Can someone help me with my math?

[rodekyll]

I'm calculating the m/c to piston sizes for a brake system.  The calipers are three-piston floaters, as opposed to fixed calipers like a standard guzzi F08.  If I consult my tables, I see that a 1" piston has a surface area of 0.7854.  Since these are floating calipers and the pistons push both pads, I multiply that surface by 2, and then by 3 for the number of pistons.  This gives me the surgace area of one caliper.  so:

0.7854 x 2 x 3 = 4.7124

I double that for two calipers and get 9.4248

Discussions here tell me the sweet spot for leverage should be in the 21/1 - 24/1 ratio.

My table tells me the area of a 19mm radial master cylinder is 0.4394. 

9.4248/0.4394 = 21.4492.

I think that might nail the ratio?  Or is my understanding of this calculation wrong?

[Kiwi_Roy]

Something tells me to multiply by 2 is wrong it would be the same pressure on the disk if the piston was just pushing on one side with a fixed pad on the other.

[rodekyll]

Yeah, I'm on shaky ground there.  I think I read somewhere that you doubled the area for a floating caliper, but I can't find the reference.  It makes sense to me both ways.    

[redrider]

Stick with the X2. You are dealing with volume of a cylinder by correlating with surface area. Proportions. My 2c.

[mgmark]

this may save some time:

[rodekyll]

Yes, that is the chart for a standard 90º master cyl.  I'm trying to correlate to a radial.  The big difference is that this chart uses a 27mm mechanical pivot and the radial is more like 18mm.  It's a big mechanical difference requiring a larger bore size in the m/c.  Some say as much as 5mm.

I need to find that chart for radial master cyls.

Here's an interesting 2-page tech talk that is very educational

http://www.mcnews.com/mcn/technical/nov2012tech.pdf

[Moto]

I'm no expert. Here is a statement that seems definitive, but a closer look at the site says that all formulas need to be verified by the user:

"System Pressure

"Pressure is a function of the required clamp load and the piston area. Remember on an opposed piston disc brake it’s only the area on one side of the disc."

source: http://www.engineeringinspiration.co.uk/brakecalcs.html#pf

That statement agrees with my opinion, which is based on my memory of having thought about this once before, but not based on any analysis I can produce!

Here is an excellent explanation, which I have extracted from an even better, longer passage, with illustrations (cited below):

"The diagram above shows the two types of caliper, floating on the left, and fixed on the right. To find out how much force the pads are getting find the total piston area ON ONE SIDE of the caliper. For a floating caliper that is just the total piston area. For a fixed caliper use only the inboard or outboard pistons. Then use that force on both brake pads.

"Why pressure from only one side?

"This can be a little confusing and some very smart people have gotten caught. But it really isn’t that hard to understand if we shift our thinking a bit. First, ignore whether or not the caliper is fixed or floating; it doesn’t matter, and in a minute you’ll see why.

"Imagine that you are holding a book between your two open hands. Now squeeze on the book. If your left hand is pushing with 25 pounds of force, then your right hand has to push back with that much force or the book will move across your chest. Each of your hands is a brake pad, and the book is the rotor. The force between the pad and the rotor is only 25 pounds, not 50. And it is the force between pad and rotor that determines the frictional force generated on the brake rotor.

"In this respect there is no difference between floating and fixed calipers. The other side just acts to keep the rotor centered in the caliper, one moves the whole caliper, the other moves the other pistons.

"But wait! There are still two pads, one on each side. If we replace one pad with a ball bearing surface the force on the rotor from the brake caliper would be reduce about half. So we have to count both pads, but using the force generated by one side of the pistons."

source: https://enderw88.wordpress.com/automotive-theory/brake-system-theory/

The article you linked is indeed interesting. It seems to me that the calculations that were done for the radial master cylinder were exactly those for a conventional cylinder, plus a final adjustment based on the ratios of their lever-arms, which in their case was 3/2. This seems simple enough, if you know or can measure those lengths.

Moto

[rodekyll]

The Brembo RCS has that pivot distance adjustable to either 18mm or 20mm.  The chart is set for 27mm + some lever distance assumption about where the hand is that may or may not correspond to the lever length of the brembo.  So while the chart can help ballpark me (there are only so many radial sizes out there) it's not going to be exact -- and I don't care. 

What I care about is do I want a 15mm, 17mm or 19mm radial master cylinder to pump up two floating calipers with three pistons each.  Each piston is 25mm.  To determine the area of these calipers for purposes of sizing the master cylinder, do I count 6 surfaces or 12?


"To find out how much force the pads are getting find the total piston area ON ONE SIDE of the caliper. For a floating caliper that is just the total piston area. For a fixed caliper use only the inboard or outboard pistons. Then use that force on both brake pads."


This says we take the area of half of the fixed pistons, or the total area of floating pistons to determine force.  That implies, per the chart, that to equate a floating piston's total area to that of a fixed caliper, we must double the number of piston surfaces.  So the article tells me to use 12 as the factor for my 6-puck floater.

Right?


       

[pikipiki]

Your 3 x 1" (25.4mm) pots will have same surface area as one 44m pot or 2 31mm pots.
knowing this refer back to your chart for 31mm (2), or taking the mean value between 30mm (2) and 32mm (2) would be close enough.
compare this figure against what you have calculated from first principles to validate your results.

If you have 3x2 active 1" pots then the line of the chart 44(2) provided above.

[rodekyll]

Thank you.  :)

[Moto]

Pikipiki's advice is right, but doesn't take the difference in pivot distances into account, of course. His advice leads to about a 13mm master cylinder, but accounting for the decreased pivot distance this becomes about [EDIT: 11mm 16mm], as I will explain.

So the article tells me to use 12 as the factor for my 6-puck floater.

Right?
       

Wrong.  The answer is 6.

The key short answer is found in this statement in the passage I quoted: "First, ignore whether or not the caliper is fixed or floating; it doesn't matter, and in a minute you'll see why."

The other quotation (the first one I cited) that you referred to is calculating the "total force" exerted on the caliper, which includes the force on both sides. The idea for both the fixed and floating calipers is to find the force exerted by one side's pistons, and then double that. But this is unnecessarily complicated for the problem you posed.

The rule of thumb you mentioned -- look for a hydraulic advantage between 21 and 24 -- already takes into account the force from the opposing piston, which is the same for both fixed and floating calipers because of one of Newton's laws. If you check the table that MGMark provided, you'll see the entries are just the ratio of the areas of slave and master pistons. For example, the entry for a 41 caliper [a single caliper of 41mm, either fixed or floating] and a 15mm master cylinder is just [(41/2)^2 * pi] / [(15/2)^2 * pi] = 41^2 / 15^2 = (41/15)^2 = 7.4711.  Some of the apparent complexity goes away when the formula is reduced to the square of the ratio of caliper and master diameters. That is, hydraulic advantage = A = (C/M)^2.

Solving the simplified formula for M, we get M = C/sqrt(A). Checking this against the same example in the table, we find M = 41/sqrt(7.4711) = 15.000, confirming the result.

The diameter of a single piston equivalent to six 1-inch pistons is sqrt(6) times as large, or sqrt(6) = 2.449 inches. To get a hydraulic advantage of 22.5, the simplified formula gives a master cylinder diameter of M = 2.449/sqrt(22.5) = 0.516 inches, or 13.11mm.

This calculated value agrees closely with the hydraulic advantage of 22.91 for a 13mm master cylinder in pikipiki's suggested row 44(2) of the table.

[EDIT: The discussion from this point down is wrong, because I misunderstood the leverage setup of the pivot arms. See the correction in a later post. All the "increases" and "decreases" in the remainder of this post needed to be reversed, and the "correction" factor calculated is the inverse of the actual correction needed.]

If the rule of thumb assumes a 27mm pivot distance and the radial is set to 18mm, you need to increase the force to take account of your lessened mechanical advantage. To increase the force provided by the master cylinder you must decrease its area proportionally. This is apparently contrary to your statement, "It's a big mechanical difference requiring a larger bore size in the m/c."  But if you think about it for a moment you can see the idea. Suppose you exert 1 pound of pressure using a disk with an area of 1 sq. inch -- that amounts to 1 psi. If you reduce the area of the disk to 0.5 sq. inch, the pressure is 2 psi. And so on. So more pressure from the disk (the master cylinder piston) requires a smaller diameter. This is what is needed to offset the loss in mechanical advantage.

The ratio of force increase needed is 27/18 = 1.5. The required reduction in diameter is therefore the factor sqrt(1/1.5) = .816. Applying that to the previously calculated diameter gives 13.11(.816) = 10.70mm as the appropriate radial master cylinder diameter.

So an 11mm master cylinder is indicated, bearing in mind that these calculations don't take into account various other factors aside from the theoretical mechanical and hydraulic advantages.

Or so it seems to me. I don't see any radial Brembo master cylinders in that size. So I guess your best bet is to take the smallest one you can, if you're intent on a radial. If you take the 15mm offering, you should expect an 86 percent increase in required lever pressure compared to the rule of thumb's recommendation [ (15/11)^2 = 1.86 ].

Remember my disclaimer, under my avatar. Sorry my post is so long, but I'm already taking too much time at it.

Moto

[rodekyll]

That's a really thorough response!

The one part I think I have a problem with is where you say the bore on a radial pump needs to be SMALLER due to the shorter pivot distance.  Brembo and others are recommending I go LARGER by 3mm - 5mm.  Using that clue I've already upped my Convert's 15mm front brake to a 19mm radial for two F08's and LOVE it.  Likewise, for a single F08 on the rear I'm using a 17mm radial and also love it.  I feel like I could lock up either wheel -- or not -- at will. 

On the trike's stock EV dual front brakes I'm also using a 19mm radial.  It has a different feel and sensitivity, but it will stop the wheel instantly if I want it to.  I want the trike's rear brakes to be as strong and controllable as all my others, and so far I'm not getting it right.  I get a mediocre rear effect if I'm already using the front pump, but by itself, the rear brake is largely useless.  It might be that 3x1" pucks just aren't enough to stop 1200# of rolling junk, but with over $1k invested here I need to sort out what I've got before throwing any more money at it.

But the simplifying part is that in the range I'm investigating, the 15mm (5/8"), 17mm (11/16"), and 19mm (3/4") are the only choices.  So once the math has me in the ballpark I'm pointed at the winner.  I think I have those sizes covered now with items from my stash.  It's just a matter of reducing the time and aggravation of swapping pumps by getting it right the first time, and that's why I'm trying to do these calculations.  The plan is to pull in to a friends' house on Vashon Island and do the changover with him to pump while I bleed.  The brakes.

[Moto]

Brembo and others are recommending I go LARGER by 3mm - 5mm.

Well, maybe I'm thinking about that part wrong. I'll let others have a chance to explain my error. (Before I waste more time on my own.)

The part leading to 13mm before the leverage correction seems correct for sure.

Moto

EDIT: BTW, a quick search turned up an example of exactly the kind of thread this one shouldn't turn into: http://www.svrider.com/forum/archive/index.php/t-84403.html. Talk about panties in a bind! If my reasoning is wrong, so be it. I may need to look more closely at a diagram of the leverage around the pivot arms to understand why.

M.

[rodekyll]

Oh, yes -- I now see the 13mm writing on the wall quite clearly.  I'm really, really hoping my 17mm pump is going to be close enough and that I've got enough surface area on the pads to do the job.  I can deal with slightly 'numb' feedback as long as the braking is efficient.  My other 'small' pump is an offshore clone of some pitbike brake and could be anything from 12mm to 5/8.  Given the choice I think I'd rather be slightly large than small . . .

If I can't sort it out from this direction I'll have to look at other calipers with different piston sizes.  These are some Ford thing, so I might have choices I don't know about.

[Moto]

I see my error now. A trip to the garage to look at my Griso's setup and another trip to a webpage with a radial mc diagrammed shows the effect of the different pivot arm lengths is to give the radial setup more, not less, mechanical advantage

So the correction to the initial 13.1mm master cylinder size should have been a factor of sqrt(1.5/1) = 1.225, not sqrt(1/1.5) = .816, and the corrected size is 13.1(1.225) = 16.0mm.

Oops. Well, at least I wasn't working on the Mars Rover...

 

Nothing like actually looking at the thing you are modeling!

M.

[rodekyll]

(in Don Adam's voice)  "The old square root of the inverted ratio trick.  It gets them every time!"

If 16mm is the correct answer, I think this is probably the closing argument for the 17mm experiment.   

I'll reduce my checked baggage by two other pumps.

[Moto]

In the scale of the recommended rule of thumb values of 21-24, the 17mm should give a value of 20, close enough.

Thanks for your patience.

[rodekyll]

Thanks for sticking with it until I understood it.