Heatsinks Booster Faq

[Heatsink]

NOTE: Following requests, this is the same post with the pictures reloaded:) Sorry about the difference in picture size, this is due to the images being shrunk during abit of TF admin. Hopefully everything can be read ok! I've split this post into 2 parts aswell, because I reached the limit on the number of images you can put in 1 post!

Brake Booster Theory Part1 of 2

Q: If I have a set weight of booster material, What cross section shape would be stiffest?

Waffly bit

You want to make a brake booster, partly because you don’t want to spend any more money on your bike (it’s already cost you an arm and a leg!) and also because it could be fun to do and cool to say that you’ve got a custom booster. You’ve noticed that these are commonly made out of Aluminium or steel, and you’ve managed to get hold of a big enough off cut for one of one of these metals. Your thinking is “Well if this material is stiff enough for those boosters out in the shops it should do the trick for me.” You may have a few tools in the shed, and we’re talking basic here like a hack saw and vice, probably not a CNCing station. All that remains is to design the blighter, with the requirements that it be sufficiently stiff to counter the flex in the bike frame, and be light weight. (What's "Light weight"? 100g? Up to you to decide. Have a look at existing boosters for an idea, but to get a feel for weight remember this useful rule of thumb - 100g is roughly the weight of 1 apple )

You’ve been searching for posts about home made boosters on TF and there’s a lot of conflicting advice about what shapes are stiffest. You realise that to transform your lump of metal into many of the shapes isn’t going to be possible to achieve without some sort of fancy equipment that you’d find in an Engineering company. With your simple tools, you know that your booster will probably end up being a simple horse shoe shape hack sawed out of sheet metal. Nevertheless, you’re interested in knowing which of the fancy cross-section shapes (I-beam, tube, rectangular box etc) is the stiffest and how they compare to the simple rectangle sheet that yours is probably going to be.

The good news is you don’t need any fancy computer software (e.g. Finite Element Analysis, “FEA”) to tell you which shapes are stiffest. All you need is a calculator!

The thing that you need to compare for the various shapes is a property called the Second Moment of Area measured in mm^4 (mm to the power of 4). Another name used for the same thing is the Moment of Inertia (In case you wanted to do some looking on the internet for yourself). Clever people (Mathematicians & Engineers etc) have a habit of giving things fancy sounding names to hide the fact that what they’ve discovered is actually rather simple. In short the Moment of Inertia, or I as we’ll call it since it saves my fingers from getting worn out, is a measure of a shape’s resistance to bending. The higher it is, the better as far as we’re concerned! If you want a closer look at the ideas behind the calculations coming up, check out This site, and do a search of your own for:

"Second Moment of Area"

"Moment of Inertia + Simple Cross Sections"

You get the idea!

If you do an internet search you’ll be able to find equations for all manner of shapes. For the shapes we’re looking at, I've pulled out the equations we need.

Brake booster cross sections to compare

The idea of arranging a set weight into a variety of shapes basically means that the cross-sectional area must be the same for all our boosters we're going to compare (Remember that Mass = Vol x Density, and Vol = c/s Area x Length. So for a constant density and length, Mass is proportional to c/s Area)

I've calculated the dimensions of these shapes so that the height of the booster is always 40mm - Which is assuming that if the booster gets bigger than this it'll get in the way of your mini-seat/arse.

To make things simple I've made the assumption that all our boosters have a constant shape along their whole length. This of course will mean that the arms of the booster will stick out too much and catch on your leg, but we're only interested in doing a stiffness comparison here, and not calculating deflections and forces.

You'll notice that I've written "Ixx" instead of I in the calculations that follow. This simply describes the direction that the braking force is acting in, which is perpendicular to the line, the axis, that I've drawn on the pics ( X at both ends). If I could do subscript then the "xx" would be below the line.

You could leave this XX out if you fancied, but once I'd done the comparison calcs I thought I'd compare the stiffness if the shapes were rotated around 90 degrees, about an YY axis. The XX and YY are therefore needed to make it clear which way round the calcs are for.

Moment of Inertia Calculations

1. I-Beam

Above are some typical dimensions. The line, x ---- x is the "Neutral Axis" around which bending is happening, and the big black arrow is the force from the brakes, trying to bend the booster.

Using the equation for this shape, to find the Moment of Inertia:

2. Tube

Above are some typical dimensions which keep the surface area the same as the I-beam.

Here's the equation for a tube with the dimensions dropped in:

>>> Part 2 in next post

[Heatsink]

Brake Booster Theory Part 2 of 2

4. Sheet

<img src='http://www.trials-forum.co.uk/gallery/1096150804/gallery_201_104_1096365568.jpg' border='0' alt='user posted image' />

This is the last one

<img src='http://www.trials-forum.co.uk/gallery/1096150804/gallery_201_104_1096365593.jpg' border='0' alt='user posted image' />

We can take these results and see what they can tell us about actual boosters.

For example: Any sheet type booster could be stiffer if the same weight of material can been arranged in an I-beam shape.

Compare the shape of the Carbon Fibre boosters by Whitehurst or the Carbonique with the Shimano one! The stiffness depends on two things remember, the shape (as we've examined) and the material. It should be pointed out that the CF across these boosters doen't have the same stiffness behaviour. The thing about Carbon Fibre is that the material stiffness can be fantastic in one direction, and next to nothing in another. This depends on the way the fibres were laid out before being glued together to form the composite. Basically the fibres resist being stretched but coil up like spaghetti if compressed. There a great article which gives more details about Carbon Fibre here.

Stiffness in other direction?

Just out of interest, imagine we rotate the cross-sections around 90 degrees. How will their stiffness compare? In the calculations so far we have found Ixx, which means I (Moment of Inertia) about the XX axis. To find the stiffness when rotated around, we need to find I about the YY axis as shown on the pic below. I don't think it's worth putting the calculations up, since you've already got a flavour of how these are done in the text above. If you want to try these for yourself, you'll need to understand the "Principle of Parallel Axis" - Another posh name for something rather simple. Have a look at This site for an explanation. (The same one as listed earlier in the post)

<img src='http://www.trials-forum.co.uk/gallery/1096150804/gallery_201_104_1096366605.jpg' border='0' alt='user posted image' />

Can I apply this "Moment of Inertia" stuff to bike frame tubes?

Yep. Have a look at this webpage which uses the same method to look at frame tubing shapes.

"Beam bending theory"

This stuff really isn't that difficult to understand. Try doing some calculations for yourself and you'll get your head around it. The trick is I've found that nothing's that complicated when you're interested in it!

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More useful info on the stiffness of boosters

Well I spent a few days getting my head around shapes for stiffness by looking through old Uni notes and doing some calculations. Then I posted my results above... Now I want to simplify things into a couple of bite-sized, easy to remember morsels :D

BITE SIZE #1

Imagine the beam shown below as part of the booster (a close up of the 12 o'clock position. So close that it would appear straight), with the bend from the braking forces exaggerated so you can actually see it!

1. The Neutral Surface - this middle surface sees no stress!

<img src='http://trials-forum.farseer.net/newimages/neutralsurf1.jpg' border='0' alt='user posted image' />

2. The Stress graph - Which shows that under braking forces the inner of the horse shoe is in tension, the top compression and the very middle (The Neutral Surface) stays the same! The stress rises the further out from the Neutral surface you move. Max compressive stress at outer surface A, Max Tensile stress at inner surface B

<img src='http://trials-forum.farseer.net/newimages/neutralsurf2.jpg' border='0' alt='user posted image' />

Here's a sentence, lifted straight from my Uni notes, which summarises the whole thing in a nutshell: "Optimise the design by reducing the volume of material near the N.A."

You want the material to be slapped at the furthest out points so that it is used to resist the biggest stresses which are to be found at the extremities.

"I" has it, "l" doesn't!! Simple as....:)

BITE SIZE #2

What effects stiffness? There are 2 things that decide it: The material's stiffness & the shape it's arranged into. You're pretty much stuck with the Young's modulus of Alu and Steel (A fancy grade of steel has the same stiffness as an indentically sized mild steel equivalent - little known fact!) For a chosen material then, the only way you can actually influence the stiffness is by working out the best shape to form your booster. Of course those best shapes are often in reality too expensive or even impossible to produce. A compromise needs to be found!

This is a useful formula I found the other day in my Uni notes (A last useful after over 5 years since I graduated!) It shows clearly the effect of the two variables I've just mentioned on how bent a straight beam will become when a bending moment acts on it:

M = EI/R

Where M is the bending moment

E, youngs modulus

I is our new friend

R is the bend radius of a once straight beam

If M is set, (from braking force)

E is set

then we want as big a bend radius (Infinity = straight) as poss I.e. biggest I possible! - The wonders of old Uni notes which are suddenly interesting again.

Q: What's a moment?

A moment is Force x distance from the fixed point, I.e. if we stuck a ruler protruding off a desk with a few books on the last inch, then stuck an apple at some distance along the free end. The moment would be:

Apple Weight (Newtons) x dist (m)

In words, the force doing the bending, (the weight of the apple) multiplied by the distance from where the ruler is fixed by the books

Just thought I'd try and simplify things because I do get a bit keyboard happy :">

Steve