Source: https://jmrplens.github.io/phonometry/devices/emission/vibration-sound-power/ # Sound power from surface vibration (ISO/TS 7849) The airborne sound power a machine radiates through the structure-borne vibration of its outer surface can be estimated from the surface vibratory velocity and a **radiation factor** $\varepsilon$ (the radiation efficiency), without an acoustic measurement. The radiated power is (ISO/TS 7849-1, Formula 6) $$ P = Z_\mathrm{c} \, \langle v^2 \rangle \, S \, \varepsilon \quad [\mathrm{W}], $$ with $Z_\mathrm{c}$ the characteristic impedance of air, $\langle v^2 \rangle$ the mean-square vibratory velocity over the radiating area $S$. Expressed in levels (velocity level re $v_0 = 5\times10^{-8}\ \text{m/s}$), the A-weighted sound power level is (Formula 12 / 15) $$ L_W = L_v + 10\log_{10}\frac{S}{S_0} + 10\log_{10}\varepsilon + 10\log_{10}\frac{Z_\mathrm{c,n}}{Z_{\mathrm{c},0}}, $$ where $S_0 = 1\ \text{m}^2$, the normalized impedance $Z_\mathrm{c,n} = 411\ \text{N·s/m}^3$ and the reference $Z_{\mathrm{c},0} = 400\ \text{N·s/m}^3$ give the fixed $10\log_{10}(411/400) = 0.118\ \text{dB}$ term. This module feeds the structure-borne source and building prediction standards (ISO 9611, EN 15657, EN 12354-5). Before any levels, the radiator itself. The `radiation_efficiency` plate model that supplies a predicted radiation factor retains its geometry, and `sigma.plot_geometry()` draws the plate in its baffle to scale, here 1.5 m by 1.25 m and simply supported. *The radiator behind the radiation factor, to scale: the 1.5 m by 1.25 m simply supported plate in its rigid baffle, whose area $S$ enters $L_W$ directly while its size decides how far $\varepsilon$ falls below one under coincidence.*

Show the code for this figure

```python import matplotlib.pyplot as plt import numpy as np from phonometry import vibration # The simply supported 1.5 x 1.25 m plate of the radiation model. f = np.geomspace(50.0, 5000.0, 200) sigma = vibration.radiation_efficiency(f, 1.5, 1.25, 2100.0) sigma.plot_geometry() plt.show() ```

Show the code for this figure

```python import matplotlib.pyplot as plt import numpy as np from phonometry import emission # Surface velocity levels and a measured radiation factor per octave band. bands = np.array([125.0, 250.0, 500.0, 1000.0, 2000.0, 4000.0]) lv = np.array([78.0, 82.0, 85.0, 83.0, 79.0, 74.0]) eps = np.array([0.20, 0.45, 0.75, 0.95, 1.00, 1.00]) lw_max = emission.radiated_sound_power_level(lv, 1.6) # Part 1, eps = 1 lw_eng = emission.radiated_sound_power_level(lv, 1.6, radiation_factor=eps) # Part 2 # One line — the LW(f) spectrum of one determination as a result object: res = emission.sound_power_from_vibration(lv, area=1.6, radiation_factor=eps, frequencies=bands) res.plot() plt.show() # By hand, comparing the two parts: x = np.arange(bands.size) fig, ax = plt.subplots() ax.bar(x - 0.2, lw_max, width=0.4, label="Part 1 upper limit ($\\varepsilon$ = 1)") ax.bar(x + 0.2, lw_eng, width=0.4, label="Part 2 engineering ($\\varepsilon$ measured)") ax.set_xticks(x, [f"{b:g}" for b in bands]) ax.set(xlabel="Frequency [Hz]", ylabel="Sound power level $L_W$ [dB re 1 pW]") ax.legend() plt.show() ```

## 1. The two parts The two parts differ only in the radiation factor. **Part 1 (survey)** assumes $\varepsilon = 1$ and yields the *upper limit* $L_{W,\max}$, needing only the velocity level and the area. **Part 2 (engineering)** applies a frequency-band radiation factor $\varepsilon_j$ determined (per ISO 9614) as $\varepsilon_j = P_j/(Z_\mathrm{c,n}\,\langle v_j^2 \rangle\,S)$. ```python import numpy as np from phonometry import emission bands = np.array([250.0, 500.0, 1000.0, 2000.0]) lv = np.array([82.0, 85.0, 83.0, 79.0]) # mean velocity level per band [dB] # Part 1 upper limit (epsilon = 1): upper = emission.sound_power_from_vibration(lv, area=1.6, frequencies=bands) print(round(upper.total_level, 1)) # e.g. 89.4 dB re 1 pW # Part 2 engineering value with a measured radiation factor: eps = np.array([0.45, 0.75, 0.95, 1.00]) eng = emission.sound_power_from_vibration(lv, area=1.6, radiation_factor=eps, frequencies=bands) print(np.round(eng.sound_power_level, 1)) # per-band L_W eng.plot() # the LW(f) spectrum, as in the figure above (needs matplotlib) ``` ## 2. Velocity level, calibration and the radiation factor The velocity level is $L_v = 20\log_{10}(v/v_0)$ (Formula 3); a sinusoidal calibration acceleration converts as $L_v = 20\log_{10}\!\left(\hat{a}/(2\pi f\,v_0\sqrt{2})\right)$ (Formula 8). The radiation factor comes from an independently measured power: ```python from phonometry import emission # The standard's worked calibration EXAMPLE: 9.81 m/s^2 at 100 Hz. print(round(float(emission.velocity_level_from_acceleration(9.81, 100.0)), 1)) # 106.9 dB # Radiation factor from a measured power (ISO 9614): eps = P / (Zc S). eps = emission.radiation_factor(3.0e-4, area=2.0, mean_square_velocity=(1e-3)**2) print(round(float(eps), 3)) # 0.365 ``` Surface velocity levels from several positions are combined with the energetic mean `mean_velocity_level` (Formula 10) or its area-weighted form (Formula 11), and the correction `extraneous_velocity_correction` removes extraneous vibration per Table 2. Those positions are not free: the standard divides the radiating surface $S$ into $N$ equal cells and puts one accelerometer at the centre of each. The area table gives the initial $N$; a strongly non-uniform vibration field can call for more positions or a redistribution. What is implemented is the calculation, not the laboratory practice: the measurement clauses of both parts (instrumentation, source installation, environmental conditions — clauses 5 to 7), their measurement-uncertainty clauses and the informative annexes are not code, so the position counts and the Table 2 ladder are rules the reader keeps, not checks the functions run. Part 2's clause 8 also defines the radiation factor of a machine batch or family, averaging $\varepsilon_j$ over several machines with its standard deviation (Formulae 9 and 10); only the single-machine Formula 8 is implemented, so pass an already-averaged $\varepsilon$ for a family determination. ## 3. When the radiation-factor assumption breaks The whole method stands on one substitution: replacing the acoustic measurement by $\varepsilon$. The Part 1 value $\varepsilon = 1$ is close to the true radiation factor only above the **critical (coincidence) frequency** of plate-like parts, where bending waves travel faster than sound and the surface radiates like a piston. Below coincidence, adjacent zones of the plate move in antiphase and their radiation largely cancels: $\varepsilon$ drops far below one and falls quickly with decreasing frequency, so the survey method can overstate the low-frequency bands of a large thin casing by 10 dB and more. The same cancellation makes small sources radiate poorly (the acoustic short circuit around an unbaffled panel). Two further assumptions are easy to violate in the field: * **The measured vibration must be the machine's own.** Vibration fed in from neighbouring machinery inflates $\langle v^2 \rangle$; Table 2 prescribes the source-off check and `extraneous_velocity_correction` applies it. * **The surface must be the dominant radiator.** Airborne sound from openings, intakes or internal sources that bypasses the measured casing is invisible to a velocity survey; the method characterises the structure-borne part only. "Adjacent zones move in antiphase and their radiation largely cancels" is a statement about a phase relationship between a travelling structural wave and the air on top of it, which no spectrum can draw. The clip below drives a 10 mm steel plate in air along its whole length — a force on the plate, not a wave arriving at it, and over the whole plate rather than a patch, because a patch radiates from the patch and below coincidence that would be the only thing in the air — and runs the same scene twice, at $f_\mathrm{c}/2$ and at $2f_\mathrm{c}$, with nothing differing between the panels but the drive frequency. [Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_elastic_radiation_efficiency.webm) Below coincidence the bending wavelength is **shorter** than the acoustic one. Adjacent half-waves push and pull the same air in antiphase, and what the air shows is a skin of alternating lobes that clings to the plate and dies within a fraction of a wavelength: the pressure at the surface is of the same order as in the other panel, but a quarter-cycle out of step with the velocity, so it carries no power away — which is what $\varepsilon \ll 1$ *means*. Above coincidence the bending wave outruns sound, the trace match is satisfied, and a plane beam leaves at $\arcsin(c_0/c_\mathrm{B}) = 45°$. Each panel is annotated with the closed forms its regime obeys, from the plate's own constants: below coincidence $\lambda_\mathrm{B} = 0.40$ m is shorter than the 0.57 m of sound in air, so $\sin\theta = \lambda/\lambda_\mathrm{B}$ has no solution and the skin that stands in its place decays over $1/\sqrt{k_\mathrm{B}^2 - k_0^2} = 0.091$ m; above it $\lambda_\mathrm{B} = 0.20$ m is longer than the 0.14 m in air and the trace match sends a beam out at 45°. Nothing is measured off the scene: the ends of the plate reflect the bending wave partially, so the air carries a leftward beam on top of the rightward one and any reading is the difference of the two. Two cautions: the scene is an effectively infinite plate, so it shows below-coincidence cancellation and not the short circuit around an unbaffled panel (the same cancellation at another scale), and the clip prints no radiation factor, because any finite driving aperture radiates a little on its own and below coincidence that would be the only thing an intensity line saw. The factor belongs to the closed-form model below. Part 2 exists exactly for the radiation-factor problem: it replaces the fixed $\varepsilon = 1$ with a band-by-band $\varepsilon_j$ determined from one reference measurement of the radiated power (ISO 9614 intensity), after which the velocity survey can be repeated cheaply on nominally identical machines. ## 4. The measurement report (`.report()`) A determination ends as a *document*. `VibrationSoundPowerResult.report(path)` writes a one-page PDF fiche laid out like a sound-power test sheet: the standard-basis line naming the applied method (the ISO/TS 7849-1 survey method with a fixed radiation factor $\varepsilon = 1$, or the ISO/TS 7849-2 engineering method with a determined radiation factor), an optional metadata header (client, machine/source, test environment, instrumentation, climate, date), a per-band table (nominal octave/one-third-octave frequency, the surface vibratory velocity level $L_v$ and the band sound-power level $L_W$), the sound-power spectrum $L_W(f)$ with a nominal band axis, and a boxed A-weighted sound power level $L_{W\mathrm{A}}$ (dB re 1 pW) with the total $L_W$, the radiating area $S$ and the applied method alongside. The relevant `ReportMetadata` fields are `client`, `specimen` (the machine/source), `test_room` (the test environment), `instrumentation`, `temperature`, `relative_humidity`, `pressure`, `test_date` and the footer identity `laboratory`, `operator`, `report_id` and `notes`; the radiating area $S$ comes from the result itself. Supplying `requirement` adds a PASS/FAIL verdict against a declared A-weighted sound-power limit (lower is better). `verbose=True` adds the radiation factor $\varepsilon$ column, and `language="es"` renders the Spanish fiche (comma decimals). Rendering needs the optional `phonometry[report]` extra (reportlab), plus matplotlib for the spectrum. ```python import numpy as np from phonometry import ReportMetadata, emission freqs = np.array([125, 250, 500, 1000, 2000, 4000], float) lv = np.array([78.0, 82, 85, 83, 79, 74]) # surface velocity level [dB] eps = np.array([0.20, 0.45, 0.75, 0.95, 1.00, 1.00]) # measured radiation factor res = emission.sound_power_from_vibration( lv, area=1.6, radiation_factor=eps, frequencies=freqs, ) res.report( "vibration_sound_power.pdf", metadata=ReportMetadata( client="Example manufacturing plant", specimen="Gearbox casing (steel panel)", test_room="Machine hall (source vibration survey)", instrumentation="Piezoelectric accelerometer (ISO 16063-21 calibration), s/n 0042", laboratory="Phonometry reference example", report_id="EXAMPLE-7849", requirement=90.0, ), ) # LWA = 88.7 dB(A) re 1 pW -> declared limit 90 dB(A): 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/iso7849_vibration_power_example.pdf) *Sound power from vibration fiche (`VibrationSoundPowerResult.report`), an ISO/TS 7849-2 engineering-method determination with the measured radiation factor and the boxed LWA.* ## See also - [Sound Power](https://jmrplens.github.io/phonometry/devices/emission/sound-power/): the route chooser this method is the sixth entry of, the accuracy grades and the ISO 4871 declaration a result feeds. - [Sound Power by Intensity Scanning (ISO 9614)](https://jmrplens.github.io/phonometry/devices/emission/sound-power-intensity/): the determination that produces the $P_j$ behind a Part 2 radiation factor. - [Sound Intensity (p-p)](https://jmrplens.github.io/phonometry/devices/emission/intensity/): the probe, its residual index and the field indicators that qualify that measurement. - [Panel Sound Insulation](https://jmrplens.github.io/phonometry/buildings/design/panel-sound-insulation/): the same radiation-efficiency model, written $\sigma$ instead of $\varepsilon$, and the coincidence physics behind it. - [Structure-borne source characterisation](https://jmrplens.github.io/phonometry/buildings/design/structure-borne-power/): ISO 9611, EN 15657 and EN 12354-5, the standards that consume this kind of result. - [Mechanical mobility (ISO 7626)](https://jmrplens.github.io/phonometry/vibration/structural/mechanical-mobility/): the transducer practice behind a measured surface velocity. - API reference: [`emission.vibration_sound_power`](https://jmrplens.github.io/phonometry/reference/api/power/vibration-sound-power/). - Theory: [Point mobilities and radiation efficiency](https://jmrplens.github.io/phonometry/reference/theory/vibration/#point-mobilities-and-radiation-efficiency-cremer-5-hopkins-29): the radiation efficiency that turns a surface velocity into a radiated power. ## 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 radiation-efficiency treatment behind section 3: coincidence, the cancellation below the critical frequency and the radiation of finite plates. - International Organization for Standardization. (2009). *Acoustics — Determination of airborne sound power levels emitted by machinery using vibration measurement — Part 1: Survey method using a fixed radiation factor* (ISO/TS 7849-1:2009). [iso.org catalogue](https://www.iso.org/standard/40537.html). The upper-limit method with $\varepsilon = 1$. - International Organization for Standardization. (2009). *Acoustics — Determination of airborne sound power levels emitted by machinery using vibration measurement — Part 2: Engineering method including determination of the adequate radiation factor* (ISO/TS 7849-2:2009). [iso.org catalogue](https://www.iso.org/standard/40538.html). The engineering method with a measured band-wise radiation factor. ## Standards ISO/TS 7849-1:2009 (*survey method using a fixed radiation factor*) and ISO/TS 7849-2:2009 (*engineering method including determination of the adequate radiation factor*), *Acoustics — Determination of airborne sound power levels emitted by machinery using vibration measurement*: the radiated power $P = Z_\mathrm{c} \, \langle v^2 \rangle \, S \, \varepsilon$ (Formula 6), the velocity level and its calibration (Formulae 3, 8), the mean over the surface (Formulae 10/11), the extraneous correction (Table 2), the radiation factor (Formula 4/8) and the sound power level (Formulae 12/15). Conformance is anchored on the standard's own worked calibration example, the exact round-trip between the radiation factor and $L_W = 10\log_{10}(P/P_0)$, and the fixed impedance term.