Vibration

This page collects the theory behind human vibration: the ISO 8041-1 frequency weightings, the whole-body and hand-arm metrics of ISO 2631-1 and ISO 5349, the action and limit values of Directive 2002/44/EC, and the ISO 2631-5 multiple-shock spinal model. It is part of the theory reference.

Human vibration (ISO 8041-1, ISO 2631-1/2, ISO 5349-1/2, Directive 2002/44/EC)

Human response to vibration depends on frequency, axis and body part, so acceleration is filtered by the frequency weightings of ISO 8041-1:2017 before any metric. Each weighting is the analog cascade $H(s)=H_h(s)H_l(s)H_t(s)H_s(s)$ (Formula 5): two-pole Butterworth band-limiting high-pass and low-pass stages (Formulae 1/2), an acceleration–velocity transition (Formula 3, carrying the only non-unity gain, $K=1.024$ for Wb) and an upward step (Formula 4), with the Table 3 corner frequencies and Q factors; a corner at infinity collapses its stage to unity (Table 3 NOTEs). Wk (vertical whole-body) and Wd (horizontal) of ISO 2631-1, Wm (buildings, ISO 2631-2), Wb (rail, ISO 2631-4), Wc/We/Wj (seat-back, rotational, head) and Wh (hand-arm, ISO 5349-1) plus Wf (motion sickness) are all implemented from the exact cascade (the filter is applied as the exact complex response via FFT, magnitude and phase, not a bilinear-warped digital approximation) and the ISO 8041-1 Annex B design-goal tables (B.1–B.9) are reproduced to 0.1 %.

The weighted metrics follow ISO 2631-1:1997: running rms with linear or exponential integration (Eqs. 2/3), MTVV as its maximum (Eq. 4), the fourth-power VDV $=(\int a_w^{4}dt{)}^{1/4}$ in m/s^1.75 (Eq. 5), the crest factor with the basic method deemed adequate up to 9 (clause 6.2), and the vibration total value $a_v=\sqrt{{\sum }_j{k}_j^2{a}_{wj}^2}$ (Eq. 10). Hand-arm exposure follows ISO 5349-1:2001: $a_{hv}$ (Eq. 1, all $k=1$), daily exposure $A(8)=a_{hv}\sqrt{T/T_0}$ with $T_0=8$ h (Eq. 2), partial exposures combined in quadrature (ISO 5349-2:2001, Eqs. 1–3), and the Annex C vascular-risk model $D_y=31.8A(8{)}^{-1.06}$ for the years to 10 % white-finger prevalence. The Directive 2002/44/EC action and limit values are built in: hand-arm $A(8)$ 2.5/5.0 m/s², whole-body $A(8)$ 0.5/1.15 m/s² or VDV 9.1/21.0 m/s^1.75 (Article 3). The ISO 5349-2 worked examples are reproduced (E.2.1: 7.4 m/s² for 2.5 h → $A(8)=4.1$ m/s²; E.3 forestry, three tools → 3.6 m/s²), as are the ISO 5349-1 Table C.1 exposure-duration rows.

The Wk whole-body weighting realized from the ISO 8041-1 cascade.

Multiple shocks (ISO 2631-5)

Repeated shocks damage the lumbar spine through peak compression rather than average energy, so ISO 2631-5:2018 replaces the Wk weighting with the seat-to-spine transfer function of clause 5.2 (Formula 1: one complex zero and six complex pole pairs, unity at DC, resonance near 5 Hz, $|H|\approx 1.54$ at 5 Hz) and accumulates the positive spinal-response peaks with a sixth-power (Palmgren-Miner) dose (clause 5.3, Formulae 3/4):

$$D_z=1.07{\left(\sum _i{A}_{z,i}^6\right)}^{1/6},D_{zd}=D_z(t_d/t_m{)}^{1/6}.$$

Annex C converts the daily dose to a compressive stress $S_d=m_z{D}_{zd}$ ($m_z=0.029/0.025$ MPa per m/s² for the 82 kg male / 64 kg female), tracks the age-declining ultimate strength $S_u=6.75-S_{age}(b+i)$ and forms the cumulative stress variable $R$ (Formulae C.3/C.4), mapped to an injury probability by the Table C.1 Weibull law $\Pi =1-e^{-(R/\alpha {)}^{\beta }}$. The spinal filter is evaluated analytically in the frequency domain and validated against the Annex D 256 Hz digital-filter tabulation within the clause 5.2 tolerance; the Annex C worked example (five 40 m/s² shocks per day over 20 years) is reproduced: $D_{zd}=55.97$ m/s², $R=1.22$, $\Pi =0.37$. The Annex A finite-element spinal model (distributed by ISO as separate software) is out of scope.

See the Human Vibration guide and the Multiple-Shock Vibration guide for usage.

Point mobilities and radiation efficiency (Cremer 5, Hopkins 2.9)

The vibrational power a point force injects into a structure is $W=\frac{1}{2}|F{|}^2\mathrm{Re}\{Y\}$ (Cremer Eq. 5.23), so the driving-point mobility $Y$ (the reciprocal of the impedance) governs how much energy the structure absorbs. For infinite structures these are closed forms (Cremer Table 5.1): an infinite thin plate is a pure resistance $Z=8\sqrt{B^{\prime }{m}^{\prime \prime }}$ (real, frequency independent, with $B^{\prime }$ the bending stiffness per unit width and $m^{\prime \prime }$ the mass per unit area), an infinite beam has $Y=(1-\mathrm{j})/(4m^{\prime }{c}_B)$ (a 45-degree phase, falling as ${\omega }^{-1/2}$ through the bending wave speed $c_B$), and a longitudinal rod has $Z=\rho c_{L}S$. These supply the receiver mobility EN 12354-5 needs when no measurement exists, and are the theoretical companions of the measured ISO 7626 mobilities. How efficiently a bending plate then radiates the airborne power is its radiation efficiency $\sigma$: below the critical frequency it radiates weakly (edge and corner modes), and above it $\sigma \to (1-f_c/f{)}^{-1/2}\to 1$ (Leppington/Maidanik, Hopkins Eqs 2.227-2.230). Because $\sigma$ is exactly the radiation factor $\epsilon$ of ISO 7849, predicting it closes the sound-power- from-vibration chain without a power measurement, and it drives the resonant transmission path of the panel sound insulation theory.

See the Predicting Panel Sound Insulation guide for usage.

Receptance, mobility and accelerance of a one-degree-of-freedom resonator: the same resonance seen through the three kinematic quantities.

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. The driving-point power W = (1/2) |F|^2 Re{Y} (Eq. 5.23) and the closed-form infinite-structure mobilities and impedances (Table 5.1) of the point-mobility section.
• Griffin, M. J. (1996). Handbook of human vibration. Academic Press. ISBN 978-0-12-303041-2. The biodynamic and health-effect evidence behind the ISO 8041-1 weightings, the rms/MTVV/VDV dose measures and the spinal-injury rationale of the multiple-shock model.
• Hopkins, C. (2007). Sound insulation. Butterworth-Heinemann. ISBN 978-0-7506-6526-1. The bending-plate radiation-efficiency theory (Eqs 2.227-2.230, the Leppington/Maidanik high-frequency limit) behind the radiation-efficiency section.
• 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). The radiation factor that equals the radiation efficiency, closing the sound-power-from-vibration chain of the radiation-efficiency section (survey method, fixed radiation factor).
• 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). The engineering method of the same sound-power-from-vibration chain, determining the adequate radiation factor from measurement.
• International Organization for Standardization. (2011). Mechanical vibration and shock — Experimental determination of mechanical mobility — Part 1: Basic terms and definitions, and transducer specifications (ISO 7626-1:2011). The measured driving-point mobilities that the closed-form infinite-structure results of the point-mobility section are the theoretical companions of.
• Mansfield, N. J. (2004). Human response to vibration. CRC Press. ISBN 978-0-415-28239-0. A compact modern walkthrough of the ISO 2631-1 whole-body and ISO 5349 hand-arm evaluation chains summarised on this page.