Computers and Mathematics in Automation and Materials Science
NUMERICAL PREDICTION AND EXPERIMENTAL VALIDATION OF SOUND TRANSMISSION LOSS FOR SANDWICH PANELS
M. VISCARDI, P. NAPOLITANO Department of Industrial Engineering, University of Naples “Federico II” Via Claudio, 21 – 80125 Naples ITALY
[email protected] Abstract: - Acoustic insulation represent a very important aspects in many fields of the acoustic engineering (aerospace, automotive, trains, civil applications); many time engineers refer to this characteristic as Sound Insulation parameter or Sound Transmission Loss Index (TL). The research of materials with an high sound Insulation Vs. weight ratio is promoting the use of composite sandwich structures, that for their specific peculiarities may play an important role if compared to isotropic homogeneous structures. While numerical modeling for TL evaluation of homogeneous material has been exhaustively studied in last decades and reliable formulations are available, composite sandwich structures presents still today an open research items for many aspects. This work is mainly related to the verification and updating of available Nilson’s formulation with relation to a specific innovative sandwich composite structure that has been developed as a multifunctional structure for railway applications. The availability of experimental data have made possible the correct evaluation of the numerical model and its successive up-dating to best fit the experimental results. Keywords: Acoustic, Sound Insulation, Transmission Loss, Composite materials, sandwich structures, measuring methods
1 Introduction When a sound pressure wave knocks against a wall, the energy ( ) transported by the wave in the time unit may be considered as divided into three parts: a part of the energy is reflected by the wall ( ), another part is transmitted through the surface ( ) and the another one ( ) is dissipated into material of which the panel is made up of (Fig.1). =
+
+
.
(1)
Fig.1:Incident sound wave on a wall
Dividing both the first and the second member of the (1) by , the following equation is derived: 1=++
The latter is used to get the Sound Reduction Index (R), or transmission loss (TL) in Anglo-Saxon literature, by the relation
(2)
in which is the absorption coefficient, is the reflection coefficient and is the transmission one.
ISBN: 978-960-474-366-7
= 10 log
117
(3)
Computers and Mathematics in Automation and Materials Science
It represents the accumulated decrease in acoustic intensity as an acoustic pressure wave propagates outwards from a source. TL is defined as the difference between the acoustic energy that strikes surface, which divides two gaps, and the transmitted acoustic energy. It is demonstrated that most of the time, TL is a function of the frequency of the incident sound. In Fig.2 is shown a “classic” reference curve that describe this trend of TL.
panel with a characteristic velocity that depends by the material properties. In solid material there are two different wave types; bending waves and shear waves. The first ones are move along the panel thickness, the second ones in the transverse direction. The velocity of bending waves inside the panel can be calculated as =
2
(5)
where f is the frequency (Hz), D0 the bending stiffness of the panel's material (N*m) and m the mass for unit surface (kg/m2). An equivalent equation can be founded for shear waves = Fig.2: Transmission loss as a function of frequency;
in this case the velocity is independent from frequency and instead of D0 present the shear modulus G (N/m2) and the the panel thickness h.
In this graph three relevant zones can be distinguished. In the low frequencies zone the main contribute to TL is given by the panel stiffness. In the resonance zone there is not a clear distinction between the contributions of the mass and the stiffness of the panel. In the third one the main input to TL comes from the panel mass according to the mass law = 10 log( ) − 48
To obtain a good insulation cb and cs must be small. For a sandwich panel can be shown [1] that at low and high frequencies the bending waves dominate and at intermediate frequencies the transverse waves arise. In general, at low frequencies the whole sandwich create bending waves, while at high frequencies the skin start to bend independently.
(4)
in which “ is the panel mass for surface unit (kg/m2) and is the frequency (Hz). This zone extends as far as the frequency of coincidence, in which there is a reduction of TL; this phenomenon is due to the coincidence between the velocity of sound waves into the panel and the speed of sound into the air. Once the “coincidence frequency” has been passed, the TL trend follow the mass law again.
The bending waves in the skin has a velocity expressed as: =
2
.
(7)
where Df is the skin stiffness (N*m), mf and mc are the skin and core mass for unit surface respectively. In Fig.3 the bending waves and shear waves velocity as function of frequency are shown, calculated with (5) (6) and (7).
1.1 Coincidence frequency The coincidence frequency is a property of the structure (like resonance frequencies).
In the same figure, it is also shown the speed of sound in air at standard pressure and temperature conditions (c=344 m/s).
As described before part of the incident sound radiation is absorbed from the material and part of them is transmitted. The sound travel inside the
ISBN: 978-960-474-366-7
(6)
The velocity of sound in the panel can be obtained considering the curve coincident with cb for low frequencies, coincident with cbf for high frequencies.
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At the intermediate frequencies the curve pass from cb to cbf at the level of cs.
The combination of two laminates (or skins) with a core inside is called sandwich. The properties of the sandwich are a combination of the skins’ and core’s properties. The advantage to use the core material is the possibility to improve the stiffness for unit mass, higher is the core higher is the bending stiffness of the panel. A sandwich panel can improve some acoustical and structural requirements decreasing the mass. The skins and the core work respectively under bending and shear so a very light core can be used obtaining good properties. A scheme of a sandwich panel is shown in Fig.5.
Fig.3: waves velocity in solid materials The coincidence phenomena arise when the bending waves' velocity inside the panel is equal to the speed of sound in air (Fig.3). So from (7) substituting cbf with c and rearranging for f =
.
(8)
2 Sandwich panels Fig.5: sandwich panel scheme
A composite is a physical structure in which two different phases can be recognized (fiber and matrix).
In general the upper and lower skins can be different for thickness and/or properties. The neutral plane of the sandwich panel is given by the following equation =
(9)
where E1 and t1 are young modulus and thickness of lower skin, E2 and h young modulus and thickness of core, E3, t3 young modulus and thickness of upper skin. If E1=E3=E and t1=t3=t equation (9) become Fig.4 laminate fabrication steps
=
In Fig.4 is shown the fabrication process for a laminate panel. The union of fiber and matrix is called ply (or lamina). The properties of the ply are a weighted combinations of fiber’s and matrix’s properties. One or more ply together make the laminate. The properties of the laminate are calculated using the lamination theory that is based on the number of plies, the sequence of lamination, the fiber direction.
ISBN: 978-960-474-366-7
(10)
and the laminate is said symmetrically. The bulk modulus of the sandwich panel is: =
(ℎ −
) +
+
(11)
The used specimen is a sandwich panel realized with carbon fiber skins and balsa wood core (Fig.6).
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Computers and Mathematics in Automation and Materials Science
The core has a thickness of 37mm and the skins are 4 mm thick.
3745), sound intensity (ISO 9614) and acoustic holography measurements. In order to measure the STL of a partition it is necessary to isolate, acoustically and mechanically, the source chamber (Reverberant) from the receiving chamber (Anechoic), so that acoustic energy transmission from an environment to another occurs trough the sample only.
Fig.6: tested specimen
3 Experimental measures Experimental results will represent the reference value for TL determination. Relative measurements have been performed in the contest of a dedicated research program by the Elasis acoustic Laboratory [2]. Fig.7: reverberation room
3.1 Anechoic-reverberant room method This method is the most common and accurate to evaluate the TL of specimen. The advantage of this method is the capacity to investigate a full scale panel and not a small specimen like for Kundt’s tube. The difference between the two method is that the anechoic-reverberant room include the modal response of the panel and effect of the resonance on the TL curve. It is composed of two coupled measurement environments: a reverberation chamber and an anechoic chamber, communicating each other. The reverberation chamber (Fig.7) is an acoustic environment with all acoustic reflective surface. These characteristics allows to create a uniform sound field throughout the environment. The reverberation chamber allows to estimate the sound power emitted by a source, through the measure of the average value of the sound pressure in the chamber (ISO 3741), the absorption coefficient of materials, through the difference of the reverberation time measured in empty chamber and with the sample (I.S.O. 354), to evaluate the panels' soundproofing Sound Transmission Loss), using the anechoic chamber in combination. The anechoic chamber (Fig.8) is an acoustic environment that has all the surfaces acoustically absorbent. The acoustic wave generated by a source is completely absorbed by the walls without suffering any reflection. This characteristic allows to estimate the sound power level of a source through sound pressure level measurement (ISO
ISBN: 978-960-474-366-7
Fig.8: anechoic room The result obtained with this method is shown in Fig.9 and has been performed by ELASIS [2] within the contest of a research program involving Department on Industial Engineering and Elasis among the others [3]. 50 40 db
30 20 10 0 100
160
250 …
400
630 1000 1600
Fig.9: TL measured with the anechoic-reverberant rooms method
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Computers and Mathematics in Automation and Materials Science
4 Analytical formulation
40
The analytical approach is based on Nilsson’s equation [2]. The equation is based on mass law and adjusted to keep in account some effects due to modal behavior, coincidence and damping.
35 30 25
4.1 Basic model
20
Nilsson’s equation presents as follow: 15 ( ) + 20 = 20 48;
(13)
5 Result’s Comparison where m is the mass for unit area, f the frequency and fc the coincidence frequency calculated using (8).
Next Fig. 10 shows the comparison between analytical and experimental results.
The Δ is a term that dependent from the constrain set applied to the panel. In particular for a simply supported panel is .
.
+
45 40
(14)
dB
Δ = 1 +
∗
50
and
35 30 25
Δ = 1 +
∗ .
.
+
20
(15)
15 100
for clamped panel.
630 1000 1600
In Fig.10 some difference can be noticed between the two curves. In particular, with reference to the anechoic-reverberant room method, the analytical formulation is close at low frequencies, but shows a strong decreasing in the surrounding of coincidence frequency.
G describes the resonant transmission trough the panel ( ) ( ⁄ )
400
Fig.10: TL comparison; anechoic-reverberant room (continue); analytical (point-dotted)
Γ is a function of baffle and plate dimensions. If the plate is mounted without baffle this term is equal to 1.
{[
250
Frequency (Hz)
where δ is the damping of the panel, b and c the dimensions of the specimen.
= ∫
160
( )]
( ⁄ )
(
)}
−1
and can be solved numerically using parameter.
(16) ⁄
This behavior of the analytical model is mainly due to a damping value that differs from the real one.
as
Another consideration must be keep in account when reading Fig.10; in anechoic-reverberant room the incident acoustic field is diffusive field and the incident waves are not only orthogonal.
In Fig.9 the Nilsson’s equation calculated transmission Loss parameter, as function of frequency is illustrated.
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the anechoic-reverberant room results but it must be settled with the right coefficients that can differ on the basis of panel specifics. These differences are mainly due to the material properties. The composite structures can have a high variety of material for core or skins whit different behaviour in the coincidence neighbour and different damping. It is very hard to accommodate the large variety of material with only one equation. This not reduce the validity of the model that can be used for ulterior predictions on panel with similar materials configuration after a “coefficient calibration phase”.
5 Analytical model’s updating An updating work has been done on the analytical model. This work has mainly consisted in a coefficient adjustment to match the curve with anechoic-reverberant curve. In particular the attention has been focused on the panel’s damping coefficient and on the coincidence frequency where higher error is measured. In particular (13) is modified as following (17) ( ) + 30 = 20 (δ) + 5 1−
( ) + 10 − 10
( )+ − 47; >
50 45
The predicted TL has been calculated using the coupled (12) and (17) system, and the result is shown in Fig.11.
dB
40 30 25
45
20
40
15
35 dB
35
100
160
30 25
250 400 630 Frequency (Hz)
1000 1600
Fig.12: TL comparison; anechoic-reverberant room (continue); intensity probe (dotted); analytical (point-dotted); analytical updated (pointed)
20 15 100
160
250 400 630 Frequency (Hz)
1000 1600
Bibliography
Fig.11: analytical TL: Nilsson (dotted); updated model (continue);
[1] A.C.Nilsson, Wave propagation in and sound transmission through sandwich plates.", Journal of sound and vibration, pp. 74-75, 85-86 (1990)
In Fig.11 is shown a comparison between original Nilsson model and updated model. For frequencies lower than coincidence frequency the curve is the same. After the coincidence the updated model shows a lower loss in TL curve. This has been obtained increasing the “weight” of the damping factor and introducing an ulterior element in the equation that reduce the loss in the coincidence frequency.
[2] Elasis Research Center. Misure di Sound transmissionLoss su pannelli multistrato in camera anecoica-riverberante- Research Report- Year 2012 [3] Viscardi M., Lecce L., Dynamic characterization of an interior trim panel aimed at the active noise control in a turboprop aircraft, Proceedings of the International Seminar on Modal Analysis, Leuven- Belgium, pp. 221-232 (1996)
6 Results and Conclusion
[4] Ricci F., Viscardi M., Dynamic behaviour of metallic and composite plates under in-plane loads, Proceedings of the International Modal Analysis Conference - IMAC, 1, pp. 99-103 (2000) and Proceedings of SPIE-The International Society for Optical Engineering (ISSN 0277-786X)
Fig.12 shown a sum up of all the presented results. From Fig.12 can be appreciated the relevant changing in analytical model and how it is really close to the experimental one. In conclusion, it can be stated that the analytical method reproduce with very good approximation
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