Week 3 - Solving second order ODEs

AIM:-To simulate the motion of Pendulum and show its transicent behaviour

THEORY:- A simple Pendulum consist of a mass m suspended by a string of length l,The angle it makes with the vertical varies with time as a sine or cosine

OBJECTIVES:-

• To Simulate the motion of the Pendulum
• To show transient behaviour of Pendulum

Governing Equation and Inputs

`(d^2theta)/(dt^2)+(b)/(m)(d theta)/(dt)+(g)/(l)sin theta=0`

L=1 metre , m-1kg, b(damping cofficient)=0.05, g=9.81m/s2

FUNCTION PROGRAM

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

MAIN PROGRAM

clear all 
close all
clc

%Input Parameters
b=0.05;                 %Damping COfficient
m=1;                    %mass
l=1;                    %Length
g=9.81;                 %Gravity

%Initial position and Velocity
theta_0=[0;3];

%Time points
t_span=linspace(0,20,200);

%Solve ODE
[t,results]=ode45(@(t,theta) ode_func(t,theta,b,g,l,m), t_span, theta_0);
figure(1)
plot(t,results(:,1))
hold on
plot(t,results(:,2))
ylabel('Plot')
xlabel('Time')
legend('Displacement','Velocity')
title('Transient Behaviour')

%For pendulum
ct=1;
for i=1:length(t_span)
    Result=results(i);
    
    %starting coordinates
    x0=0;
    y0=0;
    
    %End coordinates
    x1=l*sind(Result);
    y1=-l*cosd(Result);
    
    %Pliot thread
    figure(2)
    
    plot([x0 x1],[y0 y1],'linewidth',1,'color','r') 
    hold on
    
    %Represent the ceiling
    plot([1 -1],[0 0],'linewidth',5,'color','y')
    
    %Plot a bob
    plot(x1,y1,'marker','o','markersize',15,'markerfacecolor','b')
    
    axis([-1.5 1.5 -1.5 1.5])
    pause(0.00005)
    
    M(ct)=getframe(gcf);
    hold off
    ct=ct+1;
    
   
   
end
movie(M)
videofile=VideoWriter('Pendulum','Uncompressed AVI');
open(videofile)
writeVideo(videofile,M)
close(videofile)

EXPLATION

Step 1- Specify all the input parameters

Step2 - provide initial position and velocity

Step 3 - Provide time span using linspace command

Step 4 - use ode45 command to get the results

Step 5- plot velocity vs time and displacement vs time graph

Step 6-Then the position of the bob can be find from sine and cosine

Step 7-Create ceiling and bob

Step 8-Use movie command to make the video

RESULT-TRANSIENT BEHAVIOUR OF PENDULUM

Simulation Video-https://youtu.be/2vWbNJgavLc

CONCLUSION- Successfully simulated the motion of pendulum and showed Transient behaviour of Pendulum