Displacement, velocity and acceleration measurement

Transducers for displacement (potentiometer, LVDT, capacitive, eddy-current, encoders), velocity (velocity pickups, tachogenerators, pulse methods) and acceleration (seismic accelerometers and vibrometers), with worked numericals.

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Why it matters

Motion measurement is the backbone of machine tools, robots, vehicles and condition monitoring. Displacement, velocity and acceleration are related by differentiation and integration, but in practice each is measured with a different transducer, and the seismic (spring–mass) principle decides whether one device acts as an accelerometer or a vibrometer.

Key ideas

Displacement.

  • Contact, analog: resistive potentiometers (simple, wear), LVDTs and RVDTs (frictionless, high resolution), strain-gauge based extensometers.
  • Non-contact: capacitive probes (sub-micrometre gaps), eddy-current proximity probes (shaft vibration and runout on conductive targets), optical triangulation and laser interferometers (nanometre resolution).
  • Digital: optical encoders. An incremental encoder gives N pulses per revolution on two channels A and B in quadrature (90° apart); the phase order gives direction and counting every edge of both channels gives 4N counts per revolution. An absolute encoder gives a unique code (usually Gray code, so only one bit changes between positions) for every position and keeps position after power loss. Linear scales work the same way along a line.

Velocity.

  • Linear: the moving-coil (or moving-magnet) velocity pickup — a coil moving in a magnetic field generates e = B·l·v, directly proportional to velocity. Doppler radar and laser Doppler vibrometers measure velocity without contact.
  • Angular: the DC tachogenerator gives a voltage proportional to speed with polarity showing direction (brush wear and ripple are its drawbacks). The AC tachogenerator (drag-cup type) gives an output whose amplitude is proportional to speed at the excitation frequency. Pulse methods count teeth passing a magnetic (variable-reluctance) pickup, Hall sensor or optical sensor: frequency ∝ speed. The variable-reluctance pickup's amplitude falls at low speed, so it fails near standstill. A stroboscope freezes the motion when its flash rate equals the rotation rate (or a sub-multiple — beware).
  • Two speed-counting methods: count pulses in a fixed gate time (good at high speed) or time one pulse period (good at low speed).

Acceleration — seismic transducers. A mass m on a spring k with damping c sits in a case fixed to the vibrating body. The measured quantity is the relative displacement z between mass and case. For base motion y = Y·sin ωt, with r = ω/ωn:

  • Accelerometer (r ≪ 1, stiff spring, high ωn): z ≈ (acceleration)/ωn². The output is proportional to acceleration; the usable range extends to about 0.2–0.3 of fn, flattest with ζ ≈ 0.6–0.7. Sensitivity falls as 1/ωn², so a wider bandwidth means a smaller signal.
  • Vibrometer / seismometer (r ≫ 1, soft spring, low ωn): the mass stays nearly still in space, so z ≈ −y; the output measures displacement of the body.
  • The sensing element can be piezoelectric (dynamic only, wide band), piezoresistive or strain-gauge (down to DC), capacitive MEMS (phones, airbags, tilt sensing — responds to gravity), or a servo (force-balance) accelerometer for high accuracy.

Integration and differentiation. Velocity and displacement can be obtained by integrating an accelerometer signal (integration attenuates high-frequency noise but drifts with offsets); differentiating displacement amplifies noise. For a sinusoid, v = ω·x and a = ω²·x in amplitude.

Formulas

Resolution = 360°/(4N) (incremental encoder with quadrature ×4 decoding) N = pulses per revolution per channel.

n = 60·f/N (speed from pulse frequency) n = speed (rev/min); f = pulse frequency (Hz); N = pulses per revolution.

e = B·l·v e = emf (V); B = flux density (T); l = active conductor length (m); v = velocity (m/s).

ωn = √(k/m), fn = ωn/(2π) k = spring stiffness (N/m); m = seismic mass (kg).

Z/(a/ωn²) = 1 / √[(1 − r²)² + (2ζr)²] (accelerometer amplitude ratio) Z = relative-displacement amplitude (m); a = base acceleration amplitude (m/s²); r = ω/ωn; ζ = damping ratio.

Z/Y = r² / √[(1 − r²)² + (2ζr)²] (vibrometer amplitude ratio → 1 for r ≫ 1) Y = base displacement amplitude (m).

v = ω·x, a = ω²·x (sinusoidal amplitudes)

Worked examples

Example 1 — encoder and tachogenerator. (a) A 1000-pulse/rev incremental encoder is decoded ×4. Find the angular resolution. (b) Channel A gives 500 pulses in a 10 ms gate. Find the speed. (c) A DC tachogenerator rated 7 V per 1000 rev/min is on the same shaft. Find its output.

  1. Resolution = 360°/(4 × 1000) = 0.09°.
  2. f = 500/0.010 = 50 000 Hz; n = 60 × 50 000/1000 = 3000 rev/min.
  3. V = 7 × 3000/1000 = 21 V.

Answer: 0.09°; 3000 rev/min; 21 V.

Example 2 — seismic accelerometer (GATE level). A seismic mass of 10 g sits on a spring of stiffness 4 × 10⁵ N/m. (a) Find fn and the relative displacement for a steady acceleration of 98.1 m/s² (10 g). (b) Find the amplitude error at 200 Hz for ζ = 0.7 and for ζ = 0.

  1. ωn = √(4 × 10⁵/0.01) = 6325 rad/s; fn = 6325/(2π) = 1007 Hz.
  2. z = a/ωn² = 98.1/(6325²) = 98.1 × 2.5 × 10⁻⁸ = 2.45 × 10⁻⁶ m = 2.45 µm.
  3. r = 200/1007 = 0.199.
  4. ζ = 0.7: M = 1/√[(1 − 0.0395)² + (2 × 0.7 × 0.199)²] = 1/√(0.9226 + 0.0774) = 1.000 — essentially no error.
  5. ζ = 0: M = 1/(1 − 0.0395) = 1.041 — reads 4.1 % high.

Answer: fn ≈ 1007 Hz, z = 2.45 µm; error ≈ 0 % with ζ = 0.7, +4.1 % undamped.

Common mistakes

  • Using a seismic device in the wrong frequency region: an accelerometer must have fn well above the signal; a vibrometer well below.
  • Forgetting the quadrature factor of 4 (or using it when only one edge of one channel is counted).
  • Counting stroboscope flashes at a sub-multiple of the speed and reporting half the true speed.
  • Expecting a piezoelectric accelerometer to measure gravity or a steady tilt — use a MEMS capacitive or piezoresistive type.
  • Mixing rad/s and Hz in ωn and r.
  • Integrating an accelerometer signal with an offset and getting runaway displacement drift.

For GATE IN

NAT questions on encoder resolution and speed counting, tachogenerator outputs, seismic transducer natural frequency, static deflection per g, amplitude ratio at a given r and ζ, and conversion between displacement, velocity and acceleration amplitudes of a sinusoid. MCQs ask which region a seismic instrument works in and why ζ ≈ 0.7 is chosen.

Quick check

  1. A 2500-pulse/rev encoder decoded ×4 has what resolution?
  2. Which seismic instrument uses a soft spring and low natural frequency?
  3. A vibration of 2 mm amplitude at 10 Hz has what acceleration amplitude?
  4. Why does a variable-reluctance speed pickup fail at very low speed? Answers: 1. 0.036°; 2. the vibrometer (displacement pickup); 3. (2π × 10)² × 0.002 = 7.9 m/s²; 4. its induced emf is proportional to the rate of change of flux, which falls towards zero.

Interview questions

All Sensors and Transducers interview questions

Try answering each one aloud before you open it.

  1. 1.What is a displacement sensor and how does it work?Concept

    A displacement sensor is a device used to measure the distance moved by an object. It works by converting the physical movement into an electrical signal. Common types include linear variable differential transformers (LVDTs) and potentiometers. LVDTs use electromagnetic induction, while potentiometers use a resistive element to measure displacement.

  2. 2.Explain the working principle of a velocity transducer.Concept

    For linear velocity, a moving-coil or moving-magnet pickup uses electromagnetic induction: a conductor moving through a magnetic field generates e = B·l·v, so the output voltage is directly proportional to velocity. For angular velocity, a DC tachogenerator gives a voltage proportional to speed with polarity showing direction, an AC drag-cup tachogenerator gives an amplitude proportional to speed, and pulse methods count teeth or slots passing a magnetic, Hall or optical sensor so that pulse frequency is proportional to speed.

  3. 3.What is an accelerometer and what are its applications?Concept

    An accelerometer is a device that measures acceleration forces. These forces can be static, like gravity, or dynamic, caused by movement or vibrations. Accelerometers are used in various applications, including smartphones for screen orientation, automotive systems for airbag deployment, and industrial equipment for vibration monitoring.

  4. 4.Why are LVDTs commonly used in industrial applications for displacement measurement?Application

    LVDTs are preferred in industrial applications because they offer high accuracy, reliability, and durability. They are non-contact sensors, which means they have a long lifespan and require minimal maintenance. Additionally, LVDTs provide a linear output over a wide range of displacements, making them suitable for precise measurements.

  5. 5.How does temperature affect the performance of accelerometers?Application

    Temperature can affect the performance of accelerometers by causing drift in the sensor's output. This is because the materials used in accelerometers can expand or contract with temperature changes, affecting their sensitivity and accuracy. Manufacturers often provide temperature compensation features to mitigate these effects.

  6. 6.What are the advantages of using optical sensors for displacement measurement?Application

    Optical sensors offer several advantages for displacement measurement, including high precision, non-contact operation, and immunity to electromagnetic interference. They can measure small displacements with high accuracy and are suitable for use in harsh environments where other sensors might fail.

  7. 7.Explain the seismic principle and how the same spring–mass system can act as an accelerometer or a vibrometer.Concept

    A seismic transducer is a mass on a spring and damper inside a case fixed to the vibrating body, and it measures the relative displacement between mass and case. If the natural frequency is well above the vibration frequency (stiff spring, r = ω/ωn ≪ 1), the relative displacement is proportional to the body's acceleration, a/ωn², so it is an accelerometer, usable up to about 0.2–0.3 of fn with ζ ≈ 0.7. If the natural frequency is well below the vibration frequency (soft spring, r ≫ 1), the mass stays almost still in space and the relative displacement equals the body's displacement, so it is a vibrometer.

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