Source: https://jmrplens.github.io/phonometry/reference/api/materials/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/guides/insulation-prediction/). (ISO 16251-1 does not apply here: its scope is limited to soft, locally-reacting floor coverings and explicitly excludes floating floors.)

Show the code for this figure

```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)/Δd`. The resiliently supported floor is a resonator whose natural frequency (Formula 2) and, in the laboratory arrangement, measured resonant frequency (Formula 3) are $$ f_0 = \frac{1}{2\pi}\sqrt{\frac{s'}{m'}}, \qquad f_r = \frac{1}{2\pi}\sqrt{\frac{s'_t}{m'_t}}, $$ so the **apparent** dynamic stiffness follows from the resonance (Formula 4): $$ s'_t = 4\pi^2\,m'_t\,f_r^2 . $$ 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 $f_r$ of the plate-on-specimen system is read from the response peak; Formula 4 turns it into $s'_t$. ```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 ``` ## 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'a = p₀/(d·ε)`, with `p₀` the atmospheric pressure, `d` the loaded thickness and `ε` the porosity. The standard's worked NOTE (`p₀ = 0.1 MPa`, `ε = 0.9`) is `s'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't` for `r ≥ 100 kPa·s/m²`, `s' = s't + s'a` for `10 ≤ r < 100 kPa·s/m²`, and for `r < 10 kPa·s/m²` the method only resolves `s' = s'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 `f₀(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 $m'_t$, the loaded thickness $d$ in metres shown in mm, test facility, climate), a metrics table (the resonant frequency $f_r$, the apparent stiffness $s'_t$ of Formula 4, the enclosed-gas term $s'_a$ 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 $s'_t$ (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]"`). ```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 $f_r$; 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 $f_r$ upward. The natural frequency that matters in the end is not the rig's $f_r$ 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. ## 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'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.