Acceleration, Velocity and Displacement PSD Assessment

The question most often asked in a random vibration laboratory is: how many g is this environment? The answer requires a PSD — decomposing the energy of a random signal over frequency and converting it to mean-square acceleration per unit frequency — so that it can be compared with the spectral shapes in test standards. The module does this in real time on one screen.

The problem it solves

The difficulty is not computing a spectrum but getting the units right, integrating cleanly and judging on a sound basis. PSD has the unit (quantity)²/Hz: FFT magnitude must first be squared, then normalized using the defined FFT scaling, window energy, frequency-bin width and one-sided or two-sided convention. Dividing an unqualified FFT magnitude by the frequency resolution alone does not give PSD. Velocity and displacement PSD are integrated line by line (division by ω² and ω⁴) and are easily contaminated by drift at low frequencies. Test evaluation must check both total energy (total RMS) and line-by-line exceedance while avoiding false alarms from single spikes. All of these rules are built in, refreshed during measurement, with a pass or fail result.

Functions

Screen layout

The left panel contains channel selection (8 check boxes), exponential averaging and count with restart, lower and upper analysis frequency, the limit alarm switch and standard selection, stay on top, export and refresh. On the right, a statistics table shows one row per channel: acceleration RMS (g), velocity RMS (m/s), displacement RMS (mm) and their 3σ peaks, and the status (pass on green, fail on red). Below are the three PSD plots — acceleration g²/Hz, velocity (m/s)²/Hz and displacement (mm)²/Hz — on log-log axes with red limit lines when a standard is selected. Double-click any plot to enlarge it.

The chain has four layers — single-sided PSD, frequency-domain integration to three domains, RMS and 3σ statistics, and limit evaluation — each traceable.

Step 1: single-sided PSD

The source describes transforming a windowed frame and normalizing squared spectral amplitudes to obtain PSD in g²/Hz. Correct normalization depends on the FFT convention, window energy, frequency-bin width and one-sided or two-sided convention. Those details must be verified together; the unexplained factor in the source is not reproduced here. PSD integrates over frequency to mean square, while individual estimates still depend on averaging and resolution.

Step 2: velocity and displacement PSD

Gv(f) = Ga(f) / (2πf)² Gd(f) = Ga(f) / (2πf)⁴

Unit factors (g² to (m/s²)², m² to mm²) are applied first. The DC line is set to zero and excluded, so zero drift does not enter the integration. The three spectra use log-log axes, as in test standards.

Step 3: total RMS and 3σ peak

RMS = √(Σ PSD · Δf) 3σ peak = 3 × RMS

The total RMS is the square root of the area under the PSD over the selected band. For a zero-mean Gaussian signal, about 99.7 % of instantaneous values lie within ±3 × RMS. A record can contain larger peaks, so this level is not a guaranteed maximum or a sufficient clearance design limit.

Step 4: exponential averaging

A single random spectrum fluctuates strongly because it has few statistical degrees of freedom, so averaging is required. Exponential averaging weights new frames highly while older frames decay exponentially, balancing stability and response. The count is 1–1000, and restart clears the average.

Step 5: limit lines and evaluation

When a standard is selected, its profile is loaded line by line as the acceleration limit and converted with (2πf)² and (2πf)⁴ to velocity and displacement limit lines in all three plots. Either of two rules gives a fail:

• Total energy: any total RMS exceeds the standard total by more than 1.1 times (about +0.8 dB).
• Line by line: the source describes a +3 dB threshold at three consecutive spectral points. For PSD, +3 dB is a power ratio of 10^(3/10), approximately 1.995. The implemented threshold and preset profiles require confirmation before a formal pass/fail assessment.

The result appears on red or green in the status column and updates with every frame.

Typical tasks

Source-provided illustrative display for channel 1, band 2.5–2000 Hz. The profile and alarm configuration are unverified; the shown status is not a compliance conclusion.

These four readings describe the display. A formal result also requires a verified profile, units, bandwidth, averaging and acceptance method.