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  1. Home/
  2. Shaik Faraz/
  3. Week 3 - Solving second order ODEs

Week 3 - Solving second order ODEs

Aim:  Write a program that solves the following ODE. This ODE represents the equation of motion of a simple pendulum with damping       Write a program in MatlabOctave that will simulate the pendulum motion. Simulate the motion between 0-20 sec, for angular displacement=0,angular velocity=3 rad/sec…

    • Shaik Faraz

      updated on 15 Apr 2022

    Aim:

     Write a program that solves the following ODE. This ODE represents the equation of motion of a simple pendulum with damping

         

    Write a program in MatlabOctave that will simulate the pendulum motion.

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

    use,

    L=1 metre

    m=1 kg

    b=0.05

    g=9.81 m/s^2

    When you do this you will get the position of the pendulum as a function of time. Use this information to animate the motion of the pendulum.

     

    Theory:

    In Engineering, ODE is used to describe the transient behavior of a system. A simple example is a pendulum

                                       Simple pendulum

    The way the pendulum moves depends on the Newtons second law. When this law is written down, we get a second order Ordinary Differential Equation that describes the position of the "ball" w.r.t time. 

    The second order ode of simple pendulum is defind as

    2nd order ode

    where,

    g = gravity in m/s2,

    L = length of the pendulum in m,

    m = mass of the ball in kg,

    b=damping coefficient.

    Now breaking down the 2nd order ode into two 1st order odeBreaking of 2nd order into 2 1st order ode

    Now we have defined θ1 and θ2.

                 ##   Code for solving the 2nd order ode of simple pendulum and simulating it   ##

    % First the code of the function

    function[dtheta_dt] = ode_func(t,theta,b,g,l,m)             % output of the function is dθ/dt
theta1 = theta(1);                                                       % θ1
theta2 = theta(2);                                                       % θ2
dtheta1_dt = theta2;                                                   % dθ1/dt = θ2
dtheta2_dt = -(b/m)*theta2 -(g/l)*sin(theta1);
dtheta_dt = [dtheta1_dt;dtheta2_dt];                          % dθ/dt = [dθ1/dt; dθ2/dt]
end
    2nd order ODE func code
     
    % Given values
    % b=0.05
    % g=9.81 m/s^2
    % l=1 m
    % m=1 kg
     
    % Main Code for ode
     
    % Solving 2nd order ode functions
    clear all
close all
clc​
     
    % inputs
    b = 0.05;
g = 9.81;
l = 1;
m = 1;​
     
    % initial condition
    % We have two initial conditions, one for angular displacement and one for angular velocity
    theta_0 = [0;3];                    % 0 is for displacment and 3 is for velocity
     
    % time between 0 to 20 sec
    t_span = linspace(0,10,250);
     
    % solving of ode
    % calling the ode45 functions for different values of time (t_span) and initial conditions (theta_0)
    % @(t,theta) refers that the ode function is the function of t and theta
    % ode45 is the inbuilt function which outputs t and result
    % t is times that result is given
    % reslut is output of the function i.,e displacement and velocity 
    % we have angular displacement and angular velocity at times t
    [t,result] = ode45(@(t,theta) ode_func(t,theta,b,g,l,m),t_span,theta_0);
     
    % Graph
    hold on
plot(t,result(:,1))
plot(t,result(:,2))
xlabel('Time')
ylabel('Displacement and Velocity')
legend('Angular displacement','Angular Velocity')​
     
    2nd order ODE code1
     
    % Code for animation of pendulum
    % counter for counting the iteration and saving the frames for the movie in M
    ct = 1;
     
    % creat a for loop from 1 to length of the times of result 
    for i = 1:length(result(:,1))
    % co ordinates for the fixed point of the pendulum
    x0 = 0;
y0 = 0;
​
    % co ordinates for the swinging point of the pendulum fixed to the mass
    x1 = l*sin(result(i,1));
y1 = -l*cos(result(i,1));​
     
    % stucture of pendulum
    figure(2)
plot([-5 5],[0 0],'color','k',LineWidth=5)               % fixed support
axis([-2 2 -1.5 0.5])      % specifying the axis so it does not move for diffrent frames

hold on
plot([x0 x1],[y0 y1],'color','b')            
plot(x1,y1,'marker','o','markerfacecolor','r','color','b','markersize',20)
pause (0.001)
hold off​
     
    % store all the frames in M by getframe(gct) coomand
    % advance the couter in for loop
    % End the loop
    M(ct) = getframe(gcf);
ct = ct +1;
end​
     
    % Creating a movie
    % Make a animated movei of the frames inside of M using movie
    % Use VideoWriter to creat a video file in avi formate
    % open videofile
    % write/insert the frames of M in videofile
    % close videofile
    movie(M)
videofile_pendulum_oscillation = VideoWriter('videofile_pendulum_oscillation.avi','Uncompressed AVI');
open(videofile_pendulum_oscillation)
writeVideo(videofile_pendulum_oscillation,M)
close(videofile_pendulum_oscillation)​
     
    2nd order ODE code2
     
     
    Result:
     
    Plot is formed of time vs dipalcement and velocity
    2nd order ode plot
     
    Link for the video simulating simple pendulum : https://youtu.be/itR21ANsATo
     
     
     
    Errors faced:
     
    1.  The angle of the pendulum swinging was very small, almost 0. Because, I used sind() and cosd() instead of sin() and cos()errors simple pendulum small angle
     
    2. I used sin for y and cos for x when i had to do the otherwise
    error 2nd order ode sin cos x y

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