Week 3 - Solving second order ODEs

It is the correct program for pendulum.
Editor Window = 

close all
clear all
clc

%input parameters
b = 0.04;                             %b = damping coefficient 
g = 9.81;                             %g = gravity (m/s^2)
l = 1;                                %l = length (m)
m = 0.1;                              %m = mass (kg)

% initial condition                   %initial displacement of pendulum = 0
theta_0 = [0;5];                      %initial velocity of pendulum = 5 m/s^2

% time points
t_span = linspace(0,20,500);          %t_span = time array

% solve ODE
[t, results] = ode45(@(t,theta) ode_func(t, theta, b,g,l,m),t_span, theta_0);
figure(1)
plot(t,results(:,1))
hold on
plot(t,results(:,2))
ylabel('plot')
xlabel('time')

%Animation of the pendulum

ct = 1;
figure(2)

for i = 1:length(results(:,1))
    x0 = 0;
    y0 = 0;
    x1 = l*sin(results(i,1));
    y1 = -l*cos(results(i,1));
    
    plot([-1.5 1.5],[0 0])
    hold on
    plot([x0 x1], [y0 y1], 'linewidth',1,'color','b')
    axis([-1.5 1.5 -1.5 1.5])
    plot(x1,y1,'.','markersize',50,'color','r')
    pause(0.0001)
    M(ct) = getframe(gcf)
    hold off
    ct = ct+1
end

% making the video file
movie(M)
videofile = VideoWriter('simple_pendulum.avi')
open(videofile)
writeVideo(videofile,M)
close(videofile)

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videofile = 

  VideoWriter

    General Properties:

       Filename:                 'simple_pendulum.avi'
       Path:                     'C:\Users\Kamlesh\Desktop'
       FileFormat:               'avi'
       Duration:                 0

    Video Properties:

       ColorChannels:            3
       Height:                   []
       Width:                    []
       FrameCount:               0
       FrameRate:                30
       VideoBitsPerPixel:        24
       VideoFormat:              'RGB24'
       VideoCompressionMethod:   'Motion JPEG'
       Quality:                  75

  Methods

Explaination of ODE = 

'ode45' is in-built solver in MATLAB so which first type that name. It's main inputs is 't & theta'.
 
't & theta' is dependent & independent variables of our ode system.

ode_func is (t, theta, b, g, l, m) are input parameters at we will be passing.

ode_func is nothing but it is a separate MATLAB program that we have written it is called  ode_func same name given as program name.


theta is solutin array.
theta1 & theta2 is 'theta'.

dtheta1_dt is first ode.
dtheta2_dt is second ode.

In dtheta_dt after dtheta1_dt (;)semicolon is represents column matrix.
If we tap on the 'results' there will be two columns first column refers to "Displacement" and second column refers to "velocity".

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

Srcond order ODE's
Editor window =

close all
clear all
clc

b = 0.04;                             %b = damping coefficient 
g = 9.81;                             %g = gravity (m/s^2)
l = 1;                                %l = length (m)
m = 0.1;                              %m = mass (kg)

% initial condition                   %initial displacement of pendulum = 0
theta_0 = [0;5];                      %initial velocity of pendulum = 5 m/s^2

% time points
t_span = linspace(0,10,500);          %t_span = time array

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

plot(t,results(:,1))
hold on
plot(t,results(:,2))
ylabel('plot')
xlabel('time')


Command window = 

theta1 =

     0


theta2 =

     5


theta1 =

   4.0190e-05


theta2 =

    5.0000


theta1 =

   6.0285e-05


theta2 =

    5.0000


theta1 =

   1.6076e-04


theta2 =

    4.9999


theta1 =

   1.7862e-04


theta2 =

    4.9999


theta1 =

   2.0095e-04


theta2 =

    4.9999


theta1 =

   2.0095e-04


theta2 =

    4.9999


theta1 =

   4.0190e-04


theta2 =

    4.9998


theta1 =

   5.0237e-04


theta2 =

    4.9998


theta1 =

    0.0010


theta2 =

    4.9996


theta1 =

    0.0011


theta2 =

    4.9996


theta1 =

    0.0012


theta2 =

    4.9995


theta1 =

    0.0012


theta2 =

    4.9995


theta1 =

    0.0022


theta2 =

    4.9991


theta1 =

    0.0027


theta2 =

    4.9989


theta1 =

    0.0052


theta2 =

    4.9979


theta1 =

    0.0057


theta2 =

    4.9977


theta1 =

    0.0062


theta2 =

    4.9975


theta1 =

    0.0062


theta2 =

    4.9975


theta1 =

    0.0112


theta2 =

    4.9954


theta1 =

    0.0138


theta2 =

    4.9943


theta1 =

    0.0263


theta2 =

    4.9888


theta1 =

    0.0285


theta2 =

    4.9878


theta1 =

    0.0313


theta2 =

    4.9865


theta1 =

    0.0313


theta2 =

    4.9865


theta1 =

    0.0564


theta2 =

    4.9750


theta1 =

    0.0688


theta2 =

    4.9678


theta1 =

    0.1309


theta2 =

    4.9307


theta1 =

    0.1418


theta2 =

    4.9234


theta1 =

    0.1555


theta2 =

    4.9138


theta1 =

    0.1557


theta2 =

    4.9137


theta1 =

    0.2791


theta2 =

    4.8261


theta1 =

    0.3384


theta2 =

    4.7499


theta1 =

    0.6174


theta2 =

    4.3554


theta1 =

    0.6502


theta2 =

    4.2781


theta1 =

    0.7103


theta2 =

    4.1627


theta1 =

    0.7313


theta2 =

    4.1532


theta1 =

    0.9282


theta2 =

    3.7639


theta1 =

    1.0059


theta2 =

    3.5082


theta1 =

    1.3196


theta2 =

    2.3662


theta1 =

    1.2976


theta2 =

    2.2900


theta1 =

    1.3495


theta2 =

    2.0044


theta1 =

    1.4469


theta2 =

    1.8020


theta1 =

    1.5585


theta2 =

    1.1544


theta1 =

    1.5691


theta2 =

    0.8435


theta1 =

    1.5866


theta2 =

   -0.6622


theta1 =

    1.5745


theta2 =

   -0.8833


theta1 =

    1.5336


theta2 =

   -1.2065


theta1 =

    1.5208


theta2 =

   -1.2627


theta1 =

    1.3796


theta2 =

   -2.3021


theta1 =

    1.1782


theta2 =

   -2.7486


theta1 =

    0.1715


theta2 =

   -3.9943


theta1 =

   -0.0846


theta2 =

   -3.8590


theta1 =

   -0.3612


theta2 =

   -3.6949


theta1 =

   -0.2633


theta2 =

   -3.8306


theta1 =

   -0.6453


theta2 =

   -3.4233


theta1 =

   -0.7905


theta2 =

   -2.8625


theta1 =

   -0.8332


theta2 =

   -0.9798


theta1 =

    0.0120


theta2 =

   -1.1973


theta1 =

   -0.0224


theta2 =

    0.0372


theta1 =

   -1.2016


theta2 =

    0.6120


theta1 =

   -0.4669


theta2 =

   -3.6136


theta1 =

   -0.5556


theta2 =

   -3.3989


theta1 =

   -0.8850


theta2 =

   -2.4099


theta1 =

   -0.8198


theta2 =

   -2.2895


theta1 =

   -0.8868


theta2 =

   -1.9846


theta1 =

   -1.0515


theta2 =

   -1.8725


theta1 =

   -1.1510


theta2 =

   -1.3802


theta1 =

   -1.1713


theta2 =

   -1.1194


theta1 =

   -1.2264


theta2 =

    0.0617


theta1 =

   -1.1945


theta2 =

    0.1397


theta1 =

   -1.1772


theta2 =

    0.4201


theta1 =

   -1.2171


theta2 =

    0.6095


theta1 =

   -1.1784


theta2 =

    1.1787


theta1 =

   -1.1184


theta2 =

    1.4372


theta1 =

   -0.8024


theta2 =

    2.5176


theta1 =

   -0.7354


theta2 =

    2.5385


theta1 =

   -0.6259


theta2 =

    2.7325


theta1 =

   -0.6200


theta2 =

    2.9203


theta1 =

   -0.4145


theta2 =

    3.2392


theta1 =

   -0.2865


theta2 =

    3.2500


theta1 =

    0.1957


theta2 =

    2.8335


theta1 =

    0.1409


theta2 =

    2.1951


theta1 =

    0.2952


theta2 =

    1.9420


theta1 =

    0.4769


theta2 =

    2.6904


theta1 =

   -0.4719


theta2 =

    3.1501


theta1 =

   -0.3847


theta2 =

    3.1890


theta1 =

   -0.0030


theta2 =

    3.1846


theta1 =

    0.0122


theta2 =

    2.9914


theta1 =

    0.1141


theta2 =

    2.9017


theta1 =

    0.1882


theta2 =

    3.1256


theta1 =

    0.3467


theta2 =

    2.9691


theta1 =

    0.4170


theta2 =

    2.8090


theta1 =

    0.6849


theta2 =

    2.0452


theta1 =

    0.6439


theta2 =

    1.9313


theta1 =

    0.6991


theta2 =

    1.6863


theta1 =

    0.8161


theta2 =

    1.6268


theta1 =

    0.9056


theta2 =

    1.1980


theta1 =

    0.9238


theta2 =

    0.9589


theta1 =

    0.9611


theta2 =

   -0.0818


theta1 =

    0.9134


theta2 =

   -0.1026


theta1 =

    0.8953


theta2 =

   -0.3676


theta1 =

    0.9565


theta2 =

   -0.6024


theta1 =

    0.9228


theta2 =

   -1.0380


theta1 =

    0.8785


theta2 =

   -1.2325


theta1 =

    0.6495


theta2 =

   -2.0168


theta1 =

    0.6029


theta2 =

   -1.9806


theta1 =

    0.5237


theta2 =

   -2.1279


theta1 =

    0.5190


theta2 =

   -2.3420


theta1 =

    0.3768


theta2 =

   -2.5807


theta1 =

    0.2893


theta2 =

   -2.6076


theta1 =

   -0.0662


theta2 =

   -2.4829


theta1 =

   -0.0526


theta2 =

   -2.1730


theta1 =

   -0.1569


theta2 =

   -2.0446


theta1 =

   -0.2612


theta2 =

   -2.4077


theta1 =

   -0.3824


theta2 =

   -2.2316


theta1 =

   -0.4331


theta2 =

   -2.0836


theta1 =

   -0.6216


theta2 =

   -1.3952


theta1 =

   -0.5869


theta2 =

   -1.3184


theta1 =

   -0.6218


theta2 =

   -1.1050


theta1 =

   -0.7103


theta2 =

   -1.0200


theta1 =

   -0.7672


theta2 =

   -0.6406


theta1 =

   -0.7718


theta2 =

   -0.4346


theta1 =

   -0.7524


theta2 =

    0.4353


theta1 =

   -0.7043


theta2 =

    0.4235


theta1 =

   -0.6760


theta2 =

    0.6450


theta1 =

   -0.7262


theta2 =

    0.8755


theta1 =

   -0.6828


theta2 =

    1.1807


theta1 =

   -0.6441


theta2 =

    1.3085


theta1 =

   -0.4537


theta2 =

    1.8078


theta1 =

   -0.4239


theta2 =

    1.7628


theta1 =

   -0.3634


theta2 =

    1.8478


theta1 =

   -0.3503


theta2 =

    2.0114


theta1 =

   -0.2339


theta2 =

    2.1597


theta1 =

   -0.1661


theta2 =

    2.1588


theta1 =

    0.1067


theta2 =

    1.9932


theta1 =

    0.0900


theta2 =

    1.7794


theta1 =

    0.1669


theta2 =

    1.6642


theta1 =

    0.2558


theta2 =

    1.8922


theta1 =

    0.3456


theta2 =

    1.7384


theta1 =

    0.3823


theta2 =

    1.6200


theta1 =

    0.5215


theta2 =

    1.0664


theta1 =

    0.5004


theta2 =

    1.0066


theta1 =

    0.5249


theta2 =

    0.8397


theta1 =

    0.5833


theta2 =

    0.7697


theta1 =

    0.6257


theta2 =

    0.4545


theta1 =

    0.6274


theta2 =

    0.2834


theta1 =

    0.6008


theta2 =

   -0.4264


theta1 =

    0.5585


theta2 =

   -0.4026


theta1 =

    0.5328


theta2 =

   -0.5850


theta1 =

    0.5758


theta2 =

   -0.7912


theta1 =

    0.5384


theta2 =

   -1.0289


theta1 =

    0.5070


theta2 =

   -1.1261


theta1 =

    0.3544


theta2 =

   -1.5022


theta1 =

    0.3323


theta2 =

   -1.4614


theta1 =

    0.2846


theta2 =

   -1.5240


theta1 =

    0.2721


theta2 =

   -1.6564


theta1 =

    0.1791


theta2 =

   -1.7672


theta1 =

    0.1256


theta2 =

   -1.7637


theta1 =

   -0.0908


theta2 =

   -1.6301


theta1 =

   -0.0785


theta2 =

   -1.4743


theta1 =

   -0.1384


theta2 =

   -1.3837


theta1 =

   -0.2079


theta2 =

   -1.5442


theta1 =

   -0.2801


theta2 =

   -1.4206


theta1 =

   -0.3097


theta2 =

   -1.3252


theta1 =

   -0.4226


theta2 =

   -0.8770


theta1 =

   -0.4065


theta2 =

   -0.8265


theta1 =

   -0.4264


theta2 =

   -0.6905


theta1 =

   -0.4724


theta2 =

   -0.6367


theta1 =

   -0.5068


theta2 =

   -0.3814


theta1 =

   -0.5085


theta2 =

   -0.2418


theta1 =

   -0.4882


theta2 =

    0.3353


theta1 =

   -0.4535


theta2 =

    0.3137


theta1 =

   -0.4333


theta2 =

    0.4634


theta1 =

   -0.4694


theta2 =

    0.6342


theta1 =

   -0.4401


theta2 =

    0.8276


theta1 =

   -0.4153


theta2 =

    0.9068


theta1 =

   -0.2949


theta2 =

    1.2142


theta1 =

   -0.2774


theta2 =

    1.1806


theta1 =

   -0.2397


theta2 =

    1.2330


theta1 =

   -0.2298


theta2 =

    1.3422


theta1 =

   -0.1562


theta2 =

    1.4353


theta1 =

   -0.1137


theta2 =

    1.4359


theta1 =

    0.0605


theta2 =

    1.3478


theta1 =

    0.0530


theta2 =

    1.2302


theta1 =

    0.1010


theta2 =

    1.1634


theta1 =

    0.1541


theta2 =

    1.2861


theta1 =

    0.2147


theta2 =

    1.1910


theta1 =

    0.2399


theta2 =

    1.1144


theta1 =

    0.3363


theta2 =

    0.7518


theta1 =

    0.3226


theta2 =

    0.7076


theta1 =

    0.3402


theta2 =

    0.5952


theta1 =

    0.3796


theta2 =

    0.5556


theta1 =

    0.4090


theta2 =

    0.3511


theta1 =

    0.4115


theta2 =

    0.2379


theta1 =

    0.4003


theta2 =

   -0.2323


theta1 =

    0.3724


theta2 =

   -0.2174


theta1 =

    0.3577


theta2 =

   -0.3406


theta1 =

    0.3880


theta2 =

   -0.4767


theta1 =

    0.3659


theta2 =

   -0.6398


theta1 =

    0.3463


theta2 =

   -0.7075


theta1 =

    0.2506


theta2 =

   -0.9698


theta1 =

    0.2361


theta2 =

   -0.9401


theta1 =

    0.2055


theta2 =

   -0.9864


theta1 =

    0.1983


theta2 =

   -1.0820


theta1 =

    0.1402


theta2 =

   -1.1626


theta1 =

    0.1063


theta2 =

   -1.1669


theta1 =

   -0.0344


theta2 =

   -1.1157


theta1 =

   -0.0305


theta2 =

   -1.0257


theta1 =

   -0.0693


theta2 =

   -0.9778


theta1 =

   -0.1095


theta2 =

   -1.0747


theta1 =

   -0.1608


theta2 =

   -1.0030


theta1 =

   -0.1826


theta2 =

   -0.9419


theta1 =

   -0.2660


theta2 =

   -0.6496


theta1 =

   -0.2541


theta2 =

   -0.6107


theta1 =

   -0.2699


theta2 =

   -0.5177


theta1 =

   -0.3041


theta2 =

   -0.4897


theta1 =

   -0.3296


theta2 =

   -0.3263


theta1 =

   -0.3328


theta2 =

   -0.2345


theta1 =

   -0.3287


theta2 =

    0.1491


theta1 =

   -0.3064


theta2 =

    0.1401


theta1 =

   -0.2960


theta2 =

    0.2416


theta1 =

   -0.3214


theta2 =

    0.3488


theta1 =

   -0.3051


theta2 =

    0.4872


theta1 =

   -0.2897


theta2 =

    0.5455


theta1 =

   -0.2134


theta2 =

    0.7713


theta1 =

   -0.2011


theta2 =

    0.7448


theta1 =

   -0.1762


theta2 =

    0.7861


theta1 =

   -0.1713


theta2 =

    0.8705


theta1 =

   -0.1254


theta2 =

    0.9404


theta1 =

   -0.0982


theta2 =

    0.9472


theta1 =

    0.0156


theta2 =

    0.9216


theta1 =

    0.0140


theta2 =

    0.8518


theta1 =

    0.0454


theta2 =

    0.8178


theta1 =

    0.0761


theta2 =

    0.8958


theta1 =

    0.1195


theta2 =

    0.8423


theta1 =

    0.1383


theta2 =

    0.7936


theta1 =

    0.2101


theta2 =

    0.5585


theta1 =

    0.1999


theta2 =

    0.5244


theta1 =

    0.2139


theta2 =

    0.4477


theta1 =

    0.2436


theta2 =

    0.4286


theta1 =

    0.2656


theta2 =

    0.2980


theta1 =

    0.2691


theta2 =

    0.2236


theta1 =

    0.2697


theta2 =

   -0.0894


theta1 =

    0.2519


theta2 =

   -0.0847


theta1 =

    0.2446


theta2 =

   -0.1683


theta1 =

    0.2659


theta2 =

   -0.2525


theta1 =

    0.2539


theta2 =

   -0.3696


theta1 =

    0.2417


theta2 =

   -0.4197


theta1 =

    0.1809


theta2 =

   -0.6133


theta1 =

    0.1704


theta2 =

   -0.5899


theta1 =

    0.1501


theta2 =

   -0.6264


theta1 =

    0.1470


theta2 =

   -0.7003


theta1 =

    0.1105


theta2 =

   -0.7606


theta1 =

    0.0887


theta2 =

   -0.7685


theta1 =

   -0.0035


theta2 =

   -0.7591


theta1 =

   -0.0031


theta2 =

   -0.7045


theta1 =

   -0.0287


theta2 =

   -0.6802


theta1 =

   -0.0523


theta2 =

   -0.7436


theta1 =

   -0.0888


theta2 =

   -0.7038


theta1 =

   -0.1049


theta2 =

   -0.6651


theta1 =

   -0.1663


theta2 =

   -0.4761


theta1 =

   -0.1576


theta2 =

   -0.4464


theta1 =

   -0.1699


theta2 =

   -0.3834


theta1 =

   -0.1954


theta2 =

   -0.3708


theta1 =

   -0.2083


theta2 =

   -0.2990


theta1 =

   -0.2120


theta2 =

   -0.2594


theta1 =

   -0.2248


theta2 =

   -0.0747


theta1 =

   -0.2222


theta2 =

   -0.0568


theta1 =

   -0.2223


theta2 =

   -0.0113


theta1 =

   -0.2272


theta2 =

    0.0110