4. Dynamic response and cross-axis sensitivity

Static tilt checks catch gross misalignment, but a sensor that reads correctly on a level bench can still produce corrupted data the moment the platform starts moving. Under vibration and dynamic motion, mounting errors stop behaving like a constant offset and start behaving like a frequency-dependent, axis-mixing transfer function. This chapter explains why that happens, how to recognize it in real signals, and how to quantify and correct it.

Why dynamic excitation exposes what static checks hide

An accelerometer measures the projection of the local acceleration vector onto its sensitive axis. If the sensitive axis is misaligned from the intended measurement axis by an angle θ, the measured value is a·cos θ plus a contribution from the orthogonal component a·sin θ. At rest, the orthogonal component is usually small and slowly varying, so the error looks like a small gain error or a bias. Under vibration, the orthogonal component can be large and rapidly changing, so the misalignment converts into a cross-axis signal that is proportional to the transverse acceleration — not to the intended one.

Two distinct mechanisms are at work, and they are often confused:

• Geometric cross-axis sensitivity — the sensitive axis is physically tilted relative to the mounting surface or the vehicle axis. This is a pure projection effect, largely frequency-independent up to the sensor's bandwidth, and it scales with the sine of the misalignment angle.
• Dynamic mounting response — the sensor is not rigidly coupled to the structure. Adhesive compliance, a soft gasket, a loose screw, or a long standoff creates a spring–mass system between the structure and the sensing element. This produces resonance, phase lag, and amplitude peaking that are strongly frequency-dependent and can be far larger than the geometric error.

The practical consequence: a 2° tilt gives roughly 3.5% cross-axis coupling (sin 2° ≈ 0.035), which is tolerable in many applications. A mounting resonance at 400 Hz with a quality factor of 10 can amplify transverse input by an order of magnitude at that frequency, producing cross-axis errors of 100% or more in the band that matters most for vibration monitoring.

Reading the signature in the data

The most reliable diagnostic is a coherence and cross-spectrum check between axes. Mount the sensor, excite the structure with a broadband source (instrumented hammer, shaker, or the machine's own run-up), and record all three axes simultaneously. Then compute:

• Magnitude-squared coherence between the intended axis and each transverse axis. A rigidly mounted, well-aligned sensor shows low coherence between axes except where the structure genuinely couples them. A mounting resonance shows a sharp coherence peak at the resonance frequency, because the sensor is now responding to transverse input through the mount.
• Cross-axis transfer ratio — the ratio of transverse to axial response at each frequency. For geometric misalignment this ratio is flat; for a mounting resonance it rises steeply near the resonance and rolls off above it.
• Phase relationship — a rigid mount gives near-zero or near-180° phase between axes at low frequency; a compliant mount introduces a phase sweep through resonance that is a clear fingerprint of the spring–mass behavior.

Quantifying the error budget

Before choosing a correction strategy, separate the two contributions. A practical procedure for a triaxial sensor on a vibrating platform:

1. Record a static six-face or three-position tilt sequence to estimate geometric misalignment angles θ_x, θ_y relative to the mounting reference. These give the frequency-independent cross-axis matrix.
2. Perform a broadband excitation and compute the axial-to-transverse transfer functions H_xy(f), H_xz(f).
3. Subtract the geometric projection (a constant, real-valued matrix) from the measured transfer functions. The residual is the dynamic mounting response.
4. Fit a single-degree-of-freedom model to the residual: resonance frequency f_n, damping ratio ζ, and coupling gain. If the fit residual is large, the mount is behaving nonlinearly (loose fastener, cracking adhesive) and should be repaired rather than modeled.

As a worked example with illustrative (not measured) numbers: suppose the static check yields θ_x = 1.8°, giving a geometric cross-axis term of 0.031. The broadband test shows a transverse-to-axial ratio of 0.45 at 380 Hz with a coherence peak of 0.92. The geometric term accounts for only 0.031 of that 0.45, so roughly 0.42 is dynamic mount response — a resonance near 380 Hz with an effective Q of about 8. The correction is therefore not a rotation matrix alone; the mount must be stiffened or the resonance moved well above the measurement band.

Correcting and validating

Correction follows directly from the error budget. For the geometric component, apply a calibration matrix C = (cos θ)⁻¹·R^T, where R is the rotation from the sensor frame to the measurement frame, derived from the static tilt sequence. This is a fixed 3×3 matrix applied to every sample and is cheap to implement in firmware.

For the dynamic component, software correction is only viable if the mount behaves linearly and the resonance is well separated from the signal band. A second-order inverse filter can flatten the response, but it amplifies noise above the resonance and is fragile to temperature drift in the adhesive. In most vibration-monitoring installations the correct action is mechanical: replace the compliant pad with a rigid stud or epoxy bond, shorten standoffs, and re-run the coherence test until the transverse ratio is flat across the band of interest.

Validation should use an independent excitation. After correction, re-measure the cross-axis transfer ratio and confirm that it is flat within the specification (for example, below 5% up to the sensor's rated bandwidth) and that coherence between axes drops to the structural baseline. If a residual peak remains at the old resonance frequency, the mechanical fix was incomplete — check fastener torque and bond-line thickness before touching the calibration matrix.

Practical checklist for dynamic mounting verification

The central lesson is that mounting errors have two faces: a static projection that can be calibrated away, and a dynamic compliance that cannot be calibrated reliably and must be fixed mechanically. Distinguishing them requires frequency-resolved, multi-axis data — a single-axis time trace or a bench tilt check will not reveal which mechanism dominates. Once the error budget is separated, the correction path becomes unambiguous.