So as the title says, I want to build a 32H hub with a 24H rim, but when I ask Google about it...
this blog's article comes up first (→here).
Looks like I have to figure this out myself.
Maybe in the near future this article will be the top result.
For a normal hub without paired spoke phasing,
the holes in the left and right flanges are offset by 360°/number of holes per side degrees.
Whether on the hub or rim, if we skip every other hole on one side of the hub flange
or on the rim at 24H,
the hole positions look like the diagram above.
When you add the holes on the other side too, it looks like this.
If you draw two concentric circles of different radii, use the larger circle as the rim and the smaller as the hub,
mark the hole positions, and connect rim holes to hub holes with lines (the spokes),
you'd have a wheel design diagram.
But when trying to build a 32H hub with a 24H rim using the same number of spokes on each side,
phase misalignment occurs.
To explain this phase misalignment intuitively, I find it easier to use
an unwrapped linear diagram of the arc rather than a circle with "radian sense,"
so I'll explain using that below.
This is literally the diagram I imagine in my head when dealing with different hole counts between hub and rim.
There are two types of unwrapped wheel arc diagrams.
One is the "Rim-Hub-Rim type" where you divide the rim following the hole pattern
(or alternating if there's no pattern) and arrange it alongside the hub.
The other is the "Hub-Rim-Hub type" where you split the hub body at the center and cut the rim to lay it flat.
Think of it like cutting a WO tire and unrolling it like grilled eel.
I'll use this type this time.
Normally the 0° position in radians is on the X-axis, at the 3 o'clock position on a clock,
but in my "radian sense" the 0° position isn't fixed.
It shifts each time depending on where I can imagine the phase most easily.
In the diagram above it's at the 12 o'clock position.
The clockwise direction is also opposite to standard radian convention.
Now I'll cut out just the 90° section.
Let me draw the rim holes and hub holes both on the 0° line.
This time it's a 24H rim and 32H hub,
and the 32H hub has 4H in 90° per flange which is even,
so holes will also appear on the 45° line.
I've drawn all the hub and rim holes in the 90° section.
When you unwrap this into a linear diagram, it looks like this.
I've also added the phase on the opposite side of the hub and the adjacent hole phase on the rim.
With a 24H rim you get 3 holes in 90°, and since that's odd you can't build tangent lacing,
so I extended the diagram to 180° to get 6 holes.
Though really I just repeated the 90° diagram side by side.
The hub has 16H in 180°, but with equal spokes on both sides
that's 8H right and 8H left.
Today's question is whether you can build 8H on one hub flange laced tangent to 6H on one rim side
without causing phase misalignment.
First I tried lacing 0 crosses at the 0° position.
"0 crosses = radial lacing" — that's wrong.
If the path of a 0-cross spoke doesn't lie on a radial line extending from the hub center
(hereafter "radial line"),
I don't call it radial lacing.
This spoke is perpendicular to the horizontal axis, so it "is radial lacing."
Without skipping any holes, I drew the 6H on one rim side as 0 crosses from the end.
Excluding the spoke I just mentioned, the remaining 5 spokes are 0 crosses,
but they don't lie on the radial line, so they're not radial lacing.
This is an example of "0 crosses but not radial lacing."
If I lace the 6 rim holes tangent at this phase,
the final crossing spokes end up with different spoke lengths.
This is a serious problem. I'll cover the actual harm later.
Whether Rim-Hub-Rim type or Hub-Rim-Hub type,
the middle line circulates, so if needed you can
shift it arbitrarily.
↑You can shift it like this.
Same diagram but redrawn with more horizontal space.
From here on, right rim holes are blue and left rim holes are red.
Hub holes are also blue for right flange and red for left flange.
The reason phases don't match is obviously because the hole spacing differs between rim and hub.
In a linear unwrapped diagram, the horizontal axis length equals the angle itself.
When you draw that, you get the diagram above — clearly the phases won't match.
Let me try tangent lacing again.
Using the next hub hole would make the spoke extremely angled,
so let me try skipping one hole here.
The resting holes I'll mark with double circles.
That leaves 3 hub holes to 2 rim holes.
Either way you route it,
the spoke length difference is the same, and the crossings are mirror symmetric.
↑I routed it this way.
The two crossings before are also mirror symmetric.
If I call the order "porcupine direction length → anti-porcupine direction length,"
spoke lengths are B and A, then A and B.
Applying the earlier crossing to this, it becomes C and D.
At this point, one flange would have 4 different spoke lengths,
and if you tension and true it this way,
the hub phase would rotate and distort rather than assemble with those exact spoke lengths.
(Of course, within the limits the spoke lengths allow.)
I've numbered the hub holes starting from the end.
The resting holes are 3 and 6.
Pulling the middle of the linear diagram to circulate it
and rotating just the hub side in the radian sense diagram are the same thing.
So I rotate it, but
by how much? That's
"until the angles of the 0-cross spokes from holes 4 and 5 are mirror symmetric."
Right here.
From that position, I created 3 tangent crossings.
Only the middle crossing has equal left and right spoke lengths.
This is probably the best-balanced way to build a 32H hub with a 24H rim using equal spokes on both sides.
First there's a crossing of length E spokes with no spoke length difference,
with resting holes adjacent to each,
and F and G spoke crossings next to those.
I've drawn this simplified as 2 crosses, but
if you can calculate spoke lengths, building 4 crosses or more is of course possible.
So would I actually build it this way?
Well, despite dragging this out so far, absolutely not.
I have a pretty firm rule:
"I don't build wheels where the spoke path itself (for 0-cross) or the line extending from the final crossing and the hub center (for tangent)
deviates from the radial line."
The "actual harm" I mentioned earlier but glossed over is that
spokes become abnormally loose very easily.
Even if you seal them with strong threadlock,
the poor spoke balance remains.
In a properly angled tangent crossing,
the resultant F3 of porcupine-direction spoke F1 and anti-porcupine-direction spoke F2
lies on the radial line.
(As I'll write later, this doesn't apply when spokes have different diameters.)
When hub and rim hole phases are misaligned and spoke angles change,
you end up lacing tangent with spokes of different lengths.
Also, the direction of F3 deviates from the radial line, so no good.
So can you build a 32H hub with a 24H rim keeping spoke pull direction on the radial line?
Actually, yes, you can.
↑But you can't determine hub and rim hole phase at the 0° position like this.
I switched the 12H rim holes in 180° of the 24H rim
from right-left-right-left-right-left-right-left-right-left-right-left
to right-right-left-right-right-left-right-right-left-right-right-left.
The rim hole pattern information is preserved as color coding.
I've made it right 8H, left 4H (instead of right 6H, left 6H), but
from here I position the hub holes so that
the 0 crosses from the left 4H are radial lacing.
The other half of the left flange becomes resting holes.
I'll add them as double circles.
Next I added the 8H right flange section.
I drew it as squares for easier distinction.
Since this is 8H rim against 8H hub, I use all of them.
I've drawn 3 of the 4 tangent crossing pairs.
The remaining pair is
brought in this way by phase shifting.
So you see, 2:1 lacing does work.
The resultant of the spoke pull direction in this tangent crossing
lies on the radial line, so it doesn't violate my prohibition.
0 crosses are different.
They deviate from the radial line, so no good.
But spoke lengths are all the same on one flange,
so spoke balance isn't as
this blog's article comes up first (→here).
Looks like I have to figure this out myself.
Maybe in the near future this article will be the top result.
For a normal hub without paired spoke phasing,
the holes in the left and right flanges are offset by 360°/number of holes per side degrees.
Whether on the hub or rim, if we skip every other hole on one side of the hub flange
or on the rim at 24H,
the hole positions look like the diagram above.
When you add the holes on the other side too, it looks like this.
If you draw two concentric circles of different radii, use the larger circle as the rim and the smaller as the hub,
mark the hole positions, and connect rim holes to hub holes with lines (the spokes),
you'd have a wheel design diagram.
But when trying to build a 32H hub with a 24H rim using the same number of spokes on each side,
phase misalignment occurs.
To explain this phase misalignment intuitively, I find it easier to use
an unwrapped linear diagram of the arc rather than a circle with "radian sense,"
so I'll explain using that below.
This is literally the diagram I imagine in my head when dealing with different hole counts between hub and rim.
There are two types of unwrapped wheel arc diagrams.
One is the "Rim-Hub-Rim type" where you divide the rim following the hole pattern
(or alternating if there's no pattern) and arrange it alongside the hub.
The other is the "Hub-Rim-Hub type" where you split the hub body at the center and cut the rim to lay it flat.
Think of it like cutting a WO tire and unrolling it like grilled eel.
I'll use this type this time.
Normally the 0° position in radians is on the X-axis, at the 3 o'clock position on a clock,
but in my "radian sense" the 0° position isn't fixed.
It shifts each time depending on where I can imagine the phase most easily.
In the diagram above it's at the 12 o'clock position.
The clockwise direction is also opposite to standard radian convention.
Now I'll cut out just the 90° section.
Let me draw the rim holes and hub holes both on the 0° line.
This time it's a 24H rim and 32H hub,
and the 32H hub has 4H in 90° per flange which is even,
so holes will also appear on the 45° line.
I've drawn all the hub and rim holes in the 90° section.
When you unwrap this into a linear diagram, it looks like this.
I've also added the phase on the opposite side of the hub and the adjacent hole phase on the rim.
With a 24H rim you get 3 holes in 90°, and since that's odd you can't build tangent lacing,
so I extended the diagram to 180° to get 6 holes.
Though really I just repeated the 90° diagram side by side.
The hub has 16H in 180°, but with equal spokes on both sides
that's 8H right and 8H left.
Today's question is whether you can build 8H on one hub flange laced tangent to 6H on one rim side
without causing phase misalignment.
First I tried lacing 0 crosses at the 0° position.
"0 crosses = radial lacing" — that's wrong.
If the path of a 0-cross spoke doesn't lie on a radial line extending from the hub center
(hereafter "radial line"),
I don't call it radial lacing.
This spoke is perpendicular to the horizontal axis, so it "is radial lacing."
Without skipping any holes, I drew the 6H on one rim side as 0 crosses from the end.
Excluding the spoke I just mentioned, the remaining 5 spokes are 0 crosses,
but they don't lie on the radial line, so they're not radial lacing.
This is an example of "0 crosses but not radial lacing."
If I lace the 6 rim holes tangent at this phase,
the final crossing spokes end up with different spoke lengths.
This is a serious problem. I'll cover the actual harm later.
Whether Rim-Hub-Rim type or Hub-Rim-Hub type,
the middle line circulates, so if needed you can
shift it arbitrarily.
↑You can shift it like this.
Same diagram but redrawn with more horizontal space.
From here on, right rim holes are blue and left rim holes are red.
Hub holes are also blue for right flange and red for left flange.
The reason phases don't match is obviously because the hole spacing differs between rim and hub.
In a linear unwrapped diagram, the horizontal axis length equals the angle itself.
When you draw that, you get the diagram above — clearly the phases won't match.
Let me try tangent lacing again.
Using the next hub hole would make the spoke extremely angled,
so let me try skipping one hole here.
The resting holes I'll mark with double circles.
That leaves 3 hub holes to 2 rim holes.
Either way you route it,
the spoke length difference is the same, and the crossings are mirror symmetric.
↑I routed it this way.
The two crossings before are also mirror symmetric.
If I call the order "porcupine direction length → anti-porcupine direction length,"
spoke lengths are B and A, then A and B.
Applying the earlier crossing to this, it becomes C and D.
At this point, one flange would have 4 different spoke lengths,
and if you tension and true it this way,
the hub phase would rotate and distort rather than assemble with those exact spoke lengths.
(Of course, within the limits the spoke lengths allow.)
I've numbered the hub holes starting from the end.
The resting holes are 3 and 6.
Pulling the middle of the linear diagram to circulate it
and rotating just the hub side in the radian sense diagram are the same thing.
So I rotate it, but
by how much? That's
"until the angles of the 0-cross spokes from holes 4 and 5 are mirror symmetric."
Right here.
From that position, I created 3 tangent crossings.
Only the middle crossing has equal left and right spoke lengths.
This is probably the best-balanced way to build a 32H hub with a 24H rim using equal spokes on both sides.
First there's a crossing of length E spokes with no spoke length difference,
with resting holes adjacent to each,
and F and G spoke crossings next to those.
I've drawn this simplified as 2 crosses, but
if you can calculate spoke lengths, building 4 crosses or more is of course possible.
So would I actually build it this way?
Well, despite dragging this out so far, absolutely not.
I have a pretty firm rule:
"I don't build wheels where the spoke path itself (for 0-cross) or the line extending from the final crossing and the hub center (for tangent)
deviates from the radial line."
The "actual harm" I mentioned earlier but glossed over is that
spokes become abnormally loose very easily.
Even if you seal them with strong threadlock,
the poor spoke balance remains.
In a properly angled tangent crossing,
the resultant F3 of porcupine-direction spoke F1 and anti-porcupine-direction spoke F2
lies on the radial line.
(As I'll write later, this doesn't apply when spokes have different diameters.)
When hub and rim hole phases are misaligned and spoke angles change,
you end up lacing tangent with spokes of different lengths.
Also, the direction of F3 deviates from the radial line, so no good.
So can you build a 32H hub with a 24H rim keeping spoke pull direction on the radial line?
Actually, yes, you can.
↑But you can't determine hub and rim hole phase at the 0° position like this.
I switched the 12H rim holes in 180° of the 24H rim
from right-left-right-left-right-left-right-left-right-left-right-left
to right-right-left-right-right-left-right-right-left-right-right-left.
The rim hole pattern information is preserved as color coding.
I've made it right 8H, left 4H (instead of right 6H, left 6H), but
from here I position the hub holes so that
the 0 crosses from the left 4H are radial lacing.
The other half of the left flange becomes resting holes.
I'll add them as double circles.
Next I added the 8H right flange section.
I drew it as squares for easier distinction.
Since this is 8H rim against 8H hub, I use all of them.
I've drawn 3 of the 4 tangent crossing pairs.
The remaining pair is
brought in this way by phase shifting.
So you see, 2:1 lacing does work.
The resultant of the spoke pull direction in this tangent crossing
lies on the radial line, so it doesn't violate my prohibition.
0 crosses are different.
They deviate from the radial line, so no good.
But spoke lengths are all the same on one flange,
so spoke balance isn't as
Original Japanese post: のむラボ日記 #1086