UNIVERSITY OF PITESTI SCIENTIFIC BULLETIN FACULTY OF ECHANICS AND TECHNOLOGY AUTOOTIVE erie, year XVII, no. ( 3 ) SIPLIFIED ODEL FOR EPICYCLIC GEAR INERTIAL CHARACTERISTICS Ciobotaru, Ticuşor *, Feraru, Octavian, Caravan, Alexandru ilitary Technical Academy, Romania, OND, Romania e-mail: cticuor4@yahoo.com EYWORDS planetary mechanim, inertia, gear, epicyclic ABSTRACT - For mechanical tranmiion, all component gear have the ame kinematic no matter the tage of the gear box. Conequently, the inertial characteritic of the mechanical tranmiion depend only on the quare of the gear box ratio. For an epicyclic gear box, the peed of the component depend on the manner in which the power i tranmitted through the planetary mechanim; conequently, the inertial characteritic of the tranmiion depend on the tructure of the epicyclic gear box. The paper preent a implified model for calculation of the inertia of an elementary epicyclic mechanim. The analyi include the error etimation a well a the implementation of thi implified model into the analyi of the epicyclic gear box. The uage of the planetary gear boxe become cover a larger area of driveline for heavy vehicle, and for off-road vehicle too. Accompanied by a torque converter, the planetary gear box allow an automatic match of the engine output with the motion requirement which eae the driving and improve the running performance. The complexity of the kinematic tructure of the tranmiion including torque converter and planetary gear box derive from the exitence of different moving element for each tage a well a from the complex movement of the atellite gear belonging to the elementary epicyclic mechanim. For contant input peed, the kinematic of the planetary gear box may be tudied uing variou method a graph theory (6) or by extending the traditional concept of a lever repreentation of a planetary gear et to one that include negative lever ratio (5). The method were found uitable in a erie of application a in () and (4). A comprehenive approach i preented in () and (3) having a aim the calculation of peed, torque, power flow and efficiency of the planetary gear box. All the paper cited above conider the tationary regime of the planetary mechanim characteried by contant peed. Conequently, the inertia of the element which form the planetary train i neglected. Thi hypothei proved to be too rough for olving ome pecific apect uch a the hifting proce or the acceleration performance of the vehicle. In order to ue the analyi method preented in () and (3) without neglecting the inertia of gear, a implified model of epicyclic gear mechanim wa developed, the main tage being preented below. For the elementary epicyclic mechanim with 3DOF, the kinematic i fully decribed by the Willi relation: ω+ ω ( + ) ω = () where: ω - angular peed of the external element noted with index for un gear, for planet gear and for carrier arm repectively; - the contant of the epicyclic gear mechanim: 5
z =, z where z repreent the teeth number of the gear. The relation () give by integration and by derivation repectively: ϕ + ϕ + ϕ = () ( ) ( ) + + = (3) For contant peed of the external element: dω dω d ω = ; = ; =, the torque are given by the following relation: = ; =. (4) + ( ) n planet gear : z atelit gear : z carrier arm un gear : z Figure The nodal repreentation of the epicyclic gear mechanim Figure The tructure of EI type epicyclic gear mechanim For the ituation of acceleration, the exitence of the inertia of gear produce a modified ditribution of torque among the external element. In order to pecifically analye the influence of inertia, the EI type gear mechanim i conidered (ee Figure ). The nodal repreentation i hown in Figure The nodal repreentation of the epicyclic gear mechanim emphaiing the external element uing the following notation: for the un gear, for the planet gear and for the carrier arm, repectively. For thi cae, the Lagrange equation i applied conidering the epicyclic gear mechanim a a ytem of ma point with two degree of freedom: d W W = Qj j =,. ω j ϕ j A generalized coordinate the angular diplacement of element and are adopted. Conequently, the total energy of the ytem i done by the following um: W = W+ W + W + W, (5) where the energy of the individual element are: * J ω J ω J ω W = ; W = ; W =. (6) 6
In the relation (6), the term J * repreent the inertia of the carrier arm itelf, without atellite. The atellite have a complex movement coniting of a rotation around it axle with the peed ω and a rotation around the axle of un gear with the peed ω : ( ω ω) ω =, (7) and: ( ϕ ϕ) ( ) ϕ = ; =. (8) For n atellite having the ma atellite become: and the moment of inertia J, the total energy of the J R W = n ω ω +. (9) Introducing the relation (6) and (9) in the relation (5), it reult the total energy of the epicyclic gear mechanim: * J J J ω J R W n ω ω = ω + ω + + +. () Taking into conideration the relation (7), the relation () become: 4nJ * ( Jω R ω 4nJ 4nJ J + ) W = J+ + + J + + + ω ω.() ( ) ( ) ( ) The overall inertia of the carrier arm and n atellite i noted J : * J = J + n R. Uing the above notation, the relation () become: 4n J J ω 4n J 4n J ω ω W = J+ + + J +. () ( ) ( ) ( ) From the Willi relation reult the peed of the planet gear: ( + ) ω ω ω =. Conequently, the relation () may be rewritten: 4n ( J J 4n J J + ) W = J+ + + J + + ( ) ( ) (3) 4n J ( J + ) + ω ω ( ) The derivation of the relation (3) give: d 4n ( J J 4n J J + ) W = J+ + + (4) d t ω ( ) ( ) 7
( ) ( ) d 4n J J + 4n J J + W = + + J + + d t ω ( ) ( ) (5) The elementary machine work i calculated uing the relation: δ Q= δϕ + δϕ + δϕ, (6) and, taking into conideration the relation (), it reult: ( + ) δ Q= δϕ + + δϕ. (7) Finally, the following relation reult: 4n ( J J 4n J J + ) J+ + + = (8) ( ) ( ) 4n ( ) ( ) ( J J + 4n J J + + ) + + J + + = + (9) ( ) ( ) The differential equation (8) and (9) decribe accurately the working of the epicyclic gear mechanim taking into conideration the inertia of all element which contitute the epicyclic gear. Below i dicued the conequence of the hypothei according to which the inertia of the atellite i neglected (but not their ma). Thi hypothei i baed on the obervation that the inertia moment depend on the quare of the gear radiu, and the radii of the atellite are mall compared with thoe of the un gear or the planet gear. Impoing the condition J =, the equation (8) and (9) become: J J ( + ) J+ = () J ( ) ( ) + J + ( + ) + J + = + () The equation () and () allow an approximate calculation of the reduced inertia moment. In order to evaluate the level of error, the mot ignificant ituation are analyed below. If the planet gear i blocked, it reult: = ; = ( + ) () Introducing the relation () in the relation (8), it reult the exact ditribution of torque acting on the external element: 8n J ( + ) J+ =. (3) ( ) Uing the relation () the approximate ditribution reult: ( + ) J =. (4) 8
Suppoing that the input element i the un gear, the nodal repreentation of the epicyclic gear mechanim i preented in Figure 3 where between the torque acting on the epicyclic gear mechanim the relation (4) applie: =. J J J J Figure 3 The nodal repreentation of the epicyclic gear mechanim The balance of the torque on the ramified node give, in general: + J + =. Suppoing that > (the input haft i accelerated), then: J =. (5) Taking into conideration the relation (), the relation (5) become: ( + ) J = (6) which i identical with the relation (4). The relative error generated by the hypothei according to which the inertia of the atellite i neglected i done by the relation: 8n J Δ =. (7) J + ( ) ( ) Similarly, if the un gear i blocked, the relative error i done by the following relation: 4n J Δ= (8) ( ) J J + For the uually value of the parameter of the relation (7) and (8) ( n = 4...6, =...4) and conidering the dimenion of the gear ued in the planetary gear boxe, the relative error are below 4%. The analyi preented above allow the concluion preented below. The propoed model i baed on the hypothei according to which the inertia of the atellite i neglected; thu, the rotation of the atellite around their axe i neglected. Neverthele, the ma of the atellite i taken into conideration being included into the relation of the equivalent inertia of the carrier arm: * J = J + nr. The reulting relative error i conidered acceptable. The relation which decribe the ditribution of the torque acting on the external element of the epicyclic gear mechanim are the following: 9
d ω J = (9) d ω J d ω J = (3) d ω ( ) J + The relation (9) and (3) allow the utiliation of the method and algorithm preented in (); thu, it become poible to ue the ame analyi method for both tationary and tranient regime of the planetary gear boxe. By the ue of the implified model for a given planetary gear box, the overall inertia of the gear box, reduced to the input haft, may be determined for each tage uing the nodal approach expoed in, without an unacceptable increae of the calculation volume. REFERENCES () Ahmore C., New epicyclic gearbox erie for To 4W ga turbine, Ga Turbine World, 36(4), 3-3, 6 () Ciobotaru T., Frunzeti D. Jäntchi L., A ethod for Analying Epicyclic Gearboxe, International Journal of Automotive Technology, Vol., No., pp. 67 7, ISSN 9-938, (3) Ciobotaru T., Frunzeti D., Ru I., Jäntchi L., ethod for Analying ulti-path Power Flow Tranmiion, Proceeding of the Intitution of echanical Engineer, Part B: Journal of Engineering anufacture, Vol. 4, Number 9/, p.447-454, ISSN 954-454 (Print), (4) araivanov A, Popov R., Computer Aided inematic Analyi of Planetary Gear Train of the 3k Type, Proceeding of the 3rd International Conference on anufacturing Engineering (ICEN), -3 October, Chalkidiki, Greece, 57-578, 8 (5) Raghavan., Efficient Computational Technique for Planetary Gear Train Analyi, th IFTo World Congre, Beançon (France), June8-, -5, 7 (6) Tai L. W., Enumeration of inematic Structure According to Function, CRC Pre LLC, ISBN -8493-9-8, 55-8,