- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Advances in Mechanical Engineering
Volume 2013 (2013), Article ID 120324, 15 pages
Nomographs and Feasibility Graphs for Enumeration of Ravigneaux-Type Automatic Transmissions
College of Engineering, University of Qadisiyah, Al Qadisiyah, Iraq
Received 12 October 2012; Revised 5 January 2013; Accepted 12 January 2013
Academic Editor: Marco Ceccarelli
Copyright © 2013 Essam L. Esmail. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Although there are many epicyclic-type automatic transmissions in production, the related configuration design methods are still tedious and borne to human error. A simple methodology for the systematic design of the Ravigneaux-type epicyclic gear transmissions needs to be developed. First, fundamentals and gear-shifting of four-speed and six-speed epicyclic-type automatic transmissions are illustrated to establish the design requirements. Second, based on the kinematic nomographs of the corresponding basic gear ratios, a simple clutching-sequence method is proposed and illustrated. Next, a planar-graph representation is presented to arrange the desired clutches for each possible clutching sequence into the epicyclic gear mechanism. Then, with the above methods, the systematic designs of the epicyclic gear mechanisms are given for demonstrating the feasibility of the proposed methodology. The result of this work shows that the seven-, eight-, and nine-link two-DOF Ravigneaux-type epicyclic gear mechanisms could reach four-, six-, and eight-forward speeds at most, respectively. New five-, six-, seven-, and eight-velocity automatic transmissions are enumerated from the two-ring eight- and nine-link Ravigneaux gear mechanisms. It is a major breakthrough to design completely satisfactory eight-speed automatic transmissions from the nine-link Ravigneaux gear mechanism.
For a long-time automatic transmissions with planetary gear trains are used in the automotive industry. Ravigneaux in 1940 proposed seven- and eight-link two-degree-of-freedom (DOF) epicyclic gear mechanisms [1, 2]. These epicyclic gear mechanisms are called the Ravigneaux gear mechanisms. Figure 1 shows an automatic transmission which provides three forward speeds and one reverse speed . It consists of a seven-link two-DOF Ravigneaux gear mechanism, two rotating clutches and , and two band clutches and . In the associated clutching sequence Table, indicates that the corresponding clutch is activated on the th link for that gear. The ranges of output velocities are classified into two kinds: under drive (UD) and reverse drive (RD) according to whether the velocity is between zero and the input velocity, or less than zero. A “direct drive” (DD) is equal to the input velocity.
The seven-link two-DOF-Ravigneaux gear train has been developed by nearly all automotive manufacturers as three- or four-velocity automatic transmission . It can be found in Ford , Ford , Mercedes Benz, Toyota A40, and Nissan, to name, a few three-velocity automatic transmissions. It can also be found in KM 175 and 176, Ford AOD, ZF 4 HP 14, and Borg-Warner , to name, a few four-velocity automatic transmissions . Figure 2 shows the ZF 4 HP 14 automatic transmission , which can provide four forward speeds and one reverse speed.
Sometimes, the seven-link two-DOF Ravigneaux gear train is integrated with a simple epicyclic gear train to form ten-link three-DOF epicyclic gear mechanisms to enhance the number of speeds [6, 7], providing six forward speeds. Figure 3 shows Lepelletier automatic transmission and its clutching sequence table . A widespread gear set concept is that of Lepelletier. This design is based on a single planetary gear set with rear-mounted Ravigneaux gear set. In 2003, ZF  used this gear set design to bring the first 6-speed passenger car transmission 6 HP 26 on the market (Figure 3).
Two-DOF eight-link Ravigneaux gear mechanism, shown in Figure 4(b), consists of a long pinion and two or more short pinions connecting two sun gears and two ring gears. Unfortunately, two-DOF eight-link Ravigneaux gear trains have been developed as four-velocity automatic transmission  while they can be used to produce automatic transmissions having more than four speeds. This work in part will attempt to attain maximum sequential velocity ratios for any given epicyclic gear train.
A seven-link 2-DOF Ravigneaux gear mechanism, an eight-link 2-DOF Ravigneaux gear mechanism, and a nine-link 2-DOF stepped Ravigneaux gear mechanism are shown in Figure 4, respectively.
The literature on the design of planetary gear trains includes conceptual designs, kinematic analysis, power flow and efficiency analysis, and configuration designs. However, relatively little work has been done on the enumeration of clutching sequences and configuration design of EGMs. This paper presents a systematic procedure to enumerate clutching sequences and to find feasible clutch layouts for Ravigneaux planetary gear trains.
2. Literature Review
The selection of an optimal clutching sequence cannot be solved analytically. Nadel et al. [11–13] formulated the task as a constraint satisfaction problem. Hsieh and Tsai [14, 15], Hwang and Huang , and Hsu and Huang  used algorithmic techniques. Ross and Route  and Esmail  introduced graphic techniques. Hattori et al.  used phase geometry method.
Nadel et al. applied an artificial intelligence technique to enumerate clutching sequences for EGMs made up of two basic epicyclic gear trains. The artificial intelligence technique is a powerful tool for solving the transmission design problem. However, the technique assumes that the design variables have discrete values in prescribed domains. Furthermore, it requires a search over the entire feasible solution space. This methodology is suitable for PGTs in which two simple PGTs are combined. These shortcomings inevitably reduce the efficiency of the algorithm.
Hattori et al.  proposed twenty-three phase geometric patterns for five-speed automatic transmissions; each could provide four clutching sequences. Each feasible clutching sequence obtained could be used to construct a clutch layout. However, this approach is suitable only for PGTs consisting of two sun gears, two ring gears, and one to three meshed planet gears mounted on a common arm. Ross and Route  introduced a design tool based on a lever analogy. It includes calculation of gear ratios, gear trains selection, and the construction of clutch layouts for parallel-connected PGTs. Hwang and Huang  and Hsu and Huang  used similar methods to that used by Hsieh and Tsai  to enumerate the clutching sequences of two-DOF eight-link Ravigneaux gear mechanism. Moreover, the five studies cited above are restricted to constructing clutch layouts of automatic transmissions for specific types of PGTs.
Hsieh and Tsai  used the concept of fundamental gear entities (FGEs) proposed by Chatterjee and Tsai  in conjunction with their earlier kinematic study  to determine the most efficient clutching sequence associated with automatic transmission . They applied combinatorial enumeration procedure to arrange the velocity ratios in a descending sequence. Then, they used an algorithm to enumerate clutching sequences for EGMs composed of two or more FGEs. A computer algorithm for the enumeration of clutching sequences is given by Hsieh . The algorithm needs information containing the approximate gear sizes arranged in a descending order.
Most combinatorial enumeration procedures are done through the process of generating and testing. The 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 important issue in using a generating and testing technique is the distribution of knowledge between the generator and tester. The generator produces solutions satisfying some of the constraints. The tester then selects those solutions that satisfy the rest of the constraints. While this technique is valid for solving transmission design problems, it limits the solutions to the knowledge (information) contained in the generator and tester. This inevitably reduces the efficiency of this solution technique and needs complicated computer algorithm. Usually, putting more knowledge in the generator results in a more efficient procedure. Nevertheless, Hsieh and Tsai  did not develop any effective method of arranging clutch layouts for the synthesized clutching sequences. However, the elimination of invalid clutching sequences was conducted by inspection. Identifying invalid clutching sequences by inspection is not always reliable.
Esmail  proposed a methodology, based on nomographs, for the enumeration of the associated clutching sequence table for an epicyclic gear mechanism. This method simplifies the synthesis of the clutching sequence of an epicyclic gear mechanism efficiently.
Without careful examination, a clutching sequence may be mistaken for a usable clutch layout for an automatic transmission. Hsu and Huang  proposed a planar-graph representation to arrange the desired clutches for each possible clutching sequence into the Ravigneaux gear mechanism. Hwang and Huang  introduced coded sketches for connecting clutch elements to planetary gear trains for automotive automatic transmissions.
The literature survey reveals that although some studies are made on the clutching sequence synthesis, they are all tedious. A simple methodology for the design of planetary automatic transmission needs to be developed. In this paper nomographs are used to synthesize the clutching sequences of Ravigneaux gear trains; the nomograph method is described in a series of previous papers [19, 25, 26]. Only the related topics needed in this paper will be reviewed wherever appeared.
In this paper, more efficient solution techniques are developed to overcome those shortcomings. This paper applies kinematic nomographs [19, 25] and feasibility graphs  of Ravigneaux gear trains to achieve the goal. By virtue of these solution techniques, completely satisfactory six- and eight-speed automatic transmissions are designed from the eight- and nine-link Ravigneaux gear mechanism, respectively.
3. Enumeration of All Feasible Clutching Sequences with the Aid of Nomograph
The first stage of designing epicyclic-type automatic transmissions is to synthesize all possible clutching sequences from the epicyclic-gear mechanism. Nomographs are used for this purpose.
3.1. Nomographs and Geometry Relations for Seven-Link Ravigneaux Gear Train
A nomograph is defined as three or more axes, or scales, arranged such that problems of three or more variables can be solved using a straightedge. In the particular case of EGTs, a nomograph can be constructed using three or more vertical parallel axes [19, 27].
Traditionally, the velocity ratio is used to study the velocity relations among the different links of an EGT .
Let the symbol denote the velocity ratio between links and , with reference to link where , , and are any three links in the gear train, then Since the gear mechanism, shown in Figure 4(a), is a double-planet FGE, then its nomograph can be drawn in terms of planet gear 6 as shown in Figure 5.
The term “gear ratio” is used in this paper to denote the ratio of a meshing gear pair. It is defined by the ratio of a planet gear with respect to a sun or ring gear : where and denote the numbers of teeth on the planet and the sun or ring gear, respectively, and the positive or negative sign depends on whether is a ring or sun gear.
Considering the kinematics of a fundamental circuit, the fundamental circuit equation can be written as  Equation (3) can be rewritten for ring, sun, planet gears, and carrier to obtain , , , and , respectively. The gear ratio for small sun gear that is not meshing directly with the first planet gear on which the nomograph is drawn and is meshing with the second planet gear can be found in terms of the gear ratio of the two planets as The values of the gear ratios are used to place the axes of the nomograph shown in Figure 5. The axis passes at the origin, and the axis is one unit apart from it.
The gear ratios for the Ravigneaux gear train are Considering the geometry relations of the Ravigneaux gear mechanism given in (5) through (8), the values of the gear ratios can be deduced as follows: From (4), we can write Since is greater than , then from (8) and (11) it can be shown that
3.2. Enumeration of All Feasible Clutching Sequences
Kinematic relationships among the links of this FGE can easily be visualized by observation from the nomograph shown in Figure 5. Any straight line through the input operating velocities of the EGT will intersect other axes at the operating velocities of the links representing to those axes. From the nomograph can be written as  Because there are four coaxial links and link four is pre-assigned as the output link, this gear train can provide six overall velocity ratios [27, 28]. Figure 6 shows the clutching sequence nomograph for this mechanism.
In arranging a clutching sequence, it is highly desirable to achieve a single-shift transition . In order to achieve single-shift transitions, the UDs can be further classified into two sets. The ODs and RDs are used with both of the UD sets.
A direct drive is obtained by simultaneously clutching two coaxial links of an EGT to the input power source. A reverse drive can be obtained by applying one or two of the clutches designed for the forward drives to the reverse drive.
3.3. Feasibility Graphs for an Epicyclic Gear Train to Form an Automatic Transmission
A graphical representation is proposed by  to easily and quickly detect the possibility of the arrangement of all the rotating clutches and band clutches into the gear mechanism to form an automatic transmission.
A graph that can be drawn on a plane such that none of two of its edges intersect is called planar; meaning it can form an automatic transmission. A graph that cannot be drawn on a plane without crossover between its edges is called nonplanar; meaning there is no possibility to form an automatic transmission. Here we shall call such graphs as the feasibility graphs of a gear train to form an automatic transmission. They are shown in Figure 8(a) through 8(d).
As a result, all of the four sets of clutching sequences are feasible to form automatic transmissions, and the corresponding four-velocity automatic transmissions are drawn, as shown in Figures 9(a) through 9(d), respectively.
Figure 9(a) shows one feasible clutching sequence with rotating clutches attached to links 1 and 2 and band clutches attached to links 1 and 3. This clutching sequence has been applied in most three-velocity Ravigneaux automatic transmissions [3, 6]. The other clutching sequences obtained are in agreement with those reported in the literature [5, 14, 23], except that there is no need to any information containing the approximate gear sizes arranged in a descending order.
3.4. Nomographs and Geometry Relations for Eight-Link Ravigneaux Gear Train
The gear ratios for the eight-link Ravigneaux gear train are the same as that for the seven-link given before. In addition the gear ratio for the added small ring gear can be written as follows: It can be concluded that . From (4), we can write Therefore, Since is greater than , then from (7) and (16)
3.5. Enumeration of All Feasible Clutching Sequences
Because there are five coaxial links and link four or seven can be assigned as the output link, this gear train can provide twelve overall velocity ratios for each assignment. Figure 11 shows the clutching sequence nomograph for this mechanism.
In order to achieve single-shift transitions, the UDs can be further classified into two sets. By adding a direct drive and two ODs, a total of four six-velocity sets are obtained. A direct drive is obtained by simultaneously clutching two coaxial links of an EGT to the input power source. As a result, we obtain four descending sequences of velocities as shown in Figure 12.
A reverse drive can be obtained by applying one or two of the clutches designed for the forward drives to the reverse drive.
If a reverse drive is added, three six-velocity clutching sequences can be enumerated for each set which result in twelve six-velocity clutching sequences. For the small ring to be an output link another twelve six-velocity clutching sequences can be enumerated.
The first set can realize six-speed automatic transmission, while having three clutches and three brakes only.
3.6. Feasibility Graphs for the Two-Ring Epicyclic Gear Train to Form an Automatic Transmission
Figure 13(a) reveals that only a clutch can be attached to link 2. For the first and second sets of clutching sequences shown in Figures 12(a) and 12(b), they are controlled by , and . As a result, all of the six clutching sequences that can be obtained from the first and second sets are feasible to form automatic transmissions.
Figure 13(b) reveals that only a band clutch can be attached to link 1. The third and fourth sets of clutching sequences shown in Figures 12(c) and 12(d) are controlled by , and . As a result by excluding from the clutching sequences nomograph, two clutching sequences are feasible to form five-velocity automatic transmissions.
Figure 12(d) shows a clutching sequence nomograph without reverse drive based on the fourth reduction set. By excluding from the clutching sequence nomograph, a five-velocity automatic transmission is possible. Figure 14 shows the corresponding functional representation, with a band clutch attached to link 3 for reverse drive. The input clutches and band clutch are applied for the reverse drive.
3.7. Nomographs and Geometry Relations for Nine-Link Ravigneaux Gear Train
The gear ratios for the stepped nine-link Ravigneaux gear train are the same as those for the seven- and eight-link gear trains given before, except that the gear ratios are written in terms of the plant to which it is meshing, either gear 6 or 61 of the stepped plant: In addition, the gear ratio for the added third sun gear can be written as, .
Since is greater than and is greater than , then from (7) and (18) it can be shown that Similarly and . Moreover Since is greater than and is greater than , then from (8) and (20) it can be shown that these gear ratios work oppositely; while one increases the gear ratio, the other decreases it, but, without loss of generality, we can select the most probable case where .
Because there are six coaxial links and link four or eight can be assigned as the output link, this gear train can provide twenty overall velocity ratios for each assignment. The velocity ratios are classified into three groups; underdrive, overdrive, and reverse drive. If there are more than two sets of velocity ratios in any group, they are further classified into all possible sets or subgroups based on the constraint that only one clutch can be shifted in each set. The velocity ratios in any set are arranged in a descending order, and then the corresponding clutching sequence is generated. Possible clutching sequences are generated by combining the UD, DD, OD, and RD subgroups together. Figure 16 shows the clutching sequence nomograph for the nine-link Ravigneaux gear mechanism shown in Figure 4(c).
In order to achieve single-shift transitions, the UD velocities are classified into two sets (shown in Figure 17).
For the first UD set, the clutching sequences, shown in Figure 18, are controlled by , and . As a result, all of the UD and OD clutching sequences that can be obtained from the first set are feasible to form automatic transmissions. The RDs are only obtained from one of the rotating clutches or and one of the band clutches or .
Figure 18 reveals that a clutch cannot be attached to link 1; thus, the last three reverse clutching sequences shown in Figure 17(a) are inadmissible. As a result, only thirteen clutching sequences shown in Figure 19 are feasible at this design stage.
Based on the first UD set of feasibility graph, the general clutching sequence layout for the nine-link Ravigneaux gear train is shown in Figure 20. All of the eight-velocity feasible clutching sequence layouts can be extracted from this general layout.
In order to achieve single-shift transitions, the ODs can be further classified into five sets; three sets consist of three-overdrives and two sets consist of two overdrives.
By combining the first set of UD clutching sequences with three sets of three-velocity OD clutching sequences and three sets of RDs, nine possible clutching sequences for eight-velocity automatic transmissions are feasible at this design stage as shown in Figure 21.
Moreover, the first set of UD clutching sequences can also be combined with two sets of two-velocity OD clutching sequences and three sets of RDs, and six possible clutching sequences for seven-velocity automatic transmissions are feasible at this design stage, as shown in Figure 22.
For the second set, since is essential for the UD clutching sequences, then only band clutches can be attached to links 1 and 8 as shown in Figure 23. Therefore, the clutching sequences are controlled by , , , , , , , and . The clutch has no role in the clutching sequence. As a result by excluding , three clutching sequences are feasible to form six-velocity automatic transmissions as shown in Figure 24.
In order to achieve single-shift transitions, the ODs can be further classified into three sets consisting of three-over-drives.
By combining the second set of UD clutching sequences with the three sets of three-velocity OD clutching sequences and the one RD, three possible clutching sequences for six-velocity automatic transmissions are feasible at this design stage as shown in Figure 25.
Based on the second UD set of feasibility graph, the general clutching sequence layout for the nine-link Ravigneaux gear train to form six-velocity Ravigneaux gear mechanisms is shown in Figure 26.
A novel feature of the suggested approach for constructing multiaxis nomographs is the ability to enumerate the feasible clutching sequences associated with two-ring eight-, and nine-link Ravigneaux-type automatic transmissions. It is a major breakthrough to design a completely satisfactory six-speed automatic transmission from the two-ring Ravigneaux gear train since it has only eight links. Moreover, new eight-speed automatic transmissions are enumerated from nine-link gear mechanisms. The new designs make use of the benefits of the Ravigneaux gear train and overcome the previous art difficulties. The completeness of the results cannot be confirmed since no publications exist for eight-velocity Ravigneaux gear mechanisms. The proposed methodology can be used for the systematic design of any epicyclic-type automatic transmission. Feasibility graphs have been also used as a design tool, allowing the designer to quickly select the most viable clutching sequence.
- P. Ravigneaux, “Gear box with freewheeling gear,” US patent no. 2195783, 1940.
- P. Ravigneaux, “Speed changing device,” US patent no. 2220174, 1940.
- L. W. Tsai, Mechanism Design: Enumeration of Kinematic Structures According to Function, CRC Press, Boca Raton, Fla, USA, 2001.
- E. Wilfinger and J. Thompson, Borg-Warner Australia Model 85 Automatic Transmission, SAE Paper no. 880480, Society of Automotive Engineers, 1988.
- C. H. Hsu and R. H. Huang, “Systematic design of four-speed ravigneaux-type automatic transmissions,” Journal of Chinese Society of Mechanical Engineers, vol. 30, no. 3, pp. 201–209, 2009.
- H. Naunheimer, B. Bertsche, J. Ryborz, and W. Novak, “Vehicle transmission systems: basic design principles,” in Automotive Transmissions: Fundamentals, Selection, Design and Application, Springer, Berlin, Germany, 2nd edition, 2011.
- N. Katou, T. Taniguchi, and K. Tsukamoto, Aisin AW New Six-Speed Automatic Transmission for FWD Vehicles, SAE Paper no. 2004-01-0651, 2004.
- P. A. G. Lepelletier, “Multispeed automatic transmission for automobile vehicles,” US patent no. 5106352, 1992.
- H. Scherer, ZF 6-Speed Automatic Transmission for Passenger Cars, SAE Paper no. 2003-01-0596, Society of Automotive Engineers, 2003.
- P. Ravigneaux, “Epicyclic change-speed gear,” US patent no. 2761333, 1956.
- B. A. Nadel and J. Lin, “Automobile transmission design as a constraint satisfaction problem: first results,” in Proceedings of the 7th IEEE Conference on Artificial Intelligence Applications, vol. 1, pp. 248–256, February 1991.
- B. A. Nadel and J. Lin, “Automobile transmission design as a constraint satisfaction problem: modeling the kinematic level,” Artificial Intelligence for Engineering Design, Analysis and Manufacturing, vol. 5, no. 3, pp. 137–171, 1991.
- B. A. Nadel, X. Wu, and D. Kagan, “Multiple abstraction levels in automobile transmission design: constraint satisfaction formulations and implementations,” International Journal of Expert Systems, vol. 6, no. 4, pp. 489–559, 1993.
- H. I. Hsieh and L. W. Tsai, “A methodology for enumeration of clutching sequences associated with epicyclic-type automatic transmission mechanisms,” SAE Transactions, vol. 105, no. 6, pp. 928–936, 1996.
- H. I. Hsieh and L. W. Tsai, “The selection of a most efficient clutching sequence associated with automatic transmission mechanisms,” Journal of Mechanical Design, vol. 120, no. 4, pp. 514–519, 1998.
- W. M. Hwang and Y. L. Huang, “Configuration design of six-speed automatic transmissions with two-degree-of-freedom planetary gear trains,” Transactions of the Canadian Society for Mechanical Engineering, vol. 29, no. 1, pp. 41–55, 2005.
- C. H. Hsu and R. H. Huang, “Systematic design of six-speed automatic transmissions with an eight-link two-DOF Ravigneaux gear mechanism,” Journal of Mechanical Design, vol. 131, no. 1, Article ID 011004, 8 pages, 2009.
- C. S. Ross and W. D. Route, “A method for selecting parallel-connected, planetary gear train arrangements for automotive automatic transmissions,” SAE Transactions, vol. 100, no. 6, pp. 1765–1774, 1991.
- E. L. Esmail, “Nomographs for enumeration of clutching sequences associated with epicyclic type automatic transmission mechanisms,” in Proceedings of the ASME International Mechanical Engineering Congress and Exposition (IMECE '08), Boston, Mass, USA, October-November 2008, paper no. 66409, also at Emirates Journal for Engineering Research, vol. 14, no.1, pp. 29–38, 2009.
- N. Hattori, T. Oshidari, and Y. Morimono, “Application of a new complex planetary gear set to five-speed automatic transmission gear train,” Transactions of the Society of Automotive Engineers of Japan, vol. 26, no. 1, pp. 79–82, 1995.
- G. Chatterjee and L. W. Tsai, “Enumeration of epicyclic-type transmission gear trains,” Transactions of SAE, vol. 103, no. 6, pp. 1415–1426, 1995, paper no. 941012.
- H. I. Hsieh and L. W. Tsai, “Kinematic analysis of epicyclic-type transmission mechanisms using the concept of fundamental geared entities,” Journal of Mechanical Design, vol. 118, no. 2, pp. 294–299, 1996.
- H. I. Hsieh, Enumeration and selection of clutching sequences associated with epicyclic-type transmission mechanisms [Ph.D. thesis], Department of Mechanical Engineering, University of Maryland, College Park, Md, USA, 1996.
- W. M. Hwang and Y. L. Huang, “Connecting clutch elements to planetary gear trains for automotive automatic transmissions via coded sketches,” Mechanism and Machine Theory, vol. 46, no. 1, pp. 44–52, 2011.
- E. L. Esmail, “Kinematic nomographs of epicyclic-type transmission mechanisms,” Emirates Journal for Engineering Research, vol. 12, no. 3, pp. 47–55, 2007.
- E. L. Esmail, “Nomographs for analysis of power circulation through closed epicyclic gear trains,” Emirates Journal For Engineering Research, vol. 16, no. 2, pp. 1–9, 2011.
- E. L. Esmail and H. A. Hussen, “Nomographs for kinematics, statics and power flow analysis of epicyclic gear trains,” in Proceedings of the ASME International Mechanical Engineering Congress and Exposition (IMECE '09), vol. 13, pp. 631–640, Lake Buena Vista, Fla, USA, November 2009, paper no. 10789.
- E. L. Esmail, “Teaching planetary gear trains with the aid of nomographs,” Advances in Mechanical Engineering, vol. 2013, Article ID 978418, 9 pages, 2013.