Solving second order ODEs using Matlab

Aim : To Solve a Second Order Differential Equation And Animate Using Matlab.

The second order Differential Equation

d^2(theta)/dt^2 + b/m * d(theta)/dt + g/L *sin(theta) .

In the above equation,

g = gravity in m/s2,

L = length of the pendulum in m,

m = mass of the ball in kg,

b=damping coefficient.

Inputs:

L=1 metre,

m=1 kg,

b=0.05.

g=9.81 m/s2.

Explanation:

1. A Higher order Differnetial Equation is suppressed into a number of first Order Differential Equations .

2.Here the Differential Equation is of Second Order so It is Suppressed into two First Order Differential Equations to solve a second order differential equation using Matlab.

3. Here the second Order Differential Equations tells about the rate of change of pendulum in a given time ,

so the Input Variables time , Displacement i.e theta are given :

theta_0 = [0;5];

t_span = linspace(0,10,100);

4.To solve above Differential Equation is used which is

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

Programm to solve the second order Differential Equation using Matlab is as follows:

clear all
close all
clc
%state variables

b = 0.05;
g = 9.81;
l = 1;
m =0.1;

%intial condition

theta_0 = [0;5];

% time points

t_span = linspace(0,10,100);


%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')

% Illstating Animation of pendulum.

ct = 1;
for i = 1:length(results(:,1))
    % Generating the Data.
   
   x0 = 0;
   y0 = 0;
   x1 = 1*sin(results(i,1));
   y1 = -1*cos(results(i,1));
   
   % plot the entire pendulum trajectory.

   figure(2)

   plot([-1,1],[0,0],'linewidth',2,'color','g');
   hold on
   line([x0 x1],[y0 y1],'linewidth',2,'color','r');
   hold on
   plot(x1,y1,'o','markersize',15,'markerfacecolor','b');
   axis([-1.5 1.5 -1.5 0.5]);
    pause(0.10);
   hold off
   M(ct)=getframe(gcf);

   ct = ct+1;
end

% Creating a Movie.


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

output:

Animation Youtube Link:

https://youtu.be/asM6Y0WAogU