- #1

- 479

- 32

## Summary:

- Can we approximate a bent rod by using fictitious forces and use Castigliano's method in reverse to solve for the position of those fictitious forces?

I have some threaded rods on my 3D printer that I want to straighten. After searching Youtube for a quick easy method that doesn't involve a million "guess and check" steps, I found this:

This guy seems to have concocted some method using Fourier series to straighten his rods. Not sure exactly what he's doing there, but it got me to think about solving the problem of where to apply forces and approximate their magnitudes to straighten the rod analytically. I'm thinking about creating some sort of 3-point bending apparatus so I can manually apply some bending force at a specific location on the rod to straighten it after fitting a curve to the deviation of the beam from true.

Thinking back to my sophomore mechanics of materials classes, I remember using Castigliano's theorem to solve for deflections based on loads, both fictitious and real. I know that nonlinear plastic deformation is a complex area of study, but I figured maybe I can get some sort of approximation by saying that the deformation is due to a number of fictitious loads at specific places. Better said, we can maybe say that there is some set of loads applied to a linearly elastic beam at specific locations such that the elastic beam displacement would approximate the deviation of the rod from straight within some tolerance.

In looking over the equations, I realize I'd have to assume a thin, prismatic beam. I'd also have to assume the displacement (and possibly the first couple of derivatives) was continuous, which is a definite stretch. I'm sure there are other assumptions I'd have to make in order to make the math come out.

If I were to use Castigliano's equation for this purpose, it seems I would be solving it in reverse since I know displacements already. I can see that I have 2 unknowns and only 1 equation, but I can't figure out how to constrain the other variable in a sensible way. The integral makes me think about the weak form for the finite element method, but it's been so long since my FEM theory classes that I can't remember enough to see if that would even be helpful here.

Any ideas? Maybe there a way simpler analytical method that I've just not seen for this type of problem, and although I'd like to know it too, I'm also curious to see if this approach is viable.

This guy seems to have concocted some method using Fourier series to straighten his rods. Not sure exactly what he's doing there, but it got me to think about solving the problem of where to apply forces and approximate their magnitudes to straighten the rod analytically. I'm thinking about creating some sort of 3-point bending apparatus so I can manually apply some bending force at a specific location on the rod to straighten it after fitting a curve to the deviation of the beam from true.

Thinking back to my sophomore mechanics of materials classes, I remember using Castigliano's theorem to solve for deflections based on loads, both fictitious and real. I know that nonlinear plastic deformation is a complex area of study, but I figured maybe I can get some sort of approximation by saying that the deformation is due to a number of fictitious loads at specific places. Better said, we can maybe say that there is some set of loads applied to a linearly elastic beam at specific locations such that the elastic beam displacement would approximate the deviation of the rod from straight within some tolerance.

In looking over the equations, I realize I'd have to assume a thin, prismatic beam. I'd also have to assume the displacement (and possibly the first couple of derivatives) was continuous, which is a definite stretch. I'm sure there are other assumptions I'd have to make in order to make the math come out.

If I were to use Castigliano's equation for this purpose, it seems I would be solving it in reverse since I know displacements already. I can see that I have 2 unknowns and only 1 equation, but I can't figure out how to constrain the other variable in a sensible way. The integral makes me think about the weak form for the finite element method, but it's been so long since my FEM theory classes that I can't remember enough to see if that would even be helpful here.

Any ideas? Maybe there a way simpler analytical method that I've just not seen for this type of problem, and although I'd like to know it too, I'm also curious to see if this approach is viable.