Solving second order ODEs

AIM:

To plot the graph of displacement and velocity v/s time and animate the motion of bob.

GOVERNING EQUATION:

The ODE representing the motuion of the simple pendulum with damping is shown below:

`(d^2θ)/dt^2+b/m*(dθ)/dt+g/l*sinθ=0`

Where,

b = damping coefficient

m = mass of the bob(kg)

g = acceleration due to gravity(`m/s^2`)

l = length of the string(l)

GIVEN VALUES:

l = 1m,

m = 1kg,

g = 9.81`m/s^2`

b = 0.05

OBJECTIVE:

1. To plot the graph of velocity and displacement v/s time.
2. To animate the motion of the bob.

The above figure shows the motion of the bob.

WORKFLOW OF THE PROJECT:

1. The value of l,m,b,g are given and the time varies from 0-20sec so we plot the graph of displacement and velocity v/s time.
2. Then we plot the motion of the bob.
3. At last we merge all the motion that is obtained and make the animation of that.

PROGRAMMING LANGUAGE USED- MATLAB

MATLAB CODE:

MAIN PROGRAM:

%program for solving second order differential equation
clear all
close all
clc

%input parameters
l = 1;            %length in m
m = 1;            %mass in kg
b = 0.05;         %damping coefficient
g = 9.81;         %acceleration due to gravity m/s^2

%intial condition
theta_0 = [0;3];   %[angular displacement,angular velocity]

%time plots
t_span = linspace(0,20,500);

%solving the ode
%here ode45 is the inbuilt command to solve the differential equation
[t,results] = ode45(@(t,theta) ode_function(t,theta,l,m,b,g),t_span,theta_0);

%plotting  angular displacement
figure(1)
plot(t,results(:,1))
hold on;
%plotting angular velocity
plot(t,results(:,2))
xlabel('Time');
ylabel('Plot');
plot([0,20],[0,0],'color','black');
legend('Angular displacement','Angular velocity');


%loop counter variable ct
ct = 1;
figure(2)
for i = 1:length(results(:,1))
    theta0 = results(i,1);
%fixed point of the string
x0=0;
y0=0;

%free point of the string
x1=l*sin(theta0);
y1=-l*cos(theta0);

%plotting of bob motion
plot([-1,1],[0,0],'linewidth',5,'color','black');
line([x0,x1],[y0,y1],'linewidth',3)
hold on
plot(x1,y1,'o','markers',15,'markerfacecolor','g');
axis([-2 2 -1.5 0.5]);
pause(0.01);
M(ct)=getframe(gcf);
hold off
ct = ct + 1;
end

%making of the movie from the collected frame
movie(M)
%saving the movie in avi format
videofile=VideoWriter('bobmotion.avi','Uncompressed AVI')
%opening thevideo file
open(videofile)
writeVideo(videofile,M)
%closing thevideo file
close(videofile)

FUNCTION PROGRAM:

function [dtheta_dt] = ode_function(t,theta,l,m,b,g) %initializing the function
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

CODE EXPLAINATION:

1. Initially, we initialize all the input parameters like l,m,b,g which are given in the problem
2. After that the time span over which the motion of the bob is to be observed is also defined.
3. After that we write the function program,here we should remember that the function name and file name should be same.
4. Then we solve the ode by using the inbuilt ode solver 'ode45' in matlab.
5. Then we plot the graph of angular displacement and angular velocity v/s time .
6. At last after plotting we animate the motion of the bob by using the movie command.

OUTPUT:

The plot of angular displacement and angular velocity v/s time is shown below:

The animation link of motion of bob of simple pendulum is given below:

https://youtu.be/uv9LIlEYm1w

CONCLUSION:

Thus we have seen the plot of angular displacement and angular velocity v/s time and also the animation of bob motion of simple pendulum was obtained.

ERRORS OCCURED AND IT'S SOLUTION:

Initially it was plotting all the position of the bob without deleting the previous one as shown below:

To overcome this error i used 'hold off' command on line number 55 as shown below:

After using this command i was able to obtained the output as required.