Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

47, 2, pp. 411-420, Warsaw 2009

ACOUSTIC INTENSITY VECTOR GENERATED BY

VIBRATING SET OF SMALL AREAS WITH RANDOM

AMPLITUDES

Marek S. Kozień

Cracow University of Technology, Institute of Applied Mechanics, Cracow, Poland

e-mail: kozien@mech.pk.edu.pl

The paper presents a generalisation of the hybridmethod of estimation
of sound radiated by vibrating surfaces, formulated previously for the
deterministic caseof randomvibrations.Theanalysis ismade for random
amplitudes of vibrations in a narrow frequency band. The results show
complexity of the analysis in comparison with the deterministic case.
Therefore, themethoddoesnot seemtobeefficient, like thedeterministic
one, in engineering applications.

Keywords:acoustic radiation, randomvibrations, sound intensityvector,
structural noise

1. Introduction

For several years, the author have been dealing with the problem of acoustic
radiation of vibtrating plates and shallow shells in deterministic and random
cases (Kozień andNizioł, 2005, 2006, 2007; Kozień and Saltarski, 2007, Nizioł
and Kozień, 2000, 2001).

A combination of the method of analysis of acoustic radiation by a har-
monically vibrating small plane element (Kwiek, 1968) and the method of
estimation of the sound intensity vector by knowledge of their amplitudes
(Mann et al., 1987) result in a new method of estimation, proposed by the
author and called the hybrid method (Kozień, 2005, 2006). The method was
previously formulated for a deterministic case of structural vibrations.

In the presented paper, a generalisation of the method for the case of
randomly vibrating system of small elements is discussed. The assumption
is that the amplitudes of vibrations of the elements are random processes
described by a probability density function.



412 M.S. Kozień

Due to assumptions, the analysis is valid for a relatively narrow frequency
band.

2. Theoretical background of the hybrid method in the

deterministic case

The analysis is provided for amonochromatic wave for the given frequency ω
and the analysis is performed in the complex space.
Analysis of an acoustic field generated by vibrating surfaces is based on

determination of the resultant acoustic intensity vector I in a chosen control
point in the acoustic volume. The vibrating area is previously divided into the
sub-areas.
Each vibrating element is the source of radiated sound for the given fre-

quency ω. The assumption of the method is that every sub-area is a small
surface element. The ”smallness” of the element is interpreted here with
respect to the wavelength (associated with the wave frequency), in accor-
dance with (2.1), where r0 is the radius of the sub-domain or the greatest
distance between the sub-domain centre and its boundary points (Kwiek,
1968)

r0 ≪
λ

4π
(2.1)

The well-known relationships between the angular frequency ω, wavenum-
ber k, wavelength λ and speed of sound in an acoustic medium c are given
as

k=
ω

c
=
2π

λ
(2.2)

The analysis is described in the Cartesian co-ordinate system Oxyz. The po-
sition of the chosen control point P(x,y,z) is given by the vector R=OP,
and the position of the center of the i-th sub-area Qi(xi,yi,zi) by the vector
ρi =OQi (Fig.1). The following relationship between the mentioned vectors
is valid

ri =R−ρi (2.3)

Hence, the distance between the center of the i-th sub-area and the control
point P can be obtained basing on the relationship

ri =
√
(x−xi)2+(y−yi)2+(z−zi)2 (2.4)



Acoustic intensity vector generated by vibrating set... 413

Fig. 1. Geometry of the sub-area and the control point P

For such a case, the acoustic pressure and the partial velocity vector ge-
nerated by the i-th sub-area in the control point P(x,y,z), can be obtained
basing on the following formulas (Kwiek, 1968)

pi = pi(ri,R)=−
1

2π
∆Siω

2ρ0
Ai

ri
ei(ωt−kri)

(2.5)

vi =vi(ri,R)=vri(ri,R)=Ai∆Si
(
−
ω2

2πc

1

ri
+i
ω

2π

1

r2i

)
ei(ωt−kri)

ri

ri

where ∆Si is the area of the i-th sub-area, and Ai is the amplitude of its
vibrations.
Formula (2.5)2 can bewritten in the Cartesian co-ordinate system in form

of three scalar relationships on the components of the partial velocity vector v
which is parallel to the vector ri as in the following

vix(xi,yi,zi,x,y,z) =Ai∆Si
(
−
ω2

2πc

1

ri
+i
ω

2π

1

r2i

)x−xi
ri
ei(ωt−kri)

viy(xi,yi,zi,x,y,z) =Ai∆Si
(
−
ω2

2πc

1

ri
+i
ω

2π

1

r2i

)y−yi
ri
ei(ωt−kri) (2.6)

viz(xi,yi,zi,x,y,z) =Ai∆Si
(
−
ω2

2πc

1

ri
+i
ω

2π

1

r2i

)z−zi
ri
ei(ωt−kri)

The next problem is the idea of superposition of the components of pres-
sures and partial valocities comming from the set of sub-areas in the resultant
form in the analysed point P. The following relationship, formulated previo-
usly by Mann et al. (1987) for the set of N-point acoustic sources, is applied
futher

I =
1

2

( N∑

i=1

pi

)( N∑

j=1

v
∗

j

)
(2.7)



414 M.S. Kozień

After suitablemanipulations, based on relationships (2.5)1, (2.6) and (2.7),
are obtained formulas for the components of the real and imaginary parts of
the resultant complex acoustic intensity vector in the chosen control point

Re(Ix)=K
[( n∑

i=1

aiAi

)( n∑

j=1

xjcjAj

)
+
( n∑

i=1

biAi

)( n∑

j=1

xjdjAj

)]

Re(Iy)=K
[( n∑

i=1

aiAi

)( n∑

j=1

yjcjAj

)
+
( n∑

i=1

biAi

)( n∑

j=1

yjdjAj

)]
(2.8)

Re(Iz)=K
[( n∑

i=1

aiAi

)( n∑

j=1

zjcjAj

)
+
( n∑

i=1

biAi

)( n∑

j=1

zjdjAj

)]

and
Im(Ix)=K

[( n∑

i=1

aiAi

)( n∑

j=1

xjdjAj

)
−
( n∑

i=1

biAi

)( n∑

j=1

xjcjAj

)]

Im(Iy)=K
[( n∑

i=1

aiAi

)( n∑

j=1

yjdjAj

)
−
( n∑

i=1

biAi

)( n∑

j=1

yjcjAj

)]
(2.9)

Im(Iz)=K
[( n∑

i=1

aiAi

)( n∑

j=1

zjdjAj

)
−
( n∑

i=1

biAi

)( n∑

j=1

zjcjAj

)]

The parameters K, ai, bi, ci and di, i = 1, . . . ,N standing in the above
formulas are defined as

K =
1

8π2
ρ0ω
3 ai =

∆Si

ri
cos(kri) bi =

∆Si

ri
sin(kri)

ci =
∆Si

r2i

[
kcos(kri)−

1

ri
sin(kri)

]
(2.10)

di =
∆Si

r2i

[
k sin(kri)+

1

ri
cos(kri)

]

Basing on the knowledge of the amplitudes of vibrations for each sub-
area, it is possible to obtain the resultant complex acoustic intensity vector for
agiven frequency.Moreover, the ratio betweenvalues of the real and imaginary
parts of the complex vector gives the information of the type of acoustic field
in the chosen point (nearfield, farfield). The method in the presented form
does not take into account the effects of absorption or reflection of acoustic
waves from any surfaces.
The knowledge of the acoustic intensity vector is usually enough to make

acoustic analysis, particularly in energy forms.But if an values of the acoustic
pressure in the chosen point are themost important, it can be obtained in an



Acoustic intensity vector generated by vibrating set... 415

approximate way based on the assumption of the plane acoustic wave in the
control point area, in the form

I =
p2

ρ0c
(2.11)

The other way of estimation of the acoustic pressure is application of the
formulas given in ISO 11205 (ISO, 2003) in the form

Lp =10log

√
(
10
LIx
10

)2
+
(
10
LIy

10

)2
+
(
10
LIz
10

)2
(2.12)

where LIx, LIy and LIz are levels [dB] of the components of the acoustic
intensity vector in the x, y and z directions, respectively.

3. Hybrid method for random amplitudes of vibrations

3.1. General formulation

Let us assume that the amplitude of vibrations for the i-th sub-area Ai is
a random process, usually with the zero middle value. Random processes are
defined by the probability density function fi(zi). Moreover, let us reduce
the analysis down to the narrow frequency band, so that the description of
propagation of the acoustic wave with a given central frequency band is valid.
Hence, in equations (2.5)2 and (2.6) instead of only deterministic ampli-

tudes Ai, the probability density functions fi(zi) are the input. Then the
relations are put into formula (2.7) which is multiplied with the same one
and integrated over the whole appropriate probability spaces. As a result, di-
spersion of components of the acoustic intensity vector is obtained (3.1)1 as
function of the statistical moment of the fourth order between random varia-
bles (amplitudes of the transverse displacement) Ai,Aj,Ak and Al with the
probability density functions fi, fj, fk and fl – m[Ai,Aj,Ak,Al] (3.1)2)

σ
2
I =

+∞∫

−∞

+∞∫

−∞

+∞∫

−∞

+∞∫

−∞

1

2

[ N∑

i=1

pi(zi)
][ N∑

j=1

v
∗

j(zj)
]1
2

[ N∑

k=1

pk(zk)
][ N∑

l=1

v
∗

l (zl)
]
·

·dzidzjdzkdzl
(3.1)

m[Ai,Aj,Ak,Al] =

=

+∞∫

−∞

+∞∫

−∞

+∞∫

−∞

+∞∫

−∞

zizjzkzlfi(zi,Ai)fj(zj,Aj)fk(zk,Ak)fl(zl,Al) dzidzjdzkdzl



416 M.S. Kozień

Thefinal explicite formulasare relatively complicated, andthe level of com-
plication is nonlinearly growing with the increasing number of sub-areas. For
example, for two sub-areas, the formulas fordeterminationof the x-component
of the real and imaginary parts of the complex acoustic vector have the forms

σ2Re(Ix)(x,y,z) =K
2
{
m[A1,A1,A1,A1](a

2
1c
2
1x̃
2
1+ b

2
1d
2
1x̃
2
1−a

2
1d
2
1x̃
2
1− b

2
1c
2
1x̃
2
1+

+4a1b1c1d1x̃
2
1)+m[A1,A1,A1,A2](2a

2
1c1c2x̃1x̃2+2a1a2c

2
1x̃
2
1+

+4a1b1c1d2x̃1x̃2+4a1b2c1d1x̃
2
1+4a1b1c2d1x̃1x̃2+4a2b1c1d1x̃

2
1+

+2b21d1d2x̃1x̃2+2b1b2d
2
1x̃
2
1−2a

2
1d1d2x̃1x̃2−2a1a2d

2
1x̃
2
1−2b

2
1c1c2x̃1x̃2+

−2b1b2c
2
1x̃
2
1)+m[A1,A1,A2,A2](4a1a2c1c2x̃1x̃2+4a1b2c1d2x̃1x̃2+

+a21c
2
2d2x̃

2
2+4a1b1c2d2x̃

2
2+4a1b2c2d1x̃1x̃2+a

2
2c
2
1x̃
2
1+4a2b1c1d2x̃1x̃2+

+4a2b2c1d1x̃
2
1+4a2b1c2d1x̃1x̃2+4b1b2d1d2x̃1x̃2+ b

2
1d
2
2x̃
2
2+ b

2
2d
2
1x̃
2
1

−4a1a2d1d2x̃1x̃2−a
2
1d
2
2x̃
2
2−a

2
2d
2
1x̃
2
1−4b1c1c2b2x̃1x̃2− b

2
1c
2
2x̃
2
2− b

2
2c
2
1x̃
2
1)+

+m[A1,A2,A2,A2](2a1a2c
2
2x̃
2
2+4a1b2c2d2x̃

2
2+2a

2
2c1c2x̃1x̃2+

+4a2b2c1d2x̃1x̃2+4a2b1c2d2x̃
2
2+4a2b2c2d1x̃1x̃2+2b1b2d

2
2x̃
2
2+

+2b22d1d2x̃1x̃2−2a1a2d
2
2x̃
2
2−2a

2
2d1d2x̃1x̃2−2b1b2c

2
2x̃
2
2−2b

2
2c1c2x̃1x̃2)+

+m[A2,A2,A2,A2](a
2
2c
2
2x̃
2
2+ b

2
2d
2
2x̃
2
2−a

2
2d
2
2x̃
2
2− b

2
2c
2
2x̃
2
2+4a2b2c2d2x̃

2
2)
}

(3.2)

σ2Im(Ix)(x,y,z) =K
2
{
m[A1,A1,A1,A1](−2a1b1c

2
1x̃
2
1+2a

2
1c1d1x̃

2
1+

+2a1b1d
2
1x̃
2
1−2b

2
1c1d1x̃

2
1)+m[A1,A1,A1,A2](2a

2
1c1d2x̃1x̃2+

+4a1c1d1a2x̃
2
1+4a1b1d1d2x̃1x̃2+2b1d

2
1a2x̃

2
1+2a

2
1d1c2x̃1x̃2+

−4a1b1c1c2x̃1x̃2−2b1c
2
1a2x̃

2
1−2a1c

2
1b2x̃

2
1−2b

2
1d1c2x̃1x̃2−4b1c1d1b2x̃

2
1+

−2b21c1d2x̃1x̃2+2a1d
2
1b2x̃

2
1)+m[A1,A1,A2,A2](4a1c1a2d2x̃1x̃2+

+4a1d1a2c2x̃1x̃2−4b1c1a2c2x̃1x̃2+4b1d1a2d2x̃1x̃2+4a1d1b2d2x̃1x̃2+

−4b1c1b2d2x̃1x̃2−4a1c1b2c2x̃1x̃2−4b1d1b2c2x̃1x̃2+2a
2
1c2d2x̃

2
2+

+2c1d1a
2
2x̃
2
1−2a1b1c

2
2x̃
2
2−2c

2
1a2b2x̃

2
1+2a1b1d

2
2x̃
2
2+2d

2
1a2b2x̃

2
1+

−2b21c2d2x̃
2
2−2c1d1b

2
2x̃
2
1−2d1b

2
2c2x̃1x̃2)+m[A1,A2,A2,A2](4a1a2c2d2x̃

2
2+

+2d1a
2
2c2x̃1x̃2+2a1b2d

2
2x̃
2
2+4d1a2b2d2x̃1x̃2+2c1a

2
2d2x̃1x̃2−2b1a2c

2
2x̃
2
2+

−4c1a2b2c2x̃1x̃2−4b1b2c2d2x̃
2
2−2c1b

2
2d2x̃1x̃2−2a1b2c

2
2x̃
2
2+2b1a2d

2
2x̃
2
2)+

+m[A2,A2,A2,A2](a
2
2c2d2x̃

2
2+2a2b2d

2
2x̃
2
2−2a2b2c

2
2x̃
2
2−2b

2
2c2d2x̃

2
2)
}

where x̃1 =x−x1, x̃2 =x−x2.



Acoustic intensity vector generated by vibrating set... 417

The probability density function for realistic cases can be obtained by the
assumption of a randomprocess or as a result of analysis of randomvibrations
of structures, e.g. by the finite element method.

3.2. Formulation by probability density functions

In this attempt, probability density functions of a random process which de-
scribes amplitudes of vibrations of the sub-areas are assumed. For example,
if the process is a normal (Gaussian) one with the zero middle value and
dispersion σ2i =A

2
i , the probbility density functions have the form

fi(zi)=
1

Ai
√
2π
e
−

z
2
i

2A2
i (3.3)

Then the whole formulas should be integrated over the probability spaces.

3.3. FEM analysis of random vibrations

For realistic cases, the finite element method is often applied to analysis of
vibrations of a randomly excited structure.As a result, some probability func-
tions, such as variance or covariance of amplitudes for each finite surface ele-
ment are obtained. Then, based on these functions, the analysis of radiation
is performed having in mind the discusssed general formulation.

4. Application of the method for FEM analysis of random

vibrations

Let us consider randomly excited vibrations of a square steel plate with thick-
ness of 2mm and edge length of 1m. The excitation are distributed external
surface loadings of a random type with a constant power spectral density of
0.1N/Hz. The analysis is preformed for the narrow frequency band arround
the basic natural frequency 9.86Hz. The analysis of randomvibrations is done
by finite element package Ansys. The resultant random functions are a base
for further analysis in the above described way. In the analysis, the disper-
sion of real and imaginary parts of the acoustic intensity vector component
perpendicular to the plate in chosen control points is calculed.

Based on these values, levels of dispersion of real and imaginary parts of
the acoustic pressure are estimated on the assumpion of the plane acoustic



418 M.S. Kozień

Table 1. Values of dispersion levels of the acoustic pressure (real and imagi-
nary parts)

Distance h [m] Lσ2
Re(p)
[dB] Lσ2

Im(p)
[dB]

1.0 128.0 132.7

2.0 107.8 108.7

5.0 80.5 68.9

10.0 59.7 57.4

Fig. 2. Position of the control point P

wave (2.11). These values are shown in Table 1 for a few control points whose
positions are schematically shown in Fig.2

The obtained values are realistic and give good interpretation of the aco-
ustic near- and far-field too. Unfortunately, the applied numerical procedures
of the hybrid method in random formulation are rather complicated and not
easy algorithmised.

5. Conclusions

Thepresented analysis reveals the theoretical background and somenumerical
simulations of the generalisation of the previously formulated hybrid method
for the case of random vibrations of plates. The results show the possibility
of application of the method, but the obtained formulas are complicated and
they are not easily alghorithmised. The main idea of the hybrid method for
deterministic cases is the possibility to easily estimate the acoustic intensity
or pressure with no need to model the acoustic medium. Unfortunately, this
idea does not hold in the presented random formulation.



Acoustic intensity vector generated by vibrating set... 419

References

1. ISO 11205: Acoustics – Determination of emission sound pressure levels ”in
situ” at the work station and at other specified positions using sound intensity,
2003

2. Kozień M.S., 2005, Hybrid method of evaluation of sounds radiated by vi-
brating surface elements, Journal of Theoretical and Applied Mechanics, 43, 1,
119-133

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– Mechanics, 331, Politechnika Krakowska,Kraków [in Polish]

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6. Kozień M.S., Nizioł J., 2007, Acoustic radiation of plates with randomly
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(Eds.),Warszawa [in Polish]

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[in Polish]

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averaged energy transfer in acoustic fields, Journal of the Acoustical Society
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420 M.S. Kozień

Wektor natężenia akustycznego generowany przez układ małych płaskich

elementów drgających z losowymi amplitudami

Streszczenie

Wartykule omówiono rozszerzeniemetody hybrydowej oszacowania dźwięku pro-
mieniowanego przez drgające powierzchnie, sformułowanej pierwotnie dla przypad-
ku drgań deterministycznych, na przypadek drgań losowych. Rozważono przypadek
drgań z losowo zmienną amplitudą wwąskim paśmie częstotliwości. Rezultaty analiz
pokazują złożoność uzyskanych formuł w stosunku do zagadnień deterministycznych.
Dlatego też wydaje się, że metoda ta w prezentowanym podejściu nie jest tak uży-
teczna w zastosowaniach inżynierskich, jak to ma miejsce w sformułowaniu determi-
nistycznym.

Manuscript received August 13, 2008; accepted for print November 18, 2008