Week 3 - Solving second order ODEs

Aim: To solve 2^nd order governing ordinary differential equation of a motion of a simple pendulum using MATLAB.

Theory:

A pendulum is nothing, but mass or weight suspended at fixed point (pivot point) by using string of foxed length. If there is no action of external force on pendulum, it remains in equilibrium position. But when it is displaced from its equilibrium position with the application of force, it oscillates at its mean position due to gravity. The time taken to complete one oscillation from left swing to right swing or from right swing to left swing is called as period of pendulum. Period of pendulum is given by,

`T=2π √(L/g)`

Where L is length of the string of pendulum and g is acceleration due to gravity.

Simple pendulum is a type of pendulum in which a mass (bob) is attached at the end of massless string. String is fixed at pivot which is frictionless. Simple pendulum is shown in figure below.

Figure: Simple pendulum

Code explanation:

1. Input of different parameters (damping Coefficient, mass, Acceleration due to gravity and length of the string of pendulum) are taken.

clear all
close all
clc

% Input of variables 
b=0.05;
g=9.81;
l=1;
m=1;

2. Initial conditions are defined. At time t=0, angular displacement = 0, angular velocity = 3 rad/sec.

   % Initial condition
   theta_0 = [0;3];​

3. Input for time span or period over which pendulum oscillates is taken.

   % Time points
   t_span = linspace(0,20,500);​

4. Separate program is written by using ‘function’ in MATLAB to calculate dθ`/dt, where `θ`is column vector given by `θ = [θ1; θ2 ].Here θ1 represents angular displacement and θ2 represents angular velocity. After doing so we have values of dθ1/dt` and dθ2/dt.

   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

5. Now to calculate θ1 and θ2 ‘ode 45’ command is used in MATLAB. By using ode45’ command, dθ1/dt` and dθ2/dt is solved to get angular displacement and angular velocity. Here 'results' is an array with 2 columns and rows equal to number of time values (here 500).

   % To solve ODE 
   [t, results] = ode45(@(t, theta) ode_func(t, theta, b,g,l,m), t_span, theta_0);​

6. Pivot point of pendulum is defined (x0, y0) = (0, 0) and plotted on x-y axis

   % To defined pivot point of pendulum and plot it.
   x0=0;
   y0=0;
   plot (x0, y0,'marker', 'o')

7. To extract the first column of a result and assigned it to 'y'

   % To extract the first column of a result and assigned it to 'y'
   y =  results(:,1);​

8. To create images for each time value and stored it in array ‘M’.

• For loop is used from i=1: length(t_span) where t_span is no of times the angular displacement and angular velocity is to be calculated from time t = 0 to 20.
• In the program ‘y’ denotes the angular displacement. A new variable Y1 is created to access individual value at each time from array ‘y’.
• x and y coordinate of mass of a pendulum is calculated for each time as x1 and y1
• Line is created from (x0,y0) to (X1,y1) which represents string of a pendulum.
• Bob or mass of a pendulum is created by using ‘marker’ command with suitable size of marker and green color.
• Tittle is given to image with'title' command.
• 'hold off' command is used to erase image from privious time step.
• Images are captured for each time step and stored in array ‘M’.

  ct=1;
  for i=1:length(t_span)
      Y1=y(i)
      x1 = l*sin(Y1);
      y1 = -l*cos(Y1);
      plot([x0 x1],[y0 y1],'linewidth',3,'marker','o')
      hold on
      plot(x1, y1, 'marker','o', 'color','g','markersize',30, 'markerfacecolor', 'g')
      hold off 
      title({'Simple harmonic motion'; 'of a pendulum'}, 'color','r')
      axis([-1 1 -2 0])
      pause(0.01)
      M(ct)= getframe(gcf)
      ct=ct+1;
  end​

  9. Anmation is created using 'movie(M)' command

  %  To create animation of S.H.M. of simple pendulum
  movie(M)
  videofile = VideoWriter('pendulum.avi', 'Uncompressed AVI');
  open(videofile)
  writeVideo(videofile,M)
  close(videofile)​

Complete code (without error)

clear all
close all
clc

% Input of variables 
b=0.05;
g=9.81;
l=1;
m=1;

% Initial condition
theta_0 = [0;3];

% Time points
t_span = linspace(0,20,500);

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

% To defined pivot point of pendulum and plot it.
x0=0;
y0=0;
plot (x0, y0,'marker', 'o')

% To extract the first column of a result and assigned it to 'y'
y =  results(:,1);

% To create images for each time value and stored it in array ‘M’.
ct=1;
for i=1:length(t_span)
    Y1=y(i)
    x1 = l*sin(Y1);
    y1 = -l*cos(Y1);
    plot([x0 x1],[y0 y1],'linewidth',3,'marker','o')
    hold on
    plot(x1, y1, 'marker','o', 'color','g','markersize',30, 'markerfacecolor', 'g')
    hold off 
    title({'Simple harmonic motion'; 'of a pendulum'}, 'color','r')
    axis([-1 1 -2 0])
    pause(0.01)
    M(ct)= getframe(gcf)
    ct=ct+1;
end

%  To create animation of S.H.M. of simple pendulum
movie(M)
videofile = VideoWriter('pendulum.avi', 'Uncompressed AVI');
open(videofile)
writeVideo(videofile,M)
close(videofile)
 

Errors face while coding

1. pivot point (x0,y0) is not defined

2. Varable 'Y1' is not defined to access indivisual values of y of each time step

3. Improper use of 'hold off' command to erase plot of privious time step

4. Missing of curly bracket in 'title' command

Results

M = 

  1×98 struct array with fields:

    cdata
    colormap


Y1 =

   -0.7356


M = 

  1×99 struct array with fields:

    cdata
    colormap


Y1 =

   -0.6684


M = 

  1×100 struct array with fields:

    cdata
    colormap


Y1 =

   -0.5916


M = 

  1×101 struct array with fields:

    cdata
    colormap


Y1 =

   -0.5062


M = 

  1×102 struct array with fields:

    cdata
    colormap


Y1 =

   -0.4133


M = 

  1×103 struct array with fields:

    cdata
    colormap


Y1 =

   -0.3143


M = 

  1×104 struct array with fields:

    cdata
    colormap


Y1 =

   -0.2106


M = 

  1×105 struct array with fields:

    cdata
    colormap


Y1 =

   -0.1039


M = 

  1×106 struct array with fields:

    cdata
    colormap


Y1 =

    0.0043


M = 

  1×107 struct array with fields:

    cdata
    colormap


Y1 =

    0.1122


M = 

  1×108 struct array with fields:

    cdata
    colormap


Y1 =

    0.2181


M = 

  1×109 struct array with fields:

    cdata
    colormap


Y1 =

    0.3203


M = 

  1×110 struct array with fields:

    cdata
    colormap


Y1 =

    0.4174


M = 

  1×111 struct array with fields:

    cdata
    colormap


Y1 =

    0.5080


M = 

  1×112 struct array with fields:

    cdata
    colormap


Y1 =

    0.5907


M = 

  1×113 struct array with fields:

    cdata
    colormap


Y1 =

    0.6644


M = 

  1×114 struct array with fields:

    cdata
    colormap


Y1 =

    0.7284


M = 

  1×115 struct array with fields:

    cdata
    colormap


Y1 =

    0.7817


M = 

  1×116 struct array with fields:

    cdata
    colormap


Y1 =

    0.8238


M = 

  1×117 struct array with fields:

    cdata
    colormap


Y1 =

    0.8542


M = 

  1×118 struct array with fields:

    cdata
    colormap


Y1 =

    0.8728


M = 

  1×119 struct array with fields:

    cdata
    colormap


Y1 =

    0.8793


M = 

  1×120 struct array with fields:

    cdata
    colormap


Y1 =

    0.8737


M = 

  1×121 struct array with fields:

    cdata
    colormap


Y1 =

    0.8561


M = 

  1×122 struct array with fields:

    cdata
    colormap


Y1 =

    0.8265


M = 

  1×123 struct array with fields:

    cdata
    colormap


Y1 =

    0.7854


M = 

  1×124 struct array with fields:

    cdata
    colormap


Y1 =

    0.7331


M = 

  1×125 struct array with fields:

    cdata
    colormap


Y1 =

    0.6705


M = 

  1×126 struct array with fields:

    cdata
    colormap


Y1 =

    0.5983


M = 

  1×127 struct array with fields:

    cdata
    colormap


Y1 =

    0.5174


M = 

  1×128 struct array with fields:

    cdata
    colormap


Y1 =

    0.4289


M = 

  1×129 struct array with fields:

    cdata
    colormap


Y1 =

    0.3340


M = 

  1×130 struct array with fields:

    cdata
    colormap


Y1 =

    0.2342


M = 

  1×131 struct array with fields:

    cdata
    colormap


Y1 =

    0.1309


M = 

  1×132 struct array with fields:

    cdata
    colormap


Y1 =

    0.0258


M = 

  1×133 struct array with fields:

    cdata
    colormap


Y1 =

   -0.0795


M = 

  1×134 struct array with fields:

    cdata
    colormap


Y1 =

   -0.1835


M = 

  1×135 struct array with fields:

    cdata
    colormap


Y1 =

   -0.2843


M = 

  1×136 struct array with fields:

    cdata
    colormap


Y1 =

   -0.3805


M = 

  1×137 struct array with fields:

    cdata
    colormap


Y1 =

   -0.4706


M = 

  1×138 struct array with fields:

    cdata
    colormap


Y1 =

   -0.5534


M = 

  1×139 struct array with fields:

    cdata
    colormap


Y1 =

   -0.6277


M = 

  1×140 struct array with fields:

    cdata
    colormap


Y1 =

   -0.6927


M = 

  1×141 struct array with fields:

    cdata
    colormap


Y1 =

   -0.7475


M = 

  1×142 struct array with fields:

    cdata
    colormap


Y1 =

   -0.7915


M = 

  1×143 struct array with fields:

    cdata
    colormap


Y1 =

   -0.8242


M = 

  1×144 struct array with fields:

    cdata
    colormap


Y1 =

   -0.8453


M = 

  1×145 struct array with fields:

    cdata
    colormap


Y1 =

   -0.8545


M = 

  1×146 struct array with fields:

    cdata
    colormap


Y1 =

   -0.8519


M = 

  1×147 struct array with fields:

    cdata
    colormap


Y1 =

   -0.8374


M = 

  1×148 struct array with fields:

    cdata
    colormap


Y1 =

   -0.8114


M = 

  1×149 struct array with fields:

    cdata
    colormap


Y1 =

   -0.7740


M = 

  1×150 struct array with fields:

    cdata
    colormap


Y1 =

   -0.7256


M = 

  1×151 struct array with fields:

    cdata
    colormap


Y1 =

   -0.6668


M = 

  1×152 struct array with fields:

    cdata
    colormap


Y1 =

   -0.5984


M = 

  1×153 struct array with fields:

    cdata
    colormap


Y1 =

   -0.5213


M = 

  1×154 struct array with fields:

    cdata
    colormap


Y1 =

   -0.4366


M = 

  1×155 struct array with fields:

    cdata
    colormap


Y1 =

   -0.3453


M = 

  1×156 struct array with fields:

    cdata
    colormap


Y1 =

   -0.2490


M = 

  1×157 struct array with fields:

    cdata
    colormap


Y1 =

   -0.1490


M = 

  1×158 struct array with fields:

    cdata
    colormap


Y1 =

   -0.0469


M = 

  1×159 struct array with fields:

    cdata
    colormap


Y1 =

    0.0558


M = 

  1×160 struct array with fields:

    cdata
    colormap


Y1 =

    0.1575


M = 

  1×161 struct array with fields:

    cdata
    colormap


Y1 =

    0.2564


M = 

  1×162 struct array with fields:

    cdata
    colormap


Y1 =

    0.3512


M = 

  1×163 struct array with fields:

    cdata
    colormap


Y1 =

    0.4404


M = 

  1×164 struct array with fields:

    cdata
    colormap


Y1 =

    0.5227


M = 

  1×165 struct array with fields:

    cdata
    colormap


Y1 =

    0.5969


M = 

  1×166 struct array with fields:

    cdata
    colormap


Y1 =

    0.6621


M = 

  1×167 struct array with fields:

    cdata
    colormap


Y1 =

    0.7175


M = 

  1×168 struct array with fields:

    cdata
    colormap


Y1 =

    0.7625


M = 

  1×169 struct array with fields:

    cdata
    colormap


Y1 =

    0.7966


M = 

  1×170 struct array with fields:

    cdata
    colormap


Y1 =

    0.8193


M = 

  1×171 struct array with fields:

    cdata
    colormap


Y1 =

    0.8305


M = 

  1×172 struct array with fields:

    cdata
    colormap


Y1 =

    0.8300


M = 

  1×173 struct array with fields:

    cdata
    colormap


Y1 =

    0.8178


M = 

  1×174 struct array with fields:

    cdata
    colormap


Y1 =

    0.7943


M = 

  1×175 struct array with fields:

    cdata
    colormap


Y1 =

    0.7595


M = 

  1×176 struct array with fields:

    cdata
    colormap


Y1 =

    0.7141


M = 

  1×177 struct array with fields:

    cdata
    colormap


Y1 =

    0.6584


M = 

  1×178 struct array with fields:

    cdata
    colormap


Y1 =

    0.5932


M = 

  1×179 struct array with fields:

    cdata
    colormap


Y1 =

    0.5193


M = 

  1×180 struct array with fields:

    cdata
    colormap


Y1 =

    0.4378


M = 

  1×181 struct array with fields:

    cdata
    colormap


Y1 =

    0.3498


M = 

  1×182 struct array with fields:

    cdata
    colormap


Y1 =

    0.2565


M = 

  1×183 struct array with fields:

    cdata
    colormap


Y1 =

    0.1595


M = 

  1×184 struct array with fields:

    cdata
    colormap


Y1 =

    0.0602


M = 

  1×185 struct array with fields:

    cdata
    colormap


Y1 =

   -0.0399


M = 

  1×186 struct array with fields:

    cdata
    colormap


Y1 =

   -0.1391


M = 

  1×187 struct array with fields:

    cdata
    colormap


Y1 =

   -0.2360


M = 

  1×188 struct array with fields:

    cdata
    colormap


Y1 =

   -0.3290


M = 

  1×189 struct array with fields:

    cdata
    colormap


Y1 =

   -0.4167


M = 

  1×190 struct array with fields:

    cdata
    colormap


Y1 =

   -0.4979


M = 

  1×191 struct array with fields:

    cdata
    colormap


Y1 =

   -0.5715


M = 

  1×192 struct array with fields:

    cdata
    colormap


Y1 =

   -0.6363


M = 

  1×193 struct array with fields:

    cdata
    colormap


Y1 =

   -0.6916


M = 

  1×194 struct array with fields:

    cdata
    colormap


Y1 =

   -0.7368


M = 

  1×195 struct array with fields:

    cdata
    colormap


Y1 =

   -0.7714


M = 

  1×196 struct array with fields:

    cdata
    colormap


Y1 =

   -0.7950


M = 

  1×197 struct array with fields:

    cdata
    colormap


Y1 =

   -0.8073


M = 

  1×198 struct array with fields:

    cdata
    colormap


Y1 =

   -0.8082


M = 

  1×199 struct array with fields:

    cdata
    colormap


Y1 =

   -0.7976


M = 

  1×200 struct array with fields:

    cdata
    colormap


Y1 =

   -0.7758


M = 

  1×201 struct array with fields:

    cdata
    colormap


Y1 =

   -0.7430


M = 

  1×202 struct array with fields:

    cdata
    colormap


Y1 =

   -0.6997


M = 

  1×203 struct array with fields:

    cdata
    colormap


Y1 =

   -0.6464


M = 

  1×204 struct array with fields:

    cdata
    colormap


Y1 =

   -0.5837


M = 

  1×205 struct array with fields:

    cdata
    colormap


Y1 =

   -0.5124


M = 

  1×206 struct array with fields:

    cdata
    colormap


Y1 =

   -0.4336


M = 

  1×207 struct array with fields:

    cdata
    colormap


Y1 =

   -0.3483


M = 

  1×208 struct array with fields:

    cdata
    colormap


Y1 =

   -0.2578


M = 

  1×209 struct array with fields:

    cdata
    colormap


Y1 =

   -0.1635


M = 

  1×210 struct array with fields:

    cdata
    colormap


Y1 =

   -0.0668


M = 

  1×211 struct array with fields:

    cdata
    colormap


Y1 =

    0.0307


M = 

  1×212 struct array with fields:

    cdata
    colormap


Y1 =

    0.1275


M = 

  1×213 struct array with fields:

    cdata
    colormap


Y1 =

    0.2221


M = 

  1×214 struct array with fields:

    cdata
    colormap


Y1 =

    0.3131


M = 

  1×215 struct array with fields:

    cdata
    colormap


Y1 =

    0.3991


M = 

  1×216 struct array with fields:

    cdata
    colormap


Y1 =

    0.4788


M = 

  1×217 struct array with fields:

    cdata
    colormap


Y1 =

    0.5510


M = 

  1×218 struct array with fields:

    cdata
    colormap


Y1 =

    0.6149


M = 

  1×219 struct array with fields:

    cdata
    colormap


Y1 =

    0.6696


M = 

  1×220 struct array with fields:

    cdata
    colormap


Y1 =

    0.7144


M = 

  1×221 struct array with fields:

    cdata
    colormap


Y1 =

    0.7488


M = 

  1×222 struct array with fields:

    cdata
    colormap


Y1 =

    0.7724


M = 

  1×223 struct array with fields:

    cdata
    colormap


Y1 =

    0.7851


M = 

  1×224 struct array with fields:

    cdata
    colormap


Y1 =

    0.7866


M = 

  1×225 struct array with fields:

    cdata
    colormap


Y1 =

    0.7769


M = 

  1×226 struct array with fields:

    cdata
    colormap


Y1 =

    0.7562


M = 

  1×227 struct array with fields:

    cdata
    colormap


Y1 =

    0.7246


M = 

  1×228 struct array with fields:

    cdata
    colormap


Y1 =

    0.6828


M = 

  1×229 struct array with fields:

    cdata
    colormap


Y1 =

    0.6311


M = 

  1×230 struct array with fields:

    cdata
    colormap


Y1 =

    0.5703


M = 

  1×231 struct array with fields:

    cdata
    colormap


Y1 =

    0.5011


M = 

  1×232 struct array with fields:

    cdata
    colormap


Y1 =

    0.4245


M = 

  1×233 struct array with fields:

    cdata
    colormap


Y1 =

    0.3416


M = 

  1×234 struct array with fields:

    cdata
    colormap


Y1 =

    0.2536


M = 

  1×235 struct array with fields:

    cdata
    colormap


Y1 =

    0.1618


M = 

  1×236 struct array with fields:

    cdata
    colormap


Y1 =

    0.0676


M = 

  1×237 struct array with fields:

    cdata
    colormap


Y1 =

   -0.0275


M = 

  1×238 struct array with fields:

    cdata
    colormap


Y1 =

   -0.1219


M = 

  1×239 struct array with fields:

    cdata
    colormap


Y1 =

   -0.2141


M = 

  1×240 struct array with fields:

    cdata
    colormap


Y1 =

   -0.3029


M = 

  1×241 struct array with fields:

    cdata
    colormap


Y1 =

   -0.3868


M = 

  1×242 struct array with fields:

    cdata
    colormap


Y1 =

   -0.4646


M = 

  1×243 struct array with fields:

    cdata
    colormap


Y1 =

   -0.5352


M = 

  1×244 struct array with fields:

    cdata
    colormap


Y1 =

   -0.5976


M = 

  1×245 struct array with fields:

    cdata
    colormap


Y1 =

   -0.6511


M = 

  1×246 struct array with fields:

    cdata
    colormap


Y1 =

   -0.6949


M = 

  1×247 struct array with fields:

    cdata
    colormap


Y1 =

   -0.7285


M = 

  1×248 struct array with fields:

    cdata
    colormap


Y1 =

   -0.7515


M = 

  1×249 struct array with fields:

    cdata
    colormap


Y1 =

   -0.7639


M = 

  1×250 struct array with fields:

    cdata
    colormap


Y1 =

   -0.7654


M = 

  1×251 struct array with fields:

    cdata
    colormap


Y1 =

   -0.7559


M = 

  1×252 struct array with fields:

    cdata
    colormap


Y1 =

   -0.7356


M = 

  1×253 struct array with fields:

    cdata
    colormap


Y1 =

   -0.7048


M = 

  1×254 struct array with fields:

    cdata
    colormap


Y1 =

   -0.6638


M = 

  1×255 struct array with fields:

    cdata
    colormap


Y1 =

   -0.6132


M = 

  1×256 struct array with fields:

    cdata
    colormap


Y1 =

   -0.5537


M = 

  1×257 struct array with fields:

    cdata
    colormap


Y1 =

   -0.4861


M = 

  1×258 struct array with fields:

    cdata
    colormap


Y1 =

   -0.4113


M = 

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    0.1625


M = 

  1×472 struct array with fields:

    cdata
    colormap


Y1 =

    0.2332


M = 

  1×473 struct array with fields:

    cdata
    colormap


Y1 =

    0.3001


M = 

  1×474 struct array with fields:

    cdata
    colormap


Y1 =

    0.3622


M = 

  1×475 struct array with fields:

    cdata
    colormap


Y1 =

    0.4187


M = 

  1×476 struct array with fields:

    cdata
    colormap


Y1 =

    0.4687


M = 

  1×477 struct array with fields:

    cdata
    colormap


Y1 =

    0.5114


M = 

  1×478 struct array with fields:

    cdata
    colormap


Y1 =

    0.5464


M = 

  1×479 struct array with fields:

    cdata
    colormap


Y1 =

    0.5732


M = 

  1×480 struct array with fields:

    cdata
    colormap


Y1 =

    0.5913


M = 

  1×481 struct array with fields:

    cdata
    colormap


Y1 =

    0.6005


M = 

  1×482 struct array with fields:

    cdata
    colormap


Y1 =

    0.6010


M = 

  1×483 struct array with fields:

    cdata
    colormap


Y1 =

    0.5925


M = 

  1×484 struct array with fields:

    cdata
    colormap


Y1 =

    0.5754


M = 

  1×485 struct array with fields:

    cdata
    colormap


Y1 =

    0.5496


M = 

  1×486 struct array with fields:

    cdata
    colormap


Y1 =

    0.5157


M = 

  1×487 struct array with fields:

    cdata
    colormap


Y1 =

    0.4740


M = 

  1×488 struct array with fields:

    cdata
    colormap


Y1 =

    0.4252


M = 

  1×489 struct array with fields:

    cdata
    colormap


Y1 =

    0.3701


M = 

  1×490 struct array with fields:

    cdata
    colormap


Y1 =

    0.3094


M = 

  1×491 struct array with fields:

    cdata
    colormap


Y1 =

    0.2441


M = 

  1×492 struct array with fields:

    cdata
    colormap


Y1 =

    0.1751


M = 

  1×493 struct array with fields:

    cdata
    colormap


Y1 =

    0.1035


M = 

  1×494 struct array with fields:

    cdata
    colormap


Y1 =

    0.0304


M = 

  1×495 struct array with fields:

    cdata
    colormap


Y1 =

   -0.0430


M = 

  1×496 struct array with fields:

    cdata
    colormap


Y1 =

   -0.1156


M = 

  1×497 struct array with fields:

    cdata
    colormap


Y1 =

   -0.1862


M = 

  1×498 struct array with fields:

    cdata
    colormap


Y1 =

   -0.2538


M = 

  1×499 struct array with fields:

    cdata
    colormap


Y1 =

   -0.3173


M = 

  1×500 struct array with fields:

    cdata
    colormap

>> 

Youtube link of animation of SHM of simple pendulum: https://youtu.be/LTGZ2dYK5js

Google drive link of animation of SHM of simple pendulum: https://drive.google.com/file/d/11aG_z19XWCimdvw0Y3pYpWic4B8Ut7iX/view?usp=sharing

Conclusion

Governing differential equation of simple pendulum is solved using MATLAB program. Animation is working correctly. Hence it can be concluded that solution obtained by program is correct.