Simultaneous Multi-Axis Random Vibration

Why Multi-Axis Random Vibration Is Different

Conventional qualification and analysis often apply random vibration one axis at a time. That simplification is convenient, but a real structure can experience simultaneous translation in several directions, rotational motion, multiple support inputs and cross-axis coupling. If the axes are statistically related, the response cannot in general be reproduced by solving each direction independently and combining RMS results afterwards. Simultaneous multi-axis random vibration is therefore a multiple-input problem in which the auto-spectral density of each input and the cross-spectral relationship between inputs determine structural response.

The issue is not simply “three PSDs instead of one”. Cross-spectral terms determine whether simultaneous inputs reinforce, cancel or redistribute modal response.

Cross-Spectral Density Matrix

At each frequency, a multi-axis environment can be represented by a spectral-density matrix. Diagonal terms are the auto-PSDs for individual axes or input channels. Off-diagonal terms are complex cross-spectral densities containing magnitude and phase information. For a physically realisable stationary process, the matrix is Hermitian and positive semidefinite. Those mathematical properties are useful verification checks: a matrix that violates them may contain inconsistent units, sign conventions, phase definitions or processing errors before it ever reaches the structural solver.

Input spectral matrix for three axes:

S(f) = [[Gxx, Gxy, Gxz],
        [Gyx, Gyy, Gyz],
        [Gzx, Gzy, Gzz]]

with Gyx = Gxy* for a Hermitian matrix.

Response spectral matrix:
Syy(f) = H(f) · Sxx(f) · Hᴴ(f)

Coherence and Phase Between Axes

Coherence describes the strength of the linear relationship between two inputs at each frequency; cross-spectral phase describes their relative timing. Perfectly coherent inputs can act in phase or out of phase, producing very different response. Zero coherence means the cross-spectral term is zero and inputs can be treated as statistically independent in a linear analysis. Real environments commonly lie between these limits. Assuming zero or unity coherence without evidence can be either conservative or non-conservative depending on the mode shape, support geometry and response quantity.

• Auto-PSD defines energy in each individual axis
• Coherence defines how strongly two axes are statistically related
• Cross-spectral phase controls constructive or destructive interaction
• The governing correlation can vary strongly with frequency

Coordinate Systems and Sign Convention

Multi-axis data are especially vulnerable to coordinate mistakes. Test, measurement and FE coordinate systems must be related explicitly, including axis handedness, sensor polarity and rotational orientation. A sign reversal that seems irrelevant for a single-axis PSD becomes important once cross-spectral phase is included. For equipment mounted on an inclined interface, measured platform axes may not align with structural axes, so the full spectral matrix should be transformed consistently rather than rotating only the diagonal PSD terms. The same discipline applies to multiple accelerometers whose local axes differ.

A PSD magnitude survives a sign reversal; a cross-spectrum does not. Coordinate and polarity control become first-order verification items in correlated multi-axis analysis.

Structural Response and Modal Coupling

Each mode has directional participation determined by its shape and by how base motion enters the model. A mode that appears weak in each single-axis analysis can become important under simultaneous excitation if several directions contribute coherently to the same modal coordinate. Conversely, anti-phase inputs can reduce response for one mode while increasing another. Examine modal participation and response spectra by frequency, not just final scalar RMS values. Local stresses, interface loads and equipment accelerations may be governed by different combinations of input axes.

Independent-Axis Combination — When It Is Valid

For a linear system with demonstrably uncorrelated input axes, separate random-response solutions can be combined in a mean-square sense because cross terms vanish. In that specific case, total response variance is the sum of variances from independent inputs. This is not the same as adding RMS values directly, and it is not valid when inputs are correlated. If only marginal PSDs are available and correlation is unknown, state the assumption and perform bounding correlation cases where the decision is sensitive rather than silently defaulting to SRSS.

For statistically independent linear responses:

σtotal² = σx² + σy² + σz²

σtotal = √(σx² + σy² + σz²)

This relation is not generally valid when cross-spectral terms are non-zero.

Multiple Supports and Rotational Input

Simultaneous multi-axis problems often occur together with multiple support points. Translational motion at separated supports can imply rotational base motion and differential displacement, and support inputs may have frequency-dependent coherence. A rigid-base approximation can be appropriate for compact equipment on a stiff mounting plane, but it should be justified. For large equipment, flexible decks or distributed mounting patterns, response can depend on spatial variation as well as axis correlation. In those cases the input definition may require a larger cross-spectral matrix spanning both direction and support location.

Simultaneous Multi-Axis Qualification Testing

Multi-axis shaker systems can apply several translational and rotational degrees of freedom simultaneously. The control problem is more demanding than single-axis testing because actuator interaction, fixture modes and test-article coupling affect the achieved spectral matrix. Qualification evidence should therefore include not only individual control PSDs but also relevant cross-spectra, coherence and phase where the specification defines them. Limiting or notching strategies need to preserve the intended relationship between axes; reducing one control channel independently can unintentionally change the correlation structure and therefore the response.

• Verify achieved auto-PSDs on all controlled axes
• Verify cross-spectral magnitude and phase when correlation is specified
• Monitor cross-axis response and fixture dynamics
• Check that notching preserves the intended environment
• Use analysis to place response instrumentation at coupled modes and critical interfaces

Verification and Sensitivity

A defensible analysis verifies the spectral matrix, coordinate transformations, modal basis, damping and response recovery. Sensitivity cases should vary uncertain coherence or phase assumptions where they materially affect the conclusion. Useful independent checks include recovering the single-axis limit by zeroing cross terms, reproducing perfectly correlated limiting cases, checking conjugate symmetry of the matrix, confirming that its eigenvalues are non-negative within numerical tolerance, and independently integrating response spectra to recover solver-reported RMS values.

• Spectral matrix Hermitian — Check conjugate symmetry of cross terms.
• Positive semidefinite input — No physically impossible negative spectral energy.
• Coordinate transformation verified — Include polarity and sensor orientation.
• Single-axis limiting cases recovered — Useful implementation check.
• Correlation sensitivity assessed — Especially where source data are incomplete.
• Response RMS independently integrated — Confirm solver post-processing.

Reporting the Engineering Assumptions

The report should make the input model transparent. State which axes and supports are included, the coordinate basis, whether inputs are measured or specified, how cross-spectra were derived, the coherence and phase assumptions, and how uncertain correlation was treated. Results should identify which modes and input combinations govern each critical response. A statement such as “three axes combined by SRSS” is not enough unless statistical independence has been justified. For qualification work, analysis and test definitions should use the same spectral conventions so correlation is meaningful.

Key Takeaways

• Simultaneous multi-axis vibration is a multiple-input spectral problem, not three independent PSD plots
• Cross-spectral magnitude and phase can materially change modal response
• Coordinate and polarity control are essential because phase-sensitive terms are involved
• Variance combination is valid only for demonstrably independent linear inputs
• Multi-axis test evidence should verify spectral relationships between controlled axes, not just each auto-PSD
• Use correlation sensitivity and limiting cases when the environment is incompletely characterised