Source: https://jmrplens.github.io/phonometry/guides/multiple-shock-vibration/ # Multiple-shock whole-body vibration (ISO 2631-5) Vibration that contains repeated **mechanical shocks** (off-road vehicles, high-speed marine craft, earth-moving machinery) loads the lumbar spine far more than its equivalent r.m.s. level suggests. **ISO 2631-5:2018** predicts the resulting spinal response and the risk of lumbar injury from the measured vertical seat acceleration. phonometry implements the normative **Clause 5** spinal-response model and the **Annex C** health-effect assessment. (The Annex A / Annex E finite-element model is distributed by ISO as separate software and is out of scope here.) The boundary with the basic method is explicit. ISO 2631-1 declares its r.m.s. evaluation normally sufficient up to a crest factor of 9, and offers the running r.m.s./MTVV and the VDV beyond it; ISO 2631-5 is the additional method for the regime past those, where the record contains repeated shocks. Its clause 4 then splits that regime in two: *severe* conditions with possible free fall or loss of contact with the seat and a dominant z-axis (military off-road vehicles, high-speed marine craft) use the clause 5 model implemented here, while *less severe* conditions in which the occupant stays seated throughout (tractors, forestry and earth-moving machinery on rough ground) belong to the Annex A finite-element model. In case of doubt the delineation is quantitative: when the band-limited vertical peak acceleration exceeds 9.81 m/s² (1 g, the free-fall threshold), clause 5 and Annex C apply. ## 1. Spinal response (clause 5.2) A seat-to-spine transfer function ``H(f)`` (one complex zero and six complex poles, unity transmissibility at 0 Hz and a resonance near 5 Hz) maps the measured seat acceleration to the vertical spinal response ``Az(t)`` (Formula 1/2): $$ A_z(t) = \mathcal{F}^{-1}\!\left[H(f)\,\mathcal{F}[a_z(t)]\right]. $$ The input record must be **conditioned (DC-removed)**: because `H` is unity at 0 Hz by design, a DC offset (e.g. the 1 g gravity component of a DC-coupled accelerometer) passes straight into `Az(t)` and corrupts the positive response peaks of the dose. Subtract the mean (or high-pass) before processing. ```python from phonometry import vibration # The transmissibility peaks near the ~5 Hz spinal resonance. print(round(abs(vibration.seat_to_spine_transfer([2.0])[0]), 2)) # 1.06 print(round(abs(vibration.seat_to_spine_transfer([5.0])[0]), 2)) # 1.54 ``` ## 2. Acceleration dose (clause 5.3) The **acceleration dose** combines the positive response peaks ``Az,i`` (each the maximum between two consecutive zero crossings) with a sixth-power law, so the largest shocks dominate (Formula 3): $$ D_z = 1.07\left(\sum_i A_{z,i}^{\,6}\right)^{1/6}. $$ A daily dose scales the measured dose to the daily exposure time ``td`` over the measurement time ``tm`` (Formula 4): ``Dzd = Dz * (td/tm)**(1/6)``. ```python from phonometry import vibration # Five 40 m/s2 response peaks in a day (the Annex C worked example). print(round(vibration.dose_from_peaks([40.0] * 5), 2)) # 55.97 m/s2 ``` ## 3. Injury risk (Annex C) The daily dose becomes a daily **compressive stress** ``Sd = mz * Dzd`` (Formula C.1), where ``mz`` (0.029 MPa per m/s² for an 82 kg male, 0.025 for a 64 kg female) converts acceleration to vertebral stress. The stress accumulates over the exposure years against the ageing spine's reducing ultimate strength ``Su = 6.75 - Sage*(b+i)`` (Formulae C.3/C.4): $$ R = \left[\sum_{i=0}^{n-1} \left(\frac{S_d\,N^{1/6}}{S_{u,i} - S_{\mathrm{stat}}}\right)^{6}\right]^{1/6}, $$ and a Weibull model gives the probability of lumbar injury (Formula C.5): $$ \Pi(R) = 1 - \exp\!\left[-\left(\frac{R}{\alpha}\right)^{\beta}\right]. $$ ```python from phonometry import vibration # Annex C worked example: 5 x 40 m/s2/day, 82 kg male, age 20 for 20 years, # 120 days/year. dz = vibration.dose_from_peaks([40.0] * 5) sd = vibration.compression_dose(dz) # 1.62 MPa r = vibration.injury_risk(sd, start_age=20, years=20, days_per_year=120) print(round(r, 2)) # 1.22 print(round(100 * vibration.injury_probability(r))) # 37 % risk of injury ``` From a measured time history the whole chain is one call: ```python import numpy as np from phonometry import vibration # A synthetic 10 s seat record at 256 Hz with five 60 m/s2 shocks (stand-in # for a measured az(t)). fs = 256.0 az = np.zeros(2560) az[256::512] = 60.0 result = vibration.multiple_shock_assessment( az, fs, start_age=20, years=20, days_per_year=120, sex="male", ) print(round(result.acceleration_dose, 2)) # 20.94 m/s2 print(round(result.risk, 2)) # 0.46 print(round(result.probability, 2)) # 0.03 result.plot() # the injury-probability curve with this assessment's R marked (needs matplotlib) ```

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```python import numpy as np import matplotlib.pyplot as plt from phonometry import vibration # Left: the seat-to-spine transfer function magnitude. f = np.logspace(np.log10(0.5), np.log10(80.0), 400) plt.plot(f, np.abs(vibration.seat_to_spine_transfer(f))) plt.xscale("log"); plt.show() # Right: the injury-probability curve with this assessment's R marked # (az/fs as in the snippet above). fs = 256.0 az = np.zeros(2560) az[256::512] = 60.0 vibration.multiple_shock_assessment( az, fs, start_age=20, years=20, days_per_year=120, ).plot() plt.show() ```

The `MultipleShockResult` carries the dose ``Dz``, the daily dose ``Dzd``, the compressive stress ``Sd``, the stress variable ``R``, the injury probability and the response peaks, and its `.plot()` draws the injury-probability curve with the 10/50/90 % risk thresholds of Table C.2. The model is vertical-axis only: clause 4 neglects the horizontal contributions to spinal compression by design, and the horizontal spinal model of the withdrawn 2004 edition is not reinstated, so horizontal whole-body exposure is assessed instead with the r.m.s., running-r.m.s./MTVV and VDV metrics of [Human Vibration](https://jmrplens.github.io/phonometry/reference/api/vibration/human-vibration/) (ISO 2631-1). ## 4. The health-risk report (`.report()`) The dose chain exists to be written down and read against the Annex C guidance. `MultipleShockResult.report(path)` writes a one-page PDF health-risk assessment sheet: the standard-basis line (Clause 5 spinal response and the Annex C risk model), an optional metadata header (client, subject via `specimen`, workplace/vehicle via `test_room`, and the `instrumentation` and `calibration` free-text fields of `ReportMetadata`), the exposure-scenario grid (subject sex, the age `b` at which the exposure started, the number of exposure years `n`, the number of exposure days per year `N` and the number of counted response shocks), and the dose-and-stress analysis table with the acceleration dose `Dz` (Formula 3), the daily dose `Dzd` (Formula 4), the daily compressive stress `Sd` (Formula C.1), the cumulative stress variable `R` (Formula C.3) and the probability of lumbar injury `Π` (Formula C.5). Because ISO 2631-5:2018 defines no exposure limit, the fiche carries a risk-band zone row rather than a PASS/FAIL verdict: the boxed `R` and `Π` name the Annex C risk classification, and a classification table places `R` among the Table C.2 stress variables for 10 / 50 / 90 % risk of injury (low / moderate / high / very high probability of an adverse health effect), with the injury-probability chart above it. `language="es"` renders the Spanish fiche (comma decimals). Rendering needs the optional `phonometry[report]` extra (reportlab), plus matplotlib for the chart. The relevant `ReportMetadata` fields for a multiple-shock report are `client`, `specimen` (the subject), `test_room` (the workplace or vehicle), `test_date`, `instrumentation`, `calibration`, and the footer identity `laboratory`, `operator`, `report_id` and `notes`. ```python import numpy as np from phonometry import vibration, ReportMetadata # The Annex C worked example, rebuilt from its five 40 m/s2 response peaks. peaks = np.array([40.0] * 5) dz = vibration.dose_from_peaks(peaks) sd = vibration.compression_dose(dz) r = vibration.injury_risk(sd, start_age=20, years=20, days_per_year=120) result = vibration.MultipleShockResult( sex="male", acceleration_dose=dz, daily_dose=dz, compression_dose=sd, risk=r, probability=float(vibration.injury_probability(r)), start_age=20.0, years=20, days_per_year=120.0, peaks=peaks, risk_thresholds=vibration.RISK_THRESHOLDS_MALE, ) result.report( "multiple_shock.pdf", metadata=ReportMetadata( client="Example transport operator", specimen="82 kg male operator (seated)", test_room="Off-road vehicle, driver's seat", report_id="EXAMPLE-2631-5", ), ) # R = 1.22, Pi = 37 % -> moderate probability of an adverse health effect ``` 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/iso2631_5_multiple_shock_example.pdf) *Whole-body multiple-shock health-risk fiche (`MultipleShockResult.report`), the ISO 2631-5:2018 Annex C worked example with the Table C.2 risk classification.* ## References - Griffin, M. J. (1996). *Handbook of human vibration*. Academic Press. ISBN 978-0-12-303041-2. [Publisher page](https://shop.elsevier.com/books/handbook-of-human-vibration/griffin/978-0-12-303041-2). Background on whole-body shock exposure, spinal biodynamics and the lumbar health effects that the ISO 2631-5 dose model quantifies. ## Standards ISO 2631-5:2018, *Mechanical vibration and shock — Evaluation of human exposure to whole-body vibration — Part 5: Method for evaluation of vibration containing multiple shocks*: the seat-to-spine transfer function (clause 5.2, Formula 1), the acceleration and daily dose (clause 5.3, Formulae 3-5) and the Annex C assessment of adverse health effects (Formulae C.1, C.3-C.5, Tables C.1/C.2).