I once saw an interesting math challenge: Assume X and Y are randomly chosen numbers, with each being equally likely to be any number between 0 and 1.  Take Y/X and round to the closest integer.  What is the probability that this number will be even (or odd)?

It turns out this problem involves using the a ratio distribution.  I felt that Wikipedia’s page on this subject was very poor, at the very least certainly unintuitive, so I hoped to come up with a more intuitive explanation.

(Note: This post does assume familiarity with probability distributions, using them to get cumulative distributions, averages, etc.)

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In early November of 2008, we had taken in a stray that had been roaming around our apartment complex.  She immediately made herself at home.  She was the first cat I have ever had. We named her Caramel.

Little did I know that only a few days later, at the end of the very same month, I would find another stray in our apartment parking lot.  It was already dark when I came home from work.  After I parked my car, a cat came out from under another car, meowing.  In the darkness, I thought I saw a collar.  I figured since we had good luck with our first cat, we could try to do a favor for this one.  I told FO about her.  FO went out to look at her.

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Beam Buckling 1

The intuitive way I understand beam buckling:

Put force on a beam.  It compresses (linearly).  It takes a certain amount of energy to compress it linearly.  At some point, it might have so much energy stored in compression that it may be able to relieve some by returning to a longer length, but bent.  At that point, the beam has buckled.

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Beam Bending Final

This is wrapping up, I think.  There’s not much more to do before trying to move on to more practical calculations.  I suppose I could compare numerical large angle results with the small angle analytic results; that’s the main other thing I can think of.  Or maybe I could look at other extensions, like buckling.

Anyway, this post has one more worked example, plus the derivation of the Euler-Bernoulli beam equation from the method I have been using.

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