1. Summary of the end part design process using BEND R. Bossert February 14, 2013

2. The methods used when designing with the BEND program are based on the assumption that the “bend the hard way” is the primary contributor to the stress in the cable, and therefore the focus is to eliminate strain in this direction. The steps used in the process of creating end part configuration with BEND follow. BEND, an interactive end part design program, was designed by Joe Cook and Jeff Brandt, initially to create ends for the SSC dipole, and has been used for all end design at Fermilab since then. The mathematics and programming were done by Joe Cook. This presentation includes the various steps used when designing an end with BEND. The underlying mathematics is available from two documents written by Joe Cook: “An Application of Differential Geometry to SSC Magnet End Winding”, J. Cook, April 1990. “Strain Energy Minimization in SSC Magnet Winding”, J. Cook, September 24, 1990. Documents describing the design procedure in detail are : “Coil End Part Design Procedure”, J. Brandt and A. Simmons, September 1998. “Coil Design for the SSC Collider Dipole Magnet, J. Brandt, July 1991. “Coil End Design for the LHC Dipole Magnet”, J. Brandt, May, 1996. “Coil End Design for the LHC IR Quadrupole Magnet” J. Brandt and A. Simmons, November 8, 1996. Introduction

3. Start Point End Point Step 1. The “Base Curve” is created. The base curve is described as an elastica, a curve into which the central line of a thin elastic rod of circular cross section will be bent when forces and couples are applied to its ends only. The curve then satisfies the condition that it have the minimum, possible strain energy, subject to certain constraints. It is, in our case, confined to a cylindrical surface and must satisfy certain initial and final conditions. These conditions are: The Base Curve a.The line must begin at a point on the surface of the cylinder and be pointing in the direction of the positive Z axis. This point is determined by the magnet cross section. b.The line must end at a point on the top center of the cylinder (at a value of x = 0) and be pointing in the direction of the negative x axis. This point is determined by magnetic considerations.

4. The Developable Surface Step 2. The “Developable Surface” is created. a.A set of closely spaced points (P 1, P 2, P 3... ) is placed on the base curve (50 points “per half curve” is the default). Points are not evenly spaced on the curve, but are spaced proportionally to the amount of curvature, so that the areas with the most curvature will have the most densely packed vectors. b.Vectors (V 1, V 2, V 3... ) are drawn from the points in such a way that they sweep out a surface called the “rectifying developable” of the curve. It is by definition perpendicular to the direction of curvature of the curve at every point. The cable is modeled by an infinitely thin strip in this surface along the curve. c.The surface is trimmed at the appropriate cable width (L). The trimmed edge is defined as the “free edge” of the strip.

5. Base Curve Placement The base curve is always placed at the outer cylindrical surface of the layer. This is done for two reasons: 1.This location facilitates the use of “shelves”, which were the preferred method of taking up leftover radial space in NbTi magnets. 2.When the outer surface is used as the base curve, rulings tend to be more dense at the free edge near the nose, where density of rulings is most needed. This could be achieved in other ways, but this is the most straightforward way. SSC end with shelves LHC Dipole end part with shelf

6. Initial prompts So: The program prompts the designer for the values shown below: The “starting angle” is the angle from vertical to the point at which the outer edge of the strip intersects the outer surface. The “initial edge angle” is the angle from vertical of the initial edge of the strip, which typically does not pass through the bore centerline. The starting angle and initial edge angle are rarely the same. BEND uses the starting angle to create the developable surface. r1: Inside radius of layer r2: Outside radius of layer b1: Starting angle b2: Initial edge angle a: Length from origin to end of base curve (determined by magnetic considerations).

7. Adjusting the Developable Surface Cable Rectifying Developable Vector The rectifying developable vector, at the point of transition between the curve and straight section, must intersect the bore centerline if it is to satisfy the initial conditions we have specified. The turns generally are not positioned radially with respect to the bore. It is therefore impossible to create a true developable surface which meets all of our initial conditions. We can, however, make something very close. The program is adjusted for turns which are not radial. BEND starts by aligning the initial edge of the strip with the line which defines the starting angle, forcing the strip to lie radially. Then the strip is rotated about the point on the outer surface until it is aligned with the actual cable (given by the cross section). This introduced twist is linearly distributed along the length of the strip. The result is: 1.The strain energy due to the “bend the easy way” changes. 2.The strain energy due to the twist changes. 3.A strain energy due to the “bend the hard way” is introduced (i.e., the surface has diverged from the rectifying developable.) This process results in a strip with the least possible strain energy, given the initial conditions.

8. The Guiding Strip This strip represents just one surface of one turn within the current block. All the other turns in the current block are stacked onto this strip, and this is done automatically in BEND. As they are stacked, they become farther and farther removed from the rectifying developable, and consequently have higher and higher strain. This strip is defined as the “guiding strip” for the current block being designed. “guiding strip” BEND allows the designer to place the guiding strip at any position within the current block, and will output the cable strain for the block with that position so the guiding strip position can be chosen. In practice, the guiding strip is almost always very close to the inside surface of the current block. Now the entire current block has then been created, and a set of points which describe the surfaces are output to the designer.

9. Cable Shape Changes Keystone angle change (KEY1, KEY2) Mid-thickness change (FAT1, FAT2) The cable, as it is wound around the ends, is subjected to stresses and consequently deforms. The BEND program allows the designer to input local changes in the cross section of the cables within the end. These changes are typically based on past experience and measurements of ends of previously constructed coils. Two types of cable shape changes can be made, keystone angle and cable midthickness. Both of these changes can be input to the program at two places; the nose of the turn (KEY2 and FAT2), and at a position midway between the end of the straight section and the nose (KEY1 and FAT1). These changes are applied in such a way that they smoothly change from each position to the other.

10. Other “Tweaks” There are three other features of BEND that allow the designer to modify the curve shape during the current block has been designed. Any of these methods will change the distribution of strain between the bend the hard way, bend the easy way and twist. The program will then output the strain energy at various positions within the current block before and after the feature is implemented. 1. Shift – This feature retains the base curve while allowing the designer to change the way the twist is distributed along the strip (more twist toward the nose or more toward the initial edge), with the goal of improving strain from bend the hard way. 3. Perturb – This feature allows the designer to change the shape of the base curve to fix geometry problems in the parts between groups. It is not generally recommended and in fact has only been used once, on an LHC Dipole “filler” which had a long, very thin sweeping cross section. The inside and outside surfaces crossed near the free edge, and it was necessary to change the base curve of one group to resolve the problem. 2. Blunt – This feature retains the base curve while allowing the designer to make the free edge of the group more blunt near the nose, with the goal of improving the tight radii of curvature. This feature has a modifier called Narrow with positive values concentrating the effect of Blunt toward the nose and negative values concentrating the effect toward the initial edge.