% SecondOrderStep.m
% AJS  Feb2009

% parameters
m = 2;
b = 5;
k = 2;
F = 1;

% initial conditions
y0 = 0;  ydot0 = 0;

% homogeneous solution
% m*s^2 + b*s + k = 0
s1 = (-b + sqrt(b^2-4*m*k))/(2*m);
s2 = (-b - sqrt(b^2-4*m*k))/(2*m);

omegan = sqrt(k/m);  zeta = b/2/sqrt(k*m);

% particular solution
yp = F/k;

% apply initial conditions
k = [1 1 ; s1 s2]\[y0-yp ydot0]';

% complete solution
t = [0:0.01:20];
y = k(1)*exp(s1*t) + k(2)*exp(s2*t) + yp;

% close all
figure
subplot(1,2,1)
hold on
plot(real(s1),imag(s1),'x')
plot(real(s2),imag(s2),'x')
xlabel('Re(s)')
ylabel('Im(s)')
title(['s_1 = ' num2str(s1) ', s_2 = ' num2str(s2)])
axis([-2 2 -2 2])
axis equal
grid on

subplot(1,2,2)
hold on
plot(t,y)
plot(t,yp*ones(length(t),1),'r--')
xlabel('t')
ylabel('y')
title(['\omega_n = ' num2str(omegan) ', \zeta = ' num2str(zeta)])
grid on