FDTD analysis on the sound insulation performance of wall system with narrow gaps

Transcription

1 FDTD analysis on the sound insulation performance of wall system with narrow gaps Takumi Asakura a Shinichi Sakamoto b Institute of Industrial Science, The University of Tokyo. Komaba 4-6-, Meguro-ku, Tokyo, , Japan ABSTRACT The sound insulation performance of such wall systems as windows and doors is often affected by sound wave propagating through narrow gaps which exist in their peripheral parts. In this paper, the sound insulation performance of the wall system with narrow gaps was investigated through numerical and experimental studies. Firstly, narrow gaps with typical cross sections were numerically modeled and their sound transmission characteristics were calculated by applying FDTD (Finite-Difference Time-Domain) method using irregular mesh system. To examine the calculation results, sound transmission characteristics of sash parts of a real glass sliding window was measured by sound intensity method and a good agreement was found between the numerical and experimental results. As a parametrical study, sound insulation performance of wall system composed of a wall body such as glass plates, and narrow gaps around the plate were calculated by vibro-acoustical analysis using FDTD method and the acoustical influence of leak transmission of sound through narrow gaps was discussed.. INTRODUCTION Sound environment inside residential buildings is influenced by transmitting sound through wall systems. The sound insulation performance of wall systems such as windows and doors which are installed in the building façade or in the interior wall is often affected by sound wave propagating through narrow gaps which exist in their peripheral parts. The influence of the acoustical transmission through the gaps or apertures has been theoretically investigated []. More practical case studies on sound insulation performance of such real building elements with gaps as doors and windows have been also investigated through experimental investigation [, 3]. On the other hand, research for prediction method of the sound transmission characteristics through gaps are conducted only for a gap or an aperture with a simple cross section such as straight types. In this paper, therefore, to predict the sound insulation performance of the wall system with narrow gaps, numerical and experimental study was investigated. a address: asakura@iis.u-tokyo.ac.jp b address: sakamo@iis.u-tokyo.ac.jp

2 . FDTD ANALYSIS WITH IRREGULAR MESH SYSTEM A. Theory In this paper, FDTD method with irregular mesh system was applied in order to calculate the sound propagation through narrow gaps as shown in Figure. A sound wave in - dimensional sound field is expressed as following equations. p u + ρc = 0 t x p u + ρ = 0 x t where, p is sound pressure, u is particle velocity, ρ is the density of the air and c is the speed of sound. In FDTD analysis, these equations are transformed into a discrete form in staggered grid system. The time and space differential terms in these equations are approximated by a center difference method and expressed as following equations. n n u u n n i+ + i pi = pi ρc t 3 x n t ( pi+ pi ) u = u 4 i+ i+ ρ x where, n and i denote the time step and the coordinate of the grid in x-direction, t and x are the time and spatial interval. In FDTD analysis using regular mesh system, p and u are arranged at regular intervals as shown in Figure (a) and u i+/ is calculated using p i and p i+ which are arranged at the same distance from u i+/. Figure (b) shows -dimensional sound field discretized using irregular mesh system. In this figure, the ratio of the mesh size between coarse and fine mesh is 3:. The particle velocity, u i+/, is obtained using following equation (we call this method ordinary method ). n + n t n + n + u = u pi + p i + i ρ x + x i 5 ( ) ( ) coarse fine In this method, sound reflection from the boundary point between coarse mesh area and fine mesh area occurs because the center difference is not adopted at the boundary. In order to keep the center difference method at the boundary point, the particle velocity, u i+/, at the boundary is calculated using p i and p i+ as following equation ( center difference method ). n t ( u = u pi+ pi ) 6 i+ i+ ρ x coarse This method is applicable for the situation that the mesh ratio is : (odd number). The stability condition for -dimensional sound field is expressed as following equation. c t x 7 Considering the stability condition for the calculation in fine mesh area, shorter time interval than that for the coarse mesh area is necessary and it leads to increase of the total calculation time. In order to reduce the calculation time, locally shorter time interval for the fine mesh area than that for the coarse mesh area was adopted. The sound pressure and particle velocity indicated by red characters in Figure 3 are renewed for n times while those in the coarse mesh area are renewed for one time. In case of -dimensional sound field analysis, the iteration number, n, for the calculation of fine mesh area is determined as following equation. xcoarse n 8 x fine

3 Particle velocity Sound pressure Fine mesh Coarse mesh Figure -dimensional sound field discretized by irregular mesh system. (a) Regular mesh p i u i + p i + x (b) Irregular mesh p i u i + p i + 3 x fine p i + p i + 3 x fine Sound pressure Particle velocity Figure Arrangement of sound pressure and particle velocity in regular or irregular mesh system. x u i 3 u i u i+ u i+3 u i+5 u i+7 u i+9 p i p i p i+ p i+ p i+ 3 p i+ 4 x Coarse mesh area x fine Fine mesh area Figure 3 -dimensional sound field analysis with iterative calculation for fine mesh area. B. Validation of the FDTD method with irregular mesh system In order to investigate the sound reflection from the boundary between different-sized meshes, -dimensional and -dimensional FDTD analysis were performed. The conditions and their symbols are shown in Table. The ratio of the mesh size between fine and coarse meshes were varied from :3 to : and the iteration number for the calculation of the fine mesh area was varied as 5 times and 0 times in addition to the condition with no iteration. The size of the coarse mesh was cm or 4 cm. In -dimensional sound field as shown in Figure 4, a sound source point and a receiving point were set and impulse response between the points was calculated. An example of the calculated results of -dimensional sound field analysis (The size of the coarse mesh is 4 cm, the ratio is 3: and with no iteration) is shown in Figure 5 (a). The reflection sound wave from the boundary is not seen in the figure. In Figure 5 (b), the same impulse response is shown with the range of the vertical axis magnified to 0.00 [pa]. The direct and reflection sounds are cut from the impulse response and the reflection error was calculated as the sound level difference between these cut data. The reflection error was also investigated by

4 FDTD analysis on -dimensional sound field as shown in Figure 6. The impulse responses between the source points and the 5 receiving points were calculated and the reflection error was calculated by the same method as that described above. In order to correct the difference of the sound attenuation in distance between the direct and reflection sound, correction value, L correct, was calculated as following equation and added to the reflection error. r + r3 L correct = 0 log 9 r Table Variations of the numerical parameters under investigation. Size of the coarse mesh Treatment of the boundary 4cm Center difference method cm 4cm ordinary method Sound field -dimensional FDTD -dimensional FDTD Receiving point P P P3 P4 P5 Iterative time No iteration 5 times 0 times No iteration :3 Type A- Type A- Type A-3 Type A-4 Type A-5 Type B- Type D- Type D- Type D- Type C- :5 Type B- Type D- Type C- Ratio :7 Type B-3 Type D-3 Type C-3 :9 Type B-4 Type D-4 Type C-4 : Type B-5 Type D-5 Type C-5 : Type B-6 Type D-3 Type D-4 Type D-6 Type C-6 Receiving point x fine x.8 m 6. m Coarse mesh area Fine mesh area Figure 4 -dimensional sound field discretized with irregular mesh system Direct sound Reflection sound Sound pressure [Pa] Sound pressure [Pa] (a) Time [ms] Direct sound Time [ms] Figure 5 Example of the calculated impulse response. (b) r P r 3 r Receiving points m P P P3 P4 P5 m m m m m Fine mesh area 40.0 mm 40.0 mm 3.3 mm Coarse mesh area Coarse mesh area Fine mesh area PML (Sound absorption layer) Figure 6 -dimensional sound field discretized with irregular mesh system.

5 Result The calculated reflection errors are shown in Figure 7. Figure 7 (a) shows the effect of the ratio of the mesh sizes between coarse and fine mesh area. In the condition from Type B- to Type B-6, all the results have almost the same tendency and the conditions from Type C- to Type C-6 have higher error at all frequency. In the condition from Type D- to Type D-6, all the results have less error by about 8 db at all frequency. It shows that the center difference method is effective for reducing the reflection error due to the discontinuity of the mesh size. Figure 7 (b) shows the effect of the iterative calculation in the fine mesh area. In comparison between Type B-, B-6 and Type D-3, D-4, Type D-3 and D-4 have a little higher error in the low frequency. In the condition Type D- and D-, higher error is seen in wide frequency range compared to Type B- and B-6. It shows that the reflection error is not affected by increasing the iterative time and that the reflection error is relatively low when the ratio of the mesh size is higher. Figure 7 (c) shows the difference of the reflection error among 5 receiving points. The result of Type A-5 agrees well with that of Type B-. As the receiving points are farther from the source point, the reflection error decreases in all frequency range. It suggests that the reflection error in case of oblique incidence is lower than that in case of normal incidence. 0 0 Reflection error [db] Type C-, C-, C-3, C-4, C-5, C-6 Type D-, D- Type B-, B-, B-3, B-4, B-5, B Type D-3, D-4 Type D-, D-, D-3, D-4, D-5, D-6-0 Type B-, B (a) Effect of the ratio of the mesh size (b) Effect of the iteration time Reflection error [db] 000 Reflection error [db] Type A-5, Type B- Type A-4 Type A-3 Type A- Type A- (c) Difference between 5 receiving points Figure 7 Calculation results of the reflection errors. 3. SOUND PROPAGATION THROUGH NARROW GAPS A. Measurement A window sash shown in Figure 8 is set in the separation wall between a reverberation room and an anechoic room and the transmitting sound power level, L, from the upper or lower W

6 sash parts, Area or Area, was measured by the scanning intensity method referring to ISO 586- [4]. In order to compare the calculated results with measured ones, correction values, L, which were determined under the assumption that the averaged sound pressure level in the correct receiving room is 00 db in each frequency band, was added to the sound power level as following equation. L W, correct = LW + Lcorrect 0 L correct = 00 L where, L is the averaged sound pressure level in the source room in /3 Oct. band. A cross section of the window sash is also shown in Figure 7 (b). Inside the sash parts, Area and Area, air volumes are generated as shown in the magnified figures. In the next chapter, the cross section of these air volumes are modeled in 3-dimensional FDTD analysis and the sound transmission characteristics through these gaps were calculated. (a) Elevation (b) Cross section Area 70 Area Unit: mm Figure 8 Elevation and cross section of a window sash under investigation. B. Calculation Sound transmission characteristics through narrow gaps were calculated by 3-dimensional FDTD method. A pair of irregular-shaped reverberation rooms and narrow gaps inside the separation wall between two rooms were modeled as shown in Figure 9 (a). The cross section of the narrow gaps are shown in Figure 9 (b). The width of the gap, t [mm], which is generated by rubber seals was varied as 0. mm and 0.4 mm. A sound source is set at the corner of the source room. In the source and receiving rooms, 490 receiving points were set. The separating distances between adjacent receiving points were 0.4 m and all receiving points were distributed at least.0 m away from the boundary surface and the sound source. The averaged sound energy levels in both rooms, L and L, were measured in /3 Oct. band. Then, the corrected sound energy level, L J,, propagating through the gap was estimated assuming that the averaged sound correct pressure level in the receiving room is 00 db in each frequency band. A0 L J, correct = L 6 0log0 + Lcorrect A L correct = 00 L 3 where, A 0 is m in this study, A is the equivalent sound absorption area ( A = 55. 3V ct ), V and T are the volume of the receiving room (64 m 3 ) and the reverberation time of the receiving room, respectively. The reverberation time in each /3 Oct. band was calculated by the integrated impulse response method. Area Area Area Area

7 (a) A pair of reverberation rooms Narrow gap (Type or Type) Unit: mm (b) Cross section of the modeled gaps Type Cross section Type Cross section t t 6.5 Figure 9 A pair of reverberation rooms and modeled gaps. 6 Unit: mm C. Result Figure 0 shows the comparison between measurement and calculation results. In the condition of Type (Area ), measurement and calculation results (t=0. mm) show good agreement. The sound insulation defect in 800 Hz band (indicated by arrow) is caused by Helmholtz resonance of the air volume inside the sash. In the calculation result, Type with the gap clearance, t=0.4 mm, the sound energy level increases in all frequency bands compared with Type with, t=0. mm, by about 0 db. In the condition of Type (Area ), the measurement result has peak in 630 Hz bands. While the peak is also seen in 500 Hz band in calculation results, the peak frequencies don t agree with the measurement result. The reason for this disagreement is that the complicated cross section of the wheels set inside the sash to carry the window sash construction is not modeled in the calculation. Sound energy/power level [db] Type (Area ) t = 0. mm t = 0.4 mm Sound energy/power level [db] Type (Area ) t = 0.4 mm t = 0. mm Measurement Calculation k k 4k Figure 0 Measurement and calculation results of the sound energy/power level.

8 4. SOUND PROPAGATON THROUGH NARROW GAPS AND PLATES In this chapter, sound transmission loss of a wall system composed of a sheet of glass pane and a narrow gap was calculated by vibro-acoustical analysis using FDTD method with irregular mesh system [5]. Figure shows two pairs of irregular-shaped reverberation rooms under investigation. In the separation wall, a sheet of glass pane (6mm thickness) or narrow gaps (Type and Type shown in Figure 9) were set. The averaged sound energy level in both of the source and receiving rooms were calculated by the same method described in chapter, 3-B. Sound transmission loss, TL, was calculated by following equations. S TL = L, plate L, plate+ gap + 0log0 (4) A L, plate 0 L, gap 0 0( 0 0 ) L, plate+ gap = 0 log + (5) where, S is the area of the glass pane, L, plate is the averaged sound energy level in the source room (Figure (a)), L, plate and L, gap are the averaged sound energy levels in the receiving rooms (Figure (a) and (b)) and the same value as A used in chapter, 3-B, was used. In this study, transmission loss of wall system, Case (glass pane), Case (glass pane and gap (t=0.mm)) and Case 3 (glass pane and gap (t=0.4mm)) was calculated. Figure shows the measurement and calculation results. As to transmission loss of Case, measurement data (quoted from the database by Flat Glass Manufacturers Association of Japan, in Japanese) and calculation result show good agreement. The calculation results of Case and Case 3 show lower value than that of Case between 400 Hz and.6k Hz band and dips are seen in 500 and 800 Hz. These dips are caused by Helmholtz resonance of the air volumes inside the gaps. Comparing Case to Case, transmission loss decreases by about 4 db in the middle frequency range. (a) Transmission through glass pane (b) Transmission through narrow gap Glass pane:.6 m (W).5 m (H) H W Unit: m Narrow gap (Type or Type) Figure Pairs of reverberation rooms with glass pane or narrow gap in the separation wall. 6. CONCLUDIONS Sound transmission characteristics through narrow gaps with complicated cross sections were investigated by numerical analysis and laboratory measurement. The narrow gaps with typical cross sections were modeled and their sound transmission characteristics were calculated by FDTD method using irregular mesh system. The sound transmission characteristics through these gaps were also measured by scanning intensity method and the results showed good agreement with the calculation results. As a case study, to predict sound insulation performance of wall system composed of a glass pane and a narrow gap, vibro-acoustical analysis using FDTD method was applied and the acoustical influence of leak transmission through gaps was discussed. The sound insulation performance of the wall system in the middle frequency range

9 was deteriorated by the sound energy transmitting through the gaps and the composite transmission loss was also deteriorated with the clearance width of the gap is wider. ACKNOLEDGMENT This research was financially supported by Grant-in-aid for JSPS (Japan Society for the Promotion of Science) Fellows. REFERENCES M. C. Gomperts, The sound insulation of circular and slit-shaped aperture, Acustica 4 (), -6 (964) P. T. Lewis, Effect of frame construction on the sound insulation of unsealed windows, Applied Acoustics, 5-4 (979). 3 M. A. Burgess, Resonator effects in window frames, Journal of Sound and Vibration 03(3), (985). 4 ISO 586-:000, Acoustics-Measurement of sound insulation in buildings and of building elements using sound intensity-, Part : In-situ condition. 5 T. Asakura, K. Ito and S. Sakamoto, Auralization technique for assessing the sound insulation performance of building façade., Proceedings of INTER-NOISE 008, (008) TL [db] Calculation (Case, glass pane) Calculation (Case, glass+gap(t=0.mm)) Calculation (Case 3, glass+gap(t=0.4mm)) Measurement (glass) k k 4k Figure Measurement and calculation results of transmission loss.