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Nonlinear dynamics of a planetary gear systemwith multiple clearancesTao Suna,*, HaiYan HubaDepartment of Precision Mechanical Engineering, Shanghai University, Shanghai 200072, PR ChinabInstitute of Vibration Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR ChinaReceived 19 April 2002; received in revised form 12 March 2003; accepted 16 May 2003AbstractPresented in this paper is on the nonlinear dynamics of a planetary gear system with multiple clearancestaken into account. A lateraltorsional coupled model is established with multiple backlashes, time-varyingmesh stiffness, error excitation and sun-gear shaft compliance considered. The solutions are determined byusing harmonic balance method from the equations in matrix form. The theoretical results from HBM areverified by using the numerical integration. Finally, effects of parameters are discussed.? 2003 Elsevier Ltd. All rights reserved.Keywords: Planetary gear transmission; Nonlinear vibration; Fourier transform; Harmonic balance method; Dynamic1. IntroductionPlanetary gear systems have been widely used in engineering owing to their advantages such aslittle space required, large ratio of transmission and high efficiency. One of the most popularapplications is to the automatic transmissions in automobiles.Because of machinery complexity, most of earlier studies on the planetary gear systems wereconfined to their static behaviors and sharing characteristics. Over the past two decades, thedynamics of planetary transmissions has drawn much attention. However, almost all the pub-lished studies on the planetary transmissions focused only on their linear vibration 1,2.In the advanced mechanical systems running at high speed, such as exact antennas and auto-matic weapon systems, which usually contain a number of planetary gear sets, the gear systems*Corresponding author. Fax: +86-21-56334458.E-mail address: tsun (T. Sun).0094-114X/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0094-114X(03)00093-4Mechanism and Machine Theory 38 (2003) 13711390/locate/mechmtoften undergo startup and brake interactively or run at high speed and under light load. Thecurrent studies have shown that a gear pair would likely lose contact and the tooth separationsNomenclaturebbacklashC, cdampingestatic transmission errorfnonlinear displacement functionIrotary inertiaK, kstiffnessM, mmassNdescribing functionP, pforceqdisplacementrradiusttimeTtorqueX, xdisplacementn, gtransverse displacementUangleapressure anglesdimensionless timeX, xfrequencyuphase angleSubscriptsaalternating componentsbbase circleccarrierdrelative tohhigh-sped partiordinal numberllow-speed partmmean componentpplanet gearrring gearssun gearnhorizontalgvertical1372T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 13711390occur due to the unavoidable backlash. Accordingly, the backlash, namely the clearance, tends tobring gear systems to exhibit typical nonlinear dynamical behaviors.A gear pair is bound to have some backlash, which may be either designed to provide betterlubrication and to eliminate interference or due to manufacturing errors and wear. Backlash-induced nonlinear vibrations may cause tooth separation and impacts in unload or lightly loadedgeared drives. Such impacts result in intense vibration and noise problems and large dynamicloads, which may affect reliability and life of the gear drive.Experimental studies on the dynamic behavior of a spur gear pair with backlash started almost40 years ago and still continue 35. For instance, Kubo et al. 4 observed a jump in the fre-quency response of a gear pair with backlash even though the test set-up was heavily damped.Such experimental studies, albeit limited in scope, have clearly shown that the dynamics of agear pair can hardly be predicted on the basis of a linear model. Consequently, the nonlineardynamics and mathematical models of a gear pair with backlash have been intensively studied inthe past decade. Although most of the nonlinear mathematical models used to describe the dy-namic behavior of a gear pair are somewhat similar to each other, they differ in terms of theexcitation mechanisms considered and especially the solution technique used.The nonlinearity of a gear backlash has to be modeled by a discontinuous and nondifferentiablefunction, which represents a strong nonlinear interaction in the dynamic equation of whole sys-tem. Comparin and Singh addressed this problem in 6 and pointed out that most techniquesavailable in the literatures cannot be directly applied to solving this problem. Many researchershave recognized this problem implicitly and therefore employed either digital or analog simula-tion techniques in their studies. Kahraman and Singh 7 made a detailed review of nonlineargear dynamics available in current publications. Their theoretical study also made contributionsto the nonlinear dynamics of a spur gear pair with backlash subject to the static transmissionerror.Although there is a vast body of literature concerned with nonlinear dynamics of a general gearpair with clearance, the studies on nonlinear dynamics of a planetary transmission system withmultiple clearances are still very limited. Kahraman took the possibility of tooth separation in aplanetary gear system into account 2. In his study, however, the model was not considered as anonlinear dynamic system. Instead, a step function in linear model was used to distinguish thetooth contact and the tooth separation roughly. As a result, the back collisions of teeth were notincluded in his study.Compared with above-mentioned works, the nonlinear dynamics of a planetary gear system ismuch more complicated. Such a system is inherently nonlinear owing to the multiple clearances,and includes the temporally and spatially varying system parameters. Moreover, the request forprediction and examination of dynamic behaviors is progressively urgent in the design of morequiet and reliable planetary transmissions. The published studies on the nonlinear dynamics ofgeared drive with backlash, however, focus on only a single gear pair, rather than any planetarygear systems. Accordingly, the focus of this paper is on the nonlinear dynamics of planetary gearsystems. In this paper, some contributions will be made to a number of key issues such as non-linear dynamics modeling, solution techniques to the nonlinear differential equations and dynamicbehaviors of planetary gear systems.T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 1371139013732. Dynamic model of system2.1. Model and assumptionsThe planetary gear system of concern in this study is a single-stage 2K-H type planetary gear setas shown in Fig. 1. The system consists of a high-speed part h, a low-speed part l, a sun gear s, aring gear r, n planet gears p and a carrier c. Here, n, the number of planet gears, is taken as 3throughout the paper. All the gears are mounted on their flexible shafts supported by rollingelement bearings.To establish the mathematical model of system, a few assumptions are introduced in the case ofspeed reduction in the planetary gear shown in Fig. 1 as following.(1) The inertial effects of prime mover (high-speed part) and load inertia (low-speed part) aretaken as those of lumped mass. Hence, the planetary gear system has 4 n rotational degreesof freedom (DOF), including the rotational displacements of low-speed part, high-speed part,sun gear, carrier and n planetary gears, respectively.(2) Considering the bending stiffness of the shaft of sun gear, the compliance of bearings and po-tential displacement as a rigid body caused by floating, the horizontal and vertical transverseDOF of the sun gear are included.(3) As the bending stiffness of the planet gear shaft is very large, the deflection of this shaft can beneglected. Thus, the transverse displacement of planet gear is not considered.(4) Because the ring gear itself is a part of gearbox, the displacement of ring gear as a rigid body isinsignificant and the center of ring gear is assumed not to move.Based on the above assumptions, the dynamic model of the planetary gear system is establishedas shown in Fig. 2. All symbols in Fig. 2 can be found in the nomenclature presented and furtherexplanations are given in subsequent sections. The total number of DOF in the model is n 4 2,rsclhpFig. 1. A planetary gear set.1374T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 13711390including n 4 rotational DOF and 2 transverse DOF, respectively. Obviously, this is a com-plicated lateraltorsional coupled nonlinear system with multiple clearances and time-varyingparameters.2.2. Equivalent displacementsIn the case of speed reduction, the meshing relation of the planetary gears is shown in Fig. 3.Regarding the angular displacement of each gear in driving, the direction of the revolution causedby the driving torque is assumed to be positive. Namely, the angular displacements hsand hcofthe sun gear and the carrier are in the same direction. And the angular displacements hrand hpofthe ring gear and the planet gear are reversed in the directions of hsand hc.In order to establish the equations of motion easily, both torsional and transverse displace-ments are unified on the pressure line in terms of equivalent displacements.(b)(a)Fig. 2. The dynamic model of a planetary gear set.(b)line of actionRing gearPlanet gearSun gear(a)Fig. 3. Mesh relation in a planetary gear set.T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 137113901375The equivalent transverse displacements in the pressure line direction caused by rotationaldisplacements are written as follows:xh rbshh;xs rbshs;xr rbrhr;xpi rbphpi;xc rbchc;xl rbchl;i 1;2;3;1where rb(subscripts ?s?, ?r?, ?p?) are the base circle radius of gears, while rbcis the nominal basecircle radius of the carrier defined as following:rbc rbs rbp rbr? rbp:2With regard to the direction of the equivalent displacements, the direction of deflection caused bydriving torque is assumed to be positive.It should be pointed out that in the case of speed reduction, the rotational displacement of thering gear r is set to be zero since the ring gear is fixed on gearbox in the 2K-H type planetarygear set shown in Fig. 1. That is, hr 0. However, the tooth deflection of the ring gear is includedwhen calculating the mesh stiffness.By using X to represent the relative displacements in the direction of press line, the relativedisplacements are obtained as follows according to the meshing relation shown in Fig. 3 and theequivalent displacementsXhs xh? xs;Xspi xs? xpi? xc;Xrpi xpi? xc? xr xpi? xc;Xcl xc? xli 1;2;3:9=;3To describe the plane motion of the sun gear, a kinetic Cartesian coordinate system attached to thecarrier, instead of general fixed coordinate system, is introduced. The origin-o of this coordinatesystem coincides with the center of carrier. And the coordinate axes marked as n;g rotate alongwith the carrier. Therefore, nsand gsrepresent the horizontal and vertical transverse displacementsof sun gear with regard to the movable reference frame respectively. As the variation of pressureangle caused by the tiny translation of sun gear can be neglected, the equivalent displacement in thepressure line direction derived from the transverse displacements nsand gsis represented asxsdi ?nssin/i? a gscos/i? a:4Here a is the pressure angle of gear pair, and /imeans that the ith planet gear is mounted at atheoretical position angle /ion the carrier with respect to positive direction of axis-n. If the centerof the first planet gear is assigned to positioned at angle 0, /iwill be/i 2pi ? 1=n;i 1;2;.;n;5where n represents the numbers of planet gears.3. Equations of motionIn the dynamical model shown in Fig. 2, the gear mesh is described by a nonlinear displacementfunction f. Here, f is defined as a nonlinear function in the relative gear mesh displacement ? q q andwith backlash 2?b b as parameter1376T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 13711390f? q q ? q q ?b b;? q q ?b b;0;?b b6? q q6?b b;? q q ?b b;? q q ?b b:8=;9whereMhIhr2bs;MsIsr2bs;MpiIpir2bp;MlIlr2bc;McIcr2bc 3mpcos2a;khsKhsr2bs;kclKclr2bc;PhThrbs;PlTlrbc:Here, the subscripts ?h?, ?s?, pi, ?c? and ?l? represent the high-speed part, the sun gear, the planetgear, the carrier and the low-speed part, respectively. I represents the inertia, m the actual mass, Mthe equivalent mass, C the damping coefficient, Thand Tlthe input and output torque respectively,Phand Plthe equivalent force, Khsand Kclthe torsional stiffness, khsand kclthe equivalent stiffnessin the direction of line of action, knand kgthe bearing stiffness.Eq. (9) features considerable difficulties in its solving process as follows. (1) It is a semi-definitesystem, which predicates prospective trivial solutions corresponding to rigid body motions. (2)The function f is nonlinear multivariate, and the number of variables is even different accordingto the external and internal gear pairs. (3) As both linear and nonlinear restoring forces exist in theequation, it is not possible to write out the governing equation in matrix form, while a generalsolution technique applicable to the systems of multiple degrees of freedom must be based onmatrix form.Therefore, Eq. (9) is simplified further by using a set of new variablesXhs xh? xs;Xsdi xs? xpi? xc? nssin/i? a gscos/i? a ? e espi;Xn ns;Xg gs;Xrdi xpi? xc? e erpi;Xcl xc? xli 1;2;3:9=;10The new coordinate variables defined in above not only have intuitional physical meaning, butalso eliminate the rigid body motions. Furthermore, f can be written as a set of functions in singlevariable in terms of variables given in Eq. (10). Hence, the set of simplified governing equationsis obtained by combining Eqs. (9) and (10)1378T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 13711390MhsX Xhs Chs_X Xhs?MhsMsX3i1Cspi_X Xsdi khsXhs?MhsMsX3i1kspi? fXsdi MhsMhPh;MsdiX Xsdi?MsdiMsChs_X XhsMsdiMpCspi_X XsdiX3i1MsdiMs?MsdiMcMsdims?Cspi_X Xsdi?MsdiMcCcl_X Xcl?sin/i? aMsdimsCn_n nscos/i? aMsdimsCg_ g gs?MsdiMpCrpi_X XrdiMsdiMcX3i1Crpi_X Xrdi?MsdiMskhsXhsMsdiMpkspi? fXsdi X3i1MsdiMs?MsdiMcMsdims?kspi? fXsdi?MsdiMckclXcl?sin/i? aMsdimskn_n nscos/i? aMsdimskggs?MsdiMpkrpifXrdiMsdiMcX3i1krpifXrdi ?Msdi? e e ? e espii 1;2;3;msX Xn Cn_X Xn?X3i1Cspi_X Xsdisin/i? a knXn?X3i1kspi? fXsdi ? sin/i? a 0;msX Xg Cg_X XgX3i1Cspi_X Xsdicos/i? a kgXgX3i1kspi? fXsdi ? cos/i? a 0;MrdiX Xrdi?MrdiMpCspi_X XsdiX3i1MrdiMcCspi_X XsdiMrdiMpCrpi_X XrdiX3i1MrdiMcCrpi_X Xrdi?MrdiMcCcl_X Xcl?MrdiMpkspifXsdi X3i1MrdiMckspi? fXsdi MrdiMpkrpifXrdi ?MrdiMckclXclX3i1MrdiMckrpifXrdi ?Mrdi? e e ? e erpii 1;2;3;MclX Xcl?MclMcX3i1Cspi_X Xsdi Ccl_X Xcl?MclMcX3i1Crpi_X Xrdi?MclMcX3i1kspi? fXsdi kclXcl?MclMcX3i1krpifXrdi MclMlP;11whereMhsMsMhMs Mh;MrdiMpMcMp Mc;MclMcMlMc Ml;MsdiMsMpiMcmsMpiMcms MsMcms MsMpims MpiMcMs:T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 137113901379If a displacement vector ? q q is introduced as following:? q q Xhs;Xsd1;Xsd2;Xsd3;ns;gs;Xrd1;Xrd2;Xrd3;XclfgT:12The equation of motion is given in the matrix form asM? q q ? q q C_? q q ? q q K?f f ? q q ? p p;13where the mass matrix M readsM diag Mhs;Msd1;Msd2;Msd3;ms;ms;Mrd1;Mrd2;Mrd3;Mcl?;14f ? q q is the vector of nonlinear functions in single variable with uniform shape in terms of thecoordinates transformed in Eq. (10)fi f? q qi ? q qi?b bi;? q qi?b bi;0;?b bi6? q qi6?b bi;? q qi?b bi;? q qi ?b bi;8 bi;0;?bi6qi6bi;qi bi;qi ?bi;8 1;?20cHl ?1;l 1;8:20dl? ?bi? qmi=qai:20eT. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 137113901381For i 1, 5, 6, 10, we havebi 0:20fSubstituting Eq. (20f) into Eqs. (20a)(20e) yieldsNmi 1;Nai 1:?20gThis result agrees with the physical fact that describing function of linear function is really 1.(4) Algebraic equations: Considering the mean value and the fundamental harmonics of time-varying periodic mesh stiffness in Fig. 4, the entries in stiffness matrix K in Eq. (18) can bewritten askij kmij kaijcosXs /kij:21aTo get the mean value and the harmonics of the time-varying mesh stiffness, the FFT algo-rithm is introduced to the periodic function of mesh stiffness.Hence, K is written in terms of two separate matrices for mean stiffness and alternating stiffnessK Km DK;21bwhereKm kmij?n?n;DK kaijcosXs /ij?n?n:21cEqs. (19a)(19c) can be written in the following matrix formp fpmign?1 fpaicosXs upign?1;22aq fqmign?1 fqaicosXs uign?1;22bf q fNmiqmign?1 fNaiqaicosXs uign?1:22cBy substituting Eqs. (19a)(19c) into Eq. (18) and balancing the same harmonics, one obtains thealgebraic equations of systemKmym12K1y3 K2y4 ? pm f0gn?1;Kmy3 K1ym? X2My1? XCy2? p1 f0gn?1;Kmy4 K2ym? X2My2? XCy1? p2 f0gn?1;8:23wherey1 fqaicosuign?1;y2 fqaisinuign?1;y3 fNaiqaicosuign?1;y4 fNaiqaisinuign?1;ym fNmiqmign?1;p1 fpaicosupign?1;p2 fpaisinupign?1;K1 fkaijcos/ijgn?n;K2 kaijsin/ij?n?n:24Eq. (23) includes following 3n unknown variables to be solvedqai; qmi; ui;i 1;2;.;n:25As it is impossible to solve the nonlinear algebraic equation (23) by any analytical methods, anumerical routine (DNEQBJ of IMSL 8) is used to determine the solution. The routine1382T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 13711390DNEQBJ uses a secant method with Broyden?s update to find the zeros of a set of nonlinearalgebraic equations.4.2. Solutions and validationAs the first example, a 2K-H type planetary gear system shown in Fig. 1 is studied. The geo-metric parameters of system are listed in Table 1 and other parameters are given in Table 2.Furthermore, the characteristic length is set to bc 0:01 mm. The excitations are associated withthe mesh stiffness fluctuation and errors. The frequency responses of system are determined byusing the HBM in the previous subsection. As a comparison, the frequency responses are shown inFig. 5(a)(e), together with the responses of corresponding linear systems without any backlash.These figures demonstrate that a planetary gear system with clearances taken into accountexhibits particular behaviors as following.(1) For all the response results shown in Fig. 5(a)(e), except for rigid and reduplicate frequen-cies, the system undergoes resonance at five different frequencies, all of which are near the nat-ural frequencies of corresponding linear system.(2) Similar to the studies on single gear pair by Kahraman and Singh 5, the dynamics of theplanetary gear system with clearances exhibits strong nonlinearity. As shown in Fig. 5(a)and (b), both regimes of no impacting (no tooth separation) and single-sided impacting (toothTable 1Parameters of a planetary gear setSun gearPlanet gearRing gearNumber of teeth152463Module (mm)5Pressure angle20?20?20?Amount of crowning (mm)0.38350.1610.7056Face width (mm)141414Load (Nm)505050Table 2Parameters in case studyParameterUnitValueParameterUnitValuemsKg0.417IlKgmm21980mpKg0.254ksN/m4.24e8IhKgmm2202KclNmm/Rad4.95e8IpKgmm2140KhsNmm/Rad4.58e8IcKgmm26000ksp1m ksp2m ksp3mN/m2.42e8esp1 esp2 esp3lm20ksp1a ksp2a ksp3aN/m0.43e8erp1 erp2 erp3lm20krp1m krp2m krp3mN/m2.91e8bsp1 bsp2 bsp3lm50ksp1m ksp2m ksp3mN/m0.47e8brp1 brp2 brp3lm50Damping ratio0.02T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 1371139013830.00.81.01.82.00.000.010.020.030.040.05Amplitude(a) (b) 1.82.001234AmplitudeFrequencyFrequency0.00.81.01.82.00.000.020.040.060.080.100.12AmplitudeFrequency(d) 1.82.00.51.01.52.02.53.03.54.0AmplitudeFrequency(c) Frequency(e) 0.00.20.4 1.82.00.000.020.040.060.08AmplitudeFig. 5. Frequency response of a planetary gear set (?) linear system; () nonlinear system: (a) frequency responseof Xhs, (b) frequency response of Xsd1, (c) frequency response of Xrd1, (d) frequency response of Xsand (e) frequencyresponse of Xcl.1384T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 13711390separation, but no back collision) exist in the frequency response. One can detect the dual-val-ued solutions and jump discontinuities near resonant frequencies. These phenomena predictinherent characters of nonlinearity.(3) As the motions of a MDOF system at different degrees of freedom are coupled each other,sudden discontinuities are also observed in Fig. 5(c) and (d), although there are no clearancesfor these parts. This indicates the interactions between gear pairs and other components.(4) Not all the resonant frequencies of system are dangerous for the nonlinear vibration. Thejump discontinuities of frequency response occur only near the resonance sensitive to vibra-tion of gear pairs.Next, we verify the solutions of HBM through the numerical simulations, where a variable-stepRungeKutta algorithm of fifthsixth order in 9 is used. The frequency responses obtained byusing HBM and RungeKutta algorithm are compared in Fig. 6. The two methods get anagreement although the response amplitudes at resonant frequencies are slightly different. Dis-crepancies may come from the assumption of single harmonics in HBM.Though the validity of HBM is confirmed, one should be aware of the following as alsomentioned by Kahraman in 5. (1) Several problems may appear in the implementation of nu-merical integration to the nonlinear dynamics due to clearances and caution must be exercised 6even though it is more precise. (2) The HBM is incapable of predicting any quasi-periodic orchaotic vibrations.5. Parametric studiesThe dynamic behaviors of a planetary gear set, such as the frequency response, the transitionfrequency and the existence of various impacting regimes, depend on the parameters of system tosome extent. Therefore, parametric studies are presented here.0.81.01.82.00123Alternating amplitudeFrequencyFig. 6. Comparison of frequency response Xsd1obtained by different methods: () numerical integration; () harmonicbalance method.T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 137113901385We first examine the effect of mesh stiffness. As well known, the mesh stiffness of a gear pair is atime-variant periodic function, which can be represented askt km kasinXs u:26To describe the extent of variation, a stiffness ratio is introduced ask k kakm:27Three different results of Xrd1are obtained by varyingk k only for krp1, the mesh stiffness between thering gear and planet gear-1, while other parameters remain unchanged as shown in Table 2. Fig. 7shows the jump discontinuities in all three cases no matter how largek k is taken. Whenk k is in-creased, the transition frequencies of jump become lower, the transition frequencies of jumping-upand jumping-down leave each other. This gives rise to the dual-valued solution region. Thecomparison indicates that although the existence of typical nonlinear dynamic behaviors ofa planetary gear system with clearances does not depend on the variation of stiffness, the extentof nonlinearity is affected really by the stiffness ratio. An increase of variation of stiffness ratioenhances the nonlinearity of response.Next, we examine the effect of static transmission error e, which is the excitation on theplanetary gear system in this study. Three case studies are shown in Fig. 8 for dimensionless ebeing 0.2, 2 and 4, respectively. When e is very small, say, e 0:2, the system dynamics is linearand the tooth pairs do not lose contact all the time although the system has clearances. When e isincreased to 2, the dynamics of system becomes nonlinear. The single-sided impact vibration andthe jump discontinuities are observed. When e is increased further to 4, the nonlinear behaviorsget more complicated. The double-sided impact (back collision) vibration are also observed nearthe resonant frequency, X 1:58. Moreover, as e is increased, the amplitudes of response near theresonance become larger. This is reasonable since e plays a role of alternating components in theexcitation on planetary gear system.1.82.00.00.51.01.52.02.53.03.5Alternating amplitudeFrequencyFig. 7. Frequency response of Xrd1for different mesh stiffness: (?)k k 0; (-)k k 0:2; ()k k 0:4.1386T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 137113906. ConclusionsIn this study, some advances are made in the nonlinear dynamics of a general planetary gearsystem with backlash and time-varying mesh stiffness, excited by the static transmission errors.First, a planetary gear system with multiple clearances taken into consideration is studied fromthe viewpoint of nonlinear dynamics. Second, a lateraltorsional coupled nonlinear dynamicmodel for the planetary gear set is developed and the governing equation is formulated in matrixform. Third, the frequency response of the planetary gear set under a harmonic excitation isdetermined by using the HBM and verified by using numerical simulation. Finally, the effects ofsome important parameters, such as the variation of mesh stiffness and static transmission errors,on the nonlinear dynamics are discussed. The study enriches the current literature in the nonlineardynamics of mechanical systems with multiple clearances and the dynamics of planetary gearsystems. The results of this study yield some guidelines to be instrumental to improvement of theplanetary transmission performance.The ongoing research work focuses on the load sharing characteristics of planetary transmis-sions with multiple clearances since the advantages of a planetary gear system greatly rely on thesharing characteristics. Future work will include the analysis of higher order harmonics ofplanetary gear systems with multiple clearances, meshing phase difference and unequal backlash.AcknowledgementWe thank National Natural Science Foundation of China for supporting this study.Appendix A. Entries in matrices K, C, pBoth K and C are 1010 square matrices, while ? p p is a 101 vector.1.82.00.00.51.01.52.02.53.03.54.04.55.0AmplitudeFrequencyFig. 8. Frequency response of Xsd1for different errors: (?) e 0:2; (-) e 2:0; () e 4:0.T. Sun, H.Y. Hu / Mechanism and Machine Theory 38 (2003) 137113901387(1) The entries in the stiffness matrix K are given as follows:?k k1;i khs;?k k1;i ?MhsMskspi;i 1;2;3;?k k2;i ?Msd1Mskhs;?k k2;2 ksp1;?k k2;i1Msd1Ms?Msd1McMsd1ms?kspi;i 2;3;?k k2;5 ?sin/1? aMsd1mskn;?k k2;6 cos/1? aMsd1mskg;?k k2;7Msd1Mc?Msd1Mp?krp1;?k k2;i6Msd1Mckrpi;i 2;3;?k k2;10 ?Msd1Mckcl;?k k3;1 ?Msd2Mskhs;?k k3;i1Msd2Ms?Msd2McMsd2ms?kspi;i 1;3;?k k3;3 ksp2;?k k3;5 ?sin/2? aMsd2mskn;?k k3;6 cos/2? aMsd2mskg;?k k3;i6Msd2Mckrpi;i 1;3;?k k3;8Msd2Mc?Msd2Mp?krp2;?k k3;10 ?Msd2Mckcl;?k k4;1 ?Msd3Mskhs;?k k4;i1Msd3Ms?Msd3McMsd3ms?kspi;i 1;2;?k k4;4 ksp3;?k k4;5 ?sin/3? aMsd3mskn;?k k4;6 cos/3? aMsd3mskg;?k k4;i6Msd3Mckrpi;i 1;2;?k k4;9Msd3Mc?Msd3Mp?krp3;?k k4;10 ?Msd3Mckcl;?k k5;i1 ?sin/i? akspi;i 1;2;3;?k k5;5 kn;?k k6;i1 cos/i? akspi;i 1;2;3;?k k6;6 kg;?k k7;2Mrd1Mc?Mrd1Mp?ksp1;?k k7;i1Mrd1Mckspi;i 2;3;?k k7;7 krp1;?k k7;i6Mrd1Mckrpi;i 2;3;?k k7;10 ?Mrd1Mckcl;?k k8;i1Mrd2Mckspi;i 1;3;?k k8;3Mrd2Mc?Mrd2Mp?ksp2;?k k8;8 krp2;?k k8;i6Mrd2Mckrpi;i 1;3;?k k8;10 ?Mrd1Mckcl;?k k9;i1Mrd3Mckspi;i 1;2;?k k9;4Mrd3Mc
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