TWO INPUT EPICYCLIC TYPE TRANSMISSION TRAINS WITH APPLICATION TO TANDEM BICYCLING

Transcription

1 Al-Qadisiya Journal For Engineering Sciences Vol. o. Year 9 WO IPU EPICYCLIC YPE ASMISSIO AIS WIH APPLICAIO O ADEM BICYCLIG Essam Lauibi Esmail echnical Institute of Diwaniya, Diwaniya, Iraq Shaker Sakran Hassan Associated Professor of Mechanical Engineering University of echnology, Baghdad, Iraq Abstract his paper presents a unification of kinematic and torque balance methods for the analysis of twoinput epicyclic-type transmission trains. A method for the derivation of the velocity ratio of an epicyclic gear train in a symbolic form is developed. A single kinematic equation is derived to study the kinematic characteristics of fundamental gear entities. he method relies on previous work on graph theory. It improves on existing techniques used for automatic transmission mechanisms in its ability to accurately solve the kinematics of geared mechanisms and estimating their velocity ratios and arranging them in a descending order in a simpler manner. he results are then used for assignment of the various links of a given two-input epicyclic gear mechanism. his study contributes to the development of a systematic methodology for the torque and power flow analyses of two-input epicyclic gear mechanisms with a reaction link, based upon the concept of fundamental circuit. Specifically, this study presents for the first time a complete design and analysis of two-input velocity changer to be used as a power coupling in a new generation of tandem bicycles. wo new designs are demonstrated. KEYWODS: Epicyclic gear train, Kinematics, Power-flow, andem bicycle, ransmission, wo-input, Velocity ratio.... ".

2 Al-Qadisiya Journal For Engineering Sciences Vol. o. Year 9. " ". " ". omenclature EGM Epicyclic gear mechanism EG Epicyclic gear train FC k th k Fundamental circuit FGE Fundamental gear entity, Gear ratio defined by a planet gear p with respect to a sun or ring gear x. p x p, x ± p / x, where p and x denote the numbers of teeth on the planet and the sun or ring gear, respectively, and the positive or negative signs depends on whether x is a ring or sun gear. x, y Or e Velocity ratio between links x and y with reference to link ga, k, gb, k and c, k Or i External torques exerted on links ga, gb and c of the k th Fundamental circuit or exerted on link i wo-dof wo-degree of freedom i umber of teeth on gear i c Carrier ga Gear a gb Gear b p Planet gear r ing gear s Sun gear ω Angular velocity of link i Introduction i A literature survey reveals that numerous methodologies have been proposed for the kinematic analysis of epicyclic gear trains [, 9,, 5, 8 and ]. Using graph theory, (Buchsbaum and Freudenstein, 97) investigated the structural characteristics associated with epicyclic gear trains (EGs) and proposed a systematic methodology for the structural synthesis of such mechanisms. Subsequently, (Freudenstein, 97) proposed a systematic methodology for the kinematic analysis of EGs. Freudenstein's method utilies the concept of fundamental circuits which can be readily derived from the graph of an EG. he method was elaborated in more detail by (Freudenstein and Yang, 97). Conventionally, the analysis is performed manually. ecently, there has been an increasing interest in the development of computer aided analysis programs [, 6]. he concept of fundamental circuit was further extended to force and power flow analysis for EGs by other researchers [9, and ]. (Hsieh and sai, 996b) apply similar formulations in conjunction with their kinematic study (996a) to determine the most efficiency kinematic configuration of one-degree of freedom automatic transmission mechanisms (Hsieh and sai, 998). he concept of fundamental circuit is a powerful tool for automated analysis of EGs. However, the analysis involves the solution of a set of linear equations for all the kinematic variables. he area that attracted little attention is the two-dof EGs. (Corey, ) attempted to present a solution technique for finding a total speed and force solution to two-dof EGs in the most general case possible. Unfortunately, he got confused between the rotational speeds and torques of EGs in the general case

3 Al-Qadisiya Journal For Engineering Sciences Vol. o. Year 9 and those of specific cases, which render those solution techniques physically meaningless (Esmail, 7). (Mir-asiri and Hussaini, 5) had presented a new concept of mobile robot speed control by using two degree of freedom gear transmission made up of a basic EG. A basic EG consists of a sun gear, a ring gear, and a carrier as shown in Figure. he first step in the design process of two-input mechanisms involves finding the configuration that not only provides the correct speed ratios, but also meets other dynamic and kinematic requirements. In general, in order to create a physical model of the system, all relevant physical relations such as speed, torque and power flow must be examined. he absence of torque and power flow analyses have led to flawed simulation results in (Mir-asiri and Hussaini, 5). he present paper applies the concepts of fundamental circuits and fundamental gear entities (FGEs) for the kinematic and dynamic analyses of two-input EGs. A single kinematic equation is derived to study the kinematic characteristics of FGEs. he results are then used for the estimation and comparison of various velocity ratios of an epicyclic gear mechanism, which lead to the development of a more efficient methodology for the assignment of the various links of a given twoinput epicyclic gear mechanism. Specifically, this study presents for the first time a complete design of two-input velocity changer to be used as a power coupling in a new generation of tandem bicycles. In what follows, the concepts of fundamental circuit and fundamental gear entity will be reviewed. Fundamental Circuit he fundamental circuit is defined as a circuit consisting of one geared edge and several turning pair edges connecting the endpoints of the geared edges (Buchsbaum and Freudenstein, 97). Figure (a) shows a gear pair which consists of two gears, ga and gb, and a supporting carrier, c. Figure (b) shows the corresponding schematic representation, which is a fundamental circuit. his schematic diagram consists of a block, three nodes, and three chords. he block represents a fundamental circuit. he symbol FC k denotes the k th fundamental circuit of an EG. he nodes represent the links of the fundamental circuit which transfer power or torque to the other fundamental circuits. he chords connecting the nodes to the block are labeled with and c, k which denote the external torques exerted on links ga, gb, and c. ga, k, gb, k Fundamental Gear Entity (Chatterjee and sai, 995) defined the fundamental geared entity (FGE) as a mechanism represented by a graph formed by a single second-level vertex or a chain of heavy-edge connected second-level vertices together with all the lower vertices connecting them to the root. Figure shows the canonical graph of the basic epicyclic gear train shown in the same Figure. here are two frequently used FGEs, one contains only one planet and the other contains two meshing planets. he graphs and functional schematics of a single-planet FGE and a double-planet FGE are shown in Figures. and, respectively. FGEs with more than two meshing planets and/or with multiple compound planets are not practical (Gott, 99) and will be excluded from this study. In a single-planet FGE, the coaxial links include a carrier and several first-level gears meshing with one planet gear. In a double-planet FGE, there are two meshing planets, one carrier, and several coaxial gears meshing with either one of the two planets.

4 Al-Qadisiya Journal For Engineering Sciences Vol. o. Year 9 Kinematics of Fges he velocity relationship among these coaxial links should be established in order to effectively compute the velocity ratios of a transmission mechanism. he term "gear ratio" is used in this paper to denote the ratio of a meshing gear pair, while the term "velocity ratio" is used to denote the velocity ratio between the input link and the output link of an EGM. / () p, x ± Considering the kinematics of a fundamental circuit, the fundamental circuit equation can be written as (Buchsbaum and Freudenstein, 97): p x ( ω x ω c) ( ω p ω c) p, x () Equation () can be re-written for the coaxial links of the single-planet FGE and the double-planet FGE shown in Figs. and as follows ( ) ( ) ω i ω c ω p ω c p, i () ( ω j ω c) ( ω p ω c) p, j () And ( ) ( ) ω k ω c ω p ω c p, k (5) ( ) ( ) ω l ω c ω p ω c p, l (6) Dividing Eq. () by () yields a kinematic equation relating the angular velocities of the two gears i and j, and the carrier c. Let the symbol i c, j denote the velocity ratio between links i and j with reference to link c. hat is ω ω ω ω c i c i, j j he gear ratio for the coaxial gears that are not meshing directly with the planet gear on which the velocity ratio is calculated and are meshing with the other planet gear can be found in terms of the gear ratio of the two planets ) as ( p, p c p, i p, j p, x p, p. p, x (7) (8) Dividing Eq. () by (5) yields ω ω c i c i, k p, i p, k ω k ω c (9) Where; p, k p, p. p, k, for the double-planet FGE. Eliminating ω p and ω c from the fundamental circuit equations (), () and (5), yields

5 Al-Qadisiya Journal For Engineering Sciences Vol. o. Year 9 k i, j ω i ω k ω ω j k p, i p, j p, k p, k () Similarly, eliminating ω p and ω c from the fundamental circuit equations (), (5) and (6), yields l i, k ω i ω k ω l ω l p, i p, k p, l p, l () Where; p, l p, p. p, l In general we can write, for the double-planet FGE. x, y ω x ω y ω ω p, x p, y p, p, () Using the facts that and, equation () provides a basis for the velocity ratio p, p p, c analysis of single-planet and double-planet FGEs and so on. Only this equation can be used to study the kinematic characteristics of all of the FGEs, which makes the kinematic analysis of EGs very simple. Velocity atio anges Kinematic inversion is defined as the process of selecting the input, output, and fixed links for a mechanism. Given three coaxial links x, y and the following basic characteristics relate the velocity ratios of these links.. y, x x, y. + y x, x, y All other velocity ratios of the three links due to kinematic inversions can be found by applying characteristics and as shown in able. he velocity ratio ranges are classified into three kinds; greater than one, between ero and one, or less than ero. Again, by applying the kinematic inversions, the velocity ratio ranges of an EG can be deduced as shown in able. Velocity atio elations A one-degree of freedom EG giving and taking power from the external environment requires at least an input, an output and a fixed link. hus, to function effectively it requires at least three ports. Similarly, a two-dof EG requires at least four ports, and so on. Following a similar procedure to that used by (Hsieh and sai, 996b) with major differences, we can write; Let x y, o and y, o denote two velocity ratios of an EG. Dividing x, o by y, o and after simplification, yields

6 Al-Qadisiya Journal For Engineering Sciences Vol. o. Year 9 x, o y, o y x, o () Dividing x, o by y, o and after simplification, yields x, o y, o x, y () Hence if the ranges of x y, o and y, o are known, the velocity ratio relation between x, o and y, can be deduced from Eqs. () and (). For example, if > x y, o > then o, > x o y, o > (5) Furthermore, if > then Eq. (5) can be further simplified to y, o > >. his y, o x, o equation implies that the velocity ratio x, o can assume two possible ratio ranges, > x, o, and > x, o >. From Eq. (), if > x, o, then > x, y >, and the velocity ratio relations can be deduced as relations > >. Also, if > x, o >, then > x, y and the velocity ratio x, o y, o is > > >. welve velocity ratio relations can be derived from Equations x, o y, o () and () for any four coaxial links of an EG as shown in able. hese velocity ratio relations are three more than the nine relations given in (Hsieh and sai, 996b). his is because that paper has not taken in consideration the effect of x, y on the velocity ratio relations. Procedure for Assignment of Four Coaxial Links of A wo-dof Egt Most enumeration procedures are done through the process of generating and testing. he procedure is thus divided into two parts: a generator of all possible solutions and a tester that selects only those solutions that meet the constraints. An EG can provide several velocities depending on the assignment of the input, output and fixed links. hese various velocities need to be estimated and arranged in a descending order to arrive at a proper design. Selecting three out of any four coaxial links of an EG (x, y,, o) at a time, yields four sets of three coaxial links (x, y, ),(x, y, o),(x,, o) and (y,, o). By expressing the train velocity ratios in terms of the gear ratios of its gear pairs with the help of Eq. (), the velocity ratio ranges can be estimated without knowing the exact dimensions of the epicyclic gear mechanism. Other velocity ratio ranges due to kinematic inversions can be found using able. Furthermore, the velocity ratio ranges are compared with the velocity ratio relations shown in able, to identify all other velocity ratio ranges of the EG. he results are then used to identify the proper assignment of the links of a two-input epicyclic gear mechanism.

7 Al-Qadisiya Journal For Engineering Sciences Vol. o. Year 9 orque Analysis orque Analysis of a Fundamental Circuit For a fundamental circuit of an EG, it is well known that under static equilibrium, torques acting on the three links of a fundamental circuit must be summed up to ero:, k gb, k + c, k + ga (6) It is also well known that under constant-velocity operation and in the absence of friction, the net power passing through a fundamental circuit is ero. Power is the product of torque and angular velocity (.ω): ga, k. ω ga + gb, k. ω gb + c, k. ω c (7) Solving Equations (), (6) and (7), yields: And, k gb, ga. ga, k + gb (8), k + ( gb, ga). ga, k c (9) Equations (8) and (9) express two of the external torques in terms of the third. he word "external" refers to all sources that are external to the gear pair of interest. hus, external torques come from all the links that are connected to links ga, gb and c. orque Analysis of wo-input EGs with a reaction link Due to the fact that the function of a transmission mechanism is to provide proper torques and velocities for a vehicle, it is essential to identify the relations among the velocity ratio, torque distribution and the power flow in an EGM. Under steady-state operation, it is well-known that torques exerted on links x, y,, and o about the central axis of the EG are summed to ero; that is x + y + + o () Where i denotes an external torque exerted on link i. Similarly, it is well-known that the net power flowing into the system via the four coaxial links are summed to ero; that is Eliminating yields x. x + y. ω y +. ω + o. ω o ω () from Equations () and () and making use of Eq. (), and after simplification,. o x, o. x y, o y () Similarly, eliminating o from Equations () and () and making use of Eq. (), yields ). ( x, o ). x + ( y, o y ()

8 Al-Qadisiya Journal For Engineering Sciences Vol. o. Year 9 Equations () and () relate torques exerted by links x, y, and o in terms of the velocity ratios x, o and y, o. Each-Link orque Analysis of wo-input EGs he torque and power equilibrium equations can be written for an n-link EG as follows: First, writing Equations (8) and (9) once for each fundamental circuit will result in (n-) equations. Second, at each node, applying the condition that the sum of torques on the link is equal to ero will result in n-node equations. A total of (n-) linear equations are obtained which can be written in a compact form by using a matrix-vector notation as [ ] [ ] [ in] A () B Where [ ] A B is a system of (A B) matrix whose elements are functions of the gear sie ratios (gb,ga), and [ ] denotes the vector of (B) torques. Hence, regardless of which links are assigned as the input, output, or fixed link, Eq. () can be used to solve for the (B) unknown torques in terms of the input torques[ in ]. hese methodologies are perhaps best illustrated through a design example. Case Studt: Multi-ider Human Powered Vehicle Epicyclic gear trains are considered as the most effective method for coupling the relatively inconsistent inputs of two human riders (Corey, ). What follows is the design of such a gear train using the present new methodologies. he first step in the design process is to set fourth any design constraints on the design. Design Constraints In this paper, several requirements must be met for the design to be viable. hese are the same as those in (Corey, ):. he drivers are to be seated with their backs to one another. his means that at the system's operating point, the two inputs rotate in opposite directions, ωi / ωi <.. he output link moves slightly above the nominal pedal cadence of 8 rpm pedal velocity associated with efficient riding 8 ωo.. Both riders pedal at approximately the same velocity and contribute approximately the same percentage of the output power. Plan An EG with one carrier and three coaxial gears meshing with one planet is shown in Fig. 5. Assume that the sies of the sun gears are in the descending order of gear > gear. Selecting three out of the four coaxial links (,,, 5) at a time, yields four sets of three coaxial links (,, ),(, 5, ),(, 5, ) and (,, 5). For the coaxial links (,, ), and making use of Eq. (), the velocity ratio, can be found as. (5),,., It can be shown that, >. For the coaxial links (, 5, ), and making use of Eq. (), we can find

9 Al-Qadisiya Journal For Engineering Sciences Vol. o. Year 9. 5 (6),, 5.,5 It is apparent that >, 5. In a similar way, we can find that 5 >, 5 and, >. All other velocity ratios and velocity ratio inversions can be found from able. hese are used to construct able. o search for an appropriate assignment of the four coaxial links, one can examine, in general the first and second design constraints which can be written in notation consistent with that used in able as > y, o > > x, o and > x, y (7) Where y and o rotate in the same direction and x rotates in the opposite direction. By searching in able for the velocity ratio ranges that satisfy Eq. (7), only two possible solutions can be found:. >, > > 5,, which gives x 5, y, o and,and. >, 5 > >, 5, which gives x, y, o 5 and hese two assignments are the potential candidates for achieving a proper design. After establishing a broad range of -ranges to be considered, the next task is to attempt to apply the design constraints to the governing equations of the system. Examining the third design constraint, it is convenient to write it in the form x y Or ω x x y y. ω. ω (8) y ω x (9) Obviously, since Eq. () must hold for the power contribution from each rider to be equal, and the input torques are equal and opposite, the velocity ratios are related after simplification by () x, o y, o And from Eq. () and the first and third design constraints, we get y, x () Since the human body provides most efficient power at a nominal pedal cadence of 8 rpm pedal velocity associated with efficient riding (Corey, ), it would be advantageous to select a gear ratio that would place the absolute value of the output velocity in the range of rpm, as required by the second design constraint. hat is And y, o () x, () o

10 Al-Qadisiya Journal For Engineering Sciences Vol. o. Year 9 Applying Eq. () to the first potential solution and simplifying, we get () 5, 5 Since 5 is larger than, and then 5, can never be equal to minus unity, which will fail the third design constraint requiring that, the power contribution from each rider to be equal. Applying Eq. () to the second potential solution and simplifying, we get, (.. ) (5) After more simplification, we get.. (6) Applying Equations () and () and simplifying, we get And, 5 5 ( + 5) (7), (8) Also, from geometric considerations with all gears having the same diametrical pitch, And + + (9) + 5. () he solution of Equations (6), (7), (8), (9) and () is listed in able. Plan his design adopts the compound EG shown in Fig. 6 to achieve the two-input velocity changer. Assume that the sies of the sun gears are in the descending order of gear > gear ² > gear ¹. Selecting three out of the four coaxial links (,,, ) at a time, yields four sets of three coaxial links (,, ),(,, ),(,,) and (,, ). For the coaxial links (,, ), and since they belong to the same FGE, then making use of Eq. (), we can find It can be shown that >. 5, 5, 5, 5,, > () ( 5 ) + ( 5 )

11 Al-Qadisiya Journal For Engineering Sciences Vol. o. Year 9 For the coaxial links (,, ), and since they belong to the same FGE, then making use of Eq. (), we can find 6, 6 6 () ( ) + ( ) It is apparent that >,. he velocity ratios of the remaining two sets of three coaxial links can be expressed in terms of the velocity ratios of the above two FGEs. We use the fact that >, > and >, to begin our evolution. By assigning x, y, and o in able, we can write > and, > >, from kinematic inversions. For this assignment the velocity ratio, can assume two possible ratio ranges, >, and >, >. If >,, then >, >. Also, if >, > then >,. wo sets of velocity ratio ranges are possible:. >, >, >,, >, and, >. >, >, >,, >, > and > >, and,. Any assignment of the four coaxial links that gives two known velocity ratios will give similar results. All other velocity ratios can be found from the kinematic inversions of able. wo different tables can be constructed for the above two sets. hese are shown in ables 5 and 6 (only the first will be considered). By searching in able 5 for the velocity ratio ranges that satisfy Eq. (7), two possible solutions are found:.. >, which gives x, y, o and, and, > >, > >, >, which gives x, y, o and., Again, only the first solution will be considered. Applying Eq. () and simplifying, we get After more simplification, we get. 6,. () () Applying Eq. (), we get

12 Al-Qadisiya Journal For Engineering Sciences Vol. o. Year 9,. (5),,, But. 6,,. (6) 6 5 Making use of Eq. (), we can write,, (7) + Substituting Equations (6) and (7) into Eq. (5), and simplifying, we get Applying Eq. (), we get. (8),. (9),,, e-arranging, Eq. (9) But And,, (5), (, ), 5 (5), ( + ) (5) Substituting Equations (5) and (5) into Eq. (5), and simplifying, we arrive again at Eq. (8). Also, from geometric considerations And + + (5) (5)

13 Al-Qadisiya Journal For Engineering Sciences Vol. o. Year 9 he solution of Equations (), (8), (5) and (5) is listed in able 7 for 6. orque and Power-Flow Analysis he corresponding fundamental-circuit diagram for the EG shown in Fig. 6 is sketched in Figure7. he gear ratios for this train are 5, 5.5 (55) 5, 5.75 (56) 6, 6 (57).6 6, 6 (58) Writing Eqs (8) and (9) once for each fundamental circuit, yields the following eight linear equations: 5 + 5,. (59) + ( 5,). (6) 5 + 5,. (6) + ( 5, ). (6) 6 + 6,. (6) + ( 6,). (6) 6 + 6,. (65) + ( 6, ). (66) Summing all the torques acting on each node to ero, yields the following six linear equations (67) + + in (68) (69) + r (7)

14 Al-Qadisiya Journal For Engineering Sciences Vol. o. Year o (7) + + in (7) Hence, a system of fourteen linear equations is obtained which can be solved for the fourteen unknowns in terms of the first input torque in, the second input torque in, and the gear sie ratios. Equations (59) through (7) can be arranged in a matrix form as: ) 7 ( in in o r he solution of Eq. (7) is listed in able 8. he torque flow through the EG is shown in Fig. 8. he power-flow through the EG is shown in Fig. 9. he second input power transmits through the entire gear train whereas the first input power transmits through links, 6 and only. he two riders will supplement each other and a constructive power balance is achieved through the train. ow, it is apparent that the reaction link is used in two-input EGs to control the direction of rotation of other links and thus the direction of the power-flow. Conclusions. his paper presents a unification of kinematic and torque balance methods for the analysis of two-input epicyclic-type transmission trains.. A new method for the derivation of the velocity ratio of EG in a symbolic form has been developed. A single kinematic equation has been derived to study the kinematic characteristics of FGEs.. he results have been used for the estimation and comparison of various velocity ratios of EGMs, which lead to the development of a methodology for the assignment of the various links for a given two-input EGM.. welve velocity ratio relations have been derived for any four coaxial links of an EG. hese velocity ratio relations are three more than the nine relations given in (Hsieh and sai, 996b). his is because that paper has not taken in consideration the effect of certain velocity ratio ranges on the velocity ratio relations.

15 Al-Qadisiya Journal For Engineering Sciences Vol. o. Year 9 5. his study contributes to the development of a systematic methodology for the torque and power flow analyses of two-input epicyclic gear mechanisms (EGMs) with a reaction link, based upon the concept of fundamental circuit. eaction links are used in two-input EGs to control the direction of rotation of other links and thus the direction of the power-flow. 6. Specifically, this study has presented for the first time a complete design and analysis of two-input velocity changer to be used as a power coupling in a new generation of tandem bicycles. wo new designs have been demonstrated. eferences [] Allen,.., 979, "Multi-port Models of the Kinematic and Dynamic Analysis of Gear Power ransmission", ASME J. of Mechanical Design, Volume, pp [] Buchsbaum, F., and Freudenstein, F., 97, "Synthesis of Kinematic Structure of Geared Kinematic Chains and Other Mechanisms," Journal of Mechanisms and Machine heory, Vol. 5, pp [] Chatterjee, G., and sai, L. W., 995, " Enumeration of Epicyclic-ype ransmission Gear rains, " ransactions of SAE, Vol., Sec. 6, Paper o. 9, pp [] Chatterjee, G., and sai, L. W., 996, "Computer-Aided Sketching of Epicyclic-ype Automatic ransmission Gear rains," ransactions of the ASME, Journal of Mechanical Design,, Vol. 8, o., pp. 5-. [5] Corey, C. A.,, "Epicyclic gear rain Solution echnique with Application to andem Bicycling," hesis (M. Sc.), Department of Mechanical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA. [6] Esmail, E. L., 7, "omographs for Design of Planetary Gear rains", Accepted for Publication in the Journal of King Abdulai University-Engineering Sciences, S. A. [7] Freudenstein, F., 97, "An Application of Boolean Algebra to the Motion of Epicyclic Drives," ASME Journal Engineering for Industry, Vol. 9, pp [8] Freudenstein, F., and Yang, A.., 97, "Kinematics and Statics of Coupled Epicyclic Spur Gear rains", J. of Mechanism and Machine heory, Volume 7, pp [9] Gibson, D. and Kramer, S., 98, "Symbolic otation and Kinematic Equations of Motion of the wenty-wo Basic Spur Planetary Gear rains", ASME J. of Mechanisms, ransmissions and Automation in Design, Volume 6, pp. -. [] Gott, P. G., 99, "Changing Gears," Society of Automotive Engineers, Inc., Warrendale, Pennsylvania. [] Hsieh, H. L., and sai, L. W., 996a, "Kinematic Analysis of Epicyclic-ype ransmission Mechanisms Using the Concept of Fundamental Geared Entities", ASME J. of Mechanical Design, Volume 8, pp

16 Al-Qadisiya Journal For Engineering Sciences Vol. o. Year 9 [] Hsieh, H. I., and sai, L. W., 996b,"A Methodology for Enumeration of Clutching Sequences Associated with Epicyclic-ype Automatic ransmission Mechanisms", SAE ransactions, Journal of Passenger Cars, Sec.5, pp [] Hsieh, H. L., and sai, L. W., 998, "he Selection of a Most Efficient Clutching Sequence Associated with Automatic ransmission," ASME J. of Mechanical Design, Volume, pp [] Levai,., 96, "heory of Epicyclic Gears and Epicyclic Change-Speed Gears", Doctoral Dissertation, Budapest. [5] Levai,., 968, "Structure and Analysis of Planetary Gear rains", Journal of Mechanisms, Volume, pp. -8. [6] Mruthyunjaya,.S., and avisankar,., 985, "Computeried Synthesis of the Structure of Geared Kinematic Chains," Journal of Mechanisms and Machine heory, Vol., pp [7] aim, M-., and Hussaini,S., 5," ew Intelligent ransmission Concept for Hybrid Mobile obot Speed Control,"Int. J. of Advanced obotic systems, Volume, o., pp, [8] Olson, D. G., Erdman, A. G., and iley, D.., 99, "opological Analysis of Single- Degree of Freedom Planetary Gear rains," ASME Journal of Mechanical Design, Vol., pp [9] Pennestri, E., and Freudenstein, F., 99, "A Systematic Approach to Power-Flow and Static Force Analysis in Epicyclic Spur-Gear rains", ASME J. of Mechanical Design, Volume 5, o., pp [] Pennestri, E., and Freudenstein, F., 99, "he Mechanical Efficiency of Epicyclic Gear rains", ASME Journal of Mechanical Design, Volume 5, o., pp [] Smith, D., 979,"Analysis of Epicyclic Gear rains via the Vector Loop Approach," Proc. Sixth Applied Mechanisms, Paper o.. [] Saggora, L. and Olson, D. G., 99,"A Simplified Approach for Force and Power-Flow Analysis of Compound Epicyclic Spur-Gear rains," Proc. of the ASME 99 Design echnical Confs. Advances in Design Automation, DE-Vol. -, Vol., pp [] Willis,. J., 98, "On the Kinematic of the Closed Epicyclic Differential Gears", ASME J. of Mechanical Design, Volume, pp. 7-7.

17 Al-Qadisiya Journal For Engineering Sciences Vol. o. Year 9 able : Velocity atio Inversions and anges. Velocity atios Velocity atio Inversions x, y y, x x y, y, x x y, x, y e /e -e /(-e) (e-)/e e/(e-) e> >/e> > e >/( e) >(e )/e> e/(e )> Velocity atio anges >e> /e> > e> /( e)> >(e )/e >e/(e ) >e >/e e> >/( e)> (e )/e> >e/(e )> able : Velocity ratio relations for any four coaxial links of an EG.

18 Al-Qadisiya Journal For Engineering Sciences Vol. o. Year 9 able : Velocity ratio ranges for the EG shown in Fig. 5. >, > 5,, > >, > 5, > >, 5 > >, > 5,, > >, > 5, > >, 5 >, > >, > >, >, >, >, > 5 5, > >, > >, 5 > 5, >, 5 > 5, > able : umber of teeth on the gears of the first design. 8 teeth able 5: First velocity ratio ranges for the compound EG shown in Fig. 6. >, >, > >, >, > >, >, >, >,, > >, >, > >, > >, >,, > >, >, > >, > >, >, > >, >, > > >,, able 6: Second velocity ratio ranges for the compound EG shown in Fig. 6. >, >, > >, >, > > >, >,, > >,, >, >, > >, > >, >, > >, >, > >, > >, >,, >, >, >, > >, > able 7: umber of teeth on the gears for the second design able 8: orque distribution of the EG shown in Fig. 6 when links,, and serve as the fixed, output, first input and second input links, respectively in. 8 in. in + in.5 in in. in in in 6. in + r. in o.6667 in in in in in in in in in in in

19 Al-Qadisiya Journal For Engineering Sciences Vol. o. Year 9 Figure : A basic epicyclic gear train with its canonical graph representation. Figure : A basic gear pair and its schematic representation. Figure : A typical single-planet FG.

20 Al-Qadisiya Journal For Engineering Sciences Vol. o. Year 9 Figure : A typical double-planet FG. Figure 5: Epicyclic gear train to be used in the first plan. Figure 6: Compound epicyclic gear train to be used in the second plan.

21 Al-Qadisiya Journal For Engineering Sciences Vol. o. Year 9 Figure 7: Fundamental-circuit diagram of the EG shown in Fig. 6 when links,, and serve as the fixed, output, first input and second input links, respectively. Figure 8: orque flow through the EG shown in Fig. 6 when links,, and serve as the fixed, output, first input and second input links, respectively. Figure 9: Power- flow through the EG shown in Fig. 6 when links,, and serve as the fixed, output, first input and second input links, respectively.