A ball bouncing (consider ideal elastic collisions) moves to and from about some point, but there is no equilibrium position. This motion sure is periodic... but is it oscillatory? What is the criterion for a periodic motion to be oscillatory? Consider another example... A body moves on $x$ axis as $x= \sin(t^2)$. Is this motion oscillatory as it moves to and from about the origin? If yes, then how is it oscillatory but not periodic?
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$\begingroup$ Oscillation can be thought as a motion of a point over time, $x(t)$, it can be periodic or irregular shapes. In Engineering literature, it is usual to use 'vibration' as a term. $\endgroup$– patagonicusCommented Sep 27 at 8:00
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$\begingroup$ "how is it oscillatory but not periodic?" == How is it not periodic? $\endgroup$– Miss UnderstandsCommented Sep 28 at 0:44
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I have worked on vibrations in an engineering sense for 20+ years and I know of no formal technical definition of "oscillation." If you demand it be a limit cycle (repeated motion) then that rules out time decay to zero. If you demand that the motion be sinusoidal then that rules out nonlinearity or multiple frequencies. If you just say it's motion in a particular phase space or on a "manifold" (if you like that word) then it can include degenerate cases that we don't consider to be oscillations (like sitting there with zero velocity). If any mathematical definitions exist, I would be curious to hear them.
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$\begingroup$ So would you consider a ball bouncing as oscillation? My high school teacher suggests that it's not oscillation as oscillation requires a mean equilibrium position.. $\endgroup$ Commented Sep 28 at 17:44
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$\begingroup$ I think the answer boils down to a matter of opinion. One way you could define oscillation, maybe, is a system that is described to first order by $A(t) \Sigma \sin(\omega_i t + \phi_i)$. The problem there is that you can create a Fourier series of pretty much any trajectory if you let $i \rightarrow \infty$. It might be an interesting thought experiment to come up with some better definition and see if a bouncing ball fits the criteria. $\endgroup$ Commented Sep 29 at 16:48