Problem Set-1 c Week 11Challenge : Transient Handling
Fs = 0.97Hz
Fus = 12.63Hz
Freq = 1.45Hz
Zs/Zo = 0.799
Zus/Zo = 0.868
Zo = 5cm
Zs= 4cm
Zus = 4.34cm
Code for FWD Car :
% INPUTS
%FWD Passenger Cars
m1=907.184; %weight (kg)
mf1=65*m1/100; % front weight (kg)
mr1=m1-mf1; % rear weight (kg)
Izz1=1128; % İnertia (kgm^2)
L1=2.539; % wheelbase (m)
a1=0.89; % cg to front axle (m)
b1=L1-a1; % cg to rear axle (m)
cf1=61172; %cornering stiffness front (N/rad)
cr1=50976; %cornering stiffness rear (N/rad)
u1=20; % forward speed (m/sec)
%--------------------------------------------------------------------------
Yb1=abs(cf1+cr1);
Yb2=abs(cf2+cr2);
Yr1=abs((a1*cf1-b1*cr1)/u1);
Yr2=abs((a2*cf2-b2*cr2)/u2);
Nb1=abs(a1*cf1-b1*cr1);
Nb2=abs(a2*cf2-b2*cr2);
Nr1=abs((a1^2*cf1-b1^2*cr1)/u1);
Nr2=abs((a2^2*cf2-b2^2*cr2)/u2);
%--------------------------------------------------------------------------
Wn1=sqrt((Yb1*Nr1+Nb1*m1*u1-Yr1*Nb1)/(m1*u1*Izz1));
Wn2=sqrt((Yb2*Nr2+Nb2*m2*u2-Yr2*Nb2)/(m2*u2*Izz2));
zeta1=-(m1*u1*Nr1+Yb1*Izz1)/(2*Wn1*m1*u1*Izz1);
zeta2=-(m2*u2*Nr2+Yb2*Izz2)/(2*Wn2*m2*u2*Izz2);
Wd1=Wn1*sqrt(1-zeta1^2);
Wd2=Wn2*sqrt(1-zeta2^2);
theta1=atan(sqrt(1-zeta1^2)/zeta1);
theta2=atan(sqrt(1-zeta2^2)/zeta2);
tr1=(3.14-theta1)/Wd1;
tr2=(3.14-theta2)/Wd2;
%--------------------------------------------------------------------------
%State Space Matrix
A=[-(cf1+cr1)/(m1*u1),(b1*cr1-a1*cf1)/(m1*u1)-u1
(b1*cr1-a1*cf1)/(Izz1*u1), -(a1^2*cf1+b1^2*cr1)/(Izz1*u1)];
B=[cf1/m1; a1*cf1/Izz1];
C_lat=[1, 0]; D_lat=0;
C_yaw=[0, 1]; D_yaw=0;
C_acc=A(1,:)+u1*[0,1]; D_acc=B(1);
C=[C_lat; C_yaw; C_acc];
D=[D_lat; D_yaw; D_acc];
t=0:0.01:6 ;
U=(5*3.14/180)*sin(7*2*3.14*t);
sys = ss(A,B,C,D)
Y=lsim(sys,U,t);
sys.inputname={'steering'};
sys.outputname={'lat';'yaw';'acc'};
figure(1)
bodeplot(sys({'yaw'},'steering'))
grid on
title('Yaw Gain Frequency Response')
figure(2)
step(sys)
grid on
figure(3)
%Lateral Speed
subplot(4,1,1)
plot(t,Y(:,1),'r','linewidth',2)
grid on
title('Lateral Speed')
xlabel('Time (sec)')
ylabel('m/sec')
%Yaw Rate
subplot(4,1,2)
plot(t,Y(:,2)*180/3.14,'r','linewidth',2)
grid on
title('Yaw Rate')
xlabel('Time (sec)')
ylabel('deg/sec')
%Lateral Acc
subplot(4,1,3)
plot(t,Y(:,3),'r','linewidth',2)
grid on
title('Lateral Acc')
xlabel('Time (sec)')
ylabel('m/sec^2')
%Steering İnput
subplot(4,1,4)
plot(t,U*180/3.14,'r','linewidth',2)
grid on
title('Steering angle')
xlabel('Time (sec)')
ylabel('deg')
Code for F1 car :
% INPUTS
% F1 Race Cars
m1=779.7246; %weight (kg)
mf1=40*m1/100; % front weight (kg)
mr1=m1-mf1; % rear weight (kg)
Izz1=865; % İnertia (kgm^2)
L1=2.9017; % wheelbase (m)
a1=1.7404; % cg to front axle (m)
b1=L1-a1; % cg to rear axle (m)
cf1=276800; %cornering stiffness front (N/rad)
cr1=429730; %cornering stiffness rear (N/rad)
u1=60; % forward speed (m/sec)
%--------------------------------------------------------------------------
Yb1=abs(cf1+cr1);
Yb2=abs(cf2+cr2);
Yr1=abs((a1*cf1-b1*cr1)/u1);
Yr2=abs((a2*cf2-b2*cr2)/u2);
Nb1=abs(a1*cf1-b1*cr1);
Nb2=abs(a2*cf2-b2*cr2);
Nr1=abs((a1^2*cf1-b1^2*cr1)/u1);
Nr2=abs((a2^2*cf2-b2^2*cr2)/u2);
%--------------------------------------------------------------------------
Wn1=sqrt((Yb1*Nr1+Nb1*m1*u1-Yr1*Nb1)/(m1*u1*Izz1));
Wn2=sqrt((Yb2*Nr2+Nb2*m2*u2-Yr2*Nb2)/(m2*u2*Izz2));
zeta1=-(m1*u1*Nr1+Yb1*Izz1)/(2*Wn1*m1*u1*Izz1);
zeta2=-(m2*u2*Nr2+Yb2*Izz2)/(2*Wn2*m2*u2*Izz2);
Wd1=Wn1*sqrt(1-zeta1^2);
Wd2=Wn2*sqrt(1-zeta2^2);
theta1=atan(sqrt(1-zeta1^2)/zeta1);
theta2=atan(sqrt(1-zeta2^2)/zeta2);
tr1=(3.14-theta1)/Wd1;
tr2=(3.14-theta2)/Wd2;
%--------------------------------------------------------------------------
%State Space Matrix
A=[-(cf1+cr1)/(m1*u1),(b1*cr1-a1*cf1)/(m1*u1)-u1
(b1*cr1-a1*cf1)/(Izz1*u1), -(a1^2*cf1+b1^2*cr1)/(Izz1*u1)];
B=[cf1/m1; a1*cf1/Izz1];
C_lat=[1, 0]; D_lat=0;
C_yaw=[0, 1]; D_yaw=0;
C_acc=A(1,:)+u1*[0,1]; D_acc=B(1);
C=[C_lat; C_yaw; C_acc];
D=[D_lat; D_yaw; D_acc];
t=0:0.01:6 ;
U=(5*3.14/180)*sin(7*2*3.14*t);
sys = ss(A,B,C,D)
Y=lsim(sys,U,t);
sys.inputname={'steering'};
sys.outputname={'lat';'yaw';'acc'};
figure(1)
bodeplot(sys({'yaw'},'steering'))
grid on
title('Yaw Gain Frequency Response')
figure(2)
step(sys)
grid on
figure(3)
%Lateral Speed
subplot(4,1,1)
plot(t,Y(:,1),'r','linewidth',2)
grid on
title('Lateral Speed')
xlabel('Time (sec)')
ylabel('m/sec')
%Yaw Rate
subplot(4,1,2)
plot(t,Y(:,2)*180/3.14,'r','linewidth',2)
grid on
title('Yaw Rate')
xlabel('Time (sec)')
ylabel('deg/sec')
%Lateral Acc
subplot(4,1,3)
plot(t,Y(:,3),'r','linewidth',2)
grid on
title('Lateral Acc')
xlabel('Time (sec)')
ylabel('m/sec^2')
%Steering İnput
subplot(4,1,4)
plot(t,U*180/3.14,'r','linewidth',2)
grid on
title('Steering angle')
xlabel('Time (sec)')
ylabel('deg')