Week 4.1 - Solving second order ODEs

Solution:

• From above question we had some parameters like,
  • L=1 metre,
  • m=1 kg,
  • b=0.05.
  • g=9.81 m/s2
  • and t = 0 to 20 sec,
  • Displacement = 0
  • angular velocity = 3 rad/sec
• So now we need to create psuedocode for programming,

Psuedocode and explaination for programming:

• Asusual we start with close all,clear all and clc
• Then give the input like
• b = 0.05; //damping coefficient
• g = 9.81; //gravity
• l=1; //length
• m=1; //mass
• At the initial condition the displacement was 0 and angular velocity is 3,so for that we have to create the column vector like,
  • theta_0 =[0;3];
• Then we need to create line for Time points that,here we create the t_span array that starts from 0 to 20 sec with 500 points,using linspace command.
  • t_span = linspace (0,20,500);
• Then we need to create line for ODE that,here t is time array and we get results as a column vector.
• Then here ode45 is inbuild ode solver in matlab,
• Now we need to define the function as @(t,theta) for time and theta value,
• Next we need to use the function that 'ode_func',
• It is a predefine fuction that contains the value of input parameters
  • [t,results] = ode45(@(t,theta) ode_func(t,theta,b,g,l,m),t_span,theta_0);
  • function dydt = ode_func(~, theta, b, g, l, ~)
    % ODEs for a simple pendulum
    dydt = zeros(2, 1);
    
    % First ODE: d(theta)/dt = omega
    dydt(1) = theta(2);
    
    % Second ODE: d(omega)/dt = -b*omega - (g/l)*sin(theta)
    dydt(2) = -b * theta(2) - (g / l) * sin(theta(1));
    
    end

• In this function we have the ode equations for to find the angles that theta 1 and theta 2.
• Then comming to the plotting,we need to create plotting in column vector methode like
  • figure(1)
    plot(t,results(:,1))
    hold on
    plot(t,results(:,2))
• Then we need to create xlabel,ylabel,legend and title as like below.
  • ylabel ('Plot')
    xlabel('Time')
    legend('Location', 'best');
    title('Solving 2nd order equation')
• Then comming to the animation setup
• Plot empty lines for Angular Displacement and Angular Velocity Set labels and title for the plot
• Then Create Figure 3 Plot empty line for Simple Pendulum Animation Set axis limits, labels, and title for the animation plot Preallocate video file 'pendulum_animation.avi'
• Next Open the VideoWriter object with the specified filename For each time step in t Calculate pendulum bob coordinates (x, y) based on results(i, 1)
• Then Update animation plot with the new coordinates Force MATLAB to update the figure (drawnow)
• And Capture the current frame for the video Write the frame to the video file Pause for a short duration
• Finally Close the video file Set labels and title for the animation plot in Figure 3.

MATLAB CODE:

%Solving second order ODES

close all

clear all

clc

%input

b = 0.05;

g = 9.81;

l=1;

m=1;

%Angular displacement = 0 ,Angular velocity = 3.

%Initial condition

theta_0 =[0;3];

%Time points

t_span = linspace (0,20,500);

%Solve ODE

[t,results] = ode45(@(t,theta) ode_func(t,theta,b,g,l,m),t_span,theta_0);

%Ploting

figure(1)

plot(t,results(:,1))

hold on

plot(t,results(:,2))

ylabel ('Plot')

xlabel('Time')

legend('Location', 'best');

title('Solving 2nd order equation')

% animation setup

figure(2);

h = plot(NaN, NaN, 'o-', 'LineWidth', 2);

axis([-1.5 1.5 -1.5 1.5]);

xlabel('X-axis');

ylabel('Y-axis');

title('Simple Pendulum Animation');

grid on;

hold on;

% preallocate video

vidObj = VideoWriter('pendulum_animation.avi');

open(vidObj);

% animation loop

for i = 1:length(t)

x = l * sin(results(i, 1));

y = -l * cos(results(i, 1));

set(h, 'XData', [0, x], 'YData', [0, y]);

drawnow; % Force MATLAB to update the figure

% capture frame for video

frame = getframe(gcf);

writeVideo(vidObj, frame);

% add labels and title

xlabel('Horizontal Displacement (m)');

ylabel('Vertical Displacement (m)');

title('Simple Pendulum Motion');

pause(0.01); % Adjust the pause duration if needed

end

% close video file

close(vidObj);

% add labels and title

xlabel('Horizontal Displacement (m)');

ylabel('Vertical Displacement (m)');

title('Simple Pendulum Motion');

function dydt = ode_func(~, theta, b, g, l, ~)

% ODEs for a simple pendulum

dydt = zeros(2, 1);

% First ODE: d(theta)/dt = omega

dydt(1) = theta(2);

% Second ODE: d(omega)/dt = -b*omega - (g/l)*sin(theta)

dydt(2) = -b * theta(2) - (g / l) * sin(theta(1));

end

OUTPUT PLOT:

AVI Screen shot:

ANIMATION GOOGLE DRIVE LINK:

https://drive.google.com/file/d/1unLA77FyzmCtGT5jXCXt7Q7whulVzmds/view?usp=sharing

Conclusion:

• By using this Second order ode equation in MATLAB we can creation pendulam animation.