Piecewise PSD Specifications

What Is It?

Most engineering PSD specifications are defined as piecewise log-log functions — a series of breakpoints (frequency, PSD value pairs) with straight-line interpolation on log-log axes between them. This piecewise representation is compact, easy to specify and easy to interpret. Understanding how to read, interpolate and integrate a piecewise PSD is a core skill for random vibration engineering.

Why It Matters

Piecewise PSD specifications are the standard format for vibration test standards, qualification specifications and FEA input. An engineer who cannot correctly interpret breakpoints and slopes, or who cannot integrate a piecewise PSD to obtain Grms, cannot verify that a test specification is correct or that an FEA input is properly defined.

Piecewise log-log PSD is the standard specification format. Every random vibration engineer must be able to read breakpoints, interpret slopes, interpolate and integrate to obtain Grms.

Breakpoints

A piecewise PSD is defined by a table of breakpoints — pairs of frequency and PSD value. Between consecutive breakpoints, the PSD follows a straight line on log-log axes. The breakpoints define the shape of the PSD. The frequency range spans from the first to the last breakpoint.

Example piecewise PSD (illustrative):

Breakpoint    Frequency (Hz)    PSD (g²/Hz)
   1               20              0.01
   2               50              0.05
   3              500              0.05
   4             2000              0.01

Four breakpoints define three segments:
  20-50 Hz:    rising slope
  50-500 Hz:   flat (0 dB/octave)
  500-2000 Hz: falling slope

Slopes

The slope of each segment describes how the PSD changes with frequency. On log-log axes, the slope is expressed in dB/octave — the change in PSD level (in dB) per doubling of frequency. A slope of 0 dB/octave is flat. A positive slope means the PSD rises with frequency; a negative slope means it falls. The slope can be converted to a power-law exponent for integration.

Slope in dB/octave:

S  =  10 · log₁₀(G₂/G₁) / log₂(f₂/f₁)    [dB/octave]

Slope exponent (power law):

m  =  log(G₂/G₁) / log(f₂/f₁)

Relationship:

m  =  S / (10 · log₁₀(2))  =  S / 3.01

G(f)  =  G₁ · (f / f₁)^m    [between breakpoints]

Log-Log Representation

PSD specifications are plotted on log-log axes because both frequency and PSD value span several orders of magnitude. On log-log axes, a power-law relationship (G ∝ f^m) appears as a straight line. This makes piecewise PSDs easy to visualise and interpret — the shape is a series of connected straight-line segments.

• Log-log axes — both frequency and PSD on logarithmic scales
• Power-law segments appear as straight lines
• Slope visible as the angle of each segment
• Breakpoints visible as the corners where slope changes

Interpolation

Between breakpoints, the PSD value at any frequency is found by log-log interpolation. The interpolation follows the power law G(f) = G₁ · (f/f₁)^m, where m is the slope exponent for that segment. This is not linear interpolation — it is logarithmic interpolation in both axes.

Log-log interpolation between (f₁, G₁) and (f₂, G₂):

m  =  log(G₂/G₁) / log(f₂/f₁)

G(f)  =  G₁ · (f / f₁)^m

Example:  f₁ = 20, G₁ = 0.01,  f₂ = 50, G₂ = 0.05
  m  =  log(5) / log(2.5)  =  0.699 / 0.398  =  1.76
  At f = 35 Hz:  G = 0.01 · (35/20)^1.76  =  0.01 · 2.47  =  0.0247 g²/Hz

Integration of Piecewise PSD

To obtain the overall Grms, each segment is integrated analytically and the results are summed. The integral of each segment depends on the slope exponent m. For m ≠ −1, the integral follows a power rule. For m = −1 (a slope of approximately −10 dB/octave), the integral involves a natural logarithm.

Segment integral from f₁ to f₂ with slope m:

If m ≠ −1:
  I  =  (G₁ / f₁^m) · (f₂^(m+1) − f₁^(m+1)) / (m+1)

If m = −1:
  I  =  (G₁ / f₁^m) · ln(f₂ / f₁)

Total variance:
  σ²  =  Σ I_i   (sum over all segments)

Grms:
  G_rms  =  √(σ²)

Worked Grms Calculation

A complete Grms calculation from a piecewise PSD specification demonstrates the process. Using the illustrative four-breakpoint specification from above.

Illustrative PSD:
  20 Hz: 0.01 g²/Hz
  50 Hz: 0.05 g²/Hz  (m = 1.76, rising)
  500 Hz: 0.05 g²/Hz  (m = 0, flat)
  2000 Hz: 0.01 g²/Hz  (m = −1.76, falling)

Segment 1 (20→50 Hz, m=1.76):
  I₁ = (0.01/20^1.76) · (50^2.76 − 20^2.76) / 2.76
  ≈ 0.93 g²

Segment 2 (50→500 Hz, m=0):
  I₂ = 0.05 · (500 − 50) = 22.5 g²

Segment 3 (500→2000 Hz, m=−1.76):
  I₃ ≈ 0.93 g²  (by symmetry with segment 1)

σ² = 0.93 + 22.5 + 0.93 = 24.36 g²
G_rms = √24.36 = 4.94 g

Note: values are illustrative for method demonstration.

Specification Interpretation

When interpreting a PSD specification, verify the following. The breakpoints and slopes define the correct shape. The Grms matches the expected overall level. The frequency range covers the relevant structural modes. The PSD values are in the correct units (g²/Hz, not g/√Hz). A common error is to misread a specification that uses different units or a different log-log convention.

Always verify: correct units (g²/Hz), correct frequency range, Grms matches expected level, and slopes interpreted correctly. Misreading a specification is a common and consequential error.

Key Takeaways

• Piecewise log-log PSD is the standard specification format — breakpoints with straight-line interpolation
• Slopes are expressed in dB/octave; convert to power-law exponent m for integration
• Log-log interpolation follows G(f) = G₁ · (f/f₁)^m, not linear interpolation
• Each segment is integrated analytically; segments are summed for total variance
• Always verify units, frequency range and Grms when interpreting a specification