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Measuring the stiffness of structures by vibration testing

How does Hz = stiffness?

by Julian Edgar

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If you look at the detailed specs released by road and race car manufacturers, you’ll sometimes see body stiffness expressed in Hz (Hertz). For example, they might state that the ‘body in white’ (the bare shell) has a beam stiffness of 27Hz.

But what does this mean? How can a number that shows ‘speed of vibration’ be related to body stiffness?

And, once that’s understood, can this measuring technique be applied by amateurs working at home?

Let’s take a look.

Stiffness, mass and defection

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Imagine a simple beam supported at each end. We place a weight in the middle of the beam and measure how much the beam sags – its deflection. If it is a stiff beam, it might sag by 2mm when a mass of 50kg is placed in the middle.

In this case we are quantifying the stiffness of the beam by measuring its deflection.

But we can measure stiffness in a completely different way. Rather than loading the beam with a weight, let’s instead hit it in the middle with a big hammer. The beam will vibrate a bit like a violin string, but at a much slower rate. When excited in this way, the middle of the beam might shake up and down 30 times per second. (That’s the same as saying at 30Hz.)

The stiffer the beam, the higher will be the frequency at which it shakes when excited.

Again, that’s just like the violin string – the stiffer the string (achieved by increasing its tension in this case), the higher will be its natural frequency.

The idea that we’ve just covered – analysing the stiffness of a simple structure by measuring its natural vibration frequency – is being used in many mechanical design areas. In fact, design software has the ability to estimate this characteristic when modelling a structure that is yet to be built.

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One example of where vibration measurement is being used is in the case of testing bridge stiffness. In one case, a 12lb sledgehammer was used to impact the bridge at mid-span. The FFT of the vibrations was analysed (more on FFT in a moment) and the natural frequency of the bridge in bending was calculated as being 11.4Hz.

If the bridge were to get weaker over time (for example, through the structure rusting) this frequency would get lower – it’s therefore a good way of assessing the structural integrity of a bridge without having to actually measure loaded deflection.

But what about more complex structures than beams?

Vibration modes

In the bridge, the vibrations were largely up and down - the bridge sagging in the middle just as the beam did when we placed the weight on it. But in more complex structures, the vibrations can involve different types of deflections.

Imagine a steel plate, rectangular in shape. Instead of hitting it with a hammer, we attach a vibration exciter. This exciter can generate vibrations at all different frequencies. We attach the exciter to one corner of the plate and then drive the exciter, sweeping up from a very low frequency (say 1Hz) to a higher frequency (say 50Hz).

On the basis of what we have so far seen, we’d expect the plate to get really excited at just one frequency – like the bridge did. But instead what we find is that the plate vibrates really strongly (resonates) at a number of different frequencies, not just the one we expected.

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The above diagram shows why that is the case. The plate does in fact bend in vibration as the bridge did – Mode 1. But it also can bend torsionally (in twist) – Mode 2. And furthermore, it can do two bends in beam mode – Mode 3. And then finally, it can get really excited in its contortions and bend torsionally and in beam – all at once (Mode 4)!

Each if these modes is associated with a different resonant frequency. Therefore, when more complex structures than simple beams are excited, you will get multiple resonant frequencies, each occurring in different vibration modes. However, typically the lowest resonance is associated with the simple beam deflection mode.

And there’s another thing to be aware of. If the structure is vibrating in complex modes, there will be points on the structure where the movement during vibration is zero – or very low. Huh? How can a vibration structure have bits not moving? It’s easiest to imagine if you go back to Mode 3 above. If you imagine the arched bits vibrating up and down almost as separate entities, you can see that at some points on the plate there will actually be very little movement.

Therefore, when more complex structures vibration in different modes, there will be spots (confusingly called ‘nodes’!) where little vibration can be detected.

Talking about detection, how can it be done?

Accelerometers

Rather than measuring displacement (distance of movement), measurement of vibration is best done by measuring acceleration. This can be achieved with electronic accelerometers. In the past, that would have been expensive, difficult and complex – but not any more.

Smart phones have in-built accelerometers and there is now cheap software available (‘apps’) that allow vibration to be measured and analysed. To be most effective, the app needs to log data rapidly (e.g. at 100Hz) and also be able to perform a Fast Fourier Transform (FFT) on it. (The FFT bit sounds very complex – but all it means is that the dominant frequencies can be picked out of a complex waveform.)

The cheap "Vibration" software from Diffraction Limited Design runs on iPhones and can achieve these functions.

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Here is a phone screen grab of the FFT analysis of a logged vibration of a car door panel, excited by being bumped with the end of a closed fist. Note how the vertical cursor has been positioned on a peak of 14 Hz.

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If you stiffen the structure, the ‘lowest peak’ frequency increases. Here is the panel response when tested with a stiffening layer of sound damping glued to the rear. The lowest frequency spike has risen to nearly 40 Hz.

You may well be looking at the above screen grabs from the phone and wondering: which peak is the one to measure? And that’s a valid question!

Another way of looking at these two screens (and a way that may be more accurate) is to look at the pattern of differences.

The unstiffened door panel clearly has a number of resonances –all at fairly low frequencies.

The stiffened door panel has a much more clearly defined peak resonance, and it occurs at a higher frequency.

These measurements were made with an iPhone running the Vibration app. The phone was stuck to the middle of the relatively flat door panel with Blu-Tack.

Digital scopes

Smart phones are limited in their ability to record information – so is there a better way? Well, yes and no.

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Digital handheld scopes are now available from about $250 – and about $500 for a good one. Shown here is a Siglent SHS806 that costs about $450.

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Electronic accelerometers are really cheap – here’s one that costs about $4 through eBay. It can be powered by a 3V source (just use two 1.5V batteries in series), and it outputs a signal voltage that is proportional to acceleration, measuring up to plus/minus 3g.

Monitor the output of the accelerometer with the scope (set in one-shot mode) and you have a vibration recording and analysis system.

But do you actually have the analysis bit?

Despite digital scopes having an FFT function, at the low frequencies and with the required resolution needed in vibration measurement, their FFT function is not particularly useful. Instead, you need to work out the frequency by measuring the period of the waveform on the scope. This is easy – but only if the object is vibrating in just one dominant mode.

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Here is the logged vibration of a china bowl, excited by being tapped with a wooden handle. You can see that the vibrations being monitored by the accelerometer and logged by the scope are very ‘clean’ – there’s clearly one dominant vibration mode.

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Because the vibration is occurring in only one mode, it’s easy to use the scope cursors to accurately measure the frequency – which in this case is 1.28 KHz (arrowed).

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The walls of a china bowl are obviously very stiff. As a comparison, here is the natural frequency of my desk top – 735Hz. (Note that the time base scale on the scope has been changed – that’s why the waveform looks similar despite it being much lower in frequency.)

But what about cars?

To measure the overall body stiffness in bending and torsion is too complex for the simple equipment being used here, but assessing the relative stiffness of different parts of the car body can be done.

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Comparison was made of the natural frequencies of different parts of these two car bodies – a 1960s Austin 1800 ute and a 2000s Honda Integra.

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With the accelerometer stuck to the B pillar with Blu-tack and the body excited by a rubber mallet (note: you don’t need to hit anywhere near the B pillar), this was the recorded vibration waveform. The measured frequency is 185Hz.

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In comparison, the much thinner Austin B pillar (also not supported by a glued-in rear window as was the case with the Honda) recorded a much lower frequency – 102Hz. Note though that this waveform is not nearly as ‘clean’ as – say – the china bowl.)

Here are a few other comparisons on the two cars:

Honda Insight Shell

Austin 1800 Ute Shell

Front guard

116Hz

178Hz

Centre of roof

166Hz

138Hz

From the above table note how the curved from guard of the Austin is stiffer than the relatively flat panel of the Integra. However, the converse applies with the roof panels – the Honda’s is stiffer than the Austin’s, despite the greater curvature in the Austin. This is probably because the Honda uses higher stiffness steel for this stressed panel.

We also tested the stiffness of the B pillar of two cars in road-going form, both tested with the driver’s door open. A 2006 Honda Legend recorded 476Hz and a similar age Volkswagen T5 Transporter dual cab recorded 500Hz.

Conclusion

This article has covered a lot of ground not familiar to most car enthusiasts – so let’s summarise.

Vibration measurement can be used to measure structure stiffness. It is being widely used in professional automotive engineering to do just that.

From a practical point of view, the more modes in which the structure vibrates, the harder it is to effectively make direct measurements of complete structures.

However, if you are building a tubular spaceframe or other similar structure, it may well be worth doing some vibration testing and analysing its stiffness. For example, the extra stiffness gained by the placement of brace held temporarily in place with clamps may be able to quickly quantified.

And for all of us, seeing car body stiffness measured in Hz is no longer quite the mystery it once was!

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