Torque Sensor 1: Mechanicals of Materials

Torque Sensor 2: Strain Gauge & Wheatstone Bridge

Torque Sensor 3: Instrumentation Amplifier

Torque Sensor 4: Torque Sensor Build Log 

Updated: 5/15/2016 – Initial Page Creation

So now that we have a number for the amount of strain we are measuring, how do we go about measuring it? With a strain gauge! A strain gauge is a device that measures small values in strain. It is basically a long strand of copper wire. When the material it is attached to is strained, the copper stretches and therefore the resistance of the copper increases. Here is a picture from Wikipedia showing a good description of how a strain gauge works:

Strain Gauge Theory

The strain in a shaft occurs at a 45° angle from the shaft’s axis – so the strain gauges must be mounted to a shaft along this line. Below are a few pictures showing this, and after those pictures we will discuss how to actually measure the extremely small change in resistance of the strain gauges.

Datum Electronics - strain direction

Appmeas.co.uk - strain gauges bonded to a torque shaft

Phidgets - strain gauges on a shaft for measuring torque

Now that we know how to attach the strain gauges to the shaft, we need to know how to measure the change in resistance of the strain gauge. A value called the Gauge Factor (GF) is used to determine this, and it is defined as \displaystyle GF=\frac{{\Delta R/R}}{\varepsilon } where \displaystyle {\Delta R} is the change in resistance, R is the strain gauge resistance at no load, and \displaystyle \varepsilon is the strain – in this case it is actually \displaystyle \gamma. The Gauge Factor is an inherit property of the strain gauge, for a strain gauge made of Constantan foil it is equal to 2. So solving the Gauge Factor equation for \displaystyle {\Delta R} with our previous value for strain, GF is 2, and let’s just say we got a 350 \displaystyle \Omega strain gauge, that makes a change in resistance of 28.74\displaystyle \mu \Omega – this is barely readable by any cheap electronics. Unless we use a Wheatstone Bridge! Below is an image of a Wheatstone Bridge, keep in mind the resistors are actually strain gauges.

Nasa - Wheatstone Bridge

I’m not going to go through the derivation of the next equation because frankly I don’t want to, but with 4 strain gauges with a GF of 2, the equation reduces to

\displaystyle \frac{{{V_0}}}{V} = \frac{{GF \cdot T \cdot (1 + v)}}{{E \cdot \pi \cdot {R^3}}} for a solid shaft and

\displaystyle \frac{{{{V_0}}}}{V}=\frac{{GF\cdot T\cdot R\cdot (1+v)}}{{E\cdot \pi \cdot ({{R}^{3}}-{{r}^{3}})}} for a hollow shaft.

Now typically the output of a Wheatstone bridge is given in mV/V – so the answers above need to be multiplied by \displaystyle {{10}^{3}} to match common descriptions of Wheatstone Bridges.

Anywho, let’s continue with our example. The output of the Wheatstone Bridge is \displaystyle \frac{{{{V}_{0}}}}{V}=\frac{{GF\cdot T\cdot (1+v)}}{{E\cdot \pi \cdot {{R}^{3}}}}=\frac{{2\cdot 140(Nm)\cdot \left( {1+0.29} \right)}}{{205\cdot {{{10}}^{9}}\frac{N}{{{{m}^{2}}}}\cdot \pi \cdot {{{\left( {0.006m} \right)}}^{3}}}}\cdot {{10}^{3}}\frac{{mV}}{V}=2.6\frac{{mV}}{V} – this means at 15V excitation, under full torque (140 N-m), the output will be 39 mV. In order to read this value with a microcontroller, we need to amplify the signal. Check out my page on Instrumentation Amplifiers for information on how to do that!