Tapered Headtube Secrets: how to slay the tapered headtube

Daniel_Y · February 24, 2023, 8:52pm

Mitering to a tapered headtube can be intimidating. This is the way @Neuhaus_Metalworks and I tackle the problem. It even the very challenging concave Bike Fab Supply headtube.

The Tapered Mitering Method:

Determine the effective angle Theta* (pronounced theta star) that the downtube intersects the headtube
Use the closest holesaw to the midpoint of where your DT intersects your taper. If in between sizes, use the smaller size.

How does this magic work?

The top and bottom edge of the miter notch does not care about the diameter of your hole saw. The ears of the notch care the most. The diameter of the ears are at the midpoint of where your DT intersects your taper.

A Practical Example:

This is an example uses a paragon IS41/52 headtube on a gravel bike with a 35mm downtube diameter.

The midpoint diameter is 54.7mm, making the closest holesaw 2.125in (54mm). Theta* is 60.4° (I will show you how to determine this later)

What is the difference between a theoretically perfect miter and this sketchy holesaw trick?:

The error of this method is less than .1mm. More importantly, this miter is symmetric, which is hard to achieve with hand filing and a template.

Three Ways to Determine Theta*:

CAD Method:

Update 3/2/23: BikeCAD v19.5 can now call out the effective angle! : Tapered head tubes (mitering to) | www.bikecad.ca

This is the easiest. Just draw a reference sketch that connects the two points where the downtube intersects the HT. Easy huh? Maybe another reason to start that Fusion360 2D Drawing Tutorial that everyone is avoiding

Small Angle Approximation Method:

This is the trick Nick @Neuhaus_Metalworks showed me. You physically measure the diameter of the headtube where the downtube intersects the headtube (W1 and W2). Using Theta from BikeCAD or your hand drawings: theta* = theta - (W2-W1)/2

This method works because for small angles, sin(theta) ~= theta in radians There is some more trig invovled, but I don’t want to do it.

In this example, Theta* ~= 60.65°

A More Accurate Derivation:

If you came this far, you probably already decided to use a 44mm headtube or learn Fusion360 2D drawings.

Using W1, W2, Theta, and D (your DT diameter), you can approximate Theta*:

Theta* = Theta - atan( ((W2-W1) * sin(Theta)) / (2*D) )

If you do the math, you end up with Theta* = 60.37°

Conclusions:

Method:	Theta*	Requires:
Fusion Method	60.4°	Fusion360 knowledge and HT CAD
Small Angle Aprox	60.65°	Measure W1 and W2
More Accurate Equation	60.37°	Measure W1, W2 and your DT diameter

If you found this useful and tried it on your frame, please post your results! That will resurface this thread and let more people see it.

Happy mitering!

One final reminder: do not confuse radians and degrees! Most online calculators and spreadsheets default to radians.

anon91558591 · February 24, 2023, 10:13pm

Excel defaulting to radians drives me nuts!

This is the same technique I use fwiw. After I try as hard as I can to get the customer to accept straight 44…

-Walt

liberationfab · February 25, 2023, 1:51am

Just to make sure I’m thinking about this the right way -

For method 1: Hole saw diameter of 46.3mm (who sells 46.3mm hole saws?) and angle of θ* = 56.1º
For method 2: Angle of θ* = 58.1º - (47.08mm - 44.71mm)/2 = 56.9º
For method 3: Angle of θ* = 58.1º - atan(((47.08mm - 44.71mm) * sin(58.1º)) / (2 * 28.6mm)) = 56.1º

Meriwether · February 25, 2023, 2:01am

Thanks for this @Daniel_Y !!
It’s so weird you posted this today. I have a PMW IS41/52 on current build but per usual i didn’t plan well and don’t have the right hole saw so i was going to file to fit using a bikecad miter template.

The way i did this for EC34/44 was take the bikecad template and get it close with a hole saw to get the majority of the cut then file to fit. Then to never do that again i’d figure out what saw/angle combo was needed by putting the tube in the fixture and figuring out what hole size and angle you’d need to use to make it work in a one cut operation. I haven’t figured out that “equation” for the IS head tubes so I’m gonna give these methods a try and will post results next week!

Daniel_Y · February 25, 2023, 2:15am

LGTM. It never cease to amaze me, but it looks like they make a 1 13/16 (46mm) holesaw

The good news is, this method is not very sensitive to the diameter of the saw. The angle is much more important. As long as the hole saw is greater than the top (W1) and less than the midpoint diameter, you will be pretty close.

You could always undershoot the miter diameter (closer to W1) and file the ears to fit closer, but realistically you will probably already be close enough.

liberationfab · February 25, 2023, 2:27am

Whoa, I was totally joking but that lines up perfectly! I modeled the difference in Fusion between the tapered cut and the straight cut, it’s pretty wild how minuscule it is. Thanks for getting this guide up, it’s been helpful to think about!

Daniel_Y · February 25, 2023, 2:39am

Batcountry · February 25, 2023, 4:22am

Wow, you are really good at breaking things down. That makes total sense and now I will probably even use tapered headtubes! Thanks!!

Neuhaus_Metalworks · February 25, 2023, 6:30am

There is no reason not to now.

TDotBikes · February 25, 2023, 12:44pm

Awesome! Thanks so much for posting this. I have been avoiding tapered head tubes, I’ll give it a try now.

sikocycles · February 25, 2023, 1:31pm

This is great. I always thought tapers Headtubes were black magic but kind of still is.
I will give it a try with the next gravel bike build

Mark · February 25, 2023, 3:31pm

46mm saws all day for PMW tapered HT’s.

AJAX · February 25, 2023, 3:42pm

Does anyone know if Rattlecad will calculate the Theta angle?

manzanitacycles · February 25, 2023, 4:10pm

So many ways to skin the cat!

I make the miter angle calculation using values from BikeCAD:

DT diameter at HT
Miter to deepest point of BB miter measured along top of down tube
Miter to deepest point of BB miter measured along bottom of down tube
Down tube front miter angle

The top tube miter angle is a little trickier to get.

Here’s a copy of the spreadsheet I use to make the calculations. I don’t guarantee the math is right. If someone feels like double checking the math, that’d be great.

[Edit]: Math is right but method isn’t as accurate. Use Daniel’s spreadsheet below.

I asked @BikeCAD to include these calculations in a future version of BikeCAD.

Daniel_Y · February 25, 2023, 4:30pm

Modified Angle Tapered Headtube Calculator

I created a Theta* calculator this morning for those who don’t believe in radians

View Link: Tapered Headtube Calculator
Copy Link: Copy Tapered Headtube Calculator

How To Use:

measure your variables in the real world
enter them into the calculator
look at the calculated values (bottom)

image1512×761 142 KB

I was able to implement a cool feature: Closest Holesaw. This allows you to find the next smallest fractional inch hole saw (1/16in increments) than the midpoint diameter (W3)

Thanks to @liberationfab example I was able to validate it with their parameters, and it checks out!

Carl_Snarl · February 25, 2023, 4:55pm

I’m attaching a document written by Hinmaton of Loco Machine who used to make tapered head tubes and later sold all his designs to @mark_pmw .
While i don’t follow it to the letter, it explains a couple of good methods.
tapered cope instructions.pdf (1.5 MB)

Daniel_Y · February 25, 2023, 5:00pm

Double checked your DT math method, and it checks out with my CAD.

CAD result: 29.62º
Calculated: 29.69º

Pretty accurate!

My only concern with this BikeCAD method is that it double compounds the error of your HT diameters (the DT and TT lengths are calculated based on the HT shape in BikeCAD). I have found bikeCAD lengths to be accurate to +/-.1mm, which normally is not a problem. But for angles, the atan function is pretty steep between 30º (.5 rad), meaning the output is very sensitive to input error (modeling error of a headtube).

The advantage of feeding in real-world HT diameter measurements into an equation is that you don’t compound the errors from your BikeCAD HT model.

manzanitacycles · February 25, 2023, 8:34pm

Thanks for the math check @Daniel_Y!

It looks like the @Neuhaus_Metalworks method and your spreadsheet is more accurate. Time to switch processes.

HPDX · February 26, 2023, 10:02pm

This is rad. Any thoughts on whether/how to alter this method for shaped (bi-oval/teardrop) down tubes. Input DT cross section height for D?

Doing this on the daily, but with a guess based on historical data, not real math…

Daniel_Y · February 27, 2023, 4:11am

Great question. With the caveat that we don’t use non-round downtubes: I am confident this method handles oval and tear-dropped shapes the same way. Here is why:

D actually represents the cross-section height (so I guess H would have been a more accurate variable name). The width of the DT does not affect the Theta* measurement at all.
The hole saw diameter is a function of your headtube widths only, so it is independent of your downtube’s size and shape.

The shape of your downtube/top tube does not have a big impact on the accuracy of the method.