EngMathTools

GRMS Calculator from PSD Breakpoints

Enter frequency and acceleration PSD breakpoints to integrate a random vibration spectrum. Sloped segments are treated as straight lines on a log-log plot using power-law integration.

PSD breakpoint table

Frequency (Hz)PSD (g²/Hz)

Segment results

Segmentf₁f₂PSD₁PSD₂Slope nArea (g²)% area

How GRMS is calculated

For a PSD defined by straight segments on log-log axes, each segment is represented as a power law, PSD(f) = C·fⁿ, where the exponent n comes from the two bounding breakpoints. The area of every segment is integrated and summed. Overall GRMS is the square root of that total mean-square area.

Important: This calculator assumes the breakpoint pairs describe straight line segments on a log-log PSD plot. All frequencies and PSD amplitudes must be greater than zero, and frequencies must increase from row to row.

Why not use the trapezoidal rule?

A straight segment on log-log axes is generally curved on linear axes. Power-law integration preserves the intended log-log PSD shape instead of approximating it with a linear trapezoid.

Units

If PSD is entered in g²/Hz and frequency in Hz, integrated area is in g² and the square root is GRMS. The same method also works for other compatible PSD units; the RMS result has the base unit associated with the spectral density.