Week 3 - Solving second order ODEs

Solving second order ODEs :-

Below is the Second order differential equation for simple pendulum.

Input Values:

b=0.05 (Damping coefficient)

g=9.81 (gravity in m/s^2)

l=1 (Length of pendulum in meter)

m=1 (Mass in Kg)
In intial condition Lets take,

Assume,

theta_0=[0;4];

Angular Dislplacement is 0 and Angular Velocity is 4 and

t_span=linspace(0,20,300);

Time span is 0 to 20 sec, with 300 intervals.

Now solve ODE,
In matlab ode45 is the inubuilt solver for ODE,
Here ODE is the function of t and theta , t and theta are dependent and independent variables.
Below is the syntax for ODE,
[t,result]=ode45(@(t,theta) ode_function(b,g,l,m,t,theta),t_span,theta_0)
We have called function of name ode_function into it and give a name to ode_function as [dtheta_dt]
First Simplify,
Assume,

theta=theta1................(1)

So, dtheta_dt=dtheta1_dt

dtheta1_dt=theta2.................. (2)

Then, d^2theta_dt^2=d^2theta1_dt^2

=d_dt(dtheta1_dt)

=d_dt(theta2)

d^2theta_dt^2=dtheta2_dt...........(3)

So, dtheta2_dt+(b/m)*theta2+(g/l)*sin(theta1)=0....(4)

dtheta1_dt=theta2........(5)

dtheta2_dt=-(b/m)*theta2-(g/l)*sin(theta1).......(6)

Now we have two equations,lets club into matrix form

d_dt[theta1; theta2]=[theta2; -(b/m)*theta2-(g/L)*sin(theta1)]

and dtheta_dt=[dtheta1_dt; dtheta2_dt]

To find dtheta_dt matrix, we will create ODE function

Pelasefind below ODE function as name given "ode_function''

function [dtheta_dt]=ode_function(b,g,l,m,t,theta)
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

In above this function we have to find values for result. Result has two columns, 1st for displacement and 2nd velocity.
Thats why i am assigning,

theta1=theta(1) and theta2=theta(2)

The result part we will cover next,

Now, we will find values for displacement and velocity w.r.t. time in a matrix form of table result.

Then call above ode_function into main program :-

close all
clear all
clc
b=0.05;
l=1;
m=1;
g=9.81;
theta_0=[0;4];
t_span=linspace(0,20,300);
[t,result]=ode45(@(t,theta) ode_function(b,g,l,m,t,theta),t_span,theta_0);
hold on
plot(t,result(:,1),'color','r');
plot(t,result(:,2),'color','b');
xlabel('Time')
ylabel('Displacemt/Velocity')
legend('Displacement','Velocity')
grid on

In ode45 synatx we defining value of b,g,l,m are assumed and then we define time span t_span and initial conditions by theta_0

The result of above progrm as belows:

I used hold on command to show the result into one graph or frame and legend command to differentiate displacement and velocity.

I got error when not use hold command :

In order to create animation vbideo i used counter (ct) command and movie command.

M is the variable to store all the output result figures and use for loop to complete time span length(t) i.e. 300 intervals.

close all
clear all
clc
b=0.05;
l=1;
m=1;
g=9.81;
theta_0=[0;4];
t_span=linspace(0,20,300);
[t,result]=ode45(@(t,theta) ode_function(b,g,l,m,t,theta),t_span,theta_0);
figure(1)
hold on
plot(t,result(:,1),'color','r');
plot(t,result(:,2),'color','b');
xlabel('Time')
ylabel('Displacemt/Velocity')
legend('Displacement','Velocity')
grid on
ct=1;
for i=1:length(t)
 
 
 x0=0;
 y0=0;
 
 x1=l*sin(result(i,1));
 y1=-l*cos(result(i,1));
 
 plot([-2 2],[0 0],'linewidth',3,'color','r');
 line([x0 x1],[y0 y1],'linewidth',4);
 
 axis([-2 2 -2 2])
 
 hold on
 plot(x1,y1,'o','markers',20,'markerfacecolor','g');
 
 
 grid on
 pause(0.03);
 
 
 M(ct)=getframe(gcf)
 
 ct=ct+1;
end
movie(M)
videofile=VideoWriter('Second_order_ODE.avi','uncompressed avi')
open(videofile)
writeVideo(videofile,M)
close(videofile)

I got another typographic error for wrongly used Videowriter command instead of used VideoWriter which.
Please find below link for Animation video of Simple Pendulum Single ODE.