Impact sound transmission through a floating floor on a concrete slab

Applied Acoustics 59 (2000) 353±372 www.elsevier.com/locate/apacoust

Impact sound transmission through a ¯oating ¯oor on a concrete slab Michael. A. Stewart a,*, Robert. J.M. Craik b a School of the Built Environment, Napier University, Edinburgh, UK Department of Building Engineering and Surveying, Heriot-Watt University, Edinburgh, UK

b

Received 17 June 1998; received in revised form 1 June 1999; accepted 1 June 1999

Abstract In this paper a theoretical model to predict bending wave transmission through parallel plates connected by a resilient line is presented. The model is based on the interaction of semiin®nite plates intersecting along an in®nite boundary. A full model is developed together with some approximations for the more common cases. The results of the model are then used in a statistical energy analysis (SEA) framework to predict transmission through a chipboard ¯oating ¯oor attached to battens which in turn are supported on a concrete ¯oor. Acoustic transmission through the cavities is also considered. The results show that transmission through the battens is the most important path when there is no resilient layer but that this path is insigni®cant if any resilient layer is present. A comparison between measured and predicted results gave generally good agreement except at high frequencies when a resilient layer was present. # 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction A lightweight timber ¯oor on battens separated from a concrete structural ¯oor by a resilient layer is a common form of party ¯oor construction and is deemed acceptable for sound insulation in both the Scottish and English Building Regulations [1,2]. A section through such a ¯oor can be seen in Fig. 1. In the 1970s the Building Research Establishment (BRE) conducted ®eld measurements of the sound transmission through this type of ¯oor [3]. One quarter of the ¯oors failed the impact sound insulation requirement of the Regulations and most of the failures were due * Corresponding author. 0003-682X/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0003-682X(99)00030-4

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to non standard types of interlayer such as cork pads, asbestos in metal clips and polystyrene. When a 25 mm mineral wool quilt was used the number of failures was much lower although the sample size was small. In this paper a model to predict structure-borne sound transmission through this type of ¯oor is developed to include the e€ect of the resilient layer. The wave model of Craik and Wilson [4] for bending wave transmission between line connected parallel plates is developed to include the e€ects of the sti€ness of the resilient layer using the same approach as Mees and Vermeir [5] and Craik and Osipov [6]. The inertia and sti€ness of the timber batten are included in the model. Acoustic transmission through the air space between the battens is also considered and predictions are compared with measurements using both laboratory models and a full size ¯oor. The paper begins with the development of a model to predict structural coupling between parallel plates connected by a resilient line. The results are used in a Statistical Energy Analysis (SEA) model where the e€ects of plate damping and other transmission paths are included [7]. Acoustic paths are added for comparison with the structural path. 2. Transmission between parallel plates connected along a resilient line Fig. 2 shows the connection between the chipboard and the structural concrete ¯oor considered in this work. A bending wave of unit amplitude on plate 1 is

Fig. 1. Section through a concrete ¯oor with a ¯oating chipboard ¯oor resting on a resilient layer.

Fig. 2. Schematic diagram of a ¯oor showing the plate layout used in the theoretical model.

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355

assumed to be incident on the boundary at an angle, , to the normal so that the amplitude of the incident wave, 0 , can be given by, 0 ˆ eÿik1 cos 1 x eÿik1 sin 1 y ei!t

…1†

k1 is the bending wavenumber on plate 1 which is given by [8],

k1 ˆ 4

r s 2 ! B

…2†

where s is the surface density, B is the bending sti€ness per unit width. Each plate will support a travelling wave with amplitude Ti and a near®eld wave with amplitude Tni so the displacement on each plate can be written as [7], 1 ˆ …eÿik1 cos 1 x ‡ T1 eik1 cos 1 x ‡ Tn1 ekn1 x †eÿik1 sin 1 y ei!t

…3†

2 ˆ …T2 eÿik2 cos 2 x ‡ Tn2 eÿkn2 x †eÿik2 sin 2 y ei!t 3 ˆ …T3 eik3 cos 3 x ‡ Tn3 ekn3 x †eÿik3 sin 3 y ei!t 4 ˆ …T4 eÿik4 cos 3 x ‡ Tn4 eÿkn4 x †eÿik4 sin 4 y ei!t kni is the near®eld wavenumber and is related to the bending wavenumber by [8], q kni ˆ ki …1 ‡ sin2 1 †

…4†

The angle at which waves will leave the junction on any plate can be found from Snell's law which gives [8] k1 sin…1 † ˆ ki sin…i †. 2.1. Displacement At the boundary the displacements of plates 1 and 2 are equal which gives [4] 1 ˆ 2 . Similarly the displacement of plates 3 and 4 are equal which gives 3 ˆ 4 . If the spring is expanded by an amount k then the displacements of plates 1 and 3 are related by 1 ˆ 3 ‡ k . Inserting Eq. (3) into these equations and setting x ˆ 0 gives three equations relating the displacements of the four plates as, 1 ‡ T1 ‡ Tn1 ˆ T2 ‡ Tn2

…5†

T3 ‡ Tn3 ˆ T4 ‡ Tn4

…6†

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1 ‡ T1 ‡ Tn1 ˆ T3 ‡ Tn3 ‡ k

…7†

2.2. Slope The slope of any plate is given by  ˆ @=@x. At the boundary the slopes of plates 1 and 2 are equal giving 1 ˆ 2 . Similarly the slopes of plates 3 and 4 are equal giving 3 ˆ 4 . If the batten is bonded to the resilient layer and the layer is ®rmly attached to the concrete then by including the angular displacement of the interlayer, k , the slopes of plates 1 and 3 at the boundary are related by, 1 ˆ 3 ‡ k . Substituting Eq. (3) into these equations gives, ÿik1 cos 1 ‡ ik1 cos 1 T1 ‡ kn1 Tn1 ˆ ÿik2 cos 2 T2 ÿ kn2 Tn2

…8†

ik3 cos 3 T3 ‡ kn3 Tn3 ˆ ÿik4 cos 4 T4 ÿ kn4 Tn4

…9†

ÿik1 cos 1 ‡ ik1 cos 1 T1 ‡ kn1 Tn1 ˆ ik3 cos 3 T3 ‡ kn3 Tn3 ‡ k

…10†

However, in real ¯oor construction the battens are simply resting on the resilient layer on the concrete ¯oor, in which case the relationship between the slope of the plates and the interlayer is only approximate. 2.3. Moments When considering the bending moments and forces it is helpful to divide the joint into two junctions, I and II, as shown in Fig 2. At each junction there will be moments applied from each plate which can be given by [7],  2  @  @2  ‡ 2 …11† M ˆ ÿB @x2 dy There will also be a moment due to the rotational sti€ness of the spring given by, M k ˆ B k k

…12†

If the resilient layer is assumed to behave as a locally reacting material with width b and thickness d then the sti€ness Bk can be found from the apparent dynamic Young's Modulus (E 0 ) using Bk ˆ

E 0 b3 d 12

The layer will also have a dynamic compression sti€ness K given by

…13†

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Kˆ

E0 b d

357

…14†

so that Bk ˆ Kb2 d 2 =12. Damping in this interlayer can be included by making the value of the apparent Young's modulus complex by multiplying by …1 ‡ ik † where k is the loss factor of the resilient layer. In addition  there are resisting moments due to the rotary inertia (Mc) and torsional @M sti€ness @yy of the batten giving a moment of [8], Mc ‡

@My ˆ …ÿ!2 J ‡ TT k 22 sin2 2 †2 @y

…15†

where J is the polar moment of inertia per unit length and TT is the torsional sti€ness of the batten per unit length. For clarity the moment is written in terms of 2 but could equally be written in terms of 1 . At junction I the sum of the moments will be zero giving, @My …16† M 1 ˆ M 2 ‡ Mk ‡ Mc ‡ @y which can be written in terms of the wave amplitudes as, B1 k21 …cos2 1 ‡ 1 sin2 1 † ‡ B1 k21 …cos2 1 ‡ 1 sin2 1 †T1 ÿ
ÿ B1 …k2n1 ÿ 1 k21 sin2 1 †Tn1 ˆ B2 k22 …cos2 2 ‡ 2 sin2 2 ÿ
ÿ
ÿ ÿ!2 J ‡ TT k22 sin2 2 †ik2 cos 2 T2 ÿ B2 …k 2n2 ÿ 2 k 22 sin2 2 ÿ
‡ ÿ!2 J ‡ TT k 22 sin2 2 †kn2 Tn2 ‡ Bk …1 ‡ ik †

…17†

…ÿik1 cos 1 ‡ ik1 cos 1 T1 ‡ kn1 Tn1 ÿ ik3 cos 3 T3 ÿ kn3 Tn3 † At junction II the sum of the moments will also be zero giving, M 3 ˆ M 4 ÿ Mk

…18†

which can be written as, B3 k23 …cos2 3 ‡ 3 sin2 3 †T3 ÿ B3 …k2n3 ÿ 3 k23 sin2 3 †Tn3 ˆ B4 k24 …cos2 4 ‡ 4 sin2 4 †T4 ÿ B4 …k2n4 ÿ 4 k24 sin2 4 †Tn4 ÿ Bk …1 ‡ ik †…ÿik1 cos 1 ‡ ik1 cos 1 T1 ‡ kn1 Tn1 ÿ ik3 cos 3 T3 ÿ kn3 Tn3 † …19† Maximum moment coupling occurs when the batten is rigidly attached to the concrete. However, if the batten is twisted (due to shrinkage following cutting of the timber) then there will be only partial contact between the batten and ¯oor so the

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moment coupling will be reduced and may even be zero. Calculations show that moment coupling is insigni®cant for the case of chipboard and concrete plates and therefore the choice of moment boundary condition [and details of the continuity equation which lead to Eqs. (8)±10)] is unimportant. 2.4. Forces At each junction there will be shear forces applied from each plate which can be given by [7],  3  @  @3  ‡ …2 ÿ † …20† FˆB @x3 @x@y2 The spring force acting on the junction Fk , equals Kk . Again damping in the interlayer can be included by making E complex. There will also be forces at the boundary due to the inertia of the batten (Fc ) and bending forces along the length of the batten which can be written [8], Fc ‡

@Fy ˆ …ÿ!2 m0 ‡ Bc k42 sin4 2 †2 @y

…21†

where m0 is the mass/unit length and Bc is the bending sti€ness of the batten. At junction I the sum of these forces must be zero giving, @Fy …22† F1 ˆ F2 ‡ Fk ‡ Fc ‡ @y which can be written in terms of the wave amplitudes as, ÿ
iB1 k31 cos 1 cos2 1 ‡ …2 ÿ 1 † sin2 1 ÿ
ÿ iB1 k31 cos 1 cos2 1 ‡ …2 ÿ 1 † sin2 1 T1 ÿ
‡ B1 kn1 k2n1 ÿ …2 ÿ 1 †k21 sin2 1 Tn1 ÿ
ˆ iB2 k32 cos 2 …cos2 2 ‡ …2 ÿ 2 † sin2 2 ÿ
ÿ ÿ
‡ ÿ!2 m0 ‡ Bc k42 sin4 2 † T2 ÿ B2 kn2 k2n2 ÿ …2 ÿ 2 †k22 sin2 2

…23†

ÿ …ÿ!2 m0 ‡ Bc k42 sin4 2 ††Tn2 ‡ K…1 ‡ ik †…1 ‡ T1 ‡ Tn1 ÿ T3 ÿ Tn3 † Similarly at junction II the sum of the forces is also zero giving F3 ˆ F4 ÿ Fk which can be written in terms of wave amplitudes as, ÿ
ÿ
ÿ iB3 k33 cos 3 cos2 3 ‡ …2 ÿ 3 † sin2 3 T3 ‡ B3 kn3 k2n3 ÿ …2 ÿ 3 †k23 sin2 3 ÿ
ÿ
Tn3 ˆ iB4 k34 cos 4 cos2 4 ‡ …2 ÿ 4 † sin2 4 T4 ÿ B4 kn4 k2n4 ÿ …2 ÿ 4 †k24 sin2 4 Tn4 ÿ K…1 ‡ ik †…1 ‡ T1 ‡ Tn1 ÿ T3 ÿ Tn3 † …24†

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Eqs. (5)±(10), (17), (19), (23) and (24) can be solved numerically to give the bending wave amplitudes on each plate. The transmission coecients from plate 1 to any other plate i can be obtained from [7], 1i ˆ

si k1 cos i jTi j2 s1 ki cos 1

…25†

In SEA it assumed that sound and vibration ®elds are di€use and so it is the angular averaged random incidence transmission coecient, av , that is of greatest interest. This is given by av ˆ

… =2 0

…† cos…†d

…26†

This transmission coecient can be expressed in dB as a structural transmission loss R=10 log 1=. The angular averaged transmission coecient can be used to calculate the coupling loss factor between platesÐif the sound ®eld on the source plate is di€use Ð using the equation [7], 1i ˆ

2L1i k1 S1

…27†

where L is the common boundary length and S1 is the area of plate 1.

3. Discussion The previous model can be used to calculate the angular averaged transmission coecient and includes both the e€ects of the batten and the resilient layer. If the @M @F batten terms Mc ; @yy ; Fc ; @yy are not included, the model becomes much simpler. A comparison of the structural transmission loss predictions with and without the batten terms can be seen in Fig. 3 which shows transmission between plates 1 and 3 for di€erent values of compression sti€ness, K, with and without the batten terms (batten and plate properties are shown in Table 1). At low frequency the inclusion of batten terms has no signi®cant e€ect, however, at higher frequencies there is an increase in R when the additional terms are included. At 3.15 kHz the value of R is about 5 dB greater than it is for a joint with no batten. In-plane forces generating in-plane waves (both longitudinal and transverse) could be included but this would increase the complexity of the model. However, such forces can be neglected for the work in this paper as the batten size is small in comparison to the thickness of the concrete and therefore the e€ects on the prediction will be negligible. The transmission coecients were re-computed with the rotational sti€ness …Bk † set to zero (so there was no moment coupling between the upper and lower plates)

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and the bending sti€ness term



@Fy @y



in the force boundary condition removed. The

resulting transmission loss was less than 0.1 dB from the value when all terms are present so the increase in structural transmission loss due to the additional terms shown in Fig. 3 is due to the inclusion of the inertia term …Fc† in the force boundary condition. It is therefore the mass of the batten that is important rather than its shape and sti€ness. Although the calculation of the transmission coecients is complicated in the above model it can be simpli®ed to give useful approximations. It has been observed [7] that, for transmission from the concrete to the chipboard, the normal incidence transmission coecient and the angular averaged transmission coecient are approximately equal. This is true providing the concrete has a lower critical frequency than the chipboard which is always the case in buildings. If the variables  and are de®ned as

Fig. 3. Structural transmission loss for transmission from chipboard to concrete (plates 1 and 3) showing the e€ect of changing the sti€ness per unit length (N/m2) of the interlayer and the e€ect of the inertia of a 4545 mm timber batten. Ðб, no batten; ....., with batten. Table 1 Properties of the chipboard, concrete and battens

Chipboard 1 and 2 Concrete 3 and 4 Battens

(kg/m3)

b(m)

d(m)

E(N/m2)

542 1737 525

± ± 0.044

0.022 0.200 0.044

4.0109 2.51010 10.8109

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kchip ˆ ˆ kconc ˆ

s fc chip fc conc

Bchip k2chip Bconc k2conc

ˆ

s chip fc conc s conc fc chip

361

…28†

…29†

then the normal incidence transmission coecient for a rigid joint can be given by [7] conc chip ˆ

 …1 ‡ 2 ‡ 2 ‡ 2 2 ‡ 4 † 2… ‡ †2 …1 ‡  †2

For the joints being considered in this paper  > 1 and approximated to p s chip fc conc  p ˆ conc chip ˆ 2 2s conc fc chip

…30†  1 so that this can be

…31†

For joints with a resilient layer it has been found that the transmission is mainly due to the forces and that the moment coupling can be neglected. Therefore if the batten

Fig 4. A comparison between the random incidence structural transmission loss and the normal incidence approximation for transmission between chipboard and concrete (plates 1 and 3). Ðб, random incidence; ....., normal incidence approximation [Eqs. (32) and (35)]; - - - - - - -, normal incidence rigid approximation [Eqs. (31) and (35)].

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terms are ignored and the moment sti€ness Bk is set to zero then the equations described previously can be simpli®ed and the transmission coecient at normal incidence is given by  K2  …32† conc chip ˆ  2 2 2 2  K ‡ 2 K ‡ K2 ÿ 4 2 2 KBconc k3conc 2 ÿ4KBconc k3conc ‡ 8B2conc 2 2 k6conc At low frequencies when kconc is small, this can be approximated to Eq. (31). At high frequencies when kconc is large this can be simpli®ed to pp K2 fc conc fc chip …33† conc chip ˆ 642 s conc s chip f 3 c2o As the angular averaged transmission from the concrete to the chipboard (from the low critical frequency to the high critical frequency) is approximately the same as the normal incidence transmission, the above approximations can be used directly in the SEA model where it is assumed that sound is randomly incident on any boundary. For transmission from the chipboard to concrete the consistency equation [7] k1 12 ˆ k2 21

…34†

can be used giving chip conc ˆ conc chip kconc =kchip ˆ conc chip =

…35†

Therefore predictions of transmission from the chipboard to concrete can be made using this equation and the normal incidence approximations in Eqs. (31) and (33). For the rigid case, which is also appropriate at low frequencies where there is a resilient layer, from Eqs. (31) and (35), the transmission coecient is chip conc ˆ

2

ˆ

s chip fc conc 2s conc fc chip

…36†

and for the case where there is a resilient layer present, at high frequencies the transmission coecient [from Eqs. (33) and (35)] is chip conc ˆ

K2 fc conc 642 s conc s chip f 3 c20

…37†

An estimate of the transition frequency can be found by equating Eqs. (36) and (37) to give

M.A. Stewart, R.J.M. Craik / Applied Acoustics 59 (2000) 353±372

f3 ˆ

363

K2 f c chip 322 2s chip c20

…38†

which shows the transition frequency is dependant on the properties of the layer and the chipboard but is independent of the properties of the concrete. A comparison of these simpli®ed equations with the full model is given in Fig 4. The solid lines are the full model computed without the e€ect of the battens as was shown in Fig. 3. The dotted curves are from the simpli®ed predictions using Eqs. (32) and (35) and the rigid approximation is obtained from Eqs. (31) and (35). It can be seen that these simpli®ed predictions give excellent agreement with the full model, particularly at high frequencies. Modelling the resilient layer as a series of locally reacting springs is only appropriate at lower frequencies as at higher frequencies longitudinal waves occur across the thickness of the interlayer. Gosele [9] has shown that at frequencies where longitudinal waves in air occur across the cavity depth between parallel plates a good estimate of the air sti€ness per unit area, K 0 , can be obtained from, K 0 ˆ !c1 . This approach can also be used to give an estimate of the sti€ness per unit length of the resilient layer as K ˆ !c1 b

…39†

where b is the width of the batten and  and c1 are the density and longitudinal wavespeed of the resilient layer [10]. Clearly there are many approximations in this approach since if there can be longitudinal waves in the thickness then there will also be waves across the width of the layer (which is always greater than the thickness). Thus from the perspective of the layer the junction is not along a line but occurs over an area several wavelengths wide. For closed cell foams that are typically only 5 mm thick, longitudinal waves can be neglected but for open cell foams (20 mm thick) wave e€ects are important. For the open cell foams considered in this paper the ®rst longitudinal mode (where half the wavelength is 20 mm) was as low as 500 Hz. In the transition region between the low and high frequency methods of calculating interlayer sti€ness [Eqs. (14) and (39)], neither method is strictly applicable. A reasonable transition frequency can be taken as where the sti€ness of Eqs. (14) and (39) are equal. Neglecting the small di€erences between the longitudinal wavespeed in an in®nite medium, a plate or beam, the transition frequency can be given by

Table 2 Properties of the resilient layers used Material

Thickness d mm

Sti€ness KN/m2

Damping

Closed cell foam Laminated foam Fibre glass Mineral wool

5 23 25 25

2800 77 165 86

0.14 0.17 0.05 0.04

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M.A. Stewart, R.J.M. Craik / Applied Acoustics 59 (2000) 353±372

fˆ

c1 2d

…40†

where c1 is calculated from the measured apparent dynamic Young's modulus and material density, s E0 …41† c1 ˆ  An alternative approach to modelling the joint is to use the approach of Craik and Smith [11] and model all the waves in the resilient layer. This approach allows all the waves in the layer to be included but does not address the diculty with a joint that is several wavelengths wide. A comparison between these models can be seen in Fig. 5. The Craik and Smith model has the same result as the model presented in this paper at low frequencies but has well de®ned dips at higher frequencies. Those occurring at multiples of 784.8 Hz occur where integer numbers of half longitudinal wavelengths ®t across the resilient layer and those at multiples of 429.9 Hz occur where integer numbers of half transverse wavelength ®t across the resilient layer. For the curve showing the three approximations, at low frequencies Eq. (36) is used. In the mid and high frequency ranges Eq. (37) is used with Eq. (14) and Eq. (39) (as appropriate) for the sti€ness.

Fig 5. A comparison of transmission loss and the normal incidence approximation for transmission between chipboard and concrete (plates 1 and 3). - - - - - - -, three approximations [Eqs. (36) and (37) Eq. (37) is used with low frequency and high frequency methods of calculating K]; ÐÐÐ, Craik and Smith [11]; ÐÐ, normal incidence model [Eqs. (32) and (35)]; б&±Ð, random incidence model.

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4. Laboratory measurements In order to test the theories given in the previous section, measurements were made on two chipboard plates connected along their edge by a batten and resilient layer as shown in the insert of Fig. 6. The plates were placed parallel to each other and the measurement was carried out in an anechoic chamber to eliminate airborne transmission. The properties of the resilient layers are shown in Table 2. Fig. 6 shows the acceleration level di€erence when a 5 mm thick closed cell foam interlayer with a sti€ness of 2800 kN/m is used. The agreement between measurement and prediction across the frequency range is good. Fig. 7 shows the measured and predicted acceleration level di€erence between the plates for a laminated foam interlayer consisting of a 13 mm layer of open cell foam bonded to a 10 mm layer of closed cell foam. The interlayer was modelled as a spring with a sti€ness of 77 kN/m and over most of the frequency range there is good agreement between the measured and predicted results. At high frequencies the agreement is poorer and this may be due to wave motion in the resilient layer. The high frequency approximation of Gosele [Eq. (39)] for the resilient layer gives a lower level di€erence but the agreement is not signi®cantly better and the shape of the revised curve does not match the measured results. The layer used in Fig. 6 was much thinner and so behaves as a spring over a wider range of frequencies. These results show that the wave model can be used to predict transmission between thin parallel plates coupled by a resilient line except at high frequencies with very soft resilient layers.

Fig 6. Measured and predicted acceleration level di€erence between two chipboard panels coupled by a 4545 mm batten and a 5 mm closed foam interlayer (K=2800 kN/m). ÐÐ, measured; - - - - -, predicted.

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Fig 7. Measured and predicted acceleration level di€erence between two chipboard panels coupled by a 4545 mm batten and a 23 mm laminated foam interlayer (K=77 kN/m). Ðб, measured; - - - - -, predicted using simple spring theory; ....., predicted using a sti€ness computed using Eq. (39).

5. Measurements on a full size ¯oor Measurements were carried out on a full size (54 m) ¯oor in a vertical transmission suite using a tapping machine as a vibration source. The ¯oor consisted of 22 mm chipboard ®xed to 4545 mm battens placed at 450 mm centres resting on a 200 mm thick concrete ¯oor as shown in Fig. 8. The material properties are given in Table 1. The measured structural level di€erence was determined from the mean of thirty positions with the accelerometers placed at random positions on both ¯oors. The tapping machine was also placed at random positions for each measurement which will have included positions directly over the batten. The resilient layers were closed cell foam, laminated foam, mineral wool and ®bre glass. The closed cell foam and mineral wool were laid as continuous sheets and the laminated foam and ®bre glass were laid as strips glued to the underside of the battens. When predicting the results it was assumed that the tapping machine was located so as to excite the middle two sections of chipboard and that equal amounts of power were input into each section. The predicted level di€erence was computed using SEA assuming transmission both by the structural path through the battens and by the airborne path through the cavity. The airborne path involves bending wave radiation [12] and near®eld radiation at the impact point of the tapping machine hammer [13] into the cavity followed by excitation of the concrete. Measurements were made ®rst on a ¯oor with

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367

no resilient layer and the results are shown in Fig. 9. There is good agreement between the measured and predicted level di€erence and it can be seen that except at low frequencies the transmission is dominated by the structural path. Measurements were then made on the same ¯oor but with a 5 mm closed cell foam (K=2800 kN/m) under the batten. Fig. 10 shows the acceleration level di€erence between chipboard and concrete ¯oor. In this case the theory shows that the structural path is approximately equal to the acoustic path through the cavity. Summing the two paths gives a prediction which is approximately 5 dB above the measurement at lower frequencies. At higher frequencies however the measured level di€erence is much lower than predicted. Similar measurements were made on the same ¯oor but with a 23 mm thick laminated foam (K=77 kN/m) and the results are shown in Fig. 11. In this case the layer is much softer and the predicted transmission is dominated by the acoustic path at

Fig 8. Section through the vertical sound transmission suite. The resilient layer may be continuous or only under the battens.

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Fig 9. Measured and predicted acceleration level di€erence from a chipboard ¯oor to a concrete ¯oor through 4545 mm battens with no interlayer. ÐÐ, measured; - - - - -, predicted transmission through the batten; ....., predicted transmission through the cavity; -.-.-.-.-., sum of the predictions.

Fig 10. Measured and predicted acceleration level di€erence from a chipboard ¯oor to a concrete ¯oor through 4545 mm battens supported on a 5 mm closed cell foam interlayer (K=2800 kN/m). Ðб, measured; - - - - -, predicted transmission through the batten; ....., predicted transmission through the cavity; -.-.-.-.-., sum of the predictions.

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Fig 11. Measured and predicted acceleration level di€erence from a chipboard ¯oor to a concrete ¯oor through 4545 mm battens supported on a 23 mm laminated foam interlayer (K=77 kN/m). Ðб, measured; - - - - -, predicted transmission through the batten; ....., predicted transmission through the

Fig 12. Measured acceleration level di€erence from a chipboard ¯oor to a concrete ¯oor through 4545 mm battens supported on various interlayers. Ðб, no battens; ....., closed cell foam (K=2800 kN/m); .-.-.-, ®bre glass (K=165 kN/m); - - - - -, laminated foam (K=77 kN/m); . . .&. . ., mineral wool strips (K=86 kN/m); - - -&- - -, mineral wool in a continuous layer (K =86 kN/m).

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all frequencies. Again there is reasonable agreement at low frequencies up to 500 Hz but poor agreement at higher frequencies. Measurements were also made with interlayers of semi-rigid ®bre glass and mineral wool quilt and all results showed similar discrepancies between the measured and predicted results at the higher frequencies. If transmission from the chipboard to the concrete is not by coupling through the batten then using di€erent interlayers should give similar results (there may be slight di€erences as the cavity loss factors were not exactly the same due to the di€erent absorption properties of each interlayer). However, di€erent results were observed as can be seen in Fig. 12 which cannot be explained by the predictions. It is possible that the occasions when the tapping machine was directly over the batten introduced direct coupling to the concrete. However, this mechanism would need to be over 30 dB greater than the predicted structural path to explain the measured results. Coupling is stronger when closed cell foam and semi rigid ®breglass layers are used which are both sti€er than quilt and laminated foam. This suggests that the resilient layer does a€ect transmission at high frequencies even though the wave model does not predict it. Either the wave model is not appropriate or else there are additional important paths that have not been included. For most practical situations it is sound radiation into the receiving room that is important and not structural ¯oor vibration. Fig. 13 shows the level di€erence

Fig 13. Measured and predicted transmission from acceleration of the chipboard to the sound pressure level in the room below for a closed foam interlayer (K=2800 kN/m). ÐÐÐ, measured; - - - - -, predicted transmission through the battens ....., predicted transmission through the cavity; -.-.-.-.-., sum of the predictions.

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between the vibration of the chipboard ¯oor and the sound pressure level in the receiving room for battens on closed cell foam. The result is similar to the structural level di€erence in Fig. 11 showing increasing divergence between the predicted and measured pressure level with increasing frequency. 6. Conclusions A wave model to predict bending wave transmission between parallel plates connected by a resilient line has been developed. This has been used to give simple approximations which can be used to estimate the transmission through the ¯oor for both a rigid joint and a joint with a resilient layer. These values of transmission coecient have then been used in a statistical energy analysis framework to predict the structure-borne sound transmission through a chipboard ¯oor separated from a concrete ¯oor by battens on resilient layers. The resilient layer was modelled as a series of independent springs. Laboratory experiments on isolated structural joints showed good agreement between the measured and predicted level di€erence for transmission across a structural joint except for a thick soft layer at higher frequencies. Results for a full size ¯oor showed that coupling through the batten was the dominant path when no interlayer was present and in that case there was good agreement between the measured and predicted results. However when a resilient layer was present the structural coupling is predicted as being negligible compared with acoustic coupling through the cavities between the battens. The agreement between the predicted and measured results is reasonable at low frequencies but the theoretical model signi®cantly under-estimates coupling at the higher frequencies. The model described in this paper assumes a continuous line connection between the parallel plates. As the battens are only resting on the concrete ¯oor, and the chipboard was screwed to the battens at discrete intervals, this condition may not be ful®lled and therefore alternative methods of modelling the boundary where localised stress ®elds are predicted should be considered [14]. Acknowledgement This work was funded by the Engineering and Physical Sciences Research Council. References [1] Anon. The building standards (Scotland) regulations 1990, part H Ð resistance to transmission of sound. HMSO, 1990. [2] Anon. The building regulations 1991, part E Ð resistance to the passage of sound HMSO, 1991. [3] Sewell EC, Alphey RS. Field measurements of the sound insulation of ¯oating party ¯oors with a solid concrete structural base. Building Research Establishment Current Paper 3/79, 1979. [4] Craik RJM, Wilson R. Sound transmission through parallel plates coupled along a line. Applied

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Acoustics 1997;49:353±72. [5] Mees P, Vermeir G. Structure-borne sound transmission at elastically connected plates. J Sound Vib 1993;166:55±76. [6] Craik RJM, Osipov AG. Structural isolation of walls using elastic interlayers. Applied Acoustics 1995;46:233±49. [7] Craik RJM. Statistical energy analysis of sound transmission through buildings. UK: Gower Press, 1996. [8] Cremer L, Heckl M, Ungar EE. Structure-borne sound. Berlin: Springer±Verlag, 1973. [9] Gosele K. Prediction of the sound transmission loss of double partitions (without structure-borne connections). Acustica 1980;45:218±27 (in German). [10] Stewart MA. Sound transmission through a chipboard ¯oating ¯oor supported on a concrete slab. Ph.D. thesis, Heriot-Watt University, Edinburgh, 1996. [11] Craik RJM, Smith RS. Sound transmission through double leaf partitions: part II structure-borne sound. Applied Acoustics, in preparation. [12] Leppington FG, Broadbent EG, Heron KH. The acoustic radiation eciency of rectangular panels. Proc Royal Soc Lond A 1982;382:245±71. [13] Maidanik G, Kerwin Jr EM. In¯uence on ¯uid loading on the radiation from in®nite plates below the critical frequency. J Acoust Soc Am 1966;40:1034±8. [14] Hammer P. Studies of power transmission and sound radiation for source and receiver structures coupled via large contact areas. Report TVBA-1003, Lund Institute of Technology, 1991.