Calculating Where to Push to Straighten a Rod

[timthereaper]

TL;DR 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.

 

[Baluncore]

Any ideas?

Without experience you can expect a few problems with a computational approach.

1. If the rod work hardened where it was bent, then it will be difficult to reverse the process.

2. The bend tells us the material was taken beyond the elastic limit and into the plastic region. You will need to know how far to push it again to enter the plastic zone, and so to reverse the bending process.

3. The length of the rod, and so the pitch of the thread may change when first bent, or during reversal. Rods tend to get longer.

I often straighten rods by holding them in a lathe chuck. Turn the chuck by hand to identify the location, direction and magnitude of the largest bend. Slide the straight part of the rod into the chuck. Now turn it again to identify the bend direction near the chuck. Apply a force to the free end of the rod by hand, or with a rubber mallet, it will bend near the chuck. Check the amount of bend remaining. Increase the force and repeat until the rod settles back into a straight line. Slide the newly straightened part into the chuck, and repeat until the rod is approximately straight.

You will quickly get the hang of it and should be able to sense the force change where it passes from elastic to plastic. Practice on scrap wire at first, then on thicker rod.

 

[Tom.G]

A somewhat related story.

"Way back when" cars had rear wheel drive, there was a Drive Shaft from the transmission to the rear axle. This was a steel tube about 3 inches (7.5cm) in diameter. Occassionally the drive shaft would get bent, usually from mis-use/hot-rodding. This would be noticed when you drove at certain speeds and hit a resonance; it would also wear out the rear seal on the transmission and you would see the oils spot where you parked.

The solution was:

• remove drive shaft
• take to local straightening shop
• mount in lathe
• using a dial indicator, find the high spot while turning by hand
• light oxyygen-acetylene torch with flame-spreading tip
• apply to high spot area until rather Red, not too bright, not dull
• turn drive shaft hot-spot to bottom
• place hydraulic jack under hot-spot
• over correct "a bit"
• remove jack
• rotate hot-spot to top
• wait for red to fade
• apply wet rag
• re-check bend

(Remember to replace transmission seal.)

There are a few "experience required" operations here.
1) not too bright, not dull
2) over correct "a bit"
3) wait for red to fade

Number 3) has no obvious impact at the time, other than maybe getting burned. The long term consequences can be disconcerting though. If you rush and apply the wet rag too soon, the steel drive shaft crystalizes while cooling, making it brittle. Especially if the shaft is from a large truck, a heavy load can cause the drive shaft to crack completely into two pieces.

I've only seen photos of the result for a car. The front end of the drive shaft dropped while traveling at speed. It caught on the pavement, folded, and went through the floorboard in the back seat.

Lucky driver!

Fortunately your learning curve does not have such dire consequences.

Cheers,
Tom

 

[Lnewqban]

If applying lateral force, protect the thread the best you can (ideally a nut with abundant length of thread).
Otherwise, the pressure could deform the crests of the thread on the area of application.

 

[Baluncore]

The technique shown being applied in the OP video is not really applicable in the real world. Consider the example of one short bend. It will have only a short length affected, but the Fourier components will have many harmonics to be considered, all of which must be removed to straighten the rod.

I believe a Chebychev approach over a specified length range would also be imperfect, but is more applicable than the Fourier approach. You are, after all, Chebychev optimising the straightness of the rod.
https://en.wikipedia.org/wiki/Chebyshev_polynomials#Definition

The straightening process appears to require focus on short circular or parabolic bends occurring between two points. The sensible way to identify those points is by rotating the rod in a lathe chuck, using the light gap between the bent rod and a flat face tool as an estimate of deflection. If you require greater accuracy then mount a dial gauge on the saddle.

If you are using a rotating chuck to identify the bounds of a bend, then you may as well quickly remove that bend by hand, then move the straightened rod into the next bend. (Warning; do not power the lathe with a long unsupported rod held in the chuck. You will only do that once).

The tips of screw threads are usually not injured in a chuck, if they are critical then mount a “joiner nut” in the lathe chuck to support the straight end of a threaded rod. A ball screw or buttress thread could be held directly in a lathe chuck.

 

[Whipley Snidelash]

I don’t know if this will help you but you might want to check out how metal arrows are straightened. Also the method for testing of an arrow for straightness is fairly simple and you may be able to use it to test the screws after you straighten them. That method is forming a V in one hand with two finger nails laying the arrow in the V and spinning it with your other hand, if it vibrates it’s not straight. That’s assuming the threads won’t prevent you from doing that. This is for a simple bend. Compound bends for example in an arrow are almost impossible to straighten.