Postby Chris Jeggo » 5 Oct 2018, 7:32pm
To slow a vehicle requires a horizontal force applied at the centre of gravity (CG). The actual braking force is applied at the tyres, thus generating a couple, or turning moment. In normal braking the vehicle does not topple because an opposing couple is generated by weight transference, the load (vertical load, implied rather than explicitly stated in what follows) on the front wheel increases while that on the back decreases.
Now let's do a thought experiment, which is easier than doing it out on the road. You are travelling at some speed and wish to slow down. Apply the back brake without locking the back wheel. There are two results - deceleration and some weight transference. Now hold that back brake setting and apply and gradually increase front braking. Both deceleration and weight transference increase. While the back wheel is still rolling the force its tyre exerts on the road remains constant, because we are holding constant the back brake setting. However, as the load on the back wheel decreases we will eventually reach the point where the back wheel slides, when rear wheel load multiplied by coefficient of friction (CF) reduces below rear wheel braking force. From this point on, as front braking continues to increase, weight transference continues to increase, rear wheel load continues to decrease, so rear wheel braking force, being CF times load, continues to decrease.
We still haven't stopped, so what happens next? That depends on the answer to the "topple or slide" question. For a bike (not recumbent) with decent tyres on a dry, decent road surface the answer is "topple". This happens when weight transference reaches its limit, the point where the rear wheel load reduces to zero, and therefore also the point at which the rear wheel braking force (CF times load) becomes zero. At this point the back brake has ceased to have any effect, so might as well not be on.
So in theory, and in "topple" conditions but not in "slide" conditions, maximum braking is achieved by front brake alone. But can the theoretical maximum be achieved in practice? Only to the extent that a cyclist can match the performance of a well designed anti-lock braking system, and the ABS servo loop can have a faster reaction time than the human.
If the front braking is less than maximum then clearly some braking can be achieved by the back wheel, but the total will be less than the maximum.
But maybe the "slide" condition is more interesting, because a front wheel skid is greatly to be avoided in that it is very likely to have you off pretty quickly. Just before the front wheel locks there is still some load on the back wheel so further braking can be obtained. The maximum braking force obtainable equals total vehicle weight times CF, but the necessary front-rear distribution of braking force to achieve this depends on the CF because it depends on the braking couple which depends on the total braking force. When the CF is on the point of being large enough to produce toppling rather than sliding then the situation is as above. The other extreme is black ice having a tiny CF, when the maximum achievable braking force is so small that to a first approximation weight transference can be ignored and the front-rear braking distribution needs to match the front-rear weight distribution.
The theory above does not describe the real-world situation exactly. I have ignored the difference between static and sliding friction, which are distinguished in school-level friction theory, and neither tyres nor road surfaces behave exactly like 'classical' friction materials. But even so, classical theory is a good guide to what happens in practice, and when you reach the limit you will get either toppling or sliding. If you correctly judge that the braking limit is due to toppling and you reckon you can react to the rear wheel lifting quickly enough to avoid an 'imperial crowner' then you should use front brake only to stop as quickly as possible. In other conditions, depending on how risk-averse you are, you can achieve more braking with two brakes than one, although in many conditions the contribution of the rear brake will be but a fraction of that of the front brake.
