Theoretical and Experimental Studies on Vibration Resistance of Composite Plates with Damping Coating

Abstract

This study performs both theoretical and experimental studies on the vibration resistance of composite plates with damping coating subjected to impulse excitation load. A dynamic model is first proposed and the key differential equations are derived to solve the natural frequencies, time-domain vibration response, and dynamic stiffness at any vibration response point regarding the excitation point of such a coated structure. Then, a dynamic experiment system of two plate specimens with and without DC knocked by a hammer excitation is set up. The measured data indicates that the proposed dynamic model is trustworthy for predicting natural frequencies and dynamic stiffness results. Furthermore, based on the calculated dynamic stiffness data associated with the first four modes, the anti-vibration contribution of DC is quantitatively evaluated. It can be found that the coating can indeed improve the vibration resistance of the structure by up to 74.7%. In addition, the vibration suppression effect of DC is found to be closely related to the mode order of such a structure as well as the selected boundary condition.

Full text

Journal of Modern Mechanical Enginee ring and Technology, 2022, 9, 67-75 67
E-ISSN: 2409-9848/22 © 2022 Zeal Press
Theoretical and Experimental Studies on Vibration Resistance of
Composite Plates with Damping Coating
Yichen Deng1,2, Yezhuang Jin1,2,*, Yao Yang2, Bocheng Dong2 and Zelin Li2, Hui Li2
1College of Aerospace Engineering, Shenyang Aerospace University, Shenyang 110136
2School of Mechanical Engineering & Automation, Northeastern University, Shenyang 110819
Abstract: This study performs both theoretical and experimental studies on the vibration resistance of composite plates
with damping coating subjected to impulse excitation load. A dynamic model is first pro posed and the key differenti al
equations are derived to solve the natur al frequencies, time-domain vibration response, and dynamic stiffness at any
vibration response point regarding the excitation point of such a co ated structure. Then, a dynamic experiment system of
two plate specimens with and without DC knocked by a hammer excitation is set up. The measured data indicates that
the proposed dynamic model is trustw orthy for predicting natural freq uencies and dynamic stiffness results. Furthermore,
based on the ca lculat ed dy namic stiffness data associated wit h the first four modes , the anti-vibration contribution of DC
is quantitat ively evaluated. It can be found that the coating can indeed impr ove the vibration resistance of the structure
by up to 74.7%. In addition, the vibration suppression effect of DC is found to be closely related to the mode order o f
such a structure as well as the selected boundary condition.
Keywords: Damping coating, Composite plate, Vibration resistance, Dynamic model.
1. INTRODUCTION
Composite plates (CPs) are being widely utilized in
various engineering fields due to their lightweight and
high stiffness property [1-3]. However, many vibration
damages and fatigue problems often occur in these
structures because many of them work in harsh
dynamic conditions [4-6]. Damping coating (DC) is one
of the advanced surfacing materials exhibiting an
extensive application prospect since it has the
advantages of low cost, high vibration reduction, and
strong corrosion resistance, etc [7, 8]. The publicized
literature [9, 10] has proven its excellent performance
compared to conventional coating materials. However,
the related investigations into the DC-CPs are in their
infancy, including the modeling technique, experiment
approach, optimization design, etc.
In the past few decades, lots of research effort has
been devoted to the vibration suppression performance
of various composite plates with different methods. For
example, Zhang et al. [11] conducted a series of
experiments on a composite elastomeric isolator to
evaluate its energy dissipation capacity. Also, the
deformation assumptions used in the theoretical
method are validated via finite element data. The
vibration responses of fiber-steered laminated plates
were predicted by Akbarzadeh et al. [12] based on
third-order shear deformation theory, the hybrid
*Address corres pondence to this author at the College of Aerospace
Engineerin g, Shenyang Aerospace Univer sity, Shenyang 110136;
Tel: + 861062787863; E-mail: jyz8133@163.com
Fourier-Galerkin approach, and the numerical
integration technique. He et al. [13] proposed an
analytical model to analyze the vibration absorption
capacity of fiber-reinforced laminated plates with mass-
spring-damper subsystems. The stopband behavior of
such structures was further evaluated by performing a
frequency response analysis. Based on the first-order
shear deformation theory and von Karman nonlinear
transformation theory, Shivakumar et al. [14]
established a finite element (FE) model with eight-node
isoparametric serendipity elements to predict the active
damping characteristics of the composite plates with
the patches of active fiber composites. Zu et al. [15]
proposed a new nonlinear vibration model of a thin
composite sheet in a thermal environment, in which the
complex modulus approach, the power function, and
the Ritz methods were adopted to introduce the
nonlinear thermal and amplitude fitting coefficients into
this model. Additionally, the measured natural
frequencies, damping ratios, and vibration responses
were employed to prove the good prediction capability
of such a model. The damped free vibration properties
of woven glass fiber-reinforced epoxy composite plates
were measured and simulated by Navaneeth et al. [16].
They pointed out that the frequency response
displayed an upward trend as the number of layers of
the composite plate increased.
With the advantages of the DC, it is often utilized as
surface improvement technology to achieve a better
anti-vibration capability of different coated structures.
Several articles have reported the progress in the
structural dynamics of composite plates with DC. For

68 Journal of Modern Mechanical Engineering and Technology, 2022, Vol. 9 Deng et al.
instance, based on experimental data, Mohotti et al.
[17] identified the damping behavior of composite
aluminum plates covered with polyurea damping
coating subjected to low-velocity impact loadings.
Moreover, they established a FE model by LS-DYNA to
quantify the energy absorption performance of such a
damping coating with different configurations. By using
a FE plate model, Kulhavy et al. [18] evaluated the
additional damping behaviours of fiber-reinforced
plates coated with viscoelastic neoprene and thin
rubber material. They also performed experimental
tests to validate the FE model as well as to find the
nonlinear damping phenomenon. An improved
component mode mistuning model is proposed by Xu
et al. [19] to identify the damping properties of the hard
and viscoelastic coated blisks. Gao et al. [20] also
conducts numerical studies on blisks with tuned and
mistuning coatings to evaluate the influence of
intentional mistuning hard coatings on vibration
characteristics of the structure. Wang and his
coauthors [21, 22] investigated the frequency
responses, natural frequencies, mode shapes, and
damping loss factors of a corrugated sandwich panel
with polyurea coating, in which a mixed experimental
and numerical approach was employed to quantify the
benefit of coating material in the face sheets on the
passive vibration attenuation of such a coated
structure. A long coated damping structure with
entangled metallic wire material is proposed by Zi et al.
[23] to investigate the effects of coating thickness,
coating length and the temperature on vibration
attenuation.
The aforementioned literature review reveals that
there are limited publications concerning the vibration
resistance of the DC-CP structures. In particular, no
one has utilized the dynamic stiffness results to
evaluate the anti-vibration performance of coated
composite structures based on an analytical model
developed. To cover this knowledge gap, by taking a
fiber-reinforced composite plate with DC as a research
object, the corresponding vibration suppression
analysis work is done in this study. In Section 2, a
dynamic model subjected to a pulse excitation load is
proposed, and then the solution expression of dynamic
stiffness is derived. Moreover, some detailed tests are
undertaken to validate the model as well as to
investigate the anti-vibration performance of coated
plate specimens with DC in Section 3. The theoretical
modelling and experimental techniques used in this
paper can provide a meaningful reference for studying
the vibration resistance of composite structures with an
advanced coating material.
2. THEORETICAL WORK
2.1. Model Description
A dynamic model of the DC-CP structure subjected
to a pulse excitation load is illustrated in Figure 1,
where an o-xyz coordinate system is established at the
mid-plane of the uncoated plate, with ‘1’, ‘2’, and ‘3’
denoting three principal fiber axes, respectively, and
θ
being the angle between the 1- and x-axis. The overall
length, width, and the total number of layers of fiber-
reinforced plate are represented by a, b, and n,
respectively. The DC is assumed to be sprayed evenly
Figure 1: A dynamic model of the DC-CP structure subjected to a pulse excitation load.

Studies o n Vibration Resistance of Com posite Plates with Damping Coating Journal of Modern Mechanical Engineering and Technology, 2022, Vol. 9 69
on both sides of the plate. The thickness values of the
coated and uncoated plate, and the DC are assumed to
be h, h1, and h2, respectively. The impulsive excitation
point is supposed to be at Pa (x1, y1), and the dynamic
response point is at Pb (x2, y2).
2.2. Displacement Field Assumption
The studied plate with DC is assumed as a
symmetric laminated structure. Based on the first-order
shear deformation theory [24], the displacements field
of this structure can be expressed as:
u(x,y,z,t)=u0(x,y,t)+z
!
(x,y,t)
v(x,y,z,t)=v0(x,y,t)+z
"
(x,y,t)
w(x,y,z,t)=w0(x,y,t)
(1)
where u, v, w are the displacements of the DC-CP
structure in the x, y, and z directions, u0, v0, and w0
represent the related displacements at the mid-plane, φ
and ψ are the rotations of transverse normal in the xoz
and yoz planes, respectively, and t is the time variable.
Assume that there is only a weak coupling effect
between the bending motion and the stretching motion
of the DC-CP structure. Thus, u0 and v0 can be
neglected [25]. As a result, the corresponding stress-
strain relationships are expressed as:
!
x="u
"x=z
#
,x;
!
y="v
"y=z
$
,y;
!
z="w
"z=0
%
xy ="u
"y+"v
"x=z
#
,y+
$
,x
( )
%
yz =kx&"w
"y+"v
"z
'
(
)*
+
,=kx&
$
+w,y
( )
%
xz =kx&"w
"x+"u
"z
'
(
)*
+
,=kx&
#
+w,x
( )
-
.
/
/
/
/
/
0
/
/
/
/
/
(2)
where εx, ε
y, γ
xy, γ
xz, γ
yz are the normal and shear
strains in the x, y, and z directions, respectively, and kx
= π2/12 is a shear correction factor.
Supposing the DC-CP structure is constrained by a
fully constrained boundary at the four edges, the
displacement field functions need to meet the following
conditions:
w(0, y,t)=w(a,y,t)=w(x,0,t)=w(x,b,t)=0
!w
!x(0, y,t)=!w
!x(a,y,t)=!w
!y(x,0,t)=!w
!y(x,b,t)=0
"
(0, y,t)=
"
(a,y,t)=
#
(x,0,t)=
#
(x,b,t)=0
(3)
According to the Ritz method [26], the displacement
relation can be formulated as:
!
=
"
mn P
m
#
( )
P
n
n=1
N
$
m=1
M
$
%
( )
sin
&
t
( )
'
=
(
mn P
m
#
( )
P
n
n=1
N
$
m=1
M
$
%
( )
sin
&
t
( )
w0=
)
mn P
m
#
( )
P
n
n=1
N
$
m=1
M
$
%
( )
sin
&
t
( )
(4)
where
!
mn
,
!
mn
, and
!
mn
are the undetermined
coefficients,
P
m
!
( )
and
P
n
!
( )
are the vibration mode
functions of the DC-CP structure, ω represents the
natural circular frequency, and M and N are the
numbers of truncated terms by the Rayleigh-Ritz
method. As reported by our previous study [27], M =
N = 8 is adopted to ensure the calculating precision.
Next, based on the Schmidt Orthogonalization
principle, the following vibration mode functions are
obtained to satisfy boundary conditions:
P
1
!
( )
=
" !
( )
, P
1
#
( )
=
$ #
( )
P
2
%
( )
=
%
&B2
( )
P
1
%
( )
%
P
k
%
( )
=
%
&Bk
( )
P
k&1
%
( )
&CkP
k&2
%
( )
%
=
!
,
#
,k>2
(5)
where
Bk
and
Ck
are the coefficient functions given by
Eq. (6):
Bk=
W
!
( )
P
k"1
!
( )
#
$%
&
2
!
0
1
'd
!
W
!
( )
P
k"1
!
( )
#
$%
&
2
0
1
'd
!
Ck=
W
!
( )
P
k"1
!
( )
P
k"2
!
( )
!
0
1
'd
!
W
!
( )
P
k"2
!
( )
#
$%
&
2
0
1
'd
!
,
!
=
(
,
)
(6)
where
W(
!
)
is the weight function, and
!
(
"
)
and
!
(
"
)
are the polynomial coefficients. Since the structure is
constrained with four sides, the expressions of
!
(
"
)
and
!
(
"
)
can be determined as:
! "
( )
=
"
21#
"
( )
2,
$ %
( )
=
%
21#
%
( )
2
"
=x/a,
%
=y/b
(7)
2.3. Solution of Natural Characteristics
Assuming that the cons titutive relationship
expression of the DC-CP structure is as follows:

70 Journal of Modern Mechanical Engineering and Technology, 2022, Vol. 9 Deng et al.
!
x
!
y
"
yz
"
xz
"
xy
#
$
%
%
%
&
%
%
%
'
(
%
%
%
)
%
%
%
=
Q11 Q12
Q12 Q22
Q44
Q55
Q66
*
+
,
,
,
,
,
,
,
-
.
/
/
/
/
/
/
/
0
x
0
y
1
yz
1
xz
1
xy
#
$
%
%
%
&
%
%
%
'
(
%
%
%
)
%
%
%
(8)
where for the coating material,
Q11 =Q22 =Ep
!
1"
µ
p
,Q12 =
µ
pEp
!
1"
µ
p
,Q44 =Q55 =Q66 =Ep
!
2(1+
µ
p)
re
presents the stiffness coefficients, with
Ep
!
being the
complex elastic modulus with the expression of
Ep
!=Ep1+i
"
p
( )
. Here,
Ep
and
!
p
represent the
traditional elastic modulus and loss factor of coating
material.
For fiber reinforced material, suppose that the
complex elastic moduli
Ef1
*
,
Ef 2
!
, and complex shear
moduli
Gf12
!
,
Gf13
!
,
Gf 23
!
in different fiber directions can
be defined as:
Ef1
*=Ef1 1+i
!
f1
( )
;
Ef 2
"=Ef 2 1+i
!
f 2
( )
;
Gf12
"=Gf12 1+i
!
f12
( )
;
Gf13
"=Gf13 1+i
!
f13
( )
;
Gf 23
"=Gf23 1+i
!
f23
( )
(9)
where
Ef1
and
Ef 2
are the traditional elastic moduli;
Gf12
,
Gf13
, and
Gf 23
represent the traditional shear
moduli;
!
f1
,
!
f2
,
!
f12
,
!
f13
, and
!
f 23
denote the
traditional loss factors in different fiber directions.
Then, the corresponding off-axis stress-strain
relationship of the composite plate is stated as:
!
x
!
y
"
yz
"
xz
"
xy
#
$
%
%
%
&
%
%
%
'
(
%
%
%
)
%
%
%
k
=*
Q11 Q12 0 0 0
Q12 Q22 0 0 0
0 0 Q44 0 0
0 0 0 Q55 0
0 0 0 0 Q66
+
,
-
-
-
-
-
-
-
.
/
0
0
0
0
0
0
0
*T
1
x
1
y
2
yz
2
xz
2
xy
#
$
%
%
%
&
%
%
%
'
(
%
%
%
)
%
%
%
k
(10)
where
!
is the coordinate transformation matrix with
the following expression:
!=
c2s20 0 "2cs
s2c20 0 2cs
0 0 c s 0
0 0 "s c 0
cs "cs 0 0 c2"s2
#
$
%
%
%
%
%
%
&
'
(
(
(
(
(
(
(11)
where
c=cos
!
k
;
s=sin
!
k
;
!
k
is the angle between
the 1- and x-axis in the k-th layer.
In Eq. (10),
Qij
represents the reduced stiffness
coefficients of fiber reinforced material with the
following expressions:
Q11 =Ef1
*
1!
µ
f1
µ
f2
,Q12 =Q21 =
µ
f1 Ef 2
"
1!
µ
f1
µ
f 2
=
µ
f 2 Ef 1
*
1!
µ
f1
µ
f2
Q22 =Ef 2
"
1!
µ
f1
µ
f 2
,Q44 =Gf23
",Q55 =Gf13
",Q66 =Gf12
"
(10)
where µ
f1 and µf2 are the Poisson’s ratios of fiber
layers.
The kinetic energies Tp1, Tp2 , Tf and the strain
energies Up1, Up2, Uf of the upper and lower layers of
DC and fiber reinforced material, as shown in Figure 1
are, respectively, given by:
Tp1 =1
2
!
p
"u
"t
#
$
%&
'
(
2
+"v
"t
#
$
%&
'
(
2
+"w
"t
#
$
%&
'
(
2
)
*
+
+
,
-
.
.
h
1/2
h/2
/
A
/dzdA
Tf=1
2
!
f
"u
"t
#
$
%&
'
(
2
+"v
"t
#
$
%&
'
(
2
+"w
"t
#
$
%&
'
(
2
)
*
+
+
,
-
.
.
0h
1/2
h
1/2
/
A
/dzdA
Tp2 =1
2
!
p
"u
"t
#
$
%&
'
(
2
+"v
"t
#
$
%&
'
(
2
+"w
"t
#
$
%&
'
(
2
)
*
+
+
,
-
.
.
0h/2
0h
1/2
/
A
/dzdA
(11)
Up1 =1
2
!
x
"
x+
!
y
"
y+
#
xy
$
xy +
#
yz
$
yz +
#
xz
$
xz
( )
h
1/2
h/2
%
A
%dzdA
Uf=1
2
!
x
"
x+
!
y
"
y+
#
xy
$
xy +
#
yz
$
yz +
#
xz
$
xz
( )
&h
1/2
h
1/2
%
A
%dzdA
Up2 =1
2
!
x
"
x+
!
y
"
y+
#
xy
$
xy +
#
yz
$
yz +
#
xz
$
xz
( )
&h/2
&h
1/2
%
A
%dzdA
(12)
where ρp and ρf are the corresponding densities of
polyuria coating and fiber layers, respectively.
Thus, the total kinetic energy T and strain energy U
of the DC-CP structure can be obtained as:
T=Tp1 +Tf+Tp2
(13)

Studies o n Vibration Resistance of Com posite Plates with Damping Coating Journal of Modern Mechanical Engineering and Technology, 2022, Vol. 9 71
U=Up1 +Uf+Up2
(14)
By substituting Eq. (4) into Eqs. (15-16), the
expressions of strain and kinetic energies can be
obtained by the undetermined coefficients
!
mn ,
"
mn ,
#
mn
. Furthermore, the corresponding
maximum strain energy Umax and the maximum kinetic
energy Tmax can be expressed as:
Umax =1
2qTKq
(15)
Tmax =1
2
!
2qTMq
(16)
where K and M are the stiffness and mass matrices,
respectively, ω is the circular natural frequency of the
structure studied, and q is the eigenvector
q=
!
11,
!
12 ,...,
!
mn "
#
11,
#
12 ,...,
#
mn "
$
11,
$
12 ,...,
$
mn
( )
T
.
The expression of Lagrange energy function L is
defined as:
L=Tmax !Umax
(17)
To solve the natural characteristics of the DC-CP
structure based on the Rayleigh-Ritz method, one need
to find the solution with a minimum value of L:
!L
!
"
mn
,!L
!
#
mn
,!L
!
$
mn
%
&
'(
)
*=0m=1, 2,..., M;n=1,2, ..., N
(18)
Substituting Eqs. (17-19) into Eq. (20), one has
K!
"
2M
( )
q=0
(19)
To obtain the solutions of Eq. (21), the determinant
of the coefficient matrix needs to be set as zero, i.e.
K!
"
2M=0
. Then, the natural frequency of the
structure can be solved. Also, the corresponding mode
shape can be calculated by substituting q into Eq. (4).
2.4. Solution of Dynamic Stiffness
Assuming that the expression of the impulse
excitation load F(x, y, t) and the structural vibration
response
( )X t
can be defined as:
F(x,y,t)=f(t)
!
(x"x1)
!
(y"y1)
f(t)=f0sin(
#
ft)
0
$
%
&
'
&
,
,
0(t(t1
t>t1
(20)
X(t)=Wmn(x,y)Tmn(t)
n=1
!
"
m=1
!
"
(21)
where (x1, y1) is the coordinate of excitation point Pa, f0
is the amplitude of excitation force, ωf is the excitation
frequency, t1 is the lasting time of load,
Wmn(x,y)
is the
modal shape, and Tmn indicates the modal shape
component of each order.
If the damping effect is considered, the vibration
differential equation of the DC-CP structure can be
written as:
d2Tmn t
( )
dt2+2
!
r
"
dTmn
dt+
"
( )
2
Tmn t
( )
=
P
mn t
( )
Mmn
(22)
where
P
mn (t)
and
Mmn
are the generalized force and
mass associated with the (m, n)-th mode shape, and
r
!
is the r-th modal damping ratio of the structure,
which can be obtained from the experimental data.
By applying the Duhamel integral method to solve
the differential equation when the initial condition value
is zero, each of the modal shape component
Tmn(t)
can
be obtained as:
Tmn(t)=Wmn x1,y1
( )
!
dMmn
f(
"
)e#
$
r
!
(t#
"
)sin
!
d(t#
"
)d
"
0
t
%
(23)
where
!
d=
!
1"
#
r
2
is the natural circular frequency
with consideration of damping.
After solving Eq. (25) by the numerical integration
method of Simpson, and then the calculated result is
substituted to Eq. (23). The time-domain vibration
response of the structure subjected to impulse
excitation load can be obtained by the mode
superposition approach.
Furthermore, the dynamic stiffness expression at
any vibration response point Pb regarding to the
excitation point Pa of the DC-CP structure can be
obtained as:
kba (
!
)=1FWmn x1,y1
( )
!
dMmn
f(
"
)sin
!
d(t#
"
)d
"
0
t
$
%
&
'
'
(
)
*
*
(24)
where
F[y]
denote to Fourier transform operation.
Once
kba(
!
)
is obtained, the dynamic stiffness results
can be used to evaluate the vibration resistance of the
studied structure system.

72 Journal of Modern Mechanical Engineering and Technology, 2022, Vol. 9 Deng et al.
3. EXPERIMENTAL WORK
3.1. Experimental Specimen and System
Before the formal measurement is conducted, two
plate specimens with the same geometric and material
parameters are fabricated by a company, in which
T300 carbon fiber/epoxy resin with the layout scheme
of [(0°/90°)5/0°/(0°/90°)5] is adopted and it include the
11 plies of unidirectional fiber prepregs. Then, one of
the specimens is coated with polyurea material, a kind
of DC, on the top and bottom surfaces via an SG9620
electric spray gun with a nozzle diameter of 1.8 mm, an
air pressure of 0.6 Mpa, and a coating solidification
time of about 1 min. Here, the polyurea material is
produced by combining amine groups with isocyanate
in a quick chemical reaction. The coating thickness is
0.6mm, with the additional weight being approximately
40% of the composite plate specimens. The other
material and geometric properties of the specimens
and the coating are: G
f12 = Gf13 = Gf23 = 4.6Gpa, Ef1 =
120Gpa, Ef2 = 7.9Gpa, µf1 = 0.30, ρf = 1780kg/m3, Ep =
230Mpa, µp = 0.40, ρp = 1020kg/m3.
A dynamic experiment system of the DC-CP
specimens with a hammer excitation is set up, as
shown in Figure 2. The system is mainly composed of
a DCB 086C01 modal hammer, a B&K 4517 lightweight
acceleration sensor, an LMS data acquisition
instrument with 16 channels, a mobile workstation. The
tested specimens with and without DC are effectively
clamped with its four sides via a clamping fixture. When
the calibration experiments of the acceleration sensor
and the hammer are finished, two specimens are
knocked by the hammer at the same excitation point.
Subsequently, the corresponding response signal and
pulse excitation force are recorded by the data
acquisition instrument. In this way, the dynamic
stiffness curves can be measured via the LMS
Test.Lab11A software. In the tests, Pa(x1, y1) = (50 mm,
50 mm) is set as the excitation point of each specimen,
and Pb(x2, y2) = (121, 100) is selected as the related
response point, with the measured frequency range
being within 10-1600 Hz.
Then, by adopting the half-power bandwidth
technique, the damping ratios of the composite plate
specimens without DC associated with the first four
modes can be identified, which are 0.95, 0.80, 0.88,
0.93 %, respectively. Also, the related damping results
of the coated specimen is measured, which are 1.07,
0.85, 0.95, and 1.04 %, respectively. By comparing
these damping data of uncoated and coated
specimens, it is clear that the DC material has a
significant impact on the vibration suppression ability of
this composite plate structure.
3.2. Model Validation and Evaluation of Vibration
Resistance
To further quantitatively clarify the damping
contribution of the coating material, the time history
curves of the excitation force and the vibration
response and dynamic stiffness results related to the
Figure 2: A dynamic experiment system of the DC-CP specimens with a hammer excitation.

Studies o n Vibration Resistance of Com posite Plates with Damping Coating Journal of Modern Mechanical Engineering and Technology, 2022, Vol. 9 73
first four modes of the coated and uncoated plate
specimens obtained via experimental tests and
theoretical calculations are compared in Figures 3 and
4, respectively. One can find out that the calculated
and measured data of the coated and uncoated
specimens have a relatively good agreement. The
maximum calculation error of natural frequency results
with and without coating is Rfre = 4.2 %, and the
corresponding error of dynamic stiffness results is Rk=
9.3%. Hence, it can be concluded that the proposed
model is trustworthy for forecasting the dynamic
stiffness curves of the composite plate structures, no
matter whether the DC is considered or not. The above
calculation error may come from: (1) a difference
between the geometric and material parameters of
coating adopted in the model and the actual specimen
being tested. For example, the coating thickness is
unequal for the coated specimen, but in the modelling
process, only constant thickness is considered for DC;
(2) a difference between the measured the simulated
positions of excitation points or response points.
In addition, by taking the dynamic stiffness values
related to the first four modes as the key indexes,
Table 1 gives the vibration suppression contribution of
such a coating material when it is sprayed on the
surface of the composite plate. Note that these
dynamic stiffness results are extracted from the trough
points in the theoretical and measured curves.
According to the deviation results in Table 1, one can
clearly see that when the damping coating is applied,
the dynamic stiffness values with different mode rise by
10.9 to 74.7%, which confirms that damping coating
does indeed improve the anti-vibration performance of
the composite plate. However, the damping effect is
closely related to the mode order of the structure as
well as the selected boundary condition [28].
4. CONCLUSIONS
In this work, the vibration suppression of composite
plates with damping coating is investigated based on a
dynamic model developed. Experimental verification is
then conducted to prove the effectiveness of such a
Figure 3: Vibration parameters of the uncoated specimen obtained via experimental tests and theoretical calculations: (a) time
history of excitation force, (b) time history of vibration response, and (c) dynamic stiffness curves.
Figure 4: Vibration parameters of the coated specimen obtained via experimental tests and theoretical calculations: (a) time
history of excitation force, (b) time history of vibration response, and (c) dynamic stiffness curves.

74 Journal of Modern Mechanical Engineering and Technology, 2022, Vol. 9 Deng et al.
model. Based on the calculated and experimental
results, the following findings are highlighted:
(1) There is a relatively good agreement between
the predicted and measured natural frequencies and
dynamic stiffness values. Thus, the proposed dynamic
model is trustworthy for predicting the vibration
parameters.
(2) The damping coating can contribute to the anti-
vibration performance of composite plate structure
undoubtedly, but the damping effect is closely related
to the mode order of such a structure as well as the
selected boundary condition.
(3) This study offers a practical tool for the
evaluation of the vibration resistance of composite
plates coated with DC material, which can be easily
extended to the dynamic assessment and analysis of
other coated composite structures after the related
theoretical model is updated.
ACKNOWLEDGMENT
This study was supported by the Fundamental
Research Funds for the Central Universities of China
(Grant No. N2104028).
DECLARATION OF CONFLICTING INTERESTS
The authors declare that they have no known
competing financial interests or personal relationships
that could have appeared to influence the work
reported in this paper.
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Table 1: Dynamic Stiffness Value of the Composite Plate Structures with and Without Damping Coating
Dynamic stiffness /N·m−1
Type
The 1st Mode
The 2nd Mode
The 3rd Mode
The 4th Mode
Uncoated plate (A)
2.463×104
5.526×104
1.409×105
7.393×104
Coated plate (B)
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1.563×105
1.152×105
Relative deviation ((B-A)/A) /%
64.4
74.7
10.9
55.8

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Received on 05-11-2022 Accepted on 06-12-2022 Published on 13-12-2022
DOI: https://doi.org/10.31875/2409-9848.2022.09.8
© 2022 Deng et al.; Zeal Press.
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