Splines and Curves of Least Energy

I'm afraid I don't understand what you are saying. The upper are of v-wheels are not experiencing any load when the rod is set up as shown. All forces and moments are in equilibrium. The only constraints are in the horizontal direction at the lower left and lower right v-wheels. The forces resolve into one force in the +x direction, and an equal and opposite force in the -x direction. That is a curve of least energy, and it matches Horns' method.

The Euler spiral does not match and the rod will not take that shape if you apply the same force in the same direction at the endpoints (lower left and lower right v-wheels.

When I fixed the rod into a block clamped to the bench at it's apex, and bent it so that the lower end was constrained purely in a horizontal direction by a single v-wheel (90 degree bend) it formed 1/2 of the curve Horn curve shown in the photo. That is a cantilevered beam bent through 90 degrees by a point load applied perpendicular to the end of the beam. It does not match an Euler spiral if you do it that way either.

Constrain the rod at the apex between the v-wheels (a point constraint). Constrain each end of the rod with a v-wheel equidistant from the apex such that the included angle between tangents to the rod end is between 0 and 180 degrees. The resulting curve, supported at 3 points, is composed of two mirrored portions of a Horn curve.

I'm happy to set the experiment back up and take more photos if you doubt it.

 

A semicircle of a fixed length is impossible to bend because the force at either end becomes infinite. To make that bend requires length well beyond the semicircle region.

 

My experiment with the rods pre-dated my looking into other methods for computing a curve of least energy. I initially thought an Euler spiral would be a match. When I set up the rig, I could see I wasn't looking at an Euler spiral - before I even bothered to map it onto the rod. So the hunt for a better explanation ensued. Horn's paper matched, and the approach and justification makes sense even if not backed up by experiments on his end. The curve of the rod aligns with his curve better than anything else I've researched. The math is much more complicated than an Euler spiral. No matter what, we're into numerical approximation. Close enough works, but a semi-circle isn't close to what I observed with real material. The method in my spreadsheet is a way to get close enough without dealing with elliptic integrals.

There is a lot of work to extend it into 3D and perhaps a useful application. I stand by it. It is based on an experiment, not a thought experiment.

 

I went a bit more deeply in the Horn paper, and so noted that the elastic bending least energy is obtained for a curvature linearly decreasing with x. As it is a characteristic intrinsic to the curve, I presume (but not able to demonstrate) that this result is actually not dependant of the XoY coordinates ? (the 1/R (x) function being a straight line in the current xoy, so still a straight line in any other XoY coordinates ?).

Here I see a similarity with my approach for the bottom line of a sailing dinghy, which is guided by the idea of a curvature regularly decreasing from the forefoot to the aft transom where it becomes zero, a linear decreasing being a particular case of the regular one. The connection is shown in this drawing here below, using the Fig. 8 of the Horn paper, the dinghy being shown in pink :


For this specific linear case, and when assuming that the curvature 1/R # Z'' because the angles (and so Z') of the bottom line versus the X axis are very small, the maths can be very simple. I did it with as input just the waterline length L and the hull body draft Tc : by double integration the bottom line formulation is then : Z = 3 √3 / 2 * Tc * (X^3/L^3 – X/L)

The output data, in addition to the line drawing itself, are the attack angles in the fore part (to favor the planing) , the water exit angle at transom and the position of the maximum draft Tc of the hull body, exactly at 1/√3 L ~ 0,577 L and remarquably independant of Tc. This ties in with intuition of the early designer of planing dinghy who put this maxi draft forward of the midship.

For the fun, I also look at this result for the case where the simplification 1/R = Z'' is no longer possible ( my drawing in the upper part of the figure above) and it seems that the maxi draft is still at 1/√3 L : it is not a demonstration ! , I let better mathematician than me to demonstrate this result (or another one ? ).

Here attached the computation of this simplified case.

 

@Dolfiman - That's an interesting approach for setting up the keel line. Your keel line would be approximated well by an Euler curve, a Horn curve, the method in your last spreadsheet, or maybe even a parabolic fit.

Here's a graph of both an Euler Spiral and the Horn Curve laid under a 'Natural' Cubic Spline drawn in KeyCreator. Euler and Horn were calculated over 128 intervals from my spreadsheet - these are numerical approximations (you can open the spreadsheet from another post or I can send you a copy if interested - requires enabling VBA macros). Dimensions have been normalized to a unit square (1" ). The 1.652 dimension is the maximum Y-value of the Horn Curve when approximated over 128 intervals.

Note how close the Euler Curve is to the Horn curve at low curvature. The Euler Spiral is a good approximation to the curve of least energy for the small deflections used to engineer most beams - it presumes constant shear and linear bending moment along the beam. It doesn't match up well for elastic bending of a thin strip around a tight curve. The Horn Curve does, with the fewest constraints.


The fiberglass rod, natural cubic spline, Euler curve, and Horn curve all use the same three control points, which are also interpolation points. Curvature at both ends = 0 in all cases (k =0). The natural cubic spline is often used as the basis for emulating a curve of least energy. It is a poor fit without over-constraining the curve. This includes adding more control points/interpolating nodes or by adjusting the end conditions.

Here is the with the cubic spline thru the same 3 points. The tangent vectors have been oriented and given magnitudes to force a good fit to the batten. Note that the curvature at the ends is not 0 anymore, even though our fiberglass doesn't have any additional bending moment or tension applied at the endpoints.




This is the why boatbuilders complain about designers using CAD that don't understand how actual materials behave and how to work the software to more closely approximate how the materials may actually behave in the shop. If a designer using CAD does have that understanding, there will be a LOT of time saved on setup and cutting out components for the boat. If the CAD designer has a reason for forcing a certain shape (hydrodynamics, contract spec, whatever), then the builder has to get on with matching the design, even if they think it's 'unfair' (pun intended).

Seems to me if software could more closely approximate the materials and building methods without the additional tweaking and fiddling, it might be more enjoyable for all involved.

 

I think we need to keep in mind that we use materials to create the boat and it is how the boat behaves that is the important thing.The designer will have determined what he believes to be the ideal shape for the function and it is the job of the builder to make the chosen material conform to that shape.Two good examples would be the IOR level rating classes of the 1970's and 12 metres.In both instances there were bumps and dips in several places so that the measurements entered on the official paperwork led to a boat that could be computed as having features that weren't ingredients for a fast shape,while having performance characteristics that made it go fast.Some clever designers emerged.I have heard builders mutter about the aesthetics of the designs,but they got paid for building them and if success followed-so did more orders for boats.

 

@wet feet - Yep. But if someone wants to spend lots of money to sail offshore fast AND look good, why not this?



Alright, alright...I'll try to keep subjectivity out of the discussion. But c'mon, you gotta agree Irens has a great eye!

 

I agree and boats with that kind of speed potential have an inherently clean look as they are designed for a particular purpose.It probably helps that they didn't have a committee of designers thinking up places to take a measurement that might relate to a feature that could confer speed.Box rules can work well for that.I note that the most recent Banque Populaire was designed by VPLP.

 

Did a small bit of work on my spreadsheet comparing Euler Spiral vs Horn Curve:



All curves are computed for 512 points. Euler spiral is on top. There are 2 curves below - one right on top of the other. The dark red is the Horn Curve. The mustard yellow is an Euler curve scaled such that the peak value matches the peak value of the Horn Curve - with a little extra...

... Because the pure scaling in Y of about 93% left a poor visual fit between 'Scaled Euler' and Horn in the middle, took Y-differences between the two and eyeballed cubic polynomial fit to the values - seen at right. Note it is a cubic polynomial fit to a bunch of data points, not a cubic spline fit. And its not a least squares fit, because I needed Y=0 at each end. Hence the eyeball fit.

The gap between new 'Y-scaled Euler with a Cubic Tweak' and the Horn Curve is minimal. Good enough for approximation, faster to compute, and much better than a 'Natural Cubic Spline' to model the physical behavior of the fiberglass batten/rod. After all, engineering is all about 'close enough,' right?

This project may be a wild goose chase, but it has a reason: If one can set up a minimal number of mold stations on a computer, numerically (not analytically) model a surface comprised of battens sprung through parametric points along those stations, and have it be very close to physical reality, then for a useful variety of hull shapes and building methods, the computer model should very closely match the physical set-up. Little-to-no tweaks and tricks between design and shop floor. Ability to quickly analyze hydrostatics or to export for additional analysis, printing, or machining (CFD, CAM, FEA).

 

That isn't exactly a small bit of work,more of an extensive study.

 

Chris, I think this is really important and insightful work and especially relevant where parts are CNC machined to fit together. If the sheet of metal, or ply, or wood doesn't bend as you expect, then a lot of fitting up and fairing will be the result. Your study is not a wild goose chase.

I tried using our CAD system to fit a spline curve to the Horn curve and was surprised that the coordinates were out by almost 3%. Some time ago I used the same system to draw a NACA curve and compared it with one generated by the NACA 4th-order polynomial and the fit was perfect. I saw a professor on youtube make a NACA foil rudder using a type of least energy fitting method and the profile appeared perfect.

A method of generating the least energy curve and then importing it into a CAD program would be very helpful. A solution is one that works is really all that is required. btw, can your spreadsheet be adapted to different end conditions? It seems like it can but I didn't want to go messing with it.

 

I investigated another line, the sheer line, when guided by the curvature and its least energy. I start with a generic line based on a parabolic shape of y'' , again for simplification (y'' being close to 1/R for small angles) allowing to formulate the y(x) sheer line by just double integration. But I use the exact 1/R value to compute at the end the integral ((1//R)^2 ds. The investigation to find the line of least curvature energy (i.e. the minimum of this integral) is then based on the more or less flat, convex or concav, shape of the parabolic y'' curve through the use of a coefficient k. I think the example given in the pdf attached for a Canoe L 6 m x B 1 m better illustrates the process than my explanations.

I give another example for a sailboat L 10 m x B 3,5 m, using also the rotation alfa of the generic line that Gene-Hull users know.

Of course, this type of approach to defining a sheer line does not claim to be hydrodynamically optimal, it is just an illustration of the use of the concept of least energy of the curvature.

Also attached are the files for this two examples (Canoe, Sailboat), you can play with the input data to see by yourself the influence of the parameters and to generate another example.

 

Thanks for the several recent replies.

@Dolfiman - I will look into your spreadsheets ASAP. Granted matching material properties in bending is not a condition of optimal hydrodynamics.

@wet feet - I'm already wondering if the computational simplification I made won't leave this lacking in the mid-portion approximation. That's where the fit is still not so-good. It may be better to stick with the computational lag and use the geometric method to go through the Horn Curve to n intervals. The rationale behind this is that fitting to piecewise segments of the Horn curve through a control point will require a search not only for tangency, but a scaling and match for curvature. I hope that makes sense. If not, I'll get to it!

@NirvanaManana - I trust you found the link to my previous Excel spreadsheet. I changed the file extension to 'trick' it to being a non-macro file. If it is opened in Excel, you would get a warning that it contains VBA macros. Hence my offer to delete the link if it did not follow forum protocol. Feel free to message me for another way to get you a file if this is a concern.

 

A few views of a lever-actuated, suction-cup. single-point constraint or 'duck' that should work for cylindrical battens from ~3/32" - 3/16" diameter (pultruded FRP or similar):



Th unit is actually quite compact. Suction cup unit is 2" diameter. V-wheel bearings are 16mm diameter OA. Studs on the toggle clamp spindles are M5. More to follow.

 

@NirvanaManana - I trust you found the link to my previous Excel spreadsheet. I changed the file extension to 'trick' it to being a non-macro file. If it is opened in Excel, you would get a warning that it contains VBA macros. Hence my offer to delete the link if it did not follow forum protocol. Feel free to message me for another way to get you a file if this is a concern.[/QUOTE]

Thanks - sorry I can't message you as I think I am not on here long enough. I was able to open the file no problem. The problem I have is trying to follow what is going on in the Macros. Are you minimising the total curvature value (Sum of individual curvatures) by adjusting the y values of each given X value? Are you using an Excel Solver or an external one?