Shock Index (SI) in Vibration Analysis

• Shock Index (SI) is a dimensionless metric that quantifies the temporal localization of energy in vibration signals by identifying transient, high-amplitude shocks.
• It employs methodologies such as the cumulative-energy and weighted mean-square approaches, offering robust and scale-invariant assessments of vibration shocks.
• SI is pivotal in applications like occupational health risk assessments and machinery diagnostics, where it complements RMS measurements to better characterize impulsive forces.

The Shock Index (SI), or more formally Vibration Shock Index (VSI), is a dimensionless measure quantifying the degree of time localization of a vibration signal's energy. It is specifically designed to characterize signal components that occur as transient, high-amplitude events ("shocks") and to distinguish them from signals with more uniform energy distribution. The VSI and its companion metric, the Vibration Shock Level (VSL), provide robust, scale-invariant quantification of both the presence and typical amplitude of shocks in vibration signals, allowing for improved assessment and comparison of sources that may cause injury, particularly in applications involving hand-held machines emitting impulsive forces (Johannisson et al., 2022).

1. Problem Formulation and Motivation

Standard root mean square (@@@@3@@@@) metrics compute the overall energy content of a vibration signal but treat all time samples equivalently. In signals where the majority of energy is concentrated in infrequent, high-amplitude bursts separated by low-activity intervals, the RMS value may be misleadingly low, as it is depressed by the long stretches of low acceleration. This shortcoming necessitates complementary indices. The required metrics should:

• Quantify the temporal localization of the signal’s power, distinguishing between continuous, pulsed, and noise-like signals.
• Remain invariant under rescalings of amplitude and time (i.e., $a(t) \rightarrow c \cdot a(t)$, $t \rightarrow \lambda t$).
• Be robust to measurement noise and statistical outliers.
• Include a scale-free index (VSI) and a shock amplitude metric (VSL) with the same units as the original signal.

This need arises prominently in vibration exposure risk assessments, where transient shocks are a significant hazard but are not adequately captured by RMS values (Johannisson et al., 2022).

2. Candidate Definitions of VSI and VSL

Multiple candidate formulations for VSI are based on the instantaneous power $P(t) = a(t)^2$, with varying operational and interpretive characteristics.

Excess Kurtosis of the Power Signal

Defined as: $\text{VSI}_{\text{kurt}} = \kappa - 3$ where

$\kappa = \frac{E[(P - E[P])^4]}{(\text{Var}[P])^2}$

This measure is scale-invariant and sensitive to heavy-tailed distributions in $P$, with extremes such as continuous sine-waves yielding $\kappa-3 \approx -1.5$, strong pulses $\approx +35$, and white Gaussian noise $\approx +12$. However, it is less intuitive and lacks an immediate companion for VSL.

Cumulative-Energy (“Energy-Step”) Method

For a sampled signal $\{a_n\}_{n=1}^N$:

1. Compute $P_n = a_n^2$.
2. Sort $\{P_n\}$ in ascending order to obtain $\{\tilde{P}_i\}$.
3. Define normalized cumulative energy:

$$\hat{W}(M) = \frac{\sum_{i=1}^M \tilde{P}_i}{\sum_{i=1}^N \tilde{P}_i}$$

1. With threshold $\alpha$ (recommended $\alpha = 1/2 - 1/\pi \approx 0.1817$), find $M^*$ where $\hat{W}(M^*) < \alpha$.
2. Define:

$$\text{VSI}_{\text{cum}} = \frac{M^*}{N - M^*}, \quad \text{VSL}_{\text{cum}} = \sqrt{\tilde{P}_{M^* + 1}}$$

Interpretation: If a small fraction $M^*$ of low-power samples contains a significant portion $\alpha$ of the total energy, VSI will be large, indicating strong energy localization.

Weighted Mean-Square ("Moment-Ratio") Method

Given $K$ (commonly $K=2$):

1. Define moment-weighted mean squares:

$$\text{WMS}(K) = \frac{\sum_{n=1}^N P_n \cdot P_n^K}{\sum_{n=1}^N P_n^K} = \frac{\sum a_n^{2(K+1)}}{\sum a_n^{2K}}$$

2. Compute:

$\text{RWMS} = \sqrt{\text{WMS}(K)}$

and

$\text{RMS} = \sqrt{\frac{\sum a_n^2}{N}}$

3. Define:

$$\text{VSI}_{\text{wms}} = \frac{\text{RWMS}}{\text{RMS}}, \quad \text{VSL}_{\text{wms}} = \text{RWMS}$$

Increasing $K$ emphasizes higher-power samples, with $K=0$ reducing to the conventional RMS.

3. Rationale and Comparative Evaluation

Central to both the cumulative-energy and weighted-mean-square approaches is their scale invariance and robustness to outliers. The cumulative method is grounded in the statistical observation that in highly impulsive (shocked) signals, most energy is confined to a small number of samples. This direct quantification, through percentile-based summation, provides high discrimination, is unambiguous, and mitigates the effect of individual spikes. The weighted-MS method extends the RMS metric with monomial weights: its closed-form is advantageous for streaming-data applications, and the parameter $K$ controls sensitivity trade-offs between shock detection and noise.

Empirical values for characteristic model signals demonstrate the discriminating power:

Signal Type	VSI_cum	VSL_cum/RMS	VSI_wms (K=2)	VSL_wms/RMS
Harmonic (100 Hz sine)	~1.0	~1.0	~1.29	~1.29
Pulsed (Gaussian train)	~17.7	~2.4	~5.1	~5.1
White Gaussian noise	~2.0	~1.0	~2.2	~2.2

For real-world impact tool signals, VSL is typically 2–5 times the RMS value, highlighting the inadequacy of RMS alone in such contexts (Johannisson et al., 2022).

4. Signal Models and Benchmarking

Model signals considered in the development and evaluation of VSI/VSL include:

• Continuous (harmonic): $a_c(t) = -A_c (2\pi f_c)^2 \cos(2\pi f_c t)$, $f_c = 100~\text{Hz}$, $A_c$ set for unit RMS.
• Pulsed: $a_p(t)$ is the second derivative of a periodic train of Gaussian-displacement pulses (period $T_p = 0.1~\text{s}$, width $t_p = 1/\sqrt{2\pi f_c}$), amplitude adjusted to unit RMS.
• White Gaussian noise: zero mean, unit RMS.

These signals span the dominant morphologies encountered in physical measurement environments: steady oscillation, impulsive transients, and stochastic (thermal/electronic) noise. Comparative metric evaluation on these classes establishes both methodological validity and utility for field deployment.

5. Practical Computation and Implementation

Computation of VSI/VSL proceeds as follows:

• Uniformly sample $a_n$ ($n=1 \dots N$) at interval $\Delta t$, ideally over spans covering many seconds.
• Pre-filtering is applied as per measurement standards or to extend frequency bandwidth for ultravibration.
• For the cumulative method:
  1. Compute $P_n = a_n^2$
  2. Sort to obtain $\{\tilde{P}_i\}$
  3. Compute cumulative sums until $\hat{W}(i)$ meets threshold $\alpha \approx 0.1817$
  4. Calculate VSI$_{\text{cum}}$ and VSL$_{\text{cum}}$

• For the weighted-MS method:

  • Accumulate $S_0 = \sum a_n^2$, $S_2 = \sum a_n^4$, $S_3 = \sum a_n^6$
  • At end: compute RMS, WMS(2), RWMS, VSI$_{\text{wms}}$, VSL$_{\text{wms}}$

Parameter choices:

• Cumulative method: $\alpha \approx 0.18$ (ensures VSI=1 for pure harmonic)
• Weighted-MS method: $K=2$ is standard

Both approaches are mathematically rigorous with respect to scale invariance, and sorting or moment accumulations ensure robustness to noise and outliers.

6. Interpretation and Application

A substantially elevated VSI (e.g., $\text{VSI} \gg 1$) reliably signals strong concentration of energy in a minority of temporal samples—i.e., the presence of shocks. VSL offers a typical shock amplitude, grounded in a statistical definition, rather than dependent on peak detection or RMS. Practically, these indices should be reported in parallel with RMS in measurement standards, enabling enhanced assessment of transient/impact content for sources such as impact wrenches. Their adoption is particularly pertinent in occupational health risk quantification, where transient vibrations have direct clinical significance for injury (Johannisson et al., 2022).

7. Concluding Remarks

The VSI and VSL metrics, specifically the cumulative-energy and weighted mean-square formulations, address longstanding limitations in vibration signal characterization by introducing mathematically stringent, robust, and physically meaningful quantifications for shock vibrations. Their implementation facilitates direct comparison of vibration sources and supports more nuanced standards and regulatory frameworks targeting impulsive injury mechanisms. Future work would focus on broader experimental validation and standardization in field applications.