*Corr. Author’s Address: Inner Mongolia University of Technology, College of Mechanical Engineering, China, 1642523516@qq.com 17
Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 17-31 Received for review: 2022-10-27
© 2023 The Authors. CC BY-NC 4.0 Int. Licensee: SV-JME Received revised form: 2022-12-30
DOI:10.5545/sv-jme.2022.427 Original Scientific Paper Accepted for publication: 2023-01-09
Analysis of the Dynamic Characteristics of a Gear-Rotor-Bearing 
System with External Excitation
Na, R. – Jia, K. – Miao, S. – Zhang, W. – Zhang, Q.
Risu Na – Kaifa Jia
*
 – Shujing Miao – Weiguo Zhang – Quan Zhang
Inner Mongolia University of Technology, College of Mechanical Engineering, China
The dynamic response of the rotor system in a ring die granulator is complex and difficult to solve when it operates under joint external, 
support and gear mesh forces. To solve this problem, a finite element method and extrusion theory was applied in this study to develop a 
dynamic coupling model for a hollow overhung rotor with external load excitation. A Newmark-β numerical integration method was used to 
solve for the dynamic response of the overhung rotor under multiple excitation forces. The results included time-domain response diagrams, 
frequency-domain response diagrams, phase diagrams, Poincaré section diagrams, and bifurcation diagrams. The model and the method 
were verified by testing a ring die granulator. On this basis, the dynamic response of the system is predicted according to the influence 
of different parameters. As the bearing support distance increased, the roller eccentricity decreased, the bearing clearance decreased, 
the response of the rotor system was significantly optimized, and the system tended to stabilize gradually. Therefore, this paper provides a 
theoretical basis and experimental verification for the optimization of a pelleting machine transmission structure.
Keywords: rotor dynamics, external incentives, finite element method, bearing, gear
Highlights
• Bearing clearance will affect the internal excitation of the bearing support of the system, so the influence law of different 
clearance parameters on the system is discussed. 
• The clearance of a helical gear can affect the internal excitation of the system-bearing support, so the influence law of 
different clearance parameters on the system is discussed.
• Bearing support distance will affect the dynamic response of the system, so the influence law of different support distance 
parameters on the system is discussed.
• The external load model of a ring mold pelleting machine is established, and its dynamic characteristics are explored according 
to the important parameters of production and operation. 
• The accuracy and correctness of the theoretical calculation are verified with experiments.
0  INTRODUCTION
Ring die granulators are key components used in the 
granulation processing. They are not only used in the 
feed-processing industry but are also widely used in 
emerging biomass pellet fuel molding industries. The 
dynamic response of the rotor is complex, making it 
difficult to calculate and predict. The rotor system in 
a ring die granulator is prone to rotor instability due 
to the combination of the gear mesh force, the bearing 
support force, and the external load force. However, 
studies regarding the responses of rotor systems 
experiencing random extrusion external loads are 
rarely reported. To improve the stability of this type 
of rotor and the service life of its key components, 
the dynamic response of a hollow overhung rotor 
experiencing multiple external forces was investigated 
in this study, and the effect patterns of the key 
parameters were explored. 
Schwarz et al. [1] and other research groups [2] 
to [3] used dynamic simulations and machine learning 
algorithms to study the impact of highly dynamic 
cage motion on the performance of rolling bearings. 
Smagala and Kecik [4] and Kurvinen et al. [5] used 
a two-degree-of-freedom nonlinear bearing model and 
the Hertz theory to study the effects of the number 
of balls and the rotational speed of the rotating axle 
on the dynamic response. Many scholars [6] to [9] 
have carried out extensive research on bearing failure 
dynamics and other aspects, and the results provide 
help for the ontology. 
Chen et al. [10] proposed a dynamic model of 
spur gear with straight teeth, which takes into account 
the detailed deformation of a single tooth, including 
the flexible deformation of the tooth surface, the 
deformation of the tooth body, the contact deflection 
of the local tooth surface and the deformation due to its 
neighbouring loaded teeth. Zheng et al. [11] and Zhu et 
al. [12] focused on the centrifugal force in high-speed 
working conditions, developed an analytical finite 
element method (FEM) framework to integrate the 
centrifugal field into the mesh stiffness and nonlinear 
dynamics. Shi and Li [13] proposed a dynamics 
model for hypoid gears, considering mesh stiffness 
that is dependent on dynamic mesh force. Based on 
the spectral density of excitation and the frequency 
response function of the transfer path between the 
excitation point and measurement point, Hajnayeb 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 17-31
18 Na, R. – Jia, K. – Miao, S. – Zhang, W. – Zhang, Q.
and Sun [14] proposed a new method to determine 
the influence of random manufacturing errors on 
vibration measurement on mating gear bearings. In 
the work of Liu et al. [15], mesh characteristics of the 
spur gear paired with the pitch deviation are studied 
in order to analyse the time-varying contact ratio 
and motion characteristics of the system. Xu et al. 
[16] used a loaded tooth contact analysis method and 
conducted quasi-static experiments to determine the 
coefficient matrix describing the relationship between 
the mesh force and the strain. Cirelli et al. [17] and 
[18] proposed an improved method to simulate the 
nonlinear dynamic response of spur gears. Mo et al. 
[19] and [20] discussed the effects of flexible support 
stiffness and floating components on the stability and 
reliability of a herringbone planetary gear system. 
Other researchers proposed a variety of accurate and 
effective time-varying mesh stiffness calculation 
methods, including the potential energy method, 
and discussed the effects of temperature and shock 
on the mesh stiffness [21] to [23]. In comparison, 
the calculation model of gear meshing stiffness and 
meshing error based on Fourier series is more widely 
used because of its representativeness.
Other researchers [24] to [27] proposed a 
Timoshenko beam consisting of a chain of masses 
and straight segments, accounting for the bending 
and shear deformations of the linear rotational and 
transverse springs. Zhan et al. [28] and Si et al. [29] 
studied the free vibrations of axially-loaded multi-
cracked Timoshenko beams with different boundary 
conditions, such as hinged, cramped, clamped-hinged, 
and cramped-free. Based on the Timoshenko beam 
theory and using Hamilton’s principle and Euler 
angles, Zhu et al. [30] and Zhang et al. [31] developed 
a novel mathematical model for a rotating axle with 
an efficient centrifugal term. The solution of rotor 
dynamic response based on discrete elements can not 
only deal with the influence of complex excitations 
but also has s high solving accuracy.
An extrusion mechanics model for a ring die 
was constructed as a function of vibration. Using a 
nonlinear deep groove ball bearing support mechanics 
model and a helical gear mesh mechanics model, the 
dynamic characteristics of the overhung rotor of a 
ring die granulator were analysed and used as a case 
study. The accuracy and validity of the theoretical 
model for coupled rotor dynamics were verified with 
experiments. This method takes into account the 
effects of complex load coupling and can provide 
guidance for solving and optimizing the dynamic 
response of irregular rotors.
1  COUPLED ROTOR MODEL WITH EXTERNAL EXCITATION
A ring die granulator rotor system consists of a hollow 
axle, a ring die, a gear, and a bearing, as shown 
in Fig. 1. Rotor vibration is an important cause of 
declining granulation efficiency and granulation 
quality. Multiple acting forces are significant factors 
affecting the stability of the rotor system. To improve 
the calculation efficiency and accuracy, the hoop-
locking connection between the ring die and the 
hollow axle was simplified as a rigid connection, with 
the assumption that the material of the rotating axle 
was uniform, and the keyway and the chamfering 
effects were negligible. A system dynamics model 
was developed using the Timoshenko finite element 
analysis method and the lumped mass method. The 
differential principle was used to discretize the 
variable-segment rotor and to divide the rotor system 
into 20 axle segments and 21 nodes as shown in Fig. 
2.
Fig. 1.  Drive rotor system of ring mold granulator
Fig. 2.  Finite element model of ring mold granulator rotor
According to the rotor structure of the ring 
die granulator, the finite element rotor model with 
the radial translational displacement and bending 
displacement vectors contained displacement 
vectors for a total of 84 element nodes. The element 
displacement vector for each axle segment could be 
expressed by Eq. (1). The supporting forces for the 
left and right bearings, the mesh force of the helical 
gear, and the pressing force of the ring die acted 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 17-31
19 Analysis of the Dynamic Characteristics of a Gear-Rotor-Bearing System with External Excitation
on the second, ninth, sixth, and nineteenth nodes, 
respectively.
 u
si
=
AA AA BB BB
T
xy xy
xy xy
,,,, ,,, .   



 (1)
In Eq. (1), x
A 
and y
A
 represent the x and y radial 
vibration displacements of the centre at the left end 
face, A, of the axle segment, respectively. θ
xA
 and 
θ
yA
 represent the radial angular displacements of the 
centre at the left end face, A, respectively. In a similar 
way, the displacements of the right end face, B, could 
be obtained. u
si
 represents the i
th
 element displacement 
vector of the rotor. The mass matrix, M
si
, the stiffness 
matrix, K
si
, the gyro matrix, G
si
, and the external force 
matrix, F
si
, could be obtained for the i
th
 element. The 
element matrices were assembled as shown in Fig. 3, 
and the rotor system’s global mass matrix, M, global 
stiffness matrix, K, global gyro matrix, G, and global 
external force matrix, F, were obtained.
Fig. 3.  Timoshenko beam element matrix assembly
By using Rayleigh damping to calculate the rotor 
viscous damping, matrix C could be expressed by Eq. 
(2):
 CMK  
oo
, (2)
where α
o
 and β
o
 are the proportional Rayleigh 
damping coefficients. The nodal displacement 
vector of the rotor system can be expressed by  
X = [x
1
 x
2
 ··· x
84
].
According to the energy method, the dynamic 
equation for the overhung hollow rotor with multi-
force coupling in the ring die granulator was 
established as Eq. (3):
 MX CG XK XQ
 
    . (3)
In Eq. (3), Q represents the assembled external 
force matrix for the coupled rotor system, which 
includes the gear mesh force matrix, F
g
, the bearing 
support force matrix, F
b
, the ring die extrusion force 
matrix, F
h
, and the axle external force matrix, F, 

X , 

X, and X are the acceleration, velocity, and 
displacement vectors of the rotor system, respectively, 
and ω is the rotational angular velocity of the rotor.
The structural parameters of the hollow axle of 
the SZLH420 ring die granulator are listed in Table 
1. The elastic modulus, E, was 210 GPa, the material 
density, ρ, was 7,850 kg/m³, the Poisson’s ratio, λ, was 
0.3, α
o
 and β
o
 were both 0.05, and the axle rotational 
speed, n, was 300 r/min.
Table 1.  Structural parameters of the beam
Element 
number
Outer diameter 
[mm]
Inner diameter 
[mm]
Length of element 
[mm]
1 120 80 45
2 120 80 45
3 150 80 45
4 150 80 45
5 150 80 45
6 150 80 45
7 190 80 60
8 200 80 50
9 200 80 50
10 230 80 110
11 560 210 10
12 600 240 10
13 640 300 10
14 680 420 20
15 666 420 8
16 652 420 8
17 640 420 8
18 580 420 85
19 580 420 85
20 600 420 30
2  MECHANICAL MODEL FOR THE COUPLED SYSTEM
2.1  Mechanical Model for the Deep Groove Ball Bearing 
Support
Bearings are a critical part of a rotor system, and 
they support and constrain the axles. The bearing 
parameters of the models of the two groups of deep 
groove ball bearings in the ring die granulator, Models 
6412 and 6420, are listed in Table 2. The deep groove 
ball bearings consisted of an inner and an outer ring, 
balls, and a cage. It was assumed that the contacts 
between the balls and the inner and outer rings were 
point contacts, and the effect of centrifugal force 
during rotation was ignored when calculating the 
bearing support force. A schematic diagram of one of 
the deep groove ball bearings is shown in Fig. 4, in 
which ω
in
 represents the angular velocity of the balls 
on the inner ring and ω
cage
 is the rotational angular 
velocity of the cage. The outer ring and frame had 
an interference fit with a connection speed of zero. 
Assuming that the balls were evenly distributed in the 
inner and outer races, the nonlinear restoring forces in 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 17-31
20 Na, R. – Jia, K. – Miao, S. – Zhang, W. – Zhang, Q.
the x- and y-directions of the deep groove ball bearing 
could be expressed by Eq. (4):
 F
FF F
FF F
xj xj j
j
N
j
N
yj yj j
j
N
j
N
b
b
=
b
=
=
=
b b
b b






 
 
cos
sin


1 1
1 1
 





. (4)
In Eq. (4), F
bx
 and F
by
 represent the supporting 
forces in the x- and y-directions of the bearing, 
respectively, N
b
 is the number of balls in the deep 
groove ball bearing, and θ
j
 is the rotation angle of the 
j
th
 ball at time t and can be expressed by Eqs. (5) and 
(6):
 

j
t
N
jj N    
cage
b
b
2
11 2 ,, ,, (5)
 

cage
in
b



r
Rr
. (6)
In Eqs. (5) and (6) R
b
 is the inner radius of the 
outer ring of the bearing and r is the outer radius of 
the inner ring of the bearing. The contact deformations 
between the balls and the inner and outer rings in the 
deep groove ball bearing satisfy the Hertz contact 
theory. Therefore, the contact extrusion force, F
j
, 
generated by the j
th
 ball on the race could be expressed 
by Eq. (7):
 FC H
jjj
 

b

32 /
, (7)
where C
b
 represents the Hertzian contact stiffness 
related to the material properties and δ
j
 represents the 
normal contact deformation between the j
th
 ball and 
the races. The interaction between the ball and the 
races can only generate a normal positive pressure. 
Therefore, a force exists when the normal contact 
deformation, δ
j
, between the ball and the races is 
greater than zero. To determine the contact between 
the ball and the races, the Heaviside function was 
introduced:
 H
j
j
j












10
00
. (8)
For the bearing clearance, γ
0
, and the x and y 
radial displacements, the normal contact deformation, 
δ
j
, between the ball and the races could be expressed 
by Eq. (9):
 
jj j
xy  coss in .
0
 (9)
Fig. 4.  Bearing mechanical model
Table 2.  Structural parameters of the support bearing
Bearing i 1 2
Outer radius R
b
 [mm] 55.55 79
Inner radius r [mm] 40.65 62.4
Ball number N
b
19 25
Contact stiffness C
b
 [GN/m
3/2
] 8.05
Bearing clearance γ
0
 [μm] 10
Support stiffness k
i
 [N/m] 7×108 9×108
Brace damping C
i
 [Ns/m] 500 1000
2.2  Mechanical Model for the Helical Gear Mesh
The gear was treated as a thin hard disk using the 
lumped mass method. The mechanical model for 
the gear shown in Fig. 5 was developed by analysing 
the forces acting on the gear and decomposing 
them orthogonally. The mass eccentricity caused 
by the casting error and nonuniform gear material is 
represented by e. The geometric centre of the gear 
is at O
1
 and the centre of mass is at O
g
. The gravity 
of the gear itself is represented by P. The centrifugal 
force of the eccentricity of the gear rotating mass 
is expressed by F
e
=Meω
2
. At the beginning of 
operations, the line O
1
O
g
 was parallel to the x-axis. 
For any t, the angle between O
1
O
g
 and the x-axis 
was ωt. Since the meshing stiffness calculation 
based on the cumulative potential energy method 
and the meshing error calculation considering gear 
alignment and modification factors have a wide range 
of applications and practices [21] to [23], the Fourier 
expansion calculated accordingly is as follows:
 kt kk t    =
ma
sin,  (10)
where k
m
 represents the average mesh stiffness, 
k
a
 is the mesh stiffness variation amplitude, and 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 17-31
21 Analysis of the Dynamic Characteristics of a Gear-Rotor-Bearing System with External Excitation
φ is the initial phase angle and is equal to 0. The 
comprehensive transmission error, e(t), affected by the 
tooth profile and pressure angle errors in an external 
gear transmission could be expressed by Eq. (11):
 et et  
r
sin,  (11)
where e
r
 is the comprehensive (general) transfer error 
amplitude. Then, the radial dynamic mesh force of the 
gear can be defined as the sum of the elastic restoring 
force and the damping force:
 Fk tfXt cXt
a
   
  ,. b
m

 (12)
In Eq. (12), c
m
 represents the mesh damping and 
f(X(t), b) is a nonlinear function. Assuming that the 
driving gear axle was a rigid axle and that its vibration 
angle displacement was negligible, the displacement 
function, X(t), of the driven gear axle vibration and 
mesh error is expressed by Eq. (13):
 Xt ye t () () .  (13)
The gear transmission had a certain one-sided 
backlash, b
0
. When there were x and y vibration 
displacements during the gear operation, the corrected 
backlash, b, could be expressed by Eq. (14):
 
bb a
aa
aa
   

  

 
00 00 0
00
cost an tan,
arccosc os /,
 

0 00
2
00
2
coss in .      xa y (14)
In Eq. (14), a
0
 and α
0
 represent the initial centre 
distance and the pressure angle, respectively, and a
’ 
and α
’ 
are the centre distance and pressure angle in the 
actual meshing process, respectively. The nonlinear 
function in the dynamic mesh force equation could be 
expressed by Eq. (15):
 fXtb
Xt bX tb
Xt b
Xt bX tb
 

  

  





,. 0 (15)
The friction force of the tooth surface could be 
obtained from the dynamic mesh force of the gear 
meshing. The secondary friction force of the gear can 
be expressed by Eq. (16):
 FF
fa
 , (16)
where μ is the static friction coefficient. For mesh 
points above a node, λ was assigned a value of 1, 
otherwise, it was assigned a value of 1. The gear 
forces, F
gx
 and F
gy
, in the x- and y- directions were 
obtained from Eq. (17):
 
Fe tF
Fe tFP
xf
ya
g
g
M
M
  
  


2
2
sin,
cos. (17)
Fig. 5.  Gear mechanical mode
The mass, M, of the ring die granulator gear was 
113 kg, the initial one-sided backlash, b
0
, was 10 μm, 
the static friction coefficient, μ, was 0.3, and the mass 
eccentricity, e, was 2 mm. The rest of the helical gear 
parameters are listed in Table 3.
Table 3.  Structure parameters of helical gear
Gear i 1 2
Number of teeth z
i
23 120
The modulus m
n
 [mm] 4 4
Pressure angle α [°] 20
Spiral angle β [°] 18
Tooth width B [mm] 150
k
m
 [N/m] 6.3×106
k
a
 [N/m] 3.2×105
e
r
 [mm] 2.2×10-5
C
m
 [Nm/s] 117
2.3  Mechanical Model for External Excitation
The extrusion force is an important cause of the 
vibration in a ring die granulator rotor. Developing an 
effective extrusion mechanical model is the key step 
for accurately analysing the dynamic characteristics of 
the ring die granulator rotor. As shown in Fig. 6, during 
relative motion of the ring die and the pressing roller, 
the material entered the gap between the die roller and 
the rotation of the ring die and was extruded to reach 
the extrusion density. The material extrusion area was 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 17-31
22 Na, R. – Jia, K. – Miao, S. – Zhang, W. – Zhang, Q.
divided into a compression area and an extrusion area, 
which corresponded to the central angles, θ
t
 and θ
max
, 
respectively. The extrusion force on the inner surface 
of the ring die increased as the extrusion density of 
the material increased, and it tended to be stable after 
reaching the extrusion density. The extrusion density 
of the material was largest at the location where the 
die-roller gap was a minimum, that is, the extrusion 
force was largest. Assuming that the frictional force 
generated by the extrusion of the material restrained 
the relative sliding between the pressure roller and 
the ring die, ignoring the rotational inertia force of 
the material, and treating the ring die and the pressure 
roller as rigid bodies, the force distribution diagram 
was obtained for the ring die, as shown in Figure 6. 
F
hx
 and F
hy
 represent the components of the extrusion 
force in the x- and y-directions, respectively, f
x
 and 
f
y
 are the components of the friction force in the 
x- and y-directions, respectively, e is the distance 
between the roller and the centre of the ring die, r
h
 
is the inner diameter of the ring die, r
r
 represents the 
outer diameter of the roller, d is the thickness after 
compression and extrusion, and h is the thickness of 
the uncompressed material.
Fig. 6.  Mechanical model of ring mode
In a ring die granulator, a gap is present between 
the pressing roller and the ring die to avoid rigid 
impacts caused by the vibration of the ring die. As 
the material is squeezed by the pressing roller, plastic 
deformation occurs in the extrusion area, and the 
density of the material is inversely proportional to 
the gap between the die and the roller. Because this 
gap exists, in addition to the material compressed 
into the die hole, an adhesion layer of the material 
is formed on the inner surface of the ring die after 
extrusion. Therefore, material extrusion occurs on 
the adhesion layer and the material density of the 
adhesion layer is equal to the theoretical discharge 
density. The relationship between the density of the 
material discharged from the ring die granulator and 
the ratio of the density of the initial material, γ(θ), and 
the central angle, θ, of the ring die can be expressed 
by Eq. (18):
 

 
 
h
re re d
hr
2
coss in
.
22
 (18)
During material extrusion, the extrusion force 
varies exponentially with the material density and 
can therefore be estimated according to the material 
properties. An exponential function can be used to 
describe the relationship between the extrusion force, 
P, and the material density with the central angle, θ, of 
the ring die:
 Pe
k


 
  



1
. (19)
In Eq. (19), λ and k are constants. In an actual 
granulation process, the theoretical discharge density 
of the adhesion layer is less than the actual discharge 
density and the surface extrusion force of the ring 
die is greater than the extrusion force of the adhesion 
layer. Therefore, Eq. (20) was used in this study to 
correct the surface extrusion force of the ring die:
 
  

PP 
r
rd
h
h
. (20)
The external excitation force of the ring die was 
obtained from a definite integral of the extrusion force 
on the inner surface of the ring die at the extrusion 
angles θ
t
 and θ
max
. Then, the orthogonal components, 
F
cx
 and F
cy
, of the centre stress of the ring die could be 
expressed by Eq. (21):
FP Pd
FP
cx tt
t
cy tt
t
  




br br d
br
hh
h


 




cosc os
sin
max
2
t t
Pd
t
2
 








 
br d
h
 


sin
.
max
 (21)
According to the acting angle of the ring die 
extrusion zone and the cosine theorem, the central 
angle of the extrusion circle could be expressed by Eq. 
(22):

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 17-31
23 Analysis of the Dynamic Characteristics of a Gear-Rotor-Bearing System with External Excitation
 



t
t
t

  




  












arccos
/
/
rd he r
rd he
hr
h
2
22
2
 



 

,
ln ln
. 

t
P
k
1 (22)
When γ
t
=1, the maximum extrusion central angle, 
θ
max
, could be obtained. The static friction coefficient, 
μ, was used to describe the static friction between the 
material and the ring die, so the resultant external 
force acting on the position of the ring die could be 
expressed by Eq. (23):
 
FF F
FF F
xc xc y
yc yc x
h
h









. (23)
When the ring die had x and y vibration 
displacements, the dynamic eccentricity value of the 
die roller was as shown in Eq. (24). By substituting the 
dynamic eccentricity, e
’
,
 
into the theoretical derivation 
described above, the dynamic random extrusion 
force that varied with the vibration displacement was 
obtained. 
    ee yx
22
. (24)
The eccentric distance, e, of the die roller of the 
ring die granulator was 0.1 m. The inner diameter, r
h
, 
of the ring die was 0.21 m. The outer diameter, r
r
, of 
the pressing roller was 0.103 m. The thickness, d, after 
molding and extrusion was 0.004 m. The thickness, h, 
of the uncompressed material was 0.02 m. The width, 
b, of the ring die was 0.12 m. The extrusion calculation 
parameters λ and k were 1,0000 and 3, respectively, 
and the static friction coefficient, μ, was 0.3.
3  EXPERIMENTAL VERIFICATION
To verify the accuracy and precision of the dynamic 
model, a SZLH420 ring die granulator was tested in 
actual production. Vibration at the ring die position 
was not easy to test because of overhang, rotation, and 
limited space. The test was conducted at the bearing 
position 1 (node 2). The test results were compared 
with theoretical calculation results. In the test, an 
A302 wireless acceleration measurement sensor 
from Beijing Bitron Co., Ltd. was used for signal 
acquisition, a BS951-T wireless gateway was used 
for data transmission, and a portable computer was 
used for data processing and analysis. The specific 
experimental scheme and the schematic diagram of 
the test setup are shown in Fig. 7. 
a)
b) 
Fig. 7. Testing program; a) testing program, and b) schematic 
diagram of test setup
The acceleration data in the x- and y-directions 
were collected during the experiment. The vibration 
displacement, velocity, and acceleration curves in 
the time domain were obtained by processing the 
data with data analysis software, and the vibration 
curves in the frequency domain were obtained from 
fast Fourier transforms. The results are shown in Fig. 
8a. The Newmark–β method was used to solve the 
nonlinear dynamics equation of the coupled system. 
The curves in the frequency and time domains for 
node 2 at the bearing position are shown in Fig. 8b. 
Using the stable data and comparing the experimental 
results and theoretical calculations, it was evident that 
the frequency errors and amplitudes at the peaks in the 
frequency-domain graphs between the experimental 
and theoretical results did not exceed 5 %, and the 
vibration amplitudes for the curves in the time domain 
did not exceed 10 %. The experimental results were 
affected by various excitation disturbances, such as 
those from the motor, feeder, and modulator. As a 
result, the experimentally measured time-domain and 
frequency-domain curves were more complex and 
the amplitudes were overestimated. The results show 
that the nonlinear dynamics model for the coupled 
overhung rotor established during this study is highly 
accurate and can be used for predicting system 
dynamic characteristics. 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 17-31
24 Na, R. – Jia, K. – Miao, S. – Zhang, W. – Zhang, Q.
field research, the replacement cost of a ring die 
accounts for more than 30 % of the production cost 
of the entire factory. Nonlinear vibrations at a ring die 
cause extreme extrusion or severe wear of the ring 
die, which can lead to damage of the ring die. Several 
4  ANALYSES AND PREDICTIONS OF DYNAMIC 
CHARACTERISTICS OF THE COUPLED ROTOR MODEL
Ring dies in the rotor systems of ring die granulators 
are prone to wear. According to actual production 
  
a) 
    
b) 
Fig. 8.  Comparison between vibration response test and theoretical results of ring die pelletizer system;  
a) test results, and b) theoretical results

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 17-31
25 Analysis of the Dynamic Characteristics of a Gear-Rotor-Bearing System with External Excitation
rotor parameters for the horizontal ring die granulator 
were selected for this study: a rotational speed of n 
= 300 r/min, a ring die roller eccentricity of e = 0.1 
m, a bearing clearance of γ
0
 = 10 μm, and a length of 
axle segment 7 of l
7
 = 60 mm. Time-domain curves, 
frequency-domain curves, phase diagrams, and 
Poincaré section diagrams for the dynamic response in 
both directions at the ring mode position (node 19) are 
shown in Fig. 9. The figure shows that the maximum 
vibration displacement response in the x-direction was 
approximately 2.09 mm and the maximum vibration 
displacement response in the y-direction was 2.12 
mm. These results occurred because the rotor system 
was subjected to the combined effects of gravity in 
the y-direction, the ring die force, the mesh force, 
and the bearing force. The time-domain curves were 
transformed into frequency-domain curves by fast 
Fourier transformation (FFT). The main vibration 
frequencies and amplitudes in both directions were the 
same, and the frequency division phenomenon caused 
by various external forces occurred near the low 
frequency. The phase diagrams were approximately 
closed circles and showed weak disorder and trends 
from quasi-periodic motion to chaotic motion. The 
Poincaré section was an approximate point system in 
a quasi-periodic motion. The current rotor system ran 
relatively smoothly.
4.1  Effect of the distance between the bearings on the 
system dynamics
For considerations of space and cost in the structural 
design of a granulator, the supporting distance 
between the two bearings of the overhung rotor 
should not be too large. The effect of changing 
the supporting distance between the two bearings, 
a)
b)
Fig. 9.  Displacement response diagram, FFT diagram, phase diagram and Poincaré section diagram at the actual ring mode position;  
a) x-direction, and b) y-direction
Fig. 10.  Bearing support distance variation of the rotor system is illustrated

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 17-31
26 Na, R. – Jia, K. – Miao, S. – Zhang, W. – Zhang, Q.
namely, the length of axle segment 7, on the dynamic 
response of the overhung rotor system is shown 
as a bifurcation diagram in Fig. 10. The diagram 
shows that as the distance between the two bearings 
increased, the system motion gradually changed from 
complex chaotic motion to quasi-periodic motion, 
and to period-1 motion at l
7
 = 90 mm. Therefore, an 
appropriate increase in the bearing support distance 
is beneficial for optimizing the dynamic response of 
the system at the ring die location. Figs. 11a and b 
show that when l
7
 = 120 mm, the maximum vibration 
displacement responses in the x- and y-directions were 
approximately 1.88 mm and 1.92 mm, respectively. 
The time-domain responses were significantly better 
than those in the actual granulator. The FFT curves 
show that the vibration amplitudes in both directions 
were significantly reduced, and the frequency 
division was weakened. The phase diagrams were 
a)
b)
c)
d)
Fig. 11.  Displacement response diagram, FFT diagram, phase diagram and Poincaré section diagram at the position of predicted ring mode; 
a)  axis segment element 7 is 90 mm in X-direction, b)  axis segment element 7 is 90 mm in Y-direction, c)  axis segment element 7 is 120 
mm in X-direction, and d) axis segment element 7 is 90 mm in Y-direction

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 17-31
27 Analysis of the Dynamic Characteristics of a Gear-Rotor-Bearing System with External Excitation
approximately closed circles, the turbulence was 
reduced, and the periodic motion of the system tended 
to be stable. The Poincaré sections were point systems 
with quasi-periodic motion, and the current rotor 
system ran more smoothly. Figs. 11c and d show that 
the response of the rotor system was further optimized 
when l
7
 = 80 mm. This phenomenon occurred 
because an appropriate increase in the bearing support 
distance was beneficial for optimizing the effect of the 
bending-torsional coupling of the system.
4.2  Effect of Roller Eccentricity on the System Dynamics
To avoid rigid collisions during the ring die assembly 
process, a gap is present between the die and the roller. 
The wear of the ring die during production operations 
leads to increases in the gap between the die and the 
roller, which causes decreases in the eccentricity of 
the press roller. When only the eccentricity of the 
rollers was varied, the effect of the roller eccentricity 
on the dynamic response of the overhung rotor 
system was predicted, and the resulting bifurcation 
diagrams are shown in Fig. 12. The figure shows 
that as the roller eccentricity increased, the system 
motion gradually changed from period-1 motion 
to quasi-periodic motion to chaotic motion. The 
eccentric distance affects the granulation efficiency. 
Appropriately reducing the eccentric distance of the 
roller is beneficial for optimizing the response of 
the rotor system. Figs. 13a and b show that when e 
= 0.099 m, the maximum vibration displacement 
responses were approximately 1.91 mm and 1.92 
mm in the x- and y-directions, respectively. The 
time-domain responses were considerably better than 
those of the actual granulator. The FFT curves show 
that the vibration amplitudes in both directions were 
reduced, and the frequency division was weakened. 
The phase diagrams were approximately closed 
circles, the disorder phenomenon was reduced, and 
the periodic motion of the system tended to be stable. 
The Poincaré sections were point systems with quasi-
periodic motion, and the current rotor system ran more 
smoothly. Figs. 13c and d shows that at e = 0.101 m, 
the response of the rotating system deteriorated, and 
the maximum vibration displacement responses were 
approximately 2.18 mm and 2.21 mm in the x- and 
y-directions, respectively. The time-domain responses 
were higher than the actual granulator amplitudes. 
The FFT curves show that the vibration amplitudes in 
both directions increased, the frequency division was 
more severe, the phase diagrams appeared severely 
disordered, and the system motion changed from 
quasi-periodic motion to mixed motion. The Poincaré 
sections were approximately closed cycles connected 
by points, representing quasi-periodic motion, and the 
current rotor system was unstable. This phenomenon 
was caused by the effect of eccentricity on the 
external extrusion load on the surface of the ring 
die and provides an important theoretical basis for 
adjusting the gap between the roller and the die during 
production operations.
4.3  Effect of Bearing Clearance on the System Dynamic 
Response
A strong nonlinearity exists in the bearing clearance 
during the bearing assembly process. The bearing 
clearance increases due to wear during operation. 
When only the bearing clearance was changed, 
the effect of the bearing clearance on the dynamic 
response of the overhung rotor system was predicated, 
and the resulting bifurcation diagrams are shown in 
Fig. 14. The figure shows that as the bearing clearance 
increased, the system motion changed from period-1 
motion to quasi-periodic motion to chaotic motion. 
Regular bearing lubrication and replacement can 
ensure a small bearing clearance, which is conducive 
for optimizing the response of the rotor system. Figs. 
15a and b show that when γ
0 
= 40 μm, the maximum 
vibration displacement responses were approximately 
2.19 mm and 2.22 mm in the x- and y-directions, 
respectively, and the time-domain response amplitudes 
were greater than those of the actual granulator. The 
FFT curves show that the vibration amplitudes in both 
directions increased, the low-frequency vibrations 
were noticeable, and the frequency division was 
strengthened. The phase diagrams were approximately 
closed ellipses, the turbulent phenomenon increased, 
and the periodic motion of the system tended to be 
chaotic. The Poincaré sections were approximately 
asymptotes, the system motion changed from quasi-
periodic to chaotic motion, and the current rotor 
system operation gradually became unstable. Figs. 15c 
and d show that when γ
0 
= 80 μm, the response of the 
rotating system deteriorated further, with maximum 
vibration displacement responses of approximately 
2.32 mm and 2.34 mm in the x- and y-directions, 
respectively. This phenomenon was caused by the 
bearing clearance affecting the bearing support force 
of the rotor system, and it provides an important 
theoretical basis for equipment operation and 
maintenance as well as regular part and component 
replacement.

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 17-31
28 Na, R. – Jia, K. – Miao, S. – Zhang, W. – Zhang, Q.
Fig. 12. Die roll eccentricity distance variation of the rotor system is illustrated
a)
b)
c)
d)
Fig. 13.  Displacement response diagram, FFT diagram, phase diagram and Poincaré section diagram at the position of predicted ring mode; 
a) ring roller eccentricity is 0.099 m in X-direction, b) ring roller eccentricity is 0.099 m in Y-direction, c) ring roller eccentricity is 0.101 m in 
X-direction, and d) ring roller eccentricity is 0.101m in Y-direction

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 17-31
29 Analysis of the Dynamic Characteristics of a Gear-Rotor-Bearing System with External Excitation
Fig. 14.  Bearing clearance variation of the rotor system 
a)
b)
c)
d)
Fig. 15.  Displacement response diagram, FFT diagram, phase diagram and Poincaré section diagram at the position of predicted ring mode; 
a) bearing clearance is 40 μm in X-direction, b) bearing clearance is 40 μm in Y-direction, c) bearing clearance is 80 μm in X-direction, and d) 
bearing clearance is 80 μm in Y-direction

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 17-31
30 Na, R. – Jia, K. – Miao, S. – Zhang, W. – Zhang, Q.
5  CONCLUSIONS
In this paper, experimental verification was conducted 
using a real-world operating ring die granulator, and 
the experimental results were compared with the 
theoretical calculation results to verify the validity 
and accuracy of the theoretical model. At the same 
time, the correctness and practicability of the 
extrusion mechanics model, the gear meshing model, 
the bearing support model, and the multi-excitation 
coupling model of the rotor system are also verified. 
This is helpful when solving the dynamics of complex 
structure and excited rotor. Finally, according to the 
theoretical model, for the strong nonlinear factors, 
such as the bearing support distance, the roller 
eccentricity, and the bearing clearance variation, 
dynamic response analyses were performed using 
bifurcation diagrams and the effects of their 
parameters on rotor performance were determined. 
The results of this study can be summarized into three 
primary points:
1. The increase of bearing support distance is 
beneficial to the optimization of dynamic 
response characteristics of the system. When 
bearing support distance is short, the response 
becomes complicated and chaotic motion is easy 
to appear. When the bearing distance increases 
by 120 mm, the displacement amplitudes in x- 
and y- directions decrease by 0.35 mm and 0.37 
mm, respectively, and the dynamic performance 
indexes of the system are optimized. Therefore, 
reasonable rotor structure design and bearing 
support distance selection are of great significance 
when optimizing system response and improving 
system stability.
2. The reduction of eccentricity of roll die is 
beneficial to the optimization of system response. 
Comparisons of the frequency-domain curves, 
the phase diagrams, and the Poincaré section 
diagrams showed that as the eccentric distance 
of the pressure roller decreased, the amplitude 
of the system response was reduced, the disorder 
phenomenon was weakened, and the dynamic 
performance indicators were optimized. The 
eccentric distance is a key factor affecting the 
granulation efficiency and quality. Therefore, it is 
important to select an appropriate eccentricity for 
the rollers to optimize the system response and 
improve the system productivity.
3. The reduction of bearing clearance is beneficial 
to the optimization of dynamic response 
characteristics of the system. Comparisons of the 
frequency-domain curves, the phase diagrams, 
and the Poincaré section diagrams showed that as 
the bearing clearance decreased, the amplitudes 
of the system response decreased, the disorder 
phenomenon was weakened, and the dynamic 
performance indicators were optimized. The 
bearing clearance is a key factor affecting 
the stability of the rotor system. Therefore, 
regular bearing maintenance and replacement 
is important for optimizing the response of the 
rotor system and improving the reliability of the 
system.
6  ACKNOWLEDGEMENTS
The authors acknowledge the financial support 
from the National Natural Science Foundation of 
China (Grant No. 52065050) & Basic research 
funds for universities directly under the Inner 
Mongolia Autonomous Region of China (Grant No. 
JY20220377).
8  REFERENCES
[1] Schwarz, S., Grillenberger, H., Tremmel, S., Wartzack, So. 
(2022). Prediction of rolling bearing cage dynamics using 
dynamic simulations and machine learning algorithms. 
Tribology Transactions, vol. 62, no. 2, p. 225-241, DOI:10.1080
/10402004.2021.1934618.
[2] Cai, Z., Lin, C. (2017). Dynamic model and analysis of 
nonlinear vibration characteristic of a curve-face gear drive. 
Strojniški vestnik - Journal of Mechanical Engineering, vol. 63, 
no. 3, p. 161-170, DOI:10.5545/sv-jme.2016.3859.
[3] Cui, Y., Deng, S., Niu, R., Chen, G. (2018). Vibration effect 
analysis of roller dynamic unbalance on the cage of high-speed 
cylindrical roller bearing. Journal of Sound and Vibration, no. 
434, p. 314-335, DOI:10.1016/j.jsv.2018.08.006.
[4] Smagala, A., Kecik, K. (2021). Nonlinear dynamics analysis of 
a rolling bearing. Journal Européen des Systèmes Automatisés, 
vol. 54, no. 1, p. 21-26, DOI:10.18280/JESA.540103.
[5] Kurvinen, E., Viitala, R., Choudhury, T., Heikkinen, J., Sopanen, 
J. (2020). Simulation of subcritical vibrations of a large 
flexible rotor with varying spherical roller bearing clearance 
and roundness profiles. Machines, vol. 8, no. 2, art. ID 28, 
DOI:10.3390/machines8020028.
[6] Luo, W, Yan, C., Yang, J., Liu, Y., Wu, L. (2021). Vibration 
response of defect-ball-defect of rolling bearing with 
compound defects on both inner and outer races. IOP 
Conference Series: Materials Science and Engineering, vol. 
1207, no. 1, DOI:10.1088/1757-899X/1207/1/012006.
[7] Bavi, R., Hajnayeb, A., Sedighi, H.M., Shishesaz, M. (2022). 
Simultaneous resonance and stability analysis of unbalanced 
asymmetric thin-walled composite shafts. International 
Journal of Mechanical Sciences, vol. 217, art. ID 107047, 
DOI:10.1016/J.IJMECSCI.2021.107047.
[8] Wang, P., Xu, H., Yang, Y., Ma, H., He, D., Zhao, X. (2022). 
Dynamic characteristics of ball bearing-coupling-rotor system 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 17-31
31 Analysis of the Dynamic Characteristics of a Gear-Rotor-Bearing System with External Excitation
with angular misalignment fault. Nonlinear Dynamics, vol. 
108, p. 3391-3415, DOI:10.1007/S11071-022-07451-1.
[9] Liu, J., Ding, S., Wang, L., Li, H., Xu, J. (2020). Effect of the 
bearing clearance on vibrations of a double-row planetary 
gear system. Proceedings of the Institution of Mechanical 
Engineers, Part K: Journal of Multi-body Dynamics, vol. 234, 
no. 2, DOI:10.1177/1464419319893488.
[10] Chen, Z., , J., Wang, K., Zhai, W. (2021). An improved dynamic 
model of spur gear transmission considering coupling effect 
between gear neighboring teeth. Nonlinear Dynamics, vol. 
106, p. 339-357, DOI:10.1007/S11071-021-06852-Y.
[11] Zheng X., Luo, W., Hu, Y., He, Z., Wang, Sg. (2022). Study on 
the mesh stiffness and nonlinear dynamics accounting for 
centrifugal effect of high-speed spur gears. Mechanism and 
Machine Theory, vol. 170, art. ID 104686, DOI:10.1016/j.
mechmachtheory.2021.104686.
[12] Zhu, L.-Y., Shi, J.-F., Gou, X.-F. (2020). Modeling and dynamics 
analyzing of a torsional-bending-pendular face-gear drive 
system considering multi-state engagements. Mechanism 
and Machine Theory, vol. 149, art. ID 103790, DOI:10.1016/j.
mechmachtheory.2020.10.103790.
[13] Shi, Z., Li, S. (2022). Nonlinear dynamics of hypoid gear 
with coupled dynamic mesh stiffness. Mechanism and 
Machine Theory, vol. 168, art. ID 104589, DOI:10.1016/j.
mechmachtheory.2021.104589.
[14] Hajnayeb, A., Sun, Q. (2022). Study of gear pair vibration 
caused by random manufacturing errors. Archive of Applied 
Mechanics, vol. 92, p. 1451-1463, DOI:10.1007/S00419-022-
02122-4.
[15] Liu, P., Zhu, L., Gou, X., Shi, J., Jin, G. (2021). Dynamics 
modeling and analyzing of spur gear pair with pitch deviation 
considering time-varying contact ratio under multi-state 
meshing. Journal of Sound and Vibration, vol. 513, art. ID 
116411, DOI:10.1016/j.jsv.2021.116411.
[16] Xu, H.g, Yang, Y., Ma, H., Luo, Z., Li, X., Han, Q., Wen, B. 
(2022). Vibration characteristics of bearing-rotor systems 
with inner ring dynamic misalignment. International Journal of 
Mechanical Sciences, vol. 230, art. ID 107536, DOI:10.1016/j.
ijmecsci.2022.107536.
[17] Cirelli, M., Valentini, P.P., Pennestrì, E. (2019). A study 
of the non-linear dynamic response of spur gear using a 
multibody contact based model with flexible teeth. Journal 
of Sound and Vibration, vol. 445, p. 148-167, DOI:10.1016/j.
jsv.2019.01.019.
[18] Cirelli, M., Giannini, O., Valentini, P.P., Pennestrì, E. 
(2020). Influence of tip relief in spur gears dynamic using 
multibody models with movable teeth. Mechanism and 
Machine Theory, vol. 152, art. ID 103948, DOI:10.1016/j.
mechmachtheory.2020.103948.
[19] Mo, S., Zhang, T., Jin, G.-G., Cao, X.-L., Gao, H.-J. (2020). 
Analytical investigation on load sharing characteristics of 
herringbone planetary gear train with flexible support and 
floating sun gear. Mechanism and Machine Theory, vol. 144, 
art. ID 103670, DOI:10.1016/j.mechmachtheory.2019.103670.
[20] Mo, S., Yue, Z., Feng, Z., Shi, L., Zou, Z., Dang, H. (2020). 
Analytical investigation on load-sharing characteristics 
for multi-power face gear split flow system. Journal 
of Mechanical Engineering Science, vol. 234, no. 2, 
DOI:10.1177/0954406219876954.
[21] Dai, H., Long, X., Chen, F., Xun, C. (2021). An improved 
analytical model for gear mesh stiffness calculation. 
Mechanism and Machine Theory, vol. 159, art. ID 104262, 
DOI:10.1016/j.mechmachtheory.2021.104262.
[22] Sun, Y., Ma, H., Huangfu, Y., Chen, K., Che, L.Y., Wen, B. 
(2018). A revised time-varying mesh stiffness model of 
spur gear pairs with tooth modifications. Mechanism and 
Machine Theory, vol. 129, p. 261-278, DOI:10.1016/j.
mechmachtheory.2018.08.003.
[23] Chang-Jian, C.W. (2010). Strong nonlinearity analysis for 
gearbearing system under nonlinear suspension-bifurcation 
and chaos. Nonlinear Analysis: Real World Applications, vol. 
11, no. 3, p. 1760-1774, DOI:10.1016/j.nonrwa.2009.03.027.
[24] Liu, J., Na, R., Cen, H. (2020). Modelling and simulation of 
extrusion force in biomass pelletisation by ring die pellet mill. 
International Journal of Simulation and Process Modelling, 
vol. 15, no.5, p. 484-490, DOI:10.1504/ijspm.2020.110927.
[25] Peter, S., Lammens, R.F., Steffens, K.-J. (2010). Roller 
compaction/Dry granulation: Use of the thin layer model for 
predicting densities and forces during roller compaction. 
Powder Technology, vol. 199, no. 2, p. 165-175, DOI:10.1016/j.
powtec.2010.01.002.
[26] Lei, T., Zheng, Y., Yu, R., Yan, Y., Xu, B. (2022). Dynamic 
response of slope inertia-based Timoshenko beam under a 
moving load. Applied Sciences, vol. 12, no. 6, art ID 3045, 
DOI:10.3390/APP12063045.
[27] Gómez-Silva, F., Zaera, R. (2022). Dynamic analysis and 
non-standard continualization of a Timoshenko beam lattice. 
International Journal of Mechanical Sciences, vol. 214, art. ID 
106873, DOI:10.1016/J.IJMECSCI.2021.106873.
[28] Zhan, D., Jiang, S., Niu, J., Sun, Y. (2020). Dynamics 
modeling and stability analysis of five-axis ball-end milling 
system with variable pitch tools. International Journal of 
Mechanical Sciences, vol. 182, art. ID 105774, DOI:10.1016/j.
ijmecsci.2020.105774.
[29] Si, H., Cao, L., Li, P.. (2020). Dynamic characteristics and 
stability prediction of steam turbine rotor based on mesh 
deformation. Strojniški vestnik - Journal of Mechanical 
Engineering, vol. 66, no. 3, p. 164-174, DOI:10.5545/sv-
jme.2019.6283.
[30] Zhu, H., Chen, W., Zhu, R:, Gao, J., Liao, M. (2020). Study on 
the dynamic characteristics of a rotor bearing system with 
damping rings subjected to base vibration. Journal of Vibration 
Engineering & Technologies, vol. 8, p. 121-132, DOI:10.1007/
s42417-019-00082-8.
[31] Zhang, Y., Yang, X., Zhang, W. (2020). Modeling and stability 
analysis of a flexible rotor based on the timoshenko beam 
theory. Acta Mechanica Solida Sinica, vol. 33, p. 281-293, 
DOI:10.1007/s10338-019-00146-y.