---
title: "Multiple-shock whole-body vibration (ISO 2631-5)"
description: "The ISO 2631-5:2018 spinal-response model for whole-body vibration containing multiple shocks: the seat-to-spine transfer function and acceleration dose of Clause 5, and the compressive stress, cumulative stress variable R and Weibull probability of lumbar injury of Annex C."
references:
  - type: book
    authors: ["Griffin, M. J."]
    year: 1996
    title: "Handbook of human vibration"
    publisher: "Academic Press"
    url: "https://shop.elsevier.com/books/handbook-of-human-vibration/griffin/978-0-12-303041-2"
    note: "ISBN 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."
  - type: standard
    organization: "International Organization for Standardization"
    year: 2018
    title: "Mechanical vibration and shock — Evaluation of human exposure to whole-body vibration — Part 5: Method for evaluation of vibration containing multiple shocks"
    designation: "ISO 2631-5:2018"
    url: "https://www.iso.org/standard/50905.html"
    note: "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)."
---

import ThemeImage from '../../../components/ThemeImage.astro';
import ReportPreview from '../../../components/ReportPreview.astro';

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.

<ThemeImage src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_iso2631_5.svg" alt="Flow from the vertical seat acceleration az(t) through the spinal response, the acceleration dose Dz and daily dose Dzd, the compressive stress Sd, the age-cumulated stress variable R, and the Weibull probability of lumbar injury" width="86%" loading="eager" />

## 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
```

<ThemeImage src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/multiple_shock.svg" alt="Left: the seat-to-spine transmissibility rising to about 1.6 near a 5 Hz resonance then rolling off to near zero by 80 Hz. Right: the Weibull probability of lumbar injury versus the stress variable R for male and female, with the 10, 50 and 90 percent risk levels and the Annex C male example at R = 1.22, about 37 percent" width="96%" />

<details>
<summary>Show the code for this figure</summary>

```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()
```

</details>

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](/phonometry/guides/human-vibration/) (ISO 2631-1).

## 4. The health-risk report

The dose chain exists to be written down and read against the Annex C guidance.
`MultipleShockResult.report()` 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, workplace/vehicle,
instrumentation, calibration), 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 $D_z$ (Formula 3),
the daily dose $D_{zd}$ (Formula 4), the daily compressive stress $S_d$
(Formula C.1), the cumulative stress variable $R$ (Formula C.3) and the
probability of lumbar injury $\Pi$ (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 $\Pi$
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).

The `metadata` argument accepts a `ReportMetadata` whose relevant 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: five 40 m/s2 spinal-response shocks per day,
# 82 kg male, exposed from age 20 for 20 years at 120 days/year. The record is
# rebuilt from those response peaks (the standard states the example in terms
# of the peaks), so the fiche prints the published Annex C values.
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 example fiche is regenerated with `make reports` and kept rendered in the
repository; click the preview to open the PDF.

<ReportPreview
	name="iso2631_5_multiple_shock_example"
	title="Whole-body multiple-shock health-risk example report (PDF)"
	description="One-page whole-body multiple-shock health-risk assessment fiche: a header with the transport operator, the subject and the off-road vehicle, the exposure-scenario grid (82 kg male, age 20, 20 years, 120 days per year, five counted shocks), the dose-and-stress analysis table with the acceleration dose Dz = 55.97 m/s2, the daily compressive stress Sd = 1.62 MPa, the cumulative stress variable R = 1.22 and the probability of lumbar injury 37 percent, the injury-probability curve with the Table C.2 risk levels and this assessment marked, the boxed R = 1.22 and 37 percent classed as a moderate probability of an adverse health effect, and the Annex C classification table where the moderate band (10 to 50 percent risk of injury) is the assessment's band."
	caption="Whole-body multiple-shock health-risk fiche (MultipleShockResult.report), the ISO 2631-5:2018 Annex C worked example with the dose-and-stress analysis and the Table C.2 risk classification."
/>

## See also

- API reference: [`vibration.multiple_shock_vibration`](/phonometry/reference/api/vibration/multiple-shock-vibration/).
