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Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000.
Digital Object Identifier 10.1109/ACCESS.2022.Doi Number
An Augmented Lever Analogy Method for
Kinematic Analysis of Dual-Input Planetary/
Epicyclic Gear Sets Involving Planet Gear
Xiaodong Yang1,2, Wennian Yu*1,2, Yimin Shao1, Zhiliang Xu1,2, Qiang Zeng1, Chunhui Nie1,2
and Dingqiang Peng1,2
1State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400044, PR China
2College of Mechanical and Vehicle Engineering, Chongqing University, Chongqing 400044, PR China
Corresponding author: Wennian Yu (e-mail: wennian.yu@cqu.edu.cn).
This research was funded by the National Natural Science Foundation of China (Grant No. 52035002, 52105086) and the China Postdoctoral Science
Foundation (Grant No. 2021M700583).
ABSTRACT 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.
INDEX TERMS Planetary gear set; Planet gear; Speed sequence; Lever analogy method; Dual-Input
TABLE I
NOMENCLATURE
PGSs/EGSs
Planetary/Epicyclic gear sets
LAM
Lever analogy method
ALAM
Augmented lever analogy method
ωRatio
The ratio between the rotating speed
EVT
Electrical variable transmission
SPGS
Simple planetary gear set
DPGS
Double-planet planetary gear set
PGS-i
First PGS (Front), i= 1; Second PGS (Rear), i= 2.
Zj
The teeth number of component j,j=S,R,P.
K
The characteristic parameter of PGS, K= ZR/ZS.
Tk
The load torque of component k,k=S,R,P,C.
ωk
The rotating speed of component k,k=S,R,P,C.
Lq
The lever length of planet gear q,q=P(SPGS); q=P1,
P2 (DPGS); q=b(rolling bearing)
S+S
A 2PGS arrangement with its first (PGS-1) and second
(PGS-2) gear set both being SPGS
S+D
A 2PGS arrangement with its first (PGS-1) and second
(PGS-2) gear set being SPGS and DPGS respectively
D+S
A 2PGS arrangement with its first (PGS-1) and second
(PGS-2) gear set being DPGS and SPGS respectively
D+D
A 2PGS arrangement with its first (PGS-1) and second
(PGS-2) gear set both being DPGS
Rki,j
The speed ratio between components iand jrelative to
the component k.
S,R,P,C
Sun gear, Ring gear, Planet gear and Carrier
R-C
Ring gear and carrier dual-input working condition
I. INTRODUCTION
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content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3206845
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VOLUME XX, 2022 2
Planetary/Epicyclic gear sets (PGSs/EGSs) have the
advantages of high transmission ratios, large torque-weight
ratios and compactness in comparison with ordinary gear
trains, which make them widely used in fields like
automotive, aerospace, marine, wind turbines, and machine
tools applications [1-3]. Currently, a lot of methods have
been proposed for the design and kinematic analysis of
PGSs/EGSs, such as the component analysis synthesis
method, the line graph synthesis method, the combination
solution method, the rod system conversion [4], and the
lever analogy method (LAM) [5], etc. The component
analysis synthesis method and the combined solution
method are traditional methods of analyzing PGSs/EGSs.
However, a large number of kinematic equations need to be
established when applying these methods which makes
them cumbersome and lacks universality. The line graph
synthesis method, the rod system conversion, and the LAM
are graph-based methods. Applying the graph theory to the
kinematic analysis of PGSs/EGSs, a simple and practical
LAM was first proposed by Benford and Leising [5], which
considers the PGS as an equivalent vertical lever based on
appropriate assumption and simplification to position PGS.
The vertical lever equivalent substitution reflects the
characteristics so that the position relationship of PGS
rotating members can be reflected faithfully. This makes
the analysis of complex planetary transmission mechanisms
easy and general [6].
Due to the advantages of the LAM, it is widely used to
facilitate the transmission analysis of complex PGS
mechanisms, especially for automotive automatic
transmission, e.g. electrical variable transmissions (EVTs).
Ahn [7] used the LAM to analyze the maximum speed for
the performance investigation of a dual-mode EVT. The
torque and speed of the powertrain elements and power split
ratio were analyzed using the LAM by Wang [8] and Kim [9].
Barhoumi [10, 11], Kim [12], and Kang [13] used the LAM
to explore fuel economy and acceleration performance
metrics, of compound split hybrid configurations. Yang [14]
analyzed the configurations and upshift/downshift process
kinematics of the dual-input compound power-split
mechanism (DICPSM). Liu [15] proposed a systematic
design method to synthesize the configuration scheme for
multi-row and multi-speed AT based on the LAM. Xie [16]
analyzed the speed ratio of components by using the lever
analogy method to derive the two-PGS PGTs with a high
reduction speed ratio. Zhang [17] obtained eight schemes
with a high reduction speed ratio based on the LAM. Liao [1]
presented an improved lever analogy method to simplify the
analysis for determining the speed ratios for automatic
transmissions based on the LAM. The LAM is used to
analyze the speed of the components of PGSs/EGSs in the
above research. Additionally, the other properties researches
of PGSs/EGSs transmissions, such as the power flow, torque,
and configuration, are using the LAM to analyze the
components’ speed. Ma [18] and Hong [19] studied the
power flow of series-split EVT for a plug-in hybrid vehicle
by using the LAM. Zhu [20] used the LAM to analyze the
transient torques with different power sources for a multi-
mode transmission with a single electric machine. Ho [21]
applied the graph theory and the LAM to analyze the power
flow, kinematic, and configurations of PGSs/EGSs. Ross [22]
established compound levers consisting of multiple levers in
parallel connection by considering the number and
magnitude of the required ratios. Liu [23] proposed a
systematic analysis methodology to design suitable three-
mode configurations based on the lever model. Based on a
deduction method and composite lever analogy, Peng [24]
developed an efficient synthesis method for the PGS with
two operating DOFs.
The above research analyzed the rotating speed of ring
gear, carrier, and sun gear based on the LAM, but that of
planet gear cannot be obtained since traditional LAM ignores
the kinematic information of planet gear. However, the
rotating speed and other kinematic information of planet gear
are necessary in cases, such as the power flow and efficiency
analysis for dual-input PGSs/EGSs [25,26], mesh phasing
relationships analysis [2,27], entrainment velocity of ball
bearings and PGSs/EGSs [28-31] and the limit speed and life
span of bearing ball and planet gear as the design limitation
factors [32]. Therefore, based on the advantages of the LAM,
an innovative method, which exceeds the research limitation
of the traditional LAM, is urgently needed to establish the
general lever model for all kinds of PGSs/EGSs and analyze
their kinematic information. In this paper, an extended lever
analogy method (ALAM) containing the kinematic
information of planet gears is proposed. In this method, the
three instantaneous centers theorem is used to establish the
kinematic equations among components of PGSs/EGSs,
which can create the new lever relationships of planet gear to
other components. This method can form the 4-node and 5-
node lever models for the simple planetary gear set (SPGS)
and double-planet planetary gear set (DPGS), respectively,
without rederiving the complex kinematic equation set.
Therefore, based on the proposed ALAM, the compound
lever models for all kinds of the PGSs/EGSs mechanisms
including compound PGSs/EGSs with planet gears in series
can be established by including the planet gear to intuitively
and efficiently calculate the rotating speed of the planet gears,
and to derive the kinematic relationship expressions for all
components of the PGSs/EGSs. This makes up for the lack in
analyzing kinematic of planet gear and the compound
PGSs/EGSs with planet gears in series of the traditional
LAM as it eliminates the kinematic information of planet
gear during the process of merging similar items. Thus, the
proposed ALAM is an efficient and universal method for the
kinematic analysis of the PGSs/EGSs mechanisms.
The rest of this paper is organized as follows. In Section 2,
the theoretical mechanism of the traditional LAM and the
ALAM are derived by utilizing the three instantaneous
centers theorem. In Section 3, the proposed ALAM is
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3206845
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
VOLUME XX, 2022 3
verified by taking an example of compound PGSs/EGSs with
planet gears in series. In Section 4, the analysis models of
dual-input for PGSs/EGSs are established based on the new
lever models of the proposed method. In Section 5, the speed
ratio relationships and speed sequence regions of all
components in PGSs/EGSs under different working modes
are calculated and discussed. Finally, conclusions are
summarized in Section 6.
II. THE PROPOSED EXTENDED LEVER ANALOGY
METHOD
A. THE TRADITIONAL LEVER ANALOGY METHOD
In order to reveal the reason for the lack of kinematics
analysis of planet gear and the compound PGSs/EGSs with
planet gears in series, and to provide a theoretical basis for
the derivation of ALAM, the theoretical mechanisms of
traditional LAM for the SPGS and DPGS are derived in this
Section.
The structure type of PGSs/EGSs includes the DPGS and
the SPGS, which are widely used in the transmission
system of the vehicle industry, such as the Ravigneaux
planetary gear set. Their elementary lever models [17, 24]
are shown in Fig. 1.
(a)
(b)
FIGURE 1. Mechanism diagrams and lever models of PGSs/EGSs: (a)
SPGS; (b) DPGS [10, 33].
The LAM uses the vertical lever equivalent substitution to
reflect the kinematic characteristics of PGSs/EGSs so that the
position relationship of PGSs/EGSs rotating members can be
reflected faithfully. This makes the analysis of complex
PGSs/EGSs transmission mechanisms easy and general [5, 6].
However, the theoretical mechanism of the traditional LAM
has not been revealed and the kinematic information of
planet gear can not be analyzed by the LAM.
In this paper, the traditional lever model of the PGSs/EGSs
is derived based on the Aronhold-Kennedy theorem of three
instantaneous centers [34], which reveals the theoretical
mechanism of LAM, and provides a theoretical basis for the
derivation of ALAM including the kinematic information of
planet gear.
The kinematic models of SPGS and DPGS shown in Fig. 2
are used to analyze the kinematic relationships among three
components, i.e. the sun gear, the ring gear, and the carrier.
In Fig. 2 (a), RPand ωPrepresent the base circle radius and
the rotating speed of the planet gear. Here, points A and B
denote the meshing points of the ring-planet gear pair and the
sun-planet gear pair of SPGS, respectively. Similarly, in Fig.
2 (b), RP1and RP2represent the base circle radius of planet
gear #1 and planet gear #2, respectively. ωP1and ωP2
represent the rotating speed of planet gear #1 and planet gear
#2, respectively. Meanwhile, points A’ and B’ denote the
meshing points of the ring-planet gear pair and the sun-planet
gear pair of DPGS, respectively.
(a) (b)
FIGURE 2. Kinematic model of a planetary gear set: (a) SPGS; (b) DPGS.
In Fig. 2 (a), the sun gear, the ring gear, and the carrier are
rotating in the same direction, and the planet gear is rotating
in the opposite direction. Therefore, the kinematic equations
of the model in Fig. 2 (a) can be described as follows:
PA C R P P
RA R R
PB C S P P
SB S S
V R R
V R
V R R
V R
(1)
where, VPB,VSB represent the peripheral velocity at point B of
the planet gear and the sun gear of SPGS, respectively. VPA,
VRA represent the peripheral velocity at point A of the planet
gear and the ring gear of SPGS, respectively. Based on the
identities VPA=VRA,VPB =VSB, so the equality of VPA +VPB =
VRA+VSB is established, and the following equation can be
obtained:
S S R R C R S
R R R R
(2)
By dividing both sides of Eq. (2) by RS, the expression can
be rewritten as follows:
1
S R C
K K
(3)
where Kis the gear ratio of the ring gear to the sun gear,
From Eqs. (1) ~ (3), it can be found that the model in Fig. 2
(a) can form the traditional lever model of SPGS as the
kinematic relationship among the sun gear, the carrier, and
the ring gear can be described completely by Eq. (3). But, the
kinematic information of planet gear cannot be represented in
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3206845
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
VOLUME XX, 2022 4
the traditional lever model since the terms related to the
kinematic information of planet gear are eliminated during
the process of merging similar items.
In Fig. 2 (b), the sun gear and carrier are rotating in
different directions. Planet gear #1 is rotating in the opposite
direction from the sun gear, and planet gear #2 is rotating in
the same direction as the sun gear and ring gear. Therefore,
the kinematic equations of the model in Fig. 2 (b) can be
described as follows:
' 2 2
'
' 1 1
'
' 1 1 2 2
PA C R P P
RA R R
PB C S P P
SB S S
PC P P P P
V R R
V R
V R R
V R
V R R
(4)
where VPA’and VRA’represent the peripheral velocity at point
A’ of the planet gear and the ring gear of DPGS, respectively.
VPB’and VSB’represent the peripheral velocity at point B’ of
the planet gear and the sun gear of DPGS, respectively. VPC’
represents the peripheral velocity at point C’ of planet gear
#1 and planet gear #2. Similar expressions can be obtained as
follows:
R R S S C R S
R R R R
(5)
1
R S C
K K
(6)
B. THE PROPOSED EXTENDED LEVER ANALOGY
METHOD
The traditional LAM ignores the relationship between the
planet gear and the lever model by eliminating the kinematic
information of planet gear during the process of merging
similar items. When the carrier is fixed and the sun gear is
the driving component, the kinematic models are shown in
Fig. 3 (a) and (b) for SPGS and DPGS respectively.
(a) (b)
FIGURE 3. Kinematic model of the planetary gear set with the fixed
carrier: (a) SPGS; (b) DPGS.
The kinematic relationship between planet gear, sun gear,
and ring gear are established by eliminating the terms
containing carrier speed, i.e. ωC= 0. In Fig. 3 (a), the planet
gear and the ring gear are rotating in the same direction, and
the sun gear is rotating in the opposite direction. The black
arrows represent the translation of peripheral velocity at
meshing points A and B. The deriving process of ALAM for
SPGS can be written as follow:
, 0
S S P P
C
P P R R
R R
R R
(7)
where the expression containing the planet gear information
is obtained:
2
R R S S P P
R R R
(8)
The assembly conditions of SPGS can be written as:
2
R S
P
R R
R
(9)
By substituting Eq. (9) into Eq. (8) and dividing both sides
of Eq. (8) by RS, the expression can be rewritten as follows:
/ ( 1) / ( 1)
R S P
K K K
(10)
From Fig. 3 and Eq. (10), the kinematic relationship of the
ALAM can be obtained.
In Fig. 3 (b), the sun gear, planet gear #2, and the ring gear
are rotating in the same direction, but planet gear #1 is
rotating in the opposite direction. The kinematic equations
are different from that of the first method as follows:
2 2
2 2 1 1
1 1
R R P P
P P P P
S S P P
R R
R R
R R
(11)
The above equations can be simplified as follows by
applying the motion inversion to the carrier:
2 2
1 1
, 0
R R P P
C
S S P P
R R
R R
(12)
The point location of planet gear in the ALAM can be
determined by considering the relationship between rotating
speeds and directions of SPGS components.
In Fig. 4, the white levers and nodes are the traditional
LAM, and the black levers and orange nodes are the
extended parts of the ALAM relative to the LAM (shown in
Fig. 1).
In terms of SPGS, the node of planet gear is located on the
upper outside of the lever model of LAM, as shown in Fig. 4
(a). But, in terms of DPGS, the nodes of planet gears #1 and
#2 are located on the upper and lower outsides of the lever
model of LAM, respectively, as shown in Fig. 4 (b). It can be
seen that the new nodes of planet gears exceed the research
limitation of the traditional lever model of LAM and extend
the LAM to the ALAM.
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3206845
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VOLUME XX, 2022 5
(a) (b)
FIGURE 4. The initially extended lever model: (a) SPGS; (b) DPGS.
According to Eq. (10) and Fig. 4 (a), the lever model
containing the information on the planet gear of SPGS can be
written as follows:
1
1 1
1
1 1
P P
R
P P
S
L K K
K K
L
K K
(13)
Furthermore, the expression of the length for the new lever
LPof SPGS can be described as follows:
1
1
P
K
LK
(14)
The new lever length of the planet gear in the ALAM is in
good agreement with that of Ref. [35].
Combining Eqs. (6), (11), and (12), the lever model
containing the information of planet gears of DPGS can be
written as follows:
2
2
1
1
1 2
2 1
, 0
R P
P R
SP
C
P S
P P
P P
R
R
R
R
R
R
(15)
Similarly, the expression of the length for the new lever LPi
of DPGS can be described as follows:
1
1
2
2
1
1
S
P
P
R
P
P
R
L K R
R
L K
R
(16)
The extended lever model can be obtained by substituting
equations (14) and (16) into Fig. 4 (a) and (b), respectively,
and describing the corresponding unknown lever length.
C. THE SUPERIORITY OF THE EXTENDED LEVER
ANALOGY METHOD
It is well known that the traditional method (analysis method)
is inefficient and cumbersome for analyzing the PGSs/EGSs
by analyzing the kinematic of PGSs/EGSs in terms of torque
and speed calculations. In addition, the abstraction of the
equations and the complexity involved make many engineers
unfamiliar with the specific functions and kinematic
characteristics of complex PGSs/EGSs [36]. The LAM
substitutes the torque and rotational speed relationship of the
PGSs/EGSs components with the horizontal force and speed
relationship of the lever node analogy. The kinematic
functions of PGSs/EGSs are clearly visualized without
entirely having a good command of the intricacies of
PGSs/EGSs [5, 37].
Since the proposed ALAM is derived based on the LAM,
the merits of LAM are also reflected in ALAM. The planet
gear kinematic information needs to be considered for
analyzing the planet gear life and the transmission efficiency
of dual-input PGSs/EGSs. However, the traditional LAM
cannot involves the kinematic information of planet gear. So
the ALAM, in which the kinematic information of planet
gear is included, becomes more significant. In addition, the
unified equations and corresponding intuitively lever models
are obtained readily by the proposed ALAM. Comparisons
among the LAM and the ALAM for kinematic analysis of
PGSs/EGSs in terms of containing components, operability,
and visualization are shown in TABLE II and Fig. 5.
TABLE II
COMPARISONS AMONG THE LAM AND ALAM
Information
Method
LAM
ALAM
S (Sun gear)
△
△
R (Ring gear)
△
△
C (Carrier)
△
△
P (Planet gear)
X
△
Operability
△
△
Visualization
△
△
Speed sequence of PGSs/EGSs
X
△
All types of PGSs/EGSs
X
△
*△: covered, X: uncovered.
FIGURE 5. The significant differe nce between LAM and ALAM.
The compound PGSs/EGSs with planet gears in series are
shown in Fig. 6, which are beyond the analysis scope of
traditional LAM. Lots of the derived 2K-H configurations of
Fig. 6 are obtained, which are shown in Fig. 7.
(a) S+S (b) S+D (Type I) (c) S+D (Type II)
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content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3206845
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VOLUME XX, 2022 6
(d) D+D (Type I) (e) D+D (Type II) (f) D+D (Type III)
FIGURE 6. Schematic diagrams of the configurations which are beyond
the analysis scope of LAM. Note: S+S, S+D, D+S, and D+D represent a
transmission device that includes two PGSs/EGSs (SPGS and DPGS)
and their distributions.
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
(j) (k)
FIGURE 7. The derived 2K- H configurations of Fig 5: (a), (b), (c): S+S; (d),
(e), (f), (g): S+D/D+S; (h), (i), (j), (k): D+D.
However, to establish the corresponding lever model, and
analyze the speed ratio relationships and transmission
efficiency under different schemes, it is necessary to solve
and analyze the speed relationship between the planet gear
and other components of PGSs/EGSs [2, 27, 38-40].
III. VALIDATION OF THE PROPOSED AUGMENTED
LEVER ANALOGY METHOD
Figure 8 shows NASA’s EGS mechanism [38, 41-43],
namely the configuration of Fig. 7 (b), in which there are
three links where the motor can be the input or the output.
During the integration, the lever length between the planet
gear and the ring gear is essential due to the specific structure
of this mechanism. To analyze the kinematics of this
mechanism conveniently based on the ALAM, the virtual
PGSs/EGSs, and the corresponding extended lever models
are established, as shown in Fig. 9 (a) and (b). According to
Figs. 4 and 6, the mechanism diagram and virtual extended
lever model can be integrated into a compound lever model,
as shown in Fig. 9 (c) and (d) respectively, by using the
augmented lever analogy method proposed in Ref. [33]. The
teeth of the gears are as follows: ZS1=28, ZP1= 36, ZS2= 36,
and ZP2= 28. The speed ratio ωRatio between the components
can be calculated by assuming that the virtual teeth of ring
gear [44] are ZR1=ZS1+2ZP1,ZR2=ZS2+2ZP2.
FIGURE 8. Schematic of epicyclic gear set proposed by NASA [41].
(a) (b)
(c) (d)
FIGURE 9. The mechanism and lever model for Fig. 8.
1 2 12
1, 2
2 1
92 1
149
36
100
1 81
1
28
C C S
S S
S
K
RK
(17)
121 2
2, 1 2
1 2 1
100 92
32
28 36
100
1 81
1
28
SS
S C C
C C
K K
RK
= =
(18)
1 1 2
2
49 32
81 81
S C C
S
(19)
1
1, 2
2 1
32
49
81 81
fS C
S S
S S
R
(20)
The results of the speed ratios of the mechanism in Fig. 8,
i.e. Eqs. (17) and (20), are in good agreement with those of
Ref. [45], which verifies the accuracy of the ALAM.
Compared to the traditional analysis method in Ref. [41-
43,45], the kinematic equations of ALAM are easily
visualized and understood without entirely having a good
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3206845
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
VOLUME XX, 2022 7
command of the intricacies of PGSs/EGSs. In addition, the
example verifies the feasibility of the ALAM for the
kinematic analysis of differential gear trains. The
corresponding lever models of the configurations in Fig. 6
are shown in Fig. 10. The lever models of the configurations
in Fig. 7 can be obtained by deleting the corresponding nodes
of the lever models in Fig. 10.
(a) S+S (b) S+D (Type I) (c) S+D (Type II)
(d) D+D (Type I) (e) D+D (Type II) (f) D+D (Type III)
FIGURE 10. Schematic diagrams of the configurations beyond the
analysis scope of LAM. Note: S+S, S+D, D+S, and D+D represent a
transmission device that includes two PGSs/EGSs (SPGS and DPGS)
and their distributions.
IV. DUAL-INPUT ROTATING SPEED RELATION
MODELLING
1) SPGS MODELLING EXAMPLE
Since all components in the dual-input working modes are
rotating, the rotating speed relationship of the three
components will affect the determination of the
driving/driven part, and the direction of power flow in the
system. Therefore, further discussion is required on a case-
by-case basis. In the paper, the dual-input PGSs/EGSs are
taken as a reduction device, and the working conditions are
determined. In Fig. 11 (a), the red dotted arrows indicate the
absolute rotating speed and direction of corresponding
components. The blue dotted arrows indicate the output
rotating speed and direction, which is relative to the carrier.
During an upshift, ωCrises and approaches ωR. Here, ωC-ωS
is a negative value, and ωC-ωRis a positive value. At this
time, min(K/(1+K))<ωC/ωR<1 and the transmission ratio K∈
[1.2,4] [16] of the planetary gear mechanism should be
considered as variables to solve the speed relationship of
each component under this condition. The proposed method
can also be applied to mechanisms with similar structures
and kinematic characteristics, such as rolling bearings, etc, as
shown in Fig. 11 (b). B, O, C, and I represent the balls, the
outer race, the cage, and the inner race of the rolling bearing,
respectively. Lband Ko/i respectively represent the lever
length corresponding to the balls and the radii ratio of the
outer race and inner race in rolling bearing. It is worth noting
that there is only one dual-input working mode for rolling
bearings (intermediate bearing), that is, the inner and outer
races are the input parts, and the cage is the output part. Due
to the limitation of space, this paper will not analyze the
dual-input case of rolling bearings.
(a) (b)
FIGURE 11. Extended lever model: (a) R-C dual-input of SPGS; (b) I-O
dual-input of rolling bearing.
In Fig. 11 (a), the rotating speed of planet gear was
calculated according to the two input rotating speeds, i.e.
rotating speed of the ring gear and the carrier. Based on the
ALAM, the speed ratios of the planet gear to the other three
components can be written as Eq. (21) ~ Eq. (24).
1
1
P C S P
LK
(21)
1 1
P R
P
C C
L
(22)
1 1
P R
P
R C
L
(23)
1
1 1
1
1 1
P
S
P
R
P
C
L
L
(24)
2) DPGS MODELLING EXAMPLE WITH PLANET GEAR
#1 AND #2 HAVING THE SAME TOOTH NUMBER
(i) R-C DUAL-INPUT DPGS MODELLING
Figure 12 shows the extended lever model of DPGS, and the
input components are the same as that in Fig. 12. Based on
the ALAM, the speed ratios model of planet gears to the
other three components can be expressed as Eq. (25) ~ Eq.
(29) while ZP1=ZP2.
FIGURE 12. R-C dual-input extended lever model of the DPGS.
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content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3206845
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VOLUME XX, 2022 8
1
11
P
P C S
L
K
(25)
1 1
11
P R P
C C
KL
K
(26)
1 1
11
C
P P
R R
KL
K
(27)
1
1
1
1
1 1
11
P
S
P
R P
C
K
KL
KL
K
(28)
1 2
2 1
P P
P P
Z
Z
(29)
(ii) S-C DUAL-INPUT DPGS MODELLING
Another extended lever model of DPGS with the input
components being the carrier and the sun gear is shown in
Fig. 13. Similarly, the speed ratios models of planet gears to
the other three components can be expressed as Eq. (30) ~ Eq.
(33) based on the Eq. (6) and Eq. (16) while ZP1=ZP2. In this
case, the rotating directions of the two inputs are in opposite
directions, i.e. ωCis negative and ωSis positive, as shown in
Fig. 13 (a). In the reverse operation of motors, ωCis positive
and increases from 0 to ωSduring upshift, as shown in Fig.
13 (b).
(a) (b)
FIGURE 13. S-C dual-input extended lever model of the DPGS: (a) ωC<0;
ωP1<0; (b) ωC>0; ωP1<0.
1 1P C R P
L
(30)
1 1
1S
P P
C C
L
K
(31)
1 1
1
C
P P
S S
L
K
(32)
1
1
1
1
1 1
1
P
R
P
SP
C
L
L
K
(33)
V. RESULTS AND DISCUSSION
In this section, the speed sequences between the
components of the system with the different structures and
input conditions are discussed.
A. SPGS R-C DUAL-INPUT
The relative rotating speed relationships between the planet
gear and other components for SPGS are obtained by
replacing Eq. (14) with Eqs. (21) ~ (24), as illustrated in Fig.
14. Given parameters Kand ωR/ωC, the speed ratios ωP/ωS,
ωP/ωC, and ωP/ωRcan be calculated readily, as shown in Fig.
14 (a). It can be found that the three speed ratios are
increasing with K, but decreasing with ωR/ωC. To further
describe the speed ratio relationship between ωPand ωS,ωC
and ωRin Fig. 14 (a), the difference between two ratios
ωP/ωCand ωP/ωRwith K=2.2, are demonstrated in Fig. 14
(b).
(a)
(b)
(c)
FIGURE 14. The speed ratios ωP/ωS,ωP/ωCand ωP/ωR: (a) The 3
dimension (D) view; (b) The 2D view of speed ratios ωP/ωCand ωP/ωR
with changing ωR/ωC; (c) The top view while ωRatio∈[0,1].
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3206845
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
VOLUME XX, 2022 9
Since the values of ωP/ωCand the ωP/ωRare distributed
on the upper and lower sides of ωRatio =1 respectively, like
the region between two vertical lines shown in Fig. 14 (b)
in which the values of ωP/ωCare larger than ωRatio =1, and
the values of ωP/ωRare less than ωRatio =1. Therefore, Fig.
14 (a) is divided into two parts, i.e. part I and part II. To
analyze the relationship between ωPand ωS,ωCand ωR, the
sectional top view of Fig. 14 (a) with ωRatio=[0,1], i.e. part
II, is shown in Fig. 14 (c) in which ①: 0<ωP/ωR<1,
ωP/ωC>1, ωP/ωS>1; ②: 0<ωP/ωR<1, 0<ωP/ωC<1, ωP/ωS>1;
③: 0<ωP/ωR<1, 0<ωP/ωC<1, 0<ωP/ωS<1.
TABLE III
THE ROTATING SPEED RELATIONSHI PS AND SPEED SEQUENCE OF R-C
DUAL-INPUT SPGS
Region
Speed ratio relation
Speed sequence
ωP/ωC
ωP/ωR
ωR/ωC
ωP/ωS
I
>1
>1
>1
>1
0<ωS<ωC<ωR<ωP
II①
>1
<1
>1
>1
0<ωS<ωC<ωP<ωR
II②
<1
<1
>1
>1
0<ωS<ωP<ωC<ωR
II③
<1
<1
>1
<1
0 <ωP<ωS<ωC<ωR
In Fig. 14 (c), part II is divided into three subparts and the
speed sequences of the dual-input with the ring gear and
carrier of SPGS are listed in TABLE III.
B. DPGS R-C DUAL-INPUT
Based on the Eq. (16) and Eqs. (25) ~ (29), the relative
rotating speed relationship between the planet gear and other
components for DPGS is illustrated in Fig. 15. The
relationship between speed ratios ωP1/ωR,ωP1/ωC, and ωP1/ωS,
which take ωC/ωRand ZS/ZP1as the variables, are shown in
Fig. 15 (a). It can be found that the three speed ratios are
increasing with both ωC/ωRand ZS/ZP1. The speed ratios
ωP1/ωR,ωP1/ωC, and ωP1/ωSare illustrated in Fig. 15 (b) to
(d), respectively. The different values of Kaffect the range
and changing trend of the speed ratio. The maximum and
minimum values occur at K=4 and K=1.2, respectively.
Similar to Fig. 14, Fig. 15 (a) is divided into four subparts.
Part I and II are shown in 15 (a). The other parts, the
sectional top view of Fig. 15 (a) with ωRatio=[0,1], namely
part II, are demonstrated in Fig. 15 (e) and (f) in which ①:
0<ωP1/ωC<1, ωP1/ωR>1, ωP1/ωS>1; ②: 0<ωP1/ωC<1,
0<ωP1/ωR<1, ωP1/ωS>1; ③: 0<ωP1/ωC<1, 0<ωP1/ωR<1,
0<ωP1/ωS<1. Three regions are changing between the limits
with K∈[1.2, 4].
Based on Fig. 15, the speed sequences of dual-input with
the ring gear and the carrier of DPGS are listed in Table IV.
TABLE IV
THE ROTATING SPEED RELATIONSHI PS AND SPEED SEQUENCE OF R-C
DUAL-INPUT DPGS
Region
Speed ratio relation
Speed sequence
ωP1/ωC
ωP1/ωR
ωP1/ω
S
ωR/ωC
I
>1
>1
>1
<1
ωP2<0<ωS<ωR<ωC<ωP1
II①
<1
>1
>1
<1
ωP2<0<ωS<ωR<ωP1<ωC
II②
<1
<1
>1
<1
ωP2<0<ωS<ωP1<ωR<ωC
II③
<1
<1
<1
<1
ωP2<0<ωP1<ωS<ωR<ωC
(a) (b)
(a) (b)
(e) (f)
FIGURE 15. The 3D view of speed ratios: (a) ωP1/ωS,ωP1/ωCand ωP1/ωR
while K=1.2; (b) ωP1/ωR; (c) ωP1/ωC; (d) ωP1/ωS. The top view of speed
ratios ωP1/ωS,ωP1/ωCand ωP1/ωRwhile ωRatio∈[0,1]: (e) K=4; (f) K=1.2.
C. DPGS S-C DUAL-INPUT
Similarly, the relative rotating speed relationships for DPGS
with the sun gear and the carrier as the input components can
be obtained by substituting Eq. (16) into Eqs. (30) ~ (33), as
illustrated in Fig. 16. The relationship between speed ratios
ωP1/ωR,ωP1/ωC, and ωP1/ωS, which take ωC/ωSand ZS/ZP1as
the variables with Kfrom 1.2 to 4., are shown in Fig. 16 (a)
that is a general view. There are different changing trends of
speed ratios with ωC/ωSand ZS/ZP1. The speed ratios ωP1/ωR,
ωP1/ωC, and ωP1/ωSwith K=4 and K=1.2 are illustrated in Fig.
16 (b). Fig. 16 (a) and (b) are divided into eight parts. Firstly,
they are divided into two parts by the plane of ωRatio=0. Then,
they are individually further divided into two subparts by
planes of ωRatio=1 and ωRatio=-1, respectively, as shown in Fig.
16 (c) to (e). Finally, the sectional top views of Fig. 16 (a)
with ωRatio=[0,1]∩[-1,0] are demonstrated in Fig.16 (f) to (h)
in which i: -1<ωP1/ωC<0, ωP1/ωR>1, ωP1/ωS>1; ii:
0<ωP1/ωC<1, 0<ωP1/ωR<1, ωP1/ωS>1; iii: 0<ωP1/ωC<1,
0<ωP1/ωR<1, 0<ωP1/ωS<1. Since ωP1/ωRand ωP1/ωCare
larger than 1 when K=1.2, and the values of ωP1/ωRand
ωP1/ωCfollow hyperbolic function, so the other regions in
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content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3206845
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10
Fig. 16 (f) are I①+II①, I①+II②, I①+II and I②+II②
with K=4.
(a) (b)
(c) (d)
(e) (f)
(g) (h)
FIGURE 16. The 3D view of speed ratios: (a)ωP1/ωS,ωP1/ωC, and ωP1/ωR;
(b) rear view while K=1.2 and K=4; (c) ωRatio∈[-1,0]∩[0,1]; (d) ωRatio≥ 0; (e)
ωRatio ≤ 0. (f) The top view of speed ratios ωP1/ωS,ωP1/ωCand ωP1/ωRwith
ωRatio∈R while K=4. The bottom view of speed ratios ωP1/ωS,ωP1/ωCand
ωP1/ωRwith ωRatio∈[-1,0] ∩[0,1]: (g) K=4; (h) K=1.2.
The three regions in Fig.16 (g) and (h) are changing
between the limits with K∈[1.2, 4], respectively. Based on
Fig. 16, the speed sequences of S-C dual-input DPGS are
listed in TABLE V.
TABLE V
THE ROTATING SPEED RELATIONSHI PS AND SPEED SEQUENCE OF S-C
DUAL-INPUT DPGS
Region
Speed ratio relation
Speed sequence
ωP1/ωC
ωP1/ωR
ωP1/ω
S
ωP2/ω
S
I
>1
<-1
>-1
<1
ωP1<ωC<0 <ωR<ωP2<ωS
II①
>1
<-1
<-1
>1
ωP1<ωC<0 <ωR<ωS
<ωP2
II②
i
>-1
<-1
<-1
>1
ωP1<0 <ωC<ωR<ωS<ωP2
ii
>-1
>-1
<-1
>1
iii
>-1
>-1
>-1
<1
ωP1<0 <ωC<ωR
<ωP2<ωS
II③
i
<1
>1
<-1
>1
ωP1<ωC<ωR<0 <ωS
<ωP2
ii
<1
>1
>-1
<1
ωP1<ωC<ωR<0 <ωP2<ωS
iii
<1
<1
>-1
<1
VI. CONCLUSIONS
In this paper, an augmented lever analogy method containing
the kinematic information of the planet gear was proposed.
The lever node and length models of planet gear in planetary
gear sets (PGSs/EGSs) are established by using the three
instantaneous centers theorem to analyze the peripheral
velocity of meshing points and make up for the lack in
analyzing kinematic of planet gear and the compound
PGSs/EGSs with planet gears in series of the traditional
LAM caused by eliminating the kinematic information of
planet gear during the process of merging similar items. The
different dual-input working modes for PGSs/EGSs were
carried out, and the rotating speed relationships and speed
sequence between all components of SPGS (Simple
planetary gear set) and DPGS (Double-planet planetary gear
set) including the planet gears were obtained and discussed
by divided regions. The main conclusion points of this study
are drawn as follows:
1) The augmented lever analogy method (ALAM) was
proposed by adding the nodes on the LAM to represent the
planet gears, which can establish the lever model for all kinds
of the PGSs/EGSs, and create new lever relationships of the
planet gear to other components.
2) The proposed ALAM method has more significantly
comprehensive merits including the newly unified kinematic
expressions and intuitive and efficient analysis of all the
components including the planet gears when compared to the
traditional lever analogy method (LAM) while analyzing the
kinematic information for the dual-input PGSs/EGSs.
3) The rotating speed relationships between components of
PGSs/EGSs were divided into several regions within the
variables range by taking the planes of ωRatio equal to 1 or -1.
The corresponding regions and speed sequence can be found
rapidly for PGSs/EGSs working at different modes.
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3206845
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VOLUME XX, 2022
11
4) It is worth noting that the mechanisms with similar
structures and kinematic characteristics (e.g. compound
PGSs/EGSs with planet gears in series, etc.) and planet-like
mechanisms, (e.g. the rolling bearing), can also be analyzed
by the proposed ALAM. Therefore, the proposed method is
more general.
ACKNOWLEDGMENT
This research was funded by the National Natural Science
Foundation of China (Grant No. 52035002, 52105086) and
the China Postdoctoral Science Foundation (Grant No.
2021M700583).
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content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3206845
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This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3206845
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
VOLUME XX, 2022
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Xiaodong Yang received the M.D. degree in
mechanical engineering from Henan Polytechnic
University, Jiaozuo, China, in 2015. He is
currently pursuing the Ph.D. degree in
mechanical engineering with Chongqing
University, Chongqing, China. His current
research interests mainly include mechanical
design, mechanical system dynamic modelling,
data analysis and electric vehicles powertrain
design and control.
Wennian Yu received his M.Sc. in mechatronic
engineering from Chongqing University,
Chongqing, China and his Ph.D. in mechanical
engineering from Queen’s University, Kingston,
Canada. He is currently a research fellow at
Chongqing University. His research focus is on
the dynamic modelling of gear transmission
system, condition monitoring of gear system for
diagnostics, prognostics and health management,
and remaining useful life estimation of
mechanical systems using machine learning and
deep learning algorithms.
Yimin Shao received the B.S. degree in
metallurgical machinery from the University of
Science and Technology Beijing in China, in
1992, and the Ph.D. degree in production
engineering speciality from the Gunma
University in Japan, in 1997. From 1997 to 2004,
he was a Research Associate with the Gunma
University in Japan, and a visiting Scholar with
the EU FP7 Marie Curie International Incoming
Fellow, UK in 2012. He is currently a professor
and vice director with the State Key Lab of
Mechanical Transmissions of Chongqing
University, Chongqing, China. His research fields include signal
processing, noise analysis and pattern recognition, equipment fault
diagnosis, intelligent monitoring and residual life prediction.
Zhiliang Xu received his B.S. degree in
mechanical engineering from Anhui University
of Science and Technology, China in 2014. He is
currently pursuing the Ph.D. degree in
mechanical engineering at Chongqing of
University, China. His research interests include,
mechanical system dynamic modelling,
equipment fault diagnosis, data analysis and
pattern recognition.
Qiang Zeng received his B.S. and Ph.D. degree
in mechanical engineering from Chongqing
University, China. He was sponsored by China
scholarship council’ s joint Ph.D. program as a
joint Ph.D. at University of Huddersfield. He is
currently a research associate in mechanical
engineering at the State Key Lab of Mechanical
Transmissions of Chongqing University,
Chongqing, China. His research interests include,
signal processing, equipment fault diagnosis, data
analysis and pattern recognition.
Chunhui Nie received his B.S. degree in
mechanical engineering from Chongqing
Jiaotong University, China in 2014. He is
currently pursuing the Ph.D. degree in
mechanical engineering at Chongqing of
University, China. His research interests include,
hybrid energy storage unit, hydrualic hybrid
energy recovery system, equipment fault
diagnosis, data analysis and pattern recognition.
Dingqiang Peng received the B.S. degree in
mechanical engineering from the University of
Science and Technology Beijing, China, in 2009.
He is currently pursuing the Ph.D. degree in
mechanical engineering from Chongqing of
University, China. His research interests mainly
include manufacturing automation, precision
engineering and signal processing.
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3206845
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
