Measuremet ucertaity of the soud absorptio coefficiet Aa Izewska Buildig Research Istitute, Filtrowa Str., 00-6 Warsaw, Polad a.izewska@itb.pl 6887
The stadard ISO/IEC 705:005 o the competece of testig ad calibratio laboratories requires that these laboratories shall apply procedures for estimatig the ucertaity of their measuremet results. Oe of the possibility is to evaluate the budget of ucertaity, takig ito accout all compoets that cotribute sigificat ucertaity to the fial result. I case of the soud absorptio coefficiet measuremet, carried out accordig to the stadard EN ISO 54:00, the overall ucertaity is first of all iflueced by the reverberatio times T, T ad the power atteuatio coefficiets m ad m, calculated accordig to the ISO 96- stadard ad represetig the climatic coditios i the reverberatio room. I spite of very little differece betwee the values m ad m represetig the chage of climatic coditios (usually, it is the case i laboratory, expoetial form of the coefficiet s fuctio causes that the ucertaity of measuremet results icrease with frequecy very fast. Particularly for the high frequecies, the values of ucertaity are so importat that the evaluatio of the soud absorptio coefficiet is practically ot possible. Itroductio Because of lack of ucertaity evaluatio based o iterlaboratory validatio approach, it has to be carried out by the laboratory itself. Usig the geeral methods specified i GUM[], first of all we eed to establish a relatioship betwee the mesurad Y ad other quatities X, X,...X (iput values through a fuctio f, called the measuremet equatio Y f X, X,... X ( ( Such equatio must express ot oly the physical relatio from which we ca obtai the value of mesurad Y but also it should be accompaied by a quatitative statemet of its ucertaity which arises from the ucertaities of the iput values of directly measured quatities x, x...x beig the estimates of X, X,...X. I geeral, ucertaity compoets (each of them - represeted by estimated stadard deviatio, termed stadard ucertaity ad oted u(xi or, shortly, ui are categorized accordig to the method used to evaluate them. There are two methods of evaluatig stadard ucertaity: - type A is based o ay valid statistical method, for example the stadard deviatio of the mea of a series of idepedet observatios s(xi; i such case the stadard ucertaity u(xi s(xi, - type B takes advatage of a outside source (for example, data provided i calibratio ad/or of a assumed distributio. The stadard deviatio of the estimated measuremet result y, called combied stadard ucertaity u c (y, is obtaied by combiig the idividual stadard ucertaities u(x i usig the usual statistic method of combiig stadard deviatios root sum squares accordig to the formula: u c f ( y u ( xi i x ( i ad calculatig the positive root square of the result. Equatio ( is called the law of propagatio of ucertaity ad the partial derivatives δf/δx i are referred to sesitivity coefficiets. The products of module δf/δx i ad u(x i are usually preseted i a table, as ucertaity budget. This practice is very useful to idetify the domiat terms that cotribute sigificat ucertaity to the result. The measure of ucertaity, that defies a iterval i which the value of the quatity subject to measuremet (the mesurad Y ca be cofidetly asserted to lie, is amed expaded ucertaity U (it meas that oe ca cofidetly believe that the value of Y, estimated by y, lies withi the limits y U Y y + U. The expaded ucertaity is obtaied by multiplyig a combied stadard ucertaity by a coverage factor k depedig o the desired level of cofidece ad the type of statistical distributio. U ( Y ku ( y ( Ucertaity compoets occurrig i measuremets of soud absorptio coefficiet. Reverberatio time The reverberatio time is measured i poits for each frequecy bad. The mea value T f calculated from measuremets is take as the estimated measuremet result. I this case, the stadard ucertaity of T f is equal to the experimetal stadard deviatio s(t f of the mea of a series of idepedet observatios T f, i,.... u( T f s( Tf, i s( Tf c, s (4 Studet s t distributio with freedom degrees is assumed.. Area of sample Assumig the rectagular distributio of estimated measuremet result, with the boudary of ±0,005 m, the stadard ucertaity of sample s area amouts to 0,005m u ( S m. Volume of room 0,009 (5 Assumig the rectagular distributio of estimated measuremet result, with the boudary of resolutio ±0,5 m, the stadard ucertaity of room volume amouts to 6888
0,5m u ( V m 0,887 (6.4 Eviromet factors The measuremet equatios for the evirometal factors like temperature, atmospheric pressure ad relative humidity assume the form: e 0 δe δe δe e e (7 0 + δ e + δe + δe + δe4 - the readig value of the evirometal factor i a room, - the dispersio of sesor idicatios, - the resolutio of sesor idicatios, - the error of sesor idicatios, δe 4 - the ucertaity of idicatios error. The stadard ucertaity of dispersio δe is calculated o the basis of the estimated stadard deviatio of m series of measuremets. u( e Si ( e (8 S i (e the experimetal stadard deviatio of i th series equal to S ( e i m i S ( e The stadard ucertaity of resolutio δe is evaluated with the assumptio of rectagular distributio with the boudary of resolutio supposed, respectively, as b ± 0.05 o C, ± 0.05 kpa or ±0.05%. u ( e b i m, i (9 δ (0 Cotributio i ucertaity of δe ad δe 4 are calculated from the expaded ucertaity U evaluated for ormal distributio with cofidece level 95% ad stated i the calibratio certificate of the device. U ( δe U ( δe u( δ e k ( N,95% U ( δe4 U ( δe4 u( δ e4 k N,95% ( The equatio of propagatio law of ucertaity has the form: u ( 4 e u ( δ e + u ( δe + u ( δe + u ( δe ( Ucertaity of the soud absorptio coefficiet measuremet The soud absorptio coefficiet α s of a plae absorber or a specified array of test objects is calculated usig the formula s A T α (4 S A T - the equivalet soud absorptio area of the test specime, m S - the area covered by the test specime, m The equivalet soud absorptio area of the test specime, A T, accordig to ISO 54:00 [] is give as follows A T A A (5 55,V 4V ( m m ct ct V - the volume of the empty reverberatio room, m T - the reverberatio time of the empty reverberatio room, s T - the reverberatio time of the reverberatio room after the test specime has bee itroduced, s c, c - the propagatio speed of soud i air at the temperature t ad t (respectively, m/s m,m - the power atteuatio coefficiets, calculated accordig to ISO 96-[4] usig the climatic coditios that have bee preset i the empty reverberatio room ad after the test specime has bee itroduced, m - The value of m is calculated from the atteuatio coefficiet α depedet o temperature t, atmospheric pressure p a ad relative humidity h r, accordig to the formula α m 0 lg(e The detailed aalyse of the ucertaity budget leads to a coclusio that the stadard ucertaity of soud absorptio depeds very strogly o the sesitivity coefficiets of humidity, particularly at the high frequecies (see Table. I the previous versio of stadard ISO 54:985 [] a possible chage of climatic coditios was ot take ito cosideratio (m m 0. Table presets the calculatio results of the soud absorptio coefficiet α s ad its stadard ad expaded ucertaities carried out with respect to eviromet factors ad without them. It ca be observed that i spite of very little differece betwee the values m ad m (represetig the chage of climatic coditios, the values of α s calculated accordig to [] i the frequecy bads f 50 Hz are cosiderably greater. Also, the ucertaities of measuremet results icrease with frequecy very fast. Particularly for the high frequecies, the values of ucertaities are so importat that the evaluatio of α s is practically ot possible. 6889
f, Hz α S u c (α S U (α S Sesitivity coefficiets of eviromet factors δα S / δpa δ α S / δpa δ α S / δhr δ α S / δhr δα S / δt δα S / δt 00 0,4 0,009 0,08 0,00000 0,00000-0,000 0,0008 0,0006 0,00085 5 0, 0,04 0,09 0,00000 0,00000-0,0005 0,0007 0,0006 0,006 60 0,54 0,07 0,05 0,00000 0,00000-0,00056 0,00045 0,0007 0,008 00 0,64 0,00 0,04 0,00000 0,00000-0,0008 0,00070 0,0007 0,007 50 0,8 0,0 0,048 0,00000 0,00000-0,000 0,0009 0,00059 0,000 5 0,9 0,05 0,05 0,00000 0,00000-0,0077 0,0074 0,0000 0,008 400 0,94 0,07 0,056 0,00000 0,00000-0,0067 0,0080-0,00064 0,00468 500,0 0,00 0,04 0,00000 0,00000-0,0098 0,0047-0,008 0,0057 60 0,99 0,09 0,040 0,0000-0,0000-0,0067 0,00694-0,00 0,00680 800,00 0,04 0,049 0,0000-0,0000-0,00994 0,09-0,00448 0,008 000 0,96 0,09 0,09 0,0000-0,0000-0,0568 0,0748-0,00568 0,0099 50 0,98 0,06 0,05 0,0000-0,0000-0,0478 0,079-0,0069 0,006 600 0,98 0,0 0,047 0,0000-0,0000-0,0405 0,04464-0,0085 0,04 000 0,96 0,00 0,060 0,0000-0,00004-0,06455 0,06960-0,00 0,048 500 0,99 0,045 0,09 0,00004-0,00005-0,06 0,087-0,085 0,070 50,0 0,068 0,9 0,00005-0,00007-0,6047 0,709-0,0665 0,046 4000,0 0,06 0,7 0,00007-0,000-0,570 0,74-0,079 0,089 5000,0 0,6 0, 0,00009-0,0005-0,9768 0,48-0,05 0,086 600 0,98 0,5 0,56 0,000-0,0000-0,690 0,6540-0,0455 0,058 8000 0,95 0,94 0,804 0,000-0,000-0,96576,064-0,0666 0,0766 Table. Soud absorptio α S with its ucertaities ad sesitivity coefficiets f, Hz Acc. to ISO 54:00 [] Acc. to ISO 54:985 [] Differece m -m α S u c (α S U (α S α S u c (α S U (α S α S 00 0,4 0,009 0,08 0,4 0,009 0,08 0,000-0,00 5 0, 0,04 0,09 0, 0,04 0,09 0,000-0,004 60 0,54 0,07 0,05 0,54 0,07 0,05 0,000 0,000 00 0,64 0,00 0,04 0,64 0,00 0,04 0,000 0,005 50 0,8 0,0 0,048 0,8 0,0 0,048 0,000 0,004 5 0,9 0,05 0,05 0,9 0,05 0,05 0,000 0,000 400 0,94 0,07 0,056 0,94 0,07 0,056 0,000 0,00 500,0 0,00 0,04,0 0,00 0,04 0,000 0,004 60 0,99 0,09 0,040 0,98 0,09 0,09 0,000 0,006 800,00 0,04 0,049,00 0,04 0,048 0,000 0,005 000 0,96 0,09 0,09 0,96 0,08 0,07 0,000 0,000 50 0,98 0,06 0,05 0,97 0,04 0,048 0,000 0,008 600 0,98 0,0 0,047 0,97 0,05 0,0 0,000 0,005 000 0,96 0,00 0,060 0,96 0,0 0,06 0,000 0,00 500 0,99 0,045 0,09 0,98 0,08 0,08 0,000 0,008 50,0 0,068 0,9,0 0,07 0,05 0,000 0,007 4000,0 0,06 0,7,00 0,04 0,09 0,000 0,006 5000,0 0,6 0,,0 0,0 0,07 0,000 0,00 600 0,98 0,5 0,56 0,99 0,07 0,05 0,000-0,00 8000 0,95 0,94 0,804 0,97 0,06 0,054 0,000-0,06 Table. Set of calculatio results of soud absorptio coefficiet α s ad its ucertaities coducted accordig to both versios of stadard ISO 54 [,] 6890
,0,00,00,000 0,80 0,800 alfa s 0,60 0,600 U (alfa s 0,40 0,400 0,0 0,00 0,00 00 5 60 00 50 5 400 500 60 800 000 50 600 000 500 50 4000 5000 600 8000 f[hz] alfa s ew alfa s old U(alfa s ew U(alfa s old 0,000 Fig.Compariso the calculatio results of the soud absorptio coefficiet ad its expaded ucertaity carried out accordig to both versio of ISO 96- stadard 4 Coclusio The evaluatio of the ucertaity is very importat because it testifies to the quality of measuremets. Observig the ucertaity budget for each bad of frequecy allows to fid the decidig factors. I case of soud absorptio measuremets, it could be very difficult to obtai accurate results due to the importat ucertaity at high frequecies bads. The fuctio circumscribig the power atteuatio coefficiet m calculated accordig to ISO 96-, thaks to the o-liear shape, iflueces the ucertaity of measuremet results i spite of very small chages i evirometal coditios i laboratory. Refereces [] Guide to the Expressio of Ucertaity i Measuremet. Iteratioal Orgaizatio for Stadardizatio, Geeva, First Editio, 99. Corrected ad reprited 995 [] EN ISO 54:985, Acoustics Measuremet of soud absorptio i a reverberatio room [] EN ISO 54:00, Acoustics Measuremet of soud absorptio i a reverberatio room [4] ISO 96-:99, Acoustics Atteuatio of soud durig propagatio outdoors Part : Calculatio of the absorptio of soud by the atmosphere 689