SIMPLE PENDULUM MOTION ANIMATION

Aim:To simulate the transient behaviour of a simple pendulum and to create an animation of it's motion.

THEORY:

A simple pendulum consists of a mass m,hanging to a string of length L,to a fixed point having damping co-efficient b.

The ordinary differrential eequation for the motion of simple pendulum is,

`(d^2theta)/dt^2+(b/m)(d theta)/dt+(g/L)sintheta=0`

where b= Damping co-efficient

m=mass

g=Acccelaration due to gravity i.e.,9.81m/s^2

L= length of the simple pendulum

To solve this second order ODE,we split the equation into first order ODEs

given,

 `(d^2theta)/dt^2+(b/m)(d theta)/dt+(g/L)sintheta=0`      ...(eq.1)

let,

`theta=theta1` 

then,

`(d theta)/dt=(d theta1)/dt`

&

`theta2=(d theta1)/dt`                                                       ...(eq.2)

then eq.1 becomes

`(d theta2)/dt+(b/m)theta2+(g/L)sintheta1=0`

so, `(d theta2)/dt=-(b/m)theta2-(g/L)sintheta1`                  ...(eq.3)

we can also write (eq.2) and (eq.3) as

`d/dt[(theta1),(theta2)]=[(theta2),(-(b/m)theta2-(g/L)sintheta1)]`                   ...(eq.4)

Now the (eq.4) is modified in to a matrix form.

Code explanation: 

 the constants dampping co-efficient,length of the string,accelaration due to gravity & massses are entered

%inputs
b=0.05;
m=1;
l=1;
g=9.81;

theta is entered as a column matrix & time span is entered using linspace command

%initial conditions
theta_0=[0;3];
%time span
t_span=linspace(0,20,200);

 To solve ODEs in matlab, a solver called ode45 is used,where the ode45 gets the values such as time span ,theta_0 & define it into the user defined function and returns it into [t,results]

[t,results]=ode45(@(t,theta)ode_function(t,theta,b,m,l,g),t_span,theta_0);

Then the results are plottted to get the transient behaviour of the pendulum motion.

figure(1) 
hold on
 plot(t,results(:,1))
 plot(t,results(:,2))

 xlabel('time span')
 ylabel('Theta')
 legend('Angular displacement','Angular velocity')
 

then the equation is solved with vertical to get the variables required for animation.

By solving the starting point & end point of the string,we get

x0=0

y0=0

`x1=lsintheta`

`y1=lcostheta`

Then 'for' loop is used to plot the displacement of the pendulum at  various angles of `theta`

A counter variable 'ct' is used to get every frame of displacement using 'getframe' command.

ct=1;
for i=1:length(t)
    x0=0;
    y0=0;
    x1=-l*sin(results(i,1));
    y1=-l*cos(results(i,1));
    figure(2)
   
    plot([x0,x1],[y0,y1],'linewidth',2)
    hold on
     line([-1 1],[0 0])
    plot(x1,y1,'.','MarkerSize',50)
    hold off
   
   axis([-1 1 -1 1])
   M(ct)=getframe(gcf);
   ct=ct+1;
    
end

 To create a video file, 'movie' commands are used.

movie(M)
videofile=VideoWriter('Pendulum_ODE.avi','uncompressed AVI');
open(videofile)
writeVideo(videofile,M)
close(videofile)

Errors & Error optimization:

Used 'hold on 'command before plotting the graphs

 hold on
    plot([x0,x1],[y0,y1],'linewidth',2)
 
     line([-1 1],[0 0])
    plot(x1,y1,'.','MarkerSize',50)
    hold off
   

error resulted was

This error was optimized by using 'hold on' command after the first plot.

CODING:

%Animation of a 2nd ODE equation
clear all
close all
clc

%inputs
b=0.05;
m=1;
l=1;
g=9.81;

%initial conditions
theta_0=[0;3];
%time span
t_span=linspace(0,20,200);
%solving ODE 
[t,results]=ode45(@(t,theta)ode_function(t,theta,b,m,l,g),t_span,theta_0);
%plotting
figure(1) 
hold on
 plot(t,results(:,1))
 plot(t,results(:,2))

 xlabel('time span')
 ylabel('Theta')
 legend('Angular displacement','Angular velocity')
 %animation
 %counter variable
ct=1;
for i=1:length(t)
    x0=0;
    y0=0;
    x1=-l*sin(results(i,1));
    y1=-l*cos(results(i,1));
    %plotting
    figure(2)
  
    plot([x0,x1],[y0,y1],'linewidth',2)
  hold on
     line([-1 1],[0 0])
    plot(x1,y1,'.','MarkerSize',50)
    hold off
   
   axis([-1 1 -1 1])
   %getting frames
   M(ct)=getframe(gcf);
   ct=ct+1;
    
end
%animation of the motion
movie(M)
videofile=VideoWriter('Pendulum_ODE.avi','uncompressed AVI');
open(videofile)
writeVideo(videofile,M)
close(videofile)

Function ode_function,

function [result_matrix]=ode_function(t,theta,b,m,l,g)
%initializing the values

theta1=theta(1);
theta2=theta(2);
dtheta1=theta2;
dtheta2=(-b/m)*theta2-(g/l)*sin(theta1);
% resultant matrix
result_matrix=[dtheta1;dtheta2];
end

CONCLUSION:

The Transient behaviour of the 2nd order ODE is:

The animation of the 2nd order  simple pendulum ODE is as below