Week 3 - Solving second order ODEs

Report on Pendulum challenge

AIM:

To write a code on a given ODE function and to make an animation of a pendulum.

GOVERNING EQUATIONS:

Some of the major equation and formulae used in this program coding are

d²Θ/dt² + (b/m) dΘ/dt + (g/L) sinΘ = 0

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.

OBJECTIVES:

• To create a code and animation of the given function.
• The output should show an oscillating pendulum.

GIVEN DATA:

L=1 m,

m=1 kg,

b=0.05.

g=9.81 m/s2.

Simulate the motion between 0-20 sec, for angular displacement=0,angular velocity=3 rad/sec at time t=0.

REPORT:

This program was done using MATLAB language.

Supplementary 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

Main code:

% Inputs for the program
b = 0.05;
g = 9.81;
l = 1;
m = 1;
% Initial value
theta_0 = [0;3];
% Time span
t_span = linspace(0,20,100);
% Calling the function for result
[t,results] = ode45(@(t,theta) ode_func(t,theta,b,g,l,m),t_span,theta_0);
% Plotting
figure(1)
title('result plotting')
subplot(2,1,1)
plot(t,results(:,1),'color','r')
ylabel('plot')
subplot(2,1,2)
plot(t,results(:,2),'color','b')
ylabel('plot')
xlabel('time')
figure(2)
plot(results(:,1),results(:,2),'color','g')
% Giving counter for the value
ct = 1;
% Creating a loop of values using for function
for i = 1:length(results(:,1))
     
    % Starting points/ origin points are as follows
        x0=0;
        y0=0;
        % End point rod/string is given as
        x1=l*sin(results(i,1));
        y1=-l*cos(results(i,1));
       % Plotting for animation
        figure(3)
        % Line plotting
        plot([-1 1],[0 0],'linewidth',6,'color','r');
        % String plotting
        plot([x0 x1],[y0 y1],'linewidth',2,'color','b');
        hold on
        % Marker plotting
        plot(x1,y1,'Marker','o','Markersize',10,'Markerfacecolor','g','Markeredgecolor','g');
      
        % Axis is changed to desired value
        axis([-1.5 1.5 -1.5 1.5])
        % Pause is given for getting a clear animation between intervals
        pause(0.03)
        % movie is specified here according to ct
        M(ct)= getframe(gcf);
        hold off
        % The value of ct should be changed with the loop 
        ct=ct+1;
end
% For making of animation we need to specify code, file name, open command,
% write command, close command.

movie(M)
v_f=VideoWriter('Pendulum_Challenge.avi','Uncompressed AVI')
open(v_f)
writeVideo(v_f,M)
close(v_f)

EXPLANATION:

Supplementry code:

1. Function command is done by using the syntax " [dtheta_dt] = ode_func(t,theta,b,g,l,m)"
2. Second order function is changed to two first order funcions.
3. [dtheta_dt] is function of both dtheta1_dt and dtheta2_dt.

Main code:

1. Inputs are given for required variables.
2. The initial value of theta is given.
3. The time span required is given.
4. The supplementary function is called and plotting for the results is done.
5. figure(1) is result plotting.
6. figure(2) is the plotting of found results.
7. for loop is created for values at different times.
8. plotting for animation is done.

ERRORS OCCURRED:

1. While plotting for animation the pendulum was facing up to overcome this error the y1 value was changed to negative.
2. Giving the marker to the pendulum was difficult.
3. At first, animation was not having a smooth transaction but when "sind and cosd" where changed to "sin and cos" the error did not happen.

CONCLUSION:

Thus the coding for ODE function and the making of pendulum animation was done.

ANIMATION LINK

https://drive.google.com/file/d/1Fzai4MchbuupomBQ2VQNxLlbNKmfv2tu/view?usp=sharing