An Augmented Lever Analogy Method for Kinematic Analysis of Dual-Input Planetary/Epicyclic Gear Sets Involving Planet Gear

The lever analogy method (LAM) is a translational system representation for the rotating components and is widely used for the kinematic analysis of PGSs/EGSs. However, it includes only the sun gear, ring gear, and carrier, and ignores the kinematic information of the planet gear. The planet gear kinematic information is vital for its bearing life prediction, and speed sequence, power flow, and efficiency analysis of dual-input PGSs/EGSs. The traditional LAM doesn’t work when involving the planet gear kinematic information, because the kinematic information of planet gear is eliminated during the process of merging similar items. In this paper, an augmented lever analogy method (ALAM) is proposed to make up for the lack of traditional LAM in analyzing planet gear kinematic information, and analyze the kinematic relationship between planet gear to other components for the dual-input PGSs/EGSs. In this method, the new nodes and lever lengths representing the planet gear are added to the LAM by analyzing peripheral velocity relationships at the meshing points of PGSs/EGSs. In addition, not all the dual-input compound PGSs/EGSs (e.g. the compound PGSs/EGSs with planet gears in series, etc.) can be analyzed by the traditional LAM. The proposed method can easily establish the augmented lever models for all of them and derive the corresponding kinematic expressions. The results show that the proposed ALAM has good visibility and greater versatility, and can accurately and efficiently calculate the rotating speed of planet gears for calculating the speed sequence, power flow, and efficiency of PGSs/EGSs, which can cover all kinds of the PGSs/EGSs, and greatly reduce the technical threshold and time for their kinematic analysis.

investigation of a dual-mode EVT. The torque and speed of 55 the powertrain elements and power split ratio were analyzed 56 using the LAM by Wang et al. [8] and Kim et al. [9]. 57 Barhoumi et al. [10], Barhoumi et al. [11], Kim and Kum [12], 58 and Kang et al. [13] used the LAM to explore fuel econ-59 omy and acceleration performance metrics, of compound 60 split hybrid configurations. Yang et al. [14] analyzed the 61 configurations and upshift/downshift process kinematics of 62 the dual-input compound power-split mechanism (DICPSM). 63 Liu et al. [15] proposed a systematic design method to syn-64 thesize the configuration scheme for multi-row and multi-65 speed AT based on the LAM. Xie et al. [16] analyzed the 66 speed ratio of components by using the lever analogy method 67 to derive the two-PGS PGTs with a high reduction speed 68 ratio. Zhang et al. [17] obtained eight schemes with a high 69 reduction speed ratio based on the LAM. Liao and Chen 70 [1] presented an improved lever analogy method to simplify 71 the analysis for determining the speed ratios for automatic 72 transmissions based on the LAM. The LAM is used to analyze 73 the speed of the components of PGSs/EGSs in the above 74 research. Additionally, the other properties researches of 75 PGSs/EGSs transmissions, such as the power flow, torque, 76 and configuration, are using the LAM to analyze the compo-77 nents' speed. Chao et al. [18] and Hong et al. [19] studied the 78 power flow of series-split EVT for a plug-in hybrid vehicle 79 by using the LAM. Zhu et al. [20] used the LAM to analyze 80 the transient torques with different power sources for a multi-81 mode transmission with a single electric machine. Ho and 82 Hwang [21] applied the graph theory and the LAM to analyze 83 the power flow, kinematic, and configurations of PGSs/EGSs. 84 Ross and Route [22] established compound levers consist-85 ing of multiple levers in parallel connection by considering 86 the number and magnitude of the required ratios. Liu et al. 87 [23] proposed a systematic analysis methodology to design 88 suitable three-mode configurations based on the lever model. 89 Based on a deduction method and composite lever analogy, 90 Peng et al. [24] developed an efficient synthesis method for 91 the PGS with two operating DOFs. 92 The above research analyzed the rotating speed of ring 93 gear, carrier, and sun gear based on the LAM, but that of 94 planet gear cannot be obtained since traditional LAM ignores 95 the kinematic information of planet gear. However, the rotat-96 ing speed and other kinematic information of planet gear are 97 necessary in cases, such as the power flow and efficiency 98 analysis for dual-input PGSs/EGSs [25], [26], mesh phas-99 ing relationships analysis [2], [27], entrainment velocity of 100 ball bearings and PGSs/EGSs [28], [29], [30], [31] and the 101 limit speed and life span of bearing ball and planet gear as 102 the design limitation factors [32]. Therefore, based on the 103 advantages of the LAM, an innovative method, which exceeds 104 the research limitation of the traditional LAM, is urgently 105 needed to establish the general lever model for all kinds of 106 PGSs/EGSs and analyze their kinematic information. In this 107 paper, an extended lever analogy method (ALAM) contain-108 ing the kinematic information of planet gears is proposed. 109 In this method, the three instantaneous centers theorem is 110 used to establish the kinematic equations among components 111 gears in series. In Section 4, the analysis models of dual-   derivation of ALAM including the kinematic information of 167 planet gear.

168
The kinematic models of SPGS and DPGS shown in 169 Fig. 2 are used to analyze the kinematic relationships among 170 three components, i.e. the sun gear, the ring gear, and the 171 carrier. In Fig. 2 (a), R P and ω P represent the base circle 172 radius and the rotating speed of the planet gear. Here, points 173 A and B denote the meshing points of the ring-planet gear pair 174 and the sun-planet gear pair of SPGS, respectively. Similarly, 175 in Fig. 2 (b), R P1 and R P2 represent the base circle radius 176 of planet gear #1 and planet gear #2, respectively. ω P1 and 177 ω P2 represent the rotating speed of planet gear #1 and planet 178 gear #2, respectively. Meanwhile, points A' and B' denote the 179 meshing points of the ring-planet gear pair and the sun-planet 180 gear pair of DPGS, respectively.

181
In Fig. 2 (a), the sun gear, the ring gear, and the carrier are 182 rotating in the same direction, and the planet gear is rotating 183 in the opposite direction. Therefore, the kinematic equations 184 of the model in Fig. 2 (a) can be described as follows: can be obtained: 194 By dividing both sides of Eq. (2) by R S , the expression can 195 be rewritten as follows: where K is the gear ratio of the ring gear to the sun gear, the kinematic equations of the model in Fig. 2 (b) can be 211 described as follows: where V PA and V RA represent the peripheral velocity at point 220 and the ring gear are rotating in the same direction, and the 233 sun gear is rotating in the opposite direction. The black arrows 234 represent the translation of peripheral velocity at meshing 235 points A and B. The deriving process of ALAM for SPGS 236 can be written as follow: where the expression containing the planet gear information 239 is obtained: The assembly conditions of SPGS can be written as: By substituting Eq. (9) into Eq. (8) and dividing both sides 244 of Eq. (8) by R S , the expression can be rewritten as follows: 245 Fig. 3 and Eq. (10), the kinematic relationship of the 247 ALAM can be obtained.

248
In Fig. 3 (b), the sun gear, planet gear #2, and the ring 249 gear are rotating in the same direction, but planet gear #1 is 250 rotating in the opposite direction. The kinematic equations 251 are different from that of the first method as follows: The above equations can be simplified as follows by apply-254 ing the motion inversion to the carrier: The point location of planet gear in the ALAM can be 257 determined by considering the relationship between rotating 258 speeds and directions of SPGS components.

259
In Fig. 4, the white levers and nodes are the traditional 260 LAM, and the black levers and orange nodes are the extended 261 parts of the ALAM relative to the LAM (shown in Fig. 1).

262
Furthermore, the expression of the length for the new lever 275 L P of SPGS can be described as follows: The new lever length of the planet gear in the ALAM is in 278 good agreement with that of Reference [35].

279
Combining Eqs. (6), (11), and (12), the lever model con-280 taining the information of planet gears of DPGS can be 281 written as follows: Similarly, the expression of the length for the new lever L Pi 284 of DPGS can be described as follows: The extended lever model can be obtained by substituting It is well known that the traditional method (analysis method) 292 is inefficient and cumbersome for analyzing the PGSs/EGSs 293 by analyzing the kinematic of PGSs/EGSs in terms of 294 torque and speed calculations. In addition, the abstraction 295 of the equations and the complexity involved make many 296 engineers unfamiliar with the specific functions and kine-297 matic characteristics of complex PGSs/EGSs [36]. The LAM 298 substitutes the torque and rotational speed relationship of 299 the PGSs/EGSs components with the horizontal force and 300 speed relationship of the lever node analogy. The kine-301 matic functions of PGSs/EGSs are clearly visualized with-302 out entirely having a good command of the intricacies of 303 PGSs/EGSs [5], [37].

304
Since the proposed ALAM is derived based on the LAM, 305 the merits of LAM are also reflected in ALAM. The planet 306 gear kinematic information needs to be considered for ana-307 lyzing the planet gear life and the transmission efficiency 308 of dual-input PGSs/EGSs. However, the traditional LAM 309 cannot involves the kinematic information of planet gear. 310 So the ALAM, in which the kinematic information of planet 311 gear is included, becomes more significant. In addition, the 312 unified equations and corresponding intuitively lever models   analyze the speed relationship between the planet gear and 325 other components of PGSs/EGSs [2], [27], [38], [39], [40]. 326

III. VALIDATION OF THE PROPOSED AUGMENTED LEVER 327
ANALOGY METHOD 328 Figure 8 shows NASA's EGS mechanism [38], [41], [42], 329 [43], namely the configuration of Fig. 7 (b), in which there 330 are three links where the motor can be the input or the 331 output. During the integration, the lever length between the 332 planet gear and the ring gear is essential due to the specific 333 structure of this mechanism. To analyze the kinematics of 334 this mechanism conveniently based on the ALAM, the virtual 335 PGSs/EGSs, and the corresponding extended lever models 336 are established, as shown in Fig. 9 (a) and (b). According to 337 Figs. 4 and 6, the mechanism diagram and virtual extended 338 lever model can be integrated into a compound lever model, 339 as shown in Fig. 9 (c) and (d) respectively, by using the 340 improved lever analogy method proposed in Reference [33]. 341 The teeth of the gears are as follows: Z S1 = 28, Z P1 = 36, 342 Z S2 = 36, and Z P2 = 28. The speed ratio ω Ratio between the 343 components can be calculated by assuming that the virtual 344 teeth of ring gear [44] The results of the speed ratios of the mechanism in  Therefore, further discussion is required on a case-by-case 370 basis. In the paper, the dual-input PGSs/EGSs are taken as a 371 reduction device, and the working conditions are determined.

372
In Fig. 11 (a), the red dotted arrows indicate the absolute 373 rotating speed and direction of corresponding components.

374
The blue dotted arrows indicate the output rotating speed and 375 direction, which is relative to the carrier. During an upshift, 376 ω C rises and approaches ω R . Here, ω S -ω C is a negative value, 377 and ω R -ω C is a positive value. At this time, min(K /(1+K )) < condition. The proposed method can also be applied to mech-382 anisms with similar structures and kinematic characteristics, 383 such as rolling bearings, etc, as shown in Fig. 11 (b). B, O, 384 C, and I represent the balls, the outer race, the cage, and 385 the inner race of the rolling bearing, respectively. L b and 386 K o/i respectively represent the lever length corresponding to 387 the balls and the radii ratio of the outer race and inner race 388 in rolling bearing. It is worth noting that there is only one 389 dual-input working mode for rolling bearings (intermediate 390 bearing), that is, the inner and outer races are the input parts, 391 and the cage is the output part. Due to the limitation of space, 392 this paper will not analyze the dual-input case of rolling 393 bearings.

394
In Fig. 11 (a), the rotating speed of planet gear was calcu-395 lated according to the two input rotating speeds, i.e. rotating 396 speed of the ring gear and the carrier. Based on the ALAM, 397 the speed ratios of the planet gear to the other three compo-398 nents can be written as Eq. (21) ∼ Eq. (24).  Figure 12 shows the extended lever model of DPGS, and the 407 input components are the same as that in Fig. 12 Fig. 13 (a). In the reverse operation of motors, ω C is 425 positive and increases from 0 to ω S during upshift, as shown 426 in Fig. 13 (b).

432
In this section, the speed sequences between the components 433 of the system with the different structures and input condi-434 tions are discussed. Given parameters K and ω R /ω C , the speed ratios ω P /ω S , 440 ω P /ω C , and ω P /ω R can be calculated readily, as shown in 441 Fig. 14 (a). It can be found that the three speed ratios are 442 increasing with K , but decreasing with ω R /ω C . To further 443 describe the speed ratio relationship between ω P and ω S , ω C 444 and ω R in Fig. 14 (a), the difference between two ratios ω P /ω C 445 and ω P /ω R with K = 2.2, are demonstrated in Fig. 14 (b).

446
Since the values of ω P /ω C and the ω P /ω R are distributed 447 on the upper and lower sides of ω Ratio = 1 respectively, like 448 the region between two vertical lines shown in Fig. 14 (b) in 449 which the values of ω P /ω C are larger than ω Ratio = 1, and 450 the values of ω P /ω R are less than ω Ratio = 1. Therefore, 451 Fig. 14 (a) is divided into two parts, i.e. part I and part II.

458
In Fig. 14 (c), part II is divided into three subparts and and other components for DPGS is illustrated in Fig. 15.

465
The relationship between speed ratios ω P1 /ω R , ω P1 /ω C , and 466 ω P1 /ω S , which take ω C /ω R and Z S /Z P1 as the variables, 467 are shown in Fig. 15 (a). It can be found that the three 468 speed ratios are increasing with both ω C /ω R and Z S /Z P1 .

508
The three regions in Fig.16 (g)

513
In this paper, an augmented lever analogy method containing 514 the kinematic information of the planet gear was proposed. 515 The lever node and length models of planet gear in plan-516 etary gear sets (PGSs/EGSs) were established by using the 517 three instantaneous centers theorem to analyze the peripheral 518 velocity of meshing points and make up for the lack in analyz-519 ing kinematic of planet gear and the compound PGSs/EGSs 520 with planet gears in series of the traditional LAM caused by 521 eliminating the kinematic information of planet gear during 522 the process of merging similar items. The different dual-input 523 working modes for PGSs/EGSs were carried out, and the 524 rotating speed relationships and speed sequence between all 525 components of SPGS (Simple planetary gear set) and DPGS 526 (Double-planet planetary gear set) including the planet gears 527 were obtained and discussed by divided regions. The main 528 conclusion points of this study are drawn as follows: 529 1) The augmented lever analogy method (ALAM) was 530 proposed by adding the nodes on the LAM to represent 531 the planet gears, which can establish the lever model 532 for all kinds of the PGSs/EGSs, and create new lever 533 relationships of the planet gear to other components. 534 2) The proposed ALAM method has more significantly 535 comprehensive merits including the newly unified 536 kinematic expressions and intuitive and efficient anal-537 ysis of all the components including the planet gears 538 when compared to the traditional lever analogy method 539 (LAM) while analyzing the kinematic information for 540 the dual-input PGSs/EGSs.

541
3) The rotating speed relationships between components 542 of PGSs/EGSs were divided into several regions within 543 the variables range by taking the planes of ω Ratio equal 544 to 1 or −1. The corresponding regions and speed 545 sequence can be found rapidly for PGSs/EGSs working 546 at different modes.

547
4) It is worth noting that the mechanisms with simi-548 lar structures and kinematic characteristics (e.g. com-549 pound PGSs/EGSs with planet gears in series, etc.) and 550 planet-like mechanisms, (e.g. the rolling bearing), can 551 also be analyzed by the proposed ALAM. Therefore, 552 the proposed method is more general.