10.3.1 Measuring stiffness by beam vibration

Given a rectangular beam made from the material to be tested, it is easy to measure the lowest resonance frequency (and perhaps some of the higher frequencies as well), and deduce the Young’s modulus of the material in the direction aligned with the long axis of the beam. The theory has already been covered in section 3.2.1: we know from equation (12) there that for a beam of length $L$, width $b$ and thickness $h$ made from a material with density $\rho$ and Young’s modulus $E$ the natural frequencies are given by

$$\omega^2 = \dfrac{EI}{\rho A} k^4 \tag{1}$$

where $A=bh$ is the cross-sectional area, $I=b h^3/12$ is the second moment of area, and for a free-free beam the values of the wavenumber $k$ are given by the roots of the equation

$$\cos kL \cosh kL =1 . \tag{2}$$

We also showed that these roots are given approximately by

$$k_n \approx (n+1/2) \pi/L \tag{3}$$

where $n=1,2,3…$ labels the modes in order of resonance frequency. This approximate expression is good enough for all the modes except the one we are most interested in, with $n=1$: a more accurate expression for that mode is

$$k_1 = 4.73/L . \tag{4}$$

Putting this all together, the frequency $f_1$ (in Hz) of the lowest mode is given by

$$f_1 = \sqrt{\dfrac{E}{\rho}} \dfrac{h}{L^2} \dfrac{(4.73)^2}{2 \pi \sqrt{12}} = 1.028 \sqrt{\dfrac{E}{\rho}} \dfrac{h}{L^2} . \tag{5}$$

This can be turned round to give an expression for $E$:

$$E=0.946 \rho \dfrac{f_1^2 L^4}{h^2}=0.946 \dfrac{m f_1^2 L^3}{b h^3} \tag{6}$$

where the final expression is expressed in terms of the total mass $m$ of the beam, rather than the density. This mass can be determined directly by weighing.

When using this expression it is, of course, important to express all the quantities in a consistent set of units. The standard choice would be to express all lengths in metres, and the mass in kilograms: $E$ will then be in Pascals (Pa).

So how should you set about measuring the frequency $f_1$ for your test sample? Figure 1 shows one good way to do it. A cross-grain spruce beam is being tested — in fact it is one of the ones shown in Fig. 3 of section 10.3. You need to support the beam in a way that doesn’t interfere with the vibration mode you are interested in, and that means positioning the supports on nodal lines. For the lowest mode of a free-free beam the two nodal lines are approximately 1/4 of the way in from each end. So the beam has been supported in those positions with two thin rubber bands, threaded over a section of U-channel which allows the position of the bands to be adjusted very easily. Using thin rubber bands means that the measurement will not be affected very much if the positioning isn’t perfect, because the extra stiffness and/or mass contributed by the bands is minimal.

Figure 1. One of the spruce beams from Fig. 3 of section 10.3 being used to measure stiffness by finding the lowest resonance frequency. The beam is supported in thin rubber bands placed at the nodal points of the mode. A small microphone is placed underneath, near the centre.

This mode has large amplitude at the centre of the beam, so it is being tapped (with a pencil) at that position. A small microphone has been placed underneath at the same position, close to the beam but not touching it. The signal from the microphone goes off to a computer running some kind of FFT app, which will allow you to find the resonance frequency by analysing the tap sound. Alternatively, you could achieve the same effect with a phone app, with the phone’s microphone in a similar position close to the beam.

If you would like to include a cross-check in your measurement, you can try to measure the frequencies of a few more of the vibration modes. They should all give more or less the same value of $E$, using the relevant values of $k_n$ from equation (3) to derive equations corresponding to equation (6). But you need to be careful doing these measurements. The nodal lines are different for every mode, so you need to move the rubber bands a bit nearer to the ends for successively higher modes. Also, for the even-numbered modes $n=2,4,…$ it is no good to tap and listen at the centre, because these modes all have a nodal line there. A safe place to tap is at the very end: every mode has an antinode there. But you also need to use an asymmetric position of the rubber bands so that the microphone isn’t exactly in the centre of the beam.

Finally, as will be explained further in section 10.3.3, you can use the same measurement to determine the damping factor associated with $E$. For that, you need a bit more sophistication in your FFT app. You either need to find the decay rate of the sound of each mode following the tap, or equivalently you find the half-power bandwidth of the peak in the frequency spectrum given by the FFT. Section 2.2.7 gives the details of how that bandwidth is affected by the damping.

You also need to be more careful with the experimental details if you want to get accurate values of damping. The rubber bands will contribute extra damping, and you want to minimise that. So try the test a few times, moving the bands slightly between tests. The lowest value of damping among these measurements should give the best estimate of the true damping of the wood. In any case, measuring damping is always tricky. You can expect to get reliable results for resonance frequencies to an accuracy around 1%, but for damping factors the accuracy is unlikely to be better than 10%. Luckily, this is good enough: we do not hear differences in damping very acutely, and it usually requires a change much bigger than 10% to be audible.