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inter.noise 2000The 29th International Congress and Exhibition on Noise Control Engineering27-30 August 2000, Nice, FRANCE

I-INCE Classification: 7.5

ASYMPTOTIC FORMULAS FOR ACTIVE AND REACTIVEMUTUAL POWER OF FREE VIBRATIONS OF A

CLAMPED ANNULAR PLATE

W.J. Rdzanek*, Z. Engel**, W.P. Rdzanek, Junior*

* Pedagogical University, Department of Acoustics, ul. Rejtana 16a, 35-310, Rzeszow, Poland

** University of Mining and Metallurgy, Department of Mechanics and Vibroacoustics, Al. Mickiewicza30, 30-059, Krakow, Poland

Tel.: +48 17 862-56-28 ext. 1082 / Fax: +48 17 852-67-92 / Email: rdzanek@atena.univ.rzeszow.pl

Keywords:MUTUAL SOUND POWER, ASYMPTOTIC METHODS, A KIRCHHOFF-LOVE’S MODEL, CON-TOUR INTEGRAL COMPUTATION

ABSTRACTNormalized mutual power of radiation, active and reactive, has been analyzed in this paper. The soundsource is an annular plate, clamped to a planar, rigid and infinite baffle. The plate’s vibrations areaxisymmetric and sinusoidal with respect to time. A Kirchhoff-Love’s model of a perfectly elastic platehas been used. First, the integral formulas for the mutual sound power have been presented and then theasymptotic formulas for fast numerical computations of the mutual sound power have been computed,which can be used for acoustically fast waves. The contour integral computation and stationary phasemethods have been used. The analysis presented can be used as a basis for analysis of the total powerof excited and damped vibrations.

1 - INTRODUCTIONA modal analysis has been made to get the normalized mutual sound power, active and reactive, of aclamped annular plate. Free, axisymmetric vibrations sinusoidally varying with respect to time with nofriction and damping forces have been considered. A Kirchhof-Love model of a perfectly elastic plate hasbeen used. Results of this analysis can further be used to compute the total sound power of a clampedannular plate.The frequency equation and its solution have been presented in several works (e.g. [2], [5, 6]). Thecontour integral computation and stationary phase methods have been used to transform some integralformulas to asymptotic formulas with known approximation error. And those methods are presented inseveral papers for computation of mutual sound power of a circular plate (e.g. [3, 4]) or sound power ofan annular plate [5]. A number of papers treat of several theoretical as well as experimental methods forobtaining the sound power of circular plates (e.g. [1]).

2 - INTEGRAL REPRESENTATIONThe mutual sound power lost via mode (0,m) and used to overcome resistance produced by the samesource via mode (0,n) of free vibrations of a clamped annular plate is

Πnm =12

∫

S

pnmv∗ndS, where Π(∞)nm =

√Π(∞)

n Π(∞)m = limk→∞Πnm (k) =

ρ0c

2

∫

S

vnv∗mdS (1)

pnm − the sound pressure produced by vibrating annular plate via mode (0,n) exerted on the same platevia mode (0,m), v − normal component of the amplitude distribution of the plate’s vibration velocity, v∗

− conjugate magnitude of v. We denote the normalized mutual sound power as Pnm = Pc,nm− iPb,nm =Πnm

Π(∞)nm

. Those definitions and denotations make possible to express the normalized mutual sound power

of a clamped annular plate in the form of integral

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Pnm = 4βx2nx2

mgnm

∫ π2−i∞

0

{xn

[sC

′1 (sxn)J0 (su)− C

′1 (xn)J0 (u)

]

−u[sC

′0 (sxn)J1 (su)− C

′0 (xn)J1 (u)

]}{xm

[sC

′1 (sxm)J0 (su)− C

′1 (xm) J0 (u)

]

−u[sC

′0 (sxm) J1 (su)− C

′0 (xm)J1 (u)

]} udϑ

(x4n − u4) (x4

m − u4)

(2)

where

s =r2

r1, u = βsinϑ, β = kr1

gnm =1√

s2C ′20 (sxn)− C ′20 (xn)1√

s2C ′20 (sxm)− C ′20 (xm)

xn=knr1 are the successive eigenvalues of the frequency equation, k4n = ω2

nρhB′ is a structural wavenumber,

C ′i (sxn) = Ji (sxn)− CnNi (sxn), i=0,1 and Cn is an integral constant presented in [5].

3 - ASYMPTOTIC FORMULASUsing the contour integral computing and stationary phase methods we can transform integral (2) to theasymptotic formulas for active (3) and reactive (4) mutual power

Pc,nm =(δnanm − δmamn) δ2

nδ2m

β (δ4n − δ4

m)

{1δ2n

(1√

1− δ2n

− 1√1 + δ2

n

)+

1δ2m

(1√

1− δ2m

− 1√1 + δ2

m

)}

+qnm

β√

β

√s

[α1cosw2 + α2sinw2 +

√2

s− 1(γ1 − γ4) cosw3 +

√2

s + 1(γ3 − γ2) cosw4

]

+qnm

β√

β(β1cosw1 + β2sinw1) + O

(δ2nδ2

mβ−3/2)

(3)

Pb,nm =2 (πβ)−1

δ4n − δ4

m

{δ2m

[d1

arcsinδn√1− δ2

n

− d2arcsinhδn√

1 + δ2n

]− δ2

n

[d3

arcsinδm√1− δ2

m

− d4arcsinhδm√

1 + δ2m

]}

+qnm

β√

β

√s

[α1sinw2 + α2cosw2 −

√2

s− 1(γ1sinw3 + γ4cosw3)−

√2

s + 1(γ3sinw4 + γ2cosw4)

]

+qnm

β√

β(−β1sinw1 + β2cosw1) + O

(δ2nδ2

mβ−3/2)

(4)which consist of elementary functions and expressions only with no integration elements, and

α1 = C′0 (sxn)C

′0 (sxm)− δnδmC

′1 (sxn)C

′1 (sxm)

α2 = δmC′1 (sxm) C

′0 (sxn) + δnC

′1 (sxn)C

′0 (sxm)

α±3 = C′1 (sxn)C

′0 (xm)± C

′1 (xn)C

′0 (sxm)

α±4 = C′1 (xm) C

′0 (sxn)± C

′1 (sxm) C

′0 (xn)

α5 = C′1 (sxn)C

′1 (xm) + C

′1 (sxm)C

′1 (xn)

α6 = C′0 (xm)C

′0 (sxn) + C

′0 (sxm)C

′0 (xn)

α7 = sC′1 (sxn) C

′0 (xm)− C

′1 (xn)C

′0 (xm)

γ1 = δnα−3 − δmα−4, γ2 = δnα+3 + δmα+4, γ3 = δnδmα5 − α6, γ4 = δnδmα5 + α6

β1 = C′0 (xn)C

′0 (xm)− δnδmC

′1 (xn)C

′1 (xm) , β2 = δmC

′1 (xm) C

′0 (xn) + δnC

′1 (xn)C

′0 (xm)

anm = gnmα7, amn = gmnα7

w1 = 2β +π

4, w2 = 2sβ +

π

4, w3 = (s− 1)β +

π

4, w4 = (s + 1) β +

π

4qnm =

2δ2nδ2

mgnm√π (1− δ4

n) (1− δ4m)

d1 = a′nmδm + a

′′nmδn, d2 = a

′nmδm − a

′′nmδn, d3 = a

′nmδn + a

′′nmδm, d4 = a

′nmδn − a

′′nmδm

a′nm = a

′mn = gnm

[sC

′1 (sxn)C

′1 (xm) + C

′1 (xn)C

′1 (xm)

]

a′′nm = a

′′mn = gnm

[sC

′0 (sxn)C

′0 (xm) + C

′0 (xn)C

′0 (xm)

]

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A number of asymptotic equations must be used to obtain formulas (3) and (4). The consequence of thisis that the formulas are valid for wave acoustically fast only, i.e. when acoustic wavenumber is greaterthan structural wavenumber. But the asymptotic formulas make possible fast and accurate numericalcomputations of mutual power active and reactive and can be the basis of analysis of the total soundpower of a clamped annular plate. Additionally, the formulas make possible separation and furtheranalysis of oscillating and non-oscillating parts of mutual sound power. This cannot be done usingintegral or any other formulas.

4 - CONCLUSIONSIn the case of mutual sound power of mode numbers even and odd the non-oscillating part disappearand the oscillating part has amplitude of a considerable value Fig. 1. In the case of both odd as well asboth even mode numbers the non-oscillating part is clear and the amplitude of oscillating part is a littlesmaller Fig. 2. Considered mutual interactions between free vibration modes proceed in surroundings ofa liquid or gaseous medium (e.g. air). An air column above vibrating plate absorbs the sound powerin some terms of time and develops the sound power in the rest terms of time. In the case of vacuumor a very weak air the mutual interactions, considered herein, disappear or became much weaker thanin the case of more thick gas. The air column has been used only as a transmission medium for themutual interactions between some vibration modes but no damping influence has been considered usinga Kirchhoff-Love model of a perfectly elastic plate.The modal analysis and formulas for the active and reactive sound power presented herein can be thebasis of the total power computing. To make a correct design of an acoustic device the active soundpower is not enough because of the energy aspect of the device. It does not matter if the active power issmall or big. If the reactive power is to high the sound device can be easily destroyed. This is the reasonwhy the reactive power is so important and has also been presented.

Figure 1: The normalized mutual active power of vibrating annular plate Pc,nm, where s=1.2,nm=02,04,13,15,24,35.

ACKNOWLEDGEMENTSThis paper is prepared within the KBN grants no. 7T07B 021 17 and no. 7T07B 051 18.

REFERENCES

1. S. Czarnecki, Z. Engel and R. Panuszka, Sound power and reduction efficiency of a circularplate, Archives of Acoustics, Vol. 16 (4), pp. 339-357, 1981

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Figure 2: The normalized mutual active power of vibrating annular plate Pc,nm, where s=1.2,nm=01,03,05,12,14,23,25,34,45.

2. M.-r. Lee and R. Singh, Analytical formulations for annular disk sound radiation using struc-tural modes, JASA, Vol. 95 (6), pp. 3311-3323, 1994

3. H. Levine and F.G. Leppington, A note on the acoustic power output of a circular plate,Journal of Sound and Vibration, Vol. 121 (2), pp. 269-275, 1988

4. W.J. Rdzanek, Mutual impedance of a circular plate for axially-symmetric free vibrations at highfrequency of radiating waves, Archives of Acoustics, Vol. 17 (3), pp. 439-448, 1992

5. W.P. Rdzanek, Jr and Z. Engel, Asymptotic formulas for the acoustic power output of aclamped annular plate, Applied Acoustics, Vol. 60 (1), pp. 29-43, 2000

6. S.M. Vogel and D.W. Skinner, Natural frequencies of transversely vibrating uniform annularplates, Journal of Applied Mechanics, Vol. 32, pp. 926-931, 1965