EFFECTS OF PERMEABILITY ON SOUND ABSORPTION AND SOUND INSULATION PERFORMANCE OF ACOUSTIC CEILING PANELS

Transcription

1 EFFECTS OF PERMEABILITY ON SOUND ABSORPTION AND SOUND INSULATION PERFORMANCE OF ACOUSTIC CEILING PANELS Kento Hashitsume and Daiji Takahashi Graduate School of Engineering, Kyoto University In room acoustics, it s very important to adjust reverberation properly. From the point of view of room acoustic design, the ceilings are considered for the best part for adjusting reverberation of the room. The ceiling panels are often made of rock wool and have little permeability. Although they are commercially available and well marketed, full investigations regarding acoustic properties haven t been given yet. In this study, the effects of permeability on sound absorption and sound insulation characteristics of acoustic ceiling panels are investigated. A theoretical model is proposed and discussed by changing its related parameters in each case of introducing permeability and micro-perforating. From this investigation, it was found that the sound absorption characteristics are strongly affected by change in flow resistivity in the former case and perforation ratio in the latter case, and both cases show a similar tendency. The effects of porous materials installed between the ceiling panel and a rigid back wall are also considered. At the same time, it s concerned that sound insulation performance might be deteriorated by introducing the permeability of the ceiling panels. This issue is also investigated by using simulation models of a ceiling-floor system and discussions are given to the influence of permeability on sound insulation performance of impact sound of floor. The results show that radiated power caused by a point force excitation generally increases by introducing permeability. However, at some frequencies, sound insulation performance is improved due to mitigation of the effect of mass-air-mass resonance of the ceiling-floor system. 1. Introduction Many kinds of building materials for room acoustics have been developed in order to adjust reverberation of the room properly. One of them is acoustic ceiling panel, which has been commercially available and been used widely. This is because it seems that the ceilings are the best part for adjusting reverberation in terms of room acoustics. In many cases, acoustic ceiling panels are made of rock wool and considered to have little permeability. However, full investigations concerning their acoustic properties haven't been carried out. This study discusses the effects of permeability on sound absorption performance in each case of introducing permeability and micro-perforating with the related parameters. Numerical results are discussed in comparison with the effect of difference in type of perforation. Furthermore, a ceilingfloor model is proposed to investigate the effects of permeability on sound insulation performance for problems of impact sound of floor. 1

2 2. Absorption Performance 2.1 Analytics of Acoustic Ceiling Panels in Sound Field In the case of introducing permeability Consider the case, in which the panel vibrates with a velocity v in the sound field of pressure difference p, as shown in Fig. 1(a) [1]. In the hole, the relative air velocity to the panel is v f v. Then, the flow resistance of a hole R h is given by R h = p v f v, (1) where R is the flow resistivity [N s/m 4 ] of a hole, and h is the thickness of the panel. The flow resistivity R can be expressed by the flow resistivity R of the panel with permeability as R = σr, (2) where σ is the perforation ratio. The particle velocity v is expressed as Substituting Eqs. (1) and (2) into Eq. (3) gives v = v + (v f v)σ. (3) v = v + p Rh. (4) In the case where the surface of the panel vibrating in the sound field has sound absorptivity, as + shown in Fig. 1(b) [2], the particle velocity v and v on the surface of both sides of the panel are given by replacing v with v + v,, v + v,, in Eq. (4) and substituting v, = p + Ζ +, v,, = p Ζ into them, respectively. The result is v + = v + p+ Ζ + + p+ p, (5) Rh v = v p Ζ + p+ p, (6) Rh where p ± and Ζ ± are the pressure and the impedance on the surface of the panel, respectively In the case of micro-perforating Consider next the case, in which the panel is micro-perforated [3]. In Fig. 1(b), when the panel is in the state of rest, that is, v =, the acoustic impedance of the hole is defined as Ζ = Ζ res + Ζ react = p v f, which yields Ζ res v f + Ζ react v f = Δp, (7) where Ζ res and Ζ react are the resistance and reactance term of the acoustic impedance Ζ of the hole, respectively. An expression of Ζ res and Ζ react is given by Maa [4]. When the panel vibrates, that is, v, by replacing v f in the first term at the left hand side with v f v, Eq. (7) can be rearranged as v f v = p Ζ react v. (8) Ζ Ζ + In a similar way to the derivation of Eqs. (5) and (6), the particle velocity v and v can be given by v + = ζ c v + p+ Ζ + + p+ p σ, (9) Ζ where ζ c = 1 Ζ react Ζ σ. v = ζ c v p Ζ + p+ p Ζ σ, (1) 2 ICSV23, Athens (Greece), 1-14 July 216

3 Figure 1: Cross sections of the panels. (a) A vibrating panel with rigid surface. (b) A vibrating panel with sound absorptivity on the surface. Figure 2: A schematic model of the ceiling panel backed by rigid wall and three layers in-between. Schematic model shown in Fig. 2 is applied to investigate the absorption performance of the acoustic ceiling panel. The gap between the ceiling panel and the rigid back wall is divided into three layers and filled with air or the glass wool, respectively. Consider that a plane wave p i enters with an incident angle θ. In both cases of introducing permeability and micro-perforating, the equation of motion can be written as D 4 w ρhω 2 w = [p i + p r p 1 ] z=, (11) where D is the complex bending stiffness of the panel written as D = E(1 iη)h 3 12(1 υ 2 ), E is the Young s modulus, η is the loss factor and υ is the Poisson s ratio. w is the displacement, ρ is the density of the panel, ω is the angular frequency, p r is the reflected pressure, and p j (j = 1, 2, 3) is the sound pressure of each layer, respectively. The boundary conditions are given as v i + v r z= = v +, (12) v 1 z= = v, (13) p 1 z=d1 = p 2 z=d1, (14) v 1 z=d1 = v 2 z=d1, (15) p 2 z=d2 = p 3 z=d2, (16) v 2 z=d2 = v 3 z=d2, (17) v 3 z=d3 =, (18) where v i and v r are the particle velocity of the incident and reflected wave, and v j (j = 1, 2, 3) is the particle velocity of each layer. When the layer is filled with the glass wool, its characteristic impedance and propagation constant are given by Miki s formulas [5]. The amplitude P r of the reflected pressure p r can be obtained from Eqs. (11) to (18), and the absorption coefficient α θ is expressed as 1 P r 2. The field incidence absorption coefficient α is written as α = θ L θ L α θ sin 2θ dθ sin 2θ dθ, θ L = 78. (19) 2.2 Numerical results and discussion In each case of introducing permeability and micro-perforating, the acoustic behaviour of the acoustic ceiling panel is investigated numerically by changing the value of related parameters. In this study, the field incidence absorption coefficient is calculated at a 1/48 octave band interval from 62.5Hz to 4kHz. The properties of the ceiling panel are the thickness h = 9mm, density ICSV23, Athens (Greece), 1-14 July 216 3

4 ρ = 358kg m 3, loss factor η =.1, Young s modulus E = N/m 2, Poisson s ratio υ =.3. The cavity depth of each layer is.1m In the case of introducing permeability The effect of the flow resistivity on sound absorption performance accompanied by the change of its value from to N s m 4 is shown in Fig. 3(a) when all backed layers are filled with air. The result for the case, in which the panel has no permeability, that is, the perforation ratio is zero, is also shown in Fig. 3(a). Figure 3: Field incidence absorption coefficient of the ceiling panel in the case of introducing permeability. (a) Effects of the flow resistivity as the panel is backed by air. (b)-(c) Effects of the position of glass wool: (b) R = N s m 4, (c) R = N s m 4. Figure 4: Field incidence absorption coefficient of the ceiling panel in the case of micro-perforating. (a) Effects of the perforation ratio as the panel is backed by air. (b)-(c) Effects of the position of glass wool: (b) σ = 4.91%, (c) σ =.196%. 4 ICSV23, Athens (Greece), 1-14 July 216

5 As can be observed in the graph, the field incidence absorption coefficient changed variously, and the sharp peak appears at around 63Hz when the flow resistivity exceeds N s m 4. Increasing the value, the graph approaches to that for the case, in which the perforation ratio is zero, because too high flow resistivity deteriorates the permeability. Thus the ability of the permeability is utilized effectively and high absorption performance is realized in wide frequency ranges when the flow resistivity is around N s m 4, and the panel can show intensive absorption characteristics as the flow resistivity increases. Figures 3(b) and 3(c) plot the results when the position of the glass wool is changed. It s shown in the graph that the installation of the glass wool makes the peak value high, the peak width broadens as the position of the glass wool approaches to the ceiling panel and the peak covers wider frequency ranges. When the flow resistivity is large in Fig. 3(c), each graph has little difference for high frequency ranges above 1kHz and the glass wool deteriorates the absorption performance in the frequency range of 125Hz-25Hz In the case of micro-perforating Figure 4(a) shows the effect of the perforation ratio, which is determined by both the hole diameter d and the pitch of the hole. As can be observed in the graph, the absorption characteristics show the various tendencies and the oscillation of the graph becomes noticeable with increasing the perforation ratio. Compared with Fig. 3(a), the change of the graph with decreasing the perforation ratio shows the same tendency as that with increasing the flow resistivity. Therefore it's confirmed that the large flow resistivity results in the approach of the graph to that for the case of no permeability. Figures 4(b) and 4(c) show the effect of the position of the glass wool. As with Figs. 3(b) and 3(c), the peak can become higher with inserting the glass wool and broader as its position comes closer to the ceiling panel. Compared to Figs. 3(b) and 3(c), when the perforation ratio are 4.91% and.196%, each graph shows the same tendency as that for the case, in which the flow resistivity are and N s m 4, respectively; therefore it becomes clear that the flow resistivity is effective in controlling the absorption performance. 3. Insulation Performance 3.1 Simulation models of a ceiling-floor system Figure 5 shows the simulation model of a ceiling-floor system to investigate the insulation performance of the acoustic ceiling panel. The ceiling panel and the floor slab are installed at an interval d of.3m and the intermediate layer is filled with air. The floor slab is excited by a point force q in z direction at (x, y) = (,). The equation of motion of the floor slab and the ceiling panel can be written as D 1 4 w 1 ρ 1 h 1 ω 2 w 1 = p I z= p II z= + q δ(x)δ(y), (2) D 2 4 w 2 ρ 2 h 2 ω 2 w 2 = p II z=d p III z=d, (21) where subscripts 1 and 2 mean the floor slab and the ceiling panel, respectively. By applying Fourier transform, the displacement w(x, y) can be expressed as W(α, β)e i(αx+βy) dαdβ, and q δ(x)δ(y) = Q(α, β)e i(αx+βy) dαdβ, (22) where δ is a delta function. The sound pressure in each region can be written as p II = p I = P I e i(αx+βy γz) dαdβ, (23) [P + II e iγz + P II e iγz ]e i(αx+βy) dαdβ, (24) ICSV23, Athens (Greece), 1-14 July 216 5

6 p III = P III e i(αx+βy+γz) dαdβ. (25) Consider that Eqs. (23) to (25) fulfill a wave equation, γ is determined as γ = { k2 (α 2 + β 2 ), as k 2 α 2 + β 2, i (α 2 + β 2 ) k 2, as k 2 α 2 + β 2. The boundary conditions are given by the particle velocity v = (1 iρ ω) ( p z) in each region as v I z= = iωw 1, (27) v II z=d = iωw 2 + p II z=d Ζ + (26) v II z= = iωw 1, (28) + [p II p III ] z=d Rh 2, (29) v III z=d = iωw 2 p III z=d Ζ + [p II p III ] z=d, (3) Rh 2 where Eqs. (5) and (6) are applied to the right hand side of Eqs. (29) and (3), respectively. Figure 5: Simulation model of a ceiling-floor system. Figure 6: A panel in a rectangular coordinates system representing a far field. When S, which is a part of the ceiling panel, is vibrating with a velocity v, by applying Rayleigh Integral, the radiated sound pressure is expressed as p(x, y, z) = iρ ω v(x, y ) eikr 2πR ds. (31) S Since R approaches to r (x cos φ + y sin φ) sin φ geometrically in the far field shown in Fig. 6, Eq. (31) is rearranged as where p(r, θ, φ) = iρ ω eikr 2πr v(x, y )e i(k xx +k y y ) dx dy, (32) k x = k cos φ sin θ, k y = k sin φ sin θ. (33) The intensity for the small areas ds = r 2 sin θ dθdφ in far field can be written as I(r, θ, φ) = p(r, θ, φ) 2 2ρ c, the radiated sound power is 2π π 2 P w = dφ I(r, θ, φ)r 2 sin θ dθ. (34) To calculate the sound pressure caused by the exciting point force q in Fig. 5, v(x, y ) in Eq. (32) is replaced with v III z=d, which is expressed as v III z=d = 1 p III = 1 iρ ω z z=d ρ ω γp IIIe iγd e i(αx+βy) dαdβ. (35) 6 ICSV23, Athens (Greece), 1-14 July 216

7 The sound pressure can be calculated by applying inverse Fourier transform to Eq. (35) and replacing α and β with k x and k y, respectively. p(r, θ, φ) = i2π eikr r k cos θ eikd cos θ P III (k x, k y ). (36) From Eqs. (34) and (36), the radiated sound power can be rewritten as P w = 4π3 k 2 π 2 P ρ c III (k x, k y ) 2 sin θ cos 2 θ dθ. (37) 3.2 Numerical results and discussion Figure 7 shows numerical results of the radiated power level from the ceiling panel excited by a point force, q = 1N, with the parameter of flow resistivity. In this study, the radiated power level is calculated at a 1/48 octave band interval from 31.25Hz to 2kHz. The properties of the floor slab are the thickness h 1 =.15m, density ρ 1 = kg m 3, loss factor η 1 = , Young s modulus E 1 = N/m 2, Poisson s ratio υ 1 =.2. The cavity depth is.3m. The properties of the ceiling panel are the same as Subsection 2.2. The graphs for the cases, in which the perforation ratio is zero and a single floor slab is excited by a point force, are also shown in Fig. 7. Figure 7: Radiated power level from a ceiling panel excited by a point force. The parameter is flow resistivity R. Figure 8: Effect of a ceiling panel, from the results in Fig. 7. Figure 9: Effect of permeability, from the results in Fig. 7. ICSV23, Athens (Greece), 1-14 July 216 7

8 The coincidence frequency can be written as f c = c 2 2πh 12(1 υ2 )ρ, (38) E where c is air velocity. The coincidence frequency f c of the floor slab from Eq. (38) is 121.5Hz and shows good agreement with the results in Fig. 7. When the ceiling panel is installed, the peak appears at some frequencies above f c because the frequency of the sound wave agrees with the natural frequency of the floor-ceiling system with permeability. Figure 8 presents the effect of the ceiling panel from the results in Fig. 7. For f > f c, except the case when the flow resistivity is N s m 4, the insulation performance can be improved. In contrast, for f < f c, the insulation performance is deteriorated when the flow resistivity is above N s m 4. This is attributed to the air resonance and the resonance frequency can be written as f r = c 2π ρ m 1 + m 2, (39) d m 1 m 2 where m is the density per unit area. The resonance frequency from Eq. (39) is f r = 62.9Hz and agrees with the result in Fig. 8. Figure 9 shows the effect of the permeability. For f > f c, increasing the flow resistivity, the permeability is deteriorated, so that the deterioration of the insulation performance can be prevented. On the other hand, for f < f c, decreasing the flow resistivity, the insulation performance can be improved up to about 5dB. 4. Concluding Remarks In this study, the effects of permeability on sound absorption performance of acoustic ceiling panels are investigated with related parameters changing in the case of both introducing permeability and micro-perforating. The parameters of the former and the latter are the flow resistivity and the perforation ratio, respectively. Consider the case where the glass wool is installed between the ceiling panel and the rigid back wall. The results for each case show the same tendency; therefore the flow resistivity is effective in controlling the absorption characteristics. The effects of permeability on sound insulation performance are also discussed by using simulation models of a ceiling-floor system. When the ceiling panel is installed, the insulation performance is improved at frequencies except the resonance frequency. However, decreasing the flow resistivity, the insulation performance is deteriorated. Hence it is important to choose the value of the flow resistivity properly according to the situation. REFERENCES 1 Takahashi, D., Sakagami, K. and Morimoto, M. Acoustic properties of permeable membranes, Journal of the Acoustical Society of America, 99 (5), 33 39, (1996). 2 Takahashi, D. Sound transmission through single plates with absorptive facings, Journal of the Acoustical Society of America, 83 (4), , (1988). 3 Takahashi, D. and Tanaka, M. Flexural vibration of perforated plates and porous elastic materials under acoustic loading, Journal of the Acoustical Society of America, 112 (4), , (22). 4 Maa, D. Y. Microperforated-panel wideband absorbers, Noise Control Engineering Journal, 29 (3), 77 84, (1987). 5 Miki, Y. Acoustical properties of porous materials-modifications of Delany-Bazley models, Journal of the Acoustical Society of Japan, 11, 19 24(E), (199). 8 ICSV23, Athens (Greece), 1-14 July 216