Week 3 - Solving second order ODEs

Aim: To simulate a pendulum motion by solving the following 2nd ODE equation as shown below:

In the above equation,

g = 9.81 m/s², L (length of the pendulum)= 1m, m = mass of the ball=1 kg, b=damping coefficient= 0.5. Simulation period = 20sec

angular displacement=0,angular velocity=3 rad/sec at time t=0.

Method:

• For each value in time the position of the pendulum is calculated
• An image that represents the position of the pendulum is plotted after solving the ODE (using function call) with a fixed pivot point
• Each calculated and plotted position is captured in frames to stitch to a movie (Matlab function)

%Matlab code to animate a pendulum in motion with mass, length of arm,
%velocity and acceleration , and drag/damp co-efficient known

clc
clear all
close all

%damp_coeff = b
b=0.05;
%acceleration of pendulum mass(m/s^2)=g
g=9.8;
%Length of pendulum=l
l=1;
%mass=m
m=1;

%pendulum position and velocity
theta0=[0;3];

%Time as position marker (t1)
tspan=linspace(0,20,200);

%simple ODE
[t,results]= ode45(@(t,theta) ode_func(t,theta,b,g,l,m),tspan,theta0);

Angle = results(:,1);

    ct=1;
for i=1:length(Angle)
    angle=Angle(i);
figure(ct)
%Pivot point
x0=1;
y0=2;

%ball coordinates
x1=l*sin(angle)+x0;
y1=-l*cos(angle)+y0;

%Animating pendulum
%coordinates of pendulum rod and ball-mass
axis([0,2,0,2.75])
hold on
plot([x0,x1],[y0,y1],'linewidth',3,'color','[0.9290 0.6940 0.1250]');
hold on
plot(x1,y1,'.','color','r','linewidth',15,'markersize',40);
plot(x0,y0,'o', 'color','k','linewidth',15,'markersize',2);
M(ct)= getframe(gcf);
ct=ct+1;
hold off
end

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

Function to solve the 2nd order function:

function [Theta_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);
Theta_dt=[dtheta1_dt ; dtheta2_dt];
end

Click here to view the video

Conclusion:

1. Breaking down complex equations 2nd order and above to simple form and solving is an easy way to get solutions
2. Learned to use ODE45 function predefined by MATLAB