Question

can
one have oscillator that
even for small amplitudes is not simple
harmonic? This is, can one have
nonlinear restoring force in an oscillator
even at arbitrarily small amplitudes?​

Answer 1

Answer:

A simple harmonic oscillator is merely one, mathematically significant and simple, form of oscillator, characterised by a second order differential equation. It produces a sinusoidal excursion and therefore derivative excursion. It is characterised by a ‘restorative’ force proportional to ‘displacement from equilibrium’, as referred to a mechanical oscillator. But any system with a second derivative proportional to the original parameter, which could be electrical, can form a simple harmonic oscillator, of course. Most real oscillators are not completely linear in this way anyway, and a real (non-computational) system will also have a first derivative damping term. Output waveform will not be pure sinusoid, but will contain harmonics and/or exponential damping envelope characteristics.

Systems with nonlinear restorative terms will generally be very hard or impossible to model analytically, and will require numerical approximations to predict.

But a harmonic oscillator, perfectly linear or otherwise, is only one of many forms of oscillator. A ‘relaxation’ oscillator will ‘charge’ a parameter up to a trigger level, and then ‘discharge’, and repeat. The discharge may be instant or progressive linear. The world’s most successful integrated circuit, by quantity made, the NE555 and equivalents, is an oscillator of this general type, with piece-wise linear behaviour and changes in operational regime at parameter thresholds. ‘Charge’ may be electrical charge or any parameter which undergoes cyclic increase and ‘discharge’. It could be fluid or fluid pressure in a vessel. Example: turbocharger and wastegate. A square wave oscillator is produced where a process cyclically crosses a threshold value while the measurable is subject to very high gain, gain direction reversal and limiting about the threshold. In both cases there need by no underlying SHO-type mechanism.