SOUND TRANSMISSION LOSS OF A DOUBLE-LEAF PARTITION WITH MICRO-PERFORATED PLATE INSERTION UNDER DIFFUSE FIELD INCIDENCE

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1 International Journal o Automotive and Mechanical Engineering (IJAME) ISSN: (Print); ISSN: (Online); Volume 7, pp , January-June 2013 Universiti Malaysia Pahang DOI: SOUND TRANSMISSION LOSS OF A DOUBLE-LEAF PARTITION WITH MICRO-PERFORATED PLATE INSERTION UNDER DIFFUSE FIELD INCIDENCE A. Putra *, A. Y. Ismail and Md. R. Ayob Faculty o Mechanical Engineering Universiti Teknikal Malaysia Melaka (UTeM) Hang Tuah Jaya, 76100, Melaka, Malaysia azma.putra@utem.edu.my ABSTRACT In noise control applications, a double-lea partition has been applied widely as a lightweight structure or noise insulation, such as in car doors, train bodies, and aircrat uselages. Unortunately, the insulation perormance deteriorates signiicantly at massair-mass resonance due to coupling between the panels and the air in the gap. This paper investigates the eect o a micro-perorated panel (MPP), inserted in the conventional double-panel partition, on sound transmission loss at troublesome resonant requencies. It is ound that the transmission loss improves at this resonance i the MPP is located at a distance o less than hal that o the air gap. A mathematical model is derived or the diuse ield incidence o acoustic loading. Keywords: Transmission loss; double-panel; noise; MPP; acoustics. INTRODUCTION A double-lea structure is a common structural design in many engineering applications. Body structures o cars, trains, and airplanes, as well as the walls o buildings are some examples o double-lea partitions in practice. The double-lea has been ound to be a better noise barrier compared with the single-lea. However, there remains a problem with the double-panel, which is the weak sound transmission loss (STL) perormance at low requencies due to the mass-air-mass resonance. At this requency range, this causes the double-lea to lose its superiority over the single-lea structure (Fahy and Gardonio, 2006). Several works have been published on attempts to overcome this problem. This includes the application o absorptive materials inside the gap o a double-lea to increase damping (Tang et al., 1998, Bravo et al., 2002). This treatment was ound to be eective in increase STL at low requency, including that o the massair-mass resonance. Another work suggested installing Hemholtz resonators in the air gap (Mao and Piertzko, 2005). This resonator acts like a single-degree o reedom system, the natural requency o which depends on its geometry. It is ound that by optimally tuning the resonator, the STL at resonance is improved signiicantly. Some active noise control systems have also been proposed to solve the massair-mass resonance problem. Two dierent pieces o equipment, i.e., a loudspeaker and an actuator were placed inside the gap to control actively the sound transmission (Li and Cheng, 2008). The loudspeaker reduces the transmission energy that propagates through the acoustic path, and the actuator reduces energy rom the structural path by creating a counter orce on the two panels to suppress the vibration; this in turn reduces the radiated sound into the air. By varying the distance between both the loudspeaker and 1086

2 Sound transmission loss o a double-lea partition with micro-perorated plate insertion under diuse ield incidence the actuator to the panels, both strategies gave eective reduction o sound transmission. Similarly, a long T-shaped resonator was embedded along the edge o the double panel to control actively the acoustic path inside the gap (Li et al., 2010). This was also been ound eective in increasing STL at an even broader requency range. In 1975, Dah You Maa in China ound that when a thin panel with submillimeter-sized holes was introduced and located in ront o a rigid panel with an air gap, the micro-perorated panel (MPP) acts as a sound absorber ollowing a Helmholtz resonator mechanism. The diameter o the holes must be within the range mm and the peroration ratio must be between 0.5% and 1.5% or optimum absorption (Maa, 1975). As the MPP is made rom a panel, it provides several advantages, such as being non-ibrous, non-abrasive, and non-polluting. Several works have been conducted concerning the improvement o sound absorption perormance o the MPP, e.g., by arranging it to be a double-lea (Sakagami et al., 2006), by increasing its thickness (Pretzner and Cobo, 2006), and by modiying its hole geometry (Sakagami et al., 2008). This paper proposes the insertion o an MPP within a double-panel partition. The proposed system might be important or a condition where any abrasive and toxic material, such as commercial glass wool and glass iber cannot be presented or health reasons. The next section describes the derivation o the mathematical model and the simulation results that show the eect o the MPP insertion, in terms o the location o the MPP in the gap, as well as its hole size and peroration ratio on the perormance o the sound transmission loss, particularly at the mass-air-mass resonance. Propagating Acoustic Pressure GOVERNING EQUATIONS Figure 1 shows a mechanical system consisting o double solid panels with an MPP inserted in between (abbreviated here as DL-MPP), which is excited by an oblique incidence o sound with an arbitrary angle φ. The solid panels are separated by distance D and the MPP is located at distance l rom the back solid plate. Each o the unbounded solid and MPP panels has mass per unit area M and m, respectively. The component o the incident wavenumber vector directed parallel to the panel is k z = ksinφ. As the panel is uniorm and ininite, the lexural or bending wave induced in the panel must also have the same wavenumber k z. The incidence and relected pressures upon the panel are given by: jkx cos jkz sin p x 0, z Ae (1) p i 1 jkx cos jkz sin x 0, z B e r (2) where k = ω/c and k represents the acoustic wavenumber, ω is the angular velocity, and c is the sound speed. Here and or the rest o the equations, time dependence e jωt is assumed implicitly. At x = 0, the acoustic pressure acting on the incident side o the ront panel can be written as: 1 p 1 = p i (x = 0) + p r (x = 0) = A 1 + B 1 (3) Similarly to Eqs. (1) and (2), the total pressure on the other side o the ront panel surace p 2 can be obtained. Note that or the analysis o propagating pressure in the x-direction, the term e jkzsinφ is a constant and this is implicit in Eq. (3) as well as or the remaining equations in this paper. The relation between the average surace particle 1087

3 Putra et al. /International Journal o Automotive and Mechanical Engineering 7(2013) velocity v and the sound pressure exciting the panel can be obtained by using the Euler equation v = 1/jρω(dp/dx). For both suraces o each panel, at x = 0 or the ront panel, at x = D l or the MPP, and at x = D or the back panel, this gives, respectively: z B 1 A 2 B 2 A 3 B 3 p t A 1 p 1 p 2 p 3 p 4 p 5 x v p1 v h v p3 v p2 x = 0 D l Figure 1. Schematic diagram o a DL-MPP system. z v p1 = (A 1,2 B 1,2 ) cos φ (4) z v m = (A 2,3 e jk(d l) cos φ B 2,3 e jk(d l) cos φ ) cos φ (5) z v p3 = (A 3 e jkdcos φ B 2 e jkdcos φ )cos φ (6) and p t = z v p3 cos φ (7) where v p is the velocity o the panel, v m is the mean particle velocity over the MPP surace, and z = ρc is the acoustic impedance o air, where ρ is the air density. Note that or the solid plate, the mean particle velocity on its surace is equal to the velocity o the panel v = v p. This is valid or light luid such as air, but not or heavy media such as water. For convenience, the distance between the panel is assumed much smaller compared with the acoustic wavelength (kd << 1). The cavity pressures can thereore be assumed uniorm between each gap: p 2 p 3 = A 2 + B 2 = p b (8) p4 p 5 = A 3 + B 3 = p c (9) 1088

4 Sound transmission loss o a double-lea partition with micro-perorated plate insertion under diuse ield incidence Eq. (5) can be expanded to: z v m = (A 2,3 [cos k(d l) jsin k(d l)] B 2,3 [cos k(d l) + jsin k(d l)]) cos φ (10) By using Eq. (8), Eq. (10) can be rewritten as: z v m = A 2 B 2 jk(d l) cos φ p b (11) z v m = A 3 B 3 jk(d l) cos φ p c (12) As the cavity pressure is uniorm, the pressure between each gap can thus be expressed in terms o the ront panel velocity: p b = z (v p1 v m )/jk(d l) cos φ (13) p c = z (v m v p1 )/ jkl cos φ (14) Hole Impedance and Mean Particle Velocity As the acoustic pressure impinges on the MPP, the air particles penetrate the holes and excite the remaining solid surace o the panel. The combination between the panel velocity and particle velocity inside the holes creates the mean particle velocity given by Takahashi and Tanaka (2002): v m = v p (1 σ) + σv h (15) where v h is the particle velocity inside the hole and σ is the peroration ratio. The air inside the holes moves like a moving piston owing to its inertial property. At the same time, the air also interacts with the inner surace o the holes creating a riction orce. These mechanisms are represented by the hole impedance. According to Maa (1975), this is given by: with Z o = Z o,r + Z o,i (16) Z o,r = 32υ a t[(1+x o 2 /32) 1/2 + (1.71X o /8)d o /t] /d o 2 (17) Z o,i = jωρt [1+(9 + X o 2 /2) -1/ d o /t] (18) where X o = (d o /2) (ωρ/υ a ) 1/2, d o is the hole diameter, t is the MPP thickness, and υ a is the viscosity o air, i.e., Ns/m 2. The resistive or real part o the impedance Z o,r represents the viscous eects, i.e., the interaction o the luid particle with the wall inside the holes, while the reactive or imaginary part Z o,i represents the acoustic reactance rom the inertia o the air. The net pressure on the surace o the panel is thereore expressed as: 1089

5 Putra et al. /International Journal o Automotive and Mechanical Engineering 7(2013) Z o,r (v h v p ) + Z o,i v h = Δp (19) Substituting Eq. (15) into Eq. (19) gives: v m = γv p + σδp/z o (20) where γ = 1 (σz o,i /Z o ) is the complex non-dimensional term. Sound Transmission Loss The equation o motion or the solid back panel is given by: z p3 v p3 = p c p t (21) where z p3 = j(gk4sin 4 φ Mω 2 )/ω is the bending mechanical impedance with G = Et 3 /12(1 ν), which is the bending stiness o the panel, E is the Young s modulus, and ν is the Poisson s ratio. Substituting both p t in Eq. (7) and p c in Eq. (14) into Eq. (21) and then dividing both sides by v p3 yields the panel velocity ratio: v p2 /v p3 = [1 + jkl cos φ(1 +z p3 cos φ/z )]/(γ +z p2 /Z) (22) The equation o motion or the MPP is expressed as: z p2 v p2 = Δp (23) where z p2 = j(gk4sin4φ mω 2 )/ω. Substituting Eqs. (13), (14) and (20) into Eq. (23) and again dividing both sides by v p3 yields: v p1 /v p3 = (v p2 /v p3 ) (jk(d l)z p2 z ) + [1 + jkd(1 +z p3 z )] (24) It can be seen that the velocity ratio o the solid panels v p1 /v p3 depends on the location o the MPP within the air gap. The equation o motion or the ront solid panel (similar to Eqs. (21) and (23)) is expressed as: z p1 v p1 = p 1 p b (25) By substituting Eqs. (3), (13) and (14) into Eq. (25), the ratio o incident to transmitted pressure is given as: p i /p t = ((v p1 /v p3 )[1 + jk(d l) cos φ(1 +z p1 /z )] (v p2 /v p3 ) (γ +z p2 /Z)) cos φ /j2k(d l) (26) The case or normal incidence o acoustic loading can be obtained by setting φ = 0 and the mechanical impedance o the panels can be assumed to be dominated by the mass, i.e., z p1 = z p3 = jmω and z p2 = jmω. 1090

6 Sound transmission loss o a double-lea partition with micro-perorated plate insertion under diuse ield incidence As or plane waves, the sound power W is proportional to the sound intensity I, which is simply a ratio o the squared magnitude sound pressure to the air impedance, i.e., I = p 2 /z. The transmission coeicient is thereore given by τ = p t /p i 2. For diuse ields, this can be governed by integrating over angles o incidence, which is given by (Fahy and Gardonio, 2007): t d = The transmission loss in db unit is: p /2 ò t (j)sin2j dj (27) 0 STL = 10log 10 (1/τ) (28) ANALYTICAL RESULTS Figure 2 shows the sound transmission loss or the case o normal incidence or partitions consisting o DL [1], triple-lea (TL), and DLMPP or an MPP located exactly in the middle o the two panels (l = 0.5D) or an air gap D = 100 mm. The mass-airmass resonance o the DL can be seen to occur around 170 Hz, where the STL is nearly 0 db. The presence o the MPP in this case does not aect this resonance because this corresponds to the gap o the double solid plates. By inserting another solid panel between the double-panel (TL), the second resonance appears at 280 Hz, which is due to the gap between the middle and the back panel. This worsens the problem, although the STL at mid-high requency increases signiicantly owing to the increase o mass within the system. Figure 2. Sound transmission loss o DL ( ), TL ( ) and DLMPP ( d o = 0.1 mm, σ = 1.5%) subjected to normal incidence o acoustic loading. 1091

7 Putra et al. /International Journal o Automotive and Mechanical Engineering 7(2013) As the objective is to improve the STL o the conventional double-lea partition (DL) at the mass-air-mass resonance, Figure 3 shows the STL or the DL and DLMPP or dierent locations o the MPP relative to the solid plate. This is plotted or the case o diuse ield incidence, which can be seen to yield a broader requency range o resonance eects due to the summation o resonance dips at various angles o incidences. This shows that or other locations o the MPP, as the MPP shits closer to the panel either at the ront or to the back solid panel, the STL can be observed to increase at the resonance. The additional damping, due to the viscous orce in the MPP holes, inluences the air layer in ront o the solid plate, which breaks the coupling between the solid panels and the air. Figure 3. Sound transmission loss o DL ( ) and DLMPP or dierent locations o MPP in the gap subjected to diuse ield incidence o acoustic loading (d o = 0.1 mm, σ = 1.5%). Figure 4 shows the eect o hole diameter o the MPP on the STL or a ixed MPP location at l = 0.1D. Around the resonance region, decreasing the hole diameter improves the STL, as this increases the domination o the real part o the hole impedance, which thus, provides more viscous orce or damping to the MPP. In Figure 5, the eect o the peroration ratio is investigated. It shows that increasing the peroration ratio does not give signiicant dierences to the STL around the resonance. Thereore, to beneit rom STL improvement at high requency due to added mass within the system, the lowest peroration or the MPP, i.e., σ = 0.5% is preerred. This could also save costs or the perorated panel i the smallest hole diameter is used (d o = 0.1 mm). 1092

8 Sound transmission loss o a double-lea partition with micro-perorated plate insertion under diuse ield incidence Figure 4. Sound transmission loss o DLMPP or dierent hole diameters (σ = 0.5%) subjected to diuse ield incidence (l = 0.1D). Figure 5. Sound transmission loss o DLMPP or dierent peroration ratios (d o = 0.1 mm) subjected to diuse ield incidence (l = 0.1D). 1093

9 Putra et al. /International Journal o Automotive and Mechanical Engineering 7(2013) For clarity o the analysis, the level o improvement o the TL (around the resonance) can be represented by the insertion loss, i.e., the ratio o the transmitted sound power beore and ater the MPP insertion, which is given by: æ F =10logç è t DL t DLMPP ö = STL DLMPP -STL DL (29) ø where STL DLMPP is the transmission loss o the DL-MPP and STL DL is or the doublelea. (a) (b) (c) Figure 6. Insertion loss o DLMPP system with dierent MPP parameters: (a) locations in the gap (d o = 0.1 mm, σ = 1.5%, l = 0.1D, - -l = 0.25D, l = 0.5D), (b) hole diameters (l = 0.1D, σ = 1.5%, d o = 0.1 mm, - -d o = 0.2 mm, d o = 0.4 mm), and (c) peroration ratio (l = 0.1D, d o = 0.1 mm, σ = 0.5%, - -σ = 1%, σ = 1.5%). The insertion loss o the DLMPP system is presented in Figure 6. These are the results rom Figures 3, 4 and 5. The improvement or DLMPP with dierent locations o the MPP (Figure 6(a)) can be seen to have a wider band o requency rom 200 Hz to 1 khz, reaching almost 5 db or l = 0.1D. The insertion loss or hole diameter variation is presented in Figure 6(b). It can be seen that a smaller diameter is still best or good insertion loss. Correspondingly, Figure 6(c) shows that a small peroration ratio is preerred, because varying the peroration ratio (with ixed hole diameter) gives almost no eect to the insertion loss. 1094

10 Sound transmission loss o a double-lea partition with micro-perorated plate insertion under diuse ield incidence CONCLUSIONS Models o sound transmission loss o a DLMPP system under diuse ield incidences o acoustic loading have been developed. It is ound that the MPP insertion reduces the eect o resonance ound in the conventional double-lea partition. However, this is only eective when the MPP distance is less than hal o the air gap and it improves as it approaches the solid plate. Reducing the size o the hole improves the STL at resonance, while changing the peroration ratio gives almost no eect on the STL perormance at resonance. The model can be used or designing a multi-layer partition where problems o mass-air-mass resonance might be experienced due to a predominantly low requency noise source. ACKNOWLEDGMENTS The authors grateully acknowledge the inancial support or this project by The Ministry o Higher Education Malaysia (MoHE) under Fundamental Research Grant Scheme, FRGS/2010/FKM/TK02/3-F0078. REFERENCES Bravo, J.M., Sinisterra, J., Uris, A., Llinaresand, L. and Estelles, H Inluence o air layers and damping layers between gypsum boards on sound transmission. Applied Acoustics, 63: Fahy, F.J. and Gardonio, P Sound and structural vibration: radiation, transmission and response. London: Academic Press, 2nd edition. Li, D., Zhang, X., Cheng, L. and Yu, G Eectiveness o T-shaped acoustic resonators in low-requency sound transmission control o a inite double-panel partition. Journal o Sound and Vibration, 329: Li, Y.Y. and Cheng, L Mechanism o active control o sound transmission through a linked double wall system into an acoustic cavity. Applied Acoustics, 69: Maa, D.Y Theory and design o microperorated panel sound absorbing constructions (in Chinese). Scientia Sinica, 18: Mao, Q. and Pietrzko, S Control o sound transmission through double wall partition using optimally tuned Hemholtz resonators. Applied Acoustics, 91: Pretzner, J. and Cobo, P Microperorated insertion units: An alternative strategy to design microperorated panels. Applied Acoustics, 67: Sakagami, K., Morimoto, M. and Koike, W A numerical study o double-lea microperorated panel absorbers. Applied Acoustics, 67: Sakagami, K., Morimoto, M., Yairi, M. and Minemura, A A pilot study on improving the absorptivity o a thick microperorated panel absorbers. Applied Acoustics, 60: Takahashi, D. and Tanaka, M Flexural vibration o perorated plates and porous elastic materials under acoustic loading. Journal o The Acoustical Society o America, 112: Tang, W.C., Cheng, H. and Ng, C.F Low requency sound transmission through close-itting inite sandwich panels. Applied Acoustics, 55: