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.
Another worked example.
More worked problems.
Last time, I asked if it was reasonable that we identify k as
Well, the units check out. Anyway, we’ll now do the problem a more “normal” way, then eventually relate it back to the method in the previous posts.
We earlier assumed all the hinge angles were equal, simply to get an answer as easily as possible. Now we can question that.
So let’s assume k’ in the original equation is replaced with kn, with k a constant.
Related to the guitar project, might have other applications.
Prevailing consensus is that Galileo Galilei made the first attempts at developing a theory of beams, but recent studies argue thatLeonardo da Vinci was the first to make the crucial observations. Da Vinci lacked Hooke’s law and calculus to complete the theory, whereas Galileo was held back by an incorrect assumption he made
So, if Galileo made incorrect assumptions, certainly that should be gone over, and then what the actual correct assumptions are, explained so clearly that it is obvious in retrospect? Because the hard part is the interesting part. Of course not, they just jump to the differential equation.
The neck angle on a guitar is an issue that has confused me. I think the issue is that most guitar building books are aimed at a general audience who is not assumed to have any knowledge of math, while I sometimes find that a little math can clarify things.
So, for example, when a book mentions figuring out the neck angle, they mention making a drawing to scale to be sure you have it right. Okay, but suppose I’d like to know general trends. Changing the bridge height by some amount changes the angle by…?
This is really just the old guitar design, but rescaled to fit the new neck.
I think it should work. I also need to remind myself of practical issues, eg., with the re-scaling, will the pickups still fit?