Week 3 - Solving second order ODEs

Aim:- To create an animation of simple pendulum by using second order ODE by using ODE solver function.

To create plot between Angular velocity and displacements of simple pendulum by using ODE 45

Objects :-

1. Solve ODE which represent the equation of motion of simple pendulum damping with the help of MATLAB code
2. Plotting the graph of variation of angular displacements VS time and angular velocity vs time.
3. Animation of the plot for the motion of pendulum bet 0-20sec and Angular displacements =0

Angular velocity =3red/sec at time =0

b = Damping coefficient (N- m/s)

m =Mass of the ball (kg)

g = Gravitational acceleration (m/s)

l- length of pendulum I (m)

clear all

close all

clc

Given initial condition in question

b=0.05;

g=9.81;

l=1;

m=1;

% initial condition

theta_0=[0;2];

% time point

t_span=linspace(0,20,500);

% slove 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))

xlabel('time')

ylabel('plot')

grid on

title('Angular displacement vs Time and angular velocity vs Time ')

legend('angular displacement in red','Angular velosity in red/sec')

Plot of Angular displacements vs time and angular velocity vs time.

% iteration using for loop to create

ct=1;

theta_1 = results(:,1);

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

% initial coordinates

x_start = 0;

y_start = 0;

theta_2 = theta_1(i);

% final coordinates

x_end = 1*sin(theta_2);

y_end = -1*cos(theta_2);

% ploting for the animation

figure(2)

% base of pendulum

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

hold on

% string the pendulum

line([x_start x_end],[y_start y_end],'linewidth',3,'color','b');

hold on

%bob of pendulum

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

axis ([-2 2 -2 2 ])

grid on

% pause(0.03)

hold off

M(ct)=getframe(gcf);

ct=ct+1;

end

% creating movie from capture frome

movie(M)

videofile = VideoWriter('simple_pendulum.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];

end

Missing variable in line 22 – dtheta_dt =theta2 against dtheta1_dt=theta2

You tube link

https://youtu.be/ytHx_6tkfSA

https://drive.google.com/file/d/1zosldVcFzl0EGSte5mQOig3KdaJCYB70/view?usp=sharing