Band-Gap And Vibration-Transmission Properties of a Periodic Compound Plate in Free Boundary Condition

Band-Gap And Vibration-Transmission 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 flexural-wave band-gap and vibration transmission characteristics of a periodic compound plate with free boundary condition are studied in this paper. The exact solutions of the band-gap frequency and the vibration response are obtained by solving governing equations and Bloch-Floquet periodic boundary equations. The finite-element-method (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 band-gap, the wave is attenuated with significant attenuation levels. Compared with the previous plane-wave 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 band-gap width of a symmetric or anti-symmetric excitation is broader than that of a general excitation. Therefore, when the periodic plate is under symmetric or anti-symmetric load, the attenuation performance of vibration can be improved.

Keywords Periodic compound plate; band-gap; vibration suppression; transfer matrix method

1. INTRODUCTION

   Periodic structure is composed of a number of identical elements repeated in one, two, and three dimensions [1]. Owing to the Bragg-scattering 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 band-gap 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 [5-7], band-gap calculation methods [8-10], and band-gap formation mechanisms [2, 3] have been conducted. In last two decades, phononic crystal [11, 12] and metamaterial [13-15] based on periodic theory have renewed our sight and injected new life to periodic structure.

   As plate-type structure is widely used in the engineering applications, the spatial periodicity was introduced in the plate-type 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 [21-24]. In the previous works about periodic plate, only a couple of studies were associated with periodic compound plate, where two different sub-plates repeat periodically along one direction. Sorokin [25] studied the band-gap performance of a periodic compound plate. However, in Sorokins model, only plane wave was considered and the effects of non-plane 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 non-plane wave modes have great effect on the band-gap characteristic. Therefore, the previous model cannot predict the real band-gap characteristic of a periodic compound plate when the plane wave and non-plane wave are coupled together.

   In this paper, Sorokins work is extended, with both the plane wave modes and the non-plane 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 band-gap 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 anti-symmetric excitation are respectively associated with symmetric band-gap and anti- symmetric band-gap, which are constituted by symmetric propagation modes and anti-symmetric 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.

2. THEORETICAL MODEL

   1. Vibration Transmission

      0

      0

      A finite periodic compound plate is shown in Fig. 1, which consists of alternating sub-plates repeated in x-direction. 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 filled-in 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

      I-2

      I-1 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

      zI-1 zI

      i 1 xi1 ai1

      i xi 0

      (7)

      y0 x0

      y2

      y1 x1 x2

      y3 y4 y5

      x3 x4 x5

      yI-1

      yI

      xI-1 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 (2-4, 8) into (5-7), 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,···,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 ,

      ···, c1

      , c 2 , ···, c2

      , c3 , ···, c3

      , c 4 , ···, c4

      , d 1 , d 1 , ···,

      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 , ···, d 2 , d 3 , ···, d 3 , d 4

      , ··· , 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 , ···,

      Ai 0 N ,

      Ai10 ,

      Ai11 , ···,

      Ai1N , ···,

      Aimn , ···,

      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 (2-4) 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
      I-1	C B C	I
      yI-1	yI
      xI-1	xI

      A		B
      I-1	C B C	I
      yI-1	yI
      xI-1	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

      1. B B

      2. 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:

      1. FBCs along x-direction ( 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.

   2. Band-gap Formulation

   The band-gap 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.

   1. FBCs along y-direction ( 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

   2. CBCs between the (i-1)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 2nd-mode group (21#,

   22#, and 23#), the 3rd-mode group (31#, 32#, and 33#), and the 4th-

   mode group (41# and 42#).

   1. 0 B 1 B

   2. 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 1st-mode group (), 2nd-mode group ( ), 3rd-mode group (—), and 4th-mode group (..) (BG: band-gap)

   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#), 2nd-mode 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 band-gap

   properties of the periodic compound plate can be finally determined.

3. BAND-GAP CHARACTERISTIC AND VALIDATION

   A

   A

   3

   3

   The band-gap 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 cross-stream modal shapes. As shown in Fig. 4 and Fig. 5, the dispersion

   22#, and (g) 23#), 3rd-mode group ((h) 31#, (i) 32#, and (j) 33#), and 4th-mode

   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 band-gap, 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 1st-mode group, however, it can propagate with mode 22# in the 2nd-mode 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 band-gap is the intersection of all the modal band-gaps.

   As shown in Fig. 4, the band-gap and band-pass alternate with each other. Three band-gaps 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 band-gap is in low frequency range, thus it can be used in the low frequency vibration control. The total band-gap width is 111.2 Hz, with the band-gap 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 band-gaps 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

   1st-mode group Sorokin

   in a finite periodic plate match very well with the band-gaps in an infinite periodic plate. Thus, the waves in pass bands can propagate freely along the axial direction, while the waves in band-gaps 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

   1. 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 band-gaps for (a) comparisons between present model and FEM model and (b) comparisons between the 1st-mode group in present model and Sorokin model (The shaded region in gray are band-gaps.)

      S.V. Sorokin has studied the band-gaps of a periodic compound plate where only plane-wave 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)

      1. elements

      2. 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 1st-mode 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 plane-wave model in Sorokins work has become a special case of this general plate model. As the non-plane wave modes are included in present model, the band-gap width become narrower. It can observed in Fig. 6 that the first band-gap (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 plane-wave model is divided into two narrow band-gaps (57.5 Hz 86.5 Hz and 112.1 Hz

         182.3 Hz) in the present model; the third band-gap (230.2 Hz 321.0 Hz) of the plane-wave model is changed to a smaller band-gap (277.5 Hz 289.5 Hz) in the present model. Thus, when the non-plane-wave propagation modes are considered, the band-gap performance varies significantly and the band-gap properties predicted b the plane-wave model cannot represent the realistic wave propagation situations, while the present model will provide good results.

4. 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 band-gaps.)

   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 band-gaps. Even with one element, the averaged attenuation level can reach to about 12 dB for the first band-gap and 20 dB for the second band- gap. The attenuation ability of the third band-gap 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 band-gaps 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 band-gap, from 20.8 dB to 107.9 dB for the second band-gap, and from 0.7 dB to 25.0 dB for the third band-gap. The attenuation ability of the third band-gap is weaker than that of the first two band-gaps, hence more elements are needed to achieve a good vibrational performance for the third band-gap.

5. EFFECT OF EXCITATION DISTRIBUTION

   In this section, the effect of excitation distribution on band-gap property is studied, including point and line excitations with symmetric and anti-symmetric 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 anti-symmetric excitation. The band-gaps 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 anti-symmetrically distributed. In fact, a symmetric or anti-symmetric 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 1st-mode and 3rd-mode groups) or anti-symmetrically distributed (the 2nd-mode and 4th-mode groups). As shown in Fig. 8, the symmetric band-gaps are constituted by the symmetric dispersion branches (11#, 12#, 13#,

   excitation (see Fig. 10(d)), the transmittance valleys are in coincidence with the anti-symmetric band-gaps. However when the force excitation is neither symmetric nor anti- symmetric, the transmittance valleys are in coincidence with the general band-gaps.

   14#, 31#, 32#, and 33#) and the anti-symmetric band-gaps are associated with the anti-symmetric dispersion branches (21#, 22#, 23#, 41#, and 42#). The band-gap frequencies are listed in Table 1. By comparing Fig. 4, Fig. 8, and Table 1, it is found that the general band-gaps are the intersection of symmetric band-gaps and anti-symmetric band-gaps. For example, the first general band-gap (57.5 Hz 86.5 Hz) is the intersection of the second symmetric band-gap (57.5 Hz 103.7 Hz) and the first anti-symmetric band-gap (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 band-gap widths of the symmetric and anti-symmetric band-gaps 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 band-gap width with

   111.2 Hz. Thus, when all of the waves are propagating in symmetric or anti-symmetric modes, the periodic compound plate will have a better performance in the aspect of band-gap 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 band-gaps of (a) symmetric modes and (b) anti-symmetric modes (BG: band-gap)

   Band-gap order	Symmetric band-gaps	Anti-Symmetric band-gaps	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

   Band-gap order	Symmetric band-gaps	Anti-Symmetric band-gaps	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 BAND-GAPS

   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 band-gaps (see Fig. 12(a)) and anti-symmetric line excitations associated with anti-symmetric band-gaps (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

   anti-symmetric 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 band-gaps. On the contrary, for the anti-symmetric 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 y-direction

   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 anti-symmetric 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 band-gaps; while for the anti- symmetric force, the transmission valleys can be predicted by the anti-symmetric band-gaps. A general force, neither

   Fig. 12. Vibration transmittances under line excitations of (a) symmetric distribution (case 5, case 6, and case 7) and (b) anti-symmetric 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 anti-symmetric excitation can be predicted by the anti-symmetric band-gaps. 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 anti-symmetric 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 anti-symmetric force, and only anti-symmetric mode makes positive contribution to the vibration response. Therefore, the symmetric force is associated with the symmetric modes and the anti-symmetric force is associated with the anti-symmetric modes. When the periodic plate is excited by neither symmetric nor anti- symmetric force, both the symmetric and anti-symmetric modes will make contribution to the vibration response, and the transmittance valleys can be predicted by the general band-gaps.

   From the above, the vibration suppression performance of a finite periodic compound plate is dependent on the force distributions. The symmetric or anti-symmetric force causes better performance than a general force, because the bandwidths of the symmetry band-gap and the anti-symmetry band-gap are broader than that of the general band-gap. Thus, when the periodic plate is used in the practical application, making the engine or excitation set work at a symmetric or anti-symmetric 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.

6. CONCLUSIONS

The flexural-wave band-gap 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 band-gap frequency and vibration response are obtained by theoretical derivation and are validated by FEM model. The vibration can be significantly reduced in the band-gaps. 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 anti-symmetric, makes the transmission valleys depending on the general band-gaps, which is narrower in band-gap width than symmetric or anti-symmetric band-gaps. Thus when the force is in symmetric or anti-symmetric situation, the vibration suppression performance of a periodic compound plate will be improved and more vibration can be reduced.

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