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# The Mathematicians Frame Jig.

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# The Mathematicians Frame Jig.

05-23-12, 05:56 AM
#1
Mark Kelly
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The Mathematicians Frame Jig.

Here’s an illustration of how this jig will work: If you draw a line from the rear dropouts which meets the seat tube at 90 degrees and then extend this all the way to the steering axis, you get this:

The angle of the headtube to the horizontal axis (hereinafter called the jig axis) is 90 degrees plus the difference between the HTA and STA. This varies only a few degrees either side of 90 for all practical frame designs. This is the only angle that needs to be adjusted in this jig design, everything else is orthogonal. To do this accurately we need a couple of precision XY translation stages. Normally these are very expensive but a cheap substitute is easily found in cross slide vices like these:

In a later post I’ll show you how to do the calculations using RattleCAD But for now lets skip straight to construction.
I used two pieces of steel square tube (aka RHS) welded into an L shape to hold three mounting plates:

The mounting plates are rectangular steel plates with holes for the adjustment screws. I got two of the plates welded on but that isn’t strictly necessary, the third is bolted in place which works well. A second set of plates bolts to the first so that the plates can be adjusted to be coplanar, on which more later. Here’s the jig with the second plates mounted:

The second plates are mounted on studs which bolt through the plates mounted to the frame, allowing the orientation of each plate to be adjusted. There are four studs for the hevy plates to which the vices will be mounted:

But only three for the small plate for the tail end:

The extra holes in the second set of plates are to mount the Cross slide vices and a short piece of aluminium slotted extrusion similar to that sold by 80 / 20:

Hopefully you are now getting the picture: basically the vice on the left will hold the head tube mechanism, the one on the middle will hold the seat tube and bottom bracket and the ally extrusion will hold the dummy axle.

Costs to date: \$70

Bill of materials to date:

2 x Cross slide vice \$60
Assorted nuts and bolts \$10
300mm 80 x 20 extrusion*
1 metre 40 x 40 RHS*
300mm 150 x 12 bar*
200mm 100 x 12 bar*
400mm 80 x 6 bar*

Items marked with an asterisk came from the offcuts pile. If you have to purchase these they will add to the cost. The aluminium extrusion costs around \$20 per metre so the costs aren't huge.

Next post I’ll start detailing how these various bits get held in place.

Last edited by Mark Kelly; 05-23-12 at 08:05 PM. Reason: Added costs and BOM
05-23-12, 06:08 AM
#2
Mark Kelly
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Up five minutes and a correction already.

Well, more a clarification.

Part of the reason this jig was cheap is that I built it mostly out of scrap and offcuts. It helps if you have access to an offcuts rack like this:

About half of what you can see is stainless.

BTW yes, the barrels up the top are full.
05-23-12, 10:21 AM
#3
calstar
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Looking forward to seeing the end product, thanks for posting this.

Brian
05-23-12, 10:31 AM
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Andrew R Stewart
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Mark- Once again we see another method of setting up a frame for construction. Your approach is interesting, I'll check out your posts as they pop up. As many have said before; don't trust the jig to hold alignment (HT/ST parallel, rear wheel in plane). That's the duty of torch skill and a true surface reference (flat plate). Andy.
05-23-12, 11:56 AM
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I also made a jig out of scrap, but in my case I used something like 3' x 3' x 3/4" polished aluminum plate, so I had a nice flat surface to work off. A drill press, dowl holes, and some channels blocked up & bolted to the plate worked pretty well. I splurged on a machinist protractor for checking angles & squareness.
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05-25-12, 05:24 AM
#6
Mark Kelly
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2me Tranche

I promised I’d update with details of how the various fixtures work. First up is the seat tube, but there’s a small diversion before we get there: the seat tube and BB are in line so the fixing plate for the BB fixture goes in first. The plate is a piece of 50 x 12 flat bar but like all such it shows some distortion from the rolling processes which formed it. The simple solution for this is a bastard file and a bit of elbow grease.

Once that’s in we can sort the details for the seat tube fixture. I used a tube clamp from Paragon Machine Works as the tube holder. If it were placed directly against the BB fixing plate there would be very little working room, so I decided to space it further away. The spacer needs to be dead square and of consistent thickness. A cheap source of material fabricated to good precision is key steel, available from any industrial supply. Because I wanted to retain the ability to use a larger tube clamp and they are 2¼” square rather than 2” square, I used a piece of 5/8” x ¾” key steel. I apologise for mixing metric and imperial units but bike tubes are all imperial anyway so it was inevitable.

Spacing the tube clamp by this distance makes it impossible to clamp in the vice so I replaced the jaws with two longer pieces of steel. I drilled holes in these for the vice mounts and the tube clamp plus I added a hold down:

The idea of the hold down is to press the various bits together so that they stay aligned:

The problem with this is that we are now asking the clamping action of the vice to perform three jobs at once: hold the BB fixture base, hold the tube clamps and keep the sides parallel. Doesn’t work. When you try to clamp the tube into the clamp, the distance between the centre of pressure and the vice jaws causes the holding plates to splay out so the BB fixture and / or the spacer fall out and hit you in the foot. It doesn't take long to get sick of this.

To obviate this I went back and arranged a method of holding the BB fixture to the bed of the vice. This is basically a piece of steel with two M6 threaded holes spaced 24mm apart which slips under the travel rod of the vice. The BB fixture plate bolts down with two M6 bolts:

I also arranged a clamp at the outer end of the tube clamp to equalise the force from the vice. To get the force right, set everything up so that it’s square before the tube goes in with enough torque on the hold down bolt to fix the spacer in place. Now clamp the tube in place which will cause the spacer to becomee loose (so hold it with one finger). Now do up the end clamp until the spacer is held by the mechanism. Voila, one seat tube clamp.

The sum of these various dimension fixes the bicycle centre line to jig base datum.

Height of vice to bed: . . . . 89.7mm
12mm plate after squaring : 11.65mm
3/4" spacer: . . . . . . . . . . 9.05mm
Tube clamp to centre line . .25.4mm

Total: . . . . . . . . . . . . . . .145.8mm

Cost to date** . . . . . . \$96

Bill of materials:

Tube clamp ex Paragon: . \$18***

Nuts and bolts: . . . . . . . \$8

50mm of ¾ x 5/8 key steel*

400mm 12 x 50 flat bar*

150mm 10 x 65 flat bar*

* as before these came from offcuts /scrap. Key steel runs about \$12 per 300mm in these sizes.

** includes previous total

*** add extra clamps for each size of seat tube you will use.

Next post: head tube fixture including DIY goniometer (go on, look it up)

Last edited by Mark Kelly; 05-25-12 at 08:00 AM. Reason: Correct dimensions
05-25-12, 11:32 AM
#7
MassiveD
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I made my first jig that way. It was basically a motorcycle jig with the set-up such that the head tube was vertical, this because there isn't drop on motorcycle frames. I set it up on parallel rails like a lathe bed, which is actually a pretty good donor part if you find one for sale. I mean it was a motorcycle frame jig, that is just how they are made. I later did something similar on a milling machine table. Since I design in Cad. All one has to do there to get the numbers is rotate the frame, say 17 degrees, no math required.

Pic was off the internet, and gave me the idea.
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05-25-12, 12:09 PM
#8
unterhausen
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Originally Posted by Mark Kelly
Next post: head tube fixture including DIY goniometer (go on, look it up)
I did know what a goniometer was, but I'm still not convinced I understand the difference between that and a protractor. I guess it's the pointer.
The goniometers I am familiar with have virtual centers (center of rotation is not on the device), but apparently that's not necessary.
05-25-12, 06:53 PM
#9
Andrew R Stewart
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Goniometers are used by "professional" bike fitters. Protractors are used by hobby builders. Vernier protractors are used by "professional" builders. Andy.
05-26-12, 01:48 AM
#10
MassiveD
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And master builders use venier protractors, but can't any longer see the marking, and rely on a lifetime of feel.
05-26-12, 06:48 AM
#11
Road Fan
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Originally Posted by MassiveD
And master builders use venier protractors, but can't any longer see the marking, and rely on a lifetime of feel.
Or stand back about 10 feet, take a quick look, and say "that's about right?"
05-26-12, 12:19 PM
#12
Andrew R Stewart
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My first dozen++ frames were aligned with only sighting along their HT and ST. When i got a surface plate and a dial indicator I could be cool and talk thousandths...Andy.
05-26-12, 07:14 PM
#13
MassiveD
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https://www.youtube.com/watch?v=TMA8X5pk2kI

Originally Posted by Andrew R Stewart
My first dozen++ frames were aligned with only sighting along their HT and ST. When i got a surface plate and a dial indicator I could be cool and talk thousandths...Andy.
Yeah, there is usually some two points that will be coplanar.
05-27-12, 05:24 AM
#14
Mark Kelly
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3me Tranche

The affordable Cyclus head tube reamer comes in two halves and is, naturally enough, made to clamp head tubes accurately. This seemed to me to offer the easiest way to make a head tube holder, as long as I can still use it as a reamer, which I’m going to need later. Here’s the reamer with the cutters removed and the halves separated:

That fine thread at the end is slightly unusual: 16mm x 1.5 (ISO fine). Fortunately that’s a standard size in the electrical industry so taps are easy to get. I bought one, drilled and tapped the end of a piece of 19mm square bar and fabricated a holding disc for the bottom of the head tube:

The assembly goes together like this, head tube included:

This is where I hit the limits of what I can do without a machine shop. I cannot get the two parts to align correctly. I think the 19mm bar width doesn't give enough shoulder to hold alignment properly with the 16mm thread.

This will be the first object of the group buy: a precisely machined head tube pivot holder.

So that you know what it has to do, let’s proceed with the dodgy one. As mentioned in the first post. The head tube angle is the only angle in the jig that needs to be adjustable. Precision goniometers can cost a bundle so I needed a way of fabricating a cheap imitation.
Take these three things:

1 degree = 0.01746 radians.

Sin(x) = x as x ->0

M12 thread pitch = 1.75mm .

Mix them together et voila, one DIY goniometer.

All we need is a pivoting holder and an M12 screw at a right angle, placed so the centre line of the screw is 200mm from the pivot. One complete turn will adjust the angle by 1.75 / 200 which is half a degree. Since it’s pretty easy to judge a quarter turn, this gives us a precision of 1/8th degree, which is good enough for government work. Another M12 screw on the other side completes the mechanism. I brazed nice big washers on the ends of the screws to make judging the number of turns easier.

We now need to align the centre of the head tube holder with the seat tube holder (145.8mm from datum as previous).
The 4” cross slide vice I used as a base is slightly taller than the 3” in which the seat tube clamp sits, 99.4mm. The base plate for the bracket is 11.9mm and the centre line of the 19mm bar is at 9.5mm so we need to add 25mm to get to 145.8.
I used a pair of square pieces of 12.5mm brass flat because the low friction of brass makes for a nice pivot. I tapped the lower one M12 and used it to lock the M12 thread used as the pivot on to the base plate. The top brass square rides between the lower one and the square bar.

The head tube holder / pivot assembly thus looks like this:

To adjust the angle, back off one of the screws, turn the other one the appropriate number of turns, then clamp with the first screw.

To make life easier the new design uses a shorter length of 25mm square bar and a precisely machined lower holder with a step sized to the head tube. This will give much better precision in placement. Since the goniometer mechanism doesn’t need the extra beef provided by the 25mm bar, I’m going to shorten this end and thread a “tail bar” into it. The cheapest way of doing this is a length of M12 threaded rod and a double length "connecting nut" or you could get a length of 16mm rod, tap an M12 thread into one end and use a shorter piece of M12 threaded rod as a joiner. I'll also try to incorporate some method of lateral clamping: the existing hold down seems OK but I'm not sure how it will hold up to the stresses applied during an actual build.

Cost to date: \$111

Bill of materials:

One Cyclus head tube reamer: I haven’t counted the cost of this into the build as you are going to need a head tube reamer anyway. They run about \$200.

Nuts and bolts: \$15

120mm of 12 x 100 flat bar*

500mm 20 x 20 RHS*

50mm of 25 x 12.5 flat bar (brass)*

*costs not included, see above re offcut pile.

There won’t be an update for a couple of weeks due to other commitments.

Last edited by Mark Kelly; 05-27-12 at 05:29 AM.
07-03-12, 02:49 AM
#15
Mark Kelly
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Back Again

Well a couple of weeks stretched into five due to illness but we’re back.

One of the details omitted previously was the flat bar which will hold the bottom bracket post. This was mostly because I’d lost it – the jig has been moved around a few times and somehow it got left behind. Having run out of the scrap I’d rescued, I needed a piece of steel with which to make a new one. I went down to my local metals supplier to buy some 50 x 12 bar but they didn’t have anything appropriate in less than 6 metre lengths so I bought a short piece of heavy channel and cut one side of it off.

I then filed this part flat and drilled and tapped four M12 holes. The two lower holes are alternate positions for the bottom bracket post, this extends the adjustability of the centre line to BB dimension – it can now be anything from 100mm to 250mm. To allow extra working room I decided to mount this plate 16mm behind the top plate. To get this space accurately I used two short pieces of 16mm key steel which can also be seen in the phot.

Because the two bolt holes are fairly close together I used M12 socket headed cap screws to hold the two plates together. I lined the two plates up edge on to a flat steel plate whilst bolting them together – I think I’ll loctite this part of the assembly together to make sure it keeps in alignment.

The other part that was omitted was the piece of aluminium T-Slot extrusion to which the dummy axles will be attached. This was cut to 250mm and one end notched to clear the fixings on the mounting plate. This allows adjustment for any chainstay length from 400mm to 500mm without moving the dummy axles on the plate, and 100mm beyond that if the axles are moved.

Before putting everything together, here’s how I lined up the mounting plates to make them co-planar. I have two laser line generators with magnetic bases bought for another project. I attached one to each mounting plate and aimed them at the far wall of the shed.

The plates were then adjusted using the four mounting bolts mentioned in the post above. You need to get three things right at once: the lateral displacement, the horizontal angle and the vertical angle. The last is the easiest as the two generated lines cross if the angles are wrong. Horizontal angle and lateral displacement are adjusted together and then double checked by turning the line generators around and aiming them at the other wall. One phot was taken with flash, the other without. You can see a slight degree of divergence in the line without flash – I hadn’t finished yet.

The last phot is the jig as it stands – sorry about the quality of the shot, but hopefully you can see where the remaining parts will go.

Cost to Date: \$133
Bill of materials:
300mm of 16mm Key Steel \$11.
200mm of 12 x 50 flat \$5 (plus almost half an angle grinder cutoff wheel).
Bolts \$6.

Next post we get down to the business end: the parts we need to get made to finish this thing.
07-04-12, 06:53 PM
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Omiak
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Does this jig only work for lugged construction?
07-04-12, 07:11 PM
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Mark Kelly
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I'm not sure what you mean - it's just a jig. You may choose your construction method.

For myself, I won't be using lugs, so I've tried to make sure there's enough room to get tacks in while the frame is on the jig.

BTW to get the jig parts to South Korea you'll have to find a volunteer to ship them to you.

Last edited by Mark Kelly; 07-04-12 at 07:17 PM.
07-05-12, 10:42 PM
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Originally Posted by Mark Kelly
I'm not sure what you mean - it's just a jig. You may choose your construction method.
I guess since I didn't see anything that looks like it could hold the top tube or the down tube in place I presumed that maybe these were to be held by the lugs while brazing. You'll have to pardon my ignorance I've only minimal framebuilding experience.
07-06-12, 04:59 AM
#19
Mark Kelly
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You are quite right, there is nothing on the jig itself which will hold the top tube or down tube in place, or the chainstays or seatstays for that matter.

As far as I can see they aren't needed if the tube mitres are correct. I'll be using magnetic holders to stop the tubes slipping out of place.
07-11-12, 05:07 AM
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Mark Kelly
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I decided I’d try to throw together a version of the parts we need to finish this thing so as to show you how it works.

First, the BB post.

I designed this for English threaded BB shells machined 0.5mm over nominal length, the BB shell inside diameter is 34.0mm and the length is 68.5mm. The setback of the BB post holder outlined above is 72.45mm from the datum line, with half the shell being 34.25mm we need the thicker section of the BB post to be 38.2mm long. The thinner section should be short of the actual BB shell length to allow for the end clamp, so I went for 100mm overall length as a nice round number. I squared off the end of the bar I was using, cut it to length and squared off the cut end. I then drilled the long axis of the bar and tapped each end M12. Having both the BB post and the holder tapped to M12 doesn’t work at all well so I redrilled the holder to 12mm bore.

The BB post itself needs to be about 33.9mm OD to ensure a slip fit. The design I came up with here involves cutting the corners off a piece of square bar to form an octagon with four long and four short sides.

If you start with 25mm square bar: the lines from the middle to the mid point of any long face of the octagon, up to the edge and then back to the middle form a right triangle, the hypotenuse of which is 33.9 / 2 = 16.95mm and the adjacent is 25 / 2 = 12.5 mm. The remaining side is found by Pythagoras to be 11.45 mm so the cut edge would be 1.05mm long making the short face of the octagon is 1.48mm. All you need to do is file off each corner of the 25mm square bar until the cut strip is 1.5mm wide and it will be a slip fit in the BB shell. Simple and surprisingly effective.

I started with 28.6mm square bar because that’s what was in the cutoff bin at the local steelyard, the calculation is the same with 14.3 substituted for 12.5, the short faces come out to 7.33 mm, I cut mine to 7.4mm. Way more filing involved but it worked out OK in the end.

The tricky bit was making sure all the filed faces terminated 38.2mm from the base of the post so that the shell fits snugly at the correct offset. In the end I was about .04mm off but that’s less than two thou so not bad for hand work.

The end clamp is a purging washer from Paragon with a short length of M12 allthread and a wingnut loctited on.

Encouraged by my success here I had another go at making the headtube pivot, figuring that the extra width of the 28.6 bar would give me more purchase to hold the Cyclus shaft square, and indeed it did. The only other modification was reducing one of the 12.5mm spacers to 7.75mm to accommodate the extra bar width.

Last, I had a bash at making a dummy axle from 20mm bar. I cut this to 137mm with squared ends – it seems common to oversize the dummies to allow for brazing movement. I then drilled the long axis to 7mm and redrilled a short (10mm) depth at each end to 10mm diameter. I cut two 15mm lengths of 10mm diameter tube and fixed them into these bores with Loctite bearing fix, then tapped to M8. I decided for this first cut not to try keyed axles, I thought that would be a bridge too far.

I brazed a couple of wingnuts onto 30mm lengths of M8 allthread as the holding nuts.

To hold the axle in alignment I made a couple of ally V blocks from 50mm lengths of 12 x 25 bar. The V is 14.1 mm deep so the 20mm bar sits with its axis at 25mm from the surface onto which it is clamped. I filed a small notch to mark the axle centre and two bigger areas to hold the clamping bars for the alloy plate to which it is clamped. The phot below shows this clamped to the offcut of the alloy plate.

Altogether this seems to have worked out quite well so I’m of a mind to try this out as is before commissioning the “real” versions hoping that I’ll at least learn enough to refine the design.

Next, I’ll run through the jig setup – expect maths (hence the name).

Final cost in this form: \$167 - pretty cheap for a fully adjustable frame jig.

Bill of materials: as previous plus

Nuts and bolts \$9

Steel bar \$25

Last edited by Mark Kelly; 07-11-12 at 05:35 AM.
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