Prediction of vibration transmission across finite double wall junctions using a hybrid di ff use-deterministic approach

Wannes Stalmans 1 , Cédric Van hoorickx 2 and Edwin Reynders 3

KU Leuven Kasteelpark Arenberg 40, 3001 Leuven, Belgium

ABSTRACT Predicting the sound insulation between two rooms is a complex problem since not only the direct path through the separating element but also the flanking transmission paths can largely influence the sound insulation of the system. An important parameter for calculating flanking transmission is the vibration reduction index, which relates to the transmission coe ffi cient between the connected plates. The international building acoustics standard ISO 12354-1 / 2 provides prediction formulas for the vibration reduction index of single wall junctions, but not for double wall junctions. A new hybrid di ff use-deterministic approach is proposed to calculate flanking transmission across double wall junctions. The walls and floors are modelled as di ff use subsystems while the connection between the double wall is modelled deterministically. In the derivation of the transmission coe ffi cient between two di ff use subsystems, the di ff use field reciprocity relationship is employed, such that the finite size and structural details of the junction are taken into account. The di ff use field reciprocity relationship relates the vibration transmission to the direct field dynamic sti ff ness of the di ff use subsystems (walls and floors), i.e., the dynamic sti ff ness of the equivalent infinite subsystem as observed at the junction. The new approach is applied to di ff erent types of double wall junctions to determine simplified regression formulas for practical sound insulation design.

1. INTRODUCTION

E ff ective protection from noise disturbance can be achieved by ensuring su ffi cient sound insulation in buildings. Unfortunately, this is a complex technical problem, since design details and multiple transmission paths can strongly influence the sound insulation [1]. Both the direct transmission through the element (partition wall or floor) as well as flanking transmission impacts the overall sound insulation between two rooms. In the international building acoustics standard ISO 12354-1 / 2, the vibration reduction index is used to take into account flanking transmission in the sound insulation of a system. For single wall junctions, the standard provides empirical formulas based on the surface mass ratio of the walls and / or floors as well as regression formulas determined by finite element simulations which are based on the

1 wannes.stalmans@kuleuven.be

2 cedric.vanhoorickx@kuleuven.be

3 edwin.reynders@kuleuven.be

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

ratio of bending impedances of the walls and / or floors. For double wall junctions, no such formulas are provided in the international building acoustics standard. In previous work done by Dijckmans [2] vibration transmission across double walls is calculated using wave theory. The walls, floors and connecting junction are modelled analytically using thin plate theory while the junction in assumed to be of infinite extend. Vibration transmission across the double wall junction is calculated by integrating the plane-wave transmission over all possible angles of incidence. Poblet-Puig and Guigou-Carter applied spectral finite elements to calculate vibration transmission across di ff erent types of finite junctions [3]. Amongst the di ff erent types of junctions, a set of double wall H-junctions is calculated and from these results regression formulas are determined for the vibration reduction index. A new hybrid di ff use-deterministic approach is proposed here to e ffi ciently calculate flanking transmission across double wall junctions while accounting for the finite length of the junction. With this new approach, the walls and / or floors are modelled as di ff use subsystems while the connection between the double wall junctions is modelled deterministically. This new approach is based on the the di ff use field reciprocity relationship which relates the blocked reverberant forces in a a vibrating subsystem at the junction to the direct field dynamic sti ff ness [4], i.e., the dynamic sti ff ness of the equivalent unbounded subsystem as observed at the finite junction. The direct field dynamic sti ff ness matrices in the wavenumber domain can be analytically derived for thin, isotropic, semi-infinite plates, as described by Langley and Heron [5]. A correction for ba ffl ed boundary conditions is used to transform these results to the spatial domain while taking into account the finite length of the junction [6]. Shell finite elements are used to model the connection between the double walls. In the presented work, this new approach is used to determine the structural transmission loss between the walls and floors of double wall junctions since the transmission loss can easily be related to the vibration reduction index. A set of double wall junctions is simulated to investigate the influence of di ff erent parameters on the transmission loss of di ff erent transmission paths while also determining regression formulas for practical sound insulation design.

2. METHODOLOGY

2.1. Determination of the vibration reduction index Two types of double wall junctions are considered in the presented work, Π - and H-junctions, as shown in figure 1. The junctions have a finite junction length L .

4 3

4 3

L

L

2

6 2

6

5 1

Figure 1: Π - and H-junction.

To take into account flanking transmission in the sound insulation of a system, the direction averaged vibration reduction index is used in the international standard ISO 12354-1 / 2, which is defined here as follows [7]:

K jl = 1

 R s , jl + R s , l j  (1)

2

in which R s , jl is the structural transmission loss between plates j and l . The structural transmission loss can be calculated based on the di ff use field transmission coe ffi cient ˆ τ jl between plates j and l .

R s , jl = − 10 log  ˆ τ jl  (2)

In the approach presented here, the ensemble-averaged di ff use field transmission coe ffi cient for thin plate vibration transmission is determined using the di ff use field reciprocity relationship. The derivation can be found in [6] and results in the following formula.

ˆ τ jl = 4 π

X

r , s Im  D ( l ) dir , rs   D − 1 tot Im  D ( j ) dir  D − H tot 

rs (3)

Lk j

Equation (3) shows that the ensemble-averaged di ff use field transmission coe ffi cient for thin plates depends on the junction length, the bending wavenumber of the excited plate, the direct field dynamic sti ff ness matrices of all plates that are connected to the junction and the sti ff ness matrix of the deterministic connection between the double wall junction. The direct field dynamic sti ff ness matrix is the dynamic sti ff ness of the equivalent unbounded subsystem as observed at the finite junction of a considered subsystem, which can be determined in multiple ways. In the work presented here analytical expression are used which are derived in the wavenumber domain using thin plate theory.

2.2. Direct field dynamic sti ff ness evaluation of thin plates This section discusses the direct field dynamic sti ff ness matrix of a thin, isotropic, elastic semi-infinite plate in the wavenumber domain in the local coordinate system of the plate. The global and local are illustrated for a Π -junction in figure 2.

4 3

y ′ x ′

z ′ y ′ x ′

z ′

y ′ z ′ x ′

x ′

y z x

6 2

y ′ z ′

Figure 2: Global and local coordinate systems of the plates illustrated for a Π -junction (local coordinate systems are indicated with a prime)

Since the in-plane (IP) and out-of-plane (OOP) behavior of the plate are decoupled, the direct field dynamic sti ff ness matrix is a block diagonal matrix of which the elements can be found in the work of Langley and Heron [5].

 D ( j ) ′

 (4)

dir , IP ( k x ) 0

D ( j ) ′

dir ( k x ) =

0 D ( j ) ′

dir , OOP ( k x )

The finite length of the junction is taken into account in the transformation of the direct field dynamic sti ff ness matrices of the plates connected to the junction to the spatial domain. A method using a correction for ba ffl ed boundary conditions is introduced to calculate the direct field dynamic sti ff ness matrix of a semi-infinite plate in the spatial domain. First, the displacements of the junction are expressed as a linear combination of a set of shape functions

q = h u ( x ) v ( x ) w ( x ) θ ( x ) i T = X

n q n , s φ n ( x ) (5)

in which φ n is chosen to be the n th sine function (sine functions are used since ba ffl ed boundary conditions are considered outside the junction):

 sin  n π x

L  if 0 ≤ x ≤ L 0 if x < 0 or x > L (6)

φ n ( x ) =

The shape functions φ n are transformed from the spatial domain to the wavenumber domain. Due to the ba ffl ed boundary conditions the integration can be limited from 0 to L .

L Z

L ) L − 1

0 φ n ( x ) e − i k x x d x = e i ( k x − n π

L  + 1 − e i ( k x + n π

L ) L

Φ n ( k x ) =

L  (7)

2  k x − n π

2  k x + n π

The direct field dynamic sti ff ness matrix for a finite junction length in the spatial domain can now be calculated using [8]

∞ Z

D ( j ) dir , nm = 1

−∞ Φ H n ( k x ) D ( j ) dir ( k x ) Φ m ( k x ) d k x (8)

2 π

in which Φ n is a vector with four components consisting of the shape functions in the wavenumber domain, D ( j ) dir is defined in the global coordinate system (see figure 2) and H is the Hermitian transpose. The direct field dynamic sti ff ness matrix D ( j ) dir

′ is defined in the local coordinate system of plate j and has to be transformed to the global coordinate system [5]

′ R T j (9)

D ( j ) dir = R j D ( j ) dir

where the transformation matrix R j transforms the direct field dynamic sti ff ness matrix from the local coordinate system to the global coordinate system. Numerical integration is used for evaluating the integral in equation (8). The wavenumbers are sampled linearly; the number of samples and the upper limit wavenumber value are determined based on the convergence of the solution of the transmission coe ffi cients.

2.3. Sti ff ness evaluation of the deterministic connection The connection between the double wall junctions is modelled as a deterministic plate which is illustrated in figure 3.

4

3

4

3

L

L

6

2

6

2

5

1

Figure 3: Illustration of the model of a Π - and H-junction.

Since shape functions are used to describe the displacements of the junctions, the displacements of the deterministic connection must match these shape functions (see equation (5)). This condition

is added to the virtual work equation of the domain of the deterministic connection. The sti ff ness matrix of the deterministic connection is then derived by discretization into finite elements. Due to the length of this derivation, it is not shown here. Linear shell finite elements are used for the results shown in section 3.

3. PRACTICAL DESIGN FORMULAS

The proposed approach is now applied to Π - and H-junctions, as shown in figure 1. A wide range of junctions is simulated since the goal here is to investigate the influence of di ff erent parameters on the vibration transmission across the junction and to determine practical design formulas. The materials used for the walls and floors in the simulations can be found in table 1, together with the equivalent homogeneous isotropic properties of the materials. In the junctions simulated here, all walls have the same properties, and all floors have the same properties.

Table 1: Material properties

Material ρ [kg / m 3 ] c L [m / s] ν [-]

Concrete (floor) 2200 3800 0.2 Hollowcore concrete (floor) 1575 3800 0.2 Brick (wall) 1750 2700 0.2 Aerated concrete (wall) 800 1900 0.2

The junction length is varied from 3 m to 6 m in steps of 1 m. The thickness of plate 1 (and 3) is varied separately from 0.1 m to 0.2 m to 0.3 m while the thickness of the walls is either 0.1 m or 0.14 m. Two values are used for spacing between the double walls, 0.05 m and 0.1 m. For plotting the results, the structural transmission loss is plotted as a function of the ratio of characteristic moment impedances ψ

χ . This variable is also used in the international standard for the regression curves based on finite element simulations for single wall junctions and in the work of Poblet-Puig and Guigou-Carter [3] for double wall junctions. The ratio of characteristic moment impedances can be calculated as follows

ψ χ = 4 v t

m ′ ⊥ j B 3 ⊥ j m ′ j B 3 j (10)

where m ′ j is the mass per unit area of the excited element j , m ′ ⊥ j is the mass per unit area of the element perpendicular to element j , B j is the bending sti ff ness of element j and B ⊥ j is the bending sti ff ness of the element perpendicular to element j . Using a set of junctions with the properties described above, the di ff use bending wave transmission coe ffi cient is calculated in one-third octave band centre frequencies from 50 Hz to 5000 Hz with the method described in section 2. Following ISO 12354-1, the low-frequency range is defined as the set of one-third octave bands from 50 Hz to 200 Hz, the mid-frequency range from 250 Hz to 1000 Hz and the high-frequency range from 1250 Hz to 5000 Hz.

3.1. Transmission from plate 2 to plate 3 First the transmission path between plates 2 and 3 is analysed, shown in figure 4. This transmission path does not cross the connection between the double walls. The verify the new approach, results are compared to the regression curves by Poblet-Puig and Guigou-Carter [3]. In this work, regression formulas are calculated for an H-junction based on the ratio of characteristic moment impedances while also accounting for the spacing between the walls.

3 4

3 4

2 6

2 6

1 5

Figure 4: Transmission path between plates 2 and 3 for a Π and H-junction

Due to a di ff erence in definition between equation (1) and the international standard 12354-1 / 2 a correction term is added to equation (1):



 (11)

p f c , j f c , l

K jl = 1

 R s , jl + R s , l j  + 5 log

f ref

2

in which f c , j is the critical frequency of plate j and f ref is a reference frequency of 1000 Hz.

(a) Low-frequency range (b) Mid-frequency range

(c) High-frequency range

Figure 5: Comparison between the vibration reduction index determined with the approach presented here (dots) and regression formulas (dashed line) by Poblet-Puig and Guigou-Carter [3] for an H- junction. Lighter colors correspond with shorter junctions lengths.

Figure 5 shows a comparison between the the vibration reduction index determined with the approach presented here and regression formulas by Poblet-Puig and Guigou-Carter for an H-junction. A good correspondence between the regression formulas and results found with the new approach presented here can be observed for all frequency ranges.

Now the influence of the di ff erent parameters of the junction will be investigated. Figure 6 shows the structural transmission loss as a function of the ratio of characteristic moment impedances. It is clearly visible that the spacing between the double walls has almost no influence on the transmission loss for this transmission path. This is due to the fact that this transmission path does not cross the connection between the double walls. Figure 6 also shows that in the low frequency range the junctions length has an influence on the transmission loss since at low frequencies the wavelengths in the plates are long compared to the junction length. A final conclusion that can be drawn from figure 6 is that for higher ratios of characteristic moment impedances, the transmission loss for the transmission path from plate 2 to plate 3 is higher for an H-junction than for a Π -junction. A higher ratio of characteristic moment impedances means that the walls have a higher moment impedance in comparison to the floors. The extra wall in the H-junction thus causes a higher transmission loss for the transmission path from the floor to a wall. For lower ratios of characteristic moment impedances the extra wall in the H-junction evidently has a lower influence on the transmission loss.

(a) Π -junction (b) H-junction

Figure 6: Transmission loss between plates 2 and 3 in the low-frequency range (50-200 Hz). Lighter colors correspond with shorter junctions lengths.

Since for this transmission path the spacing between the double wall does not influence the transmission loss, the spacing is not taken into account when calculating regression curves for this transmission path. The independent variable used for the regression analysis is the logarithmic ψ/χ -ratio or PC, which is the variable used by Hopkins et al. [9] for determining the regression curves for single wall junctions which can also be found in ISO 12354-1 [10]:

PC = log ψ

! (12)

χ

where ψ

χ is the ratio of characteristic moment impedances (see formula 10). Cubic polynomials are used ( A · PC 3 + B · PC 2 + C · PC + D ) to fit regression curves through the data for the transmission losses per junction length. For simplicity of use, single values are used for parameters of the regression curves which only varied slightly for di ff erent junction lengths. For the other parameters a logarithmic fit ( E + F log ( L )) is used to find the parameter of the regression curve for di ff erent junction lengths. With these regression curves, coe ffi cients of determination of 0.99 are found for both junction types, indicating a high accuracy of the fit.

R s , 23 , Π , low = 1 . 1PC 3 + 5 . 6PC 2 − 0 . 85PC + 11 . 7 − 2 . 4 log ( L ) (13)

R s , 23 , H , low = 0 . 9PC 3 + 5 . 8PC 2 + 1 . 65PC + 14 . 6 − 2 . 6 log ( L ) (14)

These regression curves can be seen together with the results in figure 7.

(a) Π -junction (b) H-junction

Figure 7: Transmission loss between plates 2 and 3 in the low-frequency range (50-200 Hz) together with the determined regression curves. Lighter colors correspond with shorter junctions lengths.

Figure 8 shows the results in the mid-frequency range. In this frequency range the junction length only has a slight influence on the transmission loss since, unlike in the low frequency range, the wavelengths are now smaller compared to the junction length. Due to this, the junction length is no longer taken into account when performing the regression analysis.

(a) Π -junction (b) H-junction

Figure 8: Transmission loss between plates 2 and 3 in the mid-frequency range (250-1000 Hz) together with the determined regression curves. Lighter colors correspond with shorter junctions lengths.

The following regression curves are found for the transmission loss in the mid-frequency range for which the coe ffi cients of determination equal 0.99. The regression curves are shown in figure 8.

R s , 23 , Π , mid = 1 . 0PC 3 + 4 . 8PC 2 − 2 . 3PC + 9 . 2 (15)

R s , 23 , H , mid = 1 . 1PC 3 + 5 . 9PC 2 + 0 . 8PC + 11 . 2 (16)

Finally, the results for the high-frequency range are shown in figure 9, together with the determined regression curves. In this frequency range the junction length also has a negligible influence on the transmission loss, like in the mid-frequency range. Thus also a single regression curve is determined per junction for the high frequency range. The coe ffi cients of determination equal 0.99 for these regression curves.

R s , 23 , Π , high = 1 . 1PC 3 + 5 . 1PC 2 − 1 . 2PC + 10 . 2 (17)

R s , 23 , H , high = 1 . 0PC 3 + 5 . 5PC 2 + 0 . 6PC + 11 . 6 (18)

(a) Π -junction (b) H-junction

Figure 9: Transmission loss between plates 2 and 3 in the high-frequency range (1250-5000 Hz) together with the determined regression curves. Lighter colors correspond with shorter junctions lengths.

3.2. Transmission from plate 2 to plate 4 The transmission path between plates 2 and 4 is now analysed, this transmission path is shown in figure 10. This transmission path does cross the connection between the double walls.

3 4

3 4

2 6

2 6

1 5

Figure 10: Transmission path between plates 2 and 4 for a Π and H-junction

Results in the low-frequency range are shown in figure 11. Here it is clearly visible that for this transmission path the spacing between the double wall does have an influence on the transmission loss. The influence is larger for higher ratios of characteristic moment impedances which is due to wall 3 (and wall 1 for an H-junction, see figure 10) acting as a blocking mass for this transmission path. For larger ratios of characteristic moment impedances the blocking mass will be larger, thus resulting in a larger transmission loss. When comparing the results for a Π -junctions and an H-junction, it is also visible that the extra moment impedance from the extra walls results in higher transmission losses. For lower ratios of characteristic moment impedances, both junctions have similar transmission losses. Since the spacing between the double walls has an influence on the transmission loss for this transmission path, this is taken into account for the regression curves. It was found that a regression curve of the following form can accurately fit the results. The ratio of characteristic moment impedances ψ

χ = 10 PC is used for the term taking into account the spacing since the influence of the spacing is larger for higher ratios of characteristic moment impedances.

R s , 24 , Π , low = A · PC 3 + B · PC 2 + C · PC + D + E log ( L ) + F · d · 10 PC (19)

(a) Π -junction (b) H-junction

Figure 11: Transmission loss between plates 2 and 4 in the low-frequency range (50-200 Hz). Lighter colors correspond with shorter junctions lengths.

The regression curves for a Π - and H-junction in the low-frequency range are given below and are shown in figure 12 together with the results. With these regression curves, coe ffi cients of determination between 0.95 and 0.98 were found.

R s , 24 , Π , low = 2 . 8 · PC 3 + 13 . 4 · PC 2 + 11 . 9 · PC + 22 . 4 − 5 . 1 log ( L ) − 30 . 0 · d · 10 PC (20)

R s , 24 , H , low = 3 . 6 · PC 3 + 18 . 0 · PC 2 + 21 . 1 · PC + 29 . 3 − 5 . 5 log ( L ) − 44 . 4 · d · 10 PC (21)

(a) Π -junction (b) H-junction

Figure 12: Transmission loss between plates 2 and 4 in the low-frequency range (50-200 Hz) together with the determined regression curves. Lighter colors correspond with shorter junctions lengths.

Figure 13 shows the results in the mid-frequency range. In this frequency range the junction length only has a minor influence on the transmission loss, the same could be observed with the transmission path from plate 2 to plate 3. The influence of the junction length is consequently not taken into account in the regression curves. Coe ffi cients of determination between 0.94 and 0.98 are found with these regression curves.

R s , 24 , Π , mid = 2 . 8 · PC 3 + 12 . 9 · PC 2 + 11 . 1 · PC + 17 . 9 − 13 . 3 · d · 10 PC (22)

R s , 24 , H , mid = 2 . 6 · PC 3 + 14 . 1 · PC 2 + 16 . 4 · PC + 23 . 0 − 27 . 2 · d · 10 PC (23)

(a) Π -junction (b) H-junction

Figure 13: Transmission loss between plates 2 and 4 in the mid-frequency range (250-1000 Hz) together with the determined regression curves. Lighter colors correspond with shorter junctions lengths.

Finally, the results for the high-frequency range are shown in figure 14, together with the determined regression curves. In this frequency range the junction length also has a negligible influence on the transmission loss, like in the mid-frequency range. In this frequency range, the spacing between the double walls also has a minor influence on the transmission loss for a Π -junction. Coe ffi cients of determination of 0.97 and 0.98 are found with these regression curves.

R s , 24 , Π , high = 1 . 9 · PC 3 + 9 . 1 · PC 2 + 6 . 3 · PC + 16 . 4 (24)

R s , 24 , H , high = 2 . 2 · PC 3 + 11 . 7 · PC 2 + 12 . 4 · PC + 20 . 6 − 17 . 0 · d · 10 PC (25)

(a) Π -junction (b) H-junction

Figure 14: Transmission loss between plates 2 and 4 in the high-frequency range (1250-5000 Hz) together with the determined regression curves. Lighter colors correspond with shorter junctions lengths.

4. CONCLUSIONS

A new hybrid di ff use-deterministic approach for calculating vibration transmission across double walls has been presented. This approach models the walls and / or floors as di ff use subsystems while the connection between the double wall junctions is modelled deterministically. The transmission coe ffi cient is calculated based the junction length, the bending wavenumber of the excited plate, the direct field dynamic sti ff ness matrices of all plates that are connected to the junction and the sti ff ness matrix of the deterministic connection between the double wall junction. Due to the computational e ffi ciency of the new approach, vibration transmission across a set of junctions could be calculated. The results from these simulation could then be used to calculate regression curves and asses the influence of di ff erent parameters on the vibration transmission for a

Π - and an H-junction. For both investigated transmission paths it was found that only in the low- frequency range, the junction length has a significant influence on the vibration transmission across the junction, since in this frequency range, the wavelengths are long compared to the length of the junction. The junctions length was therefore included in the regression curves for the low-frequency range. In the mid- and high-frequency ranges, the wavelengths are shorter, so the junction length has a smaller influence and was not included in the regression analysis. The first transmission path that was analysed did not cross the connection between the double walls. For all frequency ranges it was found that the spacing between the double walls has a negligible influence on the transmission loss for both types of junctions. The second investigated transmission path did cross the connection between the double walls and here the spacing did have a clear influence on the transmission loss of the transmission path. The influence was larger for larger ratios of characteristic moment impedances since some walls act as a blocking mass for this transmission path. For the second transmission path, the spacing between the double walls was included in the regression curves. For all frequency ranges and junctions types, regression curves are proposed which show a good fit to the data from the simulations. The new regression curves take into account the influence of the junction length and the spacing between the double walls on the vibration transmission across the junction when necessary.

ACKNOWLEDGEMENTS The research presented in this paper has been performed within the frame of the VirBAcous project (project ID 714591) “Virtual building acoustics: a robust and e ffi cient analysis and optimization framework for noise transmission reduction” funded by the European Research Council in the form of an ERC Starting Grant. The financial support is gratefully acknowledged.

REFERENCES

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