acoustic radiation by set of l-jointed vibrating plates

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
[email protected], [email protected]
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⋅1011 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.
1
2
3
4
5
6
7
8
9
10
Natural frequency [Hz]
0.7
1.1
2.1
3.9
5.1
6.7
7.6
8.6
13.0
13.5
Modal damping coeff. [-]
0.001
0.002
0.004
0.007
0.010
0.013
0.014
0.016
0.025
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]; <vn2>- average value of squared velocity, ρc0acoustic resistivity of the medium, σrad-radiation efficiency factor.
N a = ρ c02 S < vn2 > σ 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: S0- reference surface area 1 m2.
186
Kozień M., Wiciak J.
S
L p = LNa − 10 lg  
 S0 
(2)
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: F0- force amplitude,
Zpf- plate impedance at the point the force is applied [2].
Win =
 1 
1 2
F0 (ω ) Re 

2
 z f (ω ) 
(3)
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.
Sound power level,
dB
110
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
Sound Pressure Level
dB
120,00
100,00
80,00
SEA
60,00
MES
Difference
40,00
20,00
0,00
100 125 160 200 250 315 400 500 630
Frequency,
Fig. 8. Sound pressure levels
Hz
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 σrad 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.
190
Kozień M., Wiciak J.
15. Wiciak J., Panuszka R., Programy obliczeniowe do oceny propagacji
wibroakustycznej. "Mechanika" AGH, 1995, T. 14, Z.2 str. 205-220.
energii
16. Wiciak J., Acoustics Radiation Of a Plate Bonded on All Edges With Four Pairs
Actuators, Molecular and Quantum Acoustics, vol.25, p.2812-290, (2004).
17. Zienkiewicz O.C, Thomson R.L., Zhu J.Z., The FEM – its Basis and Fundamentals,
Elsevier, 2005.