Week 3 - Solving second order ODEs

We solve this Initial VAlue Problem ODE using reuction of the orer method. The goal is to simplify the order of the problem and to solve for the output for a system of equations. We input all the parameters required to define the system! The length in question should be length of the string+ distance to center of gravity. So I took the length as 0.95 and a uniform bob of radius 0.5 units( relative unit to the string).

https://youtu.be/mYiragCqqDQ

clear all
clc
%inputs
b=0.05;%damping constant

g=9.81;%acceleration due to Gravity

l=0.95;% length of the pendulum

m=0.1;%mass of the bob

%initial conditions

theta_0=[0 ; 5];%Initial conditions to solve the IVP problem

%timepoints
t_span=linspace(0,20,200);

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

plot(t,results(:,1))
figure(1)
hold on
plot(t,results(:,2))
legend('Displacement','Angular')
xlabel('time')
ylabel('Angular Displacement and Velocity')


count=1;

for i=1:length(results)
    
    
    
    x_0=0;%initial positions
    y_0=0;
    
    x_final=l*sin(results(i,1));%results(i,1) gives you location of the bob at each instant
    y_final=-l*cos(results(i,1));
    
    figure(2)
    plot([-2 2], [0 0], 'linewidth',5)
    
    hold on
    plot([x_0 x_final] ,[y_0 y_final],'linewidth',3);
    hold on
    plot(x_final,y_final,'o','markersize',10,'markerFacecolor','g')
    
    axis([-2 2 -2 2])
    pause(0.05)
    hold off
    
    fr_ame(count)=getframe(gcf);
    count=count+1;
    
end

 movie(fr_ame)       
 videofile=VideoWriter('Pendulum_skilllync.avi', 'Uncompressed AVI');
 open(videofile)
 writeVideo(videofile,fr_ame)
 close(videofile)
         
    
    

%Ode_func

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

theta_1=theta(1)
theta_2=theta(2)

diff_theta1= theta_2;
diff_theta2= -(b/m)*theta_2 -(g/l)*(sin(theta_1))

diff_theta=[diff_theta1;diff_theta2];
end