Vehicle Dynamics - Four wheel Steering

1. Deriving formula for del_r/del_f:

The steady-state representation for 4-wheel steering is,

Where,

v = Lateral Speed.

R= Yaw Rate

C_af = Front Cornering Stiffness

C_ar = Rear Cornering Stiffness

a = Distance Between Front Axle and CG

b = Distance between Rear Axle and CG

m = Mass of the vehicle

U₀ = Speed of the vehicle

I_z = Yaw moment of Inertia.

For Steady state, We consider dv/dt = 0 and dr/dt = 0. Therefore,

For Zero side slip angle, the lateral speed and yaw rate is 0

i.e. v = 0, r = 0

Therefore,

2. Plot of the ratio against forward speed:

Octave Code:

% Rear and front steer angle ratio and forward speed plot.

clear all
close all
clc

% Inputs
m = 1000; % kg, Mass
l = 2.54; % m, Wheelbase
a = 1.14; % m, Distance between front axle and CG
b = l-a; % m, Distance between rear axle and CG
Caf = 2400*57.2958; % N/rad, Front cornering stiffness
Car = 2000*57.2958; % N/rad, Rear cornering stiffness

% Speed.
u = 0:0.5:100;

% Main Loop
% Steer angle ratio.
for i = 1:length(u)
del_ratio(i) = ((a*m*u(i)^2*Caf)-(b^2*Caf*Car)+(a*b*Caf*Car))/((b*m*u(i)^2*Car)+(a^2*Caf*Car)+(a*b*Caf*Car));
end

% Plot
figure(1)
plot(u,del_ratio,'linewidth',2);
xlabel('Speed (m/s)');
ylabel('Steer angle ratio');
title('4 wheel steering');

saveas(1,'4 Wheel Steering','jpeg')

Output Plot:

Google Drive link for Octave Code:

https://drive.google.com/file/d/1UY6sWZP2dZsJVgA6t5Hb4qsmsf_wowly/view?usp=sharing

3. Conclusion:

From the plot, we get to know that the Steering angle ratio spikes exponentially until the steady state ratio of 0.96, when the vehicle is at low speed. When the speed increases, at a specific point, the steering angle ratio saturates and remains unchanged.