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What's the minimum (physics first) to get an oscillator?

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Nominal Animal:

--- Quote from: T3sl4co1l on May 23, 2023, 02:20:35 pm ---Oscillator in the simplest physics sense (a passive oscillatory system e.g. spring-and-mass), or practically speaking (a driven system with feedback)?  Or more specifically still, a combination of the two (a periodic or even ~sinusoidal oscillator)?
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And do we include more esoteric phenomena like quantum time crystals, where the lowest energy state is a non-static, repetitive one?

It all boils down to what you specifically and exactly mean by an oscillator.  In my opinion, one useful disctinction is between the phenomenon, and as an usable tool.
 _ _ _ _ _ _

For the tool view, any sufficiently regular repetitive system whose phase space (behaviour with respect to any variable that may affect it) is such that useful amounts of energy can be exchanged (injected and/or extracted) without destabilizing the system, can be used as an oscillator.

Basically all useful oscillators – all continuous-amplitude oscillators! – can be described as harmonic oscillators, damped harmonic oscillators, or driven harmonic oscillators.  Even non-harmonic oscillators can be described as damped-driven harmonic oscillators, where the damping-driving force depends on the phase of the oscillator; with the base harmonic oscillator being a near approximation, with the damping-driving force causing only relatively small changes to the period and/or amplitude.

The reasons for this are actually very interesting, and can be divided into two separate classes.

One class is that any convex shape is analogous to a sphere.  (There is a better mathematical term for this, but I can't recall it right now.)  In the oscillator case, because we are interested in repetitive state changes, we care only about the interval at which the repetition occurs.  So, whether nominal sub-states take an equal amount of time, does not matter.  In essence, we don't care that much about the overall "shape" of the oscillation.  (Insert automotive analogy about lap times with track of the same length but different shapes, or tracks of different length and shape but yielding same/similar lap times for each driver, here.)

The other class is how any periodic continuous signal, say the amplitude of any continuous-state oscillator, can be decomposed into an (infinite) set of sinusoidal signals, and how even non-continous signals can be approximated to any desired precision: Fourier series.  (Sine and cosine themselves being the solutions to any pure harmonic oscillator.)

There is a third one, too: the smaller the amplitude compared to the energies and distances involved in an oscillator, the closer it seems to match a harmonic oscillator.  However, I myself do not know why this is, or even if it is just a practical result of a noisy universe (i.e., any non-harmonic components drowned in noise in real life), so I consider it more or less just a rule of thumb.

IanB:

--- Quote from: CatalinaWOW on May 23, 2023, 02:16:42 pm ---The relaxation oscillator highlights the fuzzy line between ideal models and reality.  The conceptual oscillator can't be fully described by a differential equation because the trigger circuit is non continuous and has no meaningful derivatives.  But it can be approximated mathematically and the physical reality can be matched by those approximations.

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This is an interesting question. What is an "ideal model"? It is presumably a model where simplifications have been made to eliminate some less significant details and make it easier to work with. For example, texts on differential equations nearly always introduce the student to linear differential equations because they often permit an analytical solution to be obtained. But this is an idealization, since the real world is not often linear.

In the real world beyond college and textbooks, complex systems are simulated numerically on a computer, and (possibly gross) nonlinearities in the system can be accommodated.

For example, suppose you put an audio amplifier into a feedback loop by putting the microphone close to the speaker. The amplifier will saturate, the signal will clip like crazy, and the whole thing will be horribly nonlinear. Yet the system is made up of analog components that can be described with differential equations. There is undoubtedly no analytical solution, but a computer simulation if programmed sufficiently accurately could reproduce in a model what the feedback loop does in real life.

RoGeorge:
The question started from an LC oscillator.

At a first look, an oscillator would need an amplifier to substitute the lost energy, a limiter to keep the amplitude constant, and a positive feedback loop.  Those alone won't cut it, would only make a latch.  It needs some inertia/delay/memory (the LC tank in the starting example).

This is where I couldn't decide what was the broader concept to pick.  It can be a resonant oscillator as an LC, or it can be a forced oscillator with a comparator and something that flips the state, as in the drinking bird example.

It gets even more bamboozling when thinking about numerical oscillators.  What would mean energy conservation for a numerical oscillator, or momentum, or seeking minimum energy equilibrium, or inertia?  All these are very important and inescapable in a physical oscillator, yet they don't make much sense, and they seem totally arbitrary for a numerical oscillator.

That is why asking to identify the essential components of a physical oscillator, at first.

ejeffrey:

--- Quote from: westfw on May 23, 2023, 06:25:34 pm ---
--- Quote ---The prototype harmonic oscillator in physics discussions is a spring
--- End quote ---


How about a pendulum ?  Or does the externality of gravity complicate things?

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No, at least in my description / definition above a simple pendulum isn't any different.  In that case, gravity is just providing the spring force.  Now a pendulum clock has a mechanism to detect the motion of the arm and provide kicks, powered by a falling weight.  That I would call an oscillator.

Nominal Animal:

--- Quote from: RoGeorge on May 23, 2023, 06:43:26 pm ---The question started from an LC oscillator.

At a first look, an oscillator would need an amplifier to substitute the lost energy, a limiter to keep the amplitude constant, and a positive feedback loop.  Those alone won't cut it, would only make a latch.  It needs some inertia/delay/memory (the LC tank in the starting example).
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You end up with negative feedback.  Which gives you a harmonic oscillator.
When we include the "amplifier" and loss of energy, we get a damped-driven harmonic oscillator, where the damping is due to losses, and the driving matches the damping.  Depending on the damping and driving mechanisms, they can cause a small change in the period.

Note that while the amplitude of the harmonic oscillator depends on the energy involved, the period does not.  So, small perturbations or fluctuations in the amount of damping or driving does not affect the period of the oscillator, as long as the amplitude stays within the elastic/ergodic range (where the same amount of energy that was needed to cause the change is released when returning to the original position) and the change is smooth and slow enough.  Typically, the driving and damping are smooth and interdependent (matching each other intrinsically, both changing only slowly and smoothly, continuously, across multiple periods), so that their effect on the oscillator period is neglible, within environmental noise.


--- Quote from: RoGeorge on May 23, 2023, 06:43:26 pm ---It gets even more bamboozling when thinking about numerical oscillators.
--- End quote ---
Numerical oscillators?  Do you mean periodic functions and periodic sequences?  They are based around periodicity, and from a physics perspective, they are "pure": more like empty universes than systems of interacting parts.

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