Source: https://docs.jmrp.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://docs.jmrp.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_hz=25.0, total_mass_per_area_kg_m2=200.0, floor_mass_per_area_kg_m2=120.0, airflow_resistivity_kpa_s_m2=50.0, thickness_m=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_hz=25.0, total_mass_per_area_kg_m2=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_m=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. The resistivity is in kPa·s/m², the unit the standard thresholds it in and a thousand times the Pa·s/m² ISO 9053 reports, so it is passed by name, `airflow_resistivity_kpa_s_m2`, with the unit on the line. `floating_floor_resonance` chains the whole determination: ```python from phonometry import materials res = materials.floating_floor_resonance( resonant_frequency_hz=25.0, total_mass_per_area_kg_m2=200.0, floor_mass_per_area_kg_m2=120.0, airflow_resistivity_kpa_s_m2=50.0, thickness_m=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_hz=45.0, total_mass_per_area_kg_m2=200.0, floor_mass_per_area_kg_m2=110.0, airflow_resistivity_kpa_s_m2=50.0, thickness_m=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) ### Published layers, when there is no declared value Designing a floating floor starts from the manufacturer's $s'$, declared to EN 29052-1 for that product. When there is none, fifteen measured layers are published as `PUBLISHED_RESILIENT_LAYERS`, transcribed from Hopkins Table A3 (printed p. 610), whose caption states they were measured according to ISO 9052-1. The column heading prints $s'$, which the book's own list of symbols defines as the dynamic stiffness of the installed layer and keeps apart from the apparent $s'_\mathrm{t}$, so every row holds `dynamic_stiffness_n_m3` and its natural frequency needs nothing but the mass of the floor. Each layer is a catalogue row like the rows of every other published table: it carries the page it was read on, and the four rebond foams carry the second-level attribution the printed cell gives them. The key is `"/"`, as in every catalogue: the packaged table, `hopkins-2007-table-a3`, and then which specimen the row is, `__`, because four rock-wool rows and four glass-wool rows differ only by those two numbers and the printed table separates them by position under a name it prints once. ```python from phonometry import materials layer = materials.resilient_layer("hopkins-2007-table-a3/mineral_wool_rock_60_30") print(layer.name, layer.density_kg_m3, layer.thickness_mm) # Mineral wool, rock 60.0 30.0 print(layer.source) # Hopkins (2007) Table A3, PDF page 637 (printed p. 610) print(round(layer.dynamic_stiffness_n_m3 / 1e6, 1)) # 10.0 MN/m3, as printed print(round(layer.natural_frequency(mass_per_area_kg_m2=120.0), 1)) # 45.9 Hz under a 120 kg/m2 screed rebond = materials.PUBLISHED_RESILIENT_LAYERS["hopkins-2007-table-a3/rebond_foam_64_20"] print(rebond.attributed_to["row"]) # Hopkins and Hall (2006) print(round(rebond.natural_frequency(mass_per_area_kg_m2=120.0), 1)) # 43.6 ``` These are measured specimens and not declared product values: they are the order of magnitude to reason with while a product is chosen, not an input to a compliance calculation. The softest row in the table, 40 mm glass wool at 7 MN/m³, puts the same screed at 38.4 Hz, and the stiffest, 5 mm closed-cell polyethylene foam at 115 MN/m³, at 155.8 Hz; that spread is the whole design question. **A layer that gives only $s'_\mathrm{t}$.** A test report to EN 29052-1 gives the apparent stiffness $s'_\mathrm{t}$ of its specimen and the enclosed-gas stiffness $s'_\mathrm{a}$, and it gives $s'$ only "if possible" (clause 9 e)); a sheet that quotes $s'_\mathrm{t}$ alone gives less than the report it came from. Either way $s'_\mathrm{t}$ is not the $s'$ that Formula 2 takes: for an air-permeable layer the enclosed gas adds to it, and Hopkins notes that it often forms a significant percentage of $s'$ (printed p. 360). A `ResilientLayer` built from such a source holds it in `apparent_dynamic_stiffness_n_m3`, and its `natural_frequency` refuses to go on until it is given the lateral airflow resistivity, in Pa·s/m² as ISO 9053 reports it, and, below 100 kPa·s/m², the enclosed-gas stiffness: the $s'_\mathrm{a}$ the report states, or Formula 7 on the thickness under the test load, which clause 9 b) has the report state as well and which is not the nominal thickness a product sheet prints. With them it takes the clause 8.2 branch that resistivity picks: ```python from phonometry import io, materials report = materials.ResilientLayer( name="Example layer", source="Example Acoustics Ltd test report 26-014, p. 2", apparent_dynamic_stiffness_n_m3=6.0e6, ) try: report.natural_frequency(mass_per_area_kg_m2=120.0) except io.CatalogueError as refusal: print(refusal) # 'Example layer' gives the apparent dynamic stiffness s't of a test specimen, # 6 MN/m3, and not the dynamic stiffness s' of the installed layer [...]: pass # airflow_resistivity_pa_s_m2 and, below 100 kPa.s/m2 (100000 Pa.s/m2), # gas_stiffness_n_m3, the enclosed-gas stiffness s'a of Formula 7 [...] # d = 30 mm under the test load, from the report (clause 9 b)); an s'a the # report states (clause 9 e)) goes in as gas_stiffness_n_m3 the same way. gas = materials.enclosed_gas_stiffness(thickness_m=0.030, porosity=0.9) # 3.7 MN/m3 f0 = report.natural_frequency( mass_per_area_kg_m2=120.0, airflow_resistivity_pa_s_m2=50_000.0, # 50 kPa.s/m2: Formula 6, s' = s't + s'a gas_stiffness_n_m3=gas, ) print(round(f0, 1)) # 45.3 Hz ``` Read as if it were $s'$, the same 6 MN/m³ would have put the screed at 35.6 Hz, almost 10 Hz too low. ## 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://docs.jmrp.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://docs.jmrp.io/phonometry/buildings/design/insulation-prediction/): the floating-floor term this measurement feeds. - [Airflow Resistance](https://docs.jmrp.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, the unit `ResilientLayer.natural_frequency` takes, while `airflow_resistivity_kpa_s_m2` is in kPa·s/m². - [Resilient layers overview](https://docs.jmrp.io/phonometry/materials/resilient/): where this measurement sits among the materials guides. - API reference: [`materials.resilient.dynamic_stiffness`](https://docs.jmrp.io/phonometry/reference/api/materials/dynamic-stiffness/). - Theory: [Point mobilities and radiation efficiency](https://docs.jmrp.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.