Week 3 - Solving second order ODEs

SIMPLE PENDULUM

AIM:

• To create an animation of simple pendulum by using second order differential equation by using ode solver function.
• For simulation the second order equation of the simple pendulum is:
• To create a plot between angular velocity and displacement of simple pendulum by using ode45.

CONCEPT:

• For a simple pendulum with a damping factor, the equation of simple pendulum is

Where

            b = damping coefficient (N-m/s)

            m = mass of the bob (kg)

            g = gravitational acceleration (m/s)

            l = length of the string to connect bob (m)

• The above equation is having a differential power of order 2. To use ode45 solver which is developed on the basis of RK method which uses the simple differential equation to solve the higher order differential equations.

OBJECTIVE:

• By using ode45 solver which is an inbuilt solver in the MATLAB. The ode45 solver should require simple differential equation to solve any kind of higher order differential equation.
• By taking constants we can solve the higher order differential equation to a simple ordinary differential equation.

Let us consider

  θ = θ1 ------ (1)

dθ/dt = dθ₁/dt = θ₂ ------(2)

d²θ/dt² = dθ₂/dt -------(3)

• By using these constants, the second order equation will become as:

dθ₂/dt = -(b/m) * θ₂ – (g/l) * sinθ

• The simple differential equations are written in the form of a matrix

• By using the above equation and the ode45 solver we can animate the simple pendulum in the MATLAB by writing a code.

program for the simple pendulum:

%program for simple pendulum using ode45.
clear all
close all
clc

%input variables for the secound order ode of a simple pendulum
b = 0.05;
g = 9.81;
l = 1;
m = 0.1;

theta_0 = [0; 3];

t_span = linspace(0,20,500);

%ode 45 function to solve second order ode.
[t,results] = ode45(@(t,theta) ode_func(t,theta,b,g,l,m),t_span, theta_0 );

%plotting of angular velocity and displacement
figure(1)
plot(t,results(:,1))
hold on
plot(t,results(:,2))
grid on
title('motion of pendulum')
xlabel('time in sec')
ylabel('angular velocity(rad/s)/displacement(rad)')
legend('displacement','angular velocity')

angle = results(:,1);

%count command for iterations.
ct = 1;

for i = 1 : length(angle)
    omega = angle(i);
    figure(2)
    title('motion of pendulum')
    
    %fixed corordinates of one end of simple pendulum.
    x0 = 0;
    y0 = 0;
    
    %moving coordinates of the bob.
    x1 = l*cos((3*pi/2)+omega);
    y1 = l*sin((3*pi/2)+omega);
    
    %plotting of the bob motion.
    plot([x0 x1],[y0 y1],'linewidth',3,'color','g');
    hold on;
    plot(x1,y1,'o','markers',15,'MarkerFaceColor','k');
    hold on;
    plot([-3 3],[0 0],'linewidth',2,'color','b');
    hold off
    axis([-3 3 -2 2])
    grid on 
    pause(0.1)
    M(ct) = getframe(gcf);
    ct = ct+1;
end

%to create a animation of the simple pendulum
movie(M)
videofile = VideoWriter('simple_pendulum.avi','uncompressed AVI');
open(videofile)
writeVideo(videofile,M)
close(videofile)

ode function program:

function [dtheta_dt] = ode_func(t,theta,b,g,l,m)

theta1 = theta(1);
theta2 = theta(2);
dtheta1_dt = theta(2);
dtheta2_dt = -(b/m)*theta2 - (g/l)*sin(theta1);
dtheta_dt = [dtheta1_dt; dtheta2_dt];

end

PROGRAM EXPLANATION:

• Clear all, close all, clc are used to clear the work space, to clear the graphs that’s exists, and clear the command window.
• The inputs are for the simple pendulum are specified and the b = damping coefficient = 0.05 (n-m/s), g = acceleration due to gravity = 9.81 (m/s), l = length of the string = 1(m), m = mass of the bob = 1 (kg).
• t_span is a command used in the ode45 solver which specifies the time span of the iterations that are going to solve the ordinary differential equation.

• ode45 is a function in MATLAB that solves ordinary or partial differential equations using a numerical method. In this problem the highest order is 2. So, we have to take two constant values so that the equation is changed to simple differential equation.
• The function should take at least 2 inputs to solve the equation to the approximate result.
• The ode45 function has a syntax in which we need to specify which should be ‘t’ and ‘y’ and the output may be of (t, y, args) and a timespan for the function to work and the initial value of the equation.
• Syntax of a ode45 solver

                        [t, results] = Ode45(@ (t, y) ode (t, y, args), t_span, y0).

• Figure 1 is going to tell how the damping is done for the pendulum. The angular velocity and displacement will come to rest at end of the function.
• plotting the values for the angular velocity and the displacement by using plot command.
• The label command is used to label the X & Y axis.

• Legend command is used to label the angular velocity and the displacement.
• To animate the motion of simple pendulum, a for loop is used to create the specified set of points to keep the bob at different locations.

• x0 and y0 be the fixed point at one end of the string and x1 and y1 be the different position of the bob.
• At each and every point of the bob the ‘ct’ command which is used to count each and every position and stores the data.

• At the end of the for loop there will be set of positions for each and every point.
• Movie(M) is used to create a film of the stored data of the motion of the pendulum.
• videoWriter command specifies the name of the file and type of data is stored in it.
• Open command is used to open the file where the video is going to store.
• Writevideo command is used to create the video animation by using M file.
• Close command is used to close the video file.

• Ode45 function is which solves the simple differential equations with a certain specified command.

• In this function, we need to specify the derivatives of theta and the equations to be solved.

ERRORS:

• while using the ‘MarkerFaceColor’ I was not aware of the capital letters of the command to be specified.

• using of the results command, in the figure 1 to get the plot the output is stored in results. But using it again to take the input for the motion of the pendulum I have written it as result which MATLAB shown it as error.

OUTPUT:

• A plot is drawn between the angular acceleration and displacement of the motion of simple pendulum.

• The blue curve is the displacement where it starts from zero. And the angular velocity is starting at 3 (rad/sec).
• At the end of the t_span both the displacement and angular velocity becomes zero and motion of simple pendulum becomes zero which becomes rest.

CONCLUSION:

• We have animated the simple pendulum using the ode45 solver where the second order is reduced to a simple differential equation where it is easy to solve the given equation.
• The animation is made using the different points of the bob and the motion of the simple pendulum is created.

animation of simple pendulum:

https://drive.google.com/file/d/1e3LQvnc7fE9Z2G8u-AXLxlHnCcXvbi2e/view?usp=sharing