Week 3 - Solving second order ODEs

Here We can simulate the transient behaviour of a simple pendulum and to create an animation of it's motion. Below is the figure of simple pendulum

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)

find 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 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

CODE

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

In ode45 synatx we defining value of b,g,l,m are assumed and then we define time span

d initial conditions by theta_0

t_span an 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

or i=1:length(t)

0=0; y0=0; 

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

figure(2)

plot([-2 2],[0 0],'linewidth',3,'color','r'); line([x0 x1],[y0 y1],'linewidth',4);

axis([-2 2 -2 2])

old on

plot(x1,y1,'o','markers',20,'markerfacecolor','g'); hold off

rid on

pause(0.03);

M(ct)=getframe(gcf)

in order to fix this i use axis([-2 2 -2 2])
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