A free (undriven

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Nonlinear dynamics and chaos

Modes, Packets and Media, Ch The anharmonic oscillator: frequency, harmonics and drift

Problem

A free (undriven, undamped) anharmonic oscillator has small-amplitude angular frequency

\omega_0=1.00\times10^3\ \mathrm{rad\,s^{-1}}

(

159.2\ \mathrm{Hz}

), measured separately at a very small swing. At a larger swing it is recorded at a sampling rate of

16.384\ \mathrm{kHz}

for

32768

samples (

2^{15}

), a record length

T_{\rm rec}=2.000\ \mathrm{s}

that holds a whole number of periods of the motion, so that every line falls on a frequency bin; the resolution is

\Delta f = 1/T_{\rm rec}=0.500\ \mathrm{Hz}

. The resulting decibel spectrum shows a fundamental line at

161.0\ \mathrm{Hz}

with amplitude

A=2.00\times10^{-2}\ \mathrm{m}

, a second-harmonic line

46.0\ \mathrm{dB}

below the fundamental, and a third-harmonic line

60.0\ \mathrm{dB}

below the fundamental. Apart from a line at zero frequency, no other lines rise above the noise floor. The phases returned with the spectrum show both harmonics peaking at the same instants as the fundamental. Using the first-order results for a free oscillator with force terms

\eta x^2

and

\varepsilon x^3

in the equation of motion (second-harmonic amplitude

\eta A^2/6\omega_0^2

and third-harmonic amplitude

\varepsilon A^3/32\omega_0^2+\eta^2A^3/48\omega_0^4

, where

A

is the fundamental amplitude), determine

\eta

and

\varepsilon

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