Is the Motion of a Pendulum Described by a Harmonic Oscillator Model?

[sergiokapone]

TL;DR

This is not homework, I'm to old for. I came up with this problem, and I want to check whether it is suitable.

It is necessary to make a mechanism, the basis of which should be oscillation of the pendulum with an amplitude φ = 0.250 ± 0.002 rad. Is it possible to describe the motion of the pendulum to use a harmonic oscillator model?

 

[BvU]

Sure, why not ?
Some amplitude correction for the period might be in order ...

 

[sergiokapone]

But with the condition, the $\sin\phi < \phi$, and difference between periods is about $\frac{\Delta T}{T_0} =0.004$.

 

[BvU]

Yes. So ?

 

[sergiokapone]

I thaught, the condition of harmonicity is $\sin\phi =\phi$, and with aquaried degree of accuracy, we can't cosider the oscillations as harmonic.

 

[BvU]

How accurate do you want your description to be ?
0.25 isn't that big an angle !
And the correction for T is easily made.

There is no distinction between the lines in fig 1, 2 and 3 here ,
i.e. up to 0.75

So what is the complete set of requirements for your design ?

 

[sergiokapone]

As mentioned in first post φ = 0.250 ± 0.002.

As described in general courses of physics, the condition of harmonicity is an aprroximate equality $\sin\phi \approx \phi$. Physically, what is that mean? How closer to a strict equality it should be? That is my main question.As I think, for verifying harmonicity is sufficient to compare sinus of an angle with angle with desire accuracy. But maybe I loosing something important, I don't know.

 

[BvU]

As mentioned in first post φ = 0.250 ± 0.002.

Yes, I read that. Reason I asked is that you do not tell what is being designed or what the complete set of requirements is. φ = 0.250 ± 0.002 does not impose any constraints on the shape or the period of the oscillation. It can just as well be a block wave with exactly φ = 0.250 ± 0.002.

What do you mean with

verifying harmonicity

 

[sergiokapone]

There are an harmonic oscillatios, described with some type of differential equation, and there are non harmonic oscillators, for example, gravitational pendulums, described with differential equation with sinus type of nonlinearity. Can it be consider the gravitational pendulums as harmonic? In general case, obviously, not. At a large amplitude the osccilations is not a harmonic, but at low there is yes, and there is no sharp border between, of course. But what experimental criteria one can propose for considering oscillation as a harmonic at a some abstract conditions?

 

[BvU]

But what experimental criteria one can propose for counting oscillation as a harmonic at a some abstract conditions?

Did you look in the links I gave ?
(here is another one) One can propose to take the relative amplitude of the $(2l+1)$ (with $l=1$) harmonic as a measure of distortion of the cosinusoidal motion and specify an upper bound for it.
Or one can look at the pictures:

We can conclude that for amplitudes lower than 135^o the periodic motion exhibited by a simple pendulum is practically harmonic but its oscillations are not isochronous (the period is a function of the initial amplitude)

which basically says the bold line in his figure 4 is still sufficiently cosinusoidal ...

 

[sergiokapone]

About Belendez article. I always thaught, the harmonic oscillatios is always isochronous. But I don't understand, what does it mean "practically" harmonic. Whose practice? Does it mean if the condition $\sin\phi =\phi$ with desired accuracy is sattisfied, the oscillations are harmonic, but perhaps this is not isochronous, and we should check the accuracy for the period value?

 

[BvU]

I suppose one could determine the total harmonic distortion for the Fourier expansion in the last link (let us know the result if you do ) to find out what Belendez still finds "practically undistorted". For me the pictures 1 2 and 3 are good enough and your 0.250 is still below the $0.1\pi$ of fig. 1 !

 

[sergiokapone]

I thought so: I know the equation of my gravitational pendulum $\ddot\phi + \omega_0^2\sin\phi = 0$, and also I know the equation of harmonic oscillatior $\ddot\phi + \omega_0^2\phi = 0$. Ok, then for describing my pendulum as harmonic oscillator, I must satisfy the condition $\sin\phi = \phi$, because only with this condition differential equations will become similar. But I see the my $\sin0.250 = 0.247 < 0.250 \pm 0.002$ ($\sin$ is slightly lower than I need), so my conclusion, the ocillations are not harmonic therefore not isochronous.

PS It turns out that I am trying to determine the harmonicity of the oscillations using the differential equation (DE), but perhaps it would be more correct to do this using a solution of DE, as Belendez proposed.