Motion of a pendulum Solution of the ODE using MATLAB

Program to simulate the transient behavior of a simple pendulum;

• ODE[Ordinary Differential Equation] - A differential equation containing one or more functions of one independent variable and the derivatives of those functions.

• Second Order[ODE] - The equation consists of a second derivative and not more than that.

• Simplification of the given Pendulum ODE;

`(d^2theta)/dt^2+b/m*(d*theta)/dt+g/lsintheta = 0`

Assume `theta = theta_1`

Differentiating `theta`gives,

`(d(theta))/dt = (d(theta_1))/dt`

let, `(d(theta_1))/dt = theta2`

Now, Differentiating `(d(theta))/dt` gives,

`(d^2(theta))/dt^2 = (d^2(theta_1))/dt^2 = d/dt((d(theta_1))/dt) = d/dt(theta_2)`

Hence,

The given ODE becomes

`(d(theta_2))/dt+b/m(theta_2)+g/Lsintheta_1 = 0`

• Given values

1. L = 1m
2. m = 1 kg
3. b = 0.05
4. g = `9.81m/s^2`

• Sample working Code;

clear all
close all
clc

%given inputs of the pendulam and forces acting on the pendulam.
%where L,m,b,g are length,mass,damping coefficient and gravity respectively
L = 1;
m = 1;
b = 0.05;
g = 9.81;

%the initial conditions for the pendulum[displacement;veloctiy]
theta_0 = [0;3];

%given work time of the pendulam
P_travel = linspace(0,20,228);

%solving ode
[t, result] = ode45(@(t,theta) ode_function(t,theta,b,g,L,m), P_travel, theta_0);

%plotting the result
%plotting the curves of the Displacement
plot(t,result(:,1))
hold on
%plotting the result of the velocity
plot(t,result(:,2))
xlabel(\'time\')
ylabel(\'Displacement&angular velocity\')
legend(\'displacement\',\'angular_velocity\')
grid on
hold off

%creating the avi file(animation of pendulum travel)
ct = 1;
for i = 1:length(result(:,1))
    %the base of the pendulum attach has zero motion.
    x0 = 0;
    y0 = 0;
    
    %the free end of the pendulum in motion
    x1 = L*sin(result(i,1));
    y1 = -L*cos(result(i,1));
    
    %plotting the base attachment
    plot([-0.5 0.5],[0 0],\'linewidth\',2,\'color\',\'b\')
    hold on
    %plotting the free end
    plot([x0 x1],[y0 y1],\'linewidth\',2,\'color\',\'r\')
    %plotting the face(ball/nob) of the pendulum
    plot(x1,y1,\'o\',\'markers\',28,\'markerfacecolor\',\'g\');
    hold off
    grid on
    axis([-1.5 1.5 -1.5 1.5])
    pause(0.228)
    %creating the M array containing all the resulting frames
    M(ct) = getframe(gcf);
    ct = ct+1;
end
%creating the movie of the array M.
movie(M)
videofile = VideoWriter(\'simple_pendulum.avi\',\'Uncompressed AVI\');
open(videofile)
writeVideo(videofile,M)
close(videofile)

• Sample Function code;

%creating the function of the ode of pendulum
function [dtheta_dt] = ode_function(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

• Graphical result;

• Link redirecting to the animation of the motion of pendulum;

• Explanation;

1. The given ODE is in second-order and so in order to generate the MatLab code that solves that ODE, it is first simplified into the single order (or) simple ODE.
2. The simplification of the ODE is carried out as shown above.
3. The simplified ODE is used to create the function code to carry out the program.
4. The ode function contains the variables that are prespecified.
5. The ode function has further simplified the form of the given ODE.
6. This ode function is incorporated in the main program.
7. The program has a linspace command that helps in generating equally spaced numbers or constants between prespecified numbers.
8. Theta is given as 0 & 3, where 0 is the angular displacement and 3 is angular velocity which is given.
9. Solving the ode using \'ode45\' command which integrates the ode function and also the total travel of the pendulum.
10. plotting the results for the displacement and velocity curve.
11. labeling THE AXES USING THE \'LABEL\' COMMAND.
12. The legend command helps in naming/identifying the curves.
13. Finally creating a movie/ animation of the results by compiling all the results of the given ode between the specified time period.
14. First, creating the model of the pendulum by giving its dimensional analysis.
15. As we know that the pendulum displaces according to the time, we just integrate the results of the angular displacement to represent the motion of the pendulum.
16. At the base joint, the displacement is zero. Hence, x0 and y0 are 0.
17. At another end of the pendulum, the angular displacement occurs.
18. The pendulum has length L and it displaces against gravity. Hence, the equation is used as shown in the program.
19. Plotting the pendulum motion at each given interval creates each result.
20. The plot has the pendulum ball created by using the markers command where the \'o\' command specifies the shape and marker face color is given as one\'s perspective.
21. The \'pause\' command avoids the overwriting of the results at each interval.
22. Creating an array M which is subjected to carry all resulting frames and then creating a movie of M.
23. The attached graph sample shows the result of the ode function.
24. The provided link shows the animation of the motion of a pendulum.

• CONCLUSION;

The report gives the idea of how the ODE is solved using the MatLab program and its consequences in detail.