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22-Agric-A5 Principles of Instrumentation · May 2016

Question 4 of 7: Beam-Type Accelerometers

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. 04-Agric-A5 Principles of Instrumentation, National Exams May 2016 — a three-hour open-book exam; any non-communicating calculator is permitted. Questions 1 and 2 are compulsory (20 marks each); candidates then choose any three (3) of Questions 3-7 (20 marks each) for a 100-mark paper. All seven questions are worked here.

Reference texts. E.O. Doebelin, Measurement Systems: Application and Design, 5th ed. (calibration, standards, static/dynamic sensor characteristics, second-order step response, sampling and ADCs); J.P. Bentley, Principles of Measurement Systems, 4th ed. (accuracy vs. precision, error propagation, signal conditioning); P. Horowitz and W. Hill, The Art of Electronics, 3rd ed. (Johnson noise, CMRR, ADC architectures, anti-aliasing, op-amp signal conditioning); J. Fraden, Handbook of Modern Sensors: Physics, Designs, and Applications, 5th ed. (thermistors, thermocouples, capacitive and photo sensors).

Question 4: Beam-Type Accelerometers (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

a) A cantilever beam accelerometer works by sensing the inertial force $\vec F=-m\vec a$ that the suspended mass exerts on the beam as the device accelerates (or, equivalently at rest, the force $m\vec g$ gravity exerts on it). Force is a vector — it has both a magnitude and a direction — and critically, the beam only bends (and the strain gauges only respond) in proportion to the component of that force along the beam's own sensitive (bending) axis; the components along the other two axes produce little or no bending of that particular beam. Since the true acceleration (and hence the resulting inertial force on the mass) can point in any direction in 3-D space, it must be treated as a vector so that its projection onto the sensor's principal axis — the only quantity the beam actually responds to — can be correctly extracted.

b) Mount the three accelerometers with their principal (most sensitive) axes along three mutually orthogonal directions ($x$, $y$, $z$), so that any true acceleration vector is fully captured across the three outputs. Each individual sensor's output is then not a pure single-axis reading but a weighted sum of the true $a_x$, $a_y$, $a_z$ components, dominated by its own principal-axis sensitivity $S_{ii}$ but with small cross-axis (off-principal-axis) sensitivities $S_{ij}$ mixed in. Characterize this fully during calibration — apply known accelerations along each of the three axes in turn and record all three sensor outputs each time — to build the $3\times3$ sensitivity (cross-coupling) matrix $S$. In normal use, the three raw outputs $\vec V_{raw}$ are then corrected by inverting that matrix, $\vec a_{true}=S^{-1}\vec V_{raw}$, which mathematically removes the cross-axis contamination and recovers the true 3-axis acceleration vector far more accurately than trusting each accelerometer's own axis in isolation.

Device body Beam (long axis) Mass sensitive axis (bending) Beam long axis VERTICAL -> sensitive axis HORIZONTAL beam Beam long axis HORIZONTAL -> sensitive axis VERTICAL (case at right)
Fig. 2 — A cantilever beam accelerometer's sensitive (bending) axis is always perpendicular to the beam's own long axis. Mounting the beam long-axis vertically (left) puts the sensitive axis horizontal — perpendicular to gravity at rest.

c) The beam should be mounted with its long axis vertical — i.e. with the sensitive (bending) axis horizontal at the level (zero-tilt) position — not with the beam horizontal.

Approach. Model the output as the component of gravity resolved onto the sensor's sensitive axis as the device tilts by angle $\theta$ from level, and compare the sensitivity (slope of output vs. tilt angle) of the two candidate mounting orientations at $\theta=0$, since that is where a tilt sensor spends most of its time and must be most responsive.

  1. Sensitive axis horizontal at rest (beam mounted vertically). As the device tilts by $\theta$ from level, the sensitive axis picks up the component of gravity along it, $a_{sense}=g\sin\theta$. The sensitivity is $$\dfrac{da_{sense}}{d\theta}=g\cos\theta,$$ which equals $g$ (its maximum possible value) exactly at $\theta=0$ — the output changes as fast as physically possible for the smallest tilts away from level, which is precisely what a tilt sensor needs to detect.
  2. Sensitive axis vertical at rest (beam mounted horizontally). Here the sensitive axis is already aligned with gravity at $\theta=0$, so $a_{sense}=g\cos\theta$, and $$\dfrac{da_{sense}}{d\theta}=-g\sin\theta,$$ which is exactly zero at $\theta=0$: the output is at a flat maximum near level, so a small tilt away from level produces almost no change in output — the worst possible sensitivity precisely where a tilt sensor is used most.
  3. Conclusion. $$\boxed{\text{mount the beam VERTICALLY (sensitive axis horizontal) for maximum tilt sensitivity near level.}}$$ The horizontal-beam orientation is a poor tilt sensor near level (it behaves instead like a good sensor for confirming the device is genuinely upside-down/vertical, where its own sensitivity peaks), while the vertical-beam orientation gives the best resolution for the small everyday tilts a leveling/tilt application actually needs to measure.