Noise Attenuation by Two One Degree of Freedom Helmholtz Resonators

Transcription

1 Global Science and Technology Journal Vol. 3. No. 1. March 015 Issue. Pp.1-9 Noise Attenuation by Two One Degree of Freedom Helmholtz Resonators Md. Amin Mahmud a*, Md. Zahid Hossain b, Md. Shahriar Islam c and Mir Md. Maruf Morshed d The resonance frequency and the transmission loss due to two 1-DOF cylindrical Helmholtz resonator attached to a duct is thoroughly investigated in this literature to attenuate low frequency noise using finite element method. Primarily, the relative spacing between the two identical resonators attached to a duct is investigated in this study to find the optimum positions between the resonators. The investigation is further carried out by changing the dimensions of the geometry of the cavity and neck for the optimum relative position Finally, based on these several investigations, a conclusion is drawn for an optimized system of two identical 1-DOF Helmholtz resonators in terms of position, orientation and geometry. The results show a significant improvement in noise reduction compared to a single 1-DOF Helmholtz resonator. The results of a single 1-DOF Helmholtz resonator of a published literature have been validated both analytically and numerically in this study. Keywords: Noise attenuation, Helmholtz resonator, 1-DOF, finite element method. Field of Research: Mechanical Engineering, Acoustics, Numerical Modeling, Environmental Engineering, Sound Engineering, Aerospace Engineering. 1. Introduction The problem of the low frequency noise is inherent to vehicles, IC engines, aircrafts, pipelines, power transformers, compressors etc. This noise not only creates disturbance to the personnel working at various industries but also causes serious health problems like hearing impairment, hypertension, heart disease, birth defects, and changes in immune system. Helmholtz resonators have been studied and applied extensively for many years in various engineering fields due to its effective performance in attenuating low frequency noise level. Neglecting the spatial distribution, the classical theory of a Helmholtz Resonator as an equivalent spring mass system was developed by Rayleigh (1945). The effect of different aperture geometries on the resonance frequency of resonators was discussed by Ingard (1953). Alster (197) investigated the effect of the shape of the vessel of the resonator on the resonance frequency. Panton and Miller (1975) performed an experimental study on the end correction theory of Ingard (1953). Extensive study on theoretical, computational and experimental investigation of Helmholtz resonators with fixed volume: lumped versus distributed analysis performed by Selamet et al (1995). a*, b, c Department of MCE, Islamic University of Technology (IUT), Bangladesh. d Department of Mechanical Engineering, Jubail University College, K.S.A. a* amin1434@iut-dhaka.edu, b zahidmce@iut-dhaka.edu, c shahriar015@iut-dhaka.edu d morshed@iut-dhaka.edu

2 Multi-dimensional analytical approaches have also been employed to predict the sound attenuation in Helmholtz resonators with circular concentric cavity, Selamet et al (1997). Griffin (001) developed an analytical model of mechanically-coupled mounted 1-DOF Helmholtz resonator attached to a one dimensional duct for achieving a wide bandwidth and confirmed by experiment. An experimental investigation showed that when the center distance of two identical resonators was greater than a quarter wavelength apart, the sound transmission loss was larger than that of a single resonator, however, when two resonators at same resonant frequency were in close proximity, the two resonators interacted and lead to a decrease in the overall performance compared to that of a single resonator Soh (001). Based on Soh (001), the effect of the resonator position on the noise reduction and the relationship between two or more closely spaced identical 1-DOF resonators close to the interior wall of a chamber core cylindrical fairing was experimentally investigated by D. Li (003). Mekid and Farooqui (01) proposed by a new design methodology for one and two degrees of freedom Helmholtz resonators attached to pipelines leading to an optimized transmission loss was. For more than a single 1-DOF resonator, the placement among the resonators is very important in many aspects. If the array of 1-DOF Helmholtz resonators in same plane are placed where the crest of the wave falls, the performance of noise attenuation increases significantly. Farooqui (01). But there hasn t been any significant study on the relative spacing between the two identical Helmholtz resonators attached to a duct or pipelines in a 3-dimentional plane. So, the primary objective of this literature is to investigate the relative spacing between the two identical 1-DOF resonators attached to a duct in 3-dimensional plane by placing them at various relative positions like (closely spaced (< λ 4 ); 1st antinode ( λ 4 ) ; 1 st node ( λ ) where λ is the wavelength of the maximum working frequency to obtain maximum TL at the optimum relative position. Secondary objective is to investigate the effect of changing both the geometry (length/diameter ratio) of the cavity and neck (keeping the volume of the resonators constant) on the resonating frequency and transmission loss. Sec. discusses the Adaptation of 1-DOF Helmholtz resonator followed by Methodology in Sec.3. Sec 4 presents the results and discussion of the findings. Sec. 5 represents the validation of previous published results. The study is concluded with final remarks in Sec.6.. Adaptation of a Single 1-DOF Helmholtz Resonator There are various types of Helmholtz resonators like conical, spherical and cylindrical type. Among conical, spherical and cylindrical Helmholtz resonators, the overall noise attenuation performance of cylindrical resonator is much better, Farooqui (01). So, the cylindrical Helmholtz resonator has been chosen for the present study. The resonating frequency (f) and the transmission loss (TL) due 1-DOF Helmholtz resonator attached to duct are represented by the equation 1, Mekid and Farooqui (01) and the equation, Selamet et al (1997) f = c π 3. L n + L c A L n 3 + ( 3. L n + L c A L n 3 ) + 3A L n 3 L c (1)

3 TL = 10 log ( a n( 1 A ) tan(kl C )+tan(kl n) a d ( 1 A ) tan(kl C )+tan(kl ) 1) () c Where A = a c is the area ratio, c is the speed of the sound in the medium, k is the a n wave number; L c, L n representthe length; d c, d n the diameter ; a c, a n the area cross sections of the cavity and neck respectively and d d, a d is the diameter and the crosssectional area of the duct respectively. The dimensions for modeling a single 1-DOF resonator in this literature was adapted from (Resonator A, Table 1 of Selamet et al (1997) shown in Figure 1. Figure 1: Single 1-DOF Helmholtz Resonator. 3. Methodology 3.1 Modeling and 3-Dimentional Numerical Simulation: The main duct was considered as a square cross section of cm and 1.5 meter long. All these resonators were modeled in (ANSYS FLUID) ANSYS APDL (using uncoupled acoustic element 10 node tet fluid 1 (considering structure absent). The mesh size was (10 elements/wavelength) of the maximum working frequency 300 Hz. Then, all the meshed elements were merged and excitation was applied on the inlet and outlet port in the form of normal velocity equivalent to sound pressure level of 150 db using equation 3, Rienstra and Hirschberg (015) u = p ρ0 c0 (3) Where, u = normal velocity excitation, p = pressure value in Pascal equivalent to 150 db, ρ0 = density of air =1.041 kg/m^3 and c0= air sound speed = m/s. 150 db was considered since it is in the threshold of pain region of human being, The Fletcher- Munson curves, Fletcher et el (1933). Then, harmonic acoustic analysis was performed for those acoustics model in low frequency range (0-300Hz). The output sound pressure level was calculated from the sound pressure values found at the proximal node of the outlet by the equation 3, Rienstra and Hirschberg (015) and TL 3

4 was measured by calculating the drop of sound pressure level from input to output. Output Sound Pressure Level = 0log10 (sound pressure at output/reference pressure level) Figure : Acoustic Model of the Resonators in ANSYS 3. Investigation Criteria: (A) Case I (Relative Positions of the Two Resonators) The two 1-DOF Helmholtz resonators attached to duct were modeled by placing them at various relative spacing Z (closely spaced; (< λ 4 ), 1st antinode ; ( λ 4 ), 1st node; ( λ ) distance apart of the wavelength of the maximum working frequency 300 Hz.The dimensions of two resonators were considered exactly same as mentioned in the Figure 1 for a single 1-DOF resonator. Figure 3: Relative Position of the Two Identical Resonators (B) Case II (Varying the geometry of the cavity keeping the volume constant) 4

5 In Case I, the cavity ratio ( L c d c ) used for both the resonators was Then it was further changed to and.77 keeping the volume of the cavity and dimensions of the cavity constant as mentioned in Table 1 to investigate the resonance frequency and maximum TL. The two resonators in this case were placed at optimum spacing found from Case I. Table 1: Dimensions of the Varying Geometry of the Cavity Cases No. II L c (cm) d c (cm) d c a b c d L c (C) Case III (Varying the Geometry of the Neck of the Two Resonators) In Case I, the neck ratio ( L n d n ) of both the resonators was.10. Then, it was further changed to.5,.5 and 3 as mentioned in Table keeping the volume of the neck constant and using optimum cavity ratio found from Case II to investigate the resonance frequency and maximum TL. The two resonators in this case were also placed at optimum spacing found from Case I. Table : Dimensions of the Varying Geometry of the Neck Cases No III L n (cm) d n (cm) L n /d n A B c d Results and Discussion 4.1 Case I (Relative Position) The measured value of maximum TL found at the resonance frequency 89 Hz was db, 9.33 db and db due to the relative spacing ( λ ), (λ ) and (< 4 λ ) respectively compared to that of 50 db for a single 1-DOF resonator shown in Figure 4 4. The above numerical results show that the overall performance for the two resonators at ( λ ) distance apart is significantly higher than that of a single 1-DOF resonator attached to a duct. In case of two closely spaced identical resonators, the measured value of maximum TL at 89 Hz is also very greater than a single resonator. Another notable observation is that an antinode (increase of sound pressure level) has been created at around 70 Hz due to the interaction of the two resonators but its effect is negligible on overall noise attenuation performance after 70 Hz shown in Figure 4. 5

6 So, the above investigation suggests that the optimum relative spacing between the two identical 1-DOF resonators is ( λ ) distance apart. Figure 4: Numerical Results of Various Relative Positions Transmission Loss [db] (a) Close (b) 1st antinode (c) 1st Node Single 1-DOF Frequency [Hz] 4. Case II (Varying Geometry of the Cavity) The measured maximum value of TL at around the resonance frequency 89 Hz was db for optimum relative spacing; ( λ ), the cavity ratio ; 1.59, neck ratio of.10 ( 4.1 Case I) for both the resonators. Then, by increasing the cavity ratio to, resonance frequency increased to 91 Hz and TL increased to 118 db. For further increasing the ratio to.77, the resonance frequency decreased to 88 Hz and TL dropped to 115 db at 88 Hz shown in Figure 5. So, from this above investigation, TL can be increased by 0 db more for the cavity ratio.0 instead of

7 Figure 5: Numerical Results of Varying Geometry of the Cavity Figure 4: Numerical Results of TL for varying Cavity Transmission Loss [db] lc/dc=1.59 (Case I) lc/dc= lc/dc=.77 L n /d n =.10 unchanged Case III 0 (Varying 50 Geometry 100 of the Neck) Frequency[Hz] For this investigation the neck ratio was changed keeping the cavity ratio.0 (4. Case I). The measured value of maximum TL for two resonators at ( λ ) distance apart was 118 db at the resonance frequency (91 Hz) in Case II for the neck ratio.10. By increasing the neck ratio to.5, resonating frequency decreased to 8 Hz and TL became 90 db at 8 Hz. But, by increasing the ratio to.5, resonance frequency decreased to 80 Hz but the TL increased to 130 db at 80 Hz. For further increasing the ratio to 3, resonance frequency decreased to 7 Hz but TL increased to 10 db at 7 Hz shown in Figure 6. So, from the above investigation, TL can be increased by about 1 db more for the neck ratio.5 than that of.10. Figure 6: Numerical Results of the Varying Geometry of the Neck Transmission Loss [db] Ln/dn =.10 Ln/dn =.5 Ln/dn =.5 Ln/dn = 3 L c =.0 d c unchanged Frequency [Hz] 7

8 5. Validation of Published Results In this literature, a validation of maximum TL was performed using the dimensions of Figure 1for a single 1-DOF Helmholtz resonator. Analytical validation of TL was done for the equation, Selamet et al (1997) and numerical validation was performed for the resonator of Figure 1 using finite element method. The analytical and numerical results of TL agree satisfactorily with the numerical results of TL of published article, Selamet et al (1997) shown in Figure 7. Figure 7: Validation of Analytical and Numerical Results with Published Numerical Results (Selamet et al,1997 ) for A Single 1-DOF Resonator 6. Conclusion An optimized system for both two identical 1-DOF Hemholtz resonators in terms of relative position and geometry of both the cavity and neck has been investigated in this literature. Maximum noise attenuation of 130 db has been achieved compared to a single 1-DOF resonator at around the resonance frequnecy has been found numerically for two identical cylindrical 1-DOF Helmholtz resonators by finite element method which signifies the importance of this proposed procedure. The optimized system found from the above investigations is : relative spacing λ, cavity ratio:.0 and neck ratio:.0. 8

9 References: Alster, M. (197), Improved calculation of resonant frequencies of Helmholtz resonators, Journal of Sound and Vibration, vol. 4, no.1, pp Li, D. (003), Vibroacoustic behavior and noise control studies of advanced composite structures, University of Pittsburgh, USA. Fletcher, H., and Munson, W. A. (1933), Loudness, Its Definition, Measurement and Calculation, Bell System Technical Journal, vol. 1, no. 4, pp Farooqui, M. (01), Noise reduction in centrifugal compressors using Helmholtz resonators, King Fahad University of Petroleum and Minerals, Soudi Arabia. Griffin, S., Lane, S. A., and Huybrechts, S. (001), Coupled Helmholtz resonators for acoustic attenuation, Journal of vibration and acoustics, vol.13, no.1, pp Ingard, U. (1953), On the theory and design of acoustic resonators, Journal of the Acoustical Society of America, Vol. 5 (6), Pp Mekid, S., and Farooqui, M. (01), Design of Helmholtz resonators in one and two degrees of freedom for noise attenuation in pipelines, Acoustics Australia, vol. 40, no.3. Panton, R. L., and Miller, J. M. (1975), Resonant frequencies of cylindrical Helmholtz resonators. Journal of the Acoustical Society of America, vol. 57, no. 6, pp Rayleigh, J. W. S (1945), The Theory of sound, Volume II, Dover, New York, Art. pp. 93. Rienstra, S. W., and Hirschberg, A., (015), An Introduction to Acoustics, Eindhoven University of Technology, Netherlands. Selamet, A., Dicky, N. S., and Novak, J. M. (1995), Theoretical, computational and experimental investigation of Helmholtz resonators with fixed volume: lumped versus distributed analyses, Journal of sound and vibration, vol. 187, no., pp Selamet, A., Radavich, P. M., Dickey, N. S., and Novak, J. M. (1997), Circular concentric Helmholtz resonators, The Journal of the Acoustical Society of America, vol. 101, no. 1, pp Soh, Y. P., Yap, E. W. T., and Gan, B. H. L. (001), Industrial Resonator Muffler Design, University of Adelaide, Australia.. 9