Solving second order ODE simple pendulum and Simulating the motion of a simple pendulum for a given time period

Video for simulation of motion of a simple pendulum is attached below:

Solving 2nd order ODE\'s in matlab:

an example of second order differential equation is a simple pendulum with damping.

In order to solve a 2nd order ODE in mat lab we need to first split it into two 1st order ODE.

in this process the above 2nd order ODE is simplified into following two equations:

let θ=x(1)

x(1)\'=dθ/dt=x(2)

x(2)=dθ/dt (1st equation)

x(2)\'=d^2θ/dt^2=-(b/m)*x(2)-(g/l)*sin(x(1))(2nd equation)

L.H.S is put into an array to make it for the matlab to understand:

therefore x=[x(2),x(2)\']=[x(2);-(b/m)*x(2)-(g/l)*sin(x(1))]

using the above function as a anonymous function in matlab the damped pendulum motion is simulated in matlab.

the parameters used for simulation are:

L=1 metre,

m=1 kg,

b=0.05.

g=9.81 m/s2.

the motion is simulated btetween a time period t=o to t=20s.

the initial conditions provided are x0=(0 3).

the function x is solved in matlab as a anonymous function using ode45 code:

f=@(t,x)[x(2);-(b/m)*x(2)-(g/l)*sin(x(1))];% function

[t,x]=ode45(f,t,x0);%ode45 function

the matlab automatically takes the time interval based on the problem.

ode 45 function gives us two outputs:

one is a time intervals array, t (1x1matrix)

other is a solution array,x(2X2 matrix)

here the first column represents the displacement and second column indicates angular acceleration.

plotting the results:

here two plots are generated for both dispalcement and acceleration using subplot function:

subplot function helps us to display two plots in a single figure .

subplot function is used as follows:

%plotting
figure(1)
subplot(2,1,1)
plot(t,x(:,1),\'color\',\'r\',\'linewidth\',3)
xlabel(\'time\')
ylabel(\'anguar velocity\')
%hold on
subplot(2,1,2)
plot(t,x(:,1),\'color\',\'b\')
hold on
xlabel(\'time\')
ylabel(\'amplitude\')
plot(t,x(:,2),\'color\',\'r\')
legend(\'angle(rad)\',\'angular speed(rad/s)\');
%hold on

Finally the animation is done by drawing a pendulum for all values of t :

code for animation:

ct = 1;

for i=1:length(t)
  %origin points
   X0=0;
   Y0=0;
  %mass points 
   X1=1*sin(x(i,1));
   Y1=-1*cos(x(i,1));
   
   figure(2)
   plot([X0,X1],[Y0,Y1]);
   hold on
   plot(X1,Y1,\'.\',\'markersize\',35);
   hold on
   grid on
   axis([-1.5 1.5 -1.5 0.5])
   pause(0.003)
   hold off  
    frame(ct) = getframe(gcf);
    ct = ct+1;
    
end

%Creating animation
movie(frame)
videofile= VideoWriter(\'simple_1.avi\',\'Uncompressed AVI\'); 


open(videofile)
writeVideo(videofile, frame)
close(videofile)

Full program is attached below:

%solvng an second order ode

clear all
close all
clc
%parameters
g=9.81;
l=1;
m=1;
b=0.05;

%second order ode function(anonymous)
f=@(t,x)[x(2);-(b/m)*x(2)-(g/l)*sind(x(1))];

%conditions
t=linspace(0,20,40);
x0=[pi/2 0];

%solve second order ode
[t,x]=ode45(f,t,x0);

%plotting
figure(1)
subplot(2,1,1)
plot(t,x(:,1),\'color\',\'r\',\'linewidth\',3)
xlabel(\'time\')
ylabel(\'anguar velocity\')
%hold on
subplot(2,1,2)
plot(t,x(:,1),\'color\',\'b\')
hold on
xlabel(\'time\')
ylabel(\'amplitude\')
plot(t,x(:,2),\'color\',\'r\')
legend(\'angle(rad)\',\'angular speed(rad/s)\');
%hold on



ct = 1;

for i=1:length(t)
  %origin points
   X0=0;
   Y0=0;
  %mass points 
   X1=1*sin(x(i,1));
   Y1=-1*cos(x(i,1));
   
   figure(2)
   plot([X0,X1],[Y0,Y1]);
   hold on
   plot(X1,Y1,\'.\',\'markersize\',35);
   hold on
   grid on
   axis([-1.5 1.5 -1.5 0.5])
   pause(0.003)
   hold off  
    frame(ct) = getframe(gcf);
    ct = ct+1;
    
end

%Creating animation
movie(frame)
videofile= VideoWriter(\'simple_1.avi\',\'Uncompressed AVI\'); 


open(videofile)
writeVideo(videofile, frame)
close(videofile)