Piecewise PSD Worked Example

A worked example interpreting a piecewise PSD specification — computing slopes, interpolating between breakpoints and verifying the Grms.

Random Vibration4 min read

worked examplepiecewise PSDslopesinterpolationGrms

Problem

A PSD specification has breakpoints at: 50 Hz (0.02 g²/Hz), 500 Hz (0.05 g²/Hz), 2000 Hz (0.01 g²/Hz). Determine the slope of each segment in dB/octave and compute the Grms.

Given

50 Hz: 0.02 g²/Hz
500 Hz: 0.05 g²/Hz
2000 Hz: 0.01 g²/Hz
All values illustrative

Solution

Segment 1 (50→500 Hz):
  m = log(0.05/0.02) / log(500/50) = log(2.5) / log(10) = 0.398
  S = m × 3.01 = 0.398 × 3.01 = 1.20 dB/octave (rising)

Segment 2 (500→2000 Hz):
  m = log(0.01/0.05) / log(2000/500) = log(0.2) / log(4) = -0.699 / 0.602 = -1.16
  S = -1.16 × 3.01 = -3.49 dB/octave (falling)

Integration:
  I₁ = (0.02/50^0.398) × (500^1.398 − 50^1.398) / 1.398 ≈ 11.4 g²
  I₂ = (0.05/500^(-1.16)) × (2000^(-0.16) − 500^(-0.16)) / (-0.16) ≈ 12.5 g²

σ² = 11.4 + 12.5 = 23.9 g²
Grms = √23.9 = 4.89 g

Note: values are illustrative.