A model to calculate the sound radiation from an unbaffled perforated plate is proposed. This is
achieved by modifying an existing model of an unbaffled plate to include the effect of perforation
in terms of a continuously distributed surface impedance to represent the holes. Results are
compared with those of an idealised situation, a perforated plate in an equally perforated baffle.
At low perforation ratios, the radiation efficiency is lower for the unbaffled case but as the
perforation ratio increases, the results for both conditions become similar. The effect of
perforation increases as the perforation ratio increases and also as the hole diameter reduces.
Comparison with existing measurement is also found to give a good agreement

Putra, A.

Thompson, D.J.

English

Southampton (e-Prints Soton)

                                      19th INTERNATIONAL CONGRESS ON ACOUSTICS                          MADRID, 2-7 SEPTEMBER 2007                                                                              SOUND RADIATION FROM A PERFORATED UNBAFFLED PLATE   PACS: 43.40. r  Putra, Azma1; Thompson, David1 1Institute of Sound and Vibration Research (ISVR); University of Southampton, Southampton, GB SO17 1BJ, United Kingdom; ap@isvr.soton.ac.uk, djt@isvr.soton.ac.uk   ABSTRACT A model to calculate the sound radiation from an unbaffled perforated plate is proposed. This is achieved by modifying an existing model of an unbaffled plate to include the effect of perforation in terms of a continuously distributed surface impedance to represent the holes. Results are compared with those of an idealised situation, a perforated plate in an equally perforated baffle. At  low  perforation  ratios,  the  radiation  efficiency  is  lower  for  the  unbaffled  case  but  as  the perforation  ratio  increases,  the  results  for  both  conditions  become  similar.  The  effect  of perforation increases as the perforation ratio increases and also as the hole diameter reduces. Comparison with existing measurement is also found to give a good agreement.   INTRODUCTION  In attempting to reduce the sound radiated by vibrating plates, several schemes have been developed including introducing perforation over the area of the plate. This technique can be seen in many practical applications, for example product collection hoppers and safety guard enclosures, such as the protective cover over flywheels and belt drives.  Existing  models  of  sound  radiation  from  a  perforated  plate  [1,  2]  approximate  the  acoustic impedance of the holes by the analytical solution for wave propagation in a small tube of circular cross section, as proposed by Maa [3]. For convenience, the holes are then considered to be distributed across the plate surface allowing a continuous surface acoustic impedance to be assumed.  Takahashi  and  Tanaka  [1]  proposed  a  model  for  the  sound  radiation  in  one dimensional case from an infinite, thin elastic plate under a single normal point force excitation. Fahy and Thompson [2] started with a model of radiation by plane bending waves propagating in  an  unbounded,  uniformly  perforated  plate  to  calculate  the  radiation  efficiency  of  a  simply supported rectangular plate. The assumptions imply that the plate is effectively mounted in a similarly  perforated  rigid  baffle  so  that  the  plate  and  the  baffle  have  an  identical  acoustic impedance. The model was then extended to the situation where the plate and the baffle have different  specific  acoustic  impedances.  The  relation  between  velocities  and  pressures  was derived in the wavenumber domain as a matrix problem and then solved by matrix inversion. As a  preliminary  case,  this  was  applied  to  the  radiation  by  a  perforated  strip  piston,  in  two dimensions,  vibrating  in  an  infinite  baffle.  Good  results  were  obtained  for  the  case  of  a perforated strip piston in a perforated baffle and in a rigid baffle. However, problems are found with  this  model  at  low  frequencies  for  an  unbaffled  perforated  strip.  A  very  low  acoustic impedance of the boundary (relative to the acoustic impedance of the holes) leads to a singular or nearly singular matrix which reduces the quality of the results from the inverted matrix. In addition,  expanding  this model into the three dimensional case is found to require  intensive computational effort.   In this paper, the sound radiation from a perforated unbaffled plate is calculated by modifying Laulagnet's model [4] to introduce the perforation. At low perforation ratios, it is found that the result is lower than that from the case of a perforated baffle. However, as the perforation ratio increases, the radiation efficiency of the perforated plate in an equally perforated baffle and the perforated unbaffled plate are found to become similar.     19th INTERNATIONAL CONGRESS ON ACOUSTICS – ICA2007MADRID 2 THEORY  A brief introduction to the model is given here. More details can be found in [4,5]. Consider a flat thin unbaffled plate in Figure 1 with a surface area  p S  located in an infinite medium, excited by a harmonic force distribution  ( ) F x y ,  at angular frequency ω .    Figure 1. A perforated unbaffled plate  The equation of motion of the plate excited by the force  F  and pressure difference p   is                                4 2 ( ) ( ) ( ) ( ) B w x y m w x y F x y p x y ω ∇ , − , = , +   ,                                          ( Eq. 1)  where  B   is  the  bending  stiffness,  m   is  the  mass  per  unit  area,  ( ) w x y ,   is  the  transverse displacement of the plate and 4 4 4 4 2 2 4 4 / 2 / / x x y y ∇ = ∂ ∂ + ∂ ∂ ∂ + ∂ ∂ . The motion of the fluid inside the holes depends on the pressure difference  p    between the two sides of the plate surface. This can be found from the specific acoustic impedance of the distribution of holes  h z ,   ( )( )hfp x yzv x y  ,=,                                                            (Eq. 2)  where  ( ) f v x y ,  is the equivalent velocity of the fluid through the holes, averaged over the plate area. For a circular hole with diameter  o d , the acoustic impedance is inertial provided that  o d  is greater than about 1 mm (for smaller holes viscous effects are important). Thus by neglecting the fluid viscosity, Maa’s approximation for the impedance of the hole [3], distributed across the plate surface, can then be written as  14ohd ktz j ctπρτ    = +                                                             (Eq. 3)  where  ρ  is the air density,  c  is the speed of sound, t  is the plate thickness,  k  is the acoustic wavenumber and τ  is the perforation ratio. The distributed holes or continuous impedance is indicated by the presence of  τ  in (Eq. 3). By using Euler’s equation on the plate surface at 0 z =  and substituting (Eq. 1) and (Eq. 2) to eliminate the pressure difference  p   , gives   2 2 4 20 ( , ) ( ( , ) ( , )) ( , ) ( ( , ) ( , ) ( , )) z fhp jx y w x y w x y w x y B w x y m w x y F x yz zρωρ ω ρ ω ω =∂| = − = + ∇ − −∂   (Eq. 4)                                                                   19th INTERNATIONAL CONGRESS ON ACOUSTICS – ICA2007MADRID 3 where  f w  is the equivalent fluid displacement relative to the plane  0 z =  and  f f v j w ω = .  The pressure at point  M  in the fluid can be defined by using Kirchhoff Helmholtz (K H) integral    ( ) ( ) ( ) ( )/ ( ) / Q Q S p M p Q G Q M n G Q M p n dS = ∂ , ∂ − , ∂ ∂ ∫                            (Eq. 5)  where  Q  is a point on the plate surface and  G  is the free field Green’s function. Assuming a thin  plate,  the  normal  acceleration  ( / (1/ ) / n Q u t p n ρ ∂ ∂ = − ∂ ∂ )  is  continuous  across  the  plate. Therefore the monopole source contributions in the K H integral (second term) vanish leaving only the dipole source contributions (first term). For points  M  on the plate surface, using this and substituting (Eq. 4) yields   2 4 2 ( ) ( ( ) ( ) ( ))hjw M B w M m w M F Mzρωρ ω ω + ∇ − −                          24 2 ( )( ( ) ( ) ( ))pp SQ MG Q MB w Q m w Q F Q dSz zω∂ ,= ∇ − −∂ ∂ ∫                               (Eq. 6)  where  the  second  term  on  the  left hand  side  has  been  added  by  the  perforation.  The displacement can be considered as the summation of plate modes ( , m n ). For the case of a simply supported plate, it can be written as   1 1 1 1( ) sin( / )sin( / ) mn mn mnm n m nw x y d d m x a n y b ϕ π π∞ ∞ ∞ ∞= = = =, = = ∑∑ ∑∑                        (Eq. 7)  where  mn d  is the modal displacement amplitude,  mn ϕ  is the mode shape function and  a b ×  are the plate dimensions. The force can be written using a similar expression. The orthogonality relationship with mode ( , p q ) gives 4 ppmn pq p mp nq SSdS ϕ ϕ δ δ = ∫                                           (Eq. 8)   where δ  is the Kronecker’s delta. Applying (Eq. 8) and substituting (Eq. 7) into (Eq. 6) yields   2 2 2 2 21 1( ( ) ) ( ( ) )4ppq pq pq pq mn mn mn pqmnm n hS jd m d F m d F Czωρρω ω ω ω ω∞ ∞= =+ − − = − − ∑∑          (Eq. 9)    where  pq F   is  the  modal  excitation  force  and  pqmn C = mnpq C   are  the  acoustical  cross modal coupling terms [4] obtained as  23 0 02pqmn z x yj abC k dk dkpqmn π∞ ∞   = ϒ     ∫ ∫                               2 21 ( 1) cos(( ) 1)(( ) 1)pp m   π   π− −ϒ =/ − / −,  2 21 ( 1) cos(( ) 1)(( ) 1)qq nχχ π χ π− −  =/ − / −                 (Eq. 10)  where  x k a   = ,  y k b χ = , 2 2 2z x y k k k k = − − ;  x k ,  y k  and  z k are component of the wavenumber.  The  effect  of  damping  can  be  included  by  replacing  the  plate  natural  frequency 2mn ω   by 2 (1 ) mn j ω η +  where is η  the damping loss factor. (Eq. 9) can be simplified by taking only the self modal coupling terms of (Eq. 10). The inverse of the cross modal coupling terms 1 [ ] pqmn C−  can    19th INTERNATIONAL CONGRESS ON ACOUSTICS – ICA2007MADRID 4 then  be  replaced  by  1/ pqpq C .  Such  an  approximation  works  well  for  a  baffled  plate  [6], particularly when the average result over different force positions is considered [7]. It has been shown that this approximation can also be used for the case of an unbaffled plate, perforated or unperforated [4,5]. After re arranging (Eq. 9), its solution can be written as    22 2 2 2 2 4 1 1ˆ ( ) 1 ( ) 14 4p pq ppq pq pq pq pq pqpqpq p h h pqpqS jm S jm d d F C S z z Cωρω ω ρω ω ωω     − − + − = −                    (Eq. 11)  where  ( 4) pq p m m S = /  and  ˆ ( 4) pq p pq F S F = /  are the generalized mass and the generalized force.  The sound power W  and the mean square spatially average plate velocity 2pq v  are given by      2 21 1( ( ) )8ppq pq pq pqp qSW j m d F dωω ω  ∞ ∞  ∗    = =    = ℜ − − ∑∑                                 (Eq. 12) 22 28pq pq v dω=                                                     (Eq. 13)    Finally the total radiation efficiency σ  can be found from    21 1 p pq p qWcS vσρ∞ ∞= ==∑ ∑                                                      (Eq. 14)   RESULTS AND ANALYSIS  Results are presented for an example 0.65× 0.5× 0.003 m aluminium plate with η =0.1 excited by point forces averaged over the plate surface. Figure 2 shows the modal and total radiation efficiencies of an unbaffled perforated plate with 40 % perforation ratio and 10 mm diameter holes. The 40 dB/decade  slope at  very  low frequency,  i.e. at below the fundamental mode, shows the expected characteristic of dipole source radiation.  10110210310410 810 610 410 2100frequency [Hz]radiation efficietncy, σ  Figure 2.  Modal and average radiation efficiency of an unbaffled perforated plate   o d =10 mm, τ = 40 %: — modal radiation efficiency, ─average radiation efficiency.  Figure 3(a) and (b) present the effect of perforation defined by 10log10(W perforated/W unperforated) for constant hole diameter and constant perforation ratio, respectively. These show that, for the case of the unbaffed perforated plate, the effect of perforation is almost independent of the frequency in the fundamental and corner mode regions (< 400 Hz) and then increases in the edge mode region as the frequency increases. The effect of perforation approaches 0 dB at    19th INTERNATIONAL CONGRESS ON ACOUSTICS – ICA2007MADRID 5 high frequency. Figure 3(a) shows that sound radiation is reduced by increasing the perforation ratio, while Figure 3(b) shows that the radiated sound can be further reduced by reducing the hole size.   Figure 4(a) compares the results with those found using the model of [2] for a perforated plate in an equally perforated baffle. The unbaffled plate has a lower radiation efficiency than for an infinite  perforated  baffle  for  most  of  the  frequency  range.  However,  as  the  perforation  ratio increases, the two results become very similar. At high perforation ratio, the contribution of the baffle to the radiated sound is no longer significant. This is obvious in Figure 4(b) which shows the differences of the radiation efficiencies for no baffle and with a perforated baffle on a dB scale. It shows that for a 10 mm hole diameter and 60 % perforation ratio, the difference is only around 1 dB. Therefore with very high perforation ratio, the system with a perforated baffle can then be considered to be equivalent to the unbaffled perforated plate.  101102103104  30 25 20 15 10 50frequency [Hz]effect of perforation [dB re 1]101102103104  30 25 20 15 10 50frequency [Hz]effect of perforation [dB re 1](a)  (b)  Figure 3. Effect of perforation on sound power radiation of a perforated unbaffled plate (a)  o d =10 mm: —τ = 10 %, − −τ = 20 %,    τ = 40 %, −   −τ = 60 %  and (b) τ = 20 %: — o d =50 mm, − − o d =20 mm,     o d =5 mm, −   − o d =2 mm.  10110210310410 810 610 410 2100frequency [Hz]radiation efficiency, σ 101102103104  10 9 8 7 6 5 4 3 2 101frequency {Hz]  σ [dB re 1]                                      (a)  (b)  Figure 4. (a) Comparison of radiation efficiency of a perforated plate in a perforated baffle (blue line) and with that of an unbaffled plate (red line), (b) radiation efficiency difference between both models ( o d =10 mm:    τ = 3 %, − −τ = 10 %, −   −τ = 40 %, —τ = 60 %)   Figure 5 shows a comparison of the radiation index (10log10σ ) from the analytical calculation with that from measurements made by Pierri [8] for 0.3× 0.3× 0.0012 m steel plates. The results are  plotted  against  1/3  octave  band  centre  frequencies.  The  damping  loss  factor  in  the calculations  is  chosen  to  be  very  low,  i.e.  η =  0.001.  A  very  good  agreement  between  the    19th INTERNATIONAL CONGRESS ON ACOUSTICS – ICA2007MADRID 6 theoretical  and  the  measured  results  is  achieved,  although  below  800  Hz  the  measured radiation efficiencies are greater than the predictions by about 3   4 dB. This may be due to the background noise from the shaker used to excite the plate.  101102103104 60 50 40 30 20 100frequency [Hz]radiation index [dB re 1] 101102103104 60 50 40 30 20 100frequency [Hz]radiation index [dB re 1] (a)  (b) 101102103104 60 50 40 30 20 100frequency [Hz]radiation index [dB re 1] 101102103104 60 50 40 30 20 100frequency [Hz]radiation index [dB re 1] (c)  (d)  Figure 5. Comparison of the radiation index (—) with measured data (−□−):  (a)  o d =5.6 mm, τ = 5.7 %; (b)  o d =7.1 mm, τ = 9.4 %; (c)  o d =8.8 mm, τ = 14.1 %;  (d)  o d =15 mm, τ = 41.5 %.   CONCLUSIONS A model for the sound radiation of a perforated unbaffled plate has been proposed. It has been shown that the model gives the same result as the model of a perforated plate in a perforated baffle  at  high  perforation  ratio.  The  effect  of  perforation  is  independent  of  frequency  in  the fundamental and corner mode regions. Increasing the perforation ratio increases the reduction of the radiated sound. From the model it is found that the reduction becomes greater by also reducing  the  hole  size.  Comparison  with  existing  measured  data  also  shows  a  very  good agreement.          References: [1] D. Takahashi and M. Tanaka: Flexural vibration of perforated plates and porous elastic materials under acoustic loading. Journal of the Acoustical Society of America 112 (2002) 1456–1464  [2] F. J. Fahy, D. J. Thompson: The effect of perforation on the radiation efficiency of vibrating plates. Proceedings of the Institute of Acoustics 26 (2004) [3] D. Y. Maa: Microperforated panel wideband absorber. Noise Control Engineering Journal (1987) 77 84 [4] B. Laulagnet: Sound radiation by a simply supported unbaffled plate. Journal of the Acoustical Society of America 103 (1998) 2451 2462 [5] A. Putra: Sound radiation from a perforated plate. ISVR Technical Memorandum 973 (2007)  [6] L. Li, H. J. Gibeling: Determination of the mutual radiation resistances of a rectangular plate and their impact on the radiated sound power. Journal of Sound and Vibration 229 (2000) 1213 1233 [7] G. Xie, D. J. Thompson, C. J. C. Jones: The radiation efficiency of baffled plates and strips. Journal of Sound and Vibration 280 (2005) 181 209 [8] R. A. Pierri: Study of a dynamic absorber for reducing the vibration and noise radiation of plate like structures. Master's thesis ISVR University of Southampton, United Kingdom (1977) 

Sound radiation from a perforated unbaffled plate

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