Vibro-acoustic behavior of a multilayered

Original Article
Vibro-acoustic behavior
of a multilayered
viscoelastic sandwich
plate under a
thermal environment
Journal of Sandwich Structures and Materials
13(5) 509–537
! The Author(s) 2011
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DOI: 10.1177/1099636211400129
jsm.sagepub.com
P Jeyaraj1, Chandramouli Padmanabhan2
and N Ganesan2
Abstract
This article presents numerical simulation studies on the vibration and acoustic
response characteristics of a multilayered viscoelastic sandwich plate in a thermal environment. Initially the critical buckling temperature is obtained followed by free and
forced vibration analyses considering the pre-stress due to the imposed thermal environment in the plate. The vibration response predicted is then used to compute the
sound radiation. The critical buckling temperature and vibration response are obtained
using finite element method while sound radiation characteristics are obtained using
boundary element method. It is found that resonant amplitude of both the vibration and
acoustic response decreases with increase in temperature. The influence of core thickness, number of layers, type of thermal field, and type of viscoelastic core material on
vibration response sound radiation characteristics are studied in detail.
Keywords
FEM/BEM, multilayered sandwich plate, thermal buckling, vibration and acoustic
response
1
School of Mechanical and Building Sciences,Vellore Institute of Technology University, Chennai Campus, Tamil
nadu, India.
2
Machine Design Section, Department of Mechanical Engineering, Indian Institute of Technology Madras,
Chennai 600 036, India.
Corresponding author:
P. Jeyaraj, School of Mechanical and Building Sciences, Vellore Institute of Technology University, Chennai
Campus, Tamil nadu, India
Email: [email protected]; [email protected]
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Journal of Sandwich Structures and Materials 13(5)
Introduction
Structures are typically exposed to moisture and heat during their service life.
Thermal stresses due to aerodynamic heating may sometimes induce buckling
and dynamic instability in structures. The pre-stress eﬀect due to thermal load
will aﬀect the dynamic behavior of the structure due to the change in the stiﬀness
of the structure. The structures under thermal environment are often subjected to
mechanical time-varying harmonic excitations. So, it is important to investigate the
dynamic behavior of a structure over a wide range of temperatures. The conventional isotropic materials such as steel and aluminum have so little amount of
inherent damping and their resonant behavior makes them eﬀective sound radiators. It is possible to control this resonant behavior by sandwiching highly damped
and dynamically stiﬀ materials such as viscoelastic materials between the conventional materials.
Extensive numerical studies have been carried out to analyze the free vibration
and damping behavior of viscoelastic sandwich plates using both numerical and
experimental methods. Ungar and Kerwin [1] were the earlier researchers who
found that the modal loss factor can be obtained by calculating the ratio of the
dissipating energy to the total structural energy, using modal strain energy method.
They used complex shear modulus to represent the viscoelastic behavior of a material, which exhibits both elastic and damping characteristics. Johnson and Kienholz
[2] described ﬁnite element based modal strain energy method to obtain modal
damping ratios. They compared the results obtained with various exact solutions
and approximate governing equations. Alam and Asnanai [3] derived equations of
motion, for vibration of a general multilayered plate, consisting of an arbitrary
number of alternate stiﬀ and soft layers of orthotropic material, using variational
principles. Cupial and Niziol [4] analyzed the natural frequencies and loss factors of
a three-layered rectangular plate with a viscoelastic core layer and laminated face
layers. They obtained complex eigenvalues numerically to extract both natural
frequency and associated modal loss factor. Rikards [5] presented a sandwich composite beam and plate ﬁnite super-elements with viscoelastic layers for vibration
and damping analysis. He presented an exact method where modal loss factors are
determined as the ratio of imaginary and real parts of complex eigenvalues. He also
presented an approximate method where modal loss factors are the ratio of dissipated and elastic strain energy. Wang et al. [6] presented experimental validation of
modal analysis of sandwich plates. They included the membrane and transverse
energies in the face plates, and shear energies in the core of their analytical model.
The shear modulus of the dissipative core was assumed to be complex and varying
with frequency and temperature. Mead [7] reviewed and compared the diﬀerent
methods of measuring the loss factors of heavily damped beams and plates damped
by uniform layers of viscoelastic damping.
Pradeep and Ganesan [8] analyzed buckling, free vibration, and modal damping
behavior of multilayer rectangular viscoelastic sandwich plates with isotropic facings under thermal load using ﬁnite element method. They found that natural
frequency reduces while loss factor increases with temperature. Gupta and
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Jeyaraj et al.
511
Kumar [9] investigated thermal eﬀect on vibration of nonhomogenous viscoelastic
rectangular plate of linearly varying thickness analytically. They found that the
eﬀect of nonhomogeneity on natural frequency is signiﬁcant.
Prediction of sound radiation is important to control noise generated from
vibrating structures. Several researchers analyzed sound radiation characteristics
of isotropic/composite plates having uniform thickness subjected to time-varying
harmonic excitations. Park et al. [10] investigated the eﬀects of support properties
on the sound radiated from the plate and found that both the velocity response and
sound radiation are strongly inﬂuenced by dissipation of vibration energy at the
edges. Qiao and Huang [11] analyzed six diﬀerent boundary conditions to investigate the inﬂuence of boundary condition on the sound radiation of a plate under a
harmonically excited point force. Qiao and Huang [11] found that boundary conditions have a large eﬀect on the sound radiated from rectangular plates. Jeyaraj
et al. [12,13] studied the eﬀect of thermal loading on vibration and acoustic
response of isotropic and composite plates. They found that the overall sound
power level of an isotropic plate is signiﬁcantly aﬀected by thermal load compared
to composite plate.
The literature survey reveals that the eﬀect of thermal loading has been rarely
included during sound radiation prediction of multi layered viscoelastic sandwich
plates. The present work investigates the eﬀect of thermal loading on the vibration
and sound radiation characteristics of a multilayered viscoelastic sandwich plate
under a thermal environment subjected to time-varying harmonic excitation.
Finite element formulation
Heat transfer analysis
A two-dimensional four-noded rectangular element is used to obtain the temperature distribution on the plate. The two-dimensional steady-state heat conduction
equation without internal heat generation is:
2
@ T @2 T
K
þ
¼0
@x2 @y2
ð1Þ
where K is thermal conductivity and T is the temperature. The variational form of
the above governing equation is:
I¼
1
2
Z
V
frTgT ½KfrTgdV þ
1
2
Z
hT2 dS S1
Z
Z
hT1 dS þ
S1
qTdS
ð2Þ
S2
where frTg is temperature gradient vector, S1 is convection heat transfer boundary, S2 is heat ﬂux speciﬁed boundary, h is the convection heat transfer co-eﬃcient,
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Journal of Sandwich Structures and Materials 13(5)
T1 is the ambient temperature, and q is the heat ﬂux. Following the ﬁnite element
procedure and minimizing the above variational expression with respect to nodal
temperature fTe g, one can obtain:
ð½K1 þ ½K2 ÞfTe g ¼ P1 þ P2
ð3Þ
where, the conduction matrix is given by:
Z
½Bt T ½K½Bt dV
ð4Þ
while the convection matrix is derived as follows:
Z
½K2 ¼ h
fNt gT fNt gdS
ð5Þ
½K1 ¼
V
S1
The load vector due to convection can be obtained as:
Z
fP1 g ¼ hT1
fNt g dS
ð6Þ
S1
while the load vector due to ﬂux is generated as shown below:
Z
fNt gT dS
fP2 g ¼ q
ð7Þ
S2
In the above equations ½Bt is the temperature gradient matrix and fNt g is the
shape function matrix for temperature. Temperature ﬁeld in the domain V can be
obtained by solving Equation (3). The reader is referred to Lewis et al. [14] to get
detailed information of the ﬁnite element formulation used for heat transfer
analysis.
Structural analysis
The displacement based formulation proposed by Khatua and Cheung [15] is used
for the structural analysis. Thermal buckling, free, and forced vibration behavior of
a multilayered sandwich plate is characterized by extending their theory. It is
assumed that complex shear modulus (G) of the viscoelastic core is temperature
dependent and dissipation in the core is only due to transverse shear:
GðTÞ ¼ G ðTÞ ð1 þ iðTÞÞ
ð8Þ
where G is real part of shear modulus while is material loss factor. The transverse shear in the stiﬀ layers and the temperature rise in the core due to shear stress
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Jeyaraj et al.
513
Figure 1. Multilayer rectangular sandwich plate element [15].
dissipation are neglected. The steady-state temperature ﬁeld is assumed throughout
the analysis.
A rectangular multilayer vicoelastic sandwich plate consists of n stiﬀ-layers and
(n1) core layers. The i-th and (i + 1)-th stiﬀ-layer and j-th sandwiched core are
shown in Figure 1. The DOF associated with the k-th node is:
fk g ¼ f! x y u1 v1 . . . ui vi un vn g
ð9Þ
@w
It is assumed that transverse displacement !, bending slopes @w
and
@x
@y are
common for all layers. The number of in-plane degrees of freedom ui and vi are
equal to the number of stiﬀ layers. The array of nodal DOF is given by:
fe g ¼ ff1 g f2 g
f3 g
f4 gg
ð10Þ
The displacement ﬁeld within the element can be related to the nodal DOF as:
vi gT ¼ ½N fe g
f! u i
8 2
@ !
>
>
< @x2
2
f"gT ¼ @@x!2
>
>
:
2
@@y!2
2
@@y!2
xzðn1Þ
yzðn1Þ
@2 !
2@x@y
@2 !
2@x@y
2
@@x!2
@u1
@x
@ui
@x
2
@@y!2
@v1
@y
@vi
@y
@2 !
2@x@y
@u1
@v1
@y þ @x
@ui
@vi
@y þ @x
@un
@x
ð11Þ
xz1
yz1
xzi
yzi
@vn
@y
@un
@y
þ
9
>
>
=
@vn
@x
>
>
;
ð12Þ
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514
Journal of Sandwich Structures and Materials 13(5)
xzi and yzi are the transverse shear strains in the j-th core given by:
xzi
Cj uiþ1 ui @!
¼
þ
@x
hj
Cj
ð13Þ
yzi
Cj viþ1 vi @!
¼
þ
@y
hj
Cj
ð14Þ
Cj ¼ hj þ
t t iþ1
i
2
ð15Þ
The strains can be related to the nodal degrees of freedom by the following
relation:
f"g ¼ ½B fe g
ð16Þ
where ½B is the strain displacement matrix. The structural stiﬀness and mass matrices can be obtained as:
ZaZa
½K ¼
½BT ½D½Bdxdy
ð17Þ
a a
a Z a
Z
½NT P ½Ndxdy
½ M ¼
a
ð18Þ
a
where ½D is the property matrix and P is the mass density matrix. Since the
shear modulus of the core is complex, the structural stiﬀness matrix is complex and
can be written as:
½K ¼ ½KR þ i½KI ð19Þ
where ½KR and ½KI are the real and imaginary parts of the structural stiﬀness
matrices respectively. The thermal load vector can be written as:
Z
a
Z
a
½BT ½Df"0 gdxdy
fFth g ¼
a
ð20Þ
a
where f"0 g is the thermal free-expansion thermal strains and is given by:
f0 0
f" 0 g ¼ 0 0
0 0
0
0
0
i T i T
i T i T
i T i T
0
0
0
0 0
0 0
0 0
g
ð21Þ
where i is the co-eﬃcient of thermal expansion of the ith layer and T is the uniform temperature rise above the ambient temperature. The expression for
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Jeyaraj et al.
515
nonlinear strains is given by:
( )T
@! 2 @! 2
f"nl g ¼
,
, n times
@x
@y
ð22Þ
which can be related to the nodal DOF as:
f"nl g ¼ Bg fe g
ð23Þ
The geometric stiﬀness matrix is given by:
Z
a
Z
a
½K ¼
a
T
Bg ½0 Bg dxdy
ð24Þ
a
where Bg is the nonlinear strain displacement matrix and ½0 is the matrix of
initial stresses in the element. The reader is referred to Pradeep [16] for more details
regarding the formulation.
Analysis approach
Finite element method (FEM) is used to ﬁnd the critical buckling temperature,
eﬀects of thermal load on the natural frequencies, and vibration response of a
multilayered viscoelastic sandwich plate. The thermal load is assumed to be created
in the plate due to a uniform or linearly varying temperature distribution across the
surface of the plate. The thermal load applied on the structure will induce membrane forces, which in turn inﬂuence the lateral deﬂections associated with the
plate. The resistance to bending deformation is reduced when membrane forces
are compressive. These pre-loads on the plate due to the thermal environment are
calculated using a static analysis. The pre-stressed modal and harmonic analysis are
carried out by keeping critical buckling temperature as a parameter to analyze the
eﬀect of thermal load on the natural frequencies and vibration response, respectively. The sound power level of the plate is calculated using SYSNOISE by assuming that the entire plate is vibrating with an average rms velocity at each frequency.
The entire analysis approach is summarized in Figure 2.
When the temperature of the plate is raised from the ambient byT, thermal
stresses develop in the plate (for any boundary condition with at least one edge
restrained). This stress state (static) is used to calculate the geometric stiﬀness
matrix ½K . Following this a buckling analysis is carried out using the structural
and geometric stiﬀness matrices ½KR and ½K :
ð½KR þ i ½K Þ i ¼ 0
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ð25Þ
516
Journal of Sandwich Structures and Materials 13(5)
Thermal
Environment
(Uniform
Temperature rise
above ambient)
Prestress
eﬀect
(Stac
Analysis)
Crical
Buckling
Temperature
(Tcr)
(Buckling
Analysis)
Tcr as a parameter
Natural Frequencies
and mode shapes
(Pre-stressed modal analysis)
Vibraon response
(Pre-stressed harmonic
analysis)
Evaluaon of space averaged normal
velocity (from pre-stressed harmonic
response analysis results)
Sound radiaon calculaon (using
combined FEM/BEM method with the
help of space averaged normal velocity)
Figure 2. A flowchart of analysis approach.
where, i is the i-th eigenvalue and
is the corresponding eigenvector.
i
The product of lowest eigenvalue 1 and the temperature rise T yields the critical buckling temperature, Tcr, that is Tcr = 1 T. Physically, Tcr deﬁnes the temperature at which the plate buckles due to thermal stresses. This is valid when
the material properties are temperature independent. As the material properties
shear modulus (G) and material loss factor () of viscoelastic core are temperature
dependant, an iterative solution technique is adopted to ﬁnd the critical buckling temperature. The value of 1 is calculated for some temperature rise T.
Taking 1 T as a new reference temperature 1 is recalculated. This procedure
is repeated until 1 becomes unity. The corresponding T is the critical buckling
temperature.
Since the structure is pre-loaded due to the thermal ﬁeld, the natural frequencies
of the structure are modiﬁed as these loads produce stresses which change the
structural stiﬀness. Pre-stressed modal analysis is carried out to ﬁnd the natural
frequency of the pre-loaded structure. The natural frequency at any given temperature can be calculated by evaluating the geometric stiﬀness matrix at that temperature and by solving the eigenvalue problem as given below:
ð½KR þ ½K Þ !2k ½M f
k g ¼ 0
ð26Þ
where, ½M is the structural mass matrix, while !k is the circular natural frequency
of the pre-stressed structure, and f
k g is the corresponding mode shape. Similarly
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Jeyaraj et al.
517
k-th modal loss factor (k ) at any given temperature can be obtained from the
following equation:
k ¼
f
k gT ½KI f
k g
f
k gT ½KR þ K f
k g
ð27Þ
After the computation of the natural frequencies, loss factors and mode shapes a
pre-stressed harmonic response analysis is carried out to ﬁnd the vibration response
of the pre-loaded structure. The general equation of motion for a pre-stressed
structure is:
½M U€ þ ½C U_ þ ð½KR þ ½K Þ fUg ¼ FftÞ
ð28Þ
where FðtÞg the applied load vector (assumed time-harmonic), U€ , U_ and fUg
are the acceleration, velocity, and displacement vector of the plate. Using a set of
modal co-ordinates yk which can be deﬁned as:
fU g ¼
n
X
f
k gyk
ð29Þ
k¼1
The Equation (29) can be written using modal co-ordinates as:
½ M
n
X
k¼1
f
k gy€ k þ ½C
n
X
f
k gy_ k þ ð½KR þ ½K Þ
k¼1
n
X
f
k gyk ¼ FðtÞ
ð30Þ
k¼1
Pre multiplying the above equation by f
k gT and after applying the orthogonal
and normal conditions Equation (30) becomes:
y€ k þ 2!k k y_ k þ !2i yk ¼ Fk
ð31Þ
where Fk ¼ f
k gT FðtÞ . Since "k ¼ 2k :
y€ k þ !k k y_ k þ !2i yk ¼ Fk
ð32Þ
The vibration response is obtained by solving the above uncoupled equation.
The boundary element model (BEM) for computing the surface pressure from the
surface displacement is:
½H Pf ¼ ½G uf
ð33Þ
Where ½H and ½G are boundary integral inﬂuence matrices, Pf and uf are
acoustic pressure and displacement of the boundary element nodes of the ﬂuids
respectively at the ﬂuid-structure interface, and is density of the ﬂuid. The reader
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518
Journal of Sandwich Structures and Materials 13(5)
is referred to Wu et al. [17] for detailed information on the formulation of indirect
BEM/FEM.
Validation studies
Critical buckling temperature (Tcr)
A 0.5 m0.5 m three-layered square viscoelastic sandwich plate clamped at its
edges analyzed by Pradeep [16] is considered for the validation of critical buckling
temperature evaluation. Thickness of each layer is equal to 3 mm and the stiﬀ layers
are made of aluminum while viscoelastic core material is EC2216.
The material properties of aluminum are as follows: Young’s modulus
E = 70 GPa, Poisson’s ratio = 0.33, coeﬃcient of thermal expansion
a = 20 106/ C, and density =2700 kg/m3. The temperature-dependent
material properties of EC2216 and DYAD606 are given in Figure 3. The temperature distribution is assumed to be uniform throughout the plate surface.
Pradeep [16] used the same Khatua and Cheung [15] ﬁnite element formulation
to obtain the critical buckling temperature. The critical buckling temperature obtained using present method is 60 C which matches well with the value
of 63 C reported by Pradeep [16]. The error is due to the diﬀerence in mesh
size of (6 6) used by Pradeep [16], which is coarser than the mesh used here
(16 16).
Natural frequencies and loss factors
A rectangular viscoelastic sandwich plate with isotropic facings analyzed by Cupial
and Niziol [4] is considered for validation of natural frequencies and loss factors.
Cupial and Niziol [4] used a simpliﬁed analytical model, which did not account for
shear deformation in the face layers. Wang et al. [6] carried out experiments to
(a)
(b) 1.2
9
EC 2216 core
DYAD 606 core
9
2.5x10
9
2.0x10
9
1.5x10
9
1.0x10
EC 2216 core
DYAD 606 core
1.0
Loss factor
Shear modulus (N/m2 )
3.0x10
0.8
0.6
0.4
8
5.0x10
0.2
0.0
0.0
0
20
40 60 80 100 120 140 160
Temperature (°C)
0
20
40 60 80 100 120 140 160
Temperature (°C)
Figure 3. Temperature-dependant material properties of visco-elastic core materials EC2216
and DYAD 606 [18]: (a) shear modulus, (b) material loss factor.
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Jeyaraj et al.
519
verify the natural frequencies and associated modal loss factors with Cupial and
Niziol [4]. The stiﬀ layers are of equal thickness and made of Aluminum. The
dimensions of the plate are a = 0.3048 m; b = 0.3480 m; thickness of stiﬀ layer
ts = 0.762 mm; thickness of core layer tc = 0.254 mm; Young’s modulus of stiﬀ
layer Es = 68.9 GPa; Poisson’s ratio of stiﬀ layer s = 0.3; density of stiﬀ layer
s = 2740 kg/m3 density of core layer c = 999 kg/m3 and shear modulus of the
core Gc = 0.896(1 + 0.5i) MPa.
From Table 1, it is clear that both the natural frequencies and modal loss factors
obtained using the present method matches well with the results reported by Cupial
and Niziol [4] and Wang et al. [6]. The small error is due to the inclusion of shear
deformation between the layers in the present formulation.
Sound radiation
A three-layered viscoelastic cantilever sandwich plate with dimensions
0.5 m 0.5 m 6 mm is considered for validation of sound radiation. The stiﬀlayers are made of aluminum while the core is EC2216. The ﬁnite element formulated based on the Khatua and Cheung [15] is validated with the commercial ﬁnite
element code ANSYS for forced vibration response calculations. As the combined
FEM/BEM needs a shell element for sound radiation calculation, SHELL 99, an
eight-noded linear structural layered shell element is used to model the sandwich
plate. Table 2 shows the comparison of natural frequencies obtained using ANSYS
with present FE formulation.
A time-varying harmonic excitation of 1 N is applied at the lower right corner of
the plate and a damping ratio of 0.01 is assumed for all the modes considered in the
harmonic response analysis. The plate is assumed to be vibrating in air whose
density is:
a = 1.21 kg/m3 with a speed of sound c = 343 m / s. In the present work sound
radiation characteristics are obtained by assuming that the entire plate is vibrating
Table 1. Comparison of natural frequencies and modal loss factors with Cupial and Niziol [4]
and Wang et al. [6]
Natural frequency (Hz)
Modal loss factor
Modal indices
Cupial and
Niziol [4]
(Analytical)
Wang et al. [6]
(Experimental)
Present
Cupial and
Niziol [4]
(Analytical)
Wang et al. [6]
(Experimental)
Present
(1,1)
(1,2)
(2,1)
(2,2)
(1,3)
60
115
131
179
196
60
115
130
178
195
63
116
132
180
197
0.190
0.203
0.199
0.181
0.174
0.192
0.203
0.198
0.179
0.172
0.182
0.192
0.187
0.172
0.163
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Journal of Sandwich Structures and Materials 13(5)
Table 2. Comparison of natural frequencies (Hz) with ANSYS
Mode
Present formulation
ANSYS
1
2
3
4
5
22
53
134
171
192
22
53
135
173
195
with a spatially averaged rms velocity at a particular frequency. To check the
validity of the sound radiation calculated based on the average rms velocity, results
obtained using:
(i) usual combined FEM/BEM (ANSYS – to obtain the vibration response and
SYSNOISE – to obtain the sound power level)
(ii) average rms velocity obtained using present formulation as an input to
SYSNOISE are compared as shown in Figure 4.
To understand why the concept of replacing the actual velocity distribution with
a spatially averaged uniform rms velocity works, the alternative expression for
sound power (using Rayleigh’s Integral) as shown by Williams [18] is considered:
Z Z Z Z
0 ck2
sinðkRÞ ð!Þ ¼
U ðx, yÞdsds0
U_ ðx0 , y0 Þ
ð34Þ
kR
4
S
S
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
In the above equation R ¼ ðx x0 Þ2 þðy y0 Þ2 where ðx yÞ and ðx0 y0 Þ are
co-ordinates of two diﬀerent point on the plate, S or S’ represents the plate area, U_
is the normal velocity of the plate (assumed to be harmonic), and U_ is its complex
conjugate. The sinc function sinðkRÞ
kR in the integral indicates that the major contribution to the integral is when ðx yÞ is close to ðx0 y0 Þ and in the limit ðkRÞ ! 0
the sinc function becomes the Dirac delta function ðRÞ. With this in mind one can
then approximate the above equation as:
ð!Þ 0 ck2
4
Z Z
U_ ðx, yÞ2 : ds
ð35Þ
S
From the above equation one can clearly see that the sound power is proportional to the square of the plate velocity integrated over the whole area of the plate.
Now if one decides to replace by the spatial average of the velocity deﬁned as:
Z Z
2 1
U_ ðx, yÞ2 ds
_
U ¼
ð36Þ
S
S
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Jeyaraj et al.
521
Figure 4. Comparison of sound variation.
then the sound power simply becomes:
ð!Þ ¼
0 cSk2 _ 2
U
4
ð37Þ
The above series of equations then oﬀer the rationale as to why the simple idea
of replacing a complex velocity pattern by a uniform spatially averaged value gives
fairly accurate estimates of sound power for plates. From Figure 4 which shows the
comparison of results obtained with sound power obtained using the formula given
in Equation (37), it is clear that there exists a good agreement between the results
obtained using the three approaches except at some oﬀ-resonance frequencies.
Results and discussion
A 0.5 m 0.5 m multilayered viscoelastic sandwich plate clamped at its edges is
now considered for the detailed investigation. The facings are made of aluminum
and two diﬀerent viscoelastic core materials considered are EC2216 and
DYAD606. The material properties of aluminum are as follows: Young’s modulus
E = 70 GPa, Poisson’s ratio = 0.33, coeﬃcient of thermal expansion
a = 20 106/ C and density = 2700 kg/m3. Shear modulus (G*) and material
loss factor () of the viscoelastic core are two important factors which inﬂuence the
vibration and damping behavior of the sandwich plates. In the present work, it is
assumed that both the shear modulus and material loss factors are temperature
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522
Journal of Sandwich Structures and Materials 13(5)
dependent and variation of these properties with temperature are taken from
Nashif et al. [19]. A curve ﬁt is done so that material properties can be obtained
at any temperature in the operating range. Figure 3 shows the temperature dependent material properties of EC2216 and DYAD606 from Nashif et al. [19].
A convergence study has been carried out for both critical buckling temperature
and natural frequencies for a three-layered sandwich plate having EC2216 as a core
with tc = ts = 2 mm at room temperature. Table 3 shows the results.
Based on the convergence study the plate is modeled using a 16 16 mesh. This
mesh size also satisﬁes the six elements per wavelength requirement for numerical
vibro-acoustic analysis using combined FEM/BEM.
Thermal buckling studies
In the present work, the vibration and acoustic response of a multilayered viscoelastic sandwich plate has been analyzed by assuming that the structure is subjected
to an uniform temperature rise above the ambient temperature. Even though the
temperature rise is assumed to be uniform for all the cases analyzed to carry out
diﬀerent parameter studies, in one case the plate is analyzed for both uniform
temperature distribution and linearly varying temperature distribution in order
to analyze the inﬂuence of a diﬀerent thermal environment on vibration and acoustic response characteristics. The temperature rise applied on the plate is varied from
0 C to Tcr and corresponding variation in natural frequencies, loss factors, vibration, and acoustic response has been analyzed. To start with, the critical buckling
temperature is obtained for diﬀerent core thickness, diﬀerent temperature ﬁeld and
diﬀerent number of layers. A three-layered viscoelastic sandwich plate having a
stiﬀ layer thickness of 2 mm is analyzed by varying the core thickness as 2 mm,
4 mm, and 6 mm in order to investigate the inﬂuence of core thickness on vibration and acoustic response characteristics. A three-layered sandwich plate with
tc
ts ¼ 1 is analyzed for two diﬀerent thermal ﬁelds namely uniform temperature
distribution and linearly varying temperature distribution to investigate the inﬂuence of nature of temperature distribution on vibration and acoustic response characteristics. For linearly varying temperature distribution case, three edges
of the square plate are held at ambient temperature while temperature on
the remaining edge is varied till the thermal buckling occurs. The same plate is
analyzed for EC2216 and DYAD606 core materials. Table 4 shows the critical
buckling temperature obtained for diﬀerent parameters of a three-layered sandwich
plate.
A viscoelastic plate having dimensions 0.5 m 0.5 m 0.01 m is considered to
investigate the inﬂuence of number of layers on vibration and sound radiation
characteristics. The plate is analyzed for three, ﬁve, and seven total layers. In all
these cases it is assumed that all layers are of equal thickness. In each case thickness
of each layer is equal to thickness of plate divided by the total number of layers.
The critical buckling temperatures of three-, ﬁve-, and seven-layered plates are
119 C, 106 C, and 101 C. The critical buckling temperature reduces with increase
Downloaded from jsm.sagepub.com at Indian Inst Of Tech Madras on December 14, 2014
Critical buckling temperature Tcr ( C)
Natural frequencies (Hz)
(Modal loss factor) x 102(1,1)
(2,1)
(1,2)
(2,2)
(3,1)
(3.69)
(3.69)
(4.97)
(5.75)
459
459
663
817
496
496
726
936
(5.18)
(5.18)
(7.85)
(8.43)
62
227 (2.19)
10 10
63
236 (3.04)
66
Mesh size
454
454
656
801
(3.54)
(3.54)
(4.65)
(5.45)
61
225 (2.12)
12 12
Table 3. Convergence study for critical buckling temperature and natural frequencies
451
451
562
792
(3.46)
(3.46)
(4.52)
(5.27)
61
225 (2.08)
14 14
449
449
649
787
(3.41)
(3.41)
(4.44)
(5.17)
61
224 (2.06)
16 16
449
449
648
785
(3.40)
(3.40)
(4.42)
(5.15)
61
224 (2.05)
18 18
Jeyaraj et al.
523
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524
Journal of Sandwich Structures and Materials 13(5)
Table 4. Critical buckling temperature (Tcr, C) of three layered plates
Tcr
Core
Temperature distribution
tc
ts
EC 2216
Uniform
Linearly varying
Uniform
60
91
49
DYAD606
¼1
tc
ts
¼2
90
151
51
tc
ts
¼3
126
220
53
in number of layers as the stiﬀness of the plate reduces with increase in number of
layers. This is due to reduction in layer thickness with increase in number of layers.
Free vibration and modal damping studies
Pre-stressed modal analysis is carried out for diﬀerent values of core thickness and
diﬀerent number of layers by varying the temperature rise above the ambient temperature from 0 C to Tcr C in order to ﬁnd the inﬂuence of thermal environment
on the natural frequencies and corresponding modal loss factors and mode shapes.
The results obtained from the pre-stressed modal analysis for the three-layered
sandwich plate are given in Table 5.
Table 6 shows the natural frequency and loss factors for some of the modes of the
plate analyzed for the inﬂuence of number of layers. From Tables 5 and 6 it is clear
that the natural frequencies of the plate reduce with increase in the fraction of the
critical buckling temperature and the natural frequency of the fundamental mode
approaches zero as the temperature rise applied reaches critical buckling temperature.
This is because reduction in the stiﬀness of structure reduces with increase in temperature due to the compressive thermal stresses. This phenomenon happens irrespective
of ttcs , type of core material, type of thermal environment, and number of layers.
The mode shapes are not signiﬁcantly aﬀected by thermal environment. The membrane stress distribution pattern due to thermal load is the same for all temperatures
even though the magnitude of the membrane stresses increases with temperature.
This is due to symmetric boundary conditions associated with the plate analyzed.
Vibration and acoustic response studies
In order to analyze the vibration and acoustic response characteristics of the multilayered viscoelastic sandwich plate, a frequency range of 0–1500 Hz is chosen.
Before carrying out the harmonic response analysis, an appropriate excitation
location is chosen using the mode shapes of the plate. The location of excitation
is chosen in such a way that it does not lie on the nodal lines of modes in the
frequency range of 0–1500 Hz; this is done at room temperature but since the mode
shapes are independent of temperature the excitation location would still not coincide with any nodal lines.
Downloaded from jsm.sagepub.com at Indian Inst Of Tech Madras on December 14, 2014
Uniform
temperature
rise
EC2216 (ttcs ¼ 3)
Linear
temperature
rise
EC2216 (ttcs ¼ 2)
Uniform
temperature
rise
EC2216 (ttcs ¼ 1)
EC2216 (ttcs ¼ 1)
0Tcr
0.25Tcr
0.5Tcr
0.75Tcr
0.95Tcr
0Tcr
0.25Tcr
0.5Tcr
0.75Tcr
0.95Tcr
0Tcr
0.25Tcr
0.5Tcr
0.75Tcr
0.95Tcr
0Tcr
0.25Tcr
T
224 (0.021)
195 (0.033)
161 (0.044)
118 (0.073)
61 (1.292)
224 (0.021)
195 (0.0314)
158 (0.0442)
113 (0.0788)
50 (0.3661)
290 (0.0388)
246 (0.0591)
193 (0.0372)
135 (0.0562)
152 (0.3101)
344 (0.0274)
283 (0.0725)
Mode (1,1)
449
409
370
326
286
449
410
366
320
276
569
498
429
370
319
662
551
(0.0341)
(0.049)
(0.055)
(0.062)
(0.374)
(0.0341)
(0.0468)
(0.0549)
(0.0634)
(0.0776)
(0.0621)
(0.0833)
(0.0435)
(0.0428)
(0.0492)
(0.0421)
(0.0972)
Mode (1,2)
Natural frequencies (loss factors)
449
409
370
326
286
449
410
366
324
288
569
498
429
370
319
662
551
(0.0341)
(0.049)
(0.055)
(0.062)
(0.374)
(0.0341)
(0.0468)
(0.0549)
(0.0610)
(0.0690)
(0.0624)
(0.0832)
(0.0434)
(0.0421)
(0.0492)
(0.0421)
(0.0973)
Mode (2,1)
Table 5. Natural frequencies (Hz) and loss factors of the three layered sandwich plate
649
600
554
506
464
649
602
552
506
468
809
712
627
562
511
931
774
(0.0442)
(0.062)
(0.065)
(0.068)
(0.369)
(0.0442)
(0.0574)
(0.0622)
(0.0662)
(0.0690)
(0.0774)
(0.0983)
(0.0474)
(0.0445)
(0.0412)
(0.0517)
(0.0111)
Mode (2,2)
Downloaded from jsm.sagepub.com at Indian Inst Of Tech Madras on December 14, 2014
(continued)
787 (0.0522)
730 (0.071)
681 (0.073)
629 (0.074)
586 (0.388)
787 (0.0522)
732 (0.0688)
676 (0.0701)
622 (0.0732)
575 (0.0754)
970 (0.0089)
853 (0.0109)
756 (0.0512)
687 (0.0561)
636 (0.0408)
1105 (0.0581)
912 (0.1207)
Mode (3,1)
Jeyaraj et al.
525
Uniform
temperature
rise
Uniform
temperature
rise
DYAD606 (ttcs ¼ 1)
Table 5. Continued
0.5Tcr
0.75Tcr
0.95Tcr
0Tcr
0.25Tcr
0.5Tcr
0.75Tcr
0.95Tcr
T
474
415
353
407
356
297
235
157
225 (0.0714)
166 (0.0855)
81 (0.2611)
211 (0.032)
182 (0.0632)
149 (0.126)
106 (0.323)
32 (4.391)
(0.0856)
(0.0681)
(0.0689)
(0.0486)
(0.0862)
(0.147)
(0.277)
(0.660)
Mode (1,2)
Mode (1,1)
Natural frequencies (loss factors)
474
415
353
407
356
297
235
157
(0.0855)
(0.0684)
(0.0681)
(0.0486)
(0.0862)
(0.147)
(0.277)
(0.660)
Mode (2,1)
680
617
554
575
502
421
342
249
(0.0914)
(0.0682)
(0.0611)
(0.0026)
(0.0977)
(0.0065)
(0.0068)
(0.0068)
Mode (2,2)
807
742
679
684
595
498
409
307
(0.0902)
(0.0711)
(0.0593)
(0.0662)
(0.106)
(0.161)
(0.261)
(0.454)
Mode (3,1)
526
Journal of Sandwich Structures and Materials 13(5)
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7 layers
5 layers
3 layers
0Tcr
0.25Tcr
0.5Tcr
0.75Tcr
0.95Tcr
0Tcr
0.25Tcr
0.5Tcr
0.75Tcr
0.95Tcr
0Tcr
0.25Tcr
0.5Tcr
0.75Tcr
0.95Tcr
T
687
589
502
438
362
427
390
353
311
272
300
274
245
214
196
354 (0.0470)
295 (0.0674)
232 (0.0802)
167 (0.1053)
30 (2.4351)
212 (0.0151)
182 (0.0245)
150 (0.0338)
105 (0.0621)
41 (0.3861)
148 (0.0072)
126 (0.0106)
98 (0.0194)
57 (0.0538)
7 (1.774)
(0.0726)
(0.0907)
(0.0875)
(0.0778)
(0.0826)
(0.0254)
(0.0371)
(0.0425)
(0.0491)
(0.0592)
(0.0117)
(0.0157)
(0.0222)
(0.0277)
(0.0161)
Mode (1,2)
Mode (1,1)
Natural frequencies (loss factors)
687
589
502
438
362
427
390
353
311
272
316
274
245
214
196
(0.0726)
(0.0907)
(0.0875)
(0.0778)
(0.0826)
(0.0254)
(0.0371)
(0.0425)
(0.0491)
(0.0592)
(0.0043)
(0.0157)
(0.0222)
(0.0277)
(0.0161)
Mode (2,1)
971 (0.0882)
837 (0.1025)
728 (0.0922)
661 (0.0760)
586 (0.0703)
541(0.0183)
540 (0.0025)
540 (0.0043)
540 (0.0023)
540 (0.0022)
316 (0.0042)
325 (0.0004)
325 (0.0005)
325 (0.0005)
325 (0.0003)
Mode (2,2)
Table 6. Natural frequencies (Hz) and loss factors of the sandwich plate analyzed for influence of number of layers
1158 (0.100)
995 (0.1119)
873 (0.097)
804 (0.0777)
729 (0.0687)
541 (0.0183)
540 (0.0025)
540 (0.0042)
540 (0.0023)
540 (0.0024)
319 (0.0030)
325 (0.0004)
325 (0.0005)
325 (0.00057)
325 (0.0003)
Mode (3,1)
Jeyaraj et al.
527
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528
Journal of Sandwich Structures and Materials 13(5)
Figures 5–7 show the displacement, velocity, and average rms velocity as the
excitation frequency is varied for a three-layered viscoelastic sandwich plate having
EC2216 as core with ttcs ¼ 1. Two trends can be seen,
(i) The natural frequencies reduce with increasing temperature and
(ii) the resonant amplitudes are decreasing with increase in temperature.
Generally the pre-stress due to thermal load will reduce the stiﬀness of the
structure causing the amplitude of vibration to be increased. This is not clearly
seen in the vibration amplitudes at the resonant frequencies. Even though the prestress reduces the natural frequency, the vibration amplitude at the resonant frequency is inﬂuenced by the modal damping. As the modal damping increases signiﬁcantly with the temperature it reduces the vibration amplitude at the resonant
frequencies as seen in the response curves. The vibration response for three-layered
plate with ttcs ¼ 2 and ttcs ¼ 3 is quite similar.
Figure 8 shows the radiation eﬃciency for three-layered viscoelastic sandwich
plate having ttcs ¼ 1 as a function of frequency. Sound power is directly proportional
to the product of square of the averaged surface normal velocity and radiation
eﬃciency. The expression for critical frequency (fcr) of an isotropic plate given by
Ohlrich and Hugin [20] is:
c2 plate tplate 2
2
D
1
fcr ¼
ð38Þ
Displacement (m)
1E-5
1E-6
1E-7
ΔT/Tcr =0
ΔT/Tcr =0.5
T =60 °C
ΔT/Tcr =0.95
cr
1E-8
0
250
500
750
1000
1250
1500
Harmonic frequency (Hz)
Figure 5. Displacement at the excitation point for three-layered plate (ttcs ¼ 1).
Downloaded from jsm.sagepub.com at Indian Inst Of Tech Madras on December 14, 2014
Jeyaraj et al.
529
0.01
Velocity (m/s)
1E-3
1E-4
1E-5
ΔT/Tcr =0
ΔT/Tcr =0.5
T =60 °C
ΔT/Tcr =0.95
cr
1E-6
0
250
500
750
1000
1250
1500
Harmonic frequency (Hz)
Figure 6. Velocity at excitation point for three-layered plate (ttcs ¼ 1).
Average rms velocity (m/sec)
0.01
1E-3
1E-4
ΔT/Tcr =0
ΔT/Tcr =0.5
ΔT/Tcr =0.95
T =60 °C
cr
1E-5
0
250
500
750
1000
1250
Harmonic frequency (Hz)
Figure 7. Average rms velocity for three-layered plate (ttcs ¼ 1).
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1500
530
Journal of Sandwich Structures and Materials 13(5)
2.0
Radiation efficiency
1.5
1.0
0.5
ΔT/Tcr =0
ΔT/Tcr =0.5
T =60 °C
ΔT/Tcr =0.95
cr
0.0
0
1000
2000
3000
4000
5000
Harmonic frequency (Hz)
Figure 8. Radiation efficiency of three-layered plate (ttcs ¼ 1).
where D is the bending stiﬀness of the plate. The equivalent bending stiﬀness of the
viscoelastic sandwich plate is calculated using the expression given by Altanbach
et al. [21] and substituted in Equation [38] to obtain the critical frequency of the
sandwich plate. The critical frequency obtained for the sandwich plate analyzed is
2206 Hz. Figure 8 shows a peak around this frequency. From Figure 8 it can be
clearly seen that the radiation eﬃciency of the plate generally decreases (no significant variation in higher frequency range) with increase in temperature.
The sound power level shown in Figure 9 reﬂects the average rms velocity
response as sound radiation is directly related to normal velocity of the structure.
To analyze further, average of mean square velocity is calculated for three-layered
plate for diﬀerent values of ttcs in the entire frequency band as shown in Figure 10
and for constant bandwidth frequency bands (250 Hz) as shown in Figure 11 for
three-layered plate with ttcs ¼ 1. From Figures 10 and 11 one can see that the velocity
generally decreases with temperature. When the uniform temperature rise reaches
the critical buckling temperature, there is a marginal increase in overall average
rms velocity. From the band wise representation it is also clear that the rms velocity
is higher when there is no rise in uniform temperature (except at the lower band)
and is lower when the uniform temperature rise approaches the critical buckling
temperature. This is due to signiﬁcant increase in modal loss factor when the uniform temperature rise is nearer to the critical buckling temperature of the structure.
In the lower frequency band, vibration response is inﬂuenced by the stiﬀness of the
structure; it can be clearly seen in the displacement, velocity, and average rms
velocity responses.
Downloaded from jsm.sagepub.com at Indian Inst Of Tech Madras on December 14, 2014
Jeyaraj et al.
531
Sound power level (dB)
100
80
60
40
ΔT/Tcr =0
ΔT/Tcr =0.5
T =60 °C
ΔT/Tcr =0.95
cr
20
0
250
500
750
1000
1250
1500
Harmonic frequency (Hz)
34
Overall rms velocity (dB, ref 1x10
–6
m/s)
Figure 9. Sound power level for three-layered plate with (ttcs ¼ 1).
32
30
tc/ts=1 (Tcr=60 °C)
28
3 layerd plate
tc/ts=2 (Tcr=90 °C)
tc/ts=3 (Tcr=126 °C)
26
0.00
0.25
0.50
ΔT/Tcr
0.75
Figure 10. Overall average rms velocity for three-layered plate with (ttcs ¼ 1).
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1.00
532
Journal of Sandwich Structures and Materials 13(5)
average rms velocity (dB ref 1x10
–6
m/s)
35
ΔT/Tcr=0
ΔT/Tcr=0.25
ΔT/Tcr=0.75
ΔT/Tcr=0.95
30
ΔT/Tcr=0.5
(Tcr =60°C)
25
20
15
10
0-250
250-500 500-750 750-1000 1000-1250 1250-1500
Frequency band (Hz)
Figure 11. Overall average rms velocity in constant frequency band for three-layered plate
with (ttcs ¼ 1).
To get a clearer picture, the overall sound power level for the entire frequency
band is computed for three-layered plate with diﬀerent ttcs ratios and the results are
shown in Figure 12. Figure 13 shows the sound power level represented in constant
bandwidth frequency bands for three-layered plate with ttcs ¼ 1 this shows the shift
towards lower frequencies. The resonant amplitude of vibration and acoustic
response increases with temperature rise for an isotropic plate [12] while the resonant amplitudes are decreasing with increase in temperature rise for the viscoelastic
sandwich plate. Even though the structural stiﬀness reduces with the increase in
temperature rise, the modal loss factor reduces the resonant amplitude as it is
increasing signiﬁcantly with temperature rise for the sandwich plate. Due to this
reason, there is no signiﬁcant change in overall sound power level of the sandwich
plate also.
From Figure 14, which shows the comparison of overall sound power level for
two diﬀerent thermal ﬁelds, it is clear that there is no signiﬁcant variation in overall
sound power level. As already explained the mode shapes are not signiﬁcantly
inﬂuenced by the two diﬀerent temperature ﬁelds considered. Because of the symmetric boundary condition, the bending modes are not aﬀected by the in-plane
stresses developed due to thermal pre-load. From Table 6 one can observe that
variation in natural frequencies and loss factors with increase in temperature for
both the thermal ﬁelds are same. This indicates that the change in structural stiﬀness is the same irrespective of the nature of temperature distribution.
Figure 15 shows the inﬂuence of type of core material on overall sound power
level of a three-layered viscoelastic sandwich plate with ttcs ¼ 1. From Figure 15, it is
clear that the overall sound power level decreases with increase in temperature rise
Downloaded from jsm.sagepub.com at Indian Inst Of Tech Madras on December 14, 2014
Jeyaraj et al.
533
Overall sound power level (dB)
110
108
106
104
102
tc/ts =1 (Tcr=60°C)
tc/ts =2 (Tcr=90°C)
3 layerd plate
100
tc/ts =3 (Tcr=126°C)
98
0.00
0.25
0.50
0.75
1.00
ΔT/Tcr
Figure 12. Influence of core thickness on overall sound power level for three-layered plate.
Overall sound power level (dB)
110
ΔT/Tcr =0
ΔT/Tcr =0.25
ΔT/Tcr =0.75
ΔT/Tcr =0.95
ΔT/Tcr =0.5
T =60°C
cr
105
100
95
90
85
0-250
250-500
500-750 750-1000 1000-1250 1250-1500
Frequency band (Hz)
Figure 13. Overall sound power level in constant frequency band for three-layered plate
with (ttcs ¼ 1).
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534
Journal of Sandwich Structures and Materials 13(5)
Overall sound power level (dB)
110
109
108
107
106
Uniform temperature rise (Tcr =60 °C)
105
Linearly varying temperature (Tcr =91 °C)
104
0.00
0.25
0.50
Δ T/ T
0.75
1.00
cr
Figure 14. Influence of type of thermal environment on overall sound power level for threelayered plate with (ttcs ¼ 1).
when the core material is DYAD606 while the overall sound power level is not
signiﬁcantly aﬀected when the core material is EC2216. Generally damping due to
DYAD606 core is more compared to EC2216 due to high material loss factor ()
associated with DYAD606. There is a signiﬁcant increase in material loss factor ()
for DYAD606 compared to EC2216, in the temperature range analyzed for the
present study. This can be clearly seen in Figure 3. This reﬂects the overall sound
power level variation with temperature shown in Figure 15.
Figure 16 shows the overall sound power level for diﬀerent total number of
layers. From Figure 16 one can see that overall sound power level increases with
the total number of layers. The core thickness reduces with increase in number of
layers, which in turn reduces the stiﬀness of the plate as seen in Table 6. The overall
sound power level increases with uniform temperature rise for three-layered plate
while there is no signiﬁcant change for ﬁve- and seven-layered plate. Even though
the modal damping increases with uniform temperature rise for three-layered plate
as seen in Table 6, the overall sound power level is inﬂuenced by stiﬀness of the
plate. This reﬂects the overall sound power level shown in Figure 16, which
increases with temperature. There is no signiﬁcant change in overall sound
power level with temperature for ﬁve-layered and seven-layered plates compared
to three-layered plates.
Downloaded from jsm.sagepub.com at Indian Inst Of Tech Madras on December 14, 2014
Jeyaraj et al.
535
Overall sound power level (dB)
110
108
106
EC2216 (Tcr =60°C)
104
DYAD606 (Tcr =49°C)
102
100
98
0.00
0.25
0.50
0.75
1.00
ΔT / Tcr
Figure 15. Influence of type of core material on overall sound power level for three-layered
plate with (ttcs ¼ 1).
107
Overall sound power level (dB)
106
105
104
103
102
101
3 layers (Tcr = 119°C)
100
5 layers (Tcr = 106°C)
7 layers (Tcr = 101°C)
99
0.00
0.25
0.50
ΔT/Tcr
0.75
Figure 16. Influence of number of layers on overall sound power level.
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1.00
536
Journal of Sandwich Structures and Materials 13(5)
Conclusion
The eﬀect of a thermal environment on the vibration response and consequent
sound radiation from a multilayered viscoelastic sandwich under a thermal environment is investigated. It is found that the amplitudes of vibration and sound
power at the resonant frequencies decrease with the increase in temperature. The
resonant amplitude is less when the uniform temperature rise approaches the critical buckling temperature of the structure. Overall sound power level increases with
core thickness and number of layers. The type of thermal environment does not
aﬀect the overall sound power level signiﬁcantly while type of core material inﬂuences signiﬁcantly.
Funding
This research received no speciﬁc grant from any funding agency in the public, commercial,
or not-for-proﬁt sectors.
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