Challenge

Week 3 - Solving second order ODEs

KKhanjan Sonani28 Dec 2021

MATLAB

AIM:To write a program for simulating the transient behaviour of a simple pendulum.

OBJECTIVES:

Simulate the transient behaviour of a simple pendulum
Create an animation of its motion.

GOVERNING FORMULA:

Ordinary diffrential equation for oscillating motion of non-linear pendulum.

`frac{d^2theta}{dt^2}+frac{b}{m}*frac{d theta}{dt}+frac{g}{l}*sin theta`

Where,

g=gravity in m/s

l= length of the pemdulum in m

m=mass of the ball in kg

b=damping coeficiant

INPUTS:

L= 1m

m= 1kg

b=0.05

g=9.81m/s^2

time span= 0-20 secs

angular displacement= 0

angular velocity=3 rad/sec

STEPS WITH CODES:

First we have to defina a diffrential equation for pedulum

Function code

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

After defining the function we write the main program.

In main program the value of each is assigned.The ode45 solver in MATLAB is used to solve the ode foe pendulum with damping.The ode45 solver in MATLAB reduce the second order diffrential equation and gives the result which are then used to plot a angular displacement vs time graph and simulating of the pedulum.

MAIN PROGRAM

clear all
close all 
clc

b=0.05;
g=9.81;
l=1;
m=0.1;

%initial condition 
theta_0 = [0;3];

%time points
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);

%plot
plot(t,results(:,1),'color','r')
hold on
plot(t,results(:,2),'color','b')
xlabel('plot')  
ylabel('time')
grid on 

%for animation 
x0=0
y0=0
for i=1:length(t)
    angle=results(:,1)
    ANGLE=angle(i)
    x1=sin(ANGLE)
    y1=-cos(ANGLE)
    figure(2)
    plot(x1,y1,'.','markersize',60)
    hold on 
    plot([x0 x1],[y0 y1],'linewidth',4)
    axis([-2 2 -1.5 1.5])
    hold off
    M(i)=getframe(gcf);
end

%creating the simulation 
movie(M)
videofile = VideoWriter('pedulum_motion')
open(videofile)
writeVideo(videofile,M)
close(videofile)

RESULT:

The simulation was successs.