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:
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)
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
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 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. Nilsson, Prediction and measurement of some dynamic properties of sandwich structures with honeycomb and foam cores, J. Sound Vib. 251(3), 409-430 (2002). 13 D. Backsström and A.C. Nilsson, Modelling the vibration of sandwich beams using frequency-dependent parameters, J. Sound Vib. 300, 589-611 (2007). 14 Nilsson A, Liu B. Vibro-Acoustics, Volume 2. Second edition ed. Berlin Heidelberg: Springer-Verlag; 2016. 15 G. Maidanik, Response of ribbed panels to reverberant acoustic fields. J. Acoust. Soc. Am.34(6), 809–826 (1962) 16 F.G. Leppington, E.G. Broadbent, K.H.Heron, The acoustic radiation efficiency of rectangular panels. Proc. R. Soc. Lond. Ser. A 382(1783), 245–271 (1982) 17 L. Beranek, Noise and Vibration Control, Noise Control Engineering (1988) 18 Djoko Setyanto, Characterization and evaluation of sheet molding compound roof tiles, SCESCM 2014 19 L. Cremer and M. Heckl, Structure-borne sound, Springer-Verlag, Berlin, 1988. 20 T.D. Northwood, Transmission loss of plasterboard walls, National Research Council of Canada (1970).