*Corr. Author’s Address: Central South University of Forestry and Technology, Changsha, 410004, China, zhyu@csuft.edu.cn 569
Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 569-581 Received for review: 2024-06-24
© 2024 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2024-09-14
DOI:10.5545/sv-jme.2024.1072 Original Scientific Paper Accepted for publication: 2024-09-18
Gear Differential Flank Modification Design Method  
for Low Noise
Zhang, Y . – Zhou, H. – Duan, C. – Wang, Z. – Luo, H.
Yu Zhang
1,2,*
 – Hai Zhou
1,2
 – Chengyu Duan
1,2
 – Zhiyong Wang
1,2
 – Hong Luo
1,2
1 
Central South University of Forestry and Technology,  
Institute of Modern Mechanical Transmission Engineering Technology, China 
2 
Central South University of Forestry and Technology,  
Engineering Research Center for Forestry Equipment of Hunan Province, China
To address the limitations of existing gear tooth modification methods, a differentiated tooth modification method is proposed, where the 
modification amount of adjacent teeth varies according to a sine function. Initially, a mathematical model of the gear tooth profile varying 
according to a sine function is established. Then, using Adams software, a simulation analysis of the dynamic characteristics such as centroid 
angular acceleration, meshing force, and transmission error of gear pairs is conducted. The dynamic transmission performance of three sets 
of gear pairs—unmodified, normally modified, and differentially modified—is compared. Additionally, Simcenter 3D software is used to analyze 
the noise characteristics of these gear pairs. The results show that the differentially modified gear pairs, compared to the normally modified 
ones, have a maximum reduction of 2.47 % in sound power level at the fundamental meshing frequency amplitude. This proves that the 
differentiated modification method enhances the dynamic transmission performance of gears, offering a new method for gear vibration and 
noise reduction.
Keywords: tooth modification; low noise; angular acceleration; meshing force
Highlights
• 	 A gear modification method is proposed with modifications varying by a sine function.
• 	 A mathematical model for tooth profiles varying sinusoidally has been established.
• 	 This method reduces angular acceleration, meshing force, and transmission error.
0 INTRODUCTION
Gear transmission is one of the core components in 
automobiles, and its performance directly affects 
the comfort and safety of the passengers [1] to [3]. 
Currently, electrification represents the main trend 
in automotive development. The most significant 
difference between electric vehicles (EVs) and 
traditional internal combustion engine vehicles lies in 
the power system. The replacement of the engine with 
an electric motor has a profound impact on the gear 
transmission system, leading to the development of 
reducers that are higher revolutions per minute (RPM), 
more efficient, and capable of handling higher peak 
torques. The maximum input speed of these reducers 
has reached 15,000 rpm to 20,000 rpm, even higher, 
which is 3 to 10 times higher than that of traditional 
vehicles. Consequently, the noise generated by the gear 
transmission system during high-speed engagement 
has become a major source of noise in EVs [4] to 
[6]. Moreover, unlike internal combustion vehicles, 
electric vehicles lack engine noise to mask other 
sounds, making the noise from the reducer assembly 
more pronounced. The noise quality of EVs can even 
be worse than that of traditional vehicles, significantly 
reducing the noise, vibration and harshness (NVH) 
performance of EVs. Existing gear flank modification 
methods apply the same modification parameters 
to all teeth, using a uniform amount of modification 
for each tooth surface. However, the effectiveness 
of these traditional flank modification methods has 
reached its limit, especially when dealing with high-
speed gear pairs, where the results are not always 
ideal. Cattanei et al. [7] have proposed a method for 
reducing rotor noise by changing the spacing between 
circumferential blades according to a specific pattern, 
thereby dispersing the harmonic peak energy of blade 
impact frequencies and reducing the harmonic peaks 
of blade impact frequencies. Although the mechanism 
of rotor noise differs significantly from that of gear 
noise, this noise reduction method offers a new 
perspective for suppressing gear noise.
Vibration and noise reduction have always been 
important topics in gear design and manufacturing 
research [8] to [10]. For this purpose, a lot of effort 
has been made, among which gear flank modification 
is an important method for gear vibration and noise 
reduction [11] to [13]. Rana et al. [14] utilized a non-
contact advanced precision finishing process involving 
pulsed electrolytic dissolution to perform five types of 
flank modifications and their combinations on spur 
gears, significantly reducing the noise and vibration 
of spur gears under a specific operating condition. 
Ghosh and Chakraborty [15] studied the influence of 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 569-581
570 Zhang, Y. – Zhou, H. – Duan, C. – Wang, Z. – Luo, H.
gear flank profile modification amount on vibration 
levels and identified the optimal modification amount. 
Through optimization, the vibrations caused by profile 
and pitch errors were significantly reduced. Wang [16] 
established the connection between gear modification, 
gear tooth load distribution, and the dynamic model 
of the gear system, and proposed a multi-objective 
optimization design method for helical gears that 
considers both vibration/noise reduction and the 
uniform distribution of tooth surface loads. Tang et al. 
[17] established a parametric model of the high-speed 
train traction gear transmission system and proposed 
a noise reduction optimization design scheme that 
combines tooth direction and tooth profile under 
multiple working conditions, achieving satisfactory 
transmission performance and noise reduction effects. 
Yoon [18] proposed a new method based on using 
cubic spline curves to modify the tooth profile to 
reduce the vibration and noise of involute gears. Li 
et al. [19] proposed a 3D modified tooth surface with 
the grinding wheel moving in an axial helical ascent 
and a radial parabolic motion, and established a multi-
objective ant lion optimizer model to optimize the 
meshing performance and dynamic characteristics 
of the herringbone gear transmission system. Han at 
al. [20] proposed a method for the lead and profile 
modification tooth flank of the work gear by setting 
the movement of the A and B axes as two fourth-order 
polynomial functions of the axial feed of the work 
gear. Wang et al. [21] proposed a dual helical gear 
modification optimization method based on a three-
segment parabolic cutter profile, which significantly 
reduced the vibration and noise of the gear.
In addition to the gear flank modification 
method, optimizing the macro geometric parameters 
and structural parameters of gears, as well as 
improving the manufacturing precision of gears, are 
also common methods for gear vibration and noise 
reduction [22] to [24]. Bozca [25] and [26] minimized 
torsional vibration, transmission error, and gear 
whine noise by optimizing the geometric parameters 
of the gearbox. Tang et al. [27] and [28] proposed 
using the minimization of noise from high-speed 
train traction gear transmissions as the optimization 
objective and obtained the optimal combination of 
retrofit parameters and noise data. Hou et al. [29] 
significantly reduced the dynamic meshing force 
and dynamic response of gears by optimizing the 
method of gear web and tooth ring thickness. Yang 
et al. [30] proposed the parameter optimization of 
weight reduction holes on gear blanks to improve the 
damping characteristics of lightweight gears. Xuet al. 
[31] studied a lightweight, low-amplitude web design 
method to reduce gear weight and lower noise. Qi et 
al. [32] employed panel acoustic contribution analysis 
and the response surface methodology for structural 
acoustic analysis and multi-objective optimization of 
gearboxes, aiming to reduce the radiated noise of the 
gearboxes.
The methods of gear flank modification, 
optimization of geometric and structural parameters 
have been proven to be effective for gear vibration 
and noise reduction. However, with the continuous 
advancement of EV technology, the requirements for 
gear transmission systems have gradually increased, 
and the existing gear vibration and noise reduction 
technologies and methods are gradually encountering 
bottlenecks. The traditional gear flank modification 
method, which processes all gear teeth into the same 
tooth surface topography, sometimes cannot achieve 
the ideal noise reduction effect. Inspired by Cattanei's 
design idea [7] of varying the spacing between blades, 
if the modification parameters of all gear teeth are 
varied according to a certain rule (such as a sine 
curve), creating a differentiated design in the tooth 
surface topography of each gear tooth, it is possible 
to achieve the purpose of gear pair vibration and 
noise reduction by tuning the fluctuations during the 
rotation of the gear teeth, which represents a new 
design approach. 
To validate this innovative design approach, a 
mathematical model of gear tooth profile modification 
varying according to a sine function was established. 
Utilizing Adams software, the dynamic transmission 
characteristics of gear pairs, including centroidal 
angular acceleration, meshing force, and transmission 
error, were analyzed. Comparisons were made among 
unmodified, normally modified, and differentially 
modified gear pairs. Additionally, Simcenter 3D 
software was employed to compare the vibration 
and noise characteristics of these three sets of gear 
pairs. The findings demonstrated that differential 
modification enhances the dynamic transmission 
performance of gears and effectively reduces 
vibration and noise in gear pairs, offering a novel 
method for gear vibration and noise reduction. A 
new gear tooth modification method is proposed, in 
which each gear tooth is modified to different degrees 
and the modification amount of adjacent teeth varies 
according to a sine function. This method improves 
the dynamic transmission performance of gear pairs 
by introducing additional sidebands in the spectrum, 
thereby reducing the amplitude of the dominant gear 
meshing harmonics.

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 569-581
571 Gear Differential Flank Modification Design Method for Low Noise 
1  MATHEMATICAL MODEL OF GEAR TOOTH  
MODIFICATION PROFILE
Tooth profile modification is a primary method to 
avoid meshing interference and impact during tooth 
pair conversion, effectively reducing vibration and 
noise. Fig. 1 illustrates the parameter setting and 
modification effect of tooth profile modification. 
In this study, we employ an arc modification curve 
(radius r) for tooth profile modification, focusing on 
the modification of the gear top as illustrated in Fig. 
1a and b. The modification begins at the intersection 
point C
1
, where the modification start circle (radius 
R
1
) meets the involute gear profile. The initial arc, 
denoted as l
1
 with center O
1
, passes through C
1
 
and reaches the top point C
2
 on the gear profile. By 
rotating arc l
1
 around C
1
 by an angle Δβ, we obtain 
the modified curve l
2
 for the tooth top. The angle Δβ 
represents the extent of modification, and adjusting 
its magnitude allows for controlled modification of 
the tooth top, as shown in Fig. 1c. The method for 
modifying the tooth root is similar, as shown in Fig. 
1d, and will not be elaborated further here.
Existing methods for gear modification typically 
standardize the modification amount or topography of 
all teeth on a gear, resulting in the same modification 
effect on all tooth surfaces. This paper introduces 
a differentiated modification approach, where the 
modification amount for each tooth on a gear varies 
according to a sinusoidal function. This means that the 
topography of each tooth is unique, thereby reducing 
gear vibration and noise. The modification amount for 
each tooth is represented by the following sinusoidal 
function:
 

n
a
z
n







 0
2
2
sin, (1)
where n represents the gear tooth number, λ
(n)
 
represents the modification amount for each tooth, λ
0
 
is the average modification amount across all teeth, a 
is the threshold for modification fluctuation, and z is 
the number of teeth on the gear. 
Due to the variability in the shape of each tooth, 
a separate model is established for each tooth. The 
following details the modeling method for the tooth 
profile curve during top modification. The involute 
parametric equation can be expressed as:
 
yR R
xR R
bb
bb





coss in
sinc os
,
 
 
 (2)
where x and y represent the coordinates of the involute 
in the Cartesian coordinate system, R
b
 is the radius 
of the base circle, and θ is the sum of the involute 
expansion angle and the pressure angle. 
Fig. 1.  a) and b) tooth profile modification, and c) and d) 
coordinate system construction
The coordinates for the starting point of the 
tooth top modification, C
1
(x
1
, y
1
), can be determined 
by solving the equations of the starting modification 
circle and the involute parametric equations. Similarly, 
the coordinates for the intersection point of the initial 
modification curve l
1
 with the tooth top, C
2
(x
2
, y
2
), can 
be obtained by solving the equations for the tooth top 
circle and the involute. The initial modification arc l
1
, 
passing through points C
1
 and C
2
 with the center at 
O
1
, can be described by the following equation:
 
xi yj r
xi yj r
11
2
11
2
2
21
2
21
2
2
    
    





. (3)
Here, r represents the radius of the modification 
arc, with the center O
1
 having coordinates (i
1
, j
1
). A 
local coordinate system, S′{x′ C
1 
y′} is established 
with the modification starting point C
1
 as the origin. 
The transformation equation between the local S′ and 
global coordinates S{x O y} is as follows:
 
xxx
yyy







1
1
. (4)
To transform the center O
1
 from global 
coordinates to local coordinates O
1
 (i
1
′, j
1
′), if the 
angle between O
1
C
1 
and the x′-axis is denoted as β
1
, 
then:
 







ir
jr
11
11
cos
sin
.


 (5)

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 569-581
572 Zhang, Y. – Zhou, H. – Duan, C. – Wang, Z. – Luo, H.
If the angle between O
2
C
2 
and the x′-axis is 
denoted as β
2
, then:
  
21
  . (6)
Here, ∆β is the modification amount, since each 
tooth has a different modification amount, so let 
∆β = λ(n).
The coordinate equation of the center O
2
 of the 
modified arc l
2
 is expressed in the coordinate system 
S′:
 







ir
jr
22
22
cos
sin
.


 (7)
To transform the center O
2 
(i
2
′, j
2
′) from the local 
coordinate system S′ to the global coordinate system S 
as O
2
 (i
2
, j
2
), and subsequently derive the equation of 
the modification arc l
2
 in the global system S, which 
serves as the mathematical model for the tooth top 
modification profile, can be represented as follows:
 xi yj r
nn
    
2
2
2
2
2
. (8)
Here, x
n
∈(x
1
, x
3
), y
n
∈(y
1
, y
3
), C
3
(x
3
, y
3
) is the 
intersection of the modification arc l
2
 with the tooth 
top circle. Collate Eqs. (4) to (7), where i
2
, j
2
 are 
respectively:
 
ir
ix
r
x
jr
ix
r
n 2
1 11
1
2
1 11

 














 




cosc os
sinc os

 



















 

n
y
1
. (9)
Similarly, the mathematical model for the tooth 
root modification curve can be obtained as follows:
       xi yj r
nn 4
2
4
2
2
, (10)
where the coordinate values i
4
, j
4
 of the circle center 
O
4
 are denoted respectively:
 
ir
ix
r
x
jr
ix
r
n 4
1 34
4
4
1 34























cosc os
sinc os


 


















	








n
y
4
. (11)
where x
n
' ∈(x
4
, x
6
), y
n
' ∈(y
4
, y
6
), C
4
(x
4
, y
4
) is the 
starting coordinate for the tooth root modification, 
C
5
(x
5
, y
5
) is the intersection point of the initial 
modification arc l
3
 with the tooth root profile, O
3
(i
3
, 
j
3
) is the center of the initial modification arc l
3
, 
O
4
(i
4
, j
4
) is the center of the modification arc l
4
 after 
a rotation angle λ'
(n)
, where λ'
(n)
 is the amount of tooth 
root modification, and C
6
(x
6
, y
6
) is the intersection 
point of l
4
 with the tooth root profile.
2  DIFFERENTIAL MODIFICATION MODEL  
OF GEAR TEETH WITH SINUSOIDAL LAW
This article uses a pair of standard involute straight-
tooth cylindrical gears as a case study for the 
following modeling and simulation. The gears have 
a module of 2.5 mm, a teeth number of 32:48, a 
pressure angle of 20°, and a tooth face width of 30 
mm. Additionally, the modification arc radius is set to 
r = 5 mm, the starting radius for tooth top modification 
is R
1 
= 42 mm, and the starting radius for tooth root 
modification is R
2 
= 38 mm. The average amount of 
tooth top modification is λ
0 
= 8° with a modification 
differential fluctuation threshold of a = 10°, and the 
average amount of tooth root modification is λ'
0 
= 5° 
with a modification differential fluctuation threshold 
of a' = 6°. The parameters mentioned above were 
obtained after optimization, and the optimization 
model is shown in Fig. 2. To enhance the efficiency 
of modification during manufacturing, differential 
modification is applied only to the pinion. The amount 
of modification for each tooth on the pinion is set 
according to Eq. (1), as shown in Fig. 3, where λ
(1)
 is 
the modification amount for the tooth top of the first 
tooth, λ'
(1)
 is the modification amount for the tooth 
root of the first tooth, λ
(2)
 is for the second tooth's 
top, and λ'
(2)
 for the second tooth's root, and so on. 
The specific modification amounts for each tooth are 
shown in Table 1.
Table 1.  Setting the modification amount for each gear tooth
Gear tooth 
number
Tooth top modification 
amount [°]
Tooth root modification 
amount [°]
λ
(1)
8+5sin(0°)=8.0000 5+3sin(0°)=5.0000
λ
(2)
8+5sin(11.25°)=8.9755 5+3sin(11.25°)=5.5853
λ
(3)
8+5sin(22.5°)=9.9134 5+3sin(22.5°)=6.1481
... ... ...
λ
(30)
8+5sin(326.25°)=5.2221 5+3sin(326.25°)=3.3333
λ
(31)
8+5sin(337.5°)=6.0866 5+3sin(337.5°)=3.8520
λ
(32)
8+5sin(348.75°)=7.0245 5+3sin(348.75°)=4.4147
The radius of the tooth root fillet generated by 
the tool is 0.8 mm. Since the tooth root fillet designed 
here has little influence on the meshing of gear pairs 
and will not interfere with the design of tooth root 
fillet, the existence of tooth root fillet can be ignored 
in the establishment of the dynamic simulation 
model in order to reduce simulation time. All the 
simulation calculations in this paper are carried out 
without setting the tooth root fillet, and the calculation 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 569-581
573 Gear Differential Flank Modification Design Method for Low Noise 
results are only used for comparison and analysis 
with each case. Using the mathematical model of the 
tooth profile curve described above, a differentiated 
tooth profile curve is established in UG software, 
followed by the formation of the tooth surface and 
three-dimensional model. The modeling process is 
illustrated in Fig. 4. 
Fig. 3.  Sinusoidal design method for gear tooth modification 
amounts, a) Gear tooth modification amount arrangements,  
b) Change rule of gear tooth modification amounts 
Fig. 4.  Establishing the 3D model process for gear pairs
3  SIMULATION AND ANALYSIS  
OF GEAR DYNAMICS PERFORMANCE
3.1  Establishment of Gear Dynamics Model
The gear model assembled in UG will be imported 
into Adams software for multibody dynamics 
simulation, the construction of the gear dynamics 
model is shown in Fig. 5. The material and contact 
parameters of the gear pair are set as shown in Table 
2. During the simulation process, the step function is 
used to define the speed of the driving gear (pinion) to 
the corresponding value within 0-0.1 s, and the driven 
gear (gear) is also subjected to a torque load to the 
Fig. 2.  Optimization model for gear tooth modification parameters

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 569-581
574 Zhang, Y. – Zhou, H. – Duan, C. – Wang, Z. – Luo, H.
corresponding value. The values are kept constant for 
solving calculations within 0.1 s to 1 s.
Fig. 5.  Gear dynamics modeling
Table 2.  Parameterization of dynamic model simulation
Parameter Value
Stiffness [N/mm] 8×10
5
Coefficient of static friction 0.3
Power index [-] 1.5
Static translation coefficient [mm/s] 100
Damping [N·s/mm] 10
Coefficient of kinetic friction [-] 0.1
Depth of penetration [mm] 0.1
Friction translation coefficient [mm/s] 1000
To verify and analyze the transmission 
performance of the differentiated profile modification 
gear pair, four pairs of gears are designed with the 
same geometric parameters. The first pair of gears does 
not undergo tooth profile modification, the second and 
the third pairs of gears adopts the existing common 
gear modification method, and the same amount of 
modification is applied to the tooth flanks of both 
the pinion and gear. The fourth pair of gears adopt 
the design method of differentiated modification, 
and their modification parameters are shown in 
Table 3. The differentiated modification parameters 
of the fourth pair of gears were obtained through the 
optimization model shown in Fig. 2. The modification 
method is the same for Case 2 and Case 3. The tooth 
top in Case 2 is modified by 8° and the tooth root by 
5°, while the tooth top in Case 3 is modified by 13° 
and the tooth root by 8°, which is the same amount 
of modification for each tooth in these two sets of 
cases. The dynamic meshing performance of the gear 
pair during transmission is analyzed by simulating the 
centroid angle acceleration of the driven gear and the 
meshing force of the gear pair under the conditions 
of the driven gear load being 100 N·m and 350 N·m 
respectively, and the driving gear speed is 3000 rpm, 
5000 rpm, and 8000 rpm, respectively. 
Table 3.  Pinion parameters for tooth profile modification
Mean value  
of modification amount [°]
Fluctuation threshold  
of modification amount [°]
Top Root Top Root
Case 1 0 0 0 0
Case 2 8 5 0 0
Case 3 13 8 0 0
Case 4 8 5 10 6
3.2  Comparative Analysis of Gear Angular Acceleration
The angular acceleration of the driven gear's centroid 
was measured over time using simulation, and its 
frequency domain representation was obtained 
through Fourier transformation. Due to space 
limitations, only the frequency domain graphs are 
shown here, and subsequent analyses will follow this 
format.
From the comparative analysis of gear angular 
acceleration in Figs. 6 to 11:
1. The theoretical meshing frequencies at known 
speeds of 3000 rpm, 5000 rpm, and 8000 rpm are 
respectively 1600 Hz, 2666.67 Hz, and 4266.67 
Hz. The meshing frequencies in all spectral 
graphs correspond to the theoretical values, 
confirming the accuracy of the dynamic model. 
Fig. 6.  Comparison of angular acceleration spectrograms  
at 3000 rpm and 100 N·m
2. Under the same speed, the amplitude of angular 
acceleration at 1 to 3 times the meshing frequency 
increases with higher loads, under the same load, 
the amplitude generally tends to increase with 
higher speeds.
3.  Across all conditions, the amplitude of angular 
acceleration at the fundamental meshing 
frequency is highest for Case 1, followed by 
Case 2 and Case 3, and lowest for Case 4. This 
indicates that differentiated modification results 
in lower angular accelerations compared to 
traditional modified and unmodified gears, with 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 569-581
575 Gear Differential Flank Modification Design Method for Low Noise 
the greatest reduction at 5000 rpm of 100 Nm, 
where Case 4 is reduced by 27.74 % compared to 
Case 1, 9.28 % compared to Case 2 and 10.75 % 
compared to Case 3. 
Fig. 7.  Comparison of angular acceleration spectrograms  
at 3000 rpm and 350 N·m]
Fig. 8.  Comparison of angular acceleration spectrograms  
at 5000 rpm and 100 N·m
Fig. 9.  Comparison of angular acceleration spectrograms  
at 5000 rpm and 350 N·m
4. At twice and three times the meshing frequency, 
Case 1 also shows the highest amplitude, followed 
by Case 2 and Case 3, and Case 4 the lowest. 
At twice the meshing frequency, the maximum 
reductions for Case 4 compared to Cases 1 to 3 
are 60.43 %, 10.66 % and 4.35 % respectively, 
under conditions of 5000 rpm of 100 Nm, 3000 
rpm of 350 Nm and 8000 rpm of 100 Nm. At 
three times the meshing frequency, the maximum 
reductions for Case 4 compared to Cases 1 to 3 
are 57.6 %, 7.08 % and 3.21 %, respectively.
5. Compared to gears modified by traditional 
methods, the amplitude of angular acceleration of 
the driven gear’s centroid is reduced, indicating 
that differentiated modification methods are 
beneficial in reducing vibration and noise in gear 
pairs.
6. Compared to Cases 1 to 3, Case 4 shows varying 
degrees of increase in the amplitude of sidebands 
adjacent to the meshing frequency peaks, proving 
the correctness of the method proposed in this 
paper, which involves introducing additional 
sidebands to reduce the amplitude of gear 
meshing harmonics.
Fig. 10.  Comparison of angular acceleration spectrograms at 
8000 rpm and 100 N·m
Fig. 11.  Comparison of angular acceleration spectrograms at 
8000 rpm and 350 N·m
7. To ensure the accuracy and reliability of the 
results, angular acceleration data from 80 
different operational conditions were analyzed, 
with sampling conducted across a range of speeds 
from 1000 rpm to 10000 rpm and loads from 50 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 569-581
576 Zhang, Y. – Zhou, H. – Duan, C. – Wang, Z. – Luo, H.
4. Compared to traditionally modified gears, the 
amplitude of the meshing force of the driven gear 
is reduced in Case 4, indicating that differential 
modification methods are beneficial in reducing 
vibration and noise in gear pairs.
5. Compared to Cases 1 to 3, Case 4 also shows 
varying degrees of increase in the amplitude 
of sidebands around the peaks of meshing 
harmonics, which is beneficial for reducing the 
amplitude at these harmonic frequencies.
Fig. 12.  Comparison of engagement force spectrograms  
at 3000 rpm and 100 N·m
Fig. 13.  Comparison of engagement force spectrograms  
at 3000 rpm and 350 N·m
Fig. 14.  Comparison of engagement force spectrograms  
at 5000 rpm and 100 N·m
N·m to 400 N·m. A detailed comparison revealed 
that in these 80 conditions, compared to Case 1, 
Case 4 demonstrated a reduction in the amplitude 
of the first three meshing frequencies in 79 cases, 
achieving an effectiveness rate of 98.75 %. 
When compared to Case 2, the effectiveness rate 
reached 71.25 %. This data robustly supports the 
superior dynamic performance of differentially 
modified gear teeth over conventionally modified 
gear teeth. 
3.3  Comparative Analysis of Engagement Force
Through simulation measurements, the resultant 
forces in the x and y directions during the meshing of 
the driving and driven gears were plotted over time, 
and their frequency domain graphs were obtained 
through Fourier transformation. At the initial stage of 
gear meshing, due to the impact between the teeth, a 
large meshing force is generated. However, this stage 
has a minor impact on the overall analysis and can be 
disregarded .
From the comparative analysis of gear meshing 
forces in Fig. 12 to 17:
1. At the same speed, the amplitude of the meshing 
force at harmonic frequencies increases with 
higher loads, at the same load, the amplitude 
of the meshing force at harmonic frequencies 
generally tends to increase with higher speeds.
2. Across all conditions, the amplitude of the 
meshing force at the fundamental meshing 
frequency is highest for Case 1, followed by 
Case 2 and Case 3, and lowest for Case 4. This 
indicates that the meshing force for differentially 
modified gears is lower than that for traditionally 
modified and unmodified gears. At the condition 
of 5000 rpm and 100 N·m, the reduction reaches 
its maximum value, with Case 4 showing a 
decrease of 27.92 % compared to Case 1, 9.14 
% compared to Case 2 and 10.37 % compared to 
Case 3. 
3. The same trend is observed at the second and 
third harmonic frequencies, where Case 1 has 
the highest amplitude, followed by Case 2 and 
Case 3, and Case 4 has the lowest. At the second 
harmonic frequency, Case 4 shows the maximum 
reduction compared to Cases 1-3 at conditions of 
5000 rpm of 100 N·m, 3000 rpm of 350 Nm and 
8000 rpm of 100 Nm, with maximum reductions 
of 59.24 %, 10.43 % and 4.63 %, respectively. 
At the third harmonic frequency, the maximum 
reductions for Case 4 compared to Cases 1 to 3 
are 80.38 %, 6.74 % and 9.07 %, respectively.

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 569-581
577 Gear Differential Flank Modification Design Method for Low Noise 
Fig. 15.  Comparison of engagement force spectrograms  
at 5000 rpm and 350 N·m
Fig. 16.  Comparison of engagement force spectrograms  
at 8000 rpm and 100 N·m
Fig. 17.  Comparison of engagement force spectrograms  
at 8000 rpm and 350 N·m
3.4  Comparative Analysis of Gear Transmission Errors
Through gear dynamics simulation, transmission 
error curves of gear pairs can be obtained, because 
transmission error is independent of gear speed, 
and since the dynamics model uses rigid bodies for 
simulation, transmission error is also unaffected by 
load. It only analyzes the transmission error under 
one operating condition (5000 rpm and 100 N·m) 
for comparison, as shown in Fig. 18. Fast Fourier 
Transform (FFT) of these transmission errors yields 
the order spectra, as illustrated in Fig. 19. From the 
comparative analysis of the gear transmission errors in 
Fig. 18 and 19, it can be observed that:
1. The driven gear exhibits 48 cycles of fluctuation 
per revolution, aligning with the number of 
teeth on the driven gear. Unlike Case 1, Case 2 
and Case 3, Case 4's transmission error curve is 
not stable but matches the pattern of a complete 
sinusoidal cycle. This phenomenon confirms the 
accuracy of the differentiated gear modification 
model. 
Fig. 18.  Comparison of transmission error time domain diagrams
2. The opening sizes of the transmission error 
curves in all Cases are comparable, and the 
periodic patterns are consistent. By comparing 
the amplitude of fluctuations in the transmission 
error curves, Case 1 has the largest, followed by 
Case 2 and Cases 3, and Case 4 has the smallest, 
indicating that the meshing in Case 4 is the most 
stable. 
3. The sum of the amplitudes of the first four 
orders of transmission errors is the largest in 
Case 1, followed by Case 2 and Cases 3, and the 
smallest in Case 4. The sum of the amplitudes 
of the first four orders of transmission errors 
effectively reflects the quality of the gear profile 
modification [33]. Case 4 shows a reduction of 
38.89 % compared to Case 1, 15.79 % compared 
to Case 2 and 25.74 % compared to Case 3. This 
demonstrates that differentiated modification 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 569-581
578 Zhang, Y. – Zhou, H. – Duan, C. – Wang, Z. – Luo, H.
provides smoother transmission compared to 
traditional modification methods and unmodified 
gears.
Fig. 19.  Comparison of transmission error spectrum diagrams
4  SIMULATION AND ANALYSIS OF GEARBOX VIBRATION  
AND RADIATED NOISE
4.1  Establishment of Gearbox Finite Element Model
The excitations generated during the gear meshing 
process are transmitted through the gear shafts to the 
gearbox, causing vibrations and noise in the gearbox. 
Therefore, the following will simulate the noise of the 
gear pair based on the finite element method. In the 
pre-processing module, the gearbox is meshed, and 
its material is specified as gray cast iron. In Adams 
software, the support reaction data in the x and y 
directions of the gear pair's shafts are extracted, these 
support reaction data are used as the excitation source 
for vibration response analysis of the gearbox, and the 
radiated noise of the gearbox are solved. In Simcenter 
3D software, corresponding models for acoustic 
analysis of the gearbox are established. The thickness 
of the gearbox is 8 mm, the radius of the sound field 
is 1 m. 
Table 4.  Partial natural frequency of gearbox
Spectral  
order
Natural 
frequency [Hz]
Spectral  
order
Natural 
frequency [Hz]
6 2008 42 6393
15 3186 44 6525
29 5074 47 6977
30 5190 50 7335
55 6051 62 8035
Since this article only compares the impact 
of different cases on the vibration and noise of the 
gearbox, to prevent other factors from affecting the 
comparison results, the entire bottom of the gearbox 
is securely constrained. Modal analysis is performed 
on the gearbox to solve the first 100 modes of the 
gearbox and obtain its first 100 natural frequencies. 
As shown in Table 4, due to space limitations, only 
the natural frequencies of the gearbox that have a 
significant impact on subsequent simulation analysis 
are listed.
4.2  Comparative Analysis of Gearbox Sound Power Levels
From the comparative analysis of the sound power 
levels of the gearbox casing shown in Figs . 20 to 25, 
the following can be observed:
1. At the same rotational speed, the sound power 
level amplitudes at the first three meshing 
frequencies increase with higher loads. Due to 
the influence of the casing's natural frequencies, 
resonance occurs at frequencies close to the 
natural frequencies of the casing, resulting in 
larger amplitudes near these natural frequencies. 
For example, near the frequency of 2008 Hz, 
an increase in amplitude is observed in all 
conditions, especially at a rotational speed of 
5000 rpm. Increased amplitudes are also seen 
between frequencies of 6000 Hz to 8000 Hz. 
Additionally, at a rotational speed of 3000 rpm, 
the amplitude at twice the meshing frequency 
is higher than at the meshing frequency, and at 
rotational speeds of 5000 rpm and 8000 rpm, the 
amplitude at three times the meshing frequency is 
higher than at twice the meshing frequency due to 
this reason. 
Fig. 20.  Comparison of sound power level spectrograms  
at 3000 rpm and 100 N·m
2. Under all conditions, the sound power level 
amplitude at the meshing frequency is the highest 
for Case 1, followed by Case 2 and Case 3, and 
the lowest for Case 4. The maximum reduction 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 569-581
579 Gear Differential Flank Modification Design Method for Low Noise 
is observed at conditions of 3000 rpm of 350 
Nm and 5000 rpm of 350 Nm, where Case 4 is 
reduced by 3.43 % compared to Case 1, 2.43 % 
compared to Case 2 and 2.47 % compared to 
Case 3. 
Fig. 21.  Comparison of sound power level spectrograms  
at 3000 rpm and 350 N·m
Fig. 22.  Comparison of sound power level spectrograms  
at 5000 rpm and 100 N·m
Fig. 23.  Comparison of sound power level spectrograms  
at 5000 rpm and 350 N·m
3. At twice and three times the meshing frequency, 
Case 1 also shows the highest amplitude, followed 
by Case 2 and Case 3, and the lowest for Case 4. 
At twice the meshing frequency, the maximum 
reduction for Case 4 compared to Cases 1-3 
occurs at conditions of 5000 rpm of 100 Nm and 
3000 rpm of 350 Nm, with maximum reductions 
of 7.43 %, 2.49 % and 1.85 %, respectively. At 
three times the meshing frequency, the maximum 
reductions for Case 4 compared to Cases 1 to 3 
are 14.7 %, 2.68 % and 3.91 %, respectively.
Fig. 23.  Comparison of sound power level spectrograms  
at 5000 rpm and 350 N·m
Fig. 24.  Comparison of sound power level spectrograms  
at 8000 rpm and 100 N·m
4. All of these laws are similar to the laws of 
vibration displacement, illustrating the close 
correlation between vibration and noise. 
Compared to gears modified by traditional 
methods, the sound power level amplitudes 
of the driven gear are reduced, indicating that 
differentiated modification methods are beneficial 
in reducing the noise of gear pairs.

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)11-12, 569-581
580 Zhang, Y. – Zhou, H. – Duan, C. – Wang, Z. – Luo, H.
Fig. 25.  Comparison of sound power level spectrograms  
at 8000 rpm and 350 N·m
5  CONCLUSIONS
This paper discusses the impact of using a 
differentiated tooth modification method on the 
dynamic meshing performance of gear pairs. It 
compares this method with normal tooth modification 
gears and unmodified gears, arriving at the following 
main conclusions: 
1. A differentiated tooth modification method is 
proposed, where the modification amount of 
adjacent teeth varies according to a sine function, 
and a mathematical model of the gear tooth 
profile is established. 
2. Using transmission error, centroid angular 
acceleration, and meshing force to characterize 
the dynamic meshing performance of gear pairs, a 
comparative analysis is conducted between gears 
using the differentiated modification method, 
normally modified gears, and unmodified gears. 
The results show that differentiated modification 
can improve the dynamic meshing performance of 
gear pairs. Compared to normally modified gears, 
the sum of the amplitudes of the first four orders 
of transmission error is reduced by up to 25.74 
%, the amplitude of the angular acceleration at 
the meshing frequency is reduced by up to 10.75 
%, and the amplitude of the meshing force at the 
meshing frequency is reduced by up to 10.37 %. 
3. A comparative analysis of the noise conditions 
of three pairs of gears shows that, compared to 
normally modified gears, the sound power level 
amplitude at the meshing frequency is reduced by 
up to 2.47 %. 
4. Eighty sets of conditions were sampled for 
multibody dynamics analysis, and the simulation 
results show that gears with differentiated 
modification performed better in dynamic 
meshing performance in over 70 % of the 
conditions compared to normally modified 
gears, fully validating the effectiveness of the 
differentiated modification method in reducing 
vibration and noise.
6  ACKNOWLEDGMENTS
The research was supported by the Natural Science 
Foundation of Hunan Province (2024JJ5643 
and 2022JJ40876), the National Natural Science 
Foundation of China (U22B2084), the Hunan 
Provincial Key R&D Program (2023GK2053).
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