Keywords
Bending Stiffness; Sound Reduction Index;  Sandwich; Double Walls.
INTRODUCTION 
Exposure to noise constitutes a health  risk. There is sufficient scientific evidence that noise exposure can induce  hearing impairment, hypertension and ischemic heart disease, annoyance, sleep  disturbance [2], and decreased school performance. For other effects such as  changes in the immune system and birth defects, the evidence is limited. Most  public health impacts of noise were already identified in the first years after  second world war [3] and noise abatement is less of a scientific but primarily  a policy problem. Noise exposure is on the increase, especially in the general  living environment, both in industrialized nations and in developing world  regions. This implies that in this century noise exposure will still be a major  health problem.  About twenty years ago the Italian lawmaker  made the first norm on building acoustic [4,5], but also a large number of law  and norm were made by Europe [6, 7]. So the building and construction company in  this last ten years improved wall with different acoustic performance,  especially for dry construction. It’s already few decades that industries  working in the automotive and aeronautic building fields are very interested in  the reduction of weight without lost mechanical properties so it is increased  the use of sandwich structure in last sixty years [8]. With the expression “sandwich panel” refers  to a structure with a thick lightweight core with thin laminate bonded to each  side of either a foam or honeycomb core to join low weight with high strength. This reduction of weight often involve a  reduction of acoustic insulation so the automotive and aeronautic  building have paid attention in this problem  in the last two/three decades [9]. It is therefore essential to optimize the  acoustic properties of such structures through reasonable predictions. The ISO standard method for the  determination of the transmission loss it’s an expensive method so in recent  years new techniques for the determination of the transmission loss of panels  have been developed. Most of them depend on the geometry of the  structure as well as on the material properties of core and laminates. The method of bonding laminates to core can influence the dynamic properties of  sandwich constructions. In general, the structure of a sandwich plate is often symmetric with respect to the centreline. The E-modulus for a laminate is  typically high and much higher than the corresponding modulus for the core.
THE FLEXURAL VIBRATION
One of the methods is based on a model  which, given an orthotropic sandwich panel, requires physical and mechanical  properties of the materials and the natural frequencies of two beams,  orthogonal directions, of the plate. Based on these result the apparent bending  stiffness can easily be determined through a least mean square method applied  to the general equation describing the dynamic characteristics of a composite  material [14]. Once incorporated in a mathematical model, these simple input  data allow the prediction of bending stiffness, coincidence frequency, material  losses and sound transmission loss of the panel. In [10, 11] there is the  summary of procedure, discussed in a number of papers[12, 13], to determine the  parameters necessary to estimate the sound transmission loss.         Obviously,  this method presents some difficulties if applied to already mounted specimens,  since it is impossible to cut beams from a mounted structure. So in same case  it’s possible determined the material parameters from point mobility  measurement, this technique can also use for non-isotropic panels.   
POINT MOBILIY THEORY
Once the material parameters for a sandwich  or honeycomb panel are determined through these simple tests, also the point  mobility of the corresponding infinite plate can be calculated (as reported in  [14]). In this paper the predicted and measured mobility results for a compound  sandwich plate with a foam core are discussed. Some investigations have been  performed in order to compare the bending stiffness computed from measurements  on beams to that obtained from measurements of point mobility of a plate.  Finally, the sound transmission loss predicted from measurements on beams and  mobility measurements are compared. One of the problems to deal with is be the  assessment of the acoustic performances of such panels once they are already  mounted in their final place. In this case, it can be of importance to find a  way to characterize their dynamic and acoustic properties, such as bending  stiffness, internal losses and sound transmission loss, through non-destructive  testing. On the basis of the apparent bending stiffness of a losses, it is  possible to predict the transmission loss of the panel in a fairly simple way.  The results obtained from the mobility tests have been compared to the  measurements carried out on beam analysis and in the future the value will be  compared with the measurement carried in sound reverberation rooms according to  the UNI EN ISO standard procedure.    The model used for the post processing of  the mobility data allows parameter studies of the sound transmission loss and  of the sound radiation ratio for structures with different thickness of the  laminates and the core once the main physical data are known.
The point  mobility function Y (ω) is related to the frequency response function. If the  FFT of  a force F, that is written as, is applied to a certain point of a dynamical system and if  the resulting FT of the velocity in that particular point is , then the point mobility Y (ω) is defined as the complex  ratio between the Fourier transform of velocity and the Fourier transform of  the force measured at the same point:
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
The vibration behaviour of finite structure can be derived from that of infinite ones. In an infinite plate the bending waves can propagate indefinitely in the specimen. In a finite plate the same bending waves reach the boundaries of the plate, and are then reflected back. The velocity depends upon plate geometry and on the boundary conditions, thus the point mobility will change depending on the location and on frequency. However, a space and frequency average of the real part of the mobility for a finite structure is in the mid and high frequency region equal to the real part of the point mobility of an infinite structure of the same material and thickness:
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Consequently can be calculated as if the  structure were infinite and excited by a point force with a power spectral  density equal to the sum of the power spectral densities of all the sources  acting on the finite structure. This assertion is valid if the modal density  within a band is independent of boundary conditions, which is true for the  medium and high frequency bands.  This means the exclusion of the first few  modes corresponding to the low range frequencies.    For obtaining a space average  of the mobility which  is representative of the dynamic behaviour of the  entire panel, the mobility must be measured  over a sufficiently large number of points, randomly distributed over the  surface of the panel. Under these conditions the mobility of a finite panel can  be predicted through the mathematical formulation of the mobility for an  analogous infinite panel.                 
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX              (3)              
Dp Bending Stiffness per unit of  width
μ mass per unit of area        
Thus the bending stiffness per unit width  of the panel at the central frequency of each band is                                 
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX    (4)
An important aspect for lightweight and  homogenous structure is: if for example a small object is mounted on a  homogeneous plate, the effective mass of the structure is no longer uniformly  distributed over the surface. The natural question is then: How does the added  mass affect the vibration pattern of the plate? 
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX                              (5)  
In the low frequency region the measured  point mobility is approximately equal to the actual mobility Y0 of  the plate. However as the frequency increases the second term in the  denominator tends to increase. At the limit, the measured mobility is equal to  which is the point  mobility of the added mass.
LOSSES
The losses, as written in [11], can be divided in three parts:  transmission losses, radiation losses and internal losses. When a structure is vibrating, some of its mechanical energy is  converted into thermal energy, transmitted to adjoining structures or radiated  as noise, putting in vibration the  surrounding medium. The boundary conditions are important for the determination of the  losses. Clamped condition introduces higher losses as compared to free  condition. The  sound radiation ratio is independent of boundary conditions for f > fc  whereas in the low frequency region the radiation ratio is higher for a clamped  than for a simply supported beam. The fraction of  energy lost during each cycle of vibration is represented by the loss factor , which can assume values between 0 and 1. The total loss  factor can be expressed as the sum of these three components:                           
 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX   (6)
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXFigure 1 - Loss of the specimen in  free-free condition, calculated with Half-Bandwidth method 
This loss of  energy means a reduction on the amplitudes of the lateral displacement and a  reduction of the time during which the panel vibrates. For lightweight  structures the influence of radiation losses is not negligible and it is  necessary to consider also the viscous effect of the surrounding fluid (air).  The radiation loss factor can be defined as [13]: 
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX                              (7)
Where  represented the so  called radiation efficiency which has been investigated among others by  Maidanik [15] and Leppington [16]. The losses  depend on the panel dimensions and on the boundary conditions. The greatest radiation comes from clamped  edges.      
For an infinite (undamped) thin plate in  bending vibration, sound only radiates at frequencies above the plate’s  coincidence frequency fc.  Furthermore, the radiation efficiency asymptotically approaches 1 at high  frequencies. The equation that describes the sound radiation for f > fc:     
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX                          (8)
At the frequencies below fc, the root in  equation becomes imaginary, and the identically zero. If losses are included in  the analysis, as in Figure 2, the most important modification is that some  radiation even occurs below coincidence; the greater the damping, the greater  that radiation [17].
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXFigure 2 -  Radiation efficency for a infinitley large thin plate undergoing bending  oscillations. For a damped plate the radiation efficiency is non-zero at all  frequencies.
For a finite  plate the generate wave is reflected from the corners and edges so create  different eigenmode as seen in Figure 3
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXFigure 3 – Illustration of  eigenmodes
So the sound  radiation behaviour is described as [17] and reported in Figure 4 below.       
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXFigure 4 - Design curve  for approximating the radiation af a finite panel. Notation P is the plate’s  perimeter, S the plate’s area, λ is the sound wavelength in the fluid medium into  which the plate radiates and
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As is evident from  the Figure 4, the plate radiates as a monopole up to the first eigenfrequency.    In the frequency  range between first eigenfrequency and 3c/P the radiation is dominated by the  corners of the plate. In the region between 3c/P and fc/2 it is the edge that provide the predominant contribution. In  the region between fc/2 and fc the  entire plates begins to contribute to radiation, and the radiation efficiency  increase rapidly. Above coincidence, the entire plates radiates, and the  radiation efficiency is given by the expression derived for the infinite plate equation  (8).
The ISO 15712-1:2005 (Annex C) gives  guidance on the calculation of the total damping loss factor:
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX(9)                
This equation isn’t suitable for the specimen  because the mass for unit of surface is very low and also ignores the radiation  loss factor because it is usually insignificant, but for this material it’s  very important.
SPECIMEN DESCRIPTION  
The panel under  the test is a sandwich with a foam produced by ACELL FOAM™, it’s an open  foam, fire resistant and the production of composite panels, using the ACELL  MONOLITHIC MOULDING PROCESS™, which allow to reproduce a wide range of traditional materials as  well as new finishes.
As external laminates, ACELL uses SMC a  polyester resin base, it’s a reinforced compound used a lot in automotive  sector is used widely as a construction material because it has many  advantages, such as being lightweight, as well as having high strength, high  stiffness, and corrosion resistance. Reinforcement content in the composite is  20 - 70 percent by weight, depending on the orientation and type of the  reinforcement It used also in building sector as for example in the paper [18].
XXXXXXXXXXXXXXXXXXXXXXFigure 5 - Section of ACELL  panel
During the moulding the resin permeates in  the foam, as it’s shown in Figure 4, for 2 o 3 millimetres. In the ACELL panel it’s possible seeing  three type of layer: internal the foam, external the SMC and between them a  middle layer.    For the last layer is very difficult to  create a finite element model because it depends on the viscosity of material,  dimension and number of cell in the foam, so It has been decide to use some  methods based on the determination of the dynamic properties.
AVAILABLE RESULT
Bending Stiffness [10, 11]
         In the following Graphs are reported the  first results of modal analysis of ACELL structure (SMC+ACELL FOAM+SMC), where  the apparent bending stiffness function is obtained. It can be seen that the  typical trend in frequency for a sandwich beam is achieved.    
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXFigure 6 – Beam Test
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXFigure 7 - Bending stiffness in  frequency
PONT MOBILITY MEASUREMENTS    
The point mobility measurements were  performed with the panel suspended with two wires with low rigidity to recreate  a free-free condition. In the Figure 8 below it’s possible to see how the  measurement points were distributed across the panel surface.
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXFigure 8 -15 Points to evaluation point  mobility, the panel is in free-free condition        
The post processing of the data was carried  out by exporting the text data from the OROS NVGate software. Now it’s possible  to calculate the average mobility for the 15 measurement positions (nothing  correction has applied to taking into account the weight of the accelerometer),  in Figure 9 is reported the mobility of all 15 points.       
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXFigure 9 –  Mobility of 15 point   
The mobility value was computed starting  from the corrected and averaged mobility synthesizing the value into extended  1/3 octave bands in order to have at least 5 modes inside the frequency span  defined by each band. Once the average mobility is known, it is  possible to compute the related bending stiffness and to use this value to determine  the apparent bending stiffness through the least mean square method applied to  a set of fn, Dn and  to the equation
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX(11)
Since the modal  density in the low frequency range is low, there is some lack of points for  computing the bending stiffness. For this reason a fictitious bending stiffness  point D0 has been introduced in order to guide the curve in the very  low frequency region.    The static  bending stiffness D0 can be calculate once some geometrical and  material parameters are known through the equation:                               (12)                 El=Elastic  modulus of sandwich external laminate;    hc=thickness  of sandwich core;    hl=thickness  of sandwich external laminate;              Figure 10 – Bending stiffness of a plate  under the test    Introduction to sound reduction  index    The sound transmission loss R is in [19] derived as  function of the material parameters of the panel and on the bending stiffness  of the plate    It is convenient to introduce the coincidence  frequency fc for which the trace matching between flexural waves on the plate  and waves in the surrounding medium can occur. The frequency fc for  which kplate=kair is given by                                   (13)                         where c is the speed of sound in air and μ the mass  per area of the plate. For a sandwich plate fc is a function of frequency. As a  first approximation kplate can be set to equal κ1 or the wavenumber  for the first propagating mode of flexural waves in the plate. Alternatively D  is given by the apparent bending stiffness of the plate (Dx).    The sound transmission loss R in dB for a plate is,  according to [19] R=-10log τd, where τ is the sound transmission  coefficient for diffuse incidence. This coefficient is defined as                               (14)                 The transmission coefficient  at the angle of  incidence  is given by                               (15)                 The parameters in the Eq. (14) are: μ total mass per  unit area of plate, f frequency, ω angular frequency, φ angle of incidence of  acoustic wave, ρc wave impedance, η loss factor of structure and fc the coincidence frequency, where  D = Dx is a function of frequency obtained from bending stiffness  curve.    The sound transmission loss for an infinite sandwich  plate having the same structure as the composite beam used for the tests are  shown in Figure 11, where also it’s seen the effect of loss that reduce the  effect of coincidence frequency.         Figure 11 - Comparison with the measurement  between the beam without the radiation loss and panel with radiation loss    CONCLUSION    The  acoustic insulation of this type of single panel is very low and it doesn’t  respond to requirement of DPCM 5/12/1997 that for houses and hotels requires Rw=50 dB,  and also for the the DNV GL AS-Human comfort Offshore Standards, that requires  sound insulation of Rw =40 dB.    The  sound transmission loss of a single leaf panel is in principle only increased  by 6 dB due to the doubling of the mass of the panel for f < fc.  This type of increase of the sound transmission loss is often unacceptable due  to weight constraints, for example to obtain a value of 47 dB of insulation  with single ACELL panel it’s needed to have 50 cm of thickness. It’s necessary  to study a double wall to increase the sound insulation of partition, with a  low thickness.         The use of double structures is an  alternative to drastically increase the sound transmission loss while keeping  the weight low. A double structure is quite simply two single leaf panels  separated by an air filled cavity or a cavity with some sound absorbing  material.    When a coupled double structure is  excited on one side, the coupled structures move almost in face in the low  frequency region. As the frequency is increased the structures starts moving in  antiphase and having a large amplitude close to and at the double wall  resonance f0. Well above  this frequency, the velocity level difference between the coupled plates is  increasing as approximately 40 log( f/ f0) up to certain limit  determined by the added sound transmission loss achieved by the two panels  separated by a cavity with a certain sound absorption. The resulting sound  transmission loss of a double structure is schematically shown in Fig. 12.             Figure 12 - Schematic drawing of the sound  transmission loss of a double leaf construction         In the low frequency range, f < f0 the double structure vibrates as a single structure  with the mass equal to the total mass of the two partitions. Thus                         for              (16)                      In the  frequency region fx > f > f0 the transmission loss increases  by an added 40 log( f/ f0). Thus:                         for             (17)                      In the very high frequency range f > fx the total transmission is the sum of the transmission  for each panel plus a correction for absorption in the cavity. The result is                         for             (18)                 The double wall resonance f0  is:                               (19)                      where d  is the distance between the two plate elements.    While the frequency fx is the solution to RII(fx)= RIII(fx).    Strength and stability of the structure  is achieved by mounting studs between the plates of the double structure.  Clearly, these connections will reduce the acoustic efficiency of a double wall  construction. There will be an acoustic energy flow between the plates through  the studs. The total transmission loss will depend on the distance between  studs and type of studs used. Some measurement results are shown Figure 13  below, reveal that by changing a wooden stud to a more flexible metal stud the  sound transmission loss is increased considerable.        Figure 13 - Influence of added sound  absorption and studs on the sound transmission loss of a double wall  construction. 1 Two 13mm plasterboard panels with a spacing of 106mm; 2  Sound absorbing material added to cavity; 3 Wooden studs connecting  plates; 4 Metal studs connecting plates. From [20]         Initially  it will be used this theoretical method to study a solution for a double wall. In the future it will provide to build the double walls and test.        The  double walls will be made, as reported in Figure 14, by a layer of 15 mm of plaster, then a layer of 80 mm of brick, then 15  mm of plaster, then an air gap of 25 mm and finally an ACELL panel of 20 mm.    The  data of brick clay wall are inferred from bibliography, the typical loss factor  is equal to 0.015 [17], it is assumed Young’s Modulus equal to 3000 MPa, and  density of 650 kg/m3 that gives a mass for unit of area equal to 106  kg/m2     All  parameters of ACELL panel are obtained from the past analysis [1,10,11].        Figure 14 – Section  of double walls with different layers        In this simulation  of double walls hasn’t been considered the effect of studs between the two  walls, after the build of double walls, they will be tested.        The analysis shows for this double wall a  value of f0 equal  to 128 Hz, fx  equal to 200 Hz, while fc is equal to  1400 Hz.        Figure 15 – Sound Transmission Loss of project of  Double Walls design             The author have designed this type of partition that  could be suitabe in internal renovation of houses or hotels for different  reasons:    -        To  dicrease the thermal transmittance from 2 W/m2K to  0,6 W/m2K        -        To  increase the sound transmission loss from Rw=40 dB to Rw=57  dB    -        The possibility to recreate all  types of finishing surface    -       Improve the fire resistance characteristics  of the wall.    And also the easiness to build the test wall.    The author will grow to find solutions for naval walls.         The paper  is a description of the status of the author’s PhD ongoing research project  that aims to develop a finite facade for building sector.    Since the  study is still in progress, only available interim results have been briefly  discussed; they mainly concerns: (1) the study of dynamic properties using beam  test (2) the study of dynamic properties using point mobility (3) evaluation of  losses, (4) proposal of double walls.    Next  months’ developments will mainly concern the improving of evaluation of radiation  losses considering the finite panel, the construction  of double walls described above and in a future the validation of the program with standard test in  reverberant room.    Try to use of other compatible material as  external laminate to improve acoustic property or try to glue a thickness aluminium  laminate, to increase the mass and the bending stiffness of material.    On the contrary no changes in foam  formulation are taken into account, because of the difficult to modify the  formulation (some test were doing to try to insert damping material, but all  of them have given negative response for the difficult of mixing these materials  in the mixture of foam).    ACKNOWLEDGMENTS    The author would like to thank ACELL ITALY Srl  for providing the material and for economic support.         REFERENCES    1      Massimo  Fortini and Edoardo Alessio Piana Implementation  of a Numerical Method for the Best Fitting of the Bending Stiffness Curve to a  Set of Experimental Points, Internoise pp. 648-655 (2016).    2      Ohrstrom E. Psycho-social effects of traffic noise, J  Sound Vib 151:513-517 (1991)    3      Ward WD, Fricke JE. Proceedings of the Conference Noise as a  Public Health Hazard. Washington: American Speech and Hearing Association,  1969.    4     L. 447/1995 - Legge quadro sull’inquinamento acustico    5     DPCM 5-12-1997 –  Requisiti acustici passivi degli edifici    6      UNI EN  12354 – Valutazioni delle prestazioni acustiche di edifici  a partire dalle prestazioni di prodotti    7     Direttiva  2002/49/CE – Determinazione e gestione  del rumore ambientale    8      N.J. Hoff, Bending and buckling of  rectangular sandwich plates, in NACA TN 2225 (1950)    9      A.C. Nilsson, Wave propagation in and  sound transmission through sandwich plates. J. Sound Vib. 138(1),  73–94 (1990)    10  Fortini,  M., Milestone I, 2016    11  Fortini,  M., Milestone II, 2016    12 E. Nilsson and A.C. 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Northwood, Transmission  loss of plasterboard walls, National Research Council of Canada (1970).