Week 4.1 - Solving second order ODEs

% splitting the second order derivative into two prts

function [dtheta_dt] = second_order_ode_pendulum (t,theta,g,l,m,b)

theta1 = theta(1)

theta2 = theta(2)

% need to look the equation in two two parts

dtheta1_dt = theta2

dtheta2_dt = (-(b/m)*theta2)-((g/l)*sin(theta1))

dtheta_dt = [dtheta1_dt;dtheta2_dt]

end

% code to simulate the pendulum motion

clear

close

clc

%given parameters

g = 9.81;

l = 1;

m = 1;

b = 0.05;

% we need to represent initial conditions to the problem

theta_0 = [0 3];

% given time interval or time span

time_given = linspace(0,20,400);

% write the ODE45 solver now

[t, outputs] = ode45(@(t, theta) second_order_ode_pendulum(t, theta, g,l,m,b), time_given,theta_0);

figure(1)

plot(t, outputs(:,1))

hold on

plot(t, outputs(:,2))

xlabel ('Time')

ylabel ('Angular displacement vs Angular velocity')

legend ('Angular displacement','Angular velocity')

grid on

figure(2)

plot(outputs(:,1), outputs(:,2))

xlabel ('position')

ylabel ('Angular velocity')

legend('Angular velocity')

theta2 = outputs(:,1);

ct = 1;

for i = 1:length(theta2)

theta3 = theta2(i);

% for fixed bar and movung bar of a pendulum

x0 = 0;

y0 = 0;

x1 = 1*sin(theta3);

y1 = -1*cos(theta3);

% giving axis, color for the pendulum bob, etc.,

figure(3)

plot([-1.1 1.1],[0,0],'linewidth',3)

hold on

plot([x0 x1],[y0 y1], 'g','linewidth',3)

hold on

plot(x1,y1,'o','markers',20,'markerfacecolor','m')

xlabel('x-axis')

ylabel('y-axis')

hold off

grid on

axis ([-1.5 1.5 -1.5 0.5])

% time frame to get the the video

pause(0.003)

M(ct) = getframe(gcf);

ct = ct+1;

end

% code to run and save the video

movie(M)

videofile = VideoWriter('Simple Pendulum.avi','uncompressed AVI');

open(videofile)

writeVideo(videofile,M)

close(videofile)

Code Explanation:-

Before writing the code, it's important to start with clear all, close all and clc commands since

Clear all - removes data from the workspace, freeing up system memory

Close all- deletes the current figure or the specified figure

clc - clears all input and output from the Command Window display, giving you a "clean screen

Function Code - In this we create a function code for splitting the second order derivation into two parts dtheta1_dt and dtheta2_dt and combine those two into one as dtheta_dt

Main Code - in main code we input all the given values, then the initial conditions are coded into the code. In second step we create an array for given time using 'Linspace' command then we create Solver code in thirsd stepusing 'ODE45' code then we plot all of our findingwe plot 2 figures.

The final step of the code is to create a movie of the animation of the Pendulum, so the movie command was used. The VideoWriter command was used to assign the video format, for this project .avi was used as the format. The open command was used to open the video. writevideo command was used to write the video with plot frames. Finally, the close command was used to close the video file.

USE OF THE FOLLOWING FUNCTIONS USED WHILE PROGRAMMING

FOR LOOP- For loop is used here for the iteration of the theta values from 0 to 90 degrees I is used as the outer loop and J is used as the inner loop ,i is used for the movement of the link 1 and J is used for the movement of link 2.

PAUSE - We have paused the code so that the process can be visualised more clearly .

Ct-Ct is used for saving the frames in order to create a movie of the frames

Getframe-Similarly getframe is used to capture the frame and then by fast movement of those frames it can be converted into a movie

Movie -Movie command is used to create a movie of the frames captured .

VideoWriter -It is to create a Video file of the movie created and save in the pc hard disc.

Plot - The plot command is used to plot the graphs of the coordinates provided.

Axis- The axis command is used to create the axis upon which we are going to plot the coordinates.

Linspace-The linspace command is useful in creating an array of values specified by the user.

RESULT:- https://youtu.be/VhCkijFMlVg