Week 3 - Solving second order ODEs

INTRODUCTION:-

The Order differential equation (ODE) is used to measure the transient behaviour any respective system in this case we are considering motion of a simple pendulum as an example. motion generally works on NEWTONS second law , which usually have second order differential equation which is given below -

from 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.

PROCEDURE TO CODE:-

1. we have to give the inputs accordingly to the base , mass , gravity and length.
2. code the initial condition when `theta=0`
3. initiate the time points from 0 to 10.
4. by forming the ODE function we need to solve the ODE with respect to time and results.
5. Then we have to plot the results with respect to time.
6. Using FOR loop iteration we need create an animation .
7. In that FOR loop we have to give the initial and final co-ordinates to plot .
8. by giving the dimensions if base , string and bob to create a figure .
9. finally we need to create a movie by using movie command which is availabe in MATLAB.

MAIN CODE:-

clear all
close all
clc


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


%initial conditions
theta_0=[0;5];

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

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

%plotting
plot(t,results(:,1))
hold on
plot(t,results(:,2))
xlabel('time')
ylabel('plot')

%for loop Iteration method is used to create animation on pendulum
ct=1;
theta_1 = results(:,1);

for i= 1:length(results(:,1))
    
    % initial co-ordinates
    x_start= 0;
    y_start= 0;
    theta_2 = theta_1(i);
    
    % final co-ordinates
    x_end = 1*sin(theta_2);
    y_end = -1*cos(theta_2);
    
    % ploting for pendulum animation
    figure(2)
    % base
    plot([-1,1],[0,0], 'linewidth',4,'color','black')
    hold on
    % string
    line([x_start x_end],[y_start y_end],'linewidth',6,'color','b')
    hold on
    % bob
    plot(x_end,y_end,'marker','o','markersize', 28,'markerfacecolor','r')
    axis([-3 3 -3 3])
    pause(0.02)
    hold off
    M(ct)=getframe(gcf);
    ct=ct+1;
end
% creating movie
 movie(M)
 videofile = VideoWriter('ode_problem.avi','uncompressed AVI');
 open(videofile);
 writeVideo(videofile,M);
 close(videofile);

FUNCTION CALLING CODE:-

function [dtheta_dt]= ode_fun(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

OUTPUT:-

In this problem we are having two outputs.

1. The output obtained is velocity and displacement curves with respect to time .

2.Animated video of damped motion on simple pendulum with respect to time.

ERRORS FACED WHILE CODING:-

we need use the correct and case sensitive spellings while coding

so we need rectify once before executing the code.