Lateral brush tire model

Similar to the longitudinal brush tire model, the lateral tire model does not include the formulation of sliding zone as it is highly non-linear. However the effect of tire during translational zone can be extended to show the sliding zone of tire. The formulation is also done is three different regions of the contact patch.

1. Adhesion region

Lateral force `F_y=C_(alpha)*alpha`(N/deg)

where `C_alpha` is the cornering stiffness =`1/2(K_y * l_t^2)`

`alpha` is slip angle in degrees

`K_y` is the lateral stiffness of the tire

Aligning moment `M_z=(C_alphal_talpha)/6` (Nm/deg)

2. Critical point for peak adhesion

Critical length `l_c=(u_pF_z)/(K_y*l_t*tanalpha_c)`

Critical slip angle ``Tanalpha_c=(u_pF_z)/(K_y*l_t^2)`` `K_y*l_t^2=2C_alpha`

`tanalpha_c=(u_pF_z)/(2C_alpha`

3. Large excitation ( Sliding)

Lateral force `F_y=u_pF_z-(u_pF_z)^2/(4C_alphatanalpha)`

Aligning moment `M_z=(u_pF_z)^2/(12l_tK_ytanalpha)*[3-(2u_pF_z)/(l_t^2K_ytanalpha)]`

The equations formulated can be modelled in MATLAB to see the effect of lateral force in slip angle generation. The modelling was done to study the effect of peak friction coefficient effect on lateral force and aligning moment.

The code for lateral brush tire model

%to plot lateral force vs slip angle
%to plot aligning moment vs slip angle

clear all
close all
clc

%input values
Fz=5000; %normal load in N
lt=0.2; %length of contact patch in m
mu=0.3; %peak co-eff of friction
mu_1=0.6;
mu_2=0.9;
Ca=50000; %cornering stiffness N/rad
Ky= 2*Ca/(lt^2); %lateral static stiffness N/m^2


%define critical slip
alpha_c= mu*Fz/(2*Ca);

for j=1:300 %slip angle from 0 to 15 degrees
alpha(j)=(j-1)*0.1*pi/180;

if alpha(j)<alpha_c %adhesion region

Fy(j)=Ca*alpha(j);%lateral force
Mz(j)=Ca*lt*alpha(j)/6; %aligning moment

else

Fy(j)=mu*Fz-(mu*Fz)^2/(4*Ca*tan(alpha(j)));
Mz(j)=(mu*Fz)^2/(12*lt*Ky*tan(alpha(j)))*(3-2*mu*Fz/(lt^2*Ky*tan(alpha(j))));

end
end

subplot(2,1,1)
plot(alpha*180/pi,Mz,'b','linewidth',2)
hold on
grid on
set(gca,'fontsize',15);
title('Aligning Moment vs Slip Angle');
xlabel('Slip Angle');
ylabel('Aligning Moment Mz');


subplot(2,1,2)
plot(alpha*180/pi,Fy,'b','linewidth',2)
hold on
grid on
set(gca,'fontsize',15);
title('Lateral Force vs Slip Angle');
xlabel('Slip Angle');
ylabel('Lateral Force N');


%define critical slip
alpha_c= mu_1*Fz/(2*Ca);

for j=1:300 %slip angle from 0 to 15 degrees
alpha(j)=(j-1)*0.1*pi/180;

if alpha(j)<alpha_c %adhesion region

Fy(j)=Ca*alpha(j);%lateral force
Mz(j)=Ca*lt*alpha(j)/6; %aligning moment

else

Fy(j)=mu_1*Fz-(mu_1*Fz)^2/(4*Ca*tan(alpha(j)));
Mz(j)=(mu_1*Fz)^2/(12*lt*Ky*tan(alpha(j)))*(3-2*mu_1*Fz/(lt^2*Ky*tan(alpha(j))));

end
end

subplot(2,1,1)
plot(alpha*180/pi,Mz,'g','linewidth',2)
ylabel('Aligning Moment Mz');
hold on

subplot(2,1,2)
plot(alpha*180/pi,Fy,'g','linewidth',2)
hold on

%define critical slip
alpha_c= mu_2*Fz/(2*Ca);

for j=1:300 %slip angle from 0 to 15 degrees
alpha(j)=(j-1)*0.1*pi/180;

if alpha(j)<alpha_c %adhesion region

Fy(j)=Ca*alpha(j);%lateral force
Mz(j)=Ca*lt*alpha(j)/6; %aligning moment

else

Fy(j)=mu_2*Fz-(mu_2*Fz)^2/(4*Ca*tan(alpha(j)));
Mz(j)=(mu_2*Fz)^2/(12*lt*Ky*tan(alpha(j)))*(3-2*mu_2*Fz/(lt^2*Ky*tan(alpha(j))));

end
end

subplot(2,1,1)
plot(alpha*180/pi,Mz,'r','linewidth',2)
legend('mu=0.3','mu_1=0.6','mu_2=0.8')


subplot(2,1,2)
plot(alpha*180/pi,Fy,'r','linewidth',2)
legend('mu=0.3','mu_1=0.6','mu_2=0.8','location','southeast')
axis([0 30 0 5000])