Second order ODE

Aim:

To solve the second order of ODE(Pendulum) using matlab

Plot:

Formulae:

%2nd Order ODE
 clear all
 close all
 clc
 
 
 b=0.05;
 g=9.81;
 l=1;
 m=0.1
 
 % initial condition
 theta_0 = [0;3];

% time points
t_span = linspace(0,20,200);

% solve ODE
[t, results] = ode45(@(t,theta) ode_func(t, theta, b,g,l,m),t_span, theta_0);

plot(t,results(:,1))
hold on 
plot(t,results(:,2))
xlabel('time')
ylabel('plot')
ct=1;

for i = 1:length(results(:,1))
    x0=0; y0=0;
    angle=results(i,1);
    x1=1*sin(angle);
    y1=-1*cos(angle);
    plot([-1 1],[0,0],'linewidth',2,'color','g')
    hold on
    plot([x0 x1],[y0 y1],'linewidth',2,'color','b')
    hold on
    plot(x1,y1,'-o','markers',20,'markerfacecolor','r');
    
    axis([-1.5 1.5 -1.5 1.5])
    pause(0.03)
    hold off
    M(ct) = getframe(gcf);
    ct=ct+1;
end


movie(M)
videofile=VideoWriter('pendulumn.avi','Uncompressed AVI');
open(videofile)
writeVideo(videofile,M)
close(videofile)

%2Order ODE Function
function [dtheta_dt] = ode_func(t,theta,b,g,l,m)

theta1 = theta(1)
theta2 = theta(2)
dtheta1_dt = theta2;
dtheta2_dt = -(b/m)*theta2 - (g/1)*sin(theta1);
dtheta_dt = [dtheta1_dt; dtheta2_dt];
end

Output:

https://youtu.be/AfWLqNXhkU8

Discription:

We have to solve the second order of ODE using Matlab. The coding used in matlab are defined above. The ODE defines the movement of pendulum in fixed point. Swinging movement of pendulum in subject to gravity is measured in angle theta. The function of theta is defined in another fromula shown above using dtheta.

Conclusion:

Thus we have successfully calculated the second order of ODE in matlab.