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anybody explain Sheldon's straddle wire MA?

Posted: 7 Nov 2011, 11:14pm
by 531colin
I have always thought mathematicians were born, not made, but I'm really frustrated by my inability to understand Sheldon on the mechanical advantage you can get from the cantilever straddle wire......can anybody explain it to me?
This is Sheldon's diagram ...Image


Further down the page http://sheldonbrown.com/cantilever-geometry.html he says the MA is 1/sin yoke angle, which leaves baffled.
He also makes a table of yoke angle and MA....I have plucked some numbers from his table;
yoke angle 50deg MA = 1.31
yoke angle 40deg MA = 1.55
What do these numbers mean? I'm used to seeing MA expressed as a ratio,like 2:1, for example.

I selected those numbers simply because I can draw a yoke angle of 45deg.
But when I draw a yoke angle of 45deg, and then raise the yoke by 10mm, the cable anchors move in by 13mm each, 26mm total. (I have moved them in a straight line, although in the real world, they move in an arc.)
On that basis, a yoke angle of 45deg produces more brake pad movement than the movement in the main cable, in a ratio of 2.6 : 1, a negative MA.

My head hurts, now!

Re: anybody explain Sheldon's straddle wire MA?

Posted: 8 Nov 2011, 7:29am
by mattsccm
Ditto. For several years I have been trying to get the most out of canti's but applying Sheldons and other people ideas and always give up as I can only process 1 idea at a time. I think he tells me to have soggy levers not the nice rock hard ones I like but I just don't know.
And don't tell me to use V's.

Re: anybody explain Sheldon's straddle wire MA?

Posted: 8 Nov 2011, 9:57am
by james01
mattsccm wrote:he tells me to have soggy levers not the nice rock hard ones I like but I just don't know.
.


And he's right. "Hard" levers suggest you're pulling lots of cable at the brake pad end, great if you've got Superman's grip. Soggy levers are the equivalent of low gears, lots of movement required to achieve a small cable pull at the pad end, but it will be much more powerful.

Re: anybody explain Sheldon's straddle wire MA?

Posted: 8 Nov 2011, 10:05am
by mattsccm
Come to that conclusion but it feels all wrong.

Re: anybody explain Sheldon's straddle wire MA?

Posted: 8 Nov 2011, 10:36am
by reohn2
I'm no engineer,but logic suggests that its the "anchor angle"(AA) that matters as thats where the mechanical advantage is gained.
Wide "froglegged" canti's have always looked to have more MA to me than so called "low profile" canti's,an acute AA must have more MA than an obtuse one,though as that AA becomes more acute it must become harder to pull,there must be an optimum,logic(again) suggests that its between 90 to 100 degrees.

Re: anybody explain Sheldon's straddle wire MA?

Posted: 8 Nov 2011, 11:07am
by JonMcD
531colin wrote:What do these numbers mean? I'm used to seeing MA expressed as a ratio,like 2:1, for example.


He is just expressing the ratio in decimal form rather than fractional form. So an MA of 1.31 is 1.31:1 or 131:100.

Sheldon's "The formula is: Mechanical Advantage = 1/sin yoke angle" looks too simple to be the entire formula to calculate MA and I think what he means that the MA is proportional to 1/sin yoke angle i.e. yoke angle is just one factor in a more complex formula. For instance the MA is clearly going to be influenced by the ratio of the lengths of the two arms of the cantilever i.e. length PA:length PS in his diagram.

Re: anybody explain Sheldon's straddle wire MA?

Posted: 8 Nov 2011, 11:33am
by Cyclenut
No I can't explain Sheldon's straddle wire MA, but I can explain the MA of a cantilever brake.

In the force diagram below:

T = brake cable tension
S = straddle tension
F = force on the rim from each brake block
h = height of the block (centre) above a horizontal line through the arm pivots
a = perpendicular distance from straddle to arm pivot
b = distance along the aforementioned horizontal line, to the intersection with an extrapolation of the straddle
A = angle of the straddle to the horizontal
Canti.gif
Canti.gif (1.67 KiB) Viewed 100612 times

(1) Balance of turning moments about the arm pivot: S×a = F×h

(2) Resolution of vertical forces on straddle yoke: T = 2×S×sinA

(3) sinA = a/b

Substitute sinA in (2): T = 2×S×a/b

Substitute F×h for S×a in (2): T = 2×F×h/b

Mechanical advantage (MA) is F/T = b/2h

Sorry Sheldon, but you were wrong!

The angle of the straddle does come into it, because flatter makes b longer. But when you start to apply the brake, a flat straddle rapibly becomes less flat and b shrinks dramatically, which is why brakes that rely upon a flat straddle have a regressive action - loads of MA when you want movement, followed by movement when you want MA!

You can also see from this formula, that when the brake arms are horizontal, b is simply the length of the brake arm and the straddle angle becomes irrelevant. It follows that straddle angle becomes less important and the braking becomes more linear (which is generally a good thing) when the arms become closer to horizontal.

This formula and a consideration of how b varies as the brake is applied also demolishes the notion that there is something special about having the straddle at right-angles to the arm. But that may nevertheless be the optimum arrangement for some designs of brake and lever.

Re: anybody explain Sheldon's straddle wire MA?

Posted: 8 Nov 2011, 11:41am
by mattsccm
Excellent. Just what the hell does it all mean? :lol:
Oh how I would love to see a chart thats tells me low straddle for wide cantis or whatever. Something visual.

Re: anybody explain Sheldon's straddle wire MA?

Posted: 8 Nov 2011, 12:01pm
by fatboy
Cyclenut wrote:You can also see from this formula, that when the brake arms are horizontal, b is simply the length of the brake arm and the straddle angle becomes irrelevant. It follows that straddle angle becomes less important and the braking becomes more linear (which is generally a good thing) when the arms become closer to horizontal.

This formula and a consideration of how b varies as the brake is applied also demolishes the notion that there is something special about having the straddle at right-angles to the arm. But that may nevertheless be the optimum arrangement for some designs of brake and lever.



So what's the best way to adjust CR520 cantis? Low, medium or high yolk position?

Re: anybody explain Sheldon's straddle wire MA?

Posted: 8 Nov 2011, 12:03pm
by fatboy
I have to agree that Sheldon was wrond as he came to the conclusion that wide-angle cantis have low-MA which was intuatively wrong to me.

Re: anybody explain Sheldon's straddle wire MA?

Posted: 8 Nov 2011, 12:06pm
by Cyclenut
mattsccm wrote:Excellent. Just what the hell does it all mean? :lol:
Oh how I would love to see a chart thats tells me low straddle for wide cantis or whatever. Something visual.

Something visual? Just look at the little diagram and see what you can do to the setup of your brake to make b bigger if you want a brake that takes less strength to apply, or smaller if you want it to feel firmer.

Re: anybody explain Sheldon's straddle wire MA?

Posted: 8 Nov 2011, 12:07pm
by Cyclenut
fatboy wrote:So what's the best way to adjust CR520 cantis? Low, medium or high yolk position?

Like I said, if the arms are horizontal it makes naff-all difference.

But personally, I like my yolk near the bottom. Then I can eat most of the white first and save the best bit til last :lol:

Re: anybody explain Sheldon's straddle wire MA?

Posted: 8 Nov 2011, 12:47pm
by snibgo
Something visual: http://www.circleacycles.com/cantilevers/

It's a great way of seeing what is going on. The maths is explained in a linked paper.

Sheldon, wonderful as he was, wasn't a mathematian. One of the components of MA is 1 / sin yoke angle. If everything else was constant (which it isn't and can't be), MA would be proportional to 1 / sin yoke angle.

Re: anybody explain Sheldon's straddle wire MA?

Posted: 8 Nov 2011, 1:01pm
by fatboy
Cyclenut wrote:But personally, I like my yolk near the bottom. Then I can eat most of the white first and save the best bit til last :lol:


So you are a little-endian rather than a big-endian!

Re: anybody explain Sheldon's straddle wire MA?

Posted: 8 Nov 2011, 1:45pm
by CJ
fatboy wrote:So you are a little-endian rather than a big-endian!

For sure. I always prefer a small and elegant solution to one that's large and complex!