- Email: [email protected]
Connecting clutch elements to planetary gear trains for automotive automatic transmissions via coded sketches
Mechanism and Machine Theory 46 (2011) 44–52
Contents lists available at ScienceDirect
Mechanism and Machine Theory j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / m e c h m t
Connecting clutch elements to planetary gear trains for automotive automatic transmissions via coded sketches Wen-Miin Hwang ⁎, Yu-Lien Huang Department of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 14 January 2010 Received in revised form 13 August 2010 Accepted 29 August 2010 Available online 27 September 2010 Keywords: Planetary gear train Automatic transmission Clutching sequence Clutch layout Coded sketch
a b s t r a c t This article presents a synthesis procedure to find feasible clutch layouts for a given planetary gear train clutching sequence. The method of constructing coded sketches for a planetary gear train is first introduced. In a coded sketch, all coaxial links, excluding the output link, are arranged in a specific order called a coaxial-link sequence. Based on the coaxial-link sequence, each link is encoded with three codes: x-code, y-code, and xy-code. Two identification rules are then concluded by examining these codes. Rule one tells how to arrange a band clutch without passage interference. Rule two is used to knock out the invalid clutching sequences leading to infeasible clutch layouts. According to the proposed coded sketches and rules, a synthesis procedure is then established for the purpose of finding all feasible coded layouts in a logical way from a given clutching sequence. Coded layouts can be directly and easily transferred to general clutch layouts. Two examples are used to demonstrate the feasibility of the synthesis procedure. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Automatic transmissions (ATs) with planetary gear trains (PGTs) have long been applied in the automotive industry. The literature on the design of planetary gear trains includes conceptual designs [1–5], kinematic analysis [6–12], power flow and efficiency analysis [7,8,13–17], and configuration designs [11,18–23]. However, less work has been done on the design of clutch layouts for ATs. For parallel-connected PGTs, Ross and Route [18] introduced an AT design tool based on a lever analogy, which included the calculation of gear ratios, gear trains selection, and the construction of clutch layouts. Nadel et al. [11] formulated the design of ATs as a constraint satisfaction problem which included kinematic, topological, stick diagram, and geometric levels. This methodology was suitable for PGTs in which two simple PGTs were combined. Hattori et al. [19] proposed 23 phase geometric patterns, in which each could provide four clutching sequences for five-speed ATs. Using a phase geometry method, each feasible clutching sequence obtained could be used to construct a clutch layout. However, this approach is suitable only for the type of PGTs consisting of two sun gears, two ring gears, and one to three meshed planet gears mounted on a common arm. Moreover, the three studies referenced above are restricted to constructing clutch layouts of ATs for specific types of PGTs. Hsieh and Tsai [20] applied the concept of fundamental geared entities of PGTs to enumerate clutching sequences for ATs. They obtained four feasible four-speed clutching sequences from the PGT composed of a Simpson gear train and a simple PGT. They also enumerated several three- and four-speed clutching sequences for each of the Simpson, Ravigneaux, and Type-6206 gear trains, and some of them were used in commercial three-speed and four-speed ATs [24]. Nevertheless, they did not develop an effective method of arranging clutch layouts for the synthesized clutching sequences. Hwang and Huang [21] proposed a novel methodology to design AT configurations which was suitable for all varieties of PGTs. The methodology included screening
⁎ Corresponding author. Tel.: + 886 6 2757575x62156; fax: + 886 6 2352973. E-mail address: [email protected] (W.-M. Hwang). 0094-114X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2010.08.013
W.-M. Hwang, Y.-L. Huang / Mechanism and Machine Theory 46 (2011) 44–52
45
suitable PGTs, enumeration of clutching sequences, optimal design of the number of teeth on gears, and an evaluation of the mechanical efficiency of PGTs. Six feasible clutching sequences and three configurations of six-speed ATs were obtained from a five-coaxial-link PGT. However, the elimination of invalid clutching sequences was conducted by inspection. Accordingly, a more precise procedure to arrange the clutch layouts of ATs is still needed for all types of PGTs. Without careful examination, a clutching sequence may be mistaken for a usable clutch layout for an AT. For example, based on the Simpson gear train in Fig. 1, four clutching sequences were derived, as shown in Fig. 2 [20]. Each XLi in the clutching sequence table indicates that the corresponding clutch connected to link i is activated for that gear. Unfortunately, the four-speed clutching sequence in Fig. 2(d) is actually an infeasible one, since no clutch layouts can be properly arranged for ATs. Two tentative arrangements depicted in Fig. 3 are constructed by equipping the Simpson gear train with three rotating clutches, C1, C2, and C3, and a band clutch, B1. The remaining band clutch, B2, required to connect link 1 always intersects with other links. A clutching sequence which seems feasible with regard to gears but leads to an infeasible clutch layout is merely an invalid sequence. Identifying invalid clutching sequences by inspection is not always reliable. The purpose of this article is to propose a precise procedure to construct usable clutch layouts for ATs composed of two degrees-of-freedom PGTs. The coded sketches of a PGT are introduced first. Each coaxial link of a PGT, excluding the output link, is encoded with three codes, x-, y-, and xy-codes, according to a specific order. Based on these codes, two rules are then proposed, that are helpful for synthesizing feasible coded layouts in a logical way. The first rule provides a non-interference condition for arranging band clutches. The second one is used to knock out the invalid clutching sequences. According to the proposed coded sketches and rules, a precise synthesis procedure is then proposed. The preceding part of the procedure is used to identify and eliminate invalid clutching sequences. The latter can synthesize all feasible coded layouts corresponding to valid clutching sequences. Afterwards, all feasible coded layouts can be directly transformed into traditional clutch layouts. Two examples are provided to show the feasibility of the procedure. 2. Coded sketches of a planetary gear train In the case of a two degrees-of-freedom PGT with n coaxial links, when one link is assigned as the output link, the remaining n − 1 links can be encoded with three codes: (a) x-codes: xn − 1, xn − 2,…, x1, (b) y-codes: y1, y2,…, yn − 1, and (c) xy-codes: xn − 1y1, xn − 2y2,…,
Fig. 1. Simpson gear train.
Fig. 2. Clutching sequences derived from Simpson gear train [20].
Fig. 3. Two infeasible clutch layouts for an invalid clutching sequence.
46
W.-M. Hwang, Y.-L. Huang / Mechanism and Machine Theory 46 (2011) 44–52
x2yn − 2, x1yn − 1. The three codes are displayed in three separate boxes used for installing clutches. The sketch connecting the n − 1 coaxial links to three coded boxes with mutually parallel passages is defined as the coded sketch of a PGT. Take the four-coaxial-link PGT in Fig. 4 [25] as an example. Since arm A is used as the output link, it is attached with a rightward arrow along the central axis of the PGT. The remaining coaxial links, S1, R1, and R2, are extended with mutually parallel passages around the planet gears and toward the left side of the PGT, as shown in Fig. 5(a). Three boxes designated as x, xy, and y are then placed at the left side of the PGT. The two upper boxes, x and y, show the regions for installing band clutches, while the lower box, xy, is for rotating clutches. The coaxial links, S1, R1, and R2, and the boxes are connected by drawing mutually parallel passages between them, as shown in Fig. 5(b). The nodes connecting box x and the three passages are assigned as x-codes from xn − 1 to x1. Hence, x-codes x3, x2, and x1 correspond to coaxial links S1, R1, and R2, respectively. The nodes connecting box y and the three passages are named as y-codes from y1 to yn − 1. Therefore, y-codes y1, y2, and y3 correspond to links S1, R1, and R2, respectively. Combining x-codes and y-codes into box xy gives xy-codes from xn − 1y1 to x1yn − 1. As a result, xy-codes x3y1, x2y2, and x1y3 corresponding to coaxial links S1, R1, and R2, respectively, are employed. Thus, the coded sketch of PGT in Fig. 4 is constructed, as shown in Fig. 5(b). Note that the sequences of the coaxial links corresponding to the x-codes, y-codes, and xy-codes are the same: S1–R1–R2. For simplicity, the sequence of coaxial links in every box is called a coaxial-link sequence. When the coaxial links are connected to boxes xy and y for a clutching sequence, their horizontal passages may intersect with the vertical passages connecting to boxes x and xy. Any intersections will make the clutch arrangement fail. Therefore, passage intersections are useful for checking whether or not the clutching sequence leads to a usable clutch layout. Take the four-speed
Fig. 4. A planetary gear train with four coaxial links [25].
Fig. 5. The illustration of the construction of a coded sketch for the PGT in Fig. 4. (a) A PGT and three associated boxes. (b) All mutually parallel passages connecting coaxial links and boxes.
W.-M. Hwang, Y.-L. Huang / Mechanism and Machine Theory 46 (2011) 44–52
47
clutching sequence in Fig. 6 [25] as an example, which is derived from the PGT in Fig. 4. Since coaxial links R1 and S1 need to engage with rotating clutch C1 and band clutch B2, respectively, the horizontal passage connecting link R1 with box xy interferes with the vertical passage connecting link S1 with box x, as shown in Fig. 7(a). It is obvious that the band clutch B2 engaging with link S1 cannot be placed at box x. According to the coded sketch in Fig. 5(b), the xy-code for link R1 is x2y2, and the x-code for link S1 is x3. The subscript of x3 is greater than that of x2 in x2y2. This indicates that when link R1 with an xy-code of x2y2 needs to engage with a rotating clutch, other coaxial links which have subscripts for their x-codes greater than two, which is the subscript of x in x2y2, cannot be engaged with a band clutch at box x. In the same way, other coaxial links which have subscripts for their y-codes greater than that of y2 in x2y2, cannot be engaged with a band clutch at box y. For example, as shown in Fig. 7(b), link R2, whose y-code is y3, cannot be engaged with a band clutch at box y because the subscript of y3 is greater than that of y2 in x2y2, which is the xy-code of link R1. Therefore, the non-interference condition for connecting rotating and band clutches with a PGT can be concluded as follows: Rule 1. For a clutching sequence derived from an n-coaxial-link PGT, if there are a total of e (e ≤ n − 1) coaxial links with xy-codes xayα, xbyβ, …, xeyε, respectively, that need to be engaged with rotating clutches, the non-interference condition for a coaxial link with x-code xi to be engaged with a band clutch at box x is: i ≤ Minða; b;…; eÞ: The non-interference condition for a coaxial link with y-code yj to be engaged with a band clutch at box y is: j ≤ Minðα; β;…; εÞ:
ð1Þ ð2Þ
As shown in Fig. 8, links S1 and R2 with xy-codes x3y1 and x1y3, respectively, need to be engaged with rotating clutches. Link R1 with x-code x2 and y-code y2 needs to be engaged with a band clutch. According to Rule 1, i = 2, Min(a, b) = Min(3, 1) = 1. The situation of i ≤ Min(a, b) does not exist. Therefore, link R1 cannot engage with a band clutch at box x. In the same way, j = 2, Min(α,
Fig. 6. A clutching sequence derived from the PGT in Fig. 4 [25].
Fig. 7. Interference of passages.
48
W.-M. Hwang, Y.-L. Huang / Mechanism and Machine Theory 46 (2011) 44–52
Fig. 8. The sandwich set SC1(RB1)RC2.
β) = Min(1, 3) = 1. The situation of j ≤ Min(α, β) does not exist, either. Thus, link R1 cannot engage with a band clutch at box y, either. According to this example, a conclusion is stated as follows: Rule 2. Coaxial links p and q, which are not adjacent to each other in a coaxial-link sequence, need to be engaged with rotating clutches. If there exists a coaxial link r located between links p and q that is required to be engaged with a band clutch, this situation will lead to an infeasible clutch layout based on the corresponding coded sketch. The coaxial-link set of p, q, and r is defined as a sandwich set, which can be expressed as pC(rB)qC. The superscripts C and B mean that links p and q need to be engaged with rotating clutches and link r with a band clutch. If the coaxial-link sequence of a coded sketch is u–v–x–y–z, 10 sandwich sets can be found by using the theory of combinations. They are uC(vB)xC, uC(vB)yC, uC(vB)zC, uC(xB)yC, uC(xB)zC, uC(yB)zC, vC(xB)yC, vC(xB)zC, vC(yB)zC, and xC(yB)zC. According to Rule 2, if any one of these sandwich sets exists in the clutching sequence discussed, its coded sketch is invalid. For example, the only sandwich set for the coded sketch in Fig. 5(b) is SC1(RB1)RC2. From the clutching sequence in Fig. 6, it is clear that links R1, R2, and S1 need to be engaged with rotating clutches, and links R2 and S1 with band clutches. The sandwich set SC1(RB1)RC2 does not exist in the clutching sequence. Therefore, the coded sketch in Fig. 5(b) constructed from the clutching sequence shown in Fig. 6 is a valid sketch. It will lead the clutching sequence to at least one feasible clutch layout. When depicting a coded sketch, the output link is always drawn with an arrow to the right and has no interference with other coaxial links. This means that a clutching sequence may be depicted in several possible coded sketches. Taking the PGT in Fig. 4 as an example, since arm A can be pulled with an arrow to the right in different ways, three coded sketches are constructed, as shown in Figs. 5(b) and 9. Compared to the coded sketch in Fig. 5(b), the other two in Fig. 9 have different coaxial-link sequences, R2–S1–R1 and R1–R2–S1. They also contain sandwich sets RC2(SB1)RC1 and RC1(RB2)SC1, respectively. Because the two sandwich sets exist in the clutching sequence shown in Fig. 6, both are invalid sketches. 3. The synthesis procedure for usable coded layouts For a given clutching sequence, using the coded sketches of its corresponding PGT and the proposed two rules, a precise synthesis procedure is presented herein to find all feasible coded layouts. If the clutching sequence leads to no feasible clutch
Fig. 9. Two other coded sketches for the PGT in Fig. 4.
W.-M. Hwang, Y.-L. Huang / Mechanism and Machine Theory 46 (2011) 44–52
49
layouts, this procedure can still check it. The steps for knocking out an invalid clutching sequence and finding coded layouts for a valid clutching sequence are described as follows: 1. Construct all coded sketches for a given PGT and its clutching sequence. 2. List all sandwich sets for each coded sketch. 3. If one of the sandwich sets of a coded sketch exists in the given clutching sequence, the coded sketch is identified as an invalid one. If all coded sketches of the PGT are recognized to be invalid sketches for the clutching sequence, the clutching sequence will lead only to infeasible clutch layouts. Thus, the clutching sequence should be knocked out. 4. For each coded sketch which does not fail in Step 3, determine which xy-codes are active. An active xy-code means that its corresponding coaxial link needs to be engaged with a rotating clutch based on the given clutching sequence. 5. For a coded sketch, if the active xy-codes obtained from Step 4 are xayα, xbyβ, …, and xeyε, in which Min(xayα, xbyβ, …, xeyε)x = Min(a, b, …, e) = s and Min(xayα, xbyβ, …, xeyε)y = Min(α, β, …, ε) = t, record (xs,yt) in a table similar to Table 1. 6. According to Rule 1, determine which x-codes and y-codes are active for each link that needs to be engaged with a band clutch. If the subscript of its corresponding x- or y-code is not greater than s (for the x-codes) or t (for the y-codes) of the active xy-codes, this x- or y-code is active. 7. Eliminate all inactive xy-, x-, and y-codes and passages connected to them. A coded layout is thus established. 8. For the given clutching sequence, repeat Steps 4 to 7 for all valid coded sketches that do not fail in Step 3. All usable coded layouts are then found.
Example 1. Identification of an invalid clutching sequence. The clutching sequence in Fig. 2(d) is derived from the Simpson gear train in Fig. 1 for four-speed ATs [20]. Using the proposed procedure, it can be identified as infeasible for a usable clutch layout at Step 3. The first three steps of the procedure for this example are described as follows: 1. Besides the configuration in Fig. 1, the Simpson gear train can also be depicted as the configuration in Fig. 10. The former configuration has only one coded sketch, as shown in Fig. 11(a), while the latter has two different coded sketches, as shown in Fig. 11(b) and (c). The coaxial-link sequences for the coded sketches in Fig. 11(a), (b), and (c) are L2–L1–L4, L4–L1–L2, and L1–L4–L2, respectively. 2. The sandwich sets for the coded sketches in Fig. 11(a), (b), and (c) are LC2(LB1)LC4, LC4(LB1)LC2, and LC1(LB4)LC2, respectively. 3. Because the sandwich sets listed at Step 2 exist in the clutching sequence in Fig. 2(d), they are all invalid coded sketches according to Rule 2. The clutching sequence leads only to infeasible clutch layouts. Therefore, it is an invalid clutching sequence and should be knocked out at this step. This result is consistent with the illustrations in Fig. 3. Example 2. Construction of coded layouts for a PGT having feasible coded sketches. The clutching sequence given in Fig. 2(a) is obtained from the Simpson gear train in Fig. 1 for three-speed ATs [20]. A total of two usable coded layouts can be found by using the proposed procedure. The steps of the procedure are described as follows: 1. As illustrated in Example 1, all available coded sketches for the Simpson gear train are shown in Fig. 11. The coaxial-link sequences for the three coded sketches are L2–L1–L4, L4–L1–L2, and L1–L4–L2, respectively. 2. Each coded sketch has only one sandwich set, i.e., LC2(LB1)LC4, LC4(LB1)LC2, and LC1(LB4)LC2, respectively. 3. Sandwich set LC1(LB4)LC2 exists in the clutching sequence in Fig. 2(a); i.e., links L2 and L1 need to be engaged with rotating clutches and link L4 with a band clutch. Therefore, the coded sketch in Fig. 11(c) is an invalid sketch. The remaining two coded sketches, as shown in Fig. 11(a) and (b), are valid ones. 4. According to the clutching sequence in Fig. 2(a), links L2 and L1 need to be engaged with rotating clutches. Hence, the active xycodes for the coded sketch in Fig. 11(a) are x3y1 and x2y2, as listed in Table 1. 5. The x-component with the minimum subscript is xs, and s is identified as Min(x3y1, x2y2)x = 2; the y-component with the minimum subscript is yt, and t is Min(x3y1, x2y2)y = 1. (xs, yt) = (x2, y1) are then obtained and recorded in Table 1. 6. According to the clutching sequence in Fig. 2(a), links L4 and L1 need to be engaged with band clutches. The x- and y-codes for link L4 in the coded sketch in Fig. 11(a) are x1 and y3, while those for link L1 are x2 and y2. From Table 1, the x- and y-components
Table 1 All feasible coded layouts for the clutching sequence in Fig. 2(a). Coaxial link
Rotating clutch L2
Coded sketch
Active xy-code
Fig. 11(a) Fig. 11(b)
x3y1 x1y3
Band clutch L1
(xs, yt)
L4
x2y2 x2y2
(x2, y1) (x1, y2)
x1 y1
L1
Coded layout
Active x-/y-code x2 y2
Fig. 12(a) Fig. 12(b)
50
W.-M. Hwang, Y.-L. Huang / Mechanism and Machine Theory 46 (2011) 44–52
Fig. 10. An alternative configuration for the Simpson gear train in Fig. 1.
with the minimum subscript of the active xy-codes are x2 and y1. Based on Rule 1, x1 is an active x-code for link L4, but y3 is not; x2 is an active x-code for link L1, but y2 is not. The results are listed in Table 1. 7. Based on the coded sketch in Fig. 11(a), a feasible coded layout with the clutching sequence given in Fig. 2(a) is established by eliminating all inactive xy-, x-, and y-codes and the passages connected to them, as shown in Fig. 12(a). One of its corresponding stick diagrams or clutch layouts is depicted in Fig. 12(c). 8. Repeat Steps 4–7 for the coded sketch in Fig. 11(b). The active xy-codes and y-codes are also listed in Table 1. The generated coded layout and one of the corresponding stick diagrams are illustrated in Fig. 12(b) and (d), respectively. This example shows that: (a) All feasible coded layouts for a given clutching sequence can be found in a logical way. (b) It is easy to transfer a feasible coded layout to a final stick diagram. 4. Conclusions In contrast to detecting invalid clutching sequences by inspection, this article proposed a systematic methodology for arranging PGT clutch layouts for automotive ATs. A new concept of coded sketches was first introduced to display all possible passages connecting clutches and coaxial links. Two identification rules were concluded for identifying invalid clutching sequences in a
Fig. 11. Coded sketches for the Simpson gear train.
W.-M. Hwang, Y.-L. Huang / Mechanism and Machine Theory 46 (2011) 44–52
51
Fig. 12. Two coded layouts and the corresponding stick diagrams.
logical way. Rule 1 indicated the way to avoid passage interference when equipping band clutches. According to Rule 2, the invalid clutching sequences, which are reasonable in gears but lead to infeasible clutch layouts, can be found and knocked out at an early stage of the configuration design. Based on the proposed coded sketches and rules, a synthesis procedure was presented to look for all usable coded clutch layouts step by step from a given clutching sequence. Two examples were then given to demonstrate the feasibility of the synthesis procedure. Example 1 showed that an invalid clutching sequence can be quickly checked by the procedure. Example 2 illustrated that all feasible coded layouts for a given clutching sequence can be found by a precise method, and that the coded layouts obtained can be directly and easily transferred to AT stick diagrams. The method presented in this article provides a good tool for the automatic generation of coded PGT clutch layouts. It can be generally applied to any type of PGTs with two degrees-of-freedom. A future work after this article will be the computer-aided sketching of the PGT clutch layouts for ATs [5]. References [1] Z. Levai, Structure and analysis of planetary gear trains, Journal of Mechanisms 3 (1968) 131–148. [2] F. Buchsbaum, F. Freudenstein, Synthesis of kinematic structure of geared kinematic chains and other mechanisms, Journal of Mechanisms 5 (1970) 357–392. [3] F. Freudenstein, An application of Boolean algebra to the motion of epicyclic drives, transaction of the ASME, Journal of Engineering for Industry (1971) 176–182 February. [4] Z. Levai, Moon-and-planet gear trains, Journal of Mechanisms 6 (1971) 21–25. [5] G. Chatterjee, L.-W. Tsai, Computer-aided sketching of epicyclic-type automatic transmission gear trains, Transaction of the ASME, Journal of Mechanical Design 118 (3) (1996) 405–411. [6] Z. Levai, Varieties of the planetary gear train types, Period Polytech Mech Eng 13 (2) (1969) 171–185. [7] W.A. Tuplin, Designing compound epicyclic gear trains for maximum efficiency at high velocity ratios, Machine Design (1957) 100–104 (April 4). [8] F. Freudenstein, A.T. Yang, Kinematics and statics of a coupled epicyclic spur-gear train, Mechanism and Machine Theory 7 (1972) 263–275. [9] W.L. Cleghorn, G. Tyc, Kinematic analysis of planetary gear trains using a microcomputer, International Journal of Mechanical Engineering Education 15 (1) (1987) 57–69. [10] H.-I. Hsieh, L.-W. Tsai, Kinematic analysis of epicyclic-type transmission mechanisms using the concept of fundamental geared entities, Transaction of the ASME, Journal of Mechanical Design 118 (2) (1996) 294–299. [11] B.A. Nadel, X. Wu, D. Kagan, Multiple abstraction levels in automobile transmission design: constraint satisfaction formulations and implementations, International Journal of Expert Systems 6 (4) (1993) 489–559. [12] M. Raghavan, The analysis of planetary gear trains, Transaction of the ASME, Journal of Mechanisms and Robotics 2 (2) (2010) 021005-1–021005-5. [13] R.H. Macmillan, Epicyclic gear efficiencies, Engineer (1949) 727–728 (Dec. 23). [14] L. Saggere, D.G. Olson, A Simplified Approach for Force and Power-Flow Analysis of Compound Epicyclic Spur-Gear Trains, Advances in Design Automation, DE-vol. 44-2(2), 1992, pp. 83–89. [15] E. Pennestri, F. Freudenstein, A systematic approach to power-flow and static force analysis in epicyclic spur-gear trains, Transaction of the ASME, Journal of Mechanical Design 115 (3) (1993) 639–644. [16] E. Pennestri, F. Freudenstein, The mechanical efficiency of epicyclic gear trains, Transaction of the ASME, Journal of Mechanical Design 115 (3) (1993) 645–651. [17] E. Pennestri, P.P. Valentini, A review of formulas for the mechanical efficiency analysis of two degrees-of-freedom epicyclic gear trains, Transaction of the ASME, Journal of Mechanical Design 125 (2003) 602–608. [18] C.S. Ross, W.D. Route, A method for selecting parallel-connected, Planetary Gear Train Arrangements for Automotive Automatic Transmissions, SAE Transactions 100 (6) (1991) 1765–1774. [19] N. Hattori, T. Oshidari, Y. Morimono, Application of a new complex planetary gearset to five-speed automatic transmission gear train, Transactions of the Society of Automotive Engineers of Japan 26 (1) (1995) 79–82.
52
W.-M. Hwang, Y.-L. Huang / Mechanism and Machine Theory 46 (2011) 44–52
[20] H.-I. Hsieh, L.-W. Tsai, A methodology for enumeration of clutching sequences associated with epicyclic-type automatic transmission mechanisms, SAE Transactions, Journal of Passenger Cars 105 (6) (1996) 928–936. [21] W.M. Hwang, 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 29 (1) (2005) 41–55. [22] C.H. Hsu, R.H. Huang, Systematic design of six-speed automatic transmissions with an eight-link two-DOF Ravigneaux gear mechanism, Transactions of the ASME, Journal of Mechanical Design 13 (1) (2009) 011004-1–011004-8. [23] C.H. Hsu, R.H. Huang, Systematic design of four-speed ravigneaux-type automatic transmissions, Journal of Chinese Society of Mechanical Engineers 30 (3) (2009) 201–209. [24] L.-W. Tsai, E.R. Maki, T. Liu, N.G. Kapil, The Categorization of Planetary Gear Trains for Automation Transmissions According to Kinematic Topology, SAE Paper No. 885062, SAE XXII FISITA '88, Automotive Systems Technology: The Future, vol. P-211 (1), 1988, pp. 1.513–1.521. [25] D.T. Shying, H.C. Huang, W.M. Hwang, Structural Synthesis of Automatic Transmission, Proceedings of the 7th IFToMM International Symposium on Linkages and Computer Aided Design Methods, Bucharest, Romania, 1997, pp. 26–30, August.
Contents lists available at ScienceDirect
Mechanism and Machine Theory j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / m e c h m t
Connecting clutch elements to planetary gear trains for automotive automatic transmissions via coded sketches Wen-Miin Hwang ⁎, Yu-Lien Huang Department of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 14 January 2010 Received in revised form 13 August 2010 Accepted 29 August 2010 Available online 27 September 2010 Keywords: Planetary gear train Automatic transmission Clutching sequence Clutch layout Coded sketch
a b s t r a c t This article presents a synthesis procedure to find feasible clutch layouts for a given planetary gear train clutching sequence. The method of constructing coded sketches for a planetary gear train is first introduced. In a coded sketch, all coaxial links, excluding the output link, are arranged in a specific order called a coaxial-link sequence. Based on the coaxial-link sequence, each link is encoded with three codes: x-code, y-code, and xy-code. Two identification rules are then concluded by examining these codes. Rule one tells how to arrange a band clutch without passage interference. Rule two is used to knock out the invalid clutching sequences leading to infeasible clutch layouts. According to the proposed coded sketches and rules, a synthesis procedure is then established for the purpose of finding all feasible coded layouts in a logical way from a given clutching sequence. Coded layouts can be directly and easily transferred to general clutch layouts. Two examples are used to demonstrate the feasibility of the synthesis procedure. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Automatic transmissions (ATs) with planetary gear trains (PGTs) have long been applied in the automotive industry. The literature on the design of planetary gear trains includes conceptual designs [1–5], kinematic analysis [6–12], power flow and efficiency analysis [7,8,13–17], and configuration designs [11,18–23]. However, less work has been done on the design of clutch layouts for ATs. For parallel-connected PGTs, Ross and Route [18] introduced an AT design tool based on a lever analogy, which included the calculation of gear ratios, gear trains selection, and the construction of clutch layouts. Nadel et al. [11] formulated the design of ATs as a constraint satisfaction problem which included kinematic, topological, stick diagram, and geometric levels. This methodology was suitable for PGTs in which two simple PGTs were combined. Hattori et al. [19] proposed 23 phase geometric patterns, in which each could provide four clutching sequences for five-speed ATs. Using a phase geometry method, each feasible clutching sequence obtained could be used to construct a clutch layout. However, this approach is suitable only for the type of PGTs consisting of two sun gears, two ring gears, and one to three meshed planet gears mounted on a common arm. Moreover, the three studies referenced above are restricted to constructing clutch layouts of ATs for specific types of PGTs. Hsieh and Tsai [20] applied the concept of fundamental geared entities of PGTs to enumerate clutching sequences for ATs. They obtained four feasible four-speed clutching sequences from the PGT composed of a Simpson gear train and a simple PGT. They also enumerated several three- and four-speed clutching sequences for each of the Simpson, Ravigneaux, and Type-6206 gear trains, and some of them were used in commercial three-speed and four-speed ATs [24]. Nevertheless, they did not develop an effective method of arranging clutch layouts for the synthesized clutching sequences. Hwang and Huang [21] proposed a novel methodology to design AT configurations which was suitable for all varieties of PGTs. The methodology included screening
⁎ Corresponding author. Tel.: + 886 6 2757575x62156; fax: + 886 6 2352973. E-mail address: [email protected] (W.-M. Hwang). 0094-114X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2010.08.013
W.-M. Hwang, Y.-L. Huang / Mechanism and Machine Theory 46 (2011) 44–52
45
suitable PGTs, enumeration of clutching sequences, optimal design of the number of teeth on gears, and an evaluation of the mechanical efficiency of PGTs. Six feasible clutching sequences and three configurations of six-speed ATs were obtained from a five-coaxial-link PGT. However, the elimination of invalid clutching sequences was conducted by inspection. Accordingly, a more precise procedure to arrange the clutch layouts of ATs is still needed for all types of PGTs. Without careful examination, a clutching sequence may be mistaken for a usable clutch layout for an AT. For example, based on the Simpson gear train in Fig. 1, four clutching sequences were derived, as shown in Fig. 2 [20]. Each XLi in the clutching sequence table indicates that the corresponding clutch connected to link i is activated for that gear. Unfortunately, the four-speed clutching sequence in Fig. 2(d) is actually an infeasible one, since no clutch layouts can be properly arranged for ATs. Two tentative arrangements depicted in Fig. 3 are constructed by equipping the Simpson gear train with three rotating clutches, C1, C2, and C3, and a band clutch, B1. The remaining band clutch, B2, required to connect link 1 always intersects with other links. A clutching sequence which seems feasible with regard to gears but leads to an infeasible clutch layout is merely an invalid sequence. Identifying invalid clutching sequences by inspection is not always reliable. The purpose of this article is to propose a precise procedure to construct usable clutch layouts for ATs composed of two degrees-of-freedom PGTs. The coded sketches of a PGT are introduced first. Each coaxial link of a PGT, excluding the output link, is encoded with three codes, x-, y-, and xy-codes, according to a specific order. Based on these codes, two rules are then proposed, that are helpful for synthesizing feasible coded layouts in a logical way. The first rule provides a non-interference condition for arranging band clutches. The second one is used to knock out the invalid clutching sequences. According to the proposed coded sketches and rules, a precise synthesis procedure is then proposed. The preceding part of the procedure is used to identify and eliminate invalid clutching sequences. The latter can synthesize all feasible coded layouts corresponding to valid clutching sequences. Afterwards, all feasible coded layouts can be directly transformed into traditional clutch layouts. Two examples are provided to show the feasibility of the procedure. 2. Coded sketches of a planetary gear train In the case of a two degrees-of-freedom PGT with n coaxial links, when one link is assigned as the output link, the remaining n − 1 links can be encoded with three codes: (a) x-codes: xn − 1, xn − 2,…, x1, (b) y-codes: y1, y2,…, yn − 1, and (c) xy-codes: xn − 1y1, xn − 2y2,…,
Fig. 1. Simpson gear train.
Fig. 2. Clutching sequences derived from Simpson gear train [20].
Fig. 3. Two infeasible clutch layouts for an invalid clutching sequence.
46
W.-M. Hwang, Y.-L. Huang / Mechanism and Machine Theory 46 (2011) 44–52
x2yn − 2, x1yn − 1. The three codes are displayed in three separate boxes used for installing clutches. The sketch connecting the n − 1 coaxial links to three coded boxes with mutually parallel passages is defined as the coded sketch of a PGT. Take the four-coaxial-link PGT in Fig. 4 [25] as an example. Since arm A is used as the output link, it is attached with a rightward arrow along the central axis of the PGT. The remaining coaxial links, S1, R1, and R2, are extended with mutually parallel passages around the planet gears and toward the left side of the PGT, as shown in Fig. 5(a). Three boxes designated as x, xy, and y are then placed at the left side of the PGT. The two upper boxes, x and y, show the regions for installing band clutches, while the lower box, xy, is for rotating clutches. The coaxial links, S1, R1, and R2, and the boxes are connected by drawing mutually parallel passages between them, as shown in Fig. 5(b). The nodes connecting box x and the three passages are assigned as x-codes from xn − 1 to x1. Hence, x-codes x3, x2, and x1 correspond to coaxial links S1, R1, and R2, respectively. The nodes connecting box y and the three passages are named as y-codes from y1 to yn − 1. Therefore, y-codes y1, y2, and y3 correspond to links S1, R1, and R2, respectively. Combining x-codes and y-codes into box xy gives xy-codes from xn − 1y1 to x1yn − 1. As a result, xy-codes x3y1, x2y2, and x1y3 corresponding to coaxial links S1, R1, and R2, respectively, are employed. Thus, the coded sketch of PGT in Fig. 4 is constructed, as shown in Fig. 5(b). Note that the sequences of the coaxial links corresponding to the x-codes, y-codes, and xy-codes are the same: S1–R1–R2. For simplicity, the sequence of coaxial links in every box is called a coaxial-link sequence. When the coaxial links are connected to boxes xy and y for a clutching sequence, their horizontal passages may intersect with the vertical passages connecting to boxes x and xy. Any intersections will make the clutch arrangement fail. Therefore, passage intersections are useful for checking whether or not the clutching sequence leads to a usable clutch layout. Take the four-speed
Fig. 4. A planetary gear train with four coaxial links [25].
Fig. 5. The illustration of the construction of a coded sketch for the PGT in Fig. 4. (a) A PGT and three associated boxes. (b) All mutually parallel passages connecting coaxial links and boxes.
W.-M. Hwang, Y.-L. Huang / Mechanism and Machine Theory 46 (2011) 44–52
47
clutching sequence in Fig. 6 [25] as an example, which is derived from the PGT in Fig. 4. Since coaxial links R1 and S1 need to engage with rotating clutch C1 and band clutch B2, respectively, the horizontal passage connecting link R1 with box xy interferes with the vertical passage connecting link S1 with box x, as shown in Fig. 7(a). It is obvious that the band clutch B2 engaging with link S1 cannot be placed at box x. According to the coded sketch in Fig. 5(b), the xy-code for link R1 is x2y2, and the x-code for link S1 is x3. The subscript of x3 is greater than that of x2 in x2y2. This indicates that when link R1 with an xy-code of x2y2 needs to engage with a rotating clutch, other coaxial links which have subscripts for their x-codes greater than two, which is the subscript of x in x2y2, cannot be engaged with a band clutch at box x. In the same way, other coaxial links which have subscripts for their y-codes greater than that of y2 in x2y2, cannot be engaged with a band clutch at box y. For example, as shown in Fig. 7(b), link R2, whose y-code is y3, cannot be engaged with a band clutch at box y because the subscript of y3 is greater than that of y2 in x2y2, which is the xy-code of link R1. Therefore, the non-interference condition for connecting rotating and band clutches with a PGT can be concluded as follows: Rule 1. For a clutching sequence derived from an n-coaxial-link PGT, if there are a total of e (e ≤ n − 1) coaxial links with xy-codes xayα, xbyβ, …, xeyε, respectively, that need to be engaged with rotating clutches, the non-interference condition for a coaxial link with x-code xi to be engaged with a band clutch at box x is: i ≤ Minða; b;…; eÞ: The non-interference condition for a coaxial link with y-code yj to be engaged with a band clutch at box y is: j ≤ Minðα; β;…; εÞ:
ð1Þ ð2Þ
As shown in Fig. 8, links S1 and R2 with xy-codes x3y1 and x1y3, respectively, need to be engaged with rotating clutches. Link R1 with x-code x2 and y-code y2 needs to be engaged with a band clutch. According to Rule 1, i = 2, Min(a, b) = Min(3, 1) = 1. The situation of i ≤ Min(a, b) does not exist. Therefore, link R1 cannot engage with a band clutch at box x. In the same way, j = 2, Min(α,
Fig. 6. A clutching sequence derived from the PGT in Fig. 4 [25].
Fig. 7. Interference of passages.
48
W.-M. Hwang, Y.-L. Huang / Mechanism and Machine Theory 46 (2011) 44–52
Fig. 8. The sandwich set SC1(RB1)RC2.
β) = Min(1, 3) = 1. The situation of j ≤ Min(α, β) does not exist, either. Thus, link R1 cannot engage with a band clutch at box y, either. According to this example, a conclusion is stated as follows: Rule 2. Coaxial links p and q, which are not adjacent to each other in a coaxial-link sequence, need to be engaged with rotating clutches. If there exists a coaxial link r located between links p and q that is required to be engaged with a band clutch, this situation will lead to an infeasible clutch layout based on the corresponding coded sketch. The coaxial-link set of p, q, and r is defined as a sandwich set, which can be expressed as pC(rB)qC. The superscripts C and B mean that links p and q need to be engaged with rotating clutches and link r with a band clutch. If the coaxial-link sequence of a coded sketch is u–v–x–y–z, 10 sandwich sets can be found by using the theory of combinations. They are uC(vB)xC, uC(vB)yC, uC(vB)zC, uC(xB)yC, uC(xB)zC, uC(yB)zC, vC(xB)yC, vC(xB)zC, vC(yB)zC, and xC(yB)zC. According to Rule 2, if any one of these sandwich sets exists in the clutching sequence discussed, its coded sketch is invalid. For example, the only sandwich set for the coded sketch in Fig. 5(b) is SC1(RB1)RC2. From the clutching sequence in Fig. 6, it is clear that links R1, R2, and S1 need to be engaged with rotating clutches, and links R2 and S1 with band clutches. The sandwich set SC1(RB1)RC2 does not exist in the clutching sequence. Therefore, the coded sketch in Fig. 5(b) constructed from the clutching sequence shown in Fig. 6 is a valid sketch. It will lead the clutching sequence to at least one feasible clutch layout. When depicting a coded sketch, the output link is always drawn with an arrow to the right and has no interference with other coaxial links. This means that a clutching sequence may be depicted in several possible coded sketches. Taking the PGT in Fig. 4 as an example, since arm A can be pulled with an arrow to the right in different ways, three coded sketches are constructed, as shown in Figs. 5(b) and 9. Compared to the coded sketch in Fig. 5(b), the other two in Fig. 9 have different coaxial-link sequences, R2–S1–R1 and R1–R2–S1. They also contain sandwich sets RC2(SB1)RC1 and RC1(RB2)SC1, respectively. Because the two sandwich sets exist in the clutching sequence shown in Fig. 6, both are invalid sketches. 3. The synthesis procedure for usable coded layouts For a given clutching sequence, using the coded sketches of its corresponding PGT and the proposed two rules, a precise synthesis procedure is presented herein to find all feasible coded layouts. If the clutching sequence leads to no feasible clutch
Fig. 9. Two other coded sketches for the PGT in Fig. 4.
W.-M. Hwang, Y.-L. Huang / Mechanism and Machine Theory 46 (2011) 44–52
49
layouts, this procedure can still check it. The steps for knocking out an invalid clutching sequence and finding coded layouts for a valid clutching sequence are described as follows: 1. Construct all coded sketches for a given PGT and its clutching sequence. 2. List all sandwich sets for each coded sketch. 3. If one of the sandwich sets of a coded sketch exists in the given clutching sequence, the coded sketch is identified as an invalid one. If all coded sketches of the PGT are recognized to be invalid sketches for the clutching sequence, the clutching sequence will lead only to infeasible clutch layouts. Thus, the clutching sequence should be knocked out. 4. For each coded sketch which does not fail in Step 3, determine which xy-codes are active. An active xy-code means that its corresponding coaxial link needs to be engaged with a rotating clutch based on the given clutching sequence. 5. For a coded sketch, if the active xy-codes obtained from Step 4 are xayα, xbyβ, …, and xeyε, in which Min(xayα, xbyβ, …, xeyε)x = Min(a, b, …, e) = s and Min(xayα, xbyβ, …, xeyε)y = Min(α, β, …, ε) = t, record (xs,yt) in a table similar to Table 1. 6. According to Rule 1, determine which x-codes and y-codes are active for each link that needs to be engaged with a band clutch. If the subscript of its corresponding x- or y-code is not greater than s (for the x-codes) or t (for the y-codes) of the active xy-codes, this x- or y-code is active. 7. Eliminate all inactive xy-, x-, and y-codes and passages connected to them. A coded layout is thus established. 8. For the given clutching sequence, repeat Steps 4 to 7 for all valid coded sketches that do not fail in Step 3. All usable coded layouts are then found.
Example 1. Identification of an invalid clutching sequence. The clutching sequence in Fig. 2(d) is derived from the Simpson gear train in Fig. 1 for four-speed ATs [20]. Using the proposed procedure, it can be identified as infeasible for a usable clutch layout at Step 3. The first three steps of the procedure for this example are described as follows: 1. Besides the configuration in Fig. 1, the Simpson gear train can also be depicted as the configuration in Fig. 10. The former configuration has only one coded sketch, as shown in Fig. 11(a), while the latter has two different coded sketches, as shown in Fig. 11(b) and (c). The coaxial-link sequences for the coded sketches in Fig. 11(a), (b), and (c) are L2–L1–L4, L4–L1–L2, and L1–L4–L2, respectively. 2. The sandwich sets for the coded sketches in Fig. 11(a), (b), and (c) are LC2(LB1)LC4, LC4(LB1)LC2, and LC1(LB4)LC2, respectively. 3. Because the sandwich sets listed at Step 2 exist in the clutching sequence in Fig. 2(d), they are all invalid coded sketches according to Rule 2. The clutching sequence leads only to infeasible clutch layouts. Therefore, it is an invalid clutching sequence and should be knocked out at this step. This result is consistent with the illustrations in Fig. 3. Example 2. Construction of coded layouts for a PGT having feasible coded sketches. The clutching sequence given in Fig. 2(a) is obtained from the Simpson gear train in Fig. 1 for three-speed ATs [20]. A total of two usable coded layouts can be found by using the proposed procedure. The steps of the procedure are described as follows: 1. As illustrated in Example 1, all available coded sketches for the Simpson gear train are shown in Fig. 11. The coaxial-link sequences for the three coded sketches are L2–L1–L4, L4–L1–L2, and L1–L4–L2, respectively. 2. Each coded sketch has only one sandwich set, i.e., LC2(LB1)LC4, LC4(LB1)LC2, and LC1(LB4)LC2, respectively. 3. Sandwich set LC1(LB4)LC2 exists in the clutching sequence in Fig. 2(a); i.e., links L2 and L1 need to be engaged with rotating clutches and link L4 with a band clutch. Therefore, the coded sketch in Fig. 11(c) is an invalid sketch. The remaining two coded sketches, as shown in Fig. 11(a) and (b), are valid ones. 4. According to the clutching sequence in Fig. 2(a), links L2 and L1 need to be engaged with rotating clutches. Hence, the active xycodes for the coded sketch in Fig. 11(a) are x3y1 and x2y2, as listed in Table 1. 5. The x-component with the minimum subscript is xs, and s is identified as Min(x3y1, x2y2)x = 2; the y-component with the minimum subscript is yt, and t is Min(x3y1, x2y2)y = 1. (xs, yt) = (x2, y1) are then obtained and recorded in Table 1. 6. According to the clutching sequence in Fig. 2(a), links L4 and L1 need to be engaged with band clutches. The x- and y-codes for link L4 in the coded sketch in Fig. 11(a) are x1 and y3, while those for link L1 are x2 and y2. From Table 1, the x- and y-components
Table 1 All feasible coded layouts for the clutching sequence in Fig. 2(a). Coaxial link
Rotating clutch L2
Coded sketch
Active xy-code
Fig. 11(a) Fig. 11(b)
x3y1 x1y3
Band clutch L1
(xs, yt)
L4
x2y2 x2y2
(x2, y1) (x1, y2)
x1 y1
L1
Coded layout
Active x-/y-code x2 y2
Fig. 12(a) Fig. 12(b)
50
W.-M. Hwang, Y.-L. Huang / Mechanism and Machine Theory 46 (2011) 44–52
Fig. 10. An alternative configuration for the Simpson gear train in Fig. 1.
with the minimum subscript of the active xy-codes are x2 and y1. Based on Rule 1, x1 is an active x-code for link L4, but y3 is not; x2 is an active x-code for link L1, but y2 is not. The results are listed in Table 1. 7. Based on the coded sketch in Fig. 11(a), a feasible coded layout with the clutching sequence given in Fig. 2(a) is established by eliminating all inactive xy-, x-, and y-codes and the passages connected to them, as shown in Fig. 12(a). One of its corresponding stick diagrams or clutch layouts is depicted in Fig. 12(c). 8. Repeat Steps 4–7 for the coded sketch in Fig. 11(b). The active xy-codes and y-codes are also listed in Table 1. The generated coded layout and one of the corresponding stick diagrams are illustrated in Fig. 12(b) and (d), respectively. This example shows that: (a) All feasible coded layouts for a given clutching sequence can be found in a logical way. (b) It is easy to transfer a feasible coded layout to a final stick diagram. 4. Conclusions In contrast to detecting invalid clutching sequences by inspection, this article proposed a systematic methodology for arranging PGT clutch layouts for automotive ATs. A new concept of coded sketches was first introduced to display all possible passages connecting clutches and coaxial links. Two identification rules were concluded for identifying invalid clutching sequences in a
Fig. 11. Coded sketches for the Simpson gear train.
W.-M. Hwang, Y.-L. Huang / Mechanism and Machine Theory 46 (2011) 44–52
51
Fig. 12. Two coded layouts and the corresponding stick diagrams.
logical way. Rule 1 indicated the way to avoid passage interference when equipping band clutches. According to Rule 2, the invalid clutching sequences, which are reasonable in gears but lead to infeasible clutch layouts, can be found and knocked out at an early stage of the configuration design. Based on the proposed coded sketches and rules, a synthesis procedure was presented to look for all usable coded clutch layouts step by step from a given clutching sequence. Two examples were then given to demonstrate the feasibility of the synthesis procedure. Example 1 showed that an invalid clutching sequence can be quickly checked by the procedure. Example 2 illustrated that all feasible coded layouts for a given clutching sequence can be found by a precise method, and that the coded layouts obtained can be directly and easily transferred to AT stick diagrams. The method presented in this article provides a good tool for the automatic generation of coded PGT clutch layouts. It can be generally applied to any type of PGTs with two degrees-of-freedom. A future work after this article will be the computer-aided sketching of the PGT clutch layouts for ATs [5]. References [1] Z. Levai, Structure and analysis of planetary gear trains, Journal of Mechanisms 3 (1968) 131–148. [2] F. Buchsbaum, F. Freudenstein, Synthesis of kinematic structure of geared kinematic chains and other mechanisms, Journal of Mechanisms 5 (1970) 357–392. [3] F. Freudenstein, An application of Boolean algebra to the motion of epicyclic drives, transaction of the ASME, Journal of Engineering for Industry (1971) 176–182 February. [4] Z. Levai, Moon-and-planet gear trains, Journal of Mechanisms 6 (1971) 21–25. [5] G. Chatterjee, L.-W. Tsai, Computer-aided sketching of epicyclic-type automatic transmission gear trains, Transaction of the ASME, Journal of Mechanical Design 118 (3) (1996) 405–411. [6] Z. Levai, Varieties of the planetary gear train types, Period Polytech Mech Eng 13 (2) (1969) 171–185. [7] W.A. Tuplin, Designing compound epicyclic gear trains for maximum efficiency at high velocity ratios, Machine Design (1957) 100–104 (April 4). [8] F. Freudenstein, A.T. Yang, Kinematics and statics of a coupled epicyclic spur-gear train, Mechanism and Machine Theory 7 (1972) 263–275. [9] W.L. Cleghorn, G. Tyc, Kinematic analysis of planetary gear trains using a microcomputer, International Journal of Mechanical Engineering Education 15 (1) (1987) 57–69. [10] H.-I. Hsieh, L.-W. Tsai, Kinematic analysis of epicyclic-type transmission mechanisms using the concept of fundamental geared entities, Transaction of the ASME, Journal of Mechanical Design 118 (2) (1996) 294–299. [11] B.A. Nadel, X. Wu, D. Kagan, Multiple abstraction levels in automobile transmission design: constraint satisfaction formulations and implementations, International Journal of Expert Systems 6 (4) (1993) 489–559. [12] M. Raghavan, The analysis of planetary gear trains, Transaction of the ASME, Journal of Mechanisms and Robotics 2 (2) (2010) 021005-1–021005-5. [13] R.H. Macmillan, Epicyclic gear efficiencies, Engineer (1949) 727–728 (Dec. 23). [14] L. Saggere, D.G. Olson, A Simplified Approach for Force and Power-Flow Analysis of Compound Epicyclic Spur-Gear Trains, Advances in Design Automation, DE-vol. 44-2(2), 1992, pp. 83–89. [15] E. Pennestri, F. Freudenstein, A systematic approach to power-flow and static force analysis in epicyclic spur-gear trains, Transaction of the ASME, Journal of Mechanical Design 115 (3) (1993) 639–644. [16] E. Pennestri, F. Freudenstein, The mechanical efficiency of epicyclic gear trains, Transaction of the ASME, Journal of Mechanical Design 115 (3) (1993) 645–651. [17] E. Pennestri, P.P. Valentini, A review of formulas for the mechanical efficiency analysis of two degrees-of-freedom epicyclic gear trains, Transaction of the ASME, Journal of Mechanical Design 125 (2003) 602–608. [18] C.S. Ross, W.D. Route, A method for selecting parallel-connected, Planetary Gear Train Arrangements for Automotive Automatic Transmissions, SAE Transactions 100 (6) (1991) 1765–1774. [19] N. Hattori, T. Oshidari, Y. Morimono, Application of a new complex planetary gearset to five-speed automatic transmission gear train, Transactions of the Society of Automotive Engineers of Japan 26 (1) (1995) 79–82.
52
W.-M. Hwang, Y.-L. Huang / Mechanism and Machine Theory 46 (2011) 44–52
[20] H.-I. Hsieh, L.-W. Tsai, A methodology for enumeration of clutching sequences associated with epicyclic-type automatic transmission mechanisms, SAE Transactions, Journal of Passenger Cars 105 (6) (1996) 928–936. [21] W.M. Hwang, 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 29 (1) (2005) 41–55. [22] C.H. Hsu, R.H. Huang, Systematic design of six-speed automatic transmissions with an eight-link two-DOF Ravigneaux gear mechanism, Transactions of the ASME, Journal of Mechanical Design 13 (1) (2009) 011004-1–011004-8. [23] C.H. Hsu, R.H. Huang, Systematic design of four-speed ravigneaux-type automatic transmissions, Journal of Chinese Society of Mechanical Engineers 30 (3) (2009) 201–209. [24] L.-W. Tsai, E.R. Maki, T. Liu, N.G. Kapil, The Categorization of Planetary Gear Trains for Automation Transmissions According to Kinematic Topology, SAE Paper No. 885062, SAE XXII FISITA '88, Automotive Systems Technology: The Future, vol. P-211 (1), 1988, pp. 1.513–1.521. [25] D.T. Shying, H.C. Huang, W.M. Hwang, Structural Synthesis of Automatic Transmission, Proceedings of the 7th IFToMM International Symposium on Linkages and Computer Aided Design Methods, Bucharest, Romania, 1997, pp. 26–30, August.