SHM of a Simple pendulum

https://www.youtube.com/watch?v=9eK6gPRihUw

The program is used to analyze the vibration and system parameters for a simple pendulum.

As this pendulum motion is SHM so the equation of motion is second order differential equation and solving this manually is time-consuming. So here we use two techniques

1 State-space consideration

2 Numerical analysis of the differential equation

State-space is nothing but the breaking down higher-order DE to single order DE ie Our equation will be in XY coordinates and we are shifting this equation and analyzing it in a different plane which will help in easy identification of a solution.

In the numerical analysis, we describe an approximate solution for the DE which is taken into consideration. Here solver used is ODE 45 as this will give more number of integrtion points while solving. 

System parameters used:

b=0.5; damping
g=10; gravity
l=0.75; Length of string
m=1; mass
theta_0=[0;5]; Intial ang disp and ang velocity ie Intial condition for the system.

The above are the initial condition used in analysing the system. So by using the both methods, we will get an array of values for the simple -pendulum\'s angular velocity and angular displacement. With this system result, these are plotted and the plot is as shown below.

In addition to that, the motion of the pendulum is captured using the method of creating a series of plots by varying the angular displacement values and the position for the pendulum is given by the coordinates associated with them.

Note: Here i have used marker face color to highlight it as bob and increase the line width to highlight the string attached to the bob.

Function

function [derivation] =ode_function_pendulum(t,theta,b,g,l,m)
theta1=theta(1);
theta2=theta(2);
derivation= [theta2; ((-b/m)*theta2-((g/l)*sin(theta1)))];
end

Main PGM

%system definition%
clear all
close all
clc
b=0.5;
g=10;
l=0.75;
m=1;
theta_0=[0;5];
tspan=linspace(0,10,400);
[t,r]=ode45(@(t,theta) ode_function_pendulum(t,theta,b,g,l,m),tspan,theta_0);

 ct=1       

for i=1:length(r)
    
    
    x0=0;
    y0=0;
    
    x1=l*sin(r(i,1));
    y1=-l*cos(r(i,1));
    

    plot([x0 x1],[y0 y1],\'linewidth\',6,\'color\',\'k\')
    hold on
    plot(x1,y1,\'o\',\'MarkerSize\',30,\'Markerfacecolor\',\'red\')
    hold off
axis([-1 1 -1.5 1.5])
       pause(0.005)
    M(ct)= getframe(gcf)
    ct=ct+1;
   
  
end
 movie(M)
 videofile = VideoWriter(\'pendulumspec3\',\'uncompressed AVI\')
 open(videofile)
 writeVideo(videofile,M)
 close(videofile)