Week 3 - Solving second order ODEs

in the above equation

g= gravity in m/s^2

L= length of the pendulum in meter.

m= mass of the ball in kg.

b= damping coeffecient.

let Θ=Θ1

dΘ1/dt=Θ2

dΘ^2/dt^2=dΘ/dt

Input:

L=1 metre

m= 1kg

b=0.05

g= 9.81m/s^2

time span=0-20 sec

angular displacement=0

angular velocity=0

angular velocity=3 rad/sec2 at time t=0

Code:

% program to simulate the transient behaviour of a simple pendulam

clear all

close all

clc

%inputs

b=0.05;

g=9.81;

l=1;

m=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);

%ploting

plot(t, results(:,1))

hold on

plot(t, results(:,2))

ylabel( 'Plot' )

xlabel( 'Time in sec' )

%Iletration using for loop to create animation of simple pendulam of simple pendulam

ct=1;

theta_1= results(:,1);

for i= 1:length(results(:,1))

%initial co-ordinates

x_start=0;

y_start=0;

theta_2= theta_1(i);

%final co-ordinates

x_end= 1*sin(theta_2);

y_end= -1*cos(theta_2);

% ploting for animation

figure(2)

%base of pendulum

plot([-1,1],[0,0], 'linewidth',4, 'color', 'black')

hold on

%string of pendulum

line([x_start x_end],[y_start y_end], 'linewidth',6, 'color', 'b')

hold on

% bob of pendulum

plot(x_end,y_end, 'marker', 'o', 'markersize', 28, 'markerfacecolor','r')

axis([-2 2 -2 2])

pause(0.03)

hold off

M(ct)=getframe(gcf);

ct=ct+1;

end

Code for function program:

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

Explanation for each step for the problem:

1. in the step input and initial condition of a simple pendulum are fed.

2. then the program is written to solve the ODE in which Matlab inbuilt function (ode45) is used with constant theta t, and another function is used, which is defined seperately in function program.

3. In the next step fo result of solved ODE is plotted w.r.t time using plot command.

4. Now iteration of co-ordinate is fed using for loop to create animation.

5.Then plot command is used to plot pendulum string and bob.

Output:

1. The output obtained is graphical representation velocity curve and displacement curve w.r.t time.

Angular displacement and angular velocity plots wer generated for a time range of 20 seconds.