1099. Application of sound loudness research in structural-acoustic coupling cavity with impedance and mobility approach method

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1 1099. Applcaton of sound loudness research n structural-acoustc couplng cavty wth mpedance and moblty approach method Jxuan Yuan Shenzhen Insttute of Informaton Technology, P. R. Chna E-mal: yuanjxuan@gmal.com (Receved 2 September 2013; accepted 5 November 2013) Abstract. The dstrbuton of acoustc feld and loudness n a structural-acoustc couplng cavty wth a harmonc force exctaton on an elastc plate s studed n ths artcle usng the analytcal algorthm. Based on the theory of mpedance and moblty, the nteror acoustc feld model s establshed for the structural-acoustc coupled system under smply supported condtons. Combned wth the Zwcker s loudness model, a model of nteror acoustc feld sound loudness, whch s also n the structural-acoustc coupled system, s establshed for forecastng. The sound loudness dstrbuton of three dfferent locaton ponts n the cavty s analyzed by usng the numercal calculaton results. The effect of spatal poston on sound ntensty level, sound ntensty densty and crtcal-band level s also studed through takng the frequency selectvty of human hearng system nto account. And the research results provde analytcal method for forecastng and desgnng sound loudness n the structural-acoustc couplng cavty. Keywords: loudness, structural-acoustc couplng, mpedance and moblty approach method, harmonc force exctaton. 1. Introducton The sound qualty of machne elements s of great mportance n the desgn of hgh-qualty machnery. Nose n an enclosed cavty, such as n the nteror car, has been pad more attenton n recent years. Therefore, t s partcularly sgnfcant to control the sound qualty n an enclosed cavty. Among the parameters of sound qualty, loudness s a very mportant parameter. Many researchers have studed nose n an enclosed cavty by usng Fnte Element Method (FEM) snce 1960s. Gladwell and Zmmermann developed an energy formulaton of the structural-acoustc theory, n whch the stage was set for the applcaton of FEM to acoustc cavty analyss [1]. Early contrbutons to development of fnte element approach were made by Craggs [2], Shuku and Ishhara [3], prmarly provdng a practcal method for analyzng the acoustc of the automoble passenger compartment. And NASTRAN was used by Nefske and Wolf [4] for the study of automoble nteror nose, as well as ts applcaton nto structural-acoustc analyses of the passenger compartment was descrbed. As for the calculaton of sound loudness, on the bass of Zwcker s loudness model [5], researchers have provded some methods such as EMBSD [6], PSQM [7], PEAQ [8] and etc. However these methods stll fal to provde all of the necessary detals. These have been summarzed n [9]. Then Kabla [10] corrected the PEAQ method whch was more precse than the others. In the present paper, the corrected PEAQ approach s appled to calculate the sound loudness radated from rectangular acoustc-structure couplng plates. The mpedance and moblty approach method for the analyss of structural-acoustc systems was proposed by Km n 1999 [11]. The kernel dea s derved acoustc mode ampltude contrbuton and elastc plate mode ampltude contrbuton n cavty by usng cavty mpedance Z n, the structural moblty Y s, modal couplng coeffcents C n,m of modal synthess theory. Consequently, the sound loudness n an enclosed cavty s thus calculated. By comparng the dfferent locaton pont n the structural-acoustc couplng cavty wth a harmonc force exctaton on an elastc plate and takng the frequency selectvty of human hearng system nto account, the effects of sound loudness are studed n detal VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER VOLUME 15, ISSUE 4. ISSN

2 2. Zwcker s sound loudness model The sensaton that corresponds most closely to the sound ntensty of the stmulus s loudness. Crtcal bandwdth plays an mportant role n the loudness, and the loudness on the slopes of the frequency selectvty characterstc of human hearng system may also play an mportant role. The exctaton level versus crtcal-band rate represents the pattern that descrbes not only the nfluence of crtcal bandwdth but also that of the slopes. Therefore, t seems reasonable to use the exctaton level versus crtcal-band rate as a bass from whch the complex of the loudness may be constructed [13]. The loudness s then an ntegral of the specfc loudness over all crtcal-band rates [5]: 24 N = N dz. 0 (1) Thus, the transformaton from crtcal-band rate level nto exctaton level versus crtcal-band rate s necessary. Both the frequency related transformaton nto crtcal-band rate and ampltude related transformaton nto specfc loudness are of crucal mportance for evaluatng sound at the specfc recever, e.g. the human hearng system. The transformaton from exctaton level nto specfc loudness must be known n order to transform exctaton level versus crtcal-band rate nto specfc loudness versus crtcal-band rate. Ths transformaton s dscussed below. Steven s law [14] says that a sensaton belongng to the category of ntensty sensatons grows wth physcal ntensty accordng to a power law. Accordng to ths law, one has to assume that a relatve change n loudness s proportonal to a relatve change n ntensty. Instead of the total loudness, specfc loudness s used as the value. Accordng to the Zwcker s loudness model, the specfc loudness s gven by as follows [5]: N = 0.08(E TQ E 0 ) 0.23 [( E 0.23 ) 1], (2) E TQ where E TQ s the exctaton level at the threshold n quet approxmated by E TQ (f) = 3.64(f 1000) 0.8, E 0 s the exctaton correspondng to the reference ntensty I 0 = W/m 2. The values for these two parameters are gven n [10]. The crtcal-band concept s thus mportant to descrbe human hearng sensatons. And the concept of crtcal-band rate scale s derved, whch s based on the fact that human hearng system analyses a broad spectrum nto parts that correspond to crtcal bands. In many cases, an analytc expresson s useful to descrbe the dependence of the crtcal-band rate (and of the crtcal bandwdth) on the frequency over the whole audtory frequency range. The followng relatons are useful [5]: z = 13arctan(0.76f 1000) + 3.5arctan(f 7500) 2, (3) and: Δf G = [ (f 1000) 2 ] 0.69, (4) where z s crtcal-band rate, Δf G s crtcal bandwdth. The frequency selectvty of human hearng system can be approxmated by subdvdng the ntensty of the sound nto parts that fall nto crtcal band. In order to descrbe such approxmaton, the noton of crtcal-band ntenstes s put forward. The crtcal-band level and the exctaton level play an mportant role n many acoustc models. The crtcal-band level ntensty I G can be calculated from the followng equaton that takes nto account the frequency dependence of crtcal VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER VOLUME 15, ISSUE 4. ISSN

3 bandwdth [5]: I G = f+0.5δf G di df df f 0.5Δf G. (5) The crtcal-band rate z s useful n descrbng the characterstcs of human hearng system. Because the crtcal-band rate z s a defnte functon of frequency, I G can also be expressed n terms of the crtcal-band rates: I G = z+0.5 z 0.5 di dz. dz (6) In logarthmc scale, usng I 0 = W/m 2 as reference value, the crtcal-band level L G s defned as: L G = 10log(I G I 0 ). (7) The crtcal-band ntensty can be seen as that part of the overall unweghted sound ntensty whch falls wthn a frequency wndow that has the wdth of a crtcal band. The transformaton of the frequency nto the crtcal-band rate transfers the frequency-dependent wndow wdth nto a wndow wdth of 1 Bark, ndependent of the crtcal-band rate. Consequently, a crtcal-band wde narrow-band nose produces a crtcal-band ntensty whch s a functon of the crtcal-band rate, and whch shows the form of a trangle wth a base wdth of 2-Bark. A harmonc tone, however, produces a functon wth a rectangular shape and the wdth of 1 Bark. The ntermedate values such as the exctaton or the exctaton level, however, represent a much better approxmaton to the frequency selectvty of human hearng system. The so-called man exctaton corresponds n ths transformaton to the maxmum value of the crtcal-band level. In most cases, the exctaton level, defned by L E, s gven by [5]: L E = 10logE E 0. (8) The exctaton level can be constructed most smply from the crtcal-band level as a functon of the crtcal-band rate, by calculatng the crtcal-band level n the range of the man exctaton. In cases of an abrupt change of the ntensty densty as a functon of the crtcal-band rate, as for low-pass nose or snusodal tones, the exctaton level s dentcal to the crtcal-band level. Fg. 1. Sketch map of structural-acoustc couplng cavty ncludng force exctaton dstrbuton and sound ntensty exctaton 1900 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER VOLUME 15, ISSUE 4. ISSN

4 3. Model of mpedance and moblty approach method The mpedance and moblty approach method for the analyss of structural-acoustc systems was proposed by Km n 1999 [11]. The kernel dea s derved acoustc mode ampltude contrbuton and elastc plate mode ampltude contrbuton n cavty by usng cavty mpedance Z n, the structural moblty Y s, modal couplng coeffcents C n,m of modal synthess theory. Ths artcle analyses regulartes of loudness dstrbuton n cavty whch has sx elastc plates wth smple support. Thnk about a rectangle structural-acoustc couplng cavty, as shown n Fgure 1. In cavty there s full of ar whose densty s ρ a, and all boundary plates are elastc wth smple support. There are a force dstrbuton functon f(ω, L p ) on plate z = z and a sound ntensty functon s(ω, L c ) n the structural-acoustc couplng cavty. Let s suppose couplng response could be decomposed nto lmted uncoupled acoustc modes and structural modes, and these uncoupled modes are acoustc modes cavty wth rgd walls and structural modes n vacuum. There are n-th acoustc modes and m-th structural modes. Therefore, sound pressure p at pont L c s as follows: N p(l c, ω) = ψ n (L c )a n (ω) = Ψ T a, (9) n=1 where n-dmensonal column vector Ψ represents uncoupled acoustc modes shape functon ψ n (x) and n-dmensonal column vector a represents sound pressure modes complex ampltude, sgn T at top left corner means transposton. The acoustc modes shape functon ψ n (x) s orthogonal n each uncoupled cavtes as follows: 2 V = ψ n (L c )dv, V (10) where V represents the volume of acoustc cavty. Accordng to reference [11], snce all boundary plates are smple support elastc plates, the relatons between n -th sound pressure modes ampltude and external exctaton s as follows: a n (ω) = ρ ac a 2 V A n(ω) ( ψ n (L c )s(l c, ω)dv V 6 + ψ n (L ps )u(l ps, ω)ds), (11) where, ρ a represents ar densty n cavty, c a represents sound speed n ar, two ntegral expressons n the bracket represent n -th acoustc modal ntensty caused by s(l c, ω) and u(l ps, ω). u(l ps, ω) represents the vbraton velocty of each smple support elastc plate whch surround the cavty, as shown bellow: M u(l p, ω) = φ m (L p )b m (ω) = Φ T b, (12) m=1 where, m-dmensonal column vector Φ represents uncoupled vbraton shape modal functon φ m (y) and b represents uncoupled vbraton velocty modal complex ampltude b m (ω). The modal shapes functons ψ n (x) and φ m (y) are orthogonal n each uncoupled system, and the expresson s as below: =1 S VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER VOLUME 15, ISSUE 4. ISSN

5 2 S = φ m (L ps )ds. S (13) Then a s shown as below: a n (ω) = ρ ac a 2 V M A n(ω) (r n + C n,m 6 =1 m=1 b m (ω)), (14) where r n = ψ V n (L c )s(l c, ω)dv, the acoustc modal resonant expresson A n (ω) s as below: 1 A 1 (ω) =, 1 Ta + jω n = 1, jω A n (ω) = { ω 2 n ω 2, + j2ζ n ω n ω n 1, (15) where T a represents tme constant of frst modal, ω n represents n-th acoustc modal frequency, ζ n represents n-th acoustc modal dampng factor. C n,m represents coupled relatons between the -th uncoupled plate and acoustc modal shape functon, whch s as follows: C n,m = ψ n (L ps )φ m (L ps )ds. (16) S Then a could represent as follows: a = Z a (r + q s ), (17) where, r represents n-th modal source ntensty vector, q s = C b represents source ntensty vector caused by elastc structural vbraton, whose effect s a group of external sound source functon n elastc structure. b s m-th ampltude modal vector, C s (n m) dmenson structural-acoustc modal shape functon coupled matrx. Z a = Aρ 2 ac a V whch composed by ( n n) dmenson uncoupled acoustc modal mpedance dagonal matrx. The m-th modal ampltude of the plate z = z s as follows: b m z (ω) = B m z (ω) ρ s h z S z ( φ m (L psz )f(l psz, ω)ds S z S z φ m (L psz )p(l psz, ω)ds), (18) where ρ s represents densty of plate, h z represents thckness of plate, S z represents area of the plate z = z, p(l psz, ω) represents sound pressure dstrbuton functon affected on the plate z = z. The frst and the second ntegral formulae n the expresson represent m-th vbraton modal forces caused by f(l psz, ω) and p(l psz, ω). The mnus between two ntegral formulae s because of the opposte drectons of external forces and acoustc pressure forces. Smlarly, the m-th modal vbraton ampltude of the other elastc plate wthout external forces s as below: b m (ω) = 1 ρ s h S B m (ω) ( φ m (L ps )p(l ps, ω)ds S ), (19) 1902 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER VOLUME 15, ISSUE 4. ISSN

6 where, h represents thckness of plate, S represents area of plate. The resonant term of plate could be expressed as follows: B jω m (ω) = ω 2 m ω 2 + j2ζ m ω m ω, (20) where ω m represents m-th modal frequency of plate, ζ m represents m-th modal dampng rato of plate. The modal vbraton ampltude contrbuton vector of plate z = z s as follows: N z b 1 m (ω) z = B ρ s h z S m (ω) z (f C T n,m a n (ω)). (21) z n=1 The modal vbraton ampltude contrbuton of the other plates s: N b m (ω) = 1 B ρ s h S m (ω) ( C T n,m a n (ω)), (22) T where C n,m below: n=1 = C m,n. Consequently, the modal vbraton ampltude vector of plate z = z s as b z = Y s z (F g a z ), (23) where m-dmensonal vector F s caused by external load dstrbuton functon f(l pz, ω), g a = C z T a represents modal force vector affected on acoustc cavty system, whch s the elastc plate structural reacton exctng by acoustc pressure fluctuaton. Y z s = Bz ρ s h z S represents z modal moblty dagonal matrx of plate z = z. The modal vbraton ampltude contrbuton vector of other plates s as below: Where g a = C T a represents modal force vector affected on acoustc cavty system, whch s the elastc plate structural reacton exctng by acoustc pressure fluctuaton. Y s = B ρ s h S represents modal moblty dagonal matrx of other plates. Smlar to acoustc mpedance matrx z Z a, Y s and Y s are also dagonal matrxes. Accordng to the dervaton above, the formula s shown as below after uncouplng: a = (I + Z a C Y s C T + Z a C z Y s z C z T ) 1 Z a (r + C z Y s z F). (24) The equaton above could be adapted nto mpedance-moblty form. Suppose there s only one force exctaton on plate (r = 0), and put q s = C b nto a = Z a (r + q s ), then a = Z a C b could be obtaned. g a = C T a s combned wth a = Z a C b, the followng equaton s obtaned: g a = Z ca b, (25) where Z ca Y cs = C T Z a C represents coupled acoustc modal mpedance dagonal matrx. Smlarly, = C Y s C T can be defned as coupled structural modal moblty dagonal matrx. Therefore, a = (I + Z a C Y s C T + Z a C z Y s z C z T ) 1 Z a (r + C z Y s z F) can be rewrtten as follows: VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER VOLUME 15, ISSUE 4. ISSN

7 a = (I + Z a Y cs z + Z a Y cs ) 1 Z a (r + C z z Y s F). (26) Snce the structural moblty Y s z and couplng coeffcent C z of plate z = z s the same wth plate z = 0, the equaton above can be derved as follows: a = (I + Z a Y cs ) 1 Z a (r + C z z Y s F). (27) The equaton above s the compact calculaton form of classc theory on acoustc modal contrbuton about the couplng response, whch s n the feld of structural-acoustc couplng system wth both structural force exctaton and acoustc source exctaton. Accordng to reference [11], the standard evaluaton crteron for weak couplng, the acoustc response [I + Z a Y cs ] [I]. That s to say, Z a Y cs could be almost equals 0 n the equaton a = (I + Z a Y cs can be obtaned: z + Z a Y cs ) 1 Z a (r + C z z Y s F) n weak couplng system, therefore, the follows a = Z a (r + C z Y s z F). (28) Put a = Z a (r + C z Y z s F) nto p(l c, ω) = N n=1 ψ n (L c )a n (ω) = Ψ T a, the sound pressure value of some node n the acoustc-structural couplng feld could be calculated as below: L I = 20log 10 P(x, y, z) P log ρ 0 c. (29) 4. An example Accordng to the models above, the dstrbuton of sound loudness n the rectangle enclosed acoustc-structural couplng feld under the condton of one exctaton on one elastc plate s studed. Fg. 2. Structural-acoustc rectangle system wth one harmonc force exctaton Fgure 2 s a rectangle cavty whch s composed by fve rgd walls and one elastc plate wth smple support on ts boundary, a = 0.5 m, b = 0.4 m, h = 0.3 m. In the cavty there s fll wth ar, ts densty s ρ 0 = kg/m 3. The sound velocty s c = 340 m/s. The densty of plate s ρ s = 2790 kg/m 3, the thckness s 5 mm, Posson rato s v = 0.3, the modulus of elastcty s 1904 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER VOLUME 15, ISSUE 4. ISSN

8 E = N/m 2, the dampng loss factor s ξ = The coordnate of the exctaton pont s (0.4, 0.1, 0.3), the exctaton force s f = e jωt (N), whch range s from 1 Hz to 630 Hz. The natural frequency and natural mode of each order of the elastc plate are as below: φ m (y) = 2sn ( m 1πy 1 ) sn ( m 2πy 2 ), L 1 L 2 (30) ω m = ( D 1 ρh ) 2 m 1 π 2 [( ) + ( m 2 2π ) ], L 1 L 2 (31) whch L 1 L 2 s the length and wdth of the elastc plate, m 1 and m 2 are postve ntegers, D = Eh 3 [12ρ(1 μ 2 )] represents bendng rgdty, ρ represents densty of alumnum, h represents thckness of the plate, μ represents Posson rato. The natural frequency and natural mode of each order of the rectangle cavty are as below: ψ n (x) = e 1 e 2 e 3 cos ( n 1πx 1 L 1 ) cos ( n 2πx 2 L 2 ) cos ( n 3πx 3 L 3 ), (32) ω n = πc 0 ( n 2 1 ) + ( n 2 2 ) + ( n 2 3 ), (33) L 1 L 2 L 3 where L 1 L 2 L 3 represents the sze of the rgd walls, f n = 0, then e = 1; else e = 2, = 1,2,3. Accordng to (30) and (32), the response natural frequences of each order of the plate and cavty s shown n Table 1 and Table 2. Table 1. Each order frequency of elastc plate Order No Frequency/Hz Table 2. Each order frequency of rectangle acoustc cavty Order No Frequency /Hz Order No Frequency /Hz Accordng to reference [15], the modal couplng coeffcent of ths example can be represented as below: C n,m m 1 m 2 (( 1) = { ( 1)n 32S n1+m1 1)(( 1) n2+m2 1) f e 1 e 2 e 3 π 2 (n 2 1 m 2 1 )(n 2 2 m 2, n 2 ) 1 m 1, n 2 m 2, 0 else. (34) The rato loudness at cross secton z = 0.15 m whch frequency s 125 Hz, 250 Hz and 450 Hz are shown n Fg. 3. Accordng to Fgure 3, the rato loudness dstrbuton changes greatly n dfferent frequences. When the frequency s 125 Hz, the mnmum rato loudness pont s at pont (0.1, 0.1, 0.15). Regard ths pont as orgnal pont, t can be found that the ncreasng speed of rato loudness along x drecton s obvously faster than x drecton. But the varaton range at 125 Hz s smaller than hgher frequency. When the frequency s 250 Hz, the mnmum rato loudness pont s at pont (0.45, 0.15, 0.15) VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER VOLUME 15, ISSUE 4. ISSN

9 and ts value s only 14 sone/hz. Except for the mnmum rato loudness pont, there are other low rato loudness ponts such as pont (0.45, 0.3, 0.15), (0.28, 0.36, 0.15), (0.18, 0.4, 0.15) and (0.11, 0.044, 0.15). (a) Rato loudness dagram at 125 Hz where cross secton z = 0.15 m (b) Rato loudness dagram at 250 Hz where cross secton z = 0.15 m (c) Rato loudness dagram at 450 Hz where cross secton z = 0.15 m Fg. 3. Rato loudness dagrams where cross secton z = 0.15 m 1906 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER VOLUME 15, ISSUE 4. ISSN

10 When further ncreasng of frequency to 450 Hz, the rato loudness value s also ncreasng, but ts dstrbuton shape s greatly changed from the 250 Hz shape. Its mnmum rato loudness value s 32.5 sone/hz whch s at pont (0.35, 0.2, 0.15) and pont (0.275, 0.175, 0.15). The average rato loudness value s 36 sone/hz whch s obvously hgher than the frequency 250 Hz. Three observaton ponts whose coordnates ob 1 (0.055, 0.044, 0.15), ob 2 (0.275, 0.176, 0.25) and ob 3 (0.33, 0.22, 0.15) are chosen to study ther varaton law. (a) Curve of sound ntensty level at pont ob 1 (b) Curve of sound ntensty level at pont ob 2 (c) Curve of sound ntensty level at pont ob 3 Fg. 4. Curves of sound ntensty level at three dfferent observaton ponts VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER VOLUME 15, ISSUE 4. ISSN

11 (a) Curve of sound ntensty densty at pont ob 1 (b) Curve of sound ntensty densty at pont ob 2 (c) Curve of sound ntensty densty at pont ob 3 Fg. 5. Curves of sound ntensty densty at three dfferent observaton ponts The sound ntensty level curve of the three observaton ponts ob 1, ob 2, ob 3 s shown as Fgure 4. Compare n the fgure, a smlar tendency of all these curves can be seen: the maxmum sound ntensty level values appear when the frequency s 125 Hz, and as the frst order mode frequency of the elastc plate s Hz, t can be concluded that the maxmum sound ntensty 1908 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER VOLUME 15, ISSUE 4. ISSN

12 level value s caused by the frst order mode frequency. And wth the ncreasng of frequency, the ntensty level value decreases untl the second maxmum value appears at 355 Hz. The value of sound ntensty level s oscllatng and three peak values s obtaned at 425 Hz (the 8th order mode frequency of the cavty), 510 Hz and 567 Hz (the 13th order mode frequency of the cavty). But there are great dfferences wth the sound ntensty level curves of three observaton ponts: the mnmum sound ntensty level value of pont ob 1 s 50.5 db at 268 Hz, whch pont ob 2 s at 630 Hz, pont ob 3 s at 299 Hz. Accordng to the three curves, a concluson can be drawn that the frst order mode of elastc plate has made great contrbutons to the couplng ntensty level n low frequency range, and the mode of cavty make more n mddle and hgh frequency area. Because of the frequency selectvty of human hearng system, the sound ntensty densty di/df s mportant n the process of estmatng the sound loudness. Accordng to the sound loudness model that has been set up n secton 2, the sound ntensty densty di/df can be calculated as shown n Fg. 5. In Fgure 5, the maxmum value of the observaton ponts sound ntensty densty n enclosure couplng cavty s relatve to the maxmum sound ntensty level value, that s to say, the frst mode frequency has made great contrbuton to the acoustc-structural couplng cavty. Snce the crtcal-band rate scale s better to descrbe human hearng sensatons and the crtcal-band rate z s a defnte functon of frequency, the sound ntensty densty di/df can be expressed as di/dz by usng the crtcal-band rate scale accordng to Equaton (3). Once the sound ntensty densty di/dz s obtaned, then the crtcal-band level ntensty I G can be calculated accordng to Equaton (6). Thus, at the observaton pont, one can obtan the crtcal-band level L G caused by dfferent observaton place s as shown n Fg. 6. Because of the frequency selectvty of human hearng system, there are more local maxmum or mnmum values of crtcal-band level curves under crtcal-band rate scale n Fgure 6, whch s dfferent from sound ntensty level curves n Fgure 4. So does the dfferental value between the maxmum value and the mnmum value. The three crtcal-band level curves have the same trend except the dfferental range n Fgure 6. But when the value of the abscssa of crtcal-band level s greater than 3 Bark, there s obvously dfference n the curves, whch s not the same wth the trend of sound ntensty level n Fgure 4. The exctaton level at threshold n quet has dfferent value at dfferent frequency because of the frequency selectvty of human hearng system, whch s shown n Fg. 7. For ths characterstc of human hearng system, the sound loudness obtaned at the observaton pont has low value n low frequency range as shown n Fg. 8. On the other hand, though the effect of thckness on the sound loudness s smlar to the effect on the crtcal-band level, there are stll great dfferences. Equaton (2) s one of the most sophstcated specfc loudness calculaton methods [6]. The power spectrum of each frame s weghted by the frequency selectvely of the human hearng. The power spectral energes are then grouped nto crtcal bands. An offset s then added to the crtcal band energes to compensate for nternal nose generated n the ear. Because the sound at the observaton s harmonc tones, accordng to Equatons (1) and (2) and the transformaton of the exctaton level from the crtcal-band level descrbed n secton 2, the sound loudness at the observaton pont s calculated as shown n Fg. 8. In Fgure 8, loudness of pont ob 1 s hgher than the other observaton ponts, and sgnfcant dfference can be seen when the crtcal-band s n mddle and hgh frequency (greater than 3.8 Bark). In low-frequency area, the shapes of the loudness curve are smlar because the modes of elastc plate play a major role to the loudness values, although the observaton ponts are n dfferent locaton. Wth frequences arsng, the modes of cavty play an ncreasngly mportant role to loudness values. So the dstrbuton locaton n the cavty appears to have more effect to loudness values n mddle-hgh-frequency range. In the three observaton pont, the length of pont ob 2 and ob 3 to the exctng pont s nearer than the one of pont ob 1, but loudness value of pont ob 1 s the hgher than the other two ponts. That s to say, loudness n the cavty s rrelevant to the dstance to exctng pont. VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER VOLUME 15, ISSUE 4. ISSN

13 (a) Curve of crtcal-band level at pont ob 1 (b) Curve of crtcal-band level at pont ob 2 (c) Curve of crtcal-band level at pont ob 3 Fg. 6. Curves of crtcal-band level at three dfferent observaton ponts Exctaton Level(dB) Exctaton Level at Threshold n Quet Crtcal-band Rate(Bark) Fg. 7. Exctaton level of human hearng threshold 1910 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER VOLUME 15, ISSUE 4. ISSN

14 (a) Curve of sound loudness at pont ob 1 (b) Curve of sound loudness at pont ob 2 5. Concluson (c) Curve of sound loudness at pont ob 3 Fg. 8. Curves of sound loudness at three dfferent observaton ponts Based on the theory of mpedance and moblty approach method, ths paper has mproved the applcaton of structural-acoustc couplng model of ths theory. The dstrbuton of sound loudness n the rectangle enclosed acoustc-structural couplng feld under the condton of one exctaton on one elastc plate s studed. Because of the frequency selectvty of human hearng VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER VOLUME 15, ISSUE 4. ISSN

15 system, the characterstcs of sound loudness are greatly dfferent from the sound ntensty level obtaned n the cavty. The dfferent locaton of observaton ponts has dfferent sound loudness and the sound ntensty level. When takng the frequency selectvty of human hearng system nto account, the crtcal-band level and the sound loudness n the enclosed cavty llustrate well the nfluence of human hearng frequency characterstc on human sensaton. The result of ths study shows: (1) The dstrbuton of the sound loudness s NOT even n the cavty n dfferent frequency and dfferent locatons, but they have a smlar dstrbuton rule. (2) In low-frequency range, the man contrbuton to sound loudness n the cavty s elastc plate mode, wth the ncrease of frequency, the man contrbuton s the cavty acoustc mode n mddle-hgh-frequency. Acknowledgement Ths work s supported by the Natural Scence Foundaton of Guangdong Provnce (Grant No. S ). References [1] C. Gladwell, G. Zmmermann On energy and complementary energy formulatons of acoustc and structural vbraton problem. Journal of Sound and Vbraton, Vol. 22, Issue 3, 1966, p [2] A. Craggs The use of smple three-dmensonal acoustc fnte element for determnng the natural modes and frequences of complex shaped enclosure. Journal of Sound and Vbraton, Vol. 23, Issue 4, 1972, p [3] T. Shuku, K. Ishhara The analyss of the acoustc feld n rregular shaped room by the fnte element method. Journal of Sound and Vbraton, Vol. 29, Issue 7, 1973, p [4] Nefske D. J., Wolf J. A., Howell L. J. Structural-acoustc fnte element analyss of the automoble passenger compartment: a revew of current practce. Journal of Sound and Vbraton, Vol. 80, Issue 3, 1982, p [5] E. Zwcker, H. Fastl Psychoacoustcs: Facts and Models (2nd Ed.). Sprnger Verlag, Berln, [6] Y. Wonho Enhanced Modfed Bark Spectral Dstorton (EMBSD): An Objectve Speech Qualty Measure Based on Audble Dstorton and Cognton Model. PhD Thess, Temple Unversty, USA, [7] ITU-R Recommendaton BS. 1387: Method for Objectve Measurement of Perceved Audo Qualty (PSQM), [8] T. Thede Perceptual Audo Qualty Assessment Usng a Non-Lnear Flter Bank. PhD Thess, TU Berln, Berln, [9] ITU-R Study Group 6. Draft Revson to BS. 1387: Method for Objectve Measurement of Perceved Audo Qualty, Document 6/BL/30-E, [10] P. Kabal An Examnaton and Interpretaton of ITU-RBS.1387: Perceptual Evaluaton of Audo Qualty. TSP Lab Techncal Report, Dept. ECE, McGll Unversty, Canada, [11] S. M. Km, M. J. Brennan. A compact matrx formulaton usng the mpedance and moblty approach for the analyss of structural-acoustc systems. Journal of Sound and Vbraton, Vol. 223, Issue 1, 1999, p [12] H. Fastl Evaluaton and measurement of perceved average loudness. Internatonal Contrbutons to Psychologcal Acoustcs V, BIS Un Oldenburg, [13] B. C. Moore An Introducton to the Psychology of Hearng. Academc Press, London, [14] L. Raylegh Theory of Sound (2nd Ed.). Dover Publcatons, New York, [15] S. D. Snyder, N. Tanaka On feed forward actve control of sound and vbraton usng vbraton error sgnals. Journal of the Acoustcal Socety of Amerca, Vol. 94, 1993, p VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER VOLUME 15, ISSUE 4. ISSN