2. Mounting error taxonomy

An accelerometer measures acceleration along its own sensitive axes, not along the axes of the machine it is bolted to. Every real installation therefore introduces a coordinate transformation between the sensor frame and the machine frame, and every imperfection in that transformation becomes a measurement error. The purpose of this taxonomy is to give each physical defect a name, a geometric definition, and a characteristic signature in the data, so that a diagnosis can move from "the reading looks wrong" to "the sensor is rotated 2.3° about Z and offset 4 mm from the reference point."

Throughout this chapter the reference convention is fixed as follows. The machine frame (Xm, Ym, Zm) is defined by the structure under test, with Zm normal to the mounting surface and pointing away from it. The sensor frame (Xs, Ys, Zs) is defined by the accelerometer package markings, with Zs along the package normal. Positive rotation follows the right-hand rule about each axis. All error definitions below are expressed as the transformation that maps the sensor frame onto the machine frame.

The five error classes and what physically causes them

Misalignment is the general term for any angular departure between the sensor frame and the machine frame. It is the umbrella category; tilt, rotation, and non-orthogonality are its specific, measurable components. In practice, "misalignment" is what a technician reports, and the other terms are what an engineer quantifies.

Tilt is an angular error about an axis lying in the mounting plane — a rotation of the sensor's Zs axis away from the surface normal Zm. Its usual causes are a burr or chip under one corner of the package, a curved or painted surface, a warped adapter plate, or a mounting stud that is not perpendicular to the surface. Tilt is the most consequential error for low-frequency and static measurements because it couples gravity into a horizontal axis: a tilt angle θ produces an apparent horizontal acceleration of g·sin θ. At g = 9.81 m/s², a mere 1° of tilt injects roughly 0.171 m/s² (17.1 mg) of false DC signal into the horizontal channel — often larger than the signal of interest.

Rotation is an angular error about the surface normal Zm — the sensor is tilted correctly but clocked to the wrong heading. A 90° rotation swaps the X and Y channels outright; a small rotation (a few degrees) causes cross-axis leakage between two horizontal channels. Rotation is easy to miss because it produces no DC offset and no obvious anomaly in a single channel; it only becomes visible when two channels are compared or when a known excitation direction is applied.

Offset (position offset, or lever-arm error) is a translational displacement of the sensor's proof mass from the point at which the motion is defined. It is not an angular error at all, but it belongs in the same taxonomy because it produces the same practical outcome: a reading that does not represent the quantity of interest. If the structure rotates with angular acceleration α about an axis at distance r from the sensor, the sensor reports a spurious tangential acceleration a = α·r, plus a centripetal term ω²·r. Offset errors dominate on rotating machinery, long cantilevers, and any structure where the sensor cannot be placed at the true reference point.

Non-orthogonality is an error internal to the sensor or introduced by its mounting interface: the angle between the sensor's own sensitive axes deviates from 90°. A three-axis MEMS accelerometer may have a specified cross-axis sensitivity of 1–3%, and a poorly machined triaxial block or a stack of single-axis sensors bolted together can easily exceed that. Non-orthogonality is distinct from tilt and rotation because it cannot be corrected by re-clocking the whole sensor; it requires a full 3×3 calibration matrix.

Mounting error taxonomy for accelerometer installation Five error classes shown against a machine reference frame: tilt about an in-plane axis, rotation about the surface normal, translational offset, non-orthogonality of sensor axes, and the resulting cross-coupling into measurement channels. A · Tilt (about in-plane axis) mounting surface Zₘ Zₛ θ aₓ₋₋₋ = g·sinθ ≈ 0.171 m/s² per 1° B · Rotation (about Zₘ) Yₘ Xₘ Yₛ Xₛ φ cross-axis leakage: aₓ ≈ aₜ·sinφ C · Offset (lever arm) rotation axis sensor r aₓ₋₋₋ = α·r + ω²·r D · Non-orthogonality ideal Xₛ Yₛ β cross-axis sensitivity ≈ sinβ E · Error class → observable signature Tilt DC offset on horizontal axes; g·sinθ bias Rotation X/Y channel swap or leakage; no DC bias Offset α·r and ω²·r terms; scales with speed² Non-orthogonality Cross-axis response to single-axis excitation Combined Full 3×3 matrix M: aₛ = M·aₘ + b Diagnosis order: verify offset → remove tilt → resolve rotation → calibrate non-orthogonality

How each error appears in the data

The value of a taxonomy is that each class leaves a different fingerprint, and the fingerprint tells you which correction to apply. Consider a triaxial sensor mounted on a plate that is nominally level, with the machine Z axis vertical. In a static, unpowered condition the sensor should read approximately (0, 0, +9.81) m/s² in the machine frame.

  • Tilt signature: the horizontal channels show a non-zero DC value that does not average to zero over a slow rotation of the assembly. If X reads 0.17 m/s² and Z reads 9.81 m/s², the tilt about Y is approximately arcsin(0.17/9.81) ≈ 1.0°. Tilting the assembly by 180° about Z and re-reading reverses the sign of the horizontal offset — a quick field check that distinguishes tilt from a genuine sensor bias.
  • Rotation signature: a known excitation applied along machine X appears mostly on sensor Y. If the ratio aYs/aXs ≈ tan φ, the sensor is clocked by φ. A 90° error is unmistakable; a 3° error shows up as roughly 5% channel leakage and is easily mistaken for cross-axis sensitivity, which is why rotation must be resolved before non-orthogonality is calibrated.
  • Offset signature: the error grows with the square of rotational speed (centripetal term) or with angular acceleration (tangential term), and it vanishes when the structure is stationary. A sensor reading that is clean at 300 rpm but shows 0.8 m/s² at 1200 rpm on a machine with a 50 mm lever arm is consistent with ω²r = (125.7 rad/s)² × 0.05 m ≈ 790 m/s² — far too large, which itself indicates the lever arm must be much smaller or the reading is dominated by structural resonance; the point is that the scaling law, not the absolute number, identifies the class.
  • Non-orthogonality signature: a single-axis shaker excitation along machine X produces a repeatable, direction-independent response on sensor Y that does not change when the sensor is re-clocked. That invariance under re-clocking is the decisive test separating non-orthogonality from rotation.
Diagnostic ordering matters. Correct in this sequence: (1) quantify and remove position offset by relocating the sensor or applying the rigid-body correction; (2) measure and remove tilt using a static gravity reference; (3) resolve rotation using a known excitation direction; (4) only then estimate the residual 3×3 matrix that absorbs non-orthogonality and any remaining cross-axis terms. Calibrating non-orthogonality first will silently absorb tilt and rotation into the matrix, producing a model that fits the calibration setup but fails when the sensor is reinstalled.

Building the correction model

All five classes collapse into a single affine model that is worth writing explicitly, because it makes the separation of concerns concrete:

a_m = M · a_s + b

where
  a_s  = raw sensor vector (m/s^2), sensor frame
  a_m  = corrected vector (m/s^2), machine frame
  M    = 3x3 matrix combining rotation R, tilt T, and non-orthogonality N
         M = R · T · N   (order chosen so each factor is estimable independently)
  b    = residual bias vector (m/s^2), estimated from a static level reference

In a practical installation the matrix M is estimated from a set of static orientations. With the sensor mounted on a fixture that can be indexed to six known attitudes (Z up, Z down, X up, X down, Y up, Y down), each attitude provides a gravity vector of known direction in the machine frame. Stacking the six measurements gives 18 equations for the 9 unknowns of M plus 3 bias terms, solved by least squares. A residual RMS above roughly 0.05 m/s² after the fit usually indicates that the fixture itself is not rigid, or that one of the indexed positions was mis-set — a mounting error in the calibration setup masquerading as a sensor error.

Field checklist for classifying an installation error

Observation Most likely class Confirming test Correction
Horizontal DC offset in static condition Tilt Rotate assembly 180° about Z; offset reverses sign Shim or re-machine mounting surface; apply tilt rotation
Excitation on X appears on Y, invariant under re-clocking Non-orthogonality Re-clock sensor 90°; leakage ratio unchanged Estimate full 3×3 matrix
Excitation on X appears on Y, changes with re-clocking Rotation Apply known uniaxial excitation; compute atan(aY/aX) Re-clock sensor or apply Z rotation
Error scales with rpm² or with angular acceleration Offset Run at two speeds; fit error vs ω² Relocate sensor; apply α·r and ω²·r correction
Error present only during dynamic events, absent statically Offset or structural resonance Compare sensor at two radii; check coherence Relocate; add mounting stiffness

The taxonomy is not academic bookkeeping. Each class has a distinct physical cause, a distinct signature, and a distinct remedy, and the order in which they are corrected determines whether the resulting calibration model generalizes beyond the bench. The next step in any real diagnosis is to measure the actual geometry of the installation — surface flatness, stud perpendicularity, sensor clocking marks, and distance to the reference point — because those measurements, not the accelerometer's datasheet, set the floor on achievable accuracy.

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