SIMULATING THE MOTION OF SIMPLE PENDULUM BY SOLVING ORDINARY DIFFERENTIAL EQUATION

DETAILED REPORT ON SIMPLE PENDULUM

1. AIM

To write a program for solving the ODE representing the motion of a simple pendulum with damping and simulate the motion between 0 to 20 seconds

2. EQUATIONS/ FORMULAEUSED

Simple Pendulum – It is usually a bob of mass m suspended from a hinge (fixed point) in a way that it oscillates (to and fro motion) when displaced from its original rest position (equilibrium point). These oscillations gradually die down over a period of time.

The way the pendulum moves depends on the Newton’s second law. When this law is written down, we get a second order Ordinary Differential Equation that describes the position of the "ball" w.r.t time.

ODE representing simple pendulum,

Given Inputs,

Length of the pendulum, L=1 m,

Mass of the bob, m=1 kg,

Damping Coefficient, b=0.05.

Gravity, g=9.81 m/s2.

3. OBJECTIVES & PROCEDURE

1. The main objective of the challenge is to solve the 2^nd order ODE representing motion of a simple pendulum and create an animation of a pendulum between 0 to 20 seconds for angular displacement equal to 0 and angular velocity equal to 3 rad/sec2 at time t=0.
2. In the process, firstly the 2^nd order ODE is spilt into two 1^st order ODE so that it can be solved using the ODE 45 solver.
3. The angular displacement and angular velocity equation is defined in the function program and called using the main program.
4. The value of angular displacement (theta) is calculated at each point of time by using ‘for’ loops. Markers are used to plot hinge (fixed support) and Bob.
5. For creating animation of oscillating pendulum, each of the plots obtained at every point of time is stitched together.
6. Both animation & graph is plotted using the values obtained from angular displacement and velocity in .gif and .jpeg format respectively.

4. PROGRAM

Programming language used – Octave 5.1.1

Program

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MAIN PROGRAM
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% Solving Simple Pendulum motion 

% time
t = [0 20];

% Initial Conditions (angular displacement, angular velocity)
theta0 = [0 3];

[t, y] = ode45(‘pendulum_function’,t,theta0);

Angular_Displacement = y(:,1);

for i = 1:length(Angular_Displacement)

T = Angular_Displacement(i);

x0 = 0;
y0 = 0;

L = 1;

x1 = L*sin(T);
y1 = -L*cos(T);

figure(1)
plot(x0,y0,’square’,’markers’,20,’markerfacecolor’,’b’)
hold on 
plot ([x0 x1],[y0 y1],’linewidth’,7)
hold on
plot(x1,y1,’o’,’markers’,30,’markerfacecolor’,’r’)
hold off

axis([-1 1 -1 0]);

%Simulate the pendulum motion by stiching ech plot together
printf(‘Creating GIF- Progress... %d/%dn’, i, length(Angular_Displacement));

img = print (‘-RGBImage’);

imwrite(img, ‘animation.gif’, ‘DelayTime’, .005, ‘Compression’, ‘bzip’, ‘WriteMode’, ‘Append’);

end

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FUNCTION PROGRAM
function [ydot] = pendulum_function(t,y)

  % Gravity (m/s2)
  g = 9.81;

  % Length of the pendulum (m)
  L = 1;

  % Mass of the ball (kg)
  m = 1;

  % Damping Coefficient
  b = 0.05;

ydot = zeros(2,1);

  % Rate of change of angular displacement = Angular Velocity
ydot(1) = y(2);

  % Angular Acceleration = ((-b/m)theta(2))-((g/L)sin(theta1))
ydot(2) = ((-b/m)*y(2))-((g/L)*sin(y(1)));


end
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5. RESULTS

Graph

Youtube Link: https://youtu.be/Rxs1syz3n_I

6. CONCLUSION

The above graph shows the angular displacement and angular velocity versus time of a simple pendulum. Also the same values are used to animate the pendulum motion.

The program can be used to solve the ODE representing the transient behavior to simulate the system.