How to prove that motion is periodic but not simple harmonic?

[vcsharp2003]

@vcsharp2003
Your answer to @haruspex in post #40 is actually a good way to show that it's not SHM. Sorry that I did not see that sooner.

I put the second derivative in the following form.

$a= -\omega ^2 x -\omega ^2 (3 sin{(2\omega t)} + 15sin {(4\omega t)})$

Isn't that clearly saying $a$ is simply not $a= -\omega ^2 x$. If it were SHM, then we would have got $a= -\omega ^2 x$, but we didn't. To me that is so obvious. Based on the above fact, clearly it's not SHM.

No need of ultra-complex proofs and other logic.

 

[haruspex]

I put the second derivative in the following form.

$a= -\omega ^2 x -\omega ^2 (3 sin{(2\omega t)} + 15sin {(4\omega t)})$

Isn't that clearly saying $a$ is simply not $a= -\omega ^2 x$.

Yes, but as pointed out, you need to show there does not exist any constant, A, the given $\omega$ or otherwise, for which $\ddot x=-A^2x$. Post #40 does that.