Correlated Multiple-Input Random Vibration

What Is It?

Correlated multiple-input random vibration occurs when a structure is excited at multiple points whose motion is related — driven by the same source or connected through the same supporting structure. In this case, the response depends not only on the individual input PSDs but also on the cross-spectral densities between the inputs. The cross terms can add or subtract depending on the phase relationship, making the response potentially larger or smaller than the uncorrelated case.

Why It Matters

Many real-world multi-support excitation problems involve correlated inputs. Equipment mounted at multiple points on a vibrating structure, multi-axis shaker testing and acoustic excitation over a surface all involve correlated inputs. Treating correlated inputs as independent — and using simple SRSS combination — can under-predict or over-predict the response, depending on whether the correlation is constructive or destructive at the critical frequencies.

When inputs are correlated, the response includes cross-spectral terms that can add or subtract. Treating correlated inputs as independent can under- or over-predict the response.

Multiple Excitation Points

In a multi-input problem, the structure is excited at N points, each with its own input PSD. The total response at any location is the sum of the responses from each input, plus the cross terms that account for the correlation between inputs. The cross terms involve the cross-spectral densities between each pair of inputs.

N-input response PSD:

G_out(f)  =  Σ_i  |H_i(f)|² · G_ii(f)
           +  Σ_{i≠j}  Re[ H_i(f) · H_j*(f) · G_ij(f) ]

where:
H_i(f)    =  transfer function from input i to response location
G_ii(f)   =  input PSD at point i (auto-spectral density)
G_ij(f)   =  cross-spectral density between inputs i and j

First sum:  individual input contributions
Second sum: cross terms (correlation effects)

Cross-Spectral Terms

The cross-spectral terms in the response equation account for the correlation between inputs. Each cross term involves the cross-spectral density between two inputs and the transfer functions from those inputs to the response. The cross term can be positive (constructive — inputs add) or negative (destructive — inputs partially cancel), depending on the phase of the cross-spectrum and the transfer functions.

• Cross terms involve G_ij(f) — the cross-spectral density between inputs i and j
• Cross terms can be positive (constructive) or negative (destructive)
• Sign depends on the phase of G_ij and the phases of H_i and H_j
• Ignoring cross terms is equivalent to assuming uncorrelated inputs

Phase

The phase relationship between correlated inputs is critical. If two inputs are in phase at a frequency where both transfer functions are large, the cross term is positive and the response is larger than the uncorrelated case. If the inputs are in anti-phase, the cross term is negative and the response is smaller. The phase varies with frequency, so the effect of correlation changes across the frequency range.

Two-input case (simplified):

G_out  =  |H_x|²·G_xx + |H_y|²·G_yy + 2·|H_x|·|H_y|·|G_xy|·cos(φ)

where:
φ  =  ∠G_xy + ∠H_x − ∠H_y  (combined phase)

φ = 0:    cross term positive — constructive (maximum)
φ = ±π:   cross term negative — destructive (minimum)
φ = ±π/2: cross term zero — same as uncorrelated

Effect on Structural Response

The effect of correlation on the structural response depends on the coherence and phase at the frequencies that dominate the response. At resonant frequencies, the transfer functions are large and the cross terms can significantly increase or decrease the response. The effect is most pronounced when the coherence is high (inputs strongly correlated) and the phase is either 0 (constructive) or π (destructive).

When Correlation Matters Most

Correlation matters most when the inputs are strongly correlated (high coherence) and the critical frequency is one where the transfer functions from both inputs are large — typically a resonant frequency. At non-resonant frequencies, the transfer functions are small and the cross terms contribute little. Correlation also matters more when the inputs are close together on the structure (strong structural coupling) than when they are far apart.

Correlation matters most at resonant frequencies where both input transfer functions are large. At non-resonant frequencies, cross terms contribute little regardless of coherence.

Computational Approach

Computing the correlated multi-input response requires the full cross-spectral matrix — all auto-spectral densities (diagonal) and all cross-spectral densities (off-diagonal). The solver computes the response using the complete matrix. This is more computationally expensive than the uncorrelated case but is necessary for accurate results when inputs are correlated.

• Full cross-spectral matrix required — N² terms for N inputs
• Diagonal terms: auto-spectral densities (PSDs)
• Off-diagonal terms: cross-spectral densities
• Solver must support correlated multi-input analysis
• More computationally expensive than uncorrelated analysis

Key Takeaways

• Correlated multi-input response includes cross-spectral terms that can add or subtract
• Cross terms depend on coherence and phase between inputs at each frequency
• Constructive correlation increases response; destructive correlation decreases it
• Effect is strongest at resonant frequencies where transfer functions are large
• Full cross-spectral matrix is required — N² terms for N inputs