On computing the analytic-signal backbone of the unforced harmonic oscillator

Backbone curves are usually obtained by analyzing nonlinear oscillators (e.g., the Duffing equation) whereas the backbones of linear oscillators are incidental by-products of such studies. Nevertheless, we focus on the harmonic oscillator, a linear, homogeneous constant-coefficient second-order ordinary differential equation. The backbone of the sinusoidally forced harmonic oscillator is well known, but the backbone of the unforced harmonic oscillator seems to have received essentially no attention. Since the 1980s, various computerized methods of backbone extraction have been under development, one of which is based on analytic-signal theory, a subfield of digital signal processing that involves Hilbert transform computations. By focusing on linear-theory backbones, a prerequisite to nonlinear analysis, we aim to pinpoint some of the theoretical and computational issues that arise when dealing with analytic-signal backbones. This paper examines the Mirror Periodic Method (MPM) as a practical way to reduce end effects in Hilbert transform computations and approximate analytic-signal backbones of unforced harmonic oscillators with light damping (less than 10 percent of critical). Using numerical examples, we demonstrate that MPM works well with both displacement and acceleration data, even for short signals, meaning just one or two damped cycles of oscillation. We then apply MPM techniques to accelerometer data that were recorded as part of earthquake-engineering teaching demonstrations involving a small-scale laboratory model of a linear single-degree-of-freedom system.