 Open Access
 Total Downloads : 56
 Authors : Zhiwei Guo , Meiping Sheng
 Paper ID : IJERTV8IS090107
 Volume & Issue : Volume 08, Issue 09 (September 2019)
 Published (First Online): 20092019
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
BandGap And VibrationTransmission Properties of a Periodic Compound Plate in Free Boundary Condition
Zhiwei Guo
School of Marine Science and Technology Northwestern Polytechnical University Xian, China
Meiping Sheng
School of Marine Science and Technology Northwestern Polytechnical University Xian, China
AbstractThe flexuralwave bandgap and vibration transmission characteristics of a periodic compound plate with free boundary condition are studied in this paper. The exact solutions of the bandgap frequency and the vibration response are obtained by solving governing equations and BlochFloquet periodic boundary equations. The finiteelementmethod (FEM) validation shows that the theoretical solutions have high accuracy and have excellent agreement with the FEM results. In the pass band, the flexural wave propagates normally without attenuation, while in the bandgap, the wave is attenuated with significant attenuation levels. Compared with the previous planewave model, the present model gives more accurate results and can represent the realistic situation of a periodic plate structure. Further study shows that the transmission characteristic of a finite periodic compound plate is dependent on the excitation distribution. The bandgap width of a symmetric or antisymmetric excitation is broader than that of a general excitation. Therefore, when the periodic plate is under symmetric or antisymmetric load, the attenuation performance of vibration can be improved.
Keywords Periodic compound plate; bandgap; vibration suppression; transfer matrix method

INTRODUCTION
Periodic structure is composed of a number of identical elements repeated in one, two, and three dimensions [1]. Owing to the Braggscattering effect [2] or locally resonant effect [3], the wave filtering phenomenon exists in a periodic structure, with waves in the pass band propagating freely and waves in the bandgap being attenuated gradually [4]. Thus the periodic structure has given a new method to reduce vibration and wave propagation. Attracted by the great potential in vibration and noise control, extensive studies about periodic structures on structure types [57], bandgap calculation methods [810], and bandgap formation mechanisms [2, 3] have been conducted. In last two decades, phononic crystal [11, 12] and metamaterial [1315] based on periodic theory have renewed our sight and injected new life to periodic structure.
As platetype structure is widely used in the engineering applications, the spatial periodicity was introduced in the platetype structure to reduce noise and vibration in extensive studies, including periodically supported plate [16], periodically stiffened plate [17], a plate with periodically attached springmass resonators [18], and a plate with
various points of views [2124]. In the previous works about periodic plate, only a couple of studies were associated with periodic compound plate, where two different subplates repeat periodically along one direction. Sorokin [25] studied the bandgap performance of a periodic compound plate. However, in Sorokins model, only plane wave was considered and the effects of nonplane wave modes were neglected. Thus, the model was quite like a periodic compound beam, while the effects of Poissons ratio were considered. In fact, the nonplane wave modes have great effect on the bandgap characteristic. Therefore, the previous model cannot predict the real bandgap characteristic of a periodic compound plate when the plane wave and nonplane wave are coupled together.
In this paper, Sorokins work is extended, with both the plane wave modes and the nonplane wave modes being included in the present model, resulting that the one dimensional model in Sorokins work becomes a special case in the present model. By considering all the modes, the present model gives more accurate prediction for the bandgap and transmission characteristic of a periodic compound plate. In addition, the effect of excitation distribution on transmission characteristic of a finite periodic plate is also studied. Symmetric excitation and antisymmetric excitation are respectively associated with symmetric bandgap and anti symmetric bandgap, which are constituted by symmetric propagation modes and antisymmetric propagation modes, respectively. Although free boundary condition is considered in this paper, the present model can be easily applied to the clamped, simply supported, or even elastic boundary conditions by simply changing the boundary equations.

THEORETICAL MODEL

Vibration Transmission
0
0
A finite periodic compound plate is shown in Fig. 1, which consists of alternating subplates repeated in xdirection. A unit element is composed of cell A and cell B, with Youngs moduli EA and EB , densities A and B , Poisson ratios A and B , lengths aA and aB , thicknesses hA and hB , and widwidththss bbA aanndd bbB (( bbA bB ). The lattice constant is expressed as a aA aB . The forced response is obtained in this sub section to study the vibration transmission under a harmonic
periodically filledin scattering units [19, 20]. In addition,
there were also a number of works related to periodic plate in
force excitaion
f e jt
applied at the first cell (cell 0).
f e
f e
jt
0
w w
0 1 2 3 4 5
I2
I1 I
w w ,
i 1
i ,
B A B A
B A B A B
i 1 xi1 ai1
i xi 0 x
x i 1, 2
z0 z1 z2
z3 z4 z5
zI1 zI
i 1 xi1 ai1
i xi 0
(7)
y0 x0
y2
y1 x1 x2
y3 y4 y5
x3 x4 x5
yI1
yI
xI1 xI
,, I
excited plate
element 1 element 2 element I/2
M i 1x x a
M ix , Qi 1x Qix
x 0 x
a
x 0
x 0 x
a
x 0
i1
i1 i i1 i1 i
Fig. 1. Schematics of a finite periodic compound plate
where the bending moments and the shearing forces in the above boundary conditions can be expressed respectively as
The harmonic governing equation of the ith thin plate is
2 w
2 w 3 w
3 w
M D i v i , Q D i (2 v ) i ,
ix i
x2
i y2
ix i
x3
i x y2
D 4 w (x , y ) h 2 w (x , y ) f
i i i i i
.(8)
i i i i i i i i i i
3
3
2 w 2 w 3 w
3 w
with bending moment
Di Eihi
121i
and
M D i v i ,
Q D i (2 v ) i
2
2
iy i
y2
i x2
iy i
y3
i x2 y
4 4 x4 24 x2y2 4 y4 . The harmonic solution
i i i i i
i i i i
of (1) can be expressed as [26]
M N
By substituting (24, 8) into (57), the boundary equations can be obtained in a matrix form
Hp Qa , (9)
i i i imn im i in i i i
i i i imn im i in i i i
0 1 I
0 1 I
0 1 I
0 1 I
w (x , y ) A cos x cos
y p x , y
where p [p p p ]T and a [a a a ]T .
m0 n0
The terms p and a (i=0,1,Â·Â·Â·,I) are defined as p =[ c1 , c1 ,
4 M N i i
i i 0 i1
where
p x , y l ( y ) cl
cos
x l (x )dl cos y ,
Â·Â·Â·, c1
, c 2 , Â·Â·Â·, c2
, c3 , Â·Â·Â·, c3
, c 4 , Â·Â·Â·, c4
, d 1 , d 1 , Â·Â·Â·,
i i ib i imim i ia i in in i
iM i 0
iM i 0
iM i 0
iM i 0 i1
l 1
m0
n0
l
l d1 , d 2 , Â·Â·Â·, d 2 , d 3 , Â·Â·Â·, d 3 , d 4
, Â·Â·Â· , d 4 ] and a =[ A ,
im
ai , and in n
bi . The terms
Aimn , cim , and din
iN i 0
iN i 0
iN i 0
iN i
i 00
are unknown coefficients and the terms l (x ) and l ( y )
Ai01 , Â·Â·Â·,
Ai 0 N ,
Ai10 ,
Ai11 , Â·Â·Â·,
Ai1N , Â·Â·Â·,
Aimn , Â·Â·Â·,
AiM 0 ,
AiM 1 ,
ia i
ib i
A ], respectively.
can be obtained from the expression of is (i ) ( s a , x
or s b, y )
iMN
Thus from the boundary equation, p is expressed by a as
p H1Qa . For the governing equations, the excitation force
is (i ) i ( )
is (i ) i ( )
i ( )
i ( )
i ( )
i ( )
i ( ) , (3)
i ( ) , (3)
1 2 3 4 T
s i s i s i s i
fi in (1) is expressed by modal forces in series form
where
1 ( ) sin 2s ,
2 ( ) cos 2s , M N
is i i i
is i i i
D 4 w (x , y ) h 2 w (x , y ) F
cos
x cos
y . (10)
3 ( ) sin 3 2s , and 4 ( ) cos3
2s .
( )
i i i i i i i i i imn im i in i
is i i i
can be expanded to series as
is i i i
is i
m0 n0
Substituting (24) into (10) gives
K a Sp 2 M a Tp F , (11)
( )
cos
. (4) 1 1
is i
r 0
r isr i
where
F F
, F , , F
, F , , F
, , F
, , F
i i 00
i 01
i 0 N i10
i1N iM 0
iMN
A
B
I1
C B C
I
yI1
yI
xI1
xI
A
B
I1
C B C
I
yI1
yI
xI1
xI
FBC FBC FBC FBC FBC FBC
and F F F F T . Substituting the formulation
B
A
B
0
C B C
1
C B C
2
C B C
y0
y1
y2
x0
x1
x2
B
A
B
0
C B C
1
C B C
2
C B C
y0
y1
y2
x0
x1
x2
A B
F C F

B B

C C
y3
0 1 I
p H1Qa into (11) gives
K 2M a F (12)
1 1
x3
FBC FBC FBC FBC FBC FBC
Fig. 2. Boundary conditions of a finite periodic plate
The boundary conditions of a finite periodic plate are shown in Fig. 2, including free (FBC) and continuous (CBC) boundary conditions. The boundary equations are expressed respectively as:

FBCs along xdirection ( yi 0 and yi bi )
where K K1 SH Q and M M1 TH Q .
At a given frequency, the eigenvector a can be obtained by solving (12), and then p is also determined by (9). As a and p become known, the vibration response of each cell will
be finally determined.


Bandgap Formulation
The bandgap frequency can be determined by the analysis
M
M
iy yi 0
M
0,
0,
Q
iy
iy
yi 0
Q
0,
0
i 0,1,, I
(5)
of a unit element with BlochFloquet periodic condition. As shown in Fig. 3, a unit element consists of three types of boundary conditions, including free (FBC), continuous (CBC),
iy yi bi iy
yi bi
and periodic (PBC) boundary conditions.

FBCs along ydirection ( x0 0 and xI aI )
M0x x 0 0, Q0x x 0 0, MI x x a 0,QI x x a 0 (6)
0 0 I I I I

CBCs between the (i1)th cell and the ith cell ( xi 1 ai 1 or xi 0 )
FBC FBC
P C P
branches below 400 Hz are divided into four groups, the 1st mode group (11#, 12#, 13#, and 14#), the 2ndmode group (21#,
22#, and 23#), the 3rdmode group (31#, 32#, and 33#), and the 4th
mode group (41# and 42#).

0 B 1 B

C C
y0 y1
x0 x1
400
350
300
14#
42#
33#
41#
3rd BG
23#
FBC FBC
Fig. 3. Boundary conditions of a unit element
The boundary equations of FBC and CBC have been shown in (5) and (7), while the boundary equations of PBC can be expressed as
250
Frequency (Hz)
Frequency (Hz)
200
150
100
50
0
13#
12#
2nd BG 1st BG
21#
32#
31#
22#
11#
iqa
w0
iqa w1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
w0 x 0 e w1 ,
e , qn
0 x1 a1
M eiqa M
x0
, Q
x0 0
eiqaQ
x1
x1 a1
, (13)
Fig. 4. Dispersion curves of the 1stmode group (), 2ndmode group ( ), 3rdmode group (—), and 4thmode group (..) (BG: bandgap)
0 x x0 0
1x x1 a1
0 x x0 0
1x x1 a1
0.5
0.5
0.5
0.5
1.0
where q is wavenumber, which represents the wave
propagation and wave attenuation performance. Equation (13) can be rewritten as a matrix form
0.4
y (m)
y (m)
y (m)
y (m)
0.3
0.2
0.1
0.0
0.4
0.3
0.2
0.1
0.0
0.4
y (m)
y (m)
y (m)
y (m)
0.3
0.2
0.1
0.0
0.4
0.3
0.2
0.1
0.0
0.9
0.8
0.7
0.6
0.5
p>H qp Q q a , (14)
0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5
(a) x (m) (b) x (m) (c) x (m) (d) x (m)
0.4
0 1
0 1
where a [a a ]T
and p [p p ]T . From (14), p can be
0.5
0.4
y (m)
y (m)
0.3
0.5
0.4
y (m)
y (m)
y (m)
y (m)
0.3
0.5
0.4
0.3
0.5
0.4
y (m)
y (m)
0.3
0.3
0.2
0.1
0 1
0 1
expressed as p H q1 Qqa . By substituting (24) into
0.2 0.2 0.2 0.2
0.1 0.1 0.1 0.1
0.0
0.1
0.2
the homogeneous plate equation (set
fi 0
in (1)), it is
0.0 0.0 0.0 0.0
.0 0.1 0.2 0.3 0.4 0.5 .0 0.1 0.2 0.3 0.4 0.5 .0 0.1 0.2 0.3 0.4 0.5 .0 0.1 0.2 0.3 0.4 0.5
0.3
0
(e)
x (m)
0
(f)
x (m)
0
(g)
x (m)
0
(h)
x (m)
0.4
obtained that
a Sp 2 M a Tp 0 . (15)
0.5
0.4
y (m)
y (m)
0.3
0.2
0.5
0.4
y (m)
y (m)
0.3
0.2
0.5
0.4
y (m)
y (m)
0.3
0.2
0.5
0.4
y (m)
y (m)
0.3
0.2
0.5
0.6
0.7
0.8
1 1
1 1
Substituting p H q1 Qqa into (15) gives
K q2Mq a 0 (16)
0.1
0.0
0.0 0.1 0.2 0.3 0.4 0.5
0.1
0.0
0.0 0.1 0.2 0.3 0.4 0.5
0.1
0.0
0.0 0.1 0.2 0.3 0.4 0.5
0.1
0.0
0.0 0.1 0.2 0.3 0.4 0.5
0.9
1.0
1
1
1
1
(i)
x (m)
(j)
x (m)
(k)
x (m)
(l)
x (m)
where K K1 SHq
Qq and M M1 TH q
Qq .
Fig. 5. Modal shapes of a periodic compound plate at qn=0.25 for the 1st mode group ((a) 11#, (b) 12#, (c) 13#, and (d) 14#), 2ndmode group (I 21#, (f)
When the wavenumber q is given, the stiffness matrix K
and mass matrix M become known. The normalized wavenumber is defined as qn . Sweeping qn in one period from 1 to 1 will give rise to the dispersion curves (the
frequency as a function of qn ), and then the bandgap
properties of the periodic compound plate can be finally determined.


BANDGAP CHARACTERISTIC AND VALIDATION
A
A
3
3
The bandgap characteristic of an infinite periodic plate is studied in this section. The plate parameters are given as EA 2 GPa , EB 210 GPa , 1142 m ,
B
B
A B A
A B A
7800 m3 , 0.39 , 0.30 , a 0.15 m , aB 0.35 m , bA bB 0.5 m , and hA hB 5 mm . The dispersion curves below 400 Hz are shown in Fig. 4. As we can see in the figure, the dispersion curves of a periodic
compound plate are constituted by several dispersion
branches, with each branch corresponding to a specific wave propagation mode (see Fig. 5). These branches are divided into several modal groups according to their crossstream modal shapes. As shown in Fig. 4 and Fig. 5, the dispersion
22#, and (g) 23#), 3rdmode group ((h) 31#, (i) 32#, and (j) 33#), and 4thmode
group ((k) 41# and (l) 42#)
The dispersion branches from the same modal group are separated between each other, and the frequency gap between neighboring dispersion branches constitute the modal band gap. For each modal bandgap, the wave cannot propagate with its corresponding modal shape, while it may propagate with the modal shapes belonging to other modal groups. For example, the wave of 100 Hz cannot propagate with the modal shape in the 1stmode group, however, it can propagate with mode 22# in the 2ndmode group. For the wave with a specific frequency, if theres no any dispersion curve related to this frequency, the wave cannot propagate with any modal shape. Thus the general bandgap is the intersection of all the modal bandgaps.
As shown in Fig. 4, the bandgap and bandpass alternate with each other. Three bandgaps exist below 400 Hz, namely 57.5 Hz 86.5 Hz, 112.1 Hz 182.3 Hz, and 277.5 Hz
289.5 Hz. The first bandgap is in low frequency range, thus it can be used in the low frequency vibration control. The total bandgap width is 111.2 Hz, with the bandgap ratio exceeding 25%, which indicates that more than a quarter of flexural wave between 0 Hz and 400 Hz are suppressed. The dispersion curves are also calculated with FEM by COMSOL
Multiphysics software with the results shown in Fig. 6(a). It can be seen that the dispersion curves from present model have an excellent agreement with the results of an FEM model, with no more than 2.5 Hz difference. Thus, the theoretical model derived in this paper has an excellent accuracy.
It can be seen in Fig. 7(a) that there are three transmission valleys, where the vibrations are significantly attenuated and only very small vibration energy transmits from the first cell (excited cell) to the last one. The bandgaps in the infinite periodic plate calculated in section 3 are displayed in shaded region in gray. As shown in Fig. 7(a), the transmission valleys
400
350
Frequency (Hz)
Frequency (Hz)
300
250
200
150
100
50
Present FEM
400
350
Frequency (Hz)
Frequency (Hz)
300
250
200
150
100
50
1stmode group Sorokin
in a finite periodic plate match very well with the bandgaps in an infinite periodic plate. Thus, the waves in pass bands can propagate freely along the axial direction, while the waves in bandgaps are attenuated with considerable levels. This filtering phenomenon is beneficial to noise and vibration control.
0
1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0

qa/
(b)
0 20
1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0
Transmittance Te (dB)
Transmittance Te (dB)
qa/ 0
20
TransmittanceTe (dB)
TransmittanceTe (dB)
0
20
Fig. 6. Dispersion curves and bandgaps for (a) comparisons between present model and FEM model and (b) comparisons between the 1stmode group in present model and Sorokin model (The shaded region in gray are bandgaps.)
S.V. Sorokin has studied the bandgaps of a periodic compound plate where only planewave propagation modes
(a)
20
40
60
80
Present (4 elements)
FEM (4 elements)
0 50 100 150 200 250 300 350 400
Frequency (Hz)
40
60
80
100
120
(b)

elements

elements
4 elements
8 elements
0 50 100 150 200 250 300 350 400
Frequency (Hz)
are considered, where the dispersion curves correspond to the dispersion branches of the 1stmode group in Fig. 4. The two results are compared in Fig. 6(b) showing perfect coincidence. As shown in Fig. 6, the dispersion curves of present model (Fig. 6(a)) are more complicated than those of plane wave model (Fig. 6(b)), and the planewave model in Sorokins work has become a special case of this general plate model. As the nonplane wave modes are included in present model, the bandgap width become narrower. It can observed in Fig. 6 that the first bandgap (9.1 Hz 22.4 Hz) of the plane wave model disappears in the present model; the second band gap (58.1 Hz 182.9 Hz) of the planewave model is divided into two narrow bandgaps (57.5 Hz 86.5 Hz and 112.1 Hz
182.3 Hz) in the present model; the third bandgap (230.2 Hz 321.0 Hz) of the planewave model is changed to a smaller bandgap (277.5 Hz 289.5 Hz) in the present model. Thus, when the nonplanewave propagation modes are considered, the bandgap performance varies significantly and the bandgap properties predicted b the planewave model cannot represent the realistic wave propagation situations, while the present model will provide good results.



VIBRATION TRANSMISSION
The vibration transmission characteristic of a finite periodic compound plate (see Fig. 1) is studied in this section with the same material and geometry parameters shown in section 3. A harmonic point force is applied at x0 a0 4 and
Fig. 7. Vibration transmittance of a periodic compound plate with (a) the
comparison between present model and FEM model and (b) the performances with the variation of the number of element (The shaded region in gray are bandgaps.)
The effect of the number of element on transmission characteristic is studied with the number of element equaling one, two, four, and eight and the comparison results are plotted in Fig. 7(b). Significant attenuation levels can be observed for the first and second bandgaps. Even with one element, the averaged attenuation level can reach to about 12 dB for the first bandgap and 20 dB for the second band gap. The attenuation ability of the third bandgap is not as strong as the first two and theres a very little attenuation with one element. With the increase of the number of element, the attenuation levels in the bandgaps increase significantly. When the number of element increases from one to eight, the average attenuation levels increase from 12.7 dB to 71.5 dB for the first bandgap, from 20.8 dB to 107.9 dB for the second bandgap, and from 0.7 dB to 25.0 dB for the third bandgap. The attenuation ability of the third bandgap is weaker than that of the first two bandgaps, hence more elements are needed to achieve a good vibrational performance for the third bandgap.

EFFECT OF EXCITATION DISTRIBUTION
In this section, the effect of excitation distribution on bandgap property is studied, including point and line excitations with symmetric and antisymmetric distributions
y0 b0 4 in the first plate (cell 0). The vibration
on the plate. Further examination shows that the excitation
transmittance is defined as T 20 log10 v v , where v
distribution has great effect on the wave transmission
e I 0
0 characteristic of a finite periodic compound plate, with
and vI are the spatially averaged transverse velocities of cell 0 and cell I, respectively. The vibration transmittances of a periodic compound plate with four elements by both the present model and the FEM model are shown in Fig. 7(a). It is observed that the result from the present model has a good coincidence with that from the FEM model. Therefore, the
present vibrational model has high accuracy in calculating the vibration response.
significant difference between a symmetric excitation and an antisymmetric excitation. The bandgaps calculated in section 3 for the infinite periodic plate cannot predict the transmission valleys in a finite periodic plate when the excitation is symmetrically or antisymmetrically distributed. In fact, a symmetric or antisymmetric excitation always causes better performance than a general excitation. This phenomenon can be explained by examining corresponding dispersion curves and propagation modes.
It is observed in Fig. 5 that the modal shapes are either symmetrically distributed (the 1stmode and 3rdmode groups) or antisymmetrically distributed (the 2ndmode and 4thmode groups). As shown in Fig. 8, the symmetric bandgaps are constituted by the symmetric dispersion branches (11#, 12#, 13#,
excitation (see Fig. 10(d)), the transmittance valleys are in coincidence with the antisymmetric bandgaps. However when the force excitation is neither symmetric nor anti symmetric, the transmittance valleys are in coincidence with the general bandgaps.
14#, 31#, 32#, and 33#) and the antisymmetric bandgaps are associated with the antisymmetric dispersion branches (21#, 22#, 23#, 41#, and 42#). The bandgap frequencies are listed in Table 1. By comparing Fig. 4, Fig. 8, and Table 1, it is found that the general bandgaps are the intersection of symmetric bandgaps and antisymmetric bandgaps. For example, the first general bandgap (57.5 Hz 86.5 Hz) is the intersection of the second symmetric bandgap (57.5 Hz 103.7 Hz) and the first antisymmetric bandgap (24.7 Hz 86.5 Hz). The
b0
(a)
a0
p3
p2 p1
y0 y1
x0 x1
excited plate
b0
a0
y0 y1
x0 x1
excited plate
total bandgap widths of the symmetric and antisymmetric bandgaps are respectively 217.1 Hz and 202.4 Hz, which are
Fig. 9. Schematics of (a) point excitation at p1 (a0/4, b0/4), p2 (a0/4, b0/2),
and p3 (a0/4, 3b0/4)) and (b) line excitation at x0=a0/4)
broader than that of the general bandgap width with
111.2 Hz. Thus, when all of the waves are propagating in symmetric or antisymmetric modes, the periodic compound plate will have a better performance in the aspect of bandgap width.
20
TransmittanceTe (dB)
TransmittanceTe (dB)
0
20
40
60
20
TransmittanceTe (dB)
TransmittanceTe (dB)
0
20
40
60
400
350
Frequency (Hz)
Frequency (Hz)
300
250
200
150
100
50
0
14#
13#
12#
11#
4rd BG
d BG
d BG
3r
2nd BG 1st BG
33#
32#
31#
400
350
Frequency (Hz)
Frequency (Hz)
300
250
200
150
100
50
0
42#
41#
4rd BG 3rd BG
BG
BG
2nd
1st BG
23#
22#
21#
(a)
TransmittanceTe (dB)
TransmittanceTe (dB)
(c)
80
20
0
20
40
60
80
0 50 100 150 200 250 300 350 400
Frequency (Hz)
0 50 100 150 200 250 300 350 400
Frequency (Hz)
80
(b)
20
TransmittanceTe (dB)
TransmittanceTe (dB)
0
20
40
60
80
(d)
0 50 100 150 200 250 300 350 400
Frequency (Hz)
0 50 100 150 200 250 300 350 400
Frequency (Hz)
0.0 0.2 0.4 0.6 0.8 1.0
(a) qn
(b)
0.0 0.2 0.4 0.6 0.8 1.0
qn
Fig. 10. Vibration transmittances under point excitations of (a) case 1, (b) case 2, (c) case 3, and (d) case 4
Fig. 8. Dispersion curves and bandgaps of (a) symmetric modes and (b) antisymmetric modes (BG: bandgap)
Bandgap order
Symmetric bandgaps
AntiSymmetric bandgaps
General band gaps
First
9.622.2 Hz
24.786.5 Hz
57.586.5 Hz
Second
57.5103.7 Hz
108.2223.6 Hz
112.1182.3 Hz
Third
112.1182.3 Hz
277.5289.5 Hz
277.5289.5 Hz
Fourth
228.5311.7 Hz
316.8325.1 Hz
N/A
Five
342.4347.3 Hz
N/A
N/A
316.8325.1 Hz
Bandgap order
Symmetric bandgaps
AntiSymmetric bandgaps
General band gaps
First
9.622.2 Hz
24.786.5 Hz
57.586.5 Hz
Second
57.5103.7 Hz
108.2223.6 Hz
112.1182.3 Hz
Third
112.1182.3 Hz
277.5289.5 Hz
277.5289.5 Hz
Fourth
228.5311.7 Hz
N/A
Five
342.4347.3 Hz
N/A
N/A
TABLE I. COMPARISON OF THE BANDGAPS
Five more cases of line excitation applied at x0=a0/4 are also studied with case 5: F y0 1 , case 6:
F y0 cos2 y0 , case 7: F y0 sin 1 y0 , case 8:
F y cos y , and case 9: F y sin y , where
0 1 0 0 2 0
1 b0 and 2 b0 . The force distribution shapes are
The vibration transmittances of a finite periodic compound plate (see Fig. 1) with four elements are examined with different force excitations. The positions of point and line excitations are shown in Fig. 9. Four cases of point excitation
shown in Fig. 11 and the corresponding vibration transmittances are shown in Fig. 12. The vibration transmittances of line excitations are similar to those of the point excitations, with symmetric line excitations associated with symmetric bandgaps (see Fig. 12(a)) and antisymmetric line excitations associated with antisymmetric bandgaps (see Fig. 12(b)).
are examined with case 1: excitation at p
with F 1N , case
uniform cos(1y0)
1 1 cos( y ) sin( y ) sin( y )
2: excitation at position p2 with F2 1N , case 3: symmetric excitation at p1 and p3 with F1 1N and F3 1N , and case 4:
1.0
F(y0) (N/m)
F(y0) (N/m)
0.5
0 1 0 2 0
antisymmetric excitation at p1 and p3 with
F1 1N
and
0.0
F3 1N . The vibration transmittances are shown in Fig. 10. It can be seen that the transmittance characteristics of the finite periodic compound plate are dependent on excitation types. For a symmetric force excitation (see Fig. 10(b, c)), the transmittance valleys are in coincidence with the symmetric bandgaps. On the contrary, for the antisymmetric force
0.5
1.0
0.0 0.1 0.2 0.3 0.4 0.5
y0 (m)
Fig. 11. Force distribution shapes of line excitations along ydirection
20
transmittanceTe (dB)
transmittanceTe (dB)
0
20
40
60
80
100
(a)
Uniform cos(2y0) sin (y0)
0 50 100 150 200 250 300 350 400
Frequency (Hz)
20
TransmittanceTe (dB)
TransmittanceTe (dB)
0
20
40
60
80
100
(b)
cos(1y0) sin (y0)
0 50 100 150 200 250 300 350 400
Frequency (Hz)
can be divided into two groups, which are associated with symmetric and antisymmetric modes, resulting that the vibration transmission characteristic of a finite periodic compound plate is dependent on the force distributions. When the force is symmetric, the transmission valleys can be predicted by the symmetric bandgaps; while for the anti symmetric force, the transmission valleys can be predicted by the antisymmetric bandgaps. A general force, neither
Fig. 12. Vibration transmittances under line excitations of (a) symmetric distribution (case 5, case 6, and case 7) and (b) antisymmetric distribution (case 8 and case 9)
Therefore, for both point and line force excitations, the excitation distributions have significant effect on wave transmission. The vibration transmittance valleys of symmetric excitation can be predicted by the symmetric band gaps and those of the antisymmetric excitation can be predicted by the antisymmetric bandgaps. When the periodic plate is excited by a symmetric force, half of an anti symmetric mode makes positive contribution to the response and the other half makes equaling negative contribution, resulting that the total contribution from an antisymmetric mode is zero. However, both of the two half symmetric modes make positive contribution, causing that the total contribution from a symmetric mode is positive. Its the same reason that when the periodic plate is excited by an antisymmetric force, and only antisymmetric mode makes positive contribution to the vibration response. Therefore, the symmetric force is associated with the symmetric modes and the antisymmetric force is associated with the antisymmetric modes. When the periodic plate is excited by neither symmetric nor anti symmetric force, both the symmetric and antisymmetric modes will make contribution to the vibration response, and the transmittance valleys can be predicted by the general bandgaps.
From the above, the vibration suppression performance of a finite periodic compound plate is dependent on the force distributions. The symmetric or antisymmetric force causes better performance than a general force, because the bandwidths of the symmetry bandgap and the antisymmetry bandgap are broader than that of the general bandgap. Thus, when the periodic plate is used in the practical application, making the engine or excitation set work at a symmetric or antisymmetric excitation situation, the vibration suppression performance of the periodic plate can be improved significantly. Therefore, in order to reduce more noise and vibration, the excitation set should be installed at the middle line, and also, if there are two identical excitation sets, symmetric installment will be a good choice to reduce noise and vibration.

CONCLUSIONS
The flexuralwave bandgap characteristic of an infinite periodic compound plate and the transmission characteristic of a finite periodic compound plate are examined in this paper. The exact solutions of bandgap frequency and vibration response are obtained by theoretical derivation and are validated by FEM model. The vibration can be significantly reduced in the bandgaps. Compared with the previous plane wave model, the present model gives more accurate results and can represent the realistic situation. The dispersion curves
symmetric nor antisymmetric, makes the transmission valleys depending on the general bandgaps, which is narrower in bandgap width than symmetric or antisymmetric bandgaps. Thus when the force is in symmetric or antisymmetric situation, the vibration suppression performance of a periodic compound plate will be improved and more vibration can be reduced.
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