*Corr. Author’s Address: Lanzhou University of Technology, Gansu Lanzhou, 730050, China, lutdcb@126.com 557
Strojniški vestnik - Journal of Mechanical Engineering 67(2021)11, 557-569 Received for review: 2021-07-08
© 2021 Journal of Mechanical Engineering. All rights reserved.  Received revised form: 2021-09-30
DOI:10.5545/sv-jme.2021.7318 Original Scientific Paper Accepted for publication: 2021-10-14
A Method for Calculating Elliptic Gear Transmission Efficiency 
Based on Transmission Experiment
Dong, C.B. – Liu, Y .P. – Zhao, G.
Changbin Dong
1
 – Yongping Liu
1
 – Gang Zhao
2
1
Lanzhou University of Technology, School of Mechanical and Electric Engineering, China 
2
Hubei Key Laboratory of Mechanical Transmission and Manufacturing Engineering, China
Transmission efficiency is an important index to evaluate the transmission performance and energy consumption of gear transmission 
systems. To analyse the transmission efficiency of elliptic gears, the load torque fluctuation model of elliptic gear is established to analyse the 
influence of load torque of an elliptic gear transmission system on the torque of input and output. The torque data of input and output under 
different working conditions are obtained by conducting an elliptic gear transmission test. Finally, the transmission efficiency of the elliptic 
gear pair is obtained through the torque measurement data of the elliptic gear transmission test, and its variation law under different working 
conditions is analysed. The results show that the transmission efficiency of the elliptic gear transmission system changes constantly and 
presents an increasing trend with the increase of load torque and a decreasing trend with the increase of speed.
Keywords: elliptic gear, transmission test, load fluctuation, torque, transmission efficiency
Highlights
• 	 The torque fluctuation calculation model of an elliptic gear is analysed and established.
• 	 The elliptic gear transmission test rig is built, which can be used to obtain the corresponding dynamic data.
• 	 The dynamic transmission efficiency of an elliptic gear pair is calculated and obtained, and the influence of various operating 
parameters on the instantaneous transmission efficiency of the system is analysed.
0  INTRODUCTION
With
 
the development of mechanical products towards 
complexity and diversification, the demand for high-
performance non-uniform transmission mechanisms 
increases day by day. The transmission mechanism 
represented by non-circular gear, which can achieve 
an accurate and stable speed ratio, shows unique 
advantages in improving equipment performance and 
realizing specific transmission requirements [1] and 
[2]. As a new type of gear transmission, the elliptic 
gear is mainly used to transmit the non-uniform 
motion between two shafts, and to realize the non-
linear relationship between the rotation angles of 
driving and driven parts [3] and [4]. The non-circular 
gear transmission system represented by elliptic gear 
has a strong bearing capacity, a compact structure, 
and a variable ratio transmission. It is mainly used 
in low speed and high torque occasions, such as 
rotary vane engines, hydraulic pumps, hydraulic 
motors, printing presses, packaging machines, 
textile equipment, spacecraft frequency converters, 
tank fire control systems [5] and  [6], continuously 
variable transmissions (CVT) [7], etc. Considering 
the expansion of its applications and the key role 
and special requirements in the transmission, it 
is meaningful to carry out in-depth research on 
the transmission efficiency of the elliptic gear 
transmission system.
Many research results have been accumulated 
for the transmission efficiency analysis method of 
gear transmission systems. Diez-Ibarbia et al. [8] 
combined the tooth surface friction coefficient and 
transmission efficiency, analysed the influence of the 
selection of friction coefficient on the transmission 
efficiency of modified spur gears. Wang et al. carried 
out a detailed study on the transmission efficiency 
of gear transmission systems. They combined tooth 
surface friction with power loss and tooth surface 
elastohydrodynamic lubrication [9] and proposed a 
calculation method of sliding friction loss of involute 
helical gear pair under load [10], considering assembly 
error, machining error, deformation, and other factors. 
On this basis, the friction power loss and transmission 
efficiency of the gear transmission system under 
different working conditions and design parameters 
are analysed [11] to [12], and the relevant data are 
obtained by building an experimental platform, 
which finally verifies the rationality of the proposed 
method [13] to [14]. Petry-Johnson et al. [15] obtained 
the transmission efficiency of the system under 
the condition of high speed and variable torque by 
building a gear transmission test and analysed the 
changing trend of the meshing efficiency of the gear 
transmission system and the total efficiency of the 
gearbox under the condition of different speed and 
load torque. Xu et al. [16] proposed and established 
the calculation model of friction-related mechanical 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)11, 557-569
558 Dong, C.B. – Liu, Y.P. – Zhao, G.
efficiency loss of parallel shaft gear pair on the basis 
of comprehensively considering gear load distribution 
model, friction model and mechanical efficiency 
formula. Through gear transmission testing, the 
effects of geometric parameters, tooth profile 
modification, working conditions, surface finish, and 
lubricant performance on mechanical efficiency loss 
were analysed. Based on the elastohydrodynamic 
lubrication model, Li and Kahraman [17] proposed a 
model to predict the load-related power loss of spur 
gear pair. The instantaneous rolling and sliding shear 
in lubricating oil are determined by transient pressure 
distribution and oil film thickness distribution, so as to 
determine the mechanical power loss of gear meshing. 
Li et al. [18] determined the calculation formula 
of the meshing efficiency of the conical involute 
gear based on the piecewise equivalent method and 
analysed the influence of different gear parameters 
on the contact line length, unit load and meshing 
efficiency in the meshing period. Liu et al. [1] and 
[2] constructed a non-uniform end face movable 
tooth transmission mechanism by combining the 
end face movable tooth mechanism and non-circular 
gear pairs and analysed the influence of tooth surface 
sliding friction on the transmission efficiency through 
simulation. Liu et al. [19] analysed the influence of 
time-varying meshing angle of non-circular gear pair 
on rolling rate loss caused by sliding friction and 
rolling friction by establishing a mathematical model 
of meshing efficiency of non-circular gear under 
elastohydrodynamic lubrication.
Although the above research results have 
irreplaceable guiding significance for the analysis of 
transmission efficiency of the elliptic gear transmission 
system, most of the analysis of transmission efficiency 
of non-circular gear is only theoretical calculation 
without systematic experimental analysis. In 
contrast, the experimental analysis can better reflect 
the authenticity and has more guiding significance. 
Therefore, this paper first analyses the load fluctuation 
of the elliptic gear transmission system and extracts 
the torque data of the input and output ends by 
building the elliptic gear transmission test-bed; it then 
calculates and obtains the instantaneous transmission 
efficiency of the elliptic gear transmission system.
1  CALCULATION MODEL OF TRANSMISSION  
EFFICIENCY OF ELLIPTIC GEAR
Because of the variable ratio transmission 
characteristics of elliptic gear pairs, their instantaneous 
transmission efficiency is always changing. The 
transmission efficiency of a gear pair can reflect its 
load-carrying capacity and power loss, especially 
for elliptic gears, which are suitable for low-speed 
and high torque conditions. Therefore, it is of great 
significance to analyse its instantaneous transmission 
efficiency. In general, the transmission efficiency of 
elliptic gear system is defined as:
 
  
T
Ti
2
11 2
100,%
 (1)
where T
1
 
represents the arithmetic mean value of input 
torque,T
2
 represents the arithmetic mean value of the 
output torque, and i
12
 represents the transmission ratio 
of elliptic gear pair.
1.1  Pitch Curve of Elliptic Gear
Fig. 1 shows the pitch curve of elliptic gear. The 
centre distance of the elliptic gear pair is a. The 
rotation angles of driving and driven gears are θ
1
 and 
θ
2
, respectively. In the initial position, θ
1
 = θ
2
 = 0. The 
pitch curve equations of the driving and driven gears 
are as follows:
Fig. 1.  Schematic diagram of elliptic gear pitch curve
 r
Ae
e
1
1
2
11
1
1



  cos
,

 (2)
 r
Ae
e
2
2
2
22
1
1



  cos
,

 (3)
where r
1
 and r
2
 represent the radius of pitch curve of 
driving gear and driven gear, respectively. A is the long 
half shaft of elliptic gear. e
1
 and e
2
 are the eccentricity 
of the driving and driven gears, respectively. c is the 
distance from the centre of elliptic symmetry to the 
focus, and c = e A. θ
1
 and θ
2
 are the rotation angles of 
driving and driven gears, respectively. The following 
are the relations between the rotation angles of driving 
and driven gears of elliptic gear pair:
 

2
1
1
1
2
1
12

















ac
e
e
tant an . (4)

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)11, 557-569
559 A Method for Calculating Elliptic Gear Transmission Efficiency Based on Transmission Experiment
1.2  Torque of Elliptic Gear Pair
The power transmission of the elliptic gear pair is 
similar to that of a cylindrical gear, and its transmission 
model is shown in Fig. 2a. Due to the time-varying 
curvature radius of the pitch curve of elliptic gears, 
there are certain differences among the teeth. Even if 
the load torque T
L
 on the driven shaft is constant, the 
torque fluctuation may occur on the driving side [20]. 
Therefore, in order to analyse the transmission torque 
of the elliptic gear pair, the load torque borne by the 
driven gear can be transformed into the driving gear, 
as shown in Fig. 2b.
The constant torque applied to the driven gear 
will lead to torque imbalance on the driving gear. Due 
to the time-varying transmission ratio function of the 
elliptic gear pair, the driven gear will produce a certain 
moment of inertia in the case of variable angular 
velocity. Therefore, under the condition that the drive 
shaft rotates at a constant speed, the total torque 
required to drive the load torque can be expressed as:
 T
T
i
TI
i
L
1
2
12
22
12

 
, (5)
where T
1
 and T
2
 
are the loads on the driving and 
driven gears, respectively, T
L
 is the load torque, I
2
 
is 
the moment of inertia of the driven gear, and α
2
 is the 
angular acceleration of the driven gear. 
It can be seen from Fig. 2 that when the angle of 
the elliptic gear is θ
1
 = ω
1
 t. The design parameters of 
the two elliptical gears are the same, so e
1
 = e
2
. The 
transmission ratio of the elliptic gear pair can be 
expressed as: 
 i
a
r
ee t
e
12
1
1
2
11
1
2
1
12
1
 
 

cos
.

 (6)
Then, the angular velocity and angular 
acceleration of the driven gear can be expressed as: 
 


2
1
2
1
1
2
11
1
12



 
e
ee t cos
, (7)
 



2
2
1
2
11
2
1
1
2
11
2
21
12
 

 
 




d
dt
ee t
ee t
sin
cos
. (8)
Because the profile of elliptic gear remains 
involute, its moment of inertia can be calculated 
according to the method of cylindrical gear. The 
elliptic gear is approximately equivalent to a cam, 
and the moment of inertia per unit tooth width of the 
elliptic gear can be expressed as:
 I
ae
e







 




0
4
1
4
0
1
1
1
2
11
2
d À
cos
. (9)
Let u = 1 + e cos θ
1
, then Eq. (9) can be simplified 
as follows:
I
ae
e
M
ae e
e



















 

01
2
4
2
3 0
01
2
4
1
2
1
1
21
12 3
41
2 2
3
1
2
1

e
,
where ρ
0
 
is the density of elliptic gear and M is as 
follows:
a)    
 
  b) 
Fig. 2.  Torque transmission model of elliptic gear pair;  
a) the original elliptic gear meshing moment model, and b) the equivalent model after the load torque is applied to the driving gear
          
M
ee ee e




  







  

1
2
2
11 1
2
11
2
1
2
1
3
51 1
6
21 1 cosc os 
3 3
51
2
1
23
21
11
2
11
11
3
1
2
1
2

 





 



e
e
e
e
e
cos
cos
cos
arcs



i in
cos
cos
.
e
e
11
11
1












Strojniški vestnik - Journal of Mechanical Engineering 67(2021)11, 557-569
560 Dong, C.B. – Liu, Y.P. – Zhao, G.
2  CONSTRUCTION OF ELLIPTIC GEAR TRANSMISSION TEST
The elliptic gear test adopts a horizontal mechanism 
composed of a precision mechanical system, 
measurement and control system, measurement 
software and other parts. The names and distribution 
positions of the main components of the test bench 
are shown in Fig. 3. The schematic diagram of its 
working principle is shown in Fig. 4. The elliptical 
gear reducer is installed in the gear test platform, the 
input and output ends of the reducer are connected 
through a coupling, and the signal is transmitted to the 
controller through computer control. The controller 
transmits the implementation command to the sensor 
and the driver, provides power and load by driving the 
motor, and then transmits the power and load to the 
reducer, so that the reducer can operate normally. The 
torque sensor and the vibration sensor sequentially 
collect the experimental data required to be measured 
and transmit them back to the computer through the 
controller to obtain the required experimental data. 
The elliptic gear transmission test is shown in Fig. 5.
The main machine of elliptic gear transmission 
test is mainly composed of a mechanical system, 
including input end components, output end 
components, reducer mounting bracket, and other 
parts. Each component is installed on the 2.6 m 
long cast iron platform base with a T-groove. The 
mounting bracket of the reducer is fixed in the middle 
of the base. The left side is the input end assembly, 
and the right side is the output end assembly. The 
input end assembly includes input end motor slide 
Fig. 4.  Working principle of elliptic gear transmission test bench
Substituting Eqs. (6) to (9) into Eq. (5), we obtain:
  T
eT ee tI ee t
L
1
1
2
1
2
11
2
21 11
2
1
11 22 1

   

 () (c os() )( )sin()  
 
 (c os() )
.
12
1
2
11
3
ee t 
 (10)
Fig. 3.  The distribution of main engine components; 1 T-groove 
cast iron platform base; 2 linear guide rail; 3 Locking device;  
4 hand wheel; 5 reducer base; 6 base; 7 Test planetary reducer; 
8 Servo motor at output end; 9 Circular grating protective cover 
at output end; 10 drag chain; 11 reducer under test; 12 input end 
circular grating protective cover; 13 protective cover;  
14 input servo motor

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)11, 557-569
561 A Method for Calculating Elliptic Gear Transmission Efficiency Based on Transmission Experiment
plate (installing input servo motor and input torque 
sensor) and input end grating encoder slide plate 
(installing input grating encoder). The output end 
assembly includes output end grating encoder slide 
plate (installing output grating encoder) and output 
end loading support slide plate (installing output 
servo motor and output torque sensor). Each slide 
plate is installed on the linear guide rail, which can be 
moved left and right through the hand gear to adjust 
the position. The input end is driven by an alternating 
current (AC) servo motor, which can work in speed 
mode and torque mode.
To realize the installation and test of different 
specifications and models of precision reducer, the 
test bench has a precision mounting bracket and 
supporting multi specification precision mounting 
accessories (connecting disc, input shaft and output 
shaft) so as to improve the universality of the test 
equipment. The whole system is controlled via an 
industrial computer, equipped with high-precision 
grating sensor system, high-precision torque sensor 
system and special measurement software, so as to 
realize the automatic measurement of the performance 
of precision reducer. The special feature of the test-
bed is that it can be used for various types of reducers. 
The mounting bracket of the reducer is fixed in the 
middle of the base, and the linear guide rail is installed 
on the cast iron platform base. The movement of the 
input and output components is realized through the 
gear rack mechanism.
The left-hand gear in Fig. 5b can ensure the 
implementation of lateral movement. There are 
similar mechanisms in the input end assembly and 
output end assembly, which can ensure the sliding of 
the input end motor and grating encoder slide plate 
and realize the connection or disconnection between 
the encoder and the output shaft of the tested reducer 
and the connection or disconnection between the 
output torque sensor and the output shaft of the tested 
reducer. In Fig. 5b, the right-hand gear can realize 
the vertical movement of the output component, 
which can make the slide plate of the input end 
grating encoder slide. To realize the connection or 
disconnection between the input shaft of the measured 
reducer and the reducer, the slide plate of the input end 
motor, and the connection or disconnection between 
the grating encoder and the input end torque sensor. 
Fig. 6 shows the processed elliptic gear pair, and its 
design parameters are shown in Table 1.
a)       b) 
Fig. 5.  The elliptic gear transmission test; a) overall view of the test bench, and b) partial view of the test bench
a)             b) 
Fig. 6.  The gear used in elliptic gear transmission experiment; a) elliptic gear reducer, and b) elliptic gear pair processed

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)11, 557-569
562 Dong, C.B. – Liu, Y.P. – Zhao, G.
Table 1.  Elliptic gear design parameters
Parameter Value
Module m [mm] 1.5
Number of teeth z 53
Centre distance a [mm] 80
Tooth width B [mm] 28
Eccentricity e 0.2
Pressure angle [°] 20
Equation of pitch curves r 

38 4
10 2
.
.c os
Material Structural steel
Lubrication mode Oil lubrication
3  TORQUE FLUCTUATION OF ELLIPTIC GEAR
According to Eq. (10), the change trend of the torque 
of the driving and driven gears of the elliptic gear pair 
under different eccentricity and load torque conditions 
can be obtained, as shown in Figs. 7 and 8. 
In Fig. 7, with the increase of eccentricity, the 
torque of the driving and driven gears of the elliptic 
gear increases. The change trend of driving gear is 
more obvious. When the eccentricity is 0, the torque 
of the driving and driven gears of the elliptic gear 
pair is constant and will not fluctuate. When the 
eccentricity is not 0, the torque fluctuation of the 
driving and driven gears will occur. With the increase 
of eccentricity, the change of torque of driven gear is 
more regular.
In contrast, the change of driving gear shows the 
same trend, but there are some differences between 
different eccentricities. This is because the driven gear 
is connected to the output and the torque at the output 
is constant. The driven gears move along a specific 
transmission ratio function under the driving gear. The 
inconsistency of the tooth profile of the elliptic gear 
pair makes the torque difference between the driving 
gears and the driven gears.
Fig. 8, with the increase of load torque, also 
presents the same change trend. In contrast, the load 
a)     b) 
Fig. 7.  Effect of eccentricity on torque of a) driving and b) driven gears 
a)     b) 
Fig. 8.  Influence of load torque on a) driving and b) driven gear torque 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)11, 557-569
563 A Method for Calculating Elliptic Gear Transmission Efficiency Based on Transmission Experiment
change of driven gear is more uniform and smoother. 
The moment of inertia is produced by the non-uniform 
rotation of the elliptic gear. If the inertia moment is 
greater than the load moment, the torque of the driving 
shaft is opposite to the working driving direction, 
which will cause impact on the non-meshing tooth 
side of the gear. In Fig. 8, when the load torque is 
0, that is, under no-load condition, the minimum 
torque of the driving and driven gears of the elliptic 
gear pair will be less than 0, which is negative and 
shows a periodic trend. This shows that there is tooth 
surface separation, which will lead to the vibration 
and noise of the gear transmission system, which is 
not conducive to the normal work of the system. With 
the increase of load torque, the phenomenon that the 
minimum torque of driving gear and driven gear is 
less than 0 will disappear, which indicates that the 
existence of load will always lead to the phenomenon 
of tooth separation.
4  TRANSMISSION EFFICIENCY OF ELLIPTIC GEAR 
TRANSMISSION SYSTEM
In view of the inconsistency of elliptic gear tooth 
profile and the complexity of applicable working 
conditions, especially for reciprocating motion 
and forward-and-reverse motion (reversing device 
of pumping unit), it is necessary to analyse its 
transmission efficiency under the condition of forward 
and reverse motion so as to provide guidance for 
practical application. In the process of the experiment, 
the rated output torque is preset. When the reducer 
is loaded from 0 to rated torque by driving motor, 
the torque data of input and output are collected. 
Each group collects 99 data points. The transmission 
efficiency of the elliptic gear pair is analysed by 
setting different operating parameters (torque and 
speed).
4.1  Torque Analysis of Elliptic Gear Pair
Figs. 9 to 11 show the change trend of input shaft 
torque and output shaft torque under the condition of 
constant speed and increasing gradient of rated load 
torque. 
With the increase of rated load torque, the torque 
of the input and output ends is increasing. In contrast, 
the input and output torque in the reverse case are 
larger than that in the case of positive rotation. The 
torque fluctuation of the output is small, and the input 
torque presents periodic change. With the increase 
of rated input torque, the difference between the 
two kinds of steering gradually increases. This is 
consistent with the torque fluctuation analysis results. 
The output torque is less than the input torque: the 
torque T
1
 of the driving gear is greater than the torque 
T
2
 of the driven gear. Therefore, the correctness of the 
torque calculation method of the ellipse gear proposed 
in this paper is proved.
Figs. 12 and 13 show the change trend of input 
shaft torque and output shaft torque under clockwise 
and counterclockwise conditions when the load torque 
remains constant and the speed changes. According 
to the analysis in Fig. 9, with the increase of speed, 
the torque at the input and output end also presents 
a periodic trend. With the increase of the speed, the 
fluctuation of the torque at the input end and the 
torque at the output end is also intensified. Similar 
to the distribution trend in Figs. 8 to 10, the input 
and output torques under clockwise rotation are 
a)              b) 
Fig. 9.  The torque distribution law of elliptic gear pair when T
L
 = 5 N·m; a) n = 5 r/min, and b) n = –5 r/min

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)11, 557-569
564 Dong, C.B. – Liu, Y.P. – Zhao, G.
second is the counterclockwise rotation experiment. 
Due to the existence of friction torque, starting 
torque, and dynamic backlash between tooth profiles 
at the beginning of counterclockwise rotation after 
greater than those under counterclockwise rotation, 
which is related to the order of clockwise and 
counterclockwise rotation during the experiment. The 
first is the clockwise rotation experiment, and the 
a)             b) 
Fig. 10.  The torque distribution law of elliptic gear pair when T
L
 = 10 N·m ; a) n = 5 r/min, and b) n = –5 r/min
a)           b)
Fig. 11.  The torque distribution law of elliptic gear pair when T
L
 = 15 N·m ; a) n = 5 r/min, and b) n = –5 r/min
a)       b)
Fig. 12.  The torque distribution law of elliptic gear pair when T
L
 = 10 N·m; a) n = 10 r/min, and b) n = –10 r/min

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)11, 557-569
565 A Method for Calculating Elliptic Gear Transmission Efficiency Based on Transmission Experiment
clockwise rotation, the torque under counterclockwise 
rotation is greater than that under clockwise rotation. 
Consistent with the previous analysis results, the 
torque T
1 
of the driving gear is greater than that of the 
driven gear T
2
, which further verifies the correctness 
of the torque calculation method of the elliptic gear 
proposed in this paper. It also shows that the change 
trend of the output torque and input torque of the 
elliptic gear transmission system will not change 
significantly with the increase of the speed, but there 
is a slight difference in the value.
There is a certain error in the data collection 
process, which will cause a certain difference between 
the torque T
1
 at the input end and the torque T
2
 at 
the output end, which will affect the instantaneous 
transmission efficiency calculation of the gear 
transmission system. The way we reduce the error 
is to collect data multiple times, and the trend of 
the collected data is basically consistent with the 
theoretical analysis, which also verifies the correctness 
of our experiment.
4.2  Analysis of Transmission Efficiency of Elliptic Gear Pair
Figs. 14 to 16 show the distribution law of 
instantaneous transmission efficiency calculated 
by Eq. (1) while keeping the speed constant and the 
rated load torque T
2
 gradient changing. Under the two 
steering conditions, the instantaneous transmission 
efficiency of the elliptic gear pair changes all the 
time. In contrast, the transmission efficiency of the 
counterclockwise transient is greater than that of the 
clockwise transient. With the increase of rated load 
torque, the instantaneous transmission efficiency of 
the elliptic gear pair gradually presents a periodic 
trend, and the periodicity is more obvious.
a)     b)
Fig. 13.  The torque distribution law of elliptic gear pair when  T
L
 = 10 N·m; a) n = 20 r/min, and b) n = –20 r/min
a)         b)
Fig. 14 .  The transmission efficiency of elliptic gear when  T
L
 = 5 N·m; a) n = 5 r/min, and b) n = –5 r/min

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)11, 557-569
566 Dong, C.B. – Liu, Y.P. – Zhao, G.
Fig. 17.  The variation trend of transmission efficiency of elliptic 
gear with torque 
Fig. 17 shows the change trend of average 
transmission efficiency of elliptic gear pair with the 
a)          b)
Fig. 15.  The transmission efficiency of elliptic gear when   T
L 
=10 N·m; a) n = 5 r/min, and b) n = –5 r/min
a)      b)
Fig. 16.  The transmission efficiency of elliptic gear when     T
L
 = 15 N·m ; a) n = 5 r/min, and b) n = –5 r/min
change of load torque. With the increase of the load 
torque, the transmission efficiency of the elliptic gear 
pair shows an increasing trend. Therefore, on the 
premise of meeting the actual working conditions and 
the established motion law, increasing the load torque 
is conducive to improving the transmission efficiency 
of the elliptic gear pair.
Figs. 18 to 20 show the distribution law of 
instantaneous transmission efficiency of elliptic 
gear pair under clockwise and counterclockwise 
conditions when the load torque remains constant, 
and the speed gradient changes. It is consistent with 
the previous analysis when the speed is constant, the 
load gradient changes, and the transmission efficiency 
in counterclockwise transient is greater than that in 
clockwise transient. However, under the two steering 
conditions, there is a periodic trend, and with the 
increase of speed, the periodicity is gradually obvious. 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)11, 557-569
567 A Method for Calculating Elliptic Gear Transmission Efficiency Based on Transmission Experiment
In contrast, the counterclockwise transmission 
efficiency is greater than the forward transmission 
efficiency, and the transmission efficiency decreases 
with the increase of speed.
Fig. 21 shows the change trend of the 
average transmission efficiency of clockwise and 
counterclockwise rotation of elliptic gear pair with 
the change of speed. With the increase of rotation 
a)          b) 
Fig. 18. Transmission efficiency of elliptic gear when T
L
 = 10 N·m ; a) n = 5 r/min, and b) n = –5 r/min
a)         b)
Fig. 19.  Transmission efficiency of elliptic gear when T
L
 = 10 N·m ; a) n = 10 r/min, and b) n = –10 r/min
a)          b) 
Fig. 20.  Transmission efficiency of elliptic gear when T
L
 = 10 N·m ; a) n = 20 r/min, and b) n =–20 r/min

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)11, 557-569
568 Dong, C.B. – Liu, Y.P. – Zhao, G.
speed, the average transmission efficiency clockwise 
and counterclockwise rotation of the elliptic gear 
pair presents a decreasing trend. This is closely 
related to the non-linear change of instantaneous 
coincidence degree and transmission ratio of elliptic 
gear. Therefore, the elliptic gear pair can present 
high instantaneous transmission efficiency at low 
speed. In the process of engineering application, the 
instantaneous transmission efficiency can be improved 
by appropriately reducing the speed on the premise 
of meeting the established working conditions and 
motion rules.
Fig. 21.  The variation trend of elliptic gear transmission  
efficiency with torque
Due to the inconsistency of the tooth profile of the 
elliptic gear, the instantaneous transmission efficiency 
during the meshing process changes all the time, and 
its transmission efficiency will be lower than that of a 
general cylindrical gear. However, the elliptic pair is 
mainly suitable for low-speed, high-torque occasions 
and for some special variable-speed sports occasions. 
Therefore, under the conditions of low speed and 
high torque, its instantaneous transmission efficiency 
can reach more than 80 %, which can meet the 
requirements of working conditions.
5  CONCLUSIONS
Based on the load fluctuation model of elliptic gear, 
the test-bed of elliptic gear transmission is built, and 
the torque data values of input and output in forward 
and reverse rotation are obtained. On this basis, the 
transmission efficiency of the elliptic gear pair is 
calculated and obtained, and the influence of load 
torque and speed on the transmission efficiency of the 
system is analysed. The results show that: 
1. Due to the time-varying pitch curve radius of the 
elliptic gear pair, the tooth profile is inconsistent, 
which makes the torque at the input end and 
output end have certain differences under the 
conditions of forward rotation and reverse 
rotation, so the transmission efficiency will have 
certain differences. Under the action of load 
torque, the torque on the driving and driven gears 
of elliptic gear will increase with the increase of 
the eccentricity and load torque of elliptic gear.
2. The load torque and speed have a certain influence 
on the transmission efficiency of the elliptic gear 
transmission system. The transmission efficiency 
will decrease with the increase of speed and 
increase with the increase of load torque, which 
also shows that the non-circular gear transmission 
system represented by elliptic gear is suitable for 
low speed and high torque conditions.
3. In view of the commonness of elliptic gear and 
non-circular gear and the subordinate relationship 
between them, the calculation method, and 
experimental means of transmission efficiency of 
elliptic gear proposed in this paper can be used 
for other types of non-circular gear transmission 
systems, and can provide some theoretical 
guidance for the calculation of transmission 
efficiency.
6  ACKNOWLEDGEMENTS
The research was support by the National Natural 
Science Founding of China (No. 51765032), and the 
Open Fund of Hubei Key Laboratory of Mechanical 
Transmission and Manufacturing Engineering at 
Wuhan University of Science and Technology (Grant 
No. MTMEOF2020B03) 
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