10.7.1 Answers to the quiz

Here are some answers to the rather open-ended questions asked in section 10.7. Figure 1 shows a repeat of the first set of measured frequency response functions from that quiz. We are expecting that the resonance peaks will be the same in all three curves, except that the amplitudes will change and it is possible that some peaks will not be visible because the mode in question has a nodal line through the excitation point or the observation point.

Figure 1. A repeat of Fig. 1 from section 10.7, showing the first set of frequency response measurements to discuss.

Figure 2 shows a set of red lines marking the peak frequencies, and a set of green stars marking the peak values in the top plot. The peaks are indeed at the same frequencies in all three plots, and furthermore the spacing looks very regular. So this system has overtone frequencies which are close to harmonic spacing throughout the range of the plot. Furthermore the fundamental frequency seems to be suspiciously close to 440 Hz (notice the 10th overtone near 4400 Hz). What familiar system do we know which has this pattern of behaviour? Surely this must be a string, tuned to A 440 Hz: and indeed the measurement was made on the A string of a violin.

Figure 2. A version of Fig. 1 marked to show the peak frequencies (red lines) and the peak heights in the top plot (green stars)

The green stars in Fig. 2 tell us the next important thing. Every 5th peak has very low amplitude, suggesting that either the excitation point or the observation point was near a nodal point of the 5th and 10th modes. Furthermore, these peaks are weak in all three plots, so probably the fixed observation point was 1/5 or 2/5 of the way along the string from one end.

Next, look at Fig. 3. This is another copy of the set of frequency response functions, this time marked to highlight some clear antiresonances in the top plot. Indeed, for every gap between peaks where the pattern can be seen clearly above the noise floor, there is an antiresonance. So we might guess that this top plot is the driving point response of the string at 1/5 or 2/5: it was in fact 1/5. Now on that same figure, look down to the middle plot. This one shows no antiresonances at all: there is a shallow valley in every gap between peaks. We have seen a pattern like this before, in Fig. 11 of section 10.5, where it was associated with a measurement between a mirror-image pair of points on a symmetrical one-dimensional system (a bending beam). A string is also symmetrical about its centre point, so we might guess, correctly, that the middle curve here was obtained by exciting the string at the 4/5 point, and observing at the 1/5 point.

Figure 3. A version of Fig. 1 marked to indicate some obvious antiresonances in the top curve.

Figure 4 focusses attention on the bottom curve in the set. The peaks are marked by stars, and this time the pattern is more complicated. We already know why the 5th and 10th peaks have low amplitude, because the string is being observed near a nodal point of those modes. But Fig. 4 also shows relatively low amplitudes of the 2nd, 4th, 6th and 8th peaks. This pattern suggests that the excitation point for this measurement was near the centre of the string: so it is a measurement from the mid-point to the 1/5 point.

Figure 4. A version of Fig. 1 marked to indicate the peak heights in the bottom curve (green stars).

We have already mentioned the noise floor, which obscures some of the details we would like to have seen. Indeed, your first reaction to this set of plots might have been that the data looks quite noisy. Think again! All measurements have a noise floor at some level, but Fig. 5 highlights the fact that the top curve in the plot has a dynamic range of some 70 dB, from the highest peak down to the noise floor. This is not noisy data! In fact, rather few measurements will show a dynamic range as big as that. It is just that the peak-to-valley excursions for this driving-point response of a string are very large because the damping is low (see section 10.6.2 for an explanation of why damping matters for this peak-to-valley range), so that even a relatively low noise floor comes into play in some frequency ranges.

Figure 5. A version of Fig. 1 marked to indicate the dynamic range of the measurement.

We now repeat the exercise with the second set of measurements. Figure 6 shows a copy of the original set of plots. Figure 7 is marked with a set of red lines to indicate the major peak frequencies. Again, these frequencies agree across the set of measurements. These lines omit a couple of smaller peaks — we will come back to those shortly. This time, the pattern of frequencies looks systematic, but it does not show regular spacing: the peaks are close together at the low-frequency end, and they get progressively wider apart as frequency rises. We have met a system which fits this description: a bending beam, as described back in section 3.2.1. In that section, beams with two different combinations of boundary conditions were analysed, and in both cases the formula for the $n$th resonance frequency involved $n^2$ so that frequencies started close together and grew progressively wider apart, just as we see in this measurement. In fact, if you lay a ruler on the plot and work out the frequency ratios, you will find a rather good match to the predicted frequencies of a beam that is free at both ends (equation (20) of section 3.2.1), and that is exactly the system that was measured here.

Figure 6. A repeat of Fig. 2 from section 10.7, showing the second set of frequency response measurements to discuss.
Figure 7. A version of Fig. 6 marked to show the peak frequencies (red lines), the peak heights in the top plot (green stars) and the peak heights in the middle plot (blue stars)

Figure 7 also has two sets of stars, marking the peak heights for the top and middle plots. This time, all the peaks within the frequency range plotted are clearly visible in both plots, with rather little variation in height. There is no trace of the pattern seen in Fig. 2, with “missing” peaks indicating the position of the observation point. We can make a guess at what was done, then: the observation point this time was close to the end of the beam. Furthermore, if we look at Fig. 8 which highlights antiresonances in the top plot we see the same pattern as in Fig. 3: an antiresonance in every gap. We deduce that this top plot looks like a driving point measurement, while the middle plot looks like an end-to-end measurement: it has shallow valleys in every gap.

Figure 8. A version of Fig. 6 marked to highlight some antiresonances in the top plot.

Figure 9 marks the peaks of the bottom plot. There is a clear even-odd pattern of heights. The even-numbered peaks are by no means absent, but they are reduced by a significant number of dB relative to the odd-numbered peaks. We can deduce that the excitation point in this case was near the centre of the beam, but not exactly AT the centre.

Figure 9. A version of Fig. 6 marked to indicate the peak heights in the bottom plot (stars)

We can even work out which side of the centre it must have been, if we assume that the beam was symmetrical and had a uniform cross-section, like the simple textbook case analysed in section 3.2.1. We see in Fig. 9 that each even-numbered peak is accompanied by an antiresonance, always falling slightly lower in frequency than the peak. If you look back at the discussion of antiresonances in section 10.5, you can deduce that the number of nodal points between the excitation point and the measurement point was the same for an even-numbered peak and the odd-numbered one below it. But between the even-numbered peak and the odd-numbered one above it, an extra nodal point has appeared. If you sketch some mode shapes and think carefully about them, this tells us that the excitation point for the bottom curve must be displaced from the centre TOWARDS the measurement point: see Fig. 10 for a plot of the first three modes of a beam, marking the nodal points (green circles), an excitation point at one end (blue circles), and a possible position for the measurement point (green line). Count the nodal points between the blue circles and the green line.

Figure 10. The first three modes of a free-free bending beam, showing the nodal points (red), the excitation position at one end (blue) and an indication of the measurement position to achieve the pattern shown in Fig. 9 (green line)

Finally, Fig. 11 draws attention to a feature we have been ignoring. As well as the sequence of major peaks we have already discussed, there are two extra peaks. These have lower amplitude, and their frequencies do not fit into the orderly pattern of the others. So we should probably be looking for a different physical explanation of these peaks. Bearing in mind that the system tested was a beam, what other kinds of vibration resonance might it show, in addition to bending modes? We have met two possibilities in earlier sections: such a beam could exhibit axial vibration, and/or torsional vibration. Both these types of vibration obey a version of the same governing equation as the string that we looked at in the first set of measurements. So if the boundary conditions at both ends of the beam are the same, both would be expected to exhibit resonance frequencies that are (approximately) harmonically spaced. Sure enough, when we look at Fig. 11 we can see that the two extra peaks show a frequency ratio close to 2:1.

Figure 11. A version of Fig. 6 marked to indicate two additional small peaks, each with an associated antiresonance.

Which of the two possibilities are we likely to be seeing in the measurement? To make a sensible guess, we need to think about the way the measurement was made. The beam was tapped on one of its long sides with an impulse hammer with the aim of predominantly exciting bending vibration. Such a hammer tap on the side of a beam is quite likely to excite some torsional vibration, but much less likely to excite axial vibration, simply because of the direction the force acts. Similarly, the response was measured by an accelerometer stuck to the same face of the beam. So our best guess is that the two extra peaks marked in Fig. 10 are probably the first two torsional modes of the beam.

Notice one more thing in the top plot of this measurement set. On this decibel scale, the plot would look qualitatively very similar if it was turned upside down: for reasons that were explained in section 10.5.2, antiresonances have very similar behaviour to “upside down peaks”. This can give a pitfall when making a driving-point measurement like this. It is fatally easy to get the cables mixed up, and plug the input and output signals into the wrong channel of your datalogger system. The frequency response plots you obtain will then be input/output, not output/input. But the result still look superficially plausible, as we have just seen, so you might not spot the error.