ULTRAGARSAS Journal Ultrasound Institute Kaunas Lithuania For all aers of this ublication click: wwwndtnet/search/docsh3?mainsource=7 ISSN 39-4 ULTRAGARSAS (ULTRASOUND) Vol63 No 008 Deendence of sound insulation of building constructions on the acoustic roerties of materials D Gužas A Gailius I Girnienė 3 Šiauliai University Vilniaus St 4 LT-7685 Šiauliai Lithuania E-mail: danieliusguzas@fondaicom Vilnius Gediminas Technical University Saulėtekio al LT-03 Vilnius Lithuania E-mail: albinasgailius@yahoocom 3 Vilnius College of Construction and Design Antakalnio St 54 LT-0303 Vilnius Lithuania E-mail: ingridagirniene@yahoocom Abstract The article deals with some acoustic roerties of materials that have an imact on the sound insulation of building constructions including sound absortion (abatement) These are namely the caacity of materials to absorb ie to reduce the sound energy assing through a building construction the influence of an angle of sound wave incidence onto the artition (constructio etc Sound ressure on the area of the artition may be uniform whereas the amlitude of oscillations gets changed throughout the whole area of the artition The article exlores in what cases sound will ass most strongly through the artition of finite dimensions when sacefrequency resonance occurs Investigation showed that the sound energy absortion deended on deformation losses in that material Changes in the angle of the sound wave incidence at the same frequencies create the oortunity for manifestation of alternation of maximums and minimums in the motion of late amlitudes The roerty of sound wave incidence at the sound transmission through the late of finite dimensions confirms the rightfulness of the division of the curve of the frequency deendence of sound insulation into three ranges of frequencies with the different intensity of sound transmission Keywords: sound insulation building materials deformation of materials inner friction losses insulation roerties sound ermeability Introduction The acoustic roerties of materials redetermine the sound enetration through building constructions At first we will clarify the roerties of the materials that become manifest in the structure of materials with sound assing through constructions from those materials In our work [] the roerties of building materials redetermining the sound insulation of building constructions are indicated This research covered the density of new building materials and its imact on the degree of sound insulation In the resent article we shall rovide one of the roerties that was not mentioned the characteristic of the incident wave onto the artition and the imact of its arameters on the sound ermeability of the artition under study The distribution of the ressure zones of sound on the surface of the late under study which shows the form of its oscillations is analysed Here sace-frequency resonances are elucidated and the highest amlitude of late oscillation is identified Effect of sound absortion roerties on sound insulation increase With an aim of investigating an effect of sound absortion roerties of materials it is necessary to know the roerties of materials to be used for sound insulation imrovement on which sound energy absortion deend Primarily it is necessary to make use of the roerties of inner friction losses of homogeneous materials Inner friction losses aear due to deformation of materials It is necessary to study whether deformation changes coincide with the changes of tensions In the case of harmonic fluctuations these losses may be exressed through the ostensible arts of elastic constants If the material under study is characterized by the comlex shearing modulus ( + i ) = G + ig G = G ηs () and the modulus of elasticity (modulus of comressio ' ( + i ) = K ik K = K η K + () then it is ossible to exress the rigidity S to lane longitudinal waves by means of a formula: 4 ρ c = S = K + G (3) 3 The rigidity S is related to the sound velocity c of lane longitudinal waves We also obtain the comlex meaning S = S( + iηs ) The coefficient of losses ηs is the measure of the coefficient of absortion α of longitudinal waves: ' 4 ' K + G 3 αλ = πη s = (4) S Therefore for obtaining the comuted otimum value αλ =5 at extremely low frequency it is necessary to select the material with η s = 04 7
ISSN 39-4 ULTRAGARSAS (ULTRASOUND) Vol63 No 008 Sound wave transmission through construction Let us study the interrelationshi of the wave motion of real lates with the wave motion in the environment For solving the set tasks the use will be made of the method of the correlation of wave arameters m m n n and angles α and α which are interrelated by the following deendences []: m = k0a / π sinα sin n = k0b / π cosα sin m = k a /π sinα u n = kub /π cosα where m and n are the numbers of flexural semi-waves along the sides a and b are coefficients of wave roagating with the velocity c k 0 k n are the wave numbers Sound wave falling onto the late at a certain angle θ excites the inertial wave in the late with the same frequency as a free wave that may differ from the inertial one by its form and frequency The emerging elastic waves as to their tye may be flexural and longitudinal The resence of the edges in the late (in our case the late is rectangular) and the incidence of inertial and free waves on them lead to the formation of free flexural waves roagating from the edges at a certain angle to the sides of the late The most favourable conditions for sound transmission through the late will be in the case of the best coincidence of the wave arameters of sound and vibration fields If the frequency of the incident sound waves at the angle θ is equal to the frequency of normal oscillations of the late three cases are ossible: satial resonance where k u = k0 α = α ie m = m n = n incomlete satial resonance where the amlitude of oscillations is highest in the case of the artial coincidence of the wave arameters kux = k0 x kuy k0 y or kux k0 x kuy = k0 y α α ie m = m n n or m m n = n ordinary satial resonance characteristic of the highest amlitude of oscillations at the discreancy of the wave arameters k u k0 α α ie m m n n Let the monochromatic lane acoustical wave fall at a certain angle θ on the rectangular late with sides a and b which with the hel of hinges rests on four sides We assume the late as thin and therefore we take into consideration only its flexural oscillations We select the rectangular system of coordinates so that its beginning coincides with the lower left to of the late and we direct axes x and y along its edges Let s combine the lane X0Y with the neutral lane of the late In the case of the satial resonance k u = k0 α = α the amlitude of dislacements of the late are given by omn mn (5) m ω iη ω [ ( ) ] mn In this exression the conditions of wave coincidence have been fulfilled in full and with the condition ω = ωmn being followed the effect of the satial resonance and the maximum sound transmission is being observed The condition for the existence of the effect of satial resonance will be: k 0 = k umn / sin or c umn = c0 / sin In difference from an infinite late where each frequency is normal for a finite late with sound waves falling at some angle the satial resonance and the highest sound transmission are observed only at frequencies of normal oscillations which are of a discrete character At incomlete satial resonance where artial coincidence of the wave arameters kux = k0 x kuy k0 y α α takes lace bsin = [ ω ( + iη) ω ] c ω cosα 0 0mn ( ) ω cos α ω cos α and in the case of correlations α α a sin = k k0 [ ω ( + iη) ω ] c ω sinα 0 0m n ( ) ω sin α ω sin α (6) ux x uy k y k = 0 In the case of an ordinary sace resonance where k u k 0 α α = absin [ ω ( + iη) ω ] ( ω sin α ω sin α ) 0 ( m)( 0m n 6c ωα cosα ( ) ω cos α ω cos α ( m)( ( m)( At deducing frequency-angular characteristics of the dislacement of lates we substitute the indexes m m n n by their values After further averaging the frequency-angular deendences of sound transmission in the frequency integral Δ f we write the square of the amlitude of late dislacements: at the frequency of satial resonance = π b sin 3 ( π 3 f Δfη) / 3 3 3π f Δfη c0 cos α ( cos α cos α ) (7) (8) (9) f (0) 8
ISSN 39-4 ULTRAGARSAS (ULTRASOUND) Vol63 No 008 3 3 3π f Δfη () c0 sin α π a sin ( sin α sin α) f at the frequency of ordinary satial resonances 3 3 3π f Δfη 4 Ωc0 sin α cos α 4 4 4 π a b sin ( sin α sin α) ( cos α cos α) f () Each of these exressions characterizes the highest resonse of the late in conformity with the condition of coordination of wave arameters In the exression () the numerical coefficient Ω = 6 for ordinary resonances in the range of frequencies is higher of satial resonance and Ω = 4 in the range lower of satial resonance From comarison of correlations (Eq9 - ) we see that the resonse of the lates to the effect of a sound wave incident at the angle θ is different for each of the conditions of the correlation of wave arameters At the secific frequency for each reset angle of sound incidence θ its own only (for thin lates) satial resonance exists The value of this frequency ractically in many cases with a sufficient degree of recision may be determined according to the frequency of wave coincidence at the oblique incidence of sound on the infinite late ie according to the Cremer s formula since the sectrum of normal frequencies of the limited late at relatively high frequencies is dense enough and correction on the discrete character of its oscillations is insignificant Let s trace the character of the frequency deendence of sound transmission directed at a certain angleθ through the late of glass with the dimensions a = m and b = 08 m thickness h = 0005 m m = 3 kg/m D/m = 563 m 4 /c η = 0 3 Alying exressions Eq 9 - we construct frequency characteristics of the amlitude of dislacement of the late excited by a noise band Δ f for the angle of incidence θ = 75 Substituting in the deendences (3) and (4) k u by its ω D value we get: πm sin α = fa c u πn D 3 cos = D = Eh / v is the fb flexural rigidity of the late deending on the module of elasticity (Young) E the Poisson s coefficient v and the late thickness h In the range of frequencies of a lower satial resonance on the same frequencies the wave length in air λ 0 is higher than the wave length of a flexural wave in the late λ u and accordingly c 0 > cu k 0 < ku Here the ordinary and incomlete satial resonances are ossible The latter are observed in the range of frequencies from the α where ( ) boundary frequency of the incomlete satial frequency to the boundary frequency of the comlete satial frequency In this range for each number m and n in the late the number m and n equalling them will be found and the angle α corresonding to them The amlitude of dislacements of the oints in the late may be calculated for mean values of the angle α in each estimated band of the frequencies Δ f and the angles θ Hence m = muv = mu mv n = nuvnu nv where m u n u are the numbers of sound semi-waves in the lane of the late along the sides a and b relating to the lower frequency of the interval m u n u are the numbers relating to the higher frequency of the interval Lower of the boundary frequency of the incomlete satial resonance the movement of the late is determined by the ordinary satial resonance In the range of frequencies higher of the satial resonance on the same frequencies wave length in air becomes less than that of the flexural one λ u c 0 < cu k 0 > k u therefore only the ordinary satial resonances are manifest On those frequencies for the known numbers m and n the numbers m and n equal to them with the real angle α are not found Therefore the amlitude of oscillations of the late in this range we calculate for our examle at reset α (45 60 and 75 ) Making calculations according to Eq9 - we assume that late dislacement at the frequency of the satial resonance is equal to unity and the angle α π / Fig resents the enveloe curves of sound transmission through the given late made of glass for the angle of incidence θ = 75 versus the frequency Fig Frequency deendence of resonant sound transmission through glass with thickness h = 0005 m with dimensions of 08 m In the range of frequencies lower of the satial resonance the curve 4 is structured according to Eq 0 and the curve according to Eq The curve enveloes maximum sound transmission at the incomlete sace 9
ISSN 39-4 ULTRAGARSAS (ULTRASOUND) Vol63 No 008 resonance The enveloe curve 3 reflects the case of ordinary resonances In the range of frequencies higher than the satial resonance the enveloe curves are reresented by lines 5 6 and 7 structured according to Eq for the angles α equalling corresondingly 75 60 and 45 For α = 75 the values of dislacement are highest since that angle is closer of all to the angle of incident sound waves α = π / From the curves in Fig it is seen that in the range of frequencies lower than the satial resonance (range II) the decisive contribution into sound transmission belongs to the incomlete sound resonance (Fig ) Fig Frequency sound insulation characteristic of a duralumin late with thickness h 0003 with dimensions of 08 m: the law of mass for the angle θ = 45 exeriment Lower of the boundary frequency of incomlete satial resonances where only conditions k u k0 α α m m n n are fulfilled the contribution to sound transmission by determination belongs to ordinary satial resonances (range I) In the range higher of satial resonance (range III) the main contribution into sound transmission is also made by ordinary resonances The aroximated values of the lowest or boundary frequencies (at low frequencies) and incomlete satial resonances will be: boundary frequency of the ordinary sace resonance at m = / and n = / : fr ( m)( = c0 /( 4asinα m sin ) n mn (3) or fr ( m)( = c0 /( 4b cosα m sin ) n mn (4) boundary frequency of the incomlete sace resonances at m = m = f rm ( = c0 / asinα mn sin mn (5) or at n = n = fr ( m) n = bcos α c0 m n sin m n (6) The lowest value is taken as the estimated value of boundary frequencies that is calculated according to Eq3-6 The exact value of those boundary frequencies is determined with account taken of the correction in the discrete character of natural oscillations of the late As illustration of the values of boundary frequencies Fig resents the frequency characteristic of sound insulation of the duralumin late with the arameters a = m b = 08 m h = 0003 m m = 793 kg/m D/m = 8 m 4 /c θ = 45 α = π / obtained exerimentally The values of boundary frequencies here are constituted of f k = 00 Hz f 0 Hz and fk = 7880 Hz whereas the measured (exerimental) are equal accordingly to 00 00 and 8000 Hz From the given examle it is seen that at the directed incidence of sound the scheme of division of the curve of frequency characteristic of sound insulation into three ranges with the different degree of sound transmission is well confirmed by exerimental data and the obtained values of boundary frequencies agree with the said data At the estimation of the character of the directed sound transmission of interest is that art of the rocess where sound waves of one frequency fall onto the late at all ossible angles from 0 to π / On the examle of the same late made of glass we calculate the relative dislacements of the lates using Eq9 - Hence we assume that the fixed frequency is the frequency of the satial resonance fmn = 5000 Hz for the angle of incidence of sound θ = 45 mn Fig 3 reresents the enveloe curves of maximums of sound transmission in deendence on the angle of incidence The curves and 4 are calculated according to Eq 0 and at the change of angle θ from 0 to 45 Fig 3 Angular deendence of resonant sound transmission through glass with thickness h 00005 m with dimensions of 08 m The curve enveloes maximums of relative dislacements at the incomlete sace resonance The enveloe curve 3 belongs to the cases of ordinary resonances In the range of angles of sound incidence from 45 to 90 the enveloe curves 5 6 and 7 are calculated at α equal to 75 60 and 45 wherefrom it is seen that k 30
ISSN 39-4 ULTRAGARSAS (ULTRASOUND) Vol63 No 008 degree of intensity of sound transmission deends on the correlation of the angles α and α and the closer the angle α to the angle of incident sound α the higher the amlitude of late dislacement Construction of the frequency characteristic of sound insulation At the directed incidence of sound waves onto the rectangular thin late with the sides a and b for each angle of incidence of sound its own frequency of satial resonance exists where sound transmission is at maximum In the ranges of frequencies higher and lower of that frequency resonse of the late is different and deends on the conditions of interaction of the acoustic field of disturbance and the field of flexural oscillations of the late Investigation of the conditions for the interaction of the fields in the cases of ordinary incomlete and comlete sace resonances enables one to trace the mechanism of sound transmission ie to identify the contribution of those resonances in each calculated range of frequencies into sound transmission through the late along its amlitudes of dislacements Quantitative correlations of the amlitude of oscillations and the oscillating seed of the late [ 4] give the oortunity to exress the acoustic ower that is radiated by the late when it gets excited by a lane sound wave falling at the angle θ Hence knowing the incident and through transmitted in the resonant regime acoustic owers we obtain the exression of sound insulation in the form of fδfη R = 0 lg Λ cosθ (7) s where Λ is the numerical coefficient which covers the ermanent integrations and deends on the width of the frequency interval and the calculated area of research (Table ) s is the averaged characteristic of acoustic 76 radiation the values of which in each of the calculated ranges are found according to Table The ractical method for constructing the frequency characteristic of sound insulation of the barrier at the directed incidence of sound onto a rectangular thin late is based on the use of the deendence given by Eq7 It is seen from Eq7 that at the directed sound incidence the quantitative and qualitative measure for sound insulation of real one-layered thin lates is as follows: the weight of the barrier the frequency of the incident sound the late dimensions its flexural rigidity the loss factor and the angle of incidence of sound waves onto the late Calculations are carried out on the geometric mean frequencies of the three-octave bands for three frequency ranges: at the frequency of the satial resonance: fγ 88433(sin θ D / m) θ (8) at frequencies higher of the satial resonance f > f Γθ at frequencies lower of the satial resonance to the boundary frequency of incomlete sace resonances: Γ θ = 7( a sinθ) f (9) The rocedure for construction of the calculated deendence of sound insulation is as follows: we define the boundary frequencies f Γθ and f Γ θ according to Eq8 and 9 for the reset angle θ we find the value of sound insulation at the frequency of satial resonance according to Eq7 and Table in deendence on the angle θ according to Eq0 we find the averaged characteristics of sound radiation in the range of frequencies higher of the satial resonance we define sound insulation in that range of frequencies according to Eq7 from Eq (see Table ) on each geometric mean frequency we find the averaged characteristics of sound radiation for the range of frequencies lower of the satial resonance we define sound insulation in this range of frequencies from Eq7 Table Values of sound insulation characteristics Calculated range of frequencies Numerical coefficient Λ Averaged characteristic of sound insulation s at degree 5 30 45 60 75 Range of sace resonances (SR) f r 90-4 05 058 07 93 Range higher of sace resonance (SR) f > f r 90-4 7 Δ = 6 0 f r s = Δ (0) 3 abf 6 Δ = 3 0 6 6 Δ =56 0 Δ = 04 0 5 Δ = 836 0 Range lower of sace resonance (SR) 70-3 f r f < f r 054 f r f r s = () ( f f r ) 3
ISSN 39-4 ULTRAGARSAS (ULTRASOUND) Vol63 No 008 we construct the frequency characteristic of sound insulation of the given barrier Fig 4 Frequency characteristic of insulation of the directed sound with silicate glass: thikness h = 0005 m dimensions 08 m = 3 kg/m η = 0 0 - The frequency characteristic in Fig 4 is constructed by using the roosed ractical method (see Table ) The frequency deendence is obtained by exeriment in a big acoustic chamber The line 3 shows for comarison of the Mass Law All deendences are constructed for the angle θ = 75 Conclusions In estimating the acoustic roerties of insulating materials it is imortant to determine whether the deformation changes do not coincide with the changes of tensions In order to determine the otimum values the changes of deformations and the changes of tensions αλ should equal αλ = 5 The material for low frequencies is selected with ης = 04 3 Calculations show that change of the angle θ at the same frequency (see Fig 3) leads (in deendence on the character of interaction of wave arameters and angles of the sound and vibration fields) to the emergence of the alternating maximums and minimums of the amlitudes of late dislacement The angular characteristic of the directed sound transmission through the late of finite dimensions confirms the rightfulness of the division of the frequency deendence of sound insulation into three ranges of frequencies with the different intensity of sound transmission 4 Investigation of the conditions for the interaction of the fields in the cases of ordinary incomlete and comlete satial resonances enables one to trace the mechanism of sound transmission ie to identify the contribution of those resonances in each calculated range of frequencies into sound transmission through the late along its amlitudes of dislacements 5 It is seen from Eq7 that at the directed sound incidence the quantitative and qualitative measure for sound insulation of real one-layered thin lates is as follows: the weight of the barrier the frequency of the incident sound the late dimensions its flexural rigidity the loss factor and the angle of incidence of sound waves onto the late 6 The given ractical method makes it ossible by the method of calculation to construct the frequency characteristics of sound insulation of real thin barriers at the reset angles of sound incidence References Čiučelis A Gailius A Gužas D Alication of vibroacoustic roerties of building Materials nd International Conference Mechatronic Systems and Materials MSM Cracow Poland 3 08 3 09 006 P 0 Sedov M S Theory of inertial sound transmission through barrier constructions Higher Institutions News Series Construction and Architectures 990 No P 37 4 (in Russia 3 Cremer L Theorie der schalidannung dumer wande bei schrigem einfall Acoustische Zeitschrift 94 No 7 P 8 4 Stauskis V J Sound insulation of the barrier constructions in residential and ublic buildings and ways for its imrovement Vilnius: LIINTI 973 P 37 D Gužas A Gailius I Girnienė Statybinių konstrukcijų garso izoliacijos riklausomybė nuo medžiagų akustinių savybių Reziumė Tyrinėjamos medžiagų savybės kurios turi didžiausią įtaką statybinių konstrukcijų garso izoliacijai Ištirta medžiagos vidaus trinties nuostolių įtaka garso sloinimui kai garso energija ereina er statybines konstrukcijas Plačiau tyrinėjama garso krintančio į ertvaros bangos kritimo kamą dažnių įtaka Nustatyta kada ir kokiems dažniams esant garso izoliacija geriausia ir kada ji yra rasta t y raleidžia tam tikrų dažnių garsą į kitą konstrukcijos sienelės usę Pateikta saudai 007 0 3