Movie Generation of motion of simple pendulum

Solving Second Order ODE:

The motion of a simple pendulum is defined with the following equation.

simplication of the equation.

%Coding to animate a simple pendulum motion
clear all
close all
clc
%General pendulum specifications
b=0.05;
g=9.81;
l=1;
m=1;
%Input condition
theta_0=[0;3];
%Time points
t_span=linspace(0,20,500);
%solving second order ODE
[t,results]=ode45(@(t,theta)ode_func(t,theta,b,g,l,m),t_span,theta_0);
%Ploting change in Displacement & Velocity with respect to time
figure(1)
plot(t,results(:,1))
hold on
plot(t,results(:,2))
ylabel(\'plot\')
xlabel(\'time\')
ct=1;
%For loop for creating Animation
for i=1:length(results(:,1))
    x0=0;
    y0=0;
    x1=l*sin(results(i,1));
    y1=-l*cos(results(i,1));
    figure(2)
    plot([-1 1],[0 0],\'linewidth\',4,\'color\',\'r\');
    hold on
    line([x0 x1],[y0 y1],\'linewidth\',4,\'color\',\'k\');
    hold on
    plot(x1,y1,\'O\',\'markers\',20,\'markerfacecolor\',\'c\');
    grid on;
    axis([-1.5 1.5 -1.5 0.5])
    pause(0.01);
    hold off
    M(ct)=getframe(gcf);
    ct=ct+1
end
%Movie Generation
movie(M)
video_file=VideoWriter(\'pendulum.avi\',\'uncompressed AVI\')
open(video_file)
writeVideo(video_file,M)
close(video_file)

Code Explanation:

General pendulam specifications:

• The values for damping coefficient (b), Gravity (g), Length of the string (l) & mass of the ball (m) are taken from the question given.

Input condition:

• The displacemnt is 0 and velocity is 3 at the starting point (i.e) at t=0.

Time points:

• To specify the values for \'t\' linspace command is used .

Solving second order ODE:

• ode45 command is used to solve the ODE with respect to time and to store the values for displacement & velocity.
• The ode_func is a function used to give values for first order & second order ODE.

Calling Function:

%Defining a function to solve ODE
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

• Specify the variables theta1 & theta2
• The derivation of theta1 will be theta 2 & the derivation of theta2 will be -(b/m)*theta2 -(g/l)*sin(theta1) from the simplified equation.
• The function will give the output for displacement & velocity.

Ploting the change in displacement & velocity with respect to time:

• Ploting the first column of results will give the curve for displacement.
• Plotting the second column of results will give the curve for velocity.

For loop for creating animation:

• specify the length of itration to be the length of the column of results.
• specify the values for x0 & y0 as 0 as they are the starting coordinates.
• x1 & y1 will be denoting the the end point of the pendulum string.
• plot a straight line through the coordinates x0 & yo which will be acting as a fixed column.
• give hold on command to pause the plotes column.
• use line command to join x0 y0 with x1 y1 so as to get the string in the plot .
• The values of x1 & y1 will be changing according to the time.
• Thus the string line will be moving for each itration.
• use pause command to hold the frame for specified time.
• hold off command can be used to show a single frame at a time.
• plot the ball using \'o\' command & mention its size & color using \'markers\' and \'markerfacecolor\' commands.

Result:

Figure (1):

Figure (2):

Error 1:

• Did\'nt specify the variables before calling the function in the ode45 command.
• @(t,theta) is included before calling the function to overcome the error.

Error 2:

• While defining x1 & y1 the equations are written as

x1=l*sind(results(i,1));
y1=-l*cosd(results(i,1));

• in figure(2) there is only less movement of the string.
• This error is because the values in results is in radians.
• the commands sind & cosd are only used for the values in degrees.
• By changing the equation as

x1=l*sin(results(i,1));
y1=-l*cos(results(i,1)); The error can be overcome.