Ball in a Cube : an optimised design

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dazz
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Ball in a Cube : an optimised design

Post by dazz » Thu 14 May , 2009 10:07 am

Introduction

A search across the Internet showed that the ball-in-the-cube (also known as a Turners Cube) has been made in many different forms and variations of the theme. My aim was to make one with a completely separated ball held captive within the cube. The version presented here is different to others found on the Internet because it aimed to optimise the design and manufacture to maximise the visual effect. The desired effect was to make something that looked impossible to make. Optimisation in this case included simplifying manufacture using an ordinary manual lathe, a basic form tool, and avoiding any obvious clues revealing how the cube was made.

The simplest way to produce a ball in the cube is to make a plunge cut straight into each face of the cube with a form tool. The radius of the form tool produces the ball. When done correctly, the ball is left free in the centre of the cube. This method of manufacture is less than optimal because the interior surfaces of the cube have unsightly sharp corners that would mark the ball over time and provide obvious clues revealing how it was made. The angle that the tool plunges into cube determines the shape of the surfaces on the cube interior. Finding a way of producing a relatively smooth surface on the interior of the cube was a key aim. Achieving this required some mathematical analysis using solid geometry and trigonometry. The results of this analysis are shown in Figure 1.
Attachments
Ball in cube 3D rendered.JPG
Figure 1 A CAD view of the Ball in a Cube
Ball in cube 3D rendered.JPG (22.55 KiB) Viewed 42654 times
Last edited by dazz on Sat 03 Apr , 2021 9:19 am, edited 2 times in total.
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Dazz

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dazz
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Post by dazz » Thu 14 May , 2009 10:24 am

Motion References

Motion will be defined using standard CNC conventions. The Z axis is aligned along the axis of the spindle. Motion along the X axis is achieved with the cross slide. All angles discussed here will be referenced relative to the Z axis of the lathe, which is the same as the angle read off the top-slide. This means, for example, a plunge cut straight into the face of a cube is at zero degrees along the Z axis.

Parameters

There are four parameters that can be varied. These can be selected to produce a ball and cube with the desired appearance and dimensions.

L = the length of a side of the cube,
r = radius of the ball,
c = clearance between the ball and the cube interior,
θ = angle of the plunge cut relative to the Z axis, as shown on the cross slide graduations.

The key parameters are shown in Figure 2.

It’s also useful to define other values

w = minimum width of the tool blank required to make the form tool
d = diameter of the entry hole at the surface of the cube

The minimum width of the form tool blank is given by the equation:

w = c + r(1-sin θ)

The diameter of each entry hole cut by the form tool at the face of the cube is given by:

d = (2r/sin θ - L).tan θ


The key dimensions chosen for the ball and cube shown in Figure 2 here were:

L= 33.5mm
r = 12.7
c = 2mm
Finished hole diameter bored to 16mm

These values were chosen based on the available metal stock and CAD drawings showing an aesthetically balanced result.
Attachments
090111 Ball in cube final 2D  key measurements.jpg
Figure 2 The key design parameters of the ball in a cube shown here.
090111 Ball in cube final 2D key measurements.jpg (52.01 KiB) Viewed 42653 times
Last edited by dazz on Wed 03 Dec , 2014 8:48 am, edited 2 times in total.
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Dazz

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dazz
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Post by dazz » Thu 14 May , 2009 10:33 am

Optimising the Plunge Angle

A plunge cut taken at zero degrees relative to the Z-axis generates cylinders in each face of the cube. A plunge cut taken at a greater angle generates cones within the cube. If the cones are the right size and shape, they will allow a ball to remain captured within the cube. A total of 6 cones are required, one cut into each face of the cube. If the plunge cuts are not deep enough, the ball will not be separated from the cube. The deeper the plunge cut, the smaller the ball. There is an optimal depth of plunge cut that will just cause the ball to separate with the maximum diameter.

The angle of the plunge produces the cones, and the form of the tool produces the ball. The smoothest surface is generated on the inside of the cube when the tool just reaches around the ball far enough to touch each of the 8 intersection points on the ball where 6 cones meet. A tangent to these points will provide the maximum angle of the plunge to reach them. Using geometric analysis it can be shown that the angle is given by the equation sin θ = 1 /√3, so θ = 35.26°. A plunge cut at an angle greater than this will not be able to reach the intersection points on the ball where the plunge cuts just meet. This angle also provides the smoothest interior surface of the cube. Setting the top-slide angle to 35.2° will result in close to the smallest entry hole into the cube with sufficient reach to get to the critical intersection points on the ball.

This is illustrated in Figure 3 which shows three cones cut into a cube with the form tool. For clarity, the remaining ball is not shown. The key feature is the intersection point where the six cones meet. At the point of intersection, the cones are tangent to the same plane which simply means there is no peak or trough at that point. At this plunge angle, the interior surfaces are the smoothest achievable with simple plunge cuts.
Attachments
cube1.jpg
Figure 3 A CAD view showing the swept path of the tool leaving behind a sphere.
cube1.jpg (212.41 KiB) Viewed 42652 times
Last edited by dazz on Sun 17 Dec , 2023 0:37 am, edited 4 times in total.
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Dazz

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dazz
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Post by dazz » Thu 14 May , 2009 10:35 am

The Ball Clearance

The next parameter to choose is the separation distance between the ball and the cube interior. A zero clearance would be impractical to make because the tip of the form tool would taper to zero width and the ball wouldn’t be free to move around inside the cube. Too big a clearance will allow the ball to fall out of the cube. Between these limits, the amount of clearance is largely a matter of personal choice.
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Dazz

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dazz
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Post by dazz » Thu 14 May , 2009 10:36 am

The Tool Blank Size

The minimum width of the form tool blank is dictated by the radius of the ball plus the clearance between the ball and cube. Once again, some trigonometry is required to calculate the minimum width of the form tool. The minimum width of the form tool is given by w = c + r(1-sin θ). For a 12.7mm radius and 2mm clearance, the tool must be at least 7.416mm wide. In this case, a 25.4mm diameter ball was formed with a tool made from a 5/16 inch (7.94mm) piece of HSS tool steel.
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Dazz

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dazz
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Post by dazz » Thu 14 May , 2009 10:47 am

Making the Form Tool

The making of the form tool was a mini-project in its own right. It took longer to grind and test the tool than to machine the ball-in-the-cube. My aim was to make a tool that would produce a good machined surface. A poor finish would provide clues on how the ball-in-the-cube was produced. The form tool would probably be easy to make on the tool grinder I don’t own. I only have a standard bench grinder, so all grinding was off-hand.

The geometry of the form tool defines the tool performance and accuracy of the finished item. A jig was set up using a radius gauge and a square as shown in Photo 1. Rare earth magnets were used to hold everything in place.

The finished tool is slender with a long overhang and long cutting edge, a perfect candidate for chatter. Grinding away metal for clearance significantly reduces tool rigidity. The weakness of the thin tip of the tool is mitigated by two factors. Firstly, rigidity the concave form tool rapidly increases away from the tip. Although the tip is thin, there is plenty of metal to provide support. Secondly, as the concave form approaches the tip, the depth of cut forming the ball diminishes to zero. The greatest depth of cut on the ball coincides with the heaviest section of the tool with the best support. Most of the cutting occurs close to the tool post. Modest front and back rack angle were used to maximise tool strength. The front rake was about 5 degrees, and the back rake about 15 degrees.

The tool was tested on an aluminium rod to check performance and determine cutting data as shown in Photo 2. Initial testing produced a poor surface finish. Tests showed that a ground finish left tool marks on the work. A polished cutting tool edge was found to be essential to obtain a good finish on the work. Initially I tried using fine grade wet and dry carborundum paper to polish the tool. The paper tended to round the edge of the tool reducing the clearance angles. This was revealed during testing when the tool caused spalling of the work piece surface and required a lot of pressure to cut the metal. A jewellers loupe was very helpful to evaluate the results of tool grinding and polishing. The problem was resolved by using buffer wheel compound applied to a piece of aluminium bar. The bar was held in the lathe and the concave cutting edge on the tool was held to the bar. This greatly improved both the cutting finish and the performance of the tool. Further experiments showed that chatter-free operation could be achieved at 200rpm and 1 thou-inch/rpm cutting rate.

Tool clearance was checked by plunging the tool through a test piece made of aluminium as shown in Photo 3. The hole drilled in the test piece was the same diameter and depth as the on finished cube. Ink was applied to the back of the tool. The tool was ground until the ink was no longer scraped off during the plunge cut. The plunge cuts were rehearsed on the test piece to check that tool extension from the tool post was just sufficient. This procedure gave confidence that the tool would fit into the small space available on the final work piece.

The checks on the tool back clearance showed that the tool needed to be mounted upside down because the range of motion available on the lathe wouldn’t allow the tool to be mounted in the conventional way. This had the added benefit of improving tool stability and reducing the risk of chatter. Tool alignment in the tool post and length of tool extension was carefully measured.
Attachments
090103 close up of tool test piece plan (514 x 484) small.jpg
Photo 3 Checking the tool back clearance.
090103 close up of tool test piece plan (514 x 484) small.jpg (55.25 KiB) Viewed 42644 times
090103 close up of tool test piece plan (514 x 484) small.jpg
Photo 2 A poor finish obtained by the early version of the tool.
090103 close up of tool test piece plan (514 x 484) small.jpg (5.9 KiB) Viewed 42644 times
IMG_0069 small.JPG
Photo 1 The jig klugged together to gauge the tool.
IMG_0069 small.JPG (51.86 KiB) Viewed 42644 times
Last edited by dazz on Wed 03 Dec , 2014 8:52 am, edited 1 time in total.
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Dazz

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dazz
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Post by dazz » Thu 14 May , 2009 11:19 am

Making the Cube

The obvious way to start was with square bar stock milled to a cube. I don’t have either square stock or a mill so I started with 2 inch round stock and used the lathe to machine each face to make a cube. Before each face was cut, Photo 4 shows how alignment was carefully checked on the lathe against the completed faces to ensure that the cube was square and accurately sized.

The four-jaw chuck on my lathe is in excellent condition and the jaws held the work squarely. The cube was big enough that it could be supported against the body of the chuck rather than passing through the centre. This wasn’t intentional but it made setting up a lot easier than for a smaller cube.

After the cube was made, shallow flat bottomed holes were bored into each face of the cube. The holes were bored to their finished diameter (Photo 5). The depth was bored to match the depth of the surface of the ball. The distance between the bottom of opposite holes exactly matched the diameter of the ball. These holes served an important role during the final stage of making the ball. They allowed a finger type DTI to be used to check location of the cube before each plunge cut. Importantly, the DTI could be used without moving the carriage, which was locked down for final plunge cuts. The centre of each hole was marked with a felt tipped pen. When the last trace of ink was removed, the plunge cut was completed to the correct depth. This provided a visual confirmation that the tool reached the proper depth.
Attachments
BnC17 checking cube small.jpg
Photo 5 The cube with each face bored before the final plunge cuts.
BnC17 checking cube small.jpg (59.92 KiB) Viewed 42639 times
BnC06 clocking 4th face of blank small.jpg
Photo 4 Checking alignment of the cube in the chuck
BnC06 clocking 4th face of blank small.jpg (64.57 KiB) Viewed 42639 times
Last edited by dazz on Thu 14 May , 2009 11:57 am, edited 1 time in total.
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Dazz

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dazz
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Post by dazz » Thu 14 May , 2009 11:24 am

Setting Up

The next problem was how to accurately locate the mounted tool relative to the cube to start the plunge cut. Correctly locating the tool relative to the cube is critical for success. If the tool doesn’t start in the right place, the ball won’t be round and might not be fully separated from the cube. By the time any misalignment becomes apparent, it is too late to apply corrections and the finished work would be ruined.

A method of correctly locating a custom made form tool at a 35° angle is not something I have seen in any engineering book. After some thought, I decided that any direct measurement off the tool would be difficult and prone to error. An indirect method was applied. A bar was turned to exactly the same diameter as the finished ball. The bar was used as a proxy for the finished ball.

The first step was to mount the bar in a chuck and use a DTI to establish a reference point from the Z axis as shown in Photo 6. The bar was then removed from the chuck and placed vertically against the form tool. The cross-slide was positioned to read the same value on the DTI. This locates the tool a known distance from the Z axis along the X axis.

The work piece was mounted in a four jaw chuck. The face of the body of the chuck was used to locate the work along the Z axis. The bar was placed vertically between the form tool and the front face of the cube as shown in Photo 7. Use of the bar alone places the tool at a known distance from the surface of the cube along the Z axis.

The tool is now located at a known reference point relative to the cube in both the X and Z axis. The final step is to offset the tool along the X axis to the starting position of the plunge cut as indicated in Figure 4.

The tool offset and plunge depth are calculated as follows:

tool offset = (r + L/2).tan θ

tool plunge depth = (r + L/2)/cos θ

The “tool offset” is the shift required to move the tool from a reference point to the point from where the plunge cut begins. The “tool plunge depth” is the distance the tool is moved to complete a plunge cut from the defined starting point.

The method described above only requires one DTI and the bar to accurately locate both the reference point and the starting point for the plunge cut. It also requires that the top slide is set to the correct angle, and that the top slide lead screw will accurately plunge to the correct depth. Tape was used to mark the start and finish locations. The numbers are reminders of the values set on the dials. The plunges were rehearsed and verified before attempting any metal cutting.
Attachments
BnC25 Setting the Z offset small.jpg
Photo 7 Setting the tool Z axis
BnC25 Setting the Z offset small.jpg (55.12 KiB) Viewed 42634 times
BnC21 setting X offset of tool small.jpg
Photo 6 Setting the tool X axis
BnC21 setting X offset of tool small.jpg (59.82 KiB) Viewed 42634 times
090423 Ball in cube final 2D tool offset.jpg
Figure 4 Tool offset measurements.
090423 Ball in cube final 2D tool offset.jpg (44.87 KiB) Viewed 42634 times
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Dazz

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Post by dazz » Thu 14 May , 2009 11:38 am

Making the Plunge Cuts

At this stage, only six plunge cuts were required to complete the work piece. The cube was held in a 4 jaw chuck carefully set up to precisely centre the cube. The plunge cut was made on five of the faces as shown in Photo 8. Success was only certain after the last cut was completed. Completing the plunge cuts only requires the top-slide to be moved. The carriage and cross slide were locked down.

The polished cutting edge of the form tool produced a mirror finish on the surface of the ball.
Attachments
BnC31 first plunge cut completed small.jpg
Photo 8 The first plunge cut completed.
BnC31 first plunge cut completed small.jpg (41.5 KiB) Viewed 42633 times
Last edited by dazz on Wed 03 Dec , 2014 9:00 am, edited 1 time in total.
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Dazz

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dazz
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Post by dazz » Thu 14 May , 2009 11:40 am

Ball Support

The ball must be fully supported prior to the last plunge cut. This is to allow the last plunge to be completed without the ball breaking away prematurely, and to prevent the ball rattling around inside the cube in the close vicinity of a sharp tool. Plumbers 50/50 solder or lead is an ideal support. Once four plunge cuts have been completed, lead was poured around the heated, partly completed ball (Photo 9).

At this point, disaster struck. It was assumed that the lead would not bond with the aluminium. This assumption was based on the previous experience of failed attempts to solder aluminium without specials solders and fluxes. Unfortunately the lead bonded with about 1/3 of the freshly made aluminium ball. At this point I seriously considered throwing the project into the bin and starting again. I managed to remove most of the lead with a solder sucker, and then scraped away the remainder. The ball was then smoothed with wet & dry paper before polishing with Brasso. The end result shown in Photo 10 was a smooth ball with no trace of machining marks to provide clues as to how it was made.

Removing the bonded lead added an unnecessary amount of work and is definitely something to be avoided. The lesson here is that the aluminium surface should have been passivated before pouring the lead. Simply washing off the cutting oil and leaving the aluminium to stand for a day would be sufficient. Boiling in water would also work by creating a thin layer of aluminium oxide.
Attachments
BnC39 Machined cube small.jpg
Photo 10 The machined ball in a cube.
BnC39 Machined cube small.jpg (36.59 KiB) Viewed 42631 times
BnC36 clamps and plugs on, lead poured small.jpg
Photo 9 The ball supported with lead before the final plunge cut.
BnC36 clamps and plugs on, lead poured small.jpg (36.31 KiB) Viewed 42631 times
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Dazz

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dazz
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Post by dazz » Thu 14 May , 2009 11:49 am

Anodising

The work looked good when finished, but aluminium tends to take on a gray weathered appearance after a while. I decided to have the ball in the cube anodised. This gives a robust long lasting finish. The initial response I got from the local anodizer was that it would be impossible because electrical contact was required with both the ball and the cube. The solution was to mount the ball and cube on an adaptor plate, also made of aluminium. The plate held the ball and the cube with a number of sharp fingers as shown in Photo 11. The anodizer then clamped onto the plate to achieve a good electrical connection.

The anodizer would not accept any steel content, so aluminium pop rivets were used to hold all the pieces together. The steel heads that normally remain in the rivets were punched out.

Some aluminium alloys do not anodise very well. I had a scrap piece of the bar anodised to test the finish. I elected to have a silver/gray finish rather than coloured. Any anodising imperfections would be less likely to show up than if a coloured dye was used.
Attachments
BnC44 ready for anodising small.jpg
Photo 11 The jig used to hold the work piece and provide good electrical contacts for anodizing.
BnC44 ready for anodising small.jpg (43.1 KiB) Viewed 42630 times
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Dazz

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dazz
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Viceroy Taper/Tracer attachment, Shop made cross slide tracer attachment, VSD.
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Post by dazz » Thu 14 May , 2009 11:50 am

The End Result

The final result is shown in Photo 12. The use of solid geometric analysis provided the theoretical tools to develop the design presented here. The CAD software helped confirm the proportions to achieve a balanced visual effect. The optimisation of the design made it possible achieve a finished hole size 2/3rds the diameter of the ball. The smooth shape of interior surfaces of the cube is the other key feature of this design. The methodology applied to development of the tooling and setup allowed the practical application of the theoretical design.

Through necessity, only basic tools were required to complete the ball in the cube. A particular point to note is that only one DTI was required to complete all the dimensional checks, location and alignment.

This deceptively simple project took a lot of time to complete. Most of the effort went into design, planning and preparation before any metal was cut on the workpiece. In hindsight, this effort eliminated a number of potential problems to allow for the successful completion of the ball-in-the-cube. The end result has intrigued all of those who have seen it. Even my wife, who usually shows glazed indifference to my workshop projects, displayed a glimmer of curiosity.
Attachments
BnC53 anodised cube on holder small.jpg
Photo 12 The finished ball in a cube.
BnC53 anodised cube on holder small.jpg (19.07 KiB) Viewed 42629 times
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Dazz

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Post by Denford Admin » Thu 14 May , 2009 12:19 pm

I like the fact you've mixed the modern with the old... Brilliant :!: 8)

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Re: Ball in a Cube : an optimised design

Post by jimsehr » Sun 30 Mar , 2014 16:40 pm

Here is my 2014 version 3/8 ball in 5/8 ball in 7/8 ball in 1 inch cube done on a Logan manual lathe.


Image


jimsehr

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