Splines and Curves of Least Energy

I am using Excel Solver in the Recalc button. That is used when you change the approximation interval. 'n' in cell C4 can vary from 8-2048 in both the Euler Spiral or Horn Curve worksheets. I'll start with the Euler Spiral worksheet because it was easier to put together and the first one I did.
Orange cells are for input. gey cells with orange type are calculations. Cells below the calcs are arrays that are resized and recalculated when you use the Re-Calc button. Total curve length, S, andMaximum Curvature, kmax can be adjusted, but if you are looking for the curve that ranges through x=1 to x=o, you can leave those alone. I usually leave C7, kmin, =0. That is at x=1, and asymptotic to a vertical line

It is possible to do an ad-hoc Solver routine that sets J10 to 1 by varying C6, but it really works better to 'bit bang' C6 until you get a value at J10 close to 1. The yellow highlighting indicates we are looking at varies from theta = o - pi/2 rad. O10 is the scaling factor that normalizes x max to x=1. R10 is the normalized total energy in the curve. You can look through the cell formulas and VBA macros to figure out the est. The Re-Calc Curve button cycles through updating the values and re-drawing the charts.

I'll get to the Horn Curve later, but it's similar. The geometry is just more work and calculation time than it takes to graph an Euler Spiral.

 

Hi Chris,
I was wondering if you developed your solver any further? This is really interesting work.

 

Funny this popped up. I haven't had time to focus on this project until a couple days ago. The solver/spreadhseet stuff hasn't been touched, but I know what needs to be done next. As a visual thinker and builder, I wanted an apparatus to facilate the beam-bending experiments that started me down this rabbit hole. To that end, I'm just about done machining a set of six 'Suction Cup Spline Ducks' that can be relocated around a 4'x 8' table top:



The CAD image shows a bottom view, with one unit stuck to the screen. The pultruded fiberglass rods I've been using as 'uniform beams' slide between the pair of v-wheels, which in turn are free to rotate around a sealed bearing. The rubber-tipped toggle clamp legs can be adjusted to tune the holding force of the suction cup (which is an OEM unit used for automobile glass resin injectors).

I'll grid out the table surface, color-code the acorn nut atop the rotater, and use a contrating color for the table surface (battens are white). A digital camera will be centered overhead. The grid will be calibrated to give accurate point mapping through OpenCV. Spline interpolation will be part of this. This is half the fun of any project: as an exhibit builder by trade, I can't 'just do the maths!'

Regarding 'the maths': From a practical standpoint, interpolating a symmetrical curve to a prescribed precision is pretty trivial. You have a start point, a midpoint, and a mirrored endpoint. Curvature is 0 at each end and equal at the midpoint.

An assymetrical curve is tougher. An iterative/convergence process is probably in order. It ceratinly is for a person who's mediocre at math and finds elliptic integrals (basis of the 'Horne Curve') challenging! My thinking: use the ratio of the length of two chords defined by your three points to start. Curvature at each end will be zero. Scale one curve section against the other based on chord length. Compare curvatures at the common point. Make your next 'scaling' guess based on how far the two curvatures differ and do it again. Lather, rinse, and repeat until you get convergence to a desired precision. I'm sure it can be done analytically. Whether that leads to a faster solution on a computer vs using an iterative, numerical approximation? Dunno...

 

Ah, I was hoping you would have done the work on the asymmetrical curve and that I wouldn't have to!

I'll be defining asymmetric Curves of least energy based on three control points. Looks like I'll just have to get stuck into it. With these problems in Excel the hardest thing is to get your head around defining the problem and how to work out a way of solving it. You have done that already.

 

The superposition principle is your friend.

 

Nirvana Manana - The race is on! I'm pretty slow out of the gate

philSweet - Did you have something in mind that could help here? I can see that superposition applies to combined loads in simple beam bendingt. I'm not sure how it helps when we are trying to model an accurate 'curve of minmum energy through inflection points.' Cubic splines are based on simple beam theory, and is the basis for 'natural cubic' splines in a lot of software. The fact that it does not model real-world bending for any but small deflections is what started this project.

 

Superpostion requires a linear system. Simple beam theory is linear. The models/equations considered in this thread are nonlinear so superpostion does not work.

 

I think the 1962 book 'Flexible Bars' by R. Fritsch-Fay as being the full-on 'mechanics of materials' background for this. It can be read online at Scribd: Flexible Bars Frisch-Fay 1962 | PDF https://www.scribd.com/document/317704558/Flexible-Bars-Frisch-fay-1962 An online PDf can also be found here: https://bigoni.dicam.unitn.it/varie/flexible_bars_frisch-fay_1962.pdf.

From page 41: The portion of the curve from A-B is exactly the curve described by Horne, and the condition shown in my photos.



From page 81: An example backing up his diagrams and math, for a symmetrical condition. The shallower curves can all be derived from a segment of a steeper curve that undergoes scaling and rotation. 'Superimposing' is useful, but the principle of superposition does not apply when deflections are not a linear function of bending moment. Fritsch Fay points out the issue on page 1 of the book.

 

Here is a Masters thesis on compound bow limb design that may be applicable here :

Optimal design of the limb in compound bows

There is a link to the author's thesis on the site. He mentions Fay's work, and how difficult the math gets. He uses a 4'th order Runge-Kutta approximation method to calculate limb deflection. Much simpler, and any decent numerical methods book will explain Runge-Kutta approximation.