Dynamic stiffness of resilient materials

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). (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

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

$$f_0=\frac{1}{2\pi }\sqrt{\frac{s^{\prime }}{m^{\prime }}},f_r=\frac{1}{2\pi }\sqrt{\frac{s_t^{\prime }}{m_t^{\prime }}},$$

so the apparent dynamic stiffness follows from the resonance (Formula 4):

$$s_t^{\prime }=4{\pi }^2{m}_t^{\prime }{f}_r^2.$$

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 $s_t^{\prime }$.

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:

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:

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 $m_t^{\prime }$ used during the test, the loaded thickness $d$ (in metres, shown in mm), test facility, climate), a compact metrics table (the resonant frequency $f_r$, the apparent stiffness $s_t^{\prime }$ of Formula 4, the enclosed-gas term $s_a^{\prime }$ when it applies, the installed $s^{\prime }$ of clause 8.2 and the natural frequency $f_0$ of Formula 2) beside the $f_0(s^{\prime })$ design curve, and a boxed apparent dynamic stiffness $s_t^{\prime }$ with the installed $s^{\prime }$ and the resonance $f_r$ alongside. Clause 9 rounds every stiffness to the nearest MN/m³. It is a characterisation, so there is no pass/fail verdict; language="es" renders the Spanish fiche. The fiche always embeds the $f_0(s^{\prime })$ design curve, so it needs both the report and plot extras (pip install "phonometry[report,plot]").

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])

One-page dynamic-stiffness fiche: a metadata header with the total mass per unit area and the loaded thickness, a metrics table of the resonant frequency, the apparent, enclosed-gas and installed dynamic stiffnesses and the natural frequency beside the f0(s') design curve, and the boxed apparent dynamic stiffness s't.

Download the report (PDF)

Dynamic-stiffness fiche (DynamicStiffnessResult.report): the resonance metrics and the f0(s') design curve.

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.

What this guide covers

Covered. EN 29052-1:1992 (identical to ISO 9052-1:1989) for materials under floating floors: the dynamic stiffness per unit area of Formula 1, the resonance relations of Formulae 2-4 (apparent_dynamic_stiffness, natural_frequency), the enclosed-gas term of Formula 7 (enclosed_gas_stiffness), and the clause 8.2 airflow-resistivity regimes of Formulae 5-6 (installed_dynamic_stiffness), chained end to end by floating_floor_resonance. DynamicStiffnessResult.report renders the clause 9 test-report fiche.

Not covered. Clause 7’s measurement procedure, extracting the resonant frequency fr from the raw excitation-response signal by sinusoidal, white-noise or pulse methods and extrapolating to zero force amplitude, is not implemented: pass an already-extrapolated fr. The lateral airflow resistivity r of clause 8.2 is likewise taken as an input rather than measured; ISO 9053’s static and alternating methods are implemented separately, in materials.airflow_resistance. Clause 6’s specimen-selection requirement (at least three 200 mm × 200 mm specimens) is not enforced.

See also

• API reference: materials.dynamic_stiffness.

References

• International Organization for Standardization. (1989). Acoustics — Determination of dynamic stiffness — Part 1: Materials used under floating floors in dwellings (ISO 9052-1:1989). The international original of EN 29052-1:1992, the method this page implements: 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 (Formulae 5-6). Conformance is anchored on the standard's own numeric NOTE plus hand-computed closed-form values of the resonance relations.
• Vigran, T. E. (2008). Building acoustics. CRC Press. https://doi.org/10.1201/9781482266016Floating-floor design and the role of the resilient layer's dynamic stiffness in the impact-sound improvement. ISBN 978-0-415-42853-8.