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\begin{document}

\title{~~~ Determination of the Dynamic and Acoustic Performances of~ Composite
Materials Based on Innovative Resin Foam Cores}



\author[1]{massimo.fortini}%
\affil[1]{Dipartimento ABC. Politecnico di Milano}%


\vspace{-1em}



  
  \date{May 20, 2024}


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\selectlanguage{english}
\begin{abstract}
ABSTRACT

\subsubsection*{During the last~ few decades, various types of
lightweight structures have been introduced in~ the vehicle and
transportation industry.This trend is~ dictated by the growing demand of
higher load capacity, reduced fuel~ consumption and increased
performances. On the other~ side, lightweight structures often have poor
acoustic properties, moreover there~ is a need to define the frequency
dependent dynamic properties of such materials. Industries working in
the automotive and transportation fields are~ interested in the
prediction and determination of the sound reduction index~ given by
lightweight partitions. The investments in this research field are~
aiming to find experimental methods capable to assess the sound
insulation~ performances of small specimens instead of full scale
partitions. Using~ expensive software based on FEM-BEM methods, some
variables deriving from the~ manufacturing stage and influencing the
sound transmission index are not taken~ into account. Moreover it is
very complicated to fully characterise non~ homogeneous materials. In
the last decades few methods based on the determination of the dynamic~
properties of sandwich beams have been developed {[}1{]} and validated.
Acell Italy produces Phenolic foam using a method which has been
patented.~ The patent concerns the coupling method of the layers through
a chemical process~ able to create a highly resistant panel with a low
weight and a high~ resistance. Acell would like to improve the dynamic
and sound insulation~ properties of the panels in order to improve its
position in the
market.}

{\label{564907}}

\subsubsection*{The scope~ of the author's PhD research project is the
development of a conceptual wall~ design with a composite materials
based on Innovative Resin Foam Cores. The cornerstones of the study are:
(1) the investigation of the material; (2) the evaluation~ of the sound
reduction index of a panel specimen (3) the design of a facade made~ by
using the innovative Acell
materials.}

{\label{564907}}\par\null%
\end{abstract}%



\sloppy


\textbf{Keywords}

Bending Stiffness; Sound Reduction Index;~ Sandwich; Double Walls.

\textbf{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.

\textbf{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.~~~

\textbf{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 (\selectlanguage{greek}ω\selectlanguage{english}) 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 (\selectlanguage{greek}ω\selectlanguage{english}) 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:

XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

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)~~~~~~~~~~~~~~

D\textsubscript{p} Bending Stiffness per unit of~ width

\selectlanguage{greek}μ \selectlanguage{english}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 Y\textsubscript{0} 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.

\textbf{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 \emph{f}
\textgreater{} \emph{f}\textsubscript{c}~ 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
\emph{f} \textgreater{} \emph{f}\textsubscript{c}:~~~~~

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, \selectlanguage{greek}λ \selectlanguage{english}is the sound
wavelength in the fluid medium into~ which the plate radiates and

XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

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 \emph{f}\textsubscript{c}/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.

\textbf{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.

\textbf{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 \emph{f}\textsubscript{n},
D\textsubscript{n} and~ to the equation

\par\null

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 \selectlanguage{greek}μ \selectlanguage{english}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 \selectlanguage{greek}κ\selectlanguage{english}1 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 \selectlanguage{greek}τ\selectlanguage{english}d, where \selectlanguage{greek}τ \selectlanguage{english}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: \selectlanguage{greek}μ \selectlanguage{english}total mass
per~ unit area of plate, f frequency, \selectlanguage{greek}ω \selectlanguage{english}angular frequency, \selectlanguage{greek}φ \selectlanguage{english}angle of
incidence of~ acoustic wave, \selectlanguage{greek}ρ\selectlanguage{english}c wave impedance, \selectlanguage{greek}η \selectlanguage{english}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 \textless{} 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
\textless{} 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 \textgreater{} f \textgreater{} f0 the transmission
loss increases~ by an added 40 log( f/ f0). Thus:~~~ ~~~~~~~~~~~~~~~~~~~
~for~~~~~~~~~~~~ (17)~~~~~~~~~~~~~~~~ ~~~~ In the very high frequency
range f \textgreater{} 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
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