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Sound transmission from stiffened double laminated composite plates
Accepted Manuscript Sound transmission from stiffened double laminated composite plates Tao Fu, Zhaobo Chen, Dong Yu, Xiaoyu Wang, Wenxiang Lu PII: DOI: Reference:
S0165-2125(17)30051-3 http://dx.doi.org/10.1016/j.wavemoti.2017.04.007 WAMOT 2156
To appear in:
Wave Motion
Received date: 1 March 2017 Accepted date: 6 April 2017 Please cite this article as: T. Fu, Z. Chen, D. Yu, X. Wang, W. Lu, Sound transmission from stiffened double laminated composite plates, Wave Motion (2017), http://dx.doi.org/10.1016/j.wavemoti.2017.04.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Research Highlights
Highlights 1)First order shear deformation theory and double panel sandwich structure model are adopted to develop an analytical model for investigating the vibroacoustic characteristic of an orthogonally rib-stiffened double laminated composite plates. 2)A further comprehension was given on the underlying mechanism of sound transmission through a rib-stiffened double laminated composite plates.
*Manuscript (Clear) Click here to view linked References
1 2 3 4 5 6 7
Sound transmission from stiffened double laminated composite plates Tao Fua, Zhaobo Chena,1, Dong Yua, Xiaoyu Wangb, Wenxiang Luc a School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001,PR China. b Beijing Institute of Spacecraft System Engineering, Beijing 100094,PR China. c School of Ocean Transportation, University of Shan Dong Jiao Tong , Jinan 264200,PR China.
8 9
Abstract: An analytical model is developed to investigate the sound transmission loss from
10
orthogonally rib-stiffened double laminated composite plates structure under a plane sound wave
11
excitation, in which first order shear deformation theory is presented for laminated composite
12
plates. By using the space harmonic approach and virtual work principle, the sound transmission
13
loss is described analytically. The validity and feasibility of the model are verified by comparing
14
the present theoretical predictions with numerical results published previously. The influences of
15
structure geometrical parameters on sound transmission loss are subsequently presented. Through
16
numerical results, it can be concluded that the proposed analytical model is accurate and simple in
17
solving the vibroacoustic behavior of an orthogonally rib-stiffened double laminated composite
18
plates.
19 20 21 22
Keywords: Laminated composite plate; First order shear deformation theory; Vibroacoustic; Sound transmission;
23
1. Introduction
24
In recent years, laminated composite plates have been given more and more attention due to
25
the extensive application, and advantageous features such as high stiffness to weight and low
26
density. Thus many laminated analytical theories have been developed [1-6]. Among them, the
27
classical laminated plate theory (CLPT) and first order shear deformation theory (FSDT) are the
28
commonly used theory for the analysis of laminated composite plates, wherein CLPT is based on
29
the Love-Kirchhoff kinematic hypothesis, and only suit for thin plate because of the neglect of
30
transverse shear deformation, whereas FSDT proposed by Reissner [1] and Mindlin [2] takes into
31
account the effects of shear deformation and applies for both thick and thin laminate composite
32
plates. However, the most limitation of FSDT is that it requires shear correction factors to rectify
33
the unrealistic variation of the transverse shear strain through the thickness [3,4]. In order to
1
Corresponding author. TeL: +86 13904810712 E-mail addresses: [email protected]; [email protected] (T. Fu). 1
34
overcome the limitation of FSDT, the higher-order shear deformation theories(HOSDT) based on
35
an assumption of nonlinear stress variation through the thickness were developed by Reddy [5]
36
and Librescu [6], etc., which can neglect shear correction factors and give more accurate and
37
stable transverse shear strain. Comparing to the single laminated composite plate, however, the
38
periodically rib-stiffened composite plates have been widely used in engineering structures.
39
Further, the rib-stiffeners play an important role in the vibration and acoustic characteristics of the
40
whole structure, particularly when the bending wave length is comparable with the periodical
41
spacing of the stiffeners [7, 8]. Consequently, more attention needs to be paid on the effect of
42
rib-stiffeners.
43
To deal with the coupling connections between the rib-stiffeners and the base thin plate, Lee
44
and Kim [9] replaced the rib-stiffener as translational springs, rotational springs and lumped mass
45
when they studied the sound transmission performance of a single stiffened plate subjected to a
46
plane wave excitation. Similarly, Alba [10] studied the effect of rib-stiffener on sound
47
transmission characteristics of stiffened panel structures. Subsequently, Wang et al. [11] extended
48
the method to a double-leaf partitions connected through vertical resilient studs and investigated
49
the physical mechanisms determining sound transmission. Following the work of Wang et al, Xin
50
[12] proposed a more accurate theoretical model for sound transmission from an orthogonally
51
rib-stiffened sandwich structures. In his study, the effects of rib-stiffeners were included by
52
introducing the tensional forces, bending moments and torsional moments as well as the
53
corresponding inertial terms into the governing equations of the two face panels. In comparison
54
with the rib-stiffeners thin plate, the stiffened laminated composite plate structure increases the
55
complexity of the theoretical modeling [13]. For the stiffened laminated composite plate structure,
56
Yin et al.[14] modeled the rib-stiffener as Bernoulli-Euler beams and studied the acoustic radiation
57
from an infinite stiffened laminated composite cylindrical shell based on the classical laminated
58
composite plate theory. It should be pointed out that the torsional moments and inertial effects of
59
the rib-stiffeners were not considered in the analysis. The same approach was adopted by Cao et al.
60
[15] to investigate the acoustic characteristics of stiffened symmetric and antisymmetric laminated
61
plates. Considering only the bending moments of rib-stiffeners, they used the first order shear
62
deformation theory to describe the equations of motion for the laminated composite plate. A
63
refined theoretical model was proposed later by Xin and Lu [16] for sound radiation from a
64
rib-stiffened plate excited by external mean flow, in which the inertial effects, bending and
65
torsional moments were considered.
66
For the vibroacoustic properties of periodically rib-stiffened infinite plate structures, as
67
reported by Xin et al. [17], fourier transform method is able to handle sound radiation from a
2
68
mechanical point force driven, whereas space-harmonic approach is particularly suited for sound
69
transmission excited by a plane wave. Both of the methods described above should transform the
70
governing equations into infinite sets of simultaneous algebraic equations and then truncate these
71
into a finite range for numerical solutions, as stated in Shen [18]. However, existing literatures on
72
the vibroacoustic behavior of laminated composite plates are generally focused on sound radiation
73
and stiffened single panel structures. Little attention has been paid to the vibrations and sound
74
transmission loss from the orthogonally stiffened double composite panel structures. It is worth
75
noting that when double panel were reinforced by orthogonal stiffening members, their dynamic
76
characteristics are quite different from those of the single stiffened plate.
77
Thus in this paper, the physical process of sound transmission through an orthogonally
78
rib-stiffened double laminated composite plates structures subjected to a plane wave excitation is
79
analytically formulated and solved by employing the space harmonic approach and virtual work
80
principle, wherein the two facing plates are modeled using the first order shear deformation theory.
81
The work is validate with support of previously published results. Then, the influences of structure
82
geometrical parameters on sound transmission loss are presented and discussed. Through
83
numerical results, a further comprehension was given on the underlying mechanism of sound
84
transmission through a rib-stiffened double laminated composite plates.
85 86
Fig.1 Pressure wave impinging upon the stiffened double laminated composite plates
3
87
2. Equations of motion for the orthogonal stiffened laminated composite plates
88
The stiffened double laminated composite plates shown in Fig.1 are composed of orthotropic
89
plies with identical material properties and different plies orientations. The stiffeners periodically
90
located at x 0,l x ,2l x , and y 0,l y ,2l y ,3l y , along with x and y directions
91
respectively are uniformly distributed on the plate surface. On the source sides, the structure is
92
impinged by a plane sound wave P1 of angular frequency
93
noted as c0 .The wave makes an incident angle
94
plane makes an azimuth angle
95
wavenumber k 0 ( / c0 ) .
96
and the sound speed of air is
with the z axis and its projection on the xy
with the x axis. The wave has an amplitude I and a
The governing equations of two panel vibrations are given by [17] and shown as followings:
D1* w1 ( x, y, t ) 97
[Qym1 ( x mlx ) M Tym1 ( x mlx )]
m
[Q ( y nl ) M
n
n x1
y
( y nl y )]
n Tx1
(1)
h h P1 ( x, y, 1 ) P2 ( x, y, 1 ) 2 2
m
n
D2* w2 ( x, y, t ) [Qym2 ( x mlx ) M Tym 2 ( x mlx )] [Qxn2 ( y nl y ) M Txn 2 ( y nl y )] 98
(2)
h h P2 ( x, y, 1 d ) P3 ( x, y, 1 h2 d ) 2 2
99
* * where D1 and D2 are the linear differential operators, and () is the Dirac delta function. The
100
compatibility of displacements on the interface between the plate and the stiffeners is employed to
101
derive the governing equation of each stiffener along the x- or y- direction [21-22].
102
103
2 4 2 4w m w m n w n w EI A Qyi , EI A Qxin , i 1,2 4 2 4 2 x t x t m
GJ m
2 2 2 2w 2 m w m n w 2 n w n , I M GJ I M Txi 0 Tyi 0 xy 2 x yx 2 y m
n
m
n
m
n
m
(3)
(4)
n
104
where ( EI , EI ) , (GJ , GJ ) , (Q yi , Qxi ) and ( M Tyi , M Txi ) are the flexural stiffness,
105
torsional stiffness, equivalent line force and equivalent line moments for the x- and y-wise
106
m n m n stiffeners, respectively. ( A , A ) , ( I , I ) and ( I 0 , I 0 ) are the cross-sectional area, moment
107
of inertia and polar moment of inertia for the x- and y-wise stiffeners, respectively. For an
108
isotropic thin plate in bending, the differential operator Di* is given by
109
m
n
Di* Di (
4 4 4 2 2 ) h i x 4 x 2y 2 x 4 t 2
(5)
110
where Di E(1 j )hi 12(1 2 ) , in which Di is the bending stiffness, E is Young’s
111
modulus, is the Poisson’s ratio, and are damping loss factor and material density of the plate,
3
4
112
respectively. For the composite panel, however, this formulation is not strictly rigorous. This is
113
because the transverse displacements of the Eqs.(1) and (2) are only one degree of freedom in the
114
thin panels vibrations model, but for a composite panel, as mentioned previously in Section 1,five
115
degrees of freedom is necessary for its vibration analysis. In existing available literature, this
116
feature has been studied by Cao’s model [15], Yin et al. [14] and they have showed how to
117
calculate the dynamic stiffness of the composite panel. The same approach is used in this paper,
118
but with a plane sound wave instead of a point force excitation of the composite panel. Therefore,
119
the dynamic stiffness can be derived as shown in Appendix A.
120
Since the sandwich structure is periodic on the xy plane and excited by a harmonic plane
121
sound wave (see Fig.1), the panel responses can be expressed using space harmonic expansion
122
[23]:
w1 ( x, y, t )
123
m n
w2 ( x, y; t )
124
1, mn
e
j [( k x 2 m / l x ) x ( k y 2 n / l y ) y t ]
m n
2 , mn
e
j [( k x 2 m / l x ) x ( k y 2 n / l y ) y t ]
(6)
(7)
125
Similarly, the sound pressure inside and outside the cavity can be represented by space harmonic
126
series [17]:
127
P1 ( x, y, z; t ) Ie
P2 ( x, y, z; t ) 128
j ( k x x k y y k z z t )
m n
130 131
P3 ( x, y, z; t )
mn
B
e
j [( k x 2 m / l x ) x ( k y 2 n / l y ) y k z ,mn z t ]
(8)
e
j [( k x 2 m / l x ) x ( k y 2 n / l y ) y k z ,mn z t ]
m n
e
(9)
mn
C
mn
j [( k x 2 m / l x ) x ( k y 2 n / l y ) y k z ,mn z t ]
m n
129
A
m n
B
mn
e
j [( k x 2 m / l x ) x ( k y 2 n / l y ) y k z ,mn z t ]
(10)
where
k x k0 sin cos , k y k0 sin sin , k z k 0 cos
132
The k z ,mn is the (m,n)th space harmonic wavenumber in the z -direction, and the corresponding
133
acoustic pressure should satisfy the scalar Helmholtz equation
134
135
P1 2 2 2 2 ( 2 2 2 k0 )P2 0 x y z P 3 Substitution of Eqs.(8)-(10) into (11),the k z ,mn is given by 5
(11)
2
k z ,mn
136
2m 2 2n 2 (k x ) (k y ) lx ly c0
(12)
137
When ( c0 ) < (k x 2m l x ) (k y 2n l y ) , the pressure waves become evanescent
138
waves, and hence k z ,mn should be taken as [24]:
2
2
k z ,mn
139
2
2m 2 2n 2 j (k x ) (k y ) lx ly c0
2
(13)
140
As well, continuity conditions at fluid-panel interfaces require that [7]
141
P1 z
z 0
P 2 0 w1 ， 2 z
z h1
P 2 0 w1 ， 2 z
z h1 d
P 2 0 w2 ， 3 z
2 0 w2
(14)
z h1 h2 d
142
Substituting Eqs.(6)-(10) into Eq.(14) and due to the fact that the sums must be true for all values
143
of x and y , the pressure coefficients and displacement amplitude coefficients are related for
144
each combination (m,n) by
A00 I
145
Bmn
jk z
(h d )
2 0 [1,mne z ,mn 1 2,mne 2k z ,mn sin( k z ,mn d ) jk
146
2 01,00
Cmn
147
, Amn
jkz ,mn h1
]
2 01,mn jkz ,mn
, m 0 or n 0 jk z ,mn ( h1 d )
, Bmn '
2 0 2,mn jkz ,mn
e
2 0 [1,mne 2,mne 2k z ,mn sin( k z ,mn d )
(15) jk z ,mn h1
]
jk z ,mn ( h1 h2 d )
(16)
(17)
148
The displacement amplitude 1, mn and 2 , mn can be derived by using the virtual work
149
' principle, which are then used to calculate the sound pressure amplitudes Amn , Bmn , Bmn and C mn .
150
Once coefficient C mn is known, the sound transmission loss can be easily calculated (see Eqs.(20)
151
and (21)). As mentioned in the Introduction, the methodology for the virtual work principle has
152
been thoroughly exposed in previous papers [15-17], so an extended version of Xin’s and Cao’s
153
model is derived in this paper. Substituting Eqs.(3)-(4) and Eqs.(6)-(10) into Eqs.(1)-(2) yields: * 2 0 2 0 cos( k z ,kl d ) 2 0 K l l l l 1 x y 1,kl ( R1 R2 l n )lx1,kn x y 2 , kl jkz ,kl k z ,kl sin( k z ,kl d ) k z ,kl sin( k z ,kl d ) n
154
(Q Q )l
n
1
2
l
n
x
2 , kn
(R
m
3
R4 k m )l y1,ml
2 Il l when k 0 and l 0 x y when k 0 or l 0 0
6
(Q
m
3
Q4 k m )l y 2,ml
(18)
155
* 2 0 2 0 cos( k z ,kl d ) 2 0 l x l y1,kl ( R1 R2 l n )l x1,kn K2 l x l y 2,kl jkz ,kl k z ,kl sin( k z ,kl d ) k z ,kl sin( k z ,kl d ) n
(Q Q )l
n
156
1
2
l
n
x
2 , kn
( R
m
3
R4 k m )l y1,ml
(Q
m
3
Q4 k m )l y 2,ml 0
where
157
n R1 A1n 2 EI1n k4 , R2 GJ1n k2 I 01 2 , R3 A1m 2 EI1m l4
158
m 2 R4 GJ1m k2 I 01 , Q1 A2n 2 EI 2n k4 , Q2 GJ 2n k2 I 02n 2
159 160
(19)
m 2 Q3 A2m 2 EI 2m l4 , Q4 GJ 2m k2 I 02 , k k x 2k l x
m k x 2m l x , l k y 2l l y , n k y 2n l y
161
* For an isotropic thin plate in bending, the dynamic stiffness K i is given by Di ( k l ) . For
162
the composite panel, the dynamic stiffness K i can be derived as shown in Appendix A. According
163
to the knowledge of convergence, the infinite linear algebraic Eqs.(18)and (19) should be
164
truncated to a finite but sufficient large number of terms, i.e., m kˆ to kˆ and n lˆ to
165
lˆ (both kˆ and lˆ are positive integers).The values of kˆ and lˆ which are used in further
166
calculation have to be determined by the convergence analysis of the solution(see section 3.1),and
167
hence it can be numerically solved.
2
2 2
*
168
The vibration displacements of the two plates can obtained by solving Eqs.(18) and (19) , and
169
then the coefficient C mn of the transmitted pressure amplitudes is obtained through Eq.(17) and
170
used to calculate the transmission coefficient of the periodic model. Since the transmission
171
coefficient is a function of sound incident angles and , the transmission coefficient is defined
172
here as the ratio of the transmitted sound power to the incident sound power [25], as
173
( , )
C
m n
2 mn
Re( k z ,mnn )
2
(20)
I kz 174
Then, the sound transmission loss (STL) expressed in decibel scale (db) is obtained [17], as
1 STL 10 log 10 ( , )
175 176
(21)
3. Results and discussion
177
In this section, numerical calculations based on theoretical formulations presented above are
178
performed to explore the vibroacoustic behavior of an orthogonally rib-stiffened double laminated
179
composite plates. Table 1 presents the properties of the materials used for the plate and stiffener.
180
Considering the influence of structural damping, complex elastic modulus E (1 j ) should be
181
used for the plate and stiffener. Material 1 is used to make stiffeners, which are chosen with depth
182
d 0.08 m ,width b 0.001 m ,and stiffener spacing lx l y 0.2 m .The density of air and 7
3
1
183
the speed of sound in air are set to 1.21 kg m and 340 m s . The sound incident angle and
184
azimuth angle are set to =45° and =45°. The base plates are symmetric laminate made from
185
six layers(see Fig.1), and thickness h 0.002 m .Lamination schemes of six layers are material
186
2/material 3/material 4/material 4/material 3/material 2. The ply angle sequence is
187
(75 / 45 / 15 / 15 / 45 / 75 ) . Unless otherwise stated, the material and geometry parameters
188
are used as mentioned above.
189
Table 1 Material parameters of laminated plates Property
Material 1
Material 2
Material 3
Material 4
Young modulus- E x (GPa)
70
138
131
207
Young modulus- E y (GPa)
70
8.96
10.3
20.7
Shear modulus- Gxy (GPa)
263
7.1
6.9
6.9
Shear modulus- Gxz (GPa)
263
7.1
6.2
6.9
Shear modulus- Gyz (GPa)
263
6.2
6.2
4.1
Poisson’s ratio- xy
0.33
0.3
0.22
0.3
)
2750
1600
1500
2000
Damping loss factor-
0.001
0.001
0.001
0.001
Density- (kg m
3
190 191
Table 2 Main parameters of four-layered laminated plate
Ex
192 193
Ey
and
Ex
and
of the middle
upper
two layer (Pa)
layer
Ey and
of the
Density of each
Thickness
Poisson’s ratio
lower
layer
(m)
of each layer
( kg m
(Pa)
Case 1
2.0 10
8
2.0 10
11
Case 2
2.0 10
11
2.0 10
11
Case 3
2.0 10
11
2.0 10
8
3
)
2000
4 0.0005
0.3
2000
4 0.0005
0.3
2000
4 0.0005
0.3
3.1. Convergence check
194
Numerical results based on Eq.(21) are calculated for the sound transmission loss from a
195
plane sound wave excited. Since the algebraic equations of stiffened composite laminated plates
196
are given in series form, the first step of the numerical calculation is determining the number of
197
items to make the numerical solution convergence. Lee and Kim [9] argued that once the solution
198
converges at a given frequency, it converges in the range lower than the given frequency. For
199
convergence criteria, Lee and Kim also assumed that once the difference between the STL results
200
calculated at two successive calculations is less than a preset error band, the solution is considered
201
to have converged. Therefore, to calculate the STL, the highest frequency of interest (i.e.10kHz)
8
202
and the error band (0.01 db) are chosen.
203
To further demonstrate the convergence of the numerical solution, four different frequency of
204
interest, f=200, 3000, 5000, 10000Hz are chosen. It is seen from Fig.2 that the number of terms
205
for a converged solution increases with the frequencies. At low frequency (f=200Hz), the solution
206
converges when terms are 4, but at higher frequency (f=10000Hz), 19 terms are enough to ensure
207
the convergence. Because 10000Hz is the largest interest frequency in this paper, the number of
208
terms, 19, is sufficient for subsequent STL calculations for all frequencies below 10000Hz. 60 55 50 f=200Hz f=2000Hz f=5000Hz f=10000Hz
STL (db)
45 40 35 30 25 20
209 210 211 212 213
0
5
10
15
20 25 30 Number of terms
35
40
45
50
Fig.2 Convergence of stiffened composite laminated plates at four different frequencies. Solid line: 200 Hz, 4 terms used; dotted line: 2000 Hz, 7 terms; dashed line: 5000Hz, 11 terms; dot-dashed line: 10000 Hz, 19 terms.
3.2. Validation of the analytical model
214
To verify the validity of the present theoretical model, the predictions are compared with
215
existing results of Xin [17] for sound transmission of orthogonal stiffened plates, and the relevant
216
geometrical dimensions and material property parameters are identical as those of Xin [17]. As
217
shown in Fig.3, the present results agree well with Xin’s theoretical results, with only slight
218
divergences in individual frequency regions. Those deviations mainly result from the difference of
219
plate theory. In Xin’s theory, the influences of stiffeners on the plate vibration were approximated
220
as translational springs and rotational springs. In contrast, the present theory model the stiffeners
221
as beams, which includes the inertial effects, bending and torsional moments. To further check the
222
validity of the present theoretical model, Yin’s method [26] (classical laminated plate theory) is
9
223
used to compare with the present model. The comparison of Fig.4 demonstrates that the results
224
from the first order shear deformation laminated plate theory are in agreement well with those
225
from classical laminated plate theory used by Yin, particularly in the low and medium frequency
226
regions. The difference appears in the high frequency region above 5000Hz, which can be
227
attributed to the fact that the transverse shear and rotatory inertia are considered in the present
228
theoretical model but not in Yin’s model. Therefore, it can be seen from the comparative analysis
229
mentioned above, the proposed theoretical model is correct and effective. 80 70
Present theory Xin’s method [17]
60
STL (db)
50 40 30 20 10 0 -10 1 10
230 231 232
2
10 Frequency (Hz)
3
10
Fig.3 Comparison between present model predictions and those by Xin [17] for sound transmission loss of orthogonal stiffened plates
10
80 Yin’s method [26],classical laminated plate theory Present results,first order shear deformation laminated plate theory
70 60
STL (db)
50 40 30 20 10 0 1 10
233 234 235 236
3
4
10
10
Frequency (Hz) Fig.4 Comparison between present model predictions and those by Yin [26] for sound transmission loss of orthogonal laminated plate
3.3. Influence of symmetric and anti-symmetric plates
237 238
2
10
To better evaluate the influences of different lamination schemes, Fig.5 presents the sound transmission
loss
of
two
typical
lamination
schemes:
symmetric
(75 / 45 / 15 / 15 / 45 / 75 )
239
(75 / 45 / 15 / 15 / 45 / 75 )
240
plies. Note that, at the range below 200 Hz, the STL values of the anti-symmetric laminates are
241
almost consistent to those of the symmetric laminates. Furthermore, within the medium and high
242
frequency range, the differences of symmetric and anti-symmetric laminates are also minor. This
243
is because the different lamination schemes lead to the difference of bending and extension
244
coupling effect. For symmetric laminates, the bending and extension coupling stiffness are equal
245
to zero, while for anti-symmetric laminates, the coupling stiffness is not zero. Generally, the
246
effects of bending and extension of symmetric and anti-symmetric laminates play a minor role in
247
sound transmission performance. A similar characteristic can also be found in existing findings of
248
Cao [15] and Yin [26].
and anti-symmetric
11
70 symmetrical laminate antisymmetrical laminate
60
STL (db)
50
40
30
20
10
0 1 10
249 250 251
2
3
10
10
4
10
Frequency (Hz) Fig.5 Influence of different lamination scheme on sound transmission loss of the structure.
3.4. Influence of single layer stiffness
252
In order to investigate the influence of single layer stiffness of laminated plate on the sound
253
transmission loss, a four-layered laminated plate with different Young’s modulus is investigated to
254
evaluate such effects. The main parameters of Young’s modulus in four cases are listed in Table 2.
255
As can be seen in Fig.6, the curve of STL peaks and dips in case 3 is shifted to lower frequency
256
with comparison to that in case 2 when Young’s modulus values of the upper and lower layer
257
decreases. This is due to that the decrease of Young’s modulus of the upper and lower layer leads
258
to the reduction of laminated plate stiffness. It is also observed that the increases of middle two
259
layer Young’s modulus in case 1 and case 2 have no obvious influence on sound transmission loss.
260
These results indicate that the sound transmission loss is sensitive to the change of stiffness in the
261
upper and lower layer by comparison with middle two layer.
12
80 case 1 case 2 case 3
70 60
STL(db)
50 40 30 20 10 0 1 10
3
4
10
10
Frequency(Hz)
262 263 264 265
2
10
Fig.6 Influence of single layer stiffness on sound transmission loss of the structure.
3.5. Influence of fiber orientation angle
266
The effects of different fiber orientation angles on sound transmission characteristics are also
267
studied. In order to facilitate analysis, the six layers of composite laminated plates each adopt the
268
same
269
angles( 0 ,30 ,45 ,60 and 90 ) are chosen in the numerical calculation. It is interesting to
270
note that since the fiber orientations of laminate plate have axial symmetry about 45 , the curves
271
of 0
272
are shown in Fig.7. Similarly, the influence of fiber orientation angle of anti-symmetric laminates
273
is consistent with symmetric laminates, as shown in Fig.8. In addition, it can be seen from
274
Fig.7-Fig.8 that the STL peaks and dips of three curves are gradually shifted to low frequency as
275
the fiber orientation angle increases from 0
276
orientation angle leads to the decrease of structural stiffness, which causes a corresponding decline
277
of the natural frequency.
material
2,
as
listed
in
Table2.
Five
different
fiber
orientation
and 90 overlap the angle of 30 and 60 , and thus only three different STL curves
to 45 . This is because the increment of fiber
13
70
60
STL (db)
50
=0 =30 =45 =60 =90
40
30
20
10
0 1 10
278 279
2
3
10
10
4
10
Frequency (Hz) Fig.7 Influence of fiber orientation angle on sound transmission loss of the symmetric laminates.
70
60
STL (db)
50
=0 =30 =45 =60 =90
40
30
20
10
0 1 10
280 281
2
3
10
10
4
10
Frequency (Hz) Fig.8 Influence of fiber orientation angle on sound transmission loss of the anti-symmetric laminates.
282 283
4. Conclusions
284
In this study, first order shear deformation theory and double panel sandwich structure model
285
are adopted to develop an analytical model for investigating the vibroacoustic characteristic of an 14
286
orthogonally rib-stiffened double laminated composite plates. Numerical results show that the
287
effects of bending and extension of symmetric and anti-symmetric laminates play a minor role in
288
sound transmission performance, and the sound transmission loss is sensitive to the change of
289
stiffness in the upper and lower layer by comparison with middle two layer. Moreover, different
290
fiber orientation angle has pronounced influence on sound transmission performance. But the
291
influence of fiber orientation angle of anti-symmetric laminates is consistent with symmetric
292 293 294
laminates. The increment of fiber orientation angle leads to the decrease of structural stiffness. Acknowledgment
295
The authors are grateful to the referees for their valuable suggestions. This work presented
296
here were supported by the National Natural Science Foundation of China under the contract
297
number 11372083, and by National Basic Research Program of China under the contract number
298 299 300
613235.
301
Appendix A The displacement field of the existing FSDT is given by [19-20]
u1 ( x, y, z , t ) u ( x, y, t ) z x ( x, y, t ) v1 ( x, y, z , t ) v( x, y, t ) z y ( x, y, t )
302
(A.1)
w1 ( x, y, z , t ) w( x, y, t ) 303
where ( u, v, w, x , y ) are unknown functions to be determined, t denotes the time and ( u, v, w )
304
denotes the displacements of the mid plane. x and
305
about the y-axis and x-axis, respectively. When a plane sound wave with amplitude q acting on
306 307
the composite panel, according the mathematical statement of the Hamilton principle, the equations of motion for the laminated plate can be derived and given as [15]
308
L11 L 21 L31 L41 L51
L12 L22
L13 L23
L14 L24
L32 L42
L33 L43
L34 L44
L52
L53
L54
y are the rotations of a transverse normal
L15 u 0 L25 v 0 L35 w q L45 x 0 L55 y 0
309
where q is the acoustic pressure acting on the plate.
310
The coefficients Lij in Eq.(A.2) are listed in the following [15]:
(A.2)
311
L11 A11 m2 2 A16 m n A66 n2 I1 2 , L12 A16 m2 A26 n2 ( A12 A66 ) m n
312
L13 0 , L14 B11 m2 2 B16 m n B66 n2 I 2 2
15
313
L15 B16 m2 B26 n2 ( B12 B66 ) m n , L22 A66 m2 A22 n2 2 A26 m n I1 2
314
L23 0 , L24 B16 m2 B26 n2 ( B12 B66 ) m n
315
L25 B66 m2 B22 n2 2 B26 m n I 2 2
316
L33 A55 m2 2A45 m n A44n2 I1 2 0 2 k z ,mn , L34 jA55 m jA45 n
317
L35 jA45 m jA44 n , L44 D11 m2 D66 n2 2 D16 m n A55 I 3 2
318
L45 D16 m2 ( D12 D66 ) m n D26 n2 A45
319
L55 D66 m2 D22 n2 2 D26 m n A44 I 3 2
320
Lij L ji , (i, j 1,2,3,4,5)
n k y 2n l y , is a shear correction factor, which consider
321
where m k x 2m l x and
322
the non-uniform distribution of shear strain in the thickness direction of the plate. The selection of
323
shear correction factors is very complicated since its value depends not only on the lamination and
324
geometric parameters, but also on the loading and boundary conditions[27]. In Mindlin’s model[2],
325
the shear correction factor is
326
coefficients Lij into Eq.(A.2) and solving the matrix in Eq.(A.2),the transverse displacement w
327
is obtained and the dynamic stiffness of the panel associated with the wave propagating in the
328
structure can be derived:
2 12 and the same value is used in this paper. Substituting
K*
329
q w
(A.3)
330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345
References [1] E. Reissner, The effect of transverse shear deformation on the bending of elastic plates, Journal of Applied Mechanics, 12 (2) (1945) 69-72. [2] R. D. Mindlin, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, Journal of Applied Mechanics, 18 (1) (1951) 31-8. [3] H.-T.Thai, D.-H. Choi, A simple first-order shear deformation theory for laminated composite plates, Composite Structures, 106 (2013) 754-763. [4] J.L.Mantari, Free Vibration of single and sandwich laminated composite plates by using a simplified FSDT, Composite Structures, 132 (2015) 952-959. [5] J.N.Reddy, A simple higher-order theory for laminated composite plates, Journal of Applied Mechanics, 51 (1984) 745-752. [6] L.Librescu, On the theory of anisotropic elastic shells and plates, International Journal of Solids and Structures, 3 (1) (1967) 53-68. [7] F.X.Xin, T.J.Lu.C.Q.Chen, External mean flow influence on noise transmission through double-leaf aeroelastic plates, AIAA Journal, 47 (8) (2009) 1940-1951.
16
346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385
[8] B.R.Mace, Sound radiation from fluid loaded orthogonally stiffened plates, Journal of Sound and Vibration, 79
386 387 388 389
The list of figure
(1981) 439-452. [9] J.H.Lee, J.Kim, Analysis of sound transmission through periodically stiffened panels by space-harmonic expansion method, Journal of Sound and Vibration, 251 (2002) 341-366. [10] J.Alba, J.Ramis, Improvement of the prediction of transmission loss of double partitions with cavity absorption by minimization techniques, Journal of Sound and Vibration, 273 (2004) 793-804. [11] J.Wang, T.J.Lu, R.S.Woodhouse, Sound transmission through lightweight double leaf partitions:Theoretical modelling, Journal of Sound and Vibration, 286 (2005) 817-847. [12] F.X.Xin, T.J.Lu, Transmission loss of orthogonally rib-stiffened double-panel structures with cavity absorption, Journal of the Acoustical Society of America, 129 (4) (2011) 1919-1934. [13] J.Legault, N.Atalla, Numerical and experimental investigation of the effect of structural links on the sound transmission of a lightweight double panel structure, Journal of Sound and Vibration, 324 (2009) 712-732. [14] X.W.Yin, H.F.Cui, Acoustic radiation from a laminated composite plate excited by longitudinal and transverse mechanical drives, Journal of Applied Mechanics, 76 (2009) 1-5. [15]X.T.Cao,H.X.Hua,Z.Y.Zhang,Sound radiation from shear deformable stiffened laminated plates,Journal of Sound and Vibration,330 (2011) 4047-4063. [16]F.X.Xin,T.J.Lu,Effects of mean flow on transmission loss of orthogonally rib-stiffened aeroelastic plates,Journal of the Acoustical Society of America,133 (6) (2013) 3909-3920. [17] F.X.Xin,T.J.Lu,Analytical modeling of fluid loaded orthogonally rib-stiffened sandwich structures:Sound transmission,Journal of the Mechanics and Physics of Solids,58 (2010) 1374-1396. [18] C.Shen,F.X.Xin,L.Cheng,T.J.Lu,Sound radiation of orthogonally stiffened laminated composite plates under airborne and structure borne excitations,Composites Science and Technology,84 (2013) 51-57. [19] J.M.Whitney,The effect of transverse shear deformation on the bending of laminated plates,Journal of Composite Materials,3 (1969) 534-547. [20] C.H.Thai,L.V.Tran,D.T.Tran, Analysis of laminated composite plates using higher-order shear deformation plate theory and node-based smoothed discrete shear gap method, Applied Mathematical Modelling, 36 (2012) 5657-5677. [21] S.Timoshenko,D.H.Young,W.Weaver.Vibration problems in Engineering,Wiley,New York,1974. [22] J.Legault,N.Atalla, Sound transmission through a double panel structure periodically coupled with vibration insulators, Journal of Sound and Vibration, 329 (2010) 3082-3100. [23] L. Maxit, Wavenumber space and physical space responses of a periodically ribbed plate to a point drive:a discrete approach, Applied Acoustics,70 (4) (2009) 563-578. [24] W.R.Graham, High frequency vibration and acoustic radiation of fluid loaded plate, Proc.R.Soc.London A ,352.1995.1-43. [25] F.Fahy, P.Gardonio, Sound and structural vibration, Elsevier, Amsterdam, 2007. [26] X.W.Yin,X.J.Gu.H.F.Cui,R.Y.Shen, Acoustic radiation from a laminated composite plate reinforced by doubly periodic parallel stiffeners, Journal of Sound and Vibration, 306 (2007) 877-889. [27] H.-T.Thai,S.-E.Kim,Free vibration of laminated composite plate using two variable refined plate theory, International Journal of Mechanical Sciences,52 (2010) 626-633
Fig.1 Pressure wave impinging upon the stiffened double laminated composite plates Fig.2 Convergence of stiffened composite laminated plates at four different frequencies. Solid line: 200 Hz, 4 terms used; dotted line: 2000 Hz, 7 terms; dashed line: 5000Hz, 11 terms; dot-dashed line: 10000 Hz, 19 terms.
17
390 391 392 393 394
Fig.3 Comparison between present model predictions and those by Xin [17] for sound transmission loss of
Fig.5
Influence of different lamination scheme on sound transmission loss of the structure.
395 396 397 398
Fig.6
Influence of single layer stiffness on sound transmission loss of the structure.
Fig.7
Influence of fiber orientation angle on sound transmission loss of the symmetric laminates.
Fig.8
Influence of fiber orientation angle on sound transmission loss of the anti-symmetric laminates.
orthogonal stiffened plates Fig.4 Comparison between present model predictions and those by Yin [26] for sound transmission loss of orthogonal laminated plate
18
S0165-2125(17)30051-3 http://dx.doi.org/10.1016/j.wavemoti.2017.04.007 WAMOT 2156
To appear in:
Wave Motion
Received date: 1 March 2017 Accepted date: 6 April 2017 Please cite this article as: T. Fu, Z. Chen, D. Yu, X. Wang, W. Lu, Sound transmission from stiffened double laminated composite plates, Wave Motion (2017), http://dx.doi.org/10.1016/j.wavemoti.2017.04.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Research Highlights
Highlights 1)First order shear deformation theory and double panel sandwich structure model are adopted to develop an analytical model for investigating the vibroacoustic characteristic of an orthogonally rib-stiffened double laminated composite plates. 2)A further comprehension was given on the underlying mechanism of sound transmission through a rib-stiffened double laminated composite plates.
*Manuscript (Clear) Click here to view linked References
1 2 3 4 5 6 7
Sound transmission from stiffened double laminated composite plates Tao Fua, Zhaobo Chena,1, Dong Yua, Xiaoyu Wangb, Wenxiang Luc a School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001,PR China. b Beijing Institute of Spacecraft System Engineering, Beijing 100094,PR China. c School of Ocean Transportation, University of Shan Dong Jiao Tong , Jinan 264200,PR China.
8 9
Abstract: An analytical model is developed to investigate the sound transmission loss from
10
orthogonally rib-stiffened double laminated composite plates structure under a plane sound wave
11
excitation, in which first order shear deformation theory is presented for laminated composite
12
plates. By using the space harmonic approach and virtual work principle, the sound transmission
13
loss is described analytically. The validity and feasibility of the model are verified by comparing
14
the present theoretical predictions with numerical results published previously. The influences of
15
structure geometrical parameters on sound transmission loss are subsequently presented. Through
16
numerical results, it can be concluded that the proposed analytical model is accurate and simple in
17
solving the vibroacoustic behavior of an orthogonally rib-stiffened double laminated composite
18
plates.
19 20 21 22
Keywords: Laminated composite plate; First order shear deformation theory; Vibroacoustic; Sound transmission;
23
1. Introduction
24
In recent years, laminated composite plates have been given more and more attention due to
25
the extensive application, and advantageous features such as high stiffness to weight and low
26
density. Thus many laminated analytical theories have been developed [1-6]. Among them, the
27
classical laminated plate theory (CLPT) and first order shear deformation theory (FSDT) are the
28
commonly used theory for the analysis of laminated composite plates, wherein CLPT is based on
29
the Love-Kirchhoff kinematic hypothesis, and only suit for thin plate because of the neglect of
30
transverse shear deformation, whereas FSDT proposed by Reissner [1] and Mindlin [2] takes into
31
account the effects of shear deformation and applies for both thick and thin laminate composite
32
plates. However, the most limitation of FSDT is that it requires shear correction factors to rectify
33
the unrealistic variation of the transverse shear strain through the thickness [3,4]. In order to
1
Corresponding author. TeL: +86 13904810712 E-mail addresses: [email protected]; [email protected] (T. Fu). 1
34
overcome the limitation of FSDT, the higher-order shear deformation theories(HOSDT) based on
35
an assumption of nonlinear stress variation through the thickness were developed by Reddy [5]
36
and Librescu [6], etc., which can neglect shear correction factors and give more accurate and
37
stable transverse shear strain. Comparing to the single laminated composite plate, however, the
38
periodically rib-stiffened composite plates have been widely used in engineering structures.
39
Further, the rib-stiffeners play an important role in the vibration and acoustic characteristics of the
40
whole structure, particularly when the bending wave length is comparable with the periodical
41
spacing of the stiffeners [7, 8]. Consequently, more attention needs to be paid on the effect of
42
rib-stiffeners.
43
To deal with the coupling connections between the rib-stiffeners and the base thin plate, Lee
44
and Kim [9] replaced the rib-stiffener as translational springs, rotational springs and lumped mass
45
when they studied the sound transmission performance of a single stiffened plate subjected to a
46
plane wave excitation. Similarly, Alba [10] studied the effect of rib-stiffener on sound
47
transmission characteristics of stiffened panel structures. Subsequently, Wang et al. [11] extended
48
the method to a double-leaf partitions connected through vertical resilient studs and investigated
49
the physical mechanisms determining sound transmission. Following the work of Wang et al, Xin
50
[12] proposed a more accurate theoretical model for sound transmission from an orthogonally
51
rib-stiffened sandwich structures. In his study, the effects of rib-stiffeners were included by
52
introducing the tensional forces, bending moments and torsional moments as well as the
53
corresponding inertial terms into the governing equations of the two face panels. In comparison
54
with the rib-stiffeners thin plate, the stiffened laminated composite plate structure increases the
55
complexity of the theoretical modeling [13]. For the stiffened laminated composite plate structure,
56
Yin et al.[14] modeled the rib-stiffener as Bernoulli-Euler beams and studied the acoustic radiation
57
from an infinite stiffened laminated composite cylindrical shell based on the classical laminated
58
composite plate theory. It should be pointed out that the torsional moments and inertial effects of
59
the rib-stiffeners were not considered in the analysis. The same approach was adopted by Cao et al.
60
[15] to investigate the acoustic characteristics of stiffened symmetric and antisymmetric laminated
61
plates. Considering only the bending moments of rib-stiffeners, they used the first order shear
62
deformation theory to describe the equations of motion for the laminated composite plate. A
63
refined theoretical model was proposed later by Xin and Lu [16] for sound radiation from a
64
rib-stiffened plate excited by external mean flow, in which the inertial effects, bending and
65
torsional moments were considered.
66
For the vibroacoustic properties of periodically rib-stiffened infinite plate structures, as
67
reported by Xin et al. [17], fourier transform method is able to handle sound radiation from a
2
68
mechanical point force driven, whereas space-harmonic approach is particularly suited for sound
69
transmission excited by a plane wave. Both of the methods described above should transform the
70
governing equations into infinite sets of simultaneous algebraic equations and then truncate these
71
into a finite range for numerical solutions, as stated in Shen [18]. However, existing literatures on
72
the vibroacoustic behavior of laminated composite plates are generally focused on sound radiation
73
and stiffened single panel structures. Little attention has been paid to the vibrations and sound
74
transmission loss from the orthogonally stiffened double composite panel structures. It is worth
75
noting that when double panel were reinforced by orthogonal stiffening members, their dynamic
76
characteristics are quite different from those of the single stiffened plate.
77
Thus in this paper, the physical process of sound transmission through an orthogonally
78
rib-stiffened double laminated composite plates structures subjected to a plane wave excitation is
79
analytically formulated and solved by employing the space harmonic approach and virtual work
80
principle, wherein the two facing plates are modeled using the first order shear deformation theory.
81
The work is validate with support of previously published results. Then, the influences of structure
82
geometrical parameters on sound transmission loss are presented and discussed. Through
83
numerical results, a further comprehension was given on the underlying mechanism of sound
84
transmission through a rib-stiffened double laminated composite plates.
85 86
Fig.1 Pressure wave impinging upon the stiffened double laminated composite plates
3
87
2. Equations of motion for the orthogonal stiffened laminated composite plates
88
The stiffened double laminated composite plates shown in Fig.1 are composed of orthotropic
89
plies with identical material properties and different plies orientations. The stiffeners periodically
90
located at x 0,l x ,2l x , and y 0,l y ,2l y ,3l y , along with x and y directions
91
respectively are uniformly distributed on the plate surface. On the source sides, the structure is
92
impinged by a plane sound wave P1 of angular frequency
93
noted as c0 .The wave makes an incident angle
94
plane makes an azimuth angle
95
wavenumber k 0 ( / c0 ) .
96
and the sound speed of air is
with the z axis and its projection on the xy
with the x axis. The wave has an amplitude I and a
The governing equations of two panel vibrations are given by [17] and shown as followings:
D1* w1 ( x, y, t ) 97
[Qym1 ( x mlx ) M Tym1 ( x mlx )]
m
[Q ( y nl ) M
n
n x1
y
( y nl y )]
n Tx1
(1)
h h P1 ( x, y, 1 ) P2 ( x, y, 1 ) 2 2
m
n
D2* w2 ( x, y, t ) [Qym2 ( x mlx ) M Tym 2 ( x mlx )] [Qxn2 ( y nl y ) M Txn 2 ( y nl y )] 98
(2)
h h P2 ( x, y, 1 d ) P3 ( x, y, 1 h2 d ) 2 2
99
* * where D1 and D2 are the linear differential operators, and () is the Dirac delta function. The
100
compatibility of displacements on the interface between the plate and the stiffeners is employed to
101
derive the governing equation of each stiffener along the x- or y- direction [21-22].
102
103
2 4 2 4w m w m n w n w EI A Qyi , EI A Qxin , i 1,2 4 2 4 2 x t x t m
GJ m
2 2 2 2w 2 m w m n w 2 n w n , I M GJ I M Txi 0 Tyi 0 xy 2 x yx 2 y m
n
m
n
m
n
m
(3)
(4)
n
104
where ( EI , EI ) , (GJ , GJ ) , (Q yi , Qxi ) and ( M Tyi , M Txi ) are the flexural stiffness,
105
torsional stiffness, equivalent line force and equivalent line moments for the x- and y-wise
106
m n m n stiffeners, respectively. ( A , A ) , ( I , I ) and ( I 0 , I 0 ) are the cross-sectional area, moment
107
of inertia and polar moment of inertia for the x- and y-wise stiffeners, respectively. For an
108
isotropic thin plate in bending, the differential operator Di* is given by
109
m
n
Di* Di (
4 4 4 2 2 ) h i x 4 x 2y 2 x 4 t 2
(5)
110
where Di E(1 j )hi 12(1 2 ) , in which Di is the bending stiffness, E is Young’s
111
modulus, is the Poisson’s ratio, and are damping loss factor and material density of the plate,
3
4
112
respectively. For the composite panel, however, this formulation is not strictly rigorous. This is
113
because the transverse displacements of the Eqs.(1) and (2) are only one degree of freedom in the
114
thin panels vibrations model, but for a composite panel, as mentioned previously in Section 1,five
115
degrees of freedom is necessary for its vibration analysis. In existing available literature, this
116
feature has been studied by Cao’s model [15], Yin et al. [14] and they have showed how to
117
calculate the dynamic stiffness of the composite panel. The same approach is used in this paper,
118
but with a plane sound wave instead of a point force excitation of the composite panel. Therefore,
119
the dynamic stiffness can be derived as shown in Appendix A.
120
Since the sandwich structure is periodic on the xy plane and excited by a harmonic plane
121
sound wave (see Fig.1), the panel responses can be expressed using space harmonic expansion
122
[23]:
w1 ( x, y, t )
123
m n
w2 ( x, y; t )
124
1, mn
e
j [( k x 2 m / l x ) x ( k y 2 n / l y ) y t ]
m n
2 , mn
e
j [( k x 2 m / l x ) x ( k y 2 n / l y ) y t ]
(6)
(7)
125
Similarly, the sound pressure inside and outside the cavity can be represented by space harmonic
126
series [17]:
127
P1 ( x, y, z; t ) Ie
P2 ( x, y, z; t ) 128
j ( k x x k y y k z z t )
m n
130 131
P3 ( x, y, z; t )
mn
B
e
j [( k x 2 m / l x ) x ( k y 2 n / l y ) y k z ,mn z t ]
(8)
e
j [( k x 2 m / l x ) x ( k y 2 n / l y ) y k z ,mn z t ]
m n
e
(9)
mn
C
mn
j [( k x 2 m / l x ) x ( k y 2 n / l y ) y k z ,mn z t ]
m n
129
A
m n
B
mn
e
j [( k x 2 m / l x ) x ( k y 2 n / l y ) y k z ,mn z t ]
(10)
where
k x k0 sin cos , k y k0 sin sin , k z k 0 cos
132
The k z ,mn is the (m,n)th space harmonic wavenumber in the z -direction, and the corresponding
133
acoustic pressure should satisfy the scalar Helmholtz equation
134
135
P1 2 2 2 2 ( 2 2 2 k0 )P2 0 x y z P 3 Substitution of Eqs.(8)-(10) into (11),the k z ,mn is given by 5
(11)
2
k z ,mn
136
2m 2 2n 2 (k x ) (k y ) lx ly c0
(12)
137
When ( c0 ) < (k x 2m l x ) (k y 2n l y ) , the pressure waves become evanescent
138
waves, and hence k z ,mn should be taken as [24]:
2
2
k z ,mn
139
2
2m 2 2n 2 j (k x ) (k y ) lx ly c0
2
(13)
140
As well, continuity conditions at fluid-panel interfaces require that [7]
141
P1 z
z 0
P 2 0 w1 ， 2 z
z h1
P 2 0 w1 ， 2 z
z h1 d
P 2 0 w2 ， 3 z
2 0 w2
(14)
z h1 h2 d
142
Substituting Eqs.(6)-(10) into Eq.(14) and due to the fact that the sums must be true for all values
143
of x and y , the pressure coefficients and displacement amplitude coefficients are related for
144
each combination (m,n) by
A00 I
145
Bmn
jk z
(h d )
2 0 [1,mne z ,mn 1 2,mne 2k z ,mn sin( k z ,mn d ) jk
146
2 01,00
Cmn
147
, Amn
jkz ,mn h1
]
2 01,mn jkz ,mn
, m 0 or n 0 jk z ,mn ( h1 d )
, Bmn '
2 0 2,mn jkz ,mn
e
2 0 [1,mne 2,mne 2k z ,mn sin( k z ,mn d )
(15) jk z ,mn h1
]
jk z ,mn ( h1 h2 d )
(16)
(17)
148
The displacement amplitude 1, mn and 2 , mn can be derived by using the virtual work
149
' principle, which are then used to calculate the sound pressure amplitudes Amn , Bmn , Bmn and C mn .
150
Once coefficient C mn is known, the sound transmission loss can be easily calculated (see Eqs.(20)
151
and (21)). As mentioned in the Introduction, the methodology for the virtual work principle has
152
been thoroughly exposed in previous papers [15-17], so an extended version of Xin’s and Cao’s
153
model is derived in this paper. Substituting Eqs.(3)-(4) and Eqs.(6)-(10) into Eqs.(1)-(2) yields: * 2 0 2 0 cos( k z ,kl d ) 2 0 K l l l l 1 x y 1,kl ( R1 R2 l n )lx1,kn x y 2 , kl jkz ,kl k z ,kl sin( k z ,kl d ) k z ,kl sin( k z ,kl d ) n
154
(Q Q )l
n
1
2
l
n
x
2 , kn
(R
m
3
R4 k m )l y1,ml
2 Il l when k 0 and l 0 x y when k 0 or l 0 0
6
(Q
m
3
Q4 k m )l y 2,ml
(18)
155
* 2 0 2 0 cos( k z ,kl d ) 2 0 l x l y1,kl ( R1 R2 l n )l x1,kn K2 l x l y 2,kl jkz ,kl k z ,kl sin( k z ,kl d ) k z ,kl sin( k z ,kl d ) n
(Q Q )l
n
156
1
2
l
n
x
2 , kn
( R
m
3
R4 k m )l y1,ml
(Q
m
3
Q4 k m )l y 2,ml 0
where
157
n R1 A1n 2 EI1n k4 , R2 GJ1n k2 I 01 2 , R3 A1m 2 EI1m l4
158
m 2 R4 GJ1m k2 I 01 , Q1 A2n 2 EI 2n k4 , Q2 GJ 2n k2 I 02n 2
159 160
(19)
m 2 Q3 A2m 2 EI 2m l4 , Q4 GJ 2m k2 I 02 , k k x 2k l x
m k x 2m l x , l k y 2l l y , n k y 2n l y
161
* For an isotropic thin plate in bending, the dynamic stiffness K i is given by Di ( k l ) . For
162
the composite panel, the dynamic stiffness K i can be derived as shown in Appendix A. According
163
to the knowledge of convergence, the infinite linear algebraic Eqs.(18)and (19) should be
164
truncated to a finite but sufficient large number of terms, i.e., m kˆ to kˆ and n lˆ to
165
lˆ (both kˆ and lˆ are positive integers).The values of kˆ and lˆ which are used in further
166
calculation have to be determined by the convergence analysis of the solution(see section 3.1),and
167
hence it can be numerically solved.
2
2 2
*
168
The vibration displacements of the two plates can obtained by solving Eqs.(18) and (19) , and
169
then the coefficient C mn of the transmitted pressure amplitudes is obtained through Eq.(17) and
170
used to calculate the transmission coefficient of the periodic model. Since the transmission
171
coefficient is a function of sound incident angles and , the transmission coefficient is defined
172
here as the ratio of the transmitted sound power to the incident sound power [25], as
173
( , )
C
m n
2 mn
Re( k z ,mnn )
2
(20)
I kz 174
Then, the sound transmission loss (STL) expressed in decibel scale (db) is obtained [17], as
1 STL 10 log 10 ( , )
175 176
(21)
3. Results and discussion
177
In this section, numerical calculations based on theoretical formulations presented above are
178
performed to explore the vibroacoustic behavior of an orthogonally rib-stiffened double laminated
179
composite plates. Table 1 presents the properties of the materials used for the plate and stiffener.
180
Considering the influence of structural damping, complex elastic modulus E (1 j ) should be
181
used for the plate and stiffener. Material 1 is used to make stiffeners, which are chosen with depth
182
d 0.08 m ,width b 0.001 m ,and stiffener spacing lx l y 0.2 m .The density of air and 7
3
1
183
the speed of sound in air are set to 1.21 kg m and 340 m s . The sound incident angle and
184
azimuth angle are set to =45° and =45°. The base plates are symmetric laminate made from
185
six layers(see Fig.1), and thickness h 0.002 m .Lamination schemes of six layers are material
186
2/material 3/material 4/material 4/material 3/material 2. The ply angle sequence is
187
(75 / 45 / 15 / 15 / 45 / 75 ) . Unless otherwise stated, the material and geometry parameters
188
are used as mentioned above.
189
Table 1 Material parameters of laminated plates Property
Material 1
Material 2
Material 3
Material 4
Young modulus- E x (GPa)
70
138
131
207
Young modulus- E y (GPa)
70
8.96
10.3
20.7
Shear modulus- Gxy (GPa)
263
7.1
6.9
6.9
Shear modulus- Gxz (GPa)
263
7.1
6.2
6.9
Shear modulus- Gyz (GPa)
263
6.2
6.2
4.1
Poisson’s ratio- xy
0.33
0.3
0.22
0.3
)
2750
1600
1500
2000
Damping loss factor-
0.001
0.001
0.001
0.001
Density- (kg m
3
190 191
Table 2 Main parameters of four-layered laminated plate
Ex
192 193
Ey
and
Ex
and
of the middle
upper
two layer (Pa)
layer
Ey and
of the
Density of each
Thickness
Poisson’s ratio
lower
layer
(m)
of each layer
( kg m
(Pa)
Case 1
2.0 10
8
2.0 10
11
Case 2
2.0 10
11
2.0 10
11
Case 3
2.0 10
11
2.0 10
8
3
)
2000
4 0.0005
0.3
2000
4 0.0005
0.3
2000
4 0.0005
0.3
3.1. Convergence check
194
Numerical results based on Eq.(21) are calculated for the sound transmission loss from a
195
plane sound wave excited. Since the algebraic equations of stiffened composite laminated plates
196
are given in series form, the first step of the numerical calculation is determining the number of
197
items to make the numerical solution convergence. Lee and Kim [9] argued that once the solution
198
converges at a given frequency, it converges in the range lower than the given frequency. For
199
convergence criteria, Lee and Kim also assumed that once the difference between the STL results
200
calculated at two successive calculations is less than a preset error band, the solution is considered
201
to have converged. Therefore, to calculate the STL, the highest frequency of interest (i.e.10kHz)
8
202
and the error band (0.01 db) are chosen.
203
To further demonstrate the convergence of the numerical solution, four different frequency of
204
interest, f=200, 3000, 5000, 10000Hz are chosen. It is seen from Fig.2 that the number of terms
205
for a converged solution increases with the frequencies. At low frequency (f=200Hz), the solution
206
converges when terms are 4, but at higher frequency (f=10000Hz), 19 terms are enough to ensure
207
the convergence. Because 10000Hz is the largest interest frequency in this paper, the number of
208
terms, 19, is sufficient for subsequent STL calculations for all frequencies below 10000Hz. 60 55 50 f=200Hz f=2000Hz f=5000Hz f=10000Hz
STL (db)
45 40 35 30 25 20
209 210 211 212 213
0
5
10
15
20 25 30 Number of terms
35
40
45
50
Fig.2 Convergence of stiffened composite laminated plates at four different frequencies. Solid line: 200 Hz, 4 terms used; dotted line: 2000 Hz, 7 terms; dashed line: 5000Hz, 11 terms; dot-dashed line: 10000 Hz, 19 terms.
3.2. Validation of the analytical model
214
To verify the validity of the present theoretical model, the predictions are compared with
215
existing results of Xin [17] for sound transmission of orthogonal stiffened plates, and the relevant
216
geometrical dimensions and material property parameters are identical as those of Xin [17]. As
217
shown in Fig.3, the present results agree well with Xin’s theoretical results, with only slight
218
divergences in individual frequency regions. Those deviations mainly result from the difference of
219
plate theory. In Xin’s theory, the influences of stiffeners on the plate vibration were approximated
220
as translational springs and rotational springs. In contrast, the present theory model the stiffeners
221
as beams, which includes the inertial effects, bending and torsional moments. To further check the
222
validity of the present theoretical model, Yin’s method [26] (classical laminated plate theory) is
9
223
used to compare with the present model. The comparison of Fig.4 demonstrates that the results
224
from the first order shear deformation laminated plate theory are in agreement well with those
225
from classical laminated plate theory used by Yin, particularly in the low and medium frequency
226
regions. The difference appears in the high frequency region above 5000Hz, which can be
227
attributed to the fact that the transverse shear and rotatory inertia are considered in the present
228
theoretical model but not in Yin’s model. Therefore, it can be seen from the comparative analysis
229
mentioned above, the proposed theoretical model is correct and effective. 80 70
Present theory Xin’s method [17]
60
STL (db)
50 40 30 20 10 0 -10 1 10
230 231 232
2
10 Frequency (Hz)
3
10
Fig.3 Comparison between present model predictions and those by Xin [17] for sound transmission loss of orthogonal stiffened plates
10
80 Yin’s method [26],classical laminated plate theory Present results,first order shear deformation laminated plate theory
70 60
STL (db)
50 40 30 20 10 0 1 10
233 234 235 236
3
4
10
10
Frequency (Hz) Fig.4 Comparison between present model predictions and those by Yin [26] for sound transmission loss of orthogonal laminated plate
3.3. Influence of symmetric and anti-symmetric plates
237 238
2
10
To better evaluate the influences of different lamination schemes, Fig.5 presents the sound transmission
loss
of
two
typical
lamination
schemes:
symmetric
(75 / 45 / 15 / 15 / 45 / 75 )
239
(75 / 45 / 15 / 15 / 45 / 75 )
240
plies. Note that, at the range below 200 Hz, the STL values of the anti-symmetric laminates are
241
almost consistent to those of the symmetric laminates. Furthermore, within the medium and high
242
frequency range, the differences of symmetric and anti-symmetric laminates are also minor. This
243
is because the different lamination schemes lead to the difference of bending and extension
244
coupling effect. For symmetric laminates, the bending and extension coupling stiffness are equal
245
to zero, while for anti-symmetric laminates, the coupling stiffness is not zero. Generally, the
246
effects of bending and extension of symmetric and anti-symmetric laminates play a minor role in
247
sound transmission performance. A similar characteristic can also be found in existing findings of
248
Cao [15] and Yin [26].
and anti-symmetric
11
70 symmetrical laminate antisymmetrical laminate
60
STL (db)
50
40
30
20
10
0 1 10
249 250 251
2
3
10
10
4
10
Frequency (Hz) Fig.5 Influence of different lamination scheme on sound transmission loss of the structure.
3.4. Influence of single layer stiffness
252
In order to investigate the influence of single layer stiffness of laminated plate on the sound
253
transmission loss, a four-layered laminated plate with different Young’s modulus is investigated to
254
evaluate such effects. The main parameters of Young’s modulus in four cases are listed in Table 2.
255
As can be seen in Fig.6, the curve of STL peaks and dips in case 3 is shifted to lower frequency
256
with comparison to that in case 2 when Young’s modulus values of the upper and lower layer
257
decreases. This is due to that the decrease of Young’s modulus of the upper and lower layer leads
258
to the reduction of laminated plate stiffness. It is also observed that the increases of middle two
259
layer Young’s modulus in case 1 and case 2 have no obvious influence on sound transmission loss.
260
These results indicate that the sound transmission loss is sensitive to the change of stiffness in the
261
upper and lower layer by comparison with middle two layer.
12
80 case 1 case 2 case 3
70 60
STL(db)
50 40 30 20 10 0 1 10
3
4
10
10
Frequency(Hz)
262 263 264 265
2
10
Fig.6 Influence of single layer stiffness on sound transmission loss of the structure.
3.5. Influence of fiber orientation angle
266
The effects of different fiber orientation angles on sound transmission characteristics are also
267
studied. In order to facilitate analysis, the six layers of composite laminated plates each adopt the
268
same
269
angles( 0 ,30 ,45 ,60 and 90 ) are chosen in the numerical calculation. It is interesting to
270
note that since the fiber orientations of laminate plate have axial symmetry about 45 , the curves
271
of 0
272
are shown in Fig.7. Similarly, the influence of fiber orientation angle of anti-symmetric laminates
273
is consistent with symmetric laminates, as shown in Fig.8. In addition, it can be seen from
274
Fig.7-Fig.8 that the STL peaks and dips of three curves are gradually shifted to low frequency as
275
the fiber orientation angle increases from 0
276
orientation angle leads to the decrease of structural stiffness, which causes a corresponding decline
277
of the natural frequency.
material
2,
as
listed
in
Table2.
Five
different
fiber
orientation
and 90 overlap the angle of 30 and 60 , and thus only three different STL curves
to 45 . This is because the increment of fiber
13
70
60
STL (db)
50
=0 =30 =45 =60 =90
40
30
20
10
0 1 10
278 279
2
3
10
10
4
10
Frequency (Hz) Fig.7 Influence of fiber orientation angle on sound transmission loss of the symmetric laminates.
70
60
STL (db)
50
=0 =30 =45 =60 =90
40
30
20
10
0 1 10
280 281
2
3
10
10
4
10
Frequency (Hz) Fig.8 Influence of fiber orientation angle on sound transmission loss of the anti-symmetric laminates.
282 283
4. Conclusions
284
In this study, first order shear deformation theory and double panel sandwich structure model
285
are adopted to develop an analytical model for investigating the vibroacoustic characteristic of an 14
286
orthogonally rib-stiffened double laminated composite plates. Numerical results show that the
287
effects of bending and extension of symmetric and anti-symmetric laminates play a minor role in
288
sound transmission performance, and the sound transmission loss is sensitive to the change of
289
stiffness in the upper and lower layer by comparison with middle two layer. Moreover, different
290
fiber orientation angle has pronounced influence on sound transmission performance. But the
291
influence of fiber orientation angle of anti-symmetric laminates is consistent with symmetric
292 293 294
laminates. The increment of fiber orientation angle leads to the decrease of structural stiffness. Acknowledgment
295
The authors are grateful to the referees for their valuable suggestions. This work presented
296
here were supported by the National Natural Science Foundation of China under the contract
297
number 11372083, and by National Basic Research Program of China under the contract number
298 299 300
613235.
301
Appendix A The displacement field of the existing FSDT is given by [19-20]
u1 ( x, y, z , t ) u ( x, y, t ) z x ( x, y, t ) v1 ( x, y, z , t ) v( x, y, t ) z y ( x, y, t )
302
(A.1)
w1 ( x, y, z , t ) w( x, y, t ) 303
where ( u, v, w, x , y ) are unknown functions to be determined, t denotes the time and ( u, v, w )
304
denotes the displacements of the mid plane. x and
305
about the y-axis and x-axis, respectively. When a plane sound wave with amplitude q acting on
306 307
the composite panel, according the mathematical statement of the Hamilton principle, the equations of motion for the laminated plate can be derived and given as [15]
308
L11 L 21 L31 L41 L51
L12 L22
L13 L23
L14 L24
L32 L42
L33 L43
L34 L44
L52
L53
L54
y are the rotations of a transverse normal
L15 u 0 L25 v 0 L35 w q L45 x 0 L55 y 0
309
where q is the acoustic pressure acting on the plate.
310
The coefficients Lij in Eq.(A.2) are listed in the following [15]:
(A.2)
311
L11 A11 m2 2 A16 m n A66 n2 I1 2 , L12 A16 m2 A26 n2 ( A12 A66 ) m n
312
L13 0 , L14 B11 m2 2 B16 m n B66 n2 I 2 2
15
313
L15 B16 m2 B26 n2 ( B12 B66 ) m n , L22 A66 m2 A22 n2 2 A26 m n I1 2
314
L23 0 , L24 B16 m2 B26 n2 ( B12 B66 ) m n
315
L25 B66 m2 B22 n2 2 B26 m n I 2 2
316
L33 A55 m2 2A45 m n A44n2 I1 2 0 2 k z ,mn , L34 jA55 m jA45 n
317
L35 jA45 m jA44 n , L44 D11 m2 D66 n2 2 D16 m n A55 I 3 2
318
L45 D16 m2 ( D12 D66 ) m n D26 n2 A45
319
L55 D66 m2 D22 n2 2 D26 m n A44 I 3 2
320
Lij L ji , (i, j 1,2,3,4,5)
n k y 2n l y , is a shear correction factor, which consider
321
where m k x 2m l x and
322
the non-uniform distribution of shear strain in the thickness direction of the plate. The selection of
323
shear correction factors is very complicated since its value depends not only on the lamination and
324
geometric parameters, but also on the loading and boundary conditions[27]. In Mindlin’s model[2],
325
the shear correction factor is
326
coefficients Lij into Eq.(A.2) and solving the matrix in Eq.(A.2),the transverse displacement w
327
is obtained and the dynamic stiffness of the panel associated with the wave propagating in the
328
structure can be derived:
2 12 and the same value is used in this paper. Substituting
K*
329
q w
(A.3)
330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345
References [1] E. Reissner, The effect of transverse shear deformation on the bending of elastic plates, Journal of Applied Mechanics, 12 (2) (1945) 69-72. [2] R. D. Mindlin, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, Journal of Applied Mechanics, 18 (1) (1951) 31-8. [3] H.-T.Thai, D.-H. Choi, A simple first-order shear deformation theory for laminated composite plates, Composite Structures, 106 (2013) 754-763. [4] J.L.Mantari, Free Vibration of single and sandwich laminated composite plates by using a simplified FSDT, Composite Structures, 132 (2015) 952-959. [5] J.N.Reddy, A simple higher-order theory for laminated composite plates, Journal of Applied Mechanics, 51 (1984) 745-752. [6] L.Librescu, On the theory of anisotropic elastic shells and plates, International Journal of Solids and Structures, 3 (1) (1967) 53-68. [7] F.X.Xin, T.J.Lu.C.Q.Chen, External mean flow influence on noise transmission through double-leaf aeroelastic plates, AIAA Journal, 47 (8) (2009) 1940-1951.
16
346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385
[8] B.R.Mace, Sound radiation from fluid loaded orthogonally stiffened plates, Journal of Sound and Vibration, 79
386 387 388 389
The list of figure
(1981) 439-452. [9] J.H.Lee, J.Kim, Analysis of sound transmission through periodically stiffened panels by space-harmonic expansion method, Journal of Sound and Vibration, 251 (2002) 341-366. [10] J.Alba, J.Ramis, Improvement of the prediction of transmission loss of double partitions with cavity absorption by minimization techniques, Journal of Sound and Vibration, 273 (2004) 793-804. [11] J.Wang, T.J.Lu, R.S.Woodhouse, Sound transmission through lightweight double leaf partitions:Theoretical modelling, Journal of Sound and Vibration, 286 (2005) 817-847. [12] F.X.Xin, T.J.Lu, Transmission loss of orthogonally rib-stiffened double-panel structures with cavity absorption, Journal of the Acoustical Society of America, 129 (4) (2011) 1919-1934. [13] J.Legault, N.Atalla, Numerical and experimental investigation of the effect of structural links on the sound transmission of a lightweight double panel structure, Journal of Sound and Vibration, 324 (2009) 712-732. [14] X.W.Yin, H.F.Cui, Acoustic radiation from a laminated composite plate excited by longitudinal and transverse mechanical drives, Journal of Applied Mechanics, 76 (2009) 1-5. [15]X.T.Cao,H.X.Hua,Z.Y.Zhang,Sound radiation from shear deformable stiffened laminated plates,Journal of Sound and Vibration,330 (2011) 4047-4063. [16]F.X.Xin,T.J.Lu,Effects of mean flow on transmission loss of orthogonally rib-stiffened aeroelastic plates,Journal of the Acoustical Society of America,133 (6) (2013) 3909-3920. [17] F.X.Xin,T.J.Lu,Analytical modeling of fluid loaded orthogonally rib-stiffened sandwich structures:Sound transmission,Journal of the Mechanics and Physics of Solids,58 (2010) 1374-1396. [18] C.Shen,F.X.Xin,L.Cheng,T.J.Lu,Sound radiation of orthogonally stiffened laminated composite plates under airborne and structure borne excitations,Composites Science and Technology,84 (2013) 51-57. [19] J.M.Whitney,The effect of transverse shear deformation on the bending of laminated plates,Journal of Composite Materials,3 (1969) 534-547. [20] C.H.Thai,L.V.Tran,D.T.Tran, Analysis of laminated composite plates using higher-order shear deformation plate theory and node-based smoothed discrete shear gap method, Applied Mathematical Modelling, 36 (2012) 5657-5677. [21] S.Timoshenko,D.H.Young,W.Weaver.Vibration problems in Engineering,Wiley,New York,1974. [22] J.Legault,N.Atalla, Sound transmission through a double panel structure periodically coupled with vibration insulators, Journal of Sound and Vibration, 329 (2010) 3082-3100. [23] L. Maxit, Wavenumber space and physical space responses of a periodically ribbed plate to a point drive:a discrete approach, Applied Acoustics,70 (4) (2009) 563-578. [24] W.R.Graham, High frequency vibration and acoustic radiation of fluid loaded plate, Proc.R.Soc.London A ,352.1995.1-43. [25] F.Fahy, P.Gardonio, Sound and structural vibration, Elsevier, Amsterdam, 2007. [26] X.W.Yin,X.J.Gu.H.F.Cui,R.Y.Shen, Acoustic radiation from a laminated composite plate reinforced by doubly periodic parallel stiffeners, Journal of Sound and Vibration, 306 (2007) 877-889. [27] H.-T.Thai,S.-E.Kim,Free vibration of laminated composite plate using two variable refined plate theory, International Journal of Mechanical Sciences,52 (2010) 626-633
Fig.1 Pressure wave impinging upon the stiffened double laminated composite plates Fig.2 Convergence of stiffened composite laminated plates at four different frequencies. Solid line: 200 Hz, 4 terms used; dotted line: 2000 Hz, 7 terms; dashed line: 5000Hz, 11 terms; dot-dashed line: 10000 Hz, 19 terms.
17
390 391 392 393 394
Fig.3 Comparison between present model predictions and those by Xin [17] for sound transmission loss of
Fig.5
Influence of different lamination scheme on sound transmission loss of the structure.
395 396 397 398
Fig.6
Influence of single layer stiffness on sound transmission loss of the structure.
Fig.7
Influence of fiber orientation angle on sound transmission loss of the symmetric laminates.
Fig.8
Influence of fiber orientation angle on sound transmission loss of the anti-symmetric laminates.
orthogonal stiffened plates Fig.4 Comparison between present model predictions and those by Yin [26] for sound transmission loss of orthogonal laminated plate
18