A recurring question in random vibration analysis is: How do we determine the relative displacement between two points when the structure is subjected to a Power Spectral Density (PSD) excitation?
This quantity is often far more critical than the absolute response. Connector pin engagement, cable harness fatigue, clearance verification (sway space), optical alignment, and electronic package integrity are typically governed by relative motion rather than the motion of either point alone.
The good news is that the problem is well-defined and can be solved rigorously using standard random vibration theory. However, relying on simplified shortcuts can lead to massive over-design or catastrophic under-prediction.
Why Relative Motion Matters: The Correlation Problem
Consider two locations, $A$ and $B$, on a structure subjected to base excitation. Each point may experience significant absolute displacement.
- High Correlation: If both points move together in-phase, the actual deformation between them may be negligible.
- Low/Negative Correlation: Conversely, two points with modest individual responses can exhibit large relative motion if they participate differently in the structural modes or move completely out-of-phase.
For this reason, relative displacement is fundamentally a correlation problem, not simply a displacement magnitude problem.
The Fundamental Equation
Let $S_{AA}(f)$ and $S_{BB}(f)$ be the displacement auto-PSDs at points $A$ and $B$, respectively, and let $S_{AB}(f)$ be the cross-spectral density (cross-PSD) between them.
The relative displacement in the time domain is:
$$u_{rel}(t) = u_A(t) – u_B(t)$$
Taking the Fourier Transform and deriving the statistical expectation yields the exact formulation for the relative displacement PSD, $S_{rel}(f)$:
$$S_{rel}(f) = S_{AA}(f) + S_{BB}(f) – 2\text{Re}[S_{AB}(f)]$$
The crucial term is the real part of the cross-spectrum, $\text{Re}[S_{AB}(f)]$, also known as the co-spectrum.
Many engineers mistakenly compute a conservative approximation:
$$S_{rel}(f) \approx S_{AA}(f) + S_{BB}(f)$$
This implicitly assumes the responses are completely uncorrelated ($\text{Re}[S_{AB}(f)] = 0$). For most structural random vibration problems, this assumption is incorrect and significantly overpredicts relative displacement, forcing unnecessary design iterations.
In the limiting case where the two points move identically in phase and magnitude ($S_{AA} = S_{BB} = S_{AB}$), the equation reduces cleanly:
$$S_{rel}(f) = S_{AA}(f) + S_{AA}(f) – 2S_{AA}(f) = 0$$
Modal Superposition: Where the Cross-Spectrum Comes From
To truly understand where $S_{AB}(f)$ comes from, we must look at the modal formulation. Most random vibration FEA solutions utilize modal superposition. The displacement vector at any degree of freedom (DOF) $i$ is expressed as a linear combination of mode shapes $\phi$ and modal coordinates $q(t)$:
$$u_i(t) = \sum_{r=1}^{N} \phi_{ir} q_r(t)$$
The relative displacement between points $A$ and $B$ is:
$$u_{rel}(t) = \sum_{r=1}^{N} (\phi_{Ar} – \phi_{Br}) q_r(t)$$
If we transform this to the frequency domain to find the relative PSD, the relationship expands to include the modal cross-compliance terms:
$$S_{rel}(f) = \sum_{r=1}^{N} \sum_{s=1}^{N} (\phi_{Ar} – \phi_{Br})(\phi_{As} – \phi_{Bs}) S_{q_r q_s}(f)$$
Where $S_{q_r q_s}(f)$ is the cross-PSD of the modal coordinates between mode $r$ and mode $s$. For structures with light, proportional damping, the cross-correlation between different modes ($r \neq s$) is often negligible, simplifying the expression to a single summation:
$$S_{rel}(f) \approx \sum_{r=1}^{N} (\phi_{Ar} – \phi_{Br})^2 S_{q_r q_r}(f)$$
Crucial Modal Insights
- Modal Cancellation: Only modes for which $\phi_{Ar} \neq \phi_{Br}$ contribute to relative motion. If both points participate identically in a global mode, that mode contributes zero to the relative displacement, regardless of how massive its absolute response is.
- Local Mode Dominance: This explains why relative responses are almost always dominated by higher-order, high-frequency local modes (e.g., circuit board flexing) rather than the low-frequency global rigid-body modes that dominate acceleration profiles.
Coherence and Structural Separation
The impact of the cross-spectrum depends heavily on the physical and structural topology separating your two points of interest.
1. Closely Spaced Points
For neighboring points on a locally stiff component (e.g., two adjacent pins on a single microchip connector):
- The coherence ($\gamma^2_{AB}(f) = \frac{|S_{AB}|^2}{S_{AA}S_{BB}}$) is typically near 1.0.
- The cross-PSD $S_{AB}$ approaches the auto-PSD $S_{AA}$.
- Destructive interference occurs in the fundamental equation, keeping relative displacement very small.
2. Disparate Substructures
For points located on entirely separate structural load paths (e.g., a component on a flexible solar array wing relative to the main spacecraft bus, or two separate PCB assemblies tied via a loose wire harness):
- The modal participation differs substantially; coherence drops toward 0.
- The cross-spectral cancellation term vanishes.
- Relative motion increases dramatically, driving structural rubbing, impact, or high-cycle wire fatigue.
Bounding the Problem via Monte Carlo Phase Simulation
What happens when you don’t have access to the finite element model, and you are given only the absolute response PSD magnitudes ($S_{AA}$ and $S_{BB}$) with absolutely no cross-spectral data? This is a frequent issue when reviewing test data or black-box vendor reports.
Can we bound the relative displacement problem using a Monte Carlo simulation for the phase? Yes, but with caveats.
Because a PSD discards phase information by definition, the cross-spectrum can be rewritten using the coherence function $\gamma^2(f)$ and a relative phase angle $\theta(f)$:
$$\text{Re}[S_{AB}(f)] = \gamma(f) \sqrt{S_{AA}(f) S_{BB}(f)} \cos(\theta(f))$$
If we substitute this into our core equation, the relative displacement PSD becomes:
$$S_{rel}(f) = S_{AA}(f) + S_{BB}(f) – 2 \gamma(f) \sqrt{S_{AA}(f) S_{BB}(f)} \cos(\theta(f))$$
The Monte Carlo Framework
To mathematically bound the response without structural topology data, a Monte Carlo simulation can randomize the unknown parameters across a high number of iterations (e.g., $N = 10,000$):
- Convert to Displacement: Convert the given acceleration PSDs to displacement PSDs ($S_{disp} = S_{accel} / (2\pi f)^4$).
- Randomize Phase ($\theta$): For each frequency bin, draw the relative phase angle $\theta(f)$ from a uniform distribution between $0$ and $2\pi$ (or $-\pi$ and $\pi$).
- Randomize Coherence ($\gamma$): Draw the coherence $\gamma(f)$ from a uniform distribution between $0$ and $1$. Alternatively, to evaluate the absolute worst-case physical bound, fix $\gamma(f) = 1.0$ and force $\theta(f) = \pi$ (completely out of phase).
- Iterate and Integrate: Compute $S_{rel}(f)$ for each trial, integrate to find $\sigma_{rel}$, and track the cumulative distribution function (CDF) of the resulting RMS values.
The Practical Catch
While a Monte Carlo simulation safely maps the statistical space, a completely unconstrained phase randomization overestimates the true physical response. In real structures, phase and coherence are governed strictly by the underlying modal parameters. Real structures do not rapidly alternate between being completely in-phase and completely out-of-phase from one narrow frequency line to the next.
Therefore, while a Monte Carlo loop is excellent for finding statistical confidence intervals when data is missing, the mathematical upper bound is often overly conservative, approaching the physical envelope of a $180^\circ$ out-of-phase condition:
$$S_{rel, \text{max}}(f) = \left( \sqrt{S_{AA}(f)} + \sqrt{S_{BB}(f)} \right)^2$$
Use Monte Carlo limits as a screening tool. If your clearances or connectors survive the unconstrained Monte Carlo upper bound, they are robust. If they fail, you must hunt down the true cross-spectral FEA data.
Translational vs. Rotational Relative Motion
Analysts frequently focus solely on translational 3-axis Cartesian components ($\Delta X, \Delta Y, \Delta Z$). However, rigid-body or elastic rotation is a major driver of actual geometric relative displacement.
Consider two points separated by a physical distance vector $\vec{L}$. A rotational PSD response ($\theta$) at the base component produces a localized translational relative displacement approximately equal to:
$$u_{rel, \text{rotation}} \approx L \cdot \theta$$
For large electronic racks, optical benches, or high-gain antennas, a tiny angular micro-radian rotation over a long moment arm ($L$) can completely dwarf the local purely translational relative displacements. Therefore, when evaluating clearances, always consider the 6-DOF relative motion matrix including angular transformations:
$$\vec{u}_{rel} = \vec{u}_A – \vec{u}_B – (\vec{\theta} \times \vec{L})$$
Practical FEA Implementation & Workarounds
When setting up your finite element model, you will quickly discover that managing cross-PSDs can be computationally expensive. Here is how to handle it across major solvers.
MSC Nastran / NX Nastran
Nastran offers two main tracks:
- Method 1: Relative DOF (The Cleanest Path): Create an MPC (Multi-Point Constraint) or a scalar element (
CELAS2) to explicitly define a new dependent scalar degree of freedom representing the difference: $u_{diff} = u_A – u_B$. Request the standard PSD output (RANDOMcard) for this specific scalar grid point. Nastran handles all cross-spectral terms natively in the solver core. - Method 2: Complex Post-Processing: If you cannot modify the bulk data, request
PUNCHfiles containing the absolute PSDs ($S_{AA}, S_{BB}$) and the cross-PSDs ($S_{AB}$) using theXYPRINTorXYPUNCHcommand structure. Post-process via MATLAB or Python using the core $S_{AA} + S_{BB} – 2\text{Re}[S_{AB}]$ identity.
ANSYS Mechanical
- Modern ANSYS environments support Response PSD Combinations directly via the worksheet or APDL post-processing.
- Alternatively, use Nodal Covariance Matrices. You can issue the APDL command
RPSDwith the covariance calculation enabled to extract the joint statistical properties of two distinct nodes directly without dumping massive raw time histories or intermediate global spectral matrices.
Abaqus
- Abaqus natively calculates random response using the
*RANDOM RESPONSEstep. Because it relies heavily on a modal basis, it does not default to outputting cross-node pairs due to file size constraints. - Analysts typically must request the modal correlation matrix or write a Python script via the Abaqus ODB API to recover the modal coordinate spectral densities ($S_{q_r q_s}$) and manually sum them using the node shape coefficients.
Beyond 3$\sigma$: Rigorous Peak Relative Displacement
Once the relative displacement PSD ($S_{rel}(f)$) is solved, the root-mean-square ($\sigma_{rel}$ or $RMS_{rel}$) value is calculated by integrating the area under the curve:
$$\sigma_{rel} = \sqrt{\int_{0}^{\infty} S_{rel}(f) \, df}$$
Historically, engineers simply multiply the RMS value by 3 (the standard $3\sigma$ assumption) to determine maximum expected clearance or peak stress. This can be dangerous. For an ideal Gaussian random process, a $3\sigma$ peak implies a $99.73\%$ probability of non-exceedance per wave crest. Over a long 60-second or 3-hour qualification test, millions of cycles occur, making the statistical probability of exceeding $3\sigma$ virtually guaranteed.
To calculate a more rigorous expected peak displacement ($\bar{u}_{max}$), use the Rice Peak Statistics formulation based on the expected number of zero-crossings ($N_0$) and test duration ($T$):
$$\bar{u}_{max} = \left[ \sqrt{2 \ln(N_0 T)} + \frac{0.5772}{\sqrt{2 \ln(N_0 T)}} \right] \sigma_{rel}$$
Where $N_0$ is the expected frequency calculated from the spectral moments ($m_0, m_2$):
$$N_0 = \sqrt{\frac{\int_0^\infty f^2 S_{rel}(f) df}{\int_0^\infty S_{rel}(f) df}}$$
If your relative response is wideband (participating in many modes simultaneously), advanced empirical methods like the Dirlik Method or Wirsching-Light should be used to map the true peak probability density function rather than relying blindly on a narrow-band $3\sigma$ multiplier.
Common Engineering Checklists & Applications
When verifying your random vibration model, relative displacement tracking should be mandatory for the following components:
| Component Type | Failure Mechanism | What to Monitor |
| Connector Headers | Pin fretting, intermittent electrical opens, contact wear | Axial and shear relative displacement between plug and receptacle. |
| Circuit Card Assemblies (CCAs) | Component lead wire fatigue, solder joint cracking | Relative displacement between the center of the PCB and the stiff chassis frame walls. |
| Coaxial/Wire Harnesses | Strain-relief mechanical failure, insulation chafing | Relative displacement between the wire tie-down mount and the terminal lug destination. |
| Optical Benches | Laser misalignment, line-of-sight jitter | Relative angular rotation and differential micro-deflection between mirror mounts. |
| Sway Clearances | Structural pounding, impacting adjacent structural walls | Enveloping maximum peak relative displacement ($\text{Rice Max}$) vs. physical geometric gap width. |
Summary Reference Table
To keep your design reviews moving smoothly, use this quick cheat sheet to evaluate relative motion claims:
| Phase Relationship | Cross-Spectrum Term (Re[SAB]) | Relative PSD Outcome | Design Implication |
| In-Phase / Identical | $= S_{AA}$ | $S_{rel} \to 0$ | Safe condition. Structure is moving as a rigid body. |
| Uncorrelated | $= 0$ | $S_{rel} = S_{AA} + S_{BB}$ | Moderate risk. Over-conservative if assumed arbitrarily. |
| Out-of-Phase ($180^\circ$) | Negative ($=- | S_{AB} | $) |
Recommended References
- Lalanne, C. – Mechanical Vibration and Shock Analysis, Volume 3: Random Vibration. (An absolute masterclass on spectral density equations).
- Craig, R. R. & Kurdila, A. J. – Fundamentals of Structural Dynamics. (Excellent baseline for modal superposition matrices).
- Wirsching, P. H., Paez, T. L., & Ortiz, K. – Random Vibrations: Theory and Practice.
- NASA-HDBK-7005 – Dynamic Environmental Criteria. (Contains excellent practical case studies regarding relative displacement clearance margins for spacecraft payloads).
- Bottom Line: Never just subtract or add RMS values blindly when evaluating relative structural limits. Let the modal cross-spectrum do the heavy lifting to ensure your designs are both lightweight and survivable.*
– Tom Irvine