Monday, August 27, 2007

Difference Engineering

I've known about Babbage's Difference and Analytical Engines for a long time, but never known much about how they work. Finally I actually read a little information on the first one (in the context of Legos) and it's pretty interesting from both a mathematical and mechanical point of view. Based on the information on that page, I worked out a tiny additional step of my own (though surely Babbage himself already knew this).

First of all, the basic idea: The purpose of the Difference Engine was to pre-calculate tables of polynomials for books in the days before portable calculators. So you'd want to known 3x3 + 14x2 + 5 for all values of x from 1 to, say, 1000. As it happens, there's a clever little shortcut such that if you have a few values you can calculate the next one very simply using only addition from the previous answer (thus "Difference Engine").

So let's say f(x) = x2 + 3x.

x   f(x)   diff1  diff2 
1     4      -      -   
2    10      6      -   
3    18      8      2
4    28     10      2
5    40     12      2
See that second difference column is a constant. If we'd picked a polynomial of the 3rd degree, we'd have to work this out to the 3rd column. Fourth degree, 4th column, etc.

So to build a machine, all you need to do is set the value for x = 1 and set the Nth difference in the last column. It automatically adds that difference to the previous difference, which is cascaded upwards until f(x + 1) is arrived at. Then you go around again. All that is required is simple addition.

What isn't explained on that page (that I saw) was how to know ahead of time what that last difference is going to be. Sure, you can work out N rows to get that Nth column, but it would be nice if you could set the inputs from direct inspection of the polynomial. As a matter of fact, this is easy.

Say f(x) = ax2 + bx + c. Then the difference between two successive answers (i.e. the value in the first difference column) is going to be:

a(x+1)2 + b(x+1) + c - ax2 - bx - c
= ax2 + 2ax + a + bx + b + c - ax2 - bx - c
= 2ax + a + b

The difference between successive entries in THAT column are going to be:

2a(x+1) + a + b - 2ax - a - b
= 2ax + 2a + a + b - 2ax - a - b
= 2a

And if we look at our example, we did indeed get 2 * 1 as the constant in the last column.

Working this out for a 3rd degree polynomial gives 6a (where a is the coefficient of x3). Based on these two examples and looking at Pascal's Triangle, I predicted the value for a 4th degree would be 20. But it was actually 24.

In fact, if you work it out it should be clear that the constant difference will be a * n!, where a is the coefficient of the highest power of x and n is that power. It should be simple to prove this using induction, since all other terms always drop out and you muliply the coefficient by the power at each step on the way down.

So if I wanted to know what the constant difference was in the 5th column of differences for the polynomial 8x5 - 3x3 + 117, I just multiple 8 times 5! and get 960. I cram that, plus the initial value for x = 1 onto the machine and get cranking.

Friday, August 24, 2007

Lego Marble Pump

This is a pretty neat machine. The core of the pump mechanism is pretty genius and I built one myself based on that graphic. The problem is that the tower can only be as high as the number of marbles you have and the number of marbles your motor can lift.

The most common way of lifting balls in a contraption is with a kind of one-way elevator. But I've already built one of those. This guy has an interesting idea. It's a little hard to see in the video, but he's got the marble going both ways through the pistons. It lifts one "story" going from left to right, then another story from right to left.

That gave me an idea. First of all, why not any number of stories? And second of all, why three pistons? Probably the answer to both is: maximum height for the amount of Lego he had.

I'm thinking you could put just one piston between two towers and with a certain amount of mechanical futzing pump the marbles up to any height.

*pause for 24 hours of trying to get some kind of Lego CAD software working on any computer in my house or office to illustrate this and failing miserably*

OK, here's the basic idea: The piston in the middle has ramps inside. The marble starts off at the bottom of one of the side towers. It rolls into the piston down one of the ramps and it stopped against the wall of the other tower. The piston moves up and there's a little nook in the other tower that the marble falls into. The piston lowers. Now there's another internal ramp, this time pointed the other direction, aligned with the nook, so the marble re-enters the piston and is stopped against the wall of the first tower. Repeat.

It's a little tricky, though. The nook is the problem. The marble needs to roll out of a ramp into the nook but then roll out of the nook into a ramp. The three piston video solved this by having good timing and long ramps. I don't have long ramps or the patience to get the timing right. What I'm trying right now is a kind of teeter-totter arrangement in the nook that tips up to receive the marble and then tips down to release it.

Oh cripes. I just re-read this entry and even I was too bored by the last two paragraphs to pay attention to them. I'll just have to make it work and post a video. Or somehow get LDraw working.

Wednesday, August 15, 2007

The Temperature of What?

So I had all these posts about temperature logging and one post with the actual logged temperatures...but the temperature of what?

A solar hot box!

You probably already know what this is, but just in case you don't: It's basically a tiny greenhouse. Or like a car left in the sun at noon in August. Only it's even hotter, since it is insulated, painted black and pointed right at the sun.

Inside the box I put a jar with 250 ml of cooking oil and poked a hole in the lid for a temperature probe. That's what these temps are.

Why did I choose cooking oil? Because I didn't want evaporation to be a problem. For one thing, it would fog up the inside of the glass. For another, it would cap my max temperature at 100°C (not that that turned out to be a problem in this case). And lastly, it would change the amount of water in the bottle and I needed that to be a constant because I did some calculations with it.

Knowing the amount of oil and the temperature change (plus looking up the specific heat of vegetable oil), I can calculate the rate at which energy is entering the oil. For the above graph, I got 2.5 watts for the steepest part of the curve. However, I see that the site I just linked to has the specific heat of veg oil as 1.67 kJ/kg K and I was using 2.5. So maybe the power is really more like 3.7 watts.

Knowing the area of the collector I can also calculate the amount of power falling into the box. That's about 75 watts. So the end-to-end efficiency was only about 3-5%. Not that great.

Imagine if you put a cup of water on the table and then turn the furnace thermostat up to 90°. How much energy are you going to waste before the water gets hot? This illustrates the 3 main problems:

  1. Air passively surrounding a container of liquid isn't going to heat it very fast.
  2. There's a lot of volume of air being heated uselessly.
  3. During all this time, heat is escaping the cracks, windows, chimney, etc. In the case of the hot box, the glass front gets very hot and is radiating a lot of the energy right back out.
If a hot box is like an oven, the next version will be like a microwave. Don't heat up the air, just beam energy right into the substance.

Wednesday, August 1, 2007

Locomotive Lego Linkage

It seems like I keep having to reinvent this linkage. The basic idea is that I want to turn a crank (i.e. rotary motion) which makes something move back and forth (i.e. linear motion). You see this exact same linkage on classic train wheels, though in that case the linear motion is the primary mover and it is turning into rotary motion to move the train.

In fact, almost all piston engines do this same thing. What I keep not realizing is that you need to hold the straight part straight or it doesn't work. Not "it doesn't go straight" but not work at all.

Let me show you what I mean

I think the reason I keep forgetting this is that the movement looks so smooth. It doesn't seem like there could be that much friction there, but there must be. The piston is rubbing against the sides. Not (just) because of a tight fit, but out of sheer necessity to make the piston run straight.

There are linkages that have been invented that can convert rotary motion to linear without all the side friction. For instance, here. A 3D version of that linkage is in the Boston Museum of Science. Another picture of Peaucellier is here and the page also calls my "locomotive linkage" a "sewing machine".

Peaucellier isn't very practical because it requires so many parts, which take up space and can fail. Or can it be made simply? Can another practical, mathematically accurate, low-friction linkage be built? In Lego?