Analysis of Wheel locking while braking on different road conditions

Objective : To understand the effect of road conditions i.e. different cofficient of friction on the limiting values of decelaration of wheels,both front & rear, for locking.

Calculations :

For front wheel locking conditions, limiting value of decelaration in Gs obtained by below eqn

`(d/g)_max =[(mu*L_2/L+Kb_f *f_r)/(Kb_f-mu*h/L) ]`

For rear wheel locking conditions, limiting value of decelaration in Gs obtained by below eqn

`(d/g)_max =[(mu*L_1/L+(1-Kb_f )*f_r)/((1-Kb_f)+mu*h/L) ]`

The above eqns are obtained by deriving the max deceleration when wheels are locking neglecting Aerodynamic and traction forces

The limiting value of decelaration for front & rear wheels locking are calculated for range of braking force fraction on front wheels and for different coefficient of friction(mu) values

Assumed the typical dimensions for a passenger vehicle :

L = Wheelbase

L1 = Distance of CG from front axle

L2 = Distance of CG from rear axle

h = height of CG from ground

MATLAB CODE :

clc
clear all
%%input parameters 
%Geometry
L1 = 1.3;
L = 2.9;
L2 = L-L1;
h = 0.5;  %CG

%loads & coeffcients
fr = 0.015 ;
dbyg_f = [];
dbyg_r = [];
Kb_f_data = [];

%%Calculation of front wheel locking & Rear wheel locking

for mu = 0.3:0.1:0.6
    
for Kb_f = 0.3:0.01:0.9
dec_g_f = (mu*L2/L+Kb_f*fr)/(Kb_f-mu*h/L);
dec_g_r = (mu*L1/L+(1-Kb_f).*fr)/((1-Kb_f)+mu*h/L);
Kb_f_data = [Kb_f_data;Kb_f];
dbyg_f = [dbyg_f;dec_g_f];
dbyg_r = [dbyg_r;dec_g_r];
end

%%Plot

sz = size(Kb_f_data);
n =sz(:,1);
Kb_f1 = linspace(0.5,0.9,n);
plot(Kb_f1,dbyg_f,Kb_f1,dbyg_r);
hold on

dbyg_f = [];  %clearing the data of the variables for next plot
dbyg_r = [];
Kb_f_data = [];

end
hold off

Plot for Variation of Limiting values of d/g for wheel locking with varation coeff. of friction :

Observation :

The plot is better visualised in excel charts

> The curves of max deceleration for front and rear intersect each other for respective value of coefficient of friction(mu).

> As the coefficient of friction(mu) increases, the intersection of the curves shifts upwards & towards right.

Conclusion :

> The limiting value of deceleration for which both front & rear wheels lock, goes on increasing with the increase in coefficient of friction(mu).

> The intersection of the respective curves is important as it indicates the limiting value of deceleration for which both front & rear wheels locks which is desired. Thus, better braking can be obtained on the roads having conditions that offer better cofficient friction.

>The ABS, is hence required for avoiding the locking of wheels during sudden braking.