Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 14, No. 1, 2025 

 

265 

Dynamic	Analysis	of	Load	Distribution	Optimization	in	
Low‐Speed	Heavy‐Duty	Multi‐Stage	Planetary	Gear	
System	
Jinghao Sun, Wei Wu, Kaiqi Yan, Xincheng Zhao 

School of Mechanical Engineering, Xi’an Shiyou University, Xi’an 710065, China 

 

Abstract: A three-stage planetary gear reducer was designed for low-speed, heavy-load operating conditions, and a dynamic 
model was established after optimizing the load distribution through floating planet carriers. This model was used to study the 
nonlinear motion factors in the gear system, with a particular focus on the internal excitation caused by time-varying meshing 
stiffness. A mathematical analysis model of the mechanism was developed using MATLAB, and the time-varying meshing 
stiffness curves were derived. The analysis shows that the system exhibits periodic motion with changes in stiffness. Additionally, 
the load distribution of the system was analyzed based on stiffness variation, and the research indicates that as the stiffness 
increases, the transmission system experiences greater vibrations and impacts, leading to a decline in load distribution 
performance. Finally, a kinematic simulation model of the planetary gear system was established using ADAMS. Simulation 
results demonstrated that load distribution optimization for this transmission system is essential and provided the basis for this 
conclusion. 

Keywords: Three-stage gear reducer; floating planet carrier; load distribution optimization; time-varying meshing stiffness, 
ADAMS. 

 

1. Introduction 
With the continuous progress and development of China’s 

modern manufacturing industry, planetary gear transmissions 
have become an indispensable transmission form due to their 
high load capacity, compact structure, smooth transmission, 
and high efficiency. They are widely used in various 
mechanical fields, including the petroleum industry, lifting 
and transportation, construction machinery, aerospace, ships, 
high-speed trains, automobiles, and many industrial machines. 

A multi-stage planetary gear transmission system is a 
complex mechanical power transmission system that exhibits 
both rigidity and flexibility. From a dynamic perspective, the 
gear transmission system is a continuous elastic system, 
where the system generates dynamic responses under internal 
and external excitations during motion, enabling power 
transmission. The internal excitation is a time-varying 
excitation generated by the meshing interaction between the 
gear pairs within the system, which is the key distinguishing 
factor of gear transmission systems compared to other 
mechanical transmission systems. Singh studied a single-
stage planetary system, considering the effects of the sun gear 
support stiffness and the planet wheel needle bearing stiffness. 
Transmission system models with 4, 5, and 6 planet gears 
were established, highlighting that as the number of planet 
gears increases, the control range of the planet wheel pinhole 
positional error becomes smaller, and the system's load-
sharing characteristics improve[1].Zhang DH, based on 
reliability theory and modern mechanical optimization design 
methods, optimized the volume, overlap degree, and 
efficiency of the NGW planetary gear transmission as 
objective functions. He then used ProE software to create a 
parametric solid model for assembly[2].Li Sheng et al. used 

the Gill integration method to solve the dimensionless motion 
equations of a two-stage planetary gear system in a milling 
cutter gearbox. They analyzed the effects of factors such as 
damping and excitation frequency on the system's bifurcation 
and chaotic characteristics[3]. 

This paper focuses on the planetary gear reducer used in 
the sugar production process for the pressing stage. A 
dynamic model of the system is established to study its 
nonlinear dynamic characteristics. The system's dynamic 
performance demonstrates the superiority of the optimized 
planetary gear reducer. The research results provide a 
theoretical foundation for the load-sharing optimization 
design of gear trains.

 

2. Modeling of the Planetary Gear 
Transmission System 

2.1. Gear Train Parameter Configuration 
The gearbox designed in this paper is applied to the sugar 

production process, with operating conditions in a low-speed, 
heavy-load environment. The motor power is 600 kW, the 
input speed ranges from 500 to 1200 r/min, and the gear ratio 
is approximately 150:1, with the expected output speed 
ranging from 0 to 10 r/min. After analysis, a multi-stage NGW 
planetary gear system was selected for the gearbox. The 
NGW system consists of a sun gear, a single set of planet 
gears, and a ring gear, using an internal meshing structure. 
This structure allows for a more compact gear arrangement 
while providing a wide range of transmission ratios, making 
it suitable for the required application. The motion schematic 
of the three-stage planetary gear reducer is shown, utilizing a 
three-stage 2K-H planetary gear system. 

 



 

266 

 
Figure 1. Schematic diagram of the motion of a three-stage planetary gear reduction gearbox 

 
Finally, after applying the gear design standards, the 

parameters of the three-stage planetary gear reducer are 
presented in Table 1. 

 
Table 1. Basic parameters of the planetary gear train 

 Z am    L 
Number of 

gears nx  

First-stage sun gear 25 7 20 145  0.3387 
First-stage planet gear 44 7 20 140 3 0.1872 
First-stage ring gear 113 7 20 140  0.6820 

Second-stage sun gear 18 14 20 200  0.3813 
Second-stage planet gear 39 14 20 210 3 0.0701 
Second-stage ring gear 96 14 20 210  0.5291 
Third-stage sun gear 26 20 20 390  0.2641 

Third-stage planet gear 28 20 20 400 3 0.2688 
Third-stage ring gear 82 20 20 400  0.8017 

 

2.2. Load-sharing Structure Optimization of 
the Planetary Gear System 

2.2.1. The Principle of Load Sharing. 
Load power distribution refers to the process in which the 

n planet gears, uniformly distributed around the sun gear in a 
planetary gear system, share and transmit the load. From a 
theoretical perspective, the output power and load capacity of 
the transmission system are directly proportional to the 
number of planet gears. [4]When there is a change in the load 
distribution between the gear teeth, some components within 
the planetary gear system are allowed to float structurally, 
with the positions of the components optimized to restore a 
balanced load distribution. As illustrated, in a single-stage 
three-planet gear system, when the load is uniformly 
distributed among the planet gears, the load distribution 
generates a vector forming an equilateral triangle, which is 
the foundation of the load-sharing principle[5]. 

 

    
Figure 2. Load distribution diagram of the planetary 

transmission system 
 

This paper employs the design of floating components 
within the gear train, primarily to facilitate practical 
manufacturing and assembly processes. The strategy is 
focused on minimizing the system's sensitivity to 
manufacturing and assembly tolerances, thereby indirectly 

improving the load-sharing performance of the planetary gear 
system[6]. 

Considering the low-speed operating conditions of the 
gearbox in this study, a floating planet carrier mechanism is 
implemented for the load-sharing system[7]. The shaft is 
coupled to the planet carrier via a dual-tooth coupling, which 
minimizes the need for excessive structural support, thereby 
facilitating the floating motion of the components. 
Additionally, the relatively large mass of the planet carrier 
itself ensures a stable floating motion, thereby maintaining 
operational stability and reducing dynamic fluctuations 
within the system. 

 

 
Figure 3. Schematic diagram of the floating motion of the planet 

carrier 
 

The 3D model of the gear train after the planet carrier 
floating motion is shown in the figure 4 below. The sun gear 
is configured as a spline-shaft-gear component, with the sun 
gears connected via a coupling. 



 

267 

  
Figure 4. 3D model of the gear train 

2.2.2. Load-Sharing Coefficient Calculation 
To quantify the uniformity of load distribution in a 

planetary gear system, the load-sharing coefficient is a critical 
parameter used to assess the degree of load distribution 
between individual gears or components[8]. The load-sharing 
coefficient is defined by the following formula: 

 

 mmax max
H n

m
m

i 1

F
K

1
F

n






 

 
             (1) 

 

In the formula, max  represents the maximum load per 

unit tooth width, m  represents the average load per unit 

tooth width, and  m max
F denotes the maximum meshing force 

between the external meshing pairs of the n gears. 
The load-sharing coefficients before and after the floating 

of the planet carrier, as calculated by the system of equations, 
are shown in the figure5. 

 

      
(a)Load-sharing coefficient without floating   (b)Load-sharing coefficient with floating 

Figure 5. Diagram of load-sharing coefficient variation 
 

As illustrated in the diagram, the load-sharing coefficient 
of the transmission system with a floating planet carrier 
fluctuates around 1.23, whereas the original transmission 
system without a floating mechanism fluctuates around 1.79. 
A lower load-sharing coefficient signifies a more uniform 
distribution of the load among the planetary gears within the 
meshing pairs, thereby enhancing the load-sharing 
performance. The incorporation of a floating planet carrier 
significantly improves the load distribution uniformity in 
planetary gear systems, which is critical for reducing stress 
concentrations and extending the operational lifespan of the 
system. 

3. Dynamic Analysis of the Gear 
Transmission System 

3.1. Nonlinear Dynamics Analysis 
In the dynamic analysis of gear systems, studying gear 

stiffness and internal excitation during meshing in multi-stage 
planetary gear systems are key research focuses .For the 
planetary gear system in this study, the energy method is used 
for calculation[9].The potential energy generated during 
meshing consists of five components[10]: shear potential 
energy sU  , Hertzian contact potential energy hU  , bending 

potential energy bU  , tooth base potential energy fU  , and 

compressive potential energy aU . From the above potential 

energies, the shear stiffness sk  , Hertzian contact stiffness

hk  ,bending stiffness bk ,substrate deformation stiffness fk ,and 

axial compressive stiffness
akcan be calculated, as described by 

the following relationship: 
 

 

2 2

s 2
d

s b

0
x

2 2

b 2
b d b a

0
x

2 2

a 2
d

a a

0
x

2

f
f

2

h
h

F F
k

2U 1.2F
2 dx

2GA

F F
k

2U F d x F h
2 dx

2EI

F F
k

2U F
2 dx

2EA

F
k

2U

F
k

2U


 




      


  



 














     

(2) 

       

 
In the formula, h is the distance from the centerline to the 

meshing point, d is the distance from the meshing point to the 
base circle, E is the Young's modulus, G is the shear modulus, 

xI is the second moment of area of the gear tooth section at 

the base circle, xA is the sectional area in the direction, aF and 

bF  are the radial and tangential forces at the meshing point, 

respectively[11]. 



 

268 

Additionally, from the geometric relationship of the 
involute curve and the diagram, it can be observed that: 

 

 
 
 

b 1 b 1 1

b 1 b b

b 1 1 b 1 b

h R cos sin

x R cos sin cos

d R cos sin cos

                     


               

(3) 

   

 
In the formula,   represents the pressure angle, bR  is the 

radius of the base circle, 1 is the angle between the meshing 

force and the tooth's centerline, and b is the half-arc angle of 

the base circle. 
 

 
Figure 6. Straight gear tooth model 

 
Using the above comprehensive equation, the different 

types of stiffness for an individual tooth pair can be 
determined. 

 

 
 
 

  
 

    
 

b

1

b

1

b

1

2
h

2
b 1

a b

2
b 1

s b

2

1 b b

3
b b

2 * f
1

f

f

1 EL

k 4 1

cos sin1
da

k 2EL cos sin

1.2 1 cos cos1
da

k EL cos sin

3 1 cos sin cos cos1
da

k 2EL cos sin

u
cos L

s1

k


















   


     
    


     

              
     

 
 










 
2

* * * 2f
1

f

u
M P 1 Q tan

s

EL




















   
       
     


  

(4) 

 

In the formula, E is the Young's modulus, is the Poisson's 
ratio, L is the tooth width, fu is the distance from the meshing 

point to the tooth root circle's vertex, fs  is the arc length 

corresponding to the previous gear tooth on the tooth root 
circle, and *L *M *P *Q is the constant related to the ratio of 
the central angle of the corresponding gear to the pitch radius. 

 
Figure 7. Substrate deformation geometric relationship 

 
For gear systems with a contact ratio between 1 and 2, 

operating in a state of alternate single and double tooth 
meshing[12].the overall stiffness k of a single tooth pair can 
be expressed as: 

 
2

h s1 s2 b1 b2 f1 f2 a1 a2

F 1
k

1 1 1 1 1 1 1 1 12U
k k k k k k k k k

 
       

  

(5) 

 
In the formula, F represents the contact force between the 

gear teeth, with subscripts 1 and 2 referring to the driving gear 
and the driven gear, respectively[13]. 
 

 
Figure 8. Gear double-tooth meshing motion relationship 

 
The comprehensive stiffness of double-tooth meshing can 

be approximately analogized as the parallel connection of two 
springs, which is equivalent to the parallel connection of the 
stiffness of a single tooth pair[14].By summing the individual 
tooth stiffness expressions as outlined above, the 
comprehensive stiffness of double-tooth meshing can be 
expressed as: 

 
2

z
i 1

h,i b1,i b2,i s1,i s2,i a1,i a2,i

1
k

1 1 1 1 1 1 1

k k k k k k k



     



 

(6) 

3.2. Calculation and Analysis of Time-varying 
Meshing Stiffness in Multi-stage Systems 

By substituting the parameters of each stage gear into the 
comprehensive equation (4), the necessary data for 



 

269 

calculating various types of meshing stiffness are obtained. 
 

 
Figure 9. Hertzian contact stiffness 

 

 
Figure 10. Variation curve of the sun gear base deformation 

stiffness 
 

 
Figure 11. Single tooth mesh overall stiffness variation 

curve 
 

 
Figure 12 Time-varying mesh stiffness variation curve 

3.3. The impact of time-varying meshing 
stiffness on load distribution in the gear 
train. 

As shown in the figure, the time-varying meshing stiffness 
of the planetary gear system follows a simple harmonic 
motion function over time[15].indicating that the meshing 
stiffness consists of alternating components and an average 
component, as expressed by the following equation: 

 

 t n mk k k cos    
         

(7)  

 

In the formula, nk is the average value of the time-varying 

meshing stiffness, 
mk  is the stiffness amplitude,   is the 

excitation frequency, and is the initial phase. 

The ratio of 
mk to

nk characterizes the temporal fluctuation 

of meshing stiffness. Thus, the meshing stiffness fluctuation 
coefficient is defined as: 

 

m
k

n

k
G

k


                 

(8)  

 
By changing the meshing stiffness fluctuation coefficient 

while keeping other parameters constant, different values 
were used for calculations. The resulting load distribution 
coefficient for the planetary gear transmission system with a 
floating structure is shown in the figure13. As the fluctuation 
coefficient increases, the load distribution coefficient also 
increases. However, when the fluctuation coefficient reaches 
about 0.5 to 1, the growth of the load distribution coefficient 
slows down. This suggests that as stiffness increases, 
vibration amplitude increases, leading to greater vibrations 
and shocks, which reduces load distribution performance. 

 

 
Figure 13. The relationship between load distribution 

coefficient and time-varying meshing stiffness 
 

4. Kinematic Simulation Based on 
ADAMS 

After constructing the 3D model in SolidWorks, it is 
imported into ADAMS, leaving only the kinematic meshing 
mechanism, as shown in the figure below. A separate 
simulation is performed for each stage of the gear system. The 
3D model in Parasolid format is loaded into ADAMS, and the 
material properties of the components are defined according 



 

270 

to the design specifications. Then, constraints between the 
various components are established, as outlined in the 
following table: 

 
Table 2. constraint conditions 

 reference Constraint 
sun gear ground revolute pair 

planet gear ground revolute pair 
 ring gear ground prismatic pair 

planet carrier ground revolute pair 
sun gear planet carrier prismatic pair 

 

 
Figure14. Planetary Gear Train Simulation Structural 

Diagram 
 

The contact force between gears is then defined, and the 
contact force is generally expressed by the following 
equation[12]: 

 

 max max
N

d
K step 0 0,d ,C

F , 0dt
0, 0

      
  

，，
(9)  

 
In the equation,   represents the actual contact distance 

between the two meshing gears,d is the penetration depth,
maxC

is the damping coefficient, and step is the step function. 
From this equation, it is evident that contact force is 

generated only when the two gears collide. Therefore, the 
contact between the meshing pairs of gears is modeled using 
"entity-to-entity" collision contacts. The stiffness coefficient 
is selected, with the force exponent set to 2.2, damping set to 
10 N·s/mm, static friction coefficient set to 0.3, dynamic 
friction coefficient set to 0.1, and pe0000000000netration 
depth set to 0.1 mm. Subsequently, the driving force and load 
are defined. The input speed is set to 1200 rpm, and the output 
speed is set to 8 rpm. A rotational drive is applied to the input 
gear, and a load in the opposite direction is applied to the 
fixed-axis gear. 

 

 
Figure15. Set parameters 

 
Figure16. Sun gear and planetary gear 

Contact force amplitude-frequency curve 
 

 
Figure 17. Planetary gear and ring gear 

Contact force amplitude-frequency curve 
 

The simulation results indicate that the meshing force 
fluctuation between the sun gear and the planet gears is 
smaller than that between the planet gears and the internal 
gear. This difference is due to the centrifugal forces generated 
by the rotation of the planet gears, which are added to the 
force between the planet gears and the internal gear. 
Furthermore, as shown in the Figure16 and Figure17, both 
meshing pairs exhibit significant fluctuations, which implies 
considerable vibrations during the meshing process. These 
vibrations can have a substantial impact on the gear's lifespan, 
tooth surface strength, and other performance characteristics. 

Additionally, the simulation of each stage of the planetary 
gear system shows that each stage experiences significant 
impacts. Therefore, it is necessary to implement load 
distribution optimization in the design to reduce the effects of 
these vibrations and improve the overall performance of the 
system. 

5. Conclusion 
(1) For planetary gear reducers in low-speed, heavy-load 

environments, a multi-stage planetary gear transmission 
system with a floating planet carrier configuration was 
established based on the original design. This system model 
facilitates smoother transmission, providing a theoretical 
foundation for the research and manufacturing of planetary 
gear transmission systems. 

(2) Through nonlinear dynamic analysis of the planetary 
gear system and its meshing stiffness, the meshing motion 
was studied to obtain various time-varying meshing stiffness 
curves, which realistically reflect the actual motion of the 



 

271 

meshing pairs. The floating structure type significantly 
influences the load sharing performance of the planetary gear 
system. From the system analysis results of load sharing 
based on the stiffness of the floating components, it can be 
concluded that when manufacturing errors exist in the 
planetary gear system, the load-sharing coefficient with 
floating components is smaller than that without floating 
components. The load-sharing performance of the planetary 
gear system changes most rapidly with the variation of the 
floating components, while the change with meshing stiffness 
is relatively slower. 

(3) The kinematic parameters obtained through the 
kinematic analysis of the gear system provide essential 
support for the optimization of structural parameters, enabling 
further optimization of the planetary gear reducer and 
improving design efficiency[17].This also lays the foundation 
for subsequent dynamic analysis as well as the analysis of 
gear friction and wear. 

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