Trigonometry for Pilot Vechile Signs

Dad rang me up this morning with an interesting problem, and I'm ashamed to say it took me most of the day to get this straight in my head so I could solve it. I think I have the math correct. Basically, you have a sign that's about 600mm tall, which pivots on the roof rack at point \(A\). There's a gas strut, which pivots on the roof rack at point \(B\), and the sign at point \(C\), completing a simple triangle. With me so far?

The gas strut (or a linear actuator, which one you use is not really relevant to the mathematics behind it, other than that you probably want one with sufficient travel so the force of wind upon the sign at 100KM/h doesn't overpower it - but that's mathematics for another day), which makes up length \(a\) (opposite point \(A\), on the right slope of the triangle on my diagrams) can range in length from \(S_{min}\) to \(S_{max}\). The top of our sign must be (per VicRoads and most other state regulations) no more than 200mm back from the bottom - the sign is 600mm tall, which means the sign must be no more than \(30\unicode{xb0}\) back from vertical, or \(60\unicode{xb0}\) from the horizontal - that's for angle \(A\). We want \(A\) to go to zero when \(a\) is as close to \(S_{min}\) as we can get, for the gas strut/linac to have maximum mechanical advantage.

So what to set \(b\) (the distance between the roof pivot and strut pivot on the sign, or the left slope on my diagrams) and \(c\) (the distance between the sign pivot and strut pivot on the roof, or the bottom line of the triangle on my diagrams) to satisfy this? That's what took me so fuckin' long. We can use the sine law to determine the ratios of the other two angles/lengths based off the ratio of \(A\) to \(a\):

\(\frac{sinA}{a} = \frac{sinB}{b}\)

\(cosA = \frac{b^2+c^2-a^2}{2bc}\)

We don't really care about the other angles or lengths, but know that ideally we want \(c = b + S_{min}\). Wolfram Alpha finally came to my rescue, because I kept screwing up solving for \(b\). I think it works thusly (assuming I've made no errors transcribing it back to my blog):

\(b=\frac{-\sqrt{(cosA-1)(-2{S_{max}}^2+{S_{min}}^2cosA+{S_{min}}^2)} + S_{min}(-cosA)+S_{min}}{2(cosA-1)}\)

\(c=b+S_{min}\)

Being that it's a quadratic, there are two solutions, I picked the negative root one because it was the only one that didn't return a negative number on the inputs I put in. But basically, you provide the function with the angle \(A\) at full excursion, and the \(S_{min}\) and \(S_{max}\) values of your strut, the formulae will tell you the rest. I would link to a solution at Wolfram Alpha, but unless you pay for pro it'll exceed the computation time and won't give you an answer.

Dad was running out of daylight so we settled on values found by brute force before I finished solving it correctly, but we ended up rather close (with \(A=58\unicode{xb0}\)). I think the maths works out though - if you set \(A = 60\), \(S_{max}\) = 565, \(S_{min} = 335\), I come up with ~317mm for \(a\) (bolt to bolt up the sign frame) and ~652mm for \(c\) (bolt to bolt along the roof rack).

Feel free to email me if I've made a huge mistake, but please use small words. :(

Horsham, VIC, Australia fwaggle

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Chromium on Debian in VirtualBox woes

A while back I discovered that at some point my Chromium inside my Debian VM has stopped working satisfactorily. It ends up drawing black elements everywhere, gets stuck on top, and just generally messes up the entire VM unless you happen to have a shell handly where you can blindly type killall chromium.

I started taking a look at it this morning, as I'm working on a project in my VM and having to access it from Chrome outside the VM, and for some reason my router can't seem to decide which IP address to give to the VM so I have to keep looking for it (or try multiple potential alternatives).

The first thing I did was install the update VirtualBox has been annoying me about for a few days now, and then do the update of the additions as well. Hampered, as usual, by the fact the autorun doesn't work on my machine - I suspect it might be because I don't have sudo installed, so it took a bit to remember that the appropriate response is grabbing root privileges, changing to the /media/cdrom0 directory, and running sh VBoxLinuxAdditions.run However, none of that fixed Chromium.

I did discover that if I run chromium --disable-gpu, Chromium runs effectively (after I'd blown away my .config/chromium directory, naturally, as I'd suspected something in there may have been the culprit). However, disabling all the GPU-related things in chrome://flags doesn't fix it permanently, so I can't run it from the Gnome UI at the moment.

I'm not sure if I care enough to try and work out what's wrong.

Horsham, VIC, Australia fwaggle

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Water heater

Sabriena woke me up yesterday morning because the water heater was leaking. Huh?

I go out to have a look at it, and water is positively streaming out of copper pipe connected to the pressure relief valve. I waited until after they opened and phoned the property manager (spoke to Freya), but in the meantime we shut off the water to the entire house to stop it from wasting water (who knows how much it wasted over night).

Later on in the day we hadn't heard anything and assumed that the plumber would not be attending today, and turning the water back on to flush the toilet and so on was becoming a pain in the arse, so I found the shut-off for just the water heater and successfully isolated it.

This morning a plumber called on the phone, and I gave him the numbers off the valve so he could source an appropriate replacement. He came out just after we dropped Duncan off at school, replacement valve in hand, but upon looking at the water heater (which is quite old and dinged to shit from the previous tenants) decided he would recommend the landlord just replace the entire water heater, as apparently the valves are not cheap.

The valve has since stopped leaking, and he showed me how I can tap it gently with any sort of metal tool which will likely reseat it in the short term until the landlord decides what they're going to do. At least we can take showers today.

Horsham, VIC, Australia fwaggle

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Passport renewal

Yesterday I dropped into the post office, printed-out form in hand, to find out about renewing my passport. It doesn't expire until November, but I'll need it before then and better safe than sorry. It turns out that I needed an appointment, though they could have squeezed me in if not for the fact that I'd printed the form incorrectly.

Their form needs to be printed so that boxes for the signature and date are visible, because I have to sign inside said boxes and if I don't then the passport office (I don't think that's the proper name for it, whoever is responsible for such things) would reject it. Try as I might to get our laser printer to print at sufficient darkness for it to be visible, I couldn't get it to work. In the end, we put the form on a USB stick, went to the library, and paid $2AUD to have it printed in colour.

Then today I had an appointment for 4pm to do the renewal, where they accepted the form (they better have, too!) and despite the fact I was $279AUD poorer, it all went rather smooth. I should have a new passport in a few weeks.

Horsham, VIC, Australia fwaggle

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