Week 3 - Solving second order ODEs

SOLVING SECOUND ORDER ODE

AIM:

To solve a secound order ODE and simulate the simple pendulam motion by using MATLAB.

GIVEN DATA'S:

• L=1 metre,
• m=1 kg,
• b=0.05.
• g=9.81 m/s2.
• Angular displacement = 0
• Angular velocity = 3 rad/sec
• T = 0

EXPRESSION:

• g = Gravity in m/s2,
• l = Length of the pendulum in m,
• m = Mass of the ball in kg,
• b =Damping coefficient.

FORMULA TO BE USED :

SIMPLE PENDULAM :

THE ABOVE PICTURE WILL SHOW AS THE SIMPLE PENDULAM

A simple pendulum consists of a mass m hanging from a string of length L and fixed at a pivot point P. When displaced to an initial angle and released, the pendulum will swing back and forth with periodic motion.A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position.

MATLAB 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

The above code which is shows as function codes used in matlab. The function codes could be called by main code as a ODE 45 .

ODE MAIN PROGRAM:

clear all
close all
clc

%inputs

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

% initial conditions
theta_0 = [0;3];

% time points
time = linspace(0,20,200);

%solve ODE
[t, result] = ode45(@(t,theta)ode_func(t, theta, b, g, l, m),time, theta_0);

% PLOTTING
figure(1)
plot(t,result(:,1),'color','g')
hold on
plot(t,result(:,2),'color','b')
xlabel('time')
ylabel('Angular velocity and displacement')
legend('change in displacement','change in angular velocity')
hold off
figure(2)
plot(result(:,1),result(:,2))
xlabel('position')
ylabel('anglar velocity')

% ANIMATION
c = 1
for i=1 : length(result(:,1))
    x0 = 0
    y0 = 0
    x1 = l*sin(result(i,1))
    y1 = -l*cos(result(i,1))
    plot([-1,1],[0,0],'linewidth',4,'color','y')
    hold on
    plot([x0 x1],[y0 y1],'linewidth',2,'color','r') 
    hold on 
    plot(x1,y1,'o','markers',20,'markerfacecolor','b')
    grid on
    axis ([-2 2 -2 2])
    pause(0.03)
    hold off
    M(c) = getframe(gcf)
    c = c+1
end

movie(M)
videofile = VideoWriter('ode_main','uncompressed avi')
open(videofile)
writeVideo(videofile,M)
close(videofile)

STEPWISE EXPLAINATION:

CLEAR ALL -it clear all privious codes.

CLOSE ALL -it close all matlab works.

CLC - I'ts clear command window.

INPUTS:

• L=1 metre,
• m=1 kg,
• b=0.05.
• g=9.81 m/s2.

INITIAL CONDITION:

theta_0 = [0;3];

This is intial starting condition to be used .Than the initial starting displacement is zero and starting velocity is 3.

TIME POINTS:

time = linspace(0,20,200);

Time intervel is - (0-20s)

Then 200 values are calculated between 20s.

%SOLVE ODE:

[t, result] = ode45(@(t,theta)ode_func(t, theta, b, g, l, m),time, theta_0);

ode 45 is call function code.Others are data which is taken from Given .

PLOTTING:

Plotting which is used to plot the values.

ANIMATION:

In this function which is used to animate the simple pendulam. In this function having the line width and color of the line . The initial stage of plottig is [x0 x1],[y0 y1]

FINAL VIDEO CODE:

In this code which is used to make a animated video for the ode main file,and save as a compressed avi file.

PLOTTED IMAGE:

SIMPLE PENDULAM ANIMATED MOVIE FILE:

THIS IS THE YOUTUPE LINK FOR ANIMATTED FILE,https://youtu.be/7T3iYyNBcHY

CONCULUTION:

Finally the Simple pendulam codeing and animatted successfully by using matlab.