Simulation of transient motion of Simple Pendulum using Matlab

Objective :

To write a program to simulate the transient behaviour of a simple pendulum and to create an animation of it\'s motion using Matlab.

Introduction:

In Engineering, ODE(Ordinary Differential Equation) is used to describe the transient behavior of a system. A simple example is a pendulum.

The way the pendulum moves depends on the Newtons 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.

The differential equation used to show the motion of a damped simple pendulum is stated as below:

Where,

L=1 metre, length of the pendulum in m

m =1kg, mass of the ball in kg,

b= 0.05 , damping coefficient

g=9.81 m/s2. , gravity

In order to solve second order ODE first we have to convert it into first order ODE as follows:

Where,

θ2 = velocity function and θ1 = position function.

% simulation of transient behaviour of a pendulam

close all 
clear all
clc

% inputs

b = 0.05;
m = 1;
l = 1; 
g = 9.81;

% initial conditions
theta0 = [0;3];

% time span
t_span = linspace(0,20,300);

%solving ode
[t,results] = ode45(@(t,theta)ode_func(theta,b,g,l,m),t_span,theta0);

z = results(:,1);

% loop formation
ct = 1;
for i = 1:length(z);
    j = z(i);
  
    x1 = 0;
    y1 = 0;
    x2 = l*sin(j);
    y2 = -l*cos(j);
    
% plotting 

    plot ([x1 x2],[y1 y2],x2,y2,\'r.\',\'MarkerSize\',18);
    axis([-2 2 -2 2]);
    pause(0.01);
    M(ct) = getframe(gcf);
    ct = ct+1;
end
   
   % creating movie of our animation

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

Explanation of Script:

% Inputs

Inputs are given to all values of second order ODE.

% Initial condition

Initial conditions are specified for angular displacement and angular velocity such as 0 and 3 rad/sec

% Time span

Time span is specified for 20 sec of duration for motion of simple pendulum. Linspace command is used to split the interval of 0-20 sec. into 300 values.

% Solving ODE

ode45 is a inbuilt function which takes the input and gives output in the form of position and velocity at any given point w.r.t time.

Now here in this case a separate funtion is used to give input to the ode solver as shown :

function [d_theta_dt] = ode_func(theta,b,g,l,m);
 theta1 = theta(1);
 theta2 = theta(2);
 dtheta1_dt = theta2;
 dtheta2_dt = -(b/m)*theta2-(g/l)*sin(theta1);
 d_theta_dt = [dtheta1_dt;dtheta2_dt];
end

The function command is saved as a separate script with the same name as ode_func. This is very important that the \'Function\' command should be save with a same name as that of Function or else it may not work.

Values for x1,y1,x2,y2 is been specified and lenght of string is considered to be 1 m.

% Plotting

The values of x1,x1,y1,y2 are plotted and with help of \'Marker\' command the ball of pendulum is created of size 18 units and a colour red.

Pause of 0.01 second is given for each value of plot.

\'M(ct)\' command is used to save each and every plot in array.

axis is specified as -2 and 2 for both axis x and y.

% Animation

Command \'movie(m)\' is used to create movie of each plot stored in array M.

Videofile is created with the help of command \'VideoWriter\' and video is stored as a avi .

Script used to create a plot of displacement and velocity w.r.t time :

plot(t,results(:,1))
hold on
plot(t,results(:,2))
ylabel(\'plot\')
xlabel(\'time\')
legend(\'displacement\',\'velocity\')

Plot :

Animation of motion.