Harmonic and anharmonic behaviour of a simple oscillator

We consider a simple oscillator that exhibits harmonic and anharmonic regimes and analyse its behaviour over the complete range of possible amplitudes. The oscillator consists of a mass m fixed at the midpoint of a horizontal rope. For zero initial rope tension and small amplitude the period of oscillation, τ, varies as τ ~ m1/6. For small initial tension the oscillator crosses over from harmonic to anharmonic and finally back to harmonic behaviour with increasing amplitude. The change in period and the amplitude of the Fourier components of the waveform (position versus time) during these crossovers are examined and the regions of harmonic and anharmonic behaviour are mapped out. For large enough initial tension the oscillator shows no crossovers and the oscillations are harmonic over the complete range of possible amplitudes. Finally we consider small amplitude two-dimensional motion of this mass in a vertical plane and show the condition that the path be closed only depends on the ratio of initial tension in the rope and spring constant of the rope.