*Corr. Author’s Address: Lanzhou University of Technology, Gansu Lanzhou, 730050, China, ldw969968604@163.com and lutdcb@126.com 128
Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 128-140 Received for review: 2023-06-04
© 2024 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2023-10-17
DOI:10.5545/sv-jme.2023.673 Original Scientific Paper Accepted for publication: 2023-11-23
Service Performance Optimization and Experimental Study  
of a New W-W Type Non-circular Planetary Gear Train
Dong, C.B. – Yang, X.D. – Li, D.W. – Zhao, G. – Wan, A.R. – Liu, Y .P. – Guo, J.H.
Changbin Dong
1
 – Xudong Yang
1
 – Dawei Li
1
 – Gang Zhao
2
 – Yongping Liu
1
1 Lanzhou University of Technology, School of Mechanical and Electrical Engineering, China 
2 Wuhan University of Science and Technology, Hubei Key Laboratory of Mechanical Transmission and Manufacturing 
Engineering, China
To address the vibration problem in traditional W-W-type non-circular planetary gear trains during operation, this study proposes two new 
combinations of W-W-type non-circular planetary gear trains. The study focuses on analysing the transmission error and improving service 
performance. By utilizing the conjugate theory and the coordinate transformation theory, an accurate mathematical model for non-circular 
gears is derived. The transmission error models for the two combination forms of the non-circular planetary gear train are established using 
the incremental meshing line method. The analysis also examines the influence of eccentricity on the transmission error. Kinematic analysis 
of the multi-body dynamics model confirms that the new W-W type non-circular planetary gear train, formed by combining non-circular gears 
and cylindrical gears, exhibits superior transmission performance. Furthermore, a test analysis is conducted on the roller pumping unit test 
platform and the indicator diagrams of the pumping unit under various working conditions are obtained. The study concludes that the optimal 
matching mode for the sucker rod counterweight and motor frequency of the reversing device of the pumping unit is 20 kN and 20 Hz, and the 
dynamic balance of the pumping unit load can be achieved by adjusting the motor frequency. 
Keywords: Non-circular planetary gear train, reversing device, incremental meshing line method, transmission error, indicator diagram
Highlights
• 	 Two new W-W type non-circular planetary gear train combinations have been proposed to reduce torque fluctuations at the 
output end.
• 	 A new method for establishing the transmission error model of new W-W-type non-circular gear planetary gear trains based on 
the incremental meshing line method was proposed.
• 	 An idealized indoor pumping unit test platform was built, and indicator diagrams of non-circular gear pumping units under 
different working conditions are obtained. 
0  INTRODUCTION
As a new type of gear transmission, non-circular gear 
(NCG) is mainly used to realize nonlinear rotation 
between two transmission shafts [1]. It has the 
advantages of strong bearing capacity and variable 
ratio transmission [2] to [4] and is widely used in 
agriculture, military, aerospace, and other important 
places. This kind of functional gear device has also 
evolved in nature [5]. Burrows have found a natural 
gear structure on the hind legs of the jumping Issus 
worm. Its distribution type, transmission mode, and 
functions are consistent with those of NCG, which 
proves that NCG can also play a functional role in the 
natural behaviour of jumping insects as a biological 
gear. With the development of modern machinery 
towards high speed, heavy load, and high precision, it 
has been difficult to make a single NCG pair to meet 
the application requirements. Thanks to the time-
varying transmission characteristics of NCG, various 
transmission forms can be realized by combining 
different types of NCG into multiple non-circular 
planetary gear trains (NCPGT), which are applied 
in hydraulic motors, hydraulic pumps, astronomical 
clocks, automobile gearboxes, and other fields [6] to 
[8].
Generally, NCPGT [9] and [10] has four types, 
namely N-G-W, N-W, W-W, N-N, and each type 
can achieve different functions and can be applied 
in different situations. In large-scale agricultural 
machinery with complex movement tracks, such as 
automatic seedling picking, hole pricking, fertilization, 
etc., using an NCG rotating mechanism to replace 
the traditional crank linkage mechanism can make 
the improved mechanism have smaller inertia, more 
compact structure, and maintain high motion accuracy 
when working at medium and high speeds [11] to [13]. 
Sun et al. [14] proposed the design method of NCPGT 
in approximate multi pose transplanting mechanism. 
Zhao et al. [15] and Ye et al. [16] studied the onion 
automatic seedling delivery mechanism and vegetable 
rotary seedling taking mechanism of NCPGT. Yu et 
al. [17] proposed a planetary gear train formed by the 
combination of incomplete eccentric circular gear and 
NCG in order to solve the rigid impact of the seedling 
picking mechanism of the transmission intermittent 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 128-140
129 Service Performance Optimization and Experimental Study of a New W-W Type Non-circular Planetary Gear Train 
planetary gear train. Based on the space “8”-shaped 
transplanting track, Wang et al. [18] designed a wide 
narrow row rice bowl seedling transfer mechanism 
driven by the combination of plane NCG and bevel 
gear. The NCPGT transfer mechanism can complete 
the actions of clamping, pulling, holding, and pushing 
seedlings, which is of great significance for improving 
the efficiency of agricultural machinery. In addition to 
agricultural applications, as the key core component 
of hydraulic motors [19], NCPGT is widely used 
in metallurgy, mines, and other dangerous working 
environments to achieve low-speed and high-torque 
transmission. Considering that NCPGT often uses 
several planetary gears to share the load, which will 
cause uneven load distribution and lead to impact and 
vibration, Lin [20] proposed a new type of double-
sided internal meshing NCPGT. The gear train has 
only one planetary gear, which avoids the problem of 
uneven load that may occur due to multiple planetary 
gears. 
Due to the accurate variable ratio transmission 
characteristics of NCG, the planetary gear train 
composed of NCG and other mechanisms plays an 
irreplaceable role in many occasions with special 
motion requirements and force requirements. 
However, it is difficult to improve the transmission 
performance of NCPGT. The existing transmission 
accuracy is only applicable to agricultural machinery, 
which makes it difficult to meet the use requirements 
of high-precision occasions, thus restricting the 
industrial application and development of NCPGT. 
Therefore, based on the original W-W type NCPGT, 
this paper proposes a comparative analysis of the 
two combination modes of the new W-W type 
NCPGT and applies them to the reversing device of 
the drum pumping unit. Through the idealized rod 
production indoor test, the optimal load and motor 
frequency matching data of the NCPGT pumping unit 
are obtained, which will lay a theoretical foundation 
for improving the transmission performance of the 
NCPGT and promote its industrial application.
1  THE GENERATION MOTION  
OF CONJUGATE TOOTH PROFILE FOR NCG
Fig. 1 shows the basic principle of using gear shaper 
cutters as gear cutters to produce NCG. The principle 
of tooth profile envelope is to ensure the pure rolling 
of the pitch surface of a pair of NCG, while the 
instantaneous contact between another cylindrical 
gear and the pitch surface of this pair of pure rolling 
NCG ensures the pure rolling relationship. 
Fig. 1.  Basic principle of using cylindrical gear shaper to produce 
NCGs
The tooth profile of NCG is an involute tooth 
profile. For involute tooth profile gear transmission, 
the section of the gear shaping cutter is an involute, 
and the section equation is:
 
xu ru uu
yu ru uu
gg b
gg b
  
  





coss in
sinc os
, (1)
where r
gb
 = r
gp
 cos α. r
gb
 is the base circle radius. r
gp
 
is the indexing circle radius of the gear cutter. u
 
is the 
tooth shape parameter. Specifically, it is the sum of the 
unfolding angle and pressure angle of the involute at 
any point. α is the gear transmission pressure angle.
According to the principle of gear meshing, the 
relationship between the generating wheel and the 
generated gear satisfies a pure rolling relationship. 
Specifically, the pitch curve of the gear shaper and the 
pitch curve of the NCGs are in instantaneous tangent 
contact, so the coordinate system can be established 
through the tangent vector and normal vector of the 
NCG pitch curve to build the kinematic relationship 
between the coordinate systems. Based on the basic 
idea of the motion reversal method, if an NCG is fixed, 
the relationship between the coordinate system of the 
gear shaper cutter, the coordinate system of the NCG, 
and the auxiliary coordinate system at the contact 
point P of the gear shaper cutter can be defined.
Fig. 2 shows the coordinate conversion 
relationship when producing NCGs, which contains 
the following three coordinate systems:
(1) Fixed coordinate system S
0
(O
0
 – x
0
y
0
z
0
), which is 
fixed on an NCG to be produced. The rotation 
angle of an NCG is expressed as θ, and the pitch 
curve equation is defined as r
p
( θ).
(2) Auxiliary coordinate system S
G
(O
g
 – x
G
y
G
z
G
), 
which is determined by the normal and tangential 
directions of the NCG pitch curve and the contact 
point P of the generating tool (gear shaper), as 
well as the center of the generating wheel. The 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 128-140
130 Dong, C.B. – Yang, X.D. – Li, D.W. – Zhao, G. – Wan, A.R. – Liu, Y.P. – Guo, J.H.
centre of the generating gear is the point of the 
radius of the normal gear shaper cutter. Using 
vector coordinate transformation, the unit vector 
of the three coordinate axes of the coordinate 
system S
p
 can be expressed as:
 
in
jt
k
p
pp
pp
T
001



  
  
 







. (2)
 The unit vector of the three axes of coordinate 
system S
0 
can be expressed as:
 
i
j
k
T
T
T
0
0
0










100
010
001
. (3)
 The coordinate transformation relationship 
between coordinate system S
G
 and coordinate 
system S
0
 is as follows:
 M
nt r
nt r
0G
p0 p0 og 0
p1 p1 og 1
0
0
00 10
00 01
() .  












 (4)
(3) Gear shaper cutter coordinate system 
S
g
(O
g
 – x
g
y
g
z
g
). In Fig. 2, the angle at which the 
relative coordinate system S
p
 rotates is the angle 
of rotation of the gear shaping cutter, which 
can be calculated based on the pure rolling 
relationship of the pitch curve using the following 
equation:
 

 

g
pg
gp
p
gp
dd d



 

S
r
r
r
0
, (5)
where S
pg
( θ) represents the arc length of the NCG 
pitch curve from the starting position to the current 
position. The coordinate transformation from the 
slotting cutter coordinate system S
g
 to the relative 
coordinate system S
p
 can be obtained as:
 M
Gg g
gg
gg
00
00
00 10
00 01


















coss in
sinc os
. (6)
According to Eqs. (4) and (6), the coordinate 
transformation from the gear shaper cutter coordinate 
system to the NCG coordinate system can be obtained 
as follows:
 MM M
0g 0G Gg g
     
. (7)
Based on the analysis of gear-cutting tools 
according to the above generation motion, the tooth 
profile of NCG can be obtained. By transforming the 
tooth surface of the generation wheel into the gear 
coordinate system, the envelope equation of NCG can 
be obtained as follows:
 RM R
Eg 0g g
 ,, ,. uv uv    
 (8)
The boundary of Eq. (8) is the tooth profile surface 
of an NCG, and its meshing equation is usually:
 
Mn
0g rg g
0.     uv v ,
 (9)
Fig. 2.  Coordinate transformation relationship during the production of NCG

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 128-140
131 Service Performance Optimization and Experimental Study of a New W-W Type Non-circular Planetary Gear Train 
Due to the fixed gear coordinate system S
0
, the 
relative velocity of the meshing point is the velocity 
of that point in the production wheel coordinate 
system. The motion of the auxiliary coordinate system 
S
G
 along the NCG pitch curve and the motion of the 
generative gear can be combined to form the relative 
velocity at the meshing point, so the relative velocity 
is:
vt R
gg
T
g p
00d d   ,, . , uv uv    



  (10)
Fig. 3.  The partial digital tooth surface of the NCG
By substituting Eq. (10) into the meshing Eq. (9), 
the instantaneous meshing line between the NCG and 
the generating gear at any time can be solved, thus 
forming the tooth profile of the NCG. The digital 
tooth surface of the NCG is established using Eq. 
(10), as shown in Fig. 3.
2  TRANSMISSION ERROR MODEL  
OF NEW W-W TYPE NCPGT
3.1  The Transmission Error Model of NCG Based on 
Meshing Line Increment Method
The transmission error of a gear transmission 
system is the main factor affecting its transmission 
performance and accuracy. Transmission errors 
caused by excitation parameters such as eccentricity 
error, time-varying backlash, and load are all dynamic 
transmission errors. The main idea of the incremental 
meshing method is to equate the centre distance error 
and eccentricity error to the instantaneous meshing 
line of the gear, and convert the increment of the 
meshing line of the driving gear and the driven gear 
under different turns into the angle of the driven gear, 
thus obtaining the transmission error of the driving 
gear under different turns. 
In some cases, an NCG pair will be required to 
achieve reciprocating motion. Because the rotation 
centre of NCG is not at its geometric centre, there is an 
eccentricity error. The difference between the various 
tooth profiles of NCG results in a more pronounced 
excitation of transmission errors by the backlash 
between the teeth. There may be some differences in 
transmission between clockwise and counterclockwise 
rotation. Therefore, the meshing line increment of the 
driving and driven gears under different direction of 
rotation is considered in the analysis.
Figs. 4 and 5 show the transmission error 
analysis models of NCG pair established by the 
incremental meshing line method after considering 
the comprehensive eccentricity error and driving 
gear direction of rotation. Figs. 4a and b show the 
driving gear meshing line increment and driven 
gear meshing line increment when the driving gear 
 
Fig. 4.  Calculation model of meshing line increment when driving gear rotates counterclockwise

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 128-140
132 Dong, C.B. – Yang, X.D. – Li, D.W. – Zhao, G. – Wan, A.R. – Liu, Y.P. – Guo, J.H.
rotates counterclockwise, and Figs. 5a and b show 
the driving gear meshing line increment and driven 
gear meshing line increment when the driving gear 
rotates clockwise. Combined with Figs. 4 and 5, it 
can be determined that the increment of meshing line 
of driving gear and driven gears under two kinds of 
direction of rotation are:
   L
11 11
es in() ,  (11)
   L
22 22
es in() ,  (12)
where e
1
 and e
2 
are the eccentricity errors of the 
driving and driven gears, respectively. θ
1
 and θ
2
 are 
the rotation angles of the driving and driven gears, 
respectively. φ
1
 and φ
2
 are the initial phase angles of 
the driving and driven gears, respectively. ∆L
1
 and 
∆L
2
 are the meshing line increments of the driving and 
driven gears, respectively. α is the pressure angle.
At the initial time, the increment of the meshing 
line of the driving and driven gears caused by the 
eccentricity error is as follows:
   L
10 11
es in() ,  (13)
   L
20 22
es in() .  (14)
According to the meshing line increment method, 
when the driving gear rotates counterclockwise and 
clockwise, the theoretical no-load transmission error 
can be obtained as follows:
 
  
 







n
e
e
e
e
180
11 1
22 2
11
rcos
[s in()
sin( )
sin( )
2 22
sin( )],   (15)
 
  
 
 






s
e
e
e
180
11 1
22 2
11
rcos
[s in()
sin( )
sin( )
e e
22
sin( )].   (16)
3.2 Establishment of Transmission Error Model of New 
W-W type NCPGT
A W-W type NCPGT consists of two externally 
engaged NCGs, and its structural principle is shown in 
Fig. 6, where 1, 2, 3 and 4 are NCGs. The NCG 1 and 
4 are fixed, and H is a tie rod. The power is input from 
I, and the NCG 4 is driven to rotate by the rotation of 
the tie rod H, and the final power is output from NCG 
4.
Fig. 6.  Schematic diagram of W-W type NCPGT mechanism
Practice shows that the difference between the 
tooth profiles of NCGs leads to a certain vibration 
Fig. 5.  Calculation model of meshing line increment when driving gear rotates clockwise

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 128-140
133 Service Performance Optimization and Experimental Study of a New W-W Type Non-circular Planetary Gear Train 
of W-W gear train during operation, and there is 
also a certain fluctuation in the torque at the output 
end [21], which is not allowed for high-precision 
gear transmission systems. Therefore, on the basis of 
the traditional W-W gear train, this paper proposes 
two new combinations as shown in Fig. 7. Case A is 
composed of two externally engaged NCGs and two 
externally engaged cylindrical gears. In Fig.7a, 1, 2, 
3 and 4 are NCGs, and 5, 6, 7 and 8 are cylindrical 
gears. Case B consists of four externally engaged 
NCGs. The new W-W type NCPGT formed by the 
above combination can achieve power balance and 
stable torque at the output end. The design parameters 
of NCGs are shown in Table 1. 
Table 1.  The design parameters of NCGs
Cylindrical gear NCG
Modulus [mm] 3.5 3
Tooth number 42 47
Eccentricity (k) 0 0.3287
Tooth width [mm] 30 30
Center distance [mm] 150 150
Pressure angle [°] 20 20 (Pitch line)
Pitch curve equation r 

 

64 667
10 3287
.
.c os
To further clarify the engagement between gears, 
the two gear trains in Fig. 7 are simplified as shown 
in Fig. 8, where the red line represents Case A. In 
Fig. 8, the engagement of gears in Cases A and B is 
as follows: in Case A, the power input drives NCGs 
2 and 3 to rotate by the tie rods at the axle holes of 
NCGs 1 and 4, and NCGs 1 and 4 are fixed. At the 
same time, the cylindrical gears 5 and 8 mesh with 
the cylindrical gears 6 and 7 respectively, and then 
the power is output by the sleeves of the shaft holes 
of the cylindrical gears 6 and 7. For Case B, only 
cylindrical gears 5 to 8 are replaced with NCGs, and 
its transmission mechanism is consistent with Case A. 
Fig. 8.  Schematic diagram of two types of NCPTG simplification
The transmission error of a gear transmission 
system is the main factor affecting its transmission 
performance. During the meshing process of the 
NCG pair, due to the different tooth profiles, a 
certain displacement difference will be generated in 
the direction of the meshing line of the driving and 
driven gears [22] and [23], that is, transmission error. 
Most NCG transmission systems can realize quick 
return motion; at this time, its rotation centre is not at 
its geometric centre, thus involving eccentric errors. 
Therefore, the NCG in Cases A and B proposed in this 
paper also has eccentric errors. In view of the key role 
and special requirements of NCPGT in transmission, 
it is more practical to analyse the transmission error 
of an NCG transmission system under eccentric 
error. Based on the incremental meshing line method 
[23] to [25] and Eqs. (15) and (16), the transmission 
error models of the two cases proposed in this paper 
are established respectively. The general formula of 
a)            b) 
Fig. 7.  Schematic diagram of new W-W type NCPGT mechanism: case A, and b) case B

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 128-140
134 Dong, C.B. – Yang, X.D. – Li, D.W. – Zhao, G. – Wan, A.R. – Liu, Y.P. – Guo, J.H.
transmission error of each gear pair in Cases A and B 
is:
 
   

 


ij
jj
ii ii
jj jj
ii
r
e
e
e
180
b
 cos
sin
sin
sin



 

i
jj j
e
ij





 
sin,
, 1,3,5,7 2,4,6,8 (17)
where e is the eccentricity error. θ is the rotation angle. 
φ is the initial phase angle.
Therefore, the transmission errors of Case A and 
Case B are respectively:
 TE
CaseAN CG12 NCG34N CG56 NCG78
    , (18)
 TE
CaseBN CG12 NCG34C G56C G78
    . (19)
Through the above mathematical model of 
transmission error, the transmission error distribution 
law of Cases A and B can be obtained under different 
design parameters, and the respective advantages 
of the two NCPGT can be determined through 
transmission error.
3.3 Comparative Analysis of Transmission Performance of 
New W-W-Type NCPGT 
As the core component of the pumping unit, 
the transmission performance of the reversing 
device directly affects the quality and efficiency 
of the pumping unit. Therefore, it is necessary to 
systematically analyse the transmission performance 
of the reversing device of the proposed two NCPGTs. 
For the new W-W type NCPGT, this paper mainly 
compares and analyses the transmission error, 
output swing angle, angular velocity, kinetic energy 
and angular momentum to determine the optimal 
combination form. In Fig. 9, the transmission errors of 
Case A and Case B show a periodic change trend. As 
the NCG has eccentricity, the transmission error of the 
gear teeth under eccentric excitation has a cumulative 
effect. In contrast, the transmission errors of Case A 
are smaller, which indicates that the new W-W type 
NCPTG composed of NCGs and cylindrical gears has 
more advantages for high-precision transmission. 
Figs. 10 and 11 show the variation trend of the 
transmission error of the NCPGT corresponding 
to Cases A and B with the eccentricity (k). With 
the increase of eccentricity, the transmission errors 
of the cases show an increasing trend. However, 
the transmission error in Case B increases more 
significantly. For the reversing device of a pumping 
unit, the transmission error is directly related to its up 
and down strokes, and excessive transmission error 
will lead to increased energy loss during the up and 
down strokes of the pumping unit. Therefore, Case 
A has better transmission performance based on 
transmission error analysis.
Fig. 9.  Transmission error distribution law of two cases
Fig. 10.  Variation trend of NCPGT transmission error 
corresponding to Case A with eccentricity
To verify the above conclusions, a virtual 
prototype model (as shown in Fig. 12) was 
established for Cases A and B, respectively, and the 
working condition parameters consistent with the 
actual working condition of the pumping unit were 
set. By solving the model, the change laws of the 
contact force, frequency spectrum, output swing 
angle, angular velocity, kinetic energy, and angular 
momentum of the NCPGT corresponding to Cases A 
and B were obtained, respectively.

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 128-140
135 Service Performance Optimization and Experimental Study of a New W-W Type Non-circular Planetary Gear Train 
the impact and torque fluctuations of the system. In 
the model corresponding to Case B, the peak meshing 
force is mainly concentrated at the turning point of 
the gear train, which is around 0.72 seconds and 1.44 
seconds. From a frequency perspective, it is mainly 
concentrated in the low-frequency range within 200 
Hz. Compared to Case B, the model responding to 
Case A shows a decrease in the peak meshing force 
from 22 N to 18 N, and the change in meshing force 
during gear meshing is more uniform. Furthermore, 
the peak value tens to expand to both sides during 
the turning of the gear train. The system response 
frequency has also been wider, expanding from 
200 Hz to 400 Hz. In Fig. 13c, there are many sub-
harmonics with the same pattern in the distribution 
of the meshing force spectrum. Due to the addition 
of cylindrical gear pairs in the gear train, torque 
fluctuations in the initial stage of system dynamic 
balance will inevitably cause system vibration, which 
is necessary to reduce torque fluctuations in the initial 
stage of the system. To summarize, compared with 
Fig. 11.  Variation trend of NCPGT transmission error 
corresponding to Case B with eccentricity
a) 
b) 
Fig. 12. Virtual prototype model: a) Case A, b) Case B
To demonstrate the differences between the 
original gear group and the added gear group, we 
conducted model validation to verify the rationality 
of the model proposed in this article. Fig. 13 shows 
the dynamic meshing force and frequency spectrum 
distribution of a) the original gear set (Fig. 6), b) Case 
B, and c) Case A. The original gear model has multiple 
peaks in the meshing force amplitude due to power 
imbalance and inertia force, which will exacerbate 
a)
b)
c)
Fig. 13.  The distribution pattern of dynamic meshing force and 
spectral; a) the original gear set, b) Case B, and c) Case A

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 128-140
136 Dong, C.B. – Yang, X.D. – Li, D.W. – Zhao, G. – Wan, A.R. – Liu, Y.P. – Guo, J.H.
the above two schemes, the dynamic meshing force 
in Case A is reduced and the distribution is more 
uniform and gentle, which indicates that Case A has 
better motion characteristics and spectral distribution 
patterns.
Fig. 14.  Comparison and analysis of output swing angle of two 
cases
In Fig. 14, the output swing angle of Cases A and 
B shows a periodic change trend. The amplitude of the 
output swing angle corresponding to Case B is larger, 
but the time interval between the extreme values of 
the output swing angle is longer. For pumping units, 
a large output swing angle means that a longer stroke 
can be achieved, which is expected in the pumping 
process and is of great significance for improving the 
working efficiency of pumping units. However, the 
curve in Case B fluctuates greatly, while Case A can 
reduce the vibration caused by NCG pair engagement 
on the basis of achieving stable power because of 
the introduction of cylindrical gears to participate 
in planetary gear train engagement. The vibration 
phenomenon should be avoided as much as possible 
during the working process of the pumping unit, and 
the output swing angle can be adjusted by the sprocket 
system at the output end of the reversing device of the 
pumping unit. In contrast, the stability of the system is 
more important.
Figs. 15 to 17 show the angular velocity, kinetic 
energy and angular momentum curves of Cases A and 
B respectively. The negative value appears in Fig. 
14, which is due to the preset clockwise rotation as a 
square during multi body dynamics simulation. When 
the output end of NCPGT rotates counterclockwise, 
a negative value will appear. In Fig. 15, Case B 
has great velocity fluctuation and impact during 
reversing, while Case A's angular velocity curve is 
Fig. 15.  Comparison and analysis of angular velocity of two cases
Fig. 16.  Comparison and analysis of kinetic energy of two cases
Fig. 17.  Comparison and analysis of angular kinetic energy of two 
cases

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 128-140
137 Service Performance Optimization and Experimental Study of a New W-W Type Non-circular Planetary Gear Train 
more periodic. In Figs. 16 and 17, although Cases A 
and B have the same distribution law, the impact and 
fluctuation of Case B are larger. It is indicated that 
within a certain range, the impact of NCG gear pairs 
on the transmission performance of the system can 
be reduced by introducing cylindrical gears, thereby 
reducing the vibration and impact of the system. The 
above analysis further shows that the NCPGT formed 
by the combination of NCG and cylindrical gear has 
better transmission performance, and also verifies the 
correctness of the transmission error analysis results. 
Therefore, the Case A scheme is adopted for the 
subsequent pumping unit reversing device.
3  TRANSMISSION TEST OF PUMPING UNIT  
REVERSING DEVICE
The idealized indoor test of sucker rod pumping 
unit is realized by building a test prototype with the 
same proportion as the actual pumping unit. The test 
prototype of the pumping unit is shown in Fig. 18. Its 
working principle is as follows: the motor inputs torque 
into the NCPGT reversing device of the pumping unit 
through belt transmission. The output shaft transmits 
the torque to the drum through the sprocket drive and 
finally completes the up and down stroke movement 
of the sucker rod through the connection between the 
steel wire rope wound on the drum and the simulation 
loading device. During the experiment, an MPS wire 
displacement sensor was used to detect the real-time 
displacement of the suspension point; The DYLY-102 
tensile sensor is used to measure the suspension load. 
Finally, the indicator diagram is drawn based on the 
dynamic matching relationship between them.
 
Fig. 18.  NCPGT reversing device pumping unit; 1 hydraulic 
cylinder, 2 sucker rod, 3 balance weight, 4 DYLY-102 tensile 
sensor, 5 MPS wire displacement sensor, 6 new w-w type NCPGT, 
and 7 variable frequency motor
Fig. 19.  New W-W type NCPGT reversing device 
Fig. 19 shows the new W-W type NCPGT reversing 
device corresponding to Case A. The indicator 
diagram is an important indicator for evaluating 
the performance of the pumping unit [26]. A closed 
parallelogram can be formed by real-time collecting the 
corresponding data in a cycle of the suspension point 
load and displacement sensor, and then establishing a 
two-dimensional coordinate system with suspension 
point displacement and load as the horizontal axis 
and vertical axis respectively. However, due to the 
influence of complex environment, the measured 
indicator diagram curve often has certain fluctuations.
The balance weight and stroke times of sucker rod 
will affect the stability of pumping unit operation. 
During the test, the counterweight of the counterweight 
box and the motor frequency are given. The indicator 
diagrams of pumping units under different working 
conditions are obtained by dynamically adjusting the 
experimental data, as shown in Figs. 20 and 21.
In Fig. 20, the change of polished rod load during 
up and down strokes has great fluctuation. The reason 
is that the impact and vibration of the gear pair in 
the new W-W type NCPGT reversing device during 
meshing, and the load mismatch in the up and down 
strokes under the load frequency mode aggravates this 
situation.
Fig. 21 shows the indicator diagram of the 
pumping unit under three working conditions: load is 
20 kN, and motor frequencies are 10 Hz, 15 Hz, and 
20 Hz, respectively. There are obvious fluctuations 
at the starting point of the upstroke, which leads to 
a large impact at the starting point of the pumping 
unit. The reason is that the inertia of the sucker rod 
counterweight load is too large, and the pumping 
unit stroke is not dynamically adjusted. With further 
increase of the motor frequency, the polished rod load 
line obviously tends to be stable, showing a stable 
state, and there is basically no fluctuation. In the 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 128-140
138 Dong, C.B. – Yang, X.D. – Li, D.W. – Zhao, G. – Wan, A.R. – Liu, Y.P. – Guo, J.H.
process of the down stroke, the fluctuation amplitude 
of polished rod load decreases gradually and tends to 
be stable. It shows that the pumping unit has the best 
balance weight and pumping stroke, and adjusting the 
motor frequency is helpful to balance the polished rod 
load from the up and down strokes of the pumping 
rod. Therefore, increasing the motor frequency can 
effectively improve the performance of the pumping 
unit.
Fig. 20.  Indicator diagram of 10 kN load
Fig. 21.  Variation trend of indicator diagram of pumping unit with 
motor frequency under 20 kN load
Fig. 22 shows the indicator diagram of the 
pumping unit when the load is 30 kN and the motor 
frequencies are 15 Hz and 20 Hz, respectively. The 
polished rod load at the starting point and the end 
point of the down stroke of the pumping unit fluctuates 
slightly. The vibration of the polished rod load at the 
beginning of the upstroke has an obvious decreasing 
trend with the increase of the motor frequency, which 
indicates that increasing the motor frequency can 
significantly reduce the load impact at the upper and 
lower dead points of the pumping unit.
Fig. 22.  Variation trend of indicator diagram of pumping unit with 
motor frequency under 30 kN load
Based on the comparison and analysis of indicator 
diagrams of several groups of different tests from 
Figs. 20 to 22, it is concluded that the pumping unit 
is the most stable when the load frequency matching 
mode of the pumping unit is 20 kN and 20 Hz. At this 
time, the pumping unit has the optimal balance weight 
and the optimal pumping unit stroke. Therefore, the 
above analysis shows that the load imbalance problem 
during the operation of the pumping unit can be 
alleviated by adjusting the matching between the load 
and the motor frequency. It can also reduce the impact 
and vibration caused by the reversing of the NCPGT 
reversing device, thereby improving the efficiency of 
the pumping unit and the service life of the reversing 
device.
4  CONCLUSIONS
This paper proposes two new combination methods 
for W-W type NCPGT. Based on the incremental 
meshing line method, the transmission error models 
are established respectively. The advantages of the 
two types of gear trains are judged by the contact 
force, frequency spectrum, transmission error, output 
swing angle, angular velocity, kinetic energy, and 
angular momentum. The correctness of the analysis 
is verified by the virtual prototype model. The new 
W-W type NCPGT, which is composed of NCGs and 
cylindrical gears, has better transmission performance. 
On this basis, the new W-W type NCPGT is applied 
to the reversing device of the drum pumping unit, 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 128-140
139 Service Performance Optimization and Experimental Study of a New W-W Type Non-circular Planetary Gear Train 
and the idealized rod production laboratory test is 
carried out. The indicator diagrams under different 
load frequency matching modes are obtained, and the 
optimal load frequency matching mode is 20 kN and 
20 Hz. The test results further show that adjusting the 
motor frequency can effectively achieve the dynamic 
load balance of the test pumping unit. The research 
results in this paper can lay a theoretical foundation 
for the industrial application and improvement of the 
transmission performance of NCPGT.
5  ACKNOWLEDGMENTS 
The research was supported by the National Natural 
Science Foundation of China (Grant No.52265008), the 
Gansu Province Youth Science Foundation Project of 
China (Grant No. 23JRRA751), the Gansu Provincial 
Department of Education of China: Innovation Fund 
Project for University Teachers (Grant No. 2023A-
021), the Open Fund of Hubei Key Laboratory 
of Mechanical Transmission and Manufacturing 
Engineering at Wuhan University of Science and 
Technology (Grant No. MTMEOF2023B01) and the 
Key Research and Development Program of Hubei 
Province of China (Grant No. 2021BAA202).
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