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Sound insulation properties of two-layer baffles used in vibroacoustic protection
Applied Acoustics 156 (2019) 297–305
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Sound insulation properties of two-layer baffles used in vibroacoustic protection Krzysztof Kosała Department of Mechanics and Vibroacoustics, AGH University of Science and Technology, al. A. Mickiewicza 30, 30-059 Krakow, Poland
a r t i c l e
i n f o
Article history: Received 8 March 2019 Received in revised form 8 July 2019 Accepted 22 July 2019
Keywords: Sound insulation Calculation models Two-layer baffles Vibroacoustic protection
a b s t r a c t The subject of the research presented in the article was the modelling of acoustic properties of two-layer baffles, used in vibroacoustic protection. Baffles of this type are used, among others, in the construction of walls of the sound insulating enclosures of machines and devices. The heterogeneous single baffles that were analysed were panels made of two sound insulation layers, solid rubber and a steel plate. A sound insulation calculation model for heterogeneous single two-layer baffles used in vibroacoustic protection has been proposed. The model has been verified in relation to the laboratory tests of rubber-steel baffles of various thickness. The results of calculations were also compared with the spectral characteristics of sound insulation obtained with the use of the mass law and commercial software. Using the proposed calculation model, simulation tests of the sound insulation effectiveness of rubber-steel and rubberaluminium baffles were performed. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction In order to protect the environment against noise, many different technical solutions are used. To ensure their required sound absorbing and sound insulating parameters, they must be properly designed earlier [1–5]. In many cases, prototypes are built to check the effectiveness of noise protection [6,7]. Evaluation of the effectiveness and profitability of a specific solution in the form of, for example, a cover or sound insulating enclosure or acoustic barriers, which are constructed of baffles, is also possible using theoretical calculation models that can be used before the construction of the prototype. The basic parameter of baffles is sound insulation, which is determined in laboratory conditions. Analysis of the sound insulation of baffles has been the subject of interest of many researchers [8–16]. In addition to laboratory tests, sound insulation can be determined using calculation models, among which the bestknown is the mass law [3,17]. The mass law model is of particular importance for homogeneous single baffles as, while ignoring the coincidence phenomenon which reduces sound insulation for certain critical frequencies of baffles, the sound insulating properties of such baffles can be easily estimated. The mass law can also be applied to baffles in which two solid layers are bonded at the interface with no air space [3]. Other well-known calculation models for homogeneous baffles include the Cremer [18], London [19], Sharp
E-mail address: [email protected] https://doi.org/10.1016/j.apacoust.2019.07.028 0003-682X/Ó 2019 Elsevier Ltd. All rights reserved.
[20], Brekke [21], Arau [22], Davy [23–24], Brunskog [25] and Wang [26–28] models. A uniform model, based in part on the Davy and Sharp models, was proposed for predicting the sound insulation of homogeneous single baffles in [29], whereas for other types of baffles, such as double sandwich panels, it was proposed in [30]. The problems of predicting the sound insulation of multilayer baffles have been examined in the papers [31–33]. The basic divisions of baffles used in sound insulating or sound absorbing enclosures include single and multiple baffles [34]. Single baffles, in contrast to multiple ones, do not contain an air gap and can be constructed from homogeneous or heterogeneous materials. Single heterogeneous baffles are understood as baffles in which two or more layers, constituting different materials, closely adhere to each other. Such baffles used in the construction of walls in noise reducing enclosures usually consist of a metal plate, steel or aluminium, which is a sound insulating material, and an additional layer (or layers) that has sound absorbing or sound insulating properties. The research concerns two-layer baffles, in which a solid rubber layer is glued to the metal plate. Solid rubber, in contrast to porous rubber, also used in this type of protection, has no absorbing but sound insulating properties. The article focuses on the attempt to use existing theoretical calculation models to develop a new model that allows sound insulation to be determined for heterogeneous single baffles, consisting of two different materials constituting the sound insulation system. Verification of the proposed model was carried out with reference to laboratory tests of three rubber-steel baffles, of various
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thickness, carried out at the Department of Mechanics and Vibroacoustics of AGH. The obtained results were also compared with calculations obtained using commercial software. As part of the research described in the article, simulation tests using the proposed model to assess the sound insulation effectiveness of twolayer baffles made of solid rubber and steel and aluminium plates are shown. 2. Proposal of the sound insulation calculation model for heterogeneous single two-layer baffles used in vibroacoustic protection The sound insulation calculation model for homogeneous single baffles used in vibroacoustic protection was shown in the paper [29]. The model proposes the use of dependencies on sound insulation developed by Davy and Sharp [35,20,36], for certain specific frequency ranges. Using the better accuracy of sound insulation calculations, in relation to laboratory tests, obtained using the Davy model for lower frequencies (f 0.2fc) and using the Sharp model for higher frequencies (f > 0.2fc) [29], the model for homogeneous single baffles can also be used when dealing with heterogeneous single two-layer baffles. The sound insulation for heterogeneous single two-layer baffles used in vibroacoustic protection can be determined from the following relationship:
8 2 2 > pfMS f > þ 20log 1 10log 1 þ > q0 c 0 fc > > > > h i > < 1þa2 R ¼ 10log ln 1þa2 cos2 hL ; dB; f 0:2f c > > > 20log pfMS 5:5; dB; 0:2f < f 0:5f > c c > q0 c0 > > > : 20log pfMS þ 10log 2gf ; dB; f f c q c0 pf c
ð1Þ
0
where
pfMS a¼ q0 c 0 hL ¼ cos1
" 2 # f ; 1 fc
ð2Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffi k pffiffiffi; 2p A
ð3Þ
where: c0 is the speed of sound in air (m/s), q0 is the air density (1.19 kg/m3), MS ¼ q1 h1 þ q2 h2 is the specific mass (surface density) of the two-layer baffle (kg/m2), q1 ; q2 is the density of Material 1 and 2, respectively (kg/m3), h1, h2 are the thickness of the materials (m), fc is the coincidence frequency (Hz), A is the area of the baffle (m2), k is the wavelength of sound at the frequency of interest (m), g is the damping factor of the two-layer baffle, calculated from the equation [3]:
g¼
ðg1 E1 h1 þ g2 E2 h2 Þðh1 þ h2 Þ i h i 3 2 3 2 E1 h1 1 þ 3ð1 2v=h1 Þ þ E2 h2 1 þ 3ð1 þ 2v=h2 Þ 2
h
ð4Þ
where: g1, g2 are the damping factors of the materials, E1, E2 are Young’s modulus of the materials, (Pa). The coincidence frequency for a two-layer baffle is calculated from the dependence [3,36]:
fc ¼
1=2 c20 M S 2p B
ð5Þ
where: B is the flexural rigidity of the two-layer baffle, calculated from [3]:
h i E h 1 1 2 1 þ 3ð1 2v=h1 Þ2 12 1 m1 3 h i E2 h 2 þ 2 2 1 þ 3ð1 þ 2v=h2 Þ 12 1 m2
where: t is Poisson’s ratio and v is the neutral axis location (see Fig.4.16 in [3]), obtained from the following equation:
v¼
2
2
E1 h1 E2 h2 2ðE1 h1 þ E2 h2 Þ
In order to obtain the full characteristics of the sound insulation R between 0.5fc and fc, according to the Sharp model [20,36], it should be approximated by connecting, with a straight line, points A and B of the plot corresponding to 0.5fc and fc. Sound insulation at points A and B is calculated from:
RA ¼ 20logf c m 54; dB
ð8Þ
RB ¼ 20logf c m þ 10log g 44; dB:
ð9Þ
3. Parameters of materials used to construct two-layer baffles The basic sound insulating materials used in the construction of walls in the sound insulating enclosures of machines and devices are metal plates, such as steel and aluminium sheets, which can be used individually or create sound insulating layer baffles, e.g. with a rubber layer. Rubber-metal baffles with relatively small thickness (a few to a dozen or so mm) can provide the required effectiveness in terms of sound insulation, provided that the materials and parameters of the layers are appropriately selected [34]. The article analysed baffles in which the additional sound insulating layer was solid rubber, with the physical properties given in Table 1. Table 1 also shows the physical properties of steel and aluminium plates. Laboratory tests related to rubber-steel baffles, while the given physical properties of aluminium plates will be used in the next step for simulation calculations of two-layer baffles. Fig. 1 shows the effect of steel, aluminium and rubber plate thickness on sound insulation, specified by the sound reduction index Rw. The Rw index was calculated from the sound insulation characteristics of the baffles obtained using the sound insulation calculation model for homogeneous single baffles used in vibroacoustic protection proposed in [29]. Fig. 1 shows that the steel plate has much better sound insulation properties than the aluminium plate. The Rw values of the steel plate are about 9 dB higher than for the aluminium plate, regardless of the plate thickness. To a certain thickness of metal plates, equal to 4 mm, there is a clear increase in Rw, while for plates with a thickness greater than 4 mm, the occurrence of the coincidence phenomenon means that the increase in Rw is not so large. A solid rubber baffle with a thickness of 1–14 mm has the widest range of Rw values from the analysed baffles, ranging from 16 to 38 dB. Fig. 2 shows the effect of steel, aluminium and rubber plate thickness on the frequency of coincidences of these plates. Coincidence frequency values for steel and aluminium plates are practically the same as the plate thickness increases, from about 12 kHz (plates of 1 mm thickness) to approx. 860 Hz (plates of 14 mm thickness). In the case of a solid rubber material with a density of 1490 kg/m3 and a thickness of 1–14 mm, the frequency of coincidence decreases with increasing thickness of the plate from 1,444,583 to 103,184 Hz (Fig. 2).
Table 1 Physical properties of the tested materials.
3
B¼
ð6Þ
ð7Þ
Young’s modulus, E (GPa) Density, q (kg/m3) Poisson’s ratio, m Loss factor, g
Steel
Aluminium
Rubber
207 7850 0.3 0.01
70 2800 0.35 0.01
0.003 1490 0.001 0.3
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an increase in the weighted sound reduction index Rw from 3 to 6 dB, depending on the thickness of the solid rubber. 5. Verification of the proposed model
Fig. 1. The sound reduction index Rw as functions of steel, aluminium and rubber plate thickness.
Fig. 2. Dependencies of the thickness of steel, aluminium and rubber plates on the frequency of coincidences of these plates.
For two-layer baffles in the form of solid rubber with varying thickness of 2.5, 5 and 10 mm and a steel plate with a thickness of 1 mm, their sound insulation characteristics were determined based on the calculation model proposed in Chapter 2. The results of calculations, shown in Fig. 4, were referred to the laboratory test results, mass law and results obtained from simulations using AFMG SoundFlow commercial software [40]. The sound insulation of a multilayer structure is calculated using the physical properties of the materials recorded in the AFMG SoundFlow database and using theoretical models developed by Mechel, Bies and others [40]. In the AFMG SoundFlow program, calculations were made for a baffle with dimensions compatible with the dimensions of the samples used for laboratory tests and the dimensions of infinite baffles, which were designated as SF (1 2) and SF (inf) respectively in Figs. 4–6. Fig. 7 shows the values of calculation errors (RMSE – Root Mean Square Error) of sound insulation determined using calculation models with reference to the results obtained from laboratory tests. The proposed model showed the lowest average RMSE values of 2.8 dB (Fig. 7d). For a baffle with a 10 mm thick rubber layer, the smallest calculation errors were demonstrated by the AFMG SoundFlow program for a baffle of infinite dimensions (Fig. 7c). In the case of the tested rubber and steel baffle, with a thickness of 2.5, 5 and 10 mm rubber together with a 1 mm thick steel plate, there is no reduction in sound insulation due to the occurrence of coincidence in the considered frequency band. The proposed calculation model for two-layer baffles practically used the scope of applicability defined by the Davy model [29]. In order to carry out the verification of the proposed model for cases requiring sound insulation calculations, taking into account the phenomenon of coincidence, (using Sharp’s dependence), simulation tests were carried out and are shown in the following chapters.
4. Results of laboratory tests
6. Simulation tests of rubber and metal baffles
Experimental studies of the airborne sound insulation of twolayer baffles were carried out in the laboratory of conjugated reverberation chambers in the Department of Mechanics and Vibroacoustics at the AGH University of Science and Technology in Krakow [34]. The baffles had dimensions of 1000 2000 mm and consisted of a layer of rubber with 2.5, 5 and 10 mm thickness glued to a steel sheet with a thickness of 1 mm. Samples for testing were placed in a measuring hole of conjugated reverberation chambers adapted for testing applied baffles used in industrial vibroacoustics. All samples were placed for testing with a rubber layer on the side of the transmitting chamber. As a sealing material, a plastic mass based on natural rubber was used [29]. The tests of the airborne sound insulation were carried out in accordance with the requirements of standards: [37–39]. The test results, in the form of spectral characteristics of sound insulation R of the tested baffles in the 1/3 octave frequency bands from 63 to 5000 Hz and weighted sound reduction index Rw together with the spectrum adaptation terms C and Ctr, are shown in Fig. 3. The characteristics of sound insulation of the two-layer baffles were related to the characteristics of a homogeneous single steel plate with a thickness of 1 mm, obtained as part of earlier studies conducted at the Department of Mechanics and Vibroacoustics [34]. The use of rubber layers glued to the steel plate resulted in
6.1. Sound insulating properties of rubber and aluminium baffles with a thickness of 10 mm The analysis of the sound insulation effectiveness of a two-layer baffle consisting of a solid rubber and an aluminium plate was carried out for five panel thickness configurations while maintaining a total baffle thickness of 10 mm. In order to determine the effect of the thickness and type of plates on sound insulation, the characteristics of sound insulation R were calculated, as shown in Fig. 8, for rubber-aluminium baffles in the following plate thickness configurations: 0–10 (layer thickness: rubber equal to 0, aluminium equal to 10 mm), 2.5–7.5, 5–5, 7.5–2.5 and 10–0 (Table 2). Sound insulation R (Fig. 8) was calculated for variants of two-layer baffles according to the model proposed in the article, while for baffles: variant 0–10 and 10–0 (Table 2), calculations were made using the procedure presented in the paper [29], in which the model concerns single homogeneous baffles. When using just the aluminium plate (baffle 0–10), the value of Rw = 33 dB. Fig. 8 shows the clear effect of coincidence on the sound insulation of the baffle (fc = 1214 Hz, Table 2). Keeping the same total dimension of the baffle thickness, equal to 10 mm, reducing the thickness of the aluminium plate and attaching the rubber layer increases the sound insulation. The frequency of coincidence moved towards higher values (Fig. 8). The use of a 2.5 mm
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Fig. 3. Spectral characteristics of sound insulation R and weighted sound reduction index Rw with the spectral adaptive indices (C;Ctr) of tested rubber-steel and steel baffles [34].
Fig. 4. Comparison of sound insulation R obtained from measuring and computing for two-layer baffles consisting of rubber of 2.5 mm and steel of 1 mm.
thick rubber layer with a 7.5 mm thick aluminium plate (baffle 2.5–7.5) results in an increase in fc up to 1766 Hz. The configuration of the baffle in relation to the same layer thickness (baffle 5–5) provides fc = 3022 Hz. Solid rubber 7.5 mm thick with an aluminium plate of 2.5 mm means that fc is shifted beyond the considered frequency range, ensuring the sound insulation of the baffle Rw = 37 dB. The use of the baffle in the form of the rubber plate itself (baffle 10–0) also does not show, in the spectral characteristic shown in Fig. 8, the reduction in sound insulation caused by coincidence in the analysed frequency range. The calculated sound insulation characteristics for baffles in five variants were compared with the results obtained using the AFMG SoundFlow software, as shown in Fig. 9. The highest convergence of results (linear correlation coefficient r = 0.9995 and r = 0.9976) were obtained for the following baffles: a rubber plate with a thickness of 10 mm and a rubber plate with a thickness of 7.5 mm with an aluminium plate of 2.5 mm. Additionally, for the other baffles (Fig. 9), the two models
Fig. 5. Comparison of sound insulation R obtained from measuring and computing for two-layer baffles consisting of rubber of 5 mm and steel of 1 mm.
Fig. 6. Comparison of sound insulation R obtained from measuring and computing for two-layer baffles consisting of rubber of 10 mm and steel of 1 mm.
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Fig. 7. Comparison of calculation errors for two-layer baffles consisting of a steel plate of 1 mm thickness and rubber plates of thickness: a) 2.5, b) 5 and c) 10; d) calculation errors averaged from all tested baffles.
Fig. 8. Comparison of sound insulation R for rubber-aluminium baffles of layer thickness: 0–10, 2.5–7.5, 5–5, 7.5–2.5 and 10–0 mm.
show high similarity results. The averaged linear correlation coefficient r = 0.8875. Noticeable deviations occur in areas related to the coincidence frequency fc. The sound insulation values R for frequencies equal to or higher than fc depend on the damping factor g, which for the proposed method is given by the formula (4).
The characteristics of sound insulation, calculated using two methods (Fig. 9a, b and c) show that the baffles have the maximum reduction of the sound insulation value, associated with the coincidence phenomenon, for the same center frequencies of 1/3 octave bands, which can be read from the charts. However, it is
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Table 2 Parameters of the analysed baffles made of solid rubber and aluminium plates. Baffle designation (rubber-aluminium plate thickness)
Specific mass, Ms (kg/m2)
Flexural rigidity, B (Nm)
Coincidence frequency, fc (Hz)
Damping factor, g
0–10 2.5–7.5 5–5 7.5–2.5 10–0
28 24.72 21.45 18.17 14.9
6648 2804.7 831.4 104.5 0.25
1214 1766 3022 7844 144,458
0.01 0.0178 0.0400 0.1594 0.3
Fig. 9. Comparison of sound insulation R obtained from the calculation model and AFMG SoundFlow (marked SoundFlow) for two-layer baffles consisting of rubber and aluminium layers of thickness: a) 0–10, b) 2.5–7.5, c) 5–5, d) 7.5–2.5 and e) 10–0 mm, with the linear correlation coefficient r.
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Fig. 10. Comparison of the sound reduction indices Rw of rubber-metal baffles for different thickness of the metal plate h2.
Fig. 11. Comparison of the sound reduction index Rw of rubber-metal baffles for different thickness of a rubber plate h1 with plates: aluminium of 5 mm and steel of 6 mm thickness.
clearly noticeable that the sound isolation characteristics obtained with AFMG SoundFlow, for frequencies equal to or higher than the coincidence frequency, show lower values compared to the values obtained using the proposed model. The same phenomenon can be observed by comparing the sound insulation characteristics for homogeneous single baffles (shown in [29]) obtained from the proposed calculation model (using Davy and Sharp models, like in this article, which concern two-layer baffles) to the characteristics obtained from the AFMG SoundFlow program. In this case, however, the results obtained using the proposed model in the area of coincidence frequency coincided with the results of laboratory tests carried out (Figs. 13 and 14 in [29]). 6.2. Modelling of rubber and metal baffles with high sound insulating effects The subject of the next analysis was to investigate the maximum sound insulation that can be achieved using two-layer
rubber-steel and rubber-aluminium baffles. First, using the calculation model proposed in the article, the sound insulation characteristics R were determined, from which a single-number weighted Rw index was calculated for baffles composed of a 2.5, 5, 7.5 and 10 mm thick rubber layer and a steel and aluminium layer with thickness from 1 to 10 mm. The comparison of the sound reduction indices Rw of these baffles is shown in Fig. 10. From the graphs shown in Fig. 10, it can be noticed that, as the thickness of the steel or aluminium plate rises, the value of Rw increases but only to a certain value of the metal plate thickness: for aluminium it is about 5 mm and for steel it is about 6 mm. Further increases in the thickness of metal plates does not bring the desired effect, which results in constant Rw values as a function of the thickness of the metal plates. This is related to the occurrence of a decrease in sound insulation due to the coincidence, for which the critical frequency of the two-layer baffle reaches values below 3150 Hz.
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In the next step, the maximum sound insulation determined by the Rw index while attaching rubber layers with a thickness of 1 to 10 mm to 5 mm thick aluminium and 6 mm thick steel plates was examined. A comparison of the change in the value of the Rw indices of rubber-metal baffles as a function of the thickness of the rubber layer is shown in Fig. 11. The increase in the value of the index Rw is linear, hence it is possible to estimate approximate formulas for sound insulation. For a two-layer baffle with a steel plate of 6 mm thickness, the sound reduction index Rw can be calculated from the formula:
Rw ¼ 0:503h1 þ 41:533; dB
ð10Þ
and for a two-layer baffle with an aluminium plate of 5 mm thickness:
Rw ¼ 0:8909h1 þ 32:8; dB
ð11Þ
where: h1 – thickness of a solid rubber layer with the physical parameters given in Table 1. A solid rubber layer with a thickness of 1–10 mm attached to a 5 mm aluminium plate gives the possibility of obtaining Rw in the range from 33 to 41 dB, while a two-layer baffle with even higher sound insulation (42 Rw 46 dB) can be obtained by using a solid rubber layer on the steel plate with a thickness of 6 mm. The coincidence frequencies for two-layer baffles like a rubber plate of 1 to 10 mm thick, have the following values: 2568– 3504 Hz with an aluminium plate with a thickness of 5 mm and 2049–2314 Hz with a steel plate with a thickness of 6 mm.
7. Conclusions Based on the results of laboratory tests carried out at the Department of Mechanics and Vibroacoustics of AGH, which concerned homogeneous single baffles and confirmed that, in the case of materials used for wall construction in vibroacoustic protection, the Davy model is more accurate for lower frequencies and the Sharp model for the frequency area associated with the occurrence of the coincidence phenomenon, an attempt was made to develop a calculation model for another type of baffle. Using the theoretical models of Davy and Sharp to determine sound insulation, a new calculation model for heterogeneous single two-layer baffles, with sound insulating properties and applicable to the construction of walls in sound insulating enclosures, was proposed. Verification of the proposed model, in which the results of calculations were related to the results obtained from laboratory tests of three rubber-steel plates with different thickness of the rubber layer, yielded satisfactory results. The model showed a relatively small calculation error compared to the results obtained from commercial software. Simulation studies using the proposed calculation model have shown that the two-layer baffle can be modelled in such a way as to achieve the highest efficiency, while maintaining certain parameters ensuring that the coincidence frequency associated with the reduction in sound insulation will be in a sufficiently high frequency range. According to this principle, it was calculated that having an aluminium plate with a thickness of 5 mm and attaching a second layer to it in the form of solid rubber with a density of 1490 kg/m3, with a thickness of 1–10 mm, it is possible to obtain a baffle with a sound reduction index Rw in the range from 33 up to 41 dB. However, adding the same layer of solid rubber with a thickness of 1 to 10 mm to a steel plate with a thickness of 6 mm makes it possible to obtain Rw in the range from 42 to 46 dB. Due to the obtained results, it is valid to continue laboratory tests in order to verify the proposed model for a larger number of samples – two-layer baffles, and in particular those for which
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Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Sound insulation properties of two-layer baffles used in vibroacoustic protection Krzysztof Kosała Department of Mechanics and Vibroacoustics, AGH University of Science and Technology, al. A. Mickiewicza 30, 30-059 Krakow, Poland
a r t i c l e
i n f o
Article history: Received 8 March 2019 Received in revised form 8 July 2019 Accepted 22 July 2019
Keywords: Sound insulation Calculation models Two-layer baffles Vibroacoustic protection
a b s t r a c t The subject of the research presented in the article was the modelling of acoustic properties of two-layer baffles, used in vibroacoustic protection. Baffles of this type are used, among others, in the construction of walls of the sound insulating enclosures of machines and devices. The heterogeneous single baffles that were analysed were panels made of two sound insulation layers, solid rubber and a steel plate. A sound insulation calculation model for heterogeneous single two-layer baffles used in vibroacoustic protection has been proposed. The model has been verified in relation to the laboratory tests of rubber-steel baffles of various thickness. The results of calculations were also compared with the spectral characteristics of sound insulation obtained with the use of the mass law and commercial software. Using the proposed calculation model, simulation tests of the sound insulation effectiveness of rubber-steel and rubberaluminium baffles were performed. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction In order to protect the environment against noise, many different technical solutions are used. To ensure their required sound absorbing and sound insulating parameters, they must be properly designed earlier [1–5]. In many cases, prototypes are built to check the effectiveness of noise protection [6,7]. Evaluation of the effectiveness and profitability of a specific solution in the form of, for example, a cover or sound insulating enclosure or acoustic barriers, which are constructed of baffles, is also possible using theoretical calculation models that can be used before the construction of the prototype. The basic parameter of baffles is sound insulation, which is determined in laboratory conditions. Analysis of the sound insulation of baffles has been the subject of interest of many researchers [8–16]. In addition to laboratory tests, sound insulation can be determined using calculation models, among which the bestknown is the mass law [3,17]. The mass law model is of particular importance for homogeneous single baffles as, while ignoring the coincidence phenomenon which reduces sound insulation for certain critical frequencies of baffles, the sound insulating properties of such baffles can be easily estimated. The mass law can also be applied to baffles in which two solid layers are bonded at the interface with no air space [3]. Other well-known calculation models for homogeneous baffles include the Cremer [18], London [19], Sharp
E-mail address: [email protected] https://doi.org/10.1016/j.apacoust.2019.07.028 0003-682X/Ó 2019 Elsevier Ltd. All rights reserved.
[20], Brekke [21], Arau [22], Davy [23–24], Brunskog [25] and Wang [26–28] models. A uniform model, based in part on the Davy and Sharp models, was proposed for predicting the sound insulation of homogeneous single baffles in [29], whereas for other types of baffles, such as double sandwich panels, it was proposed in [30]. The problems of predicting the sound insulation of multilayer baffles have been examined in the papers [31–33]. The basic divisions of baffles used in sound insulating or sound absorbing enclosures include single and multiple baffles [34]. Single baffles, in contrast to multiple ones, do not contain an air gap and can be constructed from homogeneous or heterogeneous materials. Single heterogeneous baffles are understood as baffles in which two or more layers, constituting different materials, closely adhere to each other. Such baffles used in the construction of walls in noise reducing enclosures usually consist of a metal plate, steel or aluminium, which is a sound insulating material, and an additional layer (or layers) that has sound absorbing or sound insulating properties. The research concerns two-layer baffles, in which a solid rubber layer is glued to the metal plate. Solid rubber, in contrast to porous rubber, also used in this type of protection, has no absorbing but sound insulating properties. The article focuses on the attempt to use existing theoretical calculation models to develop a new model that allows sound insulation to be determined for heterogeneous single baffles, consisting of two different materials constituting the sound insulation system. Verification of the proposed model was carried out with reference to laboratory tests of three rubber-steel baffles, of various
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thickness, carried out at the Department of Mechanics and Vibroacoustics of AGH. The obtained results were also compared with calculations obtained using commercial software. As part of the research described in the article, simulation tests using the proposed model to assess the sound insulation effectiveness of twolayer baffles made of solid rubber and steel and aluminium plates are shown. 2. Proposal of the sound insulation calculation model for heterogeneous single two-layer baffles used in vibroacoustic protection The sound insulation calculation model for homogeneous single baffles used in vibroacoustic protection was shown in the paper [29]. The model proposes the use of dependencies on sound insulation developed by Davy and Sharp [35,20,36], for certain specific frequency ranges. Using the better accuracy of sound insulation calculations, in relation to laboratory tests, obtained using the Davy model for lower frequencies (f 0.2fc) and using the Sharp model for higher frequencies (f > 0.2fc) [29], the model for homogeneous single baffles can also be used when dealing with heterogeneous single two-layer baffles. The sound insulation for heterogeneous single two-layer baffles used in vibroacoustic protection can be determined from the following relationship:
8 2 2 > pfMS f > þ 20log 1 10log 1 þ > q0 c 0 fc > > > > h i > < 1þa2 R ¼ 10log ln 1þa2 cos2 hL ; dB; f 0:2f c > > > 20log pfMS 5:5; dB; 0:2f < f 0:5f > c c > q0 c0 > > > : 20log pfMS þ 10log 2gf ; dB; f f c q c0 pf c
ð1Þ
0
where
pfMS a¼ q0 c 0 hL ¼ cos1
" 2 # f ; 1 fc
ð2Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffi k pffiffiffi; 2p A
ð3Þ
where: c0 is the speed of sound in air (m/s), q0 is the air density (1.19 kg/m3), MS ¼ q1 h1 þ q2 h2 is the specific mass (surface density) of the two-layer baffle (kg/m2), q1 ; q2 is the density of Material 1 and 2, respectively (kg/m3), h1, h2 are the thickness of the materials (m), fc is the coincidence frequency (Hz), A is the area of the baffle (m2), k is the wavelength of sound at the frequency of interest (m), g is the damping factor of the two-layer baffle, calculated from the equation [3]:
g¼
ðg1 E1 h1 þ g2 E2 h2 Þðh1 þ h2 Þ i h i 3 2 3 2 E1 h1 1 þ 3ð1 2v=h1 Þ þ E2 h2 1 þ 3ð1 þ 2v=h2 Þ 2
h
ð4Þ
where: g1, g2 are the damping factors of the materials, E1, E2 are Young’s modulus of the materials, (Pa). The coincidence frequency for a two-layer baffle is calculated from the dependence [3,36]:
fc ¼
1=2 c20 M S 2p B
ð5Þ
where: B is the flexural rigidity of the two-layer baffle, calculated from [3]:
h i E h 1 1 2 1 þ 3ð1 2v=h1 Þ2 12 1 m1 3 h i E2 h 2 þ 2 2 1 þ 3ð1 þ 2v=h2 Þ 12 1 m2
where: t is Poisson’s ratio and v is the neutral axis location (see Fig.4.16 in [3]), obtained from the following equation:
v¼
2
2
E1 h1 E2 h2 2ðE1 h1 þ E2 h2 Þ
In order to obtain the full characteristics of the sound insulation R between 0.5fc and fc, according to the Sharp model [20,36], it should be approximated by connecting, with a straight line, points A and B of the plot corresponding to 0.5fc and fc. Sound insulation at points A and B is calculated from:
RA ¼ 20logf c m 54; dB
ð8Þ
RB ¼ 20logf c m þ 10log g 44; dB:
ð9Þ
3. Parameters of materials used to construct two-layer baffles The basic sound insulating materials used in the construction of walls in the sound insulating enclosures of machines and devices are metal plates, such as steel and aluminium sheets, which can be used individually or create sound insulating layer baffles, e.g. with a rubber layer. Rubber-metal baffles with relatively small thickness (a few to a dozen or so mm) can provide the required effectiveness in terms of sound insulation, provided that the materials and parameters of the layers are appropriately selected [34]. The article analysed baffles in which the additional sound insulating layer was solid rubber, with the physical properties given in Table 1. Table 1 also shows the physical properties of steel and aluminium plates. Laboratory tests related to rubber-steel baffles, while the given physical properties of aluminium plates will be used in the next step for simulation calculations of two-layer baffles. Fig. 1 shows the effect of steel, aluminium and rubber plate thickness on sound insulation, specified by the sound reduction index Rw. The Rw index was calculated from the sound insulation characteristics of the baffles obtained using the sound insulation calculation model for homogeneous single baffles used in vibroacoustic protection proposed in [29]. Fig. 1 shows that the steel plate has much better sound insulation properties than the aluminium plate. The Rw values of the steel plate are about 9 dB higher than for the aluminium plate, regardless of the plate thickness. To a certain thickness of metal plates, equal to 4 mm, there is a clear increase in Rw, while for plates with a thickness greater than 4 mm, the occurrence of the coincidence phenomenon means that the increase in Rw is not so large. A solid rubber baffle with a thickness of 1–14 mm has the widest range of Rw values from the analysed baffles, ranging from 16 to 38 dB. Fig. 2 shows the effect of steel, aluminium and rubber plate thickness on the frequency of coincidences of these plates. Coincidence frequency values for steel and aluminium plates are practically the same as the plate thickness increases, from about 12 kHz (plates of 1 mm thickness) to approx. 860 Hz (plates of 14 mm thickness). In the case of a solid rubber material with a density of 1490 kg/m3 and a thickness of 1–14 mm, the frequency of coincidence decreases with increasing thickness of the plate from 1,444,583 to 103,184 Hz (Fig. 2).
Table 1 Physical properties of the tested materials.
3
B¼
ð6Þ
ð7Þ
Young’s modulus, E (GPa) Density, q (kg/m3) Poisson’s ratio, m Loss factor, g
Steel
Aluminium
Rubber
207 7850 0.3 0.01
70 2800 0.35 0.01
0.003 1490 0.001 0.3
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an increase in the weighted sound reduction index Rw from 3 to 6 dB, depending on the thickness of the solid rubber. 5. Verification of the proposed model
Fig. 1. The sound reduction index Rw as functions of steel, aluminium and rubber plate thickness.
Fig. 2. Dependencies of the thickness of steel, aluminium and rubber plates on the frequency of coincidences of these plates.
For two-layer baffles in the form of solid rubber with varying thickness of 2.5, 5 and 10 mm and a steel plate with a thickness of 1 mm, their sound insulation characteristics were determined based on the calculation model proposed in Chapter 2. The results of calculations, shown in Fig. 4, were referred to the laboratory test results, mass law and results obtained from simulations using AFMG SoundFlow commercial software [40]. The sound insulation of a multilayer structure is calculated using the physical properties of the materials recorded in the AFMG SoundFlow database and using theoretical models developed by Mechel, Bies and others [40]. In the AFMG SoundFlow program, calculations were made for a baffle with dimensions compatible with the dimensions of the samples used for laboratory tests and the dimensions of infinite baffles, which were designated as SF (1 2) and SF (inf) respectively in Figs. 4–6. Fig. 7 shows the values of calculation errors (RMSE – Root Mean Square Error) of sound insulation determined using calculation models with reference to the results obtained from laboratory tests. The proposed model showed the lowest average RMSE values of 2.8 dB (Fig. 7d). For a baffle with a 10 mm thick rubber layer, the smallest calculation errors were demonstrated by the AFMG SoundFlow program for a baffle of infinite dimensions (Fig. 7c). In the case of the tested rubber and steel baffle, with a thickness of 2.5, 5 and 10 mm rubber together with a 1 mm thick steel plate, there is no reduction in sound insulation due to the occurrence of coincidence in the considered frequency band. The proposed calculation model for two-layer baffles practically used the scope of applicability defined by the Davy model [29]. In order to carry out the verification of the proposed model for cases requiring sound insulation calculations, taking into account the phenomenon of coincidence, (using Sharp’s dependence), simulation tests were carried out and are shown in the following chapters.
4. Results of laboratory tests
6. Simulation tests of rubber and metal baffles
Experimental studies of the airborne sound insulation of twolayer baffles were carried out in the laboratory of conjugated reverberation chambers in the Department of Mechanics and Vibroacoustics at the AGH University of Science and Technology in Krakow [34]. The baffles had dimensions of 1000 2000 mm and consisted of a layer of rubber with 2.5, 5 and 10 mm thickness glued to a steel sheet with a thickness of 1 mm. Samples for testing were placed in a measuring hole of conjugated reverberation chambers adapted for testing applied baffles used in industrial vibroacoustics. All samples were placed for testing with a rubber layer on the side of the transmitting chamber. As a sealing material, a plastic mass based on natural rubber was used [29]. The tests of the airborne sound insulation were carried out in accordance with the requirements of standards: [37–39]. The test results, in the form of spectral characteristics of sound insulation R of the tested baffles in the 1/3 octave frequency bands from 63 to 5000 Hz and weighted sound reduction index Rw together with the spectrum adaptation terms C and Ctr, are shown in Fig. 3. The characteristics of sound insulation of the two-layer baffles were related to the characteristics of a homogeneous single steel plate with a thickness of 1 mm, obtained as part of earlier studies conducted at the Department of Mechanics and Vibroacoustics [34]. The use of rubber layers glued to the steel plate resulted in
6.1. Sound insulating properties of rubber and aluminium baffles with a thickness of 10 mm The analysis of the sound insulation effectiveness of a two-layer baffle consisting of a solid rubber and an aluminium plate was carried out for five panel thickness configurations while maintaining a total baffle thickness of 10 mm. In order to determine the effect of the thickness and type of plates on sound insulation, the characteristics of sound insulation R were calculated, as shown in Fig. 8, for rubber-aluminium baffles in the following plate thickness configurations: 0–10 (layer thickness: rubber equal to 0, aluminium equal to 10 mm), 2.5–7.5, 5–5, 7.5–2.5 and 10–0 (Table 2). Sound insulation R (Fig. 8) was calculated for variants of two-layer baffles according to the model proposed in the article, while for baffles: variant 0–10 and 10–0 (Table 2), calculations were made using the procedure presented in the paper [29], in which the model concerns single homogeneous baffles. When using just the aluminium plate (baffle 0–10), the value of Rw = 33 dB. Fig. 8 shows the clear effect of coincidence on the sound insulation of the baffle (fc = 1214 Hz, Table 2). Keeping the same total dimension of the baffle thickness, equal to 10 mm, reducing the thickness of the aluminium plate and attaching the rubber layer increases the sound insulation. The frequency of coincidence moved towards higher values (Fig. 8). The use of a 2.5 mm
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Fig. 3. Spectral characteristics of sound insulation R and weighted sound reduction index Rw with the spectral adaptive indices (C;Ctr) of tested rubber-steel and steel baffles [34].
Fig. 4. Comparison of sound insulation R obtained from measuring and computing for two-layer baffles consisting of rubber of 2.5 mm and steel of 1 mm.
thick rubber layer with a 7.5 mm thick aluminium plate (baffle 2.5–7.5) results in an increase in fc up to 1766 Hz. The configuration of the baffle in relation to the same layer thickness (baffle 5–5) provides fc = 3022 Hz. Solid rubber 7.5 mm thick with an aluminium plate of 2.5 mm means that fc is shifted beyond the considered frequency range, ensuring the sound insulation of the baffle Rw = 37 dB. The use of the baffle in the form of the rubber plate itself (baffle 10–0) also does not show, in the spectral characteristic shown in Fig. 8, the reduction in sound insulation caused by coincidence in the analysed frequency range. The calculated sound insulation characteristics for baffles in five variants were compared with the results obtained using the AFMG SoundFlow software, as shown in Fig. 9. The highest convergence of results (linear correlation coefficient r = 0.9995 and r = 0.9976) were obtained for the following baffles: a rubber plate with a thickness of 10 mm and a rubber plate with a thickness of 7.5 mm with an aluminium plate of 2.5 mm. Additionally, for the other baffles (Fig. 9), the two models
Fig. 5. Comparison of sound insulation R obtained from measuring and computing for two-layer baffles consisting of rubber of 5 mm and steel of 1 mm.
Fig. 6. Comparison of sound insulation R obtained from measuring and computing for two-layer baffles consisting of rubber of 10 mm and steel of 1 mm.
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Fig. 7. Comparison of calculation errors for two-layer baffles consisting of a steel plate of 1 mm thickness and rubber plates of thickness: a) 2.5, b) 5 and c) 10; d) calculation errors averaged from all tested baffles.
Fig. 8. Comparison of sound insulation R for rubber-aluminium baffles of layer thickness: 0–10, 2.5–7.5, 5–5, 7.5–2.5 and 10–0 mm.
show high similarity results. The averaged linear correlation coefficient r = 0.8875. Noticeable deviations occur in areas related to the coincidence frequency fc. The sound insulation values R for frequencies equal to or higher than fc depend on the damping factor g, which for the proposed method is given by the formula (4).
The characteristics of sound insulation, calculated using two methods (Fig. 9a, b and c) show that the baffles have the maximum reduction of the sound insulation value, associated with the coincidence phenomenon, for the same center frequencies of 1/3 octave bands, which can be read from the charts. However, it is
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Table 2 Parameters of the analysed baffles made of solid rubber and aluminium plates. Baffle designation (rubber-aluminium plate thickness)
Specific mass, Ms (kg/m2)
Flexural rigidity, B (Nm)
Coincidence frequency, fc (Hz)
Damping factor, g
0–10 2.5–7.5 5–5 7.5–2.5 10–0
28 24.72 21.45 18.17 14.9
6648 2804.7 831.4 104.5 0.25
1214 1766 3022 7844 144,458
0.01 0.0178 0.0400 0.1594 0.3
Fig. 9. Comparison of sound insulation R obtained from the calculation model and AFMG SoundFlow (marked SoundFlow) for two-layer baffles consisting of rubber and aluminium layers of thickness: a) 0–10, b) 2.5–7.5, c) 5–5, d) 7.5–2.5 and e) 10–0 mm, with the linear correlation coefficient r.
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Fig. 10. Comparison of the sound reduction indices Rw of rubber-metal baffles for different thickness of the metal plate h2.
Fig. 11. Comparison of the sound reduction index Rw of rubber-metal baffles for different thickness of a rubber plate h1 with plates: aluminium of 5 mm and steel of 6 mm thickness.
clearly noticeable that the sound isolation characteristics obtained with AFMG SoundFlow, for frequencies equal to or higher than the coincidence frequency, show lower values compared to the values obtained using the proposed model. The same phenomenon can be observed by comparing the sound insulation characteristics for homogeneous single baffles (shown in [29]) obtained from the proposed calculation model (using Davy and Sharp models, like in this article, which concern two-layer baffles) to the characteristics obtained from the AFMG SoundFlow program. In this case, however, the results obtained using the proposed model in the area of coincidence frequency coincided with the results of laboratory tests carried out (Figs. 13 and 14 in [29]). 6.2. Modelling of rubber and metal baffles with high sound insulating effects The subject of the next analysis was to investigate the maximum sound insulation that can be achieved using two-layer
rubber-steel and rubber-aluminium baffles. First, using the calculation model proposed in the article, the sound insulation characteristics R were determined, from which a single-number weighted Rw index was calculated for baffles composed of a 2.5, 5, 7.5 and 10 mm thick rubber layer and a steel and aluminium layer with thickness from 1 to 10 mm. The comparison of the sound reduction indices Rw of these baffles is shown in Fig. 10. From the graphs shown in Fig. 10, it can be noticed that, as the thickness of the steel or aluminium plate rises, the value of Rw increases but only to a certain value of the metal plate thickness: for aluminium it is about 5 mm and for steel it is about 6 mm. Further increases in the thickness of metal plates does not bring the desired effect, which results in constant Rw values as a function of the thickness of the metal plates. This is related to the occurrence of a decrease in sound insulation due to the coincidence, for which the critical frequency of the two-layer baffle reaches values below 3150 Hz.
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In the next step, the maximum sound insulation determined by the Rw index while attaching rubber layers with a thickness of 1 to 10 mm to 5 mm thick aluminium and 6 mm thick steel plates was examined. A comparison of the change in the value of the Rw indices of rubber-metal baffles as a function of the thickness of the rubber layer is shown in Fig. 11. The increase in the value of the index Rw is linear, hence it is possible to estimate approximate formulas for sound insulation. For a two-layer baffle with a steel plate of 6 mm thickness, the sound reduction index Rw can be calculated from the formula:
Rw ¼ 0:503h1 þ 41:533; dB
ð10Þ
and for a two-layer baffle with an aluminium plate of 5 mm thickness:
Rw ¼ 0:8909h1 þ 32:8; dB
ð11Þ
where: h1 – thickness of a solid rubber layer with the physical parameters given in Table 1. A solid rubber layer with a thickness of 1–10 mm attached to a 5 mm aluminium plate gives the possibility of obtaining Rw in the range from 33 to 41 dB, while a two-layer baffle with even higher sound insulation (42 Rw 46 dB) can be obtained by using a solid rubber layer on the steel plate with a thickness of 6 mm. The coincidence frequencies for two-layer baffles like a rubber plate of 1 to 10 mm thick, have the following values: 2568– 3504 Hz with an aluminium plate with a thickness of 5 mm and 2049–2314 Hz with a steel plate with a thickness of 6 mm.
7. Conclusions Based on the results of laboratory tests carried out at the Department of Mechanics and Vibroacoustics of AGH, which concerned homogeneous single baffles and confirmed that, in the case of materials used for wall construction in vibroacoustic protection, the Davy model is more accurate for lower frequencies and the Sharp model for the frequency area associated with the occurrence of the coincidence phenomenon, an attempt was made to develop a calculation model for another type of baffle. Using the theoretical models of Davy and Sharp to determine sound insulation, a new calculation model for heterogeneous single two-layer baffles, with sound insulating properties and applicable to the construction of walls in sound insulating enclosures, was proposed. Verification of the proposed model, in which the results of calculations were related to the results obtained from laboratory tests of three rubber-steel plates with different thickness of the rubber layer, yielded satisfactory results. The model showed a relatively small calculation error compared to the results obtained from commercial software. Simulation studies using the proposed calculation model have shown that the two-layer baffle can be modelled in such a way as to achieve the highest efficiency, while maintaining certain parameters ensuring that the coincidence frequency associated with the reduction in sound insulation will be in a sufficiently high frequency range. According to this principle, it was calculated that having an aluminium plate with a thickness of 5 mm and attaching a second layer to it in the form of solid rubber with a density of 1490 kg/m3, with a thickness of 1–10 mm, it is possible to obtain a baffle with a sound reduction index Rw in the range from 33 up to 41 dB. However, adding the same layer of solid rubber with a thickness of 1 to 10 mm to a steel plate with a thickness of 6 mm makes it possible to obtain Rw in the range from 42 to 46 dB. Due to the obtained results, it is valid to continue laboratory tests in order to verify the proposed model for a larger number of samples – two-layer baffles, and in particular those for which
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