When trying to make parts of my machine stiffer I got to wondering about the relationship between a material's thickness and its resistance to bending. It is obviously not a linear relationship because as a sheet gets thicker not only is there more material to resist bending, but the outer layers have more leverage than the inner, so it must be at least a square law. I tried googling this for some time but failed to find a formula. I did find a comment on CNCzone by somebody that thought he recollected it being a fourth power law. I can believe this because we recently had two versions of a metal box made at work, one in 0.5 mm steel and the other 0.8 mm. While the thin one was quite flimsy the 60% thicker one was very solid. Any mechanical engineers out there?
I came across another fourth power law recently on the website of the company that made my XY table. If you have a servo system moving something from A to B as fast as it can go, then going twice as fast requires 16 times the power. Some video lectures, and a lot of other useful info about servo systems, are here www.neat.com/products/corner/default.asp.
The highest power law I have ever come across is that incandescent bulb lifetime is inversely proportional to the 12th power of voltage, see www.allegromicro.com/en/Products/Design/an/an295012.pdf. Can anybody beat that?