Saturday, August 14, 2010

Ringing A Bridge

Matt and Jared standing on
our experiment
When you strike a bell, it rings at a given frequency.  This frequency is called the resonant frequency and is the natural frequency at which the bell likes to ring.  Just about anything that can shake, rattle, or oscillate will have a resonant frequency.  Things like quartz crystals, wine glasses, and suspension bridges all have a resonant frequency.  The quartz crystals oscillate at frequencies high enough for accurate timekeeping in watches, the wine glasses at audible frequencies to make boring dinners more interesting, and bridges at low enough frequencies that you can feel it when you walk.  It is the resonant frequency of bridges that we decided to measure.

To make our measurements, we "borrowed" Yariv's fancy phone.  One of the nice things about fancy new phones is that most of them have internal accelerometers to detect motion.  You can do a whole bunch of fun experiments and take some pretty good data with these accelerometers (see, for example, physicist and TV star Rhett Allain's posts over at dot physics).  Placing Yariv's phone on the suspension footbridge on campus, Alemi, Matt and I took data and confused passers-by for about 15 minutes.       

The accelerometer in the phone measures acceleration in three coordinate directions: x is along the width of the bridge, y is along the length of the bridge, and z is up and down.  The raw data is shown below.  The z data is shown in blue, and x and y in green and red.

The first thing you'll notice about this data is that the z direction (blue) has big spikes in it around 180s, 300s, and 800s.  The biggest spikes are when Alemi and I jumped up and down to ring the bridge.  The smaller bumps in the blue data are the result of people walking or jogging by.  

With the raw acceleration data and knowledge of the sample rate of the accelerometer ( 90 Hz ), we can Fourier transform it to get frequencies.  Doing this to the raw data for each dimension we get the following spectrograms.  Each of the spectrograms illustrates how much of each frequency is present at each point in time.  

The most relevant direction for us is the z direction.  We see that at several points there are strong signals at all frequencies followed by longer periods where the main signal is around 1 Hz.  These events correspond to when Alemi and I jumped up and down and are analogous to ringing a bell.  The striking of the bell is just a sharp impulse (roughly a delta function) which is composed of all frequencies.  Soon after the impulse, all of the frequencies die out except for the resonant frequency, which keeps on ringing.  Just looking at this graph, it looks like the bridge resonant frequency is around 1 Hz.

We can also make similar graphs for the x and y directions.  Remember, the x direction is the width of the bridge and the y direction is the length of the bridge.  Although there is less motion in these directions, the spikes where we jumped and people walked by are still clearly visible.

Finally, we can find out how much of a particular frequency is in the whole signal.  To do this we take find the power spectrum density of the entire data set (blue is z, green is x and red is y).  The ringdown frequency of about 1 Hz we saw in the spectrograms above after the jumps is illustrated in this graph as the first blue peak.  There are also some other peaks at around 15 Hz, 25 Hz and 35 Hz.  I am not sure what they correspond to.    

To clean up this a bit, we can just take the data without the jumps in it.  Computing a new power spectrum density with just the data from about 400s - 700s, we get the following graph, which also displays a fairly prominent peak around 1 Hz.

So it seems that there is definitely something going on around 1 Hz.  Initially, I was worried that this is just the rate at which people walk and therefore it was just showing up because we had people walking the whole time.  However, the strong 1 Hz signal after each ringing in each z spectrogram seems to indicate that it is intrinsic to the bridge.  Therefore, it seems as though the resonant frequency in the z direction of the bridge is about 1 Hz.

But don't take our word for it.  If you want to do your own analysis, you can find the raw data here.  


  1. It's not like bridges, or bells for that matter, have just one resonant frequency, so the spikes at 25Hz and so on might just be the higher resonances corresponding to other modes of vibration. For an elastic structure, they won't necessarily be in any nice harmonic ratios either. It's a bridge, after all, not a harp.

  2. What are the exponents of the power laws?

  3. particle_person: You're right, they are probably just other modes. It is interesting to note that since they do not appear in the secondary plots (excising the jumps), they likely decay much faster than the 1 Hz signal.

    Forbes: I just did a couple quick fits and it looks like the exponents are -1.10 for the z data, -1.25 for the y data and -1.27 for the x data.