Gaussian Random Vibration

Technical provenance

Applicable standards / specifications

• IEC 60068-2-64 (2008+AMD1:2019) — Environmental testing — Part 2-64: Tests — Test Fh: Vibration, broadband random and guidance
• GSFC-STD-7000B (B (2021)) — General Environmental Verification Standard for GSFC Flight Programs and Projects — Applicable to relevant GSFC flight programmes; not a universal random-vibration specification.

References

• Newland, D. E. — An Introduction to Random Vibrations, Spectral & Wavelet Analysis — Background reference for PSD-based random processes and structural response.
• NASA GSFC-STD-7000B — General Environmental Verification Standard (GEVS) — Space-hardware environmental verification reference; project-specific environments govern.

What Is It?

The Gaussian (normal) distribution is the most commonly assumed amplitude distribution for random vibration. It describes the probability of the instantaneous amplitude taking any given value. For a Gaussian process, the distribution is fully defined by two parameters: the mean and the standard deviation. Most engineering random vibration analysis assumes Gaussian excitation and therefore Gaussian response.

Why It Matters

The Gaussian assumption underpins the statistical interpretation of random vibration results — sigma levels, peak factors, exceedance probabilities. If the distribution is not Gaussian, these interpretations may be incorrect. Understanding when the Gaussian assumption is reasonable and when it is not is essential for credible random vibration assessment.

The Gaussian assumption underpins sigma-level screening and peak estimation. If the amplitude distribution is not Gaussian, the statistical interpretation of the response may be wrong.

Gaussian Amplitude Distribution

A Gaussian random process has instantaneous values that follow the normal probability density function. The distribution is symmetric about the mean, with the characteristic bell-curve shape. For vibration, the mean is typically zero, so the distribution is centred at zero and characterised solely by the standard deviation (RMS).

Gaussian probability density function:

p(x)  =  (1 / (σ√(2π))) · exp( −x² / (2σ²) )

where:
σ  =  standard deviation  (= RMS for zero-mean)
x  =  instantaneous amplitude

The distribution is fully defined by σ.

Mean and Standard Deviation

For a zero-mean Gaussian process — the usual case in vibration — the mean is zero and the standard deviation equals the RMS. The standard deviation determines the spread of the distribution. A larger standard deviation means the instantaneous values are more widely distributed — larger peaks are more likely.

• Mean (μ) — average value, typically zero for vibration
• Standard deviation (σ) — spread of the distribution, equals RMS for zero-mean
• The entire distribution is defined by σ alone when μ = 0

RMS and Sigma Levels

For a Gaussian process, the probability of the instantaneous amplitude exceeding a given number of standard deviations is well defined. The 1σ level is exceeded about 32% of the time. The 2σ level about 4.6%. The 3σ level about 0.27%. These probabilities refer to the instantaneous value at any sample — not to the peak over a duration.

Peak Probability

The Gaussian distribution describes the instantaneous amplitude, not the peak over a duration. The peak over a duration depends on how many samples or cycles occur — more cycles mean a higher likely peak. This is why 3σ is not a maximum — over a long enough duration, the response will exceed 3σ. The expected peak over N cycles of a narrowband Gaussian process is approximately σ·√(2·ln(N)), which grows slowly with N.

The Gaussian distribution describes instantaneous values, not peaks over a duration. Over a long enough duration, peaks will exceed 3σ. Three-sigma is a screening level, not a maximum.

Why Gaussian Is Assumed

The Gaussian assumption is supported by the central limit theorem: when a signal is the sum of many independent random sources, the amplitude distribution tends to Gaussian regardless of the individual source distributions. Many engineering vibration environments — acoustic excitation from turbulent boundary layers, broadband machinery noise, road vehicle vibration — are the sum of many sources, making the Gaussian assumption reasonable.

• Central limit theorem — sum of many independent sources tends to Gaussian
• Many engineering environments are the sum of many sources
• Gaussian is mathematically convenient — fully defined by mean and standard deviation
• Standard random vibration analysis methods assume Gaussian input and response

Limitations of Gaussian Assumptions

The Gaussian assumption is not universally valid. Several situations can produce non-Gaussian vibration, where the standard sigma-level interpretation may be incorrect.

When to Question Gaussian

The Gaussian assumption should be questioned when the excitation is dominated by a few discrete sources, when the structural response is nonlinear (contact, gaps, plasticity), or when the environment is impulsive rather than continuous. In these cases, the actual peak distribution may have heavier tails than Gaussian — meaning peaks larger than 3σ occur more frequently than the Gaussian prediction. Time-domain analysis with rainflow counting may be more appropriate.

If the excitation is dominated by few sources, or the response is nonlinear, the amplitude distribution may not be Gaussian. Peaks may exceed the Gaussian prediction. Consider time-domain analysis.

Key Takeaways

• Gaussian distribution is fully defined by mean and standard deviation (RMS for zero-mean)
• Sigma levels (1σ, 2σ, 3σ) describe instantaneous exceedance probabilities, not peak over duration
• The central limit theorem supports the Gaussian assumption for broadband multi-source environments
• Non-Gaussian situations: few dominant sources, nonlinear response, contact/gaps, impulsive excitation
• When non-Gaussian, peaks may be larger than predicted — consider time-domain analysis