Numerical and experimental investigation of the narrow-band impact sound insulation of layered floors

Jasper Vastiau 1 , Cédric Van hoorickx 2 , and Edwin P.B. Reynders 3

KU Leuven, Department of Civil Engineering, Structural Mechanics Section Kasteelpark Arenberg 40, box 2448, B-3001, Leuven, Belgium

ABSTRACT To obtain detailed information on the modal behavior of the floor and on the influence of the impact source, narrow-band sound pressure level measurements are performed on a concrete base floor and a floating floor. Various tapping machine positions are considered and the results are averaged over several microphone positions for each tapping machine position in order to reduce the influence of the spatial distribution of the sound pressure in the receiver room. Both the finite element method and a novel prediction method for impact sound transmission, termed the modal Transfer Matrix Method, or mTMM, are used to interpret and investigate the measurement results. The mTMM can be combined with a detailed source model and accounts for the finite dimensions of the layered floors. The measurement results of the layered floor clearly indicate the importance of considering all five impact hammer locations of the tapping machine to achieve a high prediction accuracy, especially at low frequencies. The influence of the impact source and the modal behavior of the floor can lead to pronounced peaks in the radiated sound power level.

1. INTRODUCTION

Research into narrow-band impact sound insulation is the result of two driving factors: first, harmonic or narrow-band measurement results yield information regarding the influence of the floor modes on sound radiation, and related, boundary conditions and excitation positions. Second, harmonic or narrow-band measurements are well suited for validation of recently developed computational models that (approximately) account for these details, resulting in a higher prediction accuracy, especially at low frequencies. Detailed measurements of the acceleration level of a timber joist floor have indicated a 2 Hz line spectrum for the ISO tapping machine [1], rather than the common assumption of a single hammer impacting with a base frequency of 10 Hz [2]. These measurements also indicated that in general, there is a di ff erence in amplitude of about 20 dB between the 10 Hz lines and the 2 Hz lines, meaning that the 10 Hz lines are still the most important ones in the impact spectrum. However, the 2 Hz lines can become important, especially at low frequencies and for lightweight floors. Recent work has

1 jasper.vastiau@kuleuven.be

2 cedric.vanhoorickx@kuleuven.be

3 edwin.reynders@kuleuven.be

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

shown that such a 2 Hz line spectrum and a 20 dB di ff erence between the 10 Hz lines and the 2 Hz lines is also present in predictions of the radiated sound power level [3] and the sound pressure level in the receiver room. However, this has not been confirmed by measurements yet, as nearly all available impact sound measurements in literature are presented in one-third octave bands (in accordance with ISO 10140-3 [4]). In order to investigate these indications, the harmonic or narrow-band impact sound measurements of the sound pressure level in the receiver room are required. Such measurements are also required for validation of the newly developped mTMM [3, 5, 6]. This novel method accounts for both the modal behaviour of the structure, resulting fron the finite dimensions, and the 2 Hz line spectrum, resulting from a detailed source model. Unfortunately, this information is lost during the band-integration step to one-third octave bands. A finite element model is constructed for a concrete base floor to further verify the mTMM predictions. Since the FE model is able to capture the boundary conditions of the actual floor more accurately, the error of the mTMM in assuming simply supported boundaries along the entire floor thickness can be assessed. The natural frequencies of the base floor are computed from the FE model and are compared to experimentally determined values. This work is organized as follows: first, the impact sound power radiation of the concrete base floor is discussed. The measurement procedure is explained and the measurement results are interpreted using a finite element (FE) model and mTMM predictions. The experimental determination of the natural frequencies and mode shapes is explained, and the results are compared to the natural frequencies and mode shapes of the FE model. For this base floor, a comparison is also made for the radiated sound power level between measurements, FE results and mTMM predictions. Second, a thermal layer and finishing screed are installed and the impact sound radiation of the floating floor is assessed.

2. EXPERIMENTALLY AND NUMERICALLY DETERMINED NATURAL FREQUENCIES AND MODE SHAPES OF A CONCRETE BASE FLOOR

A 14 cm thick concrete base floor (test element: horizontal dimensions L x = 4 . 40 m and L y = 2 . 58 m.) is rigidly connected to a 30 cm thick concrete frame, which is 25 cm wide in both horizontal directions. This frame also has rigid connections to the adjoining concrete slabs of 30 cm thickness through a cemented joint. Measurements of the natural frequencies and mode shapes of this floor are compared to a FE model, constructed using Ansys.

2.1. Measurements A roving hammer test with impact locations in a rectangular grid pattern (cfr. Figure 1) is performed to experimentally determine the natural frequencies and mode shapes of the full structure (test element + rigidly connected concrete slabs). Repeated hammer impacts on the floor are provided by a type PCB 086B20 hammer. Accelerations are measured on 24 locations on the floor, using a combination of type Dytran 3100D24 and PCB 353B34 accelerometers. Both hammer and accelerometer signals are collected with a NI PXI aqcuisition unit, which consists of a PXI-1050 chassis with four PXI-4472B modules connected to a portable computer. Note that a simple wooden beam construction (cfr. Figure 1) avoids mass addition from the hammer operator to the floor, such that the natural frequencies and mode shapes are determined more accurately. MACEC [7] is used for post-processing of the data and system identification. The collected signals are first decimated to obtain the frequency data of interest. During the measurements, a sampling frequency of 1000 Hz (Nyquist: 500 Hz) was used, which is decimated by a factor of three in order to identify natural frequencies and mode shapes up to 166 Hz. Additionally, the DC component is removed and a high pass filter with a cut-on frequency of 10 Hz is applied to reduce the low- frequency drift. A frequency-response function (FRF) can then be estimated from the processed data using a nonparametric H 1 estimator [8]. The FRF is estimated for each hammer-accelerometer

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Figure 1: Modal testing at the BBRI laboratory (a) laboratory setup (b) plan view of hammer locations (blue line intersections) and accelerometer placement (red dots).

combination, and the corresponding estimates are gathered into an overall FRF matrix. Afterwards, the Poly-reference Least Squares Complex Frequency Domain method (pLSCF) is employed for fitting a parametric, right matrix fraction description (RMFD) model to the nonparametric FRF data. Finally, the modal characteristics can be selected from a stabilization diagram and the resulting natural frequencies are listed in Table 1.

2.2. Finite element model The entirety of the full structure (test element + adjoining concrete slabs) is included in the finite element model. This structure is decoupled from adjacent floors by means of air gaps, filled with mineral wool. As a result, the floor is only supported by the walls underneath, which is why simply supported boundary conditions are only applied at the bottom edges of the floor. However, the floor is not supported by the partition wall, so it is only supported at three sides. The mesh is constructed using 20-node hexahedral quadratic solid elements (SOLID186), with 16 elements along the total thickness of 30 cm and 8 elements along the smallest thickness of 14 cm in the middle section. Figure 2 (a) illustrates the finite element model. The block Lanczos mode-extraction method is applied for the modal analysis and the resulting natural frequencies and mode shapes are illustrated in Figure 3. A modal assurance criterion (MAC) [9] is applied to check the correspondance between the experimentally determined and computeed mode shape. The MAC values are illustrated on Figure 2 (b), showing an excellent agreement between both, with MAC values ranging from 0.77 for mode 6 to 0.93 for mode 2. Table 1 lists the natural frequencies, both computed and experimentally determined.

3. IMPACT SOUND RADIATION OF A CONCRETE BASE FLOOR

After validation of the natural frequencies and mode shapes, this section handles a second validation step, which is the resulting sound pressure level in the receiver room when the tapping machine excites the base floor. The FE results and mTMM predictions are used to interpret the measurement results of the sound pressure level.

12 9 1011

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Figure 2: (a) Finite element model, constructed using Ansys. (b) MAC values.

Mode shape Experimental [Hz] Ansys [Hz] Error

(1,1) 22.8 17.4 -23.7 %

(1,2) 30.4 30.4 0.0 %

(2,1) 60.2 57.9 -4.0 %

(2,2) 77.6 72.8 -6.2 %

(1,3) 82.4 85.8 4.1 %

(3,1) 105.4 104.2 -1.1 %

Table 1: Natural frequencies: comparison of experimentally determined values and values computed using Ansys, including relative errors.

3.1. Measurements Impact forces are provided by a battery powered magnetic Tapping Machine Nor227. An overview of the tapping machine and the measured floor is displayed in Figure 4. Three di ff erent tapping machine positions are used (cfr. Figure 5) on the test element and all hammer locations are accurately measured and listed in Table 2. At position 1, the tapping machine is placed at the center of the floor and the hammers are aligned with the x-axis. At position 2, the tapping machine is placed such that the distance between the floor edge and the closest hammer is around 50 cm. In this position, the hammers are placed along the y-axis. Position 3 is located close to a corner of the floor, with the tapping machine placed at an angle of approximately 45 ◦ with respect to the x-axis. This tapping machine position is chosen to gain insight into the error made by the assumption of simply supported boundaries in mTMM with respect to the actual boundary conditions. The sound pressure level in the receiver room is measured by means of a sound level meter NTI XL2 using a manual scanning procedure (in accordance with ISO 16283-2:2020 [10]), in order to obtain measurement results with a sampling frequency of 48 kHz. The sound pressure level is measured at two di ff erent receiver locations for each tapping machine position. Averaging times are at least 30 s in accordance with ISO 10140-4:2021. The time history of the pressure signal is then transformed to the frequency domain using a Fast Fourier Transform (FFT) and subsequently converted to a sound pressure level. The total sound pressure level for a single tapping machine position is obtained by averaging the pressure over two measurements at varying receiver locations in

(a) 17.4 Hz (1,1)

(b) 30.4 Hz (1,2)

(c) 57.9 Hz (2,1)

(d) 72.8 Hz (1,3)

(e) 85.8 Hz (2,2)

(f) 104.2 Hz (3,1)

Figure 3: FEM results for the natural frequencies and mode shapes of the entire base floor.

order to capture the spatially averaged sound pressure level. Figure 6 shows the measurement results in 1 / 48 octave bands for all three tapping machine positions. The largest peaks can be seen for tapping machine position 1 (i.e. in the middle of the floor). Even for frequencies higher than 4000 Hz, the resonance peaks at the natural frequencies are clearly distinguishable. For tapping machine position 3 at the corner of the floor, it is di ffi cult to excite the first mode (22.8 Hz), resulting in a resonance peak which is ± 20 dB lower with respect to position 1.

3.2. FEM The FE model of thefull structure is coupled to a di ff use field in the receiver room, in accordance to [11], such that the sound pressure in the receiver room can be derived from the displacements of the floor.

a

3.3. mTMM predictions A detailed source model is used in the mTMM to obtain accurate prediction results at low frequencies. The mechanical properties of the concrete base floor are listed in Table 3, including the thickness t, mass density ρ , damping ratio η , Poisson coe ffi cient ν and elasticity modulus E. These properties have been measured by the Belgian Building Research Institute (BBRI) [12], except for the damping of the

Figure 4: Laboratory setup for the impact sound measurements.

Figure 5: Tapping machine positions w.r.t. the test element.

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Figure 6: Total sound pressure level for each of the tapping machine positions in 1 / 48 octave bands; black: position 1, blue: position 2, red: position 3.

thermal layer. The hammer locations in the detailed source model are identical to the measurements, in accordance to Table 2. Both the harmonic prediction results and measurement data are then band-integrated over 1 / 48 octave bands, reducing the noise on the measurement data, allowing for a comparison between the two. Comparing the sound pressure level in 1 / 48 octave bands for the measurements, the FE model and the mTMM predictions, some significant discrepancies can be

Ly = 2,60m Ly =4,40m

Pos 1 [m] Pos 2 [m] Pos 3 [m]

Hammer 1 (2.03, 1.30) (3.46, 0.53) (3.97, 2.15)

Hammer 2 (2.13, 1.30) (3.46, 0.63) (4.04, 2.21)

Hammer 3 (2.23, 1.30) (3.46, 0.73) (4.12, 2.27)

Hammer 4 (2.33, 1.30) (3.46, 0.83) (4.19, 2.33)

Hammer 5 (2.43, 1.30) (3.46, 0.93) (4.27, 2.48)

Table 2: Tapping machine positions, including exact hammer locations.

t [m] ρ [ kg

m 3 ] η [-] ν [-] E [ N

m 2 ]

Finishing screed 0.05 1900 0.04 - 0.02 0.2 17.4 · 10 9

EPS 0.05 125 0.8 - 0.1 0.2 1.97 · 10 7 - 1.64 · 10 7

concrete base floor 0.14 2445 0.06 - 0.01 0.2 22 · 10 9

Table 3: Mechanical properties for each layer [12], used in mTMM.

noticed. In contrast to the floating floor, not all 10 Hz multiples yield accurate results, especially in the frequency range 60-600 Hz. This could possibly be attributed to the interaction of the impact spectrum and the floor modes, but also the coincidence frequency of the floor.

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Figure 7: Sound pressure level in 1 / 48 octave bands for tapping machine position 1. Blue: measurement, red dots: mTMM, black: FE model.

From the one-third octave band results in Figure 8, it can immediately be seen that the mTMM generally yields an overestimation with respect to the measurements and the FE model. This overestimation is likely caused by a di ff erence in boundary conditions between the FE model and mTMM, with the 14 cm thick plate being rigidly connected to the thicker floor in reality and in the FE model, whereas the mTMM assumes simply supported boundary conditions. Due to these rigid connections, the FE model envelopes a larger concrete structure, which allows for more mass and more geometrical damping, which both lead to lower displacements in the base floor itself. Therefore, an equivalent mTMM model is constructed, which accounts for the horizontal dimensions of the full structure instead of the test element only. The mTMM requires a constant layer thickness, so an equivalent thickness is computed using a minimization of the error on the natural frequencies with respect to the FE model, yielding an equivalent thickness of 22 cm. The equivalent mTMM model yields significantly better predictions at low frequencies (cfr. Figure 8). At high frequencies

however, the sound pressure level in the receiver room is now underestimated, since in reality the 14 cm thick test element dynamically decouples from the thicker floor.

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Figure 8: Sound pressure level in the receiver room in one-third octave bands, resulting from tapping machine impacts on the concrete base floor. Blue: measurement, red solid line: mTMM prediction, red dotted line: equivalent mTMM prediction, black: FEM (a) impact position 1 (b) impact position 2 (c) impact position 3.

4. IMPACT SOUND RADIATION OF A FLOATING FLOOR

4.1. Measurements The test element now consists of several layers: the base floor is a concrete slab with a thickness of 14 cm. A thermal layer (cement mortar with EPS-beads) with a thickness of 5 cm and a 5 cm lightweight concrete finishing screed are placed on top of the existing base floor. Comparing the di ff erent tapping machine positions, it is clear from Figure 9 that the impact location at the corner of the floor yields consistently lower sound pressure levels in the receiver room. This e ff ect is especially present at the di ff erent resonances, such as the mass-spring-mass resonance and at the natural frequencies of the floor between 40 and 70 Hz. At higher frequencies, when the modal overlap is su ffi ciently high and the distance between the tapping machine and the corner of the test element amounts to several wavelengths, the total sound pressure level converges to the results from the other impact locations. Around 80 Hz, all three impact positions yield a high sound pressure level in the receiver room. This frequency corresponds to the (1,3) mode of the full structure, or the (1,1) mode

of the test element.

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Figure 9: Total sound pressure level in 1 / 48 octave bands for each of the tapping machine positions; black: position 1, blue: position 2, red: position 3.

Taking a closer look at the low-frequency results in Figure 10 from one of the measurements for tapping machine position 1, both the impact spectrum of the tapping machine and resonance peaks resulting from natural frequencies of the floor can be distinguished. The 2 Hz line spectrum of the tapping machine impact force is clearly present in the frequency range below 50 Hz. At frequencies above 50 Hz, the frequencies that are a multiple of 10 Hz dominate the sound pressure level response in the receiver room. The measurement data also support the initial findings of other measurements on the relative acceleration level [1] and the use of a detailed source model in the mTMM predictions [3], that the sound pressure level at discrete frequencies that are multiples of 10 Hz, yield around 10-20 dB higher values at low-frequencies than the multiples of 2 Hz.

Figure 10: Low-frequent sound pressure level in 1 / 48 octave bands for tapping machine position 1.

4.2. mTMM predictions The floor layering, layer thicknesses and mechanical properties of the floating floor are listed in Table 3. If a range of values is shown, the first value is valid at 50 Hz and the second value is valid at 5000 Hz [12]. The measured and predicted sound pressure level (cfr. Figure 11) show a good agreement at multiples of 10 Hz. The mTMM predictions for 2 Hz multiples yield an underestimation with respect

L, (dB) 6 318 (Hz)

to the measurements, for frequencies up to 400 Hz. It is also clear from the comparison that the mTMM predictions underestimate both the mass-spring-mass resonance and the thickness resonance of the thermal layer. This underestimation might be the result of an overestimated damping value for the thermal layer, since a measurement of this property was inhibited by accessibility contraints.

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Figure 11: Sound pressure level in 1 / 48 octave bands for tapping machine position 2. Blue: measurement, red dots: mTMM.

The one-third octave band results are shown for all three tapping machine positions in Figure 12. The same conclusion holds for the first two positions: the mass-spring-mass resonance and the thickness resonance of the thermal layer are underestimated. In the other bands, a good agreement is obtained with respect to the measurements. For position 3 however, the mTMM yields an underestimation of the measurement result for the entire frequency range. This is caused by a di ff erence in boundary conditions in the prediction model with respect to reality. In the prediction model, all layers are assumed to be simply supported, i.e. the displacement of the floor edge is zero along the entire thickness. in reality however, the concrete base floor rigidly connected to the thicker floor, whereas the thermal insulation layer and the floating screed have free boundary conditions. For an impact location close to a corner of the floor, this means that the mTMM will underestimate the displacement of the floor and will consequently also underestimate the radiated sound power and the sound pressure level in the receiver room. The equivalent mTMM model once again yields better predictions at low frequencies, especially for impact position 3 due to the improved boundary conditions.

5. CONCLUSIONS

Measurements of the sound pressure level in the receiver room for tapping machine impacts display a clear 2 Hz line spectrum. The di ff erence between 2 Hz lines and 10 Hz lines starts around 20 dB at the lowest frequencies and decreases with increasing frequency. This e ff ect is supported by mTMM predictions exhibiting the same pattern. A comparison of the measured and predicted sound pressure level of the floating floor leads to a generally good agreement between measurements and mTMM, except at the mass-spring-mass resonance, at the first thickness resonance of the thermal layer and at specific natural frequencies, where the mTMM underestimates the sound pressure level. This can probably be attributed to a high assumed damping value of the thermal layer, as this property could not be measured due to inaccessibility of the layer. For the concrete base floor, a finite element model is used to validate the experimentally determined natural frequencies. Both the FE model and the mTMM framework are used to interpret and support

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Figure 12: Sound pressure level in one-third octave bands as the result of tapping machine impacts on the floating floor. Blue: measurement, red: mTMM prediction (a) impact position 1 (b) impact position 2 (c) impact position 3.

the measured acoustic behaviour of the concrete base floor. Even though the FE model yields more accurate results at low frequencies, the mTMM surpasses the FE model at high frequencies in terms of computational e ffi ciency. It also tends to have a better prediction accuracy in the mid-frequency range (100-500 Hz). It is also shown that impact positions close to the floor edge leads to significant predictions errors if the floor is in reality not simply supported along the entire thickness of the floor. This is clearly the case for the floating floor, where the base floor is clamped, while the thermal layer and the screed are free.The prediction error is smaller for the concrete base floor, where only a single layer is clamped, which the mTMM can predict better with its assumed simply supported boundary conditions.

ACKNOWLEDGEMENTS

This research was funded by the European Research Council (ERC) Executive Agency, in the form of an ERC Starting Grant provided to Edwin Reynders under the Horizon 2020 framework program, project 714591 VirBAcous "virtual building acoustics: a robust and e ffi cient analysis and optimization framework for noise transmission reduction". The financial support from the European Commission is gratefully acknowledged. The Belgian Building Research Institute have provided laboratory access and allowed for

experiments to be conducted on the di ff erent floor setups. The help of BBRI is gratefully acknowledged.

REFERENCES

[1] V. Wittstock. On the spectral shape of the sound generated by standard tapping machines. Acta Acustica united with Acustica , 98:301–308, 2012. [2] I.L. Vér. Impact noise isolation of composite floors. Journal of the Acoustical Society of America , 50(4):1043–1050, 1971. [3] J. Vastiau, C. Van hoorickx, and E.P.B. Reynders. A modal transfer matrix approach for the prediction of impact sound insulation. In Proceedings of the 12th European Congress and Exposition on Noise Control Engineering, Euronoise 2021 , pages 1745–1754, Madeira, Portugal, October 2021. Sociedade Portuguesa de Acústica. [4] International Organization for Standardization. ISO 10140-3: Acoustics – Laboratory measurement of sound insulation of building elements – Part 3: Measurement of impact sound insulation , 2010. [5] J. Vastiau, C. Van hoorickx, and E.P.B. Reynders. Wavenumber domain solution for impact sound prediction of finite floors with the (modal) transfer matrix method. In T. Dare, S. Bolton, P. Davies, Y. Xue, and G. Ebbitt, editors, Proceedings of the 50th International Congress and Exposition on Noise Control Engineering, Inter-Noise 2021 , pages 734–745, Washington, D.C., August 2021. [6] C. Decraene, A. Dijckmans, and E.P.B. Reynders. Fast mean and variance computation of the di ff use sound transmission through finite-sized thick and layered wall and floor systems. Journal of Sound and Vibration , 422:131–145, 2018. [7] E. Reynders, M. Schevenels, and G. De Roeck. MACEC 3.4: The Matlab toolbox for experimental and operational modal analysis. Report BWM-2021-04, Department of Civil Engineering, KU Leuven, April 2021. [8] E. Reynders. System identification methods for (operational) modal analysis: review and comparison. Archives of Computational Methods in Engineering , 19(1):51–124, 2012. [9] R.J. Allemang and D.L. Brown. A correlation coe ffi cient for modal vector analysis. In Proceedings of the 1st International Modal Analysis Conference , pages 110–116, Orlando, FL, 1982. [10] International Organization for Standardization. ISO 16283-2:2020: Acoustics - Field measurement of sound insulation in buildings and of building elements - Part 2: Impact sound insulation , 2020. [11] E.P.B. Reynders, P. Wang, C. Van hoorickx, and G. Lombaert. Prediction and uncertainty quantification of structure-borne sound radiation into a di ff use field. Journal of Sound and Vibration , 463:114984, 2019. [12] C. Crispin, D. Wuyts, and A. Dijckmans. Thickness-resonance waves in underlays of floating screed. In Proceedings of 50th International Congress and Exposition on Noise Control Engineering, Inter-Noise 2021 , Washington, D.C., August 2021.