Vibration Shock Level

• Vibration Shock Level is a rigorous metric that quantifies transient, high-amplitude vibrations using methods such as the cumulative-energy and RWMS approaches.
• It overcomes RMS limitations by focusing on short, energy-dense pulses, offering a more accurate assessment of shock severity.
• VSL is applied in fields like engineering and occupational safety to create statistically sound test envelopes and risk assessments.

The Vibration Shock Level (VSL) provides a mathematically rigorous measure of the characteristic amplitude of shock or transient vibration in time series and multivariate environmental data. VSL is specifically designed to quantify vibrational signals dominated by intermittent, high-amplitude pulses, which are poorly characterized by classical time-averaged root-mean-square (RMS) metrics. VSL can be defined both for scalar and multivariate data, with methodologies reflecting statistical, probabilistic, and signal-processing perspectives. Two principal classes of VSL are prevalent: (1) scalar VSLs for localized, high-energy transient identification in a single axis or channel, and (2) multivariate, probability-based VSLs (often also called VSL) for environmental specification in multi-axis testing and reliability contexts (Johannisson et al., 2022, Watts et al., 2024).

1. Motivation and Problem Statement

Traditional vibration metrics, notably the ISO 5349-1 RMS acceleration $a_{\mathrm{RMS}}$, systematically underestimate the severity of low-duty-cycle shock or transient events encountered in environments such as handheld power tool operation, mechanical impact systems, or space hardware qualification (Johannisson et al., 2022). RMS evaluates the squared mean, which yields diminishing values for short, intense pulses spaced widely in time. The RMS thus fails to reflect the true physiological or mechanical "risk" associated with short-lived high-amplitude shocks.

The VSL concept is introduced to address two distinct but related needs:

• A quantification of the typical amplitude of the shocks occurring in a vibration signal, robust to noise and signal scaling.
• For multi-axis cases, a statistically conservative threshold ensuring that the environmental specification envelope covers a prescribed fraction of possible extreme responses, accounting for axis correlations (Watts et al., 2024).

2. Scalar VSL: Mathematical Definitions and Properties

Two mathematically precise scalar VSL definitions are prominent (Johannisson et al., 2022):

2.1 Cumulative-Energy Method

This approach defines VSL at the threshold of the sorted energy (power) distribution. Given a sampled time record $a_n$ ($n=1,\dots,N$), instantaneous power $P_n = a_n^2$:

1. Sort $$\tilde P_{(1)} \leq \tilde P_{(2)} \leq \cdots \leq \tilde P_{(N)}$$.
2. Compute cumulative energy $W_M = \Delta t \sum_{n=1}^M \tilde P_{(n)}$.
3. For chosen $\alpha$ (usually $$\alpha = \tfrac{1}{2} - \tfrac{1}{\pi} \approx 0.18$$), find largest $M^*$ with $W_{M^*} < \alpha W_{\mathrm{tot}}$, $$W_{\mathrm{tot}} = \Delta t \sum_{n=1}^N \tilde P_{(n)}$$.
4. The VSL is

$$\mathrm{VSL}_{\mathrm{CE}} = \sqrt{ \tilde P_{(M^*+1)} }$$

This definition yields a physically interpretable "shock level": the amplitude at which the signal's highest-energy portions contribute a fixed fraction of total energy.

2.2 Weighted Mean Square (RWMS) Method

Define, for exponent $K \geq 0$:

$$\mathrm{WMS}(K) = \frac{ \int_0^T a^{2(K+1)}(t) dt }{ \int_0^T a^{2K}(t) dt }$$

The corresponding VSL is

$$\mathrm{VSL}_{\mathrm{RWMS}}(K) = \sqrt{ \mathrm{WMS}(K) }$$

The recommended value is $K=2$:

$$\mathrm{VSL}_{\mathrm{RWMS}} = \sqrt{ \frac{ \int a^6(t) dt }{ \int a^4(t) dt } }$$

This family of measures transitions smoothly from classical RMS ($K=0$) to increasingly peak-weighted amplitude characterizations as $K$ increases.

2.3 Key Properties

• Scale invariance: Both methods are scale-conformant: multiplying $a(t)$ by $c$ scales VSL by $|c|$.
• Time localization: Both methods select amplitude scales set by transient peaks, not bulk averages.
• Robustness: The energy-fraction and finite-$K$ weighting limit extreme sensitivity to outliers compared to pure max or RMS.

Table: Example Signal Responses (Johannisson et al., 2022)

Signal Type	VSL$_{\mathrm{CE}}$ ($\alpha=0.18$)	VSL$_{\mathrm{RWMS}},\,K=2$	RMS
Continuous Sine	$\sim 1.0$	$\sim 1.3$	1.0
Pulsed Train	$\sim 2.4$	$\sim 5.1$	1.0
White Gaussian	$\sim 1.0$	$\sim 2.2$	1.0

For pulsed trains, the VSL rises by a factor of 4–5 over RMS.

3. Multivariate VSL: Quantile-Based Specifications

For multi-axis shock data, the Vibration Shock Level is formalized as a multivariate normal quantile threshold, providing statistically meaningful coverage across all axes (Watts et al., 2024).

3.1 Multivariate Quantile Definition

Let $X = (X_1, ..., X_d)^\top \sim N(\mu, \Sigma)$. The VSL at probability level $\tau$ (e.g., $0.90$) is the set:

$$Q_\tau = \left\{ x \in \mathbb{R}^d : F_X(x) = \tau \right\}$$

where $F_X$ is the joint cumulative distribution function. The "critical point" $v(\tau)$ on $Q_\tau$, i.e., the highest-density point on the iso-surface $F_X(x) = \tau$, is often used as a practical, conservative specification:

$v(\tau) = \arg\max_{x \in Q_\tau} f_X(x)$

where $f_X$ is the multivariate normal PDF.

3.2 Confidence Interval Methods

Two Monte Carlo approaches enable the estimation of confidence bounds on the VSL:

• CDF-tessellation + percentile bootstrap: CDF values are computed on a finely tessellated grid in standardized space over bootstrap replicates, constructing percentile intervals for $F_X(x)$, and hence confidence bounds for $Q_\tau$ as iso-surfaces.
• Critical-point bootstrap: For each bootstrap replicate, the highest-density quantile point $v^*$ is computed and transformed back to original units, forming component-wise or joint percentile confidence intervals.

The resultant upper-bound vector $v_U(\tau)$ provides the conservative multi-axis VSL, exceeding or equaling simultaneous single-axis tolerance bounds by accounting for inter-axis correlations. Specifically, given nine repeated shock runs with tri-axial data and $\tau=0.90$, a typical VSL upper bound might be $[10.05, 18.26, 4.58]\,$G for $(X, Y, Z)$, compared to smaller univariate tolerance bounds (Watts et al., 2024).

4. Theoretical Justification and Comparison to RMS

VSL definitions rectify key deficiencies of RMS for signals dominated by localized shocks:

• RMS fails for low-duty-cycle pulses: as inter-pulse interval increases, RMS vanishes regardless of instantaneous peak. VSL methods lock onto energy-dense time regions, providing amplitude measures closely tracking actual shock severity (Johannisson et al., 2022).
• In the multivariate case, simultaneous univariate or Bonferroni-adjusted bounds can under-cover the true joint quantile set, particularly when variables are correlated. The multivariate VSL, constructed via the aforementioned quantile surfaces, offers explicit probabilistic guarantees for simultaneous axis coverage (Watts et al., 2024).

5. Recommended Practices and Methodological Steps

The literature prescribes the following protocols for computing VSL in domain applications:

1. Field Data Acquisition: Collect repeated multi-axis shock runs, compute maximum response per test/frequency bin.
2. Normality Assessment: Check d-variate normality (e.g., Mahalanobis distances, QQ-plot, Anderson–Darling) (Watts et al., 2024).
3. Parameter Choices: Set desired quantile probability $\tau$ and confidence level $\alpha$.
4. Algorithmic Execution:
   • For scalar signals: Apply the cumulative-energy ($\alpha = 0.18$) or RWMS ($K=2$) definition (Johannisson et al., 2022).
   • For multi-axis: Standardize data; apply critical-point bootstrap with $B$ repetitions; form confidence intervals (Watts et al., 2024).
5. Spec Envelope Construction: Iterate across all frequency bins to assemble a multi-axis, frequency-dependent specification envelope.
6. Interpretation: Use the one-sided upper confidence bound as the conservative VSL for test planning; compare to traditional univariate or RMS-based specifications.

6. Applications and Extension to Environmental Specification

VSL metrics underpin the design and validation of mechanical shock and vibration test procedures in engineering, occupational health, and device qualification. By offering quantifiable, reproducible, and interpretable measures that capture the elusive “severity” of highly localized transients, VSL facilitates:

• Risk assessment and regulatory compliance in human exposure standards (e.g., hand–arm vibration) (Johannisson et al., 2022).
• Conservative test envelope specification for multi-axis hardware testing, ensuring coverage of extreme field responses with quantifiable confidence (Watts et al., 2024).
• Benchmarking and comparison across devices, environments, and mitigation strategies.

This suggests that future work will focus on correlating VSL (and the related VSI for "shockiness") with physical measures of damage or injury, evaluating robustness under different acquisition scenarios, and extending the family of definitions to accommodate non-Gaussian and nonstationary data sets.

7. Limitations and Future Directions

Both the cumulative-energy and RWMS approaches rely on continuous signals with reasonable signal-to-noise ratios and may be sensitive to acquisition window and filtering choices. The multivariate VSL as implemented is dependent on the normality assumption; extensions to non-normal, heavy-tailed, or multimodal distributions may require more sophisticated modeling. The validation of VSL values against independent markers of physiological or mechanical consequence remains a key research challenge. Robustness evaluation across environments (e.g., with ultrahigh-frequency components) and linkage to accelerated biomechanical trials are noted as recommended next steps (Johannisson et al., 2022, Watts et al., 2024).