1. Pendulum Explanation-

Figure 1.1 Forces on Pendulum

1^st Step-

According to the newton’s second law of motion-

F _tangential = m*a _tangential

F _{tangential = - (m*g*sin (θ))}

_{damping force = (b/m*L)*`(d theta)/dt`}

a _tangential =R*alpha

alpha = `(d^2theta)/dt^2`

So mass on string of length L, tangential acceleration is a _tangential = L* ((d^2theta)/dt^2) (θ(theta) is in radians)’

In case of pendulum with damping the damping force will also play its role, hence equation becomes –

_{- (m*g*sin (θ)) – b*`(d theta)/dt`} = m*L*(d^2theta)/dt^2

((d^2theta)/dt^2) + (b/m*L)*`(d theta)/dt`+ (g/L)*sin (θ) = 0 (ODE to be solved)

k=(b/m*L) ; (g/L) = r

2^nd Step-

Convert second order ODE to first order ODE –

X(1) = θ

X(2) =((d theta)/dt)

`(d(X(1)))/dt`= ((d theta)/dt) = X(2) (1)

`(d(x(2)))/dt`` (d^2theta)/dt^2 = -k*X(2) - r*sin (θ) (2)

Above two equations are solved in Matlab.

Matlab Code –

%system parameters

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

k=(b/(m*L));

r=g/L;

t = [0 20];

x0 = [0 3];

%ODE using anonymous function

[t, x] = ode45(@(t,x)[x(2);-k*x(2)-r*sin(x(1))],t,x0);

plot(t,x(:,1));

grid on;

hold on;

plot(t,x(:,2),\'r\');

xlabel(\'Time(s)\');

ylabel(\'Amplitude\');

legend({\'Angle(rad)\',\'Angular_speed(rad/sec)\'});

Results –

%system parameters
m=1;L=1;b=0.05;g=9.81;
k=(b/(m*L));
r=g/L;
t = [0 20];
x0 = [0 3];
%ODE using anonymous function
[t, x] = ode45(@(t,x)[x(2);-k*x(2)-r*sin(x(1))],t,x0);
plot(t,x(:,1));
grid on;
hold on;
plot(t,x(:,2),\'r\');
xlabel(\'Time(s)\');
ylabel(\'Amplitude\');
legend({\'Angle(rad)\',\'Angular_speed(rad/sec)\'});

%system parameters
m=1;L=1;b=0.05;g=9.81;
k=(b/(m*L));
r=g/L;
t = [0 20];
x0 = [0 3];
%ODE using anonymous function
[t, x] = ode45(@(t,x)[x(2);-k*x(2)-r*sin(x(1))],t,x0);

%animation
%origin
O=[0,0];
axis(gca,\'equal\'); %aspect raito of the plot
axis([-1.5 1.5 -1.5 0.4]);
grid on;
% animation loop
ct=1;
for i = 1:length(t)
    %point where mass is attached
    p=L*[sin(x(i,1)) -cos(x(i,1))];
     %origin_circle
     O_circ = viscircles(O,0.01);
     %pendulum
     pend = line([O(1) p(1)],[O(2) p(2)],\'linewidth\',3,\'color\',\'black\');
     %ball
     ball = viscircles(p,0.05);
     %time interval to update plot
     pause(0.001);
     M(ct)= getframe(gcf);
     ct=ct+1;
     %delete previous objects if it is not the final loop
     if i<length(t)
         delete(pend);
         delete(ball);
         delete(O_circ);
     end
     %F=getframe(gcf);
end
movie(M)
videofile = VideoWriter(\'animation_pendulum.avi\',\'uncompressed AVI\');
open(videofile)
writeVideo(videofile,M)
close(videofile)

Youtube Animation link-

https://www.youtube.com/watch?v=HVWrv6jJl3s