1. Numerical Solution/Simulation
Euler Method, Applied to a Single 1st Order ODE
q1(t)=x(t), m=0 kg, c=1Ns/m, k=5 N/m

4. A particular simulation with x(0)=q1(0)=0, and
F(t)=u(t)=0, t<0, F(t)=u(t)=1 N, t>0

6. Euler Method, Applied to a Set of 1st Order ODE’s

10. q1(t)=x(t), q2(t)= x(t),
m=1 kg, c=0.5Ns/m, k=1 N/m
q1(0)=0, q2(0)=0

12. q1(0)=0 m, q2(0)= 0 m/s,
F(t)=u(t)=0, t<0, F(t)=u(t)=1 N, t>0

14. Code Modules Required to Use ODE45 in Matlab

15. q1(t)=x(t), q2(t)= x(t),
m=1 kg, c=0.5Ns/m, k=1 N/m
q1(0)=0, q2(0)=0

32. N=5000;
tend=100;
t=linspace(0,tend,N);
% Set up initial conditions for nonlinear model
ics=[ ; ];
% ics1=[ ;
% ];
%ics=[ics ics1];
for i1=1:6
xo=ics(:,i1)
% Simulate Nonlinear Model
[t,x]=ode45('InvPendNonlin',t,xo);
% Plot simulation of nonlinear model
subplot(2,1,1); plot(t,x(:,1),t,x(:,2));
%axis([0 tend ]);
xlabel('time (sec)');
ylabel('x1 (rad), x2 (rad/sec)');
title(['x1(0)=',num2str(xo(1)),'x2(0)=',num2str(xo(2))]);
grid on;
%hold on
subplot(2,1,2); plot(x(:,1),x(:,2));
%axis([ ]);
xlabel('x1 (rad)');
ylabel('x2(rad/sec)');
hold on;
pause;
end