# Vibroacoustic analysis of a clamped finite orthotropic laminated double-composite plate with an air cavity

Tóm tắt Vibroacoustic analysis of a clamped finite orthotropic laminated double-composite plate with an air cavity: ...c analysis of a clamped finite orthotropic laminated double-composite plate with an air cavity 263 ψmn = ω ( A2,mn − A1,mne−jkz H ) kz ( ejkz(h−H) − ejkz(H+h)) ; γmn = ωA2,mnejkz(H+2h)kz . (26) Substituting Eqs. (12) and (25)–(26) into Eqs. (1)–(2) and applying the orthogonal properties of...hickness. In other words, in all the frequency ranges, the double-composite plates display superior sound insulation than the sing plate. Fig. 3. STL of a finite aluminum and an orthotropic double-composite plates with incident angle ϕ = 30◦ and azimuth angle θ = 0◦ As can be seen in Fig. 3 ...= 30◦). For frequencies below about 140 Hz, the STL curve varies v ry little for four cases. At high frequencies, the change of the four curves is more obvious. Therefore, it may be concluded that the incident azimuth angle has negligible influ nce on the STL behavio of clamped finite do ble-plat...

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fluence of the incident elevation angle ϕ (with azimuth angle fixed at θ = 0◦) on the STL of the clamped finite double-plate: the incident sound waves with large elevation angles are easier to transmit through the double-plate than those with smaller elevation angles. For the case studied, the STL values decrease with increasing elevation angle for frequencies below about 90 Hz, while for frequencies above this value the overall trend is similar apart from the complicated system modal behavior. The sound insulation properties of the clamped finite double-plate partition for se- lected incident azimuth angles (θ = 15◦, 30◦, 45◦, 60◦) are compared in Fig. 6, (with the 268 Tran Ich Thinh, Pham Ngoc Thanh Vibroacoutic analysis of a finite clamped orthotropic laminated double-composite plate with an air cavity 9 Fig. 4. Comparison of STL across an orthotropic composite single-plate and double-composite plate with incident angle φ = 30o and azimuth angle θ = 0o. In addition, by neglecting the appearance of air cavity (H = 0), we observed that the curve of STL of double-composite plate is almost coincided with that of single composite plate for all range of frequency considered. Once again, that affirmed the reliability of our results for STL across a clamped double-composite plate with an air cavity. 4.4. Influence of several parameter on sound transmission loss In this section, the effects of incident angles, plate thickness and the air cavity thickness on STL of clamped double-composite plate are discussed. Influence of incident angles The effect of sound incident angles (elevation angle and azimuth angle) on the sound insulation behavior of a finite clamped orthotropic laminated composite double-plate is considered. The dimensions of the plates are chosen as follows: length of plate a = 1m, width of plate b = 1m. The plate have thickness h = 0.005 m, the thickness of the air cavity H = 0.08 m. The double-plate consists of two identical orthotropic laminated composite faceplates. Laminate configuration of bottom and upper plate is [0/90/0/90]s. The mechanical properties of composite material, the air speed of sound, the air density and the initial amplitude of the incident sound are presented in the above sections. Fig. 5. Influence of incident angle on STL of clamped composite double-plate excited by incident sound having different elevation angles φ (azimuth angle fixed at θ=0o). Fig. 5. Influence of incident angle on STL of clamped composite double-plate excited by incident sound having different elevation angles ϕ (azimuth angle fixed at θ = 0◦) 10 Tran Ich Thinh and Pham Ngoc Thanh Fig. 5 shows considerable influence of the incident elevation angle φ (with azimuth angle fixed at θ=0°) on the STL of the clamped finite double-plate: the incident sound waves with large elevation angles are easier to transmit through the double-plate than those with smaller elevation angles. For the case studied, the STL values decrease with increasing elevation angle for frequencies below about 90 Hz, while for frequencies above this value the overall trend is similar apart from the complicated system moda behavior. Fig. 6. Influence of azimuth angle on STL of clamped double-plate excited by incident sound having different azimuth angles θ (elevation angle fixed at φ=30o). The sound insulation properties of the clamped finite double-plate partition for selected incident azimuth angles (θ =15°, 30°, 45°, 60°) are compared in Fig. 6, (with the elevation angle fixed at φ=30°). For frequencies below about 140 Hz, the STL curve varies very little for four cases. At high frequencies, the change of the four curves is more obvious. Therefore, it may be concluded that the incident azimuth angle has negligible influence on the STL behavior of clamped finite double-plate. Influence of faceplate thickness (h) To quantify the influence of faceplate thickness, the STL versus frequency curve is presented in Fig. 7 for a finite double-plate. Three values of plate thickness are chosen: h = 0.002, 0.005 and 0.01 m. The dimensions of the plate are: a x b = 1m x 1m and the thickness of the air cavity is fixed at H = 0.08 m. The material properties are the same as in the subsection 3.1. Fig. 7. Effects of plate thickness on STL of clamped double-composite plate with enclosed air cavity. Three different plate thicknesses (h = 0.002, 0.005 and 0.01 m) were considered. Fig. 6. Influence of azimuth angle on STL of clamped double-plate excited by incident sound having different azimuth angles θ (elevation angle fixed at ϕ = 30◦) elevati n angle fixed at ϕ = 30◦). For frequencies below about 140 Hz, the STL curve varies v ry little for four cases. At high frequencies, the change of the four curves is more obvious. Therefore, it may be concluded that the incident azimuth angle has negligible influ nce on the STL behavio of clamped finite do ble-plate. Influence of faceplate thickness (h) To quantify the influence of faceplate thickness, the STL versus frequency curve is presented in Fig. 7 for a finite double-plate. Three values of plate thickness are chosen: h = 0.002, 0.005 and 0.01 m. The dimensions of the plate are: a× b = 1 m × 1 m and the thickness of the air cavity is fixed at H = 0.08 m. The material properties are the same as in the subsection 3.1. According to Fig. 7, the STL value increases sharply when increasing the thickness of the plate. Effect of thickness of faceplate for STL is particularly strong at frequencies Vibroacoustic analysis of a clamped finite orthotropic laminated double-composite plate with an air cavity 269 10 Tran Ich Thinh and Pham Ngoc Thanh Fig. 5 shows considerable influence of the incident elevation angle φ (with azimuth angle fixed at θ=0°) on the STL of the clamped finite double-plate: the incident sound waves with large elevation angles are easier to transmit through the double-plate than those with smaller elevation angles. For the case studied, the STL values decrease with increasing elevation angle for frequencies below about 90 Hz, while for frequencies above this value the overall trend is similar apart from the complicated system modal behavior. Fig. 6. Influence of azimuth angle on STL of clamped double-plate excited by incident sound having different azimuth angles θ (elevation angle fixed at φ=30o). The sound insulation properties of the clamped finite double-plate partition for selected incident azimuth angles (θ =15°, 30°, 45°, 60°) are compared in Fig. 6, (with the elevation angle fixed at φ=30°). For frequencies below about 140 Hz, the STL curve varies very little for four cases. At high frequencies, the change of the four curves is more obvious. Therefore, it may be concluded that the incident azimuth angle has negligible influence on the STL behavior of clamped finite double-plate. Influence of faceplate thickness (h) To quantify the influence of faceplate thickness, the STL versus frequency curve is presented in Fig. 7 for a finite double-plate. Three values of plate thickness are chosen: h = 0.002, 0.005 and 0.01 m. The dimensions of the plate are: a x b = 1m x 1m and the thickness of the air cavity is fixed at H = 0.08 m. The material properties are the same as in the subsection 3.1. Fig. 7. Effects of plate thickness on STL of clamped double-composite plate with enclosed air cavity. Three different plate thicknesses (h = 0.002, 0.005 and 0.01 m) were considered. Fig. 7. Effects of plate thickness on STL of clamped double-composite plate with enclosed air cavity. Three different plate thicknesses (h = 0.002, 0.005 and 0.01 m) were considered lower than 100 Hz. This is a very important region when designing finite dual sound insulation plates in practice. At higher frequencies, when peaks and poles appear in this mode, this is attributed to the strong interaction of individual plate behavior with the overall system performance for finite system. The position of double-plate resonances in Fig. 7 moves to higher frequencies when the thickness of the faceplate is increased. Influence of air cavity thickness (H) To demonstrate the influence of air cavity thickness on STL, the STLs are calculated for a finite double-composite plate with selected values of air cavity thickness: H = 0.02, 0.04, and 0.08 m; h1 = h2 = 0.01 m, as shown in Fig. 8. The other geometrical and material parameters are the same as section 4.2 and normal sound excitation is imposed. Vibroacoutic analysis of a finite clamped orthotropic laminated double-composite plate with an air cavity 11 According to Fig. 7, the STL value increases sharply when increasing the thickness of the plate. Effect of thickness of faceplate for STL is particularly strong at frequencies lower than 100Hz. This is a very important region when designing finite dual sound insulation plates in practice. At higher frequencies, when peaks and poles appear in this mode, this is attributed to the strong interaction of individual plate behavior with the overall system performance for finite system. The position of double-plate resonances in Figure 7 moves to higher frequencies when the thickness of the faceplate is increased. Influence of air cavity thickness (H) To demonstrate the influence of ir cavity thi kness on STL, the STLs are calculated for a finite double-composite plate with selected value of air cavity thickness: H = 0.02, 0.04, nd 0.08 m; h1 = h2 = 0.01m, as shown in Fig. 8. Th other ge metrical a d material parameters are the same as section 4.2 and normal sound excitation is imposed. Fig. 8. Effects of air cavity thickness on STL of clamped double-composite plate with enclosed air cavity. Three different air cavity thicknesses (i.e., 0.02, 0.04, and 0.08 m) were considered. As can be seen in Fig. 8, the first dip position does not depend on the air cavity thickness because it completely depends on the surface density of the plate. However, the second dip position changes drastically when the air cavity thickness increases (moving toward the lower frequency, Figure 8) due to the fact that the plate-cavity-plate resonance plays a major role in this case. Their remaining positions are almost unchanged because the plate-cavity-plate is operating synchronously. Therefore, by tailoring the thickness of air cavity, it is possible to design finite double-plate partitions with better sound insulation properties over a wide frequency range. 5. Conclusions An analytical approach has been developed to study the vibroacoustic behavior and the sound transmission loss across a clamped orthotropic laminated composite double-plate with enclosed air cavity. We get some of the following conclusions: • The theoretical predictions are in good agreement with existing experimental results. • The sound insulation capacity of double-composite plates is better than that of the aluminum double-plates when they have the same geometric parameters. • The double-composite plate has better sound insulation than the single-composite plate when they have the same geometric parameters and the same mechanical properties. • When the incident angle of the incident sound waves increases, the sound transmission loss decreases. On the other hand, the incident azimuth angle has negligible influence on the STL of a finite clamped double-composite plate. Fig. 8. Effects of air cavity thickness on STL of clamped double-composite plate with enclosed air cavity. Three different air cavity thicknesses (i.e., 0.02, 0.04, and 0.08 m) were considered 270 Tran Ich Thinh, Pham Ngoc Thanh As can be seen in Fig. 8, the first dip position does not depend on the air cavity thick- ness because it completely depends on the surface density of the plate. However, the second dip position changes drastically when the air cavity thickness increases (moving toward the lower frequency, Fig. 8) due to the fact that the plate-cavity-plate resonance plays a major role in this case. Their remaining positions are almost unchanged because the plate-cavity-plate is operating synchronously. Therefore, by tailoring the thickness of air cavity, it is possible to design finite double-plate partitions with better sound insula- tion properties over a wide frequency range. 5. CONCLUSIONS An analytical approach has been developed to study the vibroacoustic behavior and the sound transmission loss across a clamped orthotropic laminated composite double- plate with enclosed air cavity. We get some of the following conclusions: - The theoretical predictions are in good agreement with existing experimental re- sults. - The sound insulation capacity of double-composite plates is better than that of the aluminum double-plates when they have the same geometric parameters. - The double-composite plate has better sound insulation than the single-composite plate when they have the same geometric parameters and the same mechanical proper- ties. - When the incident angle of the incident sound waves increases, the sound trans- mission loss decreases. On the other hand, the incident azimuth angle has negligible influence on the STL of a finite clamped double-composite plate. - The influence of plate thickness on STL is particularly strong for the finite sys- tems at low frequencies, which is useful when designing clamped sound-proof double- composite plates. - As the thickness of the air cavity increases, the sound insulation capacity of the double-composite plate also increases but is not as strong as changing the thickness of the plate. ACKNOWLEDGEMENTS This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number: 107.02-2018.07. REFERENCES [1] J. P. Carneal and C. R. Fuller. An analytical and experimental investigation of active structural acoustic control of noise transmission through double panel systems. Journal of Sound and Vibration, 272, (3-5), (2004), pp. 749–771. https://doi.org/10.1016/s0022-460x(03)00418-8. [2] S. J. Pietrzko and Q. Mao. 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