Week 3 - Solving second order ODEs

Aim ;

Write a program and solve the given ODE and represent the equation of motion of simple pendulum with damping. Then create a animation to show them.

Explanation ;

The given ODE to simulate to the second order equation of the simple pendulum to create plot between angular velocity and displacement by using the given ODE45.

By using ODE45 to solve a given equation and run to generate the program with animation.

Formula :

d^2θ/dt^2 = b/m . dθ/dt + g/L . sinθ = 0

θ = 0

dθ/dt = dθ/dt = θ2 .........(1)

d^2θ/dt^2 = d^2θ/dt^2 = d/dt(dθ1/dt) = d/dt(θ2)..........(2)

d^2θ/dt^2 + b/m dθ/t + g/L sinθ = 0

dθ1^2/dt^2 + b/m dθ/t + g/L sinθ1 = 1

dθ2/dt + b/m θ2 + g/L sinθ1 = 0

dθ1/dt = θ2

dθ2/dt = -b/m θ2 - g/L sinθ1

d/dt[θ1] = [θ2]

d/dt[θ2] = [-b/m θ2 - g/L sinθ]

Where,

b = damping coefficient

m = mass

g = gravitational acceleration

l = length pf the string

Procedure ;

Step 1 : Start

Step 2 : Open the Matlab Application

Step 3 : Type the command in application

Step 4 : Save to run the program

Step 5 : Display the output

Step 6 : Stop

Program ;

• Program for simple pendulum :

  clear all
  close all
  clc
  
  % inputs
  b = 0.05;
  g = 9;
  l = 1;
  m = 0.1;
  
  % initial
  theta_0 = [0;4]
  
  % time
  t_span = linspace(0,20,200)
  
  % solve
  [t, result] = ode45(@(t,theta)ode_func(t,theta,b,g,l,m),t_span,theta_0)
  
  figure(1)
  plot(t,result(:,1))
  hold on;
  plot(t,result(:,2))
  videofile = VideoWriter('simple_pendulum.avi','uncompressed AVI');
  open(videofile)
  
  figure(2)
  %animation
  for i = 1: length (result(:,1))
      x0 = 0;
      y0 = 0;
      
      x1 = 1*sin(result(i,1));
      y1 = -1*cos(result(i,1));
  
      
      plot([0,0],[-1.5,1.5])
      hold on
      plot([x0 x1],[y0 y1],'linewidth',2,'color','b')
     
      plot(x1,y1,'o','markersize',20,'color','r')
      axis([-1.5 1.5 -1.5 1.5])
      pause(0.005)
      writeVideo(videofile,getframe(gcf))
      hold off
     
  end
  
  
  
  
  close(videofile)​

• program for ODE function :

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

Program Explanation ;

To start clear all, close all, clc and command window to clear all. Then inputs given to simulate and values for apply to run the half of the program, then to attached the center of the program for function of ODE to combine. After to using ODE45 to add with write remaining program to run and create the animation to make a movie with a link to attach below title for them.

Output Link ;

https://drive.google.com/file/d/1mxQwo9FlHOveA4tIEW0e1CXW4RK4SMdO/view?usp=sharing

Conclusion ;

The given program for simple pendulum with function of ODE run and animation is successfully executed.