Human Vibration

Vibration transmitted to a person is evaluated with the same measurement chain whatever its origin: the acceleration is frequency-weighted to reflect how the body responds at each frequency, reduced to a weighted r.m.s. acceleration (with dose measures for shocks and long records), condensed across axes into a single magnitude (the vibration total value for hand-arm exposure and for comfort; the highest axis value for the whole-body health assessment), and finally normalised to an 8-hour daily exposure A(8) that is compared against the action and limit values of the European directive.

The weightings themselves are defined once, in ISO 8041-1:2017, as a cascade of analog filters; ISO 2631-1 applies them to whole-body vibration, ISO 2631-2 to vibration in buildings, ISO 2631-4 to rail ride comfort, and ISO 5349-1/-2 to hand-transmitted vibration. This page covers the whole chain.

1. Frequency weightings (ISO 8041-1)

Every human-vibration weighting is the product of four analog stages evaluated at $s=j2\pi f$ (ISO 8041-1 Formulae (1)–(5)): a second-order Butterworth high-pass and low-pass band limiting, an acceleration–velocity transition carrying the overall gain $K$, and an upward step:

$$H(s)=H_h(s)H_l(s)H_t(s)H_s(s).$$

A single Table 3 parameter set $(f_1,Q_1,f_2,Q_2,f_3,f_4,Q_4,f_5,Q_5,f_6,Q_6,K)$ realises all nine weightings (Wb, Wc, Wd, We, Wf, Wh, Wj, Wk, Wm), with a corner set to infinity collapsing its stage to unity. The principal whole-body weighting is Wk (vertical, seat surface); Wd is the horizontal weighting, and Wh the hand-arm weighting.

from phonometry import vibration

# The overall weighting response at any frequencies (ISO 8041-1 Formula (5)).
resp = vibration.frequency_weighting("Wk", [1.0, 6.3096, 20.0])
print(resp.magnitude.round(3))      # [0.482 1.054 0.636]  (factors)
print(resp.magnitude_db.round(2))   # [-6.33  0.46 -3.93]  (dB)

resp.plot()   # the weighting response in dB, as in the figure below (needs matplotlib)

The factors reproduce the ISO 8041-1 Annex B design-goal tables to their four significant figures: Wk plateaus near −6 dB below 2 Hz, peaks at +0.46 dB near 6.3 Hz and rolls off above.

Show the code for this figure

import numpy as np
import matplotlib.pyplot as plt
from phonometry import vibration

result = vibration.frequency_weighting("Wk", np.geomspace(0.4, 100.0, 240))

# One line:
result.plot()
plt.show()

# By hand, from the result's fields, mirroring what WeightingResponse.plot() draws:
fig, ax = plt.subplots()
ax.semilogx(result.frequencies, result.magnitude_db, color="#1f77b4")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Weighting factor [dB]")
ax.set_title("Whole-body vertical weighting Wk (ISO 8041-1)")
plt.show()

To weight a time signal, apply_weighting applies the exact complex response in the frequency domain (so magnitude and phase match the standard), which the time-domain dose metrics below then consume.

Which weighting on which axis

The weighting is selected by posture, measurement point and axis, not by application alone. ISO 2631-1 (Tables 1 and 2, clauses 7.2.3 and 8.2.2) maps them as follows; the multiplying factors $k$ return in the vibration total value of section 3:

Posture, measurement point	Axis	Weighting	$k$ health	$k$ comfort
Seated, seat surface	x, y	Wd	1.4	1.0
Seated, seat surface	z	Wk	1.0	1.0
Seated, seat surface (rotation)	rx / ry / rz	We	—	0.63 / 0.40 / 0.20 m/rad
Seated, backrest	x	Wc	(0.8)¹	0.8
Seated, backrest	y / z	Wd	—	0.5 / 0.4
Seated, feet	x / y / z	Wk	—	0.25 / 0.25 / 0.4
Standing, floor	x, y	Wd	—	1.0
Standing, floor	z	Wk	—	1.0
Recumbent, under the pelvis	horizontal	Wd	—	1.0
Recumbent, under the pelvis	vertical	Wk	—	1.0
Recumbent, under the head	vertical	Wj	—	1.0
Motion sickness (clause 9)	vertical	Wf	—	—

¹ The health assessment of clause 7 is defined on the seat surface; the backrest x measurement with Wc, $k=0.8$ is encouraged but excluded from the Annex B severity assessment (7.2.3). The remaining weightings live in the companion parts: Wm for building occupants on all axes (ISO 2631-2), Wb for vertical rail ride comfort (ISO 2631-4), and Wh for hand-transmitted vibration on all three hand axes with every $k=1$ (ISO 5349-1).

2. Weighted acceleration and dose measures (ISO 2631-1)

The basic evaluation is the weighted r.m.s. acceleration. From a one-third-octave spectrum it is (ISO 2631-1 Eq. (9); the identical construction gives the hand-arm $a_{hw}$ of ISO 5349-1 Eq. (A.1)):

$$a_w=\sqrt{\sum _i{\left(W_i{a}_i\right)}^2},$$

with $W_i$ the weighting factor at band centre $i$ and $a_i$ the measured band acceleration. The factors are evaluated at exactly the frequencies you pass; note that the ISO tables (ISO 8041-1 Annex B, ISO 2631-1 Table 3, ISO 5349-1 Table A.2) tabulate $W_i$ at the true one-third-octave centres $10^{n/10}$ Hz (6.31, 7.943, 15.85, …), not at the nominal band labels (6.3, 8, 16, …); pass true centres when comparing against the tabulated factors.

import numpy as np
from phonometry import vibration

# A measured vertical seat spectrum (r.m.s. per one-third octave, m/s^2).
freqs = np.array([1.0, 2.0, 4.0, 8.0, 16.0, 31.5, 63.0])
accel = np.array([0.20, 0.45, 0.42, 0.25, 0.12, 0.05, 0.02])

result = vibration.weighted_acceleration(accel, freqs, "Wk")
print(round(result.overall, 3))          # 0.555  m/s^2 (a_w)
print(result.weighted.round(3))          # W_i * a_i per band

result.plot()   # unweighted vs weighted bands, as in the figure below (needs matplotlib)

Show the code for this figure

import numpy as np
import matplotlib.pyplot as plt
from phonometry import vibration

freqs = np.array([1.0, 1.25, 1.6, 2.0, 2.5, 3.15, 4.0, 5.0, 6.3, 8.0, 10.0,
                  12.5, 16.0, 20.0, 25.0, 31.5, 40.0, 63.0, 80.0])
accel = np.array([0.18, 0.24, 0.33, 0.46, 0.52, 0.55, 0.48, 0.39, 0.31, 0.26,
                  0.21, 0.17, 0.13, 0.10, 0.078, 0.060, 0.045, 0.028, 0.020])
result = vibration.weighted_acceleration(accel, freqs, "Wk")

# One line:
result.plot()
plt.show()

# By hand, mirroring what WeightedSpectrum.plot() draws:
pos = np.arange(freqs.size)
fig, ax = plt.subplots()
ax.bar(pos - 0.2, result.band_accelerations, 0.4, color="#bbbbbb",
       label="Unweighted $a_i$")
ax.bar(pos + 0.2, result.weighted, 0.4, color="#1f77b4",
       label="Weighted $W_i a_i$ (Wk)")
ax.set_xticks(pos)
ax.set_xticklabels([f"{f:g}" for f in freqs], rotation=45, ha="right")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel(r"r.m.s. acceleration [m/s$^2$]")
ax.set_title(f"Weighted acceleration ($a_w$ = {result.overall:.3f} m/s²)")
ax.legend()
plt.show()

When the r.m.s. value understates an intermittent or shock-laden exposure, ISO 2631-1 adds dose measures computed on the weighted time signal: the running r.m.s. and its maximum, the maximum transient vibration value MTVV (Eq. (4), a 1 s running r.m.s.); the fourth-power vibration dose value $\text{VDV}={\left(\int a_w^{4}dt\right)}^{1/4}$ (Eq. (5)); the motion sickness dose value $\text{MSDV}={\left(\int a_w^{2}dt\right)}^{1/2}$; and the crest factor (peak / r.m.s.), whose value above 9 signals that the basic method is inadequate.

import numpy as np
from phonometry import vibration

fs = 1000.0
raw = np.random.default_rng(0).standard_normal(int(60 * fs))   # 60 s record
a_w = vibration.apply_weighting(raw, fs, "Wk")                        # weighted signal

print(round(vibration.vibration_dose_value(a_w, fs), 3))   # 0.744  VDV [m/s^1.75]
print(round(vibration.mtvv(a_w, fs), 3))                   # 0.265  MTVV [m/s^2]
print(round(vibration.crest_factor(a_w), 2))               # 3.74   crest factor

3. Vibration total value and daily exposure A(8)

Across the three axes the vibration total value combines the axis-weighted r.m.s. accelerations with the posture multiplying factors $k_j$ (ISO 2631-1 Eq. (10); for hand-arm, ISO 5349-1 Eq. (1) with every $k=1$):

$$a_v=\sqrt{\sum _j{k}_j^2{a}_{wj}^2}.$$

from phonometry import vibration

# Health, seated: k = 1.4 / 1.4 / 1.0 (ISO 2631-1, 7.2.3).
a_v = vibration.vibration_total_value([0.35, 0.28, 0.62], k=[1.4, 1.4, 1.0])
print(round(a_v, 3))     # 0.882  m/s^2

Health or comfort? Two different readings of the same measurement. The health assessment of clause 7 stays per axis: each axis-weighted r.m.s. (with $k$ = 1.4 / 1.4 / 1.0 seated) is judged by the highest single axis value against the Annex B health guidance caution zone, a band based mainly on 4 h to 8 h exposures below which health effects are not clearly documented, inside which caution is indicated and above which risks are likely; Directive 2002/44/EC turns that guidance into the enforceable A(8) action and limit values used below. The comfort assessment of clause 8 instead combines all axes (and, where relevant, backrest, feet and rotation) into the vibration total value $a_v$ with its own $k$ set and reads it on the Annex C scale for public transport, whose deliberately overlapping bands run from “not uncomfortable” below 0.315 m/s² to “extremely uncomfortable” above 2 m/s²; the standard defines no comfort limit, since acceptable magnitudes depend on trip duration and what the passengers are trying to do. For orientation, the median perception threshold of a Wk-weighted vibration is about 0.015 m/s² peak (Annex C).

The daily exposure normalises the exposure magnitude $a$ to a reference 8-hour day ($T_0=28800$ s). For a single operation $A(8)=a\sqrt{T/T_0}$; several operations combine through their partial exposures $A_i(8)=a_i\sqrt{T_i/T_0}$ as $A(8)=\sqrt{{\sum }_i{A}_i(8{)}^2}$ (ISO 5349-1 Eqs. (2)/(3); ISO 5349-2 Eqs. (1)–(3)). Directive 2002/44/EC fixes which magnitude $a$ each kind is based on (Annex, points 1): for hand-arm vibration the vector total $a_{hv}$ (Part A), but for whole-body vibration the highest frequency-weighted axis value $\max (1.4a_{wx},1.4a_{wy},a_{wz})$ (Part B), not the vector total $a_v$ above. wbv_exposure_basis returns that dominant-axis value:

from phonometry import vibration

# Directive 2002/44/EC whole-body basis (Annex Part B): the dominant axis.
a = vibration.wbv_exposure_basis(0.35, 0.28, 0.62)
print(round(a, 3))    # 0.62  m/s^2 (max of 0.49, 0.392, 0.62; not a_v = 0.882)

daily_vibration_exposure builds the partial exposures, combines them and assesses the result against Directive 2002/44/EC: hand-arm action value A(8) = 2.5 and limit value 5 m/s²; whole-body action 0.5 and limit 1.15 m/s² (or a VDV of 9.1 / 21 m/s¹·⁷⁵):

from phonometry import vibration

# ISO 5349-2 Annex E.3: a forestry worker's three chain-saw tasks.
result = vibration.daily_vibration_exposure(
    total_values=[4.6, 6.0, 3.6],                 # a_hv per task, m/s^2
    durations_s=[2 * 3600, 1 * 3600, 2 * 3600],   # exposure time per task
    kind="hav",
    labels=["brush-saw", "felling", "stripping"],
)
print(result.partials.round(2))            # [2.3  2.12 1.8 ]  A_i(8)
print(round(result.a8, 2))                 # 3.61  m/s^2
print(result.assessment.zone)             # 'action'  (2.5 <= A(8) < 5.0)

result.plot()   # partial exposures and A(8) against the EAV/ELV, as in the figure below (needs matplotlib)

Show the code for this figure

import numpy as np
import matplotlib.pyplot as plt
from phonometry import vibration

result = vibration.daily_vibration_exposure(
    [4.6, 6.0, 3.6], [2 * 3600, 1 * 3600, 2 * 3600], kind="hav",
    labels=["brush-saw", "felling", "stripping"],
)

# One line:
result.plot()
plt.show()

# By hand, mirroring what DailyVibrationExposure.plot() draws:
labels = [*result.labels, "A(8)"]
values = [*result.partials.tolist(), result.a8]
a = result.assessment
fig, ax = plt.subplots()
ax.bar(range(len(values)), values,
       color=["#bbbbbb"] * result.partials.size + ["#1f77b4"])
ax.axhline(a.action_value, color="#2ca02c", ls="--", label=f"EAV = {a.action_value:g}")
ax.axhline(a.limit_value, color="#d62728", ls="--", label=f"ELV = {a.limit_value:g}")
ax.set_xticks(range(len(values)))
ax.set_xticklabels(labels, rotation=30, ha="right")
ax.set_ylabel(r"Daily exposure A(8) [m/s$^2$]")
ax.legend()
plt.show()

4. Exposure–response guidance

For hand-transmitted vibration, ISO 5349-1 Annex C relates the daily exposure to the group-mean lifetime $D_y$ (in years) that produces vibration-white-finger in 10 % of an exposed group, $D_y=31.8A(8{)}^{-1.06}$ (Eq. (C.1)):

from phonometry import vibration

print(round(vibration.hav_vwf_lifetime_years(7.0), 1))   # 4.0 years (Table C.1)

The standards deliberately define no safe limit; A(8) and the directive’s action and limit values are the basis for any exposure criterion. For whole-body exposure, energy_equivalent_acceleration gives the ISO 2631-1 Eq. (B.3) energy-equivalent magnitude across periods of different magnitude and duration.

5. The exposure assessment report

A daily exposure is rarely the end of the job: it exists to be written down and compared with the law. DailyVibrationExposure.report() writes a one-page PDF assessment sheet laid out like the hand-arm and whole-body exposure calculators of occupational-hygiene practice (the HSE calculators, the EU Good Practice Guides): the standard-basis line naming the applied ISO method (ISO 5349-1/-2 for hand-transmitted vibration, ISO 2631-1 for whole-body vibration) and the directive it is assessed against, a header grid (company, operator/worker, workplace, instrumentation and calibration), the per-operation exposure analysis (each operation’s vibration magnitude, the hand-arm vector total $a_{hv}$ or the whole-body Directive Part B dominant-axis value $a_{w,max}$, its daily exposure time $T_i$ and the partial exposure $A_i(8)$, closed by the daily total and the combined $A(8)$) with the contribution chart, and the boxed $A(8)$ with its exposure zone.

Because the number exists to be compared with the law, the fiche then assesses $A(8)$ against Directive 2002/44/EC (Article 3): the exposure action value (EAV) and exposure limit value (ELV) for the vibration kind (hand-arm $2.5$ / $5$ m/s²; whole-body $0.5$ / $1.15$ m/s²), each marked exceeded / not exceeded on the value exactly as displayed, with a PASS/FAIL verdict against the limit value. A printed note records that the ISO standards define no safe exposure limit and that reaching the EAV triggers the employer’s control measures and the workers’ entitlement to health surveillance. verbose=True adds each operation’s share of the daily vibration energy, and language="es" renders the Spanish fiche (comma decimals).

The metadata argument accepts a ReportMetadata whose relevant fields for a vibration exposure report are client (the company), specimen (the operator or worker whose exposure was determined), test_room (the workplace), test_date, instrumentation, calibration, and the footer identity laboratory, operator, report_id and notes.

from phonometry import vibration, ReportMetadata

# The ISO 5349-2 Annex E.3 forestry worker: brush-saw (2 h, 4.6 m/s²),
# chain-saw felling (1 h, 6.0 m/s²) and branch stripping (2 h, 3.6 m/s²).
res = vibration.daily_vibration_exposure(
    [4.6, 6.0, 3.6],
    [2 * 3600, 1 * 3600, 2 * 3600],
    kind="hav",
    labels=["Brush-saw clearance", "Chain-saw felling", "Chain-saw branch stripping"],
)

res.report(
    "a8.pdf",
    metadata=ReportMetadata(
        client="Example forestry contractor",
        specimen="Forestry worker (right hand)",
        test_room="Managed woodland, plot 12",
        instrumentation="Hand-arm vibration meter (ISO 8041-1), s/n 0042",
        report_id="EXAMPLE-5349",
    ),
)   # A(8) = 3.61 m/s² -> action zone (EAV exceeded, ELV not), verdict PASS

The example fiche is regenerated with make reports and kept rendered in the repository; click the preview to open the PDF.

One-page daily vibration exposure assessment fiche: a header with the forestry contractor, the worker, the woodland workplace and the instrumentation, the per-operation exposure-analysis table (brush-saw clearance, chain-saw felling and branch stripping with the vibration total values, daily exposure times and partial exposures A_i(8) closed by the daily-total row), the per-operation contribution chart, the boxed daily exposure A(8) = 3.61 m/s2 in the action zone, and the Directive 2002/44/EC assessment table where the 2.5 m/s2 exposure action value is exceeded and the 5 m/s2 exposure limit value is not, ending in a PASS verdict.

Download the report (PDF)

Daily vibration exposure fiche (DailyVibrationExposure.report), the ISO 5349-2 Annex E.3 hand-arm day with the exposure analysis and the Directive 2002/44/EC assessment.

What this guide covers

Covered. ISO 8041-1:2017 as far as it defines the frequency weightings: the Table 3 cascade of analog stages (Formulae (1)-(5)) that frequency_weighting and apply_weighting realise for all nine weightings, reproducing the Annex B design-goal tables. ISO 2631-1:1997 for whole-body vibration, covering the weighted r.m.s. a_w and vibration total value a_v (weighted_acceleration, vibration_total_value), the MTVV, VDV, MSDV and crest-factor dose measures, and the Annex B energy-equivalent magnitude (energy_equivalent_acceleration). ISO 2631-2:2003 (Wm), ISO 2631-4:2001 (Wb) and ISO 5349-1/-2:2001 for hand-transmitted vibration, whose vector total, daily exposure A(8) (daily_vibration_exposure, wbv_exposure_basis) and Annex C vibration-white-finger dose relation (hav_vwf_lifetime_years) are implemented too. The Directive 2002/44/EC action and limit values close the assessment, written up by DailyVibrationExposure.report().

Not covered. ISO 8041-1’s own subject, the design and type-testing of general purpose vibration meters, is not implemented: only the frequency-weighting definitions are taken from it. ISO 2631-5:2018, the multiple-shock spinal-response and injury-risk model for whole-body vibration, is a separate standard with its own module: see Multiple Shock Vibration for records containing repeated shocks, which the r.m.s. and VDV measures on this page do not substitute for.

See also

• API reference: vibration.human_vibration.

References

• European Parliament and Council. (2002). Directive 2002/44/EC on the minimum health and safety requirements regarding the exposure of workers to the risks arising from physical agents (vibration) (Directive 2002/44/EC). Official Journal of the European Union. The exposure action and limit values the daily exposure A(8) is assessed against.
• Griffin, M. J. (1996). Handbook of human vibration. Academic Press. ISBN 978-0-12-303041-2. The standard monograph on whole-body and hand-transmitted vibration: the biodynamics, discomfort and health-effect evidence behind the weightings, dose measures and exposure-response guidance on this page.
• International Organization for Standardization. (1997). Mechanical vibration and shock — Evaluation of human exposure to whole-body vibration — Part 1: General requirements (ISO 2631-1:1997). The whole-body evaluation: the weighted r.m.s. and dose measures and the vibration total value.
• International Organization for Standardization. (2001). Mechanical vibration — Measurement and evaluation of human exposure to hand-transmitted vibration — Part 1: General requirements (ISO 5349-1:2001). Hand-transmitted vibration: the general requirements of the evaluation.
• International Organization for Standardization. (2001). Mechanical vibration — Measurement and evaluation of human exposure to hand-transmitted vibration — Part 2: Practical guidance for measurement at the workplace (ISO 5349-2:2001). Hand-transmitted vibration: practical guidance for measurement at the workplace.
• International Organization for Standardization. (2001). Mechanical vibration and shock — Evaluation of human exposure to whole-body vibration — Part 4: Guidelines for the evaluation of the effects of vibration and rotational motion on passenger and crew comfort in fixed-guideway transport systems (ISO 2631-4:2001). Rail ride comfort: the Wb weighting.
• International Organization for Standardization. (2003). Mechanical vibration and shock — Evaluation of human exposure to whole-body vibration — Part 2: Vibration in buildings (1 Hz to 80 Hz) (ISO 2631-2:2003). Vibration in buildings: the Wm weighting.
• International Organization for Standardization. (2017). Human response to vibration — Measuring instrumentation — Part 1: General purpose vibration meters (ISO 8041-1:2017). The frequency-weighting definitions and tolerances the implemented weightings follow.
• Mansfield, N. J. (2004). Human response to vibration. CRC Press. ISBN 978-0-415-28239-0. A compact modern textbook on the ISO 2631-1 and ISO 5349 evaluation chains, from perception and comfort to the occupational exposure limits.