Week 3 - Solving second order ODEs

Aim:

Using Matlab : * Solve the second order ODE which represents the motion of simple pendulum with damping and

* TO Stimulate the motion of the pendulum between time_span of 0-20sec for angular displacement=0, angular velocity=3rad/sec at time t=0.

CODE AND EXPLANATION:

Program 1:To create a simple pendulum with damper:

clear all
close all
clc

% input parameters:
% b= damping co-effecient
% g= gravity
% l= length
% m= mass

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

%initial conditions we have two 1st order ODE 
%one for velocity and other for displacement
 theta_o = [0,5];
%Here initial displacement is 0 & initial velocity is 5m^2

%Time points
 t_span = linspace(0,20,300);
 
%solve ODE ;
[t,result]=ode45(@(t,theta)ode_func(t,theta,b,g,l,m),t_span,theta_o);
ct=1;
%where ode45 is an in-built solution in matlab

figure(1)
plot(t,result(:,1))
hold on
plot(t,result(:,2))
title('Motion of a simple damper pendulum')
xlabel('time')
ylabel('plot')
legend('Displacement(rad)','velocity(rad/s)')


for i=1:length(result(:,1))
    ref=result(i);
    
    %co-ordinates
    x0=0;
    y0=0;
    
    x1= 1*sind(ref);
    y1=1*(-cosd(ref));
    
    figure(2)
    plot ([x0 x1],[y0 y1],'linewidth',3,'color','b');
    axis([-1 1 -1 1]);
   
     %get individual frames to create a animated one
        %M is the frame we capture
    M(ct) = getframe(gcf);
    ct=ct+1;

    
end

%conversion of plot into video
movie(M)
videofile = VideoWriter('ODE_MAIN.avi','uncompressed AVI');
open(videofile)
writeVideo(videofile,M)
close(videofile)

Description:

Here * inorder to create an animation ,all the frames corresponds to each time_span value needs to be saved. we use For loop function with a required command to captures each frame and convert into graphical representation of a pendulum.

* ode45 is an inbuilt solution in matlab.

* Conversion of plots into video Under the videofile command: where the videoWriten with the required file name & type(.avi)

CODE AND EXPLANATION:

PROGRAM 2: Calling the ode_func in the main program:

Where * The 2nd order ODE is splitted into two 1st order ODE,then the 1st order ODE is solved individually.

* Here we call the function as ode_func.

* Calling the function ode_func into the main function.

function [dtheta_dt]=ode_func(t,theta,b,g,l,m)
 
 %solution array
  theta1 = theta(1);
  theta2 = theta(2);
 
 % 1st ODE
  dtheta1_dt=theta2;
 % 2nd ODE
  dtheta2_dt=-(b/m)*theta2-(g/l)*sin(theta1);
 
  dtheta_dt=[dtheta1_dt;dtheta2_dt];
 
end

We get the successive result of the program for damping effect on the pendulum in motion as per the assigned values.

Output:

youtube link attached :

https://youtu.be/Q9kgjzK1NQE