This study aims to carry out the frictional analysis of cam-controlled planetary gear trains through an illustrative example. First, the equations for frictional analysis proposed in the previous study are presented. Then, the results of friction force and couples are obtained by numerical analysis. The results indicate that a cam-controlled planetary gear train is of high mechanical efficiency, but will induce significant vibration

Ting-Jian Huang

Wen Hsiang Hsieh

English

Vibroengineering PROCEDIA

 362 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533  
Frictional analysis of cam-controlled planetary gear 
trains 
Wen Hsiang Hsieh1, Ting-Jian Huang2 
Department of Automation Engineering, National Formosa University,  
No. 62, Wenhua Rd., Huwei Township, Yunlin 632, Taiwan, ROC 
1Corresponding author 
E-mail: 1allen@nfu.edu.tw, 2t819329@gmail.com 
(Accepted 19 August 2015) 
Abstract. This study aims to carry out the frictional analysis of cam-controlled planetary gear 
trains through an illustrative example. First, the equations for frictional analysis proposed in the 
previous study are presented. Then, the results of friction force and couples are obtained by 
numerical analysis. The results indicate that a cam-controlled planetary gear train is of high 
mechanical efficiency, but will induce significant vibration. 
Keywords: cam-controlled, planetary gear trains, frictional analysis, kinetostatic analysis. 
1. Introduction 
A cam-controlled planetary gear train (CCPGT) refers to a planetary gear train with cam pairs 
[1]. Rothbart [2] called CCPGT as epicyclic gears and moving (or fixed) cam, and illustrated three 
kinds of CCPGTs. Chironis [3] also illustrated a CCPGT in his book. Fig. 1 show, the exploded 
view and the structural sketch of the train. It consists of a cam groove (the frame, member 1), an 
arm (member 2, the input), a planetary gear (member 3), and a sun gear (member 4, the output). 
Generally, the planetary arm rotates at constant speed, and then the planetary gear is driven to 
rotate around the sun gear and to spin around itself concurrently. Simultaneously, the planetary 
gear will produce an oscillatory motion generated by the contact of the attached roller and the cam 
groove. Therefore, the output (sun gear) will produce a non-uniform motion through the engaging 
with the planetary gear. The key advantage is that it can produce a varied non-uniform output 
motion. It is now at work in film drives, e.g. the paper feeding of automatic die cutting and creasing 
machine, etc. 
 
a) Exploded view [1] 
 
b) Structural sketch [4] 
Fig. 1. Cam-controlled planetary gear trains 
Although CCPGTs are introduced in Rothbart’s and Chironis’ books, but the approaches of 
analysis and design are not addressed. Hsieh et el. [1, 4-6] coducted a series of systematic 
investigations on CCPGTs, including kinematic and experimental studies [1], kinematic synthesis 
[4], structural synthesis [5], modeling and control [6], and kinetostatic and mechanical efficiency 
analyses [7, 8]. 
Sun gear
(output)
Arm (input)
Planetary gear
Cam (frame)
Roller
2
3
4
1 1
1
d a
b
e
c
FRICTIONAL ANALYSIS OF CAM-CONTROLLED PLANETARY GEAR TRAINS.  
WEN HSIANG HSIEH, TING-JIAN HUANG 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533 363 
The purpose of this study is to carry out the frictional analysis of cam-controlled planetary 
gear trains through an illustrative example. In Section 2, the equations of frictional analysis 
proposed in the previous study will be outlined. Then, Section 3 will be devoted to conduct the 
numerical analysis for frictional force and couples. Finally, conclusions will be drawn in  
Section 4. 
2. Friction analysis 
The approach of frictional analysis of CCPGTs had been developed in Ref. [7]. The equations 
of frictional forces and couples are summarized here. 
The normal force ܨ௡௜௝ and the friction force acting on each revolute joint can be equivalently 
expressed as a resultant force ܨ௜௝ passing through the pin center and a couple ݐ௜௝. The friction 
couple ݐ௜௝ can be expressed as: 
ݐ௜௝ = ߤ௜௝ݎ௝หܨ௜௝หܦ௜௝, (1)
where ߤ௜௝ is the coefficient of kinetic friction between ݅ and ݆, ݎ௝ is the pin radius of member ݆, ܨ௜௝ is 
the resultant force passing through the pin center, and ܦ௜௝  is a direction indicator that can be 
determined by: 
ܦ௜௝ = ൜
+1,   if  ߱௜௝ is counter clockwise,
−1,   if  ߱௜௝ is clockwise. (2)
Similarly, the forces between a cam joint with a roller follower can be formulated as those of 
a revolute joint, except ߤ௜௝ is replaced by a coefficient of rolling friction. And in the same way, the 
friction force, due to relative velocity, between the engaging teeth of two gears can be formulated 
by: 
ସ݂ଷ = ߤସଷ|ܨସଷ|ܧସଷ, (3)
where ܧସଷ is the directional indicators of the relative velocity between the contacting point of the 
two engaging teeth, and can be determined by: 
ܧ௜௝ = ൜
+1,   if  ௜ܸ௝  is positive,
−1,   if  ௜ܸ௝  is negative. (4)
Based on the equations above and the solving procedure in Ref. [7], each friction forces and 
couples can then be found by applying an adequate numerical approach. 
3. Design example 
The loading torque and the input speed are set as 1.5 N-m and 120 rpm, respectively.  
Moreover, the geometric dimensions and inertial parameters employed in this example are the 
same as listed in Table 1 of Ref. [8]. Fig. 2 shows the numerical results of the reactional forces 
acting between each part and the required input torque. It is found that the reactional forces change 
dramatically, therefore vibration will be a major concern for precision applications. Fig. 3 depicts 
that the magnitudes of the frictional force and couples are small, therefore the mechanical 
efficiency is high. There is a directional change between the arm (part 2) and the planetary gear 
(part 3), a special care must be taken during the numerical solving. 
FRICTIONAL ANALYSIS OF CAM-CONTROLLED PLANETARY GEAR TRAINS.  
WEN HSIANG HSIEH, TING-JIAN HUANG 
364 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533  
a) ܨଵଶ 
 
b) ܨଵଷ 
 
c) ܨଵସ 
 
d) ܨଷଶ 
 
e) ܨସଷ 
 
f) ܳ௜௡ 
Fig. 2. Reaction forces and the input torque 
4. Conclusions 
In this work, the friction analysis, proposed by the previous study, of the CCPGT has been 
performed. The reaction forces and the required input torque have been solved, and the fiction 
force and couples have also been obtained. The results have indicated that a cam-controlled 
planetary gear trains is of high mechanical efficiency. However, significant vibration will be 
induced, and should be minimized for precision and/or high speed applications. 
0.00 90.00 180.00 270.00 360.00
Input Angle (deg)
30.00
40.00
50.00
60.00
70.00
F1
3 (
N)
0.00 90.00 180.00 270.00 360.00
Input Angle (deg)
49.00
49.40
49.80
50.20
50.60
51.00
F4
3 (
N)
0.00 90.00 180.00 270.00 360.00
Input Angle (deg)
1.50
1.60
1.70
1.80
1.90
2.00
Inp
ut 
To
rqu
e (
N-
m)
FRICTIONAL ANALYSIS OF CAM-CONTROLLED PLANETARY GEAR TRAINS.  
WEN HSIANG HSIEH, TING-JIAN HUANG 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533 365 
a) ସ݂ଷ b) ݐଵଶ 
 
c) ݐଷଶ 
 
d) ݐଵଷ 
 
e) ݐଵସ 
Fig. 3. Friction force and couples 
Acknowledgement 
The support of the Minster of Science and Technology, Republic of China (Taiwan), under 
Grants NSC 92-2212-E-150-035 and MOST 104-2221-E-150-054 is gratefully acknowledged. 
References 
[1] Hsieh W. H. An experimental study on cam controlled planetary gear trains. Mechanism and Machine 
Theory, Vol. 42, Issue 5, 2007, p. 513-525. 
[2] Rothbart H. A. Cams: Design, Dynamics and Accuracy. Wiley, New York, 1956. 
[3] Chironis N. P. Mechanisms and Mechanical Devices Sourcebook. McGraw-Hill, New York, 1991. 
[4] Hsieh W. H. Kinematic Synthesis of cam-controlled planetary gear trains. Mechanism and Machine 
Theory, Vol. 44, Issue 5, 2009, p. 873-895. 
[5] Hsieh W. H., Chen S. J. Innovative design of cam-controlled planetary gear trains. International 
Journal of Engineering and Technology Innovation, Vol. 1, Issue 1, 2011, p. 1-11. 
[6] Hsieh W. H., Lee I. C. Modelling and control of cam-controlled planetary gear trains. International 
Journal of Modelling, Identification and Control, Vol. 12, Issue 3, 2011, p. 272-279. 
[7] Hsieh W. H. Kinetostatic and mechanical efficiency studies on cam-controlled planetary gear trains – 
part 1: theoretical analysis. Indian Journal of Engineering and Materials Sciences, Vol. 20, Issue 3, 
2013, p. 191-198. 
[8] Hsieh W. H. Kinetostatic and mechanical efficiency Studies on cam-controlled planetary gear trains 
– part 2: design and experiment. Indian Journal of Engineering and Materials Sciences, Vol. 20, 
Issue 3, 2013, p. 199-204. 
0.00 90.00 180.00 270.00 360.00
Input Angle (deg)
7.00
7.20
7.40
7.60
7.80
8.00
Fri
cti
on
 Fo
rce
 (N
)
0.00 90.00 180.00 270.00 360.00
Input Angle (deg)
-0.20
-0.18
-0.16
-0.14
-0.12
Fri
cti
on
 C
ou
ple
 (N
-m
)
0.00 90.00 180.00 270.00 360.00
Input Angle (deg)
-0.08
-0.04
0.00
0.04
0.08
Fri
cti
on
 C
ou
ple
 (N
-m
)
0.00 90.00 180.00 270.00 360.00
Input Angle (deg)
-0.03
-0.02
-0.01
0.00
Fri
cti
on
 C
ou
ple
 (N
-m
)
0.00 90.00 180.00 270.00 360.00
Input Angle (deg)
-0.05
-0.04
-0.03
-0.02
Fri
cti
on
 C
ou
ple
 (N
-m
)


Frictional analysis of cam-controlled planetary gear trains

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Frictional analysis of cam-controlled planetary gear trains

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