Calibration of the ISO tapping machine for finite-element prediction tool on a wooden-base floor

Abstract

One important challenge of the wooden constructions is to achieve a high quality of acoustic insulation, especially decreasing the impact noise in the low-frequency range. In order to avoid over-designed solutions and expensive experimental tests in the design phase, reliable prediction tools are called for. This article is an initial investigation of modeling the ISO standardized tapping machine on a cross-laminated timber floor, using finite element method. The wooden-based floor was first calibrated in terms of its dynamic properties. The influence of the material properties of the cross-laminated timber floor was discussed. The force generated by the tapping machine was then introduced in the established cross-laminated timber model. The model was finally validated by comparing the simulation results with the measured accelerations.

Keywords

Timber constructions acoustic comfort sound insulation impact noise finite element

Introduction

During the last decades, the wooden constructions have steadily increased in the markets due to their many advantages, such as sustainability, thermal performance, and so on. However, the sound insulation issue is a big challenge for these kinds of constructions. Even when these buildings fulfill the requirements of the ISO sound insulation standards,^1,2 the impact noise, especially at low-frequency range, is often a source of nuisance for the residents in the wooden buildings.^3–5 One reason for that is that the evaluation standards were initially designed for the heavy constructions. And without enough knowledge about the wooden constructions, the standards are directly applied to wooden buildings. Moreover, not enough attention is paid to the low-frequency noise in the evaluation standards, as the standardized evaluation frequency range is from 100 to 3150 Hz.⁶ With an adaptation term, the evaluation frequency range could go down to 50 Hz; however, frequencies below that value are believed to have a big influence on how inhabitants perceive the vibroacoustic performance of those buildings, as the first eigen-frequencies lie around 10–20 Hz. As consequence, the objective evaluations do not always correlate with the subjective evaluations. Until now, most improvements of the acoustic insulation of wooden constructions are based on the empirical models, engineers’ experience, and experimental tests which are always expensive and time-consuming. To that end, the use of reliable low-frequency range impact sound insulation prediction tools in design phase would be of great importance in order to decrease the price and time cost of the wooden construction.

To achieve that, the first question which arises is how to model the standardized excitation source in numerical model. The current standards^2,6,7 employ a five-hammer tapping machine as an excitation source. To describe this source, the Fourier series can be used⁸

\begin{matrix} F (t) = F_{0} + 2 \sum_{n = 1}^{\infty} F_{n} \cos (ω_{n} t) \end{matrix}

(1)

where $ω_{n} = n 2 π f_{S}$ and $F_{n}$ are the Fourier coefficients which can be obtained by

\begin{matrix} F_{n} = \frac{2}{T} \int_{0}^{T} F (t) \cos (ω_{n} t) dt = \frac{2}{T} I = 2 m v_{0} f_{S} \end{matrix}

(2)

where $I$ represents the momentum that results from the impact hammer mass $m$ striking the floor surface in question with velocity $v_{0}$ . It should be noticed that when using this method, the hammer will be considered not to bounce back when a resilient layer is added on top of the floor base. This drawback was resolved by adding an elastic layer in the analytical model.⁹ Since the impact insulation improvement of the floor depends on the load-bearing base, these two mentioned analytical models are only suitable for concrete-base floors, meaning low admittance floors. In contrast to concrete-base floors, wooden-base floors are not homogeneous structures with high impedance. Hence, corrections should be made to adjust the model to wooden floors. In another work,¹⁰ the excitation stiffness was taken in account into the analytical model for the wooden structure. But the model just considered the whole mass of the floor, whereas the participation of the floor mass varies depending on the different eigen-modes. Moreover, the energy dissipation of floor was ignored in this model. Another improvement was made by taking into consideration the local and global impedance in the model.¹¹. Nonetheless, the discrepancies induced by the unknown parameters of the relative velocity between the bounced hammer and the vibrating floor were observed when comparing the analytical results and the experimental data. This problem was addressed in Rabold et al.¹² Just a slight improvement was achieved when mimicking measurement results, even though the relative velocity between the hammer and the vibrating floor was introduced in the model. Until here, each improvement increased the complexity of the analytical model, whereas no satisfying force prediction results were achieved.

Therefore, the study reported here aims at ultimately developing a low-frequency impact sound numerical prediction tool by means of finite element models in a larger project. To achieve that, a cross-laminated timber (CLT) floor base was first investigated by means of manual calibration and stochastic calibration. Subsequently, the ISO tapping machine was introduced in the numerical model by a simplified method. The introduced modeling method was validated by means of the experimental tests.

Experimental

Specimen description

The wooden structural element under investigation is a 4 m × 1.5 m CLT panel obtained from Les Chantiers Chibougamau Ltée in Chibougamau, Province of Quebec, Canada. The specimen was made of Canadian black spruce of machine stress rated grade E1 (L 1950Fb et T No. 3/Stud), SPF (spruce-pine-fir) machine stress rated (MSR) parallel layers, and SPF Stud perpendicular layers in parallel layers and visual grade No. 3 in perpendicular layers with density of 520 kg/m³. This species accounts for 12% of the Canada’s total softwood inventory and is desirable for its straight lines, lightweight, fiber density, and dimensional stability.

Measurement set-up

At the initial stage, the 4 m × 1.5 m CLT panel was assembled with another same dimension CLT panel with help of a nailed lath. This assembled 4 m × 3 m floor was mounted on a console of a step sound laboratory which has a sound sending upper volume and a sound receiving lower volume. However, the dynamic properties of this entire CLT floor were difficult to extract due to the weak connection of the two assembled CLT slabs and intricate boundary condition set-up in the console of the step sound laboratory. As a result, only one half CLT slab was investigated to avoid the uncertainties induced by the connection between the panels. The CLT panel was placed on top of the steel-I beams at the shorter sides of the CLT (cf. Figure 1) with the two longer sides of CLT set free (SFSF boundary conditions).

Figure 1.

Connection between two CLT slabs (left) and tested CLT specimen with SFSF boundary conditions (right).

Experimental modal analysis (EMA) tests were conducted on the CLT panel in order to extract the dynamic properties of the specimen. A predefined mesh grid (cf. Figure 2) was drawn on top of the CLT to determine the positions of the excitation and the accelerometers. The resolution of mesh grid depends on the frequency range of interests. The uni-axial accelerometers (Brüel & Kjær Accelerometer Type 4507 001) were placed on the point 10, 11, 13, 17, and 24 (red dots), and the impact hammer (Brüel & Kjær impact hammer Type 51994) was roved around all the points except the shorter edges where zero displacements were assumed. The accelerometers and the impact hammer were connected to the Brüel & Kjær LAN-XI Type 3050-A-060 6 channels front-end. The acceleration signals and the force signals were collected by the Brüel & Kjær PULSE Labshop. The eigen-frequencies, the mode shapes, and the corresponding damping ratios were obtained with help of the Brüel & Kjær PULSE Reflex.

Figure 2.

Predefined mesh on CLT panel.

After the dynamic properties and the dynamic response of the CLT were extracted, the standardized ISO tapping machine was placed at point 14 on top the CLT to investigate the vibration behavior of the CLT plate. The collected information was used to develop prediction method of the ISO tapping machine. At the initial stage, only one hammer was taken into account. The other four hammers were tapped under the tapping machine during the tests. One accelerometer was placed just near the striking hammer. And another two accelerometers were attached on point 13 and point 14 to record the accelerations of the floor in order to eventually retrieve the force (Figure 3).

Figure 3.

Measurement of tapping machine on top of CLT.

Modelization and calibration

Model description

A finite-element (FE) model was developed under the environment of the commercial FE software Abaqus.¹³ The five-ply CLT described in the specimen description section was modeled by a five fully tied layers CLT model, shown in Figure 4.

Figure 4.

FE model of the 5-ply CLT.

The dimensions of the FE model were set to be 4 m × 1.5 m, same as the dimension of the CLT specimen. The material properties of each layer of CLT were first collected from the literature.¹⁴ And the material property orientation of each layer was 90° rotated with one another to model the cross-laminated layers of CLT. All the displacements and the rotations of the shorter edges at bottom of CLT model were constrained to mimic the SFSF boundary conditions. A 20-node quadratic brick, reduced integration elements were assigned to the entire model. The dimension of the mesh was 0.1 m × 0.1 m to ensure the accuracy of the highest frequency of interest. The CLT was calibrated at the first place as the accuracy of the base model influences the accuracy of the method/element introduced later in model.

CLT calibration

In order to calibrate the CLT slab, the elastic material properties were collected from literature and then tweaked until a good match of the simulated eigen-frequencies and the mode shapes was achieved. The CLT model was calibrated by means of the normalized relative frequency difference (NRFD) and modal assurance criterion (MAC), which are defined by

\begin{matrix} NRFD (%) = \frac{| f_{s i m} - f_{m e s} |}{f_{m e s}} \end{matrix}

(3)

where $f_{s i m}$ (Hz) is the simulated eigen-frequency and $f_{m e s}$ (Hz) is the measured eigen-frequency

\begin{matrix} MAC = \frac{{| {(Φ_{i}^{s i m})}^{T} (Φ_{j}^{e x p}) |}^{2}}{{(Φ_{i}^{s i m})}^{T} (Φ_{i}^{s i m}) | {(Φ_{j}^{e x p})}^{T} (Φ_{j}^{e x p})} \end{matrix}

(4)

where $Φ_{i}^{s i m}$ is the ith simulated mode shape and $Φ_{j}^{e x p}$ is the jth measured mode shape. The NRFD aims at evaluating the relative difference of the simulated and the measured eigen-frequencies, whereas the MAC focus on estimating the similarity of the simulated and the measured mode shapes. These two criterions assure the accuracy of the eigen-frequencies, meanwhile, keeping the simulated mode order in the measured mode order. The calibrated elastic properties are shown in Table.1.

Table 1.

Properties of CLT used in the calibrated FE model.

E ₁	E ₂	E ₃	G ₁₂	G ₁₃	G ₂₃	$υ$ ₁₂	$υ$ ₁₃	$υ$ ₂₃	$ρ$
8500	3800	3800	800	70	61.8	0.44	0.3	0.3	520

FE: finite-element; CLT: cross-laminated timber.

Stiffness parameters have the unit of MPa, and the density is given in kg/m³.

The simulated eigen-frequencies and the measured eigen-frequencies are reported in Table 2.

Table 2.

Measured and simulated eigen-frequencies of the bending modes and the measured corresponding damping ratios.

Mode	Measured eigen-frequency (Hz)/damping ratio (%)	Simulated eigen-frequency (Hz)
1	19.8/4.9	22.2
2	32.2/3.6	33.7
3	56.7/3.2	56.2
4	73.8/3.2	72.9
5	91.0/3.3	105.0
6	125.5/2.8	118.7
7	139.8/1.7	146.8
8	170.4/1.2	170.8

The NRFDs of the simulated and the measured eigen-frequencies are shown in Figure 5, and the MAC number is shown in Figure 6.

Figure 5.

NRFD of the simulated and the measured eigen-frequency.

Figure 6.

Cross-MAC for the FE CLT model.

From the NRFD and the MAC, it can be observed that lower than 100 Hz, all the eigen-frequencies were well captured. The NRFD stays in a relative low range. And the MAC numbers are close to one for the first four eigen-frequency. After the eigen-frequency and the eigen-mode were calibrated, the simulated frequency response function (FRF) was then investigated to increase the stability and reliability of the to-be developed model. A steady-state analysis was performed in Abaqus to obtain the FRFs of the CLT model at the point 11, 13, 17, and 24. The damping ratio extracted from the experimental tests (shown in Table 2) was introduced in the model by means of the direct modal damping.

From Figure 7, it could be seen that the first four simulated resonances (red curves) were captured compared with the measurements (blue curves). However, frequency shifts could still be observed in Figure 7. These discrepancies are caused by the relative frequency differences in the simulated eigen-frequency and the measured eigen-frequency. This implies that the uncertainties induced by the material properties should be further investigated in order to obtain a more accurate FRF. Moreover, the variation of the material properties of wood are important and can vary significantly. To that end, the variation of the elastic material properties of the CLT was considered in the CLT model through a stochastic approach. The sensitivity analysis of the material properties of CLT indicates that the Young’s modulus in the longitudinal direction of each wooden layer of CLT model has the largest influence on the output results.^15,16 Therefore, only the influences of the Young’s modulus in longitudinal direction and the shear modulus were investigated in this work, since there is no available data base of CLT material properties. The Monte Carlo simulation was performed to evaluate the influence of the Young’s modulus in longitudinal direction and shear modulus of each layer on the dynamic response of CLT model.

Figure 7.

FRFs of the point 11, 13, 17, and 24. The blue curves are the measurement results and the red curves are the simulation results.

In Figure 8, there are 40 realizations in each subfigure. The variation of the Young’s modulus in the longitudinal direction E₁ of each layer obeys the uniform distribution ranging from 7000 to 9000 MPa and shear modulus G₁₂ obeys the uniform distribution ranging from 700 to 900 MPa. The Young’s modulus and the shear modulus generated by MATLAB¹⁷ were then exported in Abaqus. Forty calculations at different points were realized by Abaqus with help of Python script. It could be observed from Figure 8 that the envelops of the variations of the Young’s modulus are larger than that of the variations of the Shear modulus, due to the fact that the Young’s modulus has more significant impact on the dynamic behavior of the CLT than the shear modulus. Consequently, a slight change of the Young’s modulus can induce an important change in the dynamic response of the CLT. It should be mentioned that all the probability distribution functions are assumed to follow a uniform distribution which may not be the reality. Thus, a more accurate material description would help improving the results. Furthermore, the simulation just variated one elastic constant each time so that it is difficult to have a global evaluation of how the different mechanical constants in different directions influence the FRFs or the vibration behavior of the floor. Therefore, it would be more reasonable to change the different material constants at one time in order to evaluate how the different material properties affect the behavior of the slab.

Figure 8.

FRFs of the point 13 and 24. The blue curves are the measurement results and the red curves (40 realizations) are the simulation results with different material properties.

ISO tapping machine modelization

Once the CLT was calibrated in terms of the eigen-frequencies, the eigen-modes as well as the FRFs, the force generated by the ISO tapping machine was introduced in the model. In Negreira and Bard,¹⁸ an alternative method to modeling the tapping machine was proposed. It was proven that the time domain method and the frequency domain method are equivalent in mimicking the tapping machine, the frequency domain approach being more efficient and less time-consuming. Following that conclusion, the frequency domain method was performed to calibrate the model with the experimental data.

In order to avoid the complicated mathematical calculation to describe the force generated by the tapping machine, another modal method¹⁹ was employed here aiming at characterizing the force in question. It is known that the acceleration of the CLT floor at point 10 and point 13 could be written as

\begin{matrix} A_{10} = H_{10, 14} \cdot F_{14} \end{matrix}

(5)

\begin{matrix} A_{13} = H_{13, 14} \cdot F_{14} \end{matrix}

(6)

where $A_{10}$ and $A_{13} 1$ are the accelerations at point 10 and point 13, $H_{10, 14}$ and $H_{13, 14}$ are the FRFs of points 10–14 and points 13 and 14, and $F_{14}$ is the force generated by one hammer of the ISO tapping machine at the point 14. Therefore, the input force generated by the tapping machine could be written as

\begin{matrix} F_{14} = \frac{A_{13}}{H_{13, 14}} \end{matrix}

(7)

where $H_{13, 14}$ could be obtained from the previous model of the CLT. After the force is obtained from equation (7), the acceleration at point 10 can be deduced by the FRF of the points 10–14 given by the CLT model and the input force at point 14. It should be noted that since the calculation of the FRF of the CLT was restricted up to 100 Hz, the measured acceleration should be filtered up to 100 Hz to keep the same frequency content as the input FRF when deriving the input force according to equation (7).

In Figure 9, the simulated acceleration of the vibration of the CLT correlates better with the measure acceleration of the CLT around 60 Hz. One of the possible reasons is that the simulated FRFs of the CLT fit better with the measured FRFs of CLT from 30 to 70 Hz. The frequency shifts around 30 Hz may be caused by the frequency shifts of the simulated FRFs lower than 30 Hz in Figure 7. Furthermore, it was found that the discrepancies of the simulated acceleration and the measured acceleration became larger compared with the discrepancies when mimicking the measured eigen-frequencies and FRFs in the previous step. It implies that the uncertainties of the previous simulation could propagate to the next step of simulation. Another source of error could be the ill-conditioned boundary condition of the CLT model when mimicking the real boundary condition (not constraining all the displacement and the rotations on the boundary line of the CLT model). Finally, another error source could also be the indirect way to estimate the input force. Therefore, when moving on to the more complex/realistic case of having the five hammers active in the tapping machine, it would be better to measure directly the input force generated by the tapping machine rather than the accelerations in order to obtain a more accurate modeling result.

Figure 9.

Acceleration of CLT at point 10 in narrow band (left) and acceleration of CLT at point 10 in 1/3 octave band (right), Measured acceleration in blue and simulated acceleration in red.

Conclusion

The CLT was calibrated by means of the eigen-frequencies, eigen-modes, and FRFs. A quite good agreement was achieved in terms of the FRFs of the CLT at different excitation positions in the low-frequency range, that is, 100 Hz. But there were still some frequency shifts in the simulation results higher than 100 Hz. The boundary conditions are one important factor on the modeling results. Therefore, further investigation should be done in order to mimic the real boundary conditions. Another important factor is the material properties which also have a significant impact on the simulation results. But due to the lack of the reliable material properties of the CLT floor and the complexity (inhomogeneity) of wood, it was difficult to achieve a good simulation of CLT at higher frequency range. Thereby, although the simulated FRFs give in broad strokes an idea of the behavior of the floor, they did not describe it in a good manner in the full-frequency range of interest. To improve that, the variations of the different elastic constants were further investigated by stochastic simulations. Moreover, the dynamic response of the wooden structure is the coupling effect of all the elastic constants in different directions so that the variations of the different elastic constants in modeling wooden structure should be taken into consideration in the future work.

The ISO standardized impact tapping machine was introduced in the previous established CLT model. The simulation results show good correlation at certain frequencies. But there were still some unignorable discrepancies in the simulations and measurements. These discrepancies were initially slightly different in the FRFs modeling but then they were propagated and accumulated in the following acceleration simulation step. Also, instead of measuring the input force, measuring the acceleration to obtain the input force could also generate more uncertainties in the simulations.

For the future work of developing the predictive tool of the tapping machine, it should first obtain an accurate base model. To achieve that, the boundary conditions and the influence of different material properties should be further investigated. Different measurement techniques may be employed depending on the frequency range of interests, in that manner controlling the input force put into the slab and thus enabling the development of different prediction tools for different frequency ranges.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors are grateful to Natural Sciences and Engineering Research Council of Canada for the financial support through its IRC and CRD programs (IRCPJ 461745-18 and RDCPJ 524504-18) as well as the industrial partners of the NSERC industrial chair on eco-responsible wood construction (CIRCERB).

ORCID iDs

Delphine Bard-Hagberg

Juan Negreira

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