Vibration Shock Index (VSI) Overview

• Vibration Shock Index (VSI) is a family of scale-invariant, quantitative descriptors that measure the temporal localization of energy in vibration signals.
• It distinguishes impulsive, shock-dominated signals from continuous vibrations using methods like the shocking coefficient, excess kurtosis, and cumulative-energy analysis.
• VSI’s dimensionless metrics facilitate cross-platform comparisons and support applications in occupational health, reliability engineering, and nonlinear wave analysis.

The Vibration Shock Index (VSI) is a family of scale-invariant, quantitative descriptors designed to characterize and discriminate the temporal localization of energy in vibration signals, particularly in the context of impulsive or shock-dominated mechanical systems. A primary application is the assessment of hazardous transient vibration exposures, as in the analysis of hand-arm vibration from impact tools. Several mathematical formalisms have been proposed for VSI, each serving as a complement to traditional amplitude metrics such as root mean square (RMS) acceleration. VSI quantifies how energy is temporally concentrated—i.e., how much of the total signal "power" is carried by short-duration, high-amplitude events—enabling precise characterization of shocks distinct from continuous or noise-like backgrounds (Johannisson et al., 2022, Kinsler, 2015).

1. Rationale and Conceptual Overview

Traditional vibration metrics, such as RMS, reflect average signal power over time but fail to distinguish between sustained, periodic excitations and signals with pronounced, short-lived pulses carrying risk of acute injury or mechanical damage. VSI provides a dimensionless, amplitude-invariant classification of signal "shock content" by quantifying the degree of temporal energy localization, enabling rigorous discrimination between continuous, pulsed, and noise-like vibration phenomena (Johannisson et al., 2022). Applications span occupational health (e.g., risk quantification from impact wrenches), reliability engineering, and nonlinear wave physics (Kinsler, 2015).

2. Definitions and Mathematical Formulations

Multiple candidate VSI definitions have been articulated, guided by the principle that a "shock" is evidenced when a small fraction of samples contributes disproportionately to the total signal energy. The most prominent methods are:

2.1 Shocking Coefficient (S) Approach

Originally developed by Kinsler, this framework quantifies waveform steepening or shock formation through a bounded shocking coefficient $S$:

$$g(t) = \frac{dA(t)/dt}{\max_{t'}|dA_0(t')/dt|}\,,\quad S(t) = \frac{2}{\pi}\arctan(|g(t)| - 1)$$

where $A(t)$ is the observed waveform and %%%%3%%%% a reference (usually initial) profile. $S=0$ for unchanged gradients, $S\to 1$ for a true shock (infinite gradient), and negative $S$ indicates net smoothing ($|g|<1$) (Kinsler, 2015).

2.2 Kurtosis-based VSI

Defined for the power signal $P(t) = [a(t)]^2$, with the excess kurtosis:

$\mathrm{VSI}_\kappa = \frac{\mu_4}{\sigma^4} - 3$

where $\mu_4$ is the fourth central moment and $\sigma^2$ is the variance. Large positive values flag heavy-tailed, pulse-rich signals; negative or small values correspond to uniform or narrow-band signals (Johannisson et al., 2022).

2.3 Cumulative-Energy (Sorted Power) VSI

After sorting power samples $\{P_n\}$ in ascending order, compute the normalized cumulative energy $\widehat{W}_M$. For threshold $\theta = \frac{1}{2} - \frac{1}{\pi} \approx 0.18$:

$$M^* = \max\{M:\widehat W_M<\theta\},\qquad \mathrm{VSI}_c = \frac{M^*}{N - M^*}$$

This directly quantifies the proportion of energy carried by the largest excursions (Johannisson et al., 2022).

2.4 Weighted-Mean-Square (WMS) VSI

For exponent $K > 0$:

$$\mathrm{WMS}(K) = \frac{\sum_{n=1}^N P_n [P_n]^K}{\sum_{n=1}^N [P_n]^K};\qquad \mathrm{VSL}_w = \sqrt{\mathrm{WMS}(K)},\qquad \mathrm{VSI}_w = \frac{\mathrm{VSL}_w}{\mathrm{RMS}}$$

This provides a tunable balance between sensitivity to peak events and computational tractability (Johannisson et al., 2022).

3. Quantitative Behavior and Signal Discrimination

Empirical evaluation demonstrates that the VSI strongly discriminates between continuous, pulsed, and stochastic noise signals. The following table summarizes values from model signals (see (Johannisson et al., 2022)):

Method	Continuous VSI	Pulsed VSI	Noise VSI
Excess kurtosis	–1.5	35.6	12
Cumulative‐energy	1.0	17.7	2.0
WMS, $K=2$	1.29	5.1	2.2

Signals with pronounced shocks (e.g., Gaussian pulse trains) yield VSI values an order of magnitude greater than smooth, continuous waveforms. White Gaussian noise presents intermediate VSI, reflecting its heavy-tailed sample statistics, but without intentional temporal localization.

4. Physical Interpretation and Application Scenarios

The core term $|g|-1$ or, more generally, the high-order statistical thresholding intrinsic to VSI definitions, measures the "excess energy" localized within short time intervals. In applications such as nonlinear optics, VSI (via $S$) has been used to quantify self-steepening and shock formation in carrier waveforms (Kinsler, 2015). In occupational health and mechanical system monitoring, VSI enables the distinction of dangerous impulsive vibrations from benign continuous signals (Johannisson et al., 2022).

The VSI, being strictly amplitude-invariant, permits comparison and classification across experiments or machine platforms, irrespective of absolute vibration magnitudes. The complementary Vibration Shock Level (VSL), derived from WMS or cumulative-energy approaches, retains physical units and is sensitive to characteristic shock amplitude, supporting risk assessment and machinery diagnostics.

5. Implementation Methodologies

Algorithmic steps for VSI and VSL calculation are straightforward and amenable to both real-time and offline processing. For discrete signals $A[i]$, core steps include gradient calculation (shocking coefficient), power computation, sorting (cumulative-energy method), and high-order moment evaluation (kurtosis method). Filtering is essential to mitigate noise-induced spurious elevations. For WMS VSI, adjustable exponent $K$ enables application-dependent tuning—e.g., higher $K$ for greater shock selectivity.

Authors recommend pre-filtering for sensor drift, ensuring adequate time-record length for statistical convergence, and application of amplitude-invariant normalization as warranted by the chosen VSI variant (Johannisson et al., 2022).

6. Comparative Advantages, Limitations, and Extensions

Advantages

• Quantitative and dimensionless: replaces subjective judgments with robust, rigorous messaging
• Scale-invariance: classification is unaffected by absolute signal level
• Discriminatory: strong contrast between shock-rich and continuous signals, especially under cumulative-energy and kurtosis metrics
• Model independence: applicable to arbitrary vibration or waveforms

Limitations

• Reference-dependence: shocking coefficient $S$ is sensitive to initial profile choice
• First-derivative focus: standard VSI variants do not capture high-frequency oscillations or curvature without modification
• Sensitivity to noise: especially for gradient-based $S$, where preprocessing is required
• Compression at high extremes: arctan mapping in $S$ may diminish dynamic range for very strong shocks

Potential Extensions

• Local normalization using sliding windows for time-varying baselines
• Multi-term VSI including higher derivatives for curvature sensitivity
• Identification of threshold regimes (e.g., $S>0.8$) for control or safety triggers
• Application to spatial fields by direct replacement $t\to x, y, z$
• Coupling VSI with spectral metrics (e.g., for high-harmonic content)

7. Practical Implications and Standardization Prospects

The VSI constitutes a promising candidate for future incorporation into vibration exposure standards (e.g., ISO 5349), filling the gap between RMS-based quantification and injury/damage risk from impulsive sources. In practical instrumentation or monitoring, the choice between cumulative-energy and WMS VSI may depend on application priorities—maximal contrast or integration with existing root-mean-square infrastructure, respectively (Johannisson et al., 2022). The use of VSI/VSL duplex metrics—one scale-invariant, one amplitude-tracking—is advocated for comprehensive risk assessment in environments with significant shock or impact vibration.

The shocking coefficient $S$ and cumulative-energy or WMS-based VSI provide a common quantitative platform for cross-disciplinary shock waveform analysis, from nonlinear physics to applied engineering (Kinsler, 2015, Johannisson et al., 2022).