Displacement, velocity and acceleration transducers form the kinematic-measurement core of the GATE Robotics and Automation (RA) 2027 “transducers for industrial instrumentation” syllabus. Because velocity is the derivative of displacement and acceleration the derivative of velocity, these three quantities are deeply linked — and examiners test both the individual devices and the relationships between them. This guide covers linear and angular displacement transducers, velocity transducers, and the seismic accelerometer, with governing equations, diagrams and worked examples.
TABLE OF CONTENTS
Linear displacement transducers
Linear displacement is the most fundamental measurement, because force, pressure, level and acceleration are all often converted to a displacement first. The main devices are:
- Resistive potentiometer: a wiper on a resistance track gives an output voltage proportional to position. Cheap and simple, but suffers from friction, wear and finite resolution.
- LVDT: the linear variable differential transformer — frictionless, high-resolution, phase-encoded output. The industry standard for precision linear displacement.
- Capacitive: area- or gap-varying capacitor for small, high-resolution displacement.
- Linear encoder / magnetostrictive: for absolute digital position over long strokes.
Angular displacement transducers
- RVDT: rotary version of the LVDT for limited angular range (typically ±40°).
- Rotary encoder: incremental (pulse counting for relative angle/speed) or absolute (unique code per position, often Gray-coded).
- Resolver & synchro: rugged electromagnetic devices giving sine/cosine outputs of shaft angle, common in aerospace and servos.
Figure 1. Incremental optical rotary encoder: light through the slots produces pulses as the shaft turns.
Velocity transducers
Velocity can be obtained by differentiating a displacement signal, but direct velocity transducers are preferred for cleaner signals. The electromagnetic (tachogenerator) type is the classic example: by Faraday’s law, a coil moving in a magnetic field generates an EMF proportional to velocity.
A DC tachogenerator gives an output voltage directly proportional to shaft angular velocity ω, with the constant kt (V per rad/s). It is widely used as the speed-feedback element in servo and motor-control loops. For linear velocity, a moving-coil (seismic velocity pickup) or laser-Doppler vibrometer is used.
Acceleration transducers (accelerometers)
An accelerometer measures acceleration using a seismic mass (proof mass) suspended by a spring and damper inside a housing. When the housing accelerates, the mass lags behind and the relative displacement of the mass — sensed by a piezoelectric, capacitive or strain-gauge element — is proportional to acceleration.
Figure 2. Seismic accelerometer: mass–spring–damper; the relative displacement of the proof mass measures acceleration.
Accelerometer as a second-order system
The mass–spring–damper obeys a second-order differential equation, so an accelerometer is a classic second-order instrument with natural frequency ωn and damping ratio ζ:
Below ωn the output faithfully tracks acceleration; a damping ratio of about ζ ≈ 0.7 gives the flattest, widest usable frequency range. A high natural frequency (stiff spring, small mass) widens the bandwidth but lowers sensitivity — a key design trade-off examiners like to probe.
Worked example
Example 1
An accelerometer has a seismic mass m = 0.01 kg and spring stiffness k = 4000 N/m. Find (a) its undamped natural frequency, and (b) the damping coefficient c for ζ = 0.7.
Solution.
(a) ωn = √(k/m) = √(4000/0.01) = √(400000) = 632.5 rad/s (≈ 100.7 Hz).
(b) c = 2ζ√(km) = 2 × 0.7 × √(4000 × 0.01) = 1.4 × 6.32 = 8.85 N·s/m.
Key formulas
FORMULA SHEET
Potentiometer output: Vo = Vs·(x/L)Tachogenerator EMF: e = kt·ω
Faraday EMF: e = −N(dφ/dt) = B·l·v
Accelerometer natural freq: ωn = √(k/m)
Damping ratio: ζ = c/(2√(km))
Kinematic links: v = dx/dt, a = dv/dt
Common mistakes to avoid
- Confusing an incremental encoder (relative, needs a reference) with an absolute encoder (unique code per position).
- Forgetting that a piezoelectric accelerometer cannot measure constant (DC) acceleration — use a capacitive/MEMS type for static tilt.
- Assuming higher natural frequency is always better — it widens bandwidth but reduces sensitivity.
- Mixing up the tachogenerator constant units — kt is V per rad/s, not V per rpm (convert carefully).
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Enroll in the GATE RA Complete CourseFrequently asked questions
What is the difference between an incremental and an absolute encoder?
An incremental encoder outputs pulses as the shaft rotates and measures change in position relative to a starting reference — if power is lost, the position must be re-referenced. An absolute encoder outputs a unique digital code for every shaft position, so it knows its exact angle immediately at power-up.
Why is a damping ratio of 0.7 preferred in accelerometers?
A damping ratio near 0.7 gives the flattest frequency response with minimal overshoot, maximising the usable bandwidth over which the output is proportional to acceleration. Lower damping causes resonant peaking; higher damping narrows the flat region.
Can a tachogenerator measure direction of rotation?
Yes — a DC tachogenerator’s output polarity reverses with the direction of rotation, so both speed (magnitude) and direction (sign) are obtained, which is why it is popular for closed-loop speed control.
How are velocity and acceleration related to displacement?
Velocity is the first time-derivative of displacement (v = dx/dt) and acceleration is the second derivative (a = d²x/dt²). In practice, differentiating amplifies noise, so direct velocity or acceleration transducers are preferred over differentiating a displacement signal.
This guide is part of the complete GATE RA 2027 Syllabus overview. Continue the sensor series with Capacitive & Inductive Sensors and Piezoelectric & Hall Effect Sensors.
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