Pascal and the proportioning valve -- SOME NEW PROGRESS

[moto]

So Pascal's Law says "that when a change in pressure is applied to an enclosed fluid, it is transmitted undiminished to all portions of the fluid and to the walls of its container." In a closed container where the effect of gravity can be considered constant, it implies that pressure, as measured by psi for example, is the same on every wall or interior surface.

As I have been trying to understand the simple Brembo proportioning valves that Guzzi fitted to various models starting around 1978, I have been clinging to Pascal's Law as my guide. With it I can work out some important facts, but am still at a loss for at least one. (I am only considering the original series of valves, without the weight-compensating lever.)

Consider these two graphs from the workshop manual for the SP III (found on Greg Bender's site):





Notice that the labels for inlets and outlets (A-D) have the same meanings on both graphs.

What does Pascal's Law have to say about the graph? First, since outlet B (front caliper) is at a different pressure from outlet D (rear caliper) at every point on the second diagram the containers of fluid they lead to (the front and rear brake lines) are not at the same pressure and are therefore not directly connected by any ports or passageways. They must be separate enclosed containers. No explanations dealing in holes of different sizes feeding the two lines, or of valves opening one line to the other at any point, can stand against Pascal's Law: either would require equal pressures in the two lines.

What mechanisms could account for the second graph? First, the offset of 4.5 bar of pressure (4.5 x 14.7 = 66 psi) between the front and rear lines when pressure is low can be achieved by inserting a free-floating piston in the front brake line, continuing to apply 4.5 bar of pressure to the input flow of fluid in that line at all times. This can be done with a suitable spring (which would need to apply only a few pounds of force since the piston can be small in diameter). It is important that the piston does not open a passageway between the two lines: it only provides a constant force of resistance to the incoming fluid as it sits in the line. The prominent fitting at B on the upper diagram must contain this mechanism. (I believe I see the bore and its piston, about 2.5mm diameter, when I look into the fitting. I don't see the spring, which I suppose attaches on the other end.)

The second feature of the second graph that needs explanation is the change in the slope of the line at 21.5 bar on the front brake outlet axis (and 26 bar on the rear, offset by that same 4.5 bar). The first thing to note is that the rate of change between the two outlet pressures is 1:1 until this "knee point." This indicates the two lines are holding the same pressure on the first piston once the spring's contribution of 4.5 bar is taken into account in the front circuit. After the knee point, the rear line enters a new regime. It is now behind another free-floating piston, this one having different diameters at its two ends (and of course different diameter bores to capture them). This piston has a smaller head on the side in the main valve chamber and a larger head in a separate chamber leading to the rear brake line. Because of that difference, the pressure exerted on the rear brake fluid is less than the pressure from the front, while the forces on the two sides are equal. Use of such a two-diameter piston is standard for this application. (Two applications of the formula P = F/A, with two different areas and the same force will show the idea.)

Obviously, a switching mechanism is needed to change the rear line from the regime where it experiences the full front line fluid's pressure to the one where it has the piston-reduced pressure. I know that some combination of diameters on the central rod and of ports in the wall of the chamber must be responsible (as it is in other proportioning valves). My question is, exactly how does this work in the Brembo valve we have? Where are the different effective areas of the two piston sides and how is the rear line being forced to get its pressure from the piston instead of from the front line at the appropriate point? (Where is the port that closes off the rear line from the front line pressure?)

Here are some photos of the valve in various states of undress, for reference:

https://www.thisoldtractor.com/moto_guzzi_tonti_brembo_brake_proportioning_valve.html

https://wildguzzi.com/forum/index.php?topic=105762.0

I guess I should probably say that I don't find the explanations provided with those photos completely satisfactory. And that I am reluctant to open the valve I recently acquired, fearing I will miss-adjust or damage it.

Thanks to any who read this far.

Moto

[fotoguzzi]

Probing the depths of Guzzidom.. good work.

[tris]

Interesting discussion that I shall follow

It's rare I'd take Wikipedia as definitive but maybe this gives some insight

https://en.m.wikipedia.org/wiki/Proportioning_valve

[moto]

I made a slight advance. The bottom graph for the SP III has a table insert that implies the relative pressure changes between the front and rear lines (holes B and D, respectively), above the knee point.

30 bar at B ~ 28 bar at D
60 bar at B ~ 33 bar at D

This is a slope of (33 -28) / (60 - 30) = 5/30 = 1/6.

(An adjustment for the 4.5 bar offset at B would make no difference, algebraically.)

The slope of 1/6 means that the rear brake circuit, after the knee point, is experiencing pressure differences one-sixth of the front circuit's. This could be achieved by a two-sided piston with the rear-brake end six times the area of the end supplied by the master cylinder. If the piston had circular ends the ratio of diameters would be sqrt(6)/sqrt(1) = 2.45:1. I haven't found likely candidate circular surfaces in my examination of the photos.

Another possibility is a stepped piston or shaft with one end forming the piston surface in the rear brake fluid chamber and the other end running through the chamber pressurized by the master cylinder and through its wall to an unpressured volume. The end of the piston or shaft would not be exposed to brake fluid pressure at all, as is the case for the central shaft in the Brembo unit, which protudes through the end with the circlip. Suppose such a shaft has two diameters, A and B, where B is the stepped one, and end A presses on the brake line fluid while B receives pressure from the master cylinder. The force of the fluid directed toward the larger diameter part of the shaft falls only on the stepped surface. The size of the step needed to form a ratio of 6:1 for the effective areas is surprisingly small. A rough calculation shows the reduced diameter B should be a little more than 0.9 of the diameter A: if it were 0.9 exactly, the area of the bearing surface lost to the reduced shaft would be 0.81 of the area at diameter A, meaning the rear would get 19% of the master cylinder pressure, instead of 1/6 = 17%.

Calculating this in full, we start by setting the diameter A = 1, for convenience; the area of its end is pi * 1^2  / 4, or pi/4 ~= 0.785. Then we require a diameter B such that the area of the step surface is 1/6 of the A surface, or 1/6 of pi/4: 

pi / 4 - pi * B^2 / 4 = 1/6 * (pi/4)

 B^2 = 1 - 1/6 = 5/6

 B = sqrt(5/6) ~= .913, a bit more than 0.9, as expected.

So we might find a slightly stepped shaft instead of two opposed pistons of very different diameters. This seems to be the likely method -- it is used in similar hydraulic devices -- and I do see about the right step in the middle of the central shaft, in this photo:



But I don't see how it is set up to do the job, if it is.

[fotoguzzi]

Why do you want to know all this stuff?
The valve works good for me and that’s all I really need to know. I respect your efforts tho. Also, I got D’s in algebra but A’s in geometry.. just how my brain works.

[moto]

I like to figure out stuff in general.

I am deciding whether to fit a proportioning valve from a Mille GT, an otherwise very similar set-up to my T3. I want to know ahead of time how it works, how it will affect my braking, and whether I will like the result. My very small knowledge of math lets me investigate this, and that makes me happy.

Thanks for your positive words. Geometry aptitude is the best indicator of a mathematical mindset, I think. I have little.

Moto

[fotoguzzi]

                         I salute you!

[John A]

I don’t think it’s a true proportioning valve and have considered getting one from Summit Racing if they have a size that works but I haven’t fooled with it much beyond that
https://www.summitracing.com/search/part-type/brake-proportioning-valves-and-distribution-blocks

[huub]

looks like one piston is pushed by the hydraulics pressure from the master cylinder , the other by the spring, resulting in a different pressure

[moto]

I don’t think it’s a true proportioning valve....

The Guzzi/Brembo valve is a true proportioning valve, with a rear/front proportion of 1/6 after the knee point. Aftermarket proportioning valves are often adjustable, but only for the knee point, not the proportioning rate. For example, here is an operation diagram for such a valve from Wilwood, with the Guzzi valve's operation added on to it by me:





The Wilwood valve in the graph has a fixed rear/front proportioning of 2/5 = 0.4, versus the Guzzi's 1/6 = 0.17. That can't be changed, although the pressure at which the valve begins its proportioning can be changed, as the diagram shows. This is done with a preload spring.

Wilwood has the best description of how these proportioning valves work that I found: https://shop.wilwood.com/blogs/news/how-does-a-proportioning-valve-work. The graph I modified came from that page. I don't know if it refers to a particular one of Wilwood's several valves.

EDIT: Looking again, I see the graph refers to the design of all their proportioning valves, which share a common proportioning rate (they say it is "less than 1/2:1," but I calculate it as 2/5 by reading their excellent graph). EDIT AGAIN: Looking at documentation for the Wilwood valves found on the Summit Racing page, the proportion rate is 0.43, not the 0.4 I eyeballed.

One thing the Wilwood valve doesn't have is the initial front line offset of 66 psi (4.5 bar). That feature is intended to avoid motorcycle front wheel skidding in low traction situations.

looks like one piston is pushed by the hydraulics pressure from the master cylinder , the other by the spring, resulting in a different pressure

That is the preload spring that determines the knee point, the transition to the proportioning regime. (See the Wilwood description page just referenced.) In the Guzzi/Brembo case the preload is not intended to be adjusted, but I suspect the slot on the apparent screw head in the photos would allow the (brave or unwary) rider to adjust the knee point at will, or by accident during reassembly.

[aklawok]

 I think you are over complicating the problem. Consider only the intended size of the brake piston as a ratio to compare it to the proportioning valve it was intended. If your bikes are of a different size, calculate a ratio front::rear by piston area, then for the bike the proportioning valve would be for. Does this sound right?

[moto]

I think you are over complicating the problem. Consider only the intended size of the brake piston as a ratio to compare it to the proportioning valve it was intended. If your bikes are of a different size, calculate a ratio front::rear by piston area, then for the bike the proportioning valve would be for. Does this sound right?

To calculate the total forces of braking applied to the two tires requires considering the issues you mention.

[moto]

If anyone has a spare proportioning valve they would like to donate to science, please PM me. It should be the kind without a lever on the outside. Condition doesn't matter. I know some of you have delinked your brakes, so might still have one in your parts stash. If you want it back after dissection, I'll send it back, reassembled, but with no guarantees. I'll cover shipping costs, at least.