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Optimization analysis of vibration reduction for aeroengine multistage blade-disk system

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In order to reduce the vibration localization of multistage blade-disk system of aeroengine compressor, the integral model and substructure model of multistage blade-disk system are established by using substructure modal synthesis method, and the dynamic characteristics of two models are analyzed and the accuracy of the substructure model is verified by comparison. Based on the substructure model, two optimization algorithms of the random wear mass and the snake optimization are proposed. The results show that the vibration amplitudes of the two models are in good agreement in the first 200 modes, and the substructure model meets the requirement of analysis accuracy. Both optimization algorithms can effectively reduce the degree of the vibration localization of the mistuned blade-disk system. The random wear mass algorithm can quickly reduce the amplitude of the blade. The snake optimization algorithm can find the global optimal solution with enough iterations. The results show that the maximum vibration amplitude and the vibration localization factor after optimization are reduced by 5.72 % and 12.0 % respectively by using the random wear mass algorithm, and that of 9.13 % and 16.7 % respectively by using the snake optimization algorithm. The analysis method presented in this paper takes into account the analysis efficiency and analysis precision, it can be used for dynamic analysis and vibration reduction optimization of large power machinery such as aeroengine and gas turbine, and provides a basis for further improving the reliability and service life of aircraft.
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JOURNAL OF VIBROENGINEERING. NOVEMBER 2024, VOLUME 26, ISSUE 7
1547
Optimization analysis of vibration reduction for
aeroengine multistage blade-disk system
Xiaomei You
1
, Haiyu Feng
2
, Hongyuan Zhang
3
, Pengbo Liu
4
1, 2, 4
School of Mechanical Engineering, Shenyang Ligong University, Shenyang, China
3
School of Automotive and Transportation, Shenyang Ligong University, Shenyang, China
1
Corresponding author
E-mail:
1
prime_queen@163.com,
2
haiyu_feng@163.com,
3
zhy_sylu@163.com,
4
193143110@qq.com
R
eceived 20 February 2024; accepted 11 September 2024; published online 25 September 2024
D
OI htt
p
s://doi.or
g
/10.21595/
j
ve.2024.24016
Copyright © 2024 Xiaomei You, et al. This is an open access article distributed under the Crea tive Commons Attribution License, which
permits unrestricted use, distribution, an d reproduction in any medium, provided the original work is properly cited.
Abstract. In order to reduce the vibration localization of multistage blade-disk system of
aeroengine compressor, the integral model and substructure model of multistage blade-disk
system are established by using substructure modal synthesis method, and the dynamic
characteristics of two models are analyzed and the accuracy of the substructure model is verified
by comparison. Based on the substructure model, two optimization algorithms of the random wear
mass and the snake optimization are proposed. The results show that the vibration amplitudes of
the two models are in good agreement in the first 200 modes, and the substructure model meets
the requirement of analysis accuracy. Both optimization algorithms can effectively reduce the
degree of the vibration localization of the mistuned blade-disk system. The random wear mass
algorithm can quickly reduce the amplitude of the blade. The snake optimization algorithm can
find the global optimal solution with enough iterations. The results show that the maximum
vibration amplitude and the vibration localization factor after optimization are reduced by 5.72 %
and 12.0 % respectively by using the random wear mass algorithm, and that of 9.13 % and 16.7 %
respectively by using the snake optimization algorithm. The analysis method presented in this
paper takes into account the analysis efficiency and analysis precision, it can be used for dynamic
analysis and vibration reduction optimization of large power machinery such as aeroengine and
gas turbine, and provides a basis for further improving the reliability and service life of aircraft.
Keywords: multistage blade-disk system, mistuned, substructure modal synthesis method,
vibration characteristics, vibration reduction optimization, aeroengine.
1. Introduction
As the heart of aircraft, the performance of the aeroengine is directly related to the performance
and safety of the aircraft [1]. The blade-disk system of compressor is a key component of
aeroengine and represents the development direction of high-performance aeroengine [2]. The
blade-disk system is composed of multistage disk, its working condition is complex and harsh,
and the coupling effect between each level of disk cannot be ignored. Therefore, it is necessary to
carry out the vibration optimization research on the blade-disk system.
In recent years, many scholars have carried out a lot of research on the vibration reduction
optimization of the mistuned blade-disk system. Yu Xueran et al. [3] established a parametric
mathematical model of the integral blade-disk system, analyzed the influence of adjusting blade
and wheel parameters on the inherent characteristics of the blade-disk system in the condition that
the design stress specifications were met, and found that the integral blade-disk system could
realize the integrated design of structural strength and vibration reduction. Lu Shan et al. [4-5]
proposed and created the mathematical model and calculation method for optimal design of
double-spoke plate structure of turbine disc. Gao Feng [6] adopted the multi-objective
optimization design method to carry out multi-objective optimization of the integral blade-disk
system with strain-dependent rigid coating, and concluded that the influence of hard coating
parameters on the vibration performance of the integral disk is mutually restricted. Fan Zhiqiang
et al. [7] improved the integral blade-disk system by optimizing and improving the performance
OPTIMIZATION ANALYSIS OF VIBRATION REDUCTION FOR AEROENGINE MULTISTAGE BLADE-DISK SYSTEM.
XIAOMEI YOU, HAIYU FENG, HONGYUAN ZHANG, PENGBO LIU
1548 ISSN PRINT 1392-8716, ISSN ONLINE 2538-8460
of the integral blade-disk system and making the strain energy distribution of the blade disk more
uniform. Dai Yiping et al. [8] established an optimization model of the mistuned blade-disk system
by using genetic algorithm, which reduces the amplitude of the system while considering dynamic
balance. The algorithm is superior to the simulated annealing method in terms of computation
time and convergence. Liu Tiejian and Zhang Liang et al. [9-10] used ant colony algorithm and
improved particle swarm optimization algorithm to optimize blade ordering, which can
significantly reduce the forced vibration amplitude of the mistuned blade-disk system and greatly
improve the vibration localization. Jing Huizhe [11] used the radial base value response surface
to replace the finite element analysis process of the vibration response of the blade disk, and used
the genetic algorithm to optimize and solve the optimal design model of the active mistuned blade
disk, and completed the vibration optimization design of the active mistuned blade disk. Zhao
Tianyu [12] used tabu genetic cat swarm algorithm to optimize the blade sequence. After
optimization, the maximum amplitude of each blade was relatively concentrated compared with
that before optimization, which was closer to the upper and lower limits of the vibration amplitude
of the harmonic system, and the degree of localization of the system vibration was reduced. Choi
et al. [13] studied blade ordering with a heuristic algorithm. Yuan Huiqun [14] used discrete
particle swarm optimization algorithm to sort the mistuned blades to achieve the purpose of
vibration reduction optimization of the mistuned blades. Thompson et al. [15] used simulated
annealing method to optimize the arrangement of the blade-disk system. Bisegna et al. [16]
optimized the design of a new passive vibration damping blade-disk system. Li Lin et al. [17-18]
studied the vibration localization suppression problem of the piezoelectric network mistuned
blade-disk system by using the lumped parameter model. Wei Wuguo et al. [19] took the basic
structure of the vane, disc and drum of the compressor rotor as the research object, optimized
variables such as shape and size, and obtained better optimization results without changing the
reliability of the structure. You Xiaomei et al. [20] established a three-stage blade-disk lumped
parameter model with interstage coupling, and used Monte Carlo sampling technique to simulate
the random mistuning mode of the stiffness of the blade-disk system. The vibration response of
the three-stage blade-disk system in the harmonic and mistuning state is obtained by analyzing the
vibration response of each vibration mode. Yang Zhengxin et al. [21] applied the concentrated
mass method to analyze the vibration response characteristics of the integral blade disk with
different types of blade mass mistuning, and obtained their amplitude-frequency characteristic
curves respectively. By comparing the amplitude-frequency response characteristics analyzed by
the finite element method, the influence law of blade mass mistuning on the vibration
characteristics of the system was summarized.
In the above researches on vibration reduction optimization of mistuned blade-disk system,
most of them are single-stage blade-disk system or two-stage blade-disk system as the research
object, and there are few researches on optimization algorithm of multistage blade-disk system,
which has certain errors with the real working condition of the blade-disk system, and often leads
to insufficient calculation accuracy. In this paper, the mistuned three-stage blade-disk system is
studied, the finite element reduction model of the mistuned multistage blade-disk system is
established, and the vibration reduction optimization algorithm is established to analyze the
vibration characteristics of the blade mass mistuned. By slightly changing the mass of the blade,
the optimal scheme of the mistuned blade-disk system is obtained without affecting the
performance of the blade-disk system. The research method in this paper not only guarantees the
analysis accuracy of multistage blade-disk system, but also improves the calculation efficiency
effectively.
2. Establishment of finite element reduction model of multistage blade-disk system
2.1. Substructure modal synthesis method
For the large and complex structure of aeroengine blade-disk system, no matter which analysis
OPTIMIZATION ANALYSIS OF VIBRATION REDUCTION FOR AEROENGINE MULTISTAGE BLADE-DISK SYSTEM.
XIAOMEI YOU, HAIYU FENG, HONGYUAN ZHANG, PENGBO LIU
JOURNAL OF VIBROENGINEERING. NOVEMBER 2024, VOLUME 26, ISSUE 7 1549
method is adopted, the calculation workload is very large. Substructure modal synthesis method
is a technique for reducing large scale models to different levels of submodels. For system models,
substructure models appear as individual units in the model. Through this reduction technique, the
scale of the analytical model can be greatly reduced. In this method, the interface degrees of
freedom connected between each substructure are defined as the main degrees of freedom, and it
is completely constrained, thus the lower order master mode set 𝜑 can be obtained, and then the
degree of freedom can be released to obtain the constrained mode set 𝜑, 𝜑 and 𝜑,which
constitutes the Ritz basis of the substructure. Based on the substructure modal synthesis method,
the substructure reduction model of the blade-disk system of an aeroengine compressor is
established by using finite element software.
For the finite element reduction model considering substructure, the general forced vibration
equation is shown in Eq. (1):
𝑀𝑢
󰇘
+𝐶𝑢󰇗+𝐾𝑢=𝐹, (1)
where 𝑀, 𝐶 and 𝐾 are the mass matrix, damping matrix and stiffness matrix of the substructure
model respectively. 𝐹 is the force vector applied to the substructure model, 𝐹=󰇣𝑓
0󰇤; 𝑢 is the
displacement vector of the model, 𝑢=󰇣𝑢
𝑢󰇤.
When the internal degree of freedom of the substructure model is 𝑢 and the boundary
retention degree of freedom is 𝑢, the mass matrix and stiffness matrix of the substructure model
are respectively:
𝑀=𝑀 𝑀
𝑀 𝑀, 𝐶=𝐶 𝐶
𝐶
𝐶

, 𝐹=𝐹 𝐹
𝐹 𝐹. (2)
Substitute 𝑀, 𝐶 and 𝐾 in Eq. (2) into Eq. (1) and further expand as shown in Eq. (3):
𝑀 𝑀
𝑀 𝑀𝑢
󰇘
𝑢
󰇘
+𝐶 𝐶
𝐶 𝐶𝑢󰇗
𝑢󰇗
+𝐾 𝐾
𝐾 𝐾󰇣𝑢
𝑢󰇤=󰇣
0󰇤, (3)
where, 𝑓 is the node force acting on the retained node of the substructure model, when the retained
degree of freedom of the substructure model is constrained, that is, 𝑢 = 0, Eq. (4) can be obtained:
𝑀𝑢
󰇘
+𝐶𝑢󰇗+𝐾𝑢=0. (4)
According to Eq. (4), the regularization mode 𝜙 of the substructure model can be solved,
which is composed of lower-order mode 𝜙 and higher-order mode 𝜙, that is, 𝜙=[𝜙,𝜙].
In addition, the regularized mode set is satisfied:
𝜙
𝑀𝜙=𝐸, 𝜙
𝐾𝜙=Λ=Λ
Λ, (5)
where: Λ is the low-order eigenvalue, Λ=diag(𝜆
,𝜆
,…,𝜆); Λis the higher-order
eigenvalue, Λ=diag(𝜆
,𝜆
,…,𝜆
); 𝐸 is the identity matrix.
The lower order modes are taken as the main index to analyze the inherent characteristics of
the blade disk system, so the higher order mode set is ignored. In this case, the lower-order mode
set of the system is selected as the primary mode set 𝜙, that is, 𝜙=𝜙.
Under the condition that only the stiffness of the multistage blade disk system and the force
on the interface degree of freedom are considered, the dynamic expression can be transformed
from Eq. (3) to Eq. (6):
OPTIMIZATION ANALYSIS OF VIBRATION REDUCTION FOR AEROENGINE MULTISTAGE BLADE-DISK SYSTEM.
XIAOMEI YOU, HAIYU FENG, HONGYUAN ZHANG, PENGBO LIU
1550 ISSN PRINT 1392-8716, ISSN ONLINE 2538-8460
𝐾 𝐾
𝐾
𝐾

󰇣𝑢
𝑢󰇤=󰇣
0󰇤. (6)
Eq. (7) can be obtained by transforming Eq. (6):
𝐾𝑢+𝐾𝑢=0. (7)
Eq. (8) can be obtained after the transfer of Eq. (7):
𝑢=−𝐾
𝐾𝑢. (8)
Suppose that the displacement vector 𝑢 at the retention of degrees of freedom of the
substructure system is the identity matrix 𝐸, and the internal degrees of freedom 𝑢 is 𝜙, the
Eq. (9) can be obtained by substituting it into Eq. (8):
𝜙=−𝐾
𝐾𝐸=−𝐾
𝐾. (9)
Let 𝜙 be the constrained mode set of the system, then the constrained mode set is:
𝜙=𝐸
𝜙=𝐸
−𝐾
𝐾. (10)
The Ritz base vector of the 𝑖-th substructure is obtained by the above derivation:
𝜙=[𝜙 𝜙]=0𝐸
𝜙𝜙
. (11)
According to Eq. (12), the transformation of the system from physical coordinates to modal
coordinates can be realized:
𝑢=𝜙𝑝. (12)
Eq. (11) is obtained by coordinate transformation:
󰇣𝑢
𝑢󰇤=0𝐸
𝜙𝜙󰇣𝑝
𝑝󰇤. (13)
Through the above coordinate transformation, the equation of motion of the system in modal
coordinates can be obtained as:
𝑀
𝑢
󰇘
+𝐶
𝑢󰇗+𝐾
𝑢=𝐹, (14)
where: 𝑀
– mass matrix in modal coordinate, 𝐶 – damping matrix in modal coordinate, 𝐾
stiffness matrix in modal coordinate, 𝐹 – force vector in modal coordinate.
Eq. (14) only considers the low-order mode, and the higher-order mode is ignored, that is, the
mode truncation, greatly reduces the scale of node degrees of freedom. If only two substructures
are considered, the Eq. (15) can be derived:
𝑀
0
0𝑀
𝑝
󰇘
𝑝
󰇘
+𝐶
0
0𝐶
𝑝󰇗
𝑝󰇗+𝐾
0
0𝐾
󰇣𝑝
𝑝󰇤=𝐹
𝐹. (15)
When the interface is rigidly connected, Eq. (16) and Eq. (17) can be obtained according to
the displace-harmonic condition and the force balance of the two substructures:
OPTIMIZATION ANALYSIS OF VIBRATION REDUCTION FOR AEROENGINE MULTISTAGE BLADE-DISK SYSTEM.
XIAOMEI YOU, HAIYU FENG, HONGYUAN ZHANG, PENGBO LIU
JOURNAL OF VIBROENGINEERING. NOVEMBER 2024, VOLUME 26, ISSUE 7
1551
𝑢

=𝑢

, (16)
𝑓

+

=0. (17)
The displacement vector 𝑝 in modal coordinates is transformed into the displacement vector 𝑝
in generalized coordinates, and the expression is shown in Eq. (18):
𝑝=𝛽𝑞, (18)
where: 𝛽 – coordinate transformation matrix, 𝑝 – displacement vector in modal coordinates, 𝑞
displacement vector in generalized coordinates.
The equation of motion in generalized coordinates is obtained by the above equation, and its
expression is shown in Eq. (19):
𝑀
𝑢󰇘+𝐶󰆻𝑢󰇗+𝐾
𝑢=𝐹, (19)
where: 𝑀
– mass matrix in generalized coordinates, 𝐶󰆻 – damping matrix in generalized
coordinates, 𝐾
– stiffness matrix in generalized coordinates, 𝐹 – force vector in generalized
coordinates.
Based on the substructure modal synthesis method, the substructure reduction model of the
blade-disk system of an aeroengine compressor is established by using finite element software and
the analysis accuracy is verified.
2.2. Establishment of substructure reduction model
The blade-disk system of an aeroengine compressor is composed of multistage blade disk. The
coupling characteristics between different levels of blade disks are considered comprehensively,
a three-dimensional solid model of the multistage blade-disk system is established. The structure
of the first three stage is shown in Fig. 1, of which the first blade disk is composed of 38 blades
and roulette, the second blade disk is composed of 53 blades and roulette, and the third blade disk
is composed of 60 blades and roulette. The multistage blade-disk system is divided into
substructures using the substructure modal synthesis method. The substructure division of the
solid model of the multistage blade-disk system is shown in Fig. 2.
Fig. 1. Solid model of multistage bladed-disk system
The material properties of the multistage blade-disk system are shown in Table 1. Each stage
blade disk is cut into corresponding blades and roulette disks. Because of the complex structure
of multistage blade-disk system, the calculation amount of finite element mesh is very large if the
higher-order element is selected. Therefore, solid185 and solid187 are used in this paper to divide
the grid of multistage blade-disk system. The meshing of the first, second and third stage blades
is dominated by solid185 hexahedral elements, and the rest of the meshing is dominated by
solid187 tetrahedral elements. The number of elements and nodes on each level of the blade disk
is shown as Table 2. The substructure analysis model of the multistage blade-disk system is shown
in Fig. 3.
OPTIMIZATION ANALYSIS OF VIBRATION REDUCTION FOR AEROENGINE MULTISTAGE BLADE-DISK SYSTEM.
XIAOMEI YOU, HAIYU FENG, HONGYUAN ZHANG, PENGBO LIU
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ISSN PRINT 1392-8716, ISSN ONLINE 2538-8460
Fig. 2. Substructure solid model of multistage bladed-disk system
Table 1. Material properties of multistage blade-disk system
Material property Blade propert
y
Roulette property
Densit
y
4400 k
g
/m
3
8200 k
g
/m
3
Elasticity modulus 1.14 GPa 1.66 GPa
Poisson's ratio 0.3 0.3
Table 2. Number of nodes and elements in a blade-disk system
Number of nodes Number of units
Primar
y
dis
k
13225 32314
Secondary dis
k
15276 34551
Tertiar
y
dis
k
37633 138876
Fig. 3. Substructure analysis model of multi-stage blade-disk system
The blade of each stage bladed disk and disk connected with it are divided into one
substructure. The first stage blade disk is divided into 38 substructures, the second stage blade
disk is divided into 52 substructures, and the third stage blade disk is divided into 60 substructures.
The finite element model of the three-stage blade-disk system after the substructure integrated
condensed super element is shown in Fig. 4.
OPTIMIZATION ANALYSIS OF VIBRATION REDUCTION FOR AEROENGINE MULTISTAGE BLADE-DISK SYSTEM.
XIAOMEI YOU, HAIYU FENG, HONGYUAN ZHANG, PENGBO LIU
JOURNAL OF VIBROENGINEERING. NOVEMBER 2024, VOLUME 26, ISSUE 7
1553
a
)
Su
p
er element
b)
Node
Fig. 4. Finite element model of multistage blade-disks after condensing super element
2.3. Precision verification of substructure reduction model
The blade-disk system of the aeroengine compressor is a high-speed rotating power system.
Whether the rotational speed is considered in the kinetic analysis will affect the analysis accuracy.
Fig. 5 shows the calculation and comparison of natural frequencies of the integral blade-disk
model and the substructure model considering the rotational speed.
As can be seen from Fig. 5, the substructure model is in good agreement with the calculation
results of the overall blade-disk model within the 200th order, indicating that the substructure
model still has a high calculation accuracy of the lower-order modes, and can meet the calculation
requirements under the premise of ignoring the higher-order modes, reducing the number of
degrees of freedom of the model and improving the calculation speed.
Fig. 5. Comparison of dynamic frequency calculation results between
the integral blade disk model and the substructure model
In addition, according to the natural frequency distribution of the substructure model, the first
200 modes can be divided into many approximate horizontal regions of “frequency-intensive
regions” and the modes of rising regions of corresponding “jump regions”, as shown in Fig. 6.
According to the “frequency-intensive region” in Fig. 6, the comparison and error between the
natural frequencies of the “frequency-intensive region” in the 8th to 25th order modes of the
integral model and the substructure model were compared, and the following Table 3 was drawn.
In order to verify the accuracy of the calculation results of the substructure model and the
overall blade-disk model, the modal shapes of the two models are compared in this paper, as shown
in Fig. 7.
It can be seen from the figure that the vibration modes of each order of the substructure model
are in good agreement with the analysis results of the overall blade-disk model, and the calculation
accuracy of the substructure model is high, which can meet the requirements of the next dynamic
calculation.
OPTIMIZATION ANALYSIS OF VIBRATION REDUCTION FOR AEROENGINE MULTISTAGE BLADE-DISK SYSTEM.
XIAOMEI YOU, HAIYU FENG, HONGYUAN ZHANG, PENGBO LIU
1554
ISSN PRINT 1392-8716, ISSN ONLINE 2538-8460
Fig. 6. Division the regions of first 200 order
Table 3. Comparison of calculation results of inherent frequency
Order Inherent fre
q
uenc
y
(
Hz
)
Relative error (%)
Integral model Substructure model
8 739.40 739.49 0.01
9 740.03 739.82
0.03
10 741.11 740.83
0.04
11 741.57 741.29
0.04
12 742.01 742.03 0.00
13 742.55 742.42
0.02
14 742.69 742.60
0.01
15 742.92 742.80
0.02
16 743.07 742.97
0.01
17 743.35 743.19
0.02
18 743.43 743.48 0.01
19 743.55 743.95 0.05
20 743.83 744.03 0.03
21 744.32 744.14
0.02
22 744.63 744.29
0.05
23 744.71 744.56
0.02
24 744.92 744.62
0.04
25 745.00 744.64
0.05
a) Integral model 144th order amplitude
of each blade
b) Substructure model 144th order amplitude
of each blade
Fig. 7. Comparison of amplitudes between the integral blade disk model and the substructure model
OPTIMIZATION ANALYSIS OF VIBRATION REDUCTION FOR AEROENGINE MULTISTAGE BLADE-DISK SYSTEM.
XIAOMEI YOU, HAIYU FENG, HONGYUAN ZHANG, PENGBO LIU
JOURNAL OF VIBROENGINEERING. NOVEMBER 2024, VOLUME 26, ISSUE 7 1555
3. Dynamic characteristics analysis of mistuned multistage blade-disk system
3.1. Finite element model of mistuned multistage blade disk
The ideal blade-disk system is cyclic symmetric, but due to processing, installation, wear and
other factors, resulting in blade mass, stiffness, natural frequency differences, that is, mistuning
[22]. The vibration energy of the blade-disk system will be concentrated in a few blades, which
will aggravate the vibration localization degree of the blade-disk system, cause fatigue damage,
and adversely affect the life of the compressor blade-disk system. Therefore, it is of great
significance to study the mode localization problem caused by the parameter mistuning of the
blade-disk system.
3.2. Model description
The finite element model of the mistuned multistage blade-disk system is established. Only
the parameters of blade and disk are different from those of the harmonic multistage blade-disk
system, such as the mass parameter and the stiffness parameter.
The mass mistuning and the stiffness mistuning of the blade-disk system are respectively
considered. The mass mistuning is Δ𝑀 and the stiffness mistuning is Δ𝐾. The expressions of the
stiffness and mass of the blade-disk system are shown in Eq. (20) and Eq. (21):
𝐾=(1+Δ𝐾)𝐾, (20)
𝑀=(1+Δ𝑀)𝑀, (21)
where: 𝐾 – the stiffness value of the mistuned multistage blade-disk system, 𝑀 – the mass value
of the mistuned multistage blade-disk system, 𝐾 – the stiffness value of the harmonic multistage
blade-disk system, 𝑀 – the mass value of the harmonic multistage blade-disk system.
Random mistuning of the mass and the stiffness is realized by Monte Carlo sampling method.
Six groups of sampling samples are set, and data are extracted in groups. The sampled data are
uniformly distributed with mean value of 0 and standard deviation of 1 %, 2 % and 5 %,
respectively, as shown in Table 4.
Table 4. Sampling of mistuning mass
Parameters Mistunin
g
condition Mean value Sam
p
le size
Δ𝐾
1 % 0 10000
2 % 0 10000
5 % 0 10000
Δ𝑀
1 % 0 10000
2 % 0 10000
5 % 0 10000
The first set of sampling results (defined as mistuning1) is shown in Fig. 8.
The sampling of mistuning1 is uniformly distributed. The mistuning mass of the primary disk
is numbered 1-38, the mistuning mass of the secondary disk is numbered 39-91, and the mistuning
mass of the tertiary disk is numbered 92-151.
3.3. Response localization characteristics of mistuned multistage blade-disk system
The forced vibration of the disk is induced by the excitation force of the mistuned disk and
determines the mode shape of the system. The multistage blade-disk system will vibrate
continuously for a long time under the influence of a certain excitation, which will cause high
cycle fatigue failure, crack and even fracture.
The expression of the exciting force on the 𝑖-th blade is shown in the following Eq. (22):
OPTIMIZATION ANALYSIS OF VIBRATION REDUCTION FOR AEROENGINE MULTISTAGE BLADE-DISK SYSTEM.
XIAOMEI YOU, HAIYU FENG, HONGYUAN ZHANG, PENGBO LIU
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ISSN PRINT 1392-8716, ISSN ONLINE 2538-8460
𝐹
=𝐹
𝑒
()
=𝐹
𝑒

𝑒

=󰇝𝐹
cos𝜙
+
𝑗
𝐹
sin𝜙
󰇞𝑒

, (22)
where: 𝐹
– excitation force amplitude, 𝜔 – excitation frequency, 𝜙
– phase angle.
The excitation force is applied as shown in Fig. 9.
Fig. 8. Results of the first sample
a) Real part exciting force
b
) Virtual part exciting force
Fig. 9. The value of the excited force of a multistage blade-disk system
The harmonic response of the mistuned multistage blade-disk system (mistuning1) is analyzed,
and the sweep frequency segment is defined as 600-1600 Hz. The vibration amplitude of the
mistuned multistage blade-disk system at each frequency is shown in Fig. 10.
As shown in Fig. 10(a) and Fig. 10(b), the black, red and blue lines in the figure correspond to
the first, second and third-order blades of the disk system respectively. The vibration amplitudes
of third-order blades corresponding to different frequencies can be obtained from Fig. 10(a), and
the blades with the highest amplitudes of the third-order blades can be obtained from Fig. 10(b).
The resonance frequency of the mistuned multistage blade-disk system occurs around 740 Hz,
970 Hz, 1050 Hz, 1320 Hz, 1400 Hz and 1500 Hz, and the maximum resonance peak value is
9.09×10
-4
mm.
4. Vibration reduction optimization analysis of mistuned multistage blade-disk system
The natural frequency of blade vibration is closely related to the stiffness and the mass of the
blade. According to Eq. (23) and Eq. (24), it can be concluded that by changing the stiffness or
the mass of the blade, the optimization of vibration reduction for a mistuned multistage blade-disk
OPTIMIZATION ANALYSIS OF VIBRATION REDUCTION FOR AEROENGINE MULTISTAGE BLADE-DISK SYSTEM.
XIAOMEI YOU, HAIYU FENG, HONGYUAN ZHANG, PENGBO LIU
JOURNAL OF VIBROENGINEERING. NOVEMBER 2024, VOLUME 26, ISSUE 7
1557
system can be achieved. Changing the stiffness of the blades can be achieved by introducing a
disturbance coefficient 𝑃
, i.e.:
𝐾
=𝐾
(1+𝑃
),𝑖=1,2,,𝑛, (23)
where, 𝐾
is the stiffness value of the 𝑖-th mistuned blades, 𝐾
is the stiffness value of the
harmonic blade, 𝑃
is the disturbance coefficient, 𝑖 is the number of the blade, 𝑛 is the number of
blades in each stage of the blade disk.
Changing the quality of the blades can be achieved by introducing a disturbance coefficient
𝑃
, i.e.:
𝑀
=𝑀
1+𝑃
,
𝑗
=1,2,,𝑛, (24)
where, 𝑀
is the mass value of the 𝑖-th mistuned blades, 𝑀
is the mass value of the harmonic
blade, 𝑃
is the disturbance coefficient, 𝑖 is the number of the blade, 𝑛 is the number of blades in
each stage of the bladed disk.
a) Maximum amplitude at each frequency of
mistuned multistage blade-disk syste
m
b) Maximum amplitude of each blade
in a mistuned multistage blade-disk syste
m
Fig. 10. Maximum amplitude of forced vibration in mistuning1 condition
a) Maximum amplitude at each frequency of a
mistuned multistage blade-disk syste
m
b) Maximum amplitude of each blade
in a mistuned multistage blade-disk syste
m
Fig. 11. Maximum amplitude of forced vibration in mistuning2 condition
In order to explore whether the change of the blade mass affects the vibration of the system, a
blade mass is randomly slightly worn (the mistuning amount is controlled within 6 %) in the above
OPTIMIZATION ANALYSIS OF VIBRATION REDUCTION FOR AEROENGINE MULTISTAGE BLADE-DISK SYSTEM.
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1558 ISSN PRINT 1392-8716, ISSN ONLINE 2538-8460
mistuning1 working condition, and the maximum amplitude of forced vibration of the system in
mistuning2 working condition is obtained under the same exciting force vector, as shown in
Fig. 10. Fig. 11(a) shows the maximum amplitude of each blade in the whole frequency band
when the excitation frequency changes from 600 Hz to 1600 Hz. Fig. 11(b) shows the maximum
amplitude of each blade of the primary, secondary and tertiary disk when the excitation frequency
changes from 600 Hz to 1600 Hz. In this frequency band, the maximum amplitude of each blade
occurs on the third-stage blade disk, and the maximum value is 8.96×10-4 mm, and the average
value of the maximum amplitude of each blade on the third-stage blade disk is 3.69×10-4 mm.
According to the above analysis, it can be seen that the maximum response amplitude of the
blade-disk system decreases from 9.09×10-4 mm to 8.96×10-4 mm after the system changes from
mistuning1 condition to mistuning2 condition through random wear of the blade mass (in
mistuning1 condition). It is found that the maximum amplitude of the forced vibration of the
mistuned multistage blade-disk system can be reduced by slightly worn the blade mass. The above
slightly worn the blade mass test can be repeated many times, and similar results can be obtained.
Therefore, it is considered that the maximum amplitude of the multistage blade-disk system can
be reduced by slightly wearing the mass of several blades.
4.1. Vibration reduction optimization algorithm of random slightly worn blade mass
The model of the multistage blade-disk system consists of three stage blade-disks with 151
blades. The parameters of each blade are defined. The finite element reduction model of the
mistuned multistage blade-disk system is established by using the substructure modal synthesis
method, and the analysis files of four substructures are generated. The dynamic analysis of the
multistage blade-disk system generated by each substructure analysis file set is carried out.
Fig. 12. Flow chart of vibration reduction optimization algorithm for random slightly worn blade mass
OPTIMIZATION ANALYSIS OF VIBRATION REDUCTION FOR AEROENGINE MULTISTAGE BLADE-DISK SYSTEM.
XIAOMEI YOU, HAIYU FENG, HONGYUAN ZHANG, PENGBO LIU
JOURNAL OF VIBROENGINEERING. NOVEMBER 2024, VOLUME 26, ISSUE 7
1559
It can be seen from literature [24] that the maximum amplitude of a blade-disk system is not
only related to its own mistuning, but also to the frequency of adjacent blades. In the analytical
model presented in this paper, adjacent blades can be blades of the same level of disks or blades
of adjacent stages. According to the above conclusions, if the parameters of blades adjacent to the
maximum amplitude are fine-tuned, that is, the amplitude of the blade-disk system will inevitably
change after the mass of other blades is slightly worn. Based on the above ideas, the results of
each iteration calculation are saved, and the mass of the blade is continuously fine-tuned at random
(and the mistuning of blade parameters is controlled within 6 %), while the parameters of other
blades remain unchanged, which can optimize the multistage blade-disk system faster. In this
paper, the independent operation of the optimization algorithm is realized by writing MATLAB
program and ANSYS command flow. The flow chart of the optimization algorithm of the
multistage blade-disk system is shown in Fig. 12.
In order to facilitate the comparison, the mistuning1 condition is optimized by using the
optimization algorithm of random slightly worn the blade mass in this paper, and the maximum
amplitude of each blade in the whole frequency band when the excitation frequency changed from
600 Hz to 1600 Hz before and after optimization is calculated. The optimization results are
verified, and it is found that the vibration of the blade-disk system could be greatly improved by
slightly worn the blade mass. The maximum amplitude of each blade before and after optimization
is shown in Fig. 13.
a
)
Before o
p
timization
b)
After o
p
timization
Fig. 13. Vibration of a mistuned multistage blade-disk system
The maximum amplitude of the system before and after optimization appeared on the third
stage blade disk, and the maximum amplitude before optimization is 9.09×10
-4
mm, and the
maximum amplitude after optimization is 8.57×10
-4
mm.
The maximum amplitude, variance and localization factor of the blade are usually used as
evaluation indexes to evaluate the quality of the vibration of the mistuned multistage blade-disk
system. The vibration localization factor proposed in literature [25] is adopted to consider the
proportional relationship between the vibration energy of the blade and the square of its
deformation. In order to facilitate calculation, the displacement change of the blade tip is used to
represent the vibration of the blade. The maximum amplitude in the blade-disk structure and the
amplitude of other blades are used to describe the degree of modal localization of the blade-disk
system. The modal localization parameters are shown in the following Eq. (25):
𝐿=
|𝑥|

1
𝑛−1𝑥
,
1
𝑛−1𝑥
,
, (25)
OPTIMIZATION ANALYSIS OF VIBRATION REDUCTION FOR AEROENGINE MULTISTAGE BLADE-DISK SYSTEM.
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1560 ISSN PRINT 1392-8716, ISSN ONLINE 2538-8460
where: 𝑛 – the number of blades, 𝑠 – the serial number of the blade with the largest amplitude,
|𝑥| – the largest amplitude.
In this paper, the Eq. (25) is used to evaluate the optimization results. The equation was
proposed by Wang Hongjian [25] of Northwestern Polytechnical University, and the author has
verified in the paper that the evaluation method is accurate and effective.
It is known that the maximum amplitude of the mistuned multistage blade-disk system is
located on the third-stage blade disk. The comparison of variances, maximum amplitudes and
vibration localization factors of each blade before and after the third-stage blade disk optimization
is shown in Table 5.
Table 5. Comparison of vibration before and after optimization
Before optimization After optimization Optimization range (%)
Variance 2.57×10-8 2.52×10-8 1.95
Maximum vibration am
p
litude 9.09×10-4 8.57×10-4 5.72
Vibration localization facto
r
2.10 1.85 12.0
According to the analysis in Table 5, the variance, maximum vibration amplitude and vibration
localization factor after optimization decreased by 1.95 %, 5.72 % and 12.0 % respectively. The
degree of vibration localization of the mistuned multistage blade-disk system is obviously
reduced, which verifies the correctness of the optimization algorithm in this paper.
In order to evaluate the effectiveness of the algorithm on the control of vibration localization
of the mistuned blade-disk system, this article adopted the calculation equation of localization
factor in the ref. [26] and verification the algorithm. Please refer to Eq. (26) for the vibration
localization factor in detail:
𝐿󰆒=𝑚𝑒𝑎𝑛(𝑋)×𝑣𝑎𝑟(𝑋), (26)
where, 𝑋 – maximum amplitude vector of each blade, 𝑚𝑒𝑎𝑛(𝑋) is the mean of maximum
amplitude of each blade, 𝑣𝑎𝑟(𝑋) is the maximum amplitude variance of each blade.
Table 6. Vibration localization factor 𝐿′ of blades before and after optimization
Before o
p
timization After o
p
timization O
p
timization ran
g
e
(
%
)
Vibration localization facto
r
7.71×10-12 7.308×10-12 5.21
The localization factor before and after optimization is calculated by the Eq. (26) and contrast
as shown in Table 6. As shown from Table 6, the localization factor after optimization is reduced
greatly by adopting the Eq. (26), the correctness of the optimization algorithm presented in this
paper is verified.
4.2. Vibration reduction optimization algorithm based on snake optimization algorithm
The Snake Optimizer (SO) is inspired by the mating behavior of snakes. In 2022, a new
meta-heuristic algorithm is proposed for the first time in literature [27], which is a new intelligent
optimization algorithm that mimics the mating behavior of snakes. In this section, the snake
optimization algorithm is introduced to analyze and optimize the vibration response characteristics
of the mistuned multistage blade-disk system.
The snake optimization algorithm, like all meta-heuristics, starts with generating a uniformly
distributed random population so that the process of optimizing the algorithm can begin. For the
convenience of comparison, the initial population here is still studied with mistuning1. Two
groups of populations are defined, male and female, and it is assumed that the leaves of the blade
disk where the maximum amplitude is located are defined as male, and the leaves of the remaining
blade disks are defined as female. The male and female expressions are shown in Eq. (27):
OPTIMIZATION ANALYSIS OF VIBRATION REDUCTION FOR AEROENGINE MULTISTAGE BLADE-DISK SYSTEM.
XIAOMEI YOU, HAIYU FENG, HONGYUAN ZHANG, PENGBO LIU
JOURNAL OF VIBROENGINEERING. NOVEMBER 2024, VOLUME 26, ISSUE 7 1561
𝑁≈𝑛,
𝑁≈𝑁−𝑛, (27)
where: 𝑁 – total number of individuals, 𝑁 – number of males, 𝑁 – number of females.
After defining the temperature and amount of food, and identifying the best individuals in each
group (i.e. the situation where the current stage reduces the amplitude of the blades most), the best
males (𝑓,) and the best females (𝑓,) and the location of food (𝑓) are got. The Temp
expression is shown by the following Eq. (28):
𝑇𝑒𝑚𝑝=exp−𝑡
𝑇, (28)
where 𝑡 represents the current number of iterations, 𝑇 represents the maximum number of
iterations. Define the quantity of food (𝑄) as expressed in the following Eq. (29):
𝑄=𝑐×exp
𝑡−𝑇
𝑇
, (29)
where: 𝑐 is constant and equals 0.5.
If 𝑄< threshold (threshold = 0.25), the snake searches for food by selecting any random
location and updating their position relative to the food (i.e. random slight wear of blade mass).
To model the exploration phase, the mass change expression of the 𝑖-th blade is shown in Eq. (30):
𝑀,(𝑡+1
)=𝑀,(𝑡)−𝑐×
𝐴
×(𝑀−𝑀)×𝑟𝑎𝑛𝑑+𝑀,
𝐴
=exp󰇧
,
𝑓,󰇨,
𝑀,(𝑡+1
)=𝑀,(𝑡)−𝑐×𝐴×(𝑀−𝑀)×𝑟𝑎𝑛𝑑+𝑀,
𝐴
=exp
󰇧
,
,
󰇨
,
(30)
where: 𝑀, – the position of the 𝑖-th male, 𝑀, – the position of the 𝑖-th female, 𝑟𝑎𝑛𝑑 – random
number, between the interval [0, 1], 𝐴 – the male’s ability to find food, 𝐴 – the female's ability
to find food, 𝑓, – the fitness of the 𝑖-th male individual, taking the maximum amplitude of the
blade as the index, the lower the maximum amplitude, the higher the fitness, 𝑓, – the fitness of
𝑖-th female individual, 𝑓, – fitness of random male individual, 𝑓, – fitness of random
female individual, 𝑐 is constant and equals 0.05.
Development phase (find the situation to reduce blade amplitude):
If 𝑄 > threshold, and if Temp > 0.6, the snake will move to the food (record the current
optimization parameters).
If 𝑄 > threshold, if Temp < 0.6, the snake will initiate the battle mode, then the optimal
male/female individual is obtained and the mating pattern shown in Eq. (31) below is performed:
𝑀,(𝑡+1
)=𝑀,±𝑐×𝑀×𝑟𝑎𝑛𝑑×󰇡𝑄×𝑀,(𝑡)−𝑀,(𝑡)󰇢,
𝑀,(𝑡+1
)=𝑀,±𝑐×𝑀×𝑟𝑎𝑛𝑑×󰇡𝑄×𝑀 ,(𝑡)−𝑀,(𝑡)󰇢,
(31)
where: 𝑀 – male mating ability, 𝑀=exp,
,,𝑀 – female mating ability,
𝑀=exp,
,, 𝑐 is constant and equals 2.
If the egg hatches, the worst male and female individuals are replaced by Eq. (32):
OPTIMIZATION ANALYSIS OF VIBRATION REDUCTION FOR AEROENGINE MULTISTAGE BLADE-DISK SYSTEM.
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𝑀
,
=𝑀

+𝑟𝑎𝑛𝑑×(𝑀

−𝑀

),
𝑀

,
=𝑀

+𝑟𝑎𝑛𝑑×(𝑀

−𝑀

). (32)
The algorithm flow is shown in Fig. 14, and the relationship between cycle times 𝑡,
temperature Temp and food quantity 𝑄 is shown in Fig. 15.
Fig. 14. Flow chart of snake optimization algorithm
Fig. 15. Graph of the number of cycles, temperature, and quantity of food
In order to facilitate comparison, the snake optimization algorithm is adopted to optimize the
OPTIMIZATION ANALYSIS OF VIBRATION REDUCTION FOR AEROENGINE MULTISTAGE BLADE-DISK SYSTEM.
XIAOMEI YOU, HAIYU FENG, HONGYUAN ZHANG, PENGBO LIU
JOURNAL OF VIBROENGINEERING. NOVEMBER 2024, VOLUME 26, ISSUE 7
1563
mistuning1 condition in Section 3.1, and the maximum amplitude of each blade in the whole
frequency band when the excitation frequency changed from 600 Hz to 1600 Hz before and after
optimization is obtained. The optimization results are verified, and it is found that the vibration
response of the blade-disk system could be greatly improved by slightly wearing the blade mass.
The maximum amplitude of each blade before and after optimization is shown in Fig. 16.
a) Before optimization
b
) After optimization
Fig. 16. Vibration of a mistuned multistage blade-disk system
The maximum amplitude of the system before and after optimization appeared on the third
stage blade disk, and the maximum amplitude before optimization is 9.09×10
-4
mm, and the
maximum amplitude after optimization is 8.26×10
-4
mm.
The maximum amplitude, variance and vibration localization factor of blades in 3.1 are also
used as evaluation indexes to evaluate the quality of vibration of the system. It is known that the
maximum amplitude of the mistuned multistage blade-disk system is located on the third-stage
blade disk, and the comparison of variances, maximum amplitudes and vibration localization
factors of each blade before and after the third-stage blade disk optimization is shown in Table 7.
According to the analysis in Table 7, the variance, maximum vibration amplitude and vibration
localization factor after optimization are reduced by 0.78 %, 9.13 % and 16.7 % respectively. The
degree of vibration localization of the mistuned multistage blade-disk system is significantly
reduced, which verifies the correctness of the optimization algorithm in this paper.
Table 7. Comparison of vibration before and after optimization
Before o
p
timization After o
p
timization O
p
timization ran
g
e
(
%
)
Variance 2.57×10
-8
2.55×10
-8
0.78
Maximum vibration amplitude 9.09×10
-4
8.26×10
-4
9.13
Vibration localization facto
r
2.10 1.75 16.7
4.3. Comparison of results of two optimization algorithms
The vibration reduction optimization results of random wear the blade mass are compared with
those based on snake optimization algorithm, as shown in Table 8.
Table 8. Comparison of two optimization algorithms
Optimization method of
random wear blade mass Snake optimizer
Variance reduction abilit
y
Stronge
r
Weake
r
Reduces maximum am
p
litude abilit
y
Weake
r
Stron
g
e
r
Reduces vibration localization factor abilit
y
Weake
r
Stron
g
e
r
Computing time Less More
The
q
ualit
y
of o
p
timal solution Locall
y
o
p
timal solution Global o
p
timal solution
OPTIMIZATION ANALYSIS OF VIBRATION REDUCTION FOR AEROENGINE MULTISTAGE BLADE-DISK SYSTEM.
XIAOMEI YOU, HAIYU FENG, HONGYUAN ZHANG, PENGBO LIU
1564 ISSN PRINT 1392-8716, ISSN ONLINE 2538-8460
The comparison results show that the two optimization algorithms have their own advantages.
The first algorithm can quickly reduce the amplitude of the blade, but its ability to reduce the
amplitude and vibration localization factor is weak, and it is easy to fall into the local optimal
solution. The second algorithm costs more computer time, but its ability to reduce amplitude and
vibration localization factor is strong, and the global optimal solution can be found when the
number of iterations is sufficient.
5. Conclusions
In this paper, the mistuned multistage blade-disk system of aeroengine compressor is studied.
Based on the substructure modal synthesis method, the finite element reduction model of the
blade-disk system is established, and the vibration reduction optimization algorithm is established.
Optimization research on the vibration reduction in the condition of the blade mass mistuning is
carried out, and the following conclusions are obtained:
1) Based on the substructure modal synthesis method, the overall structure of the blade-disk
system is divided into four substructures considering the inter-stage coupling, which not only
considers the inter-stage coupling of the multistage blade-disk system, but also improves the
calculation speed and efficiency. In the first 200 modes, the natural frequencies of the substructure
model and the overall blade-disk model have a high agreement.
2) Aiming at the problem of the mass mistuning optimization of blades, the vibration reduction
optimization method using random slight wear of a blade mass (the mistuning is controlled within
6 %) can reduce the degree of vibration localization based on the finite element reduction model.
Each optimization calculation only regenerates the substructure with the change of the mass,
which saves a lot of computer time under the premise of ensuring the calculation accuracy, and
the calculation efficiency is high. Based on the snake optimization algorithm, an optimization
model of the mistuned multistage blade-disk system is established. The blade disk with the
maximum amplitude is defined as the male population, and the remaining sub-structures are
defined as the female population. The optimal solution of the two populations is selected for
mating, and the next generation is obtained. By comparing the vibration reduction optimization
solutions, the optimal solution can be obtained through a certain number of iterations, that is, a
small amount of the blade mass is worn (the mistuning is controlled within 6 %), and the maximum
amplitude of the mistuning multistage blade-disk system is reduced.
3) The research shows that the optimization method used in this paper can effectively reduce
the degree of vibration localization of the mistuned blade-disk system, and the global optimal
solution can be found when the number of iterations is sufficient. The two optimization algorithms
proposed in this paper can both reduce the degree of vibration localization and increase the life of
aeroengine. The analysis method presented in this paper takes into account the analysis efficiency
and analysis precision, and can provide a theoretical basis for the optimization of vibration
reduction in practical engineering.
Acknowledgements
This research was funded by the research foundation of the Education Department of Liaoning
Province (JYTMS20230213).
Data availability
The datasets generated during and/or analyzed during the current study are available from the
corresponding author on reasonable request.
Author contributions
Xiaomei You: conceptualization, methodology, writing-original draft preparation. Haiyu
OPTIMIZATION ANALYSIS OF VIBRATION REDUCTION FOR AEROENGINE MULTISTAGE BLADE-DISK SYSTEM.
XIAOMEI YOU, HAIYU FENG, HONGYUAN ZHANG, PENGBO LIU
JOURNAL OF VIBROENGINEERING. NOVEMBER 2024, VOLUME 26, ISSUE 7 1565
Feng: software, validation, data curation, writing-original draft preparation. Hongyuan Zhang:
formal analysis. Pengbo Liu: writing-review and editing.
Conflict of interest
The authors declare that they have no conflict of interest.
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OPTIMIZATION ANALYSIS OF VIBRATION REDUCTION FOR AEROENGINE MULTISTAGE BLADE-DISK SYSTEM.
XIAOMEI YOU, HAIYU FENG, HONGYUAN ZHANG, PENGBO LIU
1566
ISSN PRINT 1392-8716, ISSN ONLINE 2538-8460
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Xiaomei You received the Ph.D. degrees in mechanical design and theory from
N
ortheastern University, China, in 2011. She is an Associate Professor in Shenyang
Ligong University, China. Her current research interests include multi-
b
ody dynamics
simulation of mechanical system, rotor dynamics, product integrated design method.
Haiyu Feng received the B.E. degrees in mechanical design manufacture and automation
from Shenyang Ligong University, China, in 2020. He is a graduate student majoring in
mechanics in Shenyang Ligong University, China. His current research interests include
mistuned bladed disk dynamics, optimization design.
Hongyuan Zhang received the Ph.D., degrees in mechanical design and theory from
N
ortheastern University, China, in 2018. He is an Associate Professor in Shenyang Ligong
University, China. His current research interests include mistuned bladed disk dynamics,
aerodynamic optimization design, rotor dynamics.
Pengbo Liu received the B.E. degrees in mechanical design manufacture and automation
from Shenyang Ligong University, China, in 2023. He is a graduate student majoring
Mechanical Engineering in Shenyang Ligong University, China. His current research
interests include mistuned bladed disk dynamics, optimization design.
ResearchGate has not been able to resolve any citations for this publication.
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