Week 3 - Solving second order ODEs

1. INTRODUCTION

The pendulum is a body or mass suspended from a fixed point so that it can swing back and forth under the influence of gravity. Pendulums are used to regulate the movement of clocks because the interval of time for each complete oscillation, called the period, is constant. The period depends on the length of the pendulum and also to a slight degree on the amplitude, the width of the pendulum's swing.

2. AIM

To solve the second-order ode function using Matlab which depicts the motion of the pendulum and to create an animation of the same.

3. OBJECTIVE

To write a program that solves the following ODE. This ODE represents the equation of 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.

FIXED PARAMETERS

Length (L) =1 metre,

Mass (m) =1 kg,

Damping coiefficient (b) =0.05.

Acceleration due to gravity (g) =9.81 m/s2.

For Time period = 0-20 sec

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

4. FUNCTION CODE

% ODE Function
function dtheta_dt = ode_funs(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

• In this function program, the second-order ODE is simplified to the first-order ODE by introducing the term theta1 & theta2.
• Then theta1 & theta2 is differentiated and the dtheta_dt gives a column matrix.

5. MAIN PROGRAM

% Fixed Parameters ======>

% 1.Damping Co‐eff.: 
b = 0.05;
% 2.Acceleration due to gravity(m/s^2): 
g = 9.81;
% 3.Mass(kg) : 
m = 1; 
% 4.Length(m) : 
l = 1; 

% Initial condition 
theta_0 = [0;3]; 

% Time points 
tspan = linspace(0,20,200); 
% Solving ODE ==============>>> 
[t,results] = ode45(@(t,theta) ode_fun(t,theta,b,g,l,m), tspan, theta_0); 

% Plotting==========>>> 
figure(1) 
plot(t,results(:,1)) 
hold on 
plot(t,results(:,2)) 
xlabel('Time') 
ylabel('Velocity (Red) & Displacement (Blue)') 

% Animation ============>>> 
ct = 1; 
for i = 1:length(results(:,1)) 
x0 = 0; 
y0 = 0; 
x1 = l*sin(results(i,1)+pi); 
y1 = l*cos(results(i,1)+pi); 
figure(2) 

plot([x0 x1],[y0 y1],'-o','MarkerSize',10,'MarkerFaceColor','r','MarkerEdgeColor','r','linewidth',4)
axis([-2 2 -2 2]) 
hold on 
plot([-1 1],[0 0],'linewidth',5,'color','[0 0.3 0]') 
M(ct) = getframe(gcf)
hold off 
ct = ct + 1; 
end 

% Movie & Videofile ===========>>> 
movie(M) 
v_f = VideoWriter('Pendulum_Challenge.avi','Uncompressed AVI') 
open(v_f) 
writeVideo(v_f,M) 
close(v_f)

EXPLANATION

• In the above program, the fixed parameters are length, mass, damping coefficient, and acceleration due to gravity.
• Time and angle are given as the initial condition.
• In the tspan command, 200 values are taken in between 0 to 20 times period.
• Using the ode 45 command ODE function is introduced.
• Using the plot command velocity and displacement is plotted in the y-axis against time in the x-axis.
• Using the for loop command the mass at various points and various times are marked in the graph and are plotted.
• Using the movie command the plot is saved as an uncompressed avi file.

6. PLOT

7. ANIMATION