Force, Load Cells and Vibration
Strain gauges and bridges, load cell ratings and mounting, torque sensing, accelerometers and vibration monitoring.
Strain gauges
A strain gauge is a thin metal-foil grid bonded to a surface. When the surface stretches, the grid gets longer and thinner and its resistance rises by a small amount. The change in resistance is proportional to the strain: ΔR ÷ R = GF × ε, with ε the strain (change in length over length) and GF the gauge factor, about 2 for metal foil. Strains are tiny (a few hundred microstrain, 10-6), so the resistance changes by a fraction of a percent: a delicate measurement that needs a bridge and a good amplifier. The same technology sits inside load cells, torque sensors and pressure sensors (see Pressure, Level and Flow Sensors).
Worked example: strain in a steel bar
- A force of 10 kN pulls a steel bar of cross-section 100 mm². The stress is σ = F ÷ A = 10 000 ÷ 100×10-6 = 100 MPa.
- With E = 200 GPa the strain is ε = σ ÷ E = 100 ÷ 200 000 = 500 µε (0.050 percent).
- A gauge with GF = 2.0 changes by ΔR ÷ R = 2.0 × 0.0005 = 0.0010, or 0.1 percent: 0.35 Ω on a 350 Ω gauge. Too small to measure directly with a meter, hence the bridge.
- Temperature compensation: the gauge material and the metal beneath expand differently, and the resistance itself changes with temperature, which is as big as the signal. Self-compensating gauges and a bridge with dummy gauges cancel it.
- Bonding is everything: a gauge must be glued with the right adhesive on a clean, prepared surface, protected from moisture and mechanical damage. A poor bond gives creep and hysteresis.
- Semiconductor (piezoresistive) gauges have a gauge factor of 100 or more, so a larger signal, but a stronger temperature effect and non-linearity.
The Wheatstone bridge
Four resistors form a bridge. With an excitation Vex across one pair of corners, the output is taken from the other pair. When all four are equal the output is zero. If one is a strain gauge whose resistance changes, the bridge becomes unbalanced and the output is nearly proportional to the change. Several active gauges increase the output and cancel unwanted effects:
| Configuration | Active gauges | Output | Notes |
|---|---|---|---|
| Quarter bridge | One | Vex × GF × ε ÷ 4 | Simplest; temperature is not compensated unless a dummy gauge is used |
| Half bridge | Two (opposite strains, as in bending) | Vex × GF × ε ÷ 2 | Twice the output; compensates temperature; used in bending beams |
| Full bridge | Four | Vex × GF × ε | Four times the quarter-bridge output; cancels temperature and unwanted bending or axial effects; the standard for load cells |
Worked example: bridge output
On the bar above with a 10 V excitation: a quarter bridge gives 10 × 2.0 × 0.0005 ÷ 4 = 2.50 mV; a full bridge gives 10.0 mV. The signal is a few millivolts on top of a 5 V common-mode level, so the amplifier needs high gain (hundreds), a high common-mode rejection and low offset drift: an instrumentation amplifier or a dedicated bridge input (see Sensors and Signal Conditioning).
- Six-wire connection: two sense wires measure the excitation voltage directly at the bridge so that cable resistance does not change the sensitivity. A four-wire load cell on a long cable loses sensitivity (and drifts with the cable temperature).
- Ratiometric measurement: if the converter’s reference is the same as the excitation, the result is independent of the supply voltage.
- Balance and zero: the bridge is never perfectly balanced: the offset (zero) is removed in the amplifier or by tare.
Load cells
A load cell is a metal spring element (steel, aluminum or stainless steel) with strain gauges in a full bridge, sealed and calibrated. The output is specified as sensitivity (rated output) in mV per volt of excitation at the rated load, typically 2 or 3 mV/V.
| Type | Typical capacity | Used for |
|---|---|---|
| S-beam (Z-type) | 10 kg to 10 t | Tension and compression, hanging scales, tension control |
| Shear beam | 50 kg to 10 t | Tank and platform scales, bench scales |
| Single-point (platform) | 1 to 500 kg | Small scales, packaging machines |
| Compression canister, column | 1 t to 1 000 t | Silos, hoppers, truck scales |
| Button / washer | 1 N to 100 kN | Small forces, press forces, bolt loads |
| Pancake / low-profile | 10 kg to 100 t | Tension and compression with a low height, test machines |
Worked example: signal and resolution
- A 100 kg load cell with a sensitivity of 2 mV/V on a 10 V excitation gives a full-scale output of 2 × 10 = 20 mV at 100 kg, i.e. 0.2 mV per kg.
- A 16-bit converter spanning ±25 mV (50 mV) has a step of 0.76 µV, so the full load is 26 214 counts and the resolution is 100 kg ÷ 26 214 = 3.8 g. A dedicated 24-bit bridge ADC reaches 0.01 percent of the span with ease.
- Accuracy is quoted as combined error (non-linearity, hysteresis and repeatability), typically 0.02 to 0.05 percent of full scale (OIML class C3 for legal-for-trade scales), plus creep (the output drifting under a constant load) and a temperature effect on zero and span.
- Overload: a safe overload of 150 percent of the rated load (and a breaking overload of 300 percent) is typical: size for shocks as well as the static weight.
Weighing systems
Tanks, hoppers and platforms are usually supported on several load cells. A summing box adds their signals (after trimming each to the same sensitivity) into a single bridge for the indicator. Each cell must carry a share of the weight, and the sum of their capacities must exceed the maximum load by a safe margin.
Worked example: a tank on four cells
A tank weighs 2 000 kg empty and holds up to 3 000 kg of product: the maximum load is 5 000 kg. On 4 cells each carries 1 250 kg if the load is shared evenly. With a margin of about 1.5 for an uneven distribution, wind and shocks the capacity of each cell should be at least 1 875 kg: choose 2 000 kg cells. Weighing the product alone means subtracting the tank (tare); the resolution is then set by the 8 t total capacity, so use enough resolution (a 24-bit indicator) to see small changes.
Mounting and calibration
- Load the cell along its axis only. Side loads, bending and twisting give errors and can destroy it. Use the maker’s mounting hardware (self-aligning feet, rocker pins, rod ends) and check rods or stays against sideways movement and uplift.
- A rigid, level foundation: a soft or sloping support shifts the load between cells and causes errors.
- Do not let pipes, hoses or conduits touch the tank: they carry part of the load. Use flexible connections.
- Weld with care: never pass welding current through a load cell; bridge it with a cable.
- Cable and junction box: use a shielded cable with the shield grounded at the indicator, keep it away from power cables, and use a sealed junction box.
- Calibrate with test weights (certified masses) at two or more points, with the cells installed and the system loaded as in use. If weights are impractical use a built-in electronic calibration with the cell data, accepting a lower accuracy.
- Protect from the environment: temperature gradients, wind, vibration and moisture all affect the reading; hermetically sealed cells and a stable location are best for precision.
Torque sensors
A torque sensor has strain gauges on a shaft (in a full bridge sensitive to twist), with the signal taken off by slip rings, a rotary transformer or telemetry. A reaction torque sensor measures the torque on a fixed flange without rotating parts. Torque is also related to the electrical power: T = 9.549 × P ÷ n (N·m, with P in watts and n in rpm), which you can use to check a sensor or a drive’s own torque reading (see Motion Control Basics): a motor delivering 2 200 W at 1 500 rpm produces 14.01 N·m.
Accelerometers and vibration
An accelerometer measures the acceleration of the surface it is fixed to. A piezoelectric (IEPE, “ICP”) type produces a charge from a crystal under a mass; the built-in electronics give a voltage output powered from a constant current line of 2 to 20 mA; typical sensitivity 100 mV/g with a frequency range of 1 Hz to 10 kHz. A MEMS accelerometer (capacitive) works down to DC (so it measures tilt and gravity) and is cheaper, with a lower frequency range and noise performance.
Worked example: velocity, acceleration and displacement
- Machine vibration is commonly rated by velocity (RMS, in mm/s; ISO 10816 / 20816 gives zones, with roughly 4.5 mm/s RMS as the boundary between acceptable and unsatisfactory for medium machines).
- For a sine vibration at 50 Hz, ω = 2π × 50 = 314.2 rad/s. A velocity of 4.5 mm/s RMS corresponds to an acceleration of ω × v = 314.2 × 0.0045 = 1.41 m/s² RMS = 0.144 g, and a displacement of v ÷ ω = 14 µm RMS.
- A 100 mV/g accelerometer therefore gives 14.4 mV RMS: a small signal at low frequency, which is why velocity limits are checked with a low-noise sensor and a good cable.
- Condition monitoring: a rise in the vibration at the bearing, gear-mesh or rotor frequency warns of a fault weeks early. Install sensors on the bearing housings in the radial direction, and trend the values.
- Mounting: a stud mounting gives the full frequency range, a magnet or adhesive less. A clean flat surface and a stiff bracket are required.
- Cable noise and grounding: use the cable made for the sensor, and avoid ground loops (isolated sensors).
- Shock and tilt sensing: MEMS parts are good for impact, tilt and motion detection in machines and vehicles.
Other force sensors
- Piezoelectric force sensors measure dynamic force (impact, press forces) with a very stiff element, but not a static load because the charge leaks away.
- Force-sensing resistors and thin-film sensors are cheap, thin and rough: good for presence and touch, not for measurement.
- Capacitive and optical force sensors serve in robotics and medicine, with a small deflection and high resolution.
- Motor current is a simple indirect estimate of force or torque in a drive: Servo Systems and Tuning shows how the loop uses it.
- Hydraulic and pneumatic pressure in a cylinder also gives a force through F = P × A (see Pneumatic and Hydraulic Actuators).
Troubleshooting
| Symptom | Likely cause | Check |
|---|---|---|
| Zero drifts | Temperature change, moisture in the cell or cable, creep, a loose mounting | Stabilize the temperature; check the insulation resistance; re-tighten |
| Reading is wrong at the load | Calibration, side load, a mechanical binding or pipework carrying load | Free the structure; recalibrate with test weights |
| Different readings at different positions on the platform | Corner error: the cells are not trimmed or the support is uneven | Corner adjustment, level, trim the summing box |
| Unstable or noisy reading | Vibration, noise on the cable, a poor connection, a broken wire | Filter; check wires and the shield; check the input resistance of the cell |
| Output saturated or negative | Overload, reversed wires, a damaged cell | Check the polarity, the mechanical stops, the bridge resistance (350 Ω nominal) |
| Vibration reading too high | A loose sensor, a resonance of the bracket, a cable fault | Mounting stiffness, the cable, a second sensor to confirm |
Related
- Sensors and Field Devices: the overview of this guide.
- Pressure, Level and Flow Sensors: pressure transmitters use the same strain-gauge bridge.
- Sensors and Signal Conditioning: the amplifier for a bridge, with worked numbers.
- Installing and Troubleshooting Sensors: shielding and grounding of low-level signals.
- Beams and Material Properties: stress, strain and elastic modulus.
FAQ
What does mV/V mean?
The rated output of a load cell in millivolts for each volt of excitation at full load. A 2 mV/V cell on a 10 V excitation gives 20 mV at the rated load.
Why does a load cell need six wires?
Two carry the excitation, two the signal and two sense the actual excitation at the cell. The sense wires let the indicator correct for the cable resistance, which would otherwise change the sensitivity.
How do I choose a load cell capacity?
Add the maximum load and the dead load, divide by the number of cells, and multiply by a safety factor (1.25 to 1.5 or more for shocks and uneven loading). Do not oversize too much: a cell used at a few percent of its capacity has poor resolution.
What is creep?
The slow change of a load cell’s output under a constant load, usually quoted as a percentage over 30 minutes. It matters for long weighing times and for the tare of a tank.
References
- Hottinger Brüel & Kjær (HBM), An Introduction to Measurements Using Strain Gages; Vishay Micro-Measurements, Strain Gage Technical Data and Tech Notes.
- OIML R 60, Metrological regulation for load cells; ISO 20816 (vibration measurement and evaluation of machines).
- Wilcoxon / PCB Piezotronics, Accelerometer selection and mounting guides.
Last reviewed: October 2026. Values are typical; use the manufacturer’s data and applicable standards for real measurements.