New functional evaluation methods of sound insulation system using the SEA method and an Lx evaluation criterion

Applied Acoustics 20 (1987) 275-296

New Functional Evaluation Methods of Sound Insulation System Using the SEA Method and an L x Evaluation Criterion Mitsuo Ohta and Shin'ya Kuwahara Faculty of Engineering, Hiroshima University, Shitami, Saijo-cho, Higashi-Hiroshima City, 724 (Japan) (Received 14 November 1985; accepted 2 January 1986) SUMMARY Recently, for the purpose of reducing residential environmental noise, many sound insulation systems have often been improved acoustically by changing their geometrical scales and/or acoustical characteristics. In this paper, new functional evaluation and probabilistic prediction methods for these improvements are theoretically and experimentally proposed in practical expression forms by introducing a few functional parameters. These functional parameters introduced only for the prediction are supported by many physical structural factors closely related to the well-known statistical energy analysis method, and are easily estimated in a preliminary experiment. The estimation procedures developed are based on two error criteria using the actual overall frequency band data. The used least-squares error criterion is the most fundamental method and the Lx evaluation criterion matches the actual situation of estimating the representative evaluation indices. Finally, by using musical sound as an input noise, the effectiveness of the proposed method is experimentally confirmed by applying it to some actual problems.

LIST O F SYMBOLS

E,,E,

Energy densities in transmission and reception rooms Dissipative loss factor of subsystem i Coupling loss factor from subsystem i to subsystem j Modal density of subsystem i 275 Applied Acoustics 0003-682X/87/$03.50 © Elsevier Applied Science Publishers Ltd, England, 1987. Printed in Great Britain r// ~ij ni

276

V,,V t

T,

Mitsuo Ohta, Shin'ya Kuwahara

Volumes of transmission and reception rooms Transmission coefficient of sound insulation system at center frequency f Thickness of panel Relative variable according to improvement of system Reverberation time of reception room

1 INTRODUCTION As is well known, for reducing residential environmental noise in a building exposed to road traffic, aircraft and/or factory noises, many sound insulation systems are often improved acoustically by changing their geometrical scales and/or acoustical characteristics. In this case, because of the arbitrariness of an input noise and the complexity of the human response, it is essential to predict all information on the output noise of the sound insulation system. For example, in the actual evaluation of environmental noise, the (100 - x) percentile level L x is widely used as one of the evaluation indices. In this paper some new functional evaluation and probabilistic prediction methods for this sound insulation system are theoretically and experimentally considered in practical expression forms by introducing a few functional parameters matched to the probabilistic evaluation, especially when the sound characteristics of single-wall and double-wall sound insulation systems are changed by the above improvement work. For the purpose of noise control these functional parameters should be supported by many of physically structural parameters--for instance, in close relation to the well-known statistical energy analysis (SEA). 1'2 Based on the SEA method, the following actual cases are considered as two examples of the above improvement works: (1) changing the thickness of a panel in a single wall and a double wall, and (2) changing the acoustical characteristic of a sound insulation system. However, structural factors used in the SEA method such as the coupling loss factor, the dissipative loss factor and the modal density are not analyzed individually. Rather, a few functional parameters closely connected with the change of output response caused by the improvement work should be developed. Each of these functional parameters is supported by a group of structural factors, so as to get both the physical foundation of the actual control and the flexibility of the probabilistic evaluation. 3

Evaluation of sound insulation system--SEA method and L~ criterion

277

Secondly, for estimating these functional parameters, the criterion function should be formulated carefully in advance, by paying attention to the information on higher-order statistics as well as lower-order statistics (such as mean value and variance), because of the arbitrariness of the actual random phenomena and the complexity of the human response. Two types of new methods for estimating these parameters are discussed in close connection with the practicability of methods. By using the actual data observed by use of an overall frequency passband, the well-known leastsquares error criterion is firstly applied as one of the practical estimation methods, and furthermore an Lx evaluation criterion is newly introduced based on information on the whole probability distribution form of an input noise. Finally, by using musical sound as an input noise, the validity and the effectiveness of the proposed method are also experimentally confirmed by applying it to some problems of predicting sound transmission loss and the cumulative probability distribution of an output noise for single-wall and double-wall type sound insulation systems.

2 THEORETICAL CONSIDERATIONS

2.1 Characteristic improvement of the system through changing the thickness of the panel 2.1.1 Single wall The input-output relation of a single-wall sound insulation system based on the SEA method has already been derived by Price and Crocker 4 as follows (an introductory explanation necessary to the following detailed analysis for the input-output relation is given in Appendix A): Er = I ~]2 "Jr-/~21 "q- ~23 --/123

/~, 2 I//~2 "1-~21 "]- /'/23 --~32 Vt E ~]131/l--~23 /13 -'[- /131 -'[- ~]32 Vr t

(1)

For evaluating the noise reduction based on the input-output relation (eqn (1)), these structural factors are theoretically analyzed in detail. 4,s However, several structural factors in the SEA method are sometimes determined by use of the experimental results, since physical values of these factors (especially the radiation ratio of a panel) are extremely difficult to theoretically evaluate in advance. From the above practical point of view it is especially important to introduce a few functional parameters supported by physically structural factors (as mentioned previously in the introduction) for finding out the practical prediction method.

278

Mitsuo Ohta, Shin' ya Kuwahara

Let the thickness of a panel in a single-wall sound insulation system change from t to t': t' = t(1 + 3)

(2)

where the prime sign indicates the situation after changing. For the purpose of discovering a few functional parameters let us consider how the corresponding parameters can be altered by the change of a thickness. Discovering altered parameters yields the following equations: //~3 = r/23(1 + 6) -1 Y/'32 = t/32(1 + 6) -2

r/~ = r/2(1 + 6) -1 g/31 =//31( 1 + ln(1

//'12 = t/12(1 + ~) -2

+6)/lna2)(1+6) -2

(3)

For a usual case with a 2 >> 1, we have Z'13 -- za3(1 + ln(1 + 6)/ln a2)(1 + 6) -.2 /'/'13 = / / 1 3 ( 1 + ln(1 + 6)/ln a2)(1 + 6) -z

(4)

It is noteworthy that these structural factors have been analyzed by Maidanik, 5 but involve fundamentally many factors which are originally difficult to be estimated reliably and stably. Therefore, in this case only the functional relationship among structural factors, rather than their absolute values, is especially considered for the change of a thickness. Substituting eqns (3) and (4) into eqn (1) yields the input-output relation explicitly reflecting an alteration caused by changing a thickness of the panel: '

'

-1"-q23

Er = --//23 ~__2+ //2 1 "~- //23 7123 -- 712-[-/'/21-{'-7123 --//23

q 2

q2

q21-[-~23 -- ~32

//13 [ - - ~ 2 3

¢ t /']3 -[-7131 -~-~32

/'/12 + (t/2 + q21 + r/23)r/l 3 in (1 + 6)/ln a 2 /713 Vt --~32 /732 ~3 --t-//31 --t-

E,

(5)

Vr

+ (r/2 + t/21 +//23) {r/a(26 + 62) + q31 In (1 + 6)/ln a2) Moreover, dividing eqn (5) by the first term of its numerator clarifies the alteration of the input-output relation after changing:

1 + CA2 V, Et E ~ -A +B A ~ Vr

(6)

279

Evaluation of sound insulation system--SEA method and L~ criterion

where A=

?/2 "Jr-?/21 "~ ?/23 --?/23

-?/32

q3 "31-?/31 "[- ?/32

i/det.

det.

?/2 + ?/21 + ?/2a

q12 I

--?/23

q13]

I

(7)

C-- (?/2 -Jr ?/21 -t- r/23)r/13/det. / In a 2

B = (?/2 + ?/21 "~- ?/23)?/3/det"

A1 = (2~ + 62) + ?/31/(?/3In a2) In (1 + ~)

A2-----ln(1+ b )

As the result of changing a thickness of the panel in a single wall, the effects of the altered characteristics are expressed separately as three functional parameters A, B and C composed complexly of many structural factors, and two variables Aa, A2 depending mainly on the change of thickness. The second term of A1 can be neglected on practical grounds, because it is much smaller than the first term as shown in the experimental consideration (even if its term is not ignored, however, it is possible to evaluate each structural factor of the second term as indicated before changing the thickness of the panel). From the above situation, provided these functional parameters supported by many structural factors are estimated by some estimation methods in advance, all of the statistics of output noise in a single-wall sound insulation system excited by arbitrary input noise can be predicted after a thickness of the panel is arbitrarily changed. 2.1.2 Double wall

The input-output relation of a double-wall sound insulation system has already been derived by Price and Crocker 6 as follows (an introductory explanation necessary to the following detailed analysis for the input-output relation is given in Appendix B)" q2t --?/32 0 --?/23 ?/at --?/4-3 0 --?/34 ?/4t 0 --?/35 --?/45

?/12

/

!

q2t--q32

0

0

?/13/]-q23 ?/3, -?/43 -?/s3 : / 0 -?/3, ?/4, -?/54 0

- - q 3 5 --q45

(8)

q5t

Let the thickness of a panel mounted on the transmission room side of the double-wall sound insulation system change from t to t': t' = t(1 + 6)-1

(9)

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Mitsuo Ohta, Shin'ya Kuwahara

In a similar manner, as shown in Section 2.1.1, finding out the altered parameters after changing the thickness yields the following equations: ?/~ - - ?/2(1 + 3) r/i t = ?/2t(1 + 8)

r/ix ?/2,(1 --I- 8) ?/23 = ?/23(1 + 8) ?/'13 = ?/x3(1 + A - A'/ln a2) =

(10)

where A=26+82

A ' = (1 + 3)21n(1 + 6)

(11)

The following equations are also obtained: ?/'12=?/12(1+8) 2

?/~2 = ?/32(1 + 3)2

?/~1 = ?/3x(1 + A - A'/ln a 2 ) ( 1 2 )

Substituting eqns (10)-(12) into eqn (8) yields the input-output relation after changing the thickness of the panel: ?/2t --?/32 0 --?/23 ?/3t --?/43 0 --?/34 q4t 0 -?/35 -?/45

?/12 I I ?/13 [ ~ ]

+ ?/34 --?~at r/ta(?/t2A + ?/2t(A_ N/lna2) )

qa5

E,= q2, -?/32

-?/23

0

q,5

0

?/3t -?/43 -?/53

0

-?/34

0

-?/35 - q 4 5 +

?/4t

-- ?/45

V'E t

v,

]

q4t -?/54]

I

q5tll

- - ?/54

,,

?/St' :t[,?/2 + /'121)?/32A + ?/2t?/31(A -- A'/ln a2) )

and I + C A 2 Vt

(13)

E ~ - A + BA 1 V~Et where A=

B----

0 ?/2t --?/32 0 ?/3t --?/43 --?/53 -?/23 0 --?/34 ?/4t --?/54 0 --?/35 --?/45 ?/5t

?/2t --t/32

det./det.

?/3t --?/s* ((?/2 + ?/20?/32 + ?/2t?/31)/det" -- ?/45

?/st I

C = '?/34 --?/4t /?/13(?/12 + ?/2t)/det. ?/35 ?/45'[

0

?/,2 I

A --~23 ?/3t --?/43 0 -- ?/34 ?/,*~ 0 --?/35 --?/45 (14)

Evaluation o f sound insulation system--SEA method and L~ criterion

q2t~31

AI = (26 + 62) - ((?/2 + ?/20r/32 + r/2,r/31) In a2

281

(1 + 6)2 In(1 + 6)

A2 = (26 + 62) - r/2,{(r/l 2 + ?/2,)In a2} -~(1 + 6) 2 In (1 + 6)

2.2 Characteristic improvement of the system through changing the sound absorption of the reception room 2.2.1 Changing the absorption of the reception room with a single wall Let the reverberation time of the reception r o o m change from Tr to T;: T" = T~(1 + 6 ) - '

(15)

Developing an altered parameter gives

?/~ = ?/3(1 -I- 3)

(16)

Substituting eqn (16) into eqn (1) yields the i n p u t - o u t p u t relation after changing the reverberation time of the reception room:

Er-

I q2 "Jr-q21 -Jr-?/23 q12 I --?/23 q13 ?/2 -{- ?/21 "F-?/23 --1732 I ..l_(~2 .~_/721 ..~_~23)?/3~ Vrr Et --/723 ?/3 + ?/31 -~- /732 I

and E, = V,/{(A +

(17)

where B = (?/2 + ?/21 "~/723)?/3/det. and A and det. are defined in eqn (7).

2.2.2 Changing the absorption of the reception room with a double wall In a similar m a n n e r to that shown in Section 2.2.1, changing the reverberation time of the reception r o o m with a double wall yields the altered i n p u t - o u t p u t relation: E, = VJ{(A + Bg)V,}E,

(18)

where

B = (q2t?/3tq4t -- qZ3q32q4t -- q2tq34?/43)q5/det. and A and det. are defined in eqn (14). As the result of changing the reverberation time of the reception r o o m with a single wall or a double wall, the effects of the altered characteristics are expressed separately in the form of two functional parameters A, B and a variable c5.

Mitsuo Ohta, Shin'ya Kuwahara

282

2.3 Estimation method In this section we will confine our study to only estimating the functional parameters in Section 2.1.1. We can apply it to the other cases above in a similar manner. The above relations (cf. eqns (6), (13), (17) and (18)) can be evaluated by use of experimental data within the well-known finite frequency bandwidth such as octave or one-third octave bands. However, using overall frequency band data is fundament~illy necessary to evaluate several types o f statistical indices or a whole probability distribution form (such as an Lx index) originally related to its time pattern; it is also convenient for the actual case with no use o f a bandpass filter.

2.3.1 Least-squares error criterion For using overall frequency band data affected by a background noise in the reception room, the input-output relation is directly defined as N

Ero A = ) ' o~iEti + v

(19)

i=1

where

1 + CiA 2 ~ ~i - Ai + BiA,

(20)

Subscripts OA and i indicate an overall frequency band and the ith frequency band (i = 1, 2,..., N), respectively. Now let us introduce a natural condition with mutual independency between JEt and v. Furthermore, let us assume the statistics of v are already known and, if necessary, E t and E r are simultaneously observed. The well-known least-squares error criterion is a fundamental and simplified method for estimation; it is defined as N

I(EroA-~iEti-v)21

~min

(21)

i=1

Parameters % are easily estimated by use of the well-known method of least squares:

I: l/ ~'

~2 =

F (E}I> (Et'E'2}
(E}2}

• ""

(Et2E,N>[

-

-

@>-

.

'

L(E,;E,1) (EtNEt2) ...

(E}N) J

_-- (EtN>
Evaluation of sound insulation system--SEA method and Lx criterion

283

On the other hand, especially for the actual sequential observation and the use of a digital computer, using a stochastic approximation method 7 proposed by Robbins and Monro is appropriate. That is, by first differentiating eqn (21) with respect to ~i and obtaining an extreme value, we directly have N

i=1

However, in this case an averaged value (v) is used instead of an instantaneous value v for an approx!mated evaluation, since an instantaneous value of v is unknown. Consequently, parameters ~i are estimated by repeating the following successive estimation algorithm:

+

/ 1)l L Nik)/

LE,;(k)__I (24)

where k indicates the discrete time of successive estimation. F(k) is defined as

F(k) = diag [yl(k), y2(k),..., ys(k)] and 7i(k) is a positive gain series satisfying the conditions of Robbins and Monro:

k=l

k=l

For example, let us adopt the well-known simple expression:

71(k)= Gi/k

(G/arbitrary positive number)

(25)

2.3.2 L x evaluation criterion In this section, for considering another algorithm with which the parameters ~i can be estimated sequentially, an Lx evaluation criterion (i.e. ( 1 0 0 - x ) percentile level) directly connected with the probability distribution form is newly introduced as

((i _fix, P,.(X)dXI

~i}=0

(26)

Mitsuo Ohta, Shin' ya Kuwahara

284

N where

(i A_ 1 -- xl/lO0, ~ A=) ' ~iE. and an energy density lxi is defined by i=1

lx i = 1 0 ( L x , - 120)/1o

Applying a stochastic approximation method to eqn (26) yields the successive estimation procedure:

" - ftx"k)p¢(X;k)dX

L~N(;-I-11

L~Nik)/

C t'<~(k)

I

:,x,,,(k)"

1

-~N- J

P¢(X; k) dX'J

where estimation values lxi(k) at time k are numerically calculated from observed values of E,o a. Here i the Iprobability! density' function Pc(X; k) can be expressed by use of the nth-order moment statistics: N

(¢") =

~iEti

(n = 1, 2,...) Et

i=1

based on observed values of Eti and the well-known Gram-Charlier A-type series 8 defined as 1---~-- exp ( (X2~)2 } [1 + ~-~j f l . H . ( ~ - ~ ) l P~(X) - x/2re a l

(28)

n=3

with # = (~), a 2 = ((~ -/x) z) and

k-g-// Using the definition of an Hermite polynomial expression: H.tX) = ( - 1)" exp ( T J \ ~ J

exp - T

and employing the probabilistic measure-preserving transformation with substituting Y = (X-/~)/a into eqn (28) yields the following equation: 1

P(Y)=~exp

{ y2}'+2 Ix/-'- (d)" { ~} --

exp --

Evaluation of sound insulation system--SEA method and L x criterion

285

and its cumulative distribution function: Q(Y) &

P(y) dy --at) O0

=~2~fr

e x p ~ ' _ Y 2 ~ d y _ ~ e1x p _~ ( 23

l(- T ;Y2] S fl.H.-I(Y) n=3

(29)

Consequently, the successive estimation procedure based on the Lx evaluation criterion can be obtained from eqns (27) and (29).

2.3.3 Identification of functional parameters For identifying a few functional parameters in each frequency band, by using the above estimation methods or the actual data of noise reduction, three parameters ~io, ~il and ~2 for three kinds of thickness of panels (t o, tl and t2) can be first determined. Since the thickness of the panel is already known, two variables, A1 and A2, are calculated as Alo, A11, A12 and A20, A21, A22 from eqn (7). By substituting these values into eqn (20), functional parameters Ai, B~ and C~ can be identified through solving the following equation:

(30) L~I2 ~i2A12

-- A22Vt/V,A

Ci

3 EXPERIMENTAL CONSIDERATION For confirming the effectiveness of the proposed method, these prediction methods are experimentally applied to the actual evaluation of the sound insulation improvement resulting from the change in panel thickness in single-wall and double-wall systems and on the change in reverberation time of a reception room with a single wall.

3.1 Arrangement of equipment Figure 1 shows a block diagram of the experimental arrangement for measuring a single-wall system. A musical noise recorded in advance on a data recorder has been supplied as an input noise via an amplifier to a loudspeaker in the transmission room. For realizing the situation of a residential environment, a musical sound has been adopted. Rock music

286

Mitsuo Ohta, Shin'ya Kuwahara

L°udspeake~r£ / / Transmission // Data t

//

Level t Sound Meter

Recorder

(~ Re~zli°n White Noise Generator

l/

Pass

Filter

Fig. 1. A block diagram of the experimental arrangement for measuring a single-wall sound insulation system. has been used, and its power spectra have mainly spread from 200 Hz to 3000 Hz. A white noise limited from 180 Hz to 2800 Hz has been supplied in the reception r o o m as a background noise with the level set equally to the transmitted noise level in this room. Furthermore, these noises observed in the transmission and reception rooms have been simultaneously recorded on the data recorder via sound level meters, and the noise in the transmission r o o m has been filtered by use of octave-band filters centered at 250 Hz, 500 Hz, 1 kHz and 2 kHz (i.e. N = 4 in eqn (19)).

3.2 Changing the thickness of the panel in a single-wall system First, we identified the functional parameters A, B and C based on three cases with 0"8, 1.2 and 1.5 m m thick aluminum panels clamped between the transmission and reception rooms. Moreover, we predicted the sound transmission loss and the output cumulative probability distribution for a single wall with a 2.0 m m thick aluminum panel by taking a case with a 1.2 m m thick panel as standard. For evaluating A 1 in eqn (7), let us consider numerically the structural factors by a rough calculation at a low frequency band centered at 250 Hz: q 3 1 = 3 " 7 x 10 -4, r/3=3"1 × 10 -3, a2=6-1. In this case with 6 = 0 " 6 7 ( t ' = 2-0mm), the following values are easily obtained: (26 + 62): ~/31ln(1 +

6)/q31na2 =

1:0.019

Since this case is the worst case in this experiment, we can in practice A 1 - 26 + 6 2. Accordingly, in the ith frequency band, three kinds of parameters ~i, corresponding to three kinds of panel thickness, have been estimated by use

approximate A 1 as

Evaluation of sound insulation system--SEA method and L x criterion

287

of either eqn (22) or eqn (24) under coexistence with a background noise. Moreover, three functional parameters A~, B~ and C~ have been identified by using eqn (30) with use o f already known values o f the panel thickness. Then, after calculating a predicted value of a parameter ~i based on these functional parameters and eqn (20) for a 2.0 m m thick panel case, we can theoretically predict the transmission loss: T L i = - 10 log ~i - 10 log ( A , J A v )

where A,i is the total sound absorption in the reception r o o m and A~ is the surface area o f the panel. By using either eqn (22) or eqn (24), we can also obtain the estimated parameter ai and its transmission loss for the 2"0 m m thick panel case under coexistence with a background noise. I

'~

l

I

30

C 2C

e,

I

J I

I

i

250

500

1000

I

2000 [ Hz]

Fig. 2. A comparison among theoretically predicted values, theoretically estimated values and experimentally observed values for the transmission loss of a single wall with 2.0 mm thick aluminum panel. ExperimentallyIobservedtvalues: (O) by use of the actual musical noise, (O) by use of the band-limited white noise. Theoretically evaluated values by use of the least-squares error criterion and a stochastic approximation method: ( ) predicted values, (. . . . ) estimated values. Figure 2 shows a comparison among theoretically predicted values, theoretically estimated values and experimentally observed values for the transmission loss o f a single wall with 2-0 m m thick panel by use o f the wellknown least-squares error criterion and a stochastic approximation method (see eqn (24)). Experimental values Q and O have been respectively observed for two cases with use of a musical sound noise and an octaveband white noise. Furthermore, using the above predicted and estimated parameters ~, we can predict the output probability distribution curves by use of eqn (19) for a 2.0 m m thick panel case when another, different, musical noise has been supplied in the transmission room. Figure 3 shows a comparison between theoretically predicted curves and experimentally sampled points for the

288

Mitsuo Ohta, Shin'ya Kuwahara

0 50

I

I

I

55

60

65

|

[dBl

7O

[ dB ]

70

(a) I

!

!

]

>

0 50

// /i// 55

60

65

(b) Fig. 3. A comparison between theoretically predicted curves and experimentally sampled points for the cumulative probability distribution of a single wall with 2.0mm thick aluminum panel. Theoretically predicted curves are lined as ( ) by the predicted parameters and ( ) by the estimated parameters; experimentally sampled points are marked as (Q). Experimentally observed data contaminated by a background noise are marked as (1), (2) and (3) corresponding to each case of panels with 0"8, 1.2 and 1.5 mm in thickness used in the preliminary experiment for identifying functional parameters A, B and C. Experimental result marked as (4) corresponds to a case of 2-0mm thick panel used in the preliminary experiment for estimating parameters ~ti. (a) Method of least squares; (b) stochastic approximation method. c u m u l a t i v e p r o b a b i l i t y d i s t r i b u t i o n o f the t r a n s m i t t e d s o u n d level fluctuation. A solid line and a d a s h e d line s h o w the theoretically p r e d i c t e d distribution curves b y use o f theoretically predicted p a r a m e t e r s a n d estimated p a r a m e t e r s respectively, and • shows a c u m u l a t i v e p r o b a b i l i t y distribution curve o f e x p e r i m e n t a l l y sampled points. In Fig. 3(a) the leastsquares m e t h o d (see eqn (22)) has been applied; in Fig. 3(b) a stochastic a p p r o x i m a t i o n m e t h o d (see eqn (24)) has been applied.

Evaluation of sound insulation system--SEA method and Lx criterion I

!

•i/ ¢ t 50

!

,,777F

.o

'ol

I

289

:

55

? 60

65

70

dB]

Fig. 4. A comparison between theoretically predicted curves and ex ~erimentally sampled points for the cumulative probability distribution of a single wall with 2'0mm thick panel. Experimentally observed data, in the absence of any background noise, are marked as (1), (2) and (3) corresponding to each case of panels 0.8, 1-2 and 1-5 mm in thickness used in the preliminary experiment for identifying functional parameters by use of the Lx evaluation criterion.

Figure 4 shows a comparison between theoretically predicted curves and experimentally sampled points for the cumulative probability distribution by use of the Lx evaluation criterion (summing up to n = 4 in eqn (29)). The number of data points involved in calculating Lxi of the transmitted noise and statistics of the input noise used in eqn (27) was fifty (in this case, however, the experimental data observed in the reception room have been measured in the absence of any background noise). As a concrete example of Lxi (i= 1,...,4), L 5, L10, Lso and L75 have been especially adopted. In Figs 2-4, all of the theoretically predicted curves are in good agreement with the experimentally sampled points. 3.3 Changing the thickness of a panel in a double-wall sound insulation system

Secondly, we used a double-wall sound insulation system with 1.2 mm thick aluminum panel mounted on the reception room side. We changed panels mounted on the transmission room side to 0.8, 1.2 and 1.5mm thick aluminum panels, only for identifying three functional parameters A, B and C. Moreover, we have theoretically predicted the transmission loss and the cumulative probability distribution for a double wall with 2.0 and 1.2 mm thick aluminum panels. For evaluating At and A 2 in eqn (14), let us consider numerically the structural factors by a rough calculation at a low frequency band centered at 250Hz. Here we have q 2 = 6 " 7 x 1 0 -2 , q 1 2 = l . 6 x 10 -a,

290

Mitsuo Ohta, Shin' ya Kuwahara

?]21 = 4 . 8 x 10 -3, ?]zt= 8.2 x 10 -2, ?]31=7.8× 10 -3, ?]32=1.6× 10 -3, a2 = 6"1. In this case, with 6 = - 0 . 4 ( t ' = 2.0 mm), we obtain the following evaluation: (26 +

62):

?]2tq31

(1 + 6) 2 In (1 + 6) = 1:0"0083

((?]2 -{- ?]21)?]32 + ?]2t?]31) In a2

(26 + 62):

?]2t/(?]12 +

?]2,) In a 2 = 1:0" 16

Since this case is the worst case in this experiment, we can practically a p p r o x i m a t e A 1, A 2 as A 1 = A 2 - 26 + 62. F i g u r e 5 shows a c o m p a r i s o n a m o n g theoretically predicted values, I

I

I

i

4c

i

3C

3 o

10

I

1

I

I

250

500

1000

2000

Hz]

Fig. 5. A comparison between theoretically predicted values and experimentally observed values for the transmission loss of a double wall with 2.0 and l ' 2 m m thick aluminum panels.

1

I

1

o n

/.7

E O 0 45


/-" f / 7 /

o

7

5 / ,/J2i/

50

55

60

I

65

dB]

Fig. 6. A comparison between theoretically predicted curves (by use of the least-squares method) and experimentally sampled points for the cumulative probability distribution of a double wall with 2.0 and 1"2 mm thick aluminum panels.

Evaluation of sound insulation system--SEA method and Lx criterion

291

theoretically estimated values and experimentally observed values for the transmission loss of a double wall by use of the least-squares method. Figure 6 shows a comparison between theoretically predicted curves and experimentally sampled points for the cumulative probability distribution by use of the least-squares method. In Figs 5 and 6 all of the theoretically predicted curves are in good agreement with the experimentally sampled points except at a low frequency band in Fig. 5. Part of this disagreement is probably due to the well-known peculiar resonant phenomena at this low-frequency region (as is well known, this situation is usually difficult to evaluate by use of a standard SEA method9). f

!

•7

0 50

f

I

oI

I

.a

55 I

1

60

65

!

i

(a)

[dB]

70 I

g

Jl

o~f

(3\6 (1) 61 6

55

60

8 o

45

50

[dB]

65

(b) Fig. 7. A comparison between theoretically predicted curves and experimentally sampled points for the cumulative probability distribution of a single wall after inserting two absorbing plates into the reception room. Experimentally observed data contaminated by a background noise are marked as (l), (2) and (3) corresponding to each case of no, one and two absorbing plates used in the preliminary experiment. (a) A case with use of 1.2 mm thick aluminum panel; (b) a case with use of 3.0 mm thick vinyl chloride panel.

292

Mitsuo Ohta, Shin'ya Kuwahara

3.4 Changing the acoustic property of the reception room Thirdly, as an example corresponding to the improvement work in Section 2.3, we have changed the reverberation time of the reception room with a single wall by putting sound-absorbing materials (glass wool, 900 × 600 x 50 m m 3 a plate) into the room. We have put no plate, and one absorbing plate, for identifying the two functional parameters A and B. Moreover, we have predicted the cumulative probability distribution in a case when two absorbing plates have been put in the reception room. Figure 7 shows a comparison between theoretically predicted curves and experimentally sampled points for the output cumulative probability distribution in a single wall with a 1.2mm thick aluminum panel or a 3.0 m m thick vinyl chloride panel. In these cases two functional parameters A and B have been identified by two estimated parameters ~i by use of the least-squares error criterion and a stochastic approximation method. In Fig. 7 all of the theoretically predicted curves are in good agreement with the experimentally sampled points.

4 CONCLUSIONS In this paper, new functional evaluation methods for predicting the sound transmission loss and the output cumulative probability distribution of the sound insulation system have been theoretically and experimentally proposed in close connection with the well-known statistical energy analysis method. The proposed methods have been based on purposely introducing a few functional parameters supported by many physical structural factors and in practice applying two kinds of error criteria for estimating these functional parameters. First, the altered input-output relation of the system, after an acoustical improvement by changing its geometrical scales or its acoustical characteristics, has been explicitly expressed by only a few functional parameters. These parameters have also been complexly composed of many of physical structural factors in close relation to the statistical energy analysis method. For the purpose of prediction and noise control, however, it has in practice been sufficient to estimate only a few functional parameters, and not many structural factors. Secondly, the estimation procedures have been found based on two types of error criteria based on the well-known least-squares error and L:, evaluation index. Finally, the present proposed method has been experimentally confirmed by applying it to the actual data observed in the coupled reverberant rooms, especially in three cases: (i) changing the thickness of the panel of a

Evaluation of sound insulation system--SEA method and L x criterion

293

single wall, (ii) changing the thickness o f the panel o f a double wall, and (iii) changing the reverberation time of the reception r o o m with a single wall. Since the present method is still being studied, there remain m a n y problems. For example, we must apply the proposed method to m a n y other actual acoustic systems. Furthermore, we must explicitly express the sound characteristic o f other improved acoustic systems by a few functional parameters, and must discover more simplified estimation methods for practical use derived through the approximation for this method and/or other estimation methods under several types of error criteria.

ACKNOWLEDGEMENT We would like to express our grateful thanks to Dr K. Hatakeyama, Dr H. Iwashige and Professor S. Yamaguchi for their helpful assistance.

REFERENCES 1. R. H. Lyon and G. Maidanik, Power flow between linearly coupled oscillators, J. Acoust. Soc. Amer., 34 (1962), p. 623. 2. L. L. Beranek (ed.), Noise and vibration control, McGraw-Hill, New York (1971), p. 296. 3. M. Ohta and S. Kuwahara, A functional method of evaluation for the characteristic improvement of a single-wall type sound insulation system--an identification and probabilistic prediction based on the SEA method and the least-squares error criterion, J. Acoust. Soc. Jpn., 41 (1985), p. 149 (in Japanese). 4. M. J. Crocker and A. J. Price, Sound transmission using statistical energy analysis, J. Sound Vib., 9 (1969), p. 469. 5. G. Maidanik, Response of ribbed panels to reverberant acoustic fields, J. Acoust. Soc. Amer., 34 (1962), p. 809. 6. A.J. Price and M. J. Crocker, Sound transmission through double panels using statistical energy analysis, J. Acoust. Soc. Amer., 47 (1970), p. 683. 7. H. Robbins and S. Monro, A stochastic approximation method, Ann. Math. Stat., 22 (1951), p.400. 8. H. Cramer, Mathematical methods of statistics, Princeton University Press, Princeton (1946), p. 222. 9. H. Iwashige and M. Ohta, A study on sound transmission loss of double-walls having several types of geometrical section by use of the improved statistical energy analysis method, J. Acoust. Soc. Jpn., 36 (1980), p. 447 (in Japanese).

A P P E N D I X A: S I N G L E - W A L L S O U N D I N S U L A T I O N SYSTEM As is well known, the SEA model of a single-wall sound insulation system is considered to consist of three coupled resonant subsystems: transmission

294

Mitsuo Ohta, Shin'ya Kuwahara

in

111 ! _ ~ T r a 1n s m i s4I sion~ Et

~" d II1

'HI3

11

~in

I I

HI3|

1..~Translmission1112~ ' r

$_d 112

$-d E r II~

Et

'~ d HI

i~35

[134 ..... 1145 Reception

S~Id "2

'~iid "4

~d ~I3

'~iid n5

Fig. 8. A block diagram representing the power flow. (a) Single wall consisting of three subsystems: (1) transmission room, (2) panel and (3) reception room; (b) double wall consisting of five subsystems:(1) transmission room, (2) panel, (3) air cavity, (4) panel and (5) reception room. room, panel and reception room, as shown in Fig. 8(a). In that case the power-balance equations are directly expressed as =

+

+

0 =

-

+

0 =

-

n,3

-

n.

(A.1) where rI~" is an input power to the subsystem 1, Ha is a power dissipated in the subsystem i and rIij is a power lost by the subsystem i through coupling to the subsystem j. The fundamental property in the SEA method can be given as .I-Iij = Ei(_oqi j - Ejt_o?l j i

I-I d ~ E,ifoqi

ni?]i j = rlj?l j i

(A.2)

where E i is a total energy stored in the subsystem i. In order to evaluate the noise reduction of a single-wall sound insulation system, the following input-output relation can be easily obtained:

I

F]2 -[- /721 q!- /723

--q23 E r = [ q2 + q21+ ~/23 I

-q23

/']12 [ /713

V, Et

-732

(A.3)

q3 + q31 + q32

Its detailed expression with respect to the many structural factors is shown in Appendix C.

A P P E N D I X B: D O U B L E - W A L L S O U N D I N S U L A T I O N SYSTEM As is well known, the SEA model of a double-wall sound insulation system is considered to consist of five coupled subsystems: transmission room, panel, air cavity, panel and reception room, as shown in Fig. 8(b). In a similar manner, using the power-balance equations: [-liln -- lid + [112 + [I13

0 = l i d - - I'I12 "~ 1"-123

0 = I-I~ -- 1-I13 -- I-I23 - ' I ] 3 4 Jr- []35

0 = li d - FI34 + [I35

0 = ri d - FI3, - FI¢s

(B.1)

Er

295

Evaluation o f sound insulation system---SEA method and L x criterion

yields the following i n p u t - o u t p u t equa~tion: 02t --032

E, =

0

0121 I

--023

03t --043 013 I

0

-034

04~

0

--035 --q45

°0 1[J

02t --032 --023

0

0

03t --043 --053 034-

04t

0

--

0

--035 --045

(B.2)

054

--

r/st

where 02t ~-- 02 + 021 -~ 023

03t ~-- /73 -Jr-031 + 032 "31-034 "~-035

q4t ~ ?]4 "[- ?/43 "[- 045

?/5t ~ q5 "3t-053 "1- 054

(B.3)

A P P E N D I X C: E V A L U A T I O N O F S T R U C T U R A L F A C T O R S 1 Double-wall sound insulation system

The coupling loss factors are given by (i=2, j=l

Oij = pCtTrad,i/~ppt = ~2pCtrrad.i/coPPt Oij

[.pCiTrad,i/ogpp t

qu = cApri j~4c° Vi

orj=4,

i=5)

(ff~i)

j = 3)

"cij = (1 + a2)/a 2

(C.1) (C.2)

ak = coppt/2pc

( i = 1, k = 2, j = 3 or i = 5, k = 4 , j = 3)

(C.3)

where pc is an acoustic characteristic impedance of air, p p and A v are volume density and surface area of the panel, araa, ~ and f~ are radiation ratio and critical frequency of the panel, and V~ is volume of the r o o m corresponding to the subsystem i. The dissipative loss factors are given by rh = 2.2/fT~

(i = 1, 5)

(C.4)

c/Ogppt

(i = 2, 4)

(C.5)

ni =

~cS~/ncoV o ( f < fa = c/2d) r13 = ~cS~/4co V o ( f > A )

(C.6)

where T i is a reverberation time of the r o o m corresponding to the subsystem i, and S, ~, Vg, d are respectively area of the b o u n d a r y surface, averaged sound absorption coefficient, volume, width of the air cavity in a double wall.

296

Mitsuo Ohm, Shin'ya Kuwahara

The modal densities are given by n i = o9 2 E / 2 g 2 c 3

(i = 1, 5)

(C.7)

n, = x / ~ . / 2 r c c L i t

(i = 2, 4)

(C.8)

~ ogA p/2rtc 2 n 3 = [o91 Vo/2rcZc 3

(ff~) where Czi is a velocity of longitudinal wave in the panel corresponding to the subsystem i. 2 Single-wall sound insulation system The coupling loss factors/721 (~- /'123) and r/i3 are given by eqns (C.1) and (C.3) respectively. The dissipative loss factors ?/2 and ?/3 are given by eqns (C.5) and (C.4) (i = 3 in eqn (C.4)). Furthermore, the modal densities/'/2, r/i, /73 are given by eqns (C.8) and (C.7) respectively (i = 1 or i-- 3 in eqn (C.7)).