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