N B Roozen

Q Leclère

D Urbán

T Méndez Echenagucia

P Block

M Rychtáriková

C Glorieux

Assessment of the airborne sound insulation from mobility vibration measurements; a hybrid experimental numerical approach

A new measurement procedure is proposed to assess the airborne sound insulation of a partition under diffuse sound field excitation using mobility measurements combined with a numerical procedure. The advantage of this hybrid approach is that the diffuse acoustic field does not need to be physically created, thus avoiding problems related to the generation of such fields at low frequencies. Furthermore, the acoustic properties of both the source and receiving rooms will not affect the determination of the sound reduction index R. The proposed method is especially suited for frequencies below the so called Schroeder frequency of the room, and is complementary to the standardized measurement approaches as described in ISO 10140-2:2010.

The measurement part of the proposed procedure involves the measurement of the mobility by forcing the partition along a grid of excitation points (e.g. by means of a shaker) and measuring its response along a grid of response points (e.g. by means of a scanning laser Doppler vibrometer). Using the resulting matrix of mobility transfer functions, the vibrational response of the partition excited by a diffuse acoustic field is numerically calculated, from which the radiated sound power is computed using the Rayleigh integral. Thus the reliance on source and receiving rooms used in standard sound insulation testing is removed entirely. The proposed method provides an estimate that only depends on the properties of the partition.

The method was tested in an acoustic laboratory on a single layer glass plate of 1.35 × 1.54 m 2 , as well as on a funicular shell structure with dimensions of 3 × 3 m 2 . Comparisons with analytical models and standardized ISO10140-2:2010 measurements confirm the validity of the results.

Introduction

In contemporary architectural design, the increased implementation of lightweight constructions with lower sound insulation at low frequencies, combined with the more frequent use of hi-fi systems and home cinema with relatively high power at low frequencies, has increased people's awareness of low frequency noise in their living environment. Indeed, adverse effects on the human well-being have been reported [START_REF] Berglund | Sources and effects of low-frequency noise[END_REF]. In automotive and aviation industries similar issues arise due to the use of lightweight structures.

The increase in interest in low frequency sound insulation has stimulated research activities on two fronts. On one hand, questions on the importance of the low frequency part of the spectrum in people's subjective perception of sound insulation have been tackled by listening tests [START_REF] Rychtarikova | Perceived Loudness of Neighbour Sounds Heard Through Heavy and Light-Weight Walls with Equal R-w + C50-5000[END_REF], aimed at designing single number quantities that include the frequency range below 100Hz [START_REF] Mahn | The Uncertainty of the Proposed Single Number Ratings for Airborne Sound Insulation[END_REF][START_REF] Scholl | Rating of Sound Insulation at Present and in Future. The Revision of ISO 717[END_REF]. In current standards the assessment of sound insulation in frequency range down to 50Hz is still optional [START_REF]acoustics -rating of sound insulation in buildings and of building elementspart 1: Airborne sound insulation[END_REF]. On the other hand, researchers have been giving extra attention [START_REF] Hongisto | Reproducibility of the Present and the Proposed Single-Number Quantities of Airborne Sound Insulation[END_REF][START_REF] Scrosati | Uncertainty analysis by a Round Robin Test of field measurements of sound insulation in buildings: Single numbers and low frequency bands evaluation -Airborne sound insulation[END_REF][START_REF] Machimbarrena | Uncertainty determination of in situ airborne sound insulation measurements[END_REF] to the problem of high uncertainties on values of the sound reduction index R obtained at low frequencies by airborne sound insulation measurements according to ISO10140-2:2010 [START_REF]building elements -part 2: Measurement of airborne sound insulation[END_REF]. Indeed, sound insulation measurements in the lower frequency range are difficult to perform. Due to the reduced density of room acoustic eigenmodes, microphone based measurements are strongly affected by non-uniformity of the sound pressure field at low frequencies [START_REF] Tadeu | Sound insulation provided by single and double panel walls -a comparison of analytical solutions versus experimental results[END_REF][START_REF] Hopkins | Field measurement of airborne sound insulation between rooms with non-diffuse sound fields at low frequencies[END_REF][START_REF] Duarte | Indirect measurement of acoustic power into a small room at low frequencies[END_REF], resulting in poor reproducibility between laboratories [START_REF] Bravo | Variability of low frequency sound transmission measurements[END_REF][START_REF] Mahn | The Uncertainty of the Proposed Single Number Ratings for Airborne Sound Insulation[END_REF]. Bravo et al. [START_REF] Bravo | Enhancing low frequency sound transmission measurements using a synthesis method[END_REF] succeeded to reduce the effect of room acoustic issues in the source room by using a number of loudspeakers close to the panel, which were driven independently in order to create a diffuse incident field down to the lowest frequencies of interest. An optimal set of loudspeaker excitation signals was generated by mak-ing use of measured transfer functions between locations close to the speakers and locations close to a grid of points along the building element. Bravo et al. measured the radiated power into the receiving room by means of a dense array of acoustic intensity measurements. However, it turned out that this approach was affected by room acoustic effects in the receiving room. The latter effects were avoided in an approach proposed by Roozen et al. [START_REF] Roozen | Determining radiated sound power of building structures by means of laser Doppler vibrometry[END_REF] in which the vibrational pattern of the partition was measured by means of laser Doppler scanning vibrometry. The radiated sound power was then calculated by means of a Rayleigh integral [START_REF] Koopmann | Method for computing the sound power of machines based on the Helmholtz integral[END_REF]. Although combining the excitation scheme of Bravo et al. and the detection scheme of Roozen et al., would be a logical next step, the experimental effort to combine the two approaches was found to be very large [START_REF] Roozen | Assessment of the sound reduction index of building elements by near field excitation through an array of loudspeakers and structural response measurements by laser Doppler vibrometry[END_REF].

Here, a hybrid experimental-numerical approach is proposed in which the sound transmission through the partition is determined without physically creating a diffuse field in the source room. The partition is mechanically excited along a grid of excitation points (at the 'source' side) and the response is measured along a grid of response points (at the 'receiving' side). The diffuse excitation field is described by well-known relationships for the spatial correlation of a diffuse acoustic field to numerically construct a cross spectral matrix for the excitation on the source side. Using the cross spectral matrix, the response of the structure is computed on the basis of the experimentally determined mobility transfer functions. The incident and radiated power, needed to determine the sound reduction index R are then determined from, respectively, the (virtual) exciting sound pressure levels, and the structural response to the diffuse field excitation, computed by means of the Rayleigh integral as in Roozen et al. [START_REF] Roozen | Determining radiated sound power of building structures by means of laser Doppler vibrometry[END_REF].

The computation and measurement of mobility transfer functions is a classical subject in the field of noise and vibration [START_REF] Gardonio | On the origins and development of mobility and impedance methods in structural dynamics[END_REF][START_REF] Gardonio | Mobility and impedance methods in structural dynamics[END_REF]. Also the spatial correlation functions of a diffuse acoustic field (see Section 2.1.1) are well-known. However, the use of these spatial correlation functions to numerically compute the vibrational response of the partition from a measured matrix of structural mobility transfer functions, is new, to the knowledge of the authors. In combination with the use of the Rayleigh integral the influence of the room acoustic effects of the source and receiving rooms -used in standard insulation testing -is completely removed from the sound insulation measurement.

In literature, point mobility measurements were also used by Piana [START_REF] Piana | A method for determining the sound reduction index of precast panels based on point mobility measurements[END_REF] to obtain an estimate of the sound reduction index of a single leaf panel by making use of an analytical model. Unlike in that work, in which the finite dimensions of the panel were not taken into account, our approach makes use of measured vibration data along the whole sample, avoiding the use of geometrical approximations, even for complex geometries.

Marchetto et al. [START_REF] Marchetto | Vibroacoustic response of panels under diffuse acoustic field excitation from sensitivity functions and reciprocity principles[END_REF] proposed a combination of mobility transfer functions measurements and acoustic transfer function measurements to obtain the sound reduction index of a building element. The measurement of the acoustic transfer functions requires the use of monopole and dipole sources. Although this method eliminates the effect of the source room acoustics, due to the airborne detection of the radiated sound, the results are still affected by room acoustic effects of the receiving room, especially for low frequencies.

Chazot and Guyader [START_REF] Chazot | Prediction of transmission loss of double panels with a patchmobility method[END_REF] numerically predict the sound reduction index of finite double panels, taking into account source room dimensions, absorption, excitation position, and panel location. The difference with our approach is that we determine the sound reduction index of the partition for diffuse field excitation from measured mobilities. Furthermore in the approach presented in this paper, the room acoustic effects of the source and receiving rooms are eliminated, thus yielding an estimate of the sound reduction index that only depends on the properties of the partition itself.

The remainder of the paper is organized as follows. Section 2 deals with the theory to numerically construct the cross spectrum of a diffuse sound field, and to use this to compute the spectrum of the structural vibrations of the partition from a matrix of measured mobilities. This section also explains the subsequent steps to compute the radiated sound power from the computed vibration spectrum, and to determine the sound reduction index. Validating experiments that were performed on a single layer glass plate are discussed in section 3. In section 4 measurements on a funicular shell ceiling system are discussed. The experimental results are compared both with results obtained with a numerical finite element model of the building element and with measurements according to ISO10140-2:2010. Conclusions are drawn in section 5. In the appendix, an analytical model that predicts the airborne sound reduction of a simply supported plate under diffuse acoustic excitation, including radiation damping effects, is discussed. Also a procedure to compute the -Rayleigh integral based-acoustic radiation impedance matrix is presented.

Theory

Determination of the sound reduction index R under diffuse acoustic field excitation

In the proposed method, a diffuse acoustic field is numerically synthesized, the structural response of the device is determined using the measured matrix of mobilities, and the radiated sound power computed by means of the Rayleigh integral. The following subsections describe the theory in more detail.

The synthesis of an acoustic diffuse field and the determination of the vibrational response

Considering a diffuse acoustic field [START_REF] Marchetto | Vibroacoustic response of panels under diffuse acoustic field excitation from sensitivity functions and reciprocity principles[END_REF][START_REF] Piersol | Use of coherence and phase data between 2 receivers in evaluation of noise environments[END_REF][START_REF] Nelisse | Characterization of a diffuse field in a reverberant room[END_REF][START_REF] Jacobsen | The coherence of reverberant sound fields[END_REF], the cross spectral density of the sound pressure between two positions x i and x j on a rigid wall is given by

G ij (x i , x j , ω) = 2p 2 e (ω) sin(k x i -x j ) k x i -x j (1)
where k = ω /c is the acoustic wavenumber, ω is the angular frequency and c is the speed of sound. p 2 e is the averaged quadratic pressure in the diffuse field, the factor 2 arises from the pressure doubling that occurs on a rigid surface. The assumption of rigidity is easily satisfied, as typically the impedance of a wall is orders of magnitude larger than the one of air.

The diffuse field is often mathematically described as a summation of an infinite set of uncorrelated plane waves incident upon the partition with uniformly distributed angles of incidence, as illustrated in Fig. 1. For that reason, many papers revert to the wavenumber domain (k x , k y ) (e.g. [START_REF] Marchetto | Vibroacoustic response of panels under diffuse acoustic field excitation from sensitivity functions and reciprocity principles[END_REF]). However, the description of a diffuse field by an infinite set, or in practice by a large set of uncorrelated plane waves, is computationally very expensive. In this paper it is shown that the diffuse field can be described directly in the spatial domain (x, y), with clear advantages.

In this work, the structure under test is characterized by transfer function measurements between a discrete set of forces f (ω) at N e excitation points x e and a discrete set of velocities v(ω) at N r response points x r , such that its response to a distributed load field can be formulated as follows

v(ω) = H(ω)f (ω) ( 2
)
where H is the N r × N e matrix of mobility transfer functions. This linear relationship between excita-tions and responses can be formulated using averaged cross-spectra as follows

G VV = H G FF H * (3)
where the frequency dependence ω is omitted for the sake of brevity. In order to synthesize the response of the structure to a diffuse field, the input force cross spectral matrix G F F (in units N 2 ) has to be converted into one for pressure (units Pa 2 ) in order to use as an input the cross spectral matrix of a diffuse field.

To do that, a portion of the structure's surface has to be associated with each excitation point, and the relation between the force and pressure distribution can be expressed as follows

p e = dS -1 f (4)
where p e is the N e × 1 vector of excitations in units Pa, and dS a diagonal matrix whose entries correspond to the surfaces allocated to input points, verifying Tr(dS) = S, where S is the total surface of the structure.

The response of the structure to a discretized diffuse field can then be synthesized as

G VV = 2p 2 e H dS G D dS H * (5)
where G D is a N e × N e matrix whose entries correspond to normalized diffuse field cross spectra (considering p 2 e =1/2 in Eq. ( 1)).

Determination of the radiated sound power

The acoustic pressure at the receiving side can be determined from the measured vibration velocity field v of the structure by numerically solving the Rayleigh integral. This step can be expressed in terms of the radiation impedance matrix [START_REF] Koopmann | Method for computing the sound power of machines based on the Helmholtz integral[END_REF][START_REF] Fahy | Sound and Structural Vibration: Radiation, Transmission and Response[END_REF][START_REF] Bai | Estimation of sound power of baffled planar sources using radiation matrices[END_REF] as follows (see Appendix A for an expression for Z):

p r = Zv (6)
where p r is a N r ×1 vector of acoustic pressures at the receiving side of the partition, and Z the N r ×N r radiation impedance matrix. Pressure-velocity cross spectra, that are needed to calculate the radiated power, are obtained with

G PrV = ZG VV (7)
The active sound power Π r being radiated by the partition can be computed by

Π r = Tr (dS Re (G PrV )) (8)
2.1.3 Determination of the incident sound power and sound reduction index R

In a diffuse field, the acoustic power incident upon the structure is equal to [START_REF] Jacobsen | The sound field in a reverberation room[END_REF] Π

e = S p 2 e 4ρc (9)
where p 2 e is the spatially averaged mean square pressure in the diffuse field.

Finally, the sound reduction index R can be computed from the ratio of incident and radiated power as

R = 10 log Π e Π r (10) = -10 log 8ρc S Tr (dS Re ( ZH dS G D dS H * ))
where log is the logarithmic base 10. In the particular case of the use of the same regular mesh of N points for excitation and response dofs, such that elements of dS are all equal to S/N , Equation ( 10) becomes

R = -10 log 8ρcS 2 N 3 Tr (Re ( ZH G D H * )) (11)

Discretization criteria

The accuracy of the proposed method depends upon the discretization, i.e. the number of excitation points N e and response points N r . Discretization takes place at the excitation side (Eq. 4, Eq. 5) and at the radiation side (Eq. 6 and Eq. 8). On both sides, discretization requirements as a function of both structural and acoustic wavelengths have to be met carefully. The wavenumber content of the diffuse field is characterized by a maximum wavenumber equal to 2π/λ a , corresponding to plane waves of grazing incidence. This excitation field acts on the structure, being represented by the matrix of mobility functions.

For an accurate representation of the mobility functions the discretization thus has to meet also some spatial sampling criteria with respect to the structural wavelength. On the reception side, the same criteria need to be considered, related to the correct representation of the acoustic radiation mechanism. Finally, a general rule of thumb can be written as follows

S/N e < min (λ a , λ s ) /n (12) S/N r < min (λ a , λ s ) /n (13)
where λ a and λ s are acoustic and structural wavelength, respectively, and where n is the number of points per wavelength. Following Shannon's sampling criterion, n should be greater than 2. In Finite Element modeling, a criterion of n = 6 points by wavelength is often advised so as to ensure convergence.

In the present work, the measurement time has also to be taken into account, the high frequency limit is considered to be reached when n = 3.

Considerations regarding the required experimental effort

The experimental effort to measure the N r × N e matrix of transfer functions H can be significant, especially when for reasons of fine sampling N r and N e need to be large. To reduce the experimental effort, the principle of reciprocity can be exploited, provided that the points of excitation coincide with the points where the response is measured. As illustrated in Fig. 2, the transfer function vj /Fi as indicated in the left part of the figure is the same as the transfer function vi /Fj on the basis of reciprocity of linear time-invariant systems. Using reciprocity reduces the amount of transfer functions that need to be measured from N 2 (with

F j v i v j /F = i v i /F j v j F i
N =N r = N e ) to (N 2 -N )/2 + N .
Note that thanks to reciprocity only half of the crossmobility points needed to be measured. The number of point-mobility measurements stays the same. Thus, exploiting reciprocity reduces the measurement effort by somewhat less than a factor 2.

In case the geometry of the partition is symmetric, the number of measurements can be reduced as well. For instance, the funicular shell that will be discussed in section 4, and illustrated in Fig. 7, has 3 axes of symmetry. This type of symmetry reduces the number of measurement points by approximately a factor 2 3 =8. Also in this case, the reduction in the measurement effort is somewhat smaller than this theoretical number due to the fact that points on the symmetry line should always be measured. The additional use of reciprocity properties can reduce the required number of measurement even further by approximately a factor 2.

Assessment of the sound reduction index for a single layer glass plate

This section discusses the assessment of the sound reduction index of a single layer glass plate at low frequencies, to validate the mobility-based approach.

The results are compared with an analytical model. The single layer glass plate was mechanically excited by means of a mini-shaker (see Figure 3). The response of the glass plate was measured by means of a scanning laser Doppler vibrometer. The mini-shaker was equipped with an impedance head (Bruel&Kjaer type 8001), comprising a force cell to measure the transmitted force.

Measurement set-up

In this case, symmetry of the partition, nor reciprocity properties were exploited to reduce the measurement effort, as the response measurements were automated through the use of a scanning vibrometer measurement system.

Based on Kirchhoff's thin shell theory [START_REF] Roozen | Estimation of plate material properties by means of a complex wavenumber fit using hankel's functions and the image source method[END_REF], the structural wavelength was estimated by means of the formula

k f = ω 2 ρh D 1/4 (14)
where ω denotes the exciting angular frequency, D is the dynamic flexural rigidity defined by

D = Eh 3 12 (1 -ν 2 ) ( 15
)
where h is the plate thickness, E is the Young's modulus and ν is the Poisson's ratio. Using a Young's modulus E=56e9 Pa, density ρ=2400 kg/m 3 , Poisson's ratio ν=0.24, thickness h=12e-3 m, the structural wavelength λ f = 2π/k f is estimated to be about 1 m at 100 Hz. The acoustic wavelength at 100 Hz equals 3.4 m. Thus, with S/N e =0.41 m and S/N r =0.21 m, the discretization criteria as described in Section 2.2 are fulfilled up to 100 Hz for this specific case.

Results; A comparison of measurements and simulations

Using the theory as described in section 2, the sound reduction index R of the glass plate under a diffuse field excitation, exploiting the measured matrix of mobility transfer functions H, was computed. The results are shown in Fig. 4. Following the same procedure the sound reduction index R was computed, using a numerically computed matrix of mobility transfer functions H. The In experimental circumstances, acoustic radiation damping is naturally included in the measured matrix of mobility transfer functions H. However, when performing analytical or numerical simulations, where an in-vacuo structure is being considered, obviously this damping effect is lacking. Having computed a matrix of transfer functions H "in-vacuo", the radiation damping can be taken into account afterwards as described in Appendix B. Both the analytical results with and without radiation damping are presented in Fig. 4.

Radiation damping effects are essential in order to obtain physically correct sound reduction indices. The sound reduction index should in all cases be non-negative, as it is physically impossible to radiate more acoustic energy than the incident acoustic energy. Note that in literature some papers show results that violate this physical rule, see for instance [START_REF] Sgard | A numerical model for the low frequency diffuse field sound transmission loss of double-wall sound barriers with elastic porous linings[END_REF] (see also Fig. 16 in Appendix B).

The correspondence between the shaker-based sound reduction index and the sound reduction index as predicted by the analytical model is qualitative and quantitatively reasonably good up to a frequency of about 100 Hz (see end of Section 3.1). At higher frequencies the excitation and response grids become too coarse. At very low frequencies, <15 Hz, the shaker based sound reduction index becomes noisy, because the physical response of the glass plate is dropping too much, thus decreasing the signal to noise ratio of the laser Doppler signal.

This experiment validates the approach outlined in section 2. 4 Assessment of the sound reduction index for a funicular shell

This section discusses the assessment of the sound reduction index of a funicular shell ceiling structure at low frequencies by means of the proposed method. The experimental set-up is discussed in section 4.1, followed by a discussion of the measured operational deflection shapes at resonance frequencies in section 4.2 and the measurement of the sound reduction index from the mobility data in Section 4.4.

A comparison of the mobility-based prediction of the sound reduction index of a funicular shell with the microphone based approach according to the ISO 10140-2:2010 standard is given in Section 4.3. The results are also compared with the results obtained with a numerical numerical finite element method (FEM) model. The model and its results are discussed in Section 4.5. The FEM model is also used to perform a convergence study of the grid discretization used for the funicular shell.

Measurement set-up

The funicular shell ceiling system was cast in the horizontal opening of the test suite at the acoustic laboratory of KU Leuven, having dimensions outer dimensions 3 × 3 = 9 m 2 (inner dimensions 2.8 m × 2.8 m). Figures 5 and6 give a view from above and from below the funicular shell, respectively.

The matrix of mobility transfer functions H (Eq. 2) was measured by exciting the funicular shell from above (the rib-side) by means of a shaker (see Fig. 5). Both excitation points on the ribs and excitation points in the bays between the ribs were used. The response of the funicular was measured from below (see Fig. 6) by means of a scanning laser Doppler vibrometer measurement system. Pieces of retro-reflecting tape were glued on the dome of the funicular shell to increase the optical reflection at the measurement points. The shaker was equipped with a stinger and an impedance head (Bruel&Kjaer type 8001), the latter comprising a force cell to measure the transmitted force.

To reduce the measurement effort, the symmetry of the funicular shell structure was exploited. As illustrated in Fig. 7, the funicular shell structure is symmetric with respect to the horizontal axis, the vertical axis and with respect to the diagonal. Exploitation of this symmetry reduced the number of measurement points by approximately a factor of 8. Reciprocity properties to reduce the measurement effort were not used.

The response of the dome of the funicular shell were measured along a full grid of 15 × 15 = 225 points, by means of a scanning laser Doppler measurement system (Polytec PSV500). The shaker forced only 1/8 th part of the dome on a grid of 18 points, which was extended to a full grid of 15 × 15 points, using symmetry considerations. Thus a matrix of mobility transfer functions H of size N r × N e = 225 × 225 was obtained for each frequency considered. The mobility transfer functions were measured in a frequency range up to 1000 Hz, with a frequency resolution of 0.625 Hz.

With N e =N r =225, S/N e = S/N r =0.2 m, it can be concluded that the discretization requirements related to the acoustic wavelength, as described in Section 2.2, are met up to a frequency of about 560 Hz (n=3 sampling points per acoustic wavelength). Using the estimates that are given in Table 1, requirement related to the structural wavelength λ s is fulfilled up to a frequency well above 330 Hz, probably even higher.

Operational deflection shapes and resonance frequencies

A typical set of mobility transfer functions H, as measured at one shaker excitation point, is shown in Fig. 8. The global coherence, as defined in [START_REF] Roozen | A global error estimator for the uncertainty of a multi-channel spectral analysis[END_REF], is shown as well. The global coherence is the ratio between the averaged coherent output power spectrum and the averaged output power spectrum of a SIMO (Single Input Multiple Output) system. It represents the part of the global output energy that is linearly related to the source. It is used here to avoid the superposition of several coherence curves that would be difficult to interpret.

From the measured matrix of mobility transfer functions H, the operational deflection shapes were constructed. For six resonance frequencies of the funicular shell the operational deflection shapes are shown in Fig. 9. These shapes will closely resemble the eigenmodes of the funicular shell, as it are resonant operational deflection shapes.

Measurement of the sound reduction index according to the standard ISO 10140-2:2010

The measurement of the sound reduction index is prescribed in ISO 10140-2:2010 [START_REF]building elements -part 2: Measurement of airborne sound insulation[END_REF]. The methodology measures the sound pressure level in the source (L p1 ) and receiving room (L p2 ), averaged over a sufficient number of positions. The measurements were performed at eight fixed microphone positions (the standard ISO 10140-2:2010 requires at least five).

In order to account for the reverberation of the sound in the receiving room, and the size of the test opening, a correction term 10 (log S /A) needs to be added to the (L p1 -L p2 ) level difference, to obtain

where S is the area of the free test opening in which the element is mounted, and A the equivalent sound absorption area in the receiving room. The equivalent sound absorption area A follows from the reverberation time measurements in the receiving room.

The uncertainty in the measured sound reduction index can be determined by considering the standard deviation of the sound pressure levels in the source and receiving room from position to position, σ Lp1 and σ Lp2 , respectively, as well as the standard deviation of the equivalent sound absorption in the receiving room resulting from the reverberation time standard deviation [START_REF] Wittstock | On the uncertainty of singlenumber quantities for rating airborne sound insulation[END_REF]. Given that the different contributions to the total uncertainty on R are independent, the combined uncertainty can then be estimated as σ R = σ 2 Lp1 + σ 2 Lp2 + σ 2 10 log(A) . In case of repeated measurements on N (eight) positions, it can be expected that the error on the mean reduces to σ R,x = σ R / √ N . The letters a-f refer to the rows in Table 1; b) Global coherence as defined in [START_REF] Roozen | A global error estimator for the uncertainty of a multi-channel spectral analysis[END_REF].

Following this approach, the sound reduction index R of the funicular shell was estimated as shown in Fig. 10. The mean R incremented and decremented with the standard deviation σ R , as well as incremented and decremented by the standard error σ 2 R,x are shown in this figure as well.

In Fig. 11 the individual standard deviations are shown. From this figure it is clear that the largest standard deviations are caused by the variation of the sound pressure levels in the source and receiving rooms, σ Lp1 and σ Lp2 . The standard deviation of the correction term that accounts for the reverberation time of the receiving room, σ 10 log(A) , is smaller.

In Fig. 10 and Fig. 11 it can be seen that the uncertainty in the sound reduction index R increases with decreasing frequency, being as large as almost 8 dB for some frequencies below 100 Hz.

Mobility based estimation of the sound reduction index

The mobility-based estimate of the sound reduction index, following the theory as described in Section 2, is shown in Fig. 12 dating again the proposed procedure. In the lower frequency range, large variations of the sound reduction index R can be observed for the microphone based standardized method, whilst the mobility based approach shows less strong variations as function of frequency. The larger variations that are shown by the standardized approach are believed to be caused by the breakdown of the diffuse field assumption at low frequencies. As such, the proposed method can be considered complementary to the standardized measurement approach. The proposed mobility-based estimate of the sound reduction index requires a larger measurement effort as compared to the standardized ISO10140-2 measurement approach. The measurement of the matrix of mobilities in this particular case of the funicular shell required about 1 full working day, whereas the ISO measurement approach required a measurement time of about 1 hour. The larger measurement effort of the mobility-based approach is mainly due to the sequential measurement of each mobility transfer function. Non-sequential measurement approaches would reduce the measurement effort.

Finite element model, comparison with mobility-based results and convergence considerations

For the complex geometry of the funicular shell, it is not possible to obtain an analytical solution. Therefore a numerical finite element method (FEM) model was built to validate the mobility-based estimate of the sound reduction index. The model consisted of 76,615 nodes and 38,860 shell elements.

The computed eigenmodes that were obtained with the numerical FEM model are presented in Figure 13, showing a reasonable correspondence with the experimentally obtained operational deflection shapes at the resonance frequencies (see Fig. 9 and Table 1).

Using the finite element model, mobility transfer functions were computed for an excitation and response grid of 15 × 15 points, similar to the grids used for the experiments on the funicular shell. The sound reduction index was computed on the basis of these mobilities as detailed in Section 2. To investigate the convergence behavior of the mobility-based estimate of the sound reduction index, an excitation and response grid of 30 × 30 points (resulting in a matrix of mobility transfer functions H with a dimension of N r × N e = 900 × 900, for each frequency considered) was considered as well. The sound reduction index as computed with the FEM-model, using both excitation and response grids, is shown in Fig. 14. It shows that up to a frequency of about 200 Hz the excitation and response grid of 15 × 15 points does not deviate from the results obtained with the 30 × 30 points grid, indicating that the 15 × 15 points grid is sufficiently fine for the estimation of the sound reduction index of the funicular shell up to 200 Hz.

In the finite element model the in-vacuo structure is being considered, which implies that the effect of radiation damping is not taken into account. Having computed a matrix of transfer functions H "invacuo", the radiation damping can be taken into account afterwards as described in Appendix B. Figure 15 compares the mobility-based estimate of the sound reduction index, based on the measured matrix of mobilities, and the computed mobilities of the finite element model. The latter is shown with and without the effect of radiation damping.

The results shown in Fig. 15 shows a reasonable correspondence between the R obtained from the finite element model and from the mobility measurements. This again validates the mobility based approach. The sound reduction index is shown for frequencies down to 10 Hz. The first resonance frequency of the funicular shell is about 70 Hz, causing the sound insulation below that frequency to be stiffness controlled, which is characterized by an R with a 40 dB/dec slope. Both the results obtained by the FE-model and the mobility based measurement results show a stiffness controlled behavior for frequencies below 70 Hz.

Conclusions

A new hybrid approach was proposed to measure the sound reduction index of a partition without the adverse effects of the acoustic properties of the source and receiving rooms at frequencies below the Schroeder frequency, yielding the true sound reduction index of the partition. In this new approach mobility measurements are combined with a numerical procedure.

The approach was shown to be consistent with the standardized ISO10140-2:2010 measurement approach for frequencies just above the Schroeder frequency. The ISO10140-2:2010 is based on the diffuse field assumption. Below the Schroeder frequency significant differences between the new approach and ISO10140-2:2010 were observed, which are believed to be caused by the breakdown of the diffuse field assumption at low frequencies.

The new approach is especially suited for the frequency range below the Schroeder frequency, and is as such complementary to the standardized measurement approach. It is expected that the new approach is also suitable for in-situ measurements. Furthermore, the method is unaffected by the presence of possible flanking paths.

During the experiments a matrix of mobilities was measured along a grid of excitation and response points, sequentially in time. Consequently the proposed method required a larger measurement effort as compared to the standardized ISO10140-2 measurement approach. Emerging measurement technologies, such as automated scanning laser Doppler measurement techniques, now already allow for accurate and fast mobility measurements that are required for this approach. With the advance of full-field optical measurement techniques like digital image correlation techniques [START_REF] Chen | Fullfield 3D measurement using multi-camera digital image correlation system[END_REF] and full-field digital holographic inter-ferometry techniques [START_REF] De Greef | Digital Stroboscopic Holography Setup for Deformation Measurement at Both Quasi-Static and Acoustic Frequencies[END_REF] the measurement effort can be further reduced. Alternatively, a grid of accelerometers can be used to capture the vibrational response of the partition in one single measurement session per excitation point. These full-field measurement techniques as well as the use of a grid of accelerometers would also allow to excite the partition by means of an instrumented hammer, which will make the measurement more easy to perform as compared to a shaker excitation. Next to the larger measurement effort, also a larger numerical effort is required to compute the cross spectral matrix of structural responses and to compute the radiated sound power by means of the Rayleigh integral. This numerical part can be well described in standardized procedures and be performed at every measurement facility.

Despite the larger experimental-numerical effort as compared to the standardized approach, the authors believe it is a viable and realistic alternative to standardized approaches for frequencies below the Schroeder frequency. It is expected that the approach presented in this manuscript will motivate a discussion on the normalization of measurement procedures for the lower frequency range.

Appendices A Computation of the radiation impedance matrix

Consider a baffled partition radiating into a semiinfinite acoustic domain, having a velocity field that is described by v(x e , ω). The pressure on the radiating surface can be computed by means of the Rayleigh integral [START_REF] Roozen | Determining radiated sound power of building structures by means of laser Doppler vibrometry[END_REF][START_REF] Koopmann | Method for computing the sound power of machines based on the Helmholtz integral[END_REF][START_REF] Fahy | Sound and Structural Vibration: Radiation, Transmission and Response[END_REF], which reads:

p (x r , ω) = iωρ 2π S v n (x e , ω) e -ikr r dS r ( 17
)
where r is the distance between the points in vector x e and vector x r , k is the wavenumber, ρ the density and ω the radial frequency. Note that the above integral has a singularity for r → 0. Equation 17 can be written in matrix form as

p = Zv (18)
where the matrix Z can be computed as follows [START_REF] Bai | Estimation of sound power of baffled planar sources using radiation matrices[END_REF], taking into account the singularity at r → 0:

Z = ρc     d -ikS 2π e ikr ij rij • • • -ikS 2π e ikr ji rji d • • • . . . . . . . . .     (19)
where

d = 1 2 k S π 2 -i 8 3π k S π (20)
B An analytical model to predict the airborne sound reduction of a simply supported plate under a diffuse acoustic excitation, including radiation damping effects

The analytic model for the simply supported flexural rectangular plate is considered here. The displacement w of the plate excited by a unitary point force at the coordinates (x e , y e ), for the angular frequency ω, can be described by a modal expansion [START_REF] Leclere | Vibration source identification using corrected finite difference schemes[END_REF]:

w(x, y) = (21) = 4 DL x L y i,j≥1
sin(k i x e ) sin(k j y e ) sin(k i x) sin(k j y)

(k 2 i + k 2 j ) 2 -k 4 F , with k i = iπ Lx , k j = jπ Ly , k F = ω 2 ρh D 1/4
where L x and L y are the plate's dimensions and D is the dynamic flexural rigidity defined by D = E(1+iη)h 3

12(1-ν 2 ) where h is the plate thickness, E is Young's modulus, η is the loss factor, ρ is the density and ν is Poisson's ratio. From Eq. 21 the matrix of transfer functions H, as defined by Eq. 2, can be constructed. However, this matrix of transfer functions relates the vector of velocities v to the vector of exciting forces f for the the in-vacuo structure. Below a method is presented to take the radiation damping into account as a simple modification of the matrix of transfer functions H. Recalling Eq. 2, we have The radiated pressure on the source side and on the receiving side, can be accounted for by means of an additional load on the structure, resulting in the modified relationship

v = Hf (22)
v = H (f -2 dS p r ) (23)
where p r is the radiated pressure field. Using the radiation impedance matrix for an infinite acoustic half space, Z [START_REF] Koopmann | Method for computing the sound power of machines based on the Helmholtz integral[END_REF][START_REF] Fahy | Sound and Structural Vibration: Radiation, Transmission and Response[END_REF][START_REF] Bai | Estimation of sound power of baffled planar sources using radiation matrices[END_REF] (see Appendix A), which links the parietal pressure on the source side and on the receiving side radiated to the vibrating structure having a velocity field v, the above relationship can be written as

v = H (f -2 dS Z v) (24)
and thus

(I + 2 H dS Z) v = Hf ( 25
)
where I is the identity matrix, yielding v = (I + 2 H dS Z) -1 Hf = H mod f (ω) [START_REF] Fahy | Sound and Structural Vibration: Radiation, Transmission and Response[END_REF] where H mod is the transfer function that takes into account the radiation load, assuming two infinite half spaces on either side of the structure, and thus also accounts for the radiation damping of the structure.

To validate the procedure to compute the diffuse field sound reduction index R from mobility data, a single aluminum plate as described by Sgard (JASA 2000) [START_REF] Sgard | A numerical model for the low frequency diffuse field sound transmission loss of double-wall sound barriers with elastic porous linings[END_REF], with dimensions L x =0.35 m, L y =0.22 m, h=0.001 m, E=71 GPa, ν=0.33, ρ=2814 kg/m 3 and η=0.001, was considered. A discretization mesh was used of N e =N r =20 × 20 = 400 points for both the excitation field and response field. Using the modal expansion given above, including eigenfrequencies up to twice the frequency range of interest, the response of the plate due to a diffuse acoustic excitation was computed using equations 1 and 5. Subsequent use of the equations 8 and 11, yielded the sound reduction index R according to the theoretical model. Fig. 16 shows the computed theoretical sound reduction index R, for the case structural damping is considered only, η=1%, and for the case the radiation damping is considered as well, using the modified mobility transfer matrix H mod given in Eq. 26.

The results compare well with the results published by Sgard [START_REF] Sgard | A numerical model for the low frequency diffuse field sound transmission loss of double-wall sound barriers with elastic porous linings[END_REF] (figure 2 on page 2868 of that publication) for the case that radiation damping is not taken into account. This validates the procedure to compute the diffuse field sound reduction index R from mobility data.

It is noteworthy to mention that the inclusion of radiation damping is essential to have predictions of the sound reduction index R which are non-negative for all frequencies. From a physical point of view a negative sound reduction index R is not possible. Only when the effects of radiation damping is taken into account, the energy balance is modeled correctly, making it impossible to have negative R-values. See also Section 3.2, Fig. 4 and Section 4.5, Fig. 15 for example calculations that illustrate this aspect.

Figure 1 :

 1 Figure 1: The diffuse acoustic field (DAF), envisaged as an infinite set of uncorrelated plane waves.

Figure 2 :

 2 Figure 2: Reciprocity principle.

Figure 3 :

 3 Figure 3: Shaker test set up of the single layer glass plate of size 0.012 x 1.35 x 1.54 m.

Figure 4 :

 4 Figure 4: Sound reduction index R of a single layer glass plate of size 0.012 x 1.35 x 1.54 m, mobilitybased result (blue solid curve), analytical model without radiation damping (magenta dotted curve), analytical model with radiation damping (red dashed curve).

Figure 5 :

 5 Figure 5: Funicular shell shaker setup, at the top of the funicular shell. Shaker excitation shown for some of the tested positions: a) on a rib near the center of the shell; b) in the bays between ribs; c) on a rib near the corner of the shell.

Figure 6 :

 6 Figure 6: Funicular shell mounted in the opening of the transmission loss facility, with retro reflecting stickers glued on the dome, at the bottom.

Figure 7 :

 7 Figure 7: Exploiting symmetry of the funicular shell.

Figure 8 :

 8 Figure 8: Measured mobilities of the funicular shell: a) Transfer functions H as measured at one shaker excitation point (gray curves) and the spatially mean mobility transfer function (black thick solid curve).The letters a-f refer to the rows in Table1; b) Global coherence as defined in[START_REF] Roozen | A global error estimator for the uncertainty of a multi-channel spectral analysis[END_REF].

Figure 9 :

 9 Figure 9: Measured funicular shell operational deflection shapes for different frequencies: a) 70 Hz; b) 107 Hz; c) 155 Hz; d) 178 Hz; e) 286 Hz; f) 327 Hz.

Figure 10 :Figure 11 :

 1011 Figure 10: Sound reduction index R of the funicular shell, as measured according to ISO 10104-2:2010, from 50 to 1000 Hz. The interval between the dotted and full gray lines indicate the estimated standard deviation between measurements on different microphone positions and the standard error on their mean respectively.

Figure 12 :

 12 Figure 12: Sound reduction index R of the funicular shell, measurements according to ISO 10140-2:2010 versus mobility-based estimate. a) Results presented in 1/3 rd -octave bands, b) results presented in 1/24 thoctave bands for the ISO method, and in FFT narrow bands for the mobility based result.

Figure 13 :

 13 Figure 13: FE-predicted funicular shell eigenmodes with eigenfrequencies: a) 71 Hz; b) 106 Hz; c) 146 Hz; d) 161 Hz; e) 268 Hz; f) 296 Hz.

Figure 14 :

 14 Figure14: Sound reduction index R of the funicular shell, using FE-predicted transfer functions, with a 15x15 mesh (solid brown curve) and a 30x30 mesh (dashed blue curve). Results are presented in FFT narrow bands.

Figure 15 :

 15 Figure 15: Sound reduction index R of the funicular shell, mobility-based (green solid curve) estimate versus FE-model result for undamped case (blue doted curve) and for the case including structural damping (5% critical) and radiation damping (dash-dotted brown curve). Results presented in FFT narrow bands.

Figure 16 :

 16 Figure 16: Diffuse field sound reduction index R of an aluminum plate, L x =0.35 m, L y =0.22 m, plate thickness h=0.001 m, computed from the theoretical mobility data of a simply supported plate (Eq. 21).

  The glass plate was positioned in the opening by a wooden frame sealed by means of rubber mastic. A matrix of mobility transfer functions H (see Eq. 2) was measured at an excitation grid of 3 × 4=12 points ( S/N e =0.41 m), and a response grid of 7 × 7=49 points ( S/N e =0.21 m). The excitation and response grid distances were uniformly distributed over the glass panel. Thus the matrix of mobility transfer functions H had a dimension of N r × N e = 12 × 49, for each frequency considered. The mobilities were measured in a frequency range from 10 Hz up to 300 Hz, with a frequency resolution of about 0.1 Hz.

A single layer glass plate was mounted in the opening of the test suite at the acoustic laboratory of KU Leuven, in which the opening and the glass plate have dimensions S= 1.35 × 1.54 = 2.08 m 2 . The single layer glass plate had a thickness of 12 mm.

Table 1 :

 1 Measured resonance frequencies and predicted eigenfrequencies of the funicular shell.

	mode (letters referring to experimental numerical difference	structural
	Fig. 8a and to sub-figures resonance	eigen-		wavelength
	in Figs. 9 and 13.	frequency	frequency
	a	70 Hz	71 Hz	1 Hz, 1.4%	λ s >3 m
	b	107 Hz	106 Hz	1 Hz, 0.9%	λ s ≈2.5 m
	c	155 Hz	146 Hz	9 Hz, 5.8%	λ s ≈2.1 m
	d	178 Hz	161 Hz	17 Hz, 9.5% λ s ≈1.5 m
	e	286 Hz	268 Hz	18 Hz, 6.3% λ s ≈1.2 m
	f	327 Hz	296 Hz	31 Hz, 9.5% λ s ≈1.2 m

Acknowledgments

This work was carried out in the framework of H2020-MSCA-RISE-2015 No. 690970 "Papabuild", and was partly funded by the Post-doctoral seed funding grant from the Partnership Council for Sustainable Construction given by the ETH Foundation.

The following people from KU Leuven are acknowledged: Stephan Solinas, Bernd Salaets, Daniel Otero and Pengchao Wang for their help in the construction of the funicular shell; Willy Bruyninckx for performing the measurements according to the standard; Ludovic Labelle for his help in setting up part of the measurement equipment; Luca Sangiuliano and Noé Geraldo Rocha de Melo Filho for their help with the scanning laser Doppler vibrometer; Edwin Reynders for coordinating the funicular cell construction works.


N B Roozen

Q Leclère

D Urbán

L Kritly

C Glorieux

Assessment of the sound reduction index of building elements by near field excitation through an array of loudspeakers and structural response measurements by laser Doppler vibrometry

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Insufficient low-frequency sound insulation between dwellings combined with the use of powerful music reproduction systems is one aspect why inhabitants are often disturbed by their neighbors. Over the last years, an increased interest in sound insulation at low frequencies can be observed. Although in current standards the assessment of sound insulation in the frequency range down to 50Hz is still optional [START_REF]acoustics -rating of sound insulation in buildings and of building elementspart 1: Airborne sound insulation[END_REF][START_REF]acoustics -field measurement of sound insulation in buildings and of building elements -part 1: Airborne sound insulation[END_REF], most studies recommend the inclusion of frequencies below 100 Hz [START_REF] Rychtarikova | Perceived Loudness of Neighbour Sounds Heard Through Heavy and Light-Weight Walls with Equal R-w + C50-5000[END_REF][START_REF] Ljunggren | Airborne sound insulation between dwellings, from 50 Hz vs 100 Hz, a Swedish field survey[END_REF]. However, it is also well known that measurents of the sound reduction index according to the standard ISO 10140:2010 [START_REF]building elements -part 2: Measurement of airborne sound insulation[END_REF] at frequencies below 100 Hz are affected by a large uncertainty [START_REF] Hongisto | Reproducibility of the Present and the Proposed Single-Number Quantities of Airborne Sound Insulation[END_REF][START_REF] Scrosati | Uncertainty analysis by a Round Robin Test of field measurements of sound insulation in buildings: Single numbers and low frequency bands evaluation -Airborne sound insulation[END_REF][START_REF] Machimbarrena | Uncertainty determination of in situ airborne sound insulation measurements[END_REF]. The reason for this uncertainty is the reduced modal density of the acoustic eigenmodes at lower frequencies, in both the source room and the receiving room of the test facility. The resulting modal behavior cause the acoustic field to deviate from the ideal diffuse field [START_REF] Tadeu | Sound insulation provided by single and double panel walls -a comparison of analytical solutions versus experimental results[END_REF][START_REF] Hopkins | Field measurement of airborne sound insulation between rooms with nondiffuse sound fields at low frequencies[END_REF][START_REF] Duarte | Indirect measurement of acoustic power into a small room at low frequencies[END_REF]. This makes the measurement results obtained in a specific testing laboratory strongly dependent on the chosen microphone positions and therefore uncertain. Moreover, as the obtained measurement result is dependent on the specific geometry of the transmission suite, it also causes a poor reproducibility between laboratories [START_REF] Bravo | Variability of low frequency sound transmission measurements[END_REF][START_REF] Mahn | The Uncertainty of the Proposed Single Number Ratings for Airborne Sound Insulation[END_REF].

In the works of Bravo, Maury and Elliott [START_REF] Bravo | Variability of low frequency sound transmission measurements[END_REF][START_REF] Bravo | Enhancing low frequency sound transmission measurements using a synthesis method[END_REF] the breakdown of the diffuse field assumption of the source room was reduced by using a number of suitably driven loudspeakers close to the building insulation element under test to create a diffuse incident field as closely as possible. To assess the acoustic power transmitted through the element under test, they measured the acoustic sound intensity at a dense array of points at the receiving side, close to the element. However, the effect of the receiving room was found not to be reduced in this way. The influence of the receiving room on the transmitted power, using the sound intensity approach, can only be reduced by the use of moving diffusers in the receiving room, or if a larger and more anechoic receiving room is used [START_REF] Bravo | Variability of low frequency sound transmission measurements[END_REF].

Roozen et al. [START_REF] Roozen | Determining radiated sound power of building structures by means of laser Doppler vibrometry[END_REF] showed that the sound power radiated by a building insulation element under test at low frequencies can be determined from the vibration pattern of the building element, using laser Doppler vibrometry (LDV) or other vibration measurement techniques. The radiated sound power was calculated from the measurement data by means of the Rayleigh integral, in which an infinite half space is assumed for the 'receiving room'. In this manner the computed sound power is not affected by the room acoustic modes of the receiving room.

In the present paper the works of Bravo et al. [START_REF] Bravo | Enhancing low frequency sound transmission measurements using a synthesis method[END_REF] and Roozen et al. [START_REF] Roozen | Determining radiated sound power of building structures by means of laser Doppler vibrometry[END_REF] are combined. The diffuse excitation field is created by a loudspeaker array closely positioned to the building element under test, and the radiated sound power is determined by means of laser Doppler vibrometry.

The paper is organized as follows. Section 2 treats the synthesis of the diffuse sound field by means of a loudspeaker array. This section also discusses the computation of the radiated sound power level, using conditioned spectral analysis. Section 3 reports about measurements on a single glazing, and validates the results by means of an analytical model. A comparison with a standardized ISO 10140:2010 [START_REF]building elements -part 2: Measurement of airborne sound insulation[END_REF] measurement is given as well. Section 4 draws the conclusions of the work.

2 Theory for the synthesis of a diffuse field and the analysis of the measurement data

The theory to synthesize spatially correlated random pressure fields is well documented in literature [START_REF] Elliott | The synthesis of spatially correlated random pressure fields[END_REF].

In this section it is briefly summarized for the sake of completeness. In addition to this brief summary, a proposal is elaborated to perform the analysis of the measurement data by means of a conditioned spectral analysis, which is different from the approach taken by Elliott, Bravo and Maury et al. [START_REF] Bravo | Enhancing low frequency sound transmission measurements using a synthesis method[END_REF][START_REF] Elliott | The synthesis of spatially correlated random pressure fields[END_REF][START_REF] Bravo | The experimental synthesis of random pressure fields: Methodology[END_REF][START_REF] Maury | The experimental synthesis of random pressure fields: Practical feasibility[END_REF].

The synthesis of spatially correlated random pressure fields

Piersol [START_REF] Piersol | Use of coherence and phase data between 2 receivers in evaluation of noise environments[END_REF] showed that the cross spectral density of the sound pressure of a diffuse acoustic field, measured between two points separated at a distance r from each other, is given by (see also [START_REF] Nelisse | Characterization of a diffuse field in a reverberant room[END_REF][START_REF] Jacobsen | The coherence of reverberant sound fields[END_REF][START_REF] Ichchou | Equivalent 'rain on the roof' loads for random spatially correlated excitations in the mid-high frequency range[END_REF][START_REF] Marchetto | Vibroacoustic response of panels under diffuse acoustic field excitation from sensitivity functions and reciprocity principles[END_REF])

S D sin(kr) kr (1)
where k = ω /c is the acoustic wavenumber, ω is the angular frequency (ω = 2πf ) and S D is the autopower spectrum level of the desired diffuse acoustic field. S D is a frequency dependent scalar, independent of the position in the room.

Let the matrix with desired cross spectral densities at the N mic control microphone positions, based on Eq. 1, be denoted by S dd . The subscript d indicates the desired diffuse field. The control microphones are positioned on the surface of the building element in a rectangular grid, between the loudspeaker array and the building element. The matrix S dd has a dimension N mic ×N mic for each frequency being considered, and the rank of this matrix is equal to N mic .

To approximate the acoustic field with the above mentioned cross spectral density as closely as possible a discrete number of loudspeakers, N LS , is used. The number of control microphone positions, N mic , is normally larger than the number of loudspeakers, N LS . However, having N LS signals available to create the desired acoustic field, that field cannot have a rank higher than N LS . Therefore, a limited number of uncorrelated principal components of the desired acoustic field are considered, following Elliott et al. [START_REF] Elliott | The synthesis of spatially correlated random pressure fields[END_REF]. For this purpose the matrix S dd is decomposed in its eigenvectors V and its eigenvalues Λ:

S dd = VΛV * (2)
where * denotes the complex conjugate. The best least-squares approximation to S dd with rank N LS can be obtained by taking the N LS terms with the largest eigenvalues:

Sdd = Ṽ Λ Ṽ * ( 3
)
where Λ is a diagonal matrix that contains the N LS largest eigenvalues, and the matrix Ṽ contains the N LS largest eigenvectors. The tilde ˜indicates that the variable is the best least-squares approximation with rank N LS . In the approximation of interest, the rank of matrix Sdd is forced to be equal to the number of loudspeakers of the loudspeaker array, N LS , being the maximal feasible rank the can be created by the array.

The N LS input signals to the loudspeaker amplifiers required to generate the desired cross spectral matrix Sdd at the control microphones can be constructed as follows. A measurement is performed to relate the N LS input signals x to the pressures y as measured at the N mic control microphone positions (for a system diagram, see Fig. 1). This results in a relationship in the frequency domain that reads

Y = HX ( 4
)
where H is a matrix of transfer functions having a dimension N mic × N LS for each frequency being considered, vector Y is the Fourier transform of the control microphone signals, having a dimension N mic × 1, for each frequency being considered, and vector X is the Fourier transform of the input signals to the loudspeaker amplifiers, having a dimension N LS × 1 for each frequency being considered. The matrix with optimal control filters W opt can be computed from where Ũ represents N LS uncorrelated random signals with unitary variances in the frequency domain. Using this matrix with optimal control filters, the optimal input signals to the loudspeaker amplifiers X can be computed with

W opt = (H * H) -1 H * Ũ Λ1 /2 (5
X = W opt Ũ (6)
The time domain signals x to drive the loudspeaker array can be obtained from the matrix X by means of an inverse Fourier transform. For a more extensive description of the synthesis of a diffuse field by means of a loudspeaker array, see Elliott et al. [START_REF] Elliott | The synthesis of spatially correlated random pressure fields[END_REF].

Analysis of measurement data by means of conditioned spectral analysis, CSA

Using the optimal loudspeaker signals x, a diffuse acoustic field with cross spectral matrix Sdd is obtained. As this excitation field consists of N LS uncorrelated principal components, as discussed in the previous section, also the structural response w of the building insulation element under test will consists of the same number of principal components. These N LS uncorrelated response components are denoted by w (k) , with k = 1...N LS . The uncorrelated response components can be extracted by means of Conditioned Spectral Analysis (CSA, after Bendat and Piersol [START_REF] Bendat | Engineering applications of correlation and spectral analysis[END_REF]) or by means of virtual source analysis (VSA, after Price and Bernhard [START_REF] Price | Virtual coherence: a digital signal processing technique for incoherent source identification[END_REF]). The advantage of both approaches is that the radiated sound powers that follow from these uncorrelated responses can be simply added to obtain the total radiated sound power, as explained by Leclere [START_REF] Leclere | Multi-channel spectral analysis of multi-pass acquisition measurements[END_REF]. In this paper, the CSA approach was taken, which will be outlined briefly below.

Having measured the cross spectra between the response signals w and reference signals x, S wx , and the auto spectra of the reference signals, S xx , the first uncorrelated contributions w (1) can be computed from

w (1) i = S wix1 / S x1x1 , i = 1...N LDV (7)
where S wixj is the (i, j)-th component of the matrix S wx , S xj xj is the (j, j)-th component of the matrix S xx , and w

i is the i-th component of the vector w (1) . In this notation the indexes range as i = 1...N LDV and j = 1...N LS .

The second uncorrelated contributions w (2) can be computed by removing the energy in the spectral matrices which is coherent with the first reference:

S (1)rem wixj = S wixj - S wix1 S x1xj S x1x1 , i = 1...N LDV , j = 1...N LS (8)
and

S (1)rem xj xj = S xj xj - |S xj x1 | 2 S x1x1 , j = 1...N LS (9)
Using these conditioned auto and cross spectra the second uncorrelated contributions, w (2) can be computed as follows w

(2) i = S (1)rem wix2 / S

(1)rem

x2x2 , i = 1...N LDV (10)
The k-th uncorrelated contributions w (k) can be computed from

w (k) i = S (k-1)rem wix k / S (k-1)rem x k x k , i = 1...N LDV , k = 2...N LS (11)
where the conditioned auto and cross spectra are computed from the previously computed conditioned spectra:

S (k)rem wixj = S (k-1)rem wixj - S (k-1)rem wix k S (k-1)rem x k xj S (k-1)rem x k x k , i = 1...N LDV , j = 1...N LS , k = 2...N LS (12)
and

S (k)rem xj xj = S (k-1)rem xj xj - |S (k-1)rem xj x k | 2 S (k-1)rem x k x k , j = 1...N LS , k = 2...N LS (13)
The multiple coherence [START_REF] Bendat | Engineering applications of correlation and spectral analysis[END_REF] can be defined by the ratio of the energy contained in the sum of the uncorrelated contributions w (k) to the total energy contained in w:

γ i = N LS k=1 |w (k) i | 2 S wiwi i = 1...N LDV (14)
Having computed all N LS uncorrelated contributions w (k) , k = 1...N LS , the sound power radiated by each of the uncorrelated responses, Π (k) rad , can be determined. In this work the radiated sound power is computed by means of the Rayleigh integral, following [START_REF] Roozen | Determining radiated sound power of building structures by means of laser Doppler vibrometry[END_REF]. Using this approach the building insulation element under test is assumed to be baffled, and the receiving acoustic domain is a semi-infinite acoustic domain. In this manner the room acoustic effects of the receiving room are eliminated. In Appendix A the numerical procedure is briefly outlined.

As w (k) , k = 1...N LS , contain the uncorrelated responses of the building element under test, extracted by CSA, the sound power radiated by each of the responses w (k) are also uncorrelated to each other. Because of this property, the total sound power radiated by the building element can be computed as the sum of the sound power components Π (k) rad :

Π rad = N LS k=1 Π (k) rad (15)
Note that it is essential to compute the (complex valued) uncorrelated responses w (k) , k = 1...N LS , to allow a correct computation of the radiated powers of each of the uncorrelated responses. The phase information contained in each of the uncorrelated responses w (k) is an essential aspect in the computation of the radiated power Π In literature it is mentioned that the accuracy of the results of the CSA approach in source identification problems strongly depends upon the elimination order of the references [START_REF] Shu | Identification of complex diesel engine noise sources based on coherent power spectrum analysis[END_REF][START_REF] Fouladi | Spectral analysis methods for vehicle interior vibro-acoustics identification[END_REF]. However, the context of the present work is not related to the separation of source contributions, it is just needed here to separate uncorrelated responses referenced to a set of arbitrary inputs. In such a case the total sound power level Π rad is not affected by the ordering of references.

Whilst driving the loudspeaker array with the optimal loudspeaker signals, x, the sound pressure levels at the grid of control points (N mic points) need to be measured. From the blocked pressure squared, spatially averaged over the N mic control points, p 2 bl , the incident sound power can be estimated by

Π inc = S 4ρc 1 2 p 2 bl = S 8ρc p 2 bl ( 16
)
where S is the surface area of the building insulation element under test and where the factor 1 /2 arises from the reflections on the rigid wall [START_REF] Waterhouse | Interference patterns in reverberant sound fields[END_REF][START_REF] Jacobsen | The diffuse sound field[END_REF]. The blocked pressure p bl is the pressure acting on the building insulation element under test, where it is assumed that the impedance of the building element under test is infinitely large (in analogy with the term 'blocked force' as introduced by Gardonio and Brennan [START_REF] Gardonio | Mobility and impedance methods in structural dynamics[END_REF]). Finally, from the incident and radiated sound powers, the sound reduction index R of the building insulation element under test can be estimated:

R = Π inc Π rad (17)
To summarize, the following data needs to be measured:

The matrix of transfer functions H between the acoustic responses y and the loudspeaker signals

x with dimension N mic × N LS ,
The cross spectra of the structural responses w and the loudspeaker signals x with dimension N LDV × N LS , whilst driving the loudspeaker array with the optimal signals x, extracted from analysis of the signals obtained in step 1,

The cross spectra of the acoustic responses y and the loudspeaker signals x, with dimension N mic × N LS , whilst driving the loudspeaker array with the optimal signals x, to allow the computation of the incident power.

Measurements

Two types of measurements were performed: measurements according to the ISO 10140-2:2010 standard [START_REF]building elements -part 2: Measurement of airborne sound insulation[END_REF], and measurements by means of the loudspeaker array placed close to the device under test, of which the theory was described in the previous section. Section 3.1 discusses the measurement set-up and Section 3.2 the results.

Measurement set-up

The measurements were performed on a single glazing with a lateral dimension of 1.35 × 1.54 m 2 and a thickness of 12 mm. The mass per unit area was measured to be 28.8 kg/m 2 . The glazing was mounted in a test opening of 1.37 × 1.555 m 2 , covered by a wooden frame and sealed by putty. The dimensions of the source and the receiving room are approximately 4.11 m (height) × 5.10 m × 4.14 m, which yields a volume of 87 m 3 each.

The text below details the set-up for the measurements by means of an array of loudspeakers and a scanning laser Doppler vibrometer.

An array of N LS =12 loudspeakers in an arrangement of 3×4 was constructed (see Fig. 3). Each loudspeaker, a 12inch woofer type Visaton W300-8, was mounted in it's own cabinet made from 18mm plywood to prevent acoustic cross-talk with neighboring speakers. The loudspeaker array cabinet, with a total lateral dimension of 1.25 × 1.50 m 2 , was put at a distance of 20 cm from the glazing, which corresponds to about half the distance between two adjacent loudspeakers. This distance is a bit closer but still of the same order of magnitude as was done in the work of Bravo [START_REF] Bravo | The experimental synthesis of random pressure fields: Methodology[END_REF].

With this loudspeaker array, an acoustic field can be controlled having a minimum acoustic wavelength twice the inter distance of the loudspeakers, following the sampling theorem of Shannon [START_REF] Jerri | The shannon sampling theorem -its various extensions and applications -a tutorial review[END_REF]. With an inter distance of 0.41 m and 0.37 m in horizontal and vertical direction, respectively, this corresponds to a theoretical maximum frequency of 415 Hz and 450 Hz, respectively, taking the speed of sound c=340 m/s.

The matrix of transfer functions H of dimension N mic × N LS for each frequency considered was mea- The control microphone grid was chosen twice as fine as compared to the loudspeaker grid in order to have enough redundancy to construct the optimal signals to drive the loudspeaker array. The maximum frequency that can be controlled by the loudspeaker array, however, is determined by the inter spacing of the loudspeakers (with a theoretical value of 415 Hz as estimated above).

Next, the glass plate was excited by the loudspeaker array using the optimal loudspeaker signals x, measuring the response of the glazing w by means of an in-house developed scanning laser Doppler measurement system, consisting of a Polytec laser head OFV-505, a Polytec controller OFV-5000, and a dual-axis scanning mirror system from Thorlabs (see Fig. 5). A measurement grid of N LDV =14 × 19=266 positions was chosen (see Fig. 2b). On the glazing retroreflecting stickers were glued (see Fig. 4) to increase the optical reflection, facilitating the laser Doppler measurement.

Using Kirchhoff's thin shell theory of an infinite plate, an approximate estimate of the structural wavelength of the glazing of about 0.5 m was obtained at the maximum frequency that can be controlled by the loudspeaker array (415 Hz). According to the sampling theorem of Shannon [START_REF] Jerri | The shannon sampling theorem -its various extensions and applications -a tutorial review[END_REF] this would require a mesh of 2•1.35/0.5=5 × 2•1.54/0.5=6 points. Taking 6 points per wavelength, a mesh of 6•1.35/0.5=16 × 6•1.54/0.5=19 would be required. A mesh of 14 × 19 was chosen.

The loudspeakers were driven by a set of amplifiers, 12 in total, and two inter-connected Roland data acquisition systems, type Octa Capture (8 input channels, 8 output) and type Studio Capture (16 input channels, 8 output) (see Fig. 6). With the inter-connected Roland data acquisition systems the structural response as well as the acoustic response were measured, including the signals that were used to drive the loudspeakers (via a loop back).

Measurement results

Estimation of the sound reduction index according to the standard

The sound reduction index of the glazing was measured according to the ISO 10140-2:2010 standard [START_REF]building elements -part 2: Measurement of airborne sound insulation[END_REF].

During this measurement the loudspeaker array was not in the source room. Instead, two loudspeakers, brand name Mackie, type SRM450 were positioned in the corners of the source room, opposite of the test opening, to excite the glazing. In our experience, these loudspeakers are capable to generate noise from the 31.5 Hz 1/3 rd octave band onwards, as well as (but with reduced level) in the 25 Hz 1/3 rd octave band. The sound pressure level in the source room L p1 was averaged over 8 microphone positions. The reverberation time T 60 and the sound pressure level in the receiving room L p2 were also averaged over 8 microphone positions. From these measurement data the sound reduction index R of the glazing was com-Figure 5: The in-house developed scanning Laser Doppler vibrometry measurement system. puted with the equation

R = L p1 -L p2 + 10 lg S A ( 18
)
where S is the area of the free test opening in which the element is mounted, and A the equivalent sound absorption area in the receiving room, which is related to T 60 according to Sabine's formula as T 60 = 0.16V /A, where V is the volume of the receiving room.

Figure 7 shows the measured sound reduction index. The figure also shows the coincidence frequency, which is estimated to be approximately 1000 Hz (12 mm thick glass plate, Young's modulus 56 GPa, density 2400 kg/m 3 ). Indeed, around the coincidence frequency the sound reduction index R shows a brad dip, as expected. Although the ISO 10140-2:2010 standard prescribes to use 1/3 rd octave bands from 50 Hz onwards, also frequency bands below 50 Hz are shown. To slightly improve the accuracy of the measurements in the low frequency range, the number of microphone positions was increased from the required 5 (ISO 10140) to 8 positions.

The uncertainty in the measured sound reduction index can be determined by considering the standard deviation of the sound pressure levels in the source and receiving room from position to position, σ Lp1 and σ Lp2 , respectively, as well as the standard deviation of the equivalent sound absorption in the receiving room resulting from the reverberation time standard deviation [START_REF] Wittstock | On the uncertainty of singlenumber quantities for rating airborne sound insulation[END_REF]. Given that the contributions of the individual terms to the total uncertainty on R are independent, the combined uncertainty can then be estimated as σ R = σ 2 Lp1 + σ 2 Lp2 + σ 2 10 lg(A) . In case of repeated measurements on N (eight) positions, it can be expected that the error on the mean reduces to σ R,x = σ R / √ N . In addition to the ISO-based results, Figure 7 also shows the measured sound reduction index incremented and decremented with its standard deviation σ R and standard error σ R,x .

Estimation of the sound reduction index by means of an array of loudspeakers and laser Doppler vibrometry

The sound reduction index of the glazing was measured by exciting the glazing by means of a diffuse sound field created by a near field array of loudspeakers and measuring the glass panel response by means of laser Doppler vibrometry. Driving the loudspeaker array by means of the optimal loudspeaker signals x, the structural responses w of the building insulation element under test and the acoustic pressure responses y close to the building insulation element under test were measured. In [START_REF] Roozen | A global error estimator for the uncertainty of a multi-channel spectral analysis[END_REF] it was shown that the measurement record length needs to be larger than the reverberation time of the structural-acoustic system to avoid serious errors in the frequency response function estimate. As the reverberation time of the source room is about 3 seconds, a measurement record length of 10 seconds was chosen, yielding a spectral resolution of 0.1 Hz. A Hanning weighting function was applied. The number of averages was 20, with 75% overlap, requiring a total measurement time of 60 seconds per point. The multiple coherence (as defined in Eqn. 14) of the structural and acoustic measurements is shown in Fig. 8.

The multiple coherence of laser Doppler velocimetry measurements (Fig. 8a) overall shows a good coherence, close to unity, except for a few measurement points (e.g. measurement points 116-118), which was caused by a bad optical reflection of the laser beam at those points. The Multiple coherence of the microphone measurement close to the glass panel (in the source room, see Fig 8b ) shows coherences close to unity as well, except for a few measurement points. Below 20 Hz the multiple coherence of laser Doppler velocimetry and the acoustic measurement drops steeply due to the fact that the signal sent to the loudspeaker array did not contain energy at frequencies below 20 Hz.

The multiple coherences shown in Fig. 8 were obtained after correcting for latencies between the two inter-connected Roland data acquisition systems. An adequate clock-synchronization appeared not to be feasible. The latency correction was possible as the loudspeaker driving signals were looped back to both Roland data acquisition systems. Without latency correction the multiple coherence is significantly lower, varying between 0.1 and 0.9 at random, for all frequencies considered.

Figure 9 shows the Rayleigh computed sound pressure amplitudes based on the 12 principal components of the glazing response, using CSA. Three frequencies were chosen to illustrate the vibrational behavior at resonant frequencies of the glazing. The fundamental frequency occurs at 28 Hz. The second mode occurs at 62 Hz and a third mode occurs at 118 Hz.

Figure 10 shows the Rayleigh computed active sound power radiated by the glazing, using Eq. 15. The total sound power clearly shows peaks at the above mentioned resonance frequencies of 28 Hz, 62 Hz and 118 Hz.

Figure 11 shows the total radiated sound power as in the previous figure, as well as the incident acoustic power calculated by means of Eq. 16. The incident acoustic power generated by the loudspeaker array Taking the ratio between the incident power to the radiated power (Eq. 17) gives the sound reduction index, as shown in Fig. 12 in terms of narrow frequency bands and in Fig. 7 in terms of 1/3 rd -octave frequency bands.

In Fig. 12 the obtained results are compared with a theoretical estimate of the sound reduction index. This estimate is based upon an analytical expression of the structural-dynamics of a simply supported plate, having dimension L x =1.35 m, L y =1.54 m, thickness h=12e-3 m, Young's modulus E=56 GPa, Poisson's ratio ν=0.24, density ρ=2400 kg/m 3 and loss factor η=0.013.

The density ρ of the glass was extracted from a weight measurement of the glass panel, and the measurement of its dimensions. The Young's modulus E was tuned in order to get a reasonable agreement between the first resonance frequency as predicted by the analytical model and the experimentally obtained first resonance frequency. The loss factor η was extracted from the half power (-3 dB) bandwidth of the measured transfer function at the first resonance frequency of the glazing. It should be noted that the damping as specified in the analytical model is valid at the first resonance frequency only. Thus, the analytical model has a limited frequency range in which it is accurate.

From the mobility of the simply-supported plate (for which theoretical expressions are readily available), an estimate of the sound reduction index R for a diffuse sound field excitation is computed (see [START_REF] Roozen | A hybrid experimental numerical approach to assess the airborne sound insulation from mobility measurements[END_REF] for details about this approach). Figure 12 shows the results for two cases: with and without taking into account the radiation damping of the plate. The results with the radiation damping taken into account correspond reasonably well with the R obtained by means of an array of loudspeakers and laser Doppler vibrometry, which validates the latter approach. Deviations up to 5 dB are observed, which are likely to be caused by the limited accuracy of the analytical model to represent the device under test.

To illustrate the effectiveness of the loudspeaker array to suppress room modes in the source room, an additional measurement was performed with classical loudspeakers in the corners of the source room. The loudspeaker array was removed from the room for this experiment. The blocking sound pressure levels at the surface of the glazing and the response of the glazing were measured, from which the incident and the radiated sound powers were computed with Eq. 16 and Eq. 15, respectively. The results are shown in Fig. 13, in which the acoustic resonance frequencies of the source room are indicated by dotted lines. The eigenfrequencies of the source room were predicted by means of a finite element model of the room, of which the results are shown in Fig. 14.

From Fig. 13 it can be seen that the acoustic pressure in the source room significantly peaks at the resonance frequencies of the source room when using the classical loudspeakers, whilst not when using the loudspeaker array. Also the response of the glazing is significantly higher at those resonance frequencies, in case of excitation by means of classical loudspeakers, which becomes evident from the higher sound powers radiated at those frequencies. In addition, the response of the glazing is also higher at its resonance frequencies (28 Hz, 62 Hz, 118 Hz), as expected. When using the loudspeaker array, the radiated sound power peaks at the first (28 Hz), second (62 Hz) and third (118 Hz) resonance frequencies of the glazing, as well as a peak at 41 Hz (see the black curves in Fig. 11 and Fig. 13b). Thus the loudspeaker array is capable to suppress the room modes in the source room, except for the room modes near 41 Hz.

Figure 7 shows the measured sound reduction index in 1/3 rd -octave bands, according to the ISO standard. When processing the ISO microphone measurement data in narrow-bands, the results as shown in Fig. 15 are obtained. This result shows that the narrow band estimate of the sound reduction index significantly drops at the eigenfrequencies of the source room eigenmodes, whilst significantly increases at frequencies just below or just above the eigenfrequencies of the source room eigenmodes. The narrow band estimate of the sound reduction index based on measurements by means of the loudspeaker array does not show this dependency on the source room eigenmodes. These is however one exception. The narrow band estimate of the sound reduction index does show a dip around the room mode eigenfrequencies near 41 Hz (see Fig. 15b), in accordance with the observation discussed in the previous paragraph. This dip is not caused by a structural resonance frequency of the glazing (the first three eigenfrequencies of the glazing occurs near 28 Hz, 62 Hz and 118 Hz). This deficiency when using the loudspeaker array can be seen in all results presented in Fig. 10, Fig. 11 and Fig. 12. It shows the inability of the loudspeaker array to suppress the source room acoustics effect at this specific frequency.

The global trend of the narrow band sound reduction index R obtained by means of an array of loudspeakers and laser Doppler vibrometry corresponds well with the analytical model (ref Fig. 12 and the previous discussion in this section). The sound reduction index R obtained by means of an array of loudspeakers and laser Doppler vibrometry in terms of 1/3 rd -octave frequency bands, however, shows significant deviations as compared to the measurements according to the ISO 10140-2:2010 standard for frequencies below 250 Hz, as shown in Fig. 7. The signature of both measurement results are the same, with dips at the 25 Hz and 62 Hz 1/3 rd -octave fre- quency bands, corresponding to the structural resonance frequencies of 28 Hz and 62 Hz, respectively. However, deviations up to 15 dB are observed. This disagreement in the quantitative results of R in the lower frequency range is likely due to the room acoustic effects of the source room (volume 87 m 3 ), causing a breakdown of the diffuse field assumption. In the 250 Hz and 315 Hz 1/3 rd -octave frequency bands the sound reduction index R obtained by means of an array of loudspeakers and laser Doppler vibrometry correspond well with the measurements according to the ISO 10140-2:2010 standard, with deviations of only 1 or 2 dB. At frequencies above 415 Hz, the loudspeaker array is not able to control the acoustic field because of insufficient spatial sampling, as discussed in Section 3.1.

Measurement effort in practice

In this section the actual measurement effort of the measurements on a single glazing, of lateral dimension In summary, the experiment took about 15 hours of measurement time (including manual positioning of the microphone), yielding about 20 GByte of data. Using multiple sensors, or a robot to replace the manually positioning of the sensor, would make the measurement easier to perform.

Note that it is required to have a record length which is at least equal to the reverberation time of the transmission facility [START_REF] Roozen | A global error estimator for the uncertainty of a multi-channel spectral analysis[END_REF], which forces the measurement time for each average to be about 10 seconds.

Conclusions

An array of loudspeakers was positioned closely to the building insulation element under test. The acoustic near field of the loudspeaker array was used to create a diffuse field, following the works of Bravo et al.. The method successfully suppressed the room acoustic modes of the source room for most of the frequencies below 150 Hz.

At the receiving side, the sound power radiated by the building element under test was computed from the measured vibration pattern of the building element. For this purpose the uncorrelated contributions to the total vibration pattern were determined using conditioned spectral analysis (CSA), followed by the application of the Rayleigh integral to compute the radiated sound power.

From the sound pressures measured at the surface of the building insulation element under test the incident sound power was determined. From the incident and radiated sound powers, the sound reduction index R was estimated. The estimation of the sound reduction index R of building elements by means of an array of loudspeakers and a scanning laser Doppler vibrometry was validated for a single glazing by means of an analytical model. The global trend corresponded well. In quantitative terms the narrow band results showed deviations up to 5 dB, which are probably due to the limited accuracy of the analytical model.

The newly proposed method is valid only in the lower frequency range due to the finite spatial sampling of the loudspeaker grid and structural response grid. For frequencies above the 200 Hz 1/3 rd octave band and up to the 630 1/3 rd octave band the results obtained with the proposed method correspond to the standardized ISO 10140:2010 measurement results for the presented case, with deviations of only 1 or 2 dB's.

At frequencies below and including the 200 Hz 1/3 rd octave band large differences occurred up to 15 dB between the ISO measurements and both the analytical model and the newly proposed measurement approach. These deviations of the ISO results are likely to be caused by a breakdown of the diffuse field assumption.

As a side note it should be remarked that the loudspeaker array was not able to eliminate the room acoustic effects for all frequencies. This affected the sound reduction index estimate of the glazing at those specific frequencies (in the present case 41 Hz), causing a deviation of the narrow band measurement result as compared to the analytical solution result.

Although the method offers clear, strong points in terms of removing room acoustic effects from the measurements, a point of concern is the measurement effort. For a loudspeaker array consisting of 12 loudspeakers, the measurements of the blocking pressure at 63 microphone control positions, and the vibration response measurement at 266 positions with an averaging time of 60 sec per position, took tens of hours in total, yielding tens of GBytes of data. Moreover, a multichannel data acquisition is required, separate amplifiers for each loudspeaker of the array, and the construction of a (heavy) loudspeaker array cabinet.

A Evaluating the Rayleigh integral to compute the radiated sound power

Consider a vibrating baffled structure radiating into a semi-infinite acoustic domain. The structure is vibrating with a velocity field that is described by v(x e , ω).

The parietal pressure at the points x r on the structure can be computed by means of the Rayleigh integral [START_REF] Roozen | Determining radiated sound power of building structures by means of laser Doppler vibrometry[END_REF][START_REF] Koopmann | Method for computing the sound power of machines based on the Helmholtz integral[END_REF][START_REF] Fahy | Sound and Structural Vibration: Radiation, Transmission and Response[END_REF]:

p (x r , ω) = iωρ 2π S v n (x e , ω) e -ikr r dS r ( 19
)
where r is the distance between the points in vector x e and vector x r , k is the wavenumber, ρ the density and ω the radial frequency. Note that the above integral has a singularity for r → 0. Equation 19 can be written in matrix form as

p = Zv ( 20
)
where the matrix Z can be computed as follows [START_REF] Bai | Estimation of sound power of baffled planar sources using radiation matrices[END_REF], taking into account the singularity at r → 0:

Z = ρc
where

d = 1 2 k S π 2 -i 8 3π k S π (22)
Since the normal component of the active acoustic intensity I along the surface S of a vibrating test wall, is given by

I (x r , ω) = 1 2 Re [p (x r , ω) v * n (x r , ω)] (23)
where Re denotes the real part of a complex quantity and the asterisk denotes the complex conjugate, the total radiated active sound power Π can be obtained by Π (ω) = S I (x r , ω) dS [START_REF] Bendat | Engineering applications of correlation and spectral analysis[END_REF]

  Assessment of the sound reduction index of building elements by near field excitation through an array of loudspeakers and structural response measurements by laser Doppler vibrometry N.B. Roozen, Quentin Leclere, D. Urbán, L. Kritly, C. Glorieux To cite this version: N.B. Roozen, Quentin Leclere, D. Urbán, L. Kritly, C. Glorieux. Assessment of the sound reduction index of building elements by near field excitation through an array of loudspeakers and structural response measurements by laser Doppler vibrometry. Applied Acoustics, 2018, 140, pp.225 -235. 10.1016/j.apacoust.2018.06.002. hal-01829079 1 Introduction

  rad by means of the Rayleigh integral.

  (a) Control microphone grid (7 × 9, indicated by crosses) (b) Laser Doppler velocimetry grid (14 × 19, indicated by dots)

Figure 2 :

 2 Figure 2: Measurement grids on the glazing, LS-array indicated by dashed lines.

Figure 3 :

 3 Figure 3: LS-array in front of glass plate.

Figure 4 :

 4 Figure 4: Glazing with retro-reflecting stickers at one side, a microphone mounted on the other side of the glass plate (traversed from one position to the other), and the loudspeaker array also on the other side of the glass plate.

Figure 6 :

 6 Figure 6: Power amplifiers (left) and data acquisition measurement systems (right).

Figure 7 :

 7 Figure 7: Sound reduction index measurement results according to the ISO 10140-2:2010 standard, including frequencies below 50 Hz (mean value as a thick dashed blue curve, mean value incremented and decremented with the standard deviation as thin dashed blue curves), compared with sound reduction index 1/3 rd octave results obtained by means of an array of loudspeakers and a scanning laser Doppler (solid magenta curve). The interval between the dash-dotted and full gray lines indicate the estimated standard deviation between measurements on different microphone positions and the standard error on their mean respectively. The coincidence frequency of the glass plate (approximately 1000 Hz) is indicated by a vertical dashed line.

Frequency

Figure 8 :

 8 Figure 8: Multiple coherence between the response data and the reference signals, based on CSA.

Figure 9 :

 9 Figure 9: Largest 12 principal components of the Rayleigh computed sound pressure distribution on the glass plate (units: Pa, spectral amplitude, frequency resolution: 0.1 Hz), computed by CSA.

Figure 10 :

 10 Figure 10: Radiated active sound power, computed from Eq. 15. The bold black curve shows the narrow band spectrum of the total sound power, computed as the sum of the individual CSA powers. The bold magenta curve shows the 1/3 rd octave spectrum. The dashed magenta curve shows the 1/12 th octave spectrum.

Figure 11 :

 11 Figure11: Narrowband incident sound power spectrum (dashed blue curve) and radiated sound power spectrum (solid black curve), glass panel excited by 12 LS array close to the glass panel, frequency resolution: 0.1 Hz, computed by means of CSA according to Eq. 16 and Eq. 15, respectively.

Figure 12 :

 12 Figure 12: Narrowband sound reduction index R obtained by means of the LS array and laser Doppler vibrometry of the glazing (blue solid curve), compared with the analytically obtained sound reduction index R of a simply-supported plate, with (red dashed curve) and without(magenta dotted curve) taking into account radiation damping.

Figure 13 :

 13 Figure13: Narrowband sound power spectrum obtained with loudspeakers in the corners of the sending room (solid blue curves) and obtained with the loudspeaker array close to the device under test (dashed black curves), a) incident sound power spectrum computed from Eq. 16, b) radiated sound power spectrum computed from Eq. 15. The acoustic resonance frequencies of the source room up to 160 Hz are indicated by vertical dotted black lines.

  (a) mode 1, 34.15 Hz (b) mode 2, 41.17 Hz (c) mode 3, 41.61 Hz (d) mode 4, 53.94 Hz (e) mode 5, 54.18 Hz (f) mode 6, 58.49 Hz (g) mode 7, 67.07 Hz (h) mode 8, 68.42 Hz

Figure 14 :

 14 Figure 14: Acoustic eigenmodes of the source room, as predicted by a finite element model of the source room. Test opening cove indicated by black lines.

Figure 15 :

 15 Figure 15: Narrowband sound reduction index R, using standardized microphone based approach (solid magenta curves) and using an array of loudspeakers and laser Doppler vibrometry (dashed blue curves). Top figure a) shows frequency range from 20 to 500 Hz, bottom figure b) shows frequency range from 20 to 100 Hz, acoustic resonance frequencies of source room indicated by vertical dotted black lines.

  1.35 × 1.54 m 2 , is summarized.

	signal of
	10 s with 3 averages took about 5 hours of mea-
	surement time, and an additional 5 hours for the
	manual positioning of the control microphone (63
	positions), yielding a few GByte of time-domain
	data, using a sampling frequency of 4.41 kHz,
	The cross spectra of the structural responses w
	and the loudspeaker signals x with dimension
	N LDV × N LS = 266 × 12, whilst driving the
	loudspeaker array with the optimal signals x,
	with a length of 10 s, 20 averages, 75 % overlap,
	took about 5 hours of measurement time, yield-
	ing about 11 GByte of time-domain data, using
	a sampling frequency of 4.41 kHz,
	The cross spectra of the acoustic responses y
	and the loudspeaker signals x, with dimension
	N mic × N LS = 63 × 12, whilst driving the loud-
	speaker array with the optimal signals x, with a
	length of 10 s, 20 averages, 75 % overlap, took
	about 1 hour of measurement time, and an addi-
	tional 4 hours for the manual positioning of the
	control microphone (63 positions), yielding about
	9 GByte of time-domain data, using a sampling
	frequency of 4.41 kHz.

The measurement of the matrix of transfer functions H between the acoustic responses y and the loudspeaker signals x with dimension N mic × N LS = 63 × 12, using a sweep excitation

Acknowledgments

This work was carried out in the framework of H2020-MSCA-RISE-2015 No. 690970 "Papabuild".

Willy Bruyninckx is acknowledged for performing the measurements according to the standard and Ludovic Labelle for his help in setting up part of the measurement equipment.

The research was conducted whilst the first author was working at KU Leuven. NBR is currently working at A&Z Acoustics, Bratislava.