Simulation of Motion of Pendulum by solving second order O D E using MATLAB

Aim:-To Simulate the motion of Pendulum by solving second order O.D.E

Objective:-

1. To convert second order O.D.E to first order in order to use the bring the variables in ode45 format.
2. Plotting the results of theta over the given time span.
3. Using the marker command to create a pendulum

Introduction:-

Motion of the pendulum is given by the following second order O.D.E

.......................................................Equation 1

where theta is the dependent variable and t is the independent variable;

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.

Assuming Inputs:-

L=1 metre,

m=1 kg,

b=0.05.

g=9.81 m/s2.

tstart=0 sec

tend=20 sec

ω(Angular Velocity)=F(θ,t)

ω=3 rad/sec at θ=0,t=0;

For converting Second order O.D.E into first order O.D.E we assume;

Θ = Θ₁ 

dθ/dt =dΘ1/dt=Θ₂

(d^2 θ)/(dt^2 )= dΘ2/dt

Substituting value of the Θ,dθ/dt,(d^2 θ)/(dt^2 ) in equation 1 we get;

dΘ2/dt+b/m*Θ₂ +g/l sinΘ₁=0

Stepwise Explanation:-

1. Inputing the intial conditions and the timespan for the motion is to be observed.
2. dtheta_dt is function of (t,theta,b,g,l,m) in which t and theta varies so;dtheta_dt is an array which represnts value of angular acceleration and angular velocity.

function [dtheta_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);
dtheta_dt=[dtheta1_dt;dtheta2_dt];
end

     3.Variables x0,y0 are fixed at the origin.For loop brings different value of theta each time which gives the position of the variables x1,y1.

     4. Plotting a straight line reference and then by using marker command making a pendulum bob at the end of the co-ordinates x1,y1.

     5.M is array which contains images of the pendulum at various angle;this pictures are then used for making a animation by using the movie command.

The programme is written below:- 

clear all
close all
clc
%Inputs
b=0.05;
g=9.81;
l=1;
m=0.1;
ct=1;
%intial condition
theta_0=[0;3];
%time interval
t_span=linspace(0,20,500);
%solve_ode
[t,results]=ode45(@(t,theta)ode_func(t,theta,b,g,l,m),t_span,theta_0);
figure(1)
plot(t,results(:,1));
figure(2)
for i=1:length(results(:,1));
    Results=results(i,1);
x0=0;
y0=0;
x1=l*sin(Results);
y1=-l*cos(Results);
plot([-1.5 1.5],[0 0],[x0 x1],[y0 y1],x1,y1,\'-o\',\'markersize\',20,\'markerfacecolor\',\'y\')
axis ([-1.5,1.5,-1.5,1]);
pause(0.3);
M(ct)=getframe(gcf);
ct=ct+1;
end
movie(M)
videofile = VideoWriter(\'pendulum_animation.avi\',\'Uncompressed AVI\')
open(videofile)
writeVideo(videofile,M)
close(videofile)

Fig.Plot of Time vs Displacement which is getting damped

Youtube link for the animation is :-https://youtu.be/ZLNseDblUqY

Errors done: -

1.Error in typing axis array in which the start end point was selected as negative .

Conclusion:- By using a O.D.E solver simulated the motion of the pendulum and created animation of the its motion by defining input variables.