Week 3 - Solving second order ODEs

Function Code

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/l)*sin(theta1);
    dtheta_dt = [dtheta1_dt; dtheta2_dt];
end


explanation
in this program the second ordee ODE is simplified to the first order ODE by introducing term theta1 and theta2


@fixed parameters
clear all 
close all
clc

b=0.05;
g=9.81;
l=1;
m=0.01;

%intial condition
theta_0 = [0;5];

%time points
t_span = linspace(0,10,500);

%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))
ylabel('Plot')
xlabel('Time')

%plotting
figure(1)
plot(t,results(:,1))
hold on
plot(t,results(:,2))
title('motion of pendulum')
xlabel('time in sec')
ylabel('angular velocity(rad/s)/displacement(rad)')
legend('displacement','angular velocity')

%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], [yo y1], '-o', 'MarkerSize',10, 'MarkerFaceColor', 'r', 'MarkerEdgeColor','r','linewidth',4)
    axis([-2 2 -2 21])
    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)

explanation
fixed parameters are lengh , mass, damping coe.and accelartion due to gravity.
time and angle given in intial condition