37

 Sustainable Marine Structures | Volume 05 | Issue 01 | March 2023

 Sustainable Marine Structures
https://ojs.nassg.org/index.php/sms

Copyright © 2023 by the author(s). Published by Nan Yang Academy of Sciences Pte Ltd. This is an open access article under the 
Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) License. (https://creativecommons.org/licenses/
by-nc/4.0/).

DOI: http://dx.doi.org/10.36956/sms.v5i1.856

*Corresponding Author:
Haichao Li,
College of Shipbuilding Engineering, Harbin Engineering University, Harbin, Heilongjiang, 150001, China;
Email: lihaichao@hrbeu.edu.cn

ARTICLE  
Vibration Isolation Characteristics of Impedance-balanced Ship 
Equipment Foundation under Unbalanced Excitation

Yuxuan Qin   Yinbing Wang   Fuzhen Pang   Zhiqi Fu   Haichao Li*

College of Shipbuilding Engineering, Harbin Engineering University, Harbin, Heilongjiang, 150001, China

ARTICLE INFO ABSTRACT

Article history
Received: 29 April 2023
Revised: 10 June 2023
Accepted: June 25 2023 
Published Online: June 30 2023

A new type of impedance-balanced ship equipment foundation structure 
based on the principle of impedance balancing using a “discontinuous 
panel-vibration isolation liquid layer-foundation structure” is proposed 
to solve the problem of poor low-frequency vibration isolation of the 
foundation under unbalanced excitation of shipboard equipment. Based 
on the finite element method, the influence of characteristic parameters 
of the foundation panel structure on its vibration reduction characteristics 
under unbalanced excitation is explored. The results show that the 
vibration isolation level of the impedance-balanced foundation is 10 dB 
higher than the traditional foundation in the low-frequency band of 10-
500 Hz when subjected to combined excitation of concentrated force and 
moment. Increasing the thickness of the impedance-balanced foundation 
panel can enhance the isolation effect. Increasing the number of sub-panels 
can effectively reduce the vibration response of the foundation panel and 
enhance the isolation performance of the foundation. The connection 
stiffness between sub-panels has a small effect on the isolation performance 
of the foundation.

Keywords:
Ship equipment foundation
Impedance-balanced design
Vibration isolation and reduction methods

1. Introduction
As the main structure connecting the equipment to the 

ship structure, the equipment foundation acts as a main 
medium for the transmission of vibration energy from 
the equipment to the outside of the ship. The impedance 
characteristic of the foundation is a critical parameter 
that characterizes the ability of the foundation structure 
to resist mechanical vibration. It is also the main index 
to reflect the vibration isolation performance of the 

equipment foundation [1,2]. Thus, the analysis of foundation 
impedance is important for the vibration reduction and 
isolation design of the foundation structure [3,4].

To enhance the vibration isolation performance of 
equipment foundations, Peng [5] proposed an impedance 
matching design method for ship foundations. This 
approach modifies the structural characteristic parameters 
and shapes of the foundation to achieve mutual matching 
between its impedance and that of the internal excitation 

http://dx.doi.org/10.36956/sms.v5i1.856


38

 Sustainable Marine Structures | Volume 05 | Issue 01 | March 2023

source, thereby achieving low radiation noise levels 
for ships. Li [6] compared the vibration acceleration and 
underwater radiation noise of the foundation structure 
before and after the installation of damping materials, 
which provided support for the low-noise optimization 
design of the surface ship structure. Wang et al. [7] found 
through research that the application of damping materials 
can reduce the value of equipment foundation vibration 
amplitude in the intermediate frequency range above 
60 Hz. Yang et al. [8,9] proposed a unified impedance 
model for ship vibration reduction that comprehensively 
optimized the structural dynamic layout of stiffness, 
damping mass and damping material to enhance the 
vibration reduction capability of the foundation structure. 
Ling and Wu [10] used the finite element method to study 
the suppression effect of multi-stage isolation mass on 
structural sound propagation. Liu et al. [11] further revealed 
the suppression mechanism of vibration damping mass on 
structural sound transmission through case analysis and 
experimental verification. Yuan et al. [12] discussed the two 
indirect force estimation approaches, and experiments 
showed that indirect approaches could be good choices 
for determining the output forces of machinery in ships. 
Chen et al. [13] established an isolation vibration system 
with distributed dynamic absorbers and found that it 
could increase the frequency range of the absorbers. 
Ye et al. [14] combined a mathematical model of particle 
damping, developed a design method of particle damping 
optimization, solved the problem of high-frequency 
vibration in the power device, proposed to combine it 
with active control technology, and verified that it can 
achieve low-medium-high broadband vibration control 
of ship equipment. Based on scaled model experiments, 
Wu et al. [15] proposed a hysteresis nonlinear foundation 
structure with MRD and verified its effectiveness in 
low-frequency line spectrum vibration. Gong et al. [16] 
discussed the effect of damping mass on the propagation 
of vibration waves in pipelines based on the impedance 
mismatch principle and numerical analysis methods, and 
the results showed that damping mass has a beneficial 
damping effect in the medium to high frequency range. To 
achieve the purpose of impedance mismatch, Qi et al. [17]  
used a modern type of vertically symmetric foundation 
with low stiffness isolators and high input impedance to 
control the coupled vibration and noise radiation of the 
propeller shaft system. Wang [18] proposed the combination 
of active or semi-active control with dynamic absorbers 
to reduce the vibration transmitted to the deck and hull 
from power equipment and piping systems. Chen et al. [19]  
derived the expression of pressure-resistant strength of 
airbag isolators from the thin shell non-distance theory 

and verified the reliability of airbag isolators. Du and 
Li [20] investigated the nonlinear vibration mechanism 
of the ship rotating machinery with an airbag isolator 
under heaving motion. Bu et al. [21] proposed a multi-
objective collaborative attitude control method for 
double-layer airbag isolator devices under flexible 
support conditions. To effectively reduce the vertical and 
horizontal stiffness of the airbag isolator, Yin et al. [22] 
adopted a composite airbag structure with serial hard-
elastic layers. Li et al. [23] designed a highly adjustable 
magnetorheological elastomer-based isolator for real-time 
adaptive control, which can change the lateral stiffness 
of the isolator under moderate magnetic field levels. For 
complex mechanical systems, Wu et al. [24] analyzed the 
quantitative relationship between the impedance of the 
two transmission channel systems and the isolation effect 
of the mechanical system and the transmission loss of 
the pipeline system based on the hammer test method. 
The experimental results show that the impedance can 
be increased and the isolation effect can be improved 
by strengthening the foundation reinforcement or 
changing the elastic element properties/layout of the 
pipeline system. Ye [25] proposed a method to improve the 
impedance of the foundation by optimizing the structural 
variables of the foundation without altering the structural 
shape of the foundation and verifying its effectiveness. 
Dario et al. [26] developed an isolation system to optimize 
the tuned mass damper inertial device, which allows the 
inertial instrument and damper to be installed in series, 
resulting in a lower mass and more effective alternative 
than conventional tuned mass dampers. Maciejewski et 
al. [27] conducted numerical simulation and optimization 
design of a vibration isolation system. They established a 
mathematical model for vibration reduction and obtained 
an optimal method for high-efficiency vibration isolation 
by simulating the dynamic behavior of the system under 
diverse working conditions.

In light of the research summarized in this review, it 
is apparent that improving the mechanical impedance 
matching or impedance mismatch by increasing the 
damping mass, changing the shape or material properties 
of the foundation structure, etc. [28], has improved the 
vibration reduction and isolation performance of the 
system. It may be necessary to increase foundation mass 
or decrease foundation stiffness in order to further extend 
the isolation frequency range of the foundation using the 
method to lower frequencies, but doing so could result 
in issues like excess foundation mass or inadequate 
stiffness, which are detrimental to the lightweight design 
of ship structures and structural safety. Therefore, this 
paper proposes a foundation impedance-balanced method, 



39

 Sustainable Marine Structures | Volume 05 | Issue 01 | March 2023

which uses a vibration isolation liquid layer instead of the 
traditional foundation support structure in the design of 
the foundation structure to transform the excitation force 
point and line transmission into surface transmission. 
To improve the vibration isolation performance of the 
foundation, the impedance of the foundation is balanced 
and distributed by interrupting the continuous propagation 
of vibration waves through the discontinuous panel 
structure.  The vibrational isolation effect of the proposed 
method is calculated using the finite element method, 
and the effect of the device foundation panel structural 
parameters on the vibrational properties is investigated.

2. Principles and Characterization Methods 
of Impedance Balancing

2.1 Impedance-balanced Principle

Assuming that the total force F and total impedance 
Z transmitted to the foundation structure by the same 
equipment are constant, the equipment is connected to the 
foundation structure through machine feet. The equipment 
excitation is transmitted only through the foot of the 
machine to the foundation structure. If the number of 
machine feet is n, the force transmitted to the foundation 
through the k-th machine foot is F k, and the input 
impedance at the k-th machine foot is Zk, so we have:

1 1 1
, 1

n n n

k k k
k k k

F F x F x
= = =

= = =∑ ∑ ∑  (1)

1 1 1
, 1

n n n

k k k
k k k

Z Z y Z y
= = =

= = =∑ ∑ ∑  (2)

where kk
F

x
F

= , kk
Z

y
Z

= , the total power flowing into the 

foundation is P.
2

1 1

n n
k

k
k k k

F
P P

Z= =
= =∑ ∑  (3)

From Equations (1)-(3), it can be concluded that:
2 2

1

n
k

k k

x F
P

y Z=
 

=  
 
∑  (4)

When k kx y= , the system can obtain the minimum input 
power, that is, when the force/impedance distribution of 
the foundation is balanced, the energy transmitted from 
the equipment to the foundation is the lowest. Figure 
1 compares the vibration response of the foundation 
structure under different loading modes. It can be seen 
that when the excitation force is the same, the vibrational 
response of the base structure is weaker when the force 
is uniformly distributed over the surface compared to 

the case of concentrated point force loading. Therefore, 
if the unbalanced excitation force of the equipment can 
be transformed into a distributed force, the transmission 
of vibration energy can be effectively reduced and 
the vibration isolation level of the foundation can be 
improved.

100 200 300 400 500

20

40

60

80

100

120

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(
d
B
(
r
e
f
=
1
0
-
6
m
/
s
2
)
)

Frequency (Hz)

 Concentrated load
 Uniform load

Figure 1. Comparison of average vibration acceleration 
levels of structures under different loading forms.

2.2 Impedance-balanced Characterization Method

This paper proposes the concept of impedance-
balanced design for the foundation, which reflects the 
level of impedance balancing of the foundation under 
the current structure form through the difference of input 
impedance curves at several points in a certain frequency 
range. The specific definition is as follows.

Assuming that n loading points are uniformly selected 
on the surface of the foundation structure, each loading 
point has an input impedance 0 ( )

iZ f  at frequency f, and 
the standard deviation Si of the n loading points on the 
foundation at frequency i is expressed as:

2
0 0

1
( )

n
i i

i
i

Z Z
S

n
=

−
=
∑  (5)

0
iZ  represents the average input impedance of each 

loading point at frequency i.
In order to eliminate the effect of high or low levels 

of data values on the dispersion of the sample values, the 
dispersion coefficient is calculated, which is defined as 
follows:

0

i
i i

S
V

Z
=  (6)

The ratio of the standard deviation to the mean of 



40

 Sustainable Marine Structures | Volume 05 | Issue 01 | March 2023

the data. Therefore, within the [a, b] frequency band, 
impedance standard deviation curve R(Si) and discrete 
coefficient curve D(Vi), average the curve values at each 
frequency point, get foundation impedance discretization 
coefficient ZDC, defined as:

1

m

i
i

V
ZDC

m
==
∑  (7)

where m is the number of analysis frequencies in the [a, b] 
frequency band.

Similarly, take the reciprocal of the foundation 
impedance discretization coefficient to obtain the 
foundation impedance-balanced coefficient ZSC, which 
represents the degree of equipment foundation impedance 
balancing, defined as:

1

1
m

i
i

m
ZSC

ZDC V
=

= =

∑  (8)

3. Impedance-balanced Ship Equipment 
Foundation Model and Vibration Reduction 
Effect

3.1 Impedance-balanced Ship Equipment Foundation 
Model

This article proposes a new type of ship equipment 
foundation structure with impedance-balanced, as shown 
in Figure 2. Replace the complete continuous panel with a 
discontinuous foundation panel, and replace the traditional 
support structure with a vibration isolation liquid layer, so 
that the surface waves transmitted to the structure surface 
undergo waveform conversion at the fluid solid interface, 
diminishing in the form of liquid surface waves, so as to 
balance the distribution of input impedance at various 
locations of the foundation panel. Keeping the impedance 
dispersion coefficient curve of the foundation at a low 
level effectively reduces the vibrational response under 
the excitation force of the device.

Equipment

Discontinuous panelFluid-filled sac

Ship 
structure Bottom 

plate

Figure 2. Schematic diagram of impedance-balanced 
foundation structure.

3.2 FEM Model of Impedance-balanced Ship 
Equipment Foundation

For complex structures such as the ship foundation, the 
finite element method is commonly used for numerical 
analysis. In order to improve the efficiency of simulation 
calculations, the paper proposes to use the structure-
acoustic coupling finite element method for analysis. 
The liquid is assumed to be ideal fluid and the influence 
of the fluid-filled sac on the isolation characteristics of 
the foundation is ignored. The sonic wave is a small 
amplitude sonic wave and there is no energy loss during 
the propagation process. The structure-acoustic coupling 
dynamic equation is expressed as follows:

impedance-balanced, as shown in Figure 2. Replace the complete continuous panel
with a discontinuous foundation panel, and replace the traditional support structure
with a vibration isolation liquid layer, so that the surface waves transmitted to the
structure surface undergo waveform conversion at the fluid solid interface,
diminishing in the form of liquid surface waves, so as to balance the distribution of
input impedance at various locations of the foundation panel. Keeping the impedance
dispersion coefficient curve of the foundation at a low level effectively reduces the
vibrational response under the excitation force of the device.

Figure 2. Schematic diagram of impedance-balanced foundation structure.
3.2 FEM Model of Impedance-balanced Ship Equipment Foundation

For complex structures such as the ship foundation, the finite element method is
commonly used for numerical analysis. In order to improve the efficiency of
simulation calculations, the paper proposes to use structure-acoustic coupling finite
element method for analysis. The liquid is assumed to be ideal fluid and the influence
of the fluid-filled sac on the isolation characteristics of the foundation is ignored. The
sonic wave is a small amplitude sonic wave and there is no energy loss during the
propagation process. The structure-acoustic coupling dynamic equation is expressed
as follows:

a a a

0 0
0 0Ta a

M C FX X XK Q
Q M C FpKP P

           
                        

 
  (9)

aM , aC and aK represent the mass matrix, damping matrix, and stiffness

matrix of the fluid, respectively. p is the acoustic pressure vector, and aK is the load

vector of the acoustic field excitation. Q is the structure-acoustic coupling matrix,

and  is the density of the fluid.

Equation (9) can be rewritten as follows utilising the modal superposition
principle and the Helmholtz equation:

 
 

2

2 2
a

T
a a a a

FXK j C M Q
pQ K j C M F

 
    

     
             

(10)

 (9)

aM , aC  and aK  represent the mass matrix, damping 
matrix, and stiffness matrix of the fluid, respectively. p is 
the acoustic pressure vector, and aK  is the load vector of 
the acoustic field excitation. Q  is the structure-acoustic 
coupling matrix, and ρ  is the density of the fluid.

Equation (9) can be rewritten as follows utilising 
the modal superposition principle and the Helmholtz 
equation:

( )
( )

2

2 2
a

T
a a a a

FXK j C M Q
pQ K j C M F

ωω ω
ρ ω ω ω ω

  + −  
=     + −       

 (10)

where ω  is the circular frequency, and j = 1, ( )F ω  
represents the modal load.

Equation (10) is the solution equation for structural 
vibration and radiated acoustic field based on structure-
acoustic coupling theory. By solving the above equation, 
the structural vibration and radiated noise under a given 
excitation load can be calculated.

E s t a b l i s h  f i n i t e  e l e m e n t  m o d e l s  f o r  t r a d i t i o n a l 
foundation and impedance-balanced foundation structures 
respectively. The computational model is shown in 
Figure 3, where the bottom boundary condition of the 
foundation is fixed on all four sides. The load is applied 
to the foundation panel in the form of a unite excitation 
force. Spring elements are used to connect each adjacent 
foundation panel. After the convergence calculation, the 
finite element mesh size of the structure was set as 0.005 
m while the acoustic mesh size was 0.01 m. The total 
number of finite element meshes is about 280,000.

The main structural parameters and material properties 
of traditional and impedance-balanced foundation are 
shown in Table 1. The main material parameters are 



41

 Sustainable Marine Structures | Volume 05 | Issue 01 | March 2023

shown in Table 2.

(a) Traditional foundation structure model

(b) Impedance-balanced foundation structure model
Figure 3. Schematic diagram of foundation structure.

Table 1. Main structural parameters of traditional and 
impedance-balanced foundation.

Panel size 2.0 m × 1.6 m
Panel 

thickness
0.02 m

Foundation plate 
thickness

0.03 m
Web plate 
thickness

0.01 m

Total height of 
foundation

0.2 m
Liquid layer 

thickness
0.2 m

Foundation material Q235 Steel
Liquid layer 

material
Water

Table 2. Main material parameters.

Material Steel Material Water

Density 7850 kg/m3 Density 1000 kg/m3

Elastic module 2.1 × 1011 Pa
Bulk 

volume
2.2 × 109 Pa

Poisson ratio 0.3

3.3 Impedance-balanced Foundation Isolation 
Effect

As shown in Figure 4, to verify the isolation effect of 
the impedance-balanced foundation, the concentrated 
force of 1 N and moment load of 1 N·m were applied 
at the midpoint and both sides of the panels of imped-
ance-balanced foundation and traditional foundation re-
spectively. The calculation and analysis frequency band 
was 10-500 Hz. Set evenly distributed assessment points 
on the surface of the panel and bottom plate on the foun-
dation, and use the average vibration acceleration level 
of the assessment points as the evaluation standard. The 
calculation results are shown in Figures 5-6.

Bracket Panel Bottom plate

Moment Concentrated force 

Observation points

Moment

Figure 4. Schematic diagrams of measurement point lay-
out and the loading situation. 

100 200 300 400 500

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-10

0

10

20

30

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L
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(
d
B
(
r
e
f
=
1
0
-
6
m
/
s
2
)
)

Frequency (Hz)

 Traditional base
 Impedance homogenized base

Figure 5. Comparison of average vibration level 
difference between two types of foundations subjected to 

concentrated force.

100 200 300 400 500

-20

-10

0

10

20

30

A
c
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L
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(
d
B
(
r
e
f
=
1
0
-
6
m
/
s
2
)
)

Frequency (Hz)

 Traditional base
 Impedance homogenized base

Figure 6. Comparison of average vibration level 
difference between two types of foundations subjected to 
the combined action of concentrated force and moment.

Figures 5-6 show that the impedance-balanced foun-



42

 Sustainable Marine Structures | Volume 05 | Issue 01 | March 2023

dation performs better than the traditional foundation in 
the frequency range of 0-500 Hz when the foundation 
structure is subjected to a concentrated force, with an 
average isolation level of 10 dB higher than that of the 
traditional foundation. However, the isolation level of the 
impedance-balanced foundation in the frequency range of 
0-100 Hz is lower than that of the conventional founda-
tion when the focused force and moment load are coupled 
to the foundation. The impedance-balanced foundation’s 
isolation level, which is more than 12 dB higher than that 
of the conventional foundation in the frequency range of 
100-500 Hz, can reach up to 30 dB at the 125 Hz frequen-
cy point. Therefore, it can be seen that the impedance-bal-
anced foundation has better isolation performance in a 
wide frequency range.

4. Analysis of the Impedance-balanced Foundation 
Isolation Characteristics

4.1 The Influence of the Structural Form of the 
Foundation Panel

The Influence of Panel Thickness on the Isolation 
Characteristics of the Impedance-balanced Foundation

Based on the above structural model, analyze and 
calculate the impact of changes in the thickness of the 
foundation panel on the vibration isolation characteristics 
of the foundation structure. To improve computational 
efficiency, the above model is further simplified into a 
shell structure model. The thickness of the foundation 
panel is 10 mm, 15 mm, and 20 mm, respectively. The 
calculation results of the average vibration acceleration 
level of the lower surface of the foundation are shown in 
Figure 7, and the difference in the foundation vibration 
level is shown in Figure 8.

 

100 200 300 400 500

60

80

100

120

A
c
c
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a
t
i
o
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L
e
v
e
l
 
(
d
B
(
r
e
f
=
1
0
-
6
m
/
s
2
)
)

Frequency (Hz)

 t=10mm
 t=15mm
 t=20mm

Figure 7. Average vibration acceleration level curve 
of assessment points with different thicknesses of 

foundation panels.

From Figures 7-8, it can be seen that as the thickness 
of the foundation panel increases, the average level of 
the assessment points of the hull structure under the 
foundation decreases overall. When the thickness is 
greater than 15 mm, further increasing the thickness does 
not reduce the system’s vibration level. As the frequency 
gradually increases, the peak acceleration response shifts 
to the high frequency. This is because the increase in 
thickness increases the stiffness of the foundation system. 
By comparing the results in Figure 8, it is found that, 
although the increase in thickness makes the foundation 
exhibits a stronger vibration isolation effect at most 
frequency points, not all frequency point positions within 
the analysis frequency band have increased vibration level 
difference, and even negative vibration level difference 
have been calculated at some frequency points. Therefore, 
i n  t h e  d e s i g n  p r o c e s s  o f  t h e  i m p e d a n c e - b a l a n c e d 
foundation, it is necessary to combine the excitation load 
characteristics of the equipment with adjusting the panel 
thickness.

 

100 200 300 400 500
-10

0

10

20

30

40

50

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L
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(
d
B
(
r
e
f
=
1
0
-
6
m
/
s
2
)
)

Frequency (Hz)

 t=10mm
 t=15mm
 t=20mm

Figure 8. Average vibration level difference curve of assess-
ment points with different thicknesses of foundation panels.

The Influence of Panel Layout on the Vibration Isolation 
Characteristics of the Impedance-balanced Foundation

With a reasonable discussion of the foundation panel, 
the elastic waves in the response plane of the panel can 
be effectively isolated and controlled, thus effectively 
increasing the vibration reduction and isolation level 
of the foundation. To investigate the influence of panel 
layout on the vibration characteristics of impedance-
balanced foundation, taking the four layout schemes 
shown in Table 3 and Figure 9 as examples, the vibration 
response of the foundation under different panel layout 
modes was calculated, and the calculation frequency band 
and loading method were the same as the previous text.



43

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The comparison of the vibration response and vibration 
level difference of different schemes of the foundation 
is shown in Figures 10-11. It can be seen that in the 
10-500 Hz frequency band, after disassembling the 
foundation panel, the vibration isolation characteristics 
of the foundation will change. In the 10-500 Hz analysis 
frequency band, Scheme 4 causes the lowest level of 
structural vibration response. In each scheme, as the 
number of sub-panels increases, the number of peaks 
in the response curve will decrease, and the curve has 
a tendency to shift towards high frequencies. From the 
vibration level difference curves of each scheme, it can be 

observed that Scheme 1 has a higher vibration isolation 
effect at 10-100 Hz, the average isolation level is about  
10 dB.

Table 3. Size parameters of fluid-filled sac under different 
layout schemes of the foundation.

Disassembly 
scheme

Number of panels/
piece

Single foundation panel 
size/m

Scheme 1 2 0.90 × 1.60

Scheme 2 3 0.60 × 1.60

Scheme 3 4 0.90 × 0.70

Scheme 4 6 0.60 × 0.70

0.9 m

1.6 mP1 P2

0.6 m

1.6 m
P1 P2 P3

(a) Scheme 1 (b) Scheme 2

0.9 m

0.7 mP1 P2

P3 P4
0.6 m

0.7 m
P1 P2 P3

P4 P5 P6

(c) Scheme 3 (d) Scheme 4

Figure 9. Schematic diagrams of different layout schemes for the foundation panel.

100 200 300 400 500

60

80

100

120

140

160

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(
d
B
(
r
e
f
=
1
0
-
6
m
/
s
2
)
)

Frequency (Hz)

 Whole plate
 Scheme1(1×2)
 Scheme2(1×3)
 Scheme3(2×2)
 Scheme4(2×3)

Figure 10. Average vibration acceleration level curve of the bottom plate with different layout schemes of the 
foundation panel.



44

 Sustainable Marine Structures | Volume 05 | Issue 01 | March 2023

100 200 300 400 500
-20

-10

0

10

20

30

40

A
c
c
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a
t
i
o
n
 
L
e
v
e
l
 
(
d
B
(
r
e
f
=
1
0
-
6
m
/
s
2
)
)

Frequency (Hz)

 Whole plate
 Scheme1(1×2)
 Scheme2(1×3)
 Scheme3(2×2)
 Scheme4(2×3)

Figure 11. Vibration level difference curve of assessment 
points with different layout schemes of foundation panels.

4.2 Influence of Foundation Panel Connection 
Stiffness

To investigate the influence of the stiffness of the 
foundation panel connection on the vibration reduction 
and isolation characteristics of the foundation, based on 
the foundation model of Scheme III mentioned above, 
the average vibration acceleration level of the foundation 
plate was calculated when the hinge stiffness between the 
foundation panels was 500 N·m2, 1000 N·m2 and 1500 
N·m2, respectively. The calculation results are shown in 
Figures 12-13.

100 200 300 400 500
40

60

80

100

120

140

160

A
c
c
e
l
e
r
a
t
i
o
n
 
L
e
v
e
l
 
(
d
B
(
r
e
f
=
1
0
-
6
m
/
s
2
)
)

Frequency (Hz)

 K=500
 K=1000
 K=1500

Figure 12. Average vibration acceleration level curve of 
the foundation plate with different connection stiffness of 

the foundation panel.

From Figures12-13, it can be seen that in the frequency 
band of 10-500 Hz, when the stiffness of the hinge 
connections among the panels increases, the acceleration 
response curves of each structure remain basically 
consistent, and the frequencies corresponding to the 

positions with larger responses in the curves remain 
basically unchanged. The maximum isolation level in the 
frequency range of 10-500 Hz is 25 dB. This is because 
the increase in the stiffness of the connections between 
the foundations has a limited effect on the overall natural 
character of the foundations. The lower the stiffness, the 
more pronounced the oscillation of the vibration level 
difference curve in the frequency band. However, as 
the stiffness increases, the change in the vibration level 
difference curve is relatively stable.

100 200 300 400 500
-10

0

10

20

30

A
c
c
e
l
e
r
a
t
i
o
n
 
L
e
v
e
l
 
(
d
B
(
r
e
f
=
1
0
-
6
m
/
s
2
)
)

Frequency (Hz)

 K=500
 K=1000
 K=1500

Figure 13. Average vibration acceleration level difference 
curve of foundation panel with different connection 

stiffness.

4.3 The Vibration Isolation Property of the Impedance-
balanced Foundation Subjected to White Noise 
Excitation

Using white noise excitation in the calculation of 
isolation performance is of significant importance. White 
noise excitation is a signal with constant average power 
spectral density across all frequencies. By containing 
components of various frequencies, the use of white noise 
excitation allows for a more comprehensive evaluation 
of the performance of the isolation system, providing a 
valuable reference for isolation design and optimization. 
Therefore, this paper conducts the calculation of 
impedance-balanced foundation isolation performance 
under white noise excitation based on the aforementioned 
model. The calculation results are shown in Figure 14.

From Figure 14, it can be observed that the impedance-
balanced foundation isolation performance, under white 
noise excitation, exhibits a similar curve trend to that 
under unit force excitation. Both show effective isolation 
in the frequency range of 10-150 Hz, with maximum 
isolation of 18 dB at 100 Hz. However, the isolation 
effectiveness is weaker in the frequency range greater 



45

 Sustainable Marine Structures | Volume 05 | Issue 01 | March 2023

than 300 Hz. Therefore, it can be concluded that the 
impedance-balanced foundation possesses favorable low-
frequency isolation performance.

100 200 300 400 500
-10

0

10

20

30

A
c
c
e
l
e
r
a
t
i
o
n
 
L
e
v
e
l
 
(
d
B
(
r
e
f
=
1
0
-
6
m
/
s
2
)
)

Frequency (Hz)

Figure 14. Average vibration acceleration level curve of 
the foundation plate with different connection stiffness of 

the foundation panel.

5. Conclusions

This paper proposes a vibration reduction method based 
on the principle of impedance balancing and designs an 
impedance-balanced foundation with a “discontinuous 
p a n e l - v i b r a t i o n  i s o l a t i o n  l i q u i d  l a y e r- f o u n d a t i o n 
structure”. Based on the structure-acoustic coupling finite 
element method, the vibration reduction characteristics 
of the impedance-balanced foundation under unbalanced 
excitation of the equipment are studied, and the influence 
of the structural parameters of the foundation panel on its 
vibration reduction characteristics is explored. The main 
conclusions are as follows:

(1) The vibration level difference of the impedance-
balanced foundation is higher than that of the traditional 
foundation with 10 dB in the low-frequency band of 10-
500 Hz subjected to the combined action of unbalanced 
excitation force and moment. 

(2) The thickness and quantity of the panel have an 
influence on the isolation performance of the impedance-
balanced foundation. Furthermore, the impedance-
balanced foundation exhibits superior vibration isolation 
performance within the low-frequency range of 10-
150 Hz. When the number of sub-panels is 4 and the 
arrangement is 2 * 2, the isolation performance of the 
foundation is better among the four layout schemes 
considered in this paper. The effect of panel connection 
stiffness on the isolation performance of the foundation is 
not obvious.

Funding

This study was funded by the National Natural Science 
Foundation of China (Grant Numbers. U2006229 and 
52101351)

Data Availability Statement

The data used to support the findings of this study are 
included within the article.

Conflict of Interest

The authors declare that there is no conflict of interest.

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