Power Spectral Density

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?

A power spectral density (PSD) describes how the power of a random signal is distributed across frequency. It is the standard engineering characterisation of random vibration environments. An acceleration PSD defines the vibration environment; a stress PSD describes the dynamic stress response. Understanding what a PSD represents — and what it does not — is essential for credible random vibration analysis.

Why It Matters

The PSD is the input to random vibration FEA. It defines the excitation environment that the structure must withstand. Misinterpreting a PSD — confusing spectral density with amplitude, or reading a PSD value as a sine amplitude — leads to incorrect analysis inputs and incorrect engineering conclusions. The PSD is also the bridge between the time domain and the frequency domain for random signals.

Physical Meaning

A PSD tells how much variance (mean-square amplitude) of a random signal is contained within each unit of frequency. The y-axis is spectral density — power per unit bandwidth. The x-axis is frequency. The area under the PSD curve over a frequency range gives the variance of the signal within that range. The square root of the total area gives the overall RMS of the signal.

G(f)  =  PSD value at frequency f    [e.g. g²/Hz]

Variance  =  ∫ G(f) df     [over the frequency range]

RMS  =  √( ∫ G(f) df )      [square root of total area]

Never read a PSD ordinate as if it were a sine amplitude. A PSD value is spectral density — variance per unit bandwidth — not an acceleration level. Only the area under the curve has physical meaning as a mean-square quantity.

Units

The units of a PSD depend on the quantity being described. An acceleration PSD has units of acceleration squared per frequency — typically g²/Hz or (m/s²)²/Hz. A stress PSD has units of stress squared per frequency — typically MPa²/Hz. The unit system must be consistent throughout the analysis. Mixing unit systems in PSD-based calculations is a common source of error.

Grms from a PSD

The overall acceleration level of a random vibration environment is expressed as Grms — the root-mean-square acceleration. It is computed as the square root of the area under the acceleration PSD curve. Grms provides a single-number characterisation of the overall severity of a random vibration environment, useful for comparing environments and for initial screening.

G_rms  =  √( ∫ G(f) df )

where:
G(f)   =  acceleration PSD  [g²/Hz]
f      =  frequency  [Hz]
G_rms  =  overall RMS acceleration  [g]

Log-Log PSD Plots

PSD plots are typically presented on log-log axes because vibration environments span several decades of frequency and several orders of magnitude in spectral density. On log-log axes, a PSD defined by straight-line segments (slopes) appears as linear segments. The slopes are often expressed in dB/octave, which describes how much the PSD changes per doubling of frequency.

• Log-log axes — frequency and PSD both on logarithmic scales
• Straight-line segments — PSD defined by breakpoints and slopes between them
• dB/octave — slope expressed in decibels per doubling of frequency
• Breakpoints — frequencies where the slope changes

Slopes and Octave Relationships

The slope of a PSD segment is often expressed in dB/octave. A slope of 0 dB/octave means the PSD is flat — constant spectral density. A slope of +3 dB/octave means the PSD doubles for each doubling of frequency. A slope of +6 dB/octave means the PSD quadruples for each doubling of frequency (proportional to f²). Understanding these slopes is essential for reading and defining PSD input environments.

Slope in dB/octave:

S  =  10 · log₁₀( G₂ / G₁ ) / log₂( f₂ / f₁ )

where:
G₁, G₂  =  PSD values at frequencies f₁, f₂

For flat PSD:  S = 0 dB/octave  →  G₂ = G₁
For +3 dB/octave:  G₂ = 2·G₁  for each octave
For +6 dB/octave:  G₂ = 4·G₁  for each octave

Frequency Resolution

The frequency resolution of a PSD affects how accurately resonant peaks are represented. A coarse frequency resolution can smear a narrow resonance over a wide band, reducing the peak value and overestimating the off-peak value. In analysis, the frequency resolution should be fine enough to resolve the narrowest resonance of interest. In measurement, the resolution is determined by the sample length and windowing.

COMMON MISTAKE: Using insufficient frequency resolution in a PSD calculation, which can smear a narrow resonance and underestimate the peak response. Ensure resolution is fine enough to resolve the narrowest mode of interest.

One-Sided vs Two-Sided PSD

Engineering PSDs are typically one-sided — they represent only positive frequencies, since negative frequencies have no physical meaning for real signals. Two-sided PSDs, used in some mathematical contexts, represent both positive and negative frequencies with half the power on each side. The relationship is G_one-sided(f) = 2 · S_two-sided(f) for f > 0. Most engineering tools and standards use one-sided PSDs, but the convention should be confirmed when comparing data from different sources.

Narrowband vs Broadband Content

A PSD can be narrowband — concentrated energy in a narrow frequency range — or broadband — energy spread across a wide range. A narrowband PSD is dominated by a single frequency or a small band and produces response that is somewhat sine-like. A broadband PSD contains energy across many frequencies and excites multiple modes simultaneously. Most engineering environments are broadband, but some have narrowband peaks superimposed on a broadband background.

PSD vs ASD Terminology

Some standards and industries use the term ASD (acceleration spectral density) instead of PSD for acceleration environments. The terms are often used interchangeably, but strictly speaking, PSD refers to power (amplitude squared) while ASD refers specifically to acceleration amplitude squared. In practice, acceleration PSD and ASD refer to the same quantity with the same units (g²/Hz). Confirm the convention used in each context.

Key Takeaways

• A PSD describes how variance is distributed across frequency — it is spectral density, not amplitude
• The area under a PSD equals variance; the square root of the area equals RMS
• PSD plots are typically log-log; slopes are expressed in dB/octave
• Frequency resolution must be fine enough to resolve the narrowest resonance
• Engineering PSDs are typically one-sided; confirm convention when comparing sources