Week 3 - Solving second order ODEs

AIM

To solve the second-order ode function using Matlab which depicts the motion of the pendulum and to create an animation of the same.

ODE fuction code

% ODE Function
function dtheta_dt = ode_funs(t,theta,b,g,l,m); 
theta1 = theta(1); 
theta2 = theta(2); 
dtheta1_dt = theta2; 
dtheta2_dt = ‐(b/m)*theta2‐(g/l)*sin(theta1);
dtheta_dt = [ dtheta1_dt ; dtheta2_dt];
end

Main program :-

% Fixed Parameters 

% Damping Co‐eff. 
b = 0.05;
% Acceleration due to gravity(m/s^2): 
g = 9.81;
% Mass(kg)  
m = 1; 
%Length(m)  
l = 1; 

 Initial condition 
theta_0 = [0;3]; 

% Time points 
tspan = linspace(0,20,200); 
% Solving ODE ==============>>> 
[t,results] = ode45(@(t,theta) ode_fun(t,theta,b,g,l,m), tspan, theta_0); 

Plotting
figure(1) 
plot(t,results(:,1)) 
hold on 
plot(t,results(:,2)) 
xlabel('Time') 
ylabel('Velocity (Red) & Displacement (Blue)') 

% Animation ============>>> 
ct = 1; 
for i = 1:length(results(:,1)) 
x0 = 0; 
y0 = 0; 
x1 = l*sin(results(i,1)+pi); 
y1 = l*cos(results(i,1)+pi); 
figure(2) 

plot([x0 x1],[y0 y1],'-o','MarkerSize',10,'MarkerFaceColor','r','MarkerEdgeColor','r','linewidth',4)
axis([-2 2 -2 2]) 
hold on 
plot([-1 1],[0 0],'linewidth',5,'color','[0 0.3 0]') 
M(ct) = getframe(gcf)
hold off 
ct = ct + 1; 
end 

% Movie & Videofile ===========>>> 
movie(M) 
v_f = VideoWriter('Pendulum_Challenge.avi','Uncompressed AVI') 
open(v_f) 
writeVideo(v_f,M) 
close(v_f)

youtube link animation