19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007 INVESTIGATION ON THE DYNAMIC BEHAVIOR OF A COMPOUND PLANETARY GEAR DRIVE

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1 9 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, -7 SEPTEMBER 007 INVESTIGATION ON THE DYNAMIC BEHAVIOR OF A COMPOUND PLANETARY GEAR DRIVE PACS: r Dhouib, Seddik ; Hbaieb, Riadh ; Chaari, Fakher 3 ; Haddar, Mohamed 4 National Shool of Engineers of Sfax; BP.W Sfax TUNISIA; dhouibseddik@yahoo.fr National Shool of Engineers of Sfax; BP.W Sfax TUNISIA; riadh_hbaieb@yahoo.fr 3 National Shool of Engineers of Sfax; BP.W Sfax TUNISIA; fakher.haari@gmail.om 4 National Shool of Engineers of Sfax; BP.W Sfax TUNISIA; mohamed.haddar@enis.rnu.tn ABSTRACT A plane model of a ompound planetary gear train is proposed to predit the dynami response of suh transmission set whih is widely used in automati vehile drive transmissions. Translational and rotational motions of the mounted gears are onsidered. The energeti Lagrange formulation is used to reover the equations of motion and the modal harateristis of the system. Profile error is modelled and taken into aount in the equation of motion. Simulations were done with a numerial example for the ases of a healthy system and defeted system. An amplitude modulation of the gearmesh signal by the defet signals ausing higher vibration levels is observed. INTRODUCTION Compound planetary gear trains CPGT are used in various transmissions suh as automobiles, aerospae appliations and heavy industry due to their substantial partiularities of ompatness, large torque-to-weight ratio and weak bearing loads. The ability to obtain different speed and torque ratios from the same CPGT system by simply hanging the assignment to input, output and stationary members makes CPGT very derisible for passenger vehile automati transmissions. In the other hand, as any gear transmission set, CPGT noise and vibration remain a problem in their appliations, whih leads researhers to fous on their ontrol in order to ensure safety and omfort. In ertain kinematis onfigurations, Gott [] used two single-planet single-stage planetary gear sets to obtain three or four forward speed ratios. However, for inreasing the performane of any mehanism suh as power density, demands on fuel eonomy and speed arrangement, using of only one CPGT are beoming quite ommon in high volume automati transmission appliations []. Major gear noise and vibration problems of the transmission unit an be eliminated, if a designer an evaluate the free vibration harateristis. The modelling of the vibratory behaviour of single stage planetary gear trains was widely treated in literature, [3]. In their study of he same system, Lin et al. [4] reovered three types of modes: translational, rotational and planet modes. The free vibration of helial planetary gears was studied by Hbaieb et al. [5]. But there is few published work available on predition of the free vibration harateristis of CPGT sets. The existing formulations employed torsional models, whih an only onsider gears defletions [6]. The movement of the bearings is not onsidered and the defets are ignored. The main objetive of this study is to develop a lumped parameters dynami model of a CPGT whih onsider the defletion of the teeth and the bearings. A profile error is modelled on sun gear. The natural and vibration modes of the CPGT are reovered and the dynami response is investigated for the healthy ase and for the profile error ase. DYNAMIC MODELLING OF A COMPOUND PLANETARY GEAR A most ommon example of CPGT alled Ravigneaux gear set [7,8] is illustrated in Figure. Here two single planet units, s-a-r and s-a-b-r are onneted to eah other though the mesh of the long planet-a and the short planet-b. s is the sun gear, r and r are the ring

2 gears. All planets are supported by a single arrier. This onfiguration offers 4 power flow onfigurations r b s a r r s r b a a a3 b3 b Figure.-Ravigneaux gear set To model this ompound planetary gear, a lumped parameters onfiguration is adopted as presented in Figure [9]. Sun (s), ring- (r), ring- (r), arrier (), N planets (a) and N planets (b), where N indiates the number of planets, are onsidered as rigid bodies with moments of inertia J s, J r, J r, J, J a, J b and having the masses m s, m r, m r, m, m a, m b. The translational and rotational motions of the arrier, ring-, ring-, sun, and planets are denoted u i, v i and θ i, i =, r, r, s, a,,an, b,,bn. These oordinates are measured with respet to a frame ( o, s, t,z ) fixed to the arrier and rotating with a onstant angular speed Ω. The rotations θ i are replaed by their orresponding translational gear mesh displaements as: so that a displaement vetor q i is defined as: q [ ] T i =u,v,ρ i i iz, where Rb i is the base irle radius for the sun, ring-, ring-, planet-ai, planet-bi, and the radius of the irle passing through planet entres for the arrier. Cirumferential planet-ai loations are speified by the fixed angles α ai and that of planet-ib are α ib whih are measured relative to the rotating basis vetor s so that α a =0 and α b =0. v, v r v r, v s t k ry k r k xx k ab k yy Planet-b Planet-a θ s k r k yy k xx α b α a Sun k s k sy k sx k sθ k θ k rθ θ Planet-b k s v a α ab v b k xx k yy θ a k r Planet-a Carrier k yy k ab k rθ k xx k r α b θ b θ r u, u r, u r u s, u a k rx s θ r u b s Ω Ring- Ring- Figure.- The ompound planetary gear model 9 th INTERNATIONAL CONGRESS ON ACOUSTICS ICA007MADRID

3 The bearings are modelled by linear springs with onstant stiffnesses k ix, k iy, i =, r, r, s and k xx, k yy for the planets. All tooth meshes (sun-planet-a, planet-a-planet-b, planet-a-ring- and planet-b-ring-.) are modelled as linear springs with time-varying stiffnesses k si (t), k abi (t), k ri (t), k ri (t), i =,..., N. Damping is not onsidered here; nevertheless, it an be introdued in parallel with gear mesh and bearing stiffnesses. Applying Lagrange formulation allows us to reover the equations of motion of eah omponent of the system. Assembling these equations in a matrix form and negleting the gyrosopi effet leads to the global equation of motion: M q + Ω G q +K() t q = T. (Eq. ) Kt () K + K (t)- Ω K. (Eq. ) = p e Ω Where q is the displaement vetor, T represents the external torques, M the mass matrix, and K P, K and K e(t) are respetively the bearing stiffness, entripetal stiffness and gearmesh Ω stiffness. MODELLING OF PROFILE ERROR A profile error is introdued on the sun gear. This manufaturing error an be modelled by onsidering an exiting funtion to be added to expression of the displaement along the lines of ations [3]. For a sun gear profile error it is expressed by : sn () = n si os( ) E t e π i f t (Eq. 3) i= Where f e is the gearmesh frequeny and esi is the amplitude of the error. This error will indue exiting fores and amplitude modulation between the gearmesh frequeny and the defet signals frequeny (the two at fe and harmonis). NUMERICAL RESULTS A numerial example for a CPGT with a fixed ring and 4 pairs of planets is presented in Table I. Eigenfrequenies are omputed and presented in Table II. A torque of 00 Nm is applied on sun gear. Table I.- Parameters of the ompound planetary gear e Sun Ring- Ring- Carrier Planet Teeth number Mass (Kg) J/ Rb i (Kg) I/ Rb i (Kg) Base radii Rb i (m) Module (mm) Gearmesh stiffnesses (N/m) k =k =k =k 8 =.0 Bearings stiffnesses (N/m) Torsional stiffnesses (N/m) Pressure angle α = 0 sp rp rp abp k = k = 0, j =, r, s, k = 0, j =,r, s 8 9 jx jy jz k =k =k = 0, k = k = 0, k = rx ry rz xx yy zz 9 k jϕ = k jψ = 0, j =,s,,...,n, k θ = 0, k sθ = 0, k rθ =k rϕ = k rψ = 0 k = 0, i =,...,N iθ 9 th INTERNATIONAL CONGRESS ON ACOUSTICS ICA007MADRID 3

4 Table II.- Parameters of the ompound planetary gear Eigenfrequenies (Hz) 0 ; 37 ; 705 ; 4 ; 89 ; 3479 ; 379 ; 3760 ; 377 ; 3783 ; ; 7008 ; 7637 ; 8500 ; 967 ; 998 ; 857 ; 88 ; 587 ; ;9994 ;0869 Healthy ase Figure 3 represents the spetrum of the displaement of sun bearing for a gearmesh frequeny fe =000 Hz. 7 x fe Amplitude ( m ) fe Frequeny / Gearmesh frequeny Figure 3.-Spetrum of the sun bearing displaement for the healthy ase and a gearmesh frequeny fe = 000 Hz It is notied the dominane of the gearmesh frequeny and its first harmoni fe. Figure 4 represents the dynami fators omputed for a set of gearmesh stiffness. The dynami fator is defined as the ration dynami load over stati load. It is represented here for the different meshes of the system..9 Dynami fator f 3 f 8 / f 6 f 8 f 4 / f f 4 f 6 f 8 (A) (B) (C) (D) Gearmesh frequeny ( Hz ) x 0 4 Figure 4.-Dynami fator for the healthy ase -A- Sun-planet-a -B- planet-a planet-b -C- planet-a -ring-a -D- planet-b -ring-b It is well observed the variation of the dynami fator for eah operating mesh frequeny. This dynami fator reahes the value of.8 for the sun-planet-a mesh. Some peaks are observed for some gearmesh frequenies near some eigenfrequenies. These peaks our at the same frequenies for the four ases of meshes. 9 th INTERNATIONAL CONGRESS ON ACOUSTICS ICA007MADRID 4

5 Profile error A profile error is introdued on the sun teeth with an amplitude of 40µm. Figure 5 represents the spetrum on the sun bearing for the same gearmesh frequeny fe = 000 Hz.5 x 0-5 fe Amplitude ( m ).5 fe Frequeny / Gearmesh frequeny Figure 5.-Spetrum of the sun bearing displaement for the profile error ase and a gearmesh frequeny fe = 000 Hz The vibration level inreased when the profile error is introdued espeially for the frequenies fe and fe. This phenomenon is indued by the amplitude modulation of the gearmesh signal by the defet signal (the two signal have the same frequeny harateristis whih are fe and the harmonis). Figure 6 shows the variation of the dynami fator for the sun-planet-a mesh. With respet of the gearmesh frequeny f 6 B A Dynami fator f 6 f 6 / Gearmesh frequeny ( Hz ) Figure 6.-Dynami fator for the sun-planet-a mesh -A- Healthy ase -B- with profile error It is notied that the dynami fator highly inreased ompared to that of the healthy ase Espeially at the proximity of some eigenfrequenies CONCLUSIONS In this paper a dynami model of a ompound planetary gear is presented. The equations of motion are omputed taking into aount the prinipal exitation soure whih is the gearmesh stiffness. This stiffness is time varying and a phasing is onsidered for the different stiffness. A 9 th INTERNATIONAL CONGRESS ON ACOUSTICS ICA007MADRID 5

6 profile error is modelled on sun teeth with an error funtion having the same frequeny to that of the gearmesh stiffness. Simulations showed the dominane of the gearmesh frequeny and its harmonis with inreasing dynami fator around some eigenfrequenies. The vibration level and the dynami fator inreased when the profile error was introdued. Referenes [] G. Gott, Changing Gears: The Development of the Automati Transmission, Soiety of Automotive Engineers, 99. [] J.R. Deniels, New Generation Drivetrains. The Eonomist Intelligene Unit, 995. [3] T. Hidaka, Y.Terauhi, M. Fuji, Analysis of dynami tooth load on planetary gear, Bulletin of the JSME, vol. 3, p , 980. [4] J. Lin, R.G. Parker, Analytial haraterization of the unique properties of planetary gear free vibration, Journal of Vibration and Aoustis, vol., pp. 36 3, 999. [5] R. Hbaieb, F. Chaari, T. Fakhfakh, M. Haddar, Three dimensional model for a helial planetary gear train vibration analysis, International Journal of Engineering Simulation, IJES, vol. 6, pp. 3 38, 005 [6] A. Kahraman, Free tortional vibration harateristis of ompound planetary gear sets, Mehanism and Mahine Theory, vol. 36, pp , 00. [7] W. H. Muller, Epiyli Drive Trains, Wayne State University Press, Detroit, 98. [8] Z. Terplan, Dimensionierung der Zahnrad-planetengetriebe, Akademiai Kiado, Budapest, 974. [9] R. Hbaieb, F. Chaari, S. Dhouib, M. Haddar, Free Vibration Charateristis of Compound Planetary Gear Sets, Proeedings of the Seond International Congress Design and Modelling of Mehanial Systems, CMSM 007, 9- Marh 007, Hammamet, Tunisia. 9 th INTERNATIONAL CONGRESS ON ACOUSTICS ICA007MADRID 6