Loaded Tooth Contact Analysis of Cycloid Planetary Gear Drives

The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015
DOI Number: 10.6567/IFToMM.14TH.WC.OS6.014
Loaded Tooth Contact Analysis of Cycloid Planetary Gear Drives
Shyi-Jeng Tsai, Ching-Hao Huang, Hsian-Yu Yeh and Wei-Jhen Huang
Department of Mechanical Engineering, National Central University, Jhong-Li, Taiwan
e-mail: sjtsai@cc.ncu.edu.tw
Abstract: An approach of loaded tooth contact analysis is
proposed for analyzing the load sharing and the load distribution
of cycloid planetary gear drives with multiple tooth pairs contact.
The mathematic equations of cycloid gear profile are derived
from kinematic relation of the gear and the pin wheel. The
analysis of tooth contact is based on instant center method. The
contact stresses of the multiple engaged tooth pairs are
calculated based on the influence coefficient method. An
analytical, simple relation for calculation of the load sharing and
the max. Hertzian contact stress with the assumption of constant
meshing stiffness is also proposed for comparison. Finally, a
study case was analyzed for load sharing and contact stress
distribution of engaged tooth pairs by both methods. The
characteristics of the contact stress of the cycloid planetary gear
drives are clearly identified.
Keywords: Cycloid planetary gear drives, Loaded tooth contact
analysis, Influence coefficient method
Nomenclature
A
matrix of influence cofficients
F
acting force
H
matrix of separation distance among engaged
flanks
P
vector of contact pressure
pitch circle radius of the pin wheel
RC
T
transmitted torque
radius of the pin
rP
e
eccentricity of the cycloidal gear reducer
h
separation distance
w
deformation
tooth number of the cycloidal disk
zC
tooth number of pin wheel
zP
reduction ratio
iC

approaching displacement

transmission angle

rotation angle of the crank shaft

curvature

generation angle for epitrochoid curve

curvature radius

pressure angle
1.
Introduction
The trends of gear transmission development such as
precision, high power density and even also high
reduction ratio can not be ignored. For example, the
transmission drive of construction machinery should be
designed as smaller as possible under requirement of high
reduction ratio. On the other hand, the demand on
precision transmission for automation machinery is also
raised recently to reduce the labor cost and to increase the
productivity. In general, the mentioned gear drives have
the common features, such like high reduction ratio, small
power density, oscillating transmission with high torque at
low speed. The conventional application of planetary gear
with involute gears can not fulfill the requirement
apparently. In contrast to the involute gears, the
differential type of planetary gear drive with cycloidal
gears has the some significant advantages, i.e.
 possibility achieve small tooth number difference (at
least one) to obtain high gear ratio;
 multiple tooth pair in contact for sharing the transmitted load to enhance the load capacity of the
transmission and the high power density, so as
suitable for application of small volume or weight;
 short teeth so as having good ability to absorb shock
of oscillating transmission, and suitable for
application of high torque at small speed.
The cycloidal planetary gear drives, or usually
known as the brand name “cyclo-drive”, haven been
already applied in many branches of industry, for example
the transmission of the pipe-jacking tunneling boring
machine (TBM) shown in Fig. 1.
Fig. 1 Pipe-jacking tunneling boring machine (TBM) and
transmission
The load analysis is an important issue for successful
application of such the drives, but there are not many
published research papers on the load, although more
articles on geometrical and kinematic design can be found.
Dong et al. [1] proposed a calculation approach for the
acting forces on the rolling bearings of the cycloidal
planetary gear drive. Blanche and Yang [2, 3] analysis the
influence of the manufacturing errors on the transmitted
load and transmission errors. Hidaka et al. [4] proposed an
analytical method based on the assumption of contact
mesh stiffness of tooth action to analyze the influence of
manufacturing errors. The analytical approach is later also
compared with FEM analysis [5]. Gorla et al. [6] develop
an simplified approach to analyze the contact stress and
also conducted an experiment to validate the theoretical
analysis results. Another often applied method for analysis
of contact stress of cycloidal gear drives is FEM, e.g.
Blagojevic et al. [7] proposed a double stage of cycloidal
gear reducer. They developed an approach for load
analysis for good load distribution and dynamic balance.
The contact stress is however analyzed by FEM. Similar
research can be also found in the works by Thube [8], Li
[9] and Kim [10].
The aim of the paper is to propose an efficient
method to calculate the loads acting in the cycloidal
planetary gear drives. It will be expected to analyze the
load sharing among the multiple contact tooth pairs, the
distribution and variation of the contact stress, as well as
the varied angular stiffness or the loaded transmission
errors by using the proposed methods. Two different
approaches for contact stress analysis of the multiple
engaged tooth pairs are presented in the paper: one is a
computerized approach based on the influence coefficient
method; another is an analytical but simple approach with
the assumption of constant meshing stiffness.
The drive analyzed in the paper consists of two
planetary stages, as the section view of the CAD model in
Fig. 2 shows. The power is transmitted through the first
planetary stage of involute gears and split into three crank
shafts, on which the planet gears are fixed. The crank
shafts drive the cycloidal disk of the second planetary
stage in a eccentric motion to transmit the power further to
the pin wheel or the carrier (see Fig. 3). In the case the
carrier is fixed and the pin wheel plays the role of power
output, here supply enough power to the primary and
secondary cutter and grinder.
Fig. 2 Section view of the cycloidal planetary gear for
pipe-jacking TBM
of various mesh stiffness of the contact tooth pairs.
Besides the loaded contact analysis method based on the
influence coefficients, an additional analytical method
based on the assumption of the equal mesh stiffness are
therefore proposed for simplified calculation.
2.1 Geometrical Relations
2.1.1 Epitrochoid profile
The tooth profile of the cycloidal disk can be derived
from the geometrical relation in Fig. 4 as the following
vector equation:
rC  RC  e i   e e i( iC )  rP e i(  ) ,
(1)
where the pressure angle  is the angle between the
contact normal MiPi and the line of centers OPi, and is
determined as
z P  e  sin( zC   ) 
.
 RC  z P  e  cos( zC   ) 

  arctan 
(2)
The initial position of the pin-wheel is based on the angle
variable  = 0. It can be also clearly identified that the
tooth profile is also the equidistant curve of the the the
epitrochoid curve with the distance rP (namely the pin
radius). The vector rC in Eq. (1) can be also represented in
the Cartesian coordinates as
xC  RC  cos  e  cos(iC   )  rP  cos(  )
(3)
yC  RC  sin   e  sin(iC   )  rP  sin(  )
(4)
2.1.2 Curvature
The curvature relation of the contact tooth pair is
important for the calculation of the Hertzian contact stress.
While the curvature of the pin is constant, the curvature 
of the epitrochoid tooth profile can be calculated
according to the definition

xC ' ( ) yC " ( )  xC " ( ) yC ' ( )
.
[ xC '2 ( )  yC '2 ( )]3 / 2
(5)
However, it is somewhat complicated to differentiate
Eq. (2) and (3) according to Eq. (4). Because of the
property of the equidistant curve of the epitrochoid curve,
the curvature radius of the cycloidal tooth profile is thus
equal to the difference of the curvature radius of the
epitrochoid curve and the pin radius. The equations of the
epitrochoid curve are
xCR  RC  cos   e  cos(iC   ),
yCR  RC  sin   e  sin(iC   ).
(6)
And after differentiating Eq. (5), the curvature radius of
the tooth profile can be obtained as
Fig. 3 Structural scheme of the cycloidal planetary stage
2.
Model of Loaded Tooth Contact Analysis
In general the mesh stiffness of the drive is not
constant. However, it is complicate to calculate the loaded
deformation and the contact stresses under consideration

RC  [1 (iC  e / RC )2  2  (iC  e / RC )  cos((iC 1) )]3/ 2
1 iC  (iC  e / RC )2  (iC  e / RC )  (1 iC )  cos[(iC 1)  ]
 rP (7)
When the curvature is equal to zero, or the curvature
radius is infinite, the point on the tooth profile is the
inflection point. The corresponding variable inf of the
inflection point must be equal to
 inf 
 1  iC  (iC  e / RC ) 2 
1
arccos 
.
iC  1
 (1  iC )  (iC  e / RC ) 
(8)
and finally
OC I C  (iC  1)  e  k  e .
(12)
The location of the instantaneous point IC is changed
with different rotation angle C. The locus of the points IC
is a center with a radius k·e. It will be easy to determine
the contact points Mi by using the line ICPi at each rotation
angle C.
2.2.2 Tooth meshing
To determine the positions of the contact points is
essential for calculation of the contact stress of the loaded
tooth pairs. The mesh analysis consists of two essential
analysis works: the positions of contact points and the
effective contact tooth pairs.
(1) Positions of contact points. When the crank shafts
rotate at an angle C, the center of the cycloidal disk
translate along the circle around the center of the pin
wheel with the angle C, and the pin wheel rotates at the
angle C / iC., as the geometrical relation shown in Fig. 5.
The position of contact point M1 can be determined from
the relation in Fig. 5, the corresponding variable 1 for the
point M1 on the tooth profile of the cycloidal disk is
Fig. 4 Geometric and kinematic relation of the cycloidal
planetary gear
1   P   C / iC ,
and for the ith tooth pair
 i  1  ( j  1)   P .
2.2 Kinematic relations
2.2.1 Instantaneous centre
The kinematic relation in Fig. 4 is so regarded that
the cycloidal disk is stationary and the pin wheel moves
relatively in two types of motions:


Translation motion: the center of the pin wheel OP
translates along the circle with the center OC, which
is also the centre of cycloidal disk. This motion is
equivalent to the revolution movement of the
cycloidal disk around the center OP of the pin wheel.
This motion is transmitted from the rotation of the
crank shafts. The transmitted angle is denoted as C.
Rotation motion: the pin wheel rotates around its
center OP with an angle P which is equal to C/iC.
Under this motion, the common normal on the
contact point M1 or Mi of the pin and cycloidal tooth is in
the direction of CP1M1 or CPiMi. Because the common
normal the instantaneous center locates also on the normal,
the instantaneous center IC must also lie on the line of the
centers OPOC. In other words, the instantaneous center IC
is the intersection point of the lines of OPOC and CP1M1.
At the moment of mesh, as shown in Fig. 4, the translation
velocity of the cycloidal disk is equal to
vC  e   C ,
(9)
and the direction is perpendicular to the line of the centers
OPOC. With the definition of the instantaneous center, the
relation of the velocities must be valid,
or
O P I C  P  e  C ,
(10)
O P I C  e  iC ,
(11)
(13)
(14)
The coordinates and the related curvature radius of the
contact point can be obtained by substituting the angle i
into Eq. (3), (4) and (7), respectively. The angle i
between the normal of contact tooth pair i and the line of
the centers OPOC can be derived from the geometrical
relation as
 i  ( i   i )   C   .
(15)
(2) Approaching distance. Because the pressure angle
of the contact tooth pair is varied during mesh, the
approaching distances of the contact tooth pairs will be
also different from each other. With assumption that the
crank shafts rotate at an additional angle  due to the
loaded deformation of all the contact tooth pairs, a
translational displacement e with an inclined angle
(/2+C) will be generated correspondingly. The angle 
between the vector E of the translational displacement and
the normal vector N can be obtained,
i 

π
 C  i  i     i .
2
2
(16)
The effective approaching distance eqi for contact tooth
pair i can be then determined as
 eqi  e    cos  i  e    sin  i  qi  e  
(17)
(3) Effective contact tooth pairs. Because of multiple
contact tooth pairs, it is also important to distinguish the
tooth pairs in contact at each mesh position for further
calculation of contact stresses. It can be classified into the
following criteria according to the motion direction of the
crank shaft:


Counterclockwise rotation direction:
the effective approaching distance eqi > 0 or
the transmission angle i < 0 or
the pressure angle   0.
Clockwise rotation direction:
the effective approaching distance eqi < 0 or
the transmission angle i  0 or
the pressure angle   0.
2.4 Influence coefficient method
2.4.1 Basic relation
The contact of any engaged tooth pair can be
regarded as the contact of two elastic bodies, as the
simplified model shown in Fig. 6 (a). The tooth profiles
are deformed due to the normal force P. The tooth will be
then approached to each other along the normal direction
with a distance 1 and 2, respectively. The point Q1,2 on
the tooth profile 1 or 2 will be also deformed in a value of
w1 and w2 under the contact pressure. In order to
distinguish whether the two points Q1 and Q2 on the
loaded teeth are coincided or not, the following equations
must be valid,
Fig. 5 Relation of loading and displacement
2.3 Method of equal mesh stiffness
The loaded deformation wPi of contact tooth pair i
can be expressed as the multiplication of the acting normal
load FPi and the compliance fPi,
wPi  FPi f Pi .
(18)
With the assumption of equal mesh stiffness, all the
compliance fPi will be equal to a constant, namely fPi = fP.
Hence the following equation is valid for contact tooth
pair i:
(19)
For n contact tooth pairs, the load equilibrium equation
can be obtained with a given torque T,
n
iC  e   ( FPi  sin  i )  T .
(20)
i 1
Together with Eq. (18) and (19), the shared load on each
individual contact tooth pair can be calculated as
FPi 
n
w1  w2  h  1   2   ,
(23)
out of contact:
w1  w2  h  1   2   ,
(24)
where h is the initial separation distance between two
engaged tooth flanks, and can be calculated from the
geometrical relations of the mathematical equations of the
flanks. Besides the relations of loaded deformation and
displacement in Eq. (22) and (23), another boundary
condition for loaded contact analysis is also essential. The
contact pressure p' on arbitrary position (x’, y’) of the
contact region A must be positive and the sum of the
contact pressure p' on each area must be also equal to the
normal force P, i.e.

A
p'(x', y')dx'dy'  P ，
p'(x',y')  0
(25)
The deformation on arbitrary position k (see Fig. 6(b)) due
to all the acting loads on the contact region can be defined
with aid of influence coefficients, namely
r
wk  w(1) k  w(2) k   f H,kj p j ,
(26)
j 1
FPi f P
 e  0 .
sin  i
T  sin  i
in contact:
.
iC  e   sin 2  i
(21)
where fH,kj is influence coefficient for the Hertzian contact
deformation on unit k due to the load acting on unit j.
Hence the set of the deformation-displacement
equations and the load equilibrium equation can be
expressed in a form of matrix equation as [11],
 A1
 0

 

 0
 s1I1 n

1
0

0
A2



0

0

An
s2I1 n
2
- q1I n 1   P1 
 H1 
 H 
- q2I n 1   P2 
 2  , (27)
 
        
 


- q p I n 1   Pp 
 Hp 
 T / e
0  e 
1
2
P
 s p I1 n
P
P
i 1
The max. contact stress of each contact tooth pair can be
also calculated according to the equation,
 Hi ,max 
FPi  E
1
1
(
 ).
2
2    b  (1  )  C  P
(22)
Fig. 6 (a) Relation of deformation and load (b) meshing
on the common tangent plane
where the approaching distance of each contact tooth pair
is equal to e·qi. The transformation coefficient qi can be
calculated according to Eq. (17).
2.4.2 Separation distance between engaged teeth
The separation hj distance between the cycloidal
flank and the circular pin in the normal direction consists
of two parts: one is the distance hMj of point Mj on the
cycloidal flank to the common plane, and the other is hCj
of point Cj on the pin to the same plane, Fig. 7.,
h j  hCj  hMj .
(28)
The distance hCj can be determined under a given distance
lMj on the tangential plane as
hCj  rp 
rp2
2
 lM
j
,
(29)
The distance lMj and the separation hMj can be determined
by using the following vector relations
rMij  t Mi  lMj ,
(30)
rMij  n Mi   hMj ,
(31)
where the vector rMij is equal to the difference of the
position vector rC(j) of point Mj and rC(i) of Mi:
rMij  rC ( j )  rC ( i ) .
(32)
On the other hand, the tangential vector tMi and the normal
vector nMi can be derived from the geometrical relations in
Fig. 7, namely,
t Mi  e
i
,
(33)

n Mi  e
with
i (  )
2 ,
     

2
(34)
.
(35)
The unknown variable j for point Mj is solved from Eq.
(30) with a given value of the distance lMj. The separation
distance hMj is thus determined with the solved j.
Fig. 7 Separation distance among the engaged flanks
3.
Numerical Example
3.1 Gearing data
The gearing data of the drive for the analysis
example is listed in Table 1. The difference of tooth
number is 1 in this case.
Table 1 Gearing data for analysis example
Items / symbols
value
Pitch circle radius of pin wheel RC
Radius of the pin rP
remarks
162.5 mm
8 mm
Eccentricity e
3.5 mm
Tooth number of the cycloidal disk zC
39
Tooth number of the pin wheel zP
40
Reduction ratio iC (Carrier fixed)
40
Thickness of the cycloidal disk t
31.5 mm
Input torque T
4000 Nm
zP/(zPzC)
3.2 Variation of load sharing and contact stress
The shared loads among the engaged tooth pairs are
calculated at the rotation angle of the crank shaft C = 0.
The load sharing factor K is defined as the portion of the
load acting on the tooth pair i to the average acting load.
The average acting load is determined from the total
normal load Fi evenly shared by the half of tooth number
zP. Thus the load sharing factor can be expressed as
K i 
z P  Fi
2   Fi
(35)
The result is illustrated in Fig. 8. It can be identified
that the results calculated both by the proposed
approaches are similar in trends but the max. load sharing
factors are quite different. The contact tooth pair #1 is the
tooth pair which engages each other at the beginning of
contact. From the result by the method of influence
coefficient (IC), contact tooth pair #3 bears the max. load,
while #4 by the method of equal mesh stiffness (EMS).
The results reveal also that the load is mainly shared to the
region of the concave flanks.
The variation of the load sharing and the contact
stress with the rotation of the input crank shaft are
illustrated in Fig. 9. The analysis results for two proposed
methods are represented: those from the method of equal
mesh stiffness (EMS) is shown in blue, and the method of
influence coefficients (IC) in red. The range of the
rotation angle C of the crank shaft in the diagram
corresponds to the profile variable , where the relation 
= C /iC is valid. With other words, the range of the angle
C is equal to 18040/39 = 184.615 in this case, while
the range of angle  = 180 for one flank side.
The max. load sharing factor calculated from the
method of influence coefficients occurs at the angle of
14.3077° with the value of 2.3179, while that from the
method of equal mesh stiffness occurs at 29.5384° with
2.005. It can be also identified that the shared load varies
continuously.
The two calculated results for contact stress from
both the methods are similar to each other. But the max.
values of the contact stress are different, although they
2.5
Load Sharing EMS
2
1,200
Contact Stress IC
Contact Stress EMS
1.5
900
1
600
0.5
300
0
-200
max. Contact Stress [MPa]
1,500
Load Sharing IC
Load Sharing Factor
occur at the similar angle about -55.385°. The max.
contact stress calculated by the method of EMS is 1320.87
N/mm2, by the method of IC is 1206.739 N/mm2.
In order to explore the contact stress on the flanks,
the variation of the contact stress and the curvature with
the rotation angle of the crank shaft are compared, as the
diagram in Fig. 10 shows. The max. curvature 0.455 mm-1
occurs at the same angle where the max. contact stress
occur. This result reveals that the main factor affecting the
contact stress is the curvature, not the shared load,
although the mesh stiffness affects the shared load. The
reason can be also identified from the calculated result by
the method of IC that the max. shared load occurs at the
rotation angle 14.3077°, where the curvature is negative
and the result contact stress is reduced.
0
-150
-100
-50
0
Rotation Angle of the Crank Shaft [deg]
Fig. 9 Variation of the load sharing and the contact stress
of the engaged tooth pair during meshing
2.5
Method of EMS
2
1200
1.5
1
0.5
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Number of Cycloid Tooth
Fig. 8 Load sharing of the engaged teeth
1000
0.5
Contact Stress -- IC
Contact Stress -- EMS
Curvature
0.4
0.3
Curvature [1/mm]
1400
Contact Stress [MPa]
Load Sharing Factor
Method of IC
800
0.2
600
0.1
400
0
200
-0.1
0
-200 -180 -160 -140 -120 -100 -80
-0.2
-60
-40
-20
0
Rotation Angle of the Crank Shaft [deg]
Fig. 10 Relation of the contact stress and the curvature
3.3 Characteristics of Contact Stress
3.3.1 Distribution on the tooth flanks
In order to explore the contact pattern, and the
distribution of the contact stress on flanks, the contact
stresses calculated by the method of IC are illustrated in
3D representation. The distributed contact stress for the
contact point near the inflection point is shown in Fig. 11,
for the contact point with the max. contact stress (max.
curvature) in Fig. 12. The inflection point on the flank
occurs at the rotation angle -30.4373°. From the results, it
is can be clearly identified that the contact pattern is
similar to a saddle form. The concentrated stresses occur
on the edges of the face-width of the cycloidal disk. The
area of the contact pattern of the tooth pair with the max.
contact stress is smaller than that on the inflection point.
Fig. 11 Contact stress distribution: tooth acting on the
inflection point
Fig. 12 Contact stress distribution: tooth acting on
the point with max. contact stress
-55.4°
-36°
-29.5°
-18.4°
-11.5°
-0.46°
Contact Stress (MPa)
1200
1000
800
600
400
0
-0.4
-0.2
0
0.2
0.4
1000
800
600
400
200
0
-0.04
-0.02
0
0.02
0.04
0.06
Minor Axis of Contact Pattern [mm]
Fig. 14 Contact stress in the profile direction: from the
contact point with max. contact stress to the contact end
3.3.3 Loaded transmission errors
The angular stiffness of the cycloidal gear drive can
be also calculated by the proposed method of influence
coefficient when the deformations of all the essential
components are considered. In the paper, only the
compliance of the contact tooth pairs due to Hertzian
contact is involved in the calculation. The rotation range
of the crank shaft for each contact cycle of the engaged
tooth pair is equal to 360/39 = 9.23077. The angular
compliance  on the input side is calculated from Eq. (27).
With expressed for output side, i.e. the angular
compliance equal to /iC, the loaded transmission error
can be obtained, as the diagram shown in Fig. 15. The
quasi sinusoid can be identified, where the transmission
errors varies in the range of 11.247 arcsec and 11.345
arcsec. The value of the error amplitude is equal to 0.098
arcsec. This small value, of course, is not the realistic
result of such the drive, but it can be thus verified that the
proposed LTCA method can be used for analysis of
angular stiffness.
-11.2
200
-0.6
-55.4°
-75.7°
-97°
-144°
-174°
-184°
1200
-0.06
0.6
Minor Axis of Contact Pattern [mm]
Fig. 13 Contact stress in the profile direction: from the
contact begin to the contact point with max. contact stress
LTE of the Pin Wheel [arcsec]
1400
1400
Conatct Stress [MPa]
3.3.2 Variation along the epitrochoid profile
It can be also found from Fig. 11 that the contact
stress of the tooth pair on the inflection point is
asymmetrically distributed in the profile direction.
Therefore the variation of such the contact stress
distribution at different rotation angle of the crank shaft is
shown in Fig. 13 for the region of stress increasing
(Region I) and Fig. 14 for the region of stress decreasing
(Region II). At contact begin, because the contact type of
the tooth pair is concave-convex, the contact pattern is
enlarged and the contact stress is reduced correspondingly.
On the other hand, the contact near the inflection point is
not only concave–convex, but also convex–convex on the
contact area. As consequence, the contact stresses
distributed asymmetrically in the profile direction, for
example the distribution curves at  = -18.4°, -29.5°, -36°
in Fig. 13. The asymmetrical distribution of the contact
stress will disappear and will be symmetrical away from
the inflection point.
The length of minor axis of the contact pattern in the
region I is increased from contact begin until to the
inflection point due to the small shared load and convexconcave contact and then decreased until to the contact
point with the max. stress due to the large shared load and
curvature. The length of minor axis in region II, on the
other hand, are at first increased and then decreased until
to the contact end. The reason is can be explained from
the relation of the curvature and the shared load. After the
point with max. contact stress, both of them are reduced.
At first the contact pattern is enlarged a little because the
reduced gradient of the curvature is larger than that of the
shared load. But after the contact point at the angle about
100°, the gradient of the curvature becomes smaller,
while the gradient of the reduced shared load is enlarged.
Consequently the length of minor axis is reduced until to
zero, see Fig. 14.
-11.225
-11.25
-11.275
-11.3
-11.325
-11.35
-11.375
-11.4
-45
-40
-35
-30
-25
-20
-15
-10
-5
Rotation Angle of the Crank Shaft [deg]
Fig. 15 Loaded transmission errors of the output shaft
0
4. Conclusion
The loaded tooth contact of the cycloidal planetary
gear is analyzed by using the method of influence
coefficients and the method of equal mesh stiffness. The
analysis results enable us to draw the following
conclusions:
 The contact stress of the pin and the epitrochoid flank
is influenced mainly by the curvature of the
epitrochoid flank. The mesh stiffness, on the other
hand, affects the load sharing among the contact
tooth pairs.
 A small shared load and a larger contact pattern can
be found in the contact region of the concave profile
of the cycloidal disk, and thus smaller contact stress
occurs. The max. contact stress occurs at the point
with max. curvature.
 The concentrated stresses occur on the edges of the
face-width of the cycloidal disk. The corresponding
contact pattern is a saddle form.
 The analytical equations of constant mesh stiffness
are suitable only for calculation of the max, contact
stress. The difference comes from the deviation of
mesh stiffness in the concave-convex contact region.
 The proposed LTCA method based on influence
coefficient can efficiently calculate the contact
pattern, the contact stress distribution, load sharing
and loaded transmission error.
 The paper presents also a possibility for LTCA of the
compound planetary gear drive with cycloidal and
involute gears, when the deformations of the other
relevant components, such as the rolling bearings, the
crank shafts, the involute planet gears and the sun
gear, are further integrated in the proposed model.
Acknowledgment
The authors would like to thank Transmission
Machinery Co., Ltd. for their financial support.
References
[1] Dong, X.; Deng, J.; Chen, J.: “Force Analysis of RV
Transmission Mechanism”, Journal of Shanghai Jiao Tong
University, Vol.30, No.5 1996, pp. 65-70, 84.
[2] Blanche, J. G., D.; Yang, C. H.: “Cycloid Drives With
Machining Tolerances”, Journal of Mechanisms,
Transmissions, and Automation in Design, No. 3, Vol. 111,
1989, pp. 337-344.
[3] Yang, D.C.H; Blanche, J.G; “Design and application
guidelines for cycloid drives with machining tolerances.”
Mechanism and Machine Theory, Volume 25, Issue 5, 1990,
pp. 487-501.
[4] Hidaka, T.; Wang, H.-Y.; et al.: “Rotational Transmission
Error of K-H-V-Planetary Gears with Cycloid Gear, 1st
Report, Analytical Method of the Rotational Transmission
Error.” Transactions of JSME, Ser. C, Vol. 60, No. 570,
1994, pp. 645-653.
[5] Ishida, T; Hidaka, T; et al.: “Bending Stress and Tooth
Contact Stress of Cycloid Gear with Thin Rims.”
Transactions of JSME, Ser. C, Vol. 62, No. 593, 1996, pp.
291-297.
[6] Gorla, C.; Davoli, P.; et al. “Theoretical and Experimental
Analysis of a Cycloidal Speed Reducer.” Journal of
Mechanical Design, Vol. 130, Issue 11, 2008.
[7] Blagojevic, M; Marjanovic, N.; et al. “A New Design of a
Two-Stage Cycloidal Speed Reducer”, Journal of
Mechanical Design, August 2011, Vol. 133/085001-1.
[8] Thube, S. V.; Bobak, T. R.: “Dynamic Analysis of a
Cycloidal Gearbox Using Finite Element Method”.
12FTM18, AGAMA Fall Technical Meeting, 2012.
[9] Li, S.: “Design and Strength Analysis Methods of the
Trochoidal Gear Reducers.” Mechanism and Machine
Theory. 81 (2104), pp. 140-154.
[10] Kim, Y.-H.; Lee, C.-S.; Ahn, H.-J.: “Torsional Rigidity of a
Cycloid Drive Considering Finite Bearing and Hertz
Contact Stiffness.” Proceedings of Proceedings of
MPT2009-Sendai, JSME International Conference on
Motion and Power Transmissions, May 13-15, 2009,
Matsushima Isles Resort, Japan.
[11] Wu, S.-H.; Tsai,S.-J.: “Contact stress analysis of skew
conical involute gear drives in approximate line contact”,
Mechanism and Machine Theory, Volume 44, Issue 9, 2009,
pp. 1658–1676.