Week 3 - Solving second order ODEs

given ODE for the motion of pendulum with damping

[d^2θ]/[dt] + [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.

given paramters to solve the equation using matlab

L=1 metre,

m=1 kg,

b=0.05.

g=9.81 m/s2.

time taken = 0 - 20 seconds

angular displacement (θ) = 0

angular velocity [dθ/dt] = 3

now the matlab program

In this program I have used a function called ode_func which is created to breakdwon the second order derivatives into linear derivatives

This is done because the command ode45 recognises only linear diiferntial equations so initially I have broke down the 2 order derivative into 1 order

In the equation

[d^2θ]/[dt] + [b/m]*[dθ/[dt] + [g/l]*sinθ = 0

θ=θ1

so

[d^2θ1]/[dt] + [b/m]*[dθ1/[dt] + [g/l]*sinθ1 = 0

then [dθ1/dt] = θ2

these are substituted to break down the equation

THE FUNCTION

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

THE PROGRAM

close all
clear all
clc

%solving  2 order ode function 
%variables
b=0.05;
l=1;
g=9.81;
m=0.1

%initial condition 
theta_0 = [0;3];

%time interval
t_span = linspace(0,20,100);



%solve ode
[t,results] = (ode45(@(t,theta) ode_func(t,theta,b,g,l,m),t_span,theta_0));
 figure(1)
plot(t,results(:,1))
hold on 
plot(t,results(:,2))
xlabel('time')
ylabel('plot') 

%displacement=results(:,1);
%velocity=results(:,2);
figure(2)
ct =1;
for i = 1:length(t_span)

    x0 = 0;
    y0 = 0;

    x1 = l*sin(results(i,1));
  
    y1 = -l*cos(results(i,1));

    % pendulum figure
    
    plot([-1.5 1.5],[0 0])
    hold on
    plot([x0 x1],[y0 y1],'linewidth',1,'Color','b')
    axis([-2 2 -2 2])
    plot(x1,y1,'o','MarkerSize',20,'Color','r')

    
  
    pause(0.01)
    M(ct) = getframe(gcf)
    hold off
    
    ct = ct+1

end

%movie
movie(M)
videofile = VideoWriter('pendulum.avi','Uncompressed AVI')
open(videofile)
writeVideo(videofile,M)
close(videofile)

PROGRAM EXPLANATION

clear all - used for clearing the variables in workspace

close all - used fo closing the existing plots , graphs

clc - clears the command window

the input paramters are given as mentioned

solving ode

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

ode45 is a inbuilt command to solve the linear order differential equations

the equation is highly dependent on θ and t(time) for integrating. so ode45(@(t,theta) is given, then the function ode_func is given as it corresponds to (t,theta,b,g,l,m).

t_span determines for how much time the function should be performed

theta_0 gives the initial condiotions of the dispacement and velocity

as it gets programmed , it is asisgned to a variable (results) and how much time it should be integrated (t)

from the function

dtheta_dt is assigned to come as results after going through ode45 operations.

FIGURE 1

figure 1 is the plot of time ,velocity and displacement

plot(t,results(:,1)) gives the plot between time and dispacement

plot(t,results(:,2)) gives the plot between time and velocity

FIGURE 2

figure 2 is done for animating the pendulum oscilating and damping for this function

It is done by using for loop

initially the time for how much length (time span ) and for what conditions it should be performed are given

x0 = 0;

y0 = 0;

x1 = l*sin(results(i,1));

y1 = -l*cos(results(i,1));

PLOTTING OF FIGURE 2

A support line is plotted between -1.5 to 1.5 on x axis

from the line at (0,0) the hanging pendulum of length L is hung . This is plotted by giving a command

plot([x0 x1],[y0 y1],'linewidth',1,'Color','b')

the pendulum having a spherical mass is hung from (0,0) along the path line L to oscilate along x1,y1 path for a specific time (t_span). This is

plot(x1,y1,'o','MarkerSize',20,'Color','r').

the figure 2 for each postion of the pendulum is attached to a variale called M and is incremented by using

M(ct) = getframe(gcf)

then the M is created as a movie

MOVIE

the variable M which has the images has commanded

movie(M)

then the video file has given AVI format in uncompressed form

then the videofile is programmed to open and play the video in the given format

then programmed to close

FIGURE 1 PLOT

The video link for this program

https://youtu.be/mbZ1CGsUXYA