Week 4.1 - Solving second order ODEs
AIM:
To write a program in Matlab, that will simulate the pendulum motion.
GOVERNING EQUATIONS:
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.
OBJECTIVES:
To write a program that solves the mentioned ODE. The ODE represents the equation of motion of a simple pendulum with damping
THEORY:
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.
BODY OF CONTEXT:
PROGRAM CODE:
% Program to solving second order ODE and make a simple pendulum
clear all; % clear all variables in workspace
close all;% close all plots
clc;%clear command window
l=1;%Initialize length to 1 metre
m=1;%Initialize length to 1 kg
g=9.81;%Initialize acc. due to gravity to 9.81m/s^2
b=0.05;%Initialize damping coefficient to 0.05
ct=1;%Initialize ct Counter to 1
%Initial Condition
theta_0=[0;3]; %Initializing displacement to 0 and velcotiy to 3 rad/s^2
t_span=linspace(0,20,500); %Initializing time upto 20 sec with 500 divisions
%ode function calling
[t output]=ode45(@(t,theta) ode_pendulum(t,theta,b,g,l,m),t_span,theta_0) %Solvinf first ODE using ode45 command
f=figure(1)%Initializing figure(1) to f
f.Position=[400 50 700 80];%Setting F position
subplot(4,1,[1,2]); %Creating a subplot
hold on %Holding on
aa=plot(t,output(:,1)) %Creating a plot with t and angular displacement and assigning to variable 'aa'
bb=plot(t,output(:,2)) %Creating a plot with t and angular Velocity and asisgning to variable 'bb'
xlabel('Time'); % Xaxis is labelled as Time
ylabel('Amplitude'); % Yaxis is labelled as Amplitude
grid on %Turning ON legend
title('Variation of Angular Displacement and Angular Veloctiy with Time'); %Assigning title name
legend([aa bb], 'Angular Displacement','Angular Velocity'); %Creating legends
%Animating simple pendulum
theta2=output(:,1) %Initializing theta2 variable with angular displacement
theta1=output(:,2) %Initializing theta1 variable with angular velocity
for i=1:length(theta2) %FOR loop for iterating from 1 to 500 numbers of different positions of theta2
figure(2); %Creating new figure
clf; %Reseting each figure for avoiding traces of previous figures plotted
theta_x=theta2(i); %theta_x variable is initialized which is assigned particular value in location of theta2 in which iteration is happenng
x0=0;%Initializing variable x0=0
y0=0;%Initializing variable y0=0
x1=l*sin(theta_x); %Initializing x1,i.e final position of bar
y1=-l*cos(theta_x); %Initializing xy,i.e final position of bar
%creating a rectangle with positions specified and setting curvature to 1pt and facecolour balck
h1=rectangle('Position',[-1 -0.1 2 0.5],'Curvature',1,'Facecolor',[0 0 0]);
hold on; %Holding ON
plot([x0 x1],[y0 y1],'linewidth',10,'color','b'); %Plotting bar of pendulum with width of 10 and blue colour
hold on;%holding ON
plot(x1,y1,'marker','o', 'markersize',20, 'markerfacecolor','r');%Plotting ball at end of bar of pendulum and giving red colour
hold on; %Holding ON
grid on;%Turning on GRID
axis([-2 3 -2 1.5]);%Setting axis and limits
set(gca,'color','y','xcolor','k','ycolor','k')%Now setting the area to get current access and colour yellow giving x label colour and y label colour as black
pause(0.00001); %Pausing each plot for 0.0001 sec
M(ct)=getframe(gcf)%Initializing M variable to capture each frames by getframe() command with gcf being figure
ct=ct+1; %increase counter value to next after each iteration
end
movie(M) %Creating movie of M variable
Videofile=VideoWriter('Simple Pendulum', 'Uncompressed AVI');%Initializing a new variable as videofile and write name and type of video
open(Videofile)%opening vidoefile
writeVideo(Videofile,M)%Writing contents of M to videofile
close(Videofile)%Close videofile
% Program to solving second order ODE and make a simple pendulum
clear all; % clear all variables in workspace
close all;% close all plots
clc;%clear command window
l=1;%Initialize length to 1 metre
m=1;%Initialize length to 1 kg
g=9.81;%Initialize acc. due to gravity to 9.81m/s^2
b=0.05;%Initialize damping coefficient to 0.05
ct=1;%Initialize ct Counter to 1
%Initial Condition
theta_0=[0;3]; %Initializing displacement to 0 and velcotiy to 3 rad/s^2
t_span=linspace(0,20,500); %Initializing time upto 20 sec with 500 divisions
%ode function calling
[t output]=ode45(@(t,theta) ode_pendulum(t,theta,b,g,l,m),t_span,theta_0) %Solvinf first ODE using ode45 command
f=figure(1)%Initializing figure(1) to f
f.Position=[400 50 700 80];%Setting F position
subplot(4,1,[1,2]); %Creating a subplot
hold on %Holding on
aa=plot(t,output(:,1)) %Creating a plot with t and angular displacement and asisgning to variable 'aa'
bb=plot(t,output(:,2)) %Creating a plot with t and angular Velocity and asisgning to variable 'bb'
xlabel('Time'); % Xaxis is labelled as Time
ylabel('Amplitude'); % Yaxis is labelled as Amplitude
grid on %Turning ON legend
title('Variation of Angular Displacement and Angular Veloctiy with Time'); %Assigning title name
legend([aa bb], 'Angular Displacement','Angular Velocity'); %Creating legends
%Animating simple pendulum
theta2=output(:,1) %Initializing theta2 variable with angular displacement
theta1=output(:,2) %Initializing theta1 variable with angular velocity
for i=1:length(theta2) %FOR loop for iterating from 1 to 500 numbers of different positions of theta2
figure(2); %Creating new figure
clf; %Reseting each figure for avoiding traces of previous figures plotted
theta_x=theta2(i); %theta_x variable is initialized which is assigned particular value in location of theta2 in which iteration is happenng
x0=0;%Initializing variable x0=0
y0=0;%Initializing variable y0=0
x1=l*sin(theta_x); %Initializing x1,i.e final position of bar
y1=-l*cos(theta_x); %Initializing xy,i.e final position of bar
%creating a rectangle with positions specified and setting curvature to 1pt and facecolour balck
h1=rectangle('Position',[-1 -0.1 2 0.5],'Curvature',1,'Facecolor',[0 0 0]);
hold on; %Holding ON
plot([x0 x1],[y0 y1],'linewidth',10,'color','b'); %Plotting bar of pendulum with width of 10 and blue colour
hold on;%holding ON
plot(x1,y1,'marker','o', 'markersize',20, 'markerfacecolor','r');%Plotting ball at end of bar of pendulum and giving red colour
hold on; %Holding ON
grid on;%Turning on GRID
axis([-2 3 -2 1.5]);%Setting axis and limits
set(gca,'color','y','xcolor','k','ycolor','k')%Now setting the area to get current access and colour yellow giving x label colour and y label colour as black
pause(0.00001); %Pausing each plot for 0.0001 sec
M(ct)=getframe(gcf)%Initializing M variable to capture each frames by getframe() command with gcf being figure
ct=ct+1; %increase counter value to next after each iteration
end
movie(M) %Creating movie of M variable
Videofile=VideoWriter('Simple Pendulum', 'Uncompressed AVI');%Initializing a new variable as videofile and write name and type of video
open(Videofile)%opening vidoefile
writeVideo(Videofile,M)%Writing contents of M to videofile
close(Videofile)%Close videofile
SYNTAX WITH EXPLANATION:
l=1;%Initialize length to 1 metre
m=1;%Initialize length to 1 kg
g=9.81;%Initialize acc. due to gravity to 9.81m/s^2
b=0.05;%Initialize damping coefficient to 0.05
ct=1;%Initialize ct Counter to 1
%Initial Condition
theta_0=[0;3]; %Initializing displacement to 0 and velcotiy to 3 rad/s^2
t_span=linspace(0,20,500); %Initializing time upto 20 sec with 500 divisions
%ode function calling
[t output]=ode45(@(t,theta) ode_pendulum(t,theta,b,g,l,m),t_span,theta_0) %Solvinf first ODE using ode45 command
f=figure(1)%Initializing figure(1) to f
f.Position=[400 50 700 80];%Setting F position
subplot(4,1,[1,2]); %Creating a subplot
hold on %Holding on
aa=plot(t,output(:,1)) %Creating a plot with t and angular displacement and assigning to variable 'aa'
bb=plot(t,output(:,2)) %Creating a plot with t and angular Velocity and asisgning to variable 'bb'
xlabel('Time'); % Xaxis is labelled as Time
ylabel('Amplitude'); % Yaxis is labelled as Amplitude
grid on %Turning ON legend
title('Variation of Angular Displacement and Angular Veloctiy with Time'); %Assigning title name
legend([aa bb], 'Angular Displacement','Angular Velocity'); %Creating legends
%Animating simple pendulum
theta2=output(:,1) %Initializing theta2 variable with angular displacement
theta1=output(:,2) %Initializing theta1 variable with angular velocity
for i=1:length(theta2) %FOR loop for iterating from 1 to 500 numbers of different positions of theta2
figure(2); %Creating new figure
clf; %Reseting each figure for avoiding traces of previous figures plotted
theta_x=theta2(i); %theta_x variable is initialized which is assigned particular value in location of theta2 in which iteration is happenng
x0=0;%Initializing variable x0=0
y0=0;%Initializing variable y0=0
x1=l*sin(theta_x); %Initializing x1,i.e final position of bar
y1=-l*cos(theta_x); %Initializing xy,i.e final position of bar
%creating a rectangle with positions specified and setting curvature to 1pt and facecolour balck
h1=rectangle('Position',[-1 -0.1 2 0.5],'Curvature',1,'Facecolor',[0 0 0]);
hold on; %Holding ON
plot([x0 x1],[y0 y1],'linewidth',10,'color','b'); %Plotting bar of pendulum with width of 10 and blue colour
hold on;%holding ON
plot(x1,y1,'marker','o', 'markersize',20, 'markerfacecolor','r');%Plotting ball at end of bar of pendulum and giving red colour
hold on; %Holding ON
grid on;%Turning on GRID
axis([-2 3 -2 1.5]);%Setting axis and limits
set(gca,'color','y','xcolor','k','ycolor','k')%Now setting the area to get current access and colour yellow giving x label colour and y label colour as black
pause(0.00001); %Pausing each plot for 0.0001 sec
M(ct)=getframe(gcf)%Initializing M variable to capture each frames by getframe() command with gcf being figure
ct=ct+1; %increase counter value to next after each iteration
end
movie(M) %Creating movie of M variable
Videofile=VideoWriter('Simple Pendulum', 'Uncompressed AVI');%Initializing a new variable as videofile and write name and type of video
open(Videofile)%opening vidoefile
writeVideo(Videofile,M)%Writing contents of M to videofile
close(Videofile)%Close videofile
FUNCTION CODE:
function [dtheta_dot]=ode_pendulum(t,theta,b,g,l,m)
%Creating function to make output of 2nd ODE to !st ODE
theta1=theta(1); %Initializing row value of theta to theta1
theta2=theta(2); %Initializing column value of theta to theta2
dtheta1_dot=theta2; %Initializing theta2 to dtheta1_dot
dtheta2_dot=-(b/m)*theta2-(g/l)*sin(theta1); %Initializing -(b/m)*theta2-(g/l)*sin(theta1) to dtheta2_dot
dtheta_dot=[dtheta1_dot;dtheta2_dot];%Output dtheta_dot is initialized with[dtheta1_dot(row);dtheta2_dot(column)]
end
function [dtheta_dot]=ode_pendulum(t,theta,b,g,l,m)
%Creating function to make output of 2nd ODE to !st ODE
theta1=theta(1); %Initializing row value of theta to theta1
theta2=theta(2); %Initializing column value of theta to theta2
dtheta1_dot=theta2; %Initializing theta2 to dtheta1_dot
dtheta2_dot=-(b/m)*theta2-(g/l)*sin(theta1); %Initializing -(b/m)*theta2-(g/l)*sin(theta1) to dtheta2_dot
dtheta_dot=[dtheta1_dot;dtheta2_dot];%Output dtheta_dot is initialized with[dtheta1_dot(row);dtheta2_dot(column)]
end
-0.3173 -1.5252
f =
Figure (1) with properties:
Number: 1
Name: ''
Color: [0.9400 0.9400 0.9400]
Position: [488 342 560 420]
Units: 'pixels'
Show all properties
aa =
Line with properties:
Color: [0 0.4470 0.7410]
LineStyle: '-'
LineWidth: 0.5000
Marker: 'none'
MarkerSize: 6
MarkerFaceColor: 'none'
XData: [0 0.0401 0.0802 0.1202 0.1603 0.2004 0.2405 0.2806 0.3206 0.3607 0.4008 … ]
YData: [0 0.1198 0.2375 0.3513 0.4594 0.5603 0.6527 0.7354 0.8074 0.8678 0.9161 … ]
Show all properties
bb =
Line with properties:
Color: [0.8500 0.3250 0.0980]
LineStyle: '-'
LineWidth: 0.5000
Marker: 'none'
MarkerSize: 6
MarkerFaceColor: 'none'
XData: [0 0.0401 0.0802 0.1202 0.1603 0.2004 0.2405 0.2806 0.3206 0.3607 0.4008 … ]
YData: [3 2.9704 2.8946 2.7748 2.6143 2.4171 2.1881 1.9323 1.6544 1.3589 1.0500 … ]
Show all properties
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M =
1×248 struct array with fields:
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M =
1×249 struct array with fields:
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M =
1×250 struct array with fields:
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M =
1×251 struct array with fields:
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M =
1×252 struct array with fields:
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M =
1×253 struct array with fields:
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M =
1×254 struct array with fields:
cdata
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M =
1×255 struct array with fields:
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M =
1×256 struct array with fields:
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M =
1×257 struct array with fields:
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M =
1×258 struct array with fields:
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M =
1×259 struct array with fields:
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M =
1×260 struct array with fields:
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M =
1×261 struct array with fields:
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M =
1×262 struct array with fields:
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M =
1×263 struct array with fields:
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M =
1×264 struct array with fields:
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M =
1×265 struct array with fields:
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M =
1×266 struct array with fields:
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M =
1×267 struct array with fields:
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M =
1×268 struct array with fields:
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M =
1×269 struct array with fields:
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M =
1×270 struct array with fields:
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M =
1×271 struct array with fields:
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M =
1×272 struct array with fields:
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M =
1×273 struct array with fields:
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M =
1×274 struct array with fields:
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M =
1×275 struct array with fields:
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M =
1×276 struct array with fields:
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M =
1×277 struct array with fields:
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M =
1×278 struct array with fields:
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M =
1×279 struct array with fields:
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M =
1×280 struct array with fields:
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M =
1×281 struct array with fields:
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M =
1×282 struct array with fields:
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M =
1×283 struct array with fields:
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M =
1×284 struct array with fields:
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M =
1×285 struct array with fields:
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M =
1×286 struct array with fields:
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M =
1×287 struct array with fields:
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M =
1×288 struct array with fields:
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M =
1×289 struct array with fields:
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M =
1×290 struct array with fields:
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M =
1×291 struct array with fields:
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M =
1×292 struct array with fields:
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M =
1×293 struct array with fields:
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M =
1×294 struct array with fields:
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M =
1×295 struct array with fields:
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M =
1×296 struct array with fields:
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M =
1×297 struct array with fields:
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M =
1×298 struct array with fields:
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M =
1×299 struct array with fields:
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M =
1×300 struct array with fields:
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M =
1×301 struct array with fields:
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M =
1×302 struct array with fields:
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M =
1×303 struct array with fields:
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M =
1×304 struct array with fields:
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M =
1×305 struct array with fields:
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M =
1×306 struct array with fields:
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M =
1×307 struct array with fields:
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M =
1×308 struct array with fields:
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M =
1×309 struct array with fields:
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M =
1×310 struct array with fields:
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M =
1×311 struct array with fields:
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M =
1×312 struct array with fields:
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M =
1×313 struct array with fields:
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M =
1×314 struct array with fields:
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M =
1×315 struct array with fields:
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M =
1×316 struct array with fields:
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M =
1×317 struct array with fields:
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M =
1×318 struct array with fields:
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M =
1×319 struct array with fields:
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M =
1×320 struct array with fields:
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M =
1×321 struct array with fields:
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M =
1×322 struct array with fields:
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M =
1×323 struct array with fields:
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M =
1×324 struct array with fields:
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M =
1×325 struct array with fields:
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M =
1×326 struct array with fields:
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M =
1×327 struct array with fields:
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M =
1×328 struct array with fields:
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M =
1×329 struct array with fields:
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M =
1×330 struct array with fields:
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M =
1×331 struct array with fields:
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M =
1×332 struct array with fields:
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M =
1×333 struct array with fields:
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M =
1×334 struct array with fields:
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M =
1×335 struct array with fields:
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M =
1×336 struct array with fields:
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M =
1×337 struct array with fields:
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M =
1×338 struct array with fields:
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M =
1×339 struct array with fields:
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M =
1×340 struct array with fields:
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M =
1×341 struct array with fields:
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M =
1×342 struct array with fields:
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M =
1×343 struct array with fields:
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M =
1×344 struct array with fields:
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M =
1×345 struct array with fields:
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M =
1×346 struct array with fields:
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M =
1×347 struct array with fields:
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M =
1×348 struct array with fields:
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M =
1×349 struct array with fields:
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M =
1×350 struct array with fields:
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M =
1×351 struct array with fields:
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M =
1×352 struct array with fields:
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M =
1×353 struct array with fields:
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M =
1×354 struct array with fields:
cdata
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M =
1×355 struct array with fields:
cdata
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M =
1×356 struct array with fields:
cdata
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M =
1×357 struct array with fields:
cdata
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M =
1×358 struct array with fields:
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M =
1×359 struct array with fields:
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M =
1×360 struct array with fields:
cdata
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M =
1×361 struct array with fields:
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M =
1×362 struct array with fields:
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M =
1×363 struct array with fields:
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M =
1×364 struct array with fields:
cdata
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M =
1×365 struct array with fields:
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M =
1×366 struct array with fields:
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M =
1×367 struct array with fields:
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M =
1×368 struct array with fields:
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M =
1×369 struct array with fields:
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M =
1×370 struct array with fields:
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M =
1×371 struct array with fields:
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M =
1×372 struct array with fields:
cdata
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M =
1×373 struct array with fields:
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M =
1×374 struct array with fields:
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M =
1×375 struct array with fields:
cdata
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M =
1×376 struct array with fields:
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M =
1×377 struct array with fields:
cdata
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M =
1×378 struct array with fields:
cdata
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M =
1×379 struct array with fields:
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M =
1×380 struct array with fields:
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M =
1×381 struct array with fields:
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M =
1×382 struct array with fields:
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M =
1×383 struct array with fields:
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M =
1×384 struct array with fields:
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M =
1×385 struct array with fields:
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M =
1×386 struct array with fields:
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M =
1×387 struct array with fields:
cdata
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M =
1×388 struct array with fields:
cdata
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M =
1×389 struct array with fields:
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M =
1×390 struct array with fields:
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M =
1×391 struct array with fields:
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M =
1×392 struct array with fields:
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M =
1×393 struct array with fields:
cdata
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M =
1×394 struct array with fields:
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M =
1×395 struct array with fields:
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M =
1×396 struct array with fields:
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M =
1×397 struct array with fields:
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M =
1×398 struct array with fields:
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M =
1×399 struct array with fields:
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M =
1×400 struct array with fields:
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M =
1×401 struct array with fields:
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M =
1×402 struct array with fields:
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M =
1×403 struct array with fields:
cdata
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M =
1×404 struct array with fields:
cdata
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M =
1×405 struct array with fields:
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M =
1×406 struct array with fields:
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M =
1×407 struct array with fields:
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M =
1×408 struct array with fields:
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M =
1×409 struct array with fields:
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M =
1×410 struct array with fields:
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M =
1×411 struct array with fields:
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M =
1×412 struct array with fields:
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M =
1×413 struct array with fields:
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M =
1×414 struct array with fields:
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M =
1×415 struct array with fields:
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M =
1×416 struct array with fields:
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M =
1×417 struct array with fields:
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M =
1×418 struct array with fields:
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M =
1×419 struct array with fields:
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M =
1×420 struct array with fields:
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M =
1×421 struct array with fields:
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M =
1×422 struct array with fields:
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M =
1×423 struct array with fields:
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M =
1×424 struct array with fields:
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M =
1×425 struct array with fields:
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M =
1×426 struct array with fields:
cdata
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M =
1×427 struct array with fields:
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M =
1×428 struct array with fields:
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M =
1×429 struct array with fields:
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M =
1×430 struct array with fields:
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M =
1×431 struct array with fields:
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M =
1×432 struct array with fields:
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M =
1×433 struct array with fields:
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M =
1×434 struct array with fields:
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M =
1×435 struct array with fields:
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M =
1×436 struct array with fields:
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M =
1×437 struct array with fields:
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M =
1×438 struct array with fields:
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M =
1×439 struct array with fields:
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M =
1×440 struct array with fields:
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M =
1×441 struct array with fields:
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M =
1×442 struct array with fields:
cdata
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M =
1×443 struct array with fields:
cdata
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M =
1×444 struct array with fields:
cdata
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M =
1×445 struct array with fields:
cdata
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M =
1×446 struct array with fields:
cdata
colormap
M =
1×447 struct array with fields:
cdata
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M =
1×448 struct array with fields:
cdata
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M =
1×449 struct array with fields:
cdata
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M =
1×450 struct array with fields:
cdata
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M =
1×451 struct array with fields:
cdata
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M =
1×452 struct array with fields:
cdata
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M =
1×453 struct array with fields:
cdata
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M =
1×454 struct array with fields:
cdata
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M =
1×455 struct array with fields:
cdata
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M =
1×456 struct array with fields:
cdata
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M =
1×457 struct array with fields:
cdata
colormap
M =
1×458 struct array with fields:
cdata
colormap
M =
1×459 struct array with fields:
cdata
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M =
1×460 struct array with fields:
cdata
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M =
1×461 struct array with fields:
cdata
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M =
1×462 struct array with fields:
cdata
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M =
1×463 struct array with fields:
cdata
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M =
1×464 struct array with fields:
cdata
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M =
1×465 struct array with fields:
cdata
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M =
1×466 struct array with fields:
cdata
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M =
1×467 struct array with fields:
cdata
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M =
1×468 struct array with fields:
cdata
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M =
1×469 struct array with fields:
cdata
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M =
1×470 struct array with fields:
cdata
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M =
1×471 struct array with fields:
cdata
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M =
1×472 struct array with fields:
cdata
colormap
M =
1×473 struct array with fields:
cdata
colormap
M =
1×474 struct array with fields:
cdata
colormap
M =
1×475 struct array with fields:
cdata
colormap
M =
1×476 struct array with fields:
cdata
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M =
1×477 struct array with fields:
cdata
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M =
1×478 struct array with fields:
cdata
colormap
M =
1×479 struct array with fields:
cdata
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M =
1×480 struct array with fields:
cdata
colormap
M =
1×481 struct array with fields:
cdata
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M =
1×482 struct array with fields:
cdata
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M =
1×483 struct array with fields:
cdata
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M =
1×484 struct array with fields:
cdata
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M =
1×485 struct array with fields:
cdata
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M =
1×486 struct array with fields:
cdata
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M =
1×487 struct array with fields:
cdata
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M =
1×488 struct array with fields:
cdata
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M =
1×489 struct array with fields:
cdata
colormap
M =
1×490 struct array with fields:
cdata
colormap
M =
1×491 struct array with fields:
cdata
colormap
M =
1×492 struct array with fields:
cdata
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M =
1×493 struct array with fields:
cdata
colormap
M =
1×494 struct array with fields:
cdata
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M =
1×495 struct array with fields:
cdata
colormap
M =
1×496 struct array with fields:
cdata
colormap
M =
1×497 struct array with fields:
cdata
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M =
1×498 struct array with fields:
cdata
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M =
1×499 struct array with fields:
cdata
colormap
M =
1×500 struct array with fields:
cdata
colormap
>>
ANIMATION OF SIMPLE PENDULUM:
Youtube link: https://youtu.be/ajiWnAjaNLo
INFERENCE:
In this program I have used ode45,set,position,rectangle commands and animation for creating animation.