Article


 

  AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES   15 (2022) 009–017 
 

   

       Contents lists available at http://qu.edu.iq 

 

Al-Qadisiyah Journal for Engineering Sciences 

  
Journal homepage: http://qu.edu.iq/journaleng/index.php/JQES  

  

 

* Corresponding author.  

E-mail address: eng.mec.mas.20.3@qu.edu.iq (Hind A. Nafeh) 

 

https://doi.org/10.30772/qjes.v15i1.807  

2411-7773/© 2021 University of Al-Qadisiyah. All rights reserved.                                This work is licensed under a Creative Commons Attribution 4.0 International License. 

    

Genetically compatible graphs for planetary gear train synthesis  

Hind A. Nafeh* and  Essam Lauibi Esmail 

Department of Mechanical Engineering, University of Al-Qadisiyah, Al-Diwaniyah, Iraq  

 

A R T I C L E I N F O 

Article history:  

Received 13 April 2022 

Received in revised form 20 May 2022 

Accepted 28 June 2022 

 

Keywords: 

Degree of Freedom 

Genetically Compatible Graphs 

Isomorphism 

Planetary Gear Trains 

Spanning Trees 

Synthesis 

Vertex Matrix  

 

A B S T R A C T 

Planetary gear trains (PGTs) are used in a wide range of mechanical applications. Graphs are used to 

represent PGTs. PGTs are classified into single-planet and multi-planet gear trains depending on the number 

of vertices in the second level of the graph. In this paper, a novel method for synthesizing PGTs up to the 

second level is developed. The first step in the synthesis process is the enumeration of spanning trees. A 

combinatorial analysis is then used to obtain the link assortment arrays and categorize them into families. 

Based on the parent graphs and spanning trees, the geared graphs are generated for each family. For 

simplicity, the steps of the newly developed genetically compatible graphing method are applied to 5 -link, 

1-DOF planetary gear trains. The synthesis steps are programmed in MATLAB. 

 

 

© 2022 University of Al-Qadisiyah. All rights reserved.    

1. Introduction

For many years, many studies have focused on planetary gear train 

synthesis and analysis [1-14]. Graph theory has aided PGT structural 

synthesis, resulting in the development of three distinct synthesis methods. 

The first method produces PGTs with 𝑁 links from 𝑁-vertex parent graphs 

by assigning geared edges to the parent graph. This is known as the non-

recursive method [1-3]. According to some researchers, this method is 

complicated to implement on a computer and requires a large number of 

parent graphs. 

The second method, known as the recursive method, generates PGTs with 

N links by adding graph components to PGTs with (𝑁 −  1) links. [4-6]. 

However, the PGTs enumerated by the recursive method are incomplete. 

The last method employs the parent-bar-linkage to enumerate PGTs. 

However, it is not currently in use. [7, 8]. PGTs with up to nine links have 

been studied using the methods described above [1-8]. For the reasons 

stated above, a method that easily provides a complete solution to the 

enumerated PGTs is required. Isomorphic graphs must be deleted as part of 

the solution. The presence of pseudo-isomorphic graphs, on the other hand, 

further complicates the isomorphism challenging task [5]. The synthesis 

results of PGTs with more than six links are in disagreement [3, 6]. To avoid 

the problem of pseudo isomorphic graphs, Chatterjee and Tsai [9, 10] 

proposed the rooted graph of PGTs. Based on spanning trees of rooted 

graphs, a method for enumerating PGTs is also proposed. Hsu and Lam [11, 

12] proposed a graph model that is identical to all pseudo isomorphic 

graphs. In addition, Hsu [13] proposed a method for detecting structural 

isomorphism. Despite the fact that Hsu [14] proposed a parent graph 

method for synthesizing PGTs, he believed that synthesizing PGTs with 

more than six links was not possible using an atlas of parent graphs. Then, 

Hsu [15] developed a new method based on acyclic graphs. 

Shanmukhasundaram et al. [16] also synthesized PGTs using acyclic 

graphs. Yang et al. [17, 18] contradicted [15 and 16] and used parent graphs 

to generate PGTs with up to nine links. Shanmukhasundaram and 

colleagues [19] examined the recursive, non-recursive, and acyclic graph- 

http://qu.edu.iq/
https://doi.org/10.30772/qjes.v15i1.807
https://doi.org/10.30772/qjes.v15i1.807
http://creativecommons.org/licenses/by/4.0/


10 HIND A. NAFEH AND ESSAM LAUIBI ESMAIL /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES   15 (2022) 009–17 

 

 

based approaches in depth. Graph theory enabled the modeling of PGT 

kinematic structure as well as the advancement of structure synthesis 

methodologies [20-28]. In this paper, geared graphs for a given number of 

links and DOF are enumerated using a method that makes use of the 

correspondence between spanning-tree graphs and parent graphs. Because 

the transfer vertices in a spanning tree graph are visible, all feasible gear 

pair connections can be identified, and geared graphs can be generated. It 

is justified that generating geared graphs from the entire collection of 

spanning-tree graphs and parent graphs results in the enumeration of an 

exhaustive and combinatorial complete set of geared graphs. The purposes 

of the work are listed as follows: 

 - To identify structural characteristics of PGTs and translate them into 

graph representation language. 

 - To overcome the problem of pseudo-isomorphism. 

 - To promote an effective method to build a structural synthesis for 

planetary gear trains with any number of links and degree of freedom.  

 - To greatly prohibit the generation of isomorphic graphs. 

For the sake of simplicity and clarity, the new method will be applied to 

PGTs with five links and 1-DOF. It applies to greater degrees of freedom 

and links. In section 2, the concepts of graph theory are defined. In section 

3, the synthesis of planetary gear trains is discussed. Finally, in section 4, 

the conclusion about the synthesis is briefed and explained.  

2. Conceptions 

Graph: - A graph (G) contains a set of vertices (V), which represent the 

number of links, jointly with a set of edges (E) which represent the joint 

between links. 

2.1.  Rooted Graph 

A rooted graph consists of many vertices. It has one vertex that is different 

from other vertices. This vertex is named the root. It is usually utilized to 

represent the basis of a mechanism or fixed link.  Fig. 1 shows a functional 

representation of the well-known Simpson gear train. 

 

 

Figure 1. Functional representation of the simpson gear train. 

 

To represent the well-known Simpson gear train, first, consider the ground 

vertex as vertex 0. Then the links are represented as vertices and distributed 

based on functional representation. After that, the dotted line is drawn for 

the revolute joint and a bold line for the gear joint. Fig. 2 denotes rooted 

graph representation of the Simpson gear train [22,23]. 

 

 

Figure 2. Rooted graph representation of the Simpson gear train. 

2.2. Spanning Tree 

A tree is a linked graph that doesn’t have loops. Fig. 3 denotes the spanning-

tree graph of the rooted graph that is shown in Fig. 2. 

 

 

Figure 3. The spanning-tree graph of the rooted graph shown in 

Figure 2. 

2.3. Parents Graph  

If all the edges of a rooted graph of PGT are assumed to be revolute or have 

the same color, then the resulting graph is the parent graph. In the parent 

graph, there is no distinction between revolute and geared edges. 

 Fig.4 shows the parent graph of the rooted graph which is shown in Fig. 2. 

By using the corresponding matrix equation, the parent graphs are 

synthesized. 

 

Nomenclature: 
      

 

A The adjacency matrix N Number of Links 
𝑎𝑖𝑗  The (𝑖, 𝑗) elements of the adjacent matrix o Root or the ground vertex 
DOF Degree of Freedom PGT Planetary gear train 

E Edges or the joint between links. V Vertices 

F Number of degrees of Freedom 𝑉2, 𝑉3, … , 𝑉𝑚 Number of binaries, ternary…m-nary vertices 
G Graph [𝑉1, 𝑉2, 𝑉3, … , 𝑉𝑚] Link assortment array 
L Number of independent loops  𝑣 Number of vertices 
𝑚 The maximal degree of vertex        



HIND A. NAFEH AND ESSAM LAUIBI ESMAIL /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES   15 (2022) 009–017                                              11 

 

 

Figure 4. parent graph of the rooted graph which is  

shown in Figure 2. 

2.4. Adjacency Matrix 

The following is a definition of an adjacency matrix element: 

𝑎𝑖𝑗 = {
1 𝑖𝑓 𝑣𝑒𝑟𝑡𝑒𝑥 𝑖 𝑖𝑠 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 𝑣𝑒𝑟𝑡𝑒𝑥 𝑗

0  𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (𝑖𝑛𝑐𝑙𝑑𝑒 𝑖 = 𝑗)               
 

Where aij  indicates the (i, j) elements of the adjacent matrix. It is a (v ∗ v) 

symmetric matrix and its diagonal elements are zeros. The degree of vertex 

is the summation of the elements of a row or column. The adjacent matrix 

has unique determination [23].  The adjacency matrix for parent graph 

shown in Fig. 4 is: 

 

𝐴 =

[
 
 
 
 
 
 
0 1 1
1 0 0
1
1
1
0
0

0
0
0
1
1

0
0
0
1
0

   

1 1 0 0
0 0 1 0
0
0
0
1
1

0
0
0
0
1

1 1
1 1
0 1
0 0
0 0]

 
 
 
 
 
 

 

2.5. Fundamental Circuit  

The addition of a chord to spanning tree results in the formation of exactly 

one circuit. A set of independent loops or fundamental circuits (FCs) are 

formed according to a combination of all circuits concerning a spanning 

tree. They serve as a foundation for the circuit space [27]. 

FC1: 0-1-5-2-0 FC2: 0-1-5-3-0 

FC3: 0-2-6-4-0 FC4: 0-2-6-3-0 

Figure 5. The four fundamental circuits of the rooted graph shown in 

Figure 2. 

3. Synthesis and analysis of planetary gear train 

The process of enumeration of PGTs is split into three steps. First step, trees 

that are suitable for the creation of rooted graphs are enumerated. Second 

step, parent graphs are enumerated. In third step, geared graphs are detected 

from parent graphs and spanning trees. The rooted graph of an N-link, F-

DOF PGT comprises (𝑁 +  1) vertices and (2 𝑁 − 𝐹 − 1) edges; hence, 

the number of independent loops can be obtained from 𝐿 =  𝑒 − 𝑣 + 1 [25-

28]. The link assortment arrays of a spanning tree and a parent graph are 

expressed as [𝑉1, 𝑉2, 𝑉3, … , 𝑉𝑚]  and [𝑉2, 𝑉3, … , 𝑉𝑚] , respectively. 

𝑉2, 𝑉3, … , 𝑉𝑚   is the number of binary, ternary…m-nary vertices in the 

spanning tree or the parent graph, respectively. The maximal degree of a 

vertex 𝑚 is obtained from 𝑚 =  𝐿 +  1. 

3.1. Topological Features of Spanning Trees 

It was shown, in a spanning tree, all geared edges are removed from the 

geared graph results. The spanning tree explains an open-loop kinematic 

chain that is consists of links which joined together via revolute joints. 

Generally, the parent graph doesn’t have a unique spanning tree. Fig. 6 

shows two spanning trees corresponding to the graph shown in Fig. 4. 

However, rooted-graph representation of PGTs rules some particular 

arrangement of the vertices. For this reason, a reliable enumeration method 

is required to synthesize the PGTs. So, a novel method to precisely 

enumerate spanning trees of N-link PGTs is presented in this paper. 

 

   

Figure 6. Two spanning trees corresponding to the graph shown in 

Figure 4. 

 

Let v denotes the vertices, and e represents the number of edges in a graph. 

Also, let Vk refer to the number of vertices of degree k. So, V1 denotes the 

number of vertices of degree one, V2 the number of vertices of degree two, 

etc. as shown in Eq. (1). 

 

𝑉1 + 𝑉2 + 𝑉3 + ⋯ + 𝑉𝑚 = 𝑣                                                                     (1) 

 

Where m refer to the maximal degree of a vertex. Since a tree of v vertices 

contain v –  1 edges and every edge have two end vertices and each of the 

Vk vertices are incident by k edges, as shown in Eq. (2). 

 

𝑉1 + 2𝑉2 + 3𝑉3 + + ⋯ + 𝑚𝑉𝑚 = 2(𝑣 –  1)                                             (2) 

 



12 HIND A. NAFEH AND ESSAM LAUIBI ESMAIL /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES   15 (2022) 009–17 

 

For example, for a 5-link, single-DOF PGT, we have v = 6, e = 8, and 

m = 4, it follows that: 

 

    𝑉1 + 𝑉2 + 𝑉3 + 𝑉4 = 6                                                                          (3) 

 

𝑉1 + 2𝑉2 + 3𝑉3 + 4𝑉4 = 10                                                                     (4) 

 

The flowchart for the enumeration of spanning tree graphs is shown in Fig. 

7. 

 

Figure 7. Flowchart for the enumeration of spanning tree graphs. 

 

By using a MATLAB program to solve Eqs. (3) and (4), we obtain the 

following link assortment arrays for the spanning trees: [4 2 1 1 1 1], [3 2 2 

1 1 1], [3 3 1 1 1 1], and [2 2 2 2 1 1]. See appendix B. 

The link assortment arrays are classified into families according to 

maximum vertex degree; the vertex with the greatest degree is chosen as 

the root. 

 

Family 1: (maximum vertex degree = 4): [4 2 1 1 1 1] 

 

Family 2: (maximum vertex degree = 3): [3 2 2 1 1 1], [3 3 1 1 1 1]  

 

Family 3: (maximum vertex degree = 2): [2 2 2 2 1 1] 

 

Table 1. Spanning trees for the 5-link, single-DOF PGT. 

Family 1 Family 3 

[4 2 1 1 1 1] [2 2 2 2 1 1] 

 
 

Family 2 

[3 2 2 1 1 1] [3 3 1 1 1 1] 

  

 

3.2. Topological features of parent graphs 

Based on equation (5) and (6), All conceivable link assortments for a given 

v-vertex graph can be derived.  

 

   𝑉2  + 𝑉3+ . . . + 𝑉𝑚 =  𝑣                                                                         (5) 

2𝑉2  + 3𝑉3 + . . . +𝑚𝑉𝑚  =  2 𝑒                                                                 (6) 

 

Taking (N =  5) for instance, we have (v =  6), (e =  8) and (m =  4). 

The link assortment Eq. is shown in Eq. (7). 

 

 𝑉2 + 𝑉3 + 𝑉4 = 6                                                                                   (7) 

2𝑉2 + 3𝑉3 + 4𝑉4 = 16                                                                             (8) 



HIND A. NAFEH AND ESSAM LAUIBI ESMAIL /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES   15 (2022) 009–017                                              13 

 

The flowchart for the enumerate of parent graphs is shown in Fig. 8. 

Five-link assortment arrays are achieved by using a MATLAB program: [4 

2 2 2 2 4], [3 3 3 2 3 2], and [3 3 2 2 3 3]. The link assortment arrays are 

also classified into families according to maximum vertex degree:   

 

 

Family 1: (maximum vertex degree = 4): [4 2 2 2 2 4] 

 

 

Family 2: (maximum vertex degree = 3): [3 3 3 2 3 2], and [3 3 2 2 3 3]. 

 

 

The second family contains two vertices in the second level, whereas the 

first family contains only one vertex. 

 

 

Figure 8. Flowchart for the enumeration of parent graphs. 

 

 

 

Table 2. parent graphs for 5-link, single-DOF PGTs. 

Family 1 Family 2 

[4 2 2 2 2 4] [3 3 3 2 3 2] [3 3 2 2 3 3] 

   

3.3. Geared graphs 

Now by comparing the parent graph and spanning trees for each family (1 

and 2), we get the following geared graphs: 

 

Table 3. Geared graphs for 5-link, single-DOF PGT. 

Family 1 

[4 2 2 2 2 4] 

[4 2 1 1 1 1] 

 

Family 2 

[3 3 3 2 3 2] [3 3 2 2 3 3] 

[3 2 2 1 1 1] [3 3 1 1 1 1] 

(a) 

(b) 

[3 3 3 2 3 2] [3 3 3 2 3 2] 

[3 2 2 1 1 1] [3 3 1 1 1 1] 

(c) (d) 

 



14 HIND A. NAFEH AND ESSAM LAUIBI ESMAIL /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES   15 (2022) 009–17 

 

The new method can be implemented on a computer using an adjacency 

matrix and a nested-do loops algorithm.  

3.4. Isomorphism 

Two graphs are isomorphic if their vertices and edges have a one-to-one 

correspondence, and their incidences are preserved. A link assortment array 

identifies each family. Isomorphism cannot exist between graphs from 

different families. In Table 3, for example, two of the four graphs in family 

2 have the same spanning tree. The link assortment arrays of the parent 

graph and spanning tree of Figures (a) and (c) are the same. The weighted 

vertex degree 𝑑𝑤  is calculated by assigning edge weights of one for 

revolute edges and two for geared edges. For example, the weighted vertex 

degree of vertices 0 through 5 of graph (a) shown in Table 3 are 3 4 4 3 5 

3, respectively. The weighted link assortment array for the graph shown in 

Fig. (a) is [3 4 4 3 5 3], whereas the weighted link assortment array for the 

graph shown in Fig. (c) is [3 4 5 2 5 3]. As a result, they aren't isomorphic.   

It should be noted that only graphs having the same link assortment arrays 

can be isomorphic. As a result, only graphs with the same link assortment 

arrays must be checked for isomorphism, and there are often few of them. 

A new algorithm based on the trail and graph marking is developed.  If the 

vertex-edge of two graphs are equivalent, the graphs are isomorphic. The 

vertices are to be identified by a three-digit number 𝑑𝑡𝑑𝑝𝑑𝑤, where 𝑑𝑡 and 

𝑑𝑝 represent the degree of vertex at the spanning tree and the parent graphs, 

respectively. The new vertex identification method takes into account not 

only the number of edges incident to the vertex but also the type of edges. 

The graphs of two PGTs are shown in Fig. 9  (a) and (b) (Fig. 14 of ref. 

[29]).  

 

 (a)  (b) 

Identification of the vertices by the three-digit number 𝒅𝒕𝒅𝒑𝒅𝒘   

(c) (d) 

A possible trail connecting all of the vertices of the first graph  (a), 

[( 666, 123, 135,234, 135, 135, 135, 333, 135, 123, 666, 258)] is 

chosen. Vertices are numbered in the same manner. 

𝑮𝑺𝟏

= [𝟒, 𝟓, 𝟔, 𝟕, 𝟏𝟏, 𝟑, 𝟐, 𝟏𝟏, 𝟗, 𝟏𝟎, 𝟏𝟏, 𝟑] 

𝐺𝑆2

= [𝟒, 𝟓, 𝟔, 𝟕, 𝟏𝟏, 𝟑, 𝟐, 𝟏𝟏, 𝟗, 𝟏𝟎, 𝟏𝟏, 𝟑] 

Geared Strings 

Figure 9. The steps for identifying isomorphism using the trail and 

graph marking method 

First, the vertices of the two graphs are identified by the three-digit number 

𝒅𝒕𝒅𝒑𝒅𝒘. Second, a possible trail connecting all of the vertices of the first 

graph (a) is chosen. Vertices are numbered in the same manner as [(666, 

123, 135, 234, 135, 135, 135, 333, 135, 123, 666, 258)]. Repetition of vertex 

and edge is allowed. Third, the two graphs are numbered according to the 

suggested trail. Finally, check the numbering of the geared strings (𝑮𝑺𝒔) 

of both geared graphs. If they are not equivalent, then the two graphs are 

not isomorphic. Because 𝑮𝑺𝟏  = 𝑮𝑺𝟐 , the two graphs are isomorphic. 

3.5.  Functional Representation 

The geared graphs shown in Table 3 can be transformed to their 

corresponding functional diagrams as shown in Appendix A. To start, each 

edge-labeling possibility of a spanning tree must be determined. Fig. 10 

shows all possibilities for labeling the edges of spanning trees. Figure A.1 

shows the graphs of 5-link 1-DOF PGTs that have only single-planet PGTs. 

Column 1: Graphs showing one of the possible distributions of internal and 

external gear pairs. Column 2: Corresponding functional representation to 

column 1. Figure A-2 shows a graph of a double-planet 5-link PGT. 

 

 

  

  

Figure 10. Possibilities for labeling the edges of spanning trees. 

 

Fig. 11 shows the labelled graph representation and the functional 

representation of the first graph in Table 3 which belongs to family 1. An 

internal gear pair will be represented by an upper-case G and an external 

gear pair will be represented by a lower-case G. 

 

4. Results 

This paper describes a method for synthesizing 5-link, 1-DOF PGTs. 

Given a set of links, all possible link assortments are found, and the graphs 

associated with each link assortment are synthesized. Table 1 shows the 

spanning trees for the 5-link, single-DOF PGT. 

 



HIND A. NAFEH AND ESSAM LAUIBI ESMAIL /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES   15 (2022) 009–017                                              15 

 

  

  

  

  

Figure 11. Five-link PGTs that have only single-planet FGEs 

To reduce the number of graphs generated and concentrate on 

methodology, the spanning trees were built with a vertex distribution up to 

the second level and a ground vertex degree greater than two. Table 2 shows 

the parent graphs for the 5-link, single-DOF PGT. Table 3 displays the 5-

link geared graphs for the link assortments [4 2 2 2 2 4], [3 3 3 2 3 2], and 

[3 3 2 2 3 3]. The spanning trees and parent graphs are used to generate 

geared graphs.  Each geared graph represents a 5-link, single-DOF PGT 

with a distinct topology and function. Figure 10, for example, shows five-

link PGTs that have only single-planet PGTs. The detailed results for the 

synthesis of graphs are shown in Appendix A. 

The results are a test of the current method's utility as a new method added 

to the previously existing technique. To automate the method, a computer 

program is created using a nested-do loops algorithm. 

5. Conclusion 

This paper presents a genetically compatible graph method to synthesize 

planetary gear trains. The new method is established on the link assortment 

arrays of spanning trees and parent graphs. Geared edges are determined by 

the fact that a parent graph contains several spanning trees. The presented 

method can be utilized for PGTs with any number of DOFs. Because this 

method makes use of a small number of spanning-tree graphs and parent 

graphs, it is possible to efficiently enumerate PGTs with any number of 

links. It is easy to program on a computer, can be used directly to perform 

PGT synthesis, and can generate an exhaustive and combinatorial complete 

set of geared graphs. The MATLAB algorithm is built to perform this 

method and the results show precise graphs without many similarities. 

REFERENCES 

 

[1] F. Buchsbaum, and F. Freudenstein. "Synthesis of kinematic structure of geared 

kinematic chains and other mechanisms." Journal of mechanisms 5, no. 3 (1970): 

357-392. 

[2] R Ravisankar, and T. S. Mruthyunjaya. "Computerized synthesis of the structure 

of geared kinematic chains." Mechanism and Machine Theory 20, no. 5 (1985): 

367-387. 

[3]  Shin, Jae Kyun, and Sundar Krishnamurty. "Standard code technique in the 

enumeration of epicyclic gear trains." Mechanism and machine theory 28, no. 3 

(1993): 347-355. 

[4] L.W.Tsai., An application of the linkage characteristic polynomial to the 

topological synthesis of Epicyclic Gear Trains, ASME Journal of Mechanisms, 

Transactions and Automation in Design, vol. 199, 329-336, (1987) 

[5] L.W. Tsai, and C.C. Lin, , The Creation of Non-fractionated Two-Degreeof- 

Freedom Epicyclic Gear Trains, ASME Journal of Mechanisms, Transmissions, 

and Automation in Design, 111, 4, (1989) 524–529. 

[6] J.U. Kim, B.M.Kwak, Application of edge permutation group to structural 

synthesis of epicyclic gear trains, Mech. Mach. Theory, 25, (199 0) 563-574. 

[7] Sohn, W. and Freudenstein, F., An Application of Dual Graphs to the Automatic 

Generation of the Kinematic Structures of Mechanism, ASME Journal of 

Mechanisms, Transmissions, and Automation in Design, 108, 3, (1986) 392–398. 

[8] Freudenstein, F., An Application of Boolean Algebra to the Motion of Epicyclic 

Drives, ASME Journal of Engineering for Industry, Series B, 93, (1971) 176 –

182. 

[9] Chatterjee, Goutam, and Lung-Wen Tsai. “Enumeration of Epicyclic-Type 

Automatic Transmission Gear Trains.” SAE Transactions 103 (1994): 1415–26. 

http://www.jstor.org/stable/44611849 

[10]  Chatterjee, Goutam, and Lung-Wen Tsai. "Computer-Aided Sketching of 

Epicyclic-Type Automatic Transmission Gear Trains." In International Design 

Engineering Technical Conferences and Computers and Information in 

Engineering Conference, vol. 12853, pp. 275-282. American Society of 

Mechanical Engineers, 1994. 

[11] Hsu, Cheng-Ho, and Kin-Tak Lam. "A new graph representation for the 

automatic kinematic analysis of planetary spur-gear trains." (1992): 196-200. 

[12] Hsu, Cheng-Ho, and Kin-Tak Lam. "Automatic analysis of kinematic structure 

of planetary gear trains." (1993): 631-638. 

[13] Hsu, Cheng-Ho. "Displacement isomorphism of planetary gear trains." 

Mechanism and Machine Theory 29, no. 4 (1994): 513-523. 

[14] Hsu, Cheng-Ho. "Synthesis of kinematic structure of planetary gear trains by 

admissible graph method." Journal of the Franklin Institute 330, no. 5 (1993): 

913-927. 

[15] Hsu CH, Hsu JJ.,  An efficient methodology for the structural synthesis of geared 

kinematic chains. Mech. Mach. Theory, 32 (8), (1997) 957–973. 

[16] V.R. Shanmukhasundaram, Y.V.D. Rao, S.P. Regalla, Enumeration of 

displacement graphs of epicyclic gear train from a given rotation graph using 

concept of building of kinematic units, Mech. Mach. Theory 134 (2019) 393–

424. 

[17] W.J. Yang , H.F. Ding , B. Zi , D. Zhang , New graph representation for planetary 

gear trains, ASME J. Mech. Des. 140 (1) (2018) 012303 . 

 [18] W.J. Yang , H.F. Ding , The complete set of one-degree-of-freedom planetary 

gear trains with up to nine links, ASME J. Mech. Des. 141 (4) (2019) 043301 . 

[19] V.R. Shanmukhasundaram, Y.V.D. Rao, S.P. Regalla, Review of structural 

synthesis algorithms for epicyclic gear trains. In: D. Sen, S. Mo han, G. 

Ananthasuresh (eds), Mechanism and Machine Science. Lecture Notes in 



16 HIND A. NAFEH AND ESSAM LAUIBI ESMAIL /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES   15 (2022) 009–17 

 

Mechanical Engineering. Springer, Singapore, 2021, https://doi.org/10.1007/ 

978- 981- 15- 4477- 4 _ 25 . 

[20] Esmail, Essam Lauibi, Nabel Kadum Abd-Ali, and Asia Abdulsattar Al-Ebadi. 

"Genetic Algorithm Optimization of Gear Teeth Numbers for Six-Velocity 

Lepelletier Automatic Transmission." Al-Qadisiyah Journal for Engineering 

Sciences 8.4 (2015): 491-503. 

[21] Esmail, Essam Lauibi "Two–Input Epicyclic–Type Transmission Trainswith 

Application To Tandem Bicycling". Al-Qadisiyah Journal for Engineering 

Sciences  (2009). 

[22] Anahed H. Juber, Essam L. Esmail,Tamather N. Ali"Graph Representation of 

Planetary Gear Trains: A Review ". Al-Qadisiyah Journal for Engineering 

Sciences (2022). 

[23] Esmail, E. L., and K. H. Salih. "Synthesis and optimization of epicyclic-type 

automatic transmissions based on nomographs." Al-Qadisiyah Journal for 

Engineering Sciences 4, no. 3 (2011): 168-185. 

[24] Esmail, Essam L., Hamed A. Hussen, and Tamather N. Ali. "Graph-Based 

Efficiency Analysis of Simpson Gear Train." In 2018 International Conference 

on Advanced Science and Engineering (ICOASE), pp. 479-484. IEEE, 2018. 

[25] Esmail, Essam L., and Anahed H. Juber. "An Application of Graph Theory for 

the Detection of Degenerate Structures in Planetary Gear Trains." In International 

Design Engineering Technical Conferences and Computers and Information in 

Engineering Conference, vol. 85444, p. V08AT08A025. American Society of 

Mechanical Engineers, 2021. 

[26] Hussen, Hamed Ali, Essam Lauibi Esmail, and Maasi F. Al-Mayali. "Structure 

synthesis of planetary gear trains using graph partitioning." In AIP Conference 

Proceedings, vol. 2386, no. 1, p. 040030. AIP Publishing LLC, 2022. 

[27] Hussen, Hamed A., and Essam L. Esmail. "Application of incidence matrix to 

topological structure and kinematic analysis of multi-planet gear trains." Results 

in Engineering 12 (2021): 100305. 

[28] N. A. Tamather, L. E., Essam, and A.N. Hind, A graph-based approach for 

detection degenerate structures in multi-planet gear trains, Cogent Engineering, 

9:1, (20220.  

[29] W.J. Yang, H.F. Ding, The perimeter loop-based method for the automatic 

isomorphism detection in planetary gear trains, ASME J. Mech. Des. 140 (12) 

(2018) 123302, doi: 10.1115/1.4041572 . 

Appendix A 

Figure A.1 shows the graphs of 5-link 1-DOF PGTs that have only single-planet PGTs. 

Column 1: Graphs showing one of the possible distributions of internal and external 

gear pairs. Column 2: Corresponding functional representation to column 1. Figure A-

2 shows a graph of a double-planet 5-link PGT. 

 

Graph representation Functional representation 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



HIND A. NAFEH AND ESSAM LAUIBI ESMAIL /AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES   15 (2022) 009–017                                              17 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 
 

 

 

  

 

 
Appendix B 

A MATLAB program using an adjacency matrix and a nested-do loops algorithm to 

solve a system of 2 linear equations in 3 unknowns: 

 

clear 

clc 

close all 

N=5; Dof=1; 

v=N+1; 

e=2*v-Dof-3; 

L=e-v+1; 

m=L+1; 

k=1; 

D=zeros(m-1,m-1); 

for V2=0:N 

    for V3=0:N 

        for V4=0:N 

                if(V2+V3+V4==v) 

                    if (2*V2+3*V3+4*V4==2*e) 

                             D(k,:)=[V2 V3 V4];    

                       k=k+1; 

                    end 

                end 

            end 

      end 

end 

disp('Vertex Degree Listing  LA =') 

disp('     V2    V3    V4  '),disp(D