Simple harmonic motion –Euler’s step method

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Ok, let’s look at

$y'' = - \frac{k}{m}y$

one more time. In this case, we need to calculate the acceleration (y´´) given the displacement, then the change of velocity and the velocity (y´) given the acceleration, and finally a new position given the velocity. I.e.

$\begin{gathered} a = - \frac{k}{m}y, \hfill \\ v \leftarrow v + a\Delta t, \hfill \\ y \leftarrow y + v\Delta t \hfill \\ \end{gathered}$

Then the above is repeated over and over again. The arrows stands for “is given by” and is used so that we don’t need to create a new variable name for each new position.

To test this we choose our k and m so that the period will be 1 s. According to the previously found solution, we have that

$\omega = \sqrt {\frac{k}{m}} = 2\pi f$

$T = 2\pi \sqrt {\frac{m}{k}}$

For T=1 s we get

${\left( {\frac{1}{{2\pi }}} \right)^2} = \frac{m}{k}$

So if we select k=10 N/m then

$m = 10{\left( {\frac{1}{{2\pi }}} \right)^2} \approx 0.2530{\text{ kg}}$

The below is done in Excel with a time step size of 0.1 s and an initial amplitude of 1 m, using the above mass.

The below is done in Excel with a time step size of 0.1 s and an initial amplitude of 1 m, using the above mass. This is the contents of the first few cells:

	D	E	F	G

3		g	9,81
4		k	10
5		m	0,253302959
6		dt	0,1
7
8	t	y	v	a
9	0	1	0	=-$F$4/$F$5*E9
10	=D9+$F$6	=E9+F10*$F$6	=F9+G9*$F$6	=-$F$4/$F$5*E10

Then row 10 is simply copied down a few hundred cells. This is the result for the first 3 s.

t	y	v	a
0.0	1.0000	0.0000	-39.4784
0.1	0.6052	-3.9478	-23.8930
0.2	-0.0285	-6.3371	1.1251
0.3	-0.6510	-6.2246	25.6989
0.4	-1.0164	-3.6547	40.1273
0.5	-0.9806	0.3580	38.7140
0.6	-0.5577	4.2294	22.0170
0.7	0.0854	6.4311	-3.3719
0.8	0.6948	6.0939	-27.4296
0.9	1.0299	3.3509	-40.6586
1.0	0.9584	-0.7149	-37.8362
1.1	0.5085	-4.4985	-20.0767
1.2	-0.1421	-6.5062	5.6088
1.3	-0.7366	-5.9453	29.0800
1.4	-1.0403	-3.0373	41.0709
1.5	-0.9334	1.0698	36.8476
1.6	-0.4579	4.7545	18.0775
1.7	0.1983	6.5623	-7.8293
1.8	0.7763	5.7793	-30.6453
1.9	1.0477	2.7148	-41.3630
2.0	0.9056	-1.4215	-35.7512
2.1	0.4059	-4.9966	-16.0254
2.2	-0.2540	-6.5991	10.0269
2.3	-0.8136	-5.5964	32.1208
2.4	-1.0521	-2.3844	41.5339
2.5	-0.8752	1.7690	34.5501
2.6	-0.3528	5.2240	13.9264
2.7	0.3089	6.6167	-12.1952
2.8	0.8486	5.3972	-33.5023
2.9	1.0533	2.0469	-41.5832
3.0	0.8422	-2.1114	-33.2478

The Excel file for this can be found here: Simple harmonic motion. It uses the decimal comma, so you might have to change that to the decimal point in the cells for the constants.

The figure below shows the same in a graphical form. As you can see the period is slightly shorter than 1 s.

With a smaller step size, 0,001 we get the following graph. It is now very close to a sine wave with amplitude 1 and period 1. Matter of fact at 1 s the y value is about 0.99997.

Up a level : Differential Equations
Previous page : Euler’s step method
Next page : The Earth orbit and Euler’s Step method

Last modified: Dec 28, 2023 @ 17:06