In a planetary gear train (PGT), the power loss by tooth friction is a function of the potential power developed within the gear train elements rather than that being transmitted through it. In the present work, we focus on the operating conditions of two-degree-of-freedom (two-DOF) PGTs. Any operating condition induces its own internal power flow pattern; this implies that tooth friction loss depends on the mechanism of power loss developed in the gearing that differs from one case to another over the entire range of operating conditions. The approach adopted in this paper stems from a unification of the kinematics and tooth friction losses of PGTs and is based on potential powers and power ratios. The range of applicability of the power relations is investigated and clearly defined, and tooth friction loss formulas obtained by their use are tabulated. A short comparison with formulas currently available in the literature is also made. The simplicity of the proposed method for analyzing two-input or two-output planetary gear trains is helpful in the design, optimization, and control of hybrid transmissions. It assists particularly in choosing correctly the appropriate operating conditions to the involved application.

References

References
1.
Kahraman
,
A.
,
Ligata
,
H.
,
Kienzle
,
K.
, and
Zini
,
D. M.
,
2004
, “
A Kinematics and Power Flow Analysis Methodology for Automatic Transmission Planetary Gear Trains
,”
ASME J. Mech. Des.
,
126
(
6
), pp.
1071
1081
.
2.
White
,
G.
,
1994
, “
Epicyclic Gears From Early Hoists and Winches—II
,”
Mech. Mach. Theory
,
29
(
2
), pp.
309
325
.
3.
White
,
G.
,
2003
, “
Derivation of High-Efficiency Two-Stage Epicyclic Gears
,”
Mech. Mach. Theory
,
38
(
2
), pp.
149
159
.
4.
Park
,
J. J.
,
Kim
,
B. S.
, and
Song
,
J. B.
,
2007
, “
Double Actuator Unit With Planetary Gear Train for a Safe Manipulator
,”
IEEE International Conference on Robotics and Automation
(
ICRA
), Rome, Italy, Apr. 10–14, pp. 1146–1151.
5.
Rabindran
,
D.
, and
Tesar
,
D.
,
2008
, “
Power Flow Analysis in Parallel Force/Velocity Actuators (PFVA): Theory and Simulations
,”
ASME
Paper No. DETC2008-49164.
6.
Chen
,
L.
,
Yin
,
C.
,
Zhu
,
F.
, and
Tang
,
L.
,
2010
, “
Scheme Design and Optimal Selection for Hybrid Electric Vehicle Planetary Gear Mechanism. Zhongguo Jixie Gongcheng/China
,”
China Mech. Eng.
,
21
(
1
), pp.
104
109
.
7.
Sheu
,
K. B.
,
2007
, “
Analysis and Evaluation of Hybrid Scooter Transmission Systems
,”
Appl. Energy
,
84
(
12
), pp.
1289
1304
.
8.
Kim
,
J.
,
Kim
,
N.
,
Hwang
,
S.
,
Hori
,
Y.
, and
Kim
,
H.
,
2009
, “
Motor Control of Input-Split Hybrid Electric Vehicles
,”
Int. J. Automot. Technol.
,
10
(
6
), pp.
733
742
.
9.
Kim
,
J.
,
Kang
,
J.
,
Choi
,
W.
,
Park
,
J.
,
Byun
,
S.
,
Jun
,
Y.
,
Kim
,
J.
,
Ko
,
J.
, and
Kim
,
H.
,
2010
, “
Control Algorithm for a Power Split Type Hybrid Electric Vehicle
,”
International Symposium on Power Electronics, Electrical Devices, Automation and Motion
(
SPEEDAM
), Pisa, Italy, June 14–16, pp. 1575–1580.
10.
Barman
,
I.
, and
Flugrad
,
D. R.
,
1992
, “
Design of an Epicyclic Transmission for Speed Control of a Turbine-Generator System
,”
Flexible Mechanisms, Dynamics, and Analysis
, Vol.
47
,
American Society of Mechanical Engineers
,
New York
, pp.
497
504
.
11.
Zhao
,
X.
, and
Maiber
,
P.
,
2003
, “
A Novel Power Splitting Drive Train for Variable Speed Wind Power Generators
,”
Renewable Energy
,
28
(
13
), pp.
2001
2011
.
12.
Esmail
,
E. L.
,
2016
, “
Meshing Efficiency Analysis of Two Degree-of-Freedom Epicyclic Gear Trains
,”
ASME J. Mech. Des.
,
138
(
8
), p.
083301
.
13.
Macmillan
,
R. H.
,
1949
, “Epicyclic Gear Efficiencies,” The Engineer, Dec. 23, pp.
727
728
.
14.
Chen
,
C.
, and
Angeles
,
J.
,
2007
, “
Virtual-Power Flow and Mechanical Gear-Mesh Tooth Friction Losses of Epicyclic Gear Trains
,”
ASME J. Mech. Des.
,
129
(
1
), pp.
107
113
.
15.
Chen
,
C.
, and
Liang
,
T. T.
,
2011
, “
Theoretic Study of Efficiency of Two-DOFs of Epicyclic Gear Transmission Via Virtual Power
,”
ASME J. Mech. Des.
,
133
(
3
), p.
031007
.
16.
Davies
,
K.
,
Chen
,
C.
, and
Chen
,
B. K.
,
2012
, “
Complete Efficiency Analysis of Epicyclic Gear Train With Two Degrees of Freedom
,”
ASME J. Mech. Des.
,
134
(
7
), p.
071006
.
17.
Freudenstein
,
F.
, and
Yang
,
A.
,
1972
, “
Kinematics, and Statics of Coupled Spur-Gear Trains
,”
Mech. Mach. Theory
,
7
(
2
), pp.
263
275
.
18.
Esmail
,
E. L.
,
2013
, “
Nomographs and Feasibility Graphs for Enumeration of Ravigneaux-Type Automatic Transmissions
,”
Adv. Mech. Eng.
, 2013, p.
120324
.
19.
Macmillan
,
R. H.
, and
Davies
,
P. B.
,
1965
, “
Analytical Study of Systems for Bifurcated Power Transmission
,”
J. Mech. Eng. Sci.
,
7
(
1
), pp.
40
47
.
20.
Laughlin
,
H.
,
Holowenko
,
A.
, and
Hall
,
A.
,
1956
, “
How to Determine Circulating Power in Controlled Epicyclic Gear Systems
,”
Mach. Des.
,
28
(
6
), pp.
132
136
.
21.
Pennestrì
,
E.
, and
Freudenstein
,
F.
,
1993
, “
The Mechanical Efficiency of Epicyclic Gear Trains
,”
ASME J. Mech. Des.
,
115
(3), pp.
645
651
.
22.
Sanger
,
D.
,
1972
, “
The Determination of Power Flow in Multiple-Path Transmission Systems
,”
Mech. Mach. Theory
,
7
(
1
), pp.
103
109
.
23.
Esmail
,
E. L.
, and
Hassan
,
S. S.
,
2010
, “
An Approach to Power-Flow and Static Force Analysis in Multi-Input Multi-Output Epicyclic-Type Transmission Trains
,”
ASME J. Mech. Des.
,
132
(
1
), p.
011009
.
24.
Macmillan
,
R. H.
,
1961
, “
Power Flow and Loss in Differential Mechanisms
,”
J. Mech. Eng. Sci.
,
3
(
1
), pp.
37
41
.
25.
Radzimovsky
,
E. I.
,
1956
, “
A Simplified Approach for Determining Tooth Friction Losses and Efficiency of Epicyclic Gear Drives
,”
Mach. Des.
,
9
, pp.
101
110
.
26.
Radzimovsky
,
E. I.
,
1959
, “
How to Find Efficiency, Speed and Tooth Friction Losses in Epicyclic Gear Drives
,”
Mach. Des.
,
11
, pp.
144
153
.
27.
Yu
,
D.
, and
Beachley
,
N.
,
1985
, “
On the Mechanical Efficiency of Differential Gearing
,”
ASME J. Mech., Transm., Autom. Des.
,
107
(
1
), pp.
61
67
.
28.
del Castillo
,
J. M.
,
2002
, “
The Analytical Expression of the Efficiency of Epicyclic Gear Trains
,”
Mech. Mach. Theory
,
37
(
2
), pp.
197
214
.
29.
Nelson
,
C. A.
, and
Cipra
,
R. J.
,
2005
, “
Simplified Kinematic Analysis of Bevel Epicyclic Gear Trains With Application to Power-Flow and Efficiency Analyses
,”
ASME J. Mech. Des.
,
127
(
2
), pp.
278
286
.
30.
Mantriota
,
G.
, and
Pennestrì
,
E.
,
2003
, “
Theoretical and Experimental Efficiency Analysis of Multi-Degrees-of-Freedom Epicyclic Gear Trains
,”
Multibody Syst. Dyn.
,
9
(
4
), pp.
389
408
.
31.
Hsieh
,
H. I.
, and
Tsai
,
L. W.
,
1998
, “
The Selection of a Most Efficient Clutching Sequences Associated With Epicyclic-Type Automatic Transmission
,”
ASME J. Mech. Des.
,
120
(
4
), pp.
514
519
.
32.
Duan
,
Q. H.
, and
Yang
,
S. R.
,
2002
, “
A Study on Power Flow and Meshing Efficiency of 3K Type Epicyclic Gear Train
,”
Mech. Sci. Technol.
,
21
(
3
), pp.
360
362
.
33.
Hsieh
,
L.-C.
, and
Chen
,
T.-H.
,
2011
, “
The Design and Efficiency Analysis of Epicyclic Gear Reducer
,”
J. Adv. Mater. Res.
,
317–319
, pp.
2226
2229
.
34.
Hsieh
,
L. C.
, and
Chen
,
T. H.
,
2012
, “
On the Meshing Efficiency of 3K-Type Epicyclic Simple Gear Reducer
,”
J. Adv. Sci. Lett.
,
12
(
1
), pp.
34
39
.
35.
Hsieh
,
L. C.
, and
Tang
,
H.-C.
,
2013
, “
On the Meshing Efficiency of 2K-2H Type Epicyclic Gear Reducer
,”
Adv. Mech. Eng.
,
2013
, p.
686187
.
36.
Ayats
,
J. G.
,
Calvet
,
J. V.
,
Canela
,
J. M.
,
Diego-Ayala
,
U.
, and
Artes
,
F. F.
,
2011
, “
Power Transmitted Through a Particular Branch in Mechanisms Comprising Epicyclic Gear Trains and Other Fixed or Variable Transmissions
,”
Mech. Mach. Theory
,
46
(11), pp. 1744–1754.
37.
Tuplin
,
W. A.
,
1957
, “
Designing Compound Epicyclic Gear Trains for Maximum Speed at High-Velocity Ratios
,”
Mach. Des.
, pp.
100
104
.
38.
Martin
,
K.
,
1981
, “
The Efficiency of Involute Spur Gears
,”
ASME J. Mech. Des.
,
103
(1), pp.
160
169
.
39.
Anderson
,
N.
, and
Loewenthal
,
S.
,
1982
, “
Design of Spur Gears for Improved Efficiency
,”
ASME J. Mech. Des.
,
104
(
4
), pp.
767
774
.
40.
Anderson
,
N.
, and
Loewenthal
,
S.
,
1983
, “Comparison of Spur Gear Efficiency Prediction Methods,” Army Research and Technology Laboratory, Cleveland, OH, Technical Report No.
NASA-CP-2210
.https://ntrs.nasa.gov/search.jsp?R=19830011870
41.
Petry-Johnson
,
T. T.
,
Kahraman
,
A.
,
Anderson
,
N. E.
, and
Chase
,
D. R.
,
2008
, “
An Experimental Investigation of Spur Gear Efficiency
,”
ASME J. Mech. Des.
,
130
(
6
), p.
062601
.
42.
Vaidyanathan
,
L. S.
,
Harianto
,
A.
, and
Kahraman
,
J.
,
2009
, “
Influence of Design Parameters on Mechanical Tooth Friction Losses of Helical Gear Pairs
,”
J. Adv. Mech. Des., Syst., Manuf.
,
3
(2), pp.
146
158
.
43.
Chiu
,
Y. P.
,
1975
, “
Approximate Calculation of Tooth Friction Loss in Involute Gears
,”
Joint ASLE-ASME Lubrication Conference
, Fiami, FL, Oct. 21–23, Paper No. 75-PTG-2.
44.
Dorey
,
R. E.
, and
McCandlish
,
D.
,
1986
, “
The Modelling of Losses in Mechanical Gear Trains for the Computer Simulation of Heavy Vehicle Transmission Systems
,”
International Conference on Integrated Engine Transmission Systems
, Bath, UK, July 8–9, pp.
69
82
.
45.
Esmail
,
E. L.
,
2012
, “
Hybrid Transmission for Mobile Robot
,”
ASME J. Mech. Des.
,
134
(
2
), p.
021001
.
46.
Maggiore
,
A.
,
1971
, “The Efficiency of Epicyclic Two DOF Gear Trains,” Atti I Congresso Nazionale di Meccanica Teorica ed Applicata, Udine, pp. 65–85.
47.
Pennestri
,
E.
,
Mariti
,
L.
,
Valentini
,
P. P.
, and
Mucino
,
V. H.
,
2012
, “
Efficiency Evaluation of Gearboxes for Parallel Hybrid Vehicles: Theory and Applications
,”
Mech. Mach. Theory
,
49
, pp.
157
176
.
48.
Monastero
,
R.
,
1976
, “The Efficiency of Series Connected Epicyclic Gear Trains,”
Rivista Associazione Tecnica dell'Automobile (ATA)
,
Torino, Italy
(in Italian).
50.
Bottiglione
,
F.
, and
Mantriota
,
G.
,
2008
, “
MG-IVT: An Infinitely Variable Transmission With Optimal Power Flows
,”
ASME J. Mech. Des.
,
130
(11), p.
112603
.
51.
Pennestri
,
E.
, and
Freudenstein
,
F.
,
1993
, “
A Systematic Approach to Power-Flow and Static-Force Analysis in Epicyclic Gear-Trains
,”
ASME J. Mech. Des.
,
115
(
3
), pp.
639
644
.
52.
Pennestrì
,
E.
, and
Valentini
,
P. P.
,
2003
, “
A Review of Formulas for the Mechanical Efficiency Analysis of Two Degrees-of-Freedom Epicyclic Gear Trains
,”
ASME J. Mech. Des.
,
125
(3), pp.
602
608
.
53.
Chen
,
C.
, and
Chen
,
J.
,
2015
, “
Efficiency Analysis of Two Degrees of Freedom Epicyclic Gear Transmission and Experimental Validation
,”
Mech. Mach. Theory
,
87
, pp.
115
130
.
54.
Laus
,
L. P.
,
Simas
,
H.
, and
Martins
,
D.
,
2012
, “
Efficiency of Gear Trains Determined Using Graph and Screw Theories
,”
Mech. Mach. Theory
,
52
, pp.
296
325
.
55.
Esmail
,
E. L.
,
2017
, “
A Universal Kinematic Analysis of Geared Mechanisms
,”
J. Braz. Soc. Mech. Sci. Eng.
,
39
(
6
), pp.
2253
2258
.
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