeneous and the inhomogeneous equation of motion, Eq. (1), cannowbefound [6,2]. Afterintroducing themodal coordinates ytaszt Rtyt )_zt _RtytRt_yt 15and inserting into Eq. (3), the terms are multiplied from the left with Lt and rearranged. The result is the decoupled modalequation ofmotion_yt&lk&264375yt Lt0M1 ft 16ByinsertingLt Pnl nLleilOtandtheoscillatingforceft peiOtintotheright handsideof (16), theequationbecomes_yt&lk&264375yt Xnl nLl0M1 peiOlOt17Now a modal solution yt yeiOlOtis assumed and inserted into (17). The transformation back to the zt coordinates isperformedusingEq. (15), andRt issubstitutedbyitsequivalentsumof Pnr nRr. Afterrearrangingtheterms, theresulting equation showstheforced response:zinhomt Xnr nXnl nRr&1iOlOlk&2666437775Ll0M1 peiOOr lt18When usinganiten andtheinitial condition zt0 z0,thesolutionto thehomogeneousequation ofmotion becomeszhomt Rt&elktt0&264375Lt0z0192.3. Sortingofeigenvalues andeigenvectorsTondthebasiseigenvectorsandbasiseigenvaluesneededtosetupRt andlkinEq. (18), theeigenvaluesandcorresponding vectorsmustbesorted.Thissortingisalso necessaryto correctlycalculate theeigenvectornormalizationfactors, as describedinalater section. TheproblemcanbedescribedinFig. 2, inwhichtheimaginarypart of theeigenvalues is plotted versus the gear mesh rotational frequency O. Both gures are zoomed to show only two families ofeigenfrequencies.In Fig.2(a),n 2Fouriercomponentsareincludedintheanalysis.Here,thesortingiseasy:the2n1lowerfrequenciesbelongtothelow-frequencyfamilycenteredaroundf1. InFig. 2(b), n 4andthesituationismorecomplicatedit isnolonger atrivial tasktodecidewhicheigenvaluesbelongtowhichfamily. Atypical eigenvaluedistribution in the complex plane is shown in Fig. 3(a). It is clear, how the different families of eigenvalues can be identiedby their real part, which is nearly identical for all members of the family [2]. This method will solve the problem visualizedinFig. 2(b). However, becauseofthetransformationintostatespacefromEqs. (1)to(3), theeigenvaluesarefoundincomplexconjugatepairs, asdepictedinFigs.3(a)and(b).Obviously,thesecannotbeseparatedbasedontheirrealpartalone. Instead, a sorting algorithm based on a rst order Taylor expansion of the eigenvalues as a function of O is used. Foragiven Ok,wherekisaninteger index,theexpected value for lisgivenbylk;expectedlk1dldODOlk1lk1lk2Ok1Ok2OkOk1 20Theeigenvalueslk;expectedpredictedbyEq. (20) arethencomparedtothoseactuallyobtainedbysolvingthehyper-eigenvalue problem, lk. The absolute difference between lk;expectedand lkis calculated, and the eigenvalues in lkare thenidentied with the eigenvalue in lk;expected, which shows the minimum difference. A few requirements must be fullled inorder touse theTaylor expansion sortingmethod:1. Thehyper-eigenvalue problem mustbesolvedforamonotonically increasing valueof O.2. At leasttwosolutions, fork2andfork1must becomputed withouttheTaylor expansion sorting.3. The values of O should not increase too much in each step in order to get a good estimation of lkfrom Eq. (20), i.e., acertain smoothnessof thefunction lOisrequired.ARTICLEINPRESSR.Pedersenetal./MechanicalSystemsandSignalProcessing24(2010)14951508 1499Whentheeigenvalueshavebeensortedwithineachfamily, thefamiliesaresortedrelativetoeachotherbasedonthemeanvalueof imaginarypart of theeigenvaluesinthefamily. Thebasiseigenvaluesarenowdenedasthecentralmember ofthefamily, basedonthe imaginarypart.3. Results3.1. PresentationofmodelThe gear pair, the attached shafts and the inertias J1J4 are shown in Fig. 4. The degrees of freedom (DOF) of the modelare the rotational angles of the four inertias and will be referred to as q q1q2q3q4

T. The model has intentionally beenkept very simple in order to keep the results easy to interpret. The basic data of the gears used in this work are presented inTable 1. The data are loosely based on the intermediate stage of a 1MW wind turbine gear box, except a zero degree helixangleandzeroproleshift onbothgearandpinionisused. ThecalculatedgearmeshstiffnessfunctionisshowninFig. 5(a). Foraconstantrotationalspeed O, theperiodicstiffnesscanbedescribedusingitscomplexFourierexpansion.Whent isthetime, theFourier seriesisdened asf t X1j 1cjeijt21ARTICLEINPRESSf1f2Im ()f1f2Im ()Angular frequency [rad/sec] Angular frequency [rad/sec]Fig. 2. Solutions tohyper-eigenvalueproblem:(a) lownorlow O;(b) highnorhigh O.00Re ()Im ()72 eigenvalues8 basis eigenvaluesf10f1Angular frequency [rad/sec] Im ()Fig.3. Complexconjugateeigenvalues:(a) eigenvaluedistribution,(b) highnorhigh O.R.Pedersenetal./MechanicalSystemsandSignalProcessing24(2010)14951508 1500In Fig. 5(b), the rst 30 Fourier components of the signals are shown, expressed as Aj2 abscj. Because of the symmetryinthecomplexFouriercomponents, wherecjconjcj, onlycjforj 1; . . . ; 30areshown. Themeanvalueofthegearmeshstiffnessc026:66Nmmmm1isnotshowninorder to illustratetheother componentsmore clearly.ARTICLEINPRESSJ4J1J2J3Fig.4. System model.Table 1Basicgeardata.Drivinggear Drivengear Unit Descriptionz 95 22 Numberofteethan20 deg Normal pressureangleb 0 deg Helixanglea 486 mm Center distancemn8 mm Normal modulex 0 0 Proleshiftb 215 225 mm ToothwidthMt74875.84 17339.67 Nm Torque051015202530Gear mesh stiffness [N (mm m)1] Gear mesh position [rad]0 2 3 4 5 6OriginalUsing 2 Fourier coeffs.Using 5 Fourier coeffs.Using 20 Fourier coeffs.1 5 10 15 20 25 3000.511.522.53Fourier component numberA = 2 abs (c) [N (mm m)1]Fig.5. (a) Gearmeshstiffness,(b) Fourierexpansionofgear meshstiffness.R.Pedersenetal./MechanicalSystemsandSignalProcessing24(2010)14951508 15013.2. StiffnessmodelvalidationThe stiffness model described in Section 2.1 must be validated. This is done by comparing to stiffness results obtainedusingthecommercialsoftwareKissSoft.ThecomparisonisshowninFig.6.InFig.6(a)isshownthegearmeshstiffness,when deformations of the gear teeth are not included in the kinematic analysis to determine tooth contact positions. In thistype of calculation, the total contact ratio for this spur gear pair equals the theoretical ea of 1.67. A good correlation betweenthe models is seen. In Fig. 6(b), the elastic deformation in the gear mesh is included. In this case, the deformations cause thetooth to touch before the theoretical start of contact, and to stay in contact a longer period of time. This leads to an increasein contact ratio beyond the theoretical value. Also in thiscase a good correlation with the KissSoft results are shown.3.3. EigenfrequenciesandmodeshapesofthesystemToidentifyeachofthe2Neigenfrequenciesandtheassociatedeigenvectors, atime-invariant(n 0)calculationhasbeen performed. The 2N modes are numbered according to Table 2. The mode shapes rjt can be evaluated at a given time tusingtheformula(7), using nandnasthelimitsforthesum. Theresult isshowninFig. 7forn 10, wherethedisplacement parts of the eigenvectors are plotted versus the gear mesh position y in the interval 0; 2p, corresponding tothetimeinterval 0; Tataconstantgearmeshrotationalfrequency O. Modes1and2therigidbodymodesdirectlyshow the gear ratio 95/22. Modes 3 through 8 are elastic modes showing torsional deformations in the shafts and the gearmesh. It can be seen how modes 36 are strongly affected by the varying stiffness, while modes 1, 2, 7, and 8 are almostconstant, r1t r1, r2t r2, r7t r7,and r8t r8.3.4. Quasi-staticeigenfrequencycalculationAs a preliminary analysis, the eigenfrequencies for the system can be calculated quasi-statically for a given time t t1.Todothis, AtinEq. (4)isevaluatedatt t1usingtherst2n1termsinEq. (6), wherenisthenumberofFourierARTICLEINPRESS051015202530Gear mesh position [rad]Gear mesh stiffness [N (mm m)1] 0 2 3 4cccth KissSoftcth, KissSoft051015202530Gear mesh position [rad]Gear mesh stiffness [N (mm m)1] 0 2 3 4ccc KissSoftc KissSoftFig. 6. Validation of stiffness c and mean stiffness cgby comparing to program KissSoft: (a) no load-induced increase in contact ratio, (b) including load-inducedincrease incontactratio.Table2Fundamentaleigenfrequenciesforn 0.Mode no. Frequency(Hz)1 02 03 3994 3995 5856 5857 32168 3216R.Pedersenetal./MechanicalSystemsandSignalProcessing24(2010)14951508 1502componentsincluded intheanalysis:At1 Xna nAaeiaOt122At1 istheninsertedintoEq. (4), andtheeigenvalueproblemissolved. Theresulting(positive) eigenfrequenciesareshowninFig. 8fortwodifferentvaluesofn. Fromthegure, itcanbeseenthatthehighesteigenfrequencyatroughly3200Hz is much more sensitive to the variations in the gear mesh stiffness than the other eigenfrequencies. Also, n has aninuence ontheextent ofthe frequency interval,inwhich thehighest eigenfrequencyislocated. Asaconclusion tothisquasi-static analysis, n can be expected to play an important role when determining the free and the forced response ofthetime-variant system.3.5. NecessarynumberofFouriercomponentsTheaccuracyofthetime-variantmodal analysisdependsonthenumberofFouriercomponents, n, includedintheanalysis. When the applied force f in Eq. (1) is constant or zero, only the changing stiffness in Kt can excite the system.From Eq. (10), it can be expected, that only frequencies up to approximately f nO will be excited. From the time-invariantmodal analysis of thesystemit is knownthat thehighest eigenfrequencyis around3215Hz. Whenthegear meshfrequency is O1357:59rad=s 216Hz, this eigenfrequency can be expected to be excited only when nZ3215=216 15.Theaccuracyofthemethodcanbedeterminedbycomparingthesolutioninthetimedomaintothecorrespondingsolution obtained by numerical integration. For this purpose, q3has been chosen, as this DOF shows the largest differencebetween the two calculation methods, as seen from Fig. 9(a). It might be interesting to represent the time variations of thedynamic mesh force (as opposed to q3in Fig. 9) since it also gives an indication on the presence of non-linearity (contactlosses whenthe mesh force is negative). Since this loss of contact isnot considered inthe present gear mesh model, thedynamicmeshforceiscalculatedasthedifferencebetweenthedisplacementsofthetwogearsmultipliedbythemeshstiffness, Ft ctq3tr3q2tr2, where r2 and r3 are the radii of the two gears. Comparison based on a single DOF is thesimplest way, since Ft offers no new information on the dynamic behavior of the system. It is clear from Fig. 9(a) how theARTICLEINPRESS0.500.511.5Mode 1, 20 /2 3/2 20.500.51Mode 3, 40 /2 3/2 20 /2 3/2 2 0 /2 3/2 20.500.511.5Mode 5, 6Gear mesh position [rad]0.500.511.5Mode 7, 8Gear mesh position [rad]q1q2q3q4Fig. 7. Modeshapesasafunction ofgearmeshposition y Ot,calculatedforn 10.R.Pedersenetal./MechanicalSystemsandSignalProcessing24(2010)14951508 1503accuracyofthemodalsolutionincreasessignicantlywhennreaches15.Anincreaseinnbeyond18doesnotincreaseaccuracy much.A zoom ofthetwo solutions isshowninFig. 9(b).3.6. Changeinfundamental frequenciesFor each number of n, the eigenvalue problem, Eq. (8), changes. It can therefore be expected that both the fundamentaleigenfrequencies (for k 0) andthehigher order parametriceigenfrequencies (for k 2 n; n; ka0) will changeas afunction of n. In the following analysis, the change in the fundamental eigenfrequencies, evaluated at the nominal speedO1357:59rad=s, is investigated. The results are shown in Fig. 10, where the change (in per cent) of the eigenfrequenciesbelongingtothethreeelasticmodes, relativetoatime-invariantmodalanalysis(n 0)areplotted. Itcanbeseenthatthere are no signicant changes in the fundamental eigenfrequencies. At no point, the frequencies change more than 0.16percent.3.7. NormalizationfactorsFor each of the 2N eigenvectors, the importance of the k th harmonic parametric vibration mode can be calculated. Thisis done byndingthe relative magnitude of the harmonic components of the eigenvector. For the k thharmonicARTICLEINPRESS0.1 0.102 0.104 0.106 0.108 0.1121012x 104Time [s]q3 (t) [rad]Numerical integrationTimevariant modal analysis0 5 10 15 20106105104Number of Fourier coefficients, nDifference [rad]q1q2q3q4Fig. 9. Comparisonof solutions: (a) maximumdifferencebetweennumerical integrationandtime-variant modal analysis, (b) comparisonof time-domainsolutions(zoom). Onlyq3tis shown.0500100015002000250030003500Eigenfrequencies [Hz]Gear mesh position [rad]0 /2 3/2 20500100015002000250030003500Eigenfrequencies [Hz]Gear mesh position [rad]0 /2 3/2 2Fig.8. Quasi-staticeigenfrequencyanalysis: (a)n 1; (b)n 18.R.Pedersenetal./MechanicalSystemsandSignalProcessing24(2010)14951508 1504component ofthejth eigenvector, rj;k,thenormalization factorNFj;kisdened asNFj;k1rTj;krj;kq with rjrj;n rj;1rj;0rj;1 rj;n26666666666643777777777775forj 1; . . . ; 2N 23InFigs.11and12,normalizationfactorsforalleightmodes areplottedversusthegearmeshfrequency O(inrad/s),forn 3and10, respectively. TheNF-axesintheplotsarelogarithmicandshowtheinterval NF 2 102; 101. NFforthefundamentalharmonic component,k 0,issettoNFj;01inallcases.Withthesedenitions,thenormalizationfactorsshowhowthefundamental part of themodeshapevectorr0isrelatedtothetime-varyingpartsrkfork n; . . . ; ndepending on the operational speed O. The k th harmonic component showing the smallest NFj;kwill be predominant inthe j th mode shape at a particular gear mesh frequency O. A number of observations can be made from the normalizationfactorplots, Figs.11and 12:1. For the rigid-body modes 1 and 2, the fundamental harmonic k 0 is predominant for all gear mesh frequencies O40.2. For all elastic modes, there is a symmetry between the positive (left column) and the negative frequencies (right columnin the plots): For the j th positive eigenfrequency, the normalization factor NFj;kis equal to NFj;kfor the correspondingnegative eigenfrequency.3. For modes 36, there exist certainmeshfrequencyranges, for whichthe time-varyingpart of the mode shape(parametricvibrations)will bemoreimportantthanthestationary. ThesefrequencyrangesareindependentofthenumberofFouriercomponentsincludedintheanalysis. Forinstance, formode3inFig.11, NFforthe 2harmoniccomponentshowsalocal minimumaroundO600rad=s. Thispotential 2harmonicresonance isfoundforallnZ2.4. Modes 7 and 8 at roughly f 73216Hz behave fundamentally different from the other elastic modes (modes 36). Non-independent frequencies withlow NFareobserved.5. For modes 7 and 8, frequency intervals exist in which NF for one or more harmonic components of the mode shape aresmallerthan1. Inthesefrequencyintervals, whichdependonthenumberofFouriercomponentsincludedbutaregenerallylocatedin O 2 0; 600 rad=s, thecontentofthetime-varyingpartofthemodeintheoverallmodeshapeislarger thanthefundamental component. Thesemodesarestronglydependent ontherelativeangular movementbetweenthe twogears,and arestronglyaffected bythetime-variant toothstiffness.ARTICLEINPRESS0 5 10 15 200.20.150.10.050Number of Fourier coefficients, nChange [per cent]Modes 7 and 8Modes 5 and 6Modes 3 and 4Fig.10. Changeinfundamental frequencies.R.Pedersenetal./MechanicalSystemsandSignalProcessing24(2010)14951508 15054. ConclusionThetheoryofmodalanalysisoftime-variantsystemshasbeenappliedtoasimplespurgearpairwithaperiodicallytime-varyinggearmeshstiffness. IthasbeenmadeclearthatalargenumberoftermsintheFourierexpansionofthesystem matrices is necessary in order to yield results of sufcient accuracy. This is a direct result of the jumps in the gearmeshstiffnessfunction. Inthecasesof exiblerotorswithnon-symmetrical crosssection, exiblerotatingdiscs, andexiblerotatingblades, averyreducednumberof Fouriercomponentsisneeded, normallyn 2, asthetime-variantcoefcientsarenormally sineand cosinefunctions.Ithasbeenshownthatthereexistregions, inwhichthehigherorderparametriccontributionstotheoverall modeshapeswillbesignicant. Forthecasestudiedhere, these effectsweremainly observed atlow gearmesh frequencies.Theelasticmodewiththehighestfrequencybehaveddifferentlyfromtheotherelasticmodes. WhilethetwolowermodesshowedparametricresonancefrequenciesthatwerelargelyindependentofthenumberofFouriercomponentsincluded in the analysis n, the parametric resonance areas of the high frequency mode strongly depended on n. Overall, thevibrations related to the higher frequency mode seemed to be more sensitive to the time-variant nature of the gear meshstiffness.The system studied in this work consisted of a single gear stage. A typical modern wind turbine gearbox consists of oneor two planetary stages followed by one or two parallel gear stages, with a total of 815 gear meshes. When applying thetheory of modal analysis of time-variant systems to such a complex system, great caremust be taken when interpretingtheresults. ThelargerthenumberofFouriercomponentsneededtoexpandtheperiodictime-variantcoefcients, thelargerthehyper-eigenvalueproblembecomes. Itmeansalso, itbecomesmorecomplicatedandcomplextophysicallyinterpretthebasicandparametricmodeshapes. Nevertheless, byexploringthedenitionofNF, theimportanceofthetime-varyingpartofthemodeshapes(parametricmodes)canbeinvestigatedandquantiedasafunctionofthegearmeshfrequency. Comparedtotime-stepintegrationschemes, themainadvantageofthetime-variantmodalanalysisisARTICLEINPRESS0 500 1000 1500102100NFMode 132101230 500 1000 1500Mode 20 500 1000 1500102100NFMode 30 500 1000 1500Mode 40 500 1000 1500102100NFMode 50 500 1000 1500Mode 60 500 1000 1500102100NF102100NF102100NF102100NF102100NFMode 70 500 1000 1500Mode 8 [rad/sec] [rad/sec]Fig.11. Normalization factors,n 3.R.Pedersenetal./MechanicalSystemsandSignalProcessing24(2010)14951508 1506that it offers an analytical solution to the vibration problem. Therefore a modal truncation is possible, which removes theneed for the very short time step used for a numerical integration of a system with high eigenfrequencies. Also the methodallowstoexpandtheanalysistotheconceptsofobservabilityandcontrollability[8], whichofferaquanticationoftheparametric vibrations.References[1] M. Ertz, A. Reister, R. 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