More Questions and Answers

From JVoccio@amsuper.com Tue May 15 08:49:41 2001

Joe,

I tried to run the code; however, I don't know what the following things are:

• guiding strip
• outer angle of guiding strip
• initial edge angle of guiding strip
• number of cables inside/outside guiding strip

I think I understand other terminologies, but could you help me understand this one? Thanks.

John Voccio

I pride myself on the large number of comment cards in the BEND source. It is a very-open-source code but, as you have noticed, the documentation is still not complete. In particular there are the four points which you have raised:

1. Guiding strip

   The central strategy of BEND is to optimize the cable configuration by finding one which minimizes the elastic strain energy subject to the given constraints, i.e. that it starts at a given point with a given direction (say at a low point of the saddle and in the z- direction), that it finishes at another given point and direction (say, for a quarter turn, at a high point of the saddle and in the minus x-direction), and that it is constrained to lie on a cylinder while going from start to finish. (Of course without constraints it is easy to find the optimal configuration: The cable would just be straight.)

   One could solve the optimization problem by a hardware simulation, by holding a piece of cable properly at both ends and up against a cylinder and watching it squirm into its minimal (potential) energy configuration. The advantages (and disadvantages) of a software simulation are well known. For parameter studies software is cheaper than hardware (when it is), and software is innately quantitative. BEND produces files which input directly into CAD systems and into NC machining systems.

   But for the rest of the cables there is an additional messy con- straint, that they lie packed together in layers. BEND simplifies this situation by just optimizing one cable, represented by the guiding strip, and then packing the rest carefully on either side of it and hoping that there are not so many of them that their configuration is not too far from optimal.

   So much for the optimization. Is it important? Aesthetically yes but actually? Well actually it is just a way of getting a smoothly turning block of cables which are precisely packed together, especially since you said that your cables were not particularly brittle. One of BEND's strong points turned out to be its packing algorithm. That it does well. The geometry is complex and messy but BEND does it well, and that is important. You want your cables to enter into the curing process firmly packed together in spite of the fact that they are twisted about into a saddle shape.

   A conceptual weakness in BEND's guiding strip strain-energy optimiza- tion (minimization) algorithm is that in its application of Euler- Kirchhoff thin rod theory, the three rigidities that resist easy- way flexure, hard-way flexure, and torsion, are taken respectively to be 0, 1, and 0. The first two rigidities, 0 and 1, are OK. In fact they are perfectly OK because only their ratios matter, but the 0 for torsional rigidity is an imperfection. I don't know how imperfect it is but the code produces smoothly twisting blocks of cables which seem to like what the NC machine has told them to do and Fermilab interests have moved on so I have left it at that.

2. Outer angle of guiding strip

   The radially outer edge of a thin flat cable is the edge which is wound on the cylinder. (For better support you may want to wind the inner edge on the cylinder instead.) Its "outer angle" is the cylin- drical polar coordinate, measured from the vertical, of its point of intersection with the x-y plane from which it starts out in the z- direction, say at a low point in the saddle.

3. Initial edge angle of guiding strip

   The initial edge of a thin flat cable is its line of cross-section with that x-y plane. The angle is the angle it makes with the ver- tical. The energy minimization will pick that angle by itself. If you would rather do the picking yourself, you input your desired angle and BEND then minimizes subject to that additional constraint.

4. Number of cables inside/outside guiding strip

   I lied to you in (1) above. The guiding strip does not represent one of the cables in your block. It is a two-dimensional rectangular strip, twisted into an optimal (rigidities = 0,1,0) configuration Your cables are not represented by such infinitely thin rectangles. They are represented by twisted flat rods of relatively small but finite thickness which are packed up against the guiding strip to be "guided" by it. Typically it is placed somewhere near the middle of the block so as to minimize the deviations from optimality which will necessarily arise in cables packed in layers away from the strip.

   So cables are packed on both sides of the strip, on the inside of the turn and its outside. Then (number of cables inside) + (number of cables outside) = (total number of cables in the block (not counting the guiding strip because it does not represent a real cable)). In general one wants the first number to be less than the second because cables tend to get more cramped on the inside, the concave side of the turn.

And you are always welcome to call me at home, (630) 964-2220. (I notice that my area code is given in the comment cards as (708) but they recently changed it to (630).)

I had a basic fundamental question for you. If we do not use any shift or blunt of the base curve, then is the equation for the elastica base curve same as an equation of the ellipse or is it some hyper-elliptical equation?

Neither SHIFT not BLUNT modify the base curve. They just cause the rulings defining the guiding strip to rotate about the base curve. Each ruling rotates about, as an axis, the tangent to the base curve at the point where it intersects that curve. In this sense the guiding strip rotates about the fixed curve on which it is "based".

As for the curve itself, it looks almost like an ellipse, or one of the hyper-ellipses, but actually it is determined by an equation from elasticity theory, not from pure geometry because there are forces involved. We are all familiar with elementary beam theory in which deformations of the beam are determined from the forces acting on it as long as the deformations are small and the beam is therefore still approximately straight. Then the differential equations determining the deformations are conveniently linear. However when the deformations become large, as in our case when the beam (cable) is twisted and bent through 90° and beyond, then one must resort to nonlinear equations, the equations of the so-called elastica. (See for example page 584 in "Mechanics of Materials", 2^nd Ed., by R. C. Hibbeler. For a much more thorough treatment of the elstica see Chapters XVIII and XIX in "A Treatise on the Mathematical Theory of Elasticity", 4^th Ed., by A. E. H. Love, Dover Publications, New York (1927).)

These equations can be derived from a variational principle, the minimization of the total elastic strain energy. I needed to modify the classical theory of the planar elastica only slightly by adding to this energy a term which measured the contribution introduced by the curvature of the plane as it is wrapped around a cylinder. In that way we can maintain the relative simplicity of the planar theory even though the elastica is now a curve in three-dimensional space.

This base curve is still just a curve. It is one-dimensional whereas the guiding strip, a rectangle smoothly twisted in space, is two-dimensional, however it is determined by the base curve as follows: A planar elastica has only one rigidity, its resistance to curvature in the plane. But our plane has been bent around a cylinder and the elastica now resists its so-called normal curvature in space where the normal curvature includes the curvature brought in by the bending of the plane. (See any elementary book on differential geometry for a precise definition of the normal curvature of a curve in space.) Now the guiding strip is attached to the curve in such a way that it contains it and is at every point along the curve perpendicular to the direction of curvature of the curve. In other words the strip, being a surface in space, has a perpendicular vector in space which is required to be parallel along its included curve to the vector of normal curvature of the curve. This is the way the strip would naturally want to bend if its rigidity against curvature in a direction perpendicular to its surface, i.e., against curvature about a line tangent to the surface, were much less than its resistance to curvature about a vector perpendicular to the surface. A strip of paper folds much more easily than it can be twisted in a tangent plane (without tearing), and so it is with cables. They can be bent much more easily about a fold line, "easy-way" bending, than they can about the line perpendicular to all possible fold lines, the "hard-way". We idealize this fact for the first approximation to the guiding strip by giving it an infinite "hard-way" rigidity with only some finite "easy-way" rigidity. It turns out that this implies that the configuration of the guiding strip is then completely determined by the fact that it must contain the base curve. The strip must be a developable surface, the rulings being its fold lines. (Again, see any elementary book on differential geometry.)

So the guiding strip has the two flexural rigidities that we would expect from the theory of the elastica in space (as opposed to the single flexural rigidity of the base curve). That theory leads us to expect also a third rigidity, the torsional rigidity, resistance to torsion about the tangent to the base curve as an axis, however BEND does not use this rigidity to compute the torsion. The orientations of the two ends of the strip, the short sides of the long rectangle, are determined by user input. This is always necessary at the initial end of the strip because its orientation is determined by the given orientation of the cable as it comes out of the straight section, and at the final end the user often has reasons for specifying the orientation, e.g., to avoid using shelves, though there is available the option of letting the cable seek its own orientation as if the torsional rigidity were zero. Then BEND computes the torsion between that given at these two endpoints by a smooth interpolation, geometrically rather than elasticity-theoretically.

The guiding strip is still not a cable. Cables have thickness. They are not "infinitesimally" thin so we use the guiding strip not as a model of a cable but only to "guide" the block (group) of cables. They are packed on either side of the guiding strip in the hope that the two cables on the outer edges of the block are still not so far away from the guiding strip that they can benefit from its energy-minimizing properties.

Dear Joe,

I needed your help with this question:

You have parameters key1, key2, fat1 and fat2 for cable shape change. Fat2 and key2 apply at the nose. Fat1 and key1 apply at the middle of the group....is that the middle of the arc length (top, bottom?) or the ruling number 25? I will appreciate your feedback regarding this.

I'm glad that you are trying to use BEND.

FAT1 and KEY1 operate at ruling number IMID.

If you do a FIND on ' IMID ' in subroutine GROUP it will get you to the comment cards

*    IMID     The number of the last point on the base curve such that
*             arc length to that point is less than or equal to half of
*             the total length of the base curve.

in the Glossary, so that your first surmise is correct: FAT1 and KEY1 apply at the middle of the base curve in the sense of its arc length, or close to it, on the base curve, not the other edge.

To make sure, FIND the DO loop 930 where the actual computation of IMID is made. It's down around line number 4481 though I am not sure about the version of BEND that you have. BLENTH is the total length of the base curve, arc length, and FAT1 and KEY1 are installed at BLENTH/2.

Actually, statement 931, right after that loop, will boost IMID up to at least 2 in the very very unlikely event that the arc length midpoint of the base curve occurs in the very first mesh interval.

Whenever you suspect that you might be able to use somebody to talk with about BEND, please feel free to get in touch with me. For example my home phone number is (630) 964-2220, with voice mail.

Dear Joe,

Here's another question that I am pondering on. Program Bend outputs a final edge angle that defines the least energy inside surface. If I artificially steepen this final edge angle, such that I still stay within the empirically determined delta L/L of less than 0.3 or if on steepening the final edge angle (to avoid using shelf) the delta L/L does not vary a lot..........In such cases, should I steepen the final edge angle to avoid using shelves. Do you have a feeling on as to if this would influence cable windability or not? I would appreciate your thoughts on the same.

As for your question about the final edge angle, it is OK to steepen it as long as the delta-L/L stays reasonable. In fact, steepening the angle may have two beneficial effects:

(1) It should decrease the curvature at the nose of the free edge, near ruling 50, which tends to be a problem, aggravated by that focusing effect you have noticed. The focusing effect is in turn a consequence of the fact that the rulings are constructed by linear extrapolation from the base curve, which is in turn a product the general rule in BEND that numerical analysis is mostly done to within first order accuracy only. This is not a bad way to go with modern compute-power. To get more accuracy, one just increases the number of mesh points and lets the computer do more work. It doesn't care. It even likes to iterate, unlike the human computors of old who often preferred higher-order schemes in order to avoid boring iteration. However, the width of the cable is not a mesh width that one can decrease. This has always bothered me. The base curve itself is computed by a higher-order scheme, second-order in fact, for which 50 mesh points is plenty, but the linear extrapolation down to the free edge --- well, that has always bothered me. My nervousness could have been cut in half by putting the base curve along the midline, as I did in program LEAD, but with BEND that would have increased my nervousness about the outer- diameter-hugging algorithm. Anyway, go ahead and steepen the final edge angle, some, and see what happens.

(2) As for windability, steepening the final edge angle may cause the cable, under longitudinal tension, to slip laterally away from the non- slip configuration prescribed by the rectifying developable, but it, if anything, causes it to slip in the SAFE direction, namely towards the mandrel, towards the axis of the cylinder where there is plenty of support, at least during curing, so maybe you shouldn't worry about it. Without a shelf? Well, go ahead anyway. Give it a (small) try. Come to think of it, one could compute the radial force induced by the z-direction tension on a steeper final edge angle. Should we compute it?

Should I get bothered by the Wrinkle warning?

Sunil,

I have now had time to look at the "wrinkle" warning. It comes from the subroutine CONCAV which is fed a strip --- rectifying, guiding, side or whatever --- and which then looks at the intersections of the corres- ponding rulings with the mandrel, the inner diameter. The z-coordinates of those intersections are supposed to be monotonically increasing. If not, a "wrinkle" is announced as a "POSSIBLE" problem and the amount of back-step is printed out. In the case of your "Dipole model-2 ..." that back-step was never more than one-third of a micro- meter. That's MICRO, not milli. Plenty small. And besides it was at rulings number 49 and 50 where the z-coordinates are almost constant anyway instead of increasing. Therefore the "POSSIBLE" problem is not a real problem at all.

So we got by that one OK. Now I've got to look at your curvature problem. That one will be harder to figure out. I don't want to just declare "numerical noise" as a cop-out.

Dear Joe,

Thanks for your comments on wrinkle warning. That makes me feel much better now. The other thing I was curious about is how do you calculate the incremental delta L over L. I can understand the calculation for overall delta L-over-L (length of outer curve-length of inner curve/length of inner curve; is that right?). How do you do it for the incremental case? Do you take the corresponding arc lengths between the ruling points? Please advice.

Thanks,
Sunil

Hi Joe,

I am really surprised to see that now I am getting a radius of curvature at point 48 as 1.16 mm (using CAD package) as opposed to 0.626 mm output be BEND. I am surprised because I do not think that I can go order of magnitude wrong in simple division.

Only thing I can think of is that yesterday my program file (which reads data into IDEAS) was not working completely to the point of building surfaces and I could not figure out why. Today that program file is working again and I do not understand how come. I also need to look into it more carefully.

Sincerely,
Sunil

Sunil,

Just a note before the long weekend:

Your file reader (which reads data into IDEAS) was reading, I think you told me, from the BEND ".ins" output. In this file the inner curve is NOT the same one used internally by BEND in computing that small curve radius. Internally the curve is the free edge of the guiding strip, the non-base-curve edge, whereas in the ".ins" file the curve used is the intersection of the entire face containing the guiding strip with the mandrel, the inner radius; so we could expect that there would be some difference in the values, though it seems to me that the focusing effect should have produced an even smaller radius instead of a bigger one.

Incidentally the reason the guiding strip is used in both of these computations instead of the values that would have been obtained from the corner files is that you have placed the guiding strip on the inside of all of the cables instead of in the middle someplace, so that is the way I coded it though Jeff says that that is an incongruity. I am going to fix it up some day.

To: Sunil, INTERNET:syadav@fnal.gov
Date: 7/6/99, 12:19 AM
Re: More on that small radius of curvature

As I mentioned before, the radius of curvature you got from your CAD system, which it got from the .ins file, was from the curve of inter- section of the inside lateral face of the group with the mandrel, whereas the unpleasantly small radius, .628 mm, put out by BEND under "ANALYSIS" was from the free edge of the guiding strip.

What was bothering me, lately, is the fact that that .628 mm was not the same as the 1.4642 mm claimed also for the 48th ruling in the previous panel but upon further investigation I have discovered that it wasn't quite the unmodified free edge but rather a modified free edge that the .628 mm referred to, modified slightly (near statement label 6000 in subroutine GROUP) to make it match better with the in- side of the inside cable in the group.

All of this happens because for ANALYSIS if there are no cables on one side of the guiding strip, no cables on which to base the .ins file, then I based the .ins file on the (slightly modified) guiding strip. Not a good idea Jeff said, so I started to fix that but then Jeff went on to other things so I never finished fixing it.

Confusing? I'll get in touch with you tomorrow, Tuesday, to talk it over.

Meanwhile trust your CAD-given radii, maybe with BLUNT=.65, instead of .3, to relieve the curve radii a little because you have some dl/l margin.

As for dl/l, it is computed in two different ways in two different parts of the code:

1. For the guiding strip, since we have both a before and an after configuration, I just took the so-called "normal strain" of the free edge as defined in, for example, page 69 of "Mechanics of Materials", 2^nd Ed., by R. C. Hibbeler, took it in the longitudinal direction.
2. For the sides of corner files, where we have no before-and-after to compare, I measured the angular spread of the rulings from the base curve.

In both cases the base curve is held fixed and the free edge is required to absorb all of the dl/l. This gives an exaggeration of the dl/l by a factor of two over what I would have gotten if I had taken the midline of the cable as the neutral line instead of the base curve.

Too brief. I'll write this out at greater length later.

Hi Joe,

I was just curious at to what determines the spacing of the 50 points on the base curve....is it based on something like the rate of change of curvature along the base curve, with more points where there is more drastic change in curvature????

Sunil,

The 50 rulings of the guiding strip intersect the base curve at intervals which are equal, not in arc length but rather in the angle which the tangent to the base curve makes with the z-axis. This has the advantage that the density of these points is higher where the base curve is curving faster and more detail is needed.

[Questions about the parameters NARROW and STIFF in BEND]

Sunil,

The parameter NARROW affects only the behavior of BLUNT. Its value is asked for only if BLUNT is not equal to zero. Therefore, because BLUNT operates only on the free edge of the guiding strip (the other edge is the base curve edge), so does NARROW.

If BLUNT > 0 the free edge is spread out to make the guiding strip more "blunt". If BLUNT < 0 the free edge is pulled back and in.

If NARROW > 0 the spreading or in-pulling effect of BLUNT is restrained near the initial end, the z=0 end of the free edge and encouraged up near the nose. If NARROW < 0, the reverse is the case: The effect of BLUNT is emphasized near the initial end and de-emphasized near the nose.

NARROW was introduced into the program in order to give the user control over the part of the free edge BLUNT should be allowed to operate on.

Now as for STIFF, or rather STIFF() because it is an array of two parameters, these parameters operate on the base curve, not on the free edge as do BLUNT and NARROW. When they are zero, their normal values, the easy-way rigidity of the guiding strip is constant. It is based on a base curve which is the classical one-dimensional elastica which has only one rigidity and its configuration is the equilibrium configuration of that elastica. That configuration will be different if the rigidity of the elastica is allowed to vary along its length instead of being constant. The two STIFF() parameters have that effect. Positive values of STIFF(1) decrease the rigidity of the elastica near its initial end, low z, making it more flimsy so it flexes more easily and has more curvature there; and increase the rigidity of the elastica near the nose, making it "stiffer" there and so decreasing its curvature at the nose. Negative values of STIFF(1) have the reverse effect.

STIFF(2) has the same kind of effect except that it modifies the rigidity of the elastica in its middle region, about half way along its length. Positive values of STIFF(2) make the elastica more flimsy along its middle, increasing its curvature there. Negative values decrease its curvature there.

So: Whereas BLUNT and NARROW move the free edge, STIFF(1) and STIFF(2) (like "perturb") move the base curve.

     "Please enter, on one line, the numbers of only those
      options that you want to change from the default.
 
		  Option                 Default     Alternate
       -------------------------------  ---------   -----------
       1  Input from an external file:  No          Yes
       2  Output to a journal file:     No          Yes
       3  Units:                        British     SI
       4  Perturb group:                No          Yes
       5  Cable cross section:          Unchanged   Change
       6  Steepen rulings:              No          Yes
       7  Automatic:                    Yes         No
       8  Vary the base curve:          No          Yes
       9  Verbose terminal I/O:         No          Yes
      10  Debug:                        No          Yes
      11  Help:                         No          Yes",
 

	 "By what ratio do you want to contract
	  the rulings towards perpendicularity?"