Week 3 - Solving second order ODEs
AIM: To evaluate the motion of simple pendulum by solving ODEusing MATLAM
THEORY:
In engineering, ODE is used to describe the transient behavior of a system. A simple examle is a pendulum. the way the pendulum moves depends on the Newtons second law. this law is writen dowen, we get a second order ordinary Differential equation that describes the position of the ballw.r.t. time.
Consider a pendulum with ball of mass (m) attached to the end string of length (l) with other end being for
Where,
θ = angular displacement
g = gravitational acceleration
t = time
b = damping coefficient
A second order differential equation should be converted into two sinlge order differential equations to slove in matlab.
θ = θ1
dθ/dt=θ2
and this leads to following differential eguations,
dθ1/dt = θ2
dθ2/dt= -(b/m).θ2-(g/l).sinθ1
Now these equation can be solved using ode45 function of MATLAM.
The parameters choosen for study are as follows:
l = 1 (meter)
m= 1kg
b = 0.05
g = 9.81(meter/s^2)
with initial conditions,
θ = 0
dθ/dt = 3
MAIN PROGRAM FOR MATLAB:
%program to slove 2nd order ode for pendulum
clear all
close all
clc
% inputs from challenge
l= 1; %length of pendulum
m=1; %mass of ball
b=0.05; %damping coefficient
g=9.81; %gravitational acceleration
%Initial condition
theta_0=[0;3];
%time points
t_span=linspace(0,20,500);
%ODE solution
[t_results]= ode45(@(t, theta) ode_func(t, theta, b, g, l, m),t_span, theta_0);
%plot results
figure(1)
plot(t, results(:,1))
hold on
plot(t, results(:,2))
xlable1('time')
ylable1('plot')
%creating animatiom
ct=1;%counter variable to keep track of frame number
for i= 1: length(results(:,1))
x0= 0;
y0=0;
x1= 1*sin(results(i,1));
y1= -1*cos(results(i,1));
figure(2)
plot([-1 1], [0 0], 'linewidth', 10 'color', 'g') % plot fixed support
axis([-2 2 -2 2])
hold on
line([x0 x1], [y0 y1], 'linewidth', 5, 'color', 'b') %plot string position
plot(x1, y1, 'o', 'markersize', 25, 'markercolor', 'r') % plot ball position
hold off
M(ct) = getframe(gcf); % stores current frame
ct = ct+1;
end
% Create movie
movie(M)
videofile = videowriter('pendulum_simulation.avi', 'Uncompressed AVI'); %create videofile
open(videofile); % open videofile
writeVideo(videofile,M) % Write frames to videofile
close(videofile) %close video file
ode_func:
function [dtheta_dt] = ode_func(t, theta, b, g, l, m)
theta1 = theta(1);
theta2 = theta(2);
theta1_dt = theta2;
theta2_dt = (b/m)*theta2-(g/l)*sin(theta1);
dtheta_dt = [dtheta1_dt; dtheta2_dt];
end not able to resovle the code