Molecular and Quantum Acoustics vol. 26, (2005) 183 ACOUSTIC RADIATION BY SET OF L-JOINTED VIBRATING PLATES Marek S. KOZIEŃ (*), Jerzy WICIAK (**) (*) Cracow University of Technology, Institute of Applied Mechanics, Al. Jana Pawła II 37, 31-864 Kraków, POLAND (**) University of Science and Technology, Department of Mechanics and Vibroacoustics Al. Mickiewicza 30, 30-095 Kraków, POLAND kozien@mech.pk.edu.pl, wiciak@agh.edu.pl Sound radiation from a L-jointed plates is investigated. Of particular interest is the acoustic pressure level at the control point at the distance of 1 m from the two plate surfaces. Two approaches were utilised, the FEM method was applied in the analysis of structural vibrations and the radiation efficiency in the analysis of acoustic phenomena. The other approach was the statistical energy analysis, referred to as SEA. Keywords: Statistical Energy Analysis, Finite Element Method, structural vibrations 1. INTRODUCTION Vibrating plates are the source of sound radiation. Such plates are used as structural elements of operator s cabs in earthmoving machines or cranes. The simplest model is made of two L-jointed plates. A full analysis requires the investigation of conjugated structural and acoustic fields, by using approximate methods: the finite element method [4][7][10][14][16][17] and boundary element method [2][3][5] in the low frequency range and the SEA method [9][15] for high frequencies. Practitioners tend to use a simplified approach which enables them to estimate the sound radiation parameters once the structural modes are known. Here the hybrid method [12] and the radiation efficiency method [2][6][11][13] are widely employed. The estimated parameter was the level of sound radiation from two L-jointed plates excited by a force harmonically varied in time. The radiation efficiency method was applied,
184 Kozień M., Wiciak J. the simplest one for this kind of analysis. Structural modes are determined using the FEM method and the SEA is employed for the sake of comparison. Numerical data are provided for illustration. The purpose of this research programme was to show the potential applications of the simplest method of analysing sound power radiation from structural elements in the shape of plates. 2. FEM ANALYSIS OF STRUCTURAL VIBRATIONS 2.1. SYSTEM S DESCRIPTION AND MODEL Of particular interest is sound radiation from a vibrating system of perpendicularly, line jointed square steel plates, 1 x 1 m. The excited plate is 2 mm thick, the other has 1 mm in thickness. The system is excited by the concentrated force 10 N, applied to the panel centre and fluctuating harmonically with time. The excitation frequency is the parameter that can be preset. The material specifications: Young modulus E=2.08 10 11 Pa, Poisson ratio ν=0.29, density ρ=7820 kg/m3, damping ratio ζ=0.0006. The computations utilise the Ansys package [1][10]. Each plate is sub-divided into 10 x 10 elements. The geometry of thus discretised system is shown in Fig. 1. The proposed modal superposition method requires the natural vibration analysis in order to find the lowest frequencies and natural vibration modes of the system. The lowest frequency values and modal damping coefficients are summarised in Table 1, the shape of the vibration modes for the specified (9-th) frequency is given in Fig. 2. The modal density of the structure seems considerable (see Fig. 4 and Fig. 5). Excited vibrations of L-jointed plates were analysed in steady-states, utilising the frequency response function. The parameter expressing the estimated sound power radiation is the surfaceaveraged velocity normal to plate surfaces, obtained for selected excitation frequencies in individual 1/3 octave bands. Fig. 1. Plate divided into finite elements and position of the exciting force
Molecular and Quantum Acoustics vol. 26, (2005) 185 Table 1. First ten mode frequencies and modal damping coefficients Mode No. Natural frequency [Hz] Modal damping coeff. [-] 1 0.7 0.001 2 1.1 0.002 3 2.1 0.004 4 3.9 0.007 5 5.1 0.010 6 6.7 0.013 7 7.6 0.014 8 8.6 0.016 9 13.0 0.025 10 13.5 0.025 Fig. 2. The shape of the 9-th natural mode of frequency 13.01 Hz 2.2. SOUND RADIATION ANALYSIS Acoustic power radiated to a half-space and acoustic pressure levels at the distance of 1 m from the plate surfaces were duly computed. The relationships given below were utilised in both cases considered in the study. The acoustic power radiated to the half-space is given by (1), where: S- source s surface area, [m 2 ]; <v n2 >- average value of squared velocity, ρc 0 - acoustic resistivity of the medium, σ rad -radiation efficiency factor. N = ρ c S < > σ 2 2 a 0 v n rad (1) The relationship (2) relating the acoustic power radiated to the surroundings to acoustic pressure yields the acoustic pressure levels at the distance of 1 m from the plate s surfaces, where: S 0 - reference surface area 1 m 2.
186 Kozień M., Wiciak J. S Lp = LNa 10lg (2) S0 Fig. 3 shows the frequency response of the system at the point the force is applied (displacement and velocity) for the frequency corresponding to the ninth mode of natural vibrations. Fig. 4 and Fig.5 shows the plot of this function (velocity) in the frequency range [0, 25 Hz] and in the frequency range [0, 250 Hz]. Results are discussed in section 4. a) b) Fig. 3. Amplitude frequency profile (displacement and velocity) at the selected point in the neighbourhood of the ninth resonance frequency: a) forced plate, b) second plate Fig. 4. Amplitude-frequency characteristic, velocity at the selected point, for the frequency range [0, 25] Hz
Molecular and Quantum Acoustics vol. 26, (2005) 187 Fig. 5. Amplitude-frequency characteristic, velocity at the selected point, for the frequency range [0, 250] Hz 3. ACOUSTIC FIELD ANALYSIS USING THE SEA METHOD Applications of the SEA methods are restricted by frequency considerations. On account of the assumptions made, the method is applicable in the high frequency range. However, the term high frequency range is not directly associated with actual frequencies but with the modal density of a structure. Results obtained using the FEM method reveal that the value of this parameter might be high even at low frequencies, it is then worthwhile to employ SEA in the analysis of sound radiation. The structural model of the system is shown in Fig. 6. Point Force Plate Flexural #1 Plate Flexural #2 Semi Infinite Fluid Fig. 6. SEA model of the system
188 Kozień M., Wiciak J. The input power supplied to the first plate is given by (3), where: F 0 - force amplitude, Z pf - plate impedance at the point the force is applied [2]. 1 W = 1 2 in F0 ( ω)re (3) 2 z f ( ω) 4. RESULTS Characteristics of sound power radiated by individual plates are shown in Fig. 7. Acoustic pressure levels at the distance of 1 m from the plate surface are given in Fig. 8. It is readily apparent that these values are close to one another, which is associated with the large modal density. The number of modes in frequency range [0, 25] Hz is 16 and in frequency range [0, 250] Hz is 250. 110 Sound power level, db 105 100 95 100 125 160 200 250 315 400 500 630 800 1000 Frequency, Hz Total Plate #1 Plate #2 Fig. 7. Sound power radiated by the plates 120,00 Sound Pressure Level db 100,00 80,00 60,00 40,00 20,00 0,00 100 125 160 200 250 315 400 500 630 SEA MES Difference Frequency, Hz Fig. 8. Sound pressure levels
Molecular and Quantum Acoustics vol. 26, (2005) 189 5. CONCLUSIONS Application of the FEM method exclusively to the analysis of structural modes makes the whole procedure practicable or vastly facilitates it. A complete description of all mechanic and acoustic aspects might exceed the computer s capacity or the time required for modelling and analyses would be prolonged. By combining the FEM approach and the radiation efficiency method sufficiently good results are obtained, fully adequate for engineering purposes. Application of a hybrid method [9] would further improve the accuracy of the solutions. Comparison of SEA results and radiation efficiency data reveals that the former might prove most useful in the high frequency range, though the modal density still remains the key parameter. When the vibrating structure displays such characteristic, the SEA method can be actually applied over the whole frequency range. REFERENCES 1. Ansys v.9 user manual, 2003 2. Beranek L. (Ed.), Noise and Vibration Control, Institute of Noise Control Engineering, Washington DC, 1988. 3. Burczyński T., Metoda elementów brzegowych w mechanice, WNT, Warszawa, 1995. 4. Cichoń Cz., Wprowadzenie do Metody Elementów Skończonych, Skrypt Politechniki Krakowskiej, Kraków, 1994 5. Ciskowski R.D., Brebbia C.A., Boundary element methods in acoustics, Computational Mechanics Publications co-published with Elsevier Applied Science, 1996. 6. Cremer L., Heckl M., Structure Borne Sound. Springer Verlag, Berlin, Heidelberg, 1988 7. Gołaś A., Metody komputerowe w akustyce wnętrz i środowiska, Wydawnictwa AGH, Kraków, 1995. 8. Lyon R., Statistical Energy Analysis of Dynamical Systems. MIT Press, 1975. 9. Lyon R., DeJong R.G.: Theory and application of Statistical Energy Analysis. Butt.-Hein., Boston, 1995. 10. Łaczek S., Wprowadzenie do systemu elementów skończonych ANSYS (Ver.5.0 i 5 ED), Wydawnictwo Politechniki Krakowskiej, Kraków, 1999. 11. Kozień M., The radiation efficiency coefficient σ rad and its dependence on some parameters for vibrating rectangular panels, Materiały Drugiej Szkoły: Metody energetyczne w wibroakustyce, Kraków-Krynica, 1993, 69-74. 12. Kozień M.S., Hybrid method of evaluation of sounds radiated by vibrating surface elements, Journal of Theoretical and Applied Mechanics, Vol.43, No.1, 2005, 119-133. 13. Kozień M. S., Nizioł J., Sound Radiation by the Viscoelastic Shallow Shells, Proc. of 21 st ICTAM04 (ed. W. Gutowski, T. Kowalski), Springer 2005, e-book SM 25_12511. 14. Kozień M. S, Wiciak J., Acoustics Radiation of a Plate With Line and Cross Type Piezoelectric Elements. Molecular and Quantum Acoustics, 24, 2003, pp. 97-108.
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