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Measuring  Forces

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If we want to measure something, we can usually buy a meter or some other tool.  We can buy ammeters, ohmmeters, voltmeters, and even devices to measure capacitance and inductance.  Then there are pressure gauges, weighing devices, instruments for measuring length, range-finders, light-meters, thermometers, frequency meters and many others.  All these may be provided with a multi-digit readout, which may fool us into thinking that our actual measurement may be more accurate than it is.  It's not the fault of the meter - it's the way we use it.  You can spend £30000 on a 6 GHz digital oscilloscope with high quality probes, but you can ruin the measurements by a simple error in applying a probe to the device Private Jet Charter under test.  Length is easy - you can buy rods, tapes and ultrasonic devices from the shops. Pressure, too, is catered for by a simple gauge from a store.

What about two other fairly basic quantities, mass and force?  On earth we "measure" mass using something that actually measures weight (a force), but is calibrated in kilograms.  Because the value of g varies around the world, measuring mass like this can produce small errors, far too small to worry people who are on a diet.  Astronauts trying to measure their "weight" have to use techniques that use the equation F = M x A, Newton's first law.  Some kind of spring may be used, with the change in length of the spring being proportional to the force, Hooke's law.  One could even use the resonant frequency of a mass on a spring, but the period varies only as the square root of the mass.

This gives us a clue to measuring force.  In the page about forces we saw that we cannot measure the forces inside an object: we can only measure the strain at the surface.  Hooke's law allows us to convert strains into stresses if we know the elastic constants of the material.  Even then, we only measure the stress at the surface.

A strain gauge uses the fact that some substances change their electrical resistance when they are stretched.  So you simply glue the material to the object under test, measure the resistance, stretch the object under test, and measure again.  Subtracting the resistances gives you the difference, and a calibration table gives you the mechanical strain.  There is a problem: the change in strain is usually very small, and the change in resistance is also very small.

We can define a gauge factor GF, which relates the change in resistance to the change in length -

GF = (dR / R) / (dL / L),

where dR and dL are the small changes in the resistance R and the length L.  dL / L is the strain, and so we have -

GF = (dR / R) / Strain,

or -

dR = R x Strain x GF.

With a strain of 0.001 and a GF of 1.0, the relative change in R is only 0.1 %.  If we want to measure the strain with an accuracy of one percent, we must measure the resistance with an error of no more than 0.001 %, which is not easy.  And we have not even considered the change in resistance with temperature.  We must try to measure the difference directly, not by measuring the total before and after.  Never, if you can possibly avoid it, measure or calculate a small quantity by subtracting two large ones

To test whether a beam in a building is level, we don't measure the height of each end above the base of the building: we use a spirit level.  So, we need an electrical spirit level.  Luckily for us, it was invented during the 19th century.  It is called the Wheatstone bridge, though it has nothing to do with bridges.

Here is a simple electrical circuit in which the current divides into two paths.

For our purpose, the current need not divide equally. Now we connect a voltmeter that forms a bridge between the two parts of the circuit.

    

The direction of the flow through the voltmeter depends on where we connect it.  If we make the connections carefully enough, we can arrange that no current at all flows through the meter.  We can now connect a very sensitive meter, knowing that the flow is zero.  If there is now a small change in the resistance to the left or right of the meter connections, in either the upper or lower branch, the meter will show a reading.  Because the meter shows the unbalance directly, not as a tiny change on a big value, the bridge method can be made exceedingly sensitive.  This is a good general principle.  If you have to measure a small difference, try to avoid making two measurements and then doing the subtraction.  Try to have the subtraction inherent in the system.

Furthermore, in this type of circuit, if the temperature changes, it will affect all parts of the circuit in such a way that the effects will tend to cancel, and the measurement will be little affected.  In the next diagram, the four arms of the circuit are labelled for convenience.

The simplest arrangement is to make one arm a strain gauge, and the three others resistors, but to reduce temperature effects, it is better if all four are identical strain gauges.

What is a strain gauge?  It is a strip of wire or foil that can be glued to the specimen under test.  To obtain a large resistance, it should be long and narrow.  Folding it up keeps it compact.  Large connections mean low resistance, and any errors caused by them will be small.

As an example, if the specimen is to be bent, A1 and B2 can be on the tension side, while A2 and B1 can be on the compression side. giving a fourfold increase in sensitivity over a single gauge.

Converting measurements into electrical signals is a popular technique because the signals are easy to transmit, record and digitize.  The receiver can be distant from the device under test, and even moving relative to the transmitter, if telemetry is used.  The signals can be taken to a digital oscilloscope, or to an ADC in a system such as PXI, or to an ADC card in a computer.  In all these cases, the data may be subjected to mathematical manipulation after recording.  An example is the use of Fourier analysis, to examine the frequency spectrum of the object under test.

This bridge method of measuring is actually a differential measurement, that is, it measures the difference between two values.  By making the difference very small, the detecting device can be made very sensitive.  The old fashioned weighing scales were a differential device - the pans contained known weights one side, and unknown weights on the other.  Such a device can easily measure down to the milligram level, and specialist devices have gone much further.  As with the electrical bridge, the detector only has to detect zero, so there is no scale to calibrate.

With a weighing device such as a spring balance, the accuracy is limited by the accuracy of the scale and the width of the pointer.  A spring balance does not actually measure in kilograms or grams, even if the scales says it does.  It actually measures the force of gravity on an object, which is translated into mass on the assumption that the acceleration due to gravity has a known value, which is around 9.8 ms-2, which is equivalent to 9.8 newtons per kilogram.

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