Correlated Input Example

Problem

A structure is excited at two supports. Each support has an input PSD of 0.02 g²/Hz. The coherence between the supports at a critical frequency is 0.8 (partially correlated). The transfer functions from each support to the response location are H1 = 5 and H2 = 3 at this frequency. Compute the response PSD for the correlated case and compare with the uncorrelated (SRSS) case.

Given

• G11 = G22 = 0.02 g²/Hz
• γ² = 0.8 at the critical frequency
• H1 = 5, H2 = 3 at this frequency

Solution

Cross-spectral density magnitude:
  |G12| = γ × √(G11 × G22) = √0.8 × 0.02 = 0.894 × 0.02 = 0.01789 g²/Hz

Uncorrelated case (SRSS, γ = 0):
  G_out = H1²×G11 + H2²×G22 = 25×0.02 + 9×0.02 = 0.50 + 0.18 = 0.68 g²/Hz

Correlated case (γ = 0.8, in-phase):
  G_out = H1²×G11 + H2²×G22 + 2×H1×H2×|G12|
  = 0.50 + 0.18 + 2×5×3×0.01789
  = 0.68 + 0.537 = 1.217 g²/Hz

The correlated response (1.217) is 1.79×
the uncorrelated response (0.68) —
a significant difference.

Note: if the inputs were in anti-phase,
the cross term would subtract, giving:
  0.68 − 0.537 = 0.143 g²/Hz
— lower than the uncorrelated case.

Values are illustrative.