134 Modal Simulation Of Gearbox Vibration With .
AD-A260 134NASAI11iIN1111;11111111111111!AVSCOMTechnical Memorandum 105702AIAA-92-3494Technical Report 92-C-018Modal Simulation of Gearbox VibrationWith Experimental CorrelationFred K. Choy and Yeefeng F. RuanThe University of AkronAkron, OhioandJames J. Zakrajsek and Fred B. OswaldNationalAeronautics and Space AdministrationLewis Research CenterClevelan4 OhioDTICSELE-CTEJAN291993Prepared for the28th Joint Propulsion Conference and Exhibitcosponsored by the AIAA, SAE, ASME, and ASEENashville, Tennessee, July 6-8, 1992WUS ARMY.SYMM S COMMAND93-01692981 28 048
MODAL SIMULATION OF GEARBOX VIBRATION WITHEXPERIMENTAL CORRELATIONFred K. Choy, Yeefeng F. RuanDepartment of Mechanical EngineeringThe University of AkronAkron, Ohio 44325Accesion ForNTIS CRA&IDTIC TABUnannouncedJustification.By .and.Distribution!James J. Zakrajsek, Fred B. OswaldNational Aeronautics and Space AdministrationLewis Research CenterCleveland, Ohio 44135Availability CodesAvail and I orSpecialDist1DTIC QUALITY INSPECHD 31Abstract{Belrotor modal displacement of (YC, Yce)A newly developed global dynamic model wasused to simulate the dynamics of a gear noise rigat the NASA Lewis Research Center. Experimental results from the test rig were used toverify the analytical model. In this model, thenumber of degrees of freedom of the system arereduced by transforming the system equations ofmotion into modal coordinates. The vibration ofthe individual gear-shaft systems is coupledthrough the gear-mesh forces. A threedimensional bearing model was used to couple thecasing structural vibration to the gear rotordynamics. The system of modal equations issolved to predict the resulting vibration at several locations on the test rig. Experimental vibration data were measured at several runningspeeds and were compared to the predictions ofthe global dynamic model. There was excellentagreement between the analytical and experimental vibration results.[CbIbearing damping matrix[J[ ]T[Cb][Cb]]T[I]TCb] [tC[[CJcasing structure damping matrix[Uj[{ jT [Cj [(D)rotor modal displacement of Z{De}casing modal displacement of (Zc, Zje)(Dr}rotor modal displacement of 9tFgear force{F(t))external and mass imbalanceexcitations{Fc(t))force acting on casing structure{FG(t))nonlinear gear forces, through gearmesh couplingNomenclature{A)rotor modal displacement of (X, 0z)(Ac)casing modal displacement of (Xc, Xje)(F 3 (t))shaft bow force [K.] {Wr}(B)rotor modal displacement of (Y, O7 )[GAJrotor angular acceleration
[(A][.T41U.[GA] [I]Ifriction coefficient between the gearitooth surface,I[4']I [.].KA][KAI"orthonormal modeorthonormal mode of casing, Iý[0cI1/2 {[KbJ .JKby]}[WI1[1[HA@]T[KA[AJ[wJ1critical speeds of each rotorcritical speeds of casinglateral cross-couplingbearing axial andstiffness-"Introduction[Large vibrations in gear transmission systemsexcessive wear and crack formation in gearteeth, which results in premature gear failure.c]T Kbcauses[K.[.[KJcasing structure stiffness[Ka]shaft bow stiffness[Mlmass matrix of rotor[MC]mass matrix of casing structureRpitch radius of gear{[W]}generalized displacement vector ofrotor{[Wc]}generalized displacement vector ofcasing{Wr}genearlized displacement vector ofrotorX,Y,Zdisplacement vectorsaangle of orientationWith the need for higher operating speeds andpower from transmission systems, the problem ofexcessive vibration becomes even more critical.In order to insure smooth and safe operation, it isnecessary to understand the dynamics of the geartransmission system.Two areas of research in the dynamics of geartransmission systems are (1) analytical simulation and (2) experimental testing. There is agreat deal of literature on the vibration analysisof a single gear stage. 1 4 Some work has beendone on multistage gear vibration, 7 "9 but verylimited work1 3 1' 1 has been done on the dynamicanalysis of gearbox vibration. Considerable efforthas been devoted to experimentally studying geardyanmics1 2 1 4 and localized vibration effects ongear teeth.1 5 A few studies have been conductedto correlate analytically predicted and experimentally measured gearbox vibrations.This paper correlates the experimental resultsobtained from the test rig at the NASA LewisoEq. (5)9generalized displacement0torsional rotational vector0,09ylateral rotational vectorsResearch Center with predictions from an analytical model developed by using the modal synthesis method.7 The major excitations of the rotorsystem include mass imbalance, shaft residualforces 1 6 and gear-toothbow, nonlinear gear-meshfrictionaleffects. 1 4 The vibratory motionbetween the rotor and the casing is coupled2
through the support bearing in the lateral andaxial directions." Gearbox mode shapes andvibration predictions from the analytical modelare compared to those obtained from experimental testing.The motions of the individual shafts are coupledto each other through the gear-mesh forces(Fig. 1) and to the casing through bearing stiffness [Kb] and damping [CbI.The equations of motion for the casing can bewritten as [Cb]{Wc - W} lKb]{WCW)Analytical ProcedureDevelopment of Equations of MotionThe equations of motion for eachsystem can be written in matrix form asIMI{*} L jrcl ear-shaft(3){(*}where [Mc] is the mass matrix of the casing struc-c [GA] XW) [Cb] {W- Wr [Kb] (W - W C[K' , (WUi(t[G,] [Kc](Wc} {IF't,represents the generalized displaceture;ment {Wj}vector of the casingstructure:MOf t) {FG(Xe)I)(1)(xc*)where [M] is the mass matrix of the rotor (iner-(wI(YCO)tia), {W} is the generalized displacement vectorconsisting of the three displacement vectorsX,Y,Z with the corresponding lateral Ox, Oy, andtorsional Ot rotational vectors as(ZC)(ZcO)[CJ is the casing structure damping matrix; [Kc]is the casing structure stiffness; and {FC(t)} isthe force acting on the casing structure. Thenonlinear gear forces for the kth individualshaft system, using nonlinear gear stiffness andgear-tooth friction,14 are given asx-force(X)) (Y(c) (W)(Z)('i)Vj Inand where [Gl is the gyroscopic force; [GA] is theFGxk rotor angular acceleration; [Chi is the bearingdirect and cross-coupling damping, [Kb] is thebearing axial and lateral cross-coupling stiffness; 1 1 {W - Wj} is the casing vibration; [K] isthe shaft bow stiffness; {W - Wr} is the shaftresidual bow; {F(t)} is the external and masst[COSimbalance excitation; and {FG(t)} is the nonlinear gear force through gear-mesh coupling.y-forceFor a multiple gear-shaft system, the equations of motion (1) are repeated for each shaft.3i 1, i okKtki [-Rcici - Re kck (Xi-Xck)cos aki (Yi-Yck)sin aki]aij (sign)(p)(sin akd](5)
nLFGyk [KtkiVRcic , iiwhere w is the rotor critical speed. Similarly, aof orthogonality conditions can be derived forthe casing equation of[Ke] {W} [Cc] {Wc} [Kc] (Wc} 0(12)-set--X(RPk9 ckwith the orthonormal mode [-fc] such that" (Yci - Yck)sin aki][sin aki (sign)(p)(cos aki)]4and torsional [I](13)c]T [Cc][# c] [pc](14)*jT [KJ [,j nCFGtk (15)where wc is the casing critical speed. Usingmodal transformation' by lettinglR{Ktki [(-Rci ci - Rck ck)i l, i wak[-P] (Xci - Xck)cos a{OI(A} (Yci -- Yd)sin alld#'(7)["II {B}[0yo] {B}[ ,] (D}*tI {Dr}Modal TransformationandThe equation of motion for the undampedItlrotor system is{AcL CJ 0(8)[#cx*l{AAcwith the average bearing support stiffness fromthe x-y direction given as[KAI .'{[Kbr] [KbY]}[ c] {Bc}(9) [Ocyo] (Bc}tocus] {Dc)and[w2 ](17)0Ics] {Dc}The orthogonality condition for the orthonormalmodes [] arem ]T M[0][ [1](10)[IT [K6 KAJ [0] (16){W} gear and (7where R is the pitch radius of theis the coefficient of friction between the geartooth surface and "SIGN" is the unity sign function to provide the sign change when the matingteeth pass the pitch point.' 4[M]{W} [[KJ [KAI]{W}(A}the equations of motion for the rotor (Eq. (1))can be transformed as(11)4
housing. The measured three-dimensional modeshapes are presented in Fig. 4. The dynamicvibration measurements consist of data collectedR] M Al]M [UA1M Al]M- [Kb - KA] { Z} [ 1T [Cb[ 4J {Zc} (18)from accelerometers placed at three of the nodeson the surface of the casing. They were chosensuch that vibration was measured in all three [W2 ] {Z} - [4]T [Kb] 14c] {Z}[@]T {F(t) FG (t) Fs(t)}-directions, X,Y, and Z. A dynamic signal analyzer was used to compute the frequency spectra ofthe vibration. The experimental frequency spectra are shown in Figs. 5(a), 6(a), and 7(a) for theX-, Y-, and Z-directions, respectively, at a numof different operating speeds.and for the casing (Eq. (3)) as[I] (Z) IUC] {ZJ [W]2 {Z}[C{j [RbI {Zc} F['b] 1T{C}M - [T[Cb [ ]{7Z}{ber[Kb][4']{Z}[4] T [Cb][LOC{JtC(19)(19Discussion and Correlation of ResultsJwhereThe measured mode shapes, shown in Fig. 4represent the major vibration modes of the gearnoise rig in the 0- to 3-kHz region. Althoughthese modes are only a small portion of the totalmodes of the system, they represent the majorportion of the total global vibration of the system. In order to produce a compatible analyticalsimulation of the test apparatus, a similar set ofmodes was predicted by a finite-element model ofthe gearbox structure. This model serves as thebasis for predicting casing vibrations in the overall global dynamic model. Of the 25 modes foundby the analytical model in the 0- to 3-kHzfrequency region, the 8 dominant modes wereused to represent the gearbox dynamic characteristics. These simulated modes are shown inFig. 8. The natural frequencies of the predictedmodes are within 5 percent of the measuredmodes (Table 2). Also, the predicted modeshapes are very similar to the experimentalmodes shapes (Fig. 4). The good correlation ofbetween the analytical model and the measurements confirms the accuracy of the dynamicrepresentation of the test gearbox using only alimited number of modes.{A}{Ac){B}{Z} {{Z}{D}Dt}(20)JDJ}Experimental StudyThe gear noise rig (Fig. 2) was used to measure the vibration, dynamic load, and noise of ageared transmission. The rig features a simplegearbox (Fig. 3) containing a pair of parallel axisgears supported by rolling element bearings. A150-kW (200-hp), variable-speed electric motorpowers the rig at one end, and an eddy-currentdynamometer applies power-absorbing torque atthe other end. The test gear parameters aregiven in Table 1.Two sets of experiments were performed onthe gearbox; (1) experimental modal analysis and(2) dynamic vibration measurements during operation. In the experimental modal analysis, modalparameters, such as system natural frequenciesand their corresponding mode shapes, were obtained through transfer function measurementsby using a two-channel, dynamic signal analyzerand modal analysis software. For this experiment, 116 nodes were selected on the gearboxFor the dynamic study of the gearbox vibration, it was found that during a slow roll (lowspeed run) of the gear-rotor assembly, a substantial residual bow (or eccentricity in the sleeveassembly) exists in the rotor system as shown byits large orbital motion in Fig. 9. Figure 9(a)5
spectra at the running speeds of 1500 and5500 rpm, respectively.represents the orbit of the driver rotor at thegear location, and Fig. 9(b) represents that of thedriven rotor. Note that the circular orbit in thedriver rotor at low speed represents the residualbow deformation of the rotor. The elliptical orbitin the driven rotor is attributed to a combinationof the residual bow effects and the vertical gearforce from the torque of the driving rotor. Inorder to analytically simulate the influence of thiseffect, a residual bow of 2 mils (0.05 mm) isincorporated into the numerical model (Eq. (1)).Figures 6 and 7 compare the predicted measured housing vibration spectra in the Y- and Zdirections, respectively. The results of the comparison are the same as those presented for thehousing vibration in the X-direction (Figs. 5(a)and (b)). Acutal values of the components in thespectra were not always in good agreement; however, the general trends for the predicted andmeasured housing vibration spectra were verysimilar. Also, as seen in Fig. 7(b) at the1500-rpm speed, the model predicts the secondand third harmonics of the gear-mesh frequency.As shown in Fig. 7(a), the measured vibrationconfirms the presence of these two harmonics atthe 1500-rpm running speed.The frequency spectra of the analytically predicted casing vibration in the X-, Y-, and Zdirections are presented in Figs. 5(b), 6(b), and7(b), respectively. As seen in Fig. 5(a), the experimental casing vibration in the X-directionshows a major vibration component at the gearmesh frequency (28 times shaft speed) at eachrotational speed. A closer examination of thisexcited frequency component shows that twomajor vibration peaks occur at the runningspeeds of 1500 rpm (at a tooth pass frequency of700 Hz) and 5500 rpm (at a tooth pass frequencyof 2560 Hz). These peaks are a result of thetooth pass frequency exciting two of the majornatural frequencies of the housing, namely the658- and 2536-Hz modes. The presence of othermodes can be seen; however, the 658- and2536-Hz modes, when excited by the gear-meshfrequency, dominate the spectra.ConclusionsA newly developed global dynamic model wasused to simulate the dynamics of a simpletransmission system. Predicted casing vibrationswere compared to measured results from the gearnoise test rig at the NASA Lewis Research Center. The conclusions of this study are summarized as follows:1. The dynamics of the housing can be accurately modeled with a limited amount of analytically predicted, experimentally verified vibrationmodes of the structure.Comparing the predicted vibration spectrawith the measured spectra reveals that, althoughthe actual amplitude values did not always agree,the general trends of the spectra were very similar. The predicted vibration spectra of the housing in the X-direction is shown in Fig. 5(b). Acomparison of Figs. 5(a) and (b) shows that thepredicted amplitude at the gear-mesh frequencyat 1500 rpm is only 3 percent above the measuredvalue. The comparison at 5500 rpm is not thatclose; the predicted amplitude is 38 percent belowthe measured value. If trends are compared, thepredicted spectra show the same gear-mesh,frequency-induced excitation of the 658- and2536-Hz modes as that found in the measured2. The global dynamic model is capable ofincluding in the analysis the effects of shaft residual bow or eccentricity.3. Absolute values of the housing vibrationpredicted by the global dynamic model did notalways agree with measured values.4. The characteristics and trends of the housingvibration spectra predicted by the global dynamicmodel are the same or very similar to those foundin the experimental data.6
ReferencesReview", Journal of Sound and Vibration,Vol. 121, No. 3, Mar. 22, 1988, pp. 383-411.1. August, R., and Kasuba, R., "Torsional Vibrations and Dynamic Loads in a BasicPlanetary Gear System", Journal of Vibration, Acoustic, Stress, and Reliability inDesign, Vol. 108, No. 3, July 1986,pp. 348-353.10. Choy, F.K., Ruan, Y.F., Zakrajsek, J.J.,Oswald, F.B., and Coy, J.J., "Analytical andExperimental Study of Vibrations in a GearTransmission", AIAA Paper-91-2019, June1991.2. Choy, F.K., Townsend, D.P., andOswald, F.B., "Dynamic Analysis ofMultimesh-Gear Helicopter Transmissions",NASA TP-2789, 1988.11. Lim, T.C., Signh, R., and Zakrajsek, J.J.,"Modal Analysis of Gear Housing and.Mounts", International Modal Analysis Conference, 7th, Vol. 2, Society of ExperimentalMechanics, Bethel, CT, 1990, pp. 1072-1078.3. Cornell, R.W., "Compliance and Stress Sensitivity of Spur Gear Teeth," Journal ofMechanical Design, Vol. 103, No. 2,Apr. 1981, pp. 447-459.12. Lewicki, D.G., and Coy, J.J., "VibrationCharacteristics of the OH-58A HelicopterMain Rotor Transmission", NASA TP-2705,1987.4. Lin, H., Houston, R.L. and Coy, J.J., "OnDynamic Loads in Parallel Shaft Transmissions 1: Modelling and Analysis," NASATM-100108, December 1987.13. Oswald, F.B., "Gear Tooth Stress Measurements on the UH-60A Helicopter Transmission", NASA TP-2698, 1987.5. Mark, W.D., "The Transfer FunctionMethod for Gear System Dynamics Appliedto Conventional and Minimum ExcitationGear Designs", NASA CR-3626, 1982.14. Rebbechi, B., Oswald, F.O., andTownsend, D.P., "Dynamic Measurements ofGear Tooth Friction and Load", NASATM-103281, 1991.6. Boyd, L.S., and Pike, J., "Epicyclic GearDynamics," AIAA Paper 87-2042, June 1986.15. Townsend, D.P., and Bamberger, E.N.,"Surface Fatigue Life of M5ONiL andAIS19310 Spur Gears and R C Bars", NASATM-104496, 1991.7. Choy, F.K., Tu, Y.K., Savage, M., andTownsend, D.P., "Vibration Signature Analysis of Multistage Gear Transmission", Journal of the Franklin Institute, Vol. 328,No. 2/3, 1991, pp. 281-299.16. Boyd, L.S., and Pike, J.A., "Epicyclic GearDynamics", AIAA Journal, Vol. 27, No. 5,May 1989, pp. 603-609.8. David, J.W., Mitchell, L.D., andDaws, J.W., "Using Transfer Matrices forParametric System Forced Response", Journal of Vibration, Acoustics, Stress and Reliability in Design, Vol. 109, No. 4, Oct. 1987,pp. 356-360.17. Choy, F.K., Townsend, D.P., andOswald, F.B., "Experimental and AnalyticalEvaluation of Dynamic Load and Vibrationof a 2240-KW Rotor craft Transmission",Journal of The Franklin Institute, Vol. 326,No. 5, 1989, pp. 721-735.9. Osguven, H.N., and Houser, D.R., "Mathematical Models Used in Gear Dynamics-A7
TABLE I. - TEST GEAR PARAMETERSStandard involute, full-depth toothGear type .Number of teeth .28Module, nun (diametrial pitch in."1 ) .3.174(8)Face width, mm (in.) .6.35(0.25)Pressure angle, deg .201.64Theoretical contact ratio .0.023 (0.0009)Driver modification amount, mm (in.) .0.025(0.0010)Driven modification amount, mm (in.) .24Driver modification start, deg .24Driven modification start, deg .1.35(0.053)Tooth-root radius, mm (in.) .AGMA class 13Gear quality .Nominal (100 percent) torque, N-m(in.-lb) . 71.77(635.25)TABLE 2.-COMPARISON OFEXPERIMENTAL MEASUREDAND ANALYTICAL MODELEDNATURAL 05123361536275230120-4.13.02.62.601.11.7
aki -ww4O--R-dDynamometer- IpiingTettag.increaser,7yxkthStag.zFigure 2.-Picture of gear noise rig.Figure 1.- Geometry of gear force simulation.Figure 3.-Test gearbox.
7 1 -Mode:M od:Frequency, 658.37 Hz2Frequency, 1048.56 HzzzyyxFrequency, 2535.95 HzFrequency, 22759.69 HzzMade:?5Mode: 6Frequency, 2722.16 HzFrequency. 296.71 HzTFigure C.-Gearbox experilmental mode shapes.10
Major modes of housingNO6 6---Gear-meshA-4frequency2C. 07.6000 .2Hzgear-meshfrequency-4500 ----]C 3 0/I251500oo. .a.,.-700-Hz geamesh frequency .,-(a)Experimental.Major modes of ncy450025 W4;000ICI1500s,0500 \1 000 1500 2000 2500 3000 3500 4000700-Hz gearFrequency, Hzmesh freqluency- (b) Analytical.Figure 5,- X-dlrectlon experimental and analytical vibration frequency spectra ofthe gearbox11
Major modes of housingS/.// --"Gest-meahfreqluency20000. .5.C700-Hz gear-mesh frequencyg e a r-m es h-055002567-Hz-oo7,-f(a) ExperEmenta.2MMajor modes of housing0\001030000050405030rmesh frquency2-'c700HZ gear.50002 1ArqecH9' . I E!, L.,Figure 40000\00101500setaofrequencyirto#--Ydm nepden n OA00503050403500Hzga-Fqnczmes-h fereqecFrqunyH(b)Mnalytical.and analytical vibration frequency spectra ofexperimentalFigure 6.- V-directionthe gearbox.
Major modes of housing . 12e0-q90frequencyNeWItk:F- 6000,1 500I/11, -- Ger-es57H," .,.700-H-z gear-mesh frequency-(a) Experimenta,Major modes of housingGaesesE- 2000 efrequency-25W0\15000-r700-Hz gear-FrqecHmesh frequency(axprmetl35F ue . Zdre
the gearbox; (1) experimental modal analysis and representation of the test gearbox using only a (2) dynamic vibration measurements during oper- limited number of modes. ation. In the experimental modal analysis, modal parameters, such as system natural frequencies For the dynamic study of the gearbox vibra-
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Field test: torque and modal characterization. Planet bearing calibration. Trunnion stiffness characterization. Phase 1. Dyno Gearbox 1: run-in and instrumentation checkout. Field Gearbox 1: loads, responses, events. Dyno Gearbox 2: run-in, limited non-torque loading. Phase 2. Dyno Gearbox 2: extensive Non-Torque Loads (NTL), dynamics, HSS .
Experimental Modal Analysis (EMA) modal model, a Finite Element Analysis (FEA) modal model, or a Hybrid modal model consisting of both EMA and FEA modal parameters. EMA mode shapes are obtained from experimental data and FEA mode shapes are obtained from an analytical finite element computer model.
LANDASAN TEORI A. Pengertian Pasar Modal Pengertian Pasar Modal adalah menurut para ahli yang diharapkan dapat menjadi rujukan penulisan sahabat ekoonomi Pengertian Pasar Modal Pasar modal adalah sebuah lembaga keuangan negara yang kegiatannya dalam hal penawaran dan perdagangan efek (surat berharga). Pasar modal bisa diartikan sebuah lembaga .
REDUCTION GEARBOX FINNØY REDUCTION GEARBOX The gearbox range covers input powers from 250 kW and up to 12 000 kW. They are divided into 3 model types: - SINGLE STAGE REDUCTION GEARBOX, with vertical or horizontal offset - GXU REDUCTION GEARBOX - TWIN
Harmonic gearbox have smaller backlash but a very high price, while a planetary gearbox have a higher load carrying capacity, a better coefficient of efficiency and a significantly lower price, all for the same overall dimensions. Because of this, it would be reasonable to use a planetary gearbox instead of a harmonic gearbox. Figure 5.
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