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Abstract

The trend of increasing the ratio of environmentally friendly solutions in the building industry is on the rise. We increasingly encounter solutions that utilise loose-form materials or, to improve thermal insulation and acoustic properties, typically multilayer lightweight systems. However, it has been proven that predicting the impact of loose-form blown-in materials on sound insulation properties can be a matter of debate. Blown-in materials with a higher bulk density tend to connect the individual layers of lightweight walls mechanically. It is not trivial to accurately determine the stiffness of the blown-in material after application in the building structure. This is especially important when creating prediction models. This paper deals with the determination of the dynamic stiffness of soft materials using the light load plate and two degrees of freedom (2-DOF) method. Since the case of evaluating resilient materials at low loads is outside the recommended range of boundary conditions as defined by EN 29052-1, this extensive work gradually analyses the relevance of applying an alternative measurement method. It discusses the effect of compression of a soft material on its actual dynamic stiffness, the effect of the excitation form of the light load plate, and presents a preliminary measurement approach. Numerical models support individual analyses and measurements. Even though testing fibrous materials in loose form is a very challenging task, a significant change in stiffness under the influence of loading has been demonstrated.

Keywords

dynamic stiffness measurement method loose material simulations light weight load plate

Introduction

Current trends in modern building design are increasingly influenced by green and sustainable solutions, which are one of the means of reducing the carbon footprint of the construction industry. Therefore, we can see an increasing application of renewable materials other than wood, for example soils and wastes as laterite soil, brick wastes, rice husk ash, burnt refuse ash, fly ash, periwinkle shell powder, earthworm cast, pulverised burned clay, fibrous materials as straw, bamboo, coconut fibres, refia palm fibres, wood shavings, wood tendons, cotton fibre, sisal fibre, asbestos fibre, sisalfibre, asbestos fibre and many others.^1,2 The implementation of bulk materials to improve acoustic, fire-resistance and thermal properties is also very popular.^3
–5 Every new material and construction solution brings new challenges in terms of proper design concerning building acoustics. A specific case is the application of blown-in fibrous materials in loose form in lightweight structures. As has already been shown, blown-in solutions have an impact on the sound insulation properties of walls.^6
–11 Just like acoustically absorbing materials such as mineral wool and the like, they fulfil the function of a sound-absorbing material for damping cavity resonance frequencies. By increasing the density, which can be achieved by increasing the blown-in batching pressure, we also increase the mass of the walls. In addition to the two advantages of the blown-in solution, a potential drawback is the mechanical coupling between the individual layers of the lightweight partition structure due to compression, which may impair the intended decoupling effect. Previously, it also acted as a spring, albeit a much softer one. As a result, the mass-spring resonance frequency remained within the relevant frequency range, that is, above 50 Hz. But how to determine the stiffness of this spring? Loose material types cover a complex group of different kinds of materials, from aggregates to the finest fibres. Therefore, we cannot generalise any existing method to all types of these materials. For example, granular loose materials in dry form are not considered, even though for each of the potential applications, they must be enclosed in a confined structure (no lateral expansion occurs when compressed). In case of materials that are compressible, soft, have non-uniform dimensions (fibres), or can hold their shape under a certain degree of compression, it is possible to speak of loose form materials that behave like a spring (in this article, we focussed on straw and cellulose material). Depending on the forming factor of loose form material, it should be possible to determine the stiffness of such material if it achieves the forming state, when the material keeps its shape (reminiscent of briquettes).¹²

In building acoustics, however, the procedures used do not directly define a method that would apply to these materials or their applications. For the assessment of materials applied from the point of view of building acoustics, we know EN 29052-1 and ISO 9052-1.^13,14 However, this standard is not directly suitable for determining the dynamic stiffness of materials in wall construction cavities, but rather for the application of materials in floors. In standard procedure, it ~~is assumed that the material is under the load of a screed~~ system, with the sample being subjected to 200 kg/m². It must not be used for materials with a load below 0.4 kPa (which is the most common application of fibrous blown-in loose form materials) and for loads above 4 kPa.A paper has already been published on this subject, which points to the need to develop a measurement procedure that takes into account the loading of samples with a low surface weight, which, in addition to the approach discussed in this article, is also a problem in predicting the stiffness of ETICS systems.¹⁵

This method would be beneficial for interpreting measurement results and refining predictive models, specifically by focussing on and enabling a direct confrontation between the dynamic stiffness of the thermal insulation and the resulting sound insulation of the external envelopes. The sound insulation is, among other factors, significantly influenced by the anchoring system and the uneven distribution of the adhesive mortar between the thermal insulation layer and the base wall. This refinement is particularly important, given that conventional prediction models do not account for the impact of compressing thermal insulation layers. However, when expressing results in one-third octave resolution, this may not constitute a fundamental concern.¹⁶ Motivated by the text above, this article presents a case study aimed at finding a way to objectively determine the actual dynamic stiffness of porous flexible materials at low loads. The article is organised as follows. In the beginning, the article focuses on the effect of load plate mass on dynamic stiffness determination. This is followed by sections where compression of resilient materials and derivation of the compression effect on the structural stiffness and flow resistivity change are discussed. In the following section, the concept of a 2-DOF mass-spring system for determining dynamic stiffness is introduced. The idea is supported by numerical verification through the results of performed parametric studies, and the applied measurement setup is described. At the end of the paper, the proposed measurement procedure is applied to two selected types of loose-form materials only.

Effect of load plate mass on dynamic stiffness determination based on single mass procedure

The method described in standard EN 29052-1¹³ was developed for materials used under floating floor dynamic stiffness determination, especially for materials with smooth surfaces used in a continuous layer. The standard proposed method may not be applied for materials with loadings lower than 0.4 kPa (materials in wall linings) or higher than 4 kPa (machinery foundation). This doubt motivated us to investigate how a change in the loading of a resilient material affects its actual stiffness. Among other things, this can be applied when trying to understand the impact of loose-form absorbing material on the acoustic properties of the wall. Based on the research presented in publications,^17,18 we assumed that the materials would exhibit lower stiffness when subjected to lower loads. In the study mentioned above, six thin, predominantly resilient layers were tested using load plates weighing 2, 4, 6, and 8.4 kg.

The presented case study in this paper focuses on the effect of the load plate’s weight on the determined dynamic stiffness of selected resilient materials commonly used in acoustic applications (see Table 1 and Figure 1).

Table 1.

Resilient layers under examination.

Material	Dimensions in m	$ρ$ in kg/m³
White EPS	$0.2 \times 0.2 \times 0.04$	14.0
Blue XPS	$0.2 \times 0.2 \times 0.04$	34.6
Mineral Wool 1	$0.2 \times 0.2 \times 0.02$	65.8
Mineral Wool 2	$0.2 \times 0.2 \times 0.01$	206.2
Wood fibre	$0.2 \times 0.2 \times 0.01$	223.6
PU elastomer	$0.1 \times 0.1 \times 0.025$	207.6

Figure 1.

Photo of chosen resilient layers. (a) White EPS (EPS); (b) Blue XPS (XPS); (c) Mineral Wool 1 (MW 1); (d) Mineral Wool 2 (MW 2); (e) Wood Fibre (WF); (f) PU Elastomer (PUE).

These materials were tested following the procedure and measuring set-up defined in EN 29052-1,¹³ with the difference that a thin plaster layer was not applied between the steel plate and the tested material. Likewise, the contact between the base and the tested material was not sealed. However, since the goal of this experiment was not to determine the exact dynamic stiffness, but to determine the effect of the load change on the relative stiffness and resonance response change, this fact should not play a fundamental role.¹⁹ The resulting apparent dynamic stiffness values were determined according to the well-known relationship:

\begin{array}{l} s_{t}^{'} = 4 π^{2} m_{t}^{'} f_{r}^{2} \end{array}

(1)

Where m’_t is the total mass per unit area in kg ⋅ m⁻² and f _r is the extrapolated resonant frequency in Hz.¹³ In comparison to,¹⁸ here presented measurements were subsequently compared with an alternative setup in which the weight of the load plate was changed from m_lp1 = 7.665 kg to m_lp2 = 1.263 kg (load plate dimensions $h \times$ 0.2 m $\times 0.2$ m; see Figures 2 and 3). Following the measurement procedure presented in,²⁰ a logarithmic sine sweep signal was used to excite the samples, and the measured values were extrapolated to zero force amplitude. The excitation source was the TIRAvib 50018 shaker charged by the TIRA E60 analogue power amplifier, and the frequency response was recorded by an acceleration sensor, Dytran 3055B2, and a force sensor, Dytran 1051V1, fed through National Instruments four-channel signal acquisition, NI 9234, mounted in NI USB-9165. The entire experiment was evaluated using an automated routine in MATLAB.

Figure 2.

Scheme of two alternatives of measurement setup used for dynamic stiffness determination of chosen resilient layers. (a) Load plate 0.2 m $\times 0.2$ m of mass m_lp1; (b) load plate 0.2 m $\times 0.2$ m of mass m_lp2.

Figure 3.

Photo of an example of a part of the measuring setup - a variation of the load plate with dimensions 0.2 m $\times 0.2$ m. The measured material in the image is designated in this article as Mineral Wool 2. (a) Load plate of mass m_lp1 = 7.665 kg; (b) load plate of mass m_lp2 = 1.263 kg.

Compression of resilient materials

Since the effect of compressibility of selected conventional materials will be discussed in the following sections, a test of changing their thickness by applying load plates of different weights and areas was performed (see Table 2). The measuring set-up consisted of a laser (optoNCDT ILD2300-2, Micro-Epsilon; Figure 4(a)) controlled by software (sensorTOOL V2.0.1). Four steel plates with additional weights were used as load plates (Figure 4(b) and Table 6). The extra weights were placed in the middle of each of the four load plates. These additional weights were placed on top of each other, and they were used to increase the weight of the load plate by 5.1 kg in 10 steps (number of weights). The compression of the measured elements was determined at a distance of 50 mm from the centre of the tested sample.

Table 2.

Load plate description.

Load plate	Mass in kg	Side length in m
No. 1	7.665	0.200
No. 2	1.263	0.200
No. 3	0.863	0.190
No. 4	0.683	0.148

Figure 4.

Compression determination measurement setup. (a) Compression test with three additional masses placed on the load plate No. 4 on the Mineral Wool 1. Laser is identifying the change of distance caused by increased load; (b) load plates 1–4.

Derivation of compression effect on the structural stiffness and air flow resistivity change

As has been shown, the interpretation of the behaviour of elastic, soft, porous materials may not always be unambiguous. In this section, using the model presented in,²¹ in this article called Kraak’s Extended Model (KEM), we will try to express the effect of compression on the resulting stiffness of solid structure of resilient material (s’_s in Pa/m) and flow resistivity ( $σ$ in Pa ⋅ s/m³). Based on the test presented in the previous section, theoretical transfer function curves were fitted depending on complex parameters to the measurements. Those parameters were used to predict s’_s and $σ$ (see Table 3). In this case, two load plates m_lp1 and m_lp2 were used. According to this model, materials MW1 and MW2 were analysed as representatives of open-cell structure materials with a relatively low flow resistivity property.

Table 3.

Quantities and their values applied in KEM based on.²¹

Quantity	Symbol	Value	Unit
Upper plate made of steel
Mass density	$ρ$	7893.7	kg/m³
Young’s modulus	E	$2 \times 10^{11}$	N/m²
Poisson’s ratio	$ν$	0.33	1
Thickness	h	0.0242	m
Air inside the underlay
Mass density	$ρ_{0}$	1.2	kg/m³
Static air pressure	p ₀	$1013 \times 10^{2}$	Pa
Adiabatic exponent	$κ$	1.4	1
Underlay of porous material
Porosity	$ϵ$	0.96	1
Structure factor	$χ$	1.0	1

The KEM discussed in²¹ takes into account a flexible upper plate instead of a rigid one. Bending waves in the upper plate are described by equation (2).

B △^{2} \hat{w} = ({m^{'}}_{t} ω^{2} - {s^{'}}_{s}) \cdot \hat{w} - \hat{p} .

(2)

△ \hat{p} = i \frac{ω γ}{κ p_{0}} \hat{p} - i \frac{ω γ}{d ε} \hat{w}

(3)

Equation (2) together with the modified Kraak’s equation (3), which is of second order, constitutes a system about two unknowns – complex amplitude of pressure $\hat{p}$ and complex amplitude of deflection $\hat{w}$ . The modification of Kraak’s original approach involves allowing for the inertia of the air inside the underlay and utilising adiabatic changes instead of isothermal ones. For such a system to have a unique solution, six boundary conditions must be fulfilled. These conditions are related to symmetries near the centre, displacement near the centre, free and open or closed boundary. The article²¹ shows that soft porous materials exhibit a significantly high and steep jump in dynamic stiffness $s^{'}$ around a particular value of flow resistivity $σ$ . A sudden change in the behaviour of air inside the pores around the mentioned critical flow resistivity explains this phenomenon. The airflow in the pores stops, and the pores transform into a kind of elastic cushion.

It has also been shown that the admittance $L_{Y}$ of a system with a flexible upper plate strongly depends on the size of the system in.²¹ Therefore, the dynamic stiffness obtained by measuring on a standardised specimen cannot be automatically applied to room-sized systems.

Numerical verification of 2-DOF mass-spring system for dynamic stiffness determination

As it will be demonstrated later, the use of excitation by a vibration source from the top of the system has its limitations when using a low-mass load plate. This motivated us to focus on a measurement setup in which the system would be excited from below via the so-called baseplate. The EN 29052-1,¹³ which is referred to several times in this article, also describes this form of excitation. However, unless the base plate is attached directly to the source (a large shaker, as is recommended in¹⁵), then it is debatable whether the equation (1) can be used. If a relatively smaller shaker is used, the base plate must be suspended from an additional spring, and it is no longer a 1-DOF system. It can be assumed that if the difference between the mass of the base plate and load plate, but especially the difference in the stiffness of the individual springs in the system, is significant (suspension vs resilient material), the effect of the second degree of freedom may be very small. In this section, we numerically focus on verifying the analytical relationship for lumped model boundary conditions, as described by Blevins and Plunkett^22,23 (see equation (4)). A simple scheme of this model is presented in the Figure 5.

\begin{array}{l} f_{i} = \frac{1}{2^{\frac{3}{2}} π} {\frac{k_{1}}{m_{1}} + \frac{k_{1}}{m_{2}} + \frac{k_{2}}{m_{2}} \\ {\mp {[{(\frac{k_{1}}{m_{1}} + \frac{k_{1}}{m_{2}} + \frac{k_{2}}{m_{2}})}^{2} - 4 \frac{k_{1} k_{2}}{m_{1} m_{2}}]}^{\frac{1}{2}}}}^{\frac{1}{2}} \end{array}

(4)

Figure 5.

Simplified 2-DOF scheme.

f _i expresses the first and second resonant frequency (depending on the sign used in the relationship) of the 2-DOF system where k₁ and k₂ are the spring stiffness in N/m, m₁ and m₂ are the masses in kg.

We then confronted this model with the SimSCAPE engineering tool for 1D physics tasks and a numerical model for 3D physics tasks based on FEM (Comsol Multiphysics). Thus, two groups of parametric studies were performed focussing on: 1. the effect of the suspension stiffness (k₂) on the resulting resonant response of the system; 2. verification of the effective area of the load plate if its area differs from the area of the tested resilient pad in a linear environment.

Parametric study 1

Three calculation methods were used in total in this parametric study. The numerical task was implemented in Simscape using a physical component–based modelling approach to simulate the basic 1D problem in the time domain. The solution is obtained numerically using Simscape’s integrated solver for differential-algebraic equations.

More specifically, a parametric study was created for the case where the 2-DOF system includes two rigid mass bodies (load plate and base plate) of constant mass m₁ = 1.263 kg and m₂ = 8.324 kg. The dynamic stiffness of the resilient layer in this case was also constant s₁ = 20 MN/m³. The stiffness of the second spring k₂ was in the range of [ $4 \times 10^{3}$ ; $1.2 \times 10^{5}$ ] with step $4 \times 10^{3}$ in N/m. The damping coefficients c₁ and c₂ were constant for both springs (10 N⋅ s/m).

In parallel, the model for comparison was created in the COMSOL Multiphysics environment. Here, the 3D geometry of the physical model was created. The load plate was defined as a steel plate with dimensions $0.2 \times 0.2 \times 0.004$ m and density of $ρ$ = 7893.7 kg/m³. The base plate was modelled as an elastic material (concrete block) of dimensions 0.2 m × 0.2 m $\times 0.01$ m and density of $ρ$ = 2081 kg/m³.

For this purpose, a numerical eigenfrequency analysis was performed to identify the natural frequencies and corresponding mode shapes of the coupled system consisting of the load and base plates, taking into account the material properties, geometry, and boundary conditions. Since this is an example focussing on the response of the system only at low frequencies without considering bending modes of the plates, a numerical mesh of density in the range from 0.002 to 0.01 m was sufficient to satisfy the Nyquist criterion (see Figure 6). Both methods mentioned above were compared to Blevins’ model.

Figure 6.

Example of mesh density of the FEM model used for case study 1. (a) Side view with schematic marking of boundary conditions; (b) axonometric view.

Parametric study 2

Since blown-in material must be placed in a rigid frame, the top plate of the experimental setup needs to be slightly smaller than the sample to ensure a gap to the surrounding frame, thereby preventing mechanical contact during the measurement. Here, we aimed to numerically demonstrate which area should be considered the effective spring area using FEM modelling. The question was whether it was necessary to consider the contact area of the resilient material and the load plate, or their adequate combination. It should be noted here that these results should be taken with caution since this problem was solved in a linear environment and the stiffness of the elastic layer was not dependent on either the load or the change in flow resistivity due to the load or the overlap. In this case, the geometrically identical model was used as described before, with the difference that the load plate side dimension varied in the range from 0.001 to 0.02 m. For this case study, all other parameters were constant (see Table 4).

Table 4.

Variables used in specific case studies.

	Case study 1	Case study 2
Load plate thickness d₁ in m	0.004	0.004
Load plate edge length a₁ in m	0.2	0.05–0.2
Base plate thickness d₂ in m	0.010	0.010
Load plate density $ρ$ ₁ in kg/m³	7893.7	7893.7
Base plate density $ρ$ ₂ in kg/m³	2081	2081
Damping coefficients c₁ and c₂ in N⋅ s/m	10	10
Resilient layer dynamic stiffness s₁ in MN/m³	20	0.2; 1; 2; 4
Suspension stiffness k₂ in N/m	4 × 10³ to 1.2 × 10⁵	4.41 × 10³

The resulting dynamic stiffness was derived from the relationship (equation (5)) as follows:

\begin{array}{l} k_{1} = \frac{16 \cdot π^{4} \cdot f^{4} - 4 \cdot π^{2} \cdot f^{2} \frac{k_{2}}{m_{2}}}{4 \cdot π^{2} \cdot f^{2} \cdot (\frac{1}{m_{1}} + \frac{1}{m_{2}}) - \frac{k_{2}}{m_{1} \cdot m_{2}}} \end{array}

(5)

Experimental application of 2-DOF mass-spring system approach for dynamic stiffness determination

In this section, two applications of the 2-DOF system according to the schemes presented in Figures 5 and 7 will be presented.

Figure 7.

Simplified 2-DOF measurement setup scheme for loose material testing.

The first experiment was performed on conventional resilient materials, as described in Table 1. The measurement set-up consisted of a base plate, which was created of a concrete block of dimensions 0.25 m × $0.25$ m × 0.07 m of mass 8.570 kg suspended by rubber ropes. The total suspension stiffness was determined as $k$ = 6.0494 × 10³ N/m. Two load plate variants were used (load plate of dimensions 0.19 m × 0.19 m × 0.003 m and mass m_lp3 = 0.868 kg; dimensions 0.25 m × 0.25 m × 0.003 m) and mass m_lp5 = 1.476 kg (see Figure 8). The excitation source, sensors, as well as DAQ were used in the same way as in the case of experiments introduced before.

Figure 8.

Photo of an example of a part of the 2-DOF measuring setup. (a) Mineral Wool 2 testing with load plate of dimensions 0.25 m × 0.25 m; (b) Straw loose material testing with load plate of dimensions 0.19 m × 0.19 m.

The second experiment was focussed on testing fibrous materials in loose form. As discussed in the authors’ previous publications, the dynamic stiffness of loose materials applied in lightweight double walls directly influences their resonant response related to the mass-spring-mass phenomenon.^10,11,24 However, the correct way to determine or estimate the stiffness of this category of materials and its boundary conditions is highly debatable. To verify one of the possible measurement procedures, two materials were selected that are increasingly used in the construction of buildings with a focus on reducing their carbon footprint (straw and cellulose material). Since the materials were in loose form, it was necessary to create a rigid frame into which the material was poured (the internal dimensions of the frame were 0.206 m × 0.206 m × 0.15 m). For this purpose, a steel frame was created, which was rigidly fixed to a base plate measuring 0.21 m × 0.21 m × 0.007 m. The total weight of this body was 9.18 kg. A load plate of dimensions 0.19 m $\times 0.19$ m × 0.003 m and mass m_lp3 = 0.868 kg was used during experiments. In this case experiment, we focussed on the influence of bulk density of loose material on its resulting dynamic stiffness (Figure 9). The effect of changing the bulk density was monitored in steps for both materials (see Table 9). To partially objectify the robustness of the measured results, each selected material was temporarily loaded to achieve five distinct levels of bulk density. Subsequently, the loose material was removed, re-loosened, and the measurement was repeated three times. As a result, each compression level was evaluated three times for each material. The compression levels showed a deviation of up to 2% in bulk density for the straw material and up to 8% for the cellulose material.

Figure 9.

An example of the gradual increase in the bulk density of cellulose through uniform compression. (a) $Δ d$ = 0 mm; (b) $Δ d$ = 53 mm; (c) $Δ d$ = 63 mm; (d) $Δ d$ = 73 mm; (e) $Δ d$ = 98 mm; (f) $Δ d$ = 108 mm.

Results

Single mass procedure - load plate mass effect

In this section, partial results of measurements are presented, focussing on the experimental investigation of the load plate weight’s influence on the determination of apparent dynamic stiffness. A simplified procedure, often also called a single-degree-of-freedom system, was used, where a mass (in this case, the load plate) oscillates on a spring (in this case, a group of conventionally used materials in practice). The dynamic stiffness determination was performed under the load of a heavy load plate (EN 29052-1¹³ compliant) and a light load plate (non-EN compliant; see Figure 10). A thin layer of plaster was not created between the load plate and the elastic material (not following EN), and the contact edge between the elastic material and the base was not sealed in any case. However, it is not expected to affect the relative differences caused by the weight of the load plate. A logarithmic sweep signal was used as the excitation signal. This form of excitation is also not explicitly described in the EN standard, but this form of excitation is now commonly used. During the experiments presented in this section, only a single specimen of each material type was tested. However, it can be reasonably assumed that the variability of the measured stiffness across a larger set of samples would exceed 5%.^25,26 Additionally, the loading duration plays a significant role. Previous studies^19,27 have demonstrated that for selected specimens subjected to loading for approximately 41 h, the stiffness increased by 4%–27.4%.

Figure 10.

Extrapolated resonant frequency determination. Marker ∘ indicates heavy load plate 0.2 m × 0.2 m of mass m_lp1 and □ load plate 0.2 m × 0.2 m of mass m_lp2; blue - White EPS; red - blue XPS; green - Mineral Wool 1; black - Mineral Wool 2; cyan - Wood Fibre; magenta - PU Elastomer.

Table 5 presents the resulting apparent dynamic stiffness values determined according to the equation (1) for the individual selected materials. This is generally not a problem for the tested materials, as they are used in specific applications with defined operational loads. However, all dynamic stiffness values provided in the datasheets are measured according to ISO 9052-1 and may therefore differ from the actual stiffness under real installation conditions. These findings are consistent with research presented in¹⁸ but also^15,28 to what extent the results are affected by the change in the flow resistivity of the damping sample due to the attraction or compression of closed pores in the material.

Table 5.

Apparent dynamic stiffness determined based on single mass approach under loading with heavy and lightweight load plates of size 0.2 m × 0.2 m.

Material	s’ in MN/m³
Material	m _lp1	m _lp2
White EPS	43.0	11.9
Blue XPS	83.8	8.7
Mineral Wool 1	2.4	0.7
Mineral Wool 2	13.3	2.4
Wood fibre	81.6	14.8
PU elastomer	41.7	29.2

Loading the sample by a load plate with a mass of m_lp2 is not a typical case in practice for floating floor systems. However, we can encounter such loads on the resilient materials or deformation caused by compression in light, multiple vertical structures or sandwich panels used on the external walls (ETICS) and in the roofing of buildings. The experiment highlights that in application scenarios where physical contact and compression with adjacent rigid layers are present, the actual material stiffness may differ significantly from the values obtained using the standardised heavy load plate method (ISO 9052-1), and thus should not be assumed to be equivalent.

Although it was possible to determine the values of s’_t in this part of the presented experiment, during the tests with the load plate of mass m_lp2, the movement of the tested sample in the horizontal direction was visually recognised when excited by a logarithmic sweep signal. This indicates that the contact between the load plate and the test sample was not continuous during the tests (the plate on the sample was slightly jumping or slamming off the spring). This effect was also recognised at excitation levels around F_rms ≈ 0.1 N. This phenomenon introduces unwanted nonlinear events into the system, which, in effect, affect the resulting value of the resonant frequency (the resonant frequency is shifted or altered, depending on different contact scenarios or modes of oscillation), depending on the boundary conditions. This phenomenon can also lead to a reduction in energy transfer. Most energy is transferred in oscillations at the resonant frequency, but when the plate jumps, the energy transfer is interrupted. This can be easily checked on the time domain record of the force transmission (see Figure 11).

Figure 11.

Time dependent logarithmic sweep signal recorded on the force sensor for the cases of heavy load plate of mass m_lp1 (blue) and load plate of mass m_lp2 (red) on the tested “Mineral Wool 2” sample. The graph also indicates the frequency change in the amplitude of the excitation force overtime on the x-axis.

The figure shows a logarithmic sweep signal recorded during experiments on the “Mineral Wool 2” resilient layer. It shows the variation of force over time for load plate m_lp1 (blue) and load plate m_lp2 (red). The x-axis above the figure also indicates the spectral waveform of the excitation signal for a better understanding of the behaviour of the system depending on the excitation frequency (amplitude-frequency dependence). The signal collected by the force sensor indicates the nonlinear nature of the response in the de facto entire excitation spectrum in the case of a light plate in comparison to the heavy one. One can recognise a frequency-dependent decrease and several peaks (at 17, 47, and 68 Hz). The reduction in the excitation force is caused by the loss of contact between the two plates and the resilient layer. The increase in force indicates the effect of the return (impact loading) of the plate. If the resilient layer has a low loss factor (there is less damping in the system), the bouncing has a larger amplitude. Another equally important note here is the similarly difficult-to-estimate effect of loading by the excitation device (shaker). With such a low weight of the load plate, even a low loading by the shaker can ultimately significantly affect the determined apparent dynamic stiffness. Replacing the shaker with a modal hammer could only be considered when using a miniature hammer, where sufficient excitation of the test sample may not be ensured. All these doubts regarding the light load plate, therefore, led us to focus further on experiments using excitation from below (double mass system).

Compression of resilient materials

Figure 12 presents the change in the thickness of the resilient layers as a function of the applied load in the range from 310 Pa to 12.7 kPa. Only the material marked PU elastomer showed a linear relationship between compression and applied load. The horizontal dashed and dotted lines in the graph indicate the load exerted by the plates used in the previous experiment. The compression of the individual resilient layers when loaded by the plate m_lp1 and m_lp2 is given in the Table 6. The table also presents the theoretical change in actual bulk density resulting from compression, which directly affects the flow resistivity of individual materials with an open pore structure.

Figure 12.

Compression of resilient materials. (a) Full measured lading range; (b) zoom on the loading range.

Table 6.

Compression under loading with heavy and lightweight load plate of size $0.2 m \times 0.2 m$ .

	$Δ$ h in mm
Material	m _lp1	m _lp2
MW 1	2.374 $\pm 0.015$	0.543 ± $0.013$
MW 2	0.889 ± $0.086$	0.285 ± $0.072$
EPS	0.180 ± $0.009$	0.098 ± $0.022$
PUE	0.092 ± $0.005$	0.013 ± $0.002$
XPS	0.108 ± $0.010$	0.010 ± $0.016$
WF	0.198 ± $0.014$	0.050 ± $0.013$
	$ρ$ in kg/m³
	m _NO	m _lp1	m _lp2
MW 1	65.83 ± $0.16$	74.70 ± $0.27$	67.67 ± $0.23$
MW 2	206.24 ± $1.03$	226.36 ± $3.37$	212.28 ± $2.97$
EPS	14.01 ± $0.02$	14.08 ± $0.21$	14.05 ± $0.20$
PUE	207.65 ± $1.12$	208.55 ± $1.25$	207.74 ± $1.23$
XPS	34.63 ± $0.42$	34.80 ± $0.50$	34.67 ± $0.49$
WF	223.60 ± $0.04$	225.68 ± $0.56$	223.89 ± $0.56$

Derivation of compression effect on the structural stiffness and air flow resistivity change

Figure 13 presents a comparison of the resulting resonance responses in the form of transfer functions $H$ in dB for resilient materials MW1 and MW2. Neglecting other imperfections of the measurements and the limits of the numerical model, we can conclude that the load increase of the resilient material resulted in an increase in flow resistivity, bulk density and stiffness s’_s according to the Table 7. However, the measurement results obtained with the light load plate experiment were fundamentally distorted, since the load plate and resilient layer were not constantly in physical contact. This was also reflected in the fitting of the measured data in the form of a limit in the flow resistivity at $23.12 \times 10^{3}$ in Pa ⋅ s/m².

Figure 13.

Comparison of the resonance response obtained by measurement and parametric fitting of the KEM model. (a) sample MW1; (b) sample MW2; Results were scaled (Similarly as in²¹).

Table 7.

Parametrically derived underlay of porous material properties applied in KEM.

MW1
Flow resistivity	σ	$63.917 \times 10^{3}$	$23.12 \times 10^{3}$	Pa ⋅ s/m²
Solid structure stiffness	s’_s	$1.93 \times 10^{6}$	$0.405 \times 10^{6}$	Pa/m
Thickness	d	0.02	0.02	m
MW2
Flow resistivity	$σ$	$105.36 \times 10^{3}$	$23.12 \times 10^{3}$	Pa ⋅ s/m²
Solid structure stiffness	s’_s	$7.448 \times 10^{6}$	$1.3 \times 10^{6}$	Pa/m
Thickness	d	0.01	0.01	m

Numerical verification of 2-DOF mass-spring system approach for dynamic stiffness determination

Based on the procedure described earlier, two numerical studies were performed. First, it was successively focussed on parametric verification of the relevance of using the Blevins relation to derive the dynamic stiffness of the resilient material in a 2-DOF system. The results of Blevins model was compared with the model created in SimScape and FEM. The parametric study showed that the results correlate very well for the given range of the assignment. It was also demonstrated that the suspension stiffness in the range from 4 × 10³ to 1.2 × 10⁵ MN/m³ has almost negligible effect on the second eigenmode (less than 0.5 Hz) of the 2-DOF system (see Figure 14) which supports the overall measurement setup approach. This raises the question of whether it is not possible to apply the 1-DOF relationship (1) for the given cases. For this reason, the parametric study was repeated for the dynamic stiffness of the resilient material for the range s₁ from 1 to 32 MN/m³ with the same variation of the suspension stiffness k₂ as in the previous cases (i.e. the range from 4 × 10³ to 1.2 × 10⁵ MN/m). It was confirmed that if it is not possible to ensure a sufficiently large ratio of the stiffnesses of the individual springs (s₂< s₁), it is necessary to take into account the calculation relationship for 2-DOF (5; see Figure 15(a)). For cases where the stiffness value s₁ was lower than 12 MN/m³ (the ratio s₁/s₂ > 2), the variation in the determined resonant frequency was greater than 1 Hz. We also focussed on the effect of the ratio of individual load plates on determining the resonant frequency. In this study, the mass m₂ = 8.324 kg was maintained. The effect of increasing the mass m₁ was investigated. For this case, it was shown that even doubling of mass m₁ resulted in a significant effect on the variation of the resonant frequency (see Figure 15(b)).

Figure 14.

Numerical verification of 2-DOF mass-spring system approach for dynamic stiffness determination. (a) Parametric study 1: 2-DOF resonance frequencies f₁ and f₂ derived based three different approaches in stiffness range in accordance to Table 4; (b) Parametric study 2: contact surface area S₁ = a₁² effect on the resulting dynamic stiffness derived from (5) based on FEM for four variations of dynamic stiffness in accordance to Table 4.

Figure 15.

Effect of changing boundary conditions in a 2-DOF system. (a) Effect of changing the stiffness of the tested resilient material for: s₁ = 1 in MN/m³ (blue); s₁ = 2 in MN/m³ (red); s₁ = 4 in MN/m³ (green); s₁ = 8 in MN/m³ (magenta); s₁ = 16 in MN/m³ (cyklamen); s₁ = 32 in MN/m³ (yellow); (b) Effect of changing the load plate mass m_lp,1 for: m_lp,1 = 1.263 in kg (blue); m_lp,1 = 2.526 in kg (red); m_lp,1 = 5.052 in kg (green); m_lp,1 = 10.104 in kg (magenta); m_lp,1 = 20.208 in kg (cyklamen); m_lp,1 = 40.416 in kg (yellow). The values corresponding to f₁ are plotted on the left, the values corresponding to f₂ are plotted on the right, with the corresponding colour indicating the stiffness of the resilient layer.

The second case study investigated the effect of altering the load plate area on determining dynamic stiffness in a linear environment. The Blevins equation (4) and the FEM model (Figure 14(b)) were used for comparison. It was shown that in the case of applying the Blevins equation, it is necessary to work with a sample area equal to the load plate area. It must be emphasised, however, that this is valid assuming that the effect of compression is neglected, which results in a change in the flow resistivity and thus the actual stiffness of the damping material. Therefore, it is recommended to achieve the smallest possible difference between the sample surface and the load plate when performing the measurement.

Experimental application of 2-DOF mass-spring system approach for dynamic stiffness determination

Selected conventional resilient materials-influence of load plate size

A series of dynamic stiffness measurements were performed on the measurement setup presented in the Figure 8(a). The force and acceleration spectrum data thus obtained were subsequently evaluated using the procedure expressed in relation (1) and (5). Following this approach, the dynamic stiffness values were also compared with the experimental results presented in Table 5.

When focussing on the comparison of the application of relations (1) and (5) in determining dynamic stiffness, it is evident that ignoring the fact that the measurement set-up is 2-DOF (i.e. the second spring in the form of a base plate suspension) leads to a systematic overestimation of the results. This also corresponds to the other findings presented in this paper, where the effect of the stiffness of the second spring on the resonant response of the system due to the tested resilient material was analysed.

Further interpretation of the findings presented in Table 8 can be divided into three parts.

1 - Testing of the PU Elastomer sample can be considered informative, since the sample area in all testing cases in this publication was 0.10 m × 0.10 m. This means that this selected resilient material was always of a smaller area compared to the load plate. Its resonant response, therefore, changed only depending on the increase in the total mass of the load plate for the given cases.

2 - Materials with assumed high airflow resistivity (White EPS, Blue EPS, Wood Fibre) show a decrease in dynamic stiffness due to the increase in the contact area between the load plates from 0.19 m × 0.19 m to 0.25 m × 0.25 m by 7% to 16%. A similar statement should also apply to the PU Elastomer material, but due to the above point in this work, we could not prove it. In this case, a slight increase in stiffness was expected since the load plate m_lp3 had a 3 % higher mass per unit area.

3 - On the contrary, materials with assumed low airflow resistivity (Mineral Wool 1 and Mineral Wool 2) show an increase in dynamic stiffness due to the increase in the contact area by 1%–16%. Here, friction in the resilient material in the region of the load plate edges can play a role. This effect can be partially caused by the difference in local stiffness at the edge of the tested material and the edge of the load plate. In addition, we must not neglect the influence of the open area of the tested sample in the horizontal plane. In the test with load plate m_lp3, the difference in areas of the resilient layer and load plate was 42 %, which may affect the reduction of the resulting stiffness. In the test with m_lp5, the areas were equal.

Table 8.

Apparent dynamic stiffness determined from 2-DOF measurement approach under loading with lightweight load plate of size 0.19 m ×0.19 m (m_lp3) and 0.25 m × 0.25 m (m_lp5). Stiffness was estimated for comparison based on equations (1) and (5).

	s’_t in MN/m³
Material	m _lp3 equation (1)	m _lp5 equation (1)	m _lp3 equation (5)	m _lp5 equation (5)
EPS	5.1	5.0	4.6	4.3
XPS	6.3	6.2	5.7	5.3
MW 1	1.0	1.1	0.9	0.9
MW 2	2.6	3.2	2.3	2.8
WF	1.9	1.7	1.7	1.5
PUE	18.6	25.8	16.9	22.0

By comparing the results presented in Tables 5 and 8, one could recognise significant deviation in determined dynamic stiffnesses. This difference can be attributed to two reasons, which become apparent when applying a load plate with a low surface weight.

1 - Jumping of the load plate, which is too strongly excited by the shaker (see Figure 11).

2 - All measurement setups were assembled “in-house.” In the 1-DOF setup, the shaker was fixed to the load plate. The shaker was suspended on rubber ropes. It wasn’t trivial to control the influence of the sample loading by the shaker. The loading introduced in this way can have a significant effect on the measured resonant frequency when a light load plate is applied.

Selected fibre loose material

Similar to the previous subsection, a 2-DOF measuring rig was used and methods for determining the actual dynamic stiffness based on equations (1) and (5) were compared. Straw and cellulose in loose form were tested. By manually pressing the upper light load plate, bulk density changes were achieved in the range from 85 to 132 kg/m³ (straw material) and in the range from 53 to 163 kg/m³ (cellulose material). The determined dynamic stiffness results are given in Table 9. As previously described, each loose form material was compressed and then released in five steps. The resulting bulk density was reached with a deviation up to 5% per compression step for the straw material and up to 14% for the cellulose material. After measuring the loose form material after compression at step five, the material was removed, loosened, and measured again. This process was repeated three times. The standard deviation in dynamic stiffness, shown in the Table 9, was calculated from the average of measurements taken across the individual compression steps.

Table 9.

Apparent dynamic stiffness determined for loose form materials from a 2-DOF measurement approach under loading with a lightweight load plate of size 0.19 m × 0.19 m. For comparison, the stiffness was estimated based on equations (1) and (5).

Straw	s’_t in MN/m³
Bulk density in kg/m³	equation (1)	u in %	equation (5)	u in %
85.4 ± $3 %$	1.20		1.10
87.2 ± $4 %$	0.94	$\pm 15 %$	0.85	$\pm 15 %$
88.5 ± $4 %$	1.26		1.15
96.9 ± $4 %$	0.94		0.86
96.9 ± $4 %$	0.95	$\pm 1 %$	0.87	$\pm 1 %$
97.7 ± $4 %$	0.96		0.88
106.1 ± $4 %$	1.19		1.09
106.1 ± $4 %$	0.94	$\pm 12 %$	0.86	$\pm 12 %$
106.1 ± $4 %$	1.00		0.92
117.4 ± $5 %$	1.43		1.31
119.7 ± $5 %$	1.36	$\pm 15 %$	1.24	$\pm 5 %$
122.1 ± $5 %$	1.79		1.36
129.9 ± $5 %$	1.68		1.53
131.3 ± $5 %$	2.65	$\pm 25 %$	2.42	$\pm 26 %$
132.7 ± 5%	1.86		1.67
Cellulose
53.6 ± 5%	0.34		0.31
58.9 ± $5 %$	0.29	$\pm 9 %$	0.26	$\pm 9 %$
62.0 ± $5 %$	0.32		0.29
71.8 ± $6 %$	0.35		0.32
78.6 ± $7 %$	0.39	$\pm 34 %$	0.35	$\pm 34 %$
84.2 ± $7 %$	0.63		0.58
84.2 $\pm 7 %$	0.59		0.54
90.6 $\pm 8 %$	0.59	$\pm 4 %$	0.54	$\pm 4 %$
98.2 $\pm 8 %$	0,63		0.58
107.1 ± $9 %$	0.94		0.86
107.1 ± $10 %$	0.94	$\pm 6 %$	0.85	$\pm 6 %$
117.8 ± $9 %$	1.05		0.95
147.3 ± $13 %$	2.29		2.09
147.3 ± $13 %$	2.37	$\pm 15 %$	2.16	$\pm 15 %$
163.6 ± $14 %$	2.99		2.73

The measured dynamic stiffness for different bulk densities is mainly influenced by the variation in the degree of homogeneity of the tested materials and the uncertainty in reaching targeted bulk density levels (Table 9). The stiffness of the straw-based material shows a fairly linear relationship with bulk density within the range of 95–130 kg/m³ Figure 16. At lower densities, the material reacts very inconsistently. During this stage, individual straw stems tend to rearrange themselves to find a stable position, making it very difficult to get a flat surface of the specimen under the light load plate. Similarly, there is a notable increase in variability of the measured data at bulk densities above 130 kg/m³. This is related to the phenomenon of reaching the so-called critical flow resistivity, which results in a sudden rise in stiffness. A similar pattern is observed for the cellulose-based material. However, in this case, the increase in stiffness happens more gradually, since the material is much more homogeneous compared to the straw-based sample. Comparing the individual columns in Table 9 confirms that there is a consistent deviation in the measurement of dynamic stiffness depending on the calculation method used. It has been shown again that using equation (1) tends to overestimate the stiffness of the damping material.

Figure 16.

Effect of bulk density variation on the loose material dynamic stiffness.

Conclusion

The application of material solutions in construction with zero or as low a carbon footprint as possible is an inseparable part of the development in the field of building construction. Here, the application of loose-form materials is increasingly common. The specific category of loose-form fibrous materials from the point of view of building acoustics or noise reduction effect has so far been very little investigated concerning their impact on the sound insulation properties of double leaf constructions from the context of stiffness increase.

The presented case study analysed the potential of using and the relevance of the measurement method using a 2-DOF measurement setup to determine the actual dynamic stiffness of fibrous materials in loose form to approach built-in boundary conditions in vertical structures. Individual chapters focussed on the theoretical framework, methodology of experimental and numerical analyses, as well as the interpretation of measured results. The basic concept of the double mass approach and its application were introduced in the introductory chapters. This was followed by a detailed methodological part, in which the experimental techniques and numerical simulations used were described. Selected conventional materials were successively experimentally analysed by applying a measuring set-up with 1-DOF and 2-DOF. The study demonstrated the need for an alternative measurement method to evaluate materials under very low loads. By evaluating the effect of changing the load plate weight from 191.375 to 31.575 kg/m² used to determine the dynamic stiffness by the 1-DOF method, six different resilient materials showed a decrease in the resulting actual stiffness of the materials by 32%–89%. These results were supported by the compressibility analysis of individual samples, which showed a decrease in bulk density with the change in the load plate by 0.1%–10%. This results in a change in the air flow resistance of open cell materials, which is associated with a decrease in stiffness. After verifying the 2-DOF method with two numerical parametric studies, experiments were performed using the 2-DOF method on selected conventional materials (the effect of the difference in area between the load plate and the sample on the obtained results was monitored) and two variants of loose form materials (the effect of increasing the bulk density of loose form material on the resulting stiffness was observed). In the analysis of conventional materials, a deviation of the results obtained by the 1-DOF and 2-DOF methods was demonstrated. In the testing case of samples with low flow resistivity, it was shown that the ratio of the sample area to the load plate results in a decrease in the actual stiffness of the tested material. The results of the tests of loose form materials, in addition to the expected smooth increase in stiffness due to the change in bulk density, showed a sudden jump in the change in stiffness, probably caused by the so-called phenomenon of achieving the so-called critical flow resistivity of the tested material. One problem here is the low resonant frequency of the lightweight plate and the high compliance of the load plate when excited from above. It also pointed out the differences in determining the actual dynamic stiffness when the relationship according to EN 29052-1 is used, even for 2-DOF systems. The main shortcomings include the idealisation of specific models, which can impact the accuracy of predictions in real-world conditions. Furthermore, it is necessary to perform more extensive experimental verification to confirm the robustness of the proposed solutions. Since 1-DOF measurements with the lightweight plate were affected by insufficient control over the influence of the weighting caused by the shaker suspended on rubber ropes. Possible directions for further research are the expansion of the models to nonlinear effects, horizontal arrangement of the measurement setup, but also the study of the interaction of the double mass with various types of dynamic loads and also in the application with other loose form materials. Although the measurement procedure discussed here enables the determination of the stiffness of blown-in materials at low compression, it remains a great challenge to experimentally simulate the complex interaction of a lightweight double structure and the blown-in material after it is pressed into the wall structure. The stiffness of the enclosing structure, together with its frame, results in a higher resonant response (apparent stiffness) than when the lightweight load plate is freely placed on a blown-in material.

This paper is not intended to demonstrate the inappropriateness of the current EN 29052-1. It is an attempt to find an effective way to study the behaviour of fibrous materials in loose form after incorporation into lightweight structures. The presented study raises several questions and problems that should be addressed in future work. The work presented here is a case study, and from the perspective of loose form materials, only two types were examined. Even among the tested materials, a high variation in the degree of material homogeneity was observed. Therefore, the conclusions of this paper should not be generalised and must be interpreted with appropriate caution. Testing additional types of loose form materials may help verify the robustness of the measurement procedure introduced here.

Footnotes

ORCID iDs

Daniel Urban

Maximilian Neusser

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors acknowledge the financial support from the AUSTRIAN SCIENCE FUND (FWF), grant number I 5503-N, (Engineered wood composites with enhanced impact sound insulation performance to improve human wellbeing), the support of all project partners of the research project “Schall.Holz.Bau II” and the project “NÖ Wirtschafts- und Tourismusfonds” in the funding track “Kooperation” under the number WST3-F2- 525823/007-2018. This research was funded in whole or in part by the Austrian Science Fund (FWF) [10.55776/I5503]. For open access purposes, the author has applied a CC BY public copyright license to any author accepted manuscript version arising from this submission. Author DU acknowledges support by the Slovak national grant VEGA 1/0205/22, KEGA 033STU-4/2024 by funding from the European Union’s Horizon Europe research and innovation programme under the HORIZON-MSCA-2021-DN-01 grant agreement No. 101072598 – “ActaReBuild.”

Declaration of conflicting interests

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

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Determination of dynamic stiffness of resilient materials using a lightweight load plate 2-DOF method

Abstract

Keywords

Introduction

Effect of load plate mass on dynamic stiffness determination based on single mass procedure

Compression of resilient materials

Derivation of compression effect on the structural stiffness and air flow resistivity change

Numerical verification of 2-DOF mass-spring system for dynamic stiffness determination

Parametric study 1

Parametric study 2

Experimental application of 2-DOF mass-spring system approach for dynamic stiffness determination

Results

Single mass procedure - load plate mass effect

Compression of resilient materials

Derivation of compression effect on the structural stiffness and air flow resistivity change

Numerical verification of 2-DOF mass-spring system approach for dynamic stiffness determination

Experimental application of 2-DOF mass-spring system approach for dynamic stiffness determination

Selected conventional resilient materials-influence of load plate size

Selected fibre loose material

Conclusion

Footnotes

ORCID iDs

Funding

Declaration of conflicting interests

References