A NEW TECHNIQUE FOR SKETCHING EPICYCLIC GEAR MECHANISMS USING ACYCLIC GRAPH

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1 International Journal of Mechanical and Production Engineering Research and Development (IJMPERD) ISSN(P): ; ISSN(E): Vol. 4, Issue 3, Jun 2014, TJPRC Pvt. Ltd. A NEW TECHNIQUE FOR SKETCHING EPICYCLIC GEAR MECHANISMS USING ACYCLIC GRAPH H. ELEASHY 1 & M. S. ELGAYYAR 2 1 Department of Mechanical Engineering, Future University in Egypt, Cairo, Egypt 2 Department of Production & Mechanical Design, Mansoura University, Mansoura, Egypt ABSTRACT In this paper, an algorithm for reverse transformation process of enumerated graphs of epicyclic gear mechanisms into a functional representation has been introduced using acyclic graph method. The displacement graph is divided into its elementary gear units which are then connected to form the functional representation of such a mechanism. Transformation of displacement graphs with up to 11-link mechanisms having at least 4 coaxial links and up to 8 coaxial links can be performed using the proposed algorithm. KEYWORDS: Sketching Epicyclic Gear, Acyclic Graph, Coaxial Links INTRODUCTION For purpose of enumerating of all possible configurations of n-link mechanism, mechanism should be represented graphically by one type of graphs representation. Results of enumeration process are a group of non-isomorphic displacement graphs each one represents a mechanism. To evaluate these mechanisms from design and manufacturing point of view, they should be in the form of functional representation. Therefore, there must be a methodology or an algorithm to transform these displacement graphs into functional representation which is considered a dimensional synthesis phase. H. Eleashy and Elgayyar [11] used displacement graph and acyclic graph to enumerate epicyclic gear mechanisms with 7 and 8-links having at least 4 coaxial links. Chattergee and Tsai [6] introduced a reverse transformation algorithm using the canonical graph representation. Here in this paper, an algorithm has been developed using the displacement graph. Displacement graph is divided into some sub graphs, each one analyzed into its elementary primitives; namely carrier, sun gear, ring gear, and planet gear. Using these primitives and type of connections for each sub graph a unit of the functional schematic of an epicyclic gear mechanism is created. These units are then connected together to form a functional representation. FUNCTIONAL REPRESENTATION Graph representation of a mechanism facilitates process of enumerating all configurations of such a mechanism and keeps its topological data. But it does not aid the designer to visualize how the mechanism looks and how it works. For example the functional schematic shown in Figure 1 (b, c) has the same graph representation shown in Figure 1 (a). However, epicyclic gear train in Figure 1(c) has three mating gears (ring gear, planet gear, and sun gear) that must have the same module and their diameters are dependent. Therefore, functional representation is an important dimensional synthesis phase after graphical enumeration phase. There are some characteristics that desired functional representation should have which can be summarized in the following subsections editor@tjprc.org

2 80 H. Eleashy & M. S. Elgayyar No Interference There should not be any crossing between links in such a functional schematic of an epicyclic gear train. Accessibility Figure 1: Functional Representations of the Same Graph Representation First level links of a graph representation of an epicyclic gear mechanism are used as input, output, and fixed links. The output link is usually connected to the next reduction unit of the system. In the functional schematic of an epicyclic gear mechanism it is necessary for the other links, that are connected to the input link or to the casing, to be arranged in such a way that they remain accessible; i.e. they can be connected to other elements of the gear train as and when required. Multi Coaxial Joints However, the coaxial links of an epicyclic gear train form a multiple joints. Therefore, the joints between a set of coaxial links can be rearranged without changing the functional characteristics of a mechanism. Few Coaxial Links It is desired from the manufacturing point of view to reduce the coaxial shafts of a functional schematic of an epicyclic gear train. That is because of the tolerance limits on such shafts are too rigid and requires more accuracy. In the functional schematic shown in Figure 2 link 1 has a carrier and a ring connected on the same shaft which is coaxial with another shaft that connects the carrier and the ring of link 3. Low Inertia Reducing the number of coaxial links usually results in the presence of overhead connections. Such a connection is represented by link 3 in the functional schematic in Figure 2 which has two ring gears attached edge-to-edge. These connections cause high moment of inertia. It also requires more material in manufacturing. Therefore the designer should reach a proper balance between minimizing the coaxial links and the presence of overhead connections. ACYCLIC GRAPH REPRESENTATION An epicyclic gear mechanism typically consists of a one degree of freedom epicyclic gear train supported by the casing on one axis. The coaxial links are mounted on concentric bearing that are housed in the casing. Only coaxial links of an epicyclic gear mechanism are used as input, output or fixed. Otherwise one of the links will have its axis moving in space and it will not be possible to connect it to any type of other reduction units or other systems. Impact Factor (JCC): Index Copernicus Value (ICV): 3.0

3 A New Technique for Sketching Epicyclic Gear Mechanisms Using Acyclic Graph 81 Figure 2: Coaxial Links and Overhead Connections in a Functional Schematic Figure 3: (a) Functional Representation (b) Corresponding Displacement Graph Representation In most of epicyclic gear mechanisms output link is never changed. Desired reduction ratios can be obtained by changing input and fixed links. The casing of an epicyclic gear mechanism is a unique link in its kinematic structure. In the acyclic graph representation, vertex representing the casing will be labeled as zero vertex. Links of an EGM are denoted as vertices, simple joints are denoted as edges and multiple joints are denoted as solid polygons. An edge connection, as a solid edge, represents a revolute joint where an edge connection, as a dashed or colored edges, represents a gear pair. Vertices that are connected by a common solid polygon represents coaxial link connected by coaxial revolute joints in the mechanism The. vertex representing the casing is labeled zero and called ground-level vertex. The vertices connected to the ground-level vertex by one revolute polygon are the first-level vertices. The vertices connected to the ground-level vertex by two solid polygons or edges are the second-level vertices. The joints connecting the coaxial links can be rearranged without affecting the functionality of the mechanism. Figure 3(a, b) represents a functional representation of an epicyclic gear mechanism of a multi-speed reduction unit and its corresponding displacement graph. Structural Characteristics In this section we will discuss the fundamental structural characteristics of an acyclic graph and the common rules between an epicyclic gear mechanism and its corresponding displacement graph [4, 10]. F1: The graph is a two-level graph, where the first-level vertices are connected to the ground-level vertex by one solid polygon. The second-level vertices are connected to the first-level vertices by one solid polygon or edge. F2: The graph, representing N-link mechanism, contains N vertices,n-3 geared edges, and one pendant vertex as the ground-level vertex. A fundamental circuit (f-circuit) is a circuit consists of three vertices, one geared edge and two revolute polygons or edges. The vertex, which is not incident to the geared edge, is called a transfer vertex. The displacement graph of N-vertices EGM is a combination of N-vertices acyclic graph and N-3 geared edges as shown in Figure 4 In such a graph, there are seen links and four geared joints. graphs. F3: The sub-graph obtained by removing all the geared edges from the graph is an N-vertex, two level acyclic F4: Adding a geared edge to the acyclic graph forms an f-circuit contains one transfer vertex. editor@tjprc.org

4 82 H. Eleashy & M. S. Elgayyar As well as an EGM has some fundamental characteristics, the corresponding displacement graph should has the following rules: F5: The graph has a t least four first-level vertices and at most three second- level vertices. Coaxial Links In the displacement graph, the first level vertices represent potential candidates for the input, output and fixed links. These links are connected to the casing by coaxial revolute joints. None of these links are connected to the casing by geared joints. Figure 4: Displacement Graph Representation (Combination of Acyclic Graph and Geared Graph ) F6: The first level vertices are connected to the root by only thin edges. F7: No geared edges can be incident to the root. As shown in Figure 3, the acyclic graph representation contains four first level vertices (1, 2, 3, and 4), that represents coaxial links. Also there are two vertices in second level (5, 6) that represents planet gears. An acyclic graph having at most six coaxial links has been used in enumeration process [9], [10]. In this study, the number of coaxial links will be generalized. It will be restricted only by fundamental rules of the acyclic graph. Open Geared Graph Open graph has one or more coaxial links in first level has no connections even revolute nor geared edges with the higher level vertices and on the other hand if one or more links in the second level has no geared connections with first level. In most cases, the graph is an open one if the pendent vertex represents one coaxial link (in first-level vertices) does not give rise to any planet (in second-level vertices). For geared graph shown in Figure 5 (a), we find the vertex set of all fundamental circuits is {2, 3, 4, 5, and 6}. So the vertex set has one pendent vertex (vertex 1). F8: The graph has (N-1) vertices in the vertex set of all the f-circuits. Impact Factor (JCC): Index Copernicus Value (ICV): 3.0

5 A New Technique for Sketching Epicyclic Gear Mechanisms Using Acyclic Graph 83 (a) (b) (a) (b) Figure 5: (a) Open Displacement Graph Figure 6: Displacement Graph Representation (b) Displacement Graph has Redundant Links Containing Degenerate Geared Graph Redundant Link A redundant link cannot be used as an input link, output link or fixed link. Removal of such a link does not change degrees of freedom of epicyclic gear mechanism. In displacement graph representation of an epicyclic gear mechanism, if there exist a sub graph that represents an m-degree of freedom epicyclic gear train, then it must have at least m+2 ports of communication with the external environment. For example, the sub graph derived from the displacement graph in Figure 5(b) by vertices (1,2,4 and 6 ) does not represent an epicyclic gear train because the carrier ( link 3) is not a member in the sub graph vertices. But sub graph formed by the vertices (1, 2 and 5) represents an epicyclic gear train contains redundant links. F9: The graph does not contain any vertices representing redundant links. Degenerate Geared Graph Degenerate geared graph contains embedded structure which has m-link gear structure of a zero-degree-of-freedom and consists of m links, m-1 revolute pairs and m-1 gear pairs. Graph of an m-link gear structure is an m-vertex geared graph which has m-1 f-circuits. For example Figure 6 (a) show a nine-vertex geared graph in which four f-circuits (1,6), (6,3), (1,7), and (7,3) from a five vertex geared structure 6,7,5,1 and 3; such that this sub graph contains five vertices, four geared edges and four revolute edges, See Figure 6 (b). So it is a degenerate geared graph. F10: The graph does not contain any embedded structures. Gear Train Primitives The functional representation of any epicyclic gear mechanism consists of some primitives which have some dimensional characteristics that should be considered during sketching of the functional schematic of a mechanism. Sun Gear The sun gear primitive is shown in Figure 7 (a). The parameters characterizing this primitive are the face width, the hub width, the hub diameter, and the gear radius. Carrier Primitive is shown in Figure 7 (b). All planet gears of an epicyclic gear train are supported on second level by the higher level axis of the carrier. For case of single planet, the height of axis of planet gear and axis of the sun gear is an editor@tjprc.org

6 84 H. Eleashy & M. S. Elgayyar important dimensional characteristic. Beside that options of whether the carrier should face left or right. For case of double planet gears, there are some additional parameters such as center distance between the two second level planet gears axes if they have external meshing. Ring Gear Figure 7: Primitives Consisting the Functional Representation span. The ring gear primitive is shown in Figure 7 (c). The most important parameters are its radius and nominal edge Planet Gear The planet gear primitive is shown in Figure 7 (d, e) representing both single planet and double planet gears. The most important parameters are the distance between the two compound planet gears. Noting that; the number of gears of the planet is determined by the number of geared edges incident on the vertex representing that planet. Simple Epicyclic Gear Train A simple planetary gear train shown in Figure 1 (b) consists of only a ring gear, a sun gear and a carrier. This structure can be made more compacted if the single planet gear is connected to both the ring gear and the sun gear as shown in Figure 1 (c). This simple gear train is considered a primitive because most epicyclic gear mechanisms contain at least one. Thus it will be used to sketch the functional representation. DETECTION OF GEAR PAIR TYPE The planet gears at the second level vertices may be a single planet gear or a multiple planet gear as shown in Figure 1 (a, b). A planet-to-sun or planet-to-ring connections represents the gear pair connections between first level vertices and second level vertices according to type of gear pair whether it is external or internal. If there is a gear pair connection between two vertices at second level representing a planet-to-planet gear edge, hence this gear pair between two planet gears may be theoretically external or internal. Figure 8 (a, b) shows external and internal planet gear pairs respectively. All gear pairs in such a displacement graph are labeled by "g" expressing external meshing and by "G" expressing internal meshing in all possible ways. This will produce a number of displacement graphs generated from the same displacement graph but each one represents a different mechanism depending on type of gear pairs involved. These generated labeled graphs may contain some isomorphic graphs that should be excluded from the considered graphs. Impact Factor (JCC): Index Copernicus Value (ICV): 3.0

7 A New Technique for Sketching Epicyclic Gear Mechanisms Using Acyclic Graph 85 Figure 9 shows some displacement graphs generated from the displacement graph of the acyclic graph This assignment gear pair type is the first step in the reverse transformation process. Figure 8: External and Internal Planet Gear Pairs Figure 9: Labeled Displacement Graphs with External and Internal Gear Pairs Elementary Gear Units The second step in the reverse transformation process is to divide the displacement graph into its elementary gear units depending on the planet gears as dividing elements. Sub graph formed by an isolated second level single planet or planet chain and all the lower level vertices connecting to them by geared edge connections or by revolute joints is called the elementary gear units. So that the elementary gear units can be categorized into two general types. The first is the single planet unit which contains one planet link and one carrier. The second is the planet chain unit if it contains two meshing planets and two carriers. Figure 10 (b, c) illustrates the two elementary gear units that can be derived from Figure 10 (a). Figure 10: Elementary Gear Units The concept of isolated second level planet or planet chain is achieved by deleting all other second level planets or planet chains and their connections to first level vertices and first level vertices themselves which have no direct connections with second level planet or planet chain under consideration. For single planet elementary gear units there is a general form for its functional schematic as shown in Figure 11. General form of functional schematic of a planet chain elementary gear unit is shown in Figure 12. It seems to be complicated because of presence of a ring gear in the middle of elementary gear unit. It may be less complicated if there is no sun gear meshing with planet at the higher level. In such a case the ring gear can be sketched at the end of functional schematic of the elementary gear unit as shown in Figure editor@tjprc.org

8 86 H. Eleashy & M. S. Elgayyar Thus the presence of sun gear, meshing with the planet at the higher level, or not produces two general forms of planet chain elementary gear units Figure 11: General Form of Single Planet Elementary Gear Unit Figure 12: General Form of Planet Chain Elementary Gear Unit Having Sun-Higher Planet Meshing Figure 13: General Form of Planet Chain Elementary Gear Unit having No Sun-Higher Planet Meshing TRANSFORMATION OF ELEMENTARY GEAR UNITS Generally, during the connection of elementary gear units there should not be any crossing of links. The common links of elementary gear units can be connected by means of coaxial shafts or overhead connections. Welding Points Each elementary gear unit has a central axis which lies on common axis of all elementary gear units. Each elementary gear unit has some points along its axis. These points are those can be connected by shafts to similar points of other elementary gear units. These points are called welding points. These links may have a common link between more than elementary gear units. The ring gear can have two welding points one by the coaxial shafts and the other by an overhead connection. Also carrier can have two welding points as shown in Figure 14 (a, b) for single planet elementary gear unit and planet chain elementary gear unit respectively. Welding points are labeled by the corresponding Impact Factor (JCC): Index Copernicus Value (ICV): 3.0

9 A New Technique for Sketching Epicyclic Gear Mechanisms Using Acyclic Graph 87 link number or label such as "R" for the ring gear having two welding points, "r" for the ring gear having one welding point, "s" for sun gear, and "c" for the carrier. See also Figure 15 (c). Figure 14: Welding Points for Single Planet and Planet Chain Elementary Gear Units Welding Points Edge Connections There are two types of connections joining welding points of elementary gear units. The first is primary edges which join welding points of the same elementary gear unit. The other is secondary edges that connect two welding points having same label of two different elementary gear units. Each secondary edge has a label which is same label of its end welding points. This secondary edge represents a shaft connection between two links. There are some rules during sketching primary and secondary edges: All secondary edges are represented as dashed path and should be below line formed by the primary edges (that represents the central axis of the functional representation) Any two welding points of the same label should be connected by a path of several primary edges or secondary edges of the same label or a combination of both. A secondary edge should not be intersected with another secondary edge or with primary edges except at its ends. There should not be a circuit formed by secondary edges of same label and the primary edges. Test for Accessibility Elementary gear units are arranged such that a point marked by a big hollow circle is drawn before and after each elementary gear unit. The welding point is accessible if any secondary edge can be drawn from that welding points to any of the marked points without crossing any other edge otherwise the welding point is not accessible. This test ensures the concept of accessibility. A link is not accessible if all its welding points are not accessible. As shown in Figure 15 (e), link 4 is not accessible in the first elementary gear unit. But in another arrangement of welding points this link can be converted to an accessible link. See Figure 15 (f) Algorithm for Connecting Elementary Gear Units Based on the above rules an algorithm will be presented in the following issues: editor@tjprc.org

10 88 H. Eleashy & M. S. Elgayyar Select one elementary gear unit such that it contains two links having common connections with another elementary gear unit. And specify its primitives such as the sun gear, the ring gear etc. Figure 15: Steps of Connecting Elementary Gear Units Selected unit is represented by means of its welding points and primary edges on its axis. Such that a ring gear and the carrier can have two welding points and the sun gear has only one welding point. Noting that two welding points representing link that have a common connection with another elementary gear unit should be placed adjacent to each other, otherwise welding point which lies between them will be inaccessible. See Figure 15 (d, e). Select another elementary gear unit to be the current one and the first elementary gear unit will be preceding one. This selected unit should have maximum number of links. This current unit is represented in the same way and at right of preceding one. Connect welding points of preceding unit and its corresponding welding points in current unit without any crossing. If there is any crossing, the order of welding points of the preceding unit can be changed. If there still any crossing a new preceding unit should be selected. So the corresponding welding points of the current unit Impact Factor (JCC): Index Copernicus Value (ICV): 3.0

11 A New Technique for Sketching Epicyclic Gear Mechanisms Using Acyclic Graph 89 should be such that the welding points "a' lies at the left of the welding point "b". See Figure 15 (f). Thus permutation of welding points or even the elementary gear units may be performed to overcome the problem of inaccessibility. Select a new elementary gear unit and represent this current one in the same way based on issues 3 and 4. See Figure 15 (g) After secondary edge connections are completed, there are some welding points that are not connected to any secondary edges. These welding points can be removed if corresponding welding points of the same label are existed in another elementary gear unit. See Figure 15 (h) Functional representation can be sketched by mapping or transforming the welding points into their associated primitives as shown in Figure 16. Figure 16: Constructing of a Functional Figure 17: Conversion of Coaxial Shafts Representation into Overhead Connection Some of the shaft connections between adjacent units can be converted into overhead connections as shown in Figure 17 (a, b). But there are specific conditions [6] for that conversion: The welding points are adjacent to each other. Both of welding points belong to a certain primitive (a carrier or a ring) One more welding points of the same label should be exists in other form of primitives at another unit. RESULTS AND DISCUSSIONS Since it is important to study the practicability of the enumerated displacement graphs from design and manufacturing view point for different application of epicyclic gear mechanisms, thus the next logical step after enumerating non-isomorphic displacement graphs is regenerating the mechanism. Acyclic graph has been used to transform enumerated displacement graphs to their corresponding functional representation. A new methodology for reverse transformation process has been introduced without using the concept of adjacency matrix. The displacement graph is divided into its elementary gear units which are then connected by means of welding point's technique to form the functional representation of such a mechanism. editor@tjprc.org

12 90 H. Eleashy & M. S. Elgayyar CONCLUSIONS In this work, the structural characteristics of epicyclic gear mechanisms have been identified. A displacement graph and an acyclic graph and their fundamental rules have been presented. Systematic procedures of reverse transformation process have been developed and illustrated to transform the enumerated displacement graphs into their corresponding functional representation using the concept of elementary gear units. These elementary units are connected using the notation of welding points and primary and secondary joints. The proposed methodology can be used to construct a data base for n-link epicyclic gear mechanisms. REFERENCES 1. Freudenstein (1971), An Application of Boolean Algebra to the Motion of Epicyclic drives. ASME Journal of Engineering for Industry, Vol. 93, series B, pp Mruthyunjaya, T.S., and Ravisankar, R., 1985, Computerized Synthesis of the Structure of Geared Kinematic Chains, Journal of Mechanisms and machine Theory, Vol. 20, pp L. W. Tsai and Lin (1988), The Creation of True Two Degree of Freedom Epicyclic Gear Trains, Number: TR 88-21, Institute for System Research, University of Maryland. 4. L. W. Tsai (1987) An Application of the Linkage Characteristic Polynomial to the Topological Synthesis of Epicyclic Gear Train. ASME Journal of Mechanisms, Transmissions, and Automation in Design, 109, No.3, Chatterjee, G. and L. W. Tsai (1994) Enumeration of Epicyclic-type Automatic Transmission Gear Trains. Transactions of SAE Technical Paper , SAE, Warrendale, PA, U.S.A. 6. L. W. Tsai (1995) An Application of Graph Theory to the Detection of Fundamental Circuits in Epicyclic Gear Trains. ASME Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 109, No.3, pp Hsieh, H. I. and L. W. Tsai (1996a) A Methodology for Enumeration of Clutching Sequences Associated with Epicyclic-type Automatic Transmission Mechanisms. Transactions of SAE Technical Paper , SAE, Warrendale, PA, U.S.A. 8. Hsieh, H. I. and L. W. Tsai (1996b) Kinematic analysis of epicyclic-type transmission mechanisms using the concept of fundamental geared entities. ASME Journal of Mechanical Design, Vol. 118, pp Hsu, C. H. (1999) Systematic Enumeration of Epicyclic Gear Mechanisms for Automotives. JSME International Journal, Series C, vol. 42, pp Hsu, C. H.and Y.C.YEH and Z.R.YANG (2001) Epicyclic Gear Mechanisms for Multi-Speed Automotive Automatic Transmissions. Proc. Natl. Sci. Counc. ROC (A), Vol. 25, No.1, pp H. ELEASHY & M. S. ELGAYYAR, Topological Synthesis of Epicyclic Gear Mechanisms Using Graphical Technique, International Journal of Mechanical and Production Engineering Research and Development (IJMPERD), Vol. 3, Issue 5, Dec 2013, Impact Factor (JCC): Index Copernicus Value (ICV): 3.0