Week 4.1 - Solving second order ODEs

%Inputs

l=1;
m=1;
b=0.05;
g=9.8;

%initial condition
theta_0 = [0;3];

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

%slove ODE
[t, results] = ode45(@(t,theta) ode_func(t, theta, b,g,l,m),t_span, theta_0);

plot(t,results(:,1))
hold on
plot(t,results(:,2))
ylabel('plot')
xlabel('time')

x0 = 0;
y0 = 0;

disp1 = results(:,1);
ct = 1;
figure(2)
for i = 1:length(disp1)
    x1 = l*sin(disp1(i,1));
    y1 = -l*cos(disp1(i,1));
    plot([x0 x1],[y0 y1])
    hold on
    plot(x1,y1,'.','Markersize',50)
    plot([-1 1],[0 0])
    axis([-2 2 -2 2])
 
    hold off
    M(ct) = getframe(gcf);
ct = ct+1;
end



movie(M)
videofile = VideoWriter("secondorder_ode.avi");
open(videofile)
writeVideo(videofile,M)
close(videofile)

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]; 

EXPLANATION:

step 1

In step one to enter the second order equaction values

there are

g = gravity in m/s2,

L = length of the pendulum in m,

m = mass of the ball in kg,

b=damping coefficient.

STEP 2

In step 2 to give the initia condition of the ode function (i.e)

%initial condition
theta_0 = [0;3];

and also to given the time

%time points

t_span = linspace(0,20,200);

STEP 3

In step 3 we can slove the second order ode (i.e)

%slove ODE
[t, results] = ode45(@(t,theta) ode_func(t, theta, b,g,l,m),t_span, theta_0);

ode_func(t, theta, b,g,l,m) in this command to take from function loop like

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];

STEP 4

i can draw thplot between plot and time

plot(t,results(:,1))
hold on
plot(t,results(:,2))
ylabel('plot')
xlabel('time')

after this commands we can given the animated plot of simple pendulum and also to create the movie file

movie(M)
videofile = VideoWriter("secondorder_ode.avi");
open(videofile)
writeVideo(videofile,M)
close(videofile)

https://drive.google.com/file/d/1NkubkGB55GxR-vFVAu7bsojr8F8TZA9U/view?usp=sharing

ERRORS:

I had struggled with one error (i.e) we can not given the x and y values