In the first part of this series, I figured out that the power structure was singing at 759 Hz in the wind. Here, the plan is to look into which component in the power structure is singing.1 As you’ll find out, the lack of good measurements leave me conclusion-less for now.
How Do We Figure This Out?
I mentioned in the original post that the singing is a result of aerodynamic flutter. This is a phenomenon where aerodynamic forces and structural forces interact in a way that results in periodic motion of the structural elements. The most famous example of this is the Tacoma Narrows Bridge which failed because of aerodynamic flutter. If you’re driving around industrial districts, you’ll also see spiral vanes on metal vent chimneys that exist to prevent destructive aerodynamic flutter from occurring.
But aerodynamic flutter is really complicated! It combines aerodynamic forces (complicated) and structural vibrations (also complicated). So going at the question of “which component is vibrating” from this arc is at best hard, and at worst not possible.
Instead, since I (right now) only want to discover which component is vibrating, looking at this from the structural vibrations only perspective is enough. It’s enough because structures like to vibrate at frequencies dictated by their mechanical properties, and so in the case of sustained aerodynamic flutter based vibrations the frequency of vibration of the structure will be at these “natural” (or harmonics of) frequencies.2
So the task comes down to figuring out the various natural frequencies of all the components in the power structure, and seeing if any of them match up to the observed 759 Hz.
The Components And Their Properties
I’ll use my best judgement to guess at what counts as separate components in the structure. Essentially, any contiguous beam that has a long span connected at either end seems like a component likely to vibrate.
My photos of the whole structure weren’t great, so I pieced together as many as I could and came up with this set of images and scale guesses:
From this, I came up with the following set of distinct beams (blue letters) and measurements (orange annotations):
If you look in the real life images, beams B, C, D, and E are all rectangular sections with pinned (bolted) supports on either end. A is a “circular” cross section, tapered from ground to top, fixed at the ground, and has multiple connections along its length.
These two groups are notable because the first, rectangular sections with pinned ends, can be looked at analytically without much issue. The second, tapered from ground to top with multiple connections along its length, is not easy to analyze!
My hope is that in the analysis of the first group, a clear “759 Hz” shows itself. I’ve started testing out Project Chrono for dynamic beam analysis which will let me do analysis of beam’s like A, but am not pursuing this here.
Beams B, C, D, and E
I’m assuming these are rectangular section beams, with pin (bolted) joint ends. I’ll assume the wall thickness of the section is constant. Using the dodgy photos I have, this means the following for physical properties:3
| Beam | Length [m] |
Section Height [mm] (in) |
Section Width [mm] (in) |
Wall Thickness [mm] (in) |
Material |
|---|---|---|---|---|---|
| B | 10.2 | 152 (6) | 102 (4) | 6.4 (0.25) | ASTM A8474 |
| C | 18.2 | 254 (10) | 152 (6) | 9.5 (0.375) | ASTM A8474 |
| D | 4.9 | 102 (4) | 51 (2) | 6.4 (0.25) | ASTM A8474 |
| E | 9.1 | 102 (4) | 51 (2) | 6.4 (0.25) | ASTM A8474 |
Analytical Beam Vibrations
Normally, strings are used as an introduction to continuous structure vibrations. This is because strings can only support tension, and so modelling the vibrations is “easy” and the modes of vibration are few.5
Beams are more complicated because they can vibrate in a at least 3 ways: longitudinally, laterally, torsionally. Also, instead of the differential element you use for modelling only being able to support a tension force at some angle, you have the whole range of forces associated with beam bending and torsion.
In this case, I’m going to start with the vibration modes (lateral vibrations) that I think have the most likely chance of being the winner.
The vibration of continuous beams is modelled by taking a differential segment of the beam, and doing a force and moment balance upon it. There are a bunch of textbooks, pdfs, and online resources that derive the general equations for beam vibrations and the boundary conditions corresponding to all the possible combinations of end fixations so I won’t cover that here.
For the case of a beam with constant 2nd moment of area along its length, homogeneous constant property material along its length, and simply supported (which is what we’ll approximate bolts as) ends, the natural frequency of the Nth lateral vibration mode is:
\[\omega_{n} = n^{2}\pi^{2}\left(\frac{E I}{\rho A l^{4}}\right)^{\frac{1}{2}}\]where:
- \(\omega_{n}\) - The frequency, in [radian/s] of a vibration mode
- \(n\) - The vibration mode, \(1, 2, 3, ...\)
- \(E\) - The young’s modulus of the beam material in Pa
- \(I\) - The 2nd moment of area of the beam parallel to the axis of curvature of the bending shape of the given lateral vibrations
- \(A\) - The cross sectional area of the beam
- \(l\) - The length of the beam from support to support
- \(\rho\) - The density of the beam material
Since the beams are not square, each needs to be looked at from two different perspectives (two different \(I\)). The table below shows the first natural frequency of vibration of both lateral directions all the beams that can be approximated by this model. These were calculated via a quick python script.
| Beam | \(\omega_{1}\) Direction 1 [Hz] |
\(\omega_{1}\) Direction 2 [Hz] |
|---|---|---|
| B | 489 | 670 |
| C | 227 | 340 |
| D | 1352 | 2403 |
| E | 392 | 697 |
Which is interesting, but definitely not conclusive! Since our measurements are so bad, it’s hard to say much other than that it likely isn’t beam D. In particular, we need to be careful about drawing conclusions because the measurements have non-linear effects on these numbers. In particular, length affects frequency to the fourth power (!) and the beam widths and thicknesses affect to the third power (via the 2nd moment of area).
Where To Next?
If I wanted to go farther into this, I could attempt to get better measurements from my images, try to track down the actual dimensions of the tower from other sources, or spend time exploring the likely measurement errors I have and which are most likely to lead in the direction of correct.
For now, this was fun but I’m not planning on pursuing this farther. It’s interesting, but really if this was important I’d go back and measure the structure more accurately. Since I’m not going to do that any time soon, and getting these results doesn’t really lead to anywhere (other than having some fun), I’m going to stop here.
If I do go back to Utah, I might go measure! And I want to keep pursuing Project Chrono since it seems like a useful tool, so I’ll probably look for more pragmatic cases where beam theory can help.
Footnotes
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Aside from “because it’s interesting!”, why would you want to do this? In this specific case I think it’s unlikely that the singing I heard represents vibrations of a magnitude that pose structural risk, but that is one reason - structures vibrating at their natural frequency are, unless designed for the case, almost certainly undergoing strains not considered in their designs. A second reason is to make sure that fatigue and vibration protections are adequate. ↩
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I understand that this is a huge jump - why the heck do structures prefer to vibrate at specific frequencies? You’ll have to trust me for now. ↩
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The educated guesses I needed to make: my “me for scale” image didn’t include a lot of the structure, so I had to guess at the initial scale. I also don’t know how the optics of my iphone distort, but they do! I have “Lens Correction” turned on, but this is only for the ultrawide and selfie cams. This is an unknown (I think it’d make beams that cross significant portions of the image estimate shorter than reality since they are compressed more at the edges?). Finally, guessing section properties are borderline impossible. I did my best to look up standard steel rectangular beam sections and use those that fit best, but these are guessed! I also guessed at the specific type of steel. ↩
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The structure wasn’t painted, so it’s a “weathering steel” (I don’t know the material science of why, but the ideas is that the steel rusts in a way that forms a protective rust layer like most other metals do instead of flaking off and allowing the rust process to eat all the way through the material.) Trade name is Cor-Ten steel in the US. Here’s the company page for it. ↩ ↩2 ↩3 ↩4
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I’ve only ever seen lateral vibration modelled, but now that I’m thinking about it I guess longitudinal waves might work too? It gets weird if you go with the “only tension” assumption, though. You’d need to pretension so that as the longitudinal wave propagates, the front “compression” side of the wave still represents a part of the string under tension. Maybe this is why the two cups and a string mic only works if you pull the string taught! ↩