Source: https://jmrplens.github.io/phonometry/materials/resilient/dynamic-stiffness/ # Dynamic stiffness of resilient materials (EN 29052-1) A **floating floor** is a heavy floating slab resting on a resilient layer; the two form a mass-spring system whose **natural frequency** governs how much the floor improves impact and airborne insulation. **EN 29052-1:1992** (identical to ISO 9052-1:1989) measures the **dynamic stiffness per unit area** $s'$ of the resilient layer from the resonance of a standard load plate on a 200 mm × 200 mm specimen. $s'$ is the input to the floating-floor term of the EN 12354-2 impact model covered in [Predicting Sound Insulation (EN 12354)](https://jmrplens.github.io/phonometry/buildings/design/insulation-prediction/). (ISO 16251-1 does not apply here: its scope is limited to soft, locally-reacting floor coverings and explicitly excludes floating floors.)

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import materials # One line — from a measured resonance, the result draws its own f0(s') # design curve with the determination marked: res = materials.floating_floor_resonance( resonant_frequency=25.0, total_mass_per_area=200.0, floor_mass_per_area=120.0, airflow_resistivity=50.0, thickness=0.020, porosity=0.9, ) res.plot() plt.show() # By hand, the same design curve for a light and a heavy floating floor: s = np.logspace(np.log10(2.0), np.log10(100.0), 300) # MN/m3 for m in (40.0, 120.0): plt.semilogx(s, materials.natural_frequency(s * 1e6, m), label=f"m' = {m:g} kg/m²") plt.xlabel("Dynamic stiffness s' [MN/m³]"); plt.ylabel("Natural frequency f₀ [Hz]") plt.legend(); plt.show() ```

## 1. Dynamic stiffness and resonance The dynamic stiffness per unit area is a dynamic force per area divided by the resulting change in thickness (Formula 1): $s' = (F/S)/\Delta d$. The resiliently supported floor is a resonator whose natural frequency (Formula 2) and, in the laboratory arrangement, measured resonant frequency (Formula 3) are $$ f_0 = \frac{1}{2\pi}\sqrt{\frac{s'}{m'}}, \qquad f_\mathrm{r} = \frac{1}{2\pi}\sqrt{\frac{s'_\mathrm{t}}{m'_\mathrm{t}}}, $$ so the **apparent** dynamic stiffness follows from the resonance (Formula 4): $$ s'_\mathrm{t} = 4\pi^2\,m'_\mathrm{t}\,f_\mathrm{r}^2 . $$ In the test arrangement the specimen lies between the rigid foundation and a load plate whose total mass per unit area, plate plus added load, is 200 kg/m² (8 kg on the 0.04 m² specimen). That load reproduces the static preload of a typical floating floor, about 2 kPa. A vertical exciter drives the plate, an accelerometer picks up its response, and the fundamental vertical resonance $f_\mathrm{r}$ of the plate-on-specimen system is read from the response peak; Formula 4 turns it into $s'_\mathrm{t}$. That extraction is the laboratory's: the Clause 7 procedures for reading $f_\mathrm{r}$ from the raw excitation-response signal — sinusoidal, white-noise or pulse excitation, with the extrapolation to zero force amplitude — are not implemented, and `apparent_dynamic_stiffness` expects the already-extrapolated $f_\mathrm{r}$. ```python from phonometry import materials # Standard 8 kg load plate on the 0.04 m2 specimen -> m't = 200 kg/m2; # the fundamental resonance is measured at 25 Hz. s_t = materials.apparent_dynamic_stiffness(resonant_frequency=25.0, total_mass_per_area=200.0) print(round(s_t / 1e6, 3)) # 4.935 MN/m3 # Installed on a 120 kg/m2 floating screed with s' = 10 MN/m3: print(round(materials.natural_frequency(10e6, 120.0), 1)) # 45.9 Hz ``` That little oscillator is small enough to draw at true scale. `plot_dynamic_stiffness_rig` puts the standard 200 mm specimen under the 8 kg load plate with the exciter and accelerometer in place. *The mass-spring system behind $s'$ at true scale: 20 mm of resilient layer under the 8 kg plate is all it takes to reproduce the 2 kPa preload of a floating floor.*

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```python import matplotlib.pyplot as plt from phonometry import materials # The standard rig: a 200 mm square specimen under the 8 kg load plate. materials.plot_dynamic_stiffness_rig() plt.show() ```

## 2. The enclosed-gas term and airflow resistivity For an air-permeable material the enclosed pore air adds a parallel stiffness from its isothermal compression (Formula 7): $s'_\mathrm{a} = p_0/(d\,\varepsilon)$, with $p_0$ the atmospheric pressure, $d$ the loaded thickness and $\varepsilon$ the porosity. The standard's worked NOTE ($p_0 = 0.1\ \text{MPa}$, $\varepsilon = 0.9$) is $s'_\mathrm{a} = 111/d$ MN/m³ for $d$ in millimetres: ```python from phonometry import materials print(round(materials.enclosed_gas_stiffness(thickness=0.020, porosity=0.9) / 1e6, 2)) # 5.56 MN/m3 (the NOTE's 111/20 = 5.55 MN/m3) ``` The dynamic stiffness of the *installed* material is then set by the lateral airflow resistivity $r$ (clause 8.2): $s' = s'_\mathrm{t}$ for $r \ge 100\ \text{kPa}\cdot\text{s/m}^2$, $s' = s'_\mathrm{t} + s'_\mathrm{a}$ for $10 \le r < 100\ \text{kPa}\cdot\text{s/m}^2$, and for $r < 10\ \text{kPa}\cdot\text{s/m}^2$ the method only resolves $s' = s'_\mathrm{t}$ when the gas term is negligible. `floating_floor_resonance` chains the whole determination: ```python from phonometry import materials res = materials.floating_floor_resonance( resonant_frequency=25.0, total_mass_per_area=200.0, floor_mass_per_area=120.0, airflow_resistivity=50.0, thickness=0.020, porosity=0.9, ) print(round(res.dynamic_stiffness / 1e6, 2), round(res.natural_frequency, 1)) # 10.49 47.1 res.plot() # the f0(s') design curve with this determination marked (needs matplotlib) ``` The `DynamicStiffnessResult` carries the apparent, enclosed-gas and installed stiffnesses, the test resonance and the installed-floor natural frequency, and its `.plot()` draws the $f_0(s')$ design curve. **Test-report fiche.** `DynamicStiffnessResult.report(path)` renders a one-page accredited dynamic-stiffness test report (EN 29052-1:1992 = ISO 9052-1:1989): a metadata header (specimen, the total mass per unit area $m'_\mathrm{t}$, the loaded thickness $d$ in metres shown in mm, test facility, climate), a metrics table (the resonant frequency $f_\mathrm{r}$, the apparent stiffness $s'_\mathrm{t}$ of Formula 4, the enclosed-gas term $s'_\mathrm{a}$ when it applies, the installed $s'$ of clause 8.2 and the natural frequency $f_0$ of Formula 2) beside the $f_0(s')$ design curve, and a boxed apparent dynamic stiffness $s'_\mathrm{t}$ (no pass/fail). Clause 9 rounds every stiffness to the nearest MN/m³; `language="es"` renders the Spanish fiche. The fiche always embeds the $f_0(s')$ design curve, so it needs both the report and plot extras (`pip install "phonometry[report,plot]"`). Clause 6's specimen selection — at least three 200 mm × 200 mm specimens — is not enforced, and neither is the Clause 9 report content: the fiche carries the excitation arrangement and the signal type only when the metadata supplies them. ```python from phonometry import ReportMetadata, materials res = materials.floating_floor_resonance( resonant_frequency=45.0, total_mass_per_area=200.0, floor_mass_per_area=110.0, airflow_resistivity=50.0, thickness=0.020, porosity=0.9, ) res.report( "dynamic_stiffness.pdf", metadata=ReportMetadata( specimen="20 mm mineral-wool resilient layer", mass_per_area=200.0, thickness=0.020, # thickness d in metres (20 mm) measurement_standard="EN 29052-1", ), ) # one-page fiche (needs phonometry[report,plot]) ``` [](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/en29052_dynamic_stiffness_example.pdf) ## 3. What the resonance method assumes, and where it bites The evaluation treats the rig as a single-degree-of-freedom system: the load plate moves as a rigid piston on a massless spring. That holds while the specimen is light against the plate and its first internal resonance sits well above $f_\mathrm{r}$; a heavy or very thick layer starts to act as a distributed system and the simple Formula 4 reading degrades. Three practical pitfalls follow from the preload: * **$s'$ is a stiffness *at the standard preload*.** Resilient layers are visibly non-linear in static load: mineral wool stiffens as it compresses, some foams soften. The 200 kg/m² load plate fixes the operating point, so the tabulated $s'$ strictly describes floors near that surface mass. Designing a much heavier screed with the same $s'$ extrapolates beyond the measurement. * **Drive small.** The tangent stiffness is defined for small dynamic strains; driving the plate hard pushes the layer into its non-linear range and shifts the apparent resonance downward. Keep the excitation at the lowest level that gives a clean peak. * **Respect the contact.** The standard seats the load plate on a thin bonding layer (a plaster paste) so the full specimen area carries the load. A dry, uneven contact concentrates the force, stiffens the response locally and biases $f_\mathrm{r}$ upward. The natural frequency that matters in the end is not the rig's $f_\mathrm{r}$ but the installed floor's $f_0$ from Formula 2: the floating floor only improves insulation well above $f_0$, which is why a low $s'$ (a soft layer under a heavy slab) is the design goal. ## See also - [Predicting Resilient-Layer Performance](https://jmrplens.github.io/phonometry/buildings/design/resilient-layers/): what the design side does with $s'$, from the mass-spring resonance to the improvement laws above it and their weighted single number. - [Predicting Sound Insulation (EN 12354)](https://jmrplens.github.io/phonometry/buildings/design/insulation-prediction/): the floating-floor term this measurement feeds. - [Airflow Resistance](https://jmrplens.github.io/phonometry/materials/absorbers/airflow-resistance/): the ISO 9053 determination of the lateral resistivity $r$ that clause 8.2 needs, reported there as $\sigma$ in Pa·s/m², while this page's `airflow_resistivity` argument is in kPa·s/m². - [Resilient layers overview](https://jmrplens.github.io/phonometry/materials/resilient/): where this measurement sits among the materials guides. - API reference: [`materials.resilient.dynamic_stiffness`](https://jmrplens.github.io/phonometry/reference/api/materials/dynamic-stiffness/). - Theory: [Point mobilities and radiation efficiency](https://jmrplens.github.io/phonometry/reference/theory/vibration/#point-mobilities-and-radiation-efficiency-cremer-5-hopkins-29): the mass-spring resonance the EN 29052-1 rig measures, seen as a mobility. ## References - Vigran, T. E. (2008). *Building acoustics*. CRC Press. ISBN 978-0-415-42853-8. [doi:10.1201/9781482266016](https://doi.org/10.1201/9781482266016). Floating-floor design and the role of the resilient layer's dynamic stiffness in the impact-sound improvement. - International Organization for Standardization. (1989). *Acoustics — Determination of dynamic stiffness — Part 1: Materials used under floating floors in dwellings* (ISO 9052-1:1989). [iso.org catalogue](https://www.iso.org/standard/16620.html). The international original of EN 29052-1, the method this page implements. ## Standards EN 29052-1:1992 (= ISO 9052-1:1989), *Acoustics — Determination of dynamic stiffness — Part 1: Materials used under floating floors in dwellings*: the dynamic stiffness per unit area (Formula 1), the resonance relations (Formulae 2-4), the enclosed-gas term (Formula 7, clause 8.2 NOTE $s'_\mathrm{a} = 111/d$ MN/m³) and the airflow-resistivity regimes (clause 8.2, Formulae 5-6). Conformance is anchored on the standard's own numeric NOTE plus hand-computed closed-form values of the resonance relations.