Week 3 - Solving second order ODEs

https://drive.google.com/file/d/1DPnqwMfjyTMoR943W6gdkjGPGCRE36Ni/view?usp=sharing

SIMPLE PENDULUM :-

A heavy point mass suspended by a weightless, perfectly flexible and inextensible string, from a perfectly rigid support is called "SIMPLE PENDULUM".

The tension in the string depends both on mg and centripetal force muThe acceleration of the simple pendulum will be given by, a = - g sing

The time period of the simple pendulum is, T = 24W

The value of the theta (0) and the rate of theta (0) at which it is changing for a simple pendulum can be determined or solved by using the set of "ordinary differential equations".

? OIDINARY DIFFERENTIAL EQUATION (ODE) :-

An ordinary differential equation is an equation containing an unknown function of one real or complex variable x. The unknown function is generally represented by a variable which therefore depends on x.

The examples for the ode are,

dy

= F(r) dy

dr = F ( 2)

d'y

= F(y) de2

In engineering, ODE is used to describe the transient behaviour of a system. A example is a simple pendulum. The following is the second order ode of a simple pendulum. The way the pendulum moves depends on the newtons second law .

simple_pendulum.m
 

%step 1: input

%Damping Coefficient

b=0.02;

%gravity of earth

g=9.81;

%length of rope/string

l=1;

%mass of ball

m=0.1;

%step 2:Initial condition for displacement and angular velocity 

theta_0 = [0;3];

%step 3: time interval points 

t_span= linspace(0, 20, 500);

%step 4: ODE results

%call ode_fun function

%ode45 gives result in rad and rad/s.

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

%step 5 plotting

%figure(1)

hold on 

plot(t, results(:,1),'linewidth',3,'color','g')

plot(t, results(:,2),'linewidth',3,'color','r')

axis([0 20 -6 8])

title('For Simple Pendulum')

xlabel('Time in sec')

ylabel('Angular Velocity(rad/s) / Displacement(rad)')

legend('Displacement', 'Angular velocity')

%step 6: animation of pendulum

amplitude= results(:,1);%store amplitude values in separate matrix for loop

p=1;

for i=1:length (amplitude)

angle=amplitude(i);

figure(2)

title('Simple Pendulum')

%fix point of pendulum

x0 = 0;

y0 = 0;

%another end of pendulum/cordinate of mass ball

x1= l*sin(angle);

y1= -l*cos(angle);

%plot

plot([-1.1 1.1], [0 0], 'Linewidth', 2); 

hold on

plot([x0 x1],[y0 y1], 'color', 'g', 'linewidth', 2)% string

plot(x1, y1, 'o', 'markers',20, 'markerfacecolor', [1 1 0])%mass ball

hold off %removing previous position

grid on

axis ([-1.5 1.5 -1.5 0.5])

pause (0.001)

M(p)= getframe(gcf);

p = p+1;

end


%making movie

movie (M)
videofile = VideoWriter('simple_pendulum.avi', 'uncompressed avi');

open(videofile)

writeVideo(videofile, M)

close(videofile)

odefuntion.m

function[dtheta_dt] = ode_fcn(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: