Source: https://jmrplens.github.io/phonometry/buildings/design/installed-structure-borne/ # Installed structure-borne sound from equipment (EN 12354-5) **EN 12354-5:2009** predicts the sound pressure level in a receiving room caused by building service equipment (pumps, fans, lifts, water installations) that injects **structure-borne sound** into the building. It closes the structural-vibroacoustics chain: the source is described by its characteristic structure-borne sound power level $L_{W\mathrm{s,c}}$, derived from the EN 15657 reception-plate measurement through the Formula (15)/(17) conversion and a mobility correction (**not** the raw plate-injected level; see [EN 15657](https://jmrplens.github.io/phonometry/buildings/design/structure-borne-power/)); the source and receiver point mobilities set how much power is actually coupled into the structure, and the building transmission carries it to the receiving room. The Annex I mobility correction `installed_power_from_reception_plate` refers the characteristic reception-plate level $L_{W\mathrm{s,n}}$ to the actual receiver, $L_{W\mathrm{s,inst}} = L_{W\mathrm{s,n}} + 10\log_{10}(Y_{\infty,i} / Y_{\infty,\mathrm{rec}})$ with $Y_{\infty,\mathrm{rec}} = 5\cdot 10^{-6}\ \text{m/(N·s)}$; with the source mobility instead it yields $L_{W\mathrm{s,c}}$ (Annex I.3, Table I.8).

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import building # EN 15657 characteristic source power and illustrative point mobilities. bands = np.array([63.0, 125.0, 250.0, 500.0, 1000.0, 2000.0, 4000.0]) lws_c = np.array([78.0, 82.0, 84.0, 81.0, 77.0, 72.0, 66.0]) ys = (2e-4 + 1e-4j) * (bands / 250.0) # source mobility yi = (3e-5 + 1e-5j) * np.ones_like(bands) # receiver mobility dc = np.array([float(building.coupling_term(a, b)) for a, b in zip(ys, yi)]) # Two transmission paths: the excited floor and a flanking wall. paths = [ {"adjustment_term": 6.0, "element_area": 12.0, "flanking_reduction_index": np.linspace(44.0, 62.0, 7)}, {"adjustment_term": 7.0, "element_area": 9.0, "flanking_reduction_index": np.linspace(46.0, 64.0, 7)}, ] res = building.installed_source_prediction(lws_c, dc, paths, frequencies=bands) res.plot() # characteristic and installed power, per-path levels and the total plt.show() ```

## 1. Coupling and installed power Only part of the characteristic power is injected into the supporting element; the loss is the **coupling term** $D_\mathrm{C}$, positive whenever the two mobilities are well mismatched, set for a point excitation by the source mobility $Y_\mathrm{s}$ and the receiver mobility $Y_i$ (Formula 19b): $$ D_{\mathrm{C},i} = 10\log_{10}\frac{|Y_\mathrm{s} + Y_i|^2}{|Y_\mathrm{s}|\,\mathrm{Re}\{Y_i\}}, $$ which reduces to $10\log_{10}(|Y_\mathrm{s}|/\mathrm{Re}\{Y_i\})$ for a **force source** (high source mobility, Formula 19c) and to $-10\log_{10}(|Y_\mathrm{s}|\,\mathrm{Re}\{Z_i\})$ for a **velocity source** (low source mobility, Formula 19d); an elastic support adds its transfer mobility $Y_k$ inside the modulus (Formula 19e). The **installed** power level is then (Formula 18b) $L_{W\mathrm{s,inst}} = L_{W\mathrm{s,c}} - D_\mathrm{C}$. The physics behind $D_\mathrm{C}$ is the classical power input of a point-excited plate: only the real part of the receiver's driving-point mobility absorbs power, and the mismatch between $Y_\mathrm{s}$ and $Y_i$ decides how much of the source's capability ever enters the structure (Hopkins 2007, Section 2.8). The mobilities themselves come from the [mechanical-mobility chain](https://jmrplens.github.io/phonometry/vibration/structural/mechanical-mobility/): measured per ISO 7626, or from the infinite-plate closed forms of [panel theory](https://jmrplens.github.io/phonometry/buildings/design/panel-sound-insulation/) when no measurement exists. A pump on a concrete slab is the textbook force source: its casing mobility is orders of magnitude above the slab's, so $D_\mathrm{C}$ collapses to Formula 19c and the injected power no longer depends on the receiver at all. ```python from phonometry import building # A near-force source (Y_s >> Y_i) on a concrete floor: dc = building.coupling_term(2e-4 + 1e-4j, 3e-5 + 1e-5j) print(round(float(dc), 2)) # 9.86 dB print(round(float(building.installed_structure_borne_power_level(82.0, dc)), 1)) # installed L_Ws ``` Nothing measures the source mobility of a boiler, so clause D.1.3 builds it out of the machine's own parts and Table D.1 gives the six closed forms it uses. `typical_element_mobility` is that table: the rows are `"mass"` $[2\pi f M]^{-1}$, `"bar_end"` $[\rho c_\mathrm{L} S]^{-1}$, `"beam"` $[7{,}6\,\rho t w\sqrt{c_\mathrm{L} t f}]^{-1}$, `"plate"` $[2{,}3\,c_\mathrm{L}\rho t^2]^{-1}$, `"pipe"` $[63\,\rho t r\sqrt{c_\mathrm{L} r f}]^{-1}$ and `"mass_spring"` $\left[\left(\frac{2\pi f\eta}{s(1+\eta^2)}\right)^2 + \left(\frac{2\pi f}{s(1+\eta^2)} - \frac{1}{2\pi f M}\right)^2\right]^{1/2}$, each in m/N.s. `TABLE_D1_QUANTITIES` is the table's "describing quantities" column, and only those may be passed; the four rows whose expression contains $f$ take `frequency` and the other two refuse it. ```python from phonometry import building # A 120 kg pump on four rubber mounts, 1,0e6 N/m each, loss factor 0,1. y_feet = building.typical_element_mobility( "mass_spring", frequency=125.0, mass=120.0, stiffness=4.0e6, loss_factor=0.1 ) print(f"{float(y_feet):.3g}") # 0.000185 m/(N.s) ``` ## 2. Transmission to the receiving room EN 12354 parts 1 and 2 handle rooms excited through the air and by the standard tapping machine; part 5 covers the sources that shake the building *directly*: pumps, fans, lifts, whirlpool baths, cisterns and the pipework that ties them into walls and floors. Once the installed power is in the structure, several elements radiate into the receiving room: the excited element itself and every element the vibration reaches across the junctions. Each excited-element/radiating-element pair $i \to j$ is a transmission path with its own adjustment term and flanking index: Each transmission path $i \to j$ gives a normalised sound pressure level from the installed power, the structure-to-airborne adjustment term $D_\mathrm{sa}$, the flanking sound reduction index $R_{ij,\mathrm{ref}}$ (EN 12354-1) and the element area (Formula 18a): $$ L_{\mathrm{n,s},ij} = L_{W\mathrm{s,inst},i} - D_{\mathrm{sa},i} - R_{ij,\mathrm{ref}} - 10\log_{10}\frac{S_i}{S_0} - 10\log_{10}\frac{A_0}{4}, $$ with $S_0 = A_0 = 10\ \text{m}^2$, and the paths combine energetically (Formula 17). $D_\mathrm{sa}$ is normally **negative** — the standard's own Annex I columns run from about $-14$ dB at 63 Hz to $-45$ dB at 2 kHz — and Formula (18a) subtracts it, so a negative value *raises* the predicted level. Annex F.2 gives the working form $D_{\mathrm{sa},i} = 10\log_{10}(400 f_{\mathrm{c},i}\sigma_i / m_i f^2)$, which is `structure_to_airborne_adjustment`; passing a positive number in its place leaves the prediction 20 to 40 dB low with no error raised. ```python import numpy as np from phonometry import building bands = np.array([250.0, 500.0, 1000.0]) res = building.installed_source_prediction( characteristic_power_level=np.array([80.0, 82.0, 78.0]), coupling_term=np.array([9.0, 10.0, 11.0]), paths=[ {"adjustment_term": np.array([-19.0, -25.0, -31.0]), "flanking_reduction_index": np.array([50.0, 52.0, 55.0]), "element_area": 12.0}, {"adjustment_term": np.array([-22.0, -28.0, -34.0]), "flanking_reduction_index": np.array([52.0, 54.0, 57.0]), "element_area": 8.0}, ], frequencies=bands, ) print(np.round(res.total_level, 1)) # total L_n,s per band print(round(res.overall_level, 1)) # band-summed level [dB] res.plot() # the per-path and total L_n,s cascade, as in the figure above (needs matplotlib) ``` The `InstalledSourceResult` carries the per-path levels, the total per band, the installed power level and `.overall_level`, and its `.plot()` draws the whole cascade. Where the receiving room is more than one junction away, clause F.1 subtracts an adjustment $\Delta K$ from the summed junction $K_{ij}$ to cover the transmission by wave types other than bending waves: `multi_junction_adjustment` returns the 4 dB it gives for two junctions and the 6 dB for three or more, with the resulting $K_{ij}$ floored at `MINIMUM_MULTI_JUNCTION_KIJ` ($-5$ dB). When the equipment itself cannot be characterised, clause D.1.2.3 substitutes a source of known force level and Table F.1 gives that level for the ISO tapping machine, in octave bands from 31,5 Hz to 4 kHz: 139, 142, 145, 148, 151, 154, 156 and 156 dB. `tapping_machine_force_level` returns them, `tapping_machine_force_level_estimate` the closed form printed beside the table (valid only up to about 1000 Hz, and the only route to one-third-octave values), and `tapping_machine_characteristic_power_level` / `tapping_machine_coupling_term` turn them into the $L_{W\mathrm{s,c}}$ and $D_\mathrm{C}$ of Formulae (D.9a) and (D.9b). The table's levels are re $10^{-6}$ N despite the "re 1 pN" its caption prints; see [the errata register](https://jmrplens.github.io/phonometry/reference/errata/). ```python import numpy as np from phonometry import building bands = np.array(building.TABLE_F1_OCTAVE_BANDS) lw_c = building.tapping_machine_characteristic_power_level( bands, building.tapping_machine_force_level() ) dc = building.tapping_machine_coupling_term(bands, 1.07e-6) # 220 mm concrete print(np.round(lw_c - dc, 1)) # [79.3 82.3 85.3 88.3 91.3 94.3 96.3 96.3] ``` ## 3. The prediction report (`.report()`) A prediction ends as a *document*. `InstalledSourceResult.report(path)` writes a one-page PDF fiche, clearly labelled a prediction and not a measurement: a prediction-basis line naming EN 12354-5:2009, an optional metadata header (client, source equipment, receiving room, instrumentation, climate, date), a per-band table (nominal octave/one-third-octave frequency, the installed structure-borne power level $L_{W\mathrm{s,inst}}$, each transmission path's normalised SPL $L_{\mathrm{n,s},ij}$ and the combined total $L_\mathrm{n,s}$), the per-path and total $L_\mathrm{n,s}(f)$ spectra, and a boxed band-summed total $L_\mathrm{n,s}$ (dB) with the installed power total and the path count. The relevant `ReportMetadata` fields are `client`, `specimen` (the source equipment), `test_room` (the receiving room), `instrumentation` and the footer identity `laboratory`, `operator`, `report_id` and `notes`. Supplying `requirement` adds a PASS/FAIL verdict against a declared upper limit on the overall $L_\mathrm{n,s}$ (lower is better). `verbose=True` adds one column per transmission path (up to five); otherwise only the installed power and the combined total are shown. `language="es"` renders the Spanish fiche. The basis strip states Formulae 18a/17 and the prediction disclaimer. Rendering needs the optional `phonometry[report]` extra (reportlab), plus matplotlib for the plot. ```python import numpy as np from phonometry import ReportMetadata, installed_source_prediction bands = np.array([63, 125, 250, 500, 1000, 2000], float) lwc = np.array([84.4, 82.5, 69.9, 67.6, 61.6, 49.9]) # characteristic power [dB] dsa = np.array([-13.6, -17.3, -17.4, -20.0, -26.9, -32.9]) paths = [ {"adjustment_term": dsa, "flanking_reduction_index": np.array([43.0, 46, 50.2, 54.7, 64.6, 73]), "element_area": 12.8}, {"adjustment_term": dsa, "flanking_reduction_index": np.array([37.0, 41.2, 35.9, 37.7, 49, 57.8]), "element_area": 12.8}, ] res = installed_source_prediction(lwc, 16.2, paths, frequencies=bands) res.report( "installed_structure_borne.pdf", metadata=ReportMetadata( client="Example dwelling refurbishment", specimen="WC flushing cistern (wall-fixed)", test_room="Receiving room: adjacent bedroom", report_id="EXAMPLE-12354-5", requirement=45.0, ), ) # overall L_n,s ~ 43 dB -> declared limit 45 dB: PASS ``` The rendered example fiche, regenerated with `make reports`, is kept in the repository. Click the preview to open the PDF: [](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/en12354_5_installed_structure_borne_example.pdf) *Installed structure-borne prediction fiche (`InstalledSourceResult.report`), an EN 12354-5 estimate of the normalised structure-borne SPL in the receiving room.* ## See also - [Structure-borne sound power of equipment (EN 15657)](https://jmrplens.github.io/phonometry/buildings/design/structure-borne-power/): the reception-plate characterisation that supplies $L_{W\mathrm{s,c}}$ and the source mobility this prediction consumes. - [Mechanical mobility and the FRF family (ISO 7626-1)](https://jmrplens.github.io/phonometry/vibration/structural/mechanical-mobility/): the measured $Y_\mathrm{s}$ and $Y_i$ behind the coupling term. - [Bending-wave transmission at plate junctions](https://jmrplens.github.io/phonometry/vibration/structural/junction-transmission/): the junction physics that carries the installed power to the flanking radiators. - [Predicting Sound Insulation (EN 12354)](https://jmrplens.github.io/phonometry/buildings/design/insulation-prediction/): the airborne and impact members of the same prediction family. ## References - Cremer, L., Heckl, M., & Petersson, B. A. T. (2005). *Structure-borne sound: Structural vibrations and sound radiation at audio frequencies* (3rd ed.). Springer. ISBN 978-3-540-22696-3. [doi:10.1007/b137728](https://doi.org/10.1007/b137728). The source-receiver mobility coupling and the structure-borne transmission across junctions behind the coupling term and the path model. - Hopkins, C. (2007). *Sound insulation*. Butterworth-Heinemann. ISBN 978-0-7506-6526-1. [doi:10.4324/9780080550473](https://doi.org/10.4324/9780080550473). Section 2.8 (driving-point impedance and mobility): the power a mechanical source injects into a plate through the source and receiver mobilities. ## Standards EN 12354-5:2009, *Building acoustics — Estimation of acoustic performance of buildings from the performance of elements — Part 5: Sound levels due to service equipment*: the coupling term (clause 4.4.3, Formulae 19a-19e), the installed structure-borne power level (Formula 18b), the structure-to- airborne adjustment term (clause 4.4.4, Formulae 20a/20b), and the normalised sound pressure level per path and its energetic combination (Formulae 18a, 17). Conformance is anchored on the coupling-term force-source limit and the standard's own Annex I worked examples: the whirlpool bath of I.2 (Table I.6a: mobility correction and path 11) and the flushing cistern of I.3 (Tables I.8/I.9: source conversion, all four transmission paths, the Formula 17 total and its 29 dB(A) closure), within the ±0.15 dB rounding of the printed one-decimal intermediates. The informative tables of the annexes are implemented as named lookups: Table D.1 (mobility of typical construction elements), Table F.1 (force level of the ISO tapping machine) with Formulae (D.9a) and (D.9b), the adjustment term of Formula (F.3) and the multi-junction $\Delta K$ of clause F.1. $R_{ij,\mathrm{ref}}$ remains an input (from measurement or EN 12354-1). Only the 2009 edition is implemented, not the EN 12354-5:2023 revision.