Solving Second order ODEs To oscillate a Simple Pendulum

AIM: To solve the second order ODE and to simulate the Motion of Simple Pendulum using Matlab.

Here, We are using equation for motion of a simple pendulum with damping

In the above equation,

g = gravity in m/s2,

L = length of the pendulum in m,

m = mass of the ball in kg,

b=damping coefficient.

CODING:

Main Program:

clear all
close all
clc

%inputs
b = 0.05;   
g = 9.81;   
m = 1;     
l = 1;      

%initial conditions
theta_0 = [0;3];     

%time points
t_span = linspace(0,10,800); 

%solve ODE
[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))
xlabel(\'time\')
ylabel(\'Angular Displacement vs Velocity\')

%simple pendulum simulation using ode
ct = 1
for i=1:length(result)
    THETA = result(i,1);
    
    %boundary conditon
    x0=0;
    y0=0;

    x1=  l*sin(THETA);
    y1= -l*cos(THETA);
    
   figure(2)
   plot([-0.6 0.6],[0 0],\'linewidth\',5,\'color\',\'b\')
   axis([-2 2 -2 2])
   hold on
   plot([x0 x1],[y0 y1],\'linewidth\',2,\'color\',\'r\')
   plot(x1,y1,\'o\',\'markersize\',30,\'markerfacecolor\',\'r\',\'markeredgecolor\',\'b\')
   pause(0.03)
   hold off
   M(ct)=getframe(gcf);
   ct=ct+1;
end
movie(M);
videofile = VideoWriter(\'simple_pendulum.avi\',\'uncompressed AVI\');
open(videofile)
writeVideo(videofile,M)
close(videofile)

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

Explanation of code:

STEP1: Clear all,close all,clc commands are used to clear all previous files and prvious content in workspace.

STEP2: Input Parameters,

L=1 metre,

m=1 kg,

b=0.05,

g=9.81 m/s2.

STEP3: Initial conditions
theta_0 = [0;3];

angular displacement=0,

angular velocity=3 rad/sec,

STEP4: Time Points

t_span = linspace(0,10,800);

• t_span is time array

we are assuming 0s - 10s

STEP5: Solving ODE

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

• ode45 is inbuilt solvent in matlab
• ode is function of t, theta
• ode function ode_func(t,theta,b,g,l,m)

Here we have different program for ode_function

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

`theta1 = theta(1)`

`theta2 = theta(2)`

`(d theta) /dt= theta2`

`(d theta2)/dt = -(b/m)*theta2-(g/l)*sin theta1`

`(d theta)/dt = [(d theta1)/dt; (d theta2)/dt]`

STEP6: Ploting figure1

• plot(t, result(:,1))
• hold on
• plot(t,result(:,2))
• xlabel(\'time\')
• ylabel(\'Angular Displacement vs Velocity\')

OUTPUT FOR ANGULAR DISPLACEMENT VS ANGULAR VELOCITY

STEP7: Pendulum simulation using ODE

• counter ct=1, counter is used to initialise the value

here for loop is used

• for i=1:length(result) value of i is going to change form 1 to length(result)

where as length(result) is number of values in result

• THETA=result(i,1) were THETA gets values form result(i,1)

STEP8: Boundary Conditon

• x0=0;
• y0=0;
• x1= l*sin(THETA);
• y1= -l*cos(THETA);

STEP9: Ploting figure2

• plot([-0.6 0.6],[0 0],\'linewidth\',5,\'color\',\'b\'), it is use to plot the line with line width of 5, and colour of blue.
• axis([-2 2 -2 2]), it defines the scale of plot -2 to 2 in x axis, 2 to 2 in y axis.
• hold on
• plot([x0 x1],[y0 y1],\'linewidth\',2,\'color\',\'r\')
• plot(x1,y1,\'o\',\'markersize\',30,\'markerfacecolor\',\'r\',\'markeredgecolor\',\'b\'), Marker is used to creat sphere.
• pause(0.03), pause each plot by 0.03 seconds.
• hold off
• M(ct)=getframe(gcf), it is used to frame the plot and saved as animation.
• ct=ct+1; counter+1 is used for incremental value.

STEP10: Saving the file

• movie(M);
• videofile = VideoWriter(\'ode_pendulum.avi\',\'uncompressed AVI\');
• open(videofile)
• writeVideo(videofile,M)
• close(videofile)

OUTPUT FOR SIMPLE PENDULUM

Animation file