Understanding Power Spectral Density
Power spectral density (PSD) describes how vibration energy is distributed across frequency, expressed as energy per unit of frequency bandwidth — units of (m/s²)²/Hz for acceleration or (mm/s)²/Hz for velocity. Where an ordinary amplitude spectrum reports the amplitude present at each frequency, PSD reports the power per hertz at each frequency, normalised by the analysis bandwidth. That single act of normalisation is what gives PSD its defining virtue: it is independent of the FFT resolution used to compute it, so spectra captured with different settings — or on different instruments — can be compared directly and fairly.
PSD comes into its own for random vibration, where energy is spread continuously across the frequency axis rather than concentrated at a handful of discrete peaks. It is the natural language for noise analysis, for environmental and qualification testing, and for any task that demands a bandwidth-independent description of a spectrum. For routine machinery fault-finding, by contrast, the familiar amplitude spectrum usually remains the more convenient view.
1. PSD vs the Amplitude Spectrum
The two displays answer different questions, and knowing which to reach for is half the skill.
Amplitude spectrum
- Shows vibration amplitude at each frequency, in everyday units such as mm/s, m/s² or mils.
- Displays sharp peaks at discrete frequencies — unbalance at 1×, bearing-fault tones, gear mesh — which is exactly what diagnostics needs.
- Its peak values depend on the FFT resolution bandwidth, so the same machine can read differently under different settings.
- The standard display for machinery diagnostics.
Power spectral density
- Shows vibration power per hertz of bandwidth, in units such as (mm/s)²/Hz or (m/s²)²/Hz.
- Represents the energy distribution across frequency rather than the height of individual lines.
- Is independent of analysis bandwidth — its defining advantage.
- The standard description for random vibration.
The relationship between them
PSD = (Amplitude)² / Δf, where Δf is the frequency resolution (the bin width).
Squaring the amplitude emphasises the largest components, and dividing by Δf removes the bandwidth dependence. The bin width itself is set by the span and line count of the transform, a relationship a FFT resolution calculator makes explicit — and which explains why a narrower Δf raises the peaks of a raw amplitude spectrum but leaves the PSD unchanged.
2. Where PSD Is Used
The applications cluster around randomness, broadband energy and the need to compare.
Random vibration analysis
This is the primary use. Random processes — flow turbulence, road inputs, seismic motion, acoustic excitation — produce continuous spectra with no discrete peaks, and PSD is the proper statistical description of how their energy is spread. Vibration-test specifications are written in PSD for exactly this reason.
Broadband noise characterisation
PSD cleanly captures broadband phenomena: cavitation noise in pumps, turbulent-flow noise in fans, aerodynamic noise, and the broadband content of bearing-defect noise that a peak-based view struggles to summarise.
Bandwidth-independent comparison
Because PSD is normalised by Δf, it lets you compare spectra taken with different FFT settings, data from different instruments or resolutions, and historical records gathered under analysis parameters nobody documented. The PSD values are directly comparable regardless of bandwidth.
Environmental and qualification testing
Vibration-test profiles are specified as PSD versus frequency, shaker-table controllers regulate to a PSD target, and product-qualification and shock-and-vibration standards are framed in the same terms — making fluency in PSD essential for anyone running or interpreting such tests.
3. Calculating PSD
The computation follows straight from the definition:
- Compute the FFT of the vibration signal.
- Square each amplitude value.
- Divide by the frequency resolution, Δf = Fmax ÷ number of lines.
- The result is the PSD in (units)²/Hz.
The units track the underlying parameter — acceleration PSD in (m/s²)²/Hz or g²/Hz, velocity PSD in (mm/s)²/Hz or (in/s)²/Hz, displacement PSD in (µm)²/Hz or (mils)²/Hz — and PSD is very often plotted on a logarithmic (dB-relative-to-reference) scale to span its wide dynamic range. Accurate PSD also depends on appropriate windowing and averaging of the time data, since random signals must be averaged over many records to yield a stable estimate.
4. Interpreting PSD Plots
The shape of a PSD curve carries diagnostic meaning of its own.
- Flat spectrum (white noise): constant PSD across frequency means equal energy per hertz everywhere — the signature of ideal broadband random vibration, and the target profile for much random-vibration testing.
- Sloped spectrum (coloured noise): PSD that varies with frequency. A rising slope concentrates energy at high frequencies; a falling slope concentrates it at low frequencies, which is common in real machinery.
- Peaks in PSD: discrete components still appear as peaks standing above the general level, and resonances show up as elevated regions, so the dominant energy contributors remain visible even within a broadband background.
5. Relationship to RMS and Total Energy
PSD connects directly to the single-number severity measures engineers rely on.
RMS = √[ ∫ PSD(f) df ]
Integrating the PSD across the full frequency range yields the mean-square value, and its square root is the overall RMS — the same quantity a tool like an overall vibration level calculator derives from a spectrum. Integrating over a narrower band gives the energy contained in that band alone, which is invaluable for judging how much each frequency region contributes to the total. This statistical framework is also the foundation of random-vibration fatigue theory: predicting fatigue life under random loading starts from the PSD, as a fatigue life calculator illustrates.
6. Advantages, and When to Choose PSD
PSD’s strengths are three. Resolution independence lets values be compared regardless of FFT settings, standardising analysis across instruments and across years of historical data. Energy representation means the curve directly depicts how vibration energy is distributed, with the squaring naturally emphasising dominant frequencies. And its statistical framework underpins random-vibration theory, enabling probabilistic analysis and fatigue prediction.
Choose PSD when analysing random vibration or noise, when comparing data captured with different bandwidths, when following a test specification written in PSD, when characterising a broadband process, or whenever the analysis is fundamentally energy-based. Stay with an amplitude spectrum — or the closely related techniques of spectral analysis — for routine machinery diagnostics, for identifying discrete fault frequencies, for trending a specific component, and wherever the amplitude value itself is the meaningful number. In day-to-day field balancing and condition monitoring with a portable analyser such as the Balancet-1A, the amplitude spectrum and 1× amplitude-and-phase remain the working tools; PSD enters when the question shifts from “which component is faulty?” to “how is broadband energy distributed, and is it comparable to last year’s data?”.
Power spectral density is a cornerstone of random-vibration analysis and the only honest way to give a bandwidth-independent description of a spectrum. Less common than the amplitude spectrum in routine diagnostics, it is indispensable for random vibration, noise characterisation, environmental testing, and any situation that calls for comparing spectra measured with different analysis parameters or on different instruments.
