While braking your car,all the wheels should lock together. If front and rear wheels are not locking together then their will be the loss of directional control or Directional stability is loss.
Here, i am going to tell you that in which proportion the braking force should be distributed to get the optimized braking which is the locking of front and rear brakes together.
If front and rear wheels lock together then:-
All braking force generated is used.
Fb(max)=Fb=μ*W
Fbf(max)=Fbf
Fbr(max)=Fbr
Where ,
Fb(max) is the force generated between the contact of ground and tire,this is basically the frictional force or adhesion force.
Fb is the force generated by the brakes at time of braking Braking force. Fb is given by ,
Fb= (Tb-ΣI*α )/r ,where Tb is the applied brake torque, I is the rotating inertia connected with the wheel being decelerated, α is the corresponding angular deceleration, and r is the rolling radius of the tire.(Fb= Fbf+Fbr)
We know
Wf=L2*W/L + h/L[Fb+fr*W] (Load on front tire while braking ) (1)
Wr=L1*W/L - h/L[Fb+fr*W] (Load on rear tire while braking) (2)
where ,
Fb= Fbr +Fbf
R=Rf+Rr=fr*W (Where R is the rolling resistance)
Fbf=kbf*Fb (3)
Fbr=kbr*Fb (4)
Where,
kbf & kbr are the proportion in which braking force should be distributed.
kbf+Kbr=1
At time of braking when all wheels lock together then
Fb(max)=Fb=μ*W (5)
Putting equation (5) in (1) & (2) we get:-
Wf=[L2+h*( μ+fr)]*W/L (6) Wr=[L1-h*( μ+fr)]*W/L (7)
Fbf(max)= μ*Wf
Fbf(max)= μ*Wr
So if we want to lock front or rear wheel then force generated by brakes which is Fb should be greater then the force generated between tire-ground contact i.e Fb(max).
If both wheels have to lock together then braking force should be distributed in that ratio which is proportion to the weight transfer.
hence,
Kbf/Kbr = Wf/Wr,
From (6) & (7) we get;
Kbf/Kbr = [ L2 + h(μ+fr) ]/[ L1 - h(μ+fr) ] (8)
The above ratio of Kbf/kbr is ideal ratio as this ratio changes when the location of COG cahnges i.e L1,L2,& h changes. It also changes when μ changes as μ varies road to road .
Conclusion
To maintain the optimized braking and make all the wheels to lock together the brake force should be distributed in the ratio of Kbf/Kbr or near to this ratio.