Simulation of pendulum by solving second order ODE using MATLAB
Objective:
To write a program that solves the following ODE. This ODE represents the equation of motion of a simple pendulum with damping, and to animate it to have a swinging pendulum
`d^2theta//dt^2+(b//m)(d theta//dt)+g//l*sintheta=0`
Procedure:
Given,
g = gravity in m/s2,
L = length of the pendulum in m,
m = mass of the ball in kg,
b=damping coefficient.
Also,
L=1 metre,
m=1 kg,
b=0.05.
g=9.81 m/s2.
Now, Simulating the motion between 0-20 sec, for angular displacement=0, angular velocity=0, at the time t=0.
First of all, Breaking the Second order ODE into simple ODE,
taking,
`theta=y1`
then,
`dy1//dt=y2`
`d^2y1//dt^2=-[(b/m)y1+g//l*sintheta]`
Now, Coding the above equations and solving it using lsode,
The function to solve \'y\' values:
function [ydot]=ode_pendulum(t,y)
%Inputs
l=1; %length in metres
m=1; %mass in kg
b=0.05; %Damping coefficient
g=9.81; %acceleration due to gravity in m/s^2
ydot=zeros(2,1);
%Change of displacement
ydot(1)=y(2);
%Change of velocity
ydot(2)=-(b/m*y(2)+g/l*sin(y(1)));
endThe Main code to simulate the equation as pendulum:
clear all
close all
clc
%Input variables
t=[0 20];%Time period
y=[0 3]; %Representation of Displacement
[y]= lsode(\'ode_pendulum\',y,t);
figure(1)
plot(t,y(:,1),\'color\',\'r\',\'linewidth\',2)
xlabel(\'Time\')
hold on
plot(t,y(:,2),\'color\',\'b\',\'linewidth\',2)
xlabel(\'Time\')
legend(\'Angular displacement\',\'Angular velocity\')
%Animation
figure(2)
ct=1
for i=1:length(y(:,1))
x0=0;
y0=0;
x1=1*sin(y(i,1));
y1=-1*cos(y(i,1));
plot([-0.5,0.5],[0,0],\'linewidth\',2,\'color\',\'y\');
line([x0,x1],[y0,y1],\'linewidth\',2,\'color\',\'b\');
hold on
plot(x1,y1,\'O\',\'markers\',25,\'MarkerFacecolor\',\'g\',\'MarkerEdgecolor\',\'k\');
axis([-1.5 1.5 -1.5 1.5]);
pause(0.02);
hold off
M(ct)=getframe(gcf);
ct=ct+1;
end
%Movie
movie(M)
videofile=VideoWriter(\'pendulum\',\'Uncompressed AVI\')
open(videofile)
writeVideo(videofile,M)
close(videofile)The Output:
%Output
y =
0 3.0000
0.0001 3.0000
0.0001 3.0000
0.0002 3.0000
0.0002 3.0000
0.0005 3.0000
0.0007 3.0000
0.0010 3.0000
0.0012 2.9999
0.0025 2.9999
0.0037 2.9998
0.0050 2.9997
0.0062 2.9996
0.0125 2.9991
0.0188 2.9985
0.0251 2.9977
0.0313 2.9968
0.0627 2.9904
0.0939 2.9808
0.1251 2.9681
0.1560 2.9521
0.2882 2.8471
0.4143 2.6871
0.5320 2.4775
0.6391 2.2251
0.7685 1.8122
0.8698 1.3481
0.9401 0.8502
0.9780 0.3338
0.9778 -0.3428
0.9219 -1.0039
0.8126 -1.6203
0.6551 -2.1583
0.4816 -2.5403
0.2844 -2.8011
0.0731 -2.9182
-0.1414 -2.8822
-0.3366 -2.7110
-0.5156 -2.4198
-0.6706 -2.0287
-0.7958 -1.5632
-0.8860 -1.0559
-0.9402 -0.5190
-0.9569 0.0305
-0.9359 0.5769
-0.8737 1.1353
-0.7723 1.6536
-0.6352 2.1081
-0.4683 2.4737
-0.2732 2.7311
-0.0643 2.8445
0.1473 2.8024
0.3503 2.6103
0.5082 2.3424
0.6465 1.9946
0.7604 1.5842
0.8468 1.1293
0.9091 0.5756
0.9304 0.0049
0.9099 -0.5638
0.8486 -1.1119
0.7604 -1.5725
0.6435 -1.9830
0.5017 -2.3249
0.3403 -2.5803
0.1424 -2.7453
-0.0622 -2.7649
-0.2628 -2.6360
-0.4489 -2.3717
-0.5858 -2.0657
-0.7019 -1.6967
-0.7937 -1.2803
-0.8588 -0.8318
-0.9001 -0.2589
-0.8978 0.3213
-0.8521 0.8879
-0.7648 1.4191
-0.6538 1.8479
-0.5161 2.2087
-0.3567 2.4816
-0.1823 2.6501
0.0130 2.7014
0.2068 2.6140
0.3893 2.3940
0.5516 2.0604
0.6709 1.6974
0.7655 1.2832
0.8326 0.8336
0.8705 0.3634
0.8760 -0.2266
0.8356 -0.8057
0.7509 -1.3495
0.6260 -1.8317
0.4839 -2.1903
0.3199 -2.4531
0.1412 -2.6017
-0.0436 -2.6259
-0.2251 -2.5248
-0.3953 -2.3060
-0.5462 -1.9844
-0.6717 -1.5818
-0.7625 -1.1491
-0.8233 -0.6818
-0.8524 -0.1959
-0.8491 0.2938
-0.8075 0.8288
-0.7273 1.3308
-0.6118 1.7768
-0.4660 2.1432
-0.3020 2.4007
-0.1237 2.5415
0.0602 2.5541
0.2404 2.4385
0.3943 2.2294
0.5314 1.9344
0.6467 1.5693
0.7363 1.1523
0.7999 0.6744
0.8296 0.1756
0.8243 -0.3271
0.7843 -0.8172
0.7155 -1.2579
0.6187 -1.6560
0.4975 -1.9944
0.3562 -2.2568
0.1865 -2.4389
0.0081 -2.4992
-0.1700 -2.4323
-0.3391 -2.2446
-0.4680 -2.0046
-0.5804 -1.7014
-0.6728 -1.3477
-0.7427 -0.9572
-0.7937 -0.4671
-0.8091 0.0381
-0.7884 0.5400
-0.7324 1.0208
-0.6544 1.4184
-0.5523 1.7696
-0.4297 2.0592
-0.2909 2.2735
-0.1223 2.4104
0.0518 2.4270
0.2226 2.3213
0.3818 2.1025
0.5021 1.8407
0.6047 1.5213
0.6863 1.1571
0.7447 0.7616
0.7822 0.2671
0.7834 -0.2356
0.7483 -0.7277
0.6785 -1.1903
0.5883 -1.5662
0.4753 -1.8861
0.3437 -2.1347
0.1985 -2.2992
0.0314 -2.3717
-0.1365 -2.3270
-0.2971 -2.1676
-0.4426 -1.9060
-0.5522 -1.6092
-0.6416 -1.2615
-0.7081 -0.8764
-0.7498 -0.4672
-0.7660 0.0341
-0.7451 0.5324
-0.6878 1.0075
-0.5969 1.4388
-0.4887 1.7750
-0.3602 2.0406
-0.2167 2.2208
-0.0642 2.3054
0.0981 2.2860
0.2550 2.1570
0.3991 1.9267
0.5240 1.6111
0.6163 1.2669
0.6853 0.8822
0.7287 0.4712
0.7452 0.0474
0.7298 -0.4546
0.6770 -0.9354
0.5889 -1.3727
0.4700 -1.7439
0.3401 -2.0057
0.1952 -2.1783
0.0417 -2.2504
-0.1132 -2.2180
-0.2622 -2.0841
-0.3987 -1.8574
-0.5165 -1.5509
-0.6111 -1.1826
-0.6770 -0.7853
-0.7154 -0.3624
-0.7251 0.0710
-0.7059 0.5006
-0.6575 0.9197
-0.5818 1.3042
-0.4814 1.6378
-0.3606 1.9049
-0.2205 2.0950
-0.0706 2.1874
0.0822 2.1752
0.2305 2.0604
0.3551 1.8752
0.4657 1.6223
0.5585 1.3134
0.6304 0.9619
0.6823 0.5487
0.7052 0.1171
0.6983 -0.3170
0.6620 -0.7384
0.6028 -1.1076
0.5211 -1.4388
0.4198 -1.7187
0.3025 -1.9350
0.1615 -2.0866
0.0133 -2.1395
-0.1350 -2.0898
-0.2764 -1.9417
-0.3876 -1.7433
-0.4851 -1.4884
-0.5658 -1.1873
-0.6273 -0.8516
-0.6721 -0.4374
-0.6876 -0.0084
-0.6733 0.4194
-0.6298 0.8301
-0.5675 1.1713
-0.4852 1.4739
-0.3856 1.7262
-0.2721 1.9176
-0.1331 2.0493
0.0116 2.0840
0.1553 2.0192
0.2911 1.8598
0.3963 1.6570
0.4877 1.4017
0.5626 1.1039
0.6187 0.7744
0.6585 0.3647
0.6692 -0.0574
0.6506 -0.4758
0.6034 -0.8747
0.5386 -1.2026
0.4545 -1.4901
0.3541 -1.7258
0.2409 -1.8995
0.1041 -2.0107
-0.0370 -2.0269
-0.1758 -1.9467
-0.3060 -1.7761
-0.4068 -1.5648
-0.4932 -1.3033
-0.5627 -1.0018
-0.6131 -0.6711
-0.6459 -0.2645
-0.6500 0.1509
-0.6250 0.5590
-0.5723 0.9438
-0.5027 1.2578
-0.4148 1.5277
-0.3115 1.7424
-0.1966 1.8922
-0.0616 1.9739
0.0758 1.9625
0.2091 1.8582
0.3322 1.6685
0.4273 1.4416
0.5070 1.1679
0.5688 0.8577
0.6109 0.5222
0.6336 0.1162
0.6274 -0.2930
0.5926 -0.6890
0.5307 -1.0556
0.4534 -1.3497
0.3589 -1.5940
0.2506 -1.7774
0.1327 -1.8913
-0.0006 -1.9301
-0.1334 -1.8779
-0.2594 -1.7378
-0.3729 -1.5189
-0.4597 -1.2678
-0.5295 -0.9746
-0.5800 -0.6506
-0.6097 -0.3069
-0.6173 0.0993
-0.5958 0.5000
-0.5462 0.8783
-0.4708 1.2174
-0.3823 1.4797
-0.2785 1.6834
-0.1635 1.8185
-0.0422 1.8783
0.0878 1.8565
0.2133 1.7482
0.3283 1.5596
0.4278 1.3022
0.5018 1.0182
0.5566 0.6998
0.5903 0.3587
0.6019 0.0069
0.5877 -0.3950
0.5449 -0.7782
0.4751 -1.1251
0.3819 -1.4186
0.2796 -1.6277
0.1656 -1.7675
0.0448 -1.8303
-0.0776 -1.8131
-0.1410 -1.7721
-0.2026 -1.7097
-0.2616 -1.6270
-0.3173 -1.5252
The Plots:UI
The Simulation obtained as follows: