Solving second order ODEs of a simple pendulum with damping

- 1. Aim: To solve an ODE equation to plot the graph and do an animation of the motion of a pendulum corresponds to the displacement value.

- 2. Objectives: 

- To solve an ODE equation.

- To generate the plot of displacement and velocity.

- To get an animation of the motion of pendulum with respect to displacement.

- 3. Method:

- 3.1 Generate the plot using ODE expression:

- ODE has a lot of application in vehicle dynamics, where suspension system can be modelled as a spring, mass and damper. We can solve the ODE and figure out how the behaviour will be.

- There are two 1st order ODE, one for displacement and 2nd for velocity.

- To solve the ODE, first, it has to be converted into a 1st order differential equation. For example, if we are having a 3rd order differential equation, then we have to generate 3 1st order equation in order to solve. 

- 3.2 Generate the motion of a pendulum and capture the movie:

- In order to generate the motion of a pendulum, as it is a continues process which is repeated until some values a ' for loop' is used in order to simulate the motion.

- Here the motion is generated by continuously changing the values of x and y component, that is the reason the iteration value for 'for loop' is taken as displacement and it is stored in 'ang_displacement'. 

- The displacement has two values in its matrix which are displacement and velocity. In order to assign the displacement values, results(;,1) is used. So that all the values of displacement are stored in 'ang_displacement'. 

- 3.3 'marker' command:

- In order to plot the bob of the pendulum 'marker' command is used. 'marker' command has its syntax need to be followed and it is shown below.

- 'markersize' is the command used to give a size to be bob and 'markerfacecolor' is used to give a color to the bob. 

- Different shapes can be given to be the marker, here as 'o' is used in order to give a round command, however other shapes can be given.

close all 
clear all 
clc

%input parameters given
b=0.05;
g=9.81;
l=1;
m=0.1;

%initial condition where 0 is displacement and 3 is velocity
theta_0 = [0;3];

% time span of 0 to 20 sec
t_span = linspace(0,20,100);
ct = 1;
% solve ode
[t,results] = ode45(@(t,theta) ode_func(t,theta), t_span, theta_0);
plot(t,results(:,1))
hold on
plot(t,results(:,2))
ylabel('plot')
xlabel('time')

% Assigning the values of displacement in the result matirx to ang_displace
ang_displacement = results(:,1);

figure(2)
% animation creation
for i = 1:length(ang_displacement)
    ANG_DISP = ang_displacement(i); %each vaalues of angular displacements are assigned to ANG_DISP
    
    x0 = 0;
    y0 = 0;
    x1 = -l*sin(ANG_DISP); 
    y1 = -l*cos(ANG_DISP);
    % "-" sign is given it order to represent the string in downward
    % direction or in 3rd quadrant
    plot([-0.5 0.5],[0 0],'color','g','linewidth',3);
    hold on
    plot([x0 x1],[y0 y1],'color','r','linewidth',3);
    hold on
    plot(x1,y1,'linewidth',3,'marker','o','markersize',15,'markerfacecolor','r');
    hold off
    axis([-1.5 1.5 -1.5 1.5])
    pause(0.001)
    M(ct) = getframe(gcf); % this cammand will capture all the frames 
    ct = ct+1;  
end

movie(M) % matlab will generate movie with all frames
videofile = VideoWriter('pendulum_simulation.avi','Uncompressed AVI');
open(videofile)
writeVideo(videofile,M)
close(videofile)

- Problems encountered: 

- 4. Learning outcome:

- Generating the animation for a pendulum.

- Introduction to the ode45 and Anonymous function.

- Use of 'for loop' for animation generation.

- 5. Conclusion:

- For the generation of an animation a 'for loop' is used which result in a continuous iteration until the condition gets false and give an output.

- While using an 'ode' command, the equations has to be solved into 1st order ODE.

Youtube video link: https://youtu.be/nlPZmqsLCGU