Force, torque and vibration measurement

Why it matters

Force, torque and vibration are the mechanical quantities that tell you whether a machine is doing its job and whether it is about to fail: weighing and filling, press and test-rig forces, engine and gearbox torque, and vibration trends for predictive maintenance. Most of these measurements reduce to strain on a well-designed elastic element or to the seismic principle.

Key ideas

Force measurement.

• Elastic elements: force produces strain or deflection in a column, beam, ring or diaphragm. Read strain with bonded gauges (load cell) or deflection with an LVDT (proving ring with LVDT, a classic calibration standard).
• Piezoelectric force sensors: very stiff, wide bandwidth, ideal for impact and dynamic forces; cannot hold a static reading.
• Hydraulic and pneumatic load cells: force acts on a piston or diaphragm and the pressure is read — no electrical supply at the sensing point.
• Force-balance (null) devices: a feedback actuator (e.g. a coil in a magnetic field) balances the applied force; the balancing current is the measurement — very accurate and linear.
• Specification terms: capacity, rated output (mV/V), combined error, creep, safe and ultimate overload.

Torque measurement.

• Strain-gauge torque shafts: torque in a circular shaft produces pure shear on the surface; the principal strains are at ±45° to the axis, equal and opposite (+ε and −ε). Four gauges at ±45° in a full bridge give maximum output, cancel bending and axial load, and compensate temperature.
• Getting the signal off a rotating shaft: slip rings (simple, noisy, wear), rotary transformers (non-contact AC excitation and signal), radio telemetry.
• Angle-of-twist (phase-shift) methods: two toothed wheels or optical discs a length L apart; torque twists the shaft by θ = T·L/(G·J), shifting the phase between two pulse trains.
• Magnetostrictive (non-contact) sensors: torque changes the permeability of a ferromagnetic shaft surface, sensed by coils.
• Reaction (cradled) dynamometers measure the reaction torque on the motor casing with a force sensor on a torque arm.
• Mechanical power is P = T·ω, so torque plus speed gives shaft power.

Vibration measurement.

• Sensors: piezoelectric accelerometers (wide band, most common), electrodynamic velocity pickups (mid-frequency, self-generating), eddy-current proximity probes (shaft displacement relative to bearings in turbomachinery), laser Doppler vibrometers (non-contact).
• Seismic principle (see the previous topic): stiff, high-fn devices measure acceleration; soft, low-fn devices measure displacement.
• Quantities: for a sinusoid x = X·sin ωt, velocity amplitude ωX and acceleration amplitude ω²X. Displacement is emphasised at low frequency, acceleration at high frequency; velocity (often in mm/s rms) is the usual machine-severity measure over the mid band. Report peak, peak-to-peak or rms explicitly — for a sine, rms = peak/√2.
• Mounting: stud mounting gives the highest usable frequency; magnets and adhesive lower the mounted resonance. Accelerometer mass must be small compared with the structure to avoid mass loading.

Formulas

τ_s = 16·T / (π·d³) (surface shear stress, solid circular shaft) τ_s = shear stress (Pa); T = torque (N·m); d = shaft diameter (m).

ε_45 = ±τ_s / (2G) (principal strains at ±45°) G = shear modulus (Pa); G = E / [2(1 + ν)].

Vo = Vex·GF·ε_45 (four ±45° gauges, full bridge) Vex = excitation (V); GF = gauge factor.

θ = T·L / (G·J), J = π·d⁴/32 θ = angle of twist (rad); L = length between sensing points (m); J = polar second moment of area (m⁴).

P = T·ω = 2π·n·T/60 P = power (W); ω = angular speed (rad/s); n = speed (rev/min).

v = ω·x, a = ω²·x, rms = peak/√2 (sinusoidal vibration)

Worked examples

Example 1 — strain-gauge torque shaft. A solid steel shaft (d = 50 mm, G = 80 GPa) transmits 1 kN·m at 1500 rev/min. Four gauges (GF = 2.0) at ±45° form a full bridge excited at 10 V. Find the surface shear stress, gauge strain, bridge output and shaft power.

1. τ_s = 16 × 1000 / (π × 0.05³) = 16 000 / 3.927 × 10⁻⁴ = 40.7 MPa.
2. ε_45 = τ_s/(2G) = 40.74 × 10⁶ / (2 × 80 × 10⁹) = 2.546 × 10⁻⁴ = 255 µε.
3. Vo = Vex·GF·ε = 10 × 2.0 × 2.546 × 10⁻⁴ = 5.09 mV.
4. P = 2π × 1500 × 1000 / 60 = 157 kW.

Answer: 40.7 MPa; 255 µε; 5.09 mV; 157 kW.

Example 2 — vibration levels (GATE level). An accelerometer of sensitivity 100 mV/g (g = 9.81 m/s²) gives 0.5 V rms on a bearing housing vibrating sinusoidally at 50 Hz. Find the rms acceleration, rms velocity and the peak-to-peak displacement.

1. a_rms = 0.5/0.1 = 5 g = 5 × 9.81 = 49.05 m/s².
2. ω = 2π × 50 = 314.16 rad/s.
3. v_rms = a_rms/ω = 49.05/314.16 = 0.156 m/s = 156 mm/s.
4. x_rms = a_rms/ω² = 49.05/98 696 = 0.497 mm; x_peak = √2 × 0.497 = 0.703 mm.
5. x_pp = 2 × 0.703 = 1.41 mm.

Answer: 49.1 m/s² rms, 156 mm/s rms, 1.41 mm peak-to-peak (a very severe vibration for most machines).

Example 3 — angle of twist. For the shaft of Example 1 with sensing discs 0.5 m apart: J = π × 0.05⁴/32 = 6.136 × 10⁻⁷ m⁴; θ = 1000 × 0.5 / (80 × 10⁹ × 6.136 × 10⁻⁷) = 0.0102 rad = 0.58°.

Common mistakes

• Mounting torque gauges along the shaft axis — pure torsion gives no axial strain; they must be at ±45°.
• Using E where G is needed (or forgetting G = E/[2(1 + ν)]).
• Mixing rms, peak and peak-to-peak values in vibration specifications.
• Using rev/min directly in P = T·ω; convert to rad/s.
• Treating a load cell's rated output (mV/V) as mV per newton.
• Expecting a piezoelectric force sensor to read a static weight.

For GATE IN

Typical NATs: shear stress and principal strain on a torque shaft and the resulting bridge output; angle of twist; power from torque and speed; converting accelerometer output into acceleration, velocity and displacement at a given frequency; load-cell output from rated mV/V. MCQs: where to place torque gauges, how to get signals off rotating shafts, which vibration sensor suits which frequency range.

Quick check

1. At what angle to the shaft axis are torque gauges mounted?
2. A shaft carries 200 N·m at 3000 rev/min. What power does it transmit?
3. A sinusoidal displacement of 0.1 mm peak at 100 Hz has what peak acceleration?
4. What device lets you excite and read a bridge on a rotating shaft without contact? Answers: 1. ±45°; 2. 62.8 kW; 3. (2π × 100)² × 0.1 × 10⁻³ = 39.5 m/s²; 4. a rotary transformer (or radio telemetry).