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Ultimately the best solution to keep a rear brake aligned on a fixed wheel bike is to use a rear disc brake and sliding dropouts.
There is a much simpler solution.
The forward opening dropouts which were a standard feature on road frames for many decades (say 1910 - 1980, at least) are angled down towards the front. This is precisely to maintain the correct angle of the brake blocks (pads if you will) as the wheel moves in the
fork end.The attached pic of my Gillott shows this angle - I could find a clearer example, but I happen to have this one conveniently to hand.
A question: for non track use, can anyone explain why it is better to have rearward opening ends for fixed?
I can see disadvantages (eg: with mudguards it's difficult to get the wheel out to change a punctured tube), but no real advantages.
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I think Condor at one point made a fixed gear frameset with horizontal dropouts precisely for this reason, as the frame was designed for commuting/audax with mudguards. I had a Surly crosscheck set up fixed like this and it worked well.
As @hollow__legs says you end up with a gap at the front of the mudguard which doesn't look great. Velo Orange make a spring loaded nut and bolt to help - it holds the mudguard close to the tyre and then you can push against the spring if you need to remove the wheel https://velo-orange.com/products/spring-thing?srsltid=AfmBOoqZPrGmxj8zFrNDAFMahqMMoeH5FiIpwL2FKOM43NBh8PJqVQup
I’ve never found it much of a hassle aligning a rear brake on a fixed, although I’ve only ever had this set up once on a Pompino.
As others have said, tension the chain, align the brake, and you’re good to go. As long as you keep the chain reasonably well tensioned you shouldn’t have any issues.
Ultimately the best solution to keep a rear brake aligned on a fixed wheel bike is to use a rear disc brake and sliding dropouts.