INTERNATIONAL JOURNAL OF RESEARCH AND INNOVATION

Subject: Mechanical Engineering IJRIME AUTOMATIC IDENTIFICATION OF DEGENERATE STRUCTURE IN 1-DOF PLANETARY GEAR TRAINS M.Amrutha 1, V.V. Kamesh (PhD) 2, V.Srinivasa Rao 3 1 Research Scholar, Department of Mechanical Engineering, Aditya Engineering College, Surampalem, Andhra Pradesh, India. 2 Professor, Department of Mechanical Engineering, Aditya Engineering College, Surampalem, Andhra Pradesh, India. 3 Professor, Department of Mechanical Engineering, Aditya Engineering College, Surampalem, Andhra Pradesh, India. Abstract Planetary gear trains find wide range of applications in various fields like power transmission in automobiles, machine tool gearboxes, robot manipulators and other transmissions to transmit motion and power from one rotating shaft to another. The present paper deals with a detailed concept regarding the automatic identification of degenerate structure in 1-DOF Planetary gear train by using computer program. Taking the coding of the graphs of different planetary gear train structures up to nine members, from that developed a circuit matrix with the help of gear pair arms. By using this categorized that circuit matrix into sub matrices based on planet gear and arm notations. This is extended in detecting degenerate structures of different planetary gear train structures. C Program is written to evaluate the results of all the planetary gear structures which contain degenerate structures. Results are in agreement with available literature. *Corresponding Author: M.Amrutha, Research Scholar, Department of Mechanical Engineering, Aditya Engineering College, Surampalem, Andhra Pradesh, India. Email: amruthamandiga30@gmail.com Year of publication: 2017 Paper Type: Review paper Review Type: peer reviewed Volume: IV, Issue: II *Citation: M.Amrutha, Research Scholar, "Automatic Identification of Degenerate Structure In 1-Dof Planetary Gear Trains" International Journal of Research and Innovation (IJRI) 4.2 (2017) 766-775. Introduction Gear transmissions are complex systems that are generally used in power transmission mechanisms of vehicles, helicopters, etc. A group of gears are utilized to impart motion from primary shaft of machine to numerous revolving parts. Sometimes, two or more than two gears are made to mesh with each other to impart motion from one shaft to another. Such a combination is called train of toothed wheels or gear train. CLASSIFICATION OF GEAR TRAINS: Gear trains are classified into fallowing types: 1. Simple gear trains 2. Compound gear trains 3. Reverted gear trains 4. Epicyclic (or planetary) gear trains In the first three types of gear trains, the axes of the shafts over which the gears are mounted are fixed relative to each other. But in case of epicyclic gear trains, the axes of the shafts on which the gears are mounted may move relative to a fixed axis. 766

EPICYCLIC (OR PLANETARY) GEAR TRAINS : The axes of the shafts, over which the gears are mounted, may move relative to a fixed axis. A simple epicyclic gear train is shown in Fig.1.1, wherever a gear A and the arm C have a common axis at O1 about which they will rotate. The gear B meshes with gear A and has its axis on the arm at O2, about which the gear B will rotate. If the arm is fixed, the gear train is simple and gear A can drive gear B or vice- versa, however if gear A is fastened and the arm is rotated about the axis of gear A (i.e. O1), then the gear B is forced to rotate upon and around gear A. Such a motion is termed epicyclic and the gear trains organized in such a manner that 1 or more of their members move upon and around another member is known as epicyclic gear trains (epi. means upon and cyclic means around). The epicyclic gear trains may be simple or compound. The epicyclic gear trains are useful for transmitting high velocity ratios with gears of moderate size in a comparatively lesser area. The epicyclic gear trains are used in the back gear of lathe, differential gears of the automobiles, hoists, pulley blocks, wrist watches etc. ring gear and additionally rotate round the central gear i.e. the sun gear. 4. Arms: Arms assembly hold planet gears along similarly as equally spaced also in an orbit. Figure shows the various elements of planetary gearing arrangement. Planetary gear is one variety of epicyclic gear which could be used where we need precise motion control. usually in industries, there are immeasurable applications where we tend to transmit higher torque in limited space as well as we require light weight unit for transmitting the power, motion and torque from driving equipment to driven equipment. In case of planetary gear system, we can increase the torque density (or) power transmission very easily by using a lot of planet gears between the sun gear and ring gear. OBJECTIVE: The main objective of the present paper is to find out of the degenerate structure of planetary gear trains of 4 links to 9 links by using computer program. LITERATURE SURVEY : Many researches contributed to the area of kinematic synthesis of planetary gear trains. In that some scholars using mathematical theorems and algorithms, and other scholars using computer programs for the analysis of kinematic synthesis. The huge amount of research efforts and scientific literature is available in the field of power transmission and synthesis of planetary gear trains. COMPONENTS OF PLANETARY GEAR SYSTEM: Planetary gear system can have following components as mentioned below. 1.Sun gear: It is the central gear that is meshing with planet gear. 2.Ring gear: It is the internal gear is that the outer gear, that is meshing with planet gears. 3. Planet gears: There will be 2 or more than two planet gears between sun gear and ring gear and these planet gears will mesh with internal gear i.e. In 2017 V.VKamesh, K.mallikarjuna rao and A.Balaji srinivasa rao [1] presented a paper on graph theory by using this a new algorithm is developed for the determination of degenerate structures in epicyclic gear trains. This algorithm is mainly depends on rotational graphs and basic circuits, by using one key element degeneracy is determined very easily when compared to other methods. Salgado and DelCastillo[2]evaluated an approach for detecting degenerate structures in epicyclic (or) planetary gear trains (PGTs). During this paper the graph illustration of the kinematic structure of PGTs, the method (or) strategy uses the thought of basic circuits and their grouping to get the detection algorithm. A computer implementation of the algorithm 767

was used to obtain the graphs of all the PGTs of up to 9 members that contain degenerate structures. Pin Liu and Zen Chen[3] in their article conferred a review regarding the concept of kinematic fractionation is introduced for planetary gear trains (PGTs) that contains structural fractionation as a degenerate case. With the concept of kinematic fractionation, kinematically independent group(s) embedded in an PGT can be identified. A composition list, that depicts the links and also the link association within the associated cluster, is employed to work out the type of fractionation. It is found that most previously enumerated structurally non-fractionated PGTs within the literature area unit kinematically fractionated. It is shown that a structurally non-fractionated PGT may be kinematically fractionated where as a structurally fractionated PGT should be kinematically fractionated. Cheng[4] presents an analytic method for the determination of the gear ratios of an epicyclic-type transmission mechanism for a given set of speed ratios, the associated speed ratio relations are derived as a criterion to detect the feasibility of any given set of speed ratios Finally, the transmission mechanism is decomposed into a system of basic geared entities such the gear ratios of all the mating gears are determined from given speed ratios in an analytic way. In 2017 V.VKamesh, K.mallikarjuna rao and A.Balaji srinivasa rao [5] developed a new method by using a parameter called Vertex Incidence Polynomial (VIP), as well as Rotation Index (RI) to synthesize epicyclic gear trains up to six links. Each gear train is represented as graph and by using these techniques very easily eliminates the isomorphism and also displacement graphs can be derived by adding the transfer vertices for each combination. These techniques are very easy when compared to other techniques for evaluation of Isomorphic graphs as well as rotational isomorphism in epicyclic gear trains. Y. V. D. Rao and A. C. Rao [6] in their paper by using a graph theory a new variety of epicyclic gear trains are generated that can be achieved in phases, initially GKS are converted into graph then graphs are converted into algebraic representation by using vertex-vertex adjacency matrix. Detection of isomorphism in PGTs is also done in this paper by using Hamming matrix as that is written by using adjacency matrix. This present paper will use for two different objectives those are checking fir isomorphism as well as compactness by using matrix Ling Xue, Hui Yang and Liu [7] in their paper presented a gear train investigation and structure for planetary gear trains by using graph theory and it is very efficient and systematic approach in the process of gear transmission design system. An evaluation of the graph based theory and it s methods for mechanical efficiency, kinematic and static flow analysis, power flow computations are represented in their paper. The graph theory method is based on the concept of basic fundamental circuit corresponding to fundamental planetary gear train. A one degree of freedom planetary gear train and 2 stage epicyclic gear trains are used to explain the application of this method. Apart from this isomorphism identification is also done in the synthesis process in that one degree of freedom planetary gear train graphs are also used. Also, the computerized methods for identification of redundant gears and degenerate structures in planetary gear trains are analyzed. Dell Castillo [] in this paper a procedure for the list of graphs one degree of freedom epicyclic gear trains is represented. As well as this list consist of only 1input shaft and only 1 output shaft, and these are obeys some functional laws that eliminates the existence of idle links and this process used for checking of isomorphism along with the resulting graphs and also it identifies the degenerate structures of those graphs It consist of total graphs of epicyclic gear trains having nine links. Ravisankar and Mruthyunjaya [9] in their article conferred a review regarding the concept of fully automation method for structural synthesis of GKC it is useful for derived the planetary gear drives. By using the graphs in this paper chains of planetary gears are represented in the form of algebraically by their vertexvertex incidence matrices. These are very advantageous and useful for implementation on a computer. The computerized or fully automation approach has been useful for the basic synthesis of 1-DOF GKC with up to 4 gear pairs are presented in this paper. Cheng and Kin [10] in their article conferred a review regarding the concept of particular sequence of steps for the computerized investigation of the kinematic structure of epicyclic gear trains with any number of DOF. In this paper a new approach of graphs are introduced to representation of kinematic structure of epicyclic gear trains. Later, one more approach is identified for the isomorphism of epicyclic gear trains. After non fractionated multi degree of freedom epicyclic gear trains can be identified from the graphs. Fin0ally a fully automated plan is developed for the automatic investigation of the structure of epicyclic gear trains. This paper results is very useful in the design and development of new epicyclic gear trains. Chieng and Hoeltzel [11] in their paper introduce a advanced computerized mechanism for converting the graph representation into Skelton diagram representation for avoiding the cross links in the system and this is used for clear idea of functional features of a mechanism,for algorithmically finding the numerically continuous mechanism sketching problem as a discrete domain problem. This new approach will consistently generate explicit, concrete sketching constraints, thereby providing a well outlined methodology for mechanism sketching. Li and Schmidt [12] in their article a tool is developed for planetary gear trains that is very helpful for the designers to generates the various structures of PGTs that tool is simple called as a grammar-based designer assistance tool for PGTs and this tool consist of 3 elements in that first one is graph generation, second one is functional sketching and the last one is concept collection module. In this paper mainly concentrated on first 2 elements and this paper is starting analysis for the grammar based designer tool and it is very useful in design field. 76

Kamesh, V V; Rao, K Mallikarjuna; Rao, A B Srinivasa [13] presented a paper in this a simple and easy technique is developed for the determination of isomorphism in planetary gear trains by using a new parameter called as 'Functional Value of Gear Train' by using this parameter the method to assess the overall influence of all links(4-6) on a single link and vice-versa, and also by using this parameter non-isomorphic graphs and rotational Isomorphism is also evaluated in very easy manner when compared to other methods computation steps is very less. Del Castillo [14] in his paper explained about it possible to deduce symbolically the efficiency. The analytical expression of the efficiency is not unique for a given planetary gear train, since it depends on the power flow structure. The number of different cases for this structure increases rapidly with the complexity of the gear train. In spite of this, it is shown that it is possible to easily derive an analytical expression thanks to the special nature of the torque and power equations satisfied by the gear train. The procedure is applied to several types of planetary gear trains. In 1997 Cheng ho hsu and yi-chang wu [15] presented a paper on automatic detection of embedded structures in planetary gear trains in this paper a computer based program is developed based on graph representation and fundamental circuits using the parameter called adjacency matrix it will automatically detects the structures very easily. STRUCTURAL REPRESENTATION: In structural representation each and every link of a mechanism is represented by a polygon whose vertices represent the kinematic pairs. In structural representation gear pair and turning pair are represented with different colours (or) different indications. Cheng ho hsu and kin-tak lam [16] developed a new methodology for graph representation of kinematic structure of planetary spur gear because of the graph representation evaluation of planetary gears are very easy and also by using graphs computer program is developed which is very useful for the synthesis of planetary spur gears with any DOF. VARIOUS REPRESENTATIONS OF PLANETARY GEAR TRAINS: Planetary gear trains are commonly used in power transmission system in order to transmit power (or) motion from one shaft to another because of its various advantageous like light weight, compactness, high speed ratios are possible with limited number of elements. These are mainly consist of gears (sun gear and planet gear ) and arm, each pair of gear has connected with a arm (or) carrier, and the distance between the 2 gear centres remains constant. Planetary gear trains are mainly represented into 3 ways: 1. Functional representation 2. Structural representation 3. Graph representation FUNCTIONAL REPRESENTATION: Functional representation of the planetary gear trains refer to the cross-sectional view drawing (or) conventional schematic structure drawing of a mechanism, by using this diagram we can represent all relationships among functional elements of the shafts, links, and joints of gear trains. For more clarity functional representations are very essential for entire system. GRAPH REPRESENTATION: In graph representation the vertices denotes link and edge denotes joints of a mechanism and mostly it has been applied to epicyclic gear trains synthesis and their analysis. It is very efficient and systematic modelling approach in the design process of gear transmission. In this representation gears joints are represented by dashed edge and revolute joints are represented by solid edge. In this representation members are arranged into 3 phases (or) rows according to their representation, arms are arranged in the upper phase, planet in the middle phase, and sun in the lower phase and then members are generates circuits based upon the above information and the circuit matrix of geared kinematic chain is given below. 769

TRANSFER VERTEX : In a fundamental circuit there is a vertex called transfer vertex with only turning pair edges incident on it. Also all the edges on one side of the transfer vertex are at the same level and edges on the other side of the transfer vertex are at a different level. The number of fundamental circuits (FC) is equal to number of gear pairs in the graph. So that FCs is equal to (n-f-1). METHOD TO DETECT THE DEGENERATE STRUCTURE IN PLANETARY GEAR TRAINS: The algorithm was proposed for the automatic identification of degenerate structure is predicated on the analysis of the circuit matrix (or) cicuit system (or) grid C and which consist of some properties in order to write the circuit matrix, and the algorithm consist of the subsequent steps : 1. The circuit system was developed based upon the given diagram (or) GKC s and which might contains all the information about it. ROTATIONAL GRAPH REPRESENTATION : Planetary gear trains are represented by using rotational graph representation. In a graph of a PGT the sub graph obtained by deleting all the gear pair edges from a tree. Any gear pairs edges added to a tree forms a fundamental circuit (FC), which is a circuit formed with the gear pair edge and several turning pair edges. 2. The circuit system C sorted out into the other systems called as sub matrix which indicated with Cm each of these new matrix (or) sub matrix (of estimated dimensions rm x 3) will be formed by the circuit grid that have their planet and arm in like way. 3. Each new system gained will be denoted by m and the amount of lines is indicated with rm and it will be the amount of circuits from which it is molded. 4. The set of elements Cmm is portrayed as the course of action of different members from each grid Cm. 5. Then find the possible combinations of the set Cmm such that the sum of the number of rows (rm) of the system Cm to which each set in the combination (or) mixes corresponding is not as much as (J-1) where J is nothing but gear pair. One hence has combination of 2 (or) more sets Cmm. 6. After this find the different sets (St) that is the union of the sets Cmm constituting each of the combinations found in the previous step, the suffix t will refer to each of these combination. 7. Finally, one calculate the amount of different element Lt of each of these sets St then if it is found that Lt minus the total number of circuits of the matrices Cm making up the combination t is equal to unity, it can be concluded that the structure formed by the members of those circuit was degenerate structure otherwise that structure was not a degenerate structure.. If the sub matrix was a single matrix then that structure was automatically not a degenerate structure. ILLUSTRATIVE EXAMPLES: By considering the algorithm following examples are illustrated as follows: 770

Example 1: (1)The Circuit Matrix C is 1 5 6 4 7 7 3 6 2 5 4 7 3 6 (2) C1 C2 C3 1 5 6 4 7 7 3 6 2 5 4 7 3 6 (3) C11 = 1 2 5 C22 = 6 4 7 C33 = 7 3 6 (1)The Circuit Matrix C is 1 4 7 1 5 9 4 7 7 5 6 9 2 6 9 3 6 9 (2) C1 C2 C3 1 4 7 1 5 6 9 9 4 7 7 5 2 6 9 3 6 9 (3) C11 = 1 9 4 7 C22 = 1 7 5 C33 = 2 3 6 9 (4) The possible combinations are all those consisting of two of the three subsets such that the number of circuits is less than 5. One must therefore analyse: Combination 1: C11 and C22 Combination 2: C11 and C33 Combination 3: C22 and C33 (5) The number of circuits involved depends on the combinations. (4) The possible combinations are all those consisting of two of the three subsets such that the number of circuits is less than 5. One must therefore analyse: Combination 1: C11 and C22 Combination 2: C11 and C33 Combination 3: C22 and C33 (5) The number of circuits involved depends on the combinations. s1 = C11 U C22= [ 1 2 5 6 4 7 ] = L1=7-4 =3 s2 = C22 U C33= [ 6 7 3 4 ] = L2=5-4 =1=DEGENERATE s3 = C11 U C33= [1 2 5 7 3 6 ] = L3=7-4 = 3 planetary gear train structure is degenerate. PROGRAM OUTPUT : Example 3: s1 = C11 U C22= [1 7 5 9 4 ]= L1=6-4= 2 s2 = C22 U C33 =[ 1 7 5 2 3 6 9 ] = L2=-5= 3 s3 = C11 U C33= [ 1 2 3 6 9 4 7 ] = L3=-5= 3 Planetary gear train structure is not a degenerate structure. Example 2: Enter Planetary Gear Train Structure Code 430a Enter no. of Links between 4 to 9 No of Gear Pairs 6 Enter 1 gear pair and arm 1 5 Enter 2 gear pair and arm 2 5 Enter 3 gear pair and arm 2 6 771

Enter 4 gear pair and arm 2 7 Enter 5 gear pair and arm 3 6 Enter 6 gear pair and arm 4 7 The Circuit Matrix C is 1 5 2 5 2 6 2 7 3 6 4 7 The sub Matrices are C1 1 5 2 5 C2 2 6 3 6 C3 2 7 4 7 Merged output of sub matrices: C11 = 1 2 5 C22 = 2 3 6 C33 = 2 4 7 L1 = C11 U C22 1 2 3 5 6 = 6 L2 = C22 U C33 2 3 4 6 7 = 6 L3 = C11 U C33 1 2 4 5 7 = 6 L11 = 4 L22 = 4 L33 = 4 s1 = L1 Max (L11, L22) = 2 s2 = L2 Max (L22, L33) = 2 s3 = L3 Max (L11, L33) = 2 430a planetary gear train structure is not a degenerate structure. VI. RESULTS In this paper it is totally consist of 34 degenerate structures in all 4-9 links. D.R.Salgado & J.M.Del Castilo proposed a algorithm for the detection of degenerate structures in planetary gear trains we are considered same algorithm, and developed a computer program by using of C language. While executing this program it can be observed that one error found in the code 9431Bf.This code is actually a degenerate structure, but in the base paper it is given as not a degenerate structure. The output results are tabulated in table 6.1. 1. In this total system, in 4 links geared kinematic chain, number of diagrams is one and that is not a degenerate structure. 2. In this total system, in 5 links geared kinematic chain, number of diagrams is one and that is also not a degenerate structure. 3. In this total system, in 6 links geared kinematic chain, it has been observed that number of diagrams are four in that all four diagrams are not a degenerate structure. 4. In this total system, in 7 links geared kinematic chain, it has been observed that numbers of diagrams are five in that all five diagrams are not a degenerate structures. 5. In this total system, in links geared kinematic chain, number of diagrams are 43 in that eight diagrams are degenerate structures and remaining structure diagrams are not degenerate structures. 6. In this total system, 9 links geared kinematic chain, number of diagrams are115. In that 26 diagrams are degenerate structures and remaining structure diagrams are not a degenerate structures. Table Output results of various planetary gear train structures No. of links (N) CONCLUSION No. of PGTs Diagrams 4 1 Nil 5 1 Nil 6 4 Nil 7 5 Nil 43 9 115 26 No. of degenerate PGTs This commenced project work describes a straightforward and effective algorithm for the automatic identification of degenerate structure in 1-DOF planetary gear trains. These structure are taken from the circuits of the graph that represents the geared kinematic chain based on these analysis circuit matrix Cm are developed, by using this circuit matrix. I am implemented a computer program by using c language and we have to execute all GKC structures of (4 links to 9 links) while executing these geared kinematic chain diagrams we can observe that total 34 degenerate structures in planetary gear trains. In this paper, a program is implemented for automatic identification of degenerate structure in 1-DOF planetary gear trains by using C language 772

REFERENCES 1. Kamesh,V.V., Mallikarjuna rao,k., and srinivasa rao, A.B., 2017 Detection of Degenerate Structure in Single Degree-of- Freedom Planetary Gear trains, Journal of Mechanical Design 139() : 03302 Paper No: MD-17-1032; doi: 10.1115/1.403672. 2. D.R. Salgado and Del Castillo, J.M., 2005 A method for detecting degenerate structures in planetary gear trains, Mechanism and Machine Theory, Vol.40 : pp.94-962. 3. Chia-Pin Liu and Dar-Zen Chen, 2000 On the Embedded kinematic fractionation of epicyclic Gear trains, Journal of Mechanical Design, Vol.122 : pp.479-43. 4. Cheng-Ho Hsu, 2002 An Analytic methodology for the kinematic synthesis of epicyclic gear mechanisms, Journal of Mechanical Design, Vol.124 : pp.574-59. 5. Kamesh,V.V., Mallikarjuna rao,k., and srinivasa rao, A.B., 2017 Topological Synthesis of Epicyclic Gear Trains Using Vertex Incidence Polynomial, Journal of Mechanical Design, 139(6) : 062304 Paper No: MD-16-157; doi: 10.1115/1.4036306. 13. Kamesh,V.V., Mallikarjuna rao,k., and srinivasa rao, A.B., 2016 A Novel Method to Detect Isomorphism in Epicyclic Gear Trains, i-manager's Journal on Future Engineering and Technology; Nagercoil 12.1 : 2-35. 14. Dell Castillo, Jose M., 2000 Symbolic computation of planetary gear train efficiency, European Congress on Computational Methods in Applied Sciences and Engineering, pp.1-20. 15. Cheng ho hsu and yi-chang wu, 1997 Automatic detection of embedded structure in planetary gear trains, Journal of mechanical design, vol.119 : pp.315-31. 16. Cheng ho hsu and kin-tak lam, 1992 A new graph representation for the automatic kinematic analysis of planetary spur gears trains, Journal of mechanical design, Vol.114 : pp.196-200. APPENDIX - I 6. Rao Y. V. D., and Rao, A. C., 200 Generation of epicyclic gear trains of one degree of freedom, Journal of Mechanical Design, Vol.130 : pp. 052604-1- 052604-. 7. Hui-Ling Xue, Geng Liu and Xia0-Hui Yang, 2015 A review of graph theory application research in gears, Journal of Mechanical Engineering Science, pp.1-1.. Dell Castillo, Jose M., Enumeration of 1-DOF planetary gear trains graphs based on functional constraints, Journal of Mechanical Design, Vol.124 : pp.723-732. 9. Ravisankar, R., and Mruthyunjaya, T.S.,195 Computerized synthesis of the structure of geared kinematic chains, Mechanism and machine theory, Vol.20(5) : pp.367-37. 10. Cheng-Ho hsu, and Kin- Tak Lam,1993 Automatic analysis of kinematic structure of planetary gear trains, Journal of Mechanical Design, Vol.115 : pp.631-63. 11. Wei-Hua Chieng and Hoeltzel, D. A., 1990 A combinatorial approach for the automatic sketching of planar kinematic chains and epicyclic gear trains, Journal of Mechanical Design, Vol.112 : pp.6-15. 12. Xin Li and Linda Schmidt, 2004 Grammarbased designer assistance tool for epicyclic gear trains, Journal of Mechanical Design, Vol.126 : pp.95-902. 773

APPENDIX II 774

AUTHORS M.Amrutha, Research Scholar, Department of Mechanical Engineering, Aditya Engineering College, Surampalem, Andhra Pradesh, India. V.V. Kamesh (PhD), Professor, Department of Mechanical Engineering, Aditya Engineering College, Surampalem, Andhra Pradesh, India. V.Srinivasa Rao, Professor, Department of Mechanical Engineering, Aditya Engineering College, Surampalem, Andhra Pradesh, India. 775