Week 3 - Solving second order ODEs

Differential equations are governing eqations for most of mechanical mechanism

ODE

this is second order differential equation

In program we have to split all higher order DE in first order ,to simplify it

To simplify the comand Function comand is usedn for making 2nd order DE in 1st

Full program according to given inputs from question.

clear all
close all
clc

%inputes
b = 0.05;   %damping coefficient
g = 9.81;   %gravity
l = 1;      %length of pendulum  
m = 1;      %mass of pendulum  

%initial condition (andular displacement and velocity)
theta_0 = [0;3];        

%time points (time limit)
t_span = linspace(0,20,500);

% solve ODE
[t, results] = ode45(@(t,theta) ode_func(t,theta,b,g,l,m),t_span,theta_0);

figure(1)

%time displacement grapg

plot(t,results(:,1))      
hold on
plot(t,results(:,2))
ylabel('plot')
xlabel('time')

ct=1;

%loop to get different orientation of pendulum
for i = 1:length(results(:,1))

    figure(2)

    %coordinates of pendulum
    x1=1*sin(results(i,1));
    y1=-1*cos(results(i,1));

    plot([-1,1],[0,0])
    hold on

    %use marker commaned to give charesterustics to bob marker
    plot(x1,y1,'o',MarkerSize=15,MarkerEdgeColor='r',MarkerFaceColor='g')

    %fixing the grid
    axis([-1.5 1.5 -1.5 1.5]);

    %ploting lines following pendulum
    plot([0,x1],[0,y1],'linewidth',3)
    plot([-1,1],[0,0],'linewidth',3)


    hold off
    
    
     pause(0.05)

    %storing frames into M
    M(ct) = getframe(gcf);
ct=ct+1;

end


%to creat animation file in avi format
movie(M);
videofile = VideoWriter('ode_pendulum.avi','Uncompressed AVI');
open(videofile);
writeVideo(videofile,M);
close(videofile);

Function is used in program to simplify main progream

Function code

%function code for Differential equation simplification 

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

PLOT for pendulum

Flinal screen after executing program

Pendelum animation file

https://youtu.be/WSa8eeyjr2U

Error faced

Error occured because of not givving enough time to complete program

solved as waited to complete animation from matlab to save video file