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I am trying to calculate the maximum braking force/deceleration for a bicycle. Note: the question assumes only the front brake is used

Free body diagram:

enter image description here enter image description here

Typo: in the image above, "$F_{a,\ max(dry)} = g X_3 / h$" is wrong, since it is an acceleration and it should be written as "$a_{max(dry)} = g X_3 / h$"

For dry tarmac where I know there is enough grip to get the rear wheel to lift I managed to calculate the max deceleration (result=6.8m/s^2). (this is the deceleration where all weight is transferred onto the front wheel (rear wheel will start lifting up at higher deceleration)).

From the moment equilibrium equation (using the front tire contact patch as the origin) I got the following equation for maximum deceleration before the rear tire starts lifting off (assuming there is enough grip): enter image description here
I used this to calculate max deceleration in dry conditions

However I am now trying to calculate the max deceleration in wet weather (lower coefficient of friction) where the front wheel slipping should be considered as an option.

I tried expressing Nf (normal force on front wheel contact patch) in terms of a (deceleration) and got the following result:

enter image description here
Which when simplified results in:

enter image description here
With this formula I can calculate Nf

Assuming that there won't be enough grip (in the wet) to make the rear wheel lift up.. I can then use Nf to determine Fbr (braking force). To determine maximum deceleration (assuming wheel slip occurs before rear lift up) I can use the equilibrium of forces in x-direction:

ΣFx=0=Fa-Fbr =ma-μNf --> a=μ*Nf/m

My questions:

  • Are my assumptions/calculations correct? If so: is there a better/faster way of doing this? To check if the rear wheel lifts up or if the tire slips first I will compare the max deceleration for slipping tire scenario and max deceleration for lifting rear wheel scenario and see which deceleration has a lower deceleration value to see which scenario will happen first

When braking some weight is transferred from the rear to the front wheel, but if the deceleration is too high the front wheel will slip. Gently increasing brake pressure will lessen the chance of front wheel slipping...

How should I go about determining max deceleration without the front wheel slipping, given a certain friction coefficient μ ? I can calculate the max braking force without slipping for a given Normal force on the front wheel but as soon as you start braking this normal force changes due to weight redistribution and I don't know how to account for this.

Thank you very much for any input! It is greatly appreciated

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  • $\begingroup$ final results: Max deceleration (dry) = +/- 6.7 m/s^2 Max deceleration (wet) = +/- 1.74 m/s^2 $\endgroup$ Jan 21 '20 at 17:37
  • $\begingroup$ Please explain whether your final result is taking into account the accepted answer or not, and add it to the question at the end $\endgroup$
    – FarO
    Jun 30 at 9:45
  • $\begingroup$ In your second image, where you write "Fa max, dry = g X3 /h" you made a mistake: it's not "Fa max, dry", but "a max, dry", since g is acceleration, X length and h length, which cancel each other and you are left with "acceleration". Your third image has the same formula but correctly marks it as acceleration. $\endgroup$
    – FarO
    Jun 30 at 9:55
  • $\begingroup$ I proposed an edit $\endgroup$
    – FarO
    Jun 30 at 9:59
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At a quick readthrough, your calculations seem to be correct and they also seem to answer the hard parts of your question. I also doubt the calculations can be made significantly simpler than your version.

I did a recalculation of the "wet condition" as follows:

  • With no lift of the rear wheel, there will moment equilibrium around the center of mass, i.e. $N_R x_2 + F_{BR}h = N_F x_3$
  • With no vertical acceleration of the rear wheel, we also have vertical equilibrium, i.e. $F_G = N_R + N_F$
  • Assume we are breaking hard enough to reach the friction coefficient of wheel and ground, $F_{BR} = \mu N_F$
  • Solving for the front wheel normal force gives $N_F = \frac{F_G x_2}{x_1-\mu h}$ as you already got. This is normal force adjusted for weight redistribution caused by braking.
  • So to calculate the corresponding deceleration, simply check the assumption that the rear wheel stays on the ground, $N_F < F_G$ and calculate $a=\frac{\mu N_F}{m}$.
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